Geometric Spin Hall Effect of Light

Der Naturwissenschaftlichen Fakult¨ at der Friedrich-Alexander-Universit¨ at Erlangen-N¨ urnberg zur Erlangung des Doktorgrades Dr. rer. nat. vorgelegt von Jan Korger aus Filderstadt

Als Dissertation genehmigt von der Naturwissenschaftlichen Fakult¨at der Friedrich-Alexander-Universit¨at Erlangen-N¨ urnberg

Tag der m¨ undlichen Pr¨ ufung: 11. April 2014

Vorsitzende(r) des Promotionsorgans: Prof. Dr. Johannes Barth Gutachter: Prof. Dr. Gerd Leuchs Gutachter: Prof. Dr. Nicolas Joly

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Geometrischer Spin-Hall-Effekt des Lichts Zusammenfassung Diese Arbeit beschreibt den Weg von der theoretischen Idee bis zum experimentellen Nachweis eines neuartigen, optischen Ph¨anomens. Erst vor kurzem, im Jahr 2009, wurde ein Strahlverschiebungseffekt vorausgesagt, der zun¨achst schwierig einzuordnen war. Dieser geometrische Spin-Hall-Effekt des Lichts, im Englischen als spin Hall effect of light oder kurz SHEL bezeichnet, f¨ uhrt zu einer Schwerpunktsverschiebung des Lichtstrahls, die u berraschenderweise nur vom Photonenspin und der Geometrie ¨ des Detektionssystems abh¨ angt. Es handelt sich um einen fundamentalen Effekt, der sich bereits bei einer theoretischen Betrachtung der Struktur des zirkular oder elliptisch polarisierten Lichtstrahls selbst zeigt. Es ist daher umso erstaunlicher, dass dieses Ph¨ anomen so lange im Verborgenen geblieben ist. Eine experimentell Beobachtung des geometrischen SHELs verlangt nach einer physikalischen Operation, die die Symmetrie des einfallenden Strahls bricht. Im Rahmen dieser Arbeit wird hierzu ein Polarisator verwendet. W¨ahrend der Effekt eines Polarisators f¨ ur senkrechten Einfall bekannt ist, verdient ein zum Strahlengang verkippter Polarisator besondere Aufmerksamkeit. Um diesen Fall zu verstehen wurden verschiedene Polarisatoren charakterisiert und ein generisches Modell erarbeitet. Dieses Modell ist geeignet eine Vielzahl an Strahlverschiebungen vorauszusagen. Die Mehrzahl der in dieser Arbeit dargestellten Experimente verwendet einen Glas-Polarisator in einem mit Immersions¨ol gef¨ ullten Tank. Der Brechungsindex des ¨ Ols entspricht dem des Glassubstrats, so dass st¨orende Oberfl¨acheneffekte vermieden werden und Polarisationseffekte in Reinform untersucht werden k¨onnen. Ohne die Immersionstechnik, dass heißt mit einem Polarisator in Luft, w¨are es nicht m¨oglich die dargestellten Ergebnisse zu wiederholen. Die Arbeit zeigt, dass der geometrische SHEL genau dann an einem Polarisator auftritt, wenn dieser zu einer vom Wellenvektor abh¨angigen Phase f¨ uhrt. Diese geometrische Phase resultiert aus dem Zusammenspiel des Drehimpulses des einfallenden Strahls mit dem Symmetriebruch, der durch den verkippten Polarisator eingef¨ uhrt wird. Den H¨ ohepunkt dieser Arbeit stellt die direkte Messung solcher polarisationsabh¨angigen Verschiebungen dar. In der experimentell untersuchten Konfiguration u ¨bersteigt die Schwerpunktsverschiebung eine Wellenl¨ange. Diese Arbeit schließt mit einer Diskussion des transversalen Drehimpulses, dem grundlegenden Konzept, das die Arbeit motiviert hat.

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Summary This thesis describes the journey from the theoretical idea to an experimental proof of a novel optical effect. Only recently, in 2009, an elusive beam shift phenomenon was predicted. This geometric spin Hall effect of light (SHEL) amounts to a displacement of a light beam, which surprisingly depends solely on the photon spin and the geometry of the detection system. In fact, this very fundamental beam shift has always been hidden in plain sight, in the structure of circularly and elliptically polarized light beams, and can be revealed theoretically by employing a non-standard, i.e. tilted reference frame. Experimentally, the observation of the geometric SHEL relies on a physical operation breaking the symmetry of a light beam. In this work, we choose to employ a polarizer for this purpose. While, the action of a polarizer is obvious for normal incidence, the structure of the light beam transmitted across a tilted polarizing interface is worth a closer look. To understand this physically, different polarizers are characterized, which leads to a generic polarizer model. This connects polarizing interfaces to a plethora of beam shifts. In particular, most experiments in this work use a glass polarizer submerged in a liquid with its refractive index matched to the polarizer’s substrate. This setup avoids detrimental effects occurring at the polarizer surface and allows to study polarization effects in a particularly pure manner. In fact, without this technique, i.e. using a polarizing interface in air, one cannot reproduce our results. It is shown that geometric SHEL occurs at a polarizer if the action of this devices leads to particular wave vector dependent phase term, which can be identified with Berry’s geometric phase. The occurrence of this phase results from the interplay between the angular momentum of the incident beam and the symmetry break induced by the tilted polarizer. The pinnacle of this work is the direct measurement of such polarizationdependent beam shifts with nanometre resolution. For the configuration studied experimentally, these displacements exceed one wavelength. This thesis concludes with a discussion of transverse angular momentum, the fundamental idea which motivated this work.

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Contents Summary

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Introduction

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1. Properties of light beams 1.1. Laser beams . . . . . . . . . . . . . . . 1.2. The light field in the angular spectrum 1.3. Understanding beam shifts . . . . . . 1.4. Linear and angular momentum of light

. . . .

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2. Well-known and novel beam shifts 2.1. Goos-H¨ anchen shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Imbert-Fedorov shift and spin Hall effect of light . . . . . . . . . . . 2.3. Angular momentum conservation and conventional beam shifts . . . 2.4. Geometric spin Hall effect of light and transverse angular momentum

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3. The geometric spin Hall effect of light at a polarizing interface 3.1. A useful generic polarizer model . . . . . . . . . . . . . . . . . . . . 3.2. Constructing a microscopic polarizer model compatible with empirical data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Beam shifts occurring at a tilted polarizer . . . . . . . . . . . . . . . 3.4. Interpretation in terms of the geometric phase . . . . . . . . . . . . . 3.5. Interpretation in terms of transverse angular momentum . . . . . . .

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Conclusion and outlook

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A. Publications A.1. Geometric Spin Hall Effect of Light at polarizing interfaces A.2. Observation of the geometric spin Hall effect of light . . . . A.3. The polarization properties of a tilted polarizer . . . . . . . A.4. Distributing entanglement with separable states . . . . . . .

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Introduction Optics is the scientific discipline studying light and its interaction with matter. In principle, any optical phenomenon can be derived from the quantum theory of the light field. Practically, many phenomena including the ones discussed in this work can be understood with a classical theory. Nevertheless, the terms photon and spin, which originate from the quantum mechanical treatment are used in the field [Haus93, Loudon00]. Most of our daily experiences with light and the technical use of optics can be understood using a surprisingly simple theory: In geometrical optics, light is thought of as a collection of thin rays propagating according to a set of geometric rules [Saleh07, §1]. Physically, light is electromagnetic radiation and its behaviour is governed by electric and magnetic fields. These vector fields are found solving a set of fundamental differential equations attributed to Maxwell. Light beams are special solutions thereof. Like the hypothetical rays, they travel along a given direction and exhibit a finite extent transverse to the direction of propagation. However, internal structure and propagation of even the most basic beams differ from ray optics. Physical light beams result from the interference of multiple elementary waves and propagate accordingly [Mandel95]. The topic of this work are beam shift phenomena. Such beam shifts are unexpected deviations from geometrical optics [Bliokh13]. A prime example for the discrepancy between wave and ray optics is the famous 1943 experiment by Goos and H¨anchen studying total internal reflection [Goos47]. Inside a glass prism, a physical light beam is not reflected geometrically at the surface as expected for a ray, but penetrates into the free space surrounding the prism [Renard64]. Here an evanescent wave carries the energy of the incident beam along the surface for a fraction of the wavelength before it is reflected. A less-known cousin of this Goos-H¨anchen effect, attributed to Imbert and Fedorov [Fedorov55, Schilling65, Imbert72], has puzzled researchers for almost 60 years. While the underlying physical problem is similar, here, the vectorial nature of electromagnetic waves manifests itself in a strong dependence on the state of polarization [Liu08]. Since polarized beams can be thought of as an ensemble of photons with a particular spin state, a polarization-dependent beam shift amounts to a coupling between the spin and spatial degrees of freedom, referred to as spin-orbit coupling [Bliokh06]. The Imbert-Fedorov shift has recently been rediscovered as the spin Hall effect of light [Onoda04, Onoda06, Bliokh06, Hosten08, M´enard09, Hermosa11, Yin13] and connected to Berry’s geometric phase [Berry84, Xiao10]. The topic of this thesis is a third kind of beam shift, the geometric spin Hall

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effect of light (geometric SHEL) [Aiello09a, Aiello10, Korger11, Korger13a]. This phenomenon is intimately connected to spin and orbital angular momentum of light beams. Both properties require a physical descriptions of the light field taking into account the vectorial nature of the problem. Geometric SHEL shares a number of common characteristics with spin Hall effects in optics and other branches of physics [Xiao10], most importantly, the connection to Berry’s phase and the dependence on the spin of the incident field. It is important to note that this geometric shift is practically independent from the physical properties of the interface where it occurs, and thus an ideal candidate to study universal features of spin-orbit coupling. This work is structured as follows: Chapter 1 reviews the physical description of light beams and develops a set of theoretical tools used throughout this thesis. The theory is constructed in an abstract manner, applicable to both conventional and novel beam shifts. Then, chapter 2 discusses examples of such beam shifts including the geometric spin Hall effect and their relation to angular momentum conservation laws. Finally, chapter 3 deals with the geometric SHEL occurring at a tilted polarizer. To this end, a physical description of such polarizing interfaces is established. Two different approaches were pursued and eventually both resulted in a suitable physical model. One attempt uses empirical data and a generic projection formula; the other one builds upon a microscopic description. These models are used to explain the observed beam shift as well as geometric aspects and fundamental limits thereof. The major results of this thesis have been submitted to peer-reviewed journals, reprinted in an appendix to this thesis: • J. Korger, A. Aiello, C. Gabriel, P. Banzer, T. Kolb, C. Marquardt, and G. Leuchs. Geometric Spin Hall Effect of Light at polarizing interfaces. arXiv:1102.1626. Applied Physics B, 102(3), 427–432, 2011 • J. Korger, A. Aiello, V. Chille, P. Banzer, C. Wittmann, N. Lindlein, C. Marquardt, and G. Leuchs. Observation of the geometric spin Hall effect of light. arXiv:1303.6974. Phys. Rev. Lett. 112, 113902, 2014 • J. Korger, T. Kolb, P. Banzer, A. Aiello, C. Wittmann, C. Marquardt, and G. Leuchs. The polarization properties of a tilted polarizer. arXiv:1308.4309. Opt. Express 21(22), 27032–27042, 2013 A more detailed discussion of the experiment has been the subject of two related Bachelor theses: • Tobias Kolb, Charakterisierung eines zum Strahlengang gekippten Polarisators, 2010. • Vanessa Chille, Experimente zum geometrischen Spin-Hall-Effekt des Lichts, 2011.

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1. Properties of light beams The topic of this work is a novel beam shift phenomenon, the geometric spin Hall effect of light at polarizing interfaces. Generally, beam shifts depend on both, the properties of the incident light field and the operation occurring at the interface. Thus, a proper physical description of both is required to predict and understand these effects. This chapter introduces the background relevant to understand beam shifts. The theoretical tools presented here will prove useful to calculate established and novel phenomena in the following chapters. First, we establish a physically sound description of Maxwell-Gaussian light beams (section 1.1). These are vectorial beams with a Gaussian envelope, compatible with Maxwell’s equations. Then, we proceed to calculate in an abstract manner how a physical operation effects the light beam and can yield to a displacement known as a beam shift (section 1.3). Finally, this chapter concludes with a discussion of linear and angular momentum (section 1.4). Spin Hall effects of light (SHEL) are closely related to the interplay between intrinsic and extrinsic angular momenta. For both the geometric SHEL and the conventional spin Hall effect of light, recent experiments [Hosten08, Korger13a] were stimulated by theoretical breakthroughs concerning the angular momentum of the light field [Onoda04, Bliokh06, Aiello08].

1.1. Laser beams Since the invention of the laser, collimated light beams are the most ubiquitous tool used in optics. The spatial structure of such laser beams are commonly described as solutions ψ(r) of the paraxial wave equation. A complete set of such solutions is given, for instance, by Laguerre-Gaussian or Hermite-Gaussian laser modes. In this work, we focus on the fundamental solution of both sets, which yields a Gaussian light beam [Saleh07, §3.1]:     2 2 2 2 A0 −ik(x + y ) A0 −(x + y )   ψ(r) = exp = exp   z + izR 2(z + izR ) z + izR w02 1 − i zzR    2 + y2) 1 + i z −(x zR A0    = exp  (1.1) 2 z + izR w2 1 + z 0

2 zR

Here, k = 2π/λ is the magnitude of the wave vector k, w0 is the beam waist, and zR = π w02 /λ is the Rayleigh length. All of these parameters depend on the

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Figure 1.1.: Reference frames used throughout this work. (a) Sketch of a Gaussian light beam described in its natural reference frame {x, y, z}, where the z is the direction of propagation. (b) Tilted reference frame {x0 , y 0 , z 0 } used to describe oblique interfaces relevant to this work. In particular, to study the geometric SHEL, we will employ a polarizer with its absorbing axis oriented in direction of x0 . wavelength λ, the natural length scale in optics. It is convenient to introduce the angular divergence θ0 = λ/(π w0 ). This dimensionless parameter θ0 is small for well-collimated beams. Thus, the Taylor expansion f (θ0 ) =

f (0) + f (1) θ0 + f (2) θ02 +O(θ03 ) ' f (0) + f (1) θ0 , | {z } |{z} | {z }

0th order

1st order

(1.2)

2nd order

of any observable f (θ0 ) depending on the light field can usually be truncated after the first-order term. In this work, we assert that the beam waist w0 ≈ 100λ exceeds the wavelength by several orders of magnitude as it is the case in our experiments. Thus, terms proportional to θ02 ≈ 10−5 become negligible. The electric field of a homogeneously polarized, monochromatic beam   i θ0 in E (r) ∝ exp (ikz) u ˆ+ zˆ (ˆ u · ∇⊥ ) ψ(r) + O(θ02 ) (1.3) 2 depends on the spatial profile ψ(r) and a unit vector u ˆ describing the state of in polarization [Haus93]. This prototype light beam E (r) will be used throughout this work to describe the incident light field. We will refrain from explicitly writing the time-dependence E(r, t) = E(r) exp(i ω t) and it is understood that the physical field is the real part Re(E) of the complex quantities given here. Unless stated otherwise, we employ a reference frame {ˆ x, y ˆ, zˆ} aligned with the light beam and identify the z-axis with the direction of propagation (Figure 1.1). The field components Ex = E · x ˆ and Ey = E · y ˆ are referred to as horizontal and vertical polarization respectively. The magnetic field B(r, t) = ωi curl E(r) exp(i ω t) of a light beam can be found from Faraday’s law of induction [Jackson98, §5.15]. Both, the magnetic and the electric fields are time-harmonic, and we will later make use of the fact that the timeaverage of the product of two such fields A1,2 (r, t) = A1,2 (r) exp(i ω t) is [Jackson98,

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§6.9] 1 T

ZT

1 Re(A1 (r, t)) Re(A2 (r, t)) d t = Re(A1 (r) A∗2 (r)). 2

(1.4)

t=0

1.2. The light field in the angular spectrum representation The representation of the electric field in the position basis (1.3) is complete and usually intuitive. For the purpose of this work, it is convenient to write the light ˜ beam in the momentum basis, that is as a function E(k) of the wave vector q  T k = k κx , κy , 1 − κ2x − κ2y . | {z }

(1.5)

κ ˆ

The connection between both representations is given by the Fourier transform ZZ ˜ κ) = 1 E(x, y, z = 0) exp(−i k (κx x + κy y)) d x d y. (1.6) E(ˆ 2π x,y

In other words, equation (1.6) expresses the light field as a superposition of plane waves exp(i k · r). Thus, if we understand the effect a physical operation ˜ in (ˆ ˜ out (ˆ E κ) → E κ) has on a generic plane wave, we can infer the action for any light field. Furthermore, for well-collimated beams, the first-order approximation κ ˆ ' (κx , κy , 1)T is usually sufficient and renders this approach efficient. Generally, the resulting light field (0,0) (1,0) (0,1) ˜ κ) + O(θ2 ) ˜jout (ˆ E κ) = (αj + αj κx + αj κy ) ψ(ˆ 0 {z } |

(1.7)

α ˜ j (ˆ κ)

(x,y)

˜ κ). The index is a function of six complex amplitudes αj and the envelope ψ(ˆ j ∈ {1, 2} denotes two orthogonal states of polarization. This generic light beam (1.7) can equivalently be expressed in position space:     out out (0,1) y (0,0) (1,0) x out out + αj Ej (r ) = αj + i αj θ0 ψ out (r out ) + O(θ02 ) (1.8) w0 w0 Throughout this work, equations (1.7) and (1.8) will be used to calculate beam shifts for a number of different physical operations. The reference frame r out can be adopted to suit the problem. If the direction of propagation does not change, as for the geometric SHEL, we choose r out = r in = r. As a side note, the plane wave representation (1.6) also explains the physical structure of polarized light beams (1.3). Since the electric field of a plane wave is

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strictly transverse to the direction of propagation κ ˆ , the orientation 

 ux u ˆ − (ˆ κ·u ˆ)ˆ κ e ˆ(ˆ κ) = p =u ˆ − (ˆ κ·u ˆ)ˆ κ + O(θ02 ) =  uy  + O(θ02 ) 1 − (ˆ κ·u ˆ) −ˆ u·κ ˆ

(1.9)

of its electric field must generally differ from the global polarization vector u ˆ. The electric field of a homogeneously polarized light beam in the angular spectrum representation ˜ κ) ˜ in (ˆ E κ) = e ˆ(ˆ κ) ψ(ˆ

(1.10)

is simply the product of the polarization vector e ˆ(ˆ κ) introduced above and the ˜ Fourier amplitude ψ(ˆ κ). The inverse Fourier transform of equation (1.10) yields our prototype light beam in real space (1.3). A rigorous discussion of light beams in the angular spectrum representation is given in [Mandel95, §5.6].

1.3. Understanding beam shifts Beam shifts are unexpected deviations from geometrical optics. In theory, such displacements are found as first-order corrections to the position of the centre of mass of a light beam’s energy density. Experimentally, beam shifts are usually found as polarization-dependent displacements, although a prominent example, the Goos-H¨ anchen effect, also occurs for unpolarized light [Goos47]. To allow for a simple uniform treatment of multiple beam shift phenomena, we calculate the energy density |E out (r)|2 = |E1out (r)|2 + |E2out (r)|2 once for the generic first-order light beam (1.8).RRIn terms of the Fourier amplitudes α ˜ j introduced above,

 r ⊥ |E out (r ⊥ )|2 d x d y RR the centre of mass r ⊥ = of our generic light beam at the beam |E out (r ⊥ )|2 d x d y waist (z = 0) is: 

  

 r ⊥ = W1 x |E1 |2 + W2 x |E2 |2 x ˆ   

 + W1 y |E1 |2 + W2 y |E2 |2 y ˆ

(1.11a)

Here, we make use of the fact that the energy density of orthogonally polarized electric fields can be calculated Both components |E1 |2 and |E2 |2

independently.  contribute to the barycentre r ⊥ of the total electric energy density and it is convenient to introduce relative weights +∞ RR

(0,0) 2 α j −∞ = Wj = +∞ 2 RR (0,0) (0,0) 2 2 α + α 2 1 |E(r)| d x d y −∞

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|Ej (r)|2 d x d y

(1.11b)

for each polarization component and the so-called relative shifts:         (0,0) (1,0) (0,0) (1,0) Im α Re α − Re α Im α

 j j j j λ x |E |2 = j (0,0) 2 2π αj   (0,0) (1,0) ∗ Im α α j j λ = (0,0) 2 2π αj   (0,0) (0,1) ∗ Im α α

 j j λ and y |E |2 = . j (0,0) 2 2π αj

(1.11c)

(1.11d)

The barycentre of our prototype light beam (1.3) coincides, by definition, with the zˆ axis. Any that effects the electric field amplitudes such that the

operation  centre of mass r ⊥ 6= 0 is displaced is referred to as a beam shift phenomenon. In particular, we study spatial beam shifts occurring independently of the distance z from the beam waist. Any such spatial displacement in real space is connected to a linear phase factor in k-space. This well-known property of the Fourier transform   manifests itself in (0,0) equations (1.11). Ignoring an irrelevant global phase (Im αi ≡ 0), we can verify that equation (1.11c) reduces to

 λ ∂φj x |E |2 = − , (1.12) j 2π ∂κx κx =0 κy =0

where φj is the argument of the electric field amplitude αj (ˆ κ) = A(ˆ κ) exp(i φj (ˆ κ)) in Fourier space. This result is completely general and applies to both well-known and novel beam shifts. In fact, a variant of equation (1.12), appeared in early theoretical work [Artmann48] connected to the Goos-H¨anchen shift.

1.4. Linear and angular momentum of light beams The electric field E(r) (as in equation (1.3)) completely describes a light beam. Furthermore, we have shown in section 1.2 that any physical interaction relevant to ˜ this work can be calculated conveniently using the Fourier transform E(k) thereof. Nevertheless, connecting the light field to other physical observables, such as linear and angular momentum, proves insightful. Making use of equation (1.4), we find the time-averaged linear momentum density [Jackson98, §6.7] p(r) =

1 0 Re(E × H ∗ ) = Re(E × B ∗ ), 2c2 2

(1.13)

which directly leads to the angular momentum density j(r) = r × p(r).

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These densities p(r) and j(r) are vector fields, that is vector-valued functions of the position r, just like the electric and magnetic fields. For light beams, the total ZZ p(r) d2 r and (1.14a) linear momentum P = angular momentum J =

ZAZ

j(r) d2 r

(1.14b)

A

per unit length is found by integrating these densities over a plane A perpendicular to the direction of propagation. The integral values P and J are conserved, and thus, obvious candidates for a physical interpretation of the densities p and j [Haus93, Allen92, Allen00]. For our prototype light beam, the linear momentum density is σ ∂|ψ(r)|2 ˆ krz p(r) ∝ k |ψ(r)|2 zˆ + Φ+ |ψ(r)|2 rˆ⊥ , | {z } 2 ∂r⊥ zR + z 2 {z } | {z } | pz

(1.15)

pd

ps

p p ˆ = (y, −x, 0)T / x2 + y 2 . Using the notion, where rˆ⊥ = (x, y, 0)T / x2 + y 2 and Φ that a quasi-monochromatic source emits photons of the energy ~ω, we can state that the integrated linear and angular momenta of our beam are ZZ ZZ 2 P = p(r) d r = pz (r) d2 r = k ~ zˆ and (1.16) J=

ZAZ

j(r) d2 r =

A

ZAZ

r × ps (r) d2 r = σ ~ zˆ

per photon.

(1.17)

A

The divergence term pd does not contribute to either integral. LG (r), there occurs an additional term of J = For Laguerre-Gaussian modes ψpl o l ~ zˆ per photon, known as orbital angular momentum [Allen92, Allen00]. Both forms of angular momentum discussed so far are intrinsic, i.e. their values are independent from the choice of the reference frame. In the context of this work, it is important to understand that the angular momentum of a light beam acquires an additional extrinsic contribution if the spatial mode ψ(r − R) is displaced with respect to the zˆ-axis: ZZ ZZ 2 J (R) = r × p(r − R) d r = (r 0 + R) × p(r 0 ) d2 r0 =

ZAZ |A

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A

0

0

2 0

J(R=0)

}

r × p(r ) d r + R × {z

|

ZZ A

p(r 0 ) d2 r0 {z

J extrinsic (R)

}

(1.18)

For a paraxial beam described in its natural reference frame, i.e. if zˆ is the optical axis, the first term J (R = 0) in equation (1.18) yields the intrinsic angular momentum [Berry98, O’Neil02], parallel to the direction zˆ of beam propagation. Only the second term J extrinsic (R) depends on the distance R from the origin of the reference frame. This extrinsic contribution to the angular momentum is perpendicular to zˆ. Both, for the conventional Imbert-Fedorov shift, as well as for our geometric spin Hall effect of light, the occurrence of such extrinsic angular momentum balances the conservation laws for energy, linear and angular momentum. Optical angular momentum is a rapidly evolving field and the literature goes way beyond the simple case of free collimated light beams described here [Marrucci11, Franke-Arnold08, Banzer13].

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2. Well-known and novel beam shifts In this chapter, we introduce a number of beam shift phenomena. First, we show how the boundary conditions at planar interfaces give rise to Goos-H¨anchen and ImbertFedorov shifts (sections 2.1 and 2.2). The physical properties of such interfaces are conveniently described using Fresnel’s formulæ and any conventional beam shift is a function of the Fresnel coefficients. Then, we connect those effects to conservation laws, in particular the conservation of angular momentum (section 2.3). Finally, we conclude with a discussion of transverse angular momentum and the original proposal of the geometric spin Hall effect of light. This work focusses on the geometric spin Hall effect occurring at a tilted polarizing interface. In the following chapter such tilted polarizers are studied, which allows to predict and measure the geometric SHEL analogously to conventional beam shifts. However, unlike conventional beam shifts, geometric SHEL is connected to transverse angular momentum and exceeds the magnitude of conventional beam shifts.

2.1. Goos-H¨ anchen shift Conventional beam shift phenomena occur for a light beam reflected from or transmitted across a planar interface between two homogeneous media. At this boundary, where the properties of the media change sharply (n1 → n2 ), electric and magnetic fields are discontinuous [Born99, Appendix VI.1]. Well-known consequences of these boundary conditions are the law of reflection, the law of refraction also known as Snell’s law, and Fresnel’s formulæ. Furthermore, all conventional beam shifts emerge from these relations. Generally, the centre of mass of a light beam reflected from or refracted at a dielectric boundary deviates from the geometrical optics approximation. In their famous experiment, Goos and H¨ anchen observed a longitudinal displacement for a light beam reflected multiple times inside a special glass prism. At that time, it was already well-known from Maxwell’s theory and experimentally confirmed, that there exists an evanescent field beyond the interface where total reflection occurs. While evanescent waves decay rapidly in direction of the surface normal, they are not negligible as energy is transported along the surface [Renard64, Lai00]. This energy flux explains the observation of the longitudinal displacement observed by Goos and H¨ anchen (Figure 2.1(a)). However, explicitly calculating the evanescent field is never required. To the contrary, Fresnel’s formulæ for the reflected field are sufficient since they already account for all boundary conditions. As a consequence, today, the term Goos-H¨anchen shift refers to any longitudinal displacement at an interface, even if no evanescent field is involved [Bliokh13].

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Figure 2.1.: Illustration of Goos-H¨ anchen and Imbert-Fedorov shifts occurring for a light beam totally reflected inside a glass prism. The reflected light field is described in a reference frame {xR , y R , z R } aligned with its direction of propagation. Here, z R is the geometric reflection of the original direction of propagation z in . Beyond this geometrical optics approximation, the position of the reflected beam does not coincide with the z R -axis. (a) For total internal reflection, any light beam undergoes a longitudinal displacement known as the Goos-H¨anchen effect. (b) If the incident light beam is polarized, additionally, a transverse shift known as the Imbert-Fedorov effect can occur. Purely s or p polarized beams are exceptions, where the transverse displacement vanishes. Quantitatively, both effects depedend on the state of polarization (Figure 2.2). In the previous sections, we have introduced the tools required to calculate the geometric SHEL. Here we connect those to conventional beam shifts using the original Goos-H¨ anchen experiment as an example. For any plane wave component of the incident field, both the reflected and refracted fields are also plane waves. The wave vectors of the outgoing waves are both in the plane of incidence, defined by the incident wave and the surface normal. If the incident wave impinges at an angle θin with respect to the surface normal, the laws of reflection and refraction require [Born99, §1.5.1] θR = −θin n1 sin(θT ) = sin(θin ). n2

and

(2.1) (2.2)

The relation between the field amplitudes is given by Fresnel’s equations [Born99, §1.5.2]. Due to their dependence on the geometry of the problem, it is convenient to express the electric field E = E k + E ⊥ in components parallel and perpendicular to the plane of incidence. The transmitted field  T    in  Ek Ek τk 0 = in T 0 τ⊥ E⊥ E⊥ 20

(2.3)

Figure 2.2.: Beam shifts occurring for a Maxwell-Gaussian light beam totally reflected inside a glass prism as a function of the incident √ state of polarization (a, 1 − a2 exp(iδ)). The approximate locations of linear (H, V, ±45◦ ) and circular (R, L) states of polarization are indicated by text labels. (a) Longitudinal displacement or Goos-H¨anchen (GH) shift. (b) Transverse displacement or Imbert-Fedorov (IF) shift. The angle of incidence θ = 60◦ is sufficiently far from the critical angle and the refractive indices n1 = 1.5 and n2 = 1 are typical for glass in the visible and infra-red regime. Both plots share the same color coding, which illustrates that the GH effect is significantly larger then the IF shift. Also, for the GH case, the dependence on the state of polarization is weak, while the IF effect depends strongly on the relative phase δ.

and the reflected one

 R    in  Ek Ek ρk 0 = R in 0 ρ E⊥ E⊥ ⊥

(2.4)

are related to the electric field vector E in of the incident wave via the so-called

21

Fresnel coefficients: τk = τ⊥ =

n2

n1 n2 ρk = n2 n1 ρ⊥ = n1

2n1 cos(θin ) cos(θin ) + n1 cos(θT ) 2n1 cos(θin ) cos(θin ) + n2 cos(θT ) cos(θin ) − n1 cos(θT ) cos(θin ) + n1 cos(θT ) cos(θin ) − n2 cos(θT ) . cos(θin ) + n2 cos(θT )

(2.5a) (2.5b) (2.5c) (2.5d)

Both the Goos-H¨ anchen and the Imbert-Fedorov shifts were first discovered for total internal reflection (n1 > n2 and θin > θcritical = arcsin(n2 /n1 )). Equations (2.5c) and (2.5d) hold true for this case if we take into account that cos(θT ) = p 1 − sin(θT )2 is imaginary [Born99, §1.5.4]. We explicitly calculate the Goos-H¨anchen shift for s polarization (ˆ u=y ˆ). For this case, consindering wave vectors in the plane of incidence suffices (κy = 0). Thus, the polarization vector (1.9) describing the incident beam reduces to e ˆ(κx , κy = 0) = y ˆ. ˜ As a consequence, the relevant electric field vector Ekˆ y is strictly transverse and changes according to the relevant Fresnel equation (2.5d): ˜ κ) e ˜ κ) y ˜ κ) ρ⊥ yˆ (2.6) ˜ in (κx , κy = 0) = ψ(ˆ ˜ out (κx , κy = 0) = ψ(ˆ E ˆ(ˆ κ) = ψ(ˆ ˆ→E For total internal reflection, the modulus square |ρ⊥ |2 = 1 is unity and the effect on the light field (2.6) amounts to a phase factor [Born99, §1.5.4]  r  n22 2   sin (θ(κx )) − n21    , ρ⊥ = exp  −2i arctan (2.7)    cos(θ(κx )) where

in in R cos(θ(κx )) = κ ˆ in · zˆ0 = cos(θin ) + sin(θin ) κin x = cos(θ ) − sin(θ ) κx .

(2.8)

Computing the Fourier transform or recalling equation (1.12), one immediately recognizes the Goos-H¨ anchen shift

R n1 sin2 (θi ) λ x =+ q . π n2 sin2 (θ ) − n2 i 1 2

(2.9)

Our results confirms the formula found 1944 by Artmann [Artmann48, §2] in response to the original work by Goos and H¨anchen [Goos47]. In modern experiments, this displacement is routinely reproduced [Gilles02]. Generally, the Goos-H¨anchen effect depends on the state of polarization (Figure 2.2(a))

22

 Note that the sign of xR in (2.9) is a consequence of the choice of our reference frame (Figure 2.1), which is not consistent across the literature. While our calculation assumes a perfect spatially coherent mode, the degree of coherence actually present in early experiments is not clear. Only recently, it was confirmed experimentally that the degree of spatial coherence does not influence such spatial beam shifts [L¨offler12]. The angular spectrum method used in this work was didactically presented by McGuirk and Carniglia [McGuirk77]. It is straightforward to generalize this method to different media, e.g. metals [Merano07], and include the case of partial reflection and transmission. The literature building upon this result is extensive [Aiello12, Bliokh13]. Technically, we have found it useful to perform all calculations in Fourier space. However, it is possible to apply the boundary conditions directly in real space [Bekshaev12], which may make these phenomena more accessible for a broader audience.

2.2. Imbert-Fedorov shift and spin Hall effect of light

 While the Goos-H¨ anchen effect describes a purely longitudinal displacement x , light

 beams interacting with an interface can also undergo a transverse shift y originally predicted by Fedorov [Fedorov55] and experimentally demonstrated by Imbert [Imbert72]. From a modern, pragmatic point of view, both conventional beam shift phenomena result analogously from the boundary conditions [Player87, Li07, Aiello08]. It was also shown experimentally that both shifts occur simultaneously for a single reflection [Pillon04]. Nevertheless, while the Goos-H¨ anchen effect was observed and theoretically explained in the 1940s [Goos47, Artmann48, Goos49], the debate about the Imbert-Fedorov shift was only settled recently [Bliokh06, Bliokh07, Aiello12, Bliokh13]. Some of the reasons, why historically the transverse shift was more difficult to asses experimentally can be seen in figure 2.2(b). The magnitude of the displacement is typically smaller than the longitudinal shift (Figure 2.2(a)) and depends strongly on the state of polarization. Consequently, the Imbert-Fedorov effect vanishes for unpolarized light. The first successful demonstration of the transverse shift [Imbert72] adopts Goos and H¨ anchen’s idea to use multiple reflections in order to enlarge the effect. However, due to the fact that the state of polarization generally changes under reflection, the experiment had to be carefully designed to preserve the handedness of the light beam. Theoretically understanding the Imbert-Fedorov shift was challenging since it depends not only on the properties of the interface, but also on the physical structure of the light beam itself. While the Goos-H¨anchen shift can be explained qualitatively with a two-dimensional scalar theory, both the Imbert-Fedorov effect as well as the geometric SHEL, require a three-dimensional vectorial treatment, which properly accounts for polarization.

23

The angular spectrum method explained in the previous section remains valid for the transverse shift. Applying the standard Fresnel formulæ (2.5) to our prototype light beam, predicts the Imbert-Fedorov shift correctly [Li07, Aiello08]. Interestingly, Fresnel’s equations can also be rewritten in a basis adopted for circular polarization, rendering the calculation of this Imbert-Fedorov shift completely analogous to the Goos-H¨ anchen case [Player87]. Careful investigation of the electric field reveals that a light beam reflected from or transmitted across an interface is not exactly equivalent to a displaced beam [Bliokh06, eq. (4)]. In fact, a linearly polarized Gaussian beam transmitted across an air-glass interface becomes a superposition of two beams with different helicities symmetrically displaced with respect to each other. Due to the striking similarity to the spin Hall effect experienced by charge carriers, the splitting has become known as the “spin Hall effect of light” (SHEL) [Onoda04, Hosten08]. This spin-dependent displacement amounts only to a fraction of the wavelength. Nevertheless, using a crossed polarization analyzer, this effect is observable [Bliokh06, Hosten08]. Hosten and Kwiat describe this polarization enhancement ingeniously as a quantum weak measurement. In their experiment, they additionally make use of beam propagation to enhance the signal [Aiello08].

2.3. Angular momentum conservation and conventional beam shifts When interacting with matter, the angular momentum carried by the light field is not necessarily conserved. Generally, energy and linear and angular momentum can be transferred from photons to other systems. However, the physical nature of the interaction and symmetry of the problem imposes constraints. Thus, one can establish particular conservation laws for a light beam propagating across different kinds of interfaces. Traditionally, beam shifts are studied for a planar interface between nonabsorbing, isotropic media. This problem is rotationally symmetric with respect to the surface normal zˆ0 . As a consequence, the normal component jn = j · zˆ0 of the light field’s angular momentum is conserved [Player87, Bliokh06, Bliokh07]. Thus, the angular momentum of the incident field, for example the spin of a circularly polarized beam, determines the normal component jn of the outgoing field, i.e. of the refracted and reflected light beams. The latter includes an extrinsic contribution connected to the position of these beams. This conservation law for jn is a necessary condition for any beam shift occurring at such an interface. In particular, for the special case of total internal reflection, the transverse Imbert-Fedorov shift can be derived solely from angular momentum conservation [Bliokh13]. If the axial symmetry is broken, the conservation of jn does not hold. In his famous experiment, Richard Beth showed that transfer of angular momentum from photons to a birefringent wave plate is indeed possible [Beth36, Beijersbergen05]. Here, the

24

light field exercises an observable torque on the anisotropic plate. The polarizer used in our experiment exhibits a comparable symmetry break. Consequently, the conservation laws governing conventional beam shifts do not apply to the polarizing interfaces. To the contrary, the geometric SHEL is connected to the conservation of transverse angular momentum.

2.4. Geometric spin Hall effect of light and transverse angular momentum Geometric spin Hall effect of light as field of research started with a discussion of linear and angular momentum in a tilted reference frame {ˆ x0 , y ˆ0 , zˆ0 }. In this context, 0 0 the intensity observed by a detector aligned with the x -y -plane was identified with the longitudinal component pz 0 = p · zˆ0 of the linear momentum density or Poynting vector [Aiello09a]. This “intensity” distribution pz 0 has the surprising property that its centre of mass (evaluated at z 0 = 0) RR 0  

0  r ⊥ pz 0 (r 0⊥ ) d2 r0 1 −Jy0 RR r⊥ p 0 = (2.10) = z Pz 0 Jx0 pz 0 d2 r0

is given directly by the transverse components Jx0 and Jy0 of the light beam’s angular momentum (equations (4) and (5) in [Aiello09a]). This fundamental connection originates from the definition of angular momentum and becomes evident if we explicitly write the components     jx0 y 0 pz 0 jy0  =  −x0 pz 0  (2.11) jz 0 x0 py0 − y 0 px0

thereof at z 0 = 0 and substitute r 0⊥ pz 0 in equation (2.10). Throughout this work, we employ a tilted reference frame with x ˆ0 = cos(θ) x ˆ− 0 sin(θ) zˆ and y ˆ =y ˆ. For our prototype light beam (1.3), that is a Maxwell-Gaussian beam propagating freely in direction of zˆ, the centroid of the linear momentum density in the tilted reference frame is

r 0⊥



pz 0

=

Jxin0 0 λ σ tan(θ) y ˆ0 , y ˆ = 4π Pzin0

(2.12)

where Jxin0 is the projection of the intrinsic spin angular momentum. Since linear and angular momenta are functions of the electric field, the beam shift (2.12) can also be found by investigating the structure of the electric field. To this end, we decompose the electric field and the corresponding energy density |E(r 0⊥ )|2 = |Ex0 (r 0⊥ )|2 + |Ey0 (r 0⊥ )|2 + |Ez 0 (r 0⊥ )|2

(2.13)

25

0

into the Cartesian components |El0 (r 0⊥ )|2 = |E(r 0⊥ ) · ˆ l |2 defined with respect to our 0 tilted reference frame ˆ l ∈ {ˆ x0 , y ˆ0 , zˆ0 }. The spatial profile of each component is approximately Gaussian. However, the respective barycentres do not coincide [Korger11]. In particular, the centre of mass of the x ˆ0 -component undergoes a displacement

 r ⊥ |E

x0 |

2

=

λσ tan(θ) y ˆ0 2π

(2.14)

resembling the one found for the Poynting vector. This motivates the use of a polarizing element to make the beam shift (2.14) visible. In the following chapter, the connection between real polarizers, polarizer models and the geometric spin Hall effect of light is discussed.

26

3. The geometric spin Hall effect of light at a polarizing interface This thesis focusses on the geometric spin Hall effect of light. In particular, we study this fundamental phenomenon occurring for a light beam propagating across a tilted polarizer. To this end, a physical description of such polarizing interfaces is vital. In textbooks, e.g. [Hecht01, §8.2] or [Born99, §15.6.3], polarizers are typically introduced as operating on plane waves. Furthermore, it is implicitly assumed that the incident wave impinges perpendicularly onto the entrance face of the polarizer. For our purpose, this approach is insufficient. As shown in chapter 1, physical light beams are superpositions of multiple plane waves and beam shifts occur if the light field acquires a wave-vector-dependent phase. In the present work, two different implementations of a polarizing interface were studied. At early stages, we worked with a polymer film sandwiched between two glass plates. Then, we proceeded with a Corning Polarcor polarizer, a glass substrate with embedded silver nano-particles, which was used for the majority of the measurements discussed in this work. Both polarizers were extensively characterized including a full Mueller matrix tomography [Kolb10, Korger13b]. Theoretically, we have employed different strategies to find suitable models compatible with our observation. One way is to start with a model that predicts infinite extinction ratios and, then, generalize this to include phenomenological parameters (section 3.1). Alternatively, one can make use of a microscopic model for the physical absorption process and describe the polarizer as an ensemble of such elementary absorbers (section 3.2). Both approaches yield useful and realistic models. The pinnacle of this chapter is the calculation of beam shifts including but not limited to the configuration studied experimentally (section 3.3). We conclude by connecting this beam shift to Berry’s geometric phase (section 3.4) and transverse angular momentum (section 3.5). Both concepts were relevant to the development of the geometric SHEL theory.

3.1. A useful generic polarizer model It is generally expected that a beam transmitted across a linear polarizer is polarized along a given axis and that the transmission through a pair of crossed polarizers vanishes. This property can be described as a projection onto an effective transmitting axis tˆ: T

E out = tˆtˆ E in .

(3.1)

27

Figure 3.1.: Visualization of the projection rule used with our generic polarizer model. Generally, the action of the polarizer is governed by a vector Pˆ , either identified with the absorbing or the transmitting axis, and its projection p1 onto the plane of the electric field. (a) For the absorbing case, the axis Pˆ is the orientation of the electric field locally absorbed at the polarizer. Since propagation in direction of z requires the fields to be in the x-y-plane, the projection p1 can be interpreted as an effective absorbing axis. Consequently, for sufficiently high extinction ratios, the transmitted field E out is directed along p2 ⊥ p1 . (b) Contrarily, for the transmitting case, we choose Pˆ perpendicular to the local direction of absorption and interpret the projection p1 as an effective transmitting axis. The prediction of both models coincide only if the direction of propagation z is perpendicular to the polarizer surface. In this chapter, we routinely make use of the dyadic product  2  tx tx ty tx tz T ty tz  tˆtˆ = ty tx t2y tz tx tz ty t2z

(3.2)

representing a projection operator. For our purpose, we have to account for the three-dimensional geometry of the problem. Interestingly, there exist two geometric polarizer models [Fainman84, Korger13b], which are closely related from an abstract mathematical point of view, but yield to completely different predictions. Fainman and Shamir suggested to describe a polarizer by a unit vector Pˆ interpreted as its transmitting axis [Fainman84, Aiello09b]. In their model, the effective transmitting axis tˆ = pˆ1 (ˆ κ) ∝ Pˆ − (ˆ κ · Pˆ ) κ ˆ is found by projecting the global vecˆ tor P onto the local plane of the electric field, i.e. the plane perpendicular to the direction of propagation κ ˆ (Figure 3.1b). However, the discussion of our experimental results [Kolb10, Chille11, Korger13a] prompted the rejection of Fainman’s transmitting model. We found that the action

28

Figure 3.2.: State of polarization transmitted across two different polarizers; each compared to the absorbing and transmitting polarizer models. We show the polarizance Stokes vector [Lu96], this is the state of polarization of the transmitted light field if the incident beam is unpolarized. The experimental data (black circles) was acquired as a part of our Mueller matrix measurements [Korger13b]. Both theoretical curves are derived from our generic model (3.3). The solid blue line depicts the case, where the absorbing axis is projected (τ1 = 0, τ2 = 1). The dashed red line is constructed analogously using the transmitting axis (τ1 = 1, τ2 = 0), which is equivalent to the model suggested by Fainman and Shamir. (a) Corning Polarcor, a glass plate embedded with metal nano-particles with their absorbing axis oriented at φ = 85.5◦ . (b) Techspec NIR linear polarizer, a polymer film layered between two glass substrates with its absorbing axis oriented at φ = 22.0◦ .

of both polarizers studied experimentally can be approximated using a similarly constructed absorbing model [Korger13b] (see figures 3.1(a) and 3.2). In this case, the unit vector Pˆ describes the orientation of local absorbers responsible for the polarization effect. Then, the projection pˆ1 thereof indicates an effective absorbing axis. And consequently, the effective transmitting axis tˆ = pˆ2 (ˆ κ) = κ ˆ × pˆ1 (ˆ κ) is perpendicular to pˆ1 . Both models introduced above, referred to as transmitting and absorbing polarizers respectively, describe polarizing elements as projectors. As a consequence, both models predict that the extinction ratio of a polarizer is only limited by the beam’s divergence. In this work, we make use of a generic polarizer model, which includes as special cases, both transmitting and absorbing polarizers. Additionally, our universal approach allows to include phenomenological parameters accounting for imperfect extinction ratios and other deviations from the simplified models (as in [Korger13a, Korger13b]).

29

In any case, the polarizing device is described by a three-dimensional unit vector ˆ P = (− cos θ sin φ, cos φ, sin θ sin φ)T projected onto the electric field plane spanned by two unit vectors pˆ1 (ˆ κ) and pˆ2 (ˆ κ). While, the physical interpretation of these vectors differs, the transmitted electric field can always be written as
in ˜ out (ˆ ˜ κ). E κ) = τ1 (θk ) pˆ1 pˆT ˆ2 pˆT (3.3) 1 + τ2 (θk ) p 2 E (ˆ

˜ is given in the angular spectrum representation (1.6). The Here, the electric fields E coefficients τ are arbitrary real-valued functions of the tilting angle θk (ˆ κ) = arccos(ˆ κ · zˆ0 ) ' arccos(cos θ + κx sin θ) ≈ θ.

(3.4)

Technically, the angle θk introduced above is the angle between the wave vector and the polarizer normal zˆ0 . For collimated light beams, the parameters κx and κy are small and higher-order terms are negligible as discussed in section 1.2. This universal approach is valid for a wide range of real and idealized polarizers. For example, if we set τ1 = 1 and τ2 = 0, equation (3.3) reduces to the transmitting model. Analogously, setting τ1 = 0 and τ2 = 1 yields the absorbing model used to qualitatively describe our observation. For our shift measurements, we have employed a polarizer with its absorbing axis Pˆ oriented horizontally. In this configuration, the modulus squared of the two phenomenological parameters can be directly measured. For horizontally polarized incident light, the transmittance is τ12 (θ) and for vertical polarization, the transmittance is τ22 (θ). In the context of our shift measurements [Korger13a], we have established the set of parameters τ2 (θ) = 1 − 1.2 exp(−12 cos θ) τ1 (θ) = 0.51 exp(−1.3 cos θ),

and

(3.5) (3.6)

to adequately describe the polarizer used. A complete Mueller matrix tomography of this device was reported in [Korger13b]. This includes a physical discussion of the result noted above.

3.2. Constructing a microscopic polarizer model compatible with empirical data Practically, the phenomenological model described in the previous section suffices to predict both intensity and centre of mass for a beam transmitted across our polarizer. Nevertheless it is insightful to connect the observed behaviour to a physical description of the interaction. The Polarcor polarizer uses polarizing layers on both faces of the glass substrate. These 25 to 50 µm thick layers are made of elongated and oriented silver nano-particles embedded in glass. For our purpose, we have chosen a product without the usual anti-reflection coating since large angles of incidence are desired for our experiment and require the use of an index-matching liquid.

30

Figure 3.3.: Illustration of a light beam propagating across a tilted polarizer with multiple layers of absorbing nano-particles oriented horizontally. Assuming the refractive index of the polarizer substrate matches its environment as in our experiment, the tilted entrance face has no effect on the field E in of the incident beam. Locally, at the position where the light interacts with a nano-particle, the electric field component parallel to the particle’s absorbing axis vanishes. Consequently, the local field E local after the interaction is no longer transverse with respect to the direction z of wave propagation. However, after propagating for a distance of multiple wavelengths, the transversality of the electric field E 1 is restored. Then, when the light beam encounters a another such particle, the process is repeated. In each step E in → E 1 → E 2 → · · · → E N = E out the the orientation of the electric field vector changes slightly. For sufficiently large N , the transmitted field is polarized almost perpendicularly to the absorbing axis. In this section, we attempt to construct a simple microscopic model compatible with our observation. The embedded nano-wires can be thought of as microscopic ˆ of these absorbers interacting locally with the electric field. The orientation A ˆ · E of the electric field which couples to the nanoabsorbers define the component A particles and is consequently scattered or absorbed. As illustrated in figure 3.3, this results in a local electric field vector E local , which is no longer transverse with respect to the wave vector k. The transmitted wave is observed to propagate along the optical axis κ ˆ in the far field. Thus, we conjecture that the longitudinal component E local · κ ˆ is lost after propagating a distance exceeding the order of the wavelength. The two steps described above can be expressed as a projection rule    in
in ˜ 1 (ˆ ˆA ˆT E ˜ (ˆ ˜ κ), E κ) = 1 − κ ˆκ ˆT 1 − A κ) = 1 − A ⊥ A T (3.7) ⊥ E (ˆ ˆ − (ˆ ˆ κ where A⊥ = A κ · A) ˆ . Generally, this differs from our generic model (3.3) since A⊥ 6= pˆ1 . An exception to this rule is normal incidence. In this case, the

31

Figure 3.4.: State of polarization transmitted across the nano-particle polarizer compared to our microscopic model. Black circles show the experimentally observed polarizance vector as in figure 3.2(a). All theoretical curves are calculated from our microscopic model (3.8) for different values of N . Assuming only one elementary absorption process, the predicted extinction ratio is much smaller than observed. One can directly see from the plot that S12 + S22 + S32 < 1 for θ > 20◦ . For larger numbers of N , the agreement with the experimental data is better, and for N → ∞ the microscopic model approaches the absorbing limit of our generic model (blue line in figure 3.2). microscopic absorbers are oriented perpendicularly to the wave vector κ ˆ and equation (3.7) coincides with both the absorbing and transmitting limits discussed in the previous section. However, when tilted, the behaviour differs and the extinction ratio decreases significantly – faster than observed. The size and density of the nano-particles embedded in our polarizer is not known precisely. A description of the manufacturing process [Borrelli93], x-ray scattering data [Polizzi98], and TEM images [Polizzi97] suggest that the nano-particle’s extent perpendicular to the elongated axis does not exceed 50 nm. On one hand, this allows for up to 1000 of such elementary absorbers, one after another interacting with a light beam propagating across a single polarizing layer. On the other hand, the propagation distance between these absorbers can easily exceed multiple wavelengths and grows if the polarizer is tilted. The latter justifies treating each absorption process individually. Thus, it is reasonable to assume that in both polarizing layers combined, a total of N elementary absorption processes (3.7) occur (as illustrated in figure 3.3): ˜ out (ˆ E κ) =

h  iN in ˆA ˆ ˜ (ˆ 1−κ ˆκ ˆ 1−A E κ)

(3.8)

This effectively increases the extinction ratio. Choosing N = 35 yields good agreement with empirical data as shown in figure 3.4. While the model (3.8) described in this section is constructed differently from our generic polarizer (3.3), both allow for a phenomenological description of our observation. For sufficiently large extinction ratios N → ∞, this microscopic model

32

agrees with our absorbing polarizer model introduced in section 3.1. The similarity between both approaches indicates that both are equally suitable for our purpose.

3.3. Beam shifts occurring at a tilted polarizer A polarizer described by the projection rule (3.3) with the axis P = (− cos θ, 0, sin θ)T oriented horizontally gives rise to a vertical displacement called the geometric spin Hall effect of light [Korger11]. We can understand this by studying how a beam of light interacts with such a polarizer. For any wave vector k ' k(κx , κy , 1), our universal polarizer model defines a basis of two unit vectors, pˆ1 = (−1, −κy tan θ, κx )T and pˆ2 = (κy tan θ, −1, κy )T . Generally, the effect of the polarizer is governed by projections onto these vectors. Applied to the electric field of the light beam (1.10), we obtain up to the first order of the k-vector components:      −beiδ κy tan θ a + beiδ κy tan θ ˜ κ) ˜ out (ˆ E κ) ' τ1  aκy tan θ  + τ2 beiδ − aκy tan θ ψ(ˆ iδ −aκx −be κy        iδ  0 be (τ1 − τ2 ) tan θ   aτ1 ˜  κ), (3.9) = beiδ τ2  +  a(τ1 − τ2 ) tan θ  κy +  0  ψ(ˆ   iδ −aτ1 κx  0 −be τ2  | {z } | {z } α(0,0)

α(0,1)

where u ˆ = (a, beiδ ) is the incident state of polarization. Here, the terms depending on κy indicate a coupling between polarization and spatial degrees of freedom and hint at the occurrence of transverse beam shifts. Applying the generic formulas (1.11) to this expression yields the centroid of the beam. The vector field E out has both horizontally and vertically polarized components. It is insightful to calculate the two corresponding beam shifts,  

 λ b sin δ τ2 (θ) r ⊥ |Ex |2 = 1− tan θ y ˆ and (3.10a) 2π a τ1 (θ)  

 λ a sin δ τ1 (θ) r ⊥ |Ey |2 = 1− tan θ y ˆ, (3.10b) 2π b τ2 (θ) individually. Experimentally, both shifts described by equations (3.10) can be observed employing an additional polarization analyser or polarizing beam splitter. If no analyser is used, both components of the energy density, |Ex |2 and |Ey |2 , contribute to the observed displacement. In this case, the total shift

 r⊥ =



 τ12 (θ)a2 τ22 (θ)b2 r + r ⊥ |Ey |2 . ⊥ 2 2 2 2 2 |E | 2 2 2 2 x τ1 (θ)a + τ2 (θ)b τ1 (θ)a + τ2 (θ)b | {z } | {z } Wx

(3.11)

Wy

33

is a weighted average of both relative shifts (3.10a) and (3.10b). The weighting factors Wx and Wy are given by the fraction of the energy in each polarization mode (equation (1.11b)). From this calculation, we learn that real polarizers with finite extinction ratios give rise to a plethora of beam shift phenomena. The predicted displacements depend on both the incident state of polarization and the polarization-dependence of the detection system (Figure 3.5). Nevertheless, these shifts have a common physical origin, that is the projection rule (3.3) and the intrinsic structure of polarized light beams. This manifests itself in the tan(θ)-dependence characteristic for the geometric spin Hall effect of light (Figure 3.6). Comparing the above-mentioned formulas or figures to the Imbert-Fedorov effect (as shown in figure 2.2) reveals the striking difference to conventional beam shift phenomena. For example, the geometric spin Hall effect of light is directly proportional to the helicity of the incident beam. Beyond the diameter of the incident beam itself, there appears to be no fundamental limit to the magnitude of the geometric SHEL (Figure 3.7).

3.4. Interpretation in terms of the geometric phase As already noted at the beginning of this chapter, our generic polarizer model includes two important special cases dubbed transmitting and absorbing polarizers. Both allow for a geometric interpretation of the transmitted state of polarization (as shown in figure 3.1). In this section, we connect those geometric aspects to the occurrence of beam shifts. This helps to see in which configuration the geometric SHEL is observable and underlines the importance of choosing a suitable polarizer model. The geometric SHEL is a consequence of the geometric phase exp(i κy tan θ), a light beam acquires when passing through a tilted polarizing element [Bliokh12a, Korger13a]. To understand the origin of this phase term, consider a plane wave with its wave vector k ' k (κy , 0, 1)T in the y-z-plane. For circular polarization, the corresponding electric field is E(k) ∼ (1, i, −i κy )T , perpendicular to k. Originally, the geometric spin Hall effect of light was connected to the transmitting polarizer model suggested by Fainman and Shamir [Fainman84, Korger11]. Here, the sought phase term occurs if the transmitting axis P is oriented horizontally. In this case, the projection of the light field onto the effective transmitting axis pˆ1 = (−1, −κy tan θ, 0)T yields (as in (3.9) with τ1 = 1 and τ2 = 0): 

   1 + iκy tan θ exp(iκy tan θ) 1 ˜ κ) ' √1  κy tan θ  ψ(ˆ ˜ κ). ˜ out (ˆ E κ) ' √  κy tan θ  ψ(ˆ 2 2 −κx −κx

(3.12)

As expected, the horizontally polarized component is dominant for well-collimated beams with small κ ˆ and has acquired a k-vector dependent phase (illustrated in

34

Figure 3.5.: Geometric spin Hall effect as a function of the incident and observed state of polarization as predicted by our phenomenological polarizer model. (a) Scheme of the setup. We assume the absorbing axis is horizontal,

 and choose a moderate tilting angle of ◦ θ = 60 . (b) Centroid y of the total energy density. (c) Centroid

 y 2 of the horizontally polarized component thereof. (d) Centroid

|Ex |

 y |Ey |2 of the vertically polarized component. Experimentally, y and

 y |Ey |2 were confirmed for circular states of polarization [Korger13a]. All plots share the same color coding.

35

Figure 3.6.: Geometric spin Hall effect of a light as a function of the tilting angle as observed with an optional analyzer in front of the detector. The relative shift when switching the polarization of √ the incident beam (λ = 795 nm) between the two states u = (a, ±i 1 − a2 ) is calculated from our phenomenological model and

compared to experimental data where available. (a) Displacement y |Ex |2 of the horizontally polarized field

 component. This beam shift contributes to the total displacement y as observed in our experiment without employing an analyzer. In principle, this “giant”

 geometric SHEL can also be observed directly. (b) Displacement y |Ey |2 of the vertically polarized field component as observed in our experiment with the optional an analyzer oriented vertically. figure 3.8). According to the Fourier transform shift theorem, the corresponding electric field E out (r) in the position space is displaced [Goodman05, Korger11]:

 λ r⊥ = tan θ y ˆ. 2π

(3.13)

For any other orientation φ of the polarizing axis, the predicted displacement is smaller and vanishes if the transmitting axis is vertical (Figure 3.9(a)). However, as we learned from early experiments employing a polymer film [Kolb10, Chille11] and confirmed for our nano-particle polarizer later, this model does not adequately describe polarization effects based on local absorption. To understand, why this is significant for beam shifts, let us describe the situation explained above with the absorbing model. Clearly, the absorbing axis Pˆ = y ˆ of a polarizer transmitting horizontally polarized light is oriented vertically. Independent of the rotation angle θ around the y-axis, for this configuration, the relevant projection onto pˆ2 = −ˆ x never yields a κy -dependent phase factor in first order (Figure 3.10(a)):   1 1 ˜ κ) ˜ out (ˆ E κ) ' √  0  ψ(ˆ (3.14) 2 −κ x

36

Figure 3.7.: Fundamental limits of the geometric spin Hall effect of light. The energy density of a circuarly polarized light beam transmitted across a tilted absorbing polarizer (θ = 89◦ ) is calculated for different values of the beam diameter w0 . The dashed red line depicts the 1/e2 -width of the incident light beam. Here, terms up to θ02 relevant for the distortion of the beam have been included. (a) For a reasonably collimated beam, the beam waist, here w0 = 30λ, is large  compared to the wavelength λ. Thus, the expected displacement y ≈ 9λ is relatively small and the transmitted beam remains approximately Gaussian. (b) At w0 = 20λ, the beam becomes visibly distorted. (c) For w0 = 10λ, one clearly sees that geometric SHEL causes a redistribution of the energy density within the boundaries of the incident beam. Here, the first-order approximation breaks down and the distortions becomes dominant.

Thus, a collimated light beam will not undergo a displacement. Nevertheless, as demonstrated by our experiment [Korger13a], absorbing polarizers are equally suitable to study the geometric spin Hall effect of light. Here, the geometric phase term occurs if the absorbing axis is oriented horizontally (Figure 3.10(b)): 

   −κy tan θ −κy tan θ 1 ˜ κ) ' √1 exp(iκy tan θ) ψ(ˆ ˜ κ). ˜ out (ˆ E κ) ' √ 1 + iκy tan θ ψ(ˆ 2 2 −κy −κy

(3.15)

For this configuration, an absorbing polarizer with a sufficiently high extinction ratio, result in a displacement

 λ r⊥ = tan θ y ˆ. 2π

(3.16)

resembling the one found originally with Fainman model. Figure 3.9(b) illustrates how the predicted shift changes for an arbitrary orientation of the absorbing axis.

37

Figure 3.8.: Light field in the angular spectrum representation before and after interacting with a transmitting polarizer oriented horizontally. (a) The phase of the electric field of a circularly polarized Gaussianq light beam as a function of the transverse wave vector k = k (κx , κy , 1 − κ2x − κ2y ). (b) Visualization of the electric field vec-

tor before and after interaction with the polarizer (τ1 = 1, τ2 = 0, Pˆ = x ˆ0 ). Note that the phase of the transmitted field depends on the orientation of the k-vector. The phase offset between k0 and k0 is the geometric phase term relevant for the geometric SHEL. (c) Electric field of the transmitted horizontally polarized light field. The color-coded phase depends linearly on κy . The Fourier transform thereof yields an electric field in position space, which is displaced with respect to the original one.

3.5. Interpretation in terms of transverse angular momentum Historically, the proposal of the geometric spin Hall effect of light was connected to the occurrence of transverse angular momentum Jx0 in a tilted reference frame (section 2.4). Here, we demonstrate the relevance of this quantity for our experiment. To this end, we choose a reference frame {ˆ x0 , y ˆ0 , zˆ0 } aligned with the polarizer such T 0 that x ˆ = P = (− cos θ, 0, sin θ) is the absorbing axis and zˆ0 is the surface normal. The angular momentum J in of the incident light field, a circularly polarized Gaussian beam, is parallel to the direction zˆ of beam propagation. Formally, in the tilted reference frame, this yields to a transverse angular momentum ˆ0 = Pzin Jxin0 = J in · x

λσ λσ sin θ = Pzin0 tan θ, 4π 4π

(3.17)

consistent with (2.10). In the last step above, we have employed the relation Pzin0 = Pzin cos θ, which results from the rotation of the reference frame. After interaction with the polarizer, the resulting light field is linearly polarized and does not carry intrinsic angular momentum: J out · zˆ = 0. However, as shown by our experiment [Korger13a] and the corresponding theory (section 3.3), the transmitted beam is displaced with respect to the optical axis of the incident beam. For the idealized case, assuming an infinite extinction ratio (τ1 = 0, τ2 =1), the centre

38

Figure 3.9.: Geometric spin Hall effect of light as a function of the orientation of the polarizer used for two special cases of our generic model. (a) Beam shifts occurring at a hypothetical transmitting polarizer (τ1 = 1, τ2 = 0) for an arbitrary orientation Pˆ = (− cos θ sin φ, cos φ, sin θ sin φ)T of its transmitting axis. Here, the displacement is most pronounced if the polarizer is oriented to transmit horizontal polarization. (b) Beam shifts occurring at an absorbing polarizer (τ1 = 0, τ2 = 1) for an arbitrary orientation Pˆ = (− cos θ sin φ, cos φ, sin θ sin φ)T of its absorbing axis. The displacement is perpendicular to its effective absorbing axis pˆ1 and the magnitude reaches its maximum if the horizontal field component is absorbed. Experimentally, we studied a configuration where the transmitted state of polarization is vertical and confirmed the prediction of the absorbing model (b). of mass is:

 λσ r out = tan(θ) y ˆ. ⊥ 2π

(3.18)

As discussed in section 1.4, this – like any beam shift – results in the occurrence of extrinsic angular momentum  

 sin(θ) 0 out λ σ 0 J out = Pzout r out x ˆ = P sin(θ)ˆ x − z ˆ . (3.19) z ⊥ 2π cos2 (θ) 39

Figure 3.10.: Light field in the angular spectrum representation before and after interacting with an absorbing polarizer. (a) The phase of the electric field of a circularly polarized Gaussianqlight beam as a function of the transverse wave vector k = k (κx , κy , 1 − κ2x − κ2y ).

(b) Visualization of the electric field vector before and after interaction with the polarizer (τ1 = 0, τ2 = 1, Pˆ = y ˆ) aligned such that horizontally polarized light is transmitted. Note that, here, the transmitted field is in-phase and no k-vector depend effect occurs. (c) Electric field of the transmitted horizontally polarized light field. The color-coded phase is completely flat. As a result, the field in position space is not displaced. (d) Visualization of the electric field vector before and after interaction with the polarizer (τ1 = 0, τ2 = 1, Pˆ = x ˆ0 ) aligned such that vertically polarized light is transmitted. As in figure 3.8, the phase of the transmitted field depends on the orientation of the k-vector. (e) Electric field of the transmitted vertically polarized light field in k-space. The color-coded phase depends linearly on κy and, thus, the transmitted field in real space is displaced in the y ˆ-direction.

Taking into account that Pzin = 2Pzout , one sees that the transverse angular momentum Jxout = J out · x ˆ0 = Pzin 0

λσ sin(θ) = Jxin0 4π

(3.20)

in the reference frame aligned with the polarizer is conserved. This shows that the beam shift, we have observed is indeed a manifestation of transverse angular momentum.

40

Figure 3.11.: Angular momentum of a light beam evaluated in a tilted reference frame before and after interaction with a tilted polarizer. The absorbing axis x0 of the polarizer is oriented horizontally at an angle of θ = 60◦ with respect to the direction z of beam propagation. The incident beam is circularly polarized and its angular momentum J in is directed along the z-axis. After interaction with the polarizer, the extrinsic angular momentum J out is perpendicular to propagation out direction z. However, the transverse angular momentum J in x0 = J x0 relevant for the geometric SHEL is conserved.

41

42

Conclusion and outlook This thesis is centred around the experimental demonstration of the geometric spin Hall effect of light. In particular, this novel beam shift phenomenon has been studied for a beam of light passing across a tilted polarizing interface. To this end, we have designed and set up a physical implementation of such an interface. A glass polarizer made of oriented, elongated metal nano-particles was found suitable for the purpose. This polarizer was submerged in a tank filled with an index-matching liquid and characterized extensively. Since the sought beam shift depends subtly on the type and orientation of the polarizer used, it was of utmost importance to establish a reasonable model compatible with the observation. In this work, three relevant models were presented: A simple geometric model provided a useful qualitative approximation for both the nanoparticle polarizer and other polymer-film based products. Our phenomenological model accurately predicted the observed beam shift. And, finally, the microscopic approach connected the observation to a physical description of the interaction. A polarizer was found suitable for our purpose if its operation on a circularly polarized beam yields a geometric phase term in Fourier space. This wave vector dependent phase was shown to result in a displacement in real space. In principle, a wide range of possible polarizers share this property including those described by the aforementioned models. Nevertheless, those models differ and our generic calculation of beam shifts stresses that it is vital to choose a suitable model in order to predict beam shifts occurring at a real-world polarizer correctly. After having established a suitable polarizer model, an experiment was set up to measure the polarization-dependent displacement of the transmitted beam. The observation confirmed the geometric spin Hall effect of light as predicted by our theory. In particular, the characteristic dependence on the incident polarization and the tangent of the tilting angle was demonstrated. The theory presented in this work shows that real-world polarizers with finite extinction ratios give rise to a plethora of beam shift effects. As a consequence of the imperfect nature of the polarizer used, the displacements observed experimentally so far are slightly smaller than expected from an idealized calculation. Interestingly, such imperfect polarizers can also incur significantly larger beam shifts if used in a different configuration (as shown in section 3.3). Using a suitable combination of incident and detected state of polarization, a “giant” geometric spin Hall effect exceeding 10 wavelengths was predicted (Figure 3.5(c)) and is waiting for experimental confirmation. While this theses focusses on collimated light beams, where transverse angular momentum manifests itself as a beam shift, related phenomena can be observed

43

with a tightly focussed spot. In fact, there exists a state of the light field with purely transverse angular momentum [Banzer13]. Here, the interplay between the polarization structure of a suitable tailored vector beam yields a distorted field in the focal plane connected to the geometric SHEL [Neugebauer13]. Despite the novelty of the topic, there exist already theoretical works building upon the geometric SHEL. For example, Kong et. al. study the effect of orbital angular momentum on this geometric Hall effect of light [Kong12]. And, Bliokh and Nori, propose a relativistic Hall effect, occurring for a light beam in free space, observed from a moving reference frame [Bliokh12b]. They note that this relativistic effect is formally equivalent to the geometric SHEL and provide a convenient geometric interpretation.

44

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A. Publications A.1. Geometric Spin Hall Effect of Light at polarizing interfaces

J. Korger, A. Aiello, C. Gabriel, P. Banzer, T. Kolb, C. Marquardt, and G. Leuchs. Geometric Spin Hall Effect of Light at polarizing interfaces. arXiv:1102.1626. Applied Physics B, 102(3), 427–432, 2011

51

52

Geometric Spin Hall Effect of Light at Polarizing Interfaces Jan Korger · Andrea Aiello · Christian Gabriel · Peter Banzer · Tobias Kolb · Christoph Marquardt · Gerd Leuchs

February 6, 2011

Abstract The geometric Spin Hall Effect of Light (geometric SHEL) amounts to a polarization-dependent positional shift when a light beam is observed from a reference frame tilted with respect to its direction of propagation. Motivated by this intriguing phenomenon, the energy density of the light beam is decomposed into its Cartesian components in the tilted reference frame. This illustrates the occurrence of the characteristic shift and the significance of the effective response function of the detector. We introduce the concept of a tilted polarizing interface and provide a scheme for its experimental implementation. A light beam passing through such an interface undergoes a shift resembling the original geometric SHEL in a tilted reference frame. This displacement is generated at the polarizer and its occurrence does not depend on the properties of the detection system. We give explicit results for this novel type of geometric SHEL and show that at grazing incidence this effect amounts to a displacement of multiple wavelengths, a shift larger than the one introduced by Goos-H¨anchen and Imbert-Fedorov effects.

PACS 42.25.Ja · 42.79.Ci · 42.25.Gy

Jan Korger · Andrea Aiello · Christian Gabriel · Peter Banzer · Tobias Kolb · Christoph Marquardt · Gerd Leuchs Max-Planck-Institute for the Science of Light, Guenther-ScharowskyStr. 1/Bau 24, 91058 Erlangen, Germany Tel: +49 9131 6877 125 Fax: +49 9131 6877 199 E-mail: [email protected] Jan Korger · Andrea Aiello · Christian Gabriel · Peter Banzer · Tobias Kolb · Christoph Marquardt · Gerd Leuchs Institute of Optics, Information and Photonics, University ErlangenNuremberg, Staudtstr. 7/B2, 91058 Erlangen, Germany

1 Introduction It is well-known that a beam of light transmitted through or reflected from a dielectric interface undergoes a polarization-dependent shift of its spatial intensity distribution. The so-called Goos-H¨anchen (GH) [1, 2] effect amounts to a longitudinal shift, i.e. a displacement in the plane of incidence, while the Imbert-Fedorov (IF) shift [3] can be observed transverse to this plane. These positional shifts are connected with angular counterparts [4, 5]. Both, the GH [6– 10] and the IF shift [11–14] have been verified experimentally in a number of configurations while also the theoretical understanding of those effects has advanced significantly [15–17]. The IF shift is also known as the Spin Hall Effect of Light (SHEL) [18–20] due to its resemblance to the Spin Hall Effect in solid state physics. It amounts to a displacement of a circularly polarized beam perpendicular to the plane of incidence, where the direction depends on the beam’s helicity or photon spin. Consequently, a linearly polarized beam will split into components of different helicity. The geometric Spin Hall Effect of Light [21] is a novel phenomenon, which like SHEL amounts to a spin-dependent shift or split of the intensity distribution of an obliquely incident light beam. This effect depends significantly on the geometric properties of the detection system and, beyond the detection process, no light-matter interaction is required. This article is structured as follows: In the following section the original geometric SHEL is reviewed and results not explicitly given in [21] are provided. In sections 3 and 4 we introduce a theoretical model for an arbitrarily oriented planar polarizing interface [22, 23] and provide a scheme for its experimental realization. Finally, we find that a light beam crossing a tilted polarizing interface undergoes a shift twice as large as the one found in the case of a tilted reference frame. Therefore, the methods presented in this article lead

2

Jan Korger et al.

x´

x xˆ + y yˆ is a two-dimensional position vector. This decomposition is depicted in Fig. 2. Analogously to (2), we decompose the barycenter hyTD iED of the energy density as

x

θ

z´ z

RR

ED

hyTD i

= RR

y|E(r⊥ )|2 dx dy = ∑ wl ∆l , |E(r⊥ )|2 dx dy l=x,y,z

Fig. 1 Geometry of the problem: A Gaussian laser beam (red) propagating in direction of zˆ 0 is observed in a plane (ˆx, yˆ ) tilted with respect to zˆ 0 . The direction yˆ = yˆ 0 is perpendicular to the drawing plane.

where we define

to a straightforward measurement of the geometric Spin Hall Effect of Light.

wl := RR

The geometric Spin Hall Effect of Light [21] occurs when a circularly polarized beam of light is observed in a plane not perpendicular to its direction of propagation. This effect amounts to a spatial shift of the intensity distribution with the intensity being defined as the flux of the Poynting vector through the detector plane. We stress that the geometric SHEL only depends on the geometry of the setup (Fig. 1), the state of polarization and the effective response function of the detector. Here we give explicit results for a fundamental Gaussian light beam traveling in direction of kˆ = sin(θ ) xˆ + cos(θ ) zˆ detected in the plane (ˆx, yˆ ). The normal zˆ to the detector surface and the propagation vector kˆ ∦ zˆ unambiguously define a plane of incidence. As shown by Aiello et al. [21], the intensity barycenter or centroid of a circularly polarized beam is shifted in direction of yˆ perpendicular to the plane of incidence. This displacement is equal to (1)

The superscript S indicates that the centroid was evaluated with respect to the Poynting vector flux while the subscript TD refers to a tilted detector. The shift depends on the helicity σ = ±1 (for left or right hand circular polarization) and is prominent at grazing incidence θ → 90◦ where it amounts to a displacement larger than the wavelength λ . In [21] it was noted that the energy density
(ED) distribution exhibits no such effect hyTD iED = 0 , which underlines the dependence on the detector response. The apparent discrepancy can be understood by decomposing the electric field energy density u(r⊥ ) = |E(r⊥ )|2 =

∑

l=x,y,z

|El (r⊥ )|2

(2)

into terms depending on one Cartesian component of the electric field in the detector reference frame (ˆx, yˆ , zˆ ) only. u(r⊥ ) is a distribution in the observation plane and r⊥ =

|El (r⊥ )|2 dx dy |E(r⊥ )|2 dx dy

(4)

as the relative weight of the field component |El |2 and RR

2 Geometric Shift in a Tilted Reference Frame

λ hyTD iS = σ tan(θ ). 4π

RR

(3)

∆l = RR

y|El (r⊥ )|2 dx dy |El (r⊥ )|2 dx dy

(5)

as the contribution of this component to the total shift. From a straightforward application of equations (4) and (5) to the electric field distribution E(r) of a circularly polarized Gaussian light beam one finds λ σ tan(θ ) + O(θ02 ), 2π ∆y = 0 + O(θ02 ),

∆x =

∆z = −

λ σ cot(θ ) + O(θ02 ), 2π

(6a) (6b) (6c)

and within the same approximation 1 cos2 (θ ), 2 1 wy = , 2 1 2 wz = sin (θ ), 2

wx =

(7a) (7b) (7c)

where θ0 = 2/(k w0 ) = λ /(π w0 ) is the angular divergence of the beam [24]. Substituting equations (6) and (7) into (3) we verify that the barycenter of a light beam’s energy density hyTD iED =

∑

l=x,y,z

wl ∆l = 0

(8)

does not shift under rotation of the reference frame. This result underlines the scalar nature of the energy density. We remind the reader that the Poynting vector flux through the detector surface sz (r⊥ ) = s(r⊥ ) · zˆ ∝ [E(r⊥ ) × B∗ (r⊥ )] · zˆ ,

(9)

where zˆ is the surface normal, is a distribution different from u(r⊥ ) and exhibits a net shift hyTD iS ∝ ∆x , where ∆x is given in (6a).

(10)

Geometric Spin Hall Effect of Light at Polarizing Interfaces

y/w0

y/w0

0

−1

1

(b)

−10

0

0 x/w0 1 2 3 103 · |Ex|2 / |E0|2

10

0

−1

0

1

(c)

y/w0

1

(a)

3

−10

0 x/w0 1

0

−1

10

2 3 4 101 · |Ey|2 / |E0|2

−10

0

0 x/w0 1

10

2 3 4 101 · |Ez|2 / |E0|2

ˆ zˆ ) = 85◦ ) on Fig. 2 Electric field energy density distribution of a Gaussian light beam (circular polarization) impinging obliquely (θ = ∠(k, a detector. The components |Ex (r⊥ )|2 , |Ey (r⊥ )|2 , and |Ez (r⊥ )|2 in the detector reference frame (ˆx, yˆ , zˆ ) are shown on color scales. |E0 |2 is a common normalization constant and w0 is the beam waist. (a) |Ex (r⊥ )|2 is clearly shifted in the positive yˆ direction. (b) |Ey (r⊥ )|2 exhibits no such shift. (c) While not visible in this pictorial representation, |Ez (r⊥ )|2 is shifted in the negative yˆ direction. Note that the relative weight wz of this component is more than two orders of magnitude larger than wx .

This connects the geometric SHEL with the fundamental question about the local response of a position-sensitive detector. The definition of the Poynting vector flux as the intensity is motivated by Poynting’s theorem. However, this choice is debatable as the theorem does not define the local Poynting vector unambiguously [25] and the definition given in equation (9) depends on the state of polarization. Contrarily, the response function of a real polarization-independent detector is more likely to be isotropic, i.e. to depend on u = |Ex |2 + |Ey |2 + |Ez |2 , and thus yield hyTD iED = 0. The shift ∆x can be detected directly if a detection scheme is used where a plane of observation (ˆx, yˆ ) can be chosen arbitrarily and the effective response function depends on |Ex |2 but not on |Ez |2 . The polarization-dependent absorption in semiconductor quantum wells [26–28] or single molecules [29, 30] can in principle be used to build a suitable detector. In the remaining part of this article we develop an alternative strategy to measure the characteristic shift caused by the geometric SHEL.

3 The Ideal Polarizer Model The operation performed by a polarizing optical element is commonly only determined for normally incident beams which can be approximated as planar wave fronts. In this case the action of a polarizer is described as a projection within a plane perpendicular to the direction of propagation 1 kˆ = |k| k. This simple model fails to describe the operation of a polarizing element when no such assumptions about the light field are made, as it is the case in this article where we deal with obliquely incident beams.

(a) Ex´

x´ k

θ

P

x

Ex´

z´

k

z (b)

incident beam

Polarizing Interface Dielectric Plate Index Matching Liquid

Fig. 3 (a) Interaction of a plane wave propagating in direction of zˆ 0 with a tilted polarizer described by Pˆ = xˆ = cos(θ ) xˆ 0 + sin(θ ) zˆ 0 . Before and after passing the polarizing element the electric field is perˆ (b) Thin film polarizer submerged in a tank of index pendicular to k. matching liquid. This scheme allows to study a tilted polarizing interface eliminating effects of physical boundaries.

To overcome this limitation, Fainman and Shamir (FS) have proposed a model describing an arbitrarily oriented ideal polarizer [22]. They introduce a three-dimensional complex-valued unit vector Pˆ and describe the action of a polarizer as follows: The electric field vector E(k) of each plane wave of the incident beam’s angular spectrum is projected onto eˆ P , a unit vector perpendicular to k: ˆ ˆ kˆ · P) ˆ kˆ × (Pˆ × k) Pˆ − k( ˆ =q eˆ P (k) =q ˆ 2 ˆ 2 1 − (kˆ · P) 1 − (kˆ · P)  ∗  ˆ eˆ P (k) ˆ · E(k) E(k) → eˆ P (k)

(11) (12)

The model is constructed such that the operation is idempotent and does not change k (Fig. 3a). A remarkable characteristic of the FS polarizer is that rotation (around yˆ ) has no effect on a single plane wave if Pˆ

4

is parallel or perpendicular to yˆ . Hence, an ideal polarizing element cannot be used to cause an imbalance between the weights wx and wz of the electric field component parallel and perpendicular to the polarizer surface. However, as we will show in section 5, the rotation gives rise to an effect on bounded beams similar to the geometric SHEL in a tilted reference frame. In the following section we will propose an experimental realization of an arbitrarily oriented polarizing interface. 4 Experimental realization of a universal polarizing interface We propose a scheme using only off-the-shelf optical components to study the interaction of a light beam with a polarizing element in any geometry (Fig. 3b). For this purpose we model a real polarizer as a composite device consisting of an infinitely thin polarizing interface sandwiched between dielectric plates with a refractive index n > 1. Since commercial thin film polarizers are typically protected from the environment by either a substrate or a coating on each face, our model is close to the realistic scenario. The interaction of the light field with an air-dielectric boundary is polarization-dependent and changes the direction of propagation kˆ of a plane wave. Those well-known effects, described by Snell’s law of refraction and Fresnel’s formulas [31], are caused by the refractive index step. At grazing incidence refraction is so severe that inside the front dielectric plate the angle between the direction of propagation and the surface normal is always significantly smaller than the corresponding angle θ in air. Therefore it is desirable to eliminate the change of the refractive index at the physical boundary. This can be done, for example, by embedding the glass-polarizer-glass system in an indexmatched environment. As a side effect this also eliminates unwanted Imbert-Fedorov and Goos-H¨anchen shifts. It is not common for vendors to specify the behavior of polarization optics under non-normal incidence. Measurements in our laboratory using the proposed scheme indicate that the FS model is suitable to describe a tilted polarizing interface. A detailed investigation will be reported elsewhere. 5 Geometric Shift at a Polarizing Interface In section 2 we described the geometric SHEL as an effect which occurs for a light beam in vacuum when the plane of observation is tilted with respect to the direction of propagation. The predicted shift depends on specific assumptions about the detection process and, therefore, cannot be easily verified. In this section we shall show that an ideal polarizer performs an operation on a light beam that amounts to

Jan Korger et al.

the characteristic geometric SHEL shift independent of the properties of the detection system. As in the case of the tilted detector, we assume the incident beam to travel in direction of kˆ =: zˆ 0 and to have a fundamental Gaussian profile with its barycenter at hx0 i = 0 and hy0 i = 0, where (ˆx0 , yˆ 0 , zˆ 0 ) is the beam’s natural reference frame and yˆ 0 coincides with yˆ (geometry as in Fig. 3). The polarizer shall be oriented along Pˆ = xˆ = cos(θ ) xˆ 0 + sin(θ ) zˆ 0 .

(13)

Using dimensionless coordinates x˜ = x0 /w0 , y˜ = y0 /w0 , z˜ = z0 /L and r0 = (x, ˜ y, ˜ z˜) where w0 denotes the beam waist and L = k w20 /2 the Rayleigh length, the fundamental solution of the paraxial scalar wave equation is:  2  1 x˜ + y˜2 0 ψ(r ) = exp − (14) 1 + i˜z 1 + i˜z Let uˆ = √12 (ˆx0 ± i yˆ 0 ) be a complex unit vector denoting left or right hand circular states of polarization. The electric field of a fundamental Gaussian beam can be written as    i θ0 i 2˜z 0 uˆ + zˆ (uˆ · ∇⊥ ) ψ(r0 ), (15) E(r ) ∝ exp 2 θ02 where θ0 = 2/(k w0 ) = λ /(π w0 ) is the angular spread of the beam and ∇⊥ = (∂x˜ , ∂y˜ ) is the transverse gradient operator [32]. To apply the FS polarizer model (12), the electric field (15) must be expressed in its angular spectrum representation E(k) and the beam after interacting with the tilted polarizing element becomes: E(r0 ) =

ZZZ

  ˆ eˆ ∗P (k) ˆ · E(k) d3 k exp(i k · r0 )ˆeP (k)

(16)

From the electric field distribution (16), we calculate the energy density of a light beam after passing through a tilted polarizer. The Cartesian components thereof are depicted in Fig. 4. Unlike in section 2, in this case the evaluation is perˆ formed in the beam reference frame (ˆx0 , yˆ 0 , zˆ 0 ) where zˆ 0 = k. Decomposing the energy density barycenter hyTP iED as in (3) one finds ∆ x0 θ0 = σ tan(θ ) + O(θ02 ), w0 2 ∆y0 = 0 + O(θ02 ), and w0 ∆z0 = 0 + O(θ02 ), w0

(17a) (17b) (17c)

where σ = ±1 is the helicity of the beam. Since we observe a collimated light beam in its natural reference frame after passing through a linear polarizer, the weights wy0 = 0 + O(θ02 ) and

(18a)

wz0 = 0 + O(θ02 )

(18b)

Geometric Spin Hall Effect of Light at Polarizing Interfaces

y´/w0

y´/w0

0

−1

0

1

(b)

−1

0 x´/w0 2

1

4 6 8 101 · |Ex´|2 / |E0|2

0

−1

0

1

(c)

y´/w0

1

(a)

5

−1

0 x´/w0

1

5 10 15 104 · |Ey´|2 / |E0|2

0

−1

−1

0

0 x´/w0

1

5 10 15 106 · |Ez´|2 / |E0|2

20

Fig. 4 Electric field energy
density distribution of a Gaussian light beam (circular polarization) after interacting with a tilted polarizer Pˆ = cos(85◦ ) xˆ 0 + sin(85◦ ) zˆ 0 . The components |Ex0 (r⊥ )|2 , |Ey0 (r⊥ )|2 , and |Ez0 (r⊥ )|2 in the beam’s natural reference frame (ˆx0 , yˆ 0 , zˆ 0 ) are shown on color scales. |E0 |2 is a common normalization constant and w0 is the beam waist. (a) |Ex0 (r⊥ )|2 is shifted as in Fig. 2a. (b), (c) |Ey0 (r⊥ )|2 and |Ez0 (r⊥ )|2 are not shifted and their relative weights are negligible.

6 Conclusion

vanish and, consequently, wx0 = 1 + O(θ02 ).

(18c)

The centroid of a circularly polarized beam transmitted across a tilted polarizer is thus hyTP iED = hyTP iS = ∆x =

λ σ tan(θ ) + O(θ02 ) 2π

(19)

and can be measured with any detector sensitive to any weighted sum of |Ex0 |2 , |Ey0 |2 , and |Ez0 |2 if the weight of the first term does not vanish. Standard detectors such as photodiodes and CCD cameras certainly meet this requirement. We stress that equation (19) resembles the original result (1). The displacement introduced by the tilted polarizer hyTP iS = 2hyTD iS

(20)

is twice the one found for the Poynting vector flux through a tilted plane of observation. Furthermore, this article gives a straightforward recipe to measure the shift. Since equation (5) from [21] is generally valid, both shifts are connected to a transverse angular momentum, which occurs when the angular momentum calculated in the local frame attached to the light beam is projected upon a global frame tilted with respect to the former. For the geometric SHEL (occurring at a tilted detector), the projection is implicitly given by the definition of intensity as the flux Sz = S · zˆ of the Poynting vector across the detector surface. Conversely, in the case of the tilted polarizing interface, as described in this section, the projection is caused by the polarizer. Therefore, we can conclude that both shifts arise because of the projection of the intrinsic longitudinal angular momentum of a circularly polarized light beam onto a tilted reference frame. We remind the reader that beyond those geometric projections, no physical interaction occurs.

First, the original geometric Spin Hall Effect of Light, as described by Aiello et al., was illustrated using an explicit decomposition of a light beam’s energy density in a tilted reference frame. We showed that in order to observe this effect, which occurs in vacuum and amounts to a polarization-dependent shift, a suitably tailored detection system is required. Then, a novel type of geometric SHEL occurring at a polarizing interface was introduced. To this end we discussed a theoretical model for an ideal polarizer and suggested an experimental implementation thereof. The light field of a collimated laser beam transmitted across such a polarizer was evaluated. In the case of the polarizing interface being tilted with respect to the direction of propagation, a beam displacement resembling the original geometric SHEL was found. This shift does not depend on the detection process and can be measured in a straightforward way by using the scheme proposed in this article. The effect derived in our work is unavoidable when a circularly polarized light beam passes through a polarizing interface tilted with respect to the direction of propagation. This underlines the importance of the geometric SHEL as polarization is a fundamental property of the light field and numerous optical devices are polarization-dependent.

Acknowledgements A.A. acknowledges support from the Alexander von Humboldt foundation. The final publication is available at www. springerlink.com.

6

Jan Korger et al.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

29. 30. 31. 32.

F. Goos, H. H¨anchen, Annalen der Physik 436(7-8), 333 (1947) K. Artmann, Annalen der Physik 2, 87 (1948) C. Imbert, Physical Review D 5(4), 787 (1972) M. Merano, A. Aiello, M.P. van Exter, J.P. Woerdman, Nature Photonics 3(6), 337 (2009) A. Aiello, M. Merano, J.P. Woerdman, Physical Review A 80(6), 061801(R) (2009) F. Bretenaker, A.L. Floch, L. Dutriaux, Physical Review Letters 68, 931 (1992) O. Emile, T. Galstyan, A.L. Floch, F. Bretenaker, Physical Review Letters 75(8), 1511 (1995) B. Jost, A. Al-Rashed, B. Saleh, Physical Review Letters 81(11), 2233 (1998) C. Bonnet, D. Chauvat, O. Emile, F. Bretenaker, A.L. Floch, L. Dutriaux, Optics Letters 26(10), 666 (2001) M. Merano, A. Aiello, G.W. ’t Hooft, M.P. van Exter, E.R. Eliel, J.P. Woerdman, Optics Express 15(24), 15928 (2007) F. Pillon, H. Gilles, S. Fahr, Applied Optics 43(9), 1863 (2004) R. Dasgupta, P. Gupta, Optics Communications 257(1), 91 (2006) C. Menzel, C. Rockstuhl, T. Paul, S. Fahr, F. Lederer, Physical Review A 77(1), 013810 (2008) J. M´enard, A.E. Mattacchione, M. Betz, H.M. van Driel, Optics Letters 34(15), 2312 (2009) P. Berman, Physical Review E 66(6), 067603 (2002) A. Aiello, J.P. Woerdman, Optics Letters 33(13), 1437 (2008) K.Y. Bliokh, I.V. Shadrivov, Y.S. Kivshar, Optics Letters 34(3), 389 (2009) M. Onoda, S. Murakami, N. Nagaosa, Physical Review Letters 93(8), 083901 (2004) K. Bliokh, Y. Bliokh, Physical Review Letters 96(7), 073903 (2006) O. Hosten, P. Kwiat, Science 319(5864), 787 (2008) A. Aiello, N. Lindlein, C. Marquardt, G. Leuchs, Physical Review Letters 103(10), 100401 (2009) Y. Fainman, J. Shamir, Applied Optics 23(18), 3188 (1984) A. Aiello, C. Marquardt, G. Leuchs, Optics Letters 34(20), 3160 (2009) L. Mandel, E. Wolf, Optical coherence and quantum optics (Cambridge University Press, Cambridge, 1995) M.V. Berry, Journal of Optics A: Pure and Applied Optics 11(9), 094001 (2009) J.S. Weiner, D.A.B. Miller, D.S. Chemla, T.C. Damen, C.A. Burrus, T.H. Wood, A.C. Gossard, W. Wiegmann, Applied Physics Letters 47(11), 1148 (1985) J.S. Weiner, D.S. Chemla, D.A.B. Miller, H.A. Haus, A.C. Gossard, W. Wiegmann, C.A. Burrus, Applied Physics Letters 47(7), 664 (1985) G.K. Rurimo, M. Schardt, S. Quabis, S. Malzer, C. Dotzler, A. Winkler, G. Leuchs, G.H. D¨ohler, D. Driscoll, M. Hanson, A.C. Gossard, S.F. Pereira, Journal of Applied Physics 100(2), 023112 (2006) B. Sick, B. Hecht, L. Novotny, Physical Review Letters 85(21), 4482 (2000) L. Novotny, M. Beversluis, K. Youngworth, T. Brown, Physical Review Letters 86(23), 5251 (2001) E. Hecht, Optik, 4th edn. (Oldenbourg, M¨unchen, 2005) H.A. Haus, American Journal of Physics 61(9), 818 (1993)

A.2. Observation of the geometric spin Hall effect of light

J. Korger, A. Aiello, V. Chille, P. Banzer, C. Wittmann, N. Lindlein, C. Marquardt, and G. Leuchs. Observation of the geometric spin Hall effect of light. arXiv:1303.6974. Phys. Rev. Lett. 112, 113902, 2014

59

60

Observation of the geometric spin Hall effect of light Jan Korger,1, 2 Andrea Aiello,1, 2, ∗ Vanessa Chille,1, 2 Peter Banzer,1, 2 Christoffer Wittmann,1, 2 Norbert Lindlein,2 Christoph Marquardt,1, 2 and Gerd Leuchs1, 2 2

1 Max Planck Institute for the Science of Light, Erlangen, Germany Institute for Optics, Information and Photonics, University Erlangen-Nuremberg, Germany (Dated: October 1, 2013)

The spin Hall effect of light (SHEL) is the photonic analogue of the spin Hall effect occurring for charge carriers in solid-state systems. This intriguing phenomenon manifests itself when a light beam refracts at an air-glass interface (conventional SHEL), or when it is projected onto an oblique plane, the latter effect being known as geometric SHEL. It amounts to a polarization-dependent displacement perpendicular to the plane of incidence. Here, we experimentally demonstrate the geometric SHEL for a light beam transmitted across an oblique polarizer. We find that the spatial intensity distribution of the transmitted beam depends on the incident state of polarization and its centroid undergoes a positional displacement exceeding one wavelength. This novel phenomenon is virtually independent from the material properties of the polarizer and, thus, reveals universal features of spin-orbit coupling.

Already in 1943, Goos and H¨anchen observed that the position of a light beam totally reflected from a glass-air interface differs from metallic reflection [1]. This is the most well-known example of a longitudinal beam shift occurring at the interface between two optical media. In honour of their seminal work, any such deviation from geometrical optics occurring in the plane of incidence, is referred to as a Goos-H¨anchen shift. Conversely, a similar shift occurring in a direction perpendicular to the plane of incidence is known as Imbert-Fedorov shift [2, 3]. These phenomena generally depend both on properties of the incident light beam and the physical properties of the interface [4]. Goos-H¨anchen and Imbert-Fedorov shifts have been observed at dielectric [5], semi-conductor [6] and metal [7] interfaces. The Imbert-Fedorov shift [2, 3, 8] is an example of the spin-Hall effect of light (SHEL), the photonic analogue of spin Hall effects occurring for charge carriers in solid-state systems [9–11]. SHEL is a consequence of the spin-orbit interaction (SOI) of light, namely the coupling between the spin and the trajectory of the optical field [9, 10, 12–22]. All electromagnetic SOI phenomena in vacuum and in locally isotropic media can be interpreted in terms of the geometric Berry phase and angular momentum dynamics [23]. For a freely propagating paraxial beam of light, SOI effects vanish unless a relevant breaking of symmetry occurs. A typical example of such a symmetry break is the interaction of the beam with an oblique surface as in ordinary light refraction processes (Figure 1). Although the resulting phenomena essentially depend on the type of the interaction with the surface [24], there are some common characteristics that reveal universality in SOI of light due to the geometry and the dynamical angular momentum aspects of the problem. Amongst the various observable effects resulting from the beamsurface interaction, the so-called geometric Hall effect of light is virtually independent from the properties of the

QWP

polarizer quadrant detector

Figure 1. Pictorial representation of the geometric SHEL. After the quarter wave-plate (QWP) the beam is circularly polarized and passes through a polarizer tilted by an angle θ. The center of the intensity distribution of the transmitted beam appears shifted with respect to the axis of the incident beam. Such a shift can be measured by a quadrant detector put behind the polarizer. The reference frame {x0 , y 0 , z 0 } is aligned with the polarizer surface.

surface [19, 25–27] and, therefore, represents the ideal candidate for studying above-mentioned universal features. It amounts to a shift of the centroid of the intensity distribution represented by Poynting-vector flow of the beam across the oblique surface of a tilted detector. Direct observation of this effect as originally proposed depends critically on the detector’s response to the light field. However, the question whether the response function of a real detector is indeed proportional to the Poynting vector density is subject to a long-standing debate [28–30]. In this work, we implement an alternative scheme [25], in which the occurrence of the beam shift is independent of the detector response. To this end, we send a circularly

2 (a)

δ

xˆ

k aˆ

k0

ˆ x'

x'

θ

x

(b) δ tanθ

x

y

aˆ

y'

y

z

z'

Figure 2. The geometric SHEL at a polarizing interface. (a) The horizontal dark grey plane represents the polarizing interface with the Cartesian reference frame {x0 , y 0 ≡ y, z 0 } attached to it. The axis x ˆ0 is taken parallel to the absorbing axis of the polarizing interface. θ ∈ [0, π/2) denotes the angle of incidence. The central wave vector k0 of the incident beam defines the direction of the axis zˆ of the frame {x, y, z} attached to the beam. It is instructive to study a wave vector k in the z-y-plane, rotated by an angle δ  1 with respect to ˆ ⊥ k, representing the direction of k0 . (b) The unit vector a the absorbed component of the incident field, lies in the common plane of k and x ˆ 0 (coloured light brown in the figure) and can be obtained, in the first-order approximation, from the rotation by the angle δ tan θ around k, of the unit vector x ˆ.

polarized beam of light across a tilted polarizing interface to demonstrate a novel kind of geometric Hall effect, in which the centroid of the resulting linearly polarized transmitted beam undergoes a spin-induced transverse shift up to several wavelengths. This is a spatial shift, which is independent of the distance the beam propagated after interaction with the polarizer. Our approach is different from the early proposal [19] in that it is observable with standard optical detectors. Furthermore, it is different from the SHEL occurring in a beam passing through an air-glass interface [10] since this geometric SHEL is practically independent of Snell’s law and the Fresnel formulas for the interface. As for conventional SHEL, the physical origin of this geometric version resides in the SOI of light. In fact, we can describe this effect in terms of the geometric phase generated by the spin-orbit interaction as follows: Think of the monochromatic beam as a superposition of many interfering plane waves with the same wavelength and different directions of propagation. When the beam passes through the polarizing interface, the plane wave components with different orientations of their wave vectors acquire different geometric phases determined by distinct local projections yielding to effective “rotations” of the

polarization vector around the wave vector. The interference of these modified plane waves produces a redistribution of the intensity spatial profile of the beam resulting in a spin-dependent transverse shift of the intensity centroid. This effect can be better understood with the help of Figure 2 that illustrates the geometry of the problem. The incident monochromatic beam is made of many plane wave components with wave vectors k spreading around the central one k0 = kˆ z , which represents the main direction of propagation of the beam, where |k0 | = k = |k|. For a well-collimated beam, the angle δ between an arbitrary wave vector k and the central one k0 is, by definition, small: δ  1. In the first-order approximation with respect to δ, we consider k = k(ˆ z cos δ − y ˆ sin δ) ∼ = ∼ k(ˆ z + κy y ˆ), where κy ≡ ky /k = − sin δ = −δ. This wave vector is not the general one, but, due to the transverse nature of the phenomenon, it is sufficient to restrict the discussion to wave vectors lying in the yz plane. The either left- (σ = +1) or right-handed (σ = −1) circular polarization of the incident √beam is determined by the unit vector u ˆ σ = (ˆ x + iσˆ y )/ 2 globally defined with respect to the axis zˆ of the beam frame {ˆ x, y ˆ, zˆ}. However, from Maxwell’s equations it follows that the divergence of the electric field of a light wave in vacuum is zero. This requires that the polarization vector ˆ eσ (k) of each plane wave component of wave vector k must necessarily be transverse, namely k · ˆ eσ (k) = 0. This requirement is clearly √ not satisfied by u ˆ σ for which one has k · u ˆ σ /k ∼ = iσκy / 2 6= 0. Anyhow, the transverse nature of the light field can be easily restored by subtracting from u ˆ σ its longitudinal component: u ˆσ → ˆ eσ (k) ∝ √ u ˆ σ − k (k · u ˆ σ ) /k 2 ∼ ˆ σ − (iσ κy / 2)ˆ z . Now that we = u have properly modeled the polarization of the incident field, let us see how it changes when the beam crosses the polarizing interface. A linear polarizer is an optical device that absorbs radiation polarized parallel to a given direction, say x ˆ0 , and transmits radiation polarized perpendicular to that direction. The electric field of each plane wave component of the beam sent through the polarizing interface, changes according to the projection rule e ˆσ (k) → e ˆσ (k) − a ˆ(ˆ a·e ˆσ (k)) = ˆt(ˆt · e ˆσ (k)), where a = x ˆ 0 − k(k · x ˆ0 )/k 2 is the effective absorbing axis with a ˆ = a/|a| ∼ = x ˆ−y ˆ κy tan θ, and tˆ = a ˆ × k/k is the effective transmitting axis. The amplitude of √ the transmitted plane wave √ is ˆt ·ˆ eσ (k) ∝ (1−iσκy tan θ)/ 2 ∼ = exp(−iσκy tan θ)/ 2, where an irrelevant κy -independent overall phase factor has been omitted. Therefore, as a result of the transmission, the amplitude √ of each plane wave component is reduced by a factor 1/ 2 and multiplied by the geometric phase term exp(−iσκy tan θ). Here, the “tan θ” behavior of the phase, characteristic of the geometric SHEL [19], is in striking contrast to the typical “cot θ” angular dependence of the conventional SHEL as, e. g., the Imbert-Fedorov shift [31]. However, the spin-orbit in-

3 teraction term σκy , expressing the coupling between the spin σ and the transverse momentum κy of the beam, is characteristic of both phenomena, thus revealing their common physical origin. According to the Fourier transform shift theorem [32], a linear phase shift in the wave vector domain introduces a translation in the space domain. Using this theorem, we can show that the geometric phase term exp(−iσκy tan θ) leads to a beam shift along the y ˆ-direction. This can be explicitly demonstrated by writing the incident circularly polarized paraxial beam as Ψin (y) = u ˆ σ ψσin (y), where only the dependence on the relevant transverse coordinate y has been displayed. With ψeσin (κy ) we denote the Fourier transform of ψσin (y). The two functions are connected by the simple relation ψσin (y) = R in ψeσ (κy ) exp(ikκy y)dκy . After transmission across the polarizing interface, the Fourier transform ψeσin (κy ) of √ ψσin (y) changes to ψeσin (κy ) exp(−iσκy tan θ)/ 2 and the output field can be written as Z 1 out ψσ (y) = √ ψeσin (κy ) exp[ikκy (y − σ tan θ/k)]dκy 2 1 (1) = √ ψσin (y − σ tan θ/k). 2 This last expression clearly shows that the scalar ampli√ tude of the output field is equal, apart from the 1/ 2 factor, to the amplitude of the input field transversally shifted by σ tan θ/k. Finally, the position of the centroid of the transmitted beam is given by R y |ψσout (y)|2 dy σ hyi = R out (2) = tan θ, k |ψσ (y)|2 dy

where k = 2π λ . Eq. (2) as derived above is valid for an ideal polarizer with a high extinction ratio. In this case, the description of the polarizer as a projection onto the effective transmission axis tˆ is adequate. At normal incidence (θ = 0) such polarizers are readily available. However, working with tilted polarizing interfaces, we have to account for the fact that the efficiency of the polarizer diminishes with larger angles θ. As a result of such loss of efficiency, the “tan θ”-dependence of the shift is modified as illustrated in Fig. 5. This is explained in detail in the Supplemental Material, where we introduce a phenomenological model for our real-world polarizer. In the experiment, we use a Corning Polarcor polarizer, made of two layers of elongated and oriented silver nanoparticles embedded in a 25 mm × 25 mm × 0.5 mm glass substrate. Directed absorption from these particles effectively polarizes the transmitted beam. In order to avoid parasitic effects from the glass surfaces (nG = 1.517), we have submerged the polarizer in a tank with index matching liquid (Cargille laser liquid 5610, nL = 1.521). Without this liquid, the effective tilting angle inside

state preparation

Q M PB WP S

M P QW

shift measurement

piezo HW

P

A

HW

It Ib

P

I+

I– S

PB

P

Stokes QW measurement

BS P HW

SMF

Figure 3. Experimental Setup: Simultaneous measurement of the incident state of polarization and the position of a light beam transmitted across a tilted polarizer. State preparation: The relative phase between horizontally and vertically polarized components is modulated using a polarizing beam splitter (PBS), a piezo mirror (M), quarter and half wave plates (QWP, HWP). The beam is spatially filtered using a single-mode fibre (SMF). Stokes measurement: The transmitted port of a beam splitter (BS) is used to monitor the state of polarization. We use the Stokes parameters S3 to distinguish left- and right-hand circular polarization. Shift measurement: The beam is propagated across our sample, a tank containing a glass polarizer and an index-matching liquid, and its position is observed using a quadrant detector. An optional PBS (A) can be employed as an analyzer in front of the detector. The photo currents I+ , I− , It , and Ib are amplified and digitally sampled for 1 s at 50 kHz.

the glass polarizer would be limited by Snell’s law to arcsin(1/nG ) ≈ 41◦ . In our setup (Figure 3), a fundamental Gaussian light beam (λ = 795 nm) is prepared with the state of polarization alternating between left- and right-hand circular. To avoid spatial jitter, the spatial mode is cleaned using a single mode fibre and no active elements are used after the fibre. We collimate the light beam using an aspheric lens (New Focus 5724-H-B) aligned such that the beam waist is at the position of the detector.

In order to simultaneously measure the beam position y and the incident state of polarization, we employ a dielectric mirror (Layertec 103210) as a non-polarizing beam splitter at an angle of incidence of 3◦ . The reflected and transmitted states of polarization coincide ex  within −I b perimental accuracy. The beam centroid y = f IItt −I b

−I− and Stokes parameter S3 = II+ are calculated from + +I− the digitized photo currents. We can measure the calibration factor f in-situ by translating the quadrant detector using a micrometre stage. Since the signal is periodic with the modulation frequency fmod = 29 Hz, we can filter technical noise in a post-processing stage. To this end, the discrete Fourier transform is computed and only spectral components with frequencies equal to fmod and higher harmonics

4

0.8

0.4

0.6

0.3

0.4

0.1

0.0

0.0

(b) 1.0

20

40 60 Tilting Angle θ [°]

80

relative shift Δy (vertical polarization)

0.8 0.6 0.4 0.2 0.0 0

20

40 60 Tilting Angle θ [°]

80

Figure 4. Polarization-dependent beam shifts occurring at a tilted polarizer. Measurement data and theoretical predictions are shown. Both series of shift measurements were repeated five times. We report the mean value and standard deviation of the mean. The dashed blue line shows the theory for a perfect polarizer, while the solid lines were calculated from our phenomenological polarizer model. (a) Overall displacement ∆T of the intensity barycentre after transmission across the polarizer. (b) Displacement ∆y of the vertically polarized intensity component solely. The two measurements ∆T and ∆y differ due to the imperfect nature of our polarizer as discussed in the SM.

geometric SHEL conventional SHEL

(c) (b)

(a)

0.2

0.2

0

shift [μm]

0.5

total shift ΔT

shift [λ]

shift [μm]

(a) 1.0

(d) (e)

0

20

40 60 Tilting Angle θ [°]

80

Figure 5. Theory for the conventional spin Hall effect of light compared to the geometric SHEL. A light beam transmitted across an interface between two media can undergo a transverse displacement known as the Imbert-Fedorov effect or conventional SHEL. Here, we plot this displacement for a left-hand circularly polarized beam (σ = +1) for two different cases and compare this with the geometric SHEL studied in this work. (a) and (b) Geometric SHEL 12 ∆T and 12 ∆y for the two configurations studied experimentally (Figure 4). (c) Geometric SHEL as predicted for an ideal polarizing interface. (d) Conventional SHEL occurring at an air-glass interface (n1 = 1, n2 = 1.5). (e) Conventional SHEL expected for the entrance face of our submerged polarizer (n1 = nL = 1.521, n2 = nG = 1.517).

the action of the sample. Thus, the shift measurements ∆(θ) = ∆R (θ) − ∆R (0◦ ) reported here are corrected with respect to the raw data. We investigate beam shifts in two different configurations. First, we measure the displacement ∆T =

 2 y |E |2 of the total transmitted energy density distriT bution |ET |2 when switching the incident state of polarization from σ = +1 to σ = −1 (Figure 4(a)). Then, we employ an additional polarization analyzer

 in front of the detector and observe the shift ∆y = 2 y |E |2 of the y

thereof are passed. We identify S3 = 0.99 ± 0.01 and S3 = −0.99 ± 0.01 with the circular states of polarization σ = +1 and σ = −1 respectively. For both states, we the

calculate  mean of all corresponding beam positions y (σ) and the

 helicity-dependent displacement ∆R = y (σ = +1) −

 y (σ = −1). An extensive characterization of statistical and systematic errors revealed that the observed position of the light beam depends slightly on the state of polarization even if no sample is present in the beam path. The magnitude of this spurious beam shift is typically much smaller than the phenomenon, we intend to study. In the Supplemental material, we discuss that this amounts to a small offset on the raw data points ∆R (θ), independent from

energy density |Ey |2 = |ET · y ˆ|2 of the vertically polarized field component solely (Figure 4(b)). These variants of the experiment coincide for polarizers with perfect extinction ratios but can differ significantly for real-world polarizers with minor deficiencies. The beam shift observed in the latter case increases proportionally to the tangent of the tilting angle, exceeding one wavelength. This characteristic “tan θ”-behaviour (and “real-world-polarizer” modification thereof) is unique to the geometric spin Hall effect of light. In both cases, the measurement agrees well with theoretical predictions using our phenomenological model. To the best of our knowledge, this is the first direct measurement of this intriguing phenomenon. The geometric spin Hall effect of light should not be confused

5 with the conventional SHEL or Imbert-Fedorov shift. The latter occurs at a physical interface and, while such interfaces are present in our experimental setup, they can only give rise to beam shifts significantly smaller than the observed effect. In Figure 5, we compare our results to the Imbert-Fedorov shift, which could occur at the polarizer substrate, for a set of realistic parameters [16, 31]. This illustrates that the beam shifts measured in this work constitute a novel spin Hall effect of light, virtually independent from surface effects. In conclusion, we have demonstrated the geometric spin Hall effect of light experimentally by propagating a circularly polarized laser beam across a suitable polarizing interface. The centre of mass of the transmitted light field was found to be displaced with respect to position of the incident beam as predicted by the theory. While a Gaussian light beam itself is invariant with respect to rotation around its axis of propagation, the geometry induced by the tilted polarizer, breaks this symmetry. The resulting displacement can be interpreted as a spin-to-orbit coupling characteristic for spin Hall effects of light.

ACKNOWLEDGEMENTS

The authors thank Christian Gabriel for fruitful discussions and for his contribution in the initial stage of the experiment.

∗

[email protected] [1] F. Goos and H. H¨ anchen, Ann. Phys. (Leipzig) 436, 333 (1947). [2] F. I. Fedorov, Dokl. Akad. Nauk. SSSR 105, 465 (1955). [3] C. Imbert, Phys. Rev. D 5, 787 (1972). [4] A. Aiello, New J. Phys. 14, 013058 (2012). [5] F. Pillon, H. Gilles, and S. Fahr, Appl. Opt. 43, 1863 (2004). [6] J.-M. M´enard, A. E. Mattacchione, M. Betz, and H. M. van Driel, Opt. Lett. 34, 2312 (2009). [7] M. Merano, A. Aiello, G. W. ’t Hooft, M. P. van Exter, E. R. Eliel, and J. P. Woerdman, Opt. Express 15, 15928 (2007). [8] H. Schilling, Ann. Phys. (Leipzig) 471, 122 (1965). [9] M. Onoda, S. Murakami, and N. Nagaosa, Phys. Rev. Lett. 93, 083901 (2004). [10] O. Hosten and P. Kwiat, Science 319, 787 (2008). [11] X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, Science 339, 1405 (2013). [12] A. V. Dooghin, N. D. Kundikova, V. S. Liberman, and B. Zel’dovich, Phys. Rev. A 45, 8204 (1992). [13] C. C. Leary, M. G. Raymer, and S. J. van Enk, Phys. Rev. A 80, 061804 (2009). [14] V. S. Liberman and B. Y. Zel’dovich, Phys. Rev. A 46, 5199 (1992).

[15] K. Y. Bliokh and Y. P. Bliokh, Phys. Lett. A 333, 181 (2004). [16] K. Y. Bliokh and Y. P. Bliokh, Phys. Rev. Lett. 96, 073903 (2006). [17] A. Aiello and J. P. Woerdman, Opt. Lett. 33, 1437 (2008). [18] K. Y. Bliokh, A. Niv, V. Kleiner, and E. Hasman, Nature Photon. 2, 748 (2008). [19] A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, Phys. Rev. Lett. 103, 100401 (2009). [20] N. B. Baranova, A. Y. Savchenko, and B. Y. Zel’dovich, JETP Lett. 59, 232 (1994). [21] K. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, Phys. Rev. Lett. 101, 030404 (2008). [22] P. Banzer, M. Neugebauer, A. Aiello, C. Marquardt, N. Lindlein, T. Bauer, and G. Leuchs, J. Europ. Opt. Soc. Rap. Public. 8, 13032 (2013). [23] K. Y. Bliokh, A. Aiello, and M. A. Alonso, in The angular momentum of light, edited by D. L. Andrews and M. Babiker (Cambridge University Press, 2012). [24] A. Bekshaev, K. Y. Bliokh, and M. Soskin, Journal of Optics 13, 053001 (2011). [25] J. Korger, A. Aiello, C. Gabriel, P. Banzer, T. Kolb, C. Marquardt, and G. Leuchs, Appl. Phys. B 102, 427 (2011). [26] A. Y. Bekshaev, J. Opt. A: Pure Appl. Opt. 11, 094003 (2009). [27] K. Y. Bliokh and F. Nori, Phys. Rev. Lett. 108, 120403 (2012). [28] M. V. Berry, J. Opt. A: Pure Appl. Opt. 11, 094001 (2009). [29] J. Durnin, C. Reece, and L. Mandel, J. Opt. Soc. Am. 71, 115 (1981). [30] J. Braat, S. van Haver, A. Janssen, and P. Dirksen, J. Europ. Opt. Soc. Rap. Public. 2, 07032 (2007). [31] B.-Y. Liu and C.-F. Li, Opt. Commun. 281, 3427 (2008). [32] J. W. Goodman, Introduction to Fourier optics, 3rd ed. (Roberts & Co, Colorado, USA, 2005). [33] Y. Fainman and J. Shamir, Appl. Opt. 23, 3188 (1984). [34] H. A. Haus, American Journal of Physics 61, 818 (1993). [35] A. Aiello, C. Marquardt, and G. Leuchs, Opt. Lett. 34, 3160 (2009).

6

1.0

Phenomenological polarizer model and characterization of our real-world polarizer

0.8

In this section, we describe a geometric polarizer model, analogously to the work by Fainman and Shamir [33], and determine empirical parameters relevant for the actual polarizer used in our experiment. Here, the interaction of an arbitrarily oriented polarizer with a plane wave is discussed while we deal with real light beams in the subsequent section on beam shifts. A polarizer is an optical device that alters the state of polarization and intensity of a plane wave without effecting its direction of propagation κ ˆ = k/k. The polarizer used within this work can be described by a real-valued unit vector Pˆa describing its absorbing axis. Projecting Pˆa onto the transverse plane of the electric field yields an effective absorbing axis Pˆa − (Pˆa · κ ˆ) κ ˆ . a ˆ(ˆ κ) = q 1 − (Pˆa · κ ˆ )2

(S1)

Thus, the electric field transmitted across an idealized polarizer is     ˜T = E ˜I − a ˜I = tˆ tˆ · E ˜I , E ˆ a ˆ·E (S2) ˜I = E ˜I (k) is the amplitude of the plane wave where E exp(i κ ˆ · r), and Pˆa × κ ˆ tˆ(ˆ κ) = a ˆ(ˆ κ) × κ ˆ= |Pˆa × κ ˆ|

(S3)

is the effective transmitting axis. However, real-world polarizers have finite extinction ratios and an experimental characterization shows that the effectiveness of our polarizer decreases when tilted (Figure S1). Thus, we have found it convenient to phenomenologically describe the transmitted light field as     ˜T = τt tˆ tˆ · E ˜I + τa a ˜I . E ˆ a ˆ·E (S4)

Here, we employ two empirical parameters, τa (θ) and τt (θ), depending on the angle θ between the propagation direction κ ˆ and the unit vector perpendicular to the surface of the polarizer. We have found the following set of parameters to be in good agreement with both the observed transmission and beam shifts: τt (θ) = 1 − 1.2 exp(−12 cos θ)

τa (θ) = 0.51 exp(−1.3 cos θ)

(S5) (S6)

A more detailed study of tilted polarizers will be published elsewhere.

shift [μm]

SUPPLEMENTAL INFORMATION

V polarization:

0.6 0.4 0.2

H polarization:

0.0 0

20

40 60 Tilting Angle θ [°]

80

Figure S1. Transmittance of light beams across the polarizer as a function of the tilting angle θ and the incident state of polarization. The polarizer is aligned such that at normal incidence (θ = 0◦ ) vertical (V) polarization is transmitted and horizontal (H) polarization is blocked. We observe how the transmittance changes when the polarizer is rotated around the vertical axes and compare the experimental data (circles) to our phenomenological model (S4) (solid lines).

Calculation of beam shifts occurring at an oblique polarizer according to our phenomenological model

In this section, we adapt the theory for the geometric spin Hall effect of light originally calculated for a different type of polarizer [25] to our phenomenological model. To this end, we express the polarizer’s absorbing axis Pˆa = x ˆ0 = cos θ x ˆ + sin θ zˆ

(S7)

in the global reference frame {ˆ x, y ˆ, zˆ}, aligned with the direction zˆ of beam propagation. The incident beam EI (r) is circularly polarized and well-collimated, i.e. it has a low divergence θ0 , with a Gaussian transverse intensity profile. This is expanded ˜I (ˆ in a plane wave basis with amplitudes E κ) such that the κ ˆ -dependent projection (S4) can be applied. As a consequence of our polarizer model, the electric field ET (r) after transmission across such a polarizer, is a superposition of two orthogonally polarized field components, Ex (r) and Ey (r). Thus, the electric field energy density distribution (at the detection plane z = 0) |ET (x, y)|2 = |ET · x ˆ|2 + |ET · y ˆ|2 + |ET · zˆ|2 | {z } | {z } | {z } =|Ex |2

=|Ey |2

(S8)

≈0

can be decomposed analogously. Geometric

SHEL manifests itself as a transverse displacement y of the transmitted light beam’s barycentre. The total electric energy density |ET |2 and both of its non-vanishing components undergo such a shift. Since this spatial displacement is independent from the

7 z coordinate, we restrict the discussion to the detection plane at z = 0. It is convenient to write the total energy density’s barycentre





 y |E |2 = wx y |E |2 + wy y |E |2 (S9) x y T

as a weighted sum of the relative shifts occurring for the horizontally and vertically polarized components respectively. Here, RR

 y u(x, y) d x d y y u = RR (S10) u(x, y) d x d y

denotes the centre of mass along the y ˆ direction calculated with respect to a scalar distribution u(x, y). The integration spans the whole detection plane. The weights RR 2 E (x, y) d x d y wx (θ) = RR x2 |E| (x, y) d x d y =

wy (θ) =

τa2 (θ) 2 τa (θ) + τt2 (θ) τt2 (θ) 2 τa (θ) + τt2 (θ)

+ O(θ02 ) + O(θ02 )

and

(S11)

(S12)

introduced in (S9) depend on the empirical parameters τt and τa (S5). Finally, we can calculate the relative shifts RR

 y |Ei |2 d x d y y |Ei |2 x,y = RR λ |Ei |2 d x d y x,y

tan θ fi (θ) + O(θ02 ), 2π where the factors τt (θ) < 0 and fx (θ) = 1 − τa (θ) τa (θ) >0 fy (θ) = 1 − τt (θ) =σ

(S13)

(S14a) (S14b)

depend critically on the performance of the polarizer. Note that, since the transmission coefficients are real and positive and 1 ≥ τt > τa (Figure S1), these relative shifts have opposite signs. Consequently, the displacement of the total energy density

 y |E |2 tan θ T =σ (wx fx + wy fy ) (S15) λ 2π | {z } 0 9

of Eq. (12b) accounts for the transmittance for crossed polarization, i.e. the fact that even if the electric field is polarized parallel to the effective absorbing axis, the absorption is not 100%. The phase of the complex parameter τa indicates that this field component is scattered with a phase determined by the orientation of the nano-particles relative to the incoming wave. For small tilting angles θ < 45◦ , the observation agrees with the prediction of the geometric absorbing model TA . Close to grazing incidence θ → 90◦ , the latter deviates, which we can understand in a physical picture. The particles embedded in our polarizer are cigar-shaped [23, 24]. Relevant for the polarization effect is the coupling of the light field to their long axes Pˆ A . By design, the wavelength is close to the resonance of the particles’ long axes. At normal incidence, the scattering and absorption is strong for states of polarization parallel to the long axis and negligible in the orthogonal case. When the polarizer is tilted, only the component of the electric field vector directed along the particles’ absorbing axis Pˆ A takes part in the interaction. Thus, the effect of a single particle decreases proportionally to cos(θ ) as the coupling becomes less efficient. The thickness of the polarizing layer guarantees that a light beam interacts with multiple particles while propagating across the device. Consequently, the observed extinction ratio is significantly larger than expected for a single particle. Our phenomenological model subsumes the sophisticated effect of this ensemble using only two functions τa (θ ) and τt (θ ), which can be directly measured. 6. Conclusion We have presented a Mueller matrix polarimeter making use of inexpensive linear polarizers and arbitrary retarding elements. Our least squares optimization approach is fast, yet accurate and precise. In particular, we have used this setup to study the effect a tilted polarizer has on the light field. Incidentally, linear polarizers are also popular as reference samples to characterize such measurement devices. Our data indicates that the combined statistical and systematic error of any matrix element is less than 0.01, while for polarimeters of comparable speed and feasibility, deviations between 0.03 and 0.10 per matrix element are typical [25]. In fact, our method is comparable with the accuracy achieved by more sophisticated calibration techniques requiring the use of multiple reference samples [26]. Finally, we have shown that a real-world polarizer, even when tilted, can be modeled geometrically. Using only the projection of the absorbing axis yielded already to an acceptable approximation for the collective action of the nano-particle ensemble. It was demonstrated that the finite extinction ratio of realistic polarizers can be taken into account phenomenologically, including configurations close to grazing incidence. We are confident that future work will connect the observation to a detailed microscopic study of such nano-particles and their interaction with the light field. Acknowledgments The authors would like to thank Norbert Lindlein and Vanessa Chille for useful discussions, and the anonymous Referees for insightful comments.

10

A.4. Distributing entanglement with separable states

Beyond the first-author publications summarized in this thesis, I was involved in a project related to the distribution of entanglement with separable quantum states of light. This work will be an essential part of Christian Peuntinger’s dissertation. C. Peuntinger, V. Chille, L. Miˇsta, N. Korolkova, M. F¨ortsch, J. Korger, C. Marquardt, and G. Leuchs. Distributing entanglement with separable states. arXiv:1304.0504. Phys. Rev. Lett. 111, 230506, 2013

81

82

Distributing entanglement with separable states Christian Peuntinger,1, 2, ∗ Vanessa Chille,1, 2, ∗ Ladislav Miˇsta, Jr.,3 Natalia Korolkova,4 Michael F¨ortsch,1, 2 Jan Korger,1, 2 Christoph Marquardt,1, 2 and Gerd Leuchs1, 2 1

Max Planck Institute for the Science of Light, G¨ unther-Scharowsky-Str. 1 / Bldg. 24, Erlangen, Germany 2 Institute of Optics, Information and Photonics, University of Erlangen-Nuremberg, Staudtstraße 7/B2, Erlangen, Germany 3 Department of Optics, Palack´ y University, 17. listopadu 12, 771 46 Olomouc, Czech Republic 4 School of Physics and Astronomy, University of St. Andrews, North Haugh, St. Andrews, Fife, KY16 9SS, Scotland (Dated: August 26, 2013) We experimentally demonstrate a protocol for entanglement distribution by a separable quantum system. In our experiment two spatially separated modes of an electromagnetic field get entangled by local operations, classical communication, and transmission of a correlated but separable mode between them. This highlights the utility of quantum correlations beyond entanglement for establishment of a fundamental quantum information resource and verifies that its distribution by a dual classical and separable quantum communication is possible. PACS numbers: 03.65.Ud, 03.67.Hk

Like a silver thread, quantum entanglement [1] runs through the foundations and breakthrough applications of quantum information theory. It cannot arise from local operations and classical communication (LOCC) and therefore represents a more intimate relationship among physical systems than we may encounter in the classical world. The ‘nonlocal’ character of entanglement manifests itself through a number of counterintuitive phenomena encompassing Einstein-Podolsky-Rosen paradox [2, 3], steering [4], Bell nonlocality [5] or negativity of entropy [6, 7]. Furthermore, it extends our abilities to process information. Here, entanglement is used as a resource which needs to be shared between remote parties. However, entanglement is not the only manifestation of quantum correlations. Notably, also separable quantum states can be used as a shared resource for quantum communication. The experiment presented in this paper highlights the quantumness of correlations in separable mixed states and the role of classical information in quantum communication by demonstrating entanglement distribution using merely a separable ancilla mode. The role of entanglement in quantum information is nowadays vividly demonstrated in a number of experiments. A pair of entangled quantum systems shared by two observers enables to teleport [8] quantum states between them with a fidelity beyond the boundary set by classical physics. Concatenated teleportations [9] can further span entanglement over large distances [10] which can be subsequently used for secure communication [11]. An a priori shared entanglement also allows to double the rate at which information can be sent through a quantum channel [12] or one can fuse bipartite entanglement into larger entangled cluster states being a ‘hardware’ for quantum computing [13].

∗

contributed equally to this work

The common feature of all entangling methods used so far is that entanglement is either produced by some global operation on the systems that are to be entangled or it results from a direct transmission of entanglement (possibly mediated by a third system) between the systems. Even entanglement swapping [9, 14], capable of establishing entanglement between the systems that do not have a common past, is not an exception to the rule because also here entanglement is directly transmitted between the participants. However, quantum mechanics admits conceptually different means of establishing entanglement which are free of transmission of entanglement. Remarkably, creation of entanglement between two observers can be disassembled into local operations and the communication of a separable quantum system between them [15]. The impossibility of entanglement creation by LOCC is not violated because communication of a quantum system is involved. The corresponding protocol exists only in a mixed-state scenario and obviously utilizes less quantum resources in comparison with the previous cases because communication of only a discordant [16–18] separable quantum system is required. In this paper, we experimentally demonstrate the entanglement distribution by a separable ancilla [15] with Gaussian states of light modes [19]. The protocol aims at entangling mode A which is in possession of a sender Alice with mode B held by a distant receiver Bob by local operations and transmission of a separable mediating mode C from Alice to Bob. This requires the parties to prepare their initial modes A, B and C in a specific correlated but fully separable Gaussian state. Once the resource state ρˆABC is established no further classical communication is needed to accomplish the protocol. To emphasize this, we attribute the state preparation process to a separate party, David. Note, that this resource state preparation is performed by LOCC only. No global quantum operation with respect to David’s separated boxes

2 ter BSAC , which output modes are denoted by A0 and C 0 . The beam splitter BSAC cannot create entanglement with mode B. Hence the state is separable with respect to B|A0 C 0 splitting. Moreover, the state also fulfils the positive partial transpose (PPT) criterion [21, 22] with respect to mode C 0 and hence is also separable across C 0 |A0 B splitting [23] as required [24]. In the final step, Alice sends mode C 0 to Bob who superimposes it with his mode B on another balanced beam splitter BSBC . The presence of the entanglement between modes A0 and B 0 is confirmed by the sufficient condition for entanglement [25, 26] ∆2norm (gˆ xA0 + x ˆB 0 )∆2norm (g pˆA0 − pˆB 0 ) < 1, FIG. 1. Sketch of the Gaussian entanglement distribution protocol. David prepares a momentum squeezed vacuum mode A, a position squeezed vacuum mode C and a vacuum mode B. He then applies random displacements (green boxes) of the x ˆ quadrature (horizontal arrow) and the pˆ quadrature (vertical arrow) as in (1), which are correlated via classical communication channel (green line). David passes the modes A and C to Alice, and mode B to Bob. Alice superimposes the modes A and C on a balanced beam splitter BSAC and communicates the separable output mode C 0 to Bob (red line connecting Alice and Bob). Bob superimposes the received mode C 0 with his mode B on another balanced beam splitter BSBC , which establishes entanglement between the output modes A0 and B 0 (black lemniscata). Note the position of the displacement on mode B. In the original protocol the displacement is performed before BSBC , which is depicted by the corresponding box with dashed green line. Equivalently, this displacement on mode B can be performed after BSBC (dashed arrow indicates the respective relocation of displacement) on mode B 0 , and even a posteriori after the measurement of mode B 0 .

is executed at the initial stage and no entanglement is present. Protocol. The protocol [19] depicted in Fig. 1 consists of three steps. Initially, a distributor David prepares modes A and C in a momentum squeezed and position squeezed vacuum state, respectively, with quadra(0) (0) tures x ˆA,C = e±r x ˆA,C , pˆA,C = e∓r pˆA,C , whereas mode (0)

B is in a vacuum state with quadratures x ˆB = x ˆB and (0) pˆB = pˆB . Here r is the squeezing parameter and the superscript “(0)” denotes the vacuum quadratures. David then exposes all the modes to suitably tailored local correlated displacements [20]: pˆA → pˆA − p, x ˆC → x ˆC + x, √ √ x ˆB → x ˆB + 2x, pˆB → pˆB + 2p.

(1)

The uncorrelated classical displacements x and p obey a zero mean Gaussian distribution with the same variance (e2r −1)/2. The state has been prepared by LOCC across A|B|C splitting and hence is fully separable. In the second step, David passes modes A, C of the resource state to Alice and mode B to Bob. Alice superimposes modes A and C on a balanced beam split-

(2)

where g is a variable gain factor. Minimizing the left hand side of Ineq. 2 with respect to g we get fulfilment of the criterion for any r > 0, which confirms successful entanglement distribution. Experiment. The experimental realization is divided in three steps: state preparation, measurement, and data processing. The corresponding setup is depicted in Fig. 2. From now on, we will work with polarization variables described by Stokes observables (see, e.g., [27, 28]) instead of quadratures. We choose the state of polarization such that mean values of Sˆ1 and Sˆ2 equal zero while hSˆ3 i  0. This configuration allows to identify the “dark” Sˆ1 -Sˆ2 plane with the quadrature phase space. Sˆθ , Sˆθ+π/2 in this plane correspond to Sˆ1 , Sˆ2 renormalized with respect to Sˆ3 ≈ S3 and can be associated with the effective quadratures x ˆ, pˆ. We use the modified version of the protocol indicated in Fig. 1 by the dashed arrow showing the alternative position of displacement in mode B: The random displacement applied by David can be performed after the beam splitter interaction of B and C 0 , even a posteriori after the measurement of mode B 0 . This is technically more convenient and emphasizes that the classical information is sufficient for the entanglement recovery after the interaction of both modes A and B with the ancilla mode C and C 0 , respectively. David prepares two identically, polarization squeezed modes [26, 27, 29, 30] and adds noise in the form of random displacements to the squeezed observables. The technical details on the generation of these modes can be found in the Supplemental Material [24]. The modulation patterns applied to modes A and C to implement the random displacements are realized using electro-optical modulators (EOMs) and are chosen such that the twomode state ρˆA0 C 0 is separable. By applying a sinusoidal voltage Vmod , the birefringence of the EOMs changes at a frequency of 18.2 MHz. In this way the state is modulated along the direction of its squeezed observable. Such two identically prepared modes A and C are interfered on a balanced beam splitter (BSAC ) with a fixed relative phase of π/2 by controlling the optical path length of one mode with a piezoelectric transducer and a locking loop. This results in equal intensities of both output modes. In the final step, Bob mixes the ancilla mode C 0 with a vacuum mode B on another balanced beam

3 splitter, and performs a measurement on the transmitted mode B 0 .

polarization squeezer A HWP

QWP

HWP

WS Down-mixer

BSAC

Amplifier

polarization squeezer C

function generator

AD sampling 10Msamples/s

BSBC vacuum mode B

HWP State Preparation

QWP

HWP

WS

Measurement Process

Data Acquisition

FIG. 2. Sketch of the experimental setup. Used abbreviations: HWP: Half-wave plate; QWP: Quarter-wave plate; EOM: Electro-optical modulator; BS: Beam splitter; WS: Wollaston prism; (State Preparation) The polarization of two polarization squeezed states (A and C) is modulated using EOMs and sinusoidal voltages from a function generator (dotted lines). The HWPs before the EOMs are used to adjust the direction of modulation to the squeezed Stokes variable, whereas the QWPs compensate for the stationary birefringence of the EOMs. The such prepared modes interfere with a relative phase of π/2 on a balanced beam splitter BSAC . In the last step of the protocol, the mode C 0 interferes with the vacuum mode B on a second balanced beam splitter BSBC . (Measurement Process) A rotatable HWP, followed by a WS and a pair of detectors, from which the difference signal is taken, allows to measure all possible Stokes observables in the Sˆ1 -Sˆ2 -plane. To determine the two-mode covariance matrix γA0 B 0 all necessary combinations of Stokes observables are measured. Removing the second beam splitter of the state preparation allows us to measure the covariance matrix of the two-mode state ρˆA0 C 0 . (Data Acquisition) To achieve displacements of the modes in the Sˆ1 -Sˆ2 -plane we electronically mix the Stokes signals with a phase matched electrical local oscillator and sample them by an analog-to-digital converter.

The states involved are Gaussian quantum states and, hence, are completely characterized by their first moments and the covariance matrix γ comprising all second moments [24]. To study the correlations between mode A0 and C 0 after BSAC , multiple pairs of Stokes observables (SˆA0 ,θ , SˆC 0 ,θ ) are measured. The covariance matrix γA0 C 0 is obtained by measuring five pairs of observables: (SˆA0 ,0◦ , SˆC 0 ,0◦ ), (SˆA0 ,90◦ , SˆC 0 ,0◦ ), (SˆA0 ,0◦ , SˆC 0 ,90◦ ), (SˆA0 ,90◦ , SˆC 0 ,90◦ ) and (SˆA0 ,45◦ , SˆC 0 ,45◦ ), which determine all its 10 independent elements. Here, θ is the angle in the Sˆ1 -Sˆ2 -plane between Sˆ0◦ and Sˆθ . For the measurements of the different Stokes observables, we use two Stokes measurement setups, each comprising a rotatable half-wave plate, a Wollaston prism and two balanced detectors. The difference signal of one pair of detectors gives one Stokes observable Sˆθ in the Sˆ1 Sˆ2 -plane, depending on the orientation of the half-wave plate. The signals are electrically down-mixed using an electric local oscillator at 18.2 MHz, which is in phase with the modulation used in the state preparation step. With this detection scheme, the modulation translates to a displacement of the states in the Sˆ1 -Sˆ2 -plane. The difference signal is low pass filtered (1.9 MHz), amplified

and then digitized using an analog-to-digital converter card (GaGe Compuscope 1610) at a sampling rate of 10M samples/s. After the measurement process we digitally low pass filter the data by an average filter with a window of 10 samples. Due to the ergodicity of the problem, we are able to create a Gaussian mixed state computationally from the data acquired as described above. By applying 80 different modulation depths Vmod to each of the EOMs we acquire a set of 6400 different modes. From these set of modes we take various amounts of samples, weighted by a two dimensional Gaussian distribution. The covariance matrix γA0 C 0 for the two-mode state after BSAC has been measured to be:   20.90 1.102 −7.796 −1.679  1.102 25.30 1.000 14.63  . (3) γA 0 C 0 =  −7.796 1.000 20.68 0.8010  −1.679 14.63 0.8010 24.65

The estimation of the statistical errors of this covariance matrix γA0 C 0 can be found in the Supplemental Material [24]. A necessary and sufficient condition for separability of a Gaussian state ρˆXY of two modes X and Y with the covariance matrix γXY is given by the PPT criterion:  2  M 0 1 (TY ) (4) γXY + iΩ2 ≥ 0, Ω2 = −1 0 i=1

(T )

where γXYY is the matrix corresponding to the partial transpose of the state ρˆXY with respect to the mode Y [24]. Effects that could possibly lead to some nonGaussianity of the utilized states are discussed in detail also in [24]. The state described by γA0 C 0 fulfils the condition (4) as the eigenvalues (39.84, 28.47, 13.85, 9.371) (T ) of (γA0CC00 + iΩ2 ) are positive, hence mode C 0 remains separable after BSAC . The measured two-mode covariance matrix of the output state γA0 B 0 is given by:   19.95 1.025 −4.758 −1.063  1.025 22.92 0.9699 9.153  γA0 B 0 =  . (5) −4.758 0.9699 9.925 0.2881  −1.063 9.153 0.2881 11.65

The statistical error of this measured covariance matrix is given in the Supplemental Material [24]. The separability is proven by the PPT criterion (eigenvalues 28.24, 21.79, 8.646, 5.756). The post-processing for the recovery of the entanglement is performed on the measured raw data of mode B 0 . Therefore, the displacement of the individual modes caused by the two modulators is calibrated. By means of this calibration, suitable displacements are applied digitally. The classical noise inherent in the mode B 0 is completely removed. A part of the classical noise associated with SˆA0 ,0◦ is subtracted from SˆB 0 ,0◦ , while the same fraction of the noise in SˆA0 ,90◦ is added to SˆB 0 ,90◦ .

4

FIG. 3. Entanglement distributed between modes A0 and B 0 . The experimental values for the criterion (2) are depicted in dependence of the gain factor g. Due to the attenuation of the mode B by 50 %, a gain factor about 0.5 yields a value smaller than 1, i.e. below the limit for entanglement (red line). The inset zooms into the interesting section around the minimum. The depicted estimated errors are so small because of the large amount of data taken.

In this way, the noise partially cancels out in the calculation of the separability criterion (2) and allows to reveal the entanglement. We chose the fraction as in (1), which is compatible with the separability of the transmitted mode C 0 from the subsystem (A0 B) in the scenario with modulation on mode B before the beam splitter BSBC . Only as Bob receives the classical information about the modulation on the initial modes A and C from David, he is able to recover the entanglement between A0 and B 0 . Bob verifies that the product entanglement criterion (2) is fulfilled as illustrated in Fig. 3. That proves the emergence of entanglement. The used gain factor g considers the slightly different detector response and the intentional loss of 50 % at Bob’s beam splitter. The clearest confirmation of entanglement 0.6922 ± 0.0002 < 1 is shown for gopt = 0.4235 ± 0.0005 (Fig. 3). This is the only step of the protocol, where entanglement emerges, thus demonstrating the remarkable possibility to entangle the remote parties Alice and Bob by sending solely a separable auxiliary mode C 0 . Discussion The performance of the protocol can be explained using the structure of the displacements (1). Entanglement distribution without sending entanglement highlights vividly the important role played by classical information in quantum information protocols. Classical information lies in our knowledge about all the correlated displacement involved. This allows the communicating parties (or David on their behalf) to adjust the displacements locally to recover through clever noise addition quantum resources initially present in the input quantum squeezed states. Mode C 0 transmitted from Alice to Bob carries on top of the sub-shot noise quadrature of the input squeezed state the displacement noise which

is anticorrelated with the displacement noise of Bob’s mode. Therefore, when the modes are interfered on Bob’s beam splitter, this noise partially cancels out in the output mode B 0 when the light quadratures of both modes add. Moreover, the residual noise in Bob’s position (momentum) quadrature is correlated (anticorrelated) with the displacement noise in Alice’s position (momentum) quadrature in mode A0 , again initially squeezed. Due to this the product of variances in criterion (2) drop below the value for separable states and thus entanglement between Alice’s and Bob’s modes emerges. The difference between the theoretically proposed protocol [19] and the experimental demonstration reported in this paper lies merely in the way how classical information is used. In the original protocol, the classical information is retained by David and he is responsible for clever tailoring of correlated noise. Bob evokes the required noise cancellation by carrying out the final part of global operation via superimposing his mode with the ancilla on BSBC . In the experimentally implemented protocol, David shares part of his information with Bob, giving Bob a possibility to get entanglement a posteriori, by using his part of classical information after the quantum operation is carried out. Thus entanglement distribution in our case is truly performed via a dual classical and quantum channel, via classical information exchange in combination with the transmission of separable quantum states. There are other interesting aspects to this protocol, which may open new, promising avenues for research. Noise introduced into the initial states by displacements contains specific classical correlations. On a more fundamental level, these displacements can be seen as correlated dissipation (including mode C into ”environment”). It is already known, that dissipation to a common reservoir can even lead to the creation of entanglement [31, 32]. Our scheme can be viewed as another manifestation of a positive role dissipation may play in quantum protocols. The presence of correlated noise results in non-zero Gaussian discord at all stages of the protocol, a more general form of quantum correlations, which are beyond entanglement [33]. The role of discord in entanglement distribution has been recently discussed theoretically [16, 17]. The requirements devised there are reflected in the particular separability properties of our global state after the interaction of modes A and C on Alice’s beam splitter. The state ρˆA0 BC 0 contains discord and entanglement across A0 |BC 0 splitting and is separable and discordant across C 0 |A0 B splitting as required by the protocol. Our work thus illustrates an interplay of entanglement and other quantum correlations, such as correlations described by discord, across different partitions of a multipartite quantum system.. ˇ L. M. acknowledges project P205/12/0694 of GACR. N. K. is grateful for the support provided by the A. von Humboldt Foundation. The project was supported by the BMBF grant “QuORep” and by the FP7 project QESSENCE. We thank Christoffer Wittmann and

5 Christian Gabriel for fruitful discussions. Note. Recently, an experiment has been presented in [34], which is based on a similar protocol. The main difference consists in the fact that it starts with entanglement which is hidden and recovered with thermal states. For this implementation no knowledge about classical

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information has to be communicated to Bob, besides the used thermal state. By contrast the setup presented in this work exhibits entanglement only at the last step of the protocol. Thus both works give good insights on different aspects of the theoretically proposed protocol [19]. Another independent demonstration of a similar protocol based on discrete variables was recently presented in [35].

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Distributing entanglement with separable states Supplementary Information Christian Peuntinger,1, 2, ∗ Vanessa Chille,1, 2, ∗ Ladislav Miˇsta, Jr.,3 Natalia Korolkova,4 Michael F¨ortsch,1, 2 Jan Korger,1, 2 Christoph Marquardt,1, 2 and Gerd Leuchs1, 2 1

I.

Max Planck Institute for the Science of Light, G¨ unther-Scharowsky-Str. 1 / Bldg. 24, Erlangen, Germany 2 Institute of Optics, Information and Photonics, University of Erlangen-Nuremberg, Staudtstraße 7/B2, Erlangen, Germany 3 Department of Optics, Palack´ y University, 17. listopadu 12, 771 46 Olomouc, Czech Republic 4 School of Physics and Astronomy, University of St. Andrews, North Haugh, St. Andrews, Fife, KY16 9SS, Scotland (Dated: August 26, 2013)

Here we have introduced the vector of quadratures ξˆ = (ˆ x1 , pˆ1 , . . . , x ˆN , pˆN ) and

PREPARATION OF POLARIZATION SQUEEZED STATES

To prepare two identically, polarization squeezed modes we use a well known technique like in [1–4]. Each of these modes is generated by launching two orthogonally polarized femtosecond pulses (∼200 fs) with balanced powers onto the two birefringent axes of a polarization maintaining fiber (FS-PM-7811, Thorlabs, 13 m). The pump source is a soliton-laser emitting light at a center wavelength of 1559 nm and a repetition rate of 80 MHz. By exploiting the optical Kerr effect of the fibers, the orthogonally polarized pulses are individually quadrature squeezed and subsequently temporally overlapped with a relative phase of π/2, resulting in a circular polarized light beam. The relative phase is actively controlled using an interferometric birefringence compensator including a piezoelectric transducer and a locking loop based on a 0.1 % tap-off signal after the fiber. In terms of Stokes observables (see [1, 5]) this results in states with zero mean values of Sˆ1 and Sˆ2 , but a bright hSˆ3 i  0 component. These states exhibit polarization squeezing at a particular angle in the Sˆ1 -Sˆ2 -plane.

II.

GAUSSIAN STATES

We implement the entanglement distribution protocol using optical modes which are systems in infinitelydimensional Hilbert state space. An N -mode system can be conveniently characterized by the quadrature operators x ˆj , pˆk , j, k = 1, 2, . . . , N satisfying the canonical commutation rules [ˆ xj , pˆk ] = iδjk which can be expressed in the compact form as [ξˆj , ξˆk ] = iΩN jk .

(1)

ΩN =

N M i=1

J,

J=



0 1 −1 0

contributed equally to this work

,

(2)

is the symplectic matrix. The present protocol relies on Gaussian quantum states. As any standard Gaussian distribution, a Gaussian state ρˆ is fully characterized by the vector of its first moments   d = Tr ρˆξˆ , (3) and by the covariance matrix γ with elements γjk = Tr[ˆ ρ{ξˆj − dj 11, ξˆk − dk 11}],

(4)

ˆ B} ˆ = AˆB ˆ +B ˆ Aˆ is the anticommutator. A real where {A, symmetric positive-definite 2N × 2N matrix γ describes a covariance matrix of a physical quantum state if and only if it satisfies the condition [6]: γ + iΩN ≥ 0.

(5)

The separability of Gaussian states can be tested using the positive partial transpose (PPT) criterion. A single mode j is separable from the remaining N − 1 modes if and only if the Gaussian state ρˆ has a positive partial transposition ρˆTj with respect to the mode j [7, 8]. On the level of the covariance matrices, the partial transposition is represented by a matrix Λj = L  (j) (j) N −1 (i) ⊕ σz , where σz = diag(1, −1) is the dii6=j=1 11

agonal Pauli z-matrix of mode j and 11(i) is the 2 × 2 identity matrix. The matrix γ (Tj ) corresponding to a partially transposed state ρˆTj reads γ (Tj ) = Λj γΛTj . In terms of the covariance matrix, one can then express the PPT criterion in the following form. A mode j is separable from the remaining N − 1 modes if and only if [7, 8] γ (Tj ) + iΩN ≥ 0.

∗



(6)

The PPT criterion (6) is a sufficient condition for separability only under the assumption of Gaussianity. In our

2 experiment, however, non-Gaussian states can be generated for which this criterion represents only a necessary condition for separability. Therefore it can fail in detecting entanglement. III.

output covariance matrix γA0 C 0 and the first moments after the beam splitter BSAC as γAC = Σ−1 U T γA0 C 0 U Σ−1 + π ˜ ⊕ 0, where

ANALYSIS OF NON-GAUSSIANITY

2

There are two sources of imperfections in our experimental set up that are potential sources of nonGaussianity. These are phase fluctuations and the modulation of the initial squeezed states before the first beam splitter. They are discussed in the following sections. A.

Phase fluctuations

π ˜=

2

1 − e2σ (1 − eσ )2 ˜ (A + α ˜) + J(A˜ + α ˜ )J T . (11) 2 2

The 2 × 2 matrices A˜ and α ˜ possess the elements A˜ij = T (U γA0 C 0 U )ij and α ˜ ij = 2(U T D0 U )ij , i, j = 1, 2. Our estimate for the variance of the phase fluctuations is σ 2 = 0.02◦ and the vector d0 of the measured mean values of the state ρˆA0 C 0 reads d0 = (−0.208, 9.876, 13.32, 1.78).

The experiment includes an interference of the modes A and C on a beam splitter, which is the first beam splitter BSAC in the protocol. Imperfect phase locking at this beam splitter might cause a phase drift resulting in a non-Gaussian character of the state ρˆA0 C 0 after the beam splitter. The phase fluctuations can be modelled by a random phase shift of mode A before the beam splitter described by a Gaussian distribution P (φ) with zero mean and variance σ 2 . Denoting the operator corresponding to a beam splitter transformation as Uˆ and the phase shift φ on mode A as VˆA (φ), the state ρˆA0 C 0 can be linked to the state ρˆAC before the onset of phase fluctuations as Z ∞ ρˆA0 C 0 = P (φ)UˆVˆA (φ)ˆ ρAC VˆA† (φ)Uˆ† dφ. (7) −∞

Hence we can express the measured covariance matrix γA0 C 0 given in Eq. (3) of the main letter, and the vector of the first moments d0 of the state ρˆA0 C 0 in terms of the covariance matrix γAC and the vector of the first moments d of the input state ρˆAC . For this it is convenient to define matrices D and D0 of the first 0 moments with elements Dij = di dj and Dij = d0i d0j , i, j = 1, . . . , 4. Using Eq. (7) and after some algebra, one gets the transformation rule for the matrix of the first moments in the form D0 = U ΣDΣU T , where U describes the beam splitter on the level of covariance maσ2 σ2 trices. Σ = diag(e− 2 , e− 2 , 1, 1) is a diagonal matrix. Similarly we get the covariance matrix γA0 C 0 = U (ΣγAC Σ + π ⊕ 0) U T , where 0 is the 2 × 2 zero matrix and 2

π=

(10)

(8)

2

1 − e−2σ (1 − e−σ )2 (A + α) + J(A + α)J T . (9) 2 2

Here the matrix A is the 2 × 2 matrix with elements Aij = (γAC )ij , i, j = 1, 2, α is the 2 × 2 matrix with elements αij = 2Dij , i, j = 1, 2, and J is defined in Eq. (2). Similar to Ref. [9] we can now invert the relation (8) and express the input covariance matrix γAC via the

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By substituting these experimental values for σ 2 and d0 in Eq. (10) and using the beam splitter with the measured transmissivity T = 0.49 we get a legitimate covariance matrix γAC before the phase fluctuations as can be easily verified by checking the condition (5). Provided that the state with the covariance matrix γAC is classical it can be expressed as a convex mixture of products of coherent states. Gaussian distributed phase fluctuations and a beam splitter preserve the structure of the state, hence the state after the first beam splitter cannot be entangled. The covariance matrix γAC determines a physical Gaussian quantum state. Moreover, the covariance matrix possesses all eigenvalues greater than one and therefore the state is not squeezed [6] which is in a full agreement with the fact that modulations of modes A and C completely destroy the squeezing. It then follows that this Gaussian state is classical and it therefore transforms to a separable state after the first beam splitter. The inversion (10) thus allows us to associate a Gaussian state before the phase fluctuations with the covariance matrix γA0 C 0 measured after the first beam splitter. The separability properties of the state after the beam splitter can then be determined from the non-classicality properties of this Gaussian state.

B.

Gaussianity of the utilized states

We have paid great attention on the modulations on modes A and C to preserve Gaussian character of the state ρˆA0 C 0 . Our success can be visually inspected at the examples in Fig. 1, which illustrates that both the modulation and the subsequent Gaussian mixing faithfully samples the required Gaussian shape. Besides this raw visual check we have also tested quantitatively Gaussianity of the involved states by measuring higher-order moments of the Stokes measurements on modes A0 and C 0 . Specifically, we have focused on the determination of the shape measures called skewness S and kurtosis K

3 plary values of skewness for various measurement settings are summarized in the Table I. TABLE I. Skewness S for Stokes measurements on modes A0 and C 0 in different measurement directions. Measurement SA0 ,0◦ SA0 ,90◦ Skewness×103 6.240 ± 0.781 −1.478 ± 0.563 Measurement SC 0 ,0◦ SC 0 ,90◦ Skewness×103 10.123 ± 0.727 1.106 ± 0.830

The skewness can vanish also for the other symmetrical distributions, which may, however, differ from a Gaussian distribution in the peak profile and the weight of tails. These differences can be captured by the kurtosis which is equal to 3 for Gaussian distributions. The exemplary values of kurtosis for various measurement settings are summarized in the Table II. FIG. 1. Histogram plots for SˆA0 ,0◦ of the Gaussian mixed state (blue) and three exemplary individual modes. This figure illustrates the preparation of the Gaussian mixed state via post processing. Exemplarily, three of the 6400 displaced individual modes are visualized by their histograms (in green, black and red colour). The normalization is chosen such that they can be depicted in the same plot as the histogram of the mixed state (blue), which is normalized to its maximum value. By merging the data for all individual modes using a weighting with a two dimensional Gaussian distribution, the mixed state is achieved. Its Gaussianity is visualized by the Gaussian fit (red curve).

defined for a random variable x as the following third and fourth standardized moments µ4 K = 4, s

µ3 S = 3, s

γA 0 C 0

20.90 ± 0.0087  1.102 ± 0.0091 = −7.796 ± 0.0069 −1.679 ± 0.0076

The tables reveal that the measured probability distributions satisfy within the experimental error the necessary Gaussianity conditions S = 0 and K = 3. More sophisticated normality tests can be performed, which is beyond the scope of the present manuscript. IV. STATISTICAL ERRORS OF THE MEASURED COVARIANCE MATRICES

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where µk = h(x − hxi)k i is the kth central moment, hxi is √ the mean value and s = µ2 is the standard deviation. Skewness characterizes the orientation and the amount of skew of a given distribution and therefore informs us about its asymmetry in the horizontal direction. Gaussian distributions possess skewness of zero. The exem-



TABLE II. Kurtosis K for Stokes measurements on modes A0 and C 0 in different measurement directions. Measurement SA0 ,0◦ SA0 ,90◦ Kurtosis 2.971 ± 2.211 × 10−3 2.986 ± 1.852 × 10−3 Measurement SC 0 ,0◦ SC 0 ,90◦ Kurtosis 2.972 ± 1.978 × 10−3 2.992 ± 1.568 × 10−3

By dividing our dataset in 10 equal in size parts we can estimate the statistical errors of our measured covariance matrices γA0 C 0 and γA0 B 0 given in Eqs. (3) and (5) of the main letter. We calculate the covariance matrix for each part and use the standard deviation as error estimation. The covariance matrix γA0 C 0 including the statistical error turns out to be

1.102 ± 0.0091 25.30 ± 0.013 1.000 ± 0.0071 14.63 ± 0.0091

−7.796 ± 0.0069 1.000 ± 0.0071 20.68 ± 0.0093 0.8010 ± 0.011

 −1.679 ± 0.0076 14.63 ± 0.0091  . 0.8010 ± 0.011  24.65 ± 0.0073

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Similarly, the covariance matrix γA0 B 0 including the statistical error reads as

4

γA 0 B 0



19.95 ± 0.011 1.025 ± 0.016 22.92 ± 0.012  1.025 ± 0.016 = −4.758 ± 0.0050 0.9699 ± 0.0047 −1.063 ± 0.0051 9.153 ± 0.0058

−4.758 ± 0.0050 0.9699 ± 0.0047 9.925 ± 0.0048 0.2881 ± 0.0047

 −1.063 ± 0.0051 9.153 ± 0.0058  . 0.2881 ± 0.0047  11.65 ± 0.0038

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We could achieve such small statistical errors by recording sufficient large datasets.

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