Stimulated Raman Scattering in Gas Filled Hollow-Core Photonic Crystal Fibres Stimulierte Raman-Streuung in mit Gas gefüllten Hohlkernfasern

June 2013 Der Naturwissenschaftlichen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg Zur Erlangung des Doktorgrades Dr. rer. nat.

vorgelegt von

Nguyen Manh Thang aus Hanoi, Vietnam

Als Dissertation genehmigt von der Naturwissenschaftlichen Fakultät at der Friedrich-Alexander- Universität at Erlangen- Nürnberg

Tag der mündlichen Prüfung: 26.9.2013 Vorsitzender des Promotionsorgans: Prof.Dr. Johannes Barth Gutachter: Prof.Dr. Philip St. J. Russell Prof.Dr. Maria Chekhova

Abstract In this thesis I use unique properties of hollow-core photonic crystal fibre (HC-PCF) to study stimulated Raman scattering (SRS) in gaseous medium. HC-PCF offers excellent abilities such as tight confinement of light and matter along diffractionless interaction length in the micron-size core, low loss and adjustable guidance bandwidth. These allow us to achieve extremely high Raman conversion efficiencies and to optimize optical processes for a desired frequency range as well as exploring SRS regimes inaccessible in conventional ways. I first give an overview of the guidance mechanisms and fabrication techniques of HC-PCF. There are two main types of HC-PCFs. Hollow-core bandgap fibre (PBG-PCF) have quite low power loss and narrow guidance bandwidth. Hollow-core photonic crystal fibres with Kagomé lattice (Kagomé-PCF) provide broadband guidance and higher loss. The light-matter nonlinear interaction efficiency in HC-PCF has been shown several orders of magnitude higher than those of previous approaches. Next, the theoretical background of SRS is described in detail through both classical and quantum mechanical pictures. Maxwell-Bloch equations governing the spatio-temporal evolution of light-gas interaction system via SRS are also derived. For application purposes, I performed a two consecutive stage pulse compression in H 2 gas-filled PBG-PCF by backward stimulated Raman scattering (BSRS). As a result, a signal pulse 20 times shorter than that of the original pump pulse was efficiently generated. Moreover, a new dynamical process generating a train of Raman pulses with flexibly controllable peak intensities have been observed in transient BSRS. We also have been able to generate a broad, mutually coherent, purely rotational Raman frequency comb by a relatively simply setup consisting of a micro-chip pump laser source and two H 2 gas-filled HC-PCFs. Lastly, I consider the effect of the collision between gaseous molecules and the fibre core on the spectral linewidth of forward stimulated Raman scattering (FSRS) in a low gas pressure range.

Zusammenfassung In dieser Arbeit nutze ich die einzigartigen Eigenschaften von photonischen Hohlkernfasern (engl.: hollow-core photonic crystal fibre: HC-PCF), um stimulierte Raman-Streuung (SRS) in gasförmigen Medien zu untersuchen. HC-PCF bieten exzellente Möglichkeiten wie beispielsweise den Einschluss von Licht und Materie auf engstem Raum und über lange Wechselwirkungslängen im mikrometer-großen Faserkern, sowie geringe Transmissionverluste und einstellbare Transmissionbänder. Dies erlaubt uns extrem hohe Raman-Konversionseffizienzen zu erreichen und die optischen Prozesse für die gewünschten Frequenzbereiche zu optimieren. Darüber hinaus können wir SRS-Bereiche erforschen, die auf herkömmliche Weise nicht zugänglich sind. Ich werde zunächst einen Überblick über die Leitungsmechanismen und die Herstellungsverfahren von photonischen Hohlkernfasern geben. Es gibt hauptsächlich zwei verschiedene HC-PCF-Typen. Bandlücken-Hohlkernfasern (engl.: hollow-core photonic bandgap fibre: PBG-PCF) haben sehr geringe Leistungsverluste und leiten in einem schmalen Frequenzband. Hohlkernfasern mit Kagomé-Gitterstruktur (KagoméPCF) erlauben breitbandige Lichtleitung, allerdings bei höheren Verlusten. Es ist bekannt, dass die Licht-Materie-Wechselwirkungseffizienz für nichtlineare Effekte in HC-PCF um mehrere Größenordnungen höher ist als mit herkömmlichen Methoden. Im Anschluss an diese Kapitel wird der theoretische Hintergrund zur SRS im Detail erklärt, ausgehend von sowohl klassischem als auch quantenmechanischem Bild. Dabei werden unter anderem die Maxwell-Bloch-Gleichungen hergeleitet, die die raumzeitliche Ausbreitung der Licht-Gas-Wechselwirkung bei SRS beschreiben. Zu Anwendungszwecken habe ich durch stimulierte Raman-Rückstreuung (engl.: backward

stimulated

Raman

scattering:

BSRS)

zwei

aufeinanderfolgende

Pulskompressionen in Wasserstoff-gefüllten PBG-PCF durchgeführt. Damit konnte auf effiziente Weise ein Signalpuls erzeugt werden, der zwanzigmal kürzer als der Ausgangpuls der Pumpquelle war. Darüber hinaus konnte ein neuer dynamischer Prozess bei der transienten BRSR beobachtet werden, welcher einen Raman-Pulszug mit flexibel kontrollierbarer Spitzenintensität erzeugt. Außerdem gelang es uns einen zugleich breiten und

kohärenten

Raman-Frequenzkamm

ausschließlich

mit

Hilfe

von

Rotationsübergängen zu erzeugen. Der verhältnismäßig einfache Aufbau besteht hauptsächlich aus einer Mikrochip-Pumplaserquelle und zwei Wasserstoff-gefüllten HCPCF. Abschließend befasse ich mich mit der spektralen Linienbreite der stimulierten Raman-Vorwärtsstreuung (engl., forward stimulated Raman scattering: FSRS) bei geringem Gasdruck, die maßgeblich von Zusammenstößen der Gasmoleküle mit der Wand des Faserkerns beeinflusst wird.

Acknowledgements Firstly, I am sincerely grateful for my supervisor Prof. Dr. Philip St.J. Russell whose give me the continuous support during my research course. Thank you for giving me an invaluable chance to work in such a highly scientific environment. Secondly, I would like to thank Amir and Andy whose spend a lot of time to explain clearly for me the dynamical processes in Raman scattering as well as nonlinear optics. Thank you for your patient reading and corrections to my thesis. You not only help me in job but also teach me the way to overcome the difficult problems in life. Amir, I learn much about your careful characteristic. The thesis could not be completed without you. Andy, thank you for sharing your plentiful knowledge in culture I have a great time with Azhar, Patrick, Sarah, Nicolai and Xiao Ming. Thank you for sharing my office and funny stories. Azhar, you are very friendly and thank you for your help about computer problem and “Taj Mahal tea” gifts. Thank Sarah for translating thesis abstract into German version. I also would like to thank my all my colleagues Tran Xuan Truong, Xin Jiang, Martin Finger, Barbara Trabold, Federico Belli, Michael Schmidberger, Anna Butsch, Oliver Schmidt, Gordon Wong, Ana Maria Cubillas, Tijmen Euser, Johannes Koehler, Micheal Frosz, David Novoa, Alessio Stefani, Thomas Weiss, Sebastian Bauerschmidt, Philipp Hoelzer, Martin Butryn, Stanislaw Doerchner and Fatma Tuemer. Finally, I would like to thank my parents, my wife and my daughter whose encourage continuously me to complete my PhD work.

Contents Chapter 1 Introduction

......................................................................................... 1

Chapter 2 Hollow-core photonic crystal fibres 2.1 Conventional fibre

............................................................................................. 4

2.2 Hollow-core photonic crystal fibre 2.3 Guidance via photonic bandgaps 2.4 Density of states 2.5 Fabrication technique

....................................................... 4

.................................................................. 5 ...................................................................... 5

............................................................................................ 8 ...................................................................................... 10

2.6 Guidance via low density of states

................................................................. 11

2.7 HC-PCF enhances the gas-based nonlinear effect

........................................... 13

Chapter 3 Theoretical background of Raman scattering 3.1 Origin of Raman scattering

................................... 18

........................................................................... 18

3.2 Spontaneous and stimulated Raman scattering 3.2.1 Spontaneous Raman scattering

............................................... 20

................................................................. 20

3.2.2 Spontaneous versus stimulated Raman scattering

...................................... 25

3.3 The coupled wave equations and stimulated Raman scattering

.......................... 27

3.3.1 Wave propagation

................................................................ 27

3.3.2 Stimulated Raman scattering

................................................................ 30

3.3.3 SRS in the language of optical phonons 3.3.4 Phase-matching diagram

..................................................... 32

................................................................ 33

3.3.5 The classical description

................................................................ 35

3.3.6 The semi-classical description

.................................................................. 40

3.3.6.1 Density matrix formalism

.................................................................. 40

3.3.6.1 Schematic of energy levels

................................................................ 43

3.3.6.1 Motion equation of density matrix 3.3.6.4 Transient regime in SRS

....................................................... 44

.................................................................... 52

Chapter 4 Backward stimulated Raman scattering in H 2 gas-filled PBG-PCF .... 55 4.1 Introduction

.................................................................................................... 55

4.2 Backward and forward Raman gain asymmetry 4.3 Motivation

............................................. 56

................................................................................................... 60

4.4 Optical pulse compression via BSRS

............................................................. 61

4.4.1 Experimental setup

............................................................... 61

4.4.2 Results and discussion

............................................................... 63

4.4.3 Dynamical analysis of reverse-pumped Raman pulse

................................ 65

4.5 Generation of like-solitary pulse train

.......................................................... 67

4.5.1 Experimental process and results

.............................................................. 67

4.6 Conclusion

.......................................................... 70

Chapter 5 Phase-coherent frequency comb generation in gas filled HC-PCFs 5.1 Introduction

................................................................................................. 71

5.2 Purely rotational frequency comb generation 5.3 Stable phase-locking charateristic in comb lines 5.4 Summary

. .71

................................................. 72 ............................................ 77

.................................................................................................. 80

Chapter 6 Raman linewidth broadening in gas filled HC-PCF 6.1 Introduction

................................................................................................. 81

6.2 Analysis of Raman linewidth change in gas medium 6.3 Experimental setup and results 6.4 Conclusion

....................................... 81

..................................................................... 85

.................................................................................................. 88

Chapter 7 Summary and outlook References

.............................. 81

.............................................................................. 89

................................................................................................................... 95

Curriculum Vitae .......................................................................................................... 102

Chapter 1 Introduction Raman scattering is a result of the interaction of light with the oscillation modes of molecules constituting the scattering medium. It can be described as the scattering of light from optical phonons, differing from acoustic phonons in Brillouin scattering [1,2]. Raman scattering is a two-photon inelastic scattering, where the frequency of scattered photons is different from that of the incident photons, with the down-shifted frequency referred to as Stokes scattering and the up-shifted frequency referred to as anti-Stokes scattering. Raman scattering can occur in various media such as solid, liquid, gases and plasma. It was first discovered in 1928 by C.V. Raman in liquids [3] and by G. Landsberg in solid [4]. It had long become important for investigating the vibronic structure of molecules and crystals. However, these initial experiments used sources with low photondensity resulting in only a spontaneous regime where the scattered light is not coherent, emitting in every direction and providing a negligible scattered efficiency only few parts in 105 of the incident radiation [1]. After the coherent light source (laser) was invented in 1960, the first experiment in a stimulated regime was also accidentally observed in 1962 by J. Woodbury [5]. SRS has notably advantageous characteristics: the high conversion efficiency to scattered frequency, high directionality, definite excitation threshold, quite narrow linewidth compared with the spontaneous regime [2,6]. These make it an excellent tool with a wide range of applications in areas such as high-resolution spectroscopy [7], optical communication, frequency shifter, pulse compression [8], comb frequency generation as well as ultrashort pulse synthesis [9,10]. Apart from the common research on the SRS in forward direction (FSRS) for frequency shifting, backward SRS (BSRS) first observed in 1966 [1] is considered as a method for amplification and generation the signal pulse of highly spatial quality from the pump beam of poor spatial quality [13,14,15,16,17]. FSRS and BSRS are different in behavior. The forward-traveling Stokes pulse just has access to the energy stored in the copropagating volume element of pump pulse envelop, the forward Stokes intensity is limited by the initial pump. On the other hand, the backward-travelling Stokes is amplified by encountering continuously with long pump pulse, resulting in a backward 1

signal intensity can be amplified to a value far in excess of the pump intensity [8]. This mechanism has a promising potential in generation of powerful ultra-short pulses [18,19,20]. For low-density media such as gases, the maximization of the SRS efficiency requires following conditions: high intensity at low power, long effective interaction length and good quality transverse beam profile. Initially, to reach the Raman threshold, the laser beam was tightly focused to a small point by lens inside a gas cell. In this simple way, the effective length of interaction is not longer than a few mm (~Rayleigh length) caused by the strong diffraction limit of the focused laser beam, which results in the SRS efficiency only a few percent [21]. Then, for increasing the effective interaction length, the laser beam was coupled into multi-pass or high-finesse Fabry-Perot cavities [24, 25], or hollow-core capillaries [22,23]. However, far better conversion efficiencies are obtained when using HC-PCF as a gas-filled novel guidance system [26]. The light is confined inside the small core of HC-PCF by means of photonic bandgap of the cladding. These structures offer unique characteristics: the free-diffraction effective interaction length, quite low loss attenuation, flexible in designing of guidance bands, small effective area (~25µm2), single-mode transverse beam profile. These excellent characteristics make HC-PCF a desired candidate for studying light-matter interactions in low-density media at very low pump power level. This approach made the Raman threshold energy drop significantly with only a single-pass interaction, much lower than that of the threshold of unwanted other nonlinear processes such as self-phase modulation, self-focusing [27]. Choosing the suitable guidance band also allows us to optimize conversion to a desirable frequency by getting rid of unwanted higher order rotational and vibrational Stokes and anti-Stokes frequencies. As a result Raman energy threshold could reduce six orders of lower than previously reported [28]. Moreover, it is possible to gain deeper insight into the different states of SRS; good overviews can be found in [29,30,31,32,33]. In this thesis, I exploited novel characteristics of HC-PCF for carrying out experimental studies in both backward SRS and forward SRS regimes. The outline of the thesis is as following:

2

Chapter 2 gives a short overview on the novel light guidance mechanisms of photonic crystal fibres (PCFs). The propagation diagram is used to analyze and compare with the conventional waveguide. Then, we will focus on two HC-PCF types including hollowcore narrowband guidance fibre (PBG-PCF) and hollow-core broadband guidance fibre (Kagomé-PCF). Finally, the advanced applications of HC-PCF in nonlinear optical interactions between the light and low-density media were also introduced. Chapter 3 introduces a theoretical background of Raman scattering. Initially we explain the physical origin of this process based on the classical picture. Coupling equations describing the spatiotemporal evolution of stimulated Raman scattering will be considered and compared from both classical and quantum viewpoints. The transient SRS regime (high coherence) important in ultrashort synthesis will also be introduced at the end of the chapter. Chapter 4 describes BSRS in H 2 gas filled PBG-PCF. Firstly, the gain asymmetry in backward and forward Raman scattering in H 2 gas medium will be analyzed. By using a two-stage compression scheme, the signal pulse 20 times shorter than the original pulse was efficiently generated. Interestingly, a train of solitary-like Raman pulses with flexibly controllable peak intensities has been also observed in transient BSRS regime. Chapter 5 presents the generation of a broad, phase-coherent, purely rotationalRaman frequency comb by a microchip pump laser source and two H 2 gas-filled HC-PCFs. Then, the doubled-frequency interferometry was used to consider the phase characteristic of the generated comb. Chapter 6 investigates the pressure dependence of the rotational Raman linewidth of hydrogen confined in the core of a PBG-PCF with a radius of 5.5µm, in which the effect of the collision between gas molecules and fibre core wall will come into play at the pressure below 1bar when the molecular mean-free path is of order of the fibre core (a few µm). Chapter 7 gives a summary and outlook for future research.

3

Chapter 2 Hollow-core photonic crystal fibres I will introduce briefly optical properties of two types of HC-PCF, i.e. photonic bandgap PCF (PBG-PCF) and kagomé-PCF. The reviewed material of this chapter is mainly based on these references [27,34,37].

2.1 Conventional fibre In order to distinguish conventional fibre clearly from HC-PCF, firstly we summarize their guiding mechanism. Conventional “step-index” fibres operate by total internal reflection (TIR). They consist of a solid core with the refractive index n 1 surrounded by an outer cladding of slightly lower refractive index n 2 n 2 (cladding index), green rays are guided when they incident on an acceptance angle. In contrast, red rays are not guided (leak into fibre’s cladding) because they are outside the acceptance cone [34]. Conventional fibre has been developed and used since the 1970s for a range of important applications such as telecommunications, imaging and high power laser. However, these fibres have some limitations: waveguide geometry and refractive index deviation of core and cladding are restricted. Fabrication of single mode fibre becomes more difficult when

4

guided wavelength gets shorter. Furthermore, for specialized applications, which require a hollow core, conventional fibres are impossible because of their dependence on TIR. 2.2 Hollow-core photonic crystal fibres HC-PCFs are a special class of the photonic crystal fibres (PCFs), which guide light in a hollow core instead of solid core, as is the case for conventional fibres, first proposed by Phillip Russell [35]. These low-loss waveguides enable new applications such as studying matter-light interactions in gas-filled or liquid-filled cores. HC-PCF guides light by means of 2D-photonic bandgaps. Photonic bandgaps are formed by a periodic wavelength-scale lattice of microscopic air holes running along the entire length of fibre, plotted illustratively in figure 2.2.

Figure 2.2: A structure of PBG-PCF with hexagonal cladding structure. It consists of a hollow core (diameter~10µm) surrounded by the cladding formed by a periodic array of air holes with diameter d~2.8 µm and pitch Λ~2.9µm (the distance between two closest air-holes), the cladding is created in a glass substrate. The appearance of photonic bandgaps can be intuitively understood in the form of “stop bands” caused by Bragg reflections [34]. However, photonic bandgaps are created by a number of gratings that consist of periodic arrays of glass rods and air holes. These gratings add up appropriately so that propagation of light is forbidden completely [36].

2.3 Guidance via photonic bandbaps

5

It is well known that when light is incident on any interface between materials, the component of the wave-vector parallel to the interface is conserved [34]. In the fibre, if the structure is invariant along its entire length, the interface of core and cladding is always parallel to the fibre axis, labeled usually as z-axis, conserved vector is called propagation constant, β . Propagation constant can be obtained by solving the Maxwell equations (as Eq. (2.1) in section 2.4 below) and gives information on the dispersion of fibres. Its maximum is nk 0 ( β ≤ nk 0 ) , with n being the refractive index of the homogeneous medium and k 0 = 2π

λ

is the vacuum wave-vector corresponding to the

wavelength λ . For a given value of β > nk 0 , light propagation is forbidden. Results in light being confined in the higher index areas by TIR. A very useful tool to describe regimes where light is able to propagate or be evanescent is the propagation diagram, described in figure 2.3. The propagation diagram shows the relation between propagation constant and light frequencies normalized to the pitch, Λ of fibre cladding. This diagram allows us to present clearly the propagation mechanisms of light in conventional fibres as well as PCFs.

Figure 2.3 Propagation diagram of a step-index fibre is presented in figure 2.3a. PCFs are presented in figure 2.3b. Where the horizontal axis shows normalized propagations β Λ, normalized frequency is presented by the vertical axis ω Λ/c. Points A, B and C and regions 1,2,3,4 are described below (also see [37]).

6

Propagation regimes of step-index fibre composed for example of a Ge-doped silica core and a pure silica cladding with slightly lower refractive index, presented in figure 2.3a: Region 1: β < n air k 0 light can propagate in all regions; air refractive index of n air ≈ 1 ; cladding index of n cladding ≈ 1.45 and solid core of n core ≈ 1.47 .

Region 2: n air k 0 < β n core k 0 no propagation with any refractive index of n. Propagation regimes of PCFs with an average refractive index of micro-structured cladding n air -glass and an air-filling fraction of 45% made of pure glass are presented in figure 2.3b. Region 1: β < n air k 0 light propagates freely in all regions of PCF, air, air-glass cladding and glass-pure core. Region 2: n air k 0 < β 1

The Eq.(3.16) gives dN S ⎛ c ⎞ = ⎜ DN P ⎟z NS ⎝ n ⎠

(3.19)

Integrate two sides of Eq.(3.19) N S (z) = N S (0) exp(g S z )

We introduce g S =

(3.20)

c DN P in Eq.(3.20) and it is called the gain coefficient of SRS. Here n

N S (0) denotes the Stokes photon occupation number at the input of the Raman medium. Raman process follows the Eq.(3.20) is called stimulated and its Stokes intensity in SRS actually experiences exponential increase with the medium length-z.

26

The relationship between SRS and spontaneous Raman scattering is expressed by the Raman gain coefficient of g S in Eq.(3.20) and the Raman cross-section of σ in Eq.(3.13). This relationship is given [1]

gS =

~ 4Nπ 3c 2 N P ⎛ ∂σ ⎞ ⎜ ⎟I P ωS2ηω P n S2 Δω ⎝ ∂Θ ⎠

(3.21)

⎛ ∂σ ⎞ Where, n S is a refractive index of the Stokes radiation, ⎜ ⎟ denotes the differential ⎝ ∂Θ ⎠ spectral cross section, where Δω is the total linewidth of the Stokes radiation, dΘ is an ~ ηcω P N P element of solid angle. I P denotes the pump intensity of I P = , where V is the Vn P effective volume of the Raman scattering, n P is the refractive index of the pump laser wavelength.

3.3 The coupled wave equations and stimulated Raman scattering

The previous section provides an overview picture of the Raman scattering. However, it can not reveal the information relating to the coherent interaction between the fields and the molecules. This information becomes especially important when SRS occurs in highly coherent regime (transient regime) where the pump pulse duration is comparable or shorter than the relaxation time of the molecular coherence. This section describes detail the coherent SRS interaction in terms of the coupled propagation approach in a nonlinear optical media. Because the coherent excitation is dominant in SRS, the applied electromagnetic fields can be treated suitably as a classical quantity [48].

3.3.1 Wave propagation

We consider a lossless nonlinear optical media with no free charge, no free current and no magnetization. The travel of light obeys the Maxwell equation is derived from [1].

27

1 ∂2 ~ ~ 1 ∂2 ~ Ρ ∇×∇×Ε + 2 2 Ε = − 2 c ∂t c ε 0 ∂t 2

(3.22)

Where, ε 0 is the electric permittivity constant and the light velocity c in vacuum. Where ~ Ρ denotes the nonlinear polarization vector of the nonlinear optical medium depending ~ nonlinearly on the electric strength vector of the classical field of Ε .

The first term in Eq.(3.22) is analyzed as follow:

(

)

~ ~ ∇ × ∇ × Ε = ∇ ∇ ⋅ Ε − ∇ 2Ε

(3.23)

~ ~ Here, we have ∇ ⋅ Ε ≈ 0 for most cases interested in nonlinear optics. For example, Ε is a transversely, infinite plane wave. More general, it often demonstrated to be small for the case of slowly varying amplitude approximation. Inserting Eq.(3.23) into Eq.(3.22) we have

1 ∂2 ~ ~ 1 ∂2 ~ ∇ 2Ε − 2 2 Ε = 2 Ρ c ∂t c ε 0 ∂t 2

(3.24a)

1 ∂2 ~ ~ D ∇ 2Ε = ε 0 c 2 ∂t 2

(3.24b)

~ ~ ~ Where the displacement field vector D = ε 0 Ε + Ρ ~ ~ ~ We split Ρ into two parts: a linear part of Ρ L (depend linearly on the field of Ε ) and a ~ nonlinear part PN (depending nonlinearly on Ε ).

~ ~L ~N ~ ~ Ρ = Ρ + Ρ = ε 0 χ 1Ε + Ρ N

(3.25)

Here χ1 is the linear electric susceptibility

28

~ ~ ~ ~ Hence D = ε 0 Ε + ε 0 χ 1Ε + Ρ N

(3.26)

We rewrite Eq.(3.26)

~ ~ ~ D = n 2ε 0 E + P N

(3.27)

where n = 1 + χ 1 is the refractive index of the medium. We substitute Eq.(3.27) into Eq.(3.24b) and obtain the general equation of wave propagation in an isotropic, dispersionless optical nonlinear medium. ∂2 ~ ~ n2 ∂2 ~ ∇2Ε − 2 2 Ε = μ 0 2 P N c ∂t ∂t

(3.28)

~ Here P N is on the right-hand side and acts as the source term of new components in

nonlinear optical interactions in general and in stimulated Raman scattering in particular. Where μ 0 = c -2 ε -10 is the magnetic permeability in vacuum. Assume the applied field of the Raman active medium consists of j linearly polarized monochromatic plane waves with the carrier frequency ω j . Their respective wavevectors k j =

n jω j

c

, where n j is the

refractive index corresponding to the ω j . The solution of Eq.(3.28) can be written as

[

]

~ 1 E = ∑ (E j (z, t )exp i(± k j z − ω j t ) + c.c ) 2 j

[

]

~N 1 P = ∑ (PjN (z, t )exp − i(ω j t ) + c.c ) 2 j

(3.29)

(3.30)

Where, PjN (z, t ) , E j (z, t ) are the temporal spatial complex envelope functions (defined as Eq.(3.5)). The signs “ ± ” represent the propagation direction of the incident waves. We take the plus (+) for forward propagation increasing the distance z, in contrast the minus (-) for backward propagation reducing the distance z. 29

Insert Eq.(3.29&3.30) into Eq.(3.28) and apply some slowly varying amplitude approximations:

∂2E j ∂t

2