UNIVERSITY OF

GLASGOW Optoelectronics Research Group

The Nonlinear Optical Properties of Semiconductors David C. Hutchings [email protected]

Dept. of Electronics and Electrical Engineering

The Nonlinear Optical Properties of Semiconductors – p. 1/39

Optical Susceptibility Tensor UNIVERSITY OF

GLASGOW Optoelectronics Research Group

h

P = ε0 χ(1) E + χ(2) EE + χ(3) EEE + . . .

i

(1)

ω

(2)

ω1 + ω2

Sum frequency generation

ω1 − ω2

Difference frequency generation, rectification

χij

χijk

ω+0

(3)

χijkl

Linear refraction and absorption

Electro-optic (Pockel’s) effect

ω+ω+ω

Third harmonic generation

ω−ω+ω

Nonlinear refraction and absorption, 4WM

ω+0+0

Kerr electro-optic effect (DC)

The Nonlinear Optical Properties of Semiconductors – p. 2/39

Contents UNIVERSITY OF

GLASGOW Optoelectronics Research Group

Structure of zincblende semiconductors χ(1) : Absorption & Refraction χ(2) : Electro-optic effect (Pockels) & Frequency

conversion, e.g. SHG χ(3) : Two-photon Absorption, Nonlinear Refraction

& 4 wave mixing Quasi-χ(3) : Carrier effects, Electro-optic effect (Kerr) & Thermal effects

The Nonlinear Optical Properties of Semiconductors – p. 3/39

Waveguide geometry UNIVERSITY OF

GLASGOW Optoelectronics Research Group

Slab or rib waveguides Assume weakly guiding Also assume nonlinearity weak such that transverse guided mode unchanged Conventional orientation with [001] growth and [110] cleavage: z x

y

The Nonlinear Optical Properties of Semiconductors – p. 4/39

Crystal Symmetries UNIVERSITY OF

GLASGOW Optoelectronics Research Group

Common compound semiconductors in photonics have a zinc-blende (cubic) structure 43m Note different layer ordering for 111 and 111 Introducing heterostructure, e.g. quantum well, breaks translational invariance in one direction The Nonlinear Optical Properties of Semiconductors – p. 5/39

Bandstructure models UNIVERSITY OF

GLASGOW Optoelectronics Research Group

Bloch form for wavefunction with periodic uk (r) ψ(r) = uk (r)eik·r

Hamiltonian contains k · p term — treat as perturbation 2 parabolic bands: scalar model Kane model with singlet conduction band and triplet valence band: vector but isotropic Kane plus next highest conduction triplet: anisotropic

The Nonlinear Optical Properties of Semiconductors – p. 6/39

AlGaAs bandstructure UNIVERSITY OF

GLASGOW Optoelectronics Research Group

Alx Ga1−x As direct bandgap for x < 0.45

The Nonlinear Optical Properties of Semiconductors – p. 7/39

k · p models

UNIVERSITY OF

GLASGOW Optoelectronics Research Group

Γ15c

∆1

Γ1c

E’c Ec Ev

v Γ15

∆0

The Nonlinear Optical Properties of Semiconductors – p. 8/39

Quantum Theory of χ(n) UNIVERSITY OF

GLASGOW Optoelectronics Research Group

Susceptibility expressions are derived using the quantum Liouville equation, with Hamiltonian H = H0 + (e/m0 )A · p.

For χ(n) use sets of (n + 1)-level systems and sum over (n + 1)! time orderings. Evaluation of χ(n) in semiconductors requires: Electronic energies (valence and conduction bands) Momentum matrix elements between electronic states Summation over all states The Nonlinear Optical Properties of Semiconductors – p. 9/39

Linear absorption and refraction UNIVERSITY OF

GLASGOW Optoelectronics Research Group

(1) χii (ω)

X e2 2 = |e · p (k)| i vc 2m20 ~ω 2 k,bands   1 1 + × Ωvc (k) − (ω + iδ) Ωvc (k) + (ω + iδ)

take limit δ → 0: α ∝ Imχ(1) & n ∝ Reχ(1)

matrix element |pvc (k)| is approximately constant.

for parabolic bands, absorption follows square-root density-of-states The Nonlinear Optical Properties of Semiconductors – p. 10/39

χ(1) for 2-level system UNIVERSITY OF

GLASGOW Optoelectronics Research Group

1 0.8 0.6 0.4 0.2 -7.5 -5 -2.5 -0.2 -0.4

2.5

5

7.5

HΩ-WL∆

The Nonlinear Optical Properties of Semiconductors – p. 11/39

Linear refraction UNIVERSITY OF

GLASGOW Optoelectronics Research Group

Dispersion with parabolic bands gives Adachi formula (



√

√ 1 + X − 1 − X)/X 2

Eg 1 ε(ω) = A f (X) + 2 Eg + ∆ f (X) = Re(2 −

λg ~ω = X= Eg λ A(x) = 6.3 + 19.0x

Eg = (1.425 + 1.155x + 0.37x2 ) eV

3/2

f (Xso )

)

+B

√ n= ε ~ω Xso = Eg + ∆ B(x) = 9.4 − 10.2x ∆ = 0.34 eV The Nonlinear Optical Properties of Semiconductors – p. 12/39

Linear refraction (300K) UNIVERSITY OF

GLASGOW Optoelectronics Research Group

3.8 3.7 3.6 3.5 3.4 3.3 800

1000

1200

1400

1600

Refractive index vs wavelength (nm) x =0, 0.1, 0.2, 0.3 The Nonlinear Optical Properties of Semiconductors – p. 13/39

Slowly varying envelope approx. UNIVERSITY OF

GLASGOW Optoelectronics Research Group

Combine Maxwell’s equations → wave equation Fourier transform: t → ω

Substitute, with k = nω/c ˆ E(ω) = E(ω, z)eikz ˆ Assume envelope E(ω, z) varies slowly in comparison to wavelength of light same as paraxial wave equation (BPM) ˆ dE α ˆ iω 2 µ0 NL −ikz =− E+ P e dz 2 2k The Nonlinear Optical Properties of Semiconductors – p. 14/39

Second-order nonlinearities UNIVERSITY OF

GLASGOW Optoelectronics Research Group

No second-order nonlinearity in media with inversion symmetry Crystal symmetry in cubic semiconductors specifies that the only non-zero tensor elements are (2)

(2)

(2)

χxyz = χyzx = χzxy

More independent tensor elements in heterostructures and at surfaces

The Nonlinear Optical Properties of Semiconductors – p. 15/39

Pockels Electro-optic effect UNIVERSITY OF

GLASGOW Optoelectronics Research Group

Pockels Electro-optic effect is 0 + ω → ω and appears as a modification to the refractive index transverse geometry, e.g. DC/RF field ⊥ surface → phase modulation for TE optical polarisation conventionally use reduced r tensor notation so that optical path length change is n3 r41 V L ∆nL = 2d

GaAs at λ =1.5 µm, r41 =1.36 pmV−1 and n =3.38

The Nonlinear Optical Properties of Semiconductors – p. 16/39

Frequency conversion by χ(2) UNIVERSITY OF

GLASGOW Optoelectronics Research Group

Optical frequency conversion (define ω3 = ω1 + ω2 ) sum frequency generation: ω1 + ω2 → ω3 second-harmonic generation: ω + ω → 2ω difference frequency generation: ω3 − ω1 → ω2 parametric amplification: ω3 − ω1 → amplifies ω1 reduced d tensor notation, d = χ(2) (ω, ω)/2.

For epitaxial GaAs at λ =4.1 µm, d14 =94 pmV−1 . For continual forward energy conversion, require Phase-matching, i.e. phase velocities of generating and generated waves must be identical. The Nonlinear Optical Properties of Semiconductors – p. 17/39

Second harmonic generation UNIVERSITY OF

GLASGOW Optoelectronics Research Group

300

40.0 1/2

kmax=0.6 (eV) 1/2 kmax=1.0 (eV) 1/2 kmax=1.5 (eV)

∆k=0

2ω Irradiance

−1

χ xyz(ω,ω) (pmV )

30.0 200

(2)

100

DR DD

20.0

10.0

∆k=/ 0 0 0.3

0.0 0.4

0.5

0.6

Photon energy (eV)

0.7

0.8

0

Lc

2Lc

3Lc

distance

4Lc

5Lc

The Nonlinear Optical Properties of Semiconductors – p. 18/39

Second harmonic generation UNIVERSITY OF

GLASGOW Optoelectronics Research Group

ω

2ω

Simplest setup to characterise nonlinear interaction type-I configuration: TE fundamental → TM SH

type-II configuration: TE and TM fundamental → TE SH

The Nonlinear Optical Properties of Semiconductors – p. 19/39

Difference Frequency Generation UNIVERSITY OF

GLASGOW Optoelectronics Research Group

ωp

ωi

ωs

ω s ω p /2

ω p /2

ωi

Channel conversion for WDM Potentially integrate pump laser on chip

The Nonlinear Optical Properties of Semiconductors – p. 20/39

Parametric Amplification UNIVERSITY OF

GLASGOW Optoelectronics Research Group

ωp

ωi

ωs

ωs

Broad-bandwidth amplifier Optical Parametric Oscillator for mid-IR — use cavity and ωs builds up from noise Potentially integrate resonator Potentially integrate pump laser on chip

The Nonlinear Optical Properties of Semiconductors – p. 21/39

χ(3) processes UNIVERSITY OF

GLASGOW Optoelectronics Research Group

third-harmonic generation ∝ χ(3) (ω, ω, ω) two-photon absorption ∆α = βI

3ω (3) β(ω) = Imχ eff (−ω, ω, ω) 2 2 2ε0 n0 c

nonlinear refraction ∆n = n2 I 3 (3) Reχ n2 (ω) = eff (−ω, ω, ω) 2 4ε0 n0 c

DC Kerr effect ∝ χ(3) (0, 0, ω) The Nonlinear Optical Properties of Semiconductors – p. 22/39

χ(3) symmetry considerations UNIVERSITY OF

GLASGOW Optoelectronics Research Group

4 independent nonzero tensor elements for bulk semiconductors 3 for single ω (3) (3) (3) χxxxx (−ω, ω, ω), χxyxy (−ω, ω, ω), χxxyy (−ω, ω, ω) Three measurements required to completely characterise each nonlinear process Breaking symmetry, e.g. heterostructure introduces many more independent nonzero tensor elements

The Nonlinear Optical Properties of Semiconductors – p. 23/39

Two-photon Absorption UNIVERSITY OF

GLASGOW Optoelectronics Research Group

−19

xxxx

xyxy

−19

4x10

(3)

2

−2

Imχ (m V )

6x10

xxyy −19

2x10

0 0.7

0.8

0.9

1.0

1.1

1.2

1.3

Photon Energy (eV)

1.4

1.5

Two-photon absorption (cm/GW)

−19

8x10

30 25 E||[111] 20

E||[011] E||[001]

15 k||[111]

10

k||[011] k||[001]

5 0 0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

Photon Energy (eV)

Calculated Imχ(3) tensor elements and spectra of β for GaAs. Scales as Eg−3 . The Nonlinear Optical Properties of Semiconductors – p. 24/39

Ultrafast Nonlinear Refraction UNIVERSITY OF

GLASGOW Optoelectronics Research Group

Three measurements completely characterise nonlinear refraction in bulk (cubic) semiconductor Strength

nL 2 [001]

Anisotropy

σ

Biref. param.

δ

n2 scales as Eg−4

∆n/I|[001]  L L L 2 n2 [001] − n2 [011] /n2 [001]  L L C n2 [001] − n2 (k k [100]) /n2 [001]

For isotropic Kleinmann: σ = 0, δ = 1/3 GaAs at the half-gap (theory): σ = −0.82, δ = 0.08

AlGaAs at the half-gap (exper.): σ = −0.54, δ = 0.18 The Nonlinear Optical Properties of Semiconductors – p. 25/39

Coupled Propagation Equations UNIVERSITY OF

GLASGOW Optoelectronics Research Group

Usual configuration in semiconductor slab waveguides has TEk[110] (cleavage planes) and TMk[001]. h  σ ∗  σ  ∗i ∂u ∂2u uu + 1 − δ − vv u + δ − i + + γu + 1 − ∂z ∂x2 2 2 h i  ∂v ∂2v σ ∗ ∗ i + − γv + 1 − δ − uu + vv v + δ − ∂z ∂x2 2

σ ∗ 2 u v =0 2 σ 2 ∗ u v =0 2

u and v are the scaled electric field amplitudes for

TE and TM. γ is proportional to the (structurally induced) birefringence nTM − nTE . The Nonlinear Optical Properties of Semiconductors – p. 26/39

Calculated χ(3) in GaAs UNIVERSITY OF

GLASGOW Optoelectronics Research Group

1.0 0.5

(a)

0.0

-19

2

-2

σ’

4x10

Re χ (m V )

(a)

-0.5 -1.0

0 -1.5

(3)

xxxx xyxy xxyy

-19

-4x10

0.4

(b)

δ’

0.2

-0.2

(b)

-0.4

-19

4x10 2

-2

Re χ (m V )

0.0

-19

-19

-2

2x10

0

2

-19

(m V )

(3)

xxxx xyxy xxyy

-4x10

(c)

4x10

0

-19

-2x10

-19

-4x10

0.0

0.5

1.0

hω (eV)

1.5

0.0

0.2

0.4

0.6

0.8

1.0

hω/Eg

Top: 14-band, bottom: 8-band (isotropic) The Nonlinear Optical Properties of Semiconductors – p. 27/39

n2 in Al0.18 Ga0.82 As UNIVERSITY OF

GLASGOW Optoelectronics Research Group

300000 -13

3x10

(pm/V)

200000

-13

2x10

150000

Reχ

(3)

2

-1

n2 (cm W )

2

250000

xxxx xxyy xyxy

-13

1x10

100000

50000

0 1.4

1.5 1.6 Wavelength (µm)

1.7

0 0.4

0.5

0.6

0.7

0.8

0.9

Photon energy (eV)

Measured dispersion of n2 in Al0.18 Ga0.82 As and calculated Reχ(3) tensor components The Nonlinear Optical Properties of Semiconductors – p. 28/39

n2 in Al0.18 Ga0.82 As UNIVERSITY OF

GLASGOW Optoelectronics Research Group

Linear

Circular

1

1

0

0

-1

-1

1

1

0

0

-1

-1 -1

-1 0

0 1

1

Deduced anistropy from measurements of n2 in Al0.18 Ga0.82 As at λ = 1.55 µm The Nonlinear Optical Properties of Semiconductors – p. 29/39

Figure-of-merit for NLR applications UNIVERSITY OF

GLASGOW Optoelectronics Research Group

light absorbed in length α−1

100.0

phase change

NLDC

2π|∆n|L/λ ∼ 2π for

|∆n|/(αλ) > 1

for χ(3) only, figureof-merit |n2 |/(βλ) ∝ |Reχ(3) |/Imχ(3)

(3) (3)

therefore require figure-of-merit

10.0

|Reχ |/Imχ

NLO applications

FP 1.0

0.1 0.5

0.6

0.7

0.8

0.9

1.0

hω/Eg

The Nonlinear Optical Properties of Semiconductors – p. 30/39

Quasi-χ(3) processes UNIVERSITY OF

GLASGOW Optoelectronics Research Group

1 0.8 0.6 0.4 0.2 -7.5 -5 -2.5 -0.2 -0.4

2.5

5

7.5

HΩ-WL∆

The Nonlinear Optical Properties of Semiconductors – p. 31/39

Carrier nonlinearities UNIVERSITY OF

GLASGOW Optoelectronics Research Group

Free carrier absorption: N ↑, α ↑

Absorption saturation (bandfilling) in passive device: N ↑, α ↓, n ↓

Gain saturation in active device: N ↓, gain↓, n ↑

Exciton absorption saturation (phase-space filling + screening): N ↑, α ↓, n ↓ N.B. effects on n are for below bandgap frequencies

The Nonlinear Optical Properties of Semiconductors – p. 32/39

Example: bandfilling UNIVERSITY OF

GLASGOW Optoelectronics Research Group

Assuming equal populations N of electrons and heavy-holes which have quasi-equilibrium Boltzmann thermal distribution, we get ∆n = σn N √ 4 π σn (ω) = − n0

2 epvc 1 m0 ωg k T B

X mrj  mrj ~(ω − ωg )  J me me kB T

j=hh,lh

The Nonlinear Optical Properties of Semiconductors – p. 33/39

Carrier nonlinearities UNIVERSITY OF

GLASGOW Optoelectronics Research Group

where J(d) =

Z

∞ 0

√ −x xe dx x−d JHdL 1.4 1.2 1 0.8 0.6 0.4 0.2

-5

-4

-3

-2

-1

d

The Nonlinear Optical Properties of Semiconductors – p. 34/39

DC Kerr effects UNIVERSITY OF

GLASGOW Optoelectronics Research Group

Franz-Keldysh effect: increases band-tail absorption Quantum Confined Stark Effect: shifts exciton resonances to longer wavelength Nonlinear absorption as excited carriers screen field (Self electro-optic effect device — SEED)

The Nonlinear Optical Properties of Semiconductors – p. 35/39

DC Kerr effects UNIVERSITY OF

GLASGOW Optoelectronics Research Group

E

Absorption

conduction band

valence band

Franz−Keldysh

Frequency

Absorption

E

QCSE

Frequency

The Nonlinear Optical Properties of Semiconductors – p. 36/39

Thermal effects UNIVERSITY OF

GLASGOW Optoelectronics Research Group

Absorbed light results in heating of medium and changes linear optical properties ∂n ∂n ∂Eg = ∂T ∂Eg ∂T

For Alx Ga1−x As ∂Eg /∂T = −(3.95 + 1.15x) × 10−4 eVK−1 .

Can differentiate Adachi formula for ∂n/∂Eg ∂n/∂Eg < 0 below bandgap giving ∂n/∂T > 0 generally e.g. 6.5 × 10−5 K−1 in GaAs at 1.5 µm The Nonlinear Optical Properties of Semiconductors – p. 37/39

Bibliography (materials) UNIVERSITY OF

GLASGOW Optoelectronics Research Group

1. S. Adachi, “GaAs, AlAs and Alx Ga1−x As: Material parameters for use in research and device applications”, J. Appl. Phys. 58, R1 (1985). 2. “Properties of Gallium Arsenide”, INSPEC (1990). 3. Ioffe Physico-Technical Institute Electronic Archive on Semiconductors http://www.ioffe.rssi.ru/SVA/NSM/ 4. E. O. Kane, J. Phys. Chem. Solids 1, 249 (1957). 5. P. Pfeffer & W. Zawadzki, “Conduction electrons in GaAs — 5-level k.p theory and polaron effects”, Phys. Rev. B 41, 1561 (1990). 6. G. Bastard, “Wave mechanics applied to semiconductor heterostructures” (Les Editions de Physique, 1989).

The Nonlinear Optical Properties of Semiconductors – p. 38/39

Bibliography (NLO) UNIVERSITY OF

GLASGOW Optoelectronics Research Group

1. P. N. Butcher & D. Cotter, “The Elements of Nonlinear Optics” (Cambridge University Press, 1990). 2. Y. R. Shen, “The Principles of Nonlinear Optics” (Wiley, 1984). 3. A. Yariv, “Quantum Electronics” (Wiley, 1989). 4. D. C. Hutchings, “Applied Nonlinear Optics”, http://userweb.elec.gla.ac.uk/d/dch/course.pdf 5. D. C. Hutchings, et al, “Kramers-Krönig relations in nonlinear optics”, Opt. and Quant. Electr. 24, 1 (1992). 6. D. C. Hutchings & B. S. Wherrett, “Theory of Anisotropy of Two-Photon Absorption in Zinc-Blende Semiconductors”, Phys. Rev. B 49, 2418 (1994). 7. D. C. Hutchings & B. S. Wherrett, “Theory of the Anisotropy of Ultrafast Nonlinear Refraction in Zinc-Blende Semiconductors”, Phys. Rev. B 52, 8150 (1995). 8. J. S. Aitchison, et al, “The Nonlinear Optical Properties of AlGaAs at the Half-Band-Gap”, IEEE J. Quantum Electron. 33, 341 (1997).

The Nonlinear Optical Properties of Semiconductors – p. 39/39