The Nonlinear Optical Properties of Semiconductors - University of

Research GroupOptoelectronics

GLASGOWUNIVERSITY OF

The Nonlinear Optical Properties ofSemiconductors

David C. Hutchings

[email protected]

Dept. of Electronics and Electrical Engineering

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



Optical Susceptibility Tensor

P = ε0

[

χ(1)E + χ(2)EE + χ(3)EEE + . . .]

χ(1)ij ω Linear refraction and absorption

χ(2)ijk ω1 + ω2 Sum frequency generation

ω1 − ω2 Difference frequency generation, rectification

ω + 0 Electro-optic (Pockel’s) effect

χ(3)ijkl ω + ω + ω Third harmonic generation

ω − ω + ω Nonlinear refraction and absorption, 4WM

ω + 0 + 0 Kerr electro-optic effect (DC)




Contents

Structure of zincblende semiconductors

χ(1): Absorption & Refraction

χ(2): Electro-optic effect (Pockels) & Frequencyconversion, e.g. SHG

χ(3): Two-photon Absorption, Nonlinear Refraction& 4 wave mixing

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




Waveguide geometry

Slab or rib waveguides

Assume weakly guiding

Also assume nonlinearity weak such thattransverse guided mode unchanged

Conventional orientation with [001] growth and [110]cleavage:

x y

z




Crystal Symmetries

Common compound semiconductors in photonicshave 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




Bandstructure models

Bloch form for wavefunction with periodic uk(r)

ψ(r) = uk(r)eik·r

Hamiltonian contains k · p term — treat asperturbation

2 parabolic bands: scalar modelKane model with singlet conduction band andtriplet valence band: vector but isotropicKane plus next highest conduction triplet:anisotropic




AlGaAs bandstructure

AlxGa1−xAs direct bandgap for x < 0.45




k · p models

∆

∆Γ

Γ

Γ

15

15

1

c

c

v

1

0

E

E

v

c

cE’




Quantum Theory of χ(n)

Susceptibility expressions are derived using thequantum Liouville equation, with HamiltonianH = H0 + (e/m0)A · p.

For χ(n) use sets of (n+ 1)-level systems and sumover (n+ 1)! time orderings.

Evaluation of χ(n) in semiconductors requires:Electronic energies (valence and conductionbands)Momentum matrix elements between electronicstatesSummation over all states




Linear absorption and refraction

χ(1)ii (ω) =

e2

2m20~ω

2

∑

k,bands

|ei · pvc(k)|2

×(

1

Ωvc(k) − (ω + iδ)+

1

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-rootdensity-of-states




χ(1) for 2-level system

-7.5 -5 -2.5 2.5 5 7.5HΩ-WL∆

-0.4

-0.2

0.2

0.4

0.6

0.8

1




Linear refraction

Dispersion with parabolic bands gives Adachi formula

ε(ω) = A

f(X) +1

2

[

Eg

Eg + ∆

]3/2

f(Xso)

+B

f(X) = Re(2 −√

1 +X −√

1 −X)/X2 n =√ε

X =~ω

Eg=λg

λXso =

~ω

Eg + ∆

A(x) = 6.3 + 19.0x B(x) = 9.4 − 10.2x

Eg = (1.425 + 1.155x+ 0.37x2) eV ∆ = 0.34 eV




Linear refraction (300K)

800 1000 1200 1400 1600

3.3

3.4

3.5

3.6

3.7

3.8

Refractive index vs wavelength (nm)x =0, 0.1, 0.2, 0.3




Slowly varying envelope approx.

Combine Maxwell’s equations → wave equation

Fourier transform: t→ ω

Substitute, with k = nω/c

E(ω) = E(ω, z)eikz

Assume envelope E(ω, z) varies slowly incomparison to wavelength of light

same as paraxial wave equation (BPM)

dE

dz= −α

2E +

iω2µ0

2kPNLe−ikz




Second-order nonlinearities

No second-order nonlinearity in media withinversion symmetry

Crystal symmetry in cubic semiconductorsspecifies that the only non-zero tensor elementsare

χ(2)xyz = χ

(2)yzx = χ

(2)zxy

More independent tensor elements inheterostructures and at surfaces




Pockels Electro-optic effect

Pockels Electro-optic effect is 0 + ω → ω andappears 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 sothat optical path length change is

∆nL =n3r41V L

2d

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




Frequency conversion by χ(2)

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, requirePhase-matching, i.e. phase velocities ofgenerating and generated waves must be identical.




Second harmonic generation

0.3 0.4 0.5 0.6 0.7 0.8Photon energy (eV)

0

100

200

300

χ(2) xyz(

ω,ω

) (p

mV

−1 )

kmax=0.6 (eV)1/2

kmax=1.0 (eV)1/2

kmax=1.5 (eV)1/2

0 Lc 2Lc 3Lc 4Lc 5Lc

distance0.0

10.0

20.0

30.0

40.0

2ω Ir

radi

ance

∆k=0

DR

DD

∆k= / 0




Second harmonic generation

ω 2ω

Simplest setup to characterise nonlinearinteraction

type-I configuration: TE fundamental → TM SH

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




Difference Frequency Generation

ωs pω /2 ωipω /2

ωp ωi

ωs

Channel conversion for WDM

Potentially integrate pump laser on chip




Parametric Amplification

ωp

ωs

ωi

ωs

Broad-bandwidth amplifier

Optical Parametric Oscillator for mid-IR — usecavity and ωs builds up from noise

Potentially integrate resonator

Potentially integrate pump laser on chip




χ(3) processes

third-harmonic generation ∝ χ(3)(ω, ω, ω)

two-photon absorption ∆α = βI

β(ω) =3ω

2ε0n20c

2Imχ

(3)eff (−ω, ω, ω)

nonlinear refraction ∆n = n2I

n2(ω) =3

4ε0n20c

Reχ(3)eff (−ω, ω, ω)

DC Kerr effect ∝ χ(3)(0, 0, ω)




χ(3) symmetry considerations

4 independent nonzero tensor elements for bulksemiconductors

3 for single ω

χ(3)xxxx(−ω, ω, ω), χ(3)

xyxy(−ω, ω, ω), χ(3)xxyy(−ω, ω, ω)

Three measurements required to completelycharacterise each nonlinear process

Breaking symmetry, e.g. heterostructureintroduces many more independent nonzerotensor elements




Two-photon Absorption

0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5Photon Energy (eV)

0

2x10−19

4x10−19

6x10−19

8x10−19

Imχ(3

) (m

2 V−

2 )

xxxx

xyxy

xxyy

0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5Photon Energy (eV)

0

5

10

15

20

25

30

Tw

o-ph

oton

abs

orpt

ion

(cm

/GW

)

E||[111]

E||[011]

E||[001]

k||[111]

k||[011]

k||[001]

Calculated Imχ(3) tensor elements and spectra of βfor GaAs.

Scales as E−3g .




Ultrafast Nonlinear Refraction

Three measurements completely characterisenonlinear refraction in bulk (cubic) semiconductor

Strength nL2 [001] ∆n/I|[001]

Anisotropy σ 2

nL2 [001] − nL

2 [011]

/nL2 [001]

Biref. param. δ

nL2 [001] − nC

2 (k ‖ [100])

/nL2 [001]

n2 scales as E−4g

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




Coupled Propagation Equations

Usual configuration in semiconductor slabwaveguides has TE‖[110] (cleavage planes) andTM‖[001].

i∂u

∂z+

∂2u

∂x2+ γu +

h

1 −

σ

2

uu∗

+

1 − δ −

σ

2

vv∗

iu +

δ −

σ

2

u∗v2= 0

i∂v

∂z+

∂2v

∂x2− γv +

h1 − δ −

σ

2

uu∗

+ vv∗

iv +

δ −

σ

2

u2v∗ = 0

u and v are the scaled electric field amplitudes forTE and TM.

γ is proportional to the (structurally induced)birefringence nTM − nTE.




Calculated χ(3) in GaAs

-4x10-19

0

4x10-19

Re

χ(3) (

m2 V

-2)

xxxxxyxyxxyy

0.0 0.5 1.0 1.5hω (eV)

-4x10-19

0

4x10-19

Re

χ(3) (

m2 V

-2)

xxxxxyxyxxyy

(b)

(a)

-1.5

-1.0

-0.5

0.0

0.5

1.0

σ’

-0.4

-0.2

0.0

0.2

0.4

δ’

0.0 0.2 0.4 0.6 0.8 1.0hω/Eg

-4x10-19

-2x10-19

0

2x10-19

4x10-19

(m2 V

-2)

(c)

(b)

(a)

Top: 14-band, bottom: 8-band (isotropic)




n2 in Al 0.18Ga0.82As

1.4 1.5 1.6 1.7Wavelength (µm)

0

1x10-13

2x10-13

3x10-13

n 2 (c

m2 W

-1)

0.4 0.5 0.6 0.7 0.8 0.9

Photon energy (eV)

0

50000

100000

150000

200000

250000

300000

Reχ

(3) (

pm/V

)2

xxxxxxyyxyxy

Measured dispersion of n2 in Al0.18Ga0.82As andcalculated Reχ(3) tensor components




n2 in Al 0.18Ga0.82As

Linear

-10

1

-1

0

1

-1

0

1

-10

1

-1

0

1Circular

-10

1

-1

0

1

-1

0

1

-10

1

-1

0

1

Deduced anistropy from measurements of n2 inAl0.18Ga0.82As at λ = 1.55 µm




Figure-of-merit for NLR applications

light absorbed inlength α−1

phase change2π|∆n|L/λ ∼ 2π forNLO applications

therefore requirefigure-of-merit|∆n|/(αλ) > 1

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

0.5 0.6 0.7 0.8 0.9 1.0hω/Eg

0.1

1.0

10.0

100.0

|Reχ

(3) |/I

mχ(3

)

NLDC

FP




Quasi-χ(3) processes

-7.5 -5 -2.5 2.5 5 7.5HΩ-WL∆

-0.4

-0.2

0.2

0.4

0.6

0.8

1




Carrier nonlinearities

Free carrier absorption: N ↑, α ↑Absorption saturation (bandfilling) in passivedevice: 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




Example: bandfilling

Assuming equal populations N of electrons andheavy-holes which have quasi-equilibrium Boltzmannthermal distribution, we get ∆n = σnN

σn(ω) = −4√π

n0

∣

∣

∣

∣

epvc

m0ωg

∣

∣

∣

∣

21

kBT

∑

j=hh,lh

mrj

meJ

(

mrj

me

~(ω − ωg)

kBT

)




Carrier nonlinearities

where

J(d) =

∫

∞

0

√xe−x dx

x− d

-5 -4 -3 -2 -1d

0.2

0.4

0.6

0.8

1

1.2

1.4

JHdL




DC Kerr effects

Franz-Keldysh effect: increases band-tailabsorption

Quantum Confined Stark Effect: shifts excitonresonances to longer wavelength

Nonlinear absorption as excited carriers screenfield (Self electro-optic effect device — SEED)




DC Kerr effects

QCSE

Frequency

Abs

orpt

ion

Frequency

Abs

orpt

ion

conductionband

valenceband

E

E

Franz−Keldysh




Thermal effects

Absorbed light results in heating of medium andchanges linear optical properties

∂n

∂T=

∂n

∂Eg

∂Eg

∂T

For AlxGa1−xAs∂Eg/∂T = −(3.95 + 1.15x) × 10−4 eVK−1.

Can differentiate Adachi formula for ∂n/∂Eg

∂n/∂Eg < 0 below bandgapgiving ∂n/∂T > 0 generallye.g. 6.5 × 10−5 K−1 in GaAs at 1.5 µm




Bibliography (materials)

1. S. Adachi, “GaAs, AlAs and AlxGa1−xAs: Material parameters for use in researchand device applications”, J. Appl. Phys. 58, R1 (1985).

2. “Properties of Gallium Arsenide”, INSPEC (1990).

3. Ioffe Physico-Technical Institute Electronic Archive on Semiconductorshttp://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 andpolaron effects”, Phys. Rev. B 41, 1561 (1990).

6. G. Bastard, “Wave mechanics applied to semiconductor heterostructures” (LesEditions de Physique, 1989).


http://www.ioffe.rssi.ru/SVA/NSM/



Bibliography (NLO)

1. P. N. Butcher & D. Cotter, “The Elements of Nonlinear Optics” (CambridgeUniversity 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. andQuant. Electr. 24, 1 (1992).

6. D. C. Hutchings & B. S. Wherrett, “Theory of Anisotropy of Two-Photon Absorptionin Zinc-Blende Semiconductors”, Phys. Rev. B 49, 2418 (1994).

7. D. C. Hutchings & B. S. Wherrett, “Theory of the Anisotropy of Ultrafast NonlinearRefraction in Zinc-Blende Semiconductors”, Phys. Rev. B 52, 8150 (1995).

8. J. S. Aitchison, et al, “The Nonlinear Optical Properties of AlGaAs at theHalf-Band-Gap”, IEEE J. Quantum Electron. 33, 341 (1997).


http://userweb.elec.gla.ac.uk/d/dch/course.pdf

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