Research GroupOptoelectronics
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The Nonlinear Optical Properties ofSemiconductors
David C. Hutchings
Dept. of Electronics and Electrical Engineering
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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)
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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
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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
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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
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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
The Nonlinear Optical Properties of Semiconductors – p. 6/39
Research GroupOptoelectronics
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AlGaAs bandstructure
AlxGa1−xAs direct bandgap for x < 0.45
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k · p models
∆
∆Γ
Γ
Γ
15
15
1
c
c
v
1
0
E
E
v
c
cE’
The Nonlinear Optical Properties of Semiconductors – p. 8/39
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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
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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
The Nonlinear Optical Properties of Semiconductors – p. 10/39
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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
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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
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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
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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
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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
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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
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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.
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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
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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
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Difference Frequency Generation
ωs pω /2 ωipω /2
ωp ωi
ωs
Channel conversion for WDM
Potentially integrate pump laser on chip
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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
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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)
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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
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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 .
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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
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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.
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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)
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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
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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
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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
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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
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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
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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
)
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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
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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)
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DC Kerr effects
QCSE
Frequency
Abs
orpt
ion
Frequency
Abs
orpt
ion
conductionband
valenceband
E
E
Franz−Keldysh
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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
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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).
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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).
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