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26. Nonlinear optics and light modulationLinear Optics and Nonlinear Optics
Linear OpticspThe optical properties, such as the refractive index and the absorption coefficient are independent of light intensity.The principle of superposition holdsThe principle of superposition holds.The frequency of light cannot be altered through the medium.Light cannot interact with light;
two beams of light in the same region of a linear opticaltwo beams of light in the same region of a linear optical medium can have no effect on each other. Thus light cannot see light.
Nonlinear optics (NLO)
The refractive index, and consequently the speed of light in an optical medium does change with the light intensitymedium, does change with the light intensity.The principle of superposition is violated.Light can alter its frequency as it passes through a nonlinear optical material (e g from red to blue!)material (e.g., from red to blue!).Light can interact with light via the medium
Thus light can control light.
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Nonlinear optics
02
Polarization :
Susceptibility :
P E
E E
ε χ=
+ + +21 2 3 LSusceptibility : E Eχ χ χ χ= + + +
χεεχεεχεεε +===→+=→+== 1 )1( 000 c
vnEEED
2 31 2 3 0 1 0 2 0 3L LP P P P E E Eε χ ε χχ ε= + + + = + + +
ε 0c
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Second-order Nonlinear optics 22 0 2P Eε χ=
Second-harmonic generation (SHG) and rectification
)2()( PP ωωω =±→∝→= )()( 2 ωω EPEE(0)
),2()(
2
22
PPP ωωω =±
Electro-optic (EO) effect (Pockell’s effect)
→∝→= )()( 2 ωω EPEEoptical Frequency doubling
Rectification
EP 22 ∝→
{ })()0( but, )()0( ,
ωω EEEEEopticalDCelectrical
>>+=
{ } { } { }{ }
DCelectricEnEEPP
EEPEEPEP
,22
222
2
)0()()0()((0),)()()(2 ,)()0()(,)0((0)
∝Δ→∝→
∝∝∝→
ωωωωωωω
Index modulation by DC E-field
Three-wave mixing
opticalopticalEEE )()( 21 ωω +=
{ } { } ,)()(2,)()(2 22
2212
12
22
ωωωω EPEP
EP
∝∝→
∝→
SHG
{ }{ })()()(
,)()()(
21212
21212
ωωωωωωωω
EEPEEP
∝−∝+ Frequency up-converter
Parametric amplifier, parametric oscillator
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Third-order Nonlinear optics 33 0 3P Eε χ=
Third-harmonic generation (THG)
{ } { })()3()()()( 32 ωωωωω EPEEP ∝∝→∝→= )()( 3 ωω EPEE { } { })()3(,)()()( 33 ωωωωω EPEEP ∝∝→∝→= )()( 3 ωω EPEEoptical
Frequency tripling
Electro-optic (EO) Kerr effectp ( )
{ })()0( but, )()0( ,
ωω EEEEEopticalDCelectrical
>>+=
22 2
DC ,
2
DC ,3 )0()()0()(electricelectric
EnEEP ∝Δ→∝→ ωω Index modulation by DC E2
Optical Kerr effect
)()()()()()( 23 ωωωωωω InEIEEP ∝Δ→∝∝ Index modulation by optical Intensity
Self-phase modulation)()( 000 nLkInnn Δ=Δ+=→Δ+= ϕϕϕ
{ } { } 00 )()( nxInxInnn >Δ→Δ+= Self-focusing, Self-guiding (Spatial solitons)
{ } { } 00 )()( nxInxInnn <Δ→Δ+= Self-defocusing
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Third-order Nonlinear optics 33 0 3P Eε χ=
Four-wave mixing
opticalopticalopticalEEEE )()()( 321 ωωω ++=
( ) terms2166,, 33321
33 =→±±±→∝→ ωωωEP ( )3213
Frequency up-converter)()()()( : 32143213 ωωωωωωω EEEPexampleOne ∝≡++→
ωωωωω 3=→==→ If THG
)()()()-( : 3*
2143213 ωωωωωωω EEEPexampleAnother ∝≡+→
ωωωωω 34321 =→==→ If THG
Degenerate four-wave mixing4321 ωωωω ===→ If
4321 ωωωω +=+→
directions oppositein traveling are themamong waves two
splane waveAssume→
)()()()( *EEEP )()()()( 43 ωωωωω EEEP ∝=→ Optical phase conjugation
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24-2. Second harmonic generation (SHG)
22 0 2P Eε χ= : Only for non-centro-symmetry crystals2 0 2χ
cosoE E tω=[GaAs. CdTe, InAs, KDP, ADP, LiNbO3, LiTaO3, …]
1 22 2
0 1 0 2cos coso o
P P P
E t E tε χ ω ε χ ω
= +
= + ( )2 1cos 1 cos 22
θ θ⎧ ⎫= +⎨ ⎬⎩ ⎭
2 20 1 0 2 0 2
1 1cos cos 22 2o o oE t E E tε χ ω ε χ ε χ ω= + +
2⎩ ⎭
1 1⎧ ⎫ ⎧ ⎫2 22 0 2 0 0 2 0 2 2
1 1( ) cos 2 (0) (2 )2 2
P t E E t P Pε χ ε χ ω ω⎧ ⎫ ⎧ ⎫= + = +⎨ ⎬ ⎨ ⎬⎩ ⎭ ⎩ ⎭
C (DC) S d h i tConstant (DC) termOptical rectification
Second harmonic term2ω
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SHG does not occur in isotropic, centrosymmetry crystals
2P E 22 0 2P Eε χ=
2 , is or If isotropic centrosymmetricχ
2-.
both E and E give the same P polarizationthat means the molecules are not polarized by the sencond effectχ
→ +
→
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Second harmonic generation
P22 2
2 0 2 0 0 2 0 2 21 1( ) cos 2 (0) (2 )2 2
P t E E t P Pε χ ε χ ω ω⎧ ⎫ ⎧ ⎫= + = +⎨ ⎬ ⎨ ⎬⎩ ⎭ ⎩ ⎭
P2(t)
E
E(t)
From Fundamentals of Photonics (Bahaa E. A. Saleh)
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Second harmonic generation
cosoE E tω=
P P P+1 2P P P= +
1 0 1 cosoP E tε χ ω=
( )21 cos 2P E tε χ ω= ( )2 0 2
2
cos 221
oP E t
E
ε χ ω
ε χ
=
+ 0 22 oEε χ+
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Second harmonic generation 2 / 2) (ω ω λ λ→ →
From Fundamentals of Photonics (Bahaa E. A. Saleh)
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Second harmonic generation
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Phase matching (index matching) in SHG
Output intensity after second harmonic generation
22sin , 2
2L kI c k k kω ωΔ⎛ ⎞∝ Δ = −⎜ ⎟
⎝ ⎠2⎜ ⎟⎝ ⎠
Optic axis
2k k kΔ = −
Phase matching : Δk=0 Direction of
Matching(Δk=0)2
2
2
2 2
k k k
n n
ω ω
ω ωω ω
Δ = −
⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
(Δk=0)
( )
2
2 2 0
c c
n n
ω ω
ω ωω
⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⎛ ⎞= − =⎜ ⎟⎝ ⎠
E-ray surface(n2ω)
( )2 cω ω ⎜ ⎟⎝ ⎠ O-ray surface
(nω)
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Frequency mixing by three-wave mixingy g y g
frequency up-converter ( )1 2 3ω ω ω+ ⇒
parametric amplifier ( )3 1 2 3 2 1 ω ω ω ω ω ω→− ⇒ − ⇒pa a et c a p e ( )3 1 2 3 2 1
parametric oscillator ( )ω ω ω ω ω ω→⇒ + − ⇒( )2 idler, or parameter, ω → 중개자
parametric oscillator ( )3 1 2 3 2 1 ω ω ω ω ω ω→⇒ + − ⇒
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Parametric interaction
1 2( ) ( )E E EE Eω ω= +
{ } { }1 1 2 2
1 1 1 2 2 2
cos cos1 1exp( ) exp( ) exp( ) exp( )2 2
o o
o o
E t E t
E i t i t E i t i t
ω ω
ω ω ω ω
= +
= + − + + −
22 0 2
2 2
P Eε χ=
( ) ( ) ( ) ( )2 22 21 1 1 2 2 2 1 3 1 3+ , + , , ω ω ω ω ω ω ω ω ω ω ω ω− = +⇒ = ==
Frequency conservation
Momentum (phase) matching
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Nonlinearity of the refractive index
0Polarization : P Eε χ=2
1 2 3 LSusceptibility : E Eχ χ χ χ= + + +
)1( +
2 221 1 ErE R= + +
)1( χ+=n
2 2on
En
rE R+ +
Second-order nonlinearity (P2) third-order nonlinearity (P3)
Linear electro-optic coefficient (r)
Pockels effect (E: DC field)
Quadratic electro-optic coefficient(R)
Kerr effect( ) Kerr effect
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Nonlinearity of the refractive index
21 1 r REE= + +2 2o
r REn
En
+ +
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Pockels effect (Linear electro-optic effect)
Phase difference (SA & FA)by inducing DC E fieldL by inducing DC E-field
2 ϕΦ = ΔSA FA
L
22o
L nπλ
⎛ ⎞= Δ⎜ ⎟
⎝ ⎠
3222
o
orL n Eπ
λ
⎝ ⎠⎛ ⎞ ⎛ ⎞≅ ⎜ ⎟ ⎜ ⎟
⎝ ⎠⎝ ⎠V
3 3
2
2 2o
o orn EL rn V
λ
π πλ λ
⎝ ⎠⎝ ⎠⎛ ⎞ ⎛ ⎞
= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
Pockels cell
h lf l t t koλ
o oλ λ⎝ ⎠ ⎝ ⎠
: half-wave plate to make 32o
HWo
Vrn
π= Φ =
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Pokels electro-optic modulator
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Pockels effect (Q-switch in laser cavity)
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Third-order nonlinear effectIn media possessing centrosymmetry, the second-order nonlinear term is absent since the polarization must reverse exactly when the p yelectric field is reversed. The dominant nonlinearity is then of third order,
33 0 3P Eε χ=
The third-order nonlinear material is called a Kerr medium.
P3
EE
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Electro-optic Kerr effect : E=V/d
Kerr cell
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All-optical Kerr effect : E=E(ω)
3 20 2
Rn n E n I⎛ ⎞Δ = ≡⎜ ⎟ 0 2( )n I n n I= +0 22n n E n IΔ ⎜ ⎟
⎝ ⎠ 0 2( )n I n n I+
S lf h d l tiSelf-phase modulationThe phase shift incurred by an optical beam of power P and cross-sectional area A, traveling a distance L in the medium,cross sectional area A, traveling a distance L in the medium,
Self-focusing (Optical Kerr lens)
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All-optical Kerr effect : Spatial solitons
I li diIn linear medium,wave is spreading.
In optical Kerr medium,
( )n I n n I= +
wave can be guided.
0 2( )n I n n I= +
Self-guided beam = spatial soliton
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24-6. Optical phase conjugation
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Phase Conjugation and Time ReversalPhase Conjugation and Time Reversal
( ) ( ) ( )[ ]kztietE −= ωψ rr Re,1Incident
( ) ( ) ( )[ ]kztietE +∗= ωψ rr Re,2Phase conjugation ( ) ( )[ ]
( ) ( ) ( )Re i t kzE t e ωψ ⎡ − − ⎤⎣ ⎦⎡ ⎤r rTime re ersal ( ) ( ) ( )2 , ReE t eψ ⎣ ⎦⎡ ⎤= ⎣ ⎦r rTime reversal
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Four-wave mixing (third-order nonlinearity)
Superposition of three waves of angular frequencies ω1, ω2, and ω3
33 0 3P Eε χ= (as sum of 63 = 216 terms)
If ω ω ω ω= +
{ }4 1 2 3
3 4 1 2
If ω ω ω ω
ω ω ω ω
= + −
→ + = +
{ }3 4 1 2
r r r r k k k k+ = +
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Four wave mixing by phase conjugation
A1Pump beam
*4 1 2 3A A A A∝
NonlinearAMedium
(FWM)
A3signal
A4 (FWM)A4PC output
A2 Pump beam
It l k lik iIt looks like a mirror,but it’s quite strange.
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Phase conjugate mirror (PCM)Conventional mirror
PCM
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Image restoration by phase conjugation
Optical reciprocity.
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24 4 Faraday effect (Linear magneto optic effect)
Some other effects for optical modulation
24-4. Faraday effect (Linear magneto-optic effect)
Rotation of the polarization plane: Faraday rotator : V; Verdet constantVBd
e dn Bd
β
λ
=
⎛ ⎞⎜ ⎟
Faraday rotator
2
1 0083
eBd
m c d
dn Bd
λ
λ
= ⎜ ⎟⎝ ⎠⎛ ⎞⎜ ⎟
d
1.0083 Bdd
λλ
= ⎜ ⎟⎝ ⎠
Optical isolatorOptical isolator
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24-5. Acousto-optic effect
AO (acousto-optic) effect : interaction of optical & acoustic wavesPhotoelasticity : change in n of crystal due to mechanical stress
( p ) pBrillouin scattering : collision between photons and acoustic phonons
K’
K
KKs
K
' sk k k= ±
: momentumconservation
': energy
sω ω ω= ±
: energy conservation
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AO effect : Bragg condition
K’Ks
K
'k k k= ±
θθK K’ : momentum conservation
sk k k±
ΦD ' ti
sω ω ω= ±
λs: energy conservation
Doppler effect for light
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AO effect : Raman-Nath regime
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(참고) Stimulated Inelastic Scattering
Stimulated Brillouin Scattering (SBS)
: interaction between photon and acoustic-phonon
photons scatter from acoustic wave
It occurs only in backward direction
Stimulated Raman Scattering (SRS)
: interaction between photon and optical-phonon (induced by incident photon)
Photon with hν energy incident on molecule having vibration frequency νm
molecules absorb energy from photon
optical phonon is induced
photon is scattered
It i f d di tiIt occurs in forward direction