19. Nonlinear Optics19. Nonlinear Optics
02
1
2 31 2 3 0 1 0 2 0 3
2 3
Polarization :
Susceptibility :
P E
E E
P P P P E E E
ε χ
χ χ χ
ε χ ε χ
χ
χ ε
=
= + + +
= + + + = + + +
Nonlinear opticsNonlinear optics
χεεχεεχεεε +===→+=→+== 1 )1(
0000 c
vnEEED
Second-order Nonlinear opticsSecond-order Nonlinear opticsSecond-harmonic generation (SHG) and rectification
(0) ),2()(
2
22
PPP ωωω =±
Electro-optic (EO) effect (Pockell’s effect)
→∝→= )()( 22 ωω EPEE
optical
{ } { } { }{ }
DCelectricEnEEPP
EEPEEPEP
EP
,22
222
2
22
)0()()0()((0),)()()(2 ,)()0()(,)0((0)
∝Δ→∝→
∝∝∝→
∝→
ωωωωωωω
{ })()0( but, )()0( ,
ωω EEEEEopticalDCelectrical
>>+=
Three-wave mixing
22 0 2P Eε χ=
opticalopticalEEE )()( 21 ωω +=
{ } { }{ }{ })()()(
,)()()( ,)()(2,)()(2
21212
21212
22
2212
12
22
ωωωωωωωω
ωωωω
EEPEEP
EPEP
EP
∝−∝+
∝∝→
∝→
SHG
Frequency up-converter
Parametric amplifier, parametric oscillator
Index modulation by DC E-field
Frequency doubling
Rectification
Third-order Nonlinear opticsThird-order Nonlinear opticsThird-harmonic generation (THG)
{ } { })()3(,)()()( 33
23 ωωωωω EPEEP ∝∝
Optical Kerr effect
→∝→= )()( 33 ωω EPEE
optical
33 0 3P Eε χ=
Self-phase modulation
Frequency tripling
)()()()()()( 23 ωωωωωω InEIEEP ∝Δ→∝∝ Index modulation by optical Intensity
)()( 000 nLkInnn Δ=Δ+=→Δ+= ϕϕϕ
{ } { } 00 )()( nxInxInnn >Δ→Δ+= Self-focusing, Self-guiding (Spatial solitons)
{ } { } 00 )()( nxInxInnn <Δ→Δ+= Self-defocusing
Electro-optic (EO) Kerr effect
{ })()0( but, )()0( ,
ωω EEEEEopticalDCelectrical
>>+=
2
DC ,
2
DC ,3 )0()()0()(electricelectric
EnEEP ∝Δ→∝→ ωω Index modulation by DC E2
Third-order Nonlinear opticsThird-order Nonlinear opticsFour-wave mixing
33 0 3P Eε χ=
opticalopticalopticalEEEE )()()( 321 ωωω ++=
( ) terms2166,, 33321
33 =→±±±→∝→ ωωωEP
Frequency up-converter
Degenerate four-wave mixing
)()()()( : 32143213 ωωωωωωω EEEPexampleOne ∝≡++→
)()()()-( : 3*
2143213 ωωωωωωω EEEPexampleAnother ∝≡+→
4321 ωωωω ===→ If
ωωωωω 3 4321 =→==→ If THG
4321 ωωωω +=+→
directions oppositein traveling are themamong waves two
splane waveAssume→
)()()()( *43 ωωωωω EEEP ∝=→ Optical phase conjugation
22 0 2P Eε χ=
1 22 2
0 1 0 2
2 20 1 0 2 0 2
cos
cos cos1 1cos cos 22 2
o
o o
o o o
E E tP P P
E t E t
E t E E t
ω
ε χ ω ε χ ω
ε χ ω ε χ ε χ ω
=
= +
= +
= + +
Constant (DC) termOptical rectification
Second harmonic term2ω
: Only for non-centro-symmetry crystals
24-2. Second harmonic generation (SHG)24-2. Second harmonic generation (SHG)
( )2 1cos 1 cos 22
θ θ⎧ ⎫= +⎨ ⎬⎩ ⎭
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⎧ ⎫ ⎧ ⎫= + = +⎨ ⎬ ⎨ ⎬⎩ ⎭ ⎩ ⎭
[GaAs. CdTe, InAs, KDP, ADP, LiNbO3, LiTaO3, …]
SHG does not occur in isotropic, centrosymmetry crystals
22 0 2P Eε χ=
2
2
,-
.
is or
If isotropic centrosymmetricboth E and E give the same P polarizationthat means the molecules are not polarized by the sencond effect
χ
χ→ +
→
Second harmonic generationSecond harmonic generation
From Fundamentals of Photonics (Bahaa E. A. Saleh)
P2
E
P2(t)
E(t)
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⎧ ⎫ ⎧ ⎫= + = +⎨ ⎬ ⎨ ⎬⎩ ⎭ ⎩ ⎭
Second harmonic generationSecond harmonic generation
cosoE E tω=
( )22 0 2
20 2
1 cos 2212
o
o
P E t
E
ε χ ω
ε χ
=
+
1 2P P P= +
1 0 1 cosoP E tε χ ω=
Second harmonic generationSecond harmonic generation
From Fundamentals of Photonics (Bahaa E. A. Saleh)
2 / 2) (ω ω λ λ→ →
Second harmonic generationSecond harmonic generation
Phase matching (index matching) in SHGPhase matching (index matching) in SHG
( )
2
2
2
2
2 2
2 0
k k k
n nc c
n nc
ω ω
ω ω
ω ω
ω ω
ω
Δ = −
⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⎛ ⎞= − =⎜ ⎟⎝ ⎠
Output intensity after second harmonic generation
22sin , 2
2L kI c k k kω ωΔ⎛ ⎞∝ Δ = −⎜ ⎟
⎝ ⎠
Phase matching : Δk=0
O-ray surface(nω)
E-ray surface(n2ω)
Optic axis
Direction ofMatching
(Δk=0)
Frequency mixing by three-wave mixingFrequency mixing by three-wave mixing
frequency up-converter
parametric amplifier
parametric oscillator
( )1 2 3ω ω ω+ ⇒
( )3 1 2 3 2 1 ω ω ω ω ω ω→− ⇒ − ⇒
( )3 1 2 3 2 1 ω ω ω ω ω ω→⇒ + − ⇒( )2 idler, or parameter, ω → 중개자
Parametric interactionParametric interaction
{ } { }
( ) ( ) ( ) ( )
1 2
1 1 2 2
1 1 1 2 2 2
22 0 2
2 2
( ) ( )cos cos
1 1exp( ) exp( ) exp( ) exp( )2 2
2 21 1 1 2 2 2 1 3 1 3+ , + , ,
o o
o o
E E EE t E t
E i t i t E i t i t
P E
ω ωω ω
ω ω ω ω
ε χ
ω ω ω ω ω ω ω ω ω ω ω ω
= +
= +
= + − + + −
− = +⇒ = =
=
=
Frequency conservation
Momentum (phase) matching
Third-order nonlinear effectThird-order nonlinear effectIn media possessing centrosymmetry, the second-order nonlinear term is absent since the polarization must reverse exactly when the electric 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
E
Optical Kerr effect
2( )n I n n I= +
Self-phase modulation
)()()()()()( 23 ωωωωωω InEIEEP ∝Δ→∝∝ Index modulation by optical Intensity
)()( 000 nLkInnn Δ=Δ+=→Δ+= ϕϕϕ
{ } { } 00 )()( nxInxInnn >Δ→Δ+= Self-focusing, Self-guiding (Spatial solitons)
{ } { } 00 )()( nxInxInnn <Δ→Δ+= Self-defocusing
2( )n I n n I= +
Self-phase modulationThe phase shift incurred by an optical beam of power P and cross-sectional area A, traveling a distance L in the medium,
Self-focusing (Optical Kerr lens)
In a linear medium,wave is spreading.
In an optical Kerr medium, wave can be self-guided.= Self-guided beam
Spatial Solitons
: Nonlinear Schrodinger equation
(Also, see 19.8)
Raman Gain
: Raman GainCoefficient
Coupling of light to the high-frequency vibrational modes of the medium,which act as an energy source providing the gain.
For low-loss media, the Raman gain may exceed the loss at reasonable levels of power, so that the medium can act as an optical amplifier.
Fiber Raman lasers
Four-wave mixing (third-order nonlinearity)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)
{ }{ }
4 1 2 3
3 4 1 2
3 4 1 2
If
k k k k
ω ω ω ω
ω ω ω ω
= + −
→ + = +
+ = +
Optical phase conjugationOptical phase conjugation
Phase conjugate mirror (PCM)
Conventional mirror
PCM
Optical phase conjugationOptical phase conjugationWhen all four waves are of the same frequency degenerated four-wave mixing (DFWM)
Assuming further that two waves (3,4) are uniform plane waves traveling in opposite directions,
Nonlinear
Medium
(DFWM)
A1Probe beam
A3Pump beam
A4 Pump beam
A2PC beam
*2 3 4 1A A A A∝ : conjugated version of wave 1
Note: Phase Conjugation and Time Reversal
( ) ( ) ( )[ ]kztietE −= ωψ rr Re,1
( ) ( ) ( )[ ]kztietE +∗= ωψ rr Re,2
( ) ( ) ( )2 , Re i t kzE t e ωψ ⎡ − − ⎤⎣ ⎦⎡ ⎤= ⎣ ⎦r r
Phase conjugation
Time reversal
Incident
Degenerate Four-Wave Mixing as a Form of Real-Time Holography
Image restoration by phase conjugationImage restoration by phase conjugation
Optical reciprocity.
19.4. Coupled-wave theory of three-wave mixing19.4. Coupled-wave theory of three-wave mixing
by equating terms on both sides at each of the frequencies w1, w2, and w3, separately,
Coupled-wave Equationsin three-wave mixing
Homework :
Mixing of Three Collinear Uniform Plane Waves
Homework :
Second-harmonic generation (SHG)
Z = 0
Second-harmonic generation (SHG)
photons of wave 1 are converted to half as many photons of wave 3.photon numbers are conserved.
Photon flux densities
Second-harmonic generation (SHG)
Efficiency of second-harmonic generation
To maximize the efficiency, we must confine the wave to the smallest possible area A and the largest possible interaction length L. This is best accomplished with waveguides (planar or channel waveguides or fibers).
Second-harmonic generation (SHG)
Effect of Phase Mismatch
For weak-coupling case
19.6. Anisotropic nonlinear media19.6. Anisotropic nonlinear media
Three-Wave Mixing in Anisotropic Second-Order Nonlinear Media
Phase Matching in Three-Wave Mixing
Thus the fundamental wave is an extraordinary wave and the second-harmonic wave is an ordinary wave.