19. Nonlinear Optics19. Nonlinear Opticsoptics.hanyang.ac.kr/~shsong/P19-Nonlinear optics.pdf · 0...

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19. Nonlinear Optics

Transcript of 19. Nonlinear Optics19. Nonlinear Opticsoptics.hanyang.ac.kr/~shsong/P19-Nonlinear optics.pdf · 0...

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19. Nonlinear Optics19. Nonlinear Optics

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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

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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

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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

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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

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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, …]

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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→ +

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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⎧ ⎫ ⎧ ⎫= + = +⎨ ⎬ ⎨ ⎬⎩ ⎭ ⎩ ⎭

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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=

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Second harmonic generationSecond harmonic generation

From Fundamentals of Photonics (Bahaa E. A. Saleh)

2 / 2) (ω ω λ λ→ →

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Second harmonic generationSecond harmonic generation

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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)

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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

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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

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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

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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

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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)

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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)

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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

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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

= + −

→ + = +

+ = +

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Optical phase conjugationOptical phase conjugation

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Phase conjugate mirror (PCM)

Conventional mirror

PCM

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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

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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

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Degenerate Four-Wave Mixing as a Form of Real-Time Holography

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Image restoration by phase conjugationImage restoration by phase conjugation

Optical reciprocity.

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19.4. Coupled-wave theory of three-wave mixing19.4. Coupled-wave theory of three-wave mixing

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by equating terms on both sides at each of the frequencies w1, w2, and w3, separately,

Coupled-wave Equationsin three-wave mixing

Homework :

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Mixing of Three Collinear Uniform Plane Waves

Homework :

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Second-harmonic generation (SHG)

Z = 0

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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

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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).

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Second-harmonic generation (SHG)

Effect of Phase Mismatch

For weak-coupling case

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19.6. Anisotropic nonlinear media19.6. Anisotropic nonlinear media

Three-Wave Mixing in Anisotropic Second-Order Nonlinear Media

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Phase Matching in Three-Wave Mixing

Thus the fundamental wave is an extraordinary wave and the second-harmonic wave is an ordinary wave.