p19 nonlinear optics

7/27/2019 P19 Nonlinear Optics

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19. Nonlinear Optics

19. Nonlinear Optics


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0

21

2 31 2 3 0 1 0 2 0 3

2 3

"

"

"

Polarization :

Susceptibil ity :

P E

E E

P P P P E E E

== + + +

= + + + = + + +

Nonlinear optics

Nonlinear optics

+===+=+== 1)1(

0

000cvnEEED


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Second-order Nonlinear optics

Second-order Nonlinear optics

Second-harmonic generation (SHG) and rectif ication

(0)

),2()(

2

22

P

PP =

Electro-optic (EO) effect (Pockells effect)

= )()( 22 EPEE optical

{ } { } { }{ }

DCelectricEnEEPP

EEPEEPEP

EP

,22

22

2

2

2

2

)0()()0()((0),

)()()(2,)()0()(,)0((0)

{ })()0(but,)()0(,

EEEEEopticalDCelectrical

>>+=

Three-wave mixing

22 0 2P E

opticaloptical EEE )()( 21 +=

{ } { }

{ }{ })()()(,)()()(

,)()(2,)()(2

21212

21212

2

2

221

2

12

2

2

EEPEEP

EPEP

EP

+

SHG

Frequency up-converter Parametric amplif ier, parametric oscillator

Index modulation by DC E-field

Frequency doubling

Rectification


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Third-order Nonlinear opticsThird-order Nonlinear optics

Third-harmonic generation (THG)

{ })()3(,)()()( 332

3 EPEEP

Optical Kerr effect

= )()( 33 EPEE optical

33 0 3P E

Self-phase modulation

Frequency tr ipling

)()()()()()(2

3 InEIEEP Index modulation by optical Intensity

)()( 000 nLkInnn =+=+=

{ } { } 00 )()( nxInxInnn >+= Self-focusing, Self-guiding (Spatial soli tons)

{ } { } 00 )()( nxInxInnn >+=

2

DC,

2

DC,3 )0()()0()( electricelectric EnEEP Index modulation by DC E2


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Third-order Nonlinear opticsThird-order Nonlinear optics

Four-wave mixing

33 0 3P E

opticalopticalopticalEEEE )()()( 321 ++=

( ) terms2166,, 333213

3 = EP

Frequency up-converter

Degenerate four-wave mixing

)()()()(: 32143213 EEEPexampleOne ++

)()()()-(: 3*

2143213 EEEPexampleAnother +

4321 ===

If

34321 === If THG

4321 +=+

directionsoppositeintraveling

arethemamongwavestwo

splane wave

Assume

)()()()( *43 EEEP = Optical phase conjugation


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22 0 2P E

1 2

2 20 1 0 2

2 20 1 0 2 0 2

cos

cos cos

1 1cos cos 22 2

o

o o

o o o

E E tP P P

E t E t

E t E E t

== += += + +

Constant (DC) term

Optical rectificationSecond harmonic term

2

: Only for non-centro-symmetry crystals

24-2. Second harmonic generation (SHG)24-2. Second harmonic generation (SHG)

( )2 1cos 1 cos 22

= +

2 22 0 2 0 0 2 0 2 2

1 1( ) cos 2 (0) (2 )

2 2P t E E t P P = + = +

[GaAs. CdTe, InAs, KDP, ADP, LiNbO3, LiTaO3, ]


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SHG does not occur in isotropic, centrosymmetry crystals

22 0 2P E

2

2

,-

.

is orIf isotropic centrosymmetricboth E and E give the same P polarization

that means the molecules are not polarized by the sencond effect

+


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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 2P t E E t P P = + = +


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cosoE E t

( )22 0 22

0 2

1 cos 22

1

2

o

o

P E t

E

=+

1 2P P P+

1 0 1 cosoP E t


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

2 / 2)(


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Phase matching (index matching) in SHGPhase matching (index matching) in SHG

( )

2

2

2

2

22

2 0

k k k

n nc c

n nc

= =

= =

Output intensity after second harmonic generation

2 2sin , 22

L kI c k k k

=

Phase matching : k=0

O-ray surface

(n)

E-ray surface

(n2)

Optic axis

Direction of

Matching

(k=0)


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Frequency mixing by three-wave mixingFrequency mixing by three-wave mixing

frequency up-converter

parametric amplifier

parametric oscil lator

( )1 2 3 +

( )3 1 2 3 2 1

( )3 1 2 3 2 1 + ( )2 idler, or parameter,


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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 1

exp( ) exp( ) exp( ) exp( )2 2

2 21 1 1 2 2 2 1 3 1 3+ , + , ,

o o

o o

E E E

E t E t

E i t i t E i t i t

P E

= += += + + +

= += ==

=

Frequency conservation Momentum (phase) matching


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Third-order nonlinear effectThird-order nonlinear effect

In 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 E

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)()()()()()(

2

3 InEIEEP

Index modulation by optical Intensity

)()( 000 nLkInnn =+=+=

{ } { } 00 )()( nxInxInnn >+= Self-focusing, Self-guiding (Spatial solitons)

{ } { } 00 )()( nxInxInnn


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2( )n I n n I+Self-phase modulation

The 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 Gain

Coefficient

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 opt ical 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 1, 2, and 3

33 0 3P E (as sum of 63 = 216 terms)

{ }{ }

4 1 2 3

3 4 1 2

3 4 1 2

G G G G

If

k k k k

= + + = +

+ = +


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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 conjugation

When 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

A4Pump 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 = rr Re,1

( ) ( ) ( )kztietE += rr Re,2

( ) ( ) ( )2 , Rei t kz

E t e

= 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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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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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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Effect of Phase Mismatch

For weak-coupl ing 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 waveand the second-harmonic wave is an ordinary wave.

p19 nonlinear optics

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