Chapter 6. Processes Resulting from the Intensity ...optics.hanyang.ac.kr/~choh/degree/[2013-1... · Hanyang Univ. Wave equation : i i i P ct E c E 4 ~ ~ ~ 2 2 2 w w w H Swhere, (r,

Nonlinear Optics Lab. Hanyang Univ.

Chapter 6. Processes Resulting from

the Intensity-Dependent Refractive Index

- Optical phase conjugation

- Self-focusing

- Optical bistability

- Two-beam coupling

- Optical solitons

Reference :

R.W. Boyd, “Nonlinear Optics”, Academic Press, INC.

- Photorefractive effect (Chapter 10)

: cannot be described by a nonlinear susceptibility c(n) for any value of n


6.1 Optical Phase Conjugation

: Generation of a time-reversed wavefront

Signal wave :

rk*

s

**

s

rk

sss

s

s

AÊ

AÊ

i

s

i

ee

ee

c.c.E),r(E~

ss tiet

Phase conjugate wave : c.c.E),r(E~ *

sc tiert where, r : amplitude reflection coefficient


rk*

s

**

ssAÊ

i

s eeProperties of phase conjugate wave :

1) *ˆse : The polarization state of circular polarized light does not change in reflection from PCM

Ex) ji/2i

00s ee ee

i) In reflection from metallic mirror [-phase shift]

ii) In reflection from PCM [-phase shift &

y component : i/2 -i/2 (- : delayed)]

2) *A s : The wavefront is reversed

)r(*)r( tt AA i

ss

i

ss eaea

3) : The incident wave is reflected back into its direction of incidencerk s i

e

ss kk


Aberration correction by optical phase conjugation

Wave equation :

0

~)r(~

2

2

2

2

t

E

cE

Solution : ..)r(),r(~ )( cceAtE tkzi

With slow varying approximation, 02)r( 2

2

22

z

AikAk

cAT

Since the equation is generally valid, so is its complex conjugate, which is given explicitly by

02)r( *

*2

2

2*2

z

AikAk

cAT

Solution : ..)r(),r(~ )(* cceAtE tkzi

c

: A wave propagating in the –z direction whose complex amplitude

is everywhere the complex of the forward-going wave



Phase Conjugation by Degenerated Four-Wave Mixing

1) Qualitative understanding

Four interacting waves : ..)r(..)r(),r(~ )r(

cceAcceEtEtki

i

ti

iii )4,3,2,1( i

Nonlinear polarization :r)(*

321

)3(*

321

)3( 32166

kkkiNL eAAAEEEP cc

(021 kk Counter-propagating waves)

r*

321

)3( 36

ik

eAAAc

New wave (k4) source term !

So, A4 is proportional to A3* (complex conjugate of A3)

and its propagation direction is –k3(in the case of perfect phase matching)


2) Rigorous treatment

Total field amplitude :4321 EEEEE

Nonlinear polarization : *2)3( )(3 EEPNL c

*

243

*

441

*

331

*

221

*

1

2

1

)3(

1 22223 EEEEEEEEEEEEEEP c

*

143

*

442

*

332

*

112

*

2

2

2

)3(


*

421

*

443

*

223

*

113

*

3

2

3

)3(


*

321

*

334

*

224

*

114

*

4

2

4

)3(


4321 ,, EEEE Neglect the 2nd order terms of E3 and E4

*

221

*

1

2

1

)3(

1 23 EEEEEP c

*

112

*

2

2

2

)3(

2 23 EEEEEP c

*

421

*

223

*

113

)3(

3 2223 EEEEEEEEEP c

*

321

*

224

*

114

)3(

4 2223 EEEEEEEEEP c


Wave equation :

ii

i Ptct

E

cE

~4~

~2

2

22

2

2

2

where, ..)r(..)r(),r(

~ )r(cceAcceEtE

tki

i

ti

iii

(1) Pump waves, A1 and A2 (slow varying approximation)

111

2

2

2

1

)3(1 ||2||6

AiAAAnc

i

dz

dAc

222

2

1

2

2

)3(2 ||2||6

AiAAAnc

i

dz

dAc

# Each wave shifts the phase of the other wave

by twice as much as it shift its own phase

# Since only the phase of the pump waves are

affected by nonlinear coupling, the quantities

|A1|2 and |A2|

2 are spatially invariant, and hence

the k1 and k2 are in fact constantSolution :zi

eAzA 1)0()( 11

zieAzA 2)0()( 22

*

3

)(

21

*

321421)0()0( AeAAAAAP

zi

(6.1.15)

: Nonlinear polarization responsible for producing the phase

conjugate wave varies spatially.

)0()0()()()0()0( 21212121 AAzAzAAA

Therefore, two pump beams should have equal intensities :


(2) Signal (A3) and conjugate waves (A4)

*

433

*

4213

2

2

2

1

)3(3 )||2|(|12

AiAiAAAAAAnc

i

dz

dAc

*

343

*

3214

2

2

2

1

)3(4 )||2|(|12

AiAiAAAAAAnc

i

dz

dAc

where, )||2|(|12 2

2

2

1

)3(

3 AAnc

i c

21

)3(12AA

nc

ic

put, zieAA 3'

33

zieAA 4'

44

'

3

'

4

'

4

'

3

Aidz

dA

Aidz

dA

)4,3(0|| '2

2

'

iii A

dz

dA


Solution :

)0(||cos

)](|sin[|

|)(

||cos

||cos)(

)0(||cos

)](|cos[|)(

||cos

||sin||)(

'*

3

'

4

'

4

'*

3

'

4

'*

3

AL

LziLA

L

zzA

AL

LzLA

L

zizA

0)('

4 LA ( conjugate wave at z=L is zero)

)0()||(tan||

)0(

||)/cos0()(

'*

3

'

4

'*

3

'*

3

ALi

A

LALA

i) )0()( '*

3

'*

3 ALA

~0)0('

4Aii)

: amplification

: depends on L|| (can exceed 100% pump wave energy)


Processes of degenerated four-wave mixing: One photon from each of the pump waves is annihilated

and one photon is added to each of the signal and conjugate waves

one photon transition two photon transition wave-vectors

Amplification of A3 and over 100% conversion of A4/A3 are possible


Experimental set-ups

0)( 21 kk 0)( 43 kk

1A2A

3A

4A

43 AA


17.5 Optical Resonator with Phase Conjugate Reflectors (A. Yariv)

189

178

167

156

145

134

123

12

),(

),(

),(),(

),(

)),((

),(),(

nm

nm

nmnm

nm

nm

nmnm

l

l

ll

R

Rl

lR

lRl

R

R ),( nml

# The self-consistence condition is satisfied

automatically every two round trips.

The phase conjugate resonator is stable

regardless of the radius of curvature R

of the mirror and the spacing l.


17.6 The ABCD Formalism of Phase Conjugate Optical Resonator

The wave incident upon the PCM :

)

2(exp)()

2(exp)(E

2

2

22

i

iiiq

krkztiε

w

rkrkztiε

rr

)

2(exp)()

2(exp)(E

2*

2

22*

r

iirq

krkztiε

w

rkrkztiε

rr

2

11

w

i-

qi

where,

Reflected conjugate wave :

*2

111

ir qw

i

q

1-0

01

DC

BA,

DC

BA*

*

Mi

ir

q

qq

Ray transfer matrix for the PCM mirror

By comparing the ABCD law for ordinary optical elements,


ABCD law at any plane following the PCM :

T

*

T

T

*

T

DC

BA

i

iout

q

qq

Example)

10

01

1R

2-

01

DC

BA

10

01

DC

BA

1R

2-

01

DC

BA

11

11

1M

Matrix after one round trip :

Matrix after two round trip :

IMM

10

01

10

01

1R

2-

01

10

01

1R

2-

012

12

: Self-consistence condition is satisfied

automatically every two round trips


17.7 Dynamic Distortion Correction within a Laser Resonator

Phase conjugate resonator

Distortion corrected beam


17.8 Holographic Analogs of Phase Conjugate Optics

1) Holography

recording reading


2) Phase conjugate optics

43 AA

Holography by phase conjugation

- Real time processing (no developing process)

- Distortion free image transmission


17.9 Imaging through a Distorted Medium

Distortion free

transmission (A2)


6.2 Self-Focusing of Light

I20 nnn 22

e)I( wrr Gaussian beam :

defocusing-self

focusing-self

: 0

: 0

2

2

n

n


Self-Trapping

: Beam spread due to diffraction is precisely compensated by the contraction due to self-focusing

Simple model for self-trapping

Critical angle for total internal reflection :nn

n

0

00cos

00

2

1

00

0

2

0

2

12

11

nn

n

n


A laser beam of diameter d will contain rays within a cone whose maximum angular extent

is of the order of magnitude of diffraction angle ;

dndnd

00

6.02

So, the condition for self-trapping :

0 d

2

0

0

0

2

1

0

6.0

2

16.02

dn

nndnn

n

21

20 I26.0

nnd I2nn

Critical laser power :

20

2

2

8

6.0I

4P

nndcr

# Independent of the beam diameter

Ex) CS2, n2=2.6x10-14 cm2/W, n0=1.7, =1m

Pcr = 33 kW


Simple model of self-focusing

2w0

zf

nn 2

10

nn 0

)2

1()()( 0

21

2

0

2

0 nnwznnz ff

2

0000

2

1

2

1)(

f

ffffz

wnznznznnz

2

1

00

0

0

2

1

00

22

nnw

nn

wz f

where, : critical angle

I2nn

21

2

02

0

21

2

00

2I

Pn

nw

n

nwz f

where, I2

1 2

0wP : total power

21

2

00 1

6.0

2

cr

f

PP

wnz


6.3 Optical Bistability

: Two different output intensities for a given input intensity

Switch in optical computing and in optical computing

Bistability in a nonlinear medium inside of a Fabry-Perot resonator

TR 22

,

Intensity reflectance and transmittance :

,where, : amplitude reflectance and transmittance

likle 2

22 AA

212 AAA

likle

22

12

1

AA (6.3.3)


1) Absorptive Bistability

In the case when only the absorption coefficient depends nonlinearly on the field intensity,

at the resonance condition, Re ikl22

Assume, 1l

)1(1

AA 1

2lR

2

A2I ii nc

12 2

II

1 (1 )

T

R l

Introducing the dimensionless parameter C,R

lRC

1

2

12

)1(

I1I

CT (6.3.7)


Assume the absorption coefficient obeys the relation valid for a two-level saturable absorber ;

SII1

0

Intracavity intensity : 2

'

22 I2II )1(

,I2I1

00

2

0

RlR

CC

CS

where,

2

2

021

I2I11II

S

CT

(6.3.7)

23 II T


2) Dispersive Bistability

In the case when only the refractive index depends nonlinearly on the field intensity,

)I(,0 fn

(6.3.3) iδikl eRe

1

A

1

AA 1

22

12

ieR2where,

20

lc

n

00 2 : linear phase shift

: nonlinear phase shiftlc

n

I2 22

2sin41

I

)cos21(

I

)1)(1(

II

22

1

2

112

TR

T

RR

T

eReR

Tiδiδ

Similarly as before,


2sin41

1

I

I22

1

2

TR

T

220 I4

l

Cn

23 II T

Documents

Chapter 6. Processes Resulting from the Intensity ...optics.hanyang.ac.kr/~choh/degree/[2013-1... · Hanyang Univ. Wave equation : i i i P ct E c E 4 ~ ~ ~ 2 2 2 w w w H Swhere, (r,