nonlinear optics lecture 6

7/27/2019 Nonlinear Optics Lecture 6

1/16

Nonlinear Optics

Lecture 6


2/16

Boundary problem for SHG in the

noncentrosymmetric media

1

(1)

(1)1

1

(1)

(1)

1, . ;

2

1, .2

1, .

2

i

ii T

i R

i i t

i i

i i t

T T

i i t

R R i

t e c c

t e c c

t e c c

k r

k r

k r

E r A

E r A

E r A

Incident wave generates regular reflected andtransmitted waves at the same frequency

according to Snells law and Fresnel formulas

It also generates 2ndorder polarization,

responsible for SH propagating inside the

medium as well as reflected from the medium

(1) (1)

(1)

(2)

(2)

21 1(1) (1) (1)

2 1

221 1(2) (2) (2)1

2 22 (1) (2)

1 1(2) (2) (2)

22 ; 2

2 216;

4

2 2;

s s

T

T

R

Ti ieff T s T

iTi

T T T

T T

RiR R R

P d A e p e k kc

p eE A e k

c ck k

E A e kc

k r k r

k r

k r

k r

Reflected wave is necessary

to satisfy boundaryconditions at the interface.

Here we assume that

nonlinear polarization

generates TE wave (s-

polarization) and neglect

birefringence


3/16

Snells law for SH

(1) (2) (2) (2) (2)

(2) (2)1 1 1

sin sin sin

sin 2 sin 2 sin

s s T T R R

T s T T R T

k k k

Usual requirement that the boundary conditions must be obeyed everywhere

along the plane interface result in the condition that components of the wave

number of all SH waves along the interface be equal to each other.

Taking into account that the wave generated by nonlinear polarization with

propagates in the same direction as the linear transmitted wave so that

we obtain the last missing relation to

determine all propagation angles

(1)2s Tk k

1 1sin sinT s R i


4/16

Equations for the fields

12(0, ,0) ,0,y xy E EiEc z y

E BMaxwell equations give:

On the incident side of the interface one has only reflected SH, so that

(2) (2) (2) (2)sin sin(2) (2) (2) (2) (2)

10 ; ( 0) 2 cosR R R Rik x ik x

R R x R R RE A e B A e

On the transmission side of the interface there are two waves propagating at two

different wave numbers so that

(1)

( 2) ( 2)

( 2) ( 2)

(1)

2 sinsin(2) (2)

1 1

sin(2) (2)

1

2 sin

1

1 1

4

( 0) 2

( 0) 2 cos

4 cos

2

sT

T T

T T

sT

ik xik x

T T

T T

ik x

x T T T

ik x

T s

T T

p e

E A e

B A e

p e


5/16

Equation for reflection and

transmission amplitudes

(2) (2)

1 1

1(2) (2) (2) (2)

1 1

1 1

(2)

1 1(2)

(2) (2)1 1 1 1

(2)

1 1(2)

1 1

4

2

4 cos

2 cos 2 cos 2

cos 2 cos4

2 2 cos 2 cos

2 cos cos4

2

R T

T T

T s

R R R T T T

T T

T s R R

T

T T T T R R

T T T

R

T T

pA A

p

A A

pA

pA

(2) (2)1 12 cos 2 coss

T T R R


6/16

Reflected wave

( 2)(2) (2)

(2)

1 1(2)

(2) (2)1 1 1 1

2

1 1

2 cos cos4

2 2 cos 2 cos

41

2 2

Ri

R R

T T T s

R

T T T T R R

effT R

E A e

pA

p p

k r


7/16

Transmitted waves

(1) (2)

(1)

(2)

1 12(2)(2) (2)

1 1 1 1

(2 )

1 12

(2) (2)1 1 1 1

1(2 )

cos 2 cos4

2 2 cos 2 cos

cos 2 cos41

2 2 cos 2 cos

4 /

T T

T

T s R Ri iT

T T T T R R

T s R Ri i kz

T T T T R R

R

pE e e

pe e

p cA

k r k r

k r

(2)1 1

(2) (2)

1 1 1

1(2 )

1

cos 2 coscos 1

2 cos 2 cos

4 /1 4

i kzT s R Rs

T T T R R

R T

effT

e

k

p c pA iz ik z

(2) (1) (2) (2) (1)

(2 )11 1

2 cos 2 cos

2 2 cos cos

T T z T T T s

z T T T s

k k k

c

k k e

e


8/16

Third order nonlinearities

(3) (3)4 4 1 2 3 1 2 3{1,2,3}

1; , , ( , ) ( , ) ( , )

2i ijkl j k l P E E E r r r

General form of the 3rdorder nonlinear polarization:

Optical Kerr effect 4 1 2 2

(3) (3)13 3 ; , , ( , ) ( , ) ( , )2

i ijkl j k l

jkl

P E E E r r r

3rdharmonic generation4

(3) (3)

1 1 1 2 2 1 2 23 ; , , ( , ) ( , ) ( , )i ijkl j k l jklP E E E

r r r

Intensity dependent refractive index

(3) (3)

1

3; , , ( , ) ( , ) ( , )

2i ijkl j k l jklP E E E

r r r


9/16

Effective dielectric constant

2 21 12 ( )1

1, 1, 1,2 2

22 ( )1

1, 1 1,2

4

0

NL

i i i

NL

i ij ij j

j

E E Pc c

E Ec

r r r

r r

In the case of optical Stark effect and intensity dependent refractive index, the

nonlinear polarization can be presented as a filed-dependent dielectric constant

For Kerr effect, the dielectric constant is modulated by a strong pump field at frequency

2, while the weak probe wave propagates at the frequency 1.

( ) (3) 1 1 2 2 2 212 ; , , ( , ) ( , )NL

ij ijkl k l

kl

E E r r

In the case of the field modulated refractive index, we deal with the self-modulation of

the index by the wave propagating at the frequency .

( ) (3)6 ; , , ( , ) ( , )NLij ijkl k l kl

E E r r


10/16

Effective refractive indexAssume that the light in linearly polarized, and lets choose one of the coordinate axes

(x) along the polarization direction. Also assume that the sample has cubic symmetry,

so that its optically isotropic and Then we can introduce intensity-

dependent effective refractive indexij ij

2(3)

0

(3)2

0

0

6 ; , , ( , )

3( , )

ij iixx x ij

x

E

n n En

r

r

In the literature, intensity dependent refractive index is often written down as

0 2n n n I

Taking into account that we find20

8n cI E

2(3)

2 2

0

24nn c

Typical values of the nonlinear refractive index coefficient are given in Table 4.1.1

of Boyds. They can range between for electronic nonlinearities and

for thermal effects

21610 cm

W

2610 cm

W


11/16

Third order susceptibility in isotropic

materials

In the most general case, when all frequencies are different and there are no additional

permutation symmetries, the isotropy condition requires any tensor of the 4thrank to have

the form of

With three independent coefficients.3rdharmonic generation:all frequency are the same so, all permutation of indexes are

equivalent, hence

Nonlinear refractive index: first two frequencies are the same, thus

Polarization in the later case becomes

(3) (3) (3) (3)

1122 1212 1221ijkl ij kl ik jl il jk

(3) (3)1122 3ijkl ij kl ik jl il jk

(3) (3) (3)1122 1221ijkl ij kl ik jl il jk

2 2(3) (3) (3) * 2 (3) (3) (3) * 2

1 1122 1221 1122 1221

3 32 ; 3

2 2i i i

P E E

E E P E E E E


12/16

Alternative notation for the 3rdorder

susceptibility

2 2(3) (3) (3) * 2 (3) (3) (3) * 2

1 1122 1221 1122 1221

(3) (3)

1122 1221

2(3) * 2

3 32 ; 3

2 2

3 ; 3

1

2

i i iP E E

A B

A B

E E P E E E E

P E E E E

( ) ( ) (3)

2(3) (3) * (3) *

1122 1122 1221

2(3) (3) (3) * *

1122 1221 1221

2 * *

(3)

1122 1221

3; ; , , ( , ) ( , )

2

3 3 3

2 2 2

3 32

2 2

1

2

3

22

eff eff

i ij j ij ijkl k l

kl

ij i j i j

ij i j i j

ij i j i j

P E E E

E E E E

E E E E

A B E E E E

A

r r

E

E

E

(3) (3)

1221; 3B

/ 6 / 3 molecular orientation

/ 1 / 2 noneresonant electrons

/ 0 / 0 electrostriction

B A B A

B A B A

B A B A


13/16

Basis of circularly polarized light

Circularly polarized light is described by combination of two beams polarized in utuallyperpendicular directions with a phase shift between them equal to /2

Left-hand circular (counterclockwise) Right-hand (clockwise)

;2 2

i i

E E

x y x y

E


14/16

Non-linear polarization in the circular basis

2(3) * 2

2 22 * 2 *

1

2

1 1;

2 2

A B P P

P A E B E P A E B E

P E E E E

E E E E

One can rewrite the polarization in the circular basis as

Where the terms quadratic in the field are given by

2 2* 2; 2E E E E E E E

Thus, circular components of the polarization become

2 2 2 2

( )

22( )

;

NL

NL

P A E A B E E P A E A B E E

P E

A E A B E


15/16

Wave equation in the circular basis

2 2 2 ( )2 1

2 2 2 2 2 2 22 1

2 2 2 2 2 2 2 ( )2 1

2 2 2 2

2 2 222 2 2 2

1 12 2

2 2

1

4

4

4

; 4 4

2

NL

NL

NL

n E PE

n c t c t

c t c t n E PE

c t c t

n EE n n n A E A B Ec t

n n B E E n

E PE

If the absence of damping, the wave equation has a solution in which amplitudes of

different polarization components are constants. In this case, the solutions are regular

plane waves, but with intensity dependent refractive indexes


16/16

Polarization rotation in the medium with

intensity dependent refractive index

/ /

/

( )

1 1; ; /

2 2

cos sin ;cos sin2

in z c in z c

in z c i i

i

E z E E A e A e

e A e A e n n n n z c

x x yiey y x

x y

nonlinear optics lecture 6

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