Download - Nonlinear Optics Lecture 6
-
7/27/2019 Nonlinear Optics Lecture 6
1/16
Nonlinear Optics
Lecture 6
-
7/27/2019 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
-
7/27/2019 Nonlinear Optics Lecture 6
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
-
7/27/2019 Nonlinear Optics Lecture 6
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
-
7/27/2019 Nonlinear Optics Lecture 6
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
-
7/27/2019 Nonlinear Optics Lecture 6
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
-
7/27/2019 Nonlinear Optics Lecture 6
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
-
7/27/2019 Nonlinear Optics Lecture 6
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
-
7/27/2019 Nonlinear Optics Lecture 6
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
-
7/27/2019 Nonlinear Optics Lecture 6
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
-
7/27/2019 Nonlinear Optics Lecture 6
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
-
7/27/2019 Nonlinear Optics Lecture 6
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
-
7/27/2019 Nonlinear Optics Lecture 6
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
-
7/27/2019 Nonlinear Optics Lecture 6
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
-
7/27/2019 Nonlinear Optics Lecture 6
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
-
7/27/2019 Nonlinear Optics Lecture 6
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