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2007 봄학기
Nonlinear Optics
(비선형 광학)
담당 교수 : 오 차 환
교 재 : A. Yariv, Optical Electronics in Modern Communications, 5th Ed.,
Oxford university Press, 1997
부교재 : R. W. Boyd, Nonlinear Optics, Academic Press, 1992
A. Yariv, P. Yeh, Optical waves in Crystals, John Wiley & Sons, 1984
Nonlinear Optics Lab.
Hanyang Univ.
Chapter 1. Electromagnetic Theory
1.0 Introduction
Propagation of plane, single-frequency electromagnetic waves in
- Homogeneous isotropic media
- Anisotropic crystal media
1.1 Complex-Function Formalism
Expression for the sinusoidally varying time functions ;
|A| i (t a ) i (t a )
a(t )|A|cos(t a ) [e
e
]Re[ Aei t ],
2
i
where A|A|e a
??
i t
 Typical expression ; a(t ) Ae
Nonlinear Optics Lab.
Hanyang Univ.
Distinction between the real and complex forms
d
1)
a(t )|A|sin( t a )iAeit
dt
|A||B|
2) a (t )b(t )
[cos( 2t a b )cos(a b )]
2
|A||B|ei ( 2ta b )
* Time averaging of sinusoidal products
T
1
|A||B|
a(t )b(t ) |A|cos(t a )|B|cos(t b )dt 
cos(a b )
T0
2
1
 Re( AB*)
2
Nonlinear Optics Lab.
Hanyang Univ.
1.2 Considerations of Energy and Power in Electromagnetic Field
Maxwell’s curl equations (in MKS units) ;
d
h  i 
t
b
e  
[ d 0ep , b0 (hm) ]
t
0 

eh  ei 
(ee)e p
2 t
t
0 

h e 
(h h) 0h  m
2 t
t
Vector identity ; ( AB)BAA B
 0  p
  0
m
- (eh)  ei   ee h h e   0h 
t  2
2
t
 t
Nonlinear Optics Lab.
Hanyang Univ.


v
s
n
Divergence theorem ; (A) dv A n da
v
s

 0  p
  0
m 
 (eh) dv (eh)n da e i  ee hh e   0h dv
v
s
v
2
t 
 t
 t  2
Total power flow
into the volume
bounded by s
: Poynting theorem
Power expended by
the field on the
moving charges
Rate of increase of
the vacuum
electromagnetic
stored energy
Power per unit
volume expended by
the field on electric
and magnetic dipoles
Nonlinear Optics Lab.
Hanyang Univ.
Dipolar dissipation in harmonic fields
The average power per unit volume expended by the field
on the medium electric polarization ;
power
p
 e
volume
t
Assume, field and polarization are parallel to each other
e(t )Re[ Eeit ]
p(t )Re[ Peit ],
where
P 0  e E
power
1

 Re[Eeiωt ]Re[iωωeiωt ]  Re[ i 0  e EE*]  0 |E|2 Re( i e )
volume
2
2
Put,  e  e 'i e "
  "
power
 0 e |E|2
volume
2

: Isotropic media
  0 Re( i ij Ei *E j )
2 i, j
: Anisotropic media
Nonlinear Optics Lab.
Hanyang Univ.
Ex) single localized electric dipole, μ (ex)
power  e 
t
DF
x x0 cos(t e )
ex E0 cost
Let, position of electron :
electric field :
power
DF

 E0 cos t  [ex0 cos(t e )] e x0 E0 cos t sin(  t e )
t

1) e 
2)  e 
2

2
: power  e x
DF
0
: power e x
DF
E0 cos 2 t
0
: The dipole(electron) continually loses power to the field
E0 cos 2 t : The field continually gives power to the dipole
 Power exchange between the field and medium via dipole interaction
Nonlinear Optics Lab.
Hanyang Univ.
1.3 Wave Propagation in Isotropic Media
Electromagnetic plane wave propagating along the z-axis in homogeneous, isotropic,
( ,  : scalar constants)
and lossless media
Put, e ex u x , h hy u y
hy hy
ex
e
 
,
 ε x
z
t
z
t
General solutions :
2
 2 hy
 2 ex
 2 ex  hy

,
ε 2
2
2
z
t
z
t


1  i (tkz )  i (tkz )
Ex e
ex ( z,t )  E x ei (t kz )  E x ei (t kz ) , hy ( z,t )  Ex e

c
1
 0
k
ε n
2
c
* wavelength :  
 2
k


E


x
* Relative amplitude : H y 
, where  


* Phase velocity : c 

Nonlinear Optics Lab.
Hanyang Univ.

Power flow in harmonic fields
Intensity (average power per unit area carried in the propagation direction by a wave) :
1
 I  |eh|ex hy  Re[ E x H y *]
2
|Ex |2 |Ex |2
1
 ikz
 ikz

ikz

ikz
Re [ Ex e  Ex e ][( Ex )*e ( Ex )*e ] 

(1.3-17)   I  
2
2
2


Electromagnetic energy density :
E  2  2 1
1
 ex  hy  Re{ Ex Ex *} Re{H y H y *}
V 2
2
22
22
E  2  2 1 2
 2
 ex  hy  {| Ex | |Ex | }
V 2
2
2
I  1  2   2 1
 |Ex | / |Ex | 
c
For positive traveling wave :
E/V 2
2

(1.3-17) 

1
 I   c |E x |2 [ W/m 2 ]
2
Nonlinear Optics Lab.
Hanyang Univ.
1.4 Wave Propagation in Crystals-The Index Ellipsoid
In general, the induced polarization is related to the electric field as
  xx  xy  xz 


P  0  E, where    yx  yy  yz 




zy
zz  : electric susceptibility tensor
 zx
 Px '  0 ( 1'1' E x '  1'2' E y '  1'3' Ez ' )

 Py '  0 (  2'1' E x '   2'2' E y '   2'3' Ez ' )

 Pz '  0 (  3'1' Ex '   3'2' E y '   3'3' Ez ' )
If we choose the principal axes, x, y, z (Diagonalization)
 Px  0 11E x

 Py  0  22 E y

 Pz  0  33 E z
 Dx  11E x

 D y   22 E y

 Dz   33 E z
11  0 (1 11 )

where  22  0 (1  22 )
  (1  )
33
 33 0
Nonlinear Optics Lab.
n   / 0
Hanyang Univ.
Secular equation
For a monochromatic plane wave ;
E E0e
 
i ( t k  r )
, H H 0e

i ( t k r )
 2E
From Maxwell’s curl equations, E  2
  t
k (k E) 2  E0
 εx

In principal coordinate,   0
0

0
εy
0
0

0
ε z 
  2 ε x k y2 k z2
 E x 
kxk y
kxkz

 
2
2
2
k ykx
 ε y k x k z
k ykz

 E y 0

2
2
2 

k
k
k
k


ε

k

k
E
z
x
z
y
z
x
y
z




Nonlinear Optics Lab.
Hanyang Univ.
Simple example (k x k , k y k z 0) : wave propagating along the x-axis
 2 ε x E x 0
 2
2
(


ε

k
) E y 0

y
 2
2
( ε z k ) E z 0
E x 0 : transverse wave !!
k  ε y , and E z 0

k  ε z , and E y 0
For nontrivial solution to exist, Det=0 ;
 2 ε x k y2 k z2
k ykx
kxk y
 2 ε y k x2 k z2
kxkz
k ykz
kzkx
kzk y
 2 ε z k x2 k y2
0
Nonlinear Optics Lab.
Hanyang Univ.
Normal surface
Simple example ( k z 0)
ky
nz /c
 n
k  sˆ , determinant equation 
c
nx /c
n y / c
nx /c
n y / c
nz /c
kx
Optic axis
2
2
 n3  2 2 2  


 2 2
n

 n1 



2
2
2
 k x k y  
 k y  
 k x k x k y 0


  c 

 c 
 
 c 

n
k k  3 
 c 
: circle
2
x
kz
2
2
y
k y2
k x2

1
n

n

 2   1 

 

 c   c 
: ellipse
Nonlinear Optics Lab.
Hanyang Univ.
Wave propagation in anisotropic media
 2D
Maxwell equations  E  2
t

i ( t k r )
, H H 0e
 n
sˆ
Define the unit vector along the propagation direction as ŝ , ( k 
c
n2
 2 ( sˆsˆE)D
c
E E0e
 
i ( t k  r )
: wave vector)
n2
D 2 [E-sˆ( sˆE)]
c
Put, =1, and ABCB( AC)C( AB)
Taking scalar product,
ŝ on both sides :
n2
sˆD 2 [ sˆE-( sˆsˆ)( sˆE)]0
c
S (poynting vector)
k
: propagation direction is perpendicular to the electric
displacement vector not to the electric field vector
Nonlinear Optics Lab.
D
E
Hanyang Univ.
Index ellipsoid
Energy density :
1
U e   ij Ei E j
2
The surface of constant energy density in D space :
2
Dx2 Dy Dz2
  2U e
x
y
z
D/ 2U e r
x2
y2
z2


1
 x / 0  y / 0  z / 0
or
x2 y2 z 2
 2  2 1
2
nx n y nz
: Index ellipsoid
Nonlinear Optics Lab.
Hanyang Univ.
Classification of anisotropic media
1) Isotropic : nx n y nz
ex) CdTe, NaCl, Diamond, GaAs, Glass, …
2) Uniaxial : nx n y nz (nz ne : extraordin ary,nx n0: ordinary)  Fast/Slow axis
(1) Positive uniaxial : nz nx
ex) Ice, Quartz, ZnS, …
(2) Negative uniaxial : nz nx
ex) KDP, ADP, LiIO3, LiNbO3, BBO, …
3) Biaxial : nx n y nz
ex) LBO, Mica, NaNO2, …
Nonlinear Optics Lab.
Hanyang Univ.
Example of index ellipsoid (positive uniaxial)
x2  y2 z 2
 2 1
2
n0
ne
z
(0,0,ne )

ŝ
propagation direction
(0,ne cos ,ne sin ) A
0
(0,n0 ,0)
y
B
x
(n0 ,0,0)
Nonlinear Optics Lab.
Hanyang Univ.
Intersection of the index ellipsoid
z
A


y2 z2
 2 1
2
n0 ne
ŝ
ne2 ( ) z 2  y 2
ne ( )
n0
0
y
z ne ( )sin , yne ( )cos
cos 2 sin 2
1

 2  2
2
n0
ne
ne ( )
Birefringence : |ne ( )n0 |
|ne (0)n0 |0, |ne (90)n0 |ne n0
Nonlinear Optics Lab.
Hanyang Univ.
Normal index surface
: The surface in which the distance of a given point from the origin is equal to
the index of refraction of a wave propagating along this direction.
1) Positive uniaxial (ne>no)
2) negative uniaxial (ne<no) 3) biaxial ( nx n y nz )
n0 z
z
ny z
y
y
n0
ne
ne
y
nx
n0
n0
Nonlinear Optics Lab.
Hanyang Univ.
nz
1.5 Jones Calculus and Its Application in Optical Systems with
Birefringence Crystals
Jones Calculus (1940, R.C. Jones) :
- The state of polarization is represented by a two-component vector
- Each optical element is represented by a 2 x 2 matrix.
- The overall transfer matrix for the whole system is obtained by multiplying
all the individual element matrices.
- The polarization state of the transmitted light is computed by multiplying
the vector representing the input beam by the overall matrix.
Examples)
 Vx 
- Polarization state : V   
V y   1 0 

- Linear polarizer (horizontal) : 
 0 0
 ei x
- Relative phase changer : 
 0

0 

i y 
e 
Report) matrix expressions
- Linear polarizers (horizontal, vertical)
- Phase retarder
- Quarter wave plate (fast horizontel, vertical)
- Half wave plate
Nonlinear Optics Lab.
Hanyang Univ.
Retardation plate (wave plate)
: Polarization-state converter (transformer)
Polarization state of incident beam :
 Vx 
V    where, Vx , Vy : complex field amplitudes
V y 
along x and y
s, f axes components :
 Vs   cos
   
Vf    sin 
sin  Vx 
Vx 
   R( ) 
cos Vy 
V y 
Polarization state of the emerging beam :

 V

 Vs  exp   ins l 
0
 s 
  
c


 

  
   

V   
0
exp   inf l  Vf 
 f
c 


Nonlinear Optics Lab.
Hanyang Univ.
Define,
l
- Difference of the phase delays :   (ns  nf )
c
1
l
- Mean absolute phase change :   (ns  nf )


 Vs i  e
 e 
 0
Vf 


i
2
 V
0  s 

i V 
e 2  f 
2
Polarization state of the emerging beam
in the xy coordinate system :
Vx   cos
   
V   sin 
 y 
 sin   Vs
 
cos Vf 

c
 Vx 
Vx 
 R( )W0 R( ) 
V  
V 
 y
 y
 cos
where, R( )
 sin
Nonlinear Optics Lab.
 i / 2
0 
sin 
i  e

, W0  e 
i / 2 
0
e
cos 


Hanyang Univ.
Transfer matrix for a retardation plate (wave plate)
W ( , )  W  R( )W0 R( )

e i (  / 2) cos 2   ei (  / 2) sin 2 
 i sin

sin( 2 )
2
 i sin

sin( 2 )
2
e i (  / 2) sin 2   ei (  / 2) cos 2 
 Transfer matrix is a unitary ( W W 1 )
: Physical properties are invariant under unitary transformation
=> If the polarization states of two beams are mutually orthogonal, they will remain
orthogonal after passing through an arbitrary wave plate.
Nonlinear Optics Lab.
Hanyang Univ.
Ex) Half wave plate
 0
 ,  /4, incident beam : V  
 1
(1.511) 
W
e i ( /2) cos 2 /4ei ( /2) sin 2 /4

isin sin(  /2)
2

isin sin(  /2)
2
e i ( /2) sin 2 /4ei ( /2) cos 2 /4
0 i

i 0
 0 i  0   i   1 
  i  : x-polarized beam
V '
 i 0  1   0   0 
Report : Problem 1.7
Nonlinear Optics Lab.
Hanyang Univ.
Ex) Quarter wave plate
 0
 /2,   /4, incident beam : V   : y-pol.
 1
1

i
1
(1.511)  W 
2 i
V '
1
1  1 i  0  1  i 
i 1

     
2  i 1  1  2  1 
2i
: left circularly
polarized beam
 1
 /2,   /4, incident beam : V   : x-pol.
 0
V '
1  1 i  1  1  1 

   
2  i 1  0  2  i 
: right circularly
polarized beam
Nonlinear Optics Lab.
Hanyang Univ.
Intensity transmission
In many cases, we need to determine the transmitted intensity, since the
combination of retardation plates and polarizers is often used to control or modulate
the transmitted optical intensity.
 Vx 


V y 
Incident beam intensity : V  

Output beam intensity :
 Vx 
V   
Vy 
 I ' V  V
Vx  V y
Vx  V y
2
2
I VV  Vx  Vy
' 2
x
2
2
Transmissivity :


2
2
' 2
y
Nonlinear Optics Lab.
Hanyang Univ.
Ex) A birefringent plate sandwiched between parallel polarizers
d
2 (ne no ) ,  /4



cos

 0 0
2

 V '
 0 1  isin 
2


isin  0   0 
2    
  1   cos 
cos 
2

2 

 (ne no )d 
 I 'cos 2 cos 2 

2



: fn. of d and 
Ex) A birefringent plate sandwiched between a pair of crossed polarizers


cos
 1 0 
2

V '
 0 0  isin 
2


isin  0    
2  i sin 
  1   2 
cos 
 0 
2 

 (ne no )d 
I 'sin 2 




Nonlinear Optics Lab.
Hanyang Univ.
Circular polarization representation
It is often more convenient to express the field in terms of “basis” vectors that are
circularly polarized ;
1
0
CCW:   and CW:  
0
1
Right circularly polarized
: constitute a complete set that can be used
to describe a field of arbitrary polarization.
Left circularly polarized
Circular representation :
1
0 V 
V  V    V      
0
1 V 
Rectangular representation :
1
 0  Vx 
V  Vx    Vy     
 0
 1  V y 
Nonlinear Optics Lab.
Hanyang Univ.
Transformation
 Vx 
V  1 1 i Vx 
   T 
 
 
V 
V  2 1  i Vy 
 y
Vx  1 1 V 
V 
 
V   i i V   S V 
 
 
 y
examples)
V  1 i  1  1
    
 
V  1  i  0  1
Vx  1 1 0 1
 
V   i i 1   i 
   
 y
Report :
1  ? 
  
0  ? 
Nonlinear Optics Lab.
 0  ?
  
 1  ?
1 ?
  
1 ?
Hanyang Univ.
Faraday rotation
In certain optical materials containing magnetic atoms or ions, the two counter-rotating,
circularly-polarized modes have different indices of refraction when an external
magnetic field is applied along the beam propagation direction.
This difference is due to the fact that the individual atomic magnetic moments process
in a unique sense about the z-axis (magnetic field direction) and thus interact differently
with the two counter-rotating modes.
D  E i 0 BE
V ( z ) V (0) i ( / c ) n  z  0  i ( / c ) n  z



e
e
V ( z )  0 
V (0)
e
 ( i / 2 )(    )
e (i / 2 )(    )
0
V (0)

 ( i / 2 )(    ) 
V
(
0
)
e
  
0
Nonlinear Optics Lab.
Hanyang Univ.
Ignoring the prefactor, exp[-(i/2)(++-],
i
V ( z ) e F ( z )


V
(
z
)
0
  
0
e
 i F ( z )
V (0)


V (0) 
1

(n  n ) z
2
2c
 Faraday rotation angle
where,  ( z )  (     ) 
F
Why (Faraday) rotation angle ?
i F ( z )
Vx ( z ) 

e

1
In rectangular representation, 

V ( z )   T
0
 y 

cos  F
sin  F
 Vx (0) 

T 
i F ( z )

V
(
0
)
e
 y 
 sin  F  Vx (0) 



cos  F Vy (0) 
0
Vx (0) 

 R( F )

V
(
0
)
 y 
Nonlinear Optics Lab.
Hanyang Univ.
Basic difference between propagation in a magnetic medium and in a dielectric
birefringent medium :
CW for +z
CW for +z
B
CW for -z
CCW for -z
<dielectric birefringent medium>
<magnetic medium>
Report : proof by calculating Jones matrix.
Nonlinear Optics Lab.
Hanyang Univ.
Optical isolator
Nonlinear Optics Lab.
Hanyang Univ.