creation of colloidal periodic structureoptics.hanyang.ac.kr › ~choh › degree › [2013-1]...

Nonlinear Optics Lab. Hanyang Univ.

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


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 ;

],ARe[][2

|A|)cos(|A|)(

)()( tititi

a eeetta aa

aie

|A|Awhere

Typical expression ; tieta A)(

??


Distinction between the real and complex forms

1) tieita

dt

d A)tsin(|A|)( a

2) )]cos()2[cos(2

|B||A|)()( babattbta

)2(|B||A| batie

* Time averaging of sinusoidal products

)cos(2

|B||A|)cos(|B|)cos(|A|

1)()(

0

bab

T

a dtttT

tbta

*)ABRe(2

1


1.2 Considerations of Energy and Power in Electromagnetic Field

Maxwell’s curl equations (in MKS units) ;

t

dih

t

be ped 0 m)(hb 0[ , ]

peeeiehett

)(

2

0

mhhhehtt

0

0 )(2

Vector identity ; BAABB)A (

ttt

mh

pehheeieh)e- 0

22( 00


Divergence theorem ;

sv

dadv nAA)(v s

n

dvttt

dadvs vv

mh

pehheeienh)eh)e 0

22(( 00

: Poynting theorem Total power flow into the volume bounded by s

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


Dipolar dissipation in harmonic fields

The average power per unit volume expended by the field on the medium electric polarization ;

t

pe

volume

power

Assume, field and polarization are parallel to each other

]Re[)( tiEete EPPetp e

ti 0where],Re[)(

)Re(||2

*]Re[2

1

volume

power 2

00 ee

tiωtiω iEEEiiωωE

]e]Re[eRe[

Put, "' eee i

20 ||2

"

volume

powerEe

)*Re(2 ,

0 ji

ji

ij EEi

: Isotropic media

: Anisotropic media


Ex) single localized electric dipole, )( exμ

power DF t

e

Let, position of electron :

electric field :

)cos(0 etxx

tEex cos0

power DF )sin(cos)]cos([cos 0000 ee ttExetex

ttE

1) :2

e

power DF tExe 2

00 cos

2) :2

e

power DF tExe 2

00 cos

: The dipole(electron) continually loses power to the field

: The field continually gives power to the dipole

Power exchange between the field and medium via dipole interaction


1.3 Wave Propagation in Isotropic Media

Electromagnetic plane wave propagating along the z-axis in homogeneous, isotropic,

and lossless media constants)scalar :,(

Put, yx uhue yx he ,

t

eε

z

h

t

h

z

e xyyx

,

2

2

2

22

2

2

,t

hε

z

h

t

e

z

e yyxx

General solutions : ,),( )()( kzti

x

kzti

xx eEeEtze )()(1),( kzti

x

kzti

xy eEeEtzh

* Phase velocity : n

c

εkc 01

* wavelength :

c

k2

2

* Relative amplitude :

where,xy

EH


Power flow in harmonic fields

Intensity (average power per unit area carried in the propagation direction by a wave) :

*]Re[2

1|| yxyx HEheheI

(1.3-17) 2

||

2

||]*)(*)[(][Re

2

1 22 xxikz

x

ikz

x

ikz

x

ikz

x

EEeEeEeEeEI

Electromagnetic energy density :

*}Re{2

1

2*}Re{

2

1

222

22

yyxxyx HHEEheV

E

(1.3-17) }|||{|2

1

22

2222 xxyx EEheV

E

For positive traveling wave : cEEV

Ixx

1||

2/||

2

1

/

22

E

]W/m[||2

1 22 xEcI


1.4 Wave Propagation in Crystals-The Index Ellipsoid

In general, the induced polarization is related to the electric field as

zzzyzx

yzyyyx

xzxyxx

E,

where0P

: electric susceptibility tensor

)(

)(

)(

''3'3''2'3''1'30'

''3'2''2'2''1'20'

''3'1''2'1''1'10'

zyxz

zyxy

zyxx

EEEP

EEEP

EEEP

If we choose the principal axes, (Diagonalization)

zz

yy

xx

EP

EP

EP

330

220

110

zyx ,,

zz

yy

xx

ED

ED

ED

33

22

11

)1(

)1(

)1(

33033

22022

11011

where

0/n


)()( , rktirkti ee

00 HHEE

Secular equation

For a monochromatic plane wave ;

From Maxwell’s curl equations, 2

2

t

EE

0)( 2 EE kk

In principal coordinate,

z

y

x

ε

ε

ε

00

00

00

0222

222

222

z

y

x

yxzyzxz

zyzxyxy

zxyxzyx

E

E

E

kkεkkkk

kkkkεkk

kkkkkkε


Simple example ( 0, zyx kkkk ) : wave propagating along the x-axis

0)(

0)(

0

22

22

2

zz

yy

xx

Ekε

Ekε

Eε

0xE : transverse wave !!

0,

0,

and

and

yz

zy

Eεk

Eεk

For nontrivial solution to exist, Det=0 ;

0222

222

222

yxzyzxz

zyzxyxy

zxyxzyx

kkεkkkk

kkkkεkk

kkkkkkε


zk

xk

yk

cnz /

cnz /

cnx /

cnx /

cny /

cny /

Normal surface

Optic axis

Simple example ( 0zk

, determinant equation

0222

2

22

2

122

2

3

yxxyyx kkk

c

nk

c

nkk

c

n

)

2

322

c

nkk yx

: circle

11

2

2

2

c

n

k

c

n

k yx

: ellipse

sc

nk ˆ


Index ellipsoid

The surface of constant energy density in D space :

e

z

z

y

y

x

x UDDD

2222

Energy density :

jiije EEU 2

1

reUD 2/

1/// 0

2

0

2

0

2

zyx

zyxor 1

2

2

2

2

2

2

zyx n

z

n

y

n

x: Index ellipsoid


Classification of anisotropic media

1) Isotropic : zyx nnn

ex) CdTe, NaCl, Diamond, GaAs, Glass, …

2) Uniaxial : zyx nnn

(1) Positive uniaxial : xz nn

ex) Ice, Quartz, ZnS, …

(2) Negative uniaxial : xz nn

ex) KDP, ADP, LiIO3, LiNbO3, BBO, …

):,:( ordinaryaryextraordin 0nnnn xez Fast/Slow axis

3) Biaxial : zyx nnn

ex) LBO, Mica, NaNO2, …


Example of index ellipsoid (positive uniaxial)

12

2

2

0

22

en

z

n

yx

)sin,cos,0( ee nn

s

x

y

z

)0,,0( 0n

)0,0,( 0n

),0,0( en

B

A

0

propagation direction


Intersection of the index ellipsoid

s

y

z

A

0

)(en

0n

222 )( yzne

12

2

2

0

2

en

z

n

y

cos)(,sin)( ee nynz

)(

1sincos22

2

2

0

2

ee nnn

Birefringence : |)(| 0nne

000 |)90(|,0|)0(| nnnnnn eee


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)

z

y

en0n

0n

2) negative uniaxial (ne<no)

z

y

0n

en

0n

3) biaxial ( )

z

y

yn

xnzn

zyx nnn


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)

- Polarization state :

- Linear polarizer (horizontal) :

- Relative phase changer :

y

x

V

VV

00

01

y

x

i

i

e

e

0

0

Report) matrix expressions

- Linear polarizers (horizontal, vertical)

- Phase retarder

- Quarter wave plate (fast horizontel, vertical)

- Half wave plate


Retardation plate (wave plate)

: Polarization-state converter (transformer)

Polarization state of incident beam :

y

x

V

VV where,

yx VV , : complex field amplitudes

along x and y

s, f axes components :

y

x

y

x

V

VR

V

V

V

V)(

cossin

sincos

f

s

Polarization state of the emerging beam :

f

s

f

s

f

s

exp0

0exp

V

V

lc

in

lc

in

V

V


Define,

- Difference of the phase delays : c

lnn

)( fs

- Mean absolute phase change : c

lnn

)(

2

1fs

f

s

2

2

f

s

0

0

V

V

e

eeV

V

i

i

i

Polarization state of the emerging beam

in the xy coordinate system :

f

s

cossin

sincos

V

V

V

V

y

x

y

x

y

x

V

VRWR

V

V)()( 0

,cossin

sincos)(

R

2/

2/

00

0i

i

i

e

eeW

where,


Transfer matrix for a retardation plate (wave plate)

2)2/(2)2/(

2)2/(2)2/(

0

cossin)2sin(2

sin

)2sin(2

sinsincos

)()(),(

ii

ii

eei

iee

RWRWW

1WWTransfer matrix is a unitary ( ) : 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.


Ex) Half wave plate

1

0 : beamincident ,4/, V

4/cos4/sin)2/sin(2

sin

)2/sin(2

sin4/sin4/cos

2)2/(2)2/(

2)2/(2)2/(

ii

ii

eei

ieeW

)115.1(

0

0

i

i

0

1

01

0

0

0' i

i

i

iV : x-polarized beam

Report : Problem 1.7


Ex) Quarter wave plate

1

0 : beamincident ,4/,2/ V

)115.1(1

1

2

1

i

iW

i

ii

i

iV

1

212

1

1

0

1

1

2

1'

: left circularly

polarized beam

: y-pol.

0

1 : beamincident ,4/,2/ V

ii

iV

1

2

1

0

1

1

1

2

1'

: right circularly

polarized beam

: x-pol.


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.

Incident beam intensity :

y

x

V

VV

22

yx VVI VV

Output beam intensity :

y

x

V

VV

2'

2'' yx VVI

Transmissivity : 22

22

yx

yx

VV

VV


Ex) A birefringent plate sandwiched between parallel polarizers

,)(2 oe

d

nn 4/

2cos

0

1

0

2cos

2sin

2sin

2cos

10

00'

i

iV

dnnI oe )(

cos2

cos' 22

: fn. of d and

Ex) A birefringent plate sandwiched between a pair of crossed polarizers

02

sin

1

0

2cos

2sin

2sin

2cos

00

01' i

i

iV

dnnI oe )(

sin' 2


Circular polarization representation

It is often more convenient to express the field in terms of “basis” vectors that are

circularly polarized ;

1

0:CW

0

1:CCW and : constitute a complete set that can be used

to describe a field of arbitrary polarization.

Right circularly polarized Left circularly polarized

Rectangular representation : Circular representation :

y

x

yx V

VVV

1

0

0

1V

V

VVV

1

0

0

1V


Transformation

y

x

y

x

V

VT

V

V

i

i

V

V

1

1

2

1

V

VS

V

V

iiV

V

y

x 11

examples)

1

1

0

1

1

1

i

i

V

V

iiiV

V

y

x 1

1

011

Report :

?

?

0

1

?

?

1

0

?

?

1

1


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.

EBiED 0

)0(

)0(

0

0

)0(

0

0

)0(

)(

)(

))(2/(

))(2/(

))(2/(

)/()/(

V

V

e

ee

eV

eV

zV

zV

i

i

i

znciznci


Ignoring the prefactor, exp[-(i/2)(++-],

)0(

)0(

0

0

)(

)(

)(

)(

V

V

e

e

zV

zV

zF

zF

i

i

anglerotationFaraday

)(2

)(2

1)(

F

znnc

z

where,

Why (Faraday) rotation angle ?

)0(

)0(

cossin

sincos

)0(

)0(

0

0

)(

)(

FF

FF

1

)(

)(

y

x

y

x

i

i

y

x

V

V

V

VT

e

eT

zV

zV

zF

zF

In rectangular representation,

)0(

)0()( F

y

x

V

VR


Basic difference between propagation in a magnetic medium and in a dielectric

birefringent medium :

<dielectric birefringent medium> <magnetic medium>

CW for +z

CW for -z

CW for +z

CCW for -z

B

Report : proof by calculating Jones matrix.


Optical isolator

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