nonlinear optics (비선형 광학)

Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.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

2007 봄학기


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

)(20

mhhhehtt

00 )(

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 theoremTotal 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 eti

0where],Re[)(

)Re(||2

*]Re[2

1

volume

power 200 ee

tiωtiω iEEEiiωωE ]e]Re[eRe[

Put, "' eee i

20 ||2

"

volume

powerEe

)*Re(2 ,

0 jiji

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 : ,),( )()( kztix

kztixx eEeEtze )()(1

),( kztix

kztixy 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

xikz

xikz

xikz

x

EEeEeEeEeEI

Electromagnetic energy density :

*}Re{2

1

2*}Re{

2

1

22222

yyxxyx HHEEheV

E

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

1

222222 xxyx EEhe

V

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

E

E

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

02222

222

1222

3

yxxyyx kkkc

nk

c

nkk

c

n

)

2

322

c

nkk yx

: circle

11

2

2

2

cn

k

cnk yx

: ellipse

sc

nk ˆ


Wave propagation in anisotropic media

Maxwell equations 2

2

t

D

E

Define the unit vector along the propagation direction as ,s sc

nk ˆ

( : wave vector)

DE )ˆˆ(2

2

ssc

n

Put, =1, and )()( BACCABCBA

)]ˆ(ˆ[2

2

E-ED ssc

n

)()( , rktirkti ee

00 HHEE

Taking scalar product, s on both sides :

0)]ˆ)(ˆˆˆ[ˆ2

2

E(-ED ssssc

ns

: propagation direction is perpendicular to the electric displacement vector not to the electric field vector E

kS

D

(poynting vector)


Index ellipsoid

The surface of constant energy density in D space :

ez

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

20

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

20

2

en

z

n

y

cos)(,sin)( ee nynz

)(

1sincos22

2

20

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

xn zn

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

0V

V

e

eeV

Vi

ii

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

ii

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

sin1

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

xyx 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

ii

znciznci


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

)0(

)0(

0

0)(

)()(

)(

V

V

e

ezV

zVzF

zF

i

i

anglerotationFaraday

)(2

)(2

1)(

F

znnc

zwhere,

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

zVzF

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

nonlinear optics (비선형 광학)

Documents