Chapter 2. The Propagation of Rays and Beamsoptics.hanyang.ac.kr/~choh/degree/[2013-1] nonlinear... · 2016. 8. 29. · 2.3 Rays in Lenslike Media Lenses : optical path across them

Nonlinear Optics Lab. Hanyang Univ.

Chapter 2. The Propagation of Rays and Beams

2.0 Introduction

Propagation of Ray through optical element : Ray (transfer) matrix

Gaussian beam propagation

2.1 Lens Waveguide

A ray can be uniquely defined by its distance from the axis (r) and its slope (r’=dr/dz).

r

r’=dr/dz

z )('

)(

zr

zr r


Paraxial ray passing through a thin lens of focal length f

f

rrr

rr

ininout

inout

''

in

in

out

out

r

r

fr

r

'11

01

'

: Ray matrix for a thin lens

Report) Derivation of ray matrices

in Table 2-1


Table 2-1 Ray Matrices



Example (a)

i

i

i

i

o

o

r

r

f

df

d

r

r

f

d

r

r

'1

1

1

'11

01

10

1

'

f

rr

drf

dr

ii

ii

'

')1(

ex) Ray on the focal plane ( ) ? fd

f

rr

fr

r

ri

i

i

o

o

'

'

'

z flens

ir

ir '

or '

ir '

f

ri

fr i'


Example (b)

i

i

i

i

o

o

r

r

f

d

f

d

r

rd

fr

r

'11

1

'10

1

11

01

'

f

r

frr

r

ri

ii

o

o

'

'

d

f f

r

f

dr

drr

ii

ii

)1('

'

ex) Ray on the lens plane ( ) ? fd

f

rr

fr

ii

i

'

'


Biperiodic lens sequence (f1, f2, d)

in

in

out

out

r

r

f

d

f

d

r

r

')1(1

1

'

s

s

s

s

r

r

f

d

f

d

f

d

f

d

r

r

')1(1

1

)1(1

1

'2211

1

1

s

s

r

r

f

d

f

d

f

d

f

d

ff

f

ddd

f

d

')1)(1()1(

11

)1(1

211121

22

sss

sss

DrCrr

BrArr

''

'

1

1

In equation form of

)2(

1

2

2

f

ddB

f

dA

211

121

11

111

f

d

f

d

f

dD

f

d

ffC

(2.1-5)


,

1' 1 sss Arr

Br 121

1' sss Arr

Br(2.1-5)

02 12 sss rbrr

21

2

12 21)(

2

1 where,

ff

d

f

d

f

dDAb

2

1

n

nBCAD (actually for all elements, 1 in this case)

trial solution :

0122 iqiq bee

isq

s err 0

iiq ebibe 21 bcos where,

general solution :

)sin(max srrs

(2.1-7), (2.1-8)

0)()( 12 sss rBCADrDAr


Stability condition

: The condition that the ray radius oscillates as a function of the cell number s

between rmax and –rmax.

: is real 1b

12

12

10

12

11

21

21

2

12

f

d

f

d

ff

d

f

d

f

d

Identical-lens waveguide (f, f, d)

f

dD

fCdBA 1,

1,,1

f

db

21cos

Stability condition : fd 40


2.2 Propagation of Rays Between Mirrors

!2/Rf

)sin(

)sin(

max

max

yn

xn

nyy

nxx

l22 (, l : integers)

stability condition :

example) 2, l=1 =/2 cos = b = 1-d/2f = 0

fd 2 (symmetric confocal)

)2/sin(max nrrn


2.3 Rays in Lenslike Media

Lenses : optical path across them is a quadratic function of the distance r from the z axis ;

f

yxikyxEyxE LR

2exp),(),(

22

phase shift

<Closely related case> Lenslike medium

)(

21),( 222

0 yxk

knyxn

<Differential equation for ray propagation>

0

r

s ray path

wave front : const

)(0)(ˆ)(rik

erErE

)rsn()r( where,

: optical path

Index of refraction :


i) 1|s|, //s

ii) Maxwell equations :

0(r)H

0(r)E

0(r)H(r)-E

0(r)E(r)H

0(r)E)(r)(E)(r)E(1 2

if =1, 22)( n

That is, n || sn ˆ

ds

rds

ˆ So, ds

rdn

22

2

1])[(

2

1)(

1)()()( n

nnnds

rd

ds

d

ds

rdn

ds

d

nds

rdn

ds

d )(

: Differential equation for ray propagation, (2.3-3)


Trivial example) In homogeneous medium : n=constant

:

,0)(2

2

basr

ds

rdn

ds

rdn

ds

d

straight line


For paraxial rays,

dz

d

ds

d

02

2

2

r

k

k

dz

rd

02

022

02

2

02

'cossin)('

'sincos)(

rzk

krz

k

k

k

kzr

rzk

k

k

krz

k

kzr

Focusing distance from the exit plane for the parallel rays :

l

k

k

k

k

nh 2

20

cot1

Report) Proof


2.4 Wave Equation in Quadratic Index Media

Gaussian beam ?

tt

HE

EH ,

Maxwell’s curl equations (isotrpic, charge free medium)

: Scalar wave equation 02

22

t

EE

Put, tiex,y,zEtzyxE )(),,,( (monochromatic wave)

=> Helmholtz equation : 0)(22 ErkE

=>

where,

)(1)( 22 ri

rk

mediumgain

medium loss

:0

:0

We limit our derivation to the case in which k2(r) is given by

where,

)0(

)0(1)0()0( 222

ikk

2

2

22 ),,( rkkkzrk 2

2

2

2

2

222 1

zrrrzt


Assume, ikzezyxE ),,(0

=> 02 2

22

2

2

2

rkk

zik

yx

Put, 2/1222

)(,]})(2

)([exp{ yxrzq

krzpi

=> q

i

dz

dp

k

k

qdz

d

q ,0)

1(

1 2

2

& slow varying approximation


0)1

(1

2

qdz

d

q=> 0qzq

is must be a complex ! => q0qAssume, is pure imaginary.

=> put, 0izzq ( : real) 0z

At z = z0,

)}0(exp{)2

exp()0(0

2

ipz

krz

Beam radius at z=0, 2/10

0 )2

(k

zw : Beam Waist

2.5 Gaussian Beams in a Homogeneous Medium

In a homogeneous medium, 02 k

Otherwise, field cannot be a form of beam.


2

0wizq at arbitrary z, q

=> 22

0

2

0

2

0

20

111

wi

Rzz

zi

zz

z

izzq

: Complex beam radius

q

i

dz

dp => )/(tan])/(1ln[)( 0

12/12

0 zzizzzip

=> )]/(tanexp[])/(1[

1)](exp[ 0

1

2/12

0

zzizz

zip


Wave field

)(2exp)/(tan[exp

)(exp

)(

),,( 2

0

1

2

2

00

zR

krizzkzi

zw

r

zw

w

E

zyxE

A

where,

2

0

2

0

2

2

0

2

0

2 11)(z

zw

nw

zwzw

: Beam radius

2

0

22

0 11)(z

zz

z

nwzzR

: Radius of curvature of the wave front

2

00

nwz : Confocal parameter(2z0) or Rayleigh range


Gaussian beam

0z0wI

Gaussian profile

02w

0/2/ nw

spread angle :

0z

Near field

(~ plane wave)

Far field

(~ spherical wave)

z


2.6 Fundamental Gaussian beam in a Lenslike Medium - ABCD law

q

iP

k

k

qq

'0

11 2

'2

For lenslike medium, from (2.4-11)

Introduce s as, s

s

q

'1 0" 2

k

kss

zk

k

k

kbz

k

k

k

kazs

zk

kbz

k

kazs

2222

22

sincos)('

cossin)(

zkkqkkzkk

zkkkkqzkkzq

/cos//sin

/sin//cos)(

2022

2202

Table 2-1 (6)

DCq

BAqq

1

12


Matrix method (Ray optics)

yi

yo

i

o

optical

elements

i

i

o

o y

DC

BAy

DC

BA: ray-transfer matrix

Transformation of the Gaussian beam – the ABCD law


ABCD law for Gaussian beam

i

i

o

o y

DC

BAy

iio

iio

DCy

BAyy

ii

ii

o

oo

DCy

BAyyR

)()( opticsGaussianqopticsrayRo

DCy

BAy

ii

ii

/

/

DCq

BAqq

1

12

2q1q

optical system

DC

BA

ABCD law for Gaussian beam :

0izzq

2

00

nwz


example) Gaussian beam focusing

1 01w

02w

1z 2z

?

?

fz

fzzzzfz

z

f

z

DC

BA

/10

//1

10

1

1/1

01

10

1

1

21212

12

)/1(/

)/()/1(

11

2121122

fzfq

fzzzzqfzq


2

01

2

2

1

2

01

2

02

11

11

w

ff

z

ww

)()/()(

)(22

01

2

1

1

2

2 fwfz

fzffz

0201 ww - If a strong positive lens is used ; => 1

01

02

f

w

fw

2

1

2

01 )(/ fzw - If => fz 2

=> dfff

w

fw N

N /,2

)2(

2

01

02

: f-number

; The smaller the f# of the lens, the smaller the beam waist at the focused spot.

Note) To satisfy this condition, the beam is expanded before being focused.


2.7 A Gaussian Beam in Lens Waveguide

Matrix for sequence of thin lenses relating a ray in plane s+1 to the plane s=1 : s

TT

Tt

DC

BA

DC

BA

sin

)1(sin)sin(

sin

)sin(

sin

)sin(

sin

)1(sin)sin(

ssDD

sCC

sBB

ssAA

T

T

T

T

21

2

12 21

2

1cos

ff

d

f

d

f

dDAwhere,

)1(sin)sin()sin(

)sin()1(sin)sin(

1

11

ssDqsC

sBqssAqs

Stability condition for the Gaussian beam :

12

12

1021

f

d

f

d

: Same as condition for

stable-ray propagation

Documents

Chapter 2. The Propagation of Rays and Beamsoptics.hanyang.ac.kr/~choh/degree/[2013-1] nonlinear... · 2016. 8. 29. · 2.3 Rays in Lenslike Media Lenses : optical path across them