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# Chapter 2. The Propagation of Rays and Beams

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(1)

### 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

(2)

f r r

r

r r

in in

out in out

' '

'  

in in out

out

### 

: Ray matrix for a thin lens

(3)

(4)

(5)

1

2

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

' 1 1 1 2 2

1

s s

r r f d f

d f

d f

d f

f

f d d f d

d

) ' 1 )(

1 ( )

1 1 ( 1

) 1 ( 1

2 1

1 1

2 1

2 2

s s

s

s s

s

1 1

### 

In equation form of

2 2

2 1

1

1 2

1

(6)

s

s 1

s

s 1

s 2

s 1

(2.1-5) 

1

2

s

s

### r

s 



2 1

2

1

2 2

1 ) 2(

1

where,

f f d f

d f D d

A b

1

AB CD (actually for all elements)

trial solution :

2iq

iq

isq s

0

i

iq

2 where, cos

### 

b

general solution :

max

s

(7)

### Stability condition

: The condition that the ray radius oscillates as a function of the cell number s between rmax and –rmax.

:  is real 

2 1

2 1

2

1 2

f D d

C f d B

A  1 , 1 ,

, 1

f b d

1 2

cos

  

(8)

max max

y n

x n

### 2 

(, l : integers)

### example)

2, l=1  =/2  cos = b = 1-d/2f = 0

### 

(symmetric confocal)

max

n

(9)

### 2.3 Rays in Lenslike Media

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



 

 

f

y ik x

y x E y

x

ER L

exp 2 ) , ( )

, (

2 2

phase shift

Index of refraction of lenslike medium :

0 2

2

2

### yxn

<Differential equation for ray propagation>

0

r

s ray path

wave front :const

ik0(r)

where,

: optical path

(10)

i)

### sˆ //   , | sˆ |  1

ii) Maxwell equations :

1

(Eˆ(r)

)

Eˆ(r)(

)2

Eˆ(r)0

if =1, (

)2

n2 That is, |

|n

s

n ˆ

So,

2 2

###  ( )

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

(11)

### Nonlinear Optics Lab . Hanyang Univ.

For paraxial rays,

dz d ds

d

2

2

2

### rd

0 2

0 2 2

0 2

2 0

2

' cos

sin )

( '

' sin

cos )

(

r k z r k

k z k k

z k r

r k z k k

r k k z z k

r





 





 





 





 

Focusing distance from the exit plane for the parallel rays : 



  l

k k k

k

h n 2

2 0

1 cot

(12)

### Gaussian beam ?

t

t

 

 

 

H

E E

H  , 

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

: Scalar wave equation

2 0

2

2

 

t

E  E

Put, E(x,y,z,t)E(x,y,z)eit (monochromatic wave)

=> Helmholtz equation :

2

2

### ( r ) E  0

=>

where,



 

 

 

 1 ( ) )

( 2

2 i r

r

k 

medium gain

medium

loss

: 0

: 0

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

where,

(0)

) 0 1 (

) 0 ( )

0

( 2

2 2





i

k k

2 2 2

2

2 2 2

2 2

2 2

2 1

z r r r

t z

 

 

 

 

(13)

### Nonlinear Optics Lab . Hanyang Univ.

Assume, E0  (x, y,z)eikz

=> 2 2 2 2 0

2 2

2  

 



 

 

kk r

ik z y

x

Put, 2 2 1/2

2

) (

, )]}

( ) 2

( [

exp{ r x y

z q z kr

p

i   

=>

### 1 (

2

2

& slow varying approximation

(14)

2

=>

### q  z  q

0

is must be a complex ! =>

Assume,

### q

0 is pure imaginary.

=> put,

0 ( : real)

0

At z = z0,

)}

0 ( exp{

2 ) exp(

) 0 (

0 2

z ip

z    kr

0

1/2

: Beam Waist

### 2.5 Gaussian Beams in a Homogeneous Medium

In a homogeneous medium,

2

###  0

Otherwise, field cannot be a form of beam.

(15)

02

at arbitrary z,

=> 2 2

0 2

0 2

0 2

0

=>

0

2

1/2

1

0

=>

2 1/2 1 0

0

(16)

Wave field

2

0 1

2 2 0

0

### zyxE

A

where,







 



 









2

0 2

0 2

2 0 2

0

2( ) 1 1

z w z

nw w z

z

w





 

 



 









2 0 2 2

0 1

1 )

( z

z z z

z nw z

R

 : Radius of curvature of the wave front

02

0

znw : Confocal parameter(2z0) or Rayleigh range

(17)

0

w0

I

Gaussian profile

0

0

###  0 z

Near field (~ plane wave)

Far field

(~ spherical wave)

(18)

### 2.6 Fundamental Gaussian beam in a Lenslike Medium - ABCD law

q P i

k k q

q 







1 2 1 ' 2 0 '

For lenslike medium,

Introduce s as,

s s q

'

1   " 20 k sk s

k z k k

b k k z

k k

a k z s

k z b k

k z a k

z s

2 2

2 2

2 2

sin cos

) ( '

cos sin

) (

##  kkkkzz  qkkkqk   kkkkz  z 

2 0

2 2

2 2

0 2

Table 2-1 (6) 

1 1 2

(19)

i

o

i

o

i i o

o

(20)

i i o

o

o i i

i i

o

i i

i i

o o

o

o

i i

i i

1 1 2

2

1

0

02

0

znw

(21)

1

01

02

1

2

?

?



 

 



 



 

 

 

 



 

f z

f z z z z f z

z f

z D

C B A

/ 1

0

/ /

1

1 0 1 1 /

1

0 1

1 0 1

1 2 1 2 1 2

1 2

1 1

2 1 2

1 1

2

2

(22)

2 01 2

2 1 2

01 2

02

2 2 01 2

1

1 2

2

###   

02

01 w

w 

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

01

02

2 1

2

01

- If =>

2

=>

N

N

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.

(23)

s

T T

T t

 

sin

) 1 ( sin ) sin(

sin ) sin(

sin ) sin(

sin

) 1 ( sin ) sin(

 

 

s s

D D

s C C

s B B

s s

A A

T T T T





2 1

2

1

2 2

2 1 cos 1

f f d f

d f D d

A where,

1

1

1

s

### A

Stability condition for the Gaussian beam :

2 1 2 1

1 0

2 1

 

 

 



 

 

f

d f

d

: Same as condition for stable-ray propagation

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