Chapter 2. The Propagation of Rays and Beams

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

) ('

) (

z r

z

 r

 r

Nonlinear Optics Lab . Hanyang Univ.

Paraxial ray passing through a thin lens of focal length f

f r r

r

r r

in in

out in out

' '

'  



in in out

out

r r r f

r

1 ' 1

0 1

'  



: Ray matrix for a thin lens

Report) Derivation of ray matrices

in Table 2-1

Nonlinear Optics Lab . Hanyang Univ.

Table 2-1 Ray Matrices

Nonlinear Optics Lab . Hanyang Univ.

Nonlinear Optics Lab . Hanyang Univ.

Biperiodic lens sequence (f

, f

, 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

' 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

Dr Cr

r

Br Ar

r

' '

'

1 1











In equation form of 

) 2

( 1

2 2

f d d

B

f A d









 

 



 



 



 

 

 



 







 

 



 



 



 







2 1

1

1 2

1

1 1

1 1 1

f d f

d f

D d

f d f

C f

Nonlinear Optics Lab . Hanyang Univ.

  ^,

'

1 r

Ar

r  B

 ^'

^{ 1}  ^r

^{ Ar}



r B

0

2

2

  

 r

br

r



 



   







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 :

0 1

2^iq

 2 be

  e

isq s

r e r 

 i

iq

b i b e

e    

 1

^

general solution :

)

max

sin(   

 r s

r

Nonlinear Optics Lab . Hanyang Univ.

Stability condition

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

:  is real 

b  1 

2 1 2 1

1 0

2 1 1

1

2 1

2 1

2

1 2

 



 



 

 

 



 















f d f

d

f f d f

d f

d

Identical-lens waveguide (f, f, d)

f D d

C f d B

A  1 , 1 ,

, 1

f b d

1 2

cos



 Stability condition : 0  d  4 f

Nonlinear Optics Lab . Hanyang Univ.

2.2 Propagation of Rays Between Mirrors

! 2 / R f 

) sin(

) sin(

max max

y n

x n

n y

y

n x

x



















 2 l

2 

stability condition :

example)

f d 2 



) 2

/

max

sin(   

 r n

r

Nonlinear Optics Lab . Hanyang Univ.

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

E_{R L}

exp 2 ) , ( )

, (

2 2

phase shift

Index of refraction of lenslike medium :

 

   

 ( )

1 2 )

,

(

x

y

k n k

y x n

<Differential equation for ray propagation>

0

r

s _{ray path}

wave front :const

E ( r )  E ˆ ( r ) e

) r sˆ n(

) r (

where,

 



 

: optical path

Nonlinear Optics Lab . Hanyang Univ.

i)

sˆ //   , | sˆ |  1

ii) Maxwell equations :

 



 































0 (r)

H ˆ

0 (r)

E ˆ

0 (r) H ˆ (r)- E ˆ

0 (r) E ˆ (r) H ˆ













_



^

^

^



^

if =1, (







s

n ˆ







ds

s ˆ dr

   ds

n dr

2 2

2 ] 1 ) 2 [(

) 1 1 (

) ( )

( )

( n

n n

n ds

dr ds

d ds

n dr ds

d                     

ds n n dr ds

d  

 ( )

Nonlinear Optics Lab . Hanyang Univ.

For paraxial rays,

dz d ds

d 

2

0

2

2

 



 



  r k k dz

r d

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

Report) Proof

Nonlinear Optics Lab . Hanyang Univ.

2.4 Wave Equation in Quadratic Index Media

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)e}^it (monochromatic wave)

=> Helmholtz equation :



E  k

( 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

( r , , z ) k kk r k   

2 2 2

2 2

2 2

2 1

z r r r

t z 

 



 



 



 









Nonlinear Optics Lab . Hanyang Univ.

Assume, E₀  (x, y,z)e^ikz

=> 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   







=>

q i dz

dp k

k q dz

d

q  1 )   0 ,  

1 (

2

& slow varying approximation

Nonlinear Optics Lab . Hanyang Univ.

0 1 ) 1 (

2

 

 dz q d

q

q  z  q

is must be a complex ! =>

q

q

=> put,

q  z  iz

z

At z = z₀,

)}

0 ( exp{

2 ) exp(

) 0 (

0 2

z ip

z    kr 



Beam radius at z=0, ₀

2

)

( k

w  z





2.5 Gaussian Beams in a Homogeneous Medium

In a homogeneous medium,

k

 0

Otherwise, field cannot be a form of beam.

Nonlinear Optics Lab . Hanyang Univ.



 w

i

z q  

at arbitrary z,

q

=> _{2 2}

0 2

0 2

0 2

0

1 1

1

i w z R

z i z z

z z iz

z

q 

 

 

 

 



q i dz

dp  

ip ( z )  ln[ 1  ( z / z

)

]

 i tan

( z / z

)

=>

exp[ tan ( / )]

] ) / ( 1 [ )] 1 (

exp[

0

z z z i

z z

ip

 



Nonlinear Optics Lab . Hanyang Univ.

Wave field

  _



 



 



 

 

 

 



 



 



) ( exp 2

) / ( tan [

) exp exp (

) ( )

, ,

(

0 1

2 2 0

0

z R i kr z

z kz

z i w

r z

w w E

z y x E

A

where,















 





 























2

0 2

0 2

2 0 2

0

2( ) 1 1

z w z

nw w z

z

w 

 : Beam radius











 



 





 























2 0 2 2

0 1

1 )

( z

z z z

z nw z

R 

 : Radius of curvature of the wave front



 ₀²

0

z  nw : Confocal parameter(2z₀) or Rayleigh range

Nonlinear Optics Lab . Hanyang Univ.

Gaussian beam

z

w0

I

Gaussian profile

2w

/

2

/   nw

 

spread angle :

 0 z

Near field (~ plane wave)

Far field

(~ spherical wave)

z

Nonlinear Optics Lab . Hanyang Univ.

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

) (









 

     

 

 ^{k k k k z z}  ^{q k k k q k}   ^{k k k k z}  ^z 

z q

/ cos

/ /

sin

/ sin

/ /

cos )

(

2 0

2 2

2 2

0 2





 



Table 2-1 (6) 

D Cq

B q Aq



 

1 1 2

Nonlinear Optics Lab . Hanyang Univ.

Matrix method (Ray optics)

y

y





optical elements

 

 



 



 



 

 

 





i i o

o

y

D C

B y A



 ^{   C D   }

B

A : ray-transfer matrix

Transformation of the Gaussian beam – the ABCD law

Nonlinear Optics Lab . Hanyang Univ.

ABCD law for Gaussian beam

 

 



 



 



 

 

 





i i o

o

y

D C

B A

y





i i

o

D Cy

B Ay

y















i i

i i

o o

o

Cy D

B Ay

R y





 

 



) (

)

( ray optics q Gaussian optics

R



D Cy

B Ay

i i

i i



 



 /

/

D Cq

B q Aq



 

1 1 2

q

q

optical system

 

 





D C

B A

ABCD law for Gaussian beam : iz

z q  





0

z  nw

Nonlinear Optics Lab . Hanyang Univ.

example) Gaussian beam focusing

₁

w

w

z

z

?

?



 











 



 







 





 



 



 



 





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

(

1 1

2 1 2

1 1

2

2

q f z f

f z z z

z q

f q z













 



Nonlinear Optics Lab . Hanyang Univ.

2 01 2

2 1 2

01 2

02

1 1 1

1 



 



 

 

 



 

 

 w f

f z w

w

) ) (

/ (

) (

) (

2 2 01 2

1

1 2

2

f

w f

z

f z f f

z 





 

  

02

01 w

w 

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

01

02





 f

w

w  f 

2 1

2

01

/ ( z f )

w   



- If =>

z

 f

=>

f f f d

w

w f 2

,

/

) 2 (

2

01

02

  









; 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.

Nonlinear Optics Lab . Hanyang Univ.

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

T T

T t

D C

B A

D C

B

A 

 

 













 





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 ( sin )

sin(

) sin(

) sin(

) 1 ( sin )

sin(

1

1

1

  





 



s s

D q

s C

s B

q s

s q

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