Download Gaussian beam

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’=dr/dz
r
r ( z)
r
r '( z )
z
Nonlinear Optics Lab.
Hanyang Univ.
Paraxial ray passing through a thin lens of focal length f
rout  rin
r ' out  r ' in 
r ' in
f
1

1
r 'out
f
rout
0 rin
1 r 'in
: 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 (f1, f2, d)
rs  1
r' s  1
1
d
1
  1 (1  d )  1
f1
f1
f2
1


d
f2
r ' out
1
1
f
d
rin
d
(1  ) r ' in
f
rs
d
(1  ) r ' s
f2
d d (1
d
)
f2
rs
1 1
d
d
d
d
 (1 )  (1 )(1 ) r 's
f1 f 2
f1
f1
f1
f2
In equation form of
rout
d
rs  1  Ars  Br ' s
r ' s  1  Crs  Dr ' s
 1 1  d 
C    1 
 f 1 f 2  f 1 
d
Bd (2 ) D  d 1 d 1 d 

 f 1  f 2 
f2
f
1 



d
A1
f2
Nonlinear Optics Lab.
Hanyang Univ.
(2.1-5) 
1
r 's  rs 1 Ars  ,
B
r' s  1 
1
rs  2  Ars  1
B
 ABCD1 (actually for all elements)

rs  22brs 1rs 0
 d d
1
d2 

where, b ( A D)1  
2
f
2
f
1
2
f
1
f
2


trial solution : rs r0 eisq
e 2iq  2beiq  1  0
 eiq bi 1b 2 e  i
where,
cos b
general solution :
rs  rmax sin( s   )
Nonlinear Optics Lab.
Hanyang Univ.
Stability condition
: The condition that the ray radius oscillates as a function of the cell number s
between rmax and –rmax.
:  is real 
b 1 
d d
d2
1  1  
1
f 2 f1 2f1f 2
 d  d 
1
  1
0  1
 2 f1  2 f 2 
Identical-lens waveguide (f, f, d)
1
d
A1, Bd , C  , D1
f
f
d
cos b1
2f
 Stability condition : 0  d  4 f
Nonlinear Optics Lab.
Hanyang Univ.
2.2 Propagation of Rays Between Mirrors
f  R / 2!
xn  xmax sin( n  x )
yn  ymax sin( n  y )
stability condition :
2  2l
(, l : integers)
example) 2, l=1  =/2  cos = b = 1-d/2f = 0
 d 2 f (symmetric confocal)
rn rmax sin( n /2 )
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 ;

x2  y 2 
ER ( x, y)  EL ( x, y) exp  ik

2
f


phase shift
Index of refraction of lenslike medium :
 k

n( x, y )  n0 1  2 ( x 2  y 2 )
 2k

<Differential equation for ray propagation>
wave front :  const
r
s
0
E (r ) Eˆ (r )e ik0 ( r )


where,  (r)n( ŝr)
ray path
Nonlinear Optics Lab.
: optical path
Hanyang Univ.
i) ŝ //  , |ŝ|1
ii) Maxwell equations :  Ĥ(r) Ê(r) 0

 Ê(r)- Ĥ(r) 0

 Ê(r) 0
 Ĥ(r) 0

  n sˆ
ˆ
s
dr
ds



(Ê(r) ) Ê(r)( ) Ê(r) 0

1
2
if =1, ( ) 2  n 2
That is, | |n
dr

ds
d dr d
dr
1
1
1
(n ) ( ) ( )  ( ) [( ) 2 ] n 2
ds ds ds
ds
n
2n
2n
So, n
d dr
(n )n : Differential equation for ray propagation, (2.3-3)
ds ds
Nonlinear Optics Lab.
Hanyang Univ.
For paraxial rays,
d
d

ds dz
d 2 r  k2 
  r  0
2
dz  k 
 k 
 k 
k
r ( z )  cos 2 z r0 
sin  2 z r '0
k2
 k 
 k 
 k 
 k 
k
r ' ( z )   2 sin  2 z r0  cos 2 z r '0
k
 k 
 k 
Focusing distance from the exit plane for the parallel rays : h 
1
n0
 k 
k
cot  2 l 
k2
 k 
Report) Proof
Nonlinear Optics Lab.
Hanyang Univ.
2.4 Wave Equation in Quadratic Index Media
Gaussian beam ?
Maxwell’s curl equations (isotrpic, charge free medium)
E
H
H  
,   E  
t
t
Put,
2E
=>  E   2  0 : Scalar wave equation
t
2
E ( x, y,z,t )E ( x,y,z)eit (monochromatic wave)
=> Helmholtz equation :  2 E k 2 (r ) E 0
 i ( r ) 
where, k (r )  1
  
2
2
 0 : loss medium

 0 : gain medium
We limit our derivation to the case in which k2(r) is given by
k 2 (r , , z )k 2 kk2 r 2
2 2 1  2
    2  2 

z r r r z 2
2
 i (0) 
where, k 2 k 2 (0) 2  (0) 1

  (0) 
2
t
Nonlinear Optics Lab.
Hanyang Univ.
Assume, E0   ( x, y, z )e  ikz
& slow varying approximation
 2 2 

2





2
ik


kk
r
0
=>  2
2
2 
z
 x y 
kr2
Put,   exp{i[ p( z ) 
]} , r  ( x 2  y 2 )1/ 2
2q( z )
=>
1 d 1 k2
dp i

(
)


0
,

2
q dz q k
dz q
Nonlinear Optics Lab.
Hanyang Univ.
2.5 Gaussian Beams in a Homogeneous Medium
In a homogeneous medium, k2 0
1 d 1
 2  ( )0 => q  z  q0
q dz q
Otherwise, field cannot be a form of beam.
 q is must be a complex ! => Assume, q0 is pure imaginary.
=> put,
q  z  iz 0
( z0 : real)
At z = z0,
kr 2
 ( z  0)  exp( 
) exp{ip (0)}
2 z0
 Beam radius at z=0,
2 z0 1/ 2
w0  (
)
: Beam Waist
k
Nonlinear Optics Lab.
Hanyang Univ.
2

w
0
q at arbitrary z, q  z  i

z0
1
1
z
1

: Complex beam radius

 2

i


i
=>
2
2
2
2
q z  iz 0 z  z0
R w
z  z0
dp
i

dz
q
2 1/ 2
1
=> ip ( z )  ln[ 1  ( z / z0 ) ]  i tan ( z / z0 )
=> exp[ ip ( z )] 
1
1
exp[
i
tan
( z / z0 )]
2 1/ 2
[1  ( z / z0 ) ]
Nonlinear Optics Lab.
Hanyang Univ.
Wave field


E0 ( x, y, z )  w0
r 2 
kr 2 
1

exp  2   exp  i[kz  tan ( z / z0 ) exp  i

EA
w
(
z
)
w
(
z
)
2
R
(
z
)







2
  z  2 



z 
2
2
2
   w0 1     : Beam radius
where, w ( z )  w0 1  
2 
z 

  nw0  

  0  


2
  nw 2  2 


z


0
0
   z 1     : Radius of curvature of the wave front
R( z )  z 1  
  z  
  z  


nw0 2
: Confocal parameter(2z0) or Rayleigh range
z0 

Nonlinear Optics Lab.
Hanyang Univ.
Gaussian beam
spread angle :
 / 2   / nw0
I
2w0
w0
z
z0
Gaussian
profile
z 0
Near field
(~ plane wave)
Far field
(~ spherical wave)
Nonlinear Optics Lab.
Hanyang Univ.
2.6 Fundamental Gaussian beam in a Lenslike Medium - ABCD law
For lenslike medium,
2
'
 1   1  k2
       0
q q k
1 s'

q s
Introduce s as,
s ( z )  a sin
s' ( z )  a

P'  
i
q
 s" s
k2
0
k
k2
k
z  b cos 2 z
k
k
Table 2-1 (6) 
k2
k
k
cos 2 z  b 2 sin
k
k
k
k2
z
k

 k /k z q  k /k sin  k /k z
q ( z )
sin  k /k z  k /k q cos k /k z 
cos
2
2
0
2
2
0
q2 
Aq1  B
Cq1  D
2
2
Nonlinear Optics Lab.
Hanyang Univ.
Transformation of the Gaussian beam – the ABCD law
Matrix method (Ray optics)
yi i
o
yo
optical
elements
 yo   A B   yi 
   C D  
 i 
 o 
A B
C D  : ray-transfer matrix


Nonlinear Optics Lab.
Hanyang Univ.
ABCD law for Gaussian beam
 yo   A B   yi 
   C D  
 i 
 o 
yo  Ayi  B i
 o  Cyi  D i
Ayi  B i
Ro 

 o Cyi  D i
Ayi /  i  B

Cyi /  i  D
yo
Ro (ray optics)  q (Gaussian optics)
optical system
ABCD law for Gaussian beam :
 A B


C D
q1
q2
Aq1  B
q2 
Cq1  D
Nonlinear Optics Lab.
q  z  iz 0
nw0 2
z0 

Hanyang Univ.
example) Gaussian beam focusing
w01
w02 ?
1
z1
A

C
B  1
 
D   0
z2 ?
z 2  1

1   1 / f
1  z 2 / f
 
0

0  1

1  0
z1 

1 
z1  z 2  z1 z 2 /
1  z1 / f
f


 q2 
(1  z 2 / f )q1  ( z1  z 2  z1 z 2 / f )
 q1 / f  (1  z1 / f )
Nonlinear Optics Lab.
Hanyang Univ.
1
1

2
2
w02
w01
2
 z1 
1  w 
1    2  01 
f 
f   

2
f 2 ( z1  f )
z2  f 
( f )
2
2
2
( z1  f )  (w01 /  )
- If a strong positive lens is used ; w01  w02
=> w02 
=> w02 
f
 f 1
w01
2 f N
2 f

, f N  f / d : f-number
 (2w01 )

; The smaller the f# of the lens, the smaller the beam waist at the focused spot.
2
- If w01 /   ( z1  f )
2
=> z2  f
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 :
At
BT
CT
DT

A
B
C
D
s
A sin( s )  sin ( s  1) 
AT 
sin 
B sin( s )
BT 
sin 
C sin( s )
CT 
sin 
D sin( s )  sin ( s  1) 
DT 
sin 


Asin( s )sin ( s1) q1  Bsin( s )
qs1 
Csin( s )q1  Dsin( s )sin ( s1) 
Stability condition for the Gaussian beam :

d 
d 
1 
  1
0  1 
 2 f1  2 f 2 
 d d
1
d2 

where, cos   A D 1  
2
 f 2 f1 2 f1 f 2 
: Same as condition for
stable-ray propagation
Nonlinear Optics Lab.
Hanyang Univ.