Download Chapter 2: Propagation of Laser Beams

Chapter 2 Propagation of Laser Beams
Lecture 1 Gaussian laser beams
2.1 Scalar wave equation
Lasers are coherent electromagnetic waves. Control of laser beams is a main topic of this
course. Laser beams usually have a concentrated profile in the transverse direction, which
diverges slowly when propagating. The beam profiles in space can be obtained by solving
the wave equation.
2
In an isotropic, linear medium with 2E    E   log     E  E   log    0
t 2
no free charge (Chapter 1):
Suppose within one wavelength the fractional change of  and  are small (slow-varying
refractive index),
Taylor series in dyadic algebra :

1
  log   λ   log   λλ :  log     1    log   1, 2 2 log   1.

2
E
 2E E
E
E
2
Also  E ~ 2 ,  2 ~ 2 ,   E ~ , E ~ .

t



 2E
We then have  E   2  0.
t
2
1
Assume any Cartesian component of the electric field varies as
( x, y, z, t )  E ( x, y, z )eit , then
 E  K (r ) E  0, with K (r)    
2
2
2
2
 2 n 2 (r)
c
2
.
Helmholtz equation
Lens-like (quadratic index) media:
Consider a cylindrically symmetric lens-like medium
n
k
n
n k
 k 
n 2 (r)  n02 1  2 r 2 , with 2  2 , k  0 , k2  2 , 2 r 2  1.
k 
k n0
c
c k

Significances of lens-like media:
1) k2 (or n2) describes the inhomogeneity of the
medium. k2 =0 returns to homogeneous medium.
2)Lens-like media simulate the behavior of many
optics, including lenses and spherical mirrors.
3)Lens-like media are an approximation of many
optical fibers.
n(r)
2
Solving the Helmholtz equation:  2 E  K 2 (r) E  0.
2
2
k2 2 
2
n
1

r   k 2  kk2 r 2 .

0
2
2
c
c
k 

Suppose the light mainly propagates in the z direction, let E ( x, y, z )   ( x, y, z )eikz .
K (r) 
2
n (r) 
2
Suppose the solution of the field is cylindrically symmetric, then
1     1 2
2
2 1  2
 
 2  2
 2
r   2
2
r r  r  r 
z
r
r r z
2
 2 1  2 
 2 
 2   ( x, y , z )e ikz  k 2  kk2 r 2   ( x, y , z )e ikz  0
r r z 
 r




 2 1  

 2 
   ' '2ik 'kk2 r 2  0
(with  ' 
)

r
r

r

z


 2 1  
 Suppose  ' '  k ' , ' does not change much in one 
 2 
  2ik 'kk2 r 2  0 

wavelengt
h.
Slow
varying
amplitude
approximat
ion
.

r
r

r








Let  be in the form   E0 exp  i  P( z ) 
2
k
1
k
   r 2  2i  k 2 r 2  '2kP'kk2 r 2  0
q
q
q

k
r 2   , then
2q( z )  
Will pick up later.
2

i
 1   1  k2

     '  0, and P'   
q
q
k
q

   

ar 2 b 0
3
2.2 Gaussian beams in a homogeneous medium
2
1 1
    '  0  q( z )  z  q0
q q
P' 
Complex beam parameter . q is a constant. 

i
i
z

 P( z )  i ln 1  
q
z  q0
 q0 
0
 the integratio n constant is unimportan t. 
 
 

k
z
kr2  
2 
  E0 exp  i  P( z ) 
r    E0 exp  i  i ln 1   

2
q
(
z
)
q
2
(
z

q
)



0
0




 
 
w02 n
.
Let q0 be purely imaginary (so that z =0 has the most intensity), and q0  i

 





 





2
2
E
z
kr

ikr
 

0
 


  E0 exp  i  i ln 1 
exp 
2
2
2
z
 

w0 n  
w n  
w0 n  
 

 
2
z

i
 i
 2 z  i 0    1  i w2 n
 
 






0
 

  
 


E0
 z 
1   2 
 w0 n 
2







 z  
 r2
ikr 2
exp i arctan  2  exp 

2
2 
2

w
n



 0 
 z 
 w0 n   

 2
 w0 1   w2 n   2 z 1   z   
 
  
   0  
4
Fundamental Gaussian laser beams:
 

w0
r2 
kr 2  
E ( x , y , z )  E0
exp  2  exp  i kz   ( z ) 

w( z )
w
(
z
)
2
R
(
z
)



 
  z  2 
  z 2 
w02 n
2
2
2
w ( z )  w0 1   2    w0 1    , z0 

  w0 n  
  z0  
2
  w2 n  2 


z


0
0
   z 1    
R( z )  z 1  

z
 
  z  
 
 z 
 z 

 

arctan
2 
 w0 n 
 z0 
 ( z )  arctan 
w0  Beam waist,
w( z )  Beam width, spot size
z0  Rayleigh range ( zR )
b  2 z0  Confocal parameter
R( z )  Radius of curvature
 ( z )  Phase shift, Gouy phase

z 

  lim  arctan


w0n
w( z ) 

z 
 Beam divergence
5
Lecture 2 Geometry of Gaussian laser beams
Geometry of the fundamental Gaussian laser beams:
1. Beam waist w0. The minimum spot size. The geometry of the beam is uniquely
determined by the size and location of w0 (suppose  and n are given).
2
 z 
2. Beam width w( z )  w0 1   2  , where the radial intensity drops to 1/e2=0.135.
 w0 n 
Also called spot size. Beam area =  w2(z). Hyperbolically depends on z. The hyperbola
can be thought as the boundary of the beam.
w02 n
3. Rayleigh range z R 
, where the beam area is doubled compared to that at the

beam waist, so the intensity is halved. Confocal parameter b=2zR can be thought as the
focal range along the beam direction, where the intensity does not drop too much. The
Rayleigh range is proportional to w02, indicating that smaller beam diffracts faster.
 z 
. Phase shift compared to a plane wave. Causes slightly
2 
 w0 n 
4. Gouy phase  ( z )  arctan 
faster local phase velocity. Total phase shift from – to  is , among which  /2
occurs in the focal range.
6
  w2 n  2 
5. Radius of curvature R( z )  z 1   0  . The radius of curvature of the wave front.
  z  
Refer to a spherical wave at a far point:

exp  ikR  exp  ik x  y  z
2
2
2



r 2 
kr 2 


 exp  ikz 1  2  exp   ikz  i


z
2
R




R(z) is infinity at z =0 and z =, where the wave fronts are flat. R(z) z when z is
large, where the beam is close to a spherical wave starting from the focal point.
6. Beam divergence   lim  arctan
z 

w( z ) 

. Half apex angle of the hyperbolic beam

z  w0 n
boundary. Inversely proportional to w0, indicating that smaller beam diffracts faster.
7
2.4 High order Gaussian beam modes in homogeneous medium
If we are not limited to

 0 , the Helmholtz equation  2E  k 2 (r )E  0 in a

homogeneous medium has the solution
 x2  y 2 
 2x   2 y 
w0
k ( x2  y 2 ) 
H m 
 exp  2
El ,m ( x, y, z )  E0
H l 
 i kz  l  m  1 ( z ) 



w( z )  w( z )   w( z ) 
w
(
z
)
2
R
(
z
)



  z  2 
w2 ( z )  w02 1   2  
  w0 n  
  w2 n  2 
R( z )  z 1   0  
  z  
 z 
 ( z )  arctan  2 
 w0 n 
Hermite polynomials
8
Hermite-Gaussian laser modes
Laguerre-Gaussian laser modes
9
Elliptical Gaussian laser beams:
E ( x , y , z )  E0
  x 2
y2  
kx2
ky2  
exp   2
 2   i kz   ( z ) 


w
(
z
)
w
(
z
)
2
R
(
z
)
2
R
(
z
)
wx ( z ) wy ( z )
  x
 
 
y
x
y
w0 x w0 y
   ( z  z )  2 
x
wx2 ( z )  w02x 1  
 
2
  w0 x n  
 (z  z ) 2 


y
wy2 ( z )  w02y 1  


2

w
n
0y
 
 
  w2 n  2 
0x
Rx ( z )  ( z  z x )1  
 

(
z

z
)
x  
 

  w2 n  2 


0y
R y ( z )  ( z  z y )1  
 
   ( z  z y )  
Elliptical Gaussian laser beams
are often seen in the lab.
(z  zy ) 
 (z  zx )  1
1
 ( z )  arctan 

arctan



2
2
2

w
n
2

w
n
0x
0y




10
11
Lecture 3 Ray equation
The electromagnetic wave in space is given by
 (r)= const
E(r, t )  E0 (r) expit   (r)
Suppose the fractional change of n (and thus E0) is small
in one wavelength, then
2
Phase :  (r ) 

n(r)
P
dr
2
dr O
O n(r)ds  O  (r)  dr   (r)   n(r) ds .
P
dr
Optical path length :  (r )   n(r )ds,  (r )  n(r ) .
O
ds
Ray dr is defined to be along , and |dr|=ds. Here s is the distance traveled by the wave.
P
P
  (r ) 
d 
dr  d
 dr 
1


n
(
r
)



(
r
)





(
r
)





(
r
)



f



f

 f


 n (r )

2
ds 
ds  ds
ds




  (r )
n 2 ( r )


 n(r )
2 n (r )
2 n (r )
2
2
Ray equation: d n(r ) dr   n(r )
ds 
ds 
Compare to
d  dr 
 m   (V )
dt  dt 
Geometrical optics is a simplification to wave
optics, essentially like classical mechanics to
quantum mechanics.
12
Application of the ray equation
Paraxial rays in a medium with n = n(x). Suppose the incident ray is in the x-z plan.
x

d 
dr 
n
(
r
)


n
(
r
)

ds 
ds 

z
Paraxial rays along z  ds  dz 
d 
dx  dn
d 2 x 1 dn
  n( x )  
 2 
dz 
dz  dx
dz
n dx
Example 1: Optical mirages.
n  n0 (1  x) (x  1)
d 2 x 1 dn
1 2




x

z  bz  c
2
dz
n dx
2
b, c are determined by initial conditions
(e.g., the injection height and direction).
dT
dn
 0,
0
dx
dx
Inferior mirage
(road, desert)
dT
dn
 0,
0
dx
dx
Superior mirage
(sea, airplane)
13
Example 2: Optical fiber with quadratic index media.


n 2 (r )  n02 1   2 r 2 , ( 2 r 2  1,   k 2 k )


Let' s consider rays in the x-z plane, n 2 ( x)  n02 1   2 x 2 .
d 2 x 1 dn
2
x




2
n dx
dz
 x( z )  a cos z  b sin z  A sin z  B 
 x ( 0)  a
Ray matrix of a quadratic index medium:
x ( z )  a cos z  b sin z   x ' ( z )  a sin z  b cos z
 x ' (0)  b

 1 sin z  x (0)   A B  x (0) 
 x ( z )   cos z

 
  
  


cos z  x ' (0)   C D  x ' (0) 
 x ' ( z )     sin z
Will pick up later
in ray matrices.
14
*(Reading) Lagrangian optics:
Lagrangian optics is a branch of geometrical optics which deals with the propagation of
rays in inhomogeneous media.
Fermat’s principle (the principle of least time):
2
2
P
 dx   dy 
  n( x, y, z )ds    n( x, y, z ) 1       dz    n( x, y, z ) 1  x'2  y '2 dz  0
S
S
S
 dz   dz 
P
P
Optical Lagrangian : L( x, y, x' , y ' , z )  n( x, y, z ) 1  x'2  y '2


d L L d L L

,

Optical Lagrangian equations :
dz

x
'

x
dz

y
'
y

Compare to
t2
  L(qk , qk , t )dt  0.
t1


d
nx'

2
2 n 


1

x
'

y
'

d  dx  n
dz 1  x'2  y '2
x 
2
2

n  
L  n ( x, y , z ) 1  x '  y ' 

 ds  ds  x
ds
1  x '2  y '2  
dz 
d 
dr 
Optical Lagrangian equations are
n
(
r
)
 n(r )



equivalent to the ray equation.
ds 
ds 
d L L

dz x' x
15
Lecture 4 Ray matrices
Ray matrices of optical elements:
The propagation of a ray through many optical elements can be described by using ray
matrices.
dr
A ray at position z is uniquely characterized by its height r(z) and its slope r ' ( z )  .
dz
 r ( z) 
We therefore use a column matrix (a vector) to represent the ray: ~
.
r ( z )  
r
'
(
z
)


An optical element transfers an input ray vector into an output ray vector.

 f 
 f 


r

f
(
r
,
r
'
)

f
(
0
,
0
)

r

2
1 1
 r  1  r '  r1 ' 
 1 0
 1 0


r '  g (r , r ' )  g (0,0)   g  r   g  r ' 
1 1
 r  1  r '  1
2
 1 0
 1 0

r1 '
r1
r2
r2 '
In paraxial optics the optical element is represented by a 22 matrix:
 r2   A B  r1 
   
 
r
'
C
D
 r '1 
 2 
Note that there may be different definitions
of ray vectors, e.g. (nr', r). The ray
matrices will accordingly be different.
16
Examples of ray matrices.
I). A thin lens with focal length f .
r2  r1

1
  r2   1
r  
r '2  r '1  1   r '2    f

f
0  r 
 1 
1  r ' 
 1 
II). Refraction
r2  r1


n2 sin r '2  n1 sin r '1  n2 r '2  n1r '1 
 r2   1 n0  r1 
   0 1  


 r '2  
n2  r '1 
r2 '
r1 r2
n1
r1 '
n2
17
Ray matrices of some common optical elements and media:
1 d 


0 1 
 1
 1

 f
1

0

0

1

0
n1 

n2 
18
 1
 n2  n1

 n2 R
1
2

R
0
n1 

n2 
Note that there may
be different sign
conventions for R
and for the z axis
after reflection.
0

1




 cos k 2 l 
 k 




  k 2 sin  k 2 l 
 k 

k



 k2  
k
sin 
l  
k2
k



 k2  
cos
l 

k

 
19
Ray matrices of combined optical elements:
Example: Thin lens combination. Two lens separated by d.
1
 A B   1

  
C
D

  f 2
0 1 d  1
0
 1


1  0 1  
1
 f1


d


1
d 

f1


d
d 
 1 1



1

 f
f 2 
 1 f 2 f1 f 2
d
Significance of the ray matrix of a lens-like media:



 cos k2 l 
 k 
 A B 



  
 k 
k
C D 
 2 sin  2 l 

k
 k 

 k2  
k
sin 
l  
k2
 k 

 k2  
cos
l 

k

 
1) Free propagation, lenses, and spherical mirrors (cases 1, 2, and 5) are equivalent to a
specific lens-like medium. E.g, a lens corresponds to l0, and k2=k/fl.
2) Single refraction (cases 3 and 4) may be equivalent to a combination of two
successive lens-like media.
20
Lecture 5 The ABCD law
2.3 Fundamental Gaussian beams in a lens-like medium
Fundamental Gaussian beams in a lens-like medium: E ( x, y, z )   ( x, y, z )e ikz
 

k
  E0 exp  i  P( z ) 
r 2  ,
2 q( z )  
 
2
 1   1  k2
i
    '  0, and P'   .
q
q q k
 k

 1   1  k2
1
k
 dx

 arctan x  
  2 tan  2 z  c    tan z  c .
    '  0,   2
q( z )
k
 x 1

q q k
 k

2
Considerin g
1
  tan z1  c , then at z2  z1  l ,
q( z1 )
1
tan z1  c   tan l
  tan z2  c    tan z1  c   l   
q ( z2 )
1  tan z1  c  tan l
1
  sin l
cos l  q( z1 )   1 sin l Aq( z1 )  B
q( z1 )

 q ( z2 ) 

1
  sin l  q( z1 )  cos l Cq( z1 )  D
 1 sin l 
 cos l
q( z1 )
cos l 
 q2 
Aq1  B
Cq1  D
21
Physical significance of the complex beam parameter q(z):
1) In a lens-like medium, define
1
1


i
,
2
q ( z ) R( z ) n0 w ( z )
  kr 2  


r2 
kr 2 
then   exp  i 
   exp  2  exp  i

 w ( z) 
 2 R( z ) 
  2q ( z )  
• The real part of 1/q(z) specifies the radius of curvature of the wave front R(z).
• The imaginary part of 1/q(z) specifies the spot size of the beam w(z).
nw02
z
2) In a homogeneous medium, q( z )  q0  z  i

• The real part of q(z) specifies the location of the beam waist z=0.
• The imaginary part of q(z) specifies the beam waist w0.
22
Another way to derive the transformation of the complex beam parameter:
Ray equation in a lens-like medium. Consider rays in an r-z plane.
k
 k

n 2 ( r )  n02 1  2 r 2 , ( with 2 r 2  1).
k 
k

ds dz d
d 
dr 
dr  dn
d 2 r k2
r" k 2

n
(
r
)


n
(
r
)

n
(
r
)



r

0

 0
2




ds 
ds 
dz 
dz  dr
dz
k
r
k
2


 1   1  k2


 
 '   0. 
q
(
z
)
transform
s
in
space
 r / r'   r / r'  k


r


2
using the same rule as that for 
Aq1  B
 1   1  k2

r' 
 q2 
    '  0


Cq1  D
q q k



 r2   A B  r1   r 
Ar / r ' 1  B

   

     

 r '2   C D  r '1   r '  2 C r / r ' 2  D

The ABCD law: The transformation of the complex beam parameter q(z) through an
optical system can be described by:
q2 
Aq1  B
1 C  D / q1
A B
or

 is the ray matrix of the optical system.
, where 
C
D
Cq1  D
q2 A  B / q1


23
Applications of the ABCD law:
Importance of the ABCD law: The law enables us to trace the Gaussian beam parameter
q(z) through a sequence of optical elements. The beam spot size w(z), the beam radius R(z),
the beam waist w0 and its location can be obtained at any point along the beam.
I) Thin lens:
 1
M   1

 f
0

1

Aq1  B
1
1 1

    w2  w1
Cq1  D
q2 q1 f  
1 1
 1


1
1

 R

i
R1 f
  2
q( z ) R( z ) nw2 ( z )
q2 
Compare to the lens equation
1 1 1
 
s o si
f
II) Focusing a Gaussian laser beam:
A Gaussian laser beam with waist size w01 is incident at its waist on a thin lens with focal
length f. Please find the location and size of the waist of the output beam.
w03=?
l =?
24
Planes 1, 2, 3 are defined in the figure. We want to find w03 and l.
Strategy: We find q2 (q at plane 2) first. The real part of q2 then specifies the location l
of the output beam waist. The imaginary part of q2 specifies the beam waist w03.

1
1

 
 i


i
2
2 

q1 R1 nw01
nw01
1
1
 

  i

2

Aq1  B
1
1 1
q2
f
nw01
q2 

 

Cq1  D
q2 q1 f 

2

nw03
q2  i
 l (This is a key equation.) 


f

l

2



f


f 
 1 


f

1

i
 nw2 
2 

2

01 

nw01  
nw03
1
i
l 
 


2
1

w01



f

 i
w

03
2

1  
2 
f
nw01
2 2


nw


nw
01 

01 

1  

f




Please note that to find w03 and l it is not necessary to calculate q3 (as our textbook does).
They are already encoded in q2. This solution also applies to the focusing of a laser beam
far from its beam waist.
25
II) Eigen mode of Gaussian beams in a quadratic-index medium:
Let us try to find a Gaussian laser beam whose beam parameter q(z) does not change in a
quadratic-index medium specified by k2.
 ( D  A)  ( D  A) 2  4CB 
Aq  B
2
q
 Cq  ( D  A)q  B  0  q 

Cq  D
2C



 k2 
 k2  

k
 cos



l
sin
l





k2
A B 

 k 
 k 

  


 k2 
 k2  
k2
C D 


sin 
l 
cos
l 


k

 k 
 k  

 R  
1
k
 i 2
 
q
k
1
1

 
2 k  4
 1  4
1 1
  w 


  
 i
k
k

n
n
 2
 0 2
q R n0 w2  
In the eigen mode the spot size w is independent of z. This is caused by the focusing
nature of the medium, which cancels the diffraction nature of the beam.
26