•Paraxial ray tracing of optical systems
–Gaussian beams
ECE 5616 OE System Design
Gaussian beams
Solution of scalar paraxial wave equation (Helmholtz equation) is a
Gaussian beam, given by:
2
2

w0  w2  z   jkz  jk 2 R  z   j  z 
e
E r   A0
w z 
Size
 z 
w z   w0 1   
 z0 
2
where
Radius of curvature
Gouy phase
  z0  2 
R z   z 1    
  z  
 z 
 z   tan  
 z0 
1
 z 
w 
 z0 
w0
 z 
R 
 z0 
z0
 z 

 z0 
 
Note that R(z) does not obey ray tracing sign convention.
Unfortunately there’s no particularly good way to fix this.
Robert McLeod
Saleh & Teich chapter 3, M&M Appendix 2
93
•Paraxial ray tracing of optical systems
–Gaussian beams
ECE 5616 OE System Design
Gaussian beams
Detailed view
Real part of E vs. radius and z
w z 
w z  / 2
At z = z0,
w z0   2w0
I(0,z0) = I(0,0)/2
R z0   2z0 (min value)
Any constant times w(z) is a ray path. Note that the rays are
converging and diverging spherical wave except near the
focus where they bend. Ergo rays do not always travel in
straight lines – the region near the focus violates the slowlyvarying envelope approximation.
Conversion formulas
w0 

 1
z0 

 0
Robert McLeod
0 
 1
 1

 w0
 z0
z0 
  2  2 w0
  w0 

0
 0
94
•Paraxial ray tracing of optical systems
–Gaussian beams
ECE 5616 OE System Design
Gaussian beam parameter
q(z)
The complete Gaussian beam expression normalized to intensity
 1 


E r   A0 


w
0 

D/2
w0
e
w z 

2
w2  z 
 jkz  jk
2
2Rz 
D is number of
transverse dimensions
=1,2
Define the complex radius of curvature
1
1


j
qz  Rz 
 w2 z 
What is q(z)?
qz  
 j  z 
1
1
  z0  2 
z 1    
  z  
 z  j z0 
 2
2 
 z  z0 
 z  j z0
j
1
  z 2 
z0 1    
  z0  
argq   tan 1
1
z0
z
2
wz 


 1  z  
z0
 z0 
w0
q
Can now write Gaussian above as
 1 


E r   j A0 


w
0 

D/2
z0
e
qz 
 jk
2
2qz 
 jkz
Note that phase of j/q(z) is 
Robert McLeod
95
•Paraxial ray tracing of optical systems
–Gaussian beams
ECE 5616 OE System Design
How does q change
with transfer and refraction?
Free space:
Start with expression for q(z)
q1  z1  j z0
q2  z 2  j z0  q1  z 2  z1 
so q2 = q1 + z
Thin lens
1
1


j
qz  Rz 
 w2 z 
Start with expression for 1/q(z)
1 1 1
 
R R f
Thin lens equation expressed as change in
curvature of wave
NOTE HOW GAUSSIAN BEAM SIGN
CONVENTION HAS CHANGED THE SIGN
1 1 1
 

q q f
Apply to 1/q
q 
q
 q f 1
Robert McLeod
Solve for q
96
•Paraxial ray tracing of optical systems
–Gaussian beams
ECE 5616 OE System Design
ABCD approach to q
Remember the ABCD matrices for thin lens refraction and free-space
transfer
 1
Rk  
  k
0
1 
1 t k 
Tk  

0
1


and define the evolution equation for q
Aq  B
q 
Cq  D
Check for free space:
1q  t k
q 
 q  t k
0q  1
Check for thin lens:
q 
q
1q  0

 k q  1  q f  1
Which says, rather remarkably, that we can model the propagation of
a Gaussian beam through a paraxial optical system using ray
matrices. But there’s a better way to do so (at least I like it’s better).
Robert McLeod
97
•Paraxial ray tracing of optical systems
–Gaussian beams
ECE 5616 OE System Design
Representation of
Gaussian beams by complex rays (1)
Define the following three rays. Note their suggestive names and
relationship to the Gaussian beam.
Paraxial ray trajectory form
 z   w0
z   z  0
w0
0
Chief ray
Waist location
Waist location
f
Define the complex ray trajectory
  z     z   j  z 
Paraxial image plane

Waist ray
ABCD vector form
 w0 
0   
0
0
0   
 0 
This is Greynolds’ definition and yields the
proper form of q. Arnaud’s definition yields q*.
You can then show that this ray contains q(z)
 z 
y  j y
 
d  dz u  j u

Robert McLeod
z 0  jw0
0
Ray heights over ray slopes
 z  j z0  q  z 
J. Arnaud, Applied Optics, V24, N4, p. 538, 15 Feb 1985
A. W. Greynolds, SPIE V 560, p. 33, 1985
M&M A2.5
E.g. at z=0
98
•Paraxial ray tracing of optical systems
–Gaussian beams
ECE 5616 OE System Design
Representation of
Gaussian beams by complex rays (2)
First we note:
n 
 y u   y  u   
n 
Lagrange invariant
(Nspots=1)
By brute for tracing of the rays, we can find the following Gaussian
parameters based on the two rays at that point:
 0  u2  u2
w z  
Which gives all other beam parameters
y2  z   y2  z 
1/e field radius at this z
We could use these two and the expressions for the Gaussian beam
parameters to generate the complete Gaussian, but this would be a bit
tedious. A more elegant way is to use the complex ray formalism:
 z 
d  dz
y  zu   j  y  zu 

u  ju
qz  
At plane z ≠0
Which, apart from the on-axis phase k0S gives the full Gaussian
beam at this plane z.
Robert McLeod
R. Herloski, S. Marshall, R. Antos, Applied Optics, V22, N8, p 1168,
15 Apr 1983
99
•Paraxial ray tracing of optical systems
–Gaussian beams
ECE 5616 OE System Design
Representation of
Gaussian beams by complex rays (3)
On-axis examples:
1)  = 1 m, w0 = 10 , f = 500 m, 1 f – 1 f system.
2)  = 1 m, w0 = 10 , f = 500 m, 2 f – 2 f system.
3)  = 1 m, w0 = 10 , f = 500 m, 3 f – 3/2 f system.
Notes
• In (1), second waist is at Fourier plane, as expected.
• In (2), second waist occurs before image plane, as expected.
• In (3), as distance to lens increases, waist moves to paraxial image plane
Robert McLeod
100
•Paraxial ray tracing of optical systems
–Gaussian beams
ECE 5616 OE System Design
Representation of
Gaussian beams by complex rays (5)
Off-axis examples:
1)  = 1 m, w0 = 10 , f = 500 m, 1 f – 1 f system.
2)  = 1 m, w0 = 10 , f = 500 m, 2 f – 2 f system.
3)  = 1 m, w0 = 10 , f = 500 m, 3 f – 3/2 f system.
Notes
• In (1), waist is centered at zero (as expected of Fourier trransform)
• In (2), image is at -10 m, expected from magnification M=-1.
• This type of problem is not possible with the ABCD formalism.
Robert McLeod
101
•Paraxial ray tracing of optical systems
–Gaussian beams
ECE 5616 OE System Design
Do Gaussian beams obey
paraxial imaging? (1/3)
f=10
w z 
z 
M=-1 imaging
Object at waist
 z 
2
1
-40
-30
-20
-10
10
20
30
-1
-2
f=10
Real object
w z 
z 
Image not at waist
2
 z 
Demagnifying imaging
1
-40
-30
-20
-10
10
20
30
-1
-2
Real image
Answer: Yes. The image is also a Gaussian E field distribution in amplitude and
any point on the object down from the peak by some value, say 1/e for the point
w(z), will image to the point on the image down from the peak by the same value.
Shown above only for real objects conjugate to real images ( -t ≥ f ).
Robert McLeod
102
•Paraxial ray tracing of optical systems
–Gaussian beams
ECE 5616 OE System Design
Do Gaussian beams obey
paraxial imaging? (2/3)
How about for virtual objects?
Virtual object
f=10
4
w z 
z 
2
Real
image
 z 
-40
-30
-20
-10
10
20
30
-2
-4
Answer: Still works.
Robert McLeod
103
•Paraxial ray tracing of optical systems
–Gaussian beams
ECE 5616 OE System Design
Do Gaussian beams obey
paraxial imaging? (3/3)
How about for virtual images?
7.5
f=10
5
w z 
z 
Virtual image
2.5
 z 
-40
-30
-20
-10
Real
object
10
20
30
-2.5
-5
-7.5
Conclusion: All parts of the object Gaussian image correctly to the appropriate parts of
the image Gaussian including both real and virtual objects and images.
Corollary: If you apply paraxial imaging to the object Gaussian over all z, you generate
the image Gaussian over all z. Gaussian beams obey paraxial imaging exactly.
Robert McLeod
104
•Design of ideal imaging systems with geometrical optics
–Gaussian beam propagation
ECE 5616 OE System Design
Design example
Collimator lens
z0
d
2L
M  T L R T d 
1  L

 
q L  
ABCD from start to center
L  d 1  L 

1  d

L  d 1  L   jz0 1  L 
1  d  j z0
 z02  d 1  d 
ReqL  0  L  2 2
 z0  1  d 2
qL  j
Robert McLeod
z0
 j z0 NEW
2
2 2
 z0  1  d 
q at center starting
with q = j z0
Where is waist?
What is new
Rayleigh range?
105