Transcript Ray tracing and ABCD matrix

Ray tracing and ABCD matrix
Optics, Eugene Hecht, Chpt. 6
Basics of ray tracing
• Consider 2D projection
• Ray uniquely defined by position and angle
– Make components of vector
• Paraxial approximation -- express angle as slope = y ’
f=y’
y
z
Example: Propagate distance L
• Angle (slope) unchanged
• Position depends on initial position and slope
(y1, y1’)
(y0, y0’)
 y1   1 L  y0 
   
 
 y1   0 1  y0 
Example: Go through lens
• Position unchanged
• Angle (slope) change depends on position & focal length
(y0, y0’)
 y1   1
   
 y1    1 / f
(y1, y1’)
0  y0 
 
1  y0 
ABCD matrix
 y1   A B  y0 
   
 
 y1   C D  y0 
• Generalize
• Can cascade to make single matrix for system
• Example: go through lens and propagate distance L = f
(y0, y0’)
 y2   1
   
 y2    1 / f
(y1, y1’)
0  y1   1
   
1  y1    1 / f
(y2, y2’)
0  1 f  y0   1

   
1  0 1  y0    1 / f
f  y0 
 
0  y0 
Example: Fourier transform
• Propagate distance f, go through lens, propagate f
• Position and angle swap
– note scale factors f and -1/f
(y1, y1’)
(y0, y0’)
(y2, y2’)
f
 y3   0
   
 y3    1 / f
f
f  y0 
 
0  y0 
(y3, y3’)
Example -- 4 f imaging
• Cascade previous example
(y1, y1’)
(y0, y0’)
(y2, y2’)
f
f
(y3, y3’)
(y4, y4’)
f
 y3    1 0  y0 
   
 
 y3   0  1 y0 
(y5, y5’)
f
(y6, y6’)
Example -- dielectric interface
• Snell’s law -- angle changes, position fixed
n1
n2
0  y 
 y1   1
 0 
  
n
0 1  y 


y
 1 
n2  0 
Other examples
• Easy to generate ABCD matrices
GRIN lens
Optical resonator
• Condition for stable cavity
• Return to initial state after integer N round trips
N
 A B
 1 0



 
 C D  roundtrip  0 1 