Ray tracing and ABCD matrix

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Ray tracing and ABCD matrix

• Optics, Eugene Hecht, Chpt. 6

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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

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Example Propagate distance L

• Angle (slope) unchanged
• Position depends on initial position and slope

(y1, y1)
(y0, y0)
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Example Go through lens

• Position unchanged
• Angle (slope) change depends on position focal
  length

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ABCD matrix

• Generalize
• Can cascade to make single matrix for system
• Example go through lens and propagate distance L
  f

(y1, y1)
(y0, y0)
(y2, y2)
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Example Fourier transform

• Propagate distance f, go through lens, propagate
  f
• Position and angle swap
• note scale factors f and -1/f

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Example -- 4 f imaging

• Cascade previous example

(y1, y1)
(y3, y3)
(y0, y0)
(y4, y4)
(y2, y2)
(y5, y5)
f
f
f
f
(y6, y6)
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Example -- dielectric interface

• Snells law -- angle changes, position fixed

n1
n2
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Other examples

• Easy to generate ABCD matrices

GRIN lens
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Optical resonator

• Condition for stable cavity
• Return to initial state after integer N round
  trips