Abstract

The design parameters of the prism scanner appropriate to systems design are investigated analytically using a thin prism approximation and a more exact thick prism analysis. The results are specialized to the case of the spiral mode but the method is applicable to other modes.

Design equations and curves are derived for the spiral scanner using the thin prism approximation. By more exact analysis, it is shown that the thin prism analysis is adequate for preliminary design. The results are checked experimentally and are in excellent agreement.

© 1960 Optical Society of America

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References

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  1. F. R. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill Book Company, Inc., New York, 1950), 2nd ed., p. 31.
  2. This is average frame time; maximum time between successive scans at the edge of center is twice the average frame time.
  3. A negative angle refers to an angle of incidence above the normal line.

Jenkins, F. R.

F. R. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill Book Company, Inc., New York, 1950), 2nd ed., p. 31.

White, H. E.

F. R. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill Book Company, Inc., New York, 1950), 2nd ed., p. 31.

Other (3)

F. R. Jenkins and H. E. White, Fundamentals of Optics (McGraw-Hill Book Company, Inc., New York, 1950), 2nd ed., p. 31.

This is average frame time; maximum time between successive scans at the edge of center is twice the average frame time.

A negative angle refers to an angle of incidence above the normal line.

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Figures (10)

Fig. 1
Fig. 1

Sketch of the spiral scan produced by the prism scanner with constant prism velocity and 10 revolutions per cycle.

Fig. 2
Fig. 2

Thin prism geometry and its vector equivalent.

Fig. 3
Fig. 3

Resultant deviation due to a pair of prisms from the vectorial addition of the deviation of each.

Fig. 4
Fig. 4

Relative position of the instantaneous field of view on the Nth revolution (normalized parameters).

Fig. 5
Fig. 5

Position of the instantaneous field of view vs the angle (2β) between the prism pair.

Fig. 6
Fig. 6

Scan redundancy as a function of the rotational position of the field of view.

Fig. 7
Fig. 7

Dwell time vs the position of the instantaneous field of view.

Fig. 8
Fig. 8

For analytical purposes, prism 1 is considered stationary.

Fig. 9
Fig. 9

Effect of prism 2 on the rays deviated by prism 1.

Fig. 10
Fig. 10

Difference between thick prism analysis and the thin prism approximation for prism angles of 10 deg and index of refraction 3.

Equations (49)

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n a sin ϕ 1 = n sin ϕ , n a sin ϕ 2 = n sin ϕ 2 ,
R ( n - 1 ) α 1             ( thin prism in air ) ,
ρ = ( R 1 2 + R 2 2 + 2 R 1 R 2 cos 2 β ) 1 2 ,
β = 1 2 ( θ 2 - θ 1 ) .
ρ = 2 R cos β .
R 1 = R 2 θ ˙ R = const θ ˙ 1 = const θ ˙ 2 = const .
ρ = 2 R cos β
ρ ˙ = - 2 R β ˙ sin β .
θ ˙ R = 1 2 ( θ ˙ 1 + θ ˙ 2 ) = β ˙ + θ ˙ 1
β ˙ = 1 2 ( θ ˙ 2 - θ ˙ 1 ) .
d t = d ρ / 2 R β sin β d ρ = 2 R sin d β 0 t f d t = 0 1 2 π d β β ˙ .
t f = π / 2 β ˙ = π / ( θ ˙ 2 - θ ˙ 1 ) .
β ˙ = π / 2 T f
β = π t / 2 t f
ρ ˙ = ( - π R / t f ) sin ( π t / 2 t f ) .
θ ˙ R = 1 2 ( θ ˙ 1 + θ ˙ 2 ) = 2 N π / t f .
θ ˙ 1 = ( π / t f ) ( 2 N - 1 2 )
θ ˙ 2 = ( π / t f ) ( 2 N + 1 2 ) .
θ R = 2 N π t / t f = 2 n π
t = n t f / N .
ρ = 2 R cos ( π t / 2 t f ) = 2 R cos ( π n / 2 N ) .
d ρ / d n = ( - R π / N ) sin ( π n / 2 N ) ,
N = R π Δ n / Δ ρ = ( R π / W ) rev .
r = Δ n = W / R π sin [ 1 2 π · ( n / N ) ] .
r / ( W / π R ) = 1 / sin ( π n / 2 N ) .
V ρ = [ ( ρ ˙ ) 2 + ( ρ θ ˙ R ) 2 ] 1 2 .
cos β = ρ / 2 R sin β = [ 1 - ( ρ / 2 R ) 2 ] 1 2 ρ ˙ = ( π R / t f ) [ 1 - ( ρ / 2 R ) 2 ] 1 2 V ρ = { ( π R / T f ) 2 [ 1 - ( ρ / 2 R ) 2 ] + ( 2 N π ρ / T f ) 2 } 1 2 = ( π R / t f ) [ 1 - ( ρ / 2 R ) 2 + 16 N 2 ( ρ / 2 R ) 2 ] 1 2 = ( π R / t f ) [ 1 + ( ρ / 2 R ) 2 ( 16 N 2 - 1 ) ] 1 2 .
t d = L / V ρ = ( L T f / π R ) [ 1 + ( ρ / 2 R ) 2 ( 16 N 2 - 1 ) ] - 1 2 .
t d / ( L T f / π R ) = 1 / 4 N ( ρ / 2 R ) .
R ( n - 1 ) α .
t f = π / ( θ ˙ 2 - θ ˙ 1 ) .
θ ˙ 1 = ( π / t f ) ( 2 N - 1 2 )
θ ˙ 2 = ( π / T f ) ( 2 N + 1 2 ) ,
ρ = 2 R cos ( π n / 2 N ) .
r / ( W / π R ) = 1 / sin ( π n / 2 N ) ,
N = R π / W
t d / ( L t f / 4 π R N ) = 1 / ( ρ / 2 R )
lim ρ / 2 R 0 t d L T f / 4 π R N = 4 N .
n = 3.00 α = 10° .
n sin ϕ 2 = n α sin ϕ 2 .
sin ϕ 2 c = n α / n .
sin ϕ 2 = ( n / n α ) sin ϕ 2 = ( n / n α ) sin α = 0.52095
ϕ 2 = 31.4° .
δ 1 = ϕ 2 - α = 21.4° .
θ 2 + θ 1 = α
sin θ 1 = n sin ( α - θ 2 )
sin θ 2 = n sin θ 2
θ 1 = α - δ 1 cos 2 β .
ρ = [ ( δ 1 sin 2 β ) 2 + θ 2 2 ] 1 2 .