Abstract

The National Bureau of Standards has been making optical calculations for well over a year, using both their new high-speed SEAC and the commercially available IBM Card Programmed Electronic Calculator.

Formulas especially suitable for machine computation are presented here. These enable a general ray to be traced through a rotationally symmetrical system of spherical or nonspherical surfaces. Formulas are also given for computing first- and third-order coefficients for any centered optical system.

The formulas are derived from well-known forms, but they are transformed so that none of the quantities used in the machine ever becomes indeterminate or infinite. The formulas for first- and third-order coefficients are simpler than those frequently given and are very convenient for hand calculation.

© 1951 Optical Society of America

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References

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  1. R. Glazebrook, A Dictionary of Applied Physics (Macmillan and Company, Ltd., London, 1923), Volume IV, p. 290.
  2. F. Wachendorf, Optik 5, 80 (1949).
  3. K. Schwarzschild, Untersuchungen zur Geometrischen Optik (Wiedmannsche Buchhandlung, Berlin, 1905), Part I.
  4. M. Born, Optik (Verlag. Julius Springer, Berlin, 1933). This book contains a condensed copy of Schwarzschild’s work.
    [Crossref]
  5. A. E. Conrady, Applied Optics and Optical Design (Oxford University Press, London, 1929).

1949 (1)

F. Wachendorf, Optik 5, 80 (1949).

Born, M.

M. Born, Optik (Verlag. Julius Springer, Berlin, 1933). This book contains a condensed copy of Schwarzschild’s work.
[Crossref]

Conrady, A. E.

A. E. Conrady, Applied Optics and Optical Design (Oxford University Press, London, 1929).

Glazebrook, R.

R. Glazebrook, A Dictionary of Applied Physics (Macmillan and Company, Ltd., London, 1923), Volume IV, p. 290.

Schwarzschild, K.

K. Schwarzschild, Untersuchungen zur Geometrischen Optik (Wiedmannsche Buchhandlung, Berlin, 1905), Part I.

Wachendorf, F.

F. Wachendorf, Optik 5, 80 (1949).

Optik (1)

F. Wachendorf, Optik 5, 80 (1949).

Other (4)

K. Schwarzschild, Untersuchungen zur Geometrischen Optik (Wiedmannsche Buchhandlung, Berlin, 1905), Part I.

M. Born, Optik (Verlag. Julius Springer, Berlin, 1933). This book contains a condensed copy of Schwarzschild’s work.
[Crossref]

A. E. Conrady, Applied Optics and Optical Design (Oxford University Press, London, 1929).

R. Glazebrook, A Dictionary of Applied Physics (Macmillan and Company, Ltd., London, 1923), Volume IV, p. 290.

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Figures (3)

Fig. 1
Fig. 1

Physical significance of the quantities used in ray tracing. (This schematic diagram shows a meridian ray, but the formulas apply generally.)

Fig. 2
Fig. 2

This diagram illustrates a step in the process of locating the intersection of the ray with an aspheric surface.

Fig. 3
Fig. 3

Significance of quantities used in paraxial ray tracing.

Equations (42)

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μ 1 N / N 1 ;
e = t X - ( x X + y Y + z Z ) ,
M 1 x = x + e X - t ,
M 1 2 = x 2 + y 2 + z 2 - e 2 + t 2 - 2 t x ,
ξ 1 = [ X 2 - c 1 ( c 1 M 1 2 - 2 M 1 x ) ] 1 2 ,
L = e + ( c 1 M 1 2 - 2 M 1 x ) / ( X + ξ 1 ) ,
x 1 = x + L X - t , y 1 = y + L Y , z 1 = z + L Z . }
ξ 1 = [ 1 - μ 1 2 ( 1 - ξ 1 2 ) ] 1 2 ,
g 1 = ξ 1 - μ 1 ξ 1 ,
X 1 = μ 1 X - g 1 c 1 x 1 + g 1 , Y 1 = μ 1 Y - g 1 c 1 y 1 , Z 1 = μ 1 Z - g 1 c 1 z 1 . }
c 1 ( x 1 2 + y 1 2 + z 1 2 ) - 2 x 1 = 0.
X 1 2 + Y 1 2 + Z 1 2 = 1
( R + U ) 2 ,             4 ,             c K ,
x = f ( y , z ) .
x = r - ( r 2 - s 2 ) 1 2 ,
x = { [ 1 - ( 1 - c 2 s 2 ) 1 2 ] / c } + [ A 2 s 2 + A 4 s 4 + ] = f .
[ 1 - ( 1 - c 2 s 2 ) 1 2 ] / c c s 2 / [ 1 + ( 1 - c 2 s 2 ) 1 2 ] .
- f / y = - y { c ( 1 - c 2 s 2 ) - 1 2 + 2 A 2 + 4 A 4 s 2 + } - f / z = - z { c ( 1 - c 2 s 2 ) - 1 2 + 2 A 2 + 4 A 4 s 2 + }
l = ( 1 - c 2 s 2 ) 1 2 ,
m = - y [ c + l ( 2 A 2 + 4 A 4 s 2 + ) ] ,
n = - z [ c + l ( 2 A 2 + 4 A 4 s 2 + ) ] .
l 0 ( x - x ¯ 0 ) + m 0 ( y - y 0 ) + n 0 ( z - z 0 ) = 0.
( x - x 0 ) / X = ( y - y 0 ) / Y = ( z - z 0 ) / Z .
x 1 = G 0 X + x 0 , y 1 = G 0 Y + y 0 , z 1 = G 0 Z + z 0 ,
G 0 = l 0 ( x ¯ 0 - x 0 ) / ( X l 0 + Y m 0 + Z n 0 ) .
l 2 + m 2 + n 2 O 2 1.
ξ ¯ 1 = X l + Y m + Z n , ξ ¯ 1 = [ O 2 ( 1 - μ 1 2 ) + μ 1 2 ξ ¯ 1 2 ] 1 2 , g ¯ 1 = ( ξ ¯ 1 - μ 1 ξ ¯ 1 ) / O 2 ,
X 1 = μ 1 X + g ¯ 1 l , Y 1 = μ 1 Y + g ¯ 1 m , Z 1 = μ 1 Z + g ¯ 1 n .
I N [ u y p - u p y ] .
2 ( C + γ ) + ( u - 1 u - 1 p - u u p ) = 0 , i p D - i E = 0 , i δ - i p = 0.
a = y ( Δ N - 1 - μ Δ N ) i , b = y ( Δ N - 1 - μ Δ N ) i p ,
B K over all the surfaces .
σ = H / h , τ = K / h , η = V / v , ζ = W / v .
H = h [ ( 1 + E o + 2 C p + F q ) σ + ( D o + 2 F p + B q ) η ] ,
K = h [ ( 1 + E o + 2 C p + F q ) τ + ( D o + 2 F p + B q ) ζ ] ,
o = σ 2 + τ 2 , p = σ η + τ ζ , q = η 2 + ζ 2 .
- u d = v , u p d = h .
- ( tan θ y ) d = V , - ( tan θ z ) d = W , ( tan α y ) d = H , ( tan α z ) d = K .
η = V / v = ( tan θ y ) / u , ζ = W / v = ( tan θ z ) / u , σ = H / h = ( tan α y ) / u p , τ = K / h = ( tan α z ) / u p .
σ = H / h , τ = K / h .
x = 1 2 c s 2 + ( 1 8 c 3 + K ) s 4 + ,
B = w y 4 , β = - w y p 4 , F = w y 3 y p , φ = - w t p 3 y , C = w y 2 y p 2 , γ = - w y p 2 y 2 , D = w y 2 y p 2 , δ = - w y p 2 y 2 , E = w y y p 3 , = - w y p y 3 , P = 0.