Abstract

When the performance of an optical system is evaluated by tracing rays from a point source through the system, it is necessary in some cases to use rays which effectively transport equal amounts of energy per unit time (isoenergetic rays). The procedure given in the optical literature for generating isoenergetic rays is to distribute the ray intersections with the entrance pupil with uniform density per unit area. It is shown here that this procedure is not rigorously correct, and a correct procedure for generating isoenergetic rays is presented.

© 1985 Optical Society of America

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References

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  1. P. J. Sands, “Aberration Coefficients and Best Focus,” J. Opt. Soc. Am. 63, 582 (1973).
    [CrossRef]
  2. T. B. Andersen, “Evaluating rms Spot Radii by Ray Tracing,” Appl. Opt. 21, 1241 (1982).
    [CrossRef] [PubMed]
  3. D. G. Burkhard, D. L. Shealy, “Analytical Illuminance Calculation in a Multi-Interface Optical System,” Opt. Acta 22, 485 (1975).
    [CrossRef]
  4. J. W. Foreman, “Computation of rms Spot Radii by Ray Tracing,” Appl. Opt. 13, 2585 (1974).
    [CrossRef] [PubMed]
  5. M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958), pp. 106 and 387.Even though Herzberger does not state explicitly in the text whether rays are to be entered at the vertices or at the centers of the equilateral triangles, it appears from Fig. 11.7b on p. 106 that he intended for rays to be placed at the vertices of the triangles.
  6. A. Cox, A System of Optical Design (Focal Press, New York, 1964), pp. 378 and 379.

1982 (1)

1975 (1)

D. G. Burkhard, D. L. Shealy, “Analytical Illuminance Calculation in a Multi-Interface Optical System,” Opt. Acta 22, 485 (1975).
[CrossRef]

1974 (1)

1973 (1)

Andersen, T. B.

Burkhard, D. G.

D. G. Burkhard, D. L. Shealy, “Analytical Illuminance Calculation in a Multi-Interface Optical System,” Opt. Acta 22, 485 (1975).
[CrossRef]

Cox, A.

A. Cox, A System of Optical Design (Focal Press, New York, 1964), pp. 378 and 379.

Foreman, J. W.

Herzberger, M.

M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958), pp. 106 and 387.Even though Herzberger does not state explicitly in the text whether rays are to be entered at the vertices or at the centers of the equilateral triangles, it appears from Fig. 11.7b on p. 106 that he intended for rays to be placed at the vertices of the triangles.

Sands, P. J.

Shealy, D. L.

D. G. Burkhard, D. L. Shealy, “Analytical Illuminance Calculation in a Multi-Interface Optical System,” Opt. Acta 22, 485 (1975).
[CrossRef]

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

Opt. Acta (1)

D. G. Burkhard, D. L. Shealy, “Analytical Illuminance Calculation in a Multi-Interface Optical System,” Opt. Acta 22, 485 (1975).
[CrossRef]

Other (2)

M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958), pp. 106 and 387.Even though Herzberger does not state explicitly in the text whether rays are to be entered at the vertices or at the centers of the equilateral triangles, it appears from Fig. 11.7b on p. 106 that he intended for rays to be placed at the vertices of the triangles.

A. Cox, A System of Optical Design (Focal Press, New York, 1964), pp. 378 and 379.

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Figures (13)

Fig. 1
Fig. 1

Sphere of radius R used to generate isoenergetic rays.

Fig. 2
Fig. 2

Geometry for generating elements of equal area on the sphere of radius R.

Fig. 3
Fig. 3

Coordinate systems used in the isoenergetic ray analysis. Although a circular entrance pupil is shown, the analysis is independent of the shape of the pupil.

Fig. 4
Fig. 4

Projected zone boundaries and lune boundaries in the plane of the entrance pupil, and ray intersections with the plane of the entrance pupil, for d = 5 (arbitrary units), δ = 20°, θmax = 42.34°, N1 = 6, N2 = 10.

Fig. 5
Fig. 5

Suggested geometry for an isoenergetic ray trace of an optical system with a circular entrance pupil.

Fig. 6
Fig. 6

Suggested geometry for an isoenergetic ray trace of an optical system with a square entrance pupil.

Fig. 7
Fig. 7

Isoenergetic ray geometry for the case of a point source at infinity.

Fig. 8
Fig. 8

Square grid method.

Fig. 9
Fig. 9

Equilateral triangle method with rays placed at the vertices of the triangles.

Fig. 10
Fig. 10

Equilateral triangle method with rays placed at the geometrical centers of the triangles.

Fig. 11
Fig. 11

Polar method using unequal radial spacing between annuli. The angular sector width is the same in all annuli.

Fig. 12
Fig. 12

Polar method with equal radial spacing between annuli. The angular sector width varies from one annulus to the next.

Fig. 13
Fig. 13

Recipolar method of Cox in which rays are placed on circles with equal radial spacing. The angular spacing of the rays varies from one circle to the next.

Tables (1)

Tables Icon

Table I Parameters for the Example Shown in Fig. 4; the Quantities a, b, and xcen are in the Same Arbitrary Units of Length as d

Equations (42)

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θ m = cos 1 ( 1 m h / R ) = cos 1 [ 1 m ( 1 cos θ max ) / N 1 ] ( m = 0 , 1 , 2 , , N 1 ) ,
φ n = n Δ φ = 2 π n / N 2 ( n = 0 , 1 , 2 , , N 2 ) .
θ = ( θ m + θ m 1 ) / 2 ,
φ = ( φ n + φ n 1 ) / 2 .
O x = sin θ cos φ cos δ + cos θ sin δ ,
O y = sin θ sin φ ,
O z = sin θ cos φ sin δ + cos θ cos δ .
d k d tan δ i + r = x i + y j ,
d tan δ + r O x = x , r O y = y , d + r O z = 0 .
d tan δ + r ( sin θ cos φ cos δ + cos θ sin δ ) = x ,
r sin θ sin φ = y ,
d r ( sin θ cos φ sin δ cos θ cos δ ) = 0 .
x = Q 1 d sin θ cos φ sec δ ,
y = Q 1 d sin θ sin φ .
x max = d sin θ m sec δ / cos ( θ m + δ ) ,
x min = d sin θ m sec δ / cos ( θ m δ ) ,
a = ( x max x min ) / 2 = d sin θ m sec δ [ sec ( θ m + δ ) + sec ( θ m δ ) ] / 2 ,
x cen = ( x max + x min ) / 2 = d sin θ m sec δ [ sec ( θ m + δ ) sec ( θ m δ ) ] / 2 .
y max = d ( 1 tan 2 θ m tan 2 δ ) 1 / 2 tan θ m sec δ , y min = y max ,
b = y max = d ( 1 tan 2 θ m tan 2 δ ) 1 / 2 × tan θ m sec δ .
[ ( x x cen ) / a ] 2 + ( y / b ) 2 = 1 .
a = b = d tan θ m , x cen = 0 ,
M = cos δ tan φ n .
θ max = tan 1 [ R p cos δ / ( d sec δ R p sin δ ) ] .
s / 2 = Q 1 d sin θ max cos φ sec δ ,
s / 2 = Q 1 d sin θ max sin φ .
φ = tan 1 ( sec δ ) .
sin φ = sec δ ( 1 + sec 2 δ ) 1 / 2 , cos φ = ( 1 + sec 2 δ ) 1 / 2 ,
θ max = tan 1 [ ( 2 + tan 2 δ ) 1 / 2 / ( 2 d / s tan δ ) ] .
Γ 2 = N 1 i = 1 N R i 2 ,
N = i = 1 N r n i ,
i = 1 N R i 2 i = 1 N r n i r i 2 ,
Γ 2 = i = 1 N r n i r i 2 / i = 1 N r n i .
Γ 2 = i = 1 N r P i r i 2 / i = 1 N r P i = P 1 i = 1 N r P i r i 2 ,
u m = ( 2 m 1 ) s / 2 , υ n = ( 2 n 1 ) s / 2 , ( m , n = 1 , 2 , 3 , ) .
υ n = n s / 2 , u m = { 2 ( m 1 ) + [ 1 + ( 1 ) n ] / 2 } 3 s / 2 , ( m , n = 1 , 2 , 3 , ) .
υ n = n s / 2 u m = [ m 1 / 2 ( 1 / 6 ) ( 1 ) { m + [ 1 + ( 1 ) n ] / 2 } ] 3 s / 2 , ( m , n = 1 , 2 , 3 , ) .
R 2 = 2 R 1 .
u m , n = [ ( R m + R m 1 ) / 2 ] cos [ ( 2 n 1 ] Δ θ / 2 ] , υ m , n = [ ( R m + R m 1 ) / 2 ] sin [ ( 2 n 1 ] Δ θ / 2 ] , ( m , n = 1 , 2 , 3 , ) ,
Δ θ 2 = 3 1 Δ θ 1 ,
u m , n = [ ( R m + R m 1 ) / 2 ] cos [ ( 2 n 1 ) Δ θ m / 2 ] , υ m , n = [ ( R m + R m 1 ) / 2 ] sin [ ( 2 n 1 ) Δ θ m / 2 ] , ( m , n = 1 , 2 , 3 , ) ,
u m , n = R m cos [ 2 n 1 ] Δ θ m / 2 ] , υ m , n = R m sin [ 2 n 1 ] Δ θ m / 2 ] , ( m , n = 1 , 2 , 3 , ) ,

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