Abstract

Computing formulas are presented for tracing skew rays through optical systems containing cylindrical and toric surfaces of arbitrary orientation and position. Particular attention is given to the treatment of diffraction gratings and a vector form of the diffraction law is suggested.

© 1962 Optical Society of America

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References

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  1. D. Feder, J. Opt. Soc. Am. 41, 630 (1951).
    [Crossref]
  2. W. Allen and J. Snyder, J. Opt. Soc. Am. 42, 243 (1952).
    [Crossref]
  3. A. Murray, J. Opt. Soc. Am. 44, 672 (1954).
    [Crossref]
  4. M. Herzberger, Modern Geometrical Optics (Interscience Publishers, Inc., New York, 1958), Chap. V.
  5. G. Rosendahl, J. Opt. Soc. Am. 51, 1 (1961).
    [Crossref]
  6. H. Yoshinaga, B. Okazaki, and S. Tatsuoka, J. Opt. Soc. Am. 50, 437 (1960).
    [Crossref]
  7. M. Herzberger, J. Opt. Soc. Am. 41, 805 (1951).
    [Crossref]
  8. R. Minkowski, Astrophys. J. 96, 305 (1942).
  9. T. di Francia, Contributed article on the Ronchi Test, “Optical image evaluation,” NBS Circ. 526,  165 (1954).
  10. J. Guild, The Interference Systems of Crossed Diffraction Gratings (Oxford University Press, New York, 1956), p. 23.

1961 (1)

1960 (1)

1954 (2)

A. Murray, J. Opt. Soc. Am. 44, 672 (1954).
[Crossref]

T. di Francia, Contributed article on the Ronchi Test, “Optical image evaluation,” NBS Circ. 526,  165 (1954).

1952 (1)

1951 (2)

1942 (1)

R. Minkowski, Astrophys. J. 96, 305 (1942).

Allen, W.

di Francia, T.

T. di Francia, Contributed article on the Ronchi Test, “Optical image evaluation,” NBS Circ. 526,  165 (1954).

Feder, D.

Guild, J.

J. Guild, The Interference Systems of Crossed Diffraction Gratings (Oxford University Press, New York, 1956), p. 23.

Herzberger, M.

M. Herzberger, J. Opt. Soc. Am. 41, 805 (1951).
[Crossref]

M. Herzberger, Modern Geometrical Optics (Interscience Publishers, Inc., New York, 1958), Chap. V.

Minkowski, R.

R. Minkowski, Astrophys. J. 96, 305 (1942).

Murray, A.

Okazaki, B.

Rosendahl, G.

Snyder, J.

Tatsuoka, S.

Yoshinaga, H.

Astrophys. J. (1)

R. Minkowski, Astrophys. J. 96, 305 (1942).

J. Opt. Soc. Am. (6)

NBS Circ. 526 (1)

T. di Francia, Contributed article on the Ronchi Test, “Optical image evaluation,” NBS Circ. 526,  165 (1954).

Other (2)

J. Guild, The Interference Systems of Crossed Diffraction Gratings (Oxford University Press, New York, 1956), p. 23.

M. Herzberger, Modern Geometrical Optics (Interscience Publishers, Inc., New York, 1958), Chap. V.

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Figures (5)

Fig. 1
Fig. 1

Reference coordinate system ( X ¯ , Y ¯ , Z ¯ ) and local system (X,Y,Z), showing incident ray and computed distances, s0 and sf.

Fig. 2
Fig. 2

Generation of (X,Y,Z) system from ( X ¯ , Y ¯ , Z ¯ ) system after translation of origin to O. ( X ¯ , Y ¯ , Z ¯ ) ( X ¯ , Y ¯ , Z ¯ ) by rotation α; ( X ¯ , Y ¯ , Z ¯ ) ( X ¯ , Y ¯ , Z ¯ ) by rotation β; ( X ¯ , Y ¯ , Z ¯ ) ( X , Y , Z ) by rotation γ.

Fig. 3
Fig. 3

Relationship of κ and c to geometrical parameters of the conics represented by Eq. (25). The dashed lines in the figure for κ < 0 represent asymptotes.

Fig. 4
Fig. 4

Generation of toric surface.

Fig. 5
Fig. 5

Mode of grating ruling generation. In Case I, the rulings are defined by the surface intersections with parallel planes. In Case II, the rulings are defined by the intersections with concentric cylinders.

Tables (1)

Tables Icon

Table I Surface obtained for various values of κ

Equations (73)

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F ( X , Y , Z ) = 0 ,
R = ( cos γ sin γ 0 sin γ cos γ 0 0 0 1 ) ( 1 0 0 0 cos β sin β 0 sin β cos β ) ( cos α 0 sin α 0 1 0 sin α 0 cos α ) = ( ( cos α cos γ + sin α sin β sin γ ) cos β sin γ ( sin α cos γ + cos α sin β sin γ ) ( cos α sin γ sin α sin β cos γ ) cos β cos γ ( sin α sin γ cos α sin β cos γ ) sin α cos β sin β cos α cos β ) .
( X 0 Y 0 Z 0 ) = R ( X ¯ 0 x ¯ 0 Y ¯ 0 y ¯ 0 Z ¯ 0 z ¯ 0 ) ,
( k l m ) = R ( k ¯ l ¯ m ¯ ) .
X = X 0 + k s , Y = Y 0 + l s , Z = Z 0 + m s ,
s 0 = Z 0 / m ,
X 1 = X 0 + k s 0 , Y 1 = Y 0 + l s 0 .
X = X 1 + k s , Y = Y 1 + l s , Z = m s .
s j + 1 = s j F ( X j , Y j , Z j ) / F ( X j , Y j , Z j )
X j = X 1 + k s j , Y j = Y 1 + l s j , Z j = m s j ,
F ( X j , Y j , Z j ) = d F / d s ( s = s j ) = ( F x ) j k + ( F y ) j l + ( F z ) j m .
s 1 = 0 ,
| s f s f 1 | < ,
p = s 0 + s f .
K = ( F x ) f , L = ( F y ) f , M = ( F z ) f .
F ( X , Y , Z ) = Z c ρ 2 / [ 1 + ( 1 κ c 2 ρ 2 ) 1 2 ] i = 1 N α i ρ 2 i = 0 ,
ρ 2 < 1 / ( κ c 2 ) .
F x = X E , F y = Y E , F z = 1 ,
E = c / ( 1 κ c 2 ρ 2 ) 1 2 + 2 j = 1 N j α j ρ 2 ( j 1 ) .
F = Z ρ / tan θ = 0 ,
F x = X / ρ tan θ , F y = Y / ρ tan θ , F z = 1.
κ = tan 2 θ .
error in F < | κ c | 1 ,
error in d F / d ρ < | 2 κ c 2 ρ 2 | 1 .
F = Z 1 2 c ( ρ 2 + κ Z 2 ) = 0.
F x = c X , F y = c Y , F z = 1 κ c Z .
Z = f ( Y ) .
X 2 + ( Z R ) 2 = [ R f ( Y ) ] 2 ,
Z = f ( Y ) + ( 1 / 2 R ) [ X 2 + Z 2 f 2 ( Y ) ] .
c R = 1 / R ,
F ( X , Y , Z ) = Z f ( Y ) 1 2 c R [ X 2 + Z 2 f 2 ( Y ) ] = 0 ,
F x = c R X , F y = [ c R f ( Y ) 1 ] ( d f / d Y ) , F z = 1 c R Z .
f ( Y ) = c Y 2 / [ 1 + ( 1 κ c 2 Y 2 ) 1 2 ] + j = 1 N α j Y 2 j .
d f / d Y = = c Y / ( 1 κ c 2 Y 2 ) 1 2 + 2 j = 1 N j α j Y 2 ( j 1 ) .
N S × r = N S × r .
S = μ S + Γ r ,
k = μ k + Γ K , l = μ l + Γ L , m = μ m + Γ M .
Γ 2 + 2 a Γ + b = 0 ,
a = μ ( k K + l L + m M ) / ( K 2 + L 2 + M 2 ) ,
b = ( μ 2 1 ) / ( K 2 + L 2 + M 2 ) .
V ( Γ ) = Γ 2 + 2 a Γ + b ,
Γ n + 1 = Γ n V ( Γ n ) / V ( Γ n ) .
V ( Γ n ) = 2 ( Γ n + a ) ,
Γ n + 1 = ( Γ n 2 b ) / 2 ( Γ n + a ) .
Γ 1 = b / 2 a .
Γ = 2 a .
k = k 2 a K , l = l 2 a L , m = m 2 a M .
S × r = μ S × r + ( n λ / N d ) q ,
Λ = n λ / N d .
( S μ S + Λ p ) × r = 0 ,
S = μ S Λ p + Γ r .
k = μ k Λ u + Γ K , l = μ l Λ υ + Γ L , m = μ m Λ w + Γ M .
Γ 2 + 2 a Γ + b = 0 ,
b = [ μ 2 1 + Λ 2 2 μ Λ ( k μ + l υ + m w ) ] / ( K 2 + L 2 + M 2 ) .
Γ n + 1 = ( Γ n 2 b ) / 2 ( Γ n + a ) .
Γ 1 = b / 2 a .
Γ 1 = b / 2 a 2 a .
d p = g ( X ) .
q x = ( p × r ) x = L w M υ = 0.
p · r = K u + L υ + M w = 0 ,
p · p = u 2 + υ 2 + w 2 = 1.
u = 1 / [ 1 + K 2 / ( L 2 + M 2 ) ] 1 2 , υ = K L u / ( L 2 + M 2 ) , w = K M u / ( L 2 + M 2 ) .
d = d p / u = g ( X ) / u .
d r = g ( ρ ) ,
ρ = ( X 2 + Y 2 ) 1 2 .
X ( L w M υ ) + Y ( M u K w ) = 0.
u = [ M 2 X + L ( L X K Y ) ] / G , υ = [ M 2 Y K ( L X K Y ) ] / G , w = M ( K X + L Y ) / G ,
G = { ( K 2 + L 2 + M 2 ) [ M 2 ρ 2 + ( L X K Y ) 2 ] } 1 2 .
d = d r / ρ · p ,
d = ρ d r / ( X u + Y υ ) = ρ g ( ρ ) / ( X u + Y υ ) .
( X ¯ Y ¯ Z ¯ ) = R 1 ( X + x ¯ 0 Y + y ¯ 0 Z + z ¯ 0 )
( k ¯ l ¯ m ¯ ) = R 1 ( k l m ) .
R 1 = ( ( cos α cos γ + sin α sin β sin γ ) ( cos α sin γ sin α sin β cos γ ) sin α cos β cos β sin γ cos β cos γ sin β ( sin α cos γ + cos α sin β sin γ ) ( sin α sin γ cos α sin β cos γ ) cos α cos β ) .