Abstract

Generalized ray tracing equations are derived to include special cases in optics such as uncentered spherical interfaces, prisms, mirrors, and centered aspheric surfaces. These equations were constructed for use with the IBM Card Programmed Electronic Calculator. The method has been applied to complicated optical systems involving uncentered lenses in addition to reflecting and refracting prisms. A method is given for the design of an aspheric surface, and a design curve of an aspheric optical system is given as an example.

© 1952 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. A. Allen and R. H. Stark, J. Opt. Soc. Am. 41, 636 (1951).
    [Crossref]
  2. A. E. Conrady, Applied Optics and Optical Design (Oxford University Press, New York, 1929), first edition, Part 1, p. 413.
  3. L. Silberstein, Simplified Method of Tracing Rays Through Any Optical System of Lenses, Prisms, and Mirrors (Longmans, Green and Company, New York, 1918).
  4. D. P. Feder, J. Opt. Soc. Am. 41, 630 (1951).
    [Crossref]
  5. R. Glazebrook, Dictionary of Applied Physics (Macmillan and Company, Ltd., London, 1923), Vol. IV, p. 287.
  6. R. K. Luneburg, Mathematical Theory of Optics (Brown University lecture notes, 1944), p. 410.
  7. W. Wallin, Theory of Optical Image Formation (University of California at Los Angeles lecture notes, U. S. Naval Ordnance Test Station, 1951), p. 17.
  8. D. P. Feder, J. Research Natl. Bur. Standards 45, 61 (1950).
    [Crossref]
  9. Henry Margenau and George Moseley Murphy, The Mathematics of Physics and Chemistry (D. Van Nostrand Company, Inc., New York, 1947), first edition, p. 475.
  10. A. Bouwers, Achievements in Optics (Elsevier Publishing Company, Inc., New York, 1950), second edition, p. 40.
  11. D. D. Maksutov, J. Opt. Soc. Am. 34, 270 (1944).
  12. F. A. Lucy, J. Opt. Soc. Am. 30, 251 (1940).
    [Crossref]
  13. H. S. Friedman, J. Opt. Soc. Am. 37, 480 (1947).
    [Crossref]
  14. William Edmund Milne, Numerical Calculus (Princeton University Press, Princeton, New Jersey, 1949), p. 242.

1951 (2)

1950 (1)

D. P. Feder, J. Research Natl. Bur. Standards 45, 61 (1950).
[Crossref]

1947 (1)

1944 (2)

D. D. Maksutov, J. Opt. Soc. Am. 34, 270 (1944).

R. K. Luneburg, Mathematical Theory of Optics (Brown University lecture notes, 1944), p. 410.

1940 (1)

Allen, W. A.

Bouwers, A.

A. Bouwers, Achievements in Optics (Elsevier Publishing Company, Inc., New York, 1950), second edition, p. 40.

Conrady, A. E.

A. E. Conrady, Applied Optics and Optical Design (Oxford University Press, New York, 1929), first edition, Part 1, p. 413.

Edmund Milne, William

William Edmund Milne, Numerical Calculus (Princeton University Press, Princeton, New Jersey, 1949), p. 242.

Feder, D. P.

D. P. Feder, J. Opt. Soc. Am. 41, 630 (1951).
[Crossref]

D. P. Feder, J. Research Natl. Bur. Standards 45, 61 (1950).
[Crossref]

Friedman, H. S.

Glazebrook, R.

R. Glazebrook, Dictionary of Applied Physics (Macmillan and Company, Ltd., London, 1923), Vol. IV, p. 287.

Lucy, F. A.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (Brown University lecture notes, 1944), p. 410.

Maksutov, D. D.

Margenau, Henry

Henry Margenau and George Moseley Murphy, The Mathematics of Physics and Chemistry (D. Van Nostrand Company, Inc., New York, 1947), first edition, p. 475.

Moseley Murphy, George

Henry Margenau and George Moseley Murphy, The Mathematics of Physics and Chemistry (D. Van Nostrand Company, Inc., New York, 1947), first edition, p. 475.

Silberstein, L.

L. Silberstein, Simplified Method of Tracing Rays Through Any Optical System of Lenses, Prisms, and Mirrors (Longmans, Green and Company, New York, 1918).

Stark, R. H.

Wallin, W.

W. Wallin, Theory of Optical Image Formation (University of California at Los Angeles lecture notes, U. S. Naval Ordnance Test Station, 1951), p. 17.

J. Opt. Soc. Am. (5)

J. Research Natl. Bur. Standards (1)

D. P. Feder, J. Research Natl. Bur. Standards 45, 61 (1950).
[Crossref]

Mathematical Theory of Optics (1)

R. K. Luneburg, Mathematical Theory of Optics (Brown University lecture notes, 1944), p. 410.

Other (7)

W. Wallin, Theory of Optical Image Formation (University of California at Los Angeles lecture notes, U. S. Naval Ordnance Test Station, 1951), p. 17.

Henry Margenau and George Moseley Murphy, The Mathematics of Physics and Chemistry (D. Van Nostrand Company, Inc., New York, 1947), first edition, p. 475.

A. Bouwers, Achievements in Optics (Elsevier Publishing Company, Inc., New York, 1950), second edition, p. 40.

R. Glazebrook, Dictionary of Applied Physics (Macmillan and Company, Ltd., London, 1923), Vol. IV, p. 287.

A. E. Conrady, Applied Optics and Optical Design (Oxford University Press, New York, 1929), first edition, Part 1, p. 413.

L. Silberstein, Simplified Method of Tracing Rays Through Any Optical System of Lenses, Prisms, and Mirrors (Longmans, Green and Company, New York, 1918).

William Edmund Milne, Numerical Calculus (Princeton University Press, Princeton, New Jersey, 1949), p. 242.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Intrinsic geometry of generalized skew ray tracing through an uncentered interface. The third dimension is implied.

Fig. 2a
Fig. 2a

Base lines surveyed through optical system from points O1, O2, respectively. Each successive segment of the base line is formed by dropping a perpendicular to the next interface. 2b. Base line of Fig. 2a forced to become tractable by means of shrewd choice of point O3 and use of fictitious interface I.

Fig. 3a
Fig. 3a

Schematic drawing of an instrument containing an aspheric surface designed by use of a calculator following the method outlined in this paper. This type of optical system is usually referred to as a hybrid Schmidt-Maksutov catadioptric corrector. 3b. Spherical aberration curve of instrument illustrated in Fig. 2a.

Tables (1)

Tables Icon

Table I Iterative scheme for solving Eq. (50).*

Equations (108)

Equations on this page are rendered with MathJax. Learn more.

c R - 1 ,
μ N / N .
m q = t u + P - T ,
P = T + m q - t u ,
P - m q = T - t u .
m = t q · u - q · T ,
P · u = T · u + m q · u - t ,
P 2 = T 2 - m 2 + t 2 - 2 t T · u .
- R = P + n q - u c - 1 .
R 2 c - 2 = P 2 + n 2 + c - 2 - 2 P · u c - 1 - 2 n q · u c - 1 .
c n 2 - 2 q · u n + c P 2 - 2 P · u = 0.
n = c - 1 ( q · u - ξ ) ,
ξ 2 ( q · u ) 2 - c ( c P 2 - 2 P · u ) .
n = ( c P 2 - 2 P · u ) ( q · u + ξ ) - 1 .
l m + n ,
T = T + l q - t u .
R + n q = u c - 1 - P .
q × R = q × S ,
S = u c - 1 - P .
c - 2 - ( q · R ) 2 = S 2 - ( q · S ) 2 .
c 2 ( q · R ) 2 = ( q · u ) 2 - c ( c P 2 - 2 P · u ) .
ξ 2 c 2 ( q · R ) 2 .
ξ 2 c 2 ( q · R ) 2 .
ξ 2 = 1 - μ 2 ( 1 - ξ 2 ) .
N q × R = N q × R .
N | q R q · R R · R | = N | q R q · R R · R | .
μ ( c - 2 q - q · RR ) = c - 2 q - q · RR .
q = μ q + ( c q · R - μ c q · R ) c R .
n c R = u - c T .
q = μ q + g n ,
g = ξ - μ ξ .
T q } initial data             c μ t u } interface data
m = t q · u - q · T
P · u = T · u + m q · u - t
P 2 = T 2 - m 2 + t 2 - 2 t T · u
ξ 2 = ( q · u ) 2 - c ( c P 2 - 2 P · u )
n = ( c P 2 - 2 P · u ) ( q · u + ξ ) - 1
l = m + n
T = T + l q - t u
ξ 2 = 1 - μ 2 ( 1 - ξ 2 )
g = ξ - μ ξ
q = [ μ q + g ( u - c T ) ] N N - 1
c T 2 - 2 T · u = 0
q 2 = 1
L = Σ N l .
T x = + 0.200000000 T y = + 1.500000000 T z = - 0.300000000 q x = + 0.800000000 q y = - 0.200000000 q z = - 0.565685425 } initial data c = + 0.333333333 μ = + 0.657894736 t = + 2.408327000 u x = + 0.924500000 u y = + 0.092450000 u z = - 0.369800956 } interface data m     = + 2.21076341 P · u = + 0.08286371 P 2    = + 1.19965430 ξ 2     = + 0.787407520 n         = + 0.12882345 l            = + 2.33958686 T x = - 0.154828823 T y = + 0.809423796 T z = - 0.732868560 } transfer data ξ 2 = + 0.907984558 g      = + 0.369092857 q x = + 0.886590872 q y = - 0.197041600 q z = - 0.418486837 } transfer data c T 2 - 2 T · u = - 0.000000038 q 2 = + 0.999999999 L = + 2.33958686 } check equations .
U ( S 2 ) Σ i A i S 2 i ,
S 2 T y 2 + T z 2 ,
A 2 i - 1 = B 2 i - 1 2 i - 1 ,
A 2 i = B 2 i B 2 i 2 i - 1 .
V ( S 2 ) c S 2 [ 1 + ( 1 - c 2 S 2 ) 1 2 ] - 1 .
F ( l ) - T x + U ( S 2 ) + V ( S 2 ) = 0 ,
T = T + l q .
F l = F s S l ,
F s = U s + V s - q x S l - 1 ,
U s = Σ i 2 i A i i S 2 i - 1 ,
V s = c S ( 1 - c 2 S 2 ) - 1 2 ,
S l = ( T y q y + T z q z ) S - 1 .
l 1 = l 0 + Δ l 0 ,
Δ l 0 = - F ( l 0 ) / F l ( l 0 ) .
H ( T x , T y , T z ) - T x + U ( S 2 ) + V ( S 2 ) = 0.
n = n 1 i + n 2 j + n 3 k ,
n 1 = H x D - 1 ,
n 2 = H y D - 1 ,
n 3 = H z D - 1 ,
D 2 = H x 2 + H y 2 + H z 2 .
H x = - 1 ,
H y = H s S y ,
H z = H s S z ,
H s = U s + V s .
S y = T y S - 1 ,
S z = T z S - 1 .
ξ = q · n ,
ξ 2 = 1 - μ 2 ( 1 - ξ 2 ) ,
g = ξ - μ ξ ,
q = ( μ q + g n ) N N - 1 ,
T x T y T z q x q y q z } initial data             c μ B 1 B 2 B 3 B 4 B 5 } interface data
l 0 = 0 ( initial guess )
T x = T x + l 0 q x T y = T y + l 0 q y T z = T z + l 0 q z } transfer data
S 2 = T y 2 + T z 2
U ( S 2 ) = B 1 S 2 + B 2 B 2 S 4 + B 3 3 S 6 + B 4 B 4 3 S 8 + B 5 5 S 10 +
V ( S 2 ) = c S 2 [ 1 + ( 1 - c 2 S 2 ) 1 2 ] - 1
F ( l 0 ) = - T x + U ( S 2 ) + V ( S 2 )
S l = ( T y q y + T z q z ) S - 1
U s = 2 B 1 S + 4 B 2 B 2 S 3 + 6 B 3 3 S 5 + 8 B 4 B 4 3 S 7 + 10 B 5 5 S 9 +
V s = c S ( 1 - c 2 S 2 ) - 1 2
F s = U s + V s - q x S l
F l = F s S l
Δ l 0 = - F ( l 0 ) / F l ( l 0 )
l 1 = l 0 + Δ l 0             ( improved root ) .
T x T y T z } transfer data .
S y = T y S - 1
S z = T z S - 1
H s = U s + V s
H x = - 1
H y = H s S y
H z = H s S z
D 2 = H x 2 + H y 2 + H z 2
n 1 = H x D - 1
n 2 = H y D - 1
n 3 = H z D - 1
ξ = q x n 1 + q y n 2 + q z n 3
ξ = [ 1 - μ 2 ( 1 - ξ 2 ) ] 1 2
g = ξ - μ ξ
q x = ( μ q x + g n 1 ) N N - 1 q y = ( μ q y + g n 2 ) N N - 1 q z = ( μ q z + g n 3 ) N N - 1 } transfer data
n 1 2 + n 2 2 + n 3 2 = 1 q x 2 + q y 2 + q z 2 = 1 } check equations .
T x = - 0.069272121 T y = + 0.388515783 T z = + 0.300941051 q x = - 0.690810728 q y = - 0.506323402 q z = - 0.516156130 } initial data c = - 0.562482300 μ = - 1.000000000 B 1 = - 0.01328406 B 2 = + 0.54001919 B 3 = - 2.08368495 B 4 = + 2.69618610 B 5 = - 2.44932430 } interface data .
S y = + 0.791452766 S z = + 0.611230336 H s = - 0.261196474 H x = - 1.000000000 H y = - 0.206724672 H z = - 0.159651209 D 2 = + 1.068223597 n 1     = - 0.967539973 n 2     = - 0.200014382 n 3     = - 0.154468924 ξ         = + 0.849389037 ξ     = + 0.849389036 g     = + 1.698778073 q x = + 0.952824962 q y = - 0.166543356 q z = - 0.253747709 } transfer data n 2 = + 1.000000000 q 2 = + 0.999999997 L = + 0.007149078 } check equations .