Abstract

A ray trace scheme for the automatic generation of optical aberration polynomials to arbitrary orders pioneered by T. B. Andersen is successfully applied to the diffraction analysis of rotationally symmetric optical systems on a desk-top computer. The diffraction-based optical transfer functions at various field positions are computed using the relatively new Winograd Fourier transform algorithm. The coding includes aspherical and reflective surfaces.

© 1983 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Heshmaty-Manesh, Ph.D. Thesis, U. London (1978).
  2. T. B. Andersen, Appl. Opt. 19, 3800 (1980).
    [CrossRef] [PubMed]
  3. T. B. Andersen, Appl. Opt. 20, 2754 (1981).
    [CrossRef] [PubMed]
  4. T. B. Andersen, Appl. Opt. 20, 3263 (1981).
    [CrossRef] [PubMed]
  5. T. B. Andersen, Appl. Opt. 20, 3723 (1981).
    [CrossRef] [PubMed]
  6. T. B. Andersen, Appl. Opt. 21, 1817 (1982).
    [CrossRef] [PubMed]
  7. M. Herzberger, J. Opt. Soc. Am. 37, 485 (1947).
    [CrossRef] [PubMed]
  8. W. T. Welford, Aberrations of the Symmetrical Optical Systems (Academic, New York, 1974).
  9. S. Winograd, Math. Comp. 32, 175 (1978).
    [CrossRef]
  10. E. O. Brigham, Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).
  11. J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965).
    [CrossRef]
  12. D. Heshmaty-Manesh, S. C. Tam, Appl. Opt. 21, 3273 (1982).
    [CrossRef] [PubMed]
  13. D. Heshmaty-Manesh, S. C. Tam, Proc. Soc. Photo-Opt. Instrum. Eng. 369, (1983), Article 13.
  14. Y. Doi, Y. Sakai, K. Sado, U.S. Pat.3,961,845, 8June1976.
  15. R. Kingslake, Appl. Opt. 15, 2948 (1976).
  16. M. J. Kidger, Opt. Acta 25, 665 (1978).
    [CrossRef]
  17. C. F. Gerald, Applied Numerical Analysis (Addison-Wesley, Reading, Mass., 1978).

1983 (1)

D. Heshmaty-Manesh, S. C. Tam, Proc. Soc. Photo-Opt. Instrum. Eng. 369, (1983), Article 13.

1982 (2)

1981 (3)

1980 (1)

1978 (2)

M. J. Kidger, Opt. Acta 25, 665 (1978).
[CrossRef]

S. Winograd, Math. Comp. 32, 175 (1978).
[CrossRef]

1976 (1)

R. Kingslake, Appl. Opt. 15, 2948 (1976).

1965 (1)

J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965).
[CrossRef]

1947 (1)

Andersen, T. B.

Brigham, E. O.

E. O. Brigham, Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Cooley, J. W.

J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965).
[CrossRef]

Doi, Y.

Y. Doi, Y. Sakai, K. Sado, U.S. Pat.3,961,845, 8June1976.

Gerald, C. F.

C. F. Gerald, Applied Numerical Analysis (Addison-Wesley, Reading, Mass., 1978).

Herzberger, M.

Heshmaty-Manesh, D.

D. Heshmaty-Manesh, S. C. Tam, Proc. Soc. Photo-Opt. Instrum. Eng. 369, (1983), Article 13.

D. Heshmaty-Manesh, S. C. Tam, Appl. Opt. 21, 3273 (1982).
[CrossRef] [PubMed]

D. Heshmaty-Manesh, Ph.D. Thesis, U. London (1978).

Kidger, M. J.

M. J. Kidger, Opt. Acta 25, 665 (1978).
[CrossRef]

Kingslake, R.

R. Kingslake, Appl. Opt. 15, 2948 (1976).

Sado, K.

Y. Doi, Y. Sakai, K. Sado, U.S. Pat.3,961,845, 8June1976.

Sakai, Y.

Y. Doi, Y. Sakai, K. Sado, U.S. Pat.3,961,845, 8June1976.

Tam, S. C.

D. Heshmaty-Manesh, S. C. Tam, Proc. Soc. Photo-Opt. Instrum. Eng. 369, (1983), Article 13.

D. Heshmaty-Manesh, S. C. Tam, Appl. Opt. 21, 3273 (1982).
[CrossRef] [PubMed]

Tukey, J. W.

J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965).
[CrossRef]

Welford, W. T.

W. T. Welford, Aberrations of the Symmetrical Optical Systems (Academic, New York, 1974).

Winograd, S.

S. Winograd, Math. Comp. 32, 175 (1978).
[CrossRef]

Appl. Opt. (7)

J. Opt. Soc. Am. (1)

Math. Comp. (2)

J. W. Cooley, J. W. Tukey, Math. Comp. 19, 297 (1965).
[CrossRef]

S. Winograd, Math. Comp. 32, 175 (1978).
[CrossRef]

Opt. Acta (1)

M. J. Kidger, Opt. Acta 25, 665 (1978).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

D. Heshmaty-Manesh, S. C. Tam, Proc. Soc. Photo-Opt. Instrum. Eng. 369, (1983), Article 13.

Other (5)

Y. Doi, Y. Sakai, K. Sado, U.S. Pat.3,961,845, 8June1976.

C. F. Gerald, Applied Numerical Analysis (Addison-Wesley, Reading, Mass., 1978).

E. O. Brigham, Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).

D. Heshmaty-Manesh, Ph.D. Thesis, U. London (1978).

W. T. Welford, Aberrations of the Symmetrical Optical Systems (Academic, New York, 1974).

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 (4)

Fig. 1
Fig. 1

Flow chart for OTF computation using Andersen’s technique and the WFTA.

Fig. 2
Fig. 2

Scheme for data setup prior to 2-D DFT by the WFTA.

Fig. 3
Fig. 3

Lens drawing showing the zoom lens in Ref. 14 at its extreme positions.

Fig. 4
Fig. 4

Pupil map in the form of Twyman-Green fringes and MTF curves.

Tables (4)

Tables Icon

Table I Lens Data for the Zoom Lens in Ref. 14 at Its Mean Position

Tables Icon

Table II Coefficients of Generated Polynomials Sp, Tp, Vp, Wp, Kp (Partial Output up to 9.° Order)

Tables Icon

Table III Comparison of Ray Trace Results

Tables Icon

Table IV MTF Values for the Mean Zoom Position

Equations (30)

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

ρ i = x i 2 + y i 2 , ψ i = ξ i 2 + η i 2 , κ i = x i ξ i + y i η i .
f i ( ρ ) = j = 1 a i j ρ j ,
[ x ˜ i y ˜ i ] = [ x i y i ] + f i ( x ˜ i , y ˜ i ) [ ξ i η i ] ,
ρ ˜ i = ρ i + ψ i f i ( ρ ˜ i ) 2 + 2 κ i f i ( ρ ˜ i ) .
α ˆ i = γ i ( f i x , f i y , 1 ) ,
γ i = [ 1 + ( f i x ) 2 + ( f i y ) 2 ] 1 / 2 = [ 1 + 4 ρ ˜ i ( d f i d ρ ) 2 ] 1 / 2 .
N i = ( 1 + ψ i ) 1 / 2 ,
cos θ i = γ i N i { 1 2 [ κ i + ψ i f i ( ρ ˜ i ) ] d f i d ρ } .
cos θ i = ( 1 μ i 2 + μ i 2 cos 2 θ i ) 1 / 2 ,
N i + 1 = μ i N i + γ i ( cos θ i μ i cos θ i ) ,
[ ξ i + 1 η i + 1 ] = β i [ x i y i ] + ( μ i N i N i + 1 β i f i ) [ ξ i η i ] ,
β i = 2 γ i N i + 1 ( cos θ i μ i cos θ i ) d f i d ρ .
[ x i + 1 y i + 1 ] = [ x ˜ i y ˜ i ] + [ d i f i ( x ˜ i , y ˜ i ) ] [ ξ i + 1 η i + 1 ] = [ 1 β i ( d i f i ) ] [ x i y i ] + { [ f i + ( d i f i ) ( μ i N i N i + 1 β i f i ) ] } [ ξ i η i ] .
[ x i + 1 y i + 1 ξ i + 1 η i + 1 ] = A i + 1 [ x i y i ξ i η i ] ,
[ x p y p ξ p η p ] = [ S p ( ρ o , ψ o , κ o ) T p ( ρ o , ψ o , κ o ) V p ( ρ o , ψ o , κ o ) W p ( ρ 0 , ψ o , κ o ) ] [ x o y o ξ o η o ] .
S p = S p + V p Δ , T p = T p + W p Δ , V p = V p , W p = W p .
K p = i = 0 p d i + ( μ i N i + 1 N i 1 ) f i ( ρ ˜ i ) μ o μ 1 μ i N i + 1 .
MAT A = B + C , MAT B = A * ( λ ) , and a = DOT ( A , B ) .
A ( u , υ ) = F ( x , y ) · exp [ i 2 π ( u x + υ y ) ] d x d y ,
F ( x , y ) = exp [ i ϕ ( x , y ) ] inside the pupil = 0 outside the pupil
I ( u , υ ) = | A ( u , υ ) | 2 .
D ( s , t ) = I ( u , υ ) · exp [ i 2 π ( u s + υ t ) ] d u d υ I ( u , υ ) d u d υ .
A ( l N T 1 , m N T 1 ) = k = 0 N 1 m = 0 N 1 F ( k T 1 , m T 1 ) exp [ i 2 π ( k l + m n ) / N ] ,
D ( l N T 2 , m N T 2 ) = k = 0 N 1 m = 0 N 1 I ( k T 2 , m T 2 ) exp [ i 2 π ( k l + m n ) / N ] ,
N 1 { 1,2,4,8,16 } , N 2 { 1,3,9 } , N 3 { 1,5 } , and N 4 { 1,7 } ,
A k = j = 0 N 1 w k j · a j ,
A = w · a .
[ A 0 A 1 A 3 A 4 A 2 ] = [ 1 1 1 1 1 1 w 1 w 3 w 4 w 2 1 w 3 w 4 w 2 w 1 1 w 4 w 2 w 1 w 3 1 w 2 w 1 w 3 w 4 ] · [ a 0 a 1 a 3 a 4 a 2 ] ,
s 1 = a 1 + a 4 , s 2 = a 1 a 4 , s 3 = a 3 + a 2 , s 4 = a 3 a 2 , s 5 = s 1 + s 3 , s 6 = s 1 s 3 , s 7 = s 2 + s 4 , s 8 = s 5 + a 0 , u = 2 π / 5 , m 0 = s 8 , m 1 = s 5. ( cos u + cos 2 u 2 ) / 2 , m 2 = s 6. ( cos u cos 2 u ) / 2 , m 3 = i . s 2. ( sin u + sin 2 u ) , m 4 = i . s 7. sin 2 u , m 5 = i . s 4. ( sin u sin 2 u ) , s 9 = m 0 + m 1 , s 10 = s 9 + m 2 , s 11 = s 9 m 2 , s 12 = m 3 m 4 , s 13 = m 4 + m 5 , s 14 = s 10 + s 12 , s 15 = s 10 s 12 , s 16 = s 11 + s 13 , s 17 = s 11 s 13.
A 0 = m 0 , A 1 = s 14 , A 2 = s 16 , A 3 = s 17 , A 4 = s 15.

Metrics