Abstract

The development of electronic computers has brought greater possibilities in evaluation methods for lens systems, and the theory of transfer functions is now being applied to practical lens design. In this paper, the evaluation by spot diagrams is discussed. A method of plotting spot diagrams is proposed, utilizing the line printer of a computer as the X-Y plotter. The geometrical optical intensity distribution Ig (x,y) given by the spot diagram is expressed in the form

Ig(x,y)=N-1i=1Nδ(x-xi,y-yi).

The convolution t (x,y) with the turbidity r (x,y) of the image receiver, the total impulse response, is given by

t(x,y)=N-1i=1Nr(x-xi,y-yi).

Intensity distributions calculated by the computer are compared with the measured ones for an actual lens, and the agreement is satisfactory. This procedure can also be applied to the case of the transfer function. The Strehl definition t (0,0) is calculated for several cases of primary and secondary spherical aberrations, and it is confirmed that there are two extremum positions of the spot diagram in some cases.

© 1963 Optical Society of America

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References

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  1. G. Black, Proc. Phys. Soc. B68, 729 (1955).
  2. D. P. Feder, J. Opt. Soc. Am. 47, 902 (1957); J. Meiron, H. M. Loebenstein, J. Opt. Soc. Am. 47, 1104 (1957).
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    [CrossRef]
  4. M. Herzberger, J. Opt. Soc. Am. 37, 485 (1947).
    [CrossRef] [PubMed]
  5. R. E. Hopkins, Report Inst. Optics, Univ. Rochester (1955).
  6. T. Suzuki et al., Program of 1963Spring Meeting of Society of Applied Physics, Japan, p. 65.
  7. K. Miyamoto, J. Appl. Phys. Japan 26, 421 (1957); K. Miyamoto, J. Opt. Soc. Am. 48, 57 (1958).
    [CrossRef]
  8. H. H. Hopkins, Proc. Phys. Soc. B70, 1162 (1957).
  9. K. Miyamoto, J. Opt. Soc. Am. 48, 567 (1957).
    [CrossRef]
  10. H. Kubota, K. Miyamoto, K. Murata, Optik 17, 143 (1960).
  11. E. H. Linfoot, Recent Advances in Optics (Oxford Univ. Press, New York, 1955).
  12. K. Sayanagi, Proceedings of the Conference on Optical Instruments and Techniques (Chapman and Hall Ltd., London, 1962), p. 95.
  13. F. A. Lucy, J. Opt. Soc. Am. 46, 699 (1956).
    [CrossRef]
  14. R. E. Hopkins, Proceedings of the Conference on Optical Instruments and Techniques (Chapman and Hall Ltd., London, 1962), p. 65.
  15. P. B. Fellget, E. H. Linfoot, Trans. Roy. Soc. A247, 367 (1955).
  16. E. H. Linfoot, J. Opt. Soc. Am. 46, 740 (1956); E. H. Linfoot, Opt. Acta 5, 1 (1958).
    [CrossRef]
  17. G. Kuwabara, J. Opt. Soc. Am. 45, 309, 625 (1955).
    [CrossRef]

1960 (1)

H. Kubota, K. Miyamoto, K. Murata, Optik 17, 143 (1960).

1957 (4)

D. P. Feder, J. Opt. Soc. Am. 47, 902 (1957); J. Meiron, H. M. Loebenstein, J. Opt. Soc. Am. 47, 1104 (1957).
[CrossRef]

K. Miyamoto, J. Appl. Phys. Japan 26, 421 (1957); K. Miyamoto, J. Opt. Soc. Am. 48, 57 (1958).
[CrossRef]

H. H. Hopkins, Proc. Phys. Soc. B70, 1162 (1957).

K. Miyamoto, J. Opt. Soc. Am. 48, 567 (1957).
[CrossRef]

1956 (2)

1955 (3)

G. Kuwabara, J. Opt. Soc. Am. 45, 309, 625 (1955).
[CrossRef]

P. B. Fellget, E. H. Linfoot, Trans. Roy. Soc. A247, 367 (1955).

G. Black, Proc. Phys. Soc. B68, 729 (1955).

1954 (1)

1947 (1)

Black, G.

G. Black, Proc. Phys. Soc. B68, 729 (1955).

Eldert, C.

Feder, D. P.

Fellget, P. B.

P. B. Fellget, E. H. Linfoot, Trans. Roy. Soc. A247, 367 (1955).

Herzberger, M.

Hopkins, H. H.

H. H. Hopkins, Proc. Phys. Soc. B70, 1162 (1957).

Hopkins, R. E.

R. E. Hopkins, Report Inst. Optics, Univ. Rochester (1955).

R. E. Hopkins, Proceedings of the Conference on Optical Instruments and Techniques (Chapman and Hall Ltd., London, 1962), p. 65.

Kubota, H.

H. Kubota, K. Miyamoto, K. Murata, Optik 17, 143 (1960).

Kuwabara, G.

G. Kuwabara, J. Opt. Soc. Am. 45, 309, 625 (1955).
[CrossRef]

Linfoot, E. H.

E. H. Linfoot, J. Opt. Soc. Am. 46, 740 (1956); E. H. Linfoot, Opt. Acta 5, 1 (1958).
[CrossRef]

P. B. Fellget, E. H. Linfoot, Trans. Roy. Soc. A247, 367 (1955).

E. H. Linfoot, Recent Advances in Optics (Oxford Univ. Press, New York, 1955).

Lucy, F. A.

Miyamoto, K.

H. Kubota, K. Miyamoto, K. Murata, Optik 17, 143 (1960).

K. Miyamoto, J. Appl. Phys. Japan 26, 421 (1957); K. Miyamoto, J. Opt. Soc. Am. 48, 57 (1958).
[CrossRef]

K. Miyamoto, J. Opt. Soc. Am. 48, 567 (1957).
[CrossRef]

Murata, K.

H. Kubota, K. Miyamoto, K. Murata, Optik 17, 143 (1960).

Rosen, S.

Sayanagi, K.

K. Sayanagi, Proceedings of the Conference on Optical Instruments and Techniques (Chapman and Hall Ltd., London, 1962), p. 95.

Suzuki, T.

T. Suzuki et al., Program of 1963Spring Meeting of Society of Applied Physics, Japan, p. 65.

J. Appl. Phys. Japan (1)

K. Miyamoto, J. Appl. Phys. Japan 26, 421 (1957); K. Miyamoto, J. Opt. Soc. Am. 48, 57 (1958).
[CrossRef]

J. Opt. Soc. Am. (7)

Optik (1)

H. Kubota, K. Miyamoto, K. Murata, Optik 17, 143 (1960).

Proc. Phys. Soc. (2)

G. Black, Proc. Phys. Soc. B68, 729 (1955).

H. H. Hopkins, Proc. Phys. Soc. B70, 1162 (1957).

Trans. Roy. Soc. (1)

P. B. Fellget, E. H. Linfoot, Trans. Roy. Soc. A247, 367 (1955).

Other (5)

R. E. Hopkins, Proceedings of the Conference on Optical Instruments and Techniques (Chapman and Hall Ltd., London, 1962), p. 65.

E. H. Linfoot, Recent Advances in Optics (Oxford Univ. Press, New York, 1955).

K. Sayanagi, Proceedings of the Conference on Optical Instruments and Techniques (Chapman and Hall Ltd., London, 1962), p. 95.

R. E. Hopkins, Report Inst. Optics, Univ. Rochester (1955).

T. Suzuki et al., Program of 1963Spring Meeting of Society of Applied Physics, Japan, p. 65.

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Figures (6)

Fig. 1
Fig. 1

The flow diagram for plotting the spot diagram.

Fig. 2
Fig. 2

Spot diagram plotted by line printer.

Fig. 3
Fig. 3

Intensity distribution of line image on axis of an actual lens (Gauss type). Full lines are values calculated by the formula (2) with r (x,y) = (2 π)−1/2σ−1 exp (−x2/2σ2), σ being 0.0035. I notations are measured values. x notations are the convolutions of measured intensity and r (x,y). Figs. 3 (a), (b), and (c) are for the defocused image planes shifting −0.14 mm and 0.12 mm and −0.1 mm from the Gauss plane, respectively. Minus sign corresponds to the direction toward the lens.

Fig. 4
Fig. 4

Transfer function of an actual lens (Sonnar type) vs the spatial frequency variable s (line/mm). Full lines are values calculated by the formula (3) and the circles are measured values. Δ in the figures is the distance from Gauss image plane (Kubota, Miyamoto, and Murata).10

Fig. 5
Fig. 5

The primary and secondary longitudinal spherical aberration S = 4S1ρ2(ρ2ρ02), which corresponds to Eq. (9) for the lateral spherical aberration.

Fig. 6
Fig. 6

Strehl definition of lens with the spherical aberration given by Fig. 5 vs position of image plane on axis. a and b are the Strehl definition of the point image and the line image, respectively.

Equations (12)

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I g ( x , y ) = N - 1 i = 1 N δ ( x - x i , y - y i ) .
t ( x , y ) = N - 1 i = 1 N r ( x - x i , y - y i ) .
I g ( x , y ) = N - 1 i = 1 N δ ( x - x i , y - y i ) ,
t ( x , y ) = I 0 ( x , y ) r ( x - x , y - y ) d x d y = N - 1 i = 1 N r ( x - x i , y - y i ) .
R g ( s , t ) = I g ( x , y ) exp [ - 2 π i ( s x + t y ) ] d x d y = N - 1 k = 1 N exp [ - 2 π i ( s x k + t y k ) ] .
R g ( s , t ) = exp { - 2 π i [ s · x ( u , v ) + t · y ( u , v ) ] } d u d v ,
x = f ( / u ) ϕ ( u , v ) ,             y = f ( / v ) ϕ ( u , v ) ,
R . G . = N - 1 i = 1 N ( x i 2 + y i 2 ) .
I . C . = N - 1 i [ ( x i 2 + y i 2 ) 1 / 2 + Δ r ] 1 / 2 ,
relative structural content : T τ τ 1 2 0 2 ¯ d u d v ; fidelity : ϕ [ 1 - 1 - τ τ 1 2 ] 0 2 ¯ d u d v ; correlation quality : Q τ τ 1 0 2 ¯ d u d v ;
t ( 0 , 0 ) = N - 1 i = 1 N r ( - x i , - y i ) .
l = ( S 1 / 2 F ) [ 4 ρ 2 ( ρ 2 - ρ 0 2 ) - a ] ρ ,             p 1

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