Abstract

The method of obtaining the sine-wave response of a lens, both theoretically and experimentally, is described, and it is shown that the line spread-function can be derived from the sine-wave response alone if the spread function is symmetrical. If the spread-function is not symmetrical, it can be computed from the sine-wave response only when the phase function is also known, and an experimental method of obtaining this function is described. The mathematical procedure for the reciprocal inversion of the spread-function and the sine-wave response is confirmed experimentally.

© 1958 Optical Society of America

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References

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  1. Lamberts, Higgins, and Wolfe, J. Opt. Soc. Am. 48, 487 (1958).
    [CrossRef]
  2. P. Lindberg, Optica Acta 1, 80 (1954).
    [CrossRef]
  3. Ingelstam, Djurle, and Sjögren, J. Opt. Soc. Am. 46, 707 (1956).
    [CrossRef]
  4. R. V. Churchill, Fourier Series and Boundary Value Problems (McGraw-Hill Book Company, Inc., New York, 1941), pp. 88–92.

1958 (1)

1956 (1)

1954 (1)

P. Lindberg, Optica Acta 1, 80 (1954).
[CrossRef]

J. Opt. Soc. Am. (2)

Optica Acta (1)

P. Lindberg, Optica Acta 1, 80 (1954).
[CrossRef]

Other (1)

R. V. Churchill, Fourier Series and Boundary Value Problems (McGraw-Hill Book Company, Inc., New York, 1941), pp. 88–92.

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Figures (13)

Fig. 1
Fig. 1

Schematic view of apparatus for measuring the sine-wave response of a lens. TO, sinusoidal test object; L, lens under test; SS, scanning slit; P, phototube; R, recorder.

Fig. 2
Fig. 2

Type of test object used for determining the sine-wave response. The mean height of the pattern is b0 and the amplitude of the variation is b1. Test object moves in the direction ξ.

Fig. 3
Fig. 3

Lens bench for the photoelectric scanning of the aerial image formed by a lens. TO, test object (illuminated slit, edge, or other appropriate pattern); SS, scanning slit; P, multiplier phototube and recorder. The lens L under test is mounted on the tangent bar T and the test object and the scanning slit are rotated simultaneously with the lens so that the entire field of the lens can be explored.

Fig. 4
Fig. 4

Graphical combination of sine and cosine Fourier transforms A#c and A#s to give the sine-wave response |A#|.

Fig. 5
Fig. 5

A, spread-function of a certain lens as measured; B, spread-function as deduced from the sine-wave response shown in Fig. 6, the phase angle ϕ being assumed to be the same for all frequencies.

Fig. 6
Fig. 6

Sine-wave response (“amplitude”) of lens whose spread function is shown at A in Fig. 5 deduced from this spread-function. Measured values are shown by circles. The phase curve was deduced from measurements as described in text.

Fig. 7
Fig. 7

Apparatus for measuring cosine Fourier transform. The lens L under test images slit S on test object TO, which consists of a pair of identical sinusoidal patterns moving in opposite directions.

Fig. 8
Fig. 8

Diagram showing action of double sinusoidal pattern. The two patterns are in phase at A and E and are at intermediate relative phases at B, C, and D. S1 is half-way between reference ends of the patterns and S2 is λ/4 away.

Fig. 9
Fig. 9

Curves of phase angle ϕ as a function of frequency ν for various positions P0P3 of the scanning slit. Curve P0 corresponds to the position S1 in Fig. 8 half-way between reference ends of the sinusoidal patterns.

Fig. 10
Fig. 10

Diagram to show how an even component A1(x) can be formed by adding a function A(x) to its mirror image A(−x) and an odd component A2(x) by adding a function A(x) to its negative mirror image −A(−x).

Fig. 11
Fig. 11

Sine-wave response |A#| (left) and Fourier transforms of it A#c and A#s (right) for a certain lens. A single sinusoidal test object was used to obtain |A#| and a double test object for A#c; A#s was computed from the other two.

Fig. 12
Fig. 12

Phase angle as a function of frequency for the lens of Fig. 11. The curve represents the mean of the two determinations ● and ▲.

Fig. 13
Fig. 13

Upper graph, double cosine transform A#c# and double sine transform A#s# of sine-wave response of lens of Fig. 11. Lower graph, sine-wave response of lens as calculated from A#c, A#s, and ϕ (broken line) (sum of A#c# and A#s#) and as measured directly (solid line).

Equations (25)

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G ( ξ ) = b 0 + b 1 cos 2 π ν ξ ,
F ( x ) = - A ( ξ ) G ( x - ξ ) d ξ .
F ( x ) = b 0 - A ( ξ ) d ξ + b 1 - A ( ξ ) cos 2 π ν ( x - ξ ) d ξ .
F ( x ) = b 0 + b 1 A # c cos 2 π ν x + b 1 A # s sin 2 π ν x .
A # c ( ν ) = - A ( ξ ) cos 2 π ν ξ d ξ ,
A # s ( ν ) = - A ( ξ ) sin 2 π ν ξ d ξ ,
A # = ( A # c 2 + A # s 2 ) 1 2 .
A # c = A # cos ϕ
A # s = A # sin ϕ ,
F ( x ) = b 0 + b 1 A # cos ( 2 π ν x - ϕ ) .
F ( x ) + F ( - x ) = 2 b 0 + 2 b 1 A # c cos 2 π ν x .
Δ ϕ = k Δ x / λ = k ν Δ x .
A 1 ( x ) = A 1 ( - x ) .
A 1 ( x ) = 1 2 [ A ( x ) + A ( - x ) ] .
A 2 ( x ) = - A 2 ( - x ) .
A 2 ( x ) = 1 2 [ A ( x ) - A ( - x ) ] .
A ( x ) = A 1 ( x ) + A 2 ( x ) .
A 1 ( x ) = 2 0 cos 2 π ν x - A 1 ( ξ ) cos 2 π ν ξ d ξ d ν = 2 0 ( A 1 ) # c cos 2 π ν x d ν .
( A 1 ) # c = A # c .
A 1 ( x ) = 2 0 A # c cos 2 π ν x d ν .
A 1 ( x ) = A # c # .
A 2 ( x ) = 2 0 sin 2 π ν x - A ( ξ ) sin 2 π ν ξ d ξ d ν = 2 0 ( A 2 ) # s sin 2 π ν x d ν .
( A 2 ) # s = A # s .
A 2 ( x ) = 2 0 A # s sin 2 π ν x d ν = A # s # ,
A ( x ) = A # c # + A # s # .