Abstract

It is shown that when a lens produces an image of a point source the three-dimensional diffraction pattern which results is the three-dimensional Fourier transform of a generalization of the lens aperture. This implies similar Fourier relations in one and two dimensions. An explicit form of the former is derived which demonstrates that the amplitude distribution on an arbitrarily directed line through the focus is the Fourier transform of the projection of the generalized aperture onto that line. These relations hold in aberrant as well as in ideal systems. Some examples are worked out by using the one-dimensional relation.

© 1964 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd., Oxford, England, 1959).
  2. I. N. Sneddon, Fourier Transforms (McGraw-Hill Book Company, Inc., New York, 1951).
  3. D. Gabor, Progress in Optics, edited by E. Wolf (North-Holland Publishing Company, Amsterdam, The Netherlands, 1961), p. 117.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd., Oxford, England, 1959).

Gabor, D.

D. Gabor, Progress in Optics, edited by E. Wolf (North-Holland Publishing Company, Amsterdam, The Netherlands, 1961), p. 117.

Sneddon, I. N.

I. N. Sneddon, Fourier Transforms (McGraw-Hill Book Company, Inc., New York, 1951).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd., Oxford, England, 1959).

Other (3)

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd., Oxford, England, 1959).

I. N. Sneddon, Fourier Transforms (McGraw-Hill Book Company, Inc., New York, 1951).

D. Gabor, Progress in Optics, edited by E. Wolf (North-Holland Publishing Company, Amsterdam, The Netherlands, 1961), p. 117.

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Figures (6)

Fig. 1
Fig. 1

Spherical converging wave: notation.

Fig. 2
Fig. 2

Aberrant converging wave.

Fig. 3
Fig. 3

The l,ϕ coordinate system, showing the Fourier-related aperture distribution B(l)r and image-amplitude distribution U(L)r. The figure is oversimplified in that in general U(L)r is complex, and so is B(l)r if the system is aberrant.

Fig. 4
Fig. 4

Circular aperture, showing the aperture and image-amplitude distributions on a line in the focal plane. The disturbance is in phase over the entire focal plane, and therefore all along the r axis. In this special case the amplitude can be displayed on a two-dimensional graph.

Fig. 5
Fig. 5

Annular aperture, showing aperture and image-intensity distributions along the lens axis. Here there are phase differences along the r axis; hence, we plot intensity rather than amplitude.

Fig. 6
Fig. 6

Gaussian aperture, showing aperture and image-intensity distributions along the lens axis. As in Fig. 5 we plot intensity rather than image amplitude because the latter has phase differences along the r axis.

Equations (29)

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U ( P ) = - i λ A e - i k f f W e i k s s d S ,
U ( P ) = - i λ A Ω W e - i k q · R d Ω .
U ( P ) = - i λ Ω A ( q ) e - i k q · R d Ω ,
F ( R ) = ( 2 π ) - 3 / 2 All Q space A ( Q ) exp ( - i k Q · R ) d V Q ,
F ( R ) = ( 2 π ) - 3 / 2 Q = 0 Q = Ω A ( Q ) exp ( - i k Q · R Q 2 d Ω d Q .
F ( R ) = ( 2 π ) - 3 / 2 Ω A ( q ) e - i k q · R d Ω ,
U ( P ) = - i ( 2 π ) 3 2 F ( R ) / λ ,
s - f - Φ ( q ) - q · R ,
U ( P ) = - i λ Ω A ( q ) e i k [ Φ ( q ) - q · R ] d Ω ,
U ( P ) = - i λ 0 2 π 0 π A ( q ) e - i k q · R sin θ d θ d ϕ .
U ( L ) r = - i λ - 1 1 0 2 π A ( l , ϕ ) r e - i k L l d ϕ d l .
U ( L ) r = - i λ - 1 1 e - i k L l 0 2 π A ( l , ϕ ) r d ϕ d l ,
U ( L ) r = - i λ - 1 1 B ( l ) r e - i k L l d l
B ( l ) r = 0 2 π A ( l , ϕ ) r d ϕ .
F ( L ) = ( 2 π ) - 1 2 - B ( L ) e - i k L l d l ,
U ( L ) r = - i ( 2 π ) 1 2 F ( L ) r / λ ,
U ( L ) = - i k A 0 X 2 [ J 1 ( k X L ) / k X L ] ,
B ( l ) = 2 A 0 ( X 2 - l 2 ) , - X < l < X B ( l ) = 0 l < - X ,             X < l ,
cos Y - cos X = a ,             and             1 2 ( cos Y + cos X ) = z ,
U ( L ) = - i ( 2 A 0 / L ) e - i k L z sin ( k L a / 2 ) ,
I ( L ) = [ ( 2 A 0 / L ) sin ( k L a / 2 ) ] 2 .
I ( L ) = [ k A 0 X 2 2 sin ( k L X 2 / 4 ) k L X 2 / 4 ] 2 .
T ( q ) = T ( G ) = e - α sin 2 G ,
A ( q ) = A 0 e - α sin 2 G ,
B ( l ) = 2 π A 0 e - α sin 2 θ = 2 π A 0 e - α ( 1 - cos 2 θ ) = 2 π A 0 e - α ( 1 - l 2 ) ,
B ( l ) = 2 π A 0 e - 2 α ( 1 - l ) .
U ( L ) = - i k A 0 [ e - i k L / ( i k L - 2 α ) ] ,
I ( L ) = ( k A 0 ) 2 [ 1 / ( k 2 L 2 + 4 α 2 ) ] .
I ( L ) = ( k A 0 / 2 α ) 2 e - k 2 L 2 / 2 π α .