Abstract

A one-dimensional function may be represented spatially as an aperture in an opaque screen where the aperture’s width is proportional to the function. When such an area-modulated screen is the input to a standard coherent optical fourier transformer, the amplitude of the light along one axis on the output plane is proportional to the fourier transform of the original function. Continuous and sampled area modulation for single and multichannel operation are discussed. Experimental agreement with the expected results is obtained. Hybrid configurations combining area and optical-density modulation are discussed.

© 1971 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  2. T. S. Gray, J. Franklin Inst. 212, 77 (1931).
    [CrossRef]
  3. H. C. Montgomery, Bell Syst. Tech. J. 17, 406 (1938).
  4. J. F. Schouten, Philips Tech. Rev. 3, 298 (1938).
  5. R. J. Drexler, Some General Comments on Optical Correlation Techniques, TIS Rept. R65ELS-86 (General Electric, 1965).
  6. P. Jacquinot, B. Roizen-Dossier, “Apodisation,” in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1964), Vol. 3, pp. 41ff.
    [CrossRef]
  7. A. W. Lohmann, D. P. Paris, Appl. Opt. 7, 651 (1968).
    [CrossRef] [PubMed]

1968

1938

H. C. Montgomery, Bell Syst. Tech. J. 17, 406 (1938).

J. F. Schouten, Philips Tech. Rev. 3, 298 (1938).

1931

T. S. Gray, J. Franklin Inst. 212, 77 (1931).
[CrossRef]

Drexler, R. J.

R. J. Drexler, Some General Comments on Optical Correlation Techniques, TIS Rept. R65ELS-86 (General Electric, 1965).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Gray, T. S.

T. S. Gray, J. Franklin Inst. 212, 77 (1931).
[CrossRef]

Jacquinot, P.

P. Jacquinot, B. Roizen-Dossier, “Apodisation,” in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1964), Vol. 3, pp. 41ff.
[CrossRef]

Lohmann, A. W.

Montgomery, H. C.

H. C. Montgomery, Bell Syst. Tech. J. 17, 406 (1938).

Paris, D. P.

Roizen-Dossier, B.

P. Jacquinot, B. Roizen-Dossier, “Apodisation,” in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1964), Vol. 3, pp. 41ff.
[CrossRef]

Schouten, J. F.

J. F. Schouten, Philips Tech. Rev. 3, 298 (1938).

Appl. Opt.

Bell Syst. Tech. J.

H. C. Montgomery, Bell Syst. Tech. J. 17, 406 (1938).

J. Franklin Inst.

T. S. Gray, J. Franklin Inst. 212, 77 (1931).
[CrossRef]

Philips Tech. Rev.

J. F. Schouten, Philips Tech. Rev. 3, 298 (1938).

Other

R. J. Drexler, Some General Comments on Optical Correlation Techniques, TIS Rept. R65ELS-86 (General Electric, 1965).

P. Jacquinot, B. Roizen-Dossier, “Apodisation,” in Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1964), Vol. 3, pp. 41ff.
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

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Figures (12)

Fig. 1
Fig. 1

Representation of transparencies for four types of continuous area modulation.

Fig. 2
Fig. 2

Optical system for obtaining the fourier transform of area-modulated signals.

Fig. 3
Fig. 3

Representations of (b) variable height and (c) variable diameter sampled area modulated where (a) is the given signal being simulated.

Fig. 4
Fig. 4

Optical system for fourier transformation of multichannel area-modulated signals.

Fig. 5
Fig. 5

Configuration for the experiments.

Fig. 6
Fig. 6

Photometric scan along f y = 0 of spectrum of the variable-height sampled area-modulation signal.

Fig. 7
Fig. 7

Photographs of the output spectrum for a 22.2 cycle/cm sine wave written as a biased bilateral area modulation where (b) is an enlargement of (a).

Fig. 8
Fig. 8

Photometric scan along f y = 0 of spectrum of 22.2 cycle/cm sine wave written as a biased bilateral area modulation.

Fig. 9
Fig. 9

Photograph of the output spectrum or a 59.4 cycle/cm sine wave written as an unbiased area modulation with a phase plate.

Fig. 10
Fig. 10

Photometric scan along f y = 0 of spectrum of 59.4 cycle/cm sine wave written as an unbiased area modulation with a phase plate.

Fig. 11
Fig. 11

Photograph of the spectrum of the variable-height sampled area-modulation signal.

Fig. 12
Fig. 12

Photograph of the multichannel spectrum for a three-channel input.

Tables (1)

Tables Icon

Table I Parameters for Experiments for Various Forms of Area Modulation

Equations (33)

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t 1 ( x , y ) = rect x L rect { y 2 [ B + b a ( x ) ] }
t 2 ( x , y ) = rect x L rect { y - [ b a ( x ) - B ] / 2 B + b a ( x ) } ,
rect y c = { 1 y c / 2 0 otherwise ,
t 3 ( x , y ) = rect ( x L ) rect [ y - b a ( x ) / 2 b a ( x ) ] sgn ( y ) ,
sgn ( y ) = { 1 y > 0 0 y = 0 - 1 y < 0.
u t ( t ) = a t ( t ) exp [ j α t ( t ) ] ,
t 4 ( x , y ) = rect ( x L ) rect ( y 2 b a ( x ) ) [ B 1 + b 1 c ( x ) ] ,
c ( x ) = cos [ 2 π f x c x + α ( x ) + θ ( x ) ] ,
θ ( x ) = { 0 a ( x ) 0 - π a ( x ) < 0 ,
a ( x ) exp [ j θ ( x ) ] = a ( x ) ,
B 1 + b 1 c ( x ) = { 1 c ( x ) 0 0 c ( x ) < 0.
G ( f x , f y ) = k 1 - t ( x , y ) exp [ - j 2 π ( f x x + f y y ) ] d x d y ,
G 1 ( f x , 0 ) = 2 k 1 - rect ( x L ) [ B + b a ( x ) ] exp ( - j 2 π f x x ) d x ,
G 2 ( f x , 0 ) = ( 1 / 2 ) G 1 ( f x , 0 ) ,
G 3 ( f x , 0 ) = b k 1 - rect ( x L ) a ( x ) exp ( - j 2 π f x x ) d x ,
G 4 ( f x , 0 ) = k 1 - rect ( x L ) × [ B 1 2 b a ( x ) + b b 1 u ( x ) exp ( j 2 π f x c x ) + b b 1 u * ( x ) exp ( - j 2 π f x c x ) ] × exp ( - j 2 π f x x ) d x ,
sinc { 2 [ B + b a ( x ) ] f y } ,
sinc { [ B + b a ( x ) ] f y } exp { - j π [ b a ( x ) - B ] f y } ,
sinc [ b a ( x ) f y ] exp [ - j π b a ( x ) f y ] ,
sinc [ 2 b a ( x ) f y ] ,
exp [ - j 2 π q ( x ) f y ] .
t b ( x , y ) = n = - rect [ y B + b a ( n x s ) ] rect ( x - n x s p ) rect ( n x s L ) ,
t c ( x , y ) = n = - circ { [ ( x - n x s ) 2 + y 2 ] 1 2 r ( n x s ) } rect ( n x s L ) ,
G b ( f x , 0 ) = k 1 p x s sinc p f x m = - A 2 ( f x - m x s ) ,
A 2 ( f x ) = - [ B + b a ( x ) ] rect ( x L ) exp ( - j 2 π f x x ) d x .
G c ( f x , 0 ) k 1 π / x b A 2 ( f x ) ,
t h ( x , y ) = { [ B + b a ( x ) - g ( x ) ] rect ( y - W y / 2 W y ) + g ( x ) rect ( y + W y / 2 W y ) } rect ( x / L ) ,
g ( x ) = { 0 B + b a ( x ) 1 1 B + b a ( x ) > 1.
G h ( f x , 0 ) = k 1 W y - [ B + b a ( x ) ] rect ( x / L ) exp ( - j 2 π f x x ) d x ,
A n ( f x ) sinc | D λ f 3 ( y 3 - n h f 3 f 1 ) | ,
f x = x 2 / ( λ d 1 ) ,
s ( x ) = a 1 cos 2 π f 1 x + a 2 cos 2 π f 2 x + a 3 cos 2 π f 3 x ,
f 1 = 25 cycles / cm , a 1 = 0.25 , f 2 = 35 cycles / cm , a 2 = 0.45 , f 3 = 60 cycles / cm , a 3 = 0.30 ,

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