Abstract

A self-Fourier function (SFF), according to Caola [ J. Phys. A 24, L1143 ( 1991)], is a function that is its own Fourier transform. The Gaussian and Dirac combs are well-known examples. Many more SFF’s have been discovered recently by Caola. This discovery might bear some fruit in optics, since the Fourier transform is perhaps the most important theoretical tool in wave optics. We show that Caola discovered all SFF’s. Furthermore, we study other self-transform functions, which are also tied to some transformations that play a role in coherent optics.

© 1992 Optical Society of America

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References

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  1. M. J. Caola, “Self-Fourier functions,”J. Phys. A 24, L1143–L1144 (1991).
    [CrossRef]
  2. M. J. Caola, r/o Sowerby Research Centre, British Aerospace plc., FPC 267, Filton, Bristol, B512 7QW, UK (personal communication, 1992).
  3. R. N. Bracewell, “Discrete Hartley transform,”J. Opt. Soc. Am. 73, 1832–1835 (1983).
    [CrossRef]
  4. R. N. Bracewell, H. Bartelt, A. W. Lohmann, N. Streibl, “Optical analog computers for the Hartley transform,” Appl. Opt. 24, 1401–1402 (1985).
    [CrossRef] [PubMed]
  5. L. Mertz, Transformations in Optics (Wiley, New York, 1965).
  6. K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. 28, pp. 3–110.
  7. A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik (Stuttgart) 79, 41–45 (1988).
  8. A. W. Lohmann, J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
    [CrossRef] [PubMed]
  9. A. W. Lohmann, “Ein neues Dualitatsprinzip in der Optik,” Optik (Stuttgart) 11, 478–488 (1954). An English version did appear in Optik (Stuttgart) 89, 93–97 (1992).

1991 (1)

M. J. Caola, “Self-Fourier functions,”J. Phys. A 24, L1143–L1144 (1991).
[CrossRef]

1990 (1)

1988 (1)

A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik (Stuttgart) 79, 41–45 (1988).

1985 (1)

1983 (1)

1954 (1)

A. W. Lohmann, “Ein neues Dualitatsprinzip in der Optik,” Optik (Stuttgart) 11, 478–488 (1954). An English version did appear in Optik (Stuttgart) 89, 93–97 (1992).

Bartelt, H.

Bracewell, R. N.

Caola, M. J.

M. J. Caola, “Self-Fourier functions,”J. Phys. A 24, L1143–L1144 (1991).
[CrossRef]

M. J. Caola, r/o Sowerby Research Centre, British Aerospace plc., FPC 267, Filton, Bristol, B512 7QW, UK (personal communication, 1992).

Lohmann, A. W.

A. W. Lohmann, J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
[CrossRef] [PubMed]

A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik (Stuttgart) 79, 41–45 (1988).

R. N. Bracewell, H. Bartelt, A. W. Lohmann, N. Streibl, “Optical analog computers for the Hartley transform,” Appl. Opt. 24, 1401–1402 (1985).
[CrossRef] [PubMed]

A. W. Lohmann, “Ein neues Dualitatsprinzip in der Optik,” Optik (Stuttgart) 11, 478–488 (1954). An English version did appear in Optik (Stuttgart) 89, 93–97 (1992).

Mertz, L.

L. Mertz, Transformations in Optics (Wiley, New York, 1965).

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. 28, pp. 3–110.

Streibl, N.

Thomas, J. A.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

J. Phys. A (1)

M. J. Caola, “Self-Fourier functions,”J. Phys. A 24, L1143–L1144 (1991).
[CrossRef]

Optik (Stuttgart) (2)

A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik (Stuttgart) 79, 41–45 (1988).

A. W. Lohmann, “Ein neues Dualitatsprinzip in der Optik,” Optik (Stuttgart) 11, 478–488 (1954). An English version did appear in Optik (Stuttgart) 89, 93–97 (1992).

Other (3)

M. J. Caola, r/o Sowerby Research Centre, British Aerospace plc., FPC 267, Filton, Bristol, B512 7QW, UK (personal communication, 1992).

L. Mertz, Transformations in Optics (Wiley, New York, 1965).

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. 28, pp. 3–110.

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Equations (52)

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f F ( u ) = f ( u ) .
f F ( u ) = f ( x ) exp ( - 2 π i u x ) d x ,
f ( x ) = f F ( u ) exp ( 2 π i u x ) d u .
f ( x ) = exp ( - π x 2 ) ,             f F ( u ) = exp ( - π u 2 ) ,
f ( x ) = n δ ( x - n ) ,             f F ( u ) = n δ ( u - n ) .
F ( x ) = g ( x ) + g F ( x ) + g ( - x ) + g F ( - x ) .
F F ( u ) = g F ( u ) + g ( - u ) + g F ( - u ) + g ( u ) .
F 1 = rect ( x ) + sinc ( x )
F 5 = exp ( - π x 2 / σ 2 ) + σ exp ( - π x 2 σ 2 ) .
T [ T [ T ] ] F ( x ) = T N [ F ( x ) ] = F ( x ) .
g ( x ) + g F ( x ) + g ( - x ) + g F ( - x ) = F ( x ) .
g ( x ) = F ( x ) / 4.
g ( x ) + g F ( x ) + g ( - x ) + g F ( - x ) = F ( x ) 4 + F ( x ) 4 + F ( x ) 4 + F ( x ) 4 = F ( x ) .
FOU { F F ( x ) } = FOU { F ( x ) } F ( - x ) = F F ( x ) = F ( x ) .
g ( x ) = F ( x ) / 4 + h 1 ( x ) ,             h 1 ( x ) = - h 1 ( - x ) .
g F ( x ) = F ( x ) / 4 + h 2 ( x ) ,             h 2 ( x ) = - h 2 ( - x ) .
g ( x ) = F ( x ) / 4 + h 1 ( x ) + h 2 F ( x ) ,
f ( x ) = f ( x + 1 ) ,
f ( x ) = m A m exp ( 2 π i m x ) ,
A m = - 0.5 0.5 f ( x ) exp ( - 2 π i m x ) d x ,
A m 0             only if             - N / 2 + 1 m N / 2.
x n = n / N ,             - N / 2 + 1 n N / 2 ,
A m = 1 N n f ( n N ) exp ( - 2 π i n m N ) .
f ( m N ) 1 N n f ( n N ) exp ( - 2 π m i n N ) = N A m ,
N A n 1 N m N A m exp ( - 2 π i n m N ) = f ( - n N ) .
N A m = f F ( m / N ) .
F ( m N ) = f ( m N ) + f F ( m N ) + f ( - m N ) + f F ( - m N ) .
F ( m / N ) = F F ( m / N ) .
f B ( u ) = f ( x ) CAS ( 2 π u x ) d x ,
CAS ( 2 π u x ) = cos ( 2 π u x ) + sin ( 2 π u x ) .
f ( x ) = f B ( u ) CAS ( 2 π u x ) d u .
F ( x ) = f ( x ) + f B ( x ) .
T [ T [ f ( x ) ] ] = T N [ f ( x ) ] = f ( x ) .
F ( x ) = g ( x ) + T [ g ( x ) ] + T 2 [ g ( x ) ] + + T N - 1 [ g ( x ) ] .
T [ F ( x ) ] = T [ g ] + T 2 [ F ] + + T N [ g ] .
g ( x ) g F ( x ) .
g F ( x ) g F ( x ) + exp ( 2 π i u 0 x ) .
g F + exp ( 2 π i u 0 x ) g F + exp ( ) 2 = + g F * ( x ) exp ( 2 π i u 0 x ) .
g F * exp ( ) exp ( - 2 π i u 0 x ) g F * exp ( 2 π i u 0 x ) = g F * ( x ) .
g ( x ) g A ( x ) = [ g F ( x ) ] * , g A ( x ) = [ g ( y ) exp ( - 2 π i x y ) d y ] * = [ g ( y ) ] * exp ( 2 π i x y ) d y .
[ g A ( z ) * exp ( 2 π i z x ) d z = [ g ( y ) ] * * exp [ 2 π i z ( x - y ) d y d z = [ g ( x ) ] * * = g ( x ) .
F ( x ) = g ( x ) + [ g F ( x ) ] * = g ( x ) + g A ( x ) .
g M ( u ) = g ( x ) exp [ i π ( x - u ) 2 ] d x ,
g ( x ) = g M ( u ) exp [ - i π ( u - x ) 2 ] d u .
g M ( x ) = [ g M ( x ) ] * ;
F ( x ) = g ( x ) + g M ( x ) = g ( x ) + [ g M ( x ) ] * .
g [ x ; z = m ( Z 0 / N ) ] = g m ( x ) ,             m = 0 , 1 , , N .
T [ g m ( x ) ] = g m + 1 ( x ) ,             m = 0 , 1 , .
g ( x ; Z 0 ) = g ( x ; 0 ) .
T N [ g ( x ; z = 0 ) ] = g ( x ; z = Z 0 ) = g ( x ; z = 0 ) .
F ( x ) = g 0 ( x ) + g 1 ( x ) + + g N - 1 ( x ) .
Th ( x ) exp ( - 2 π i u x ) d x = Th F ( u ) .

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