Abstract

A heuristic development of the operation of a square root filter (one whose amplitude transmittance is proportional to the square root of position in frequency space) is presented. The output of a coherent optical processor utilizing this filter for phase image inputs is shown to be an intensity which is directly proportional to the derivative of the input phase function. The results of a digital filtering experiment which confirms this linear relationship are presented. The use of this filter in phase image visualization systems is also discussed.

© 1978 Optical Society of America

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References

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  1. R. A. Sprague, B. J. Thompson, Appl. Opt. 11, 1469 (1972).
    [CrossRef] [PubMed]
  2. B. A. Horwitz, Dissertation, University of Rochester (1976).
  3. R. Hoffman, L. Gross, Appl. Opt. 14, 1169 (1975).
    [CrossRef] [PubMed]
  4. P. DeSantis, F. Gori, G. Guattari, C. Palma, Appl. Opt. 15, 2385 (1976).
    [CrossRef]
  5. B. A. Horwitz, Opt. Commun. 17, 231 (1976).
    [CrossRef]
  6. K. B. Oldham, J. Spanier, The Fractional Calculus (Academic, New York, 1974).
  7. R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965), p. 113.
  8. See, for example, bandwidth calculations for FM signals in M. Schwartz, Information Transmission, Modulation, and Noise, (McGraw-Hill, New York, 1970), pp. 238–246.
  9. G. E. Sommargren, B. J. Thompson, Appl. Opt. 12, 2130 (1973).
    [CrossRef] [PubMed]
  10. M. Abramowitz, I. A. Stegun, Eds. Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1970), pp. 300, 301.

1976 (2)

1975 (1)

1973 (1)

1972 (1)

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965), p. 113.

DeSantis, P.

Gori, F.

Gross, L.

Guattari, G.

Hoffman, R.

Horwitz, B. A.

B. A. Horwitz, Opt. Commun. 17, 231 (1976).
[CrossRef]

B. A. Horwitz, Dissertation, University of Rochester (1976).

Oldham, K. B.

K. B. Oldham, J. Spanier, The Fractional Calculus (Academic, New York, 1974).

Palma, C.

Schwartz, M.

See, for example, bandwidth calculations for FM signals in M. Schwartz, Information Transmission, Modulation, and Noise, (McGraw-Hill, New York, 1970), pp. 238–246.

Sommargren, G. E.

Spanier, J.

K. B. Oldham, J. Spanier, The Fractional Calculus (Academic, New York, 1974).

Sprague, R. A.

Thompson, B. J.

Appl. Opt. (4)

Opt. Commun. (1)

B. A. Horwitz, Opt. Commun. 17, 231 (1976).
[CrossRef]

Other (5)

K. B. Oldham, J. Spanier, The Fractional Calculus (Academic, New York, 1974).

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965), p. 113.

See, for example, bandwidth calculations for FM signals in M. Schwartz, Information Transmission, Modulation, and Noise, (McGraw-Hill, New York, 1970), pp. 238–246.

M. Abramowitz, I. A. Stegun, Eds. Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1970), pp. 300, 301.

B. A. Horwitz, Dissertation, University of Rochester (1976).

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Figures (5)

Fig. 1
Fig. 1

Input phase functions and derivatives: (a) cosine input; (b) integrated sawtooth input; (c) integrated semicircles.

Fig. 2
Fig. 2

Resultant intensities for square root and ramp filtering—cosine input.

Fig. 3
Fig. 3

Resultant intensities for square root and ramp filtering—integrated sawtooth inputs.

Fig. 4
Fig. 4

Resultant intensities for square root and ramp filtering—integrated semicircles input.

Fig. 5
Fig. 5

Comparison of square root filter and clear aperture impulse responses.

Equations (23)

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F ( k x ) = 1 ( 2 k o ) 1 / 2 ( k x + k o ) 1 / 2 rect ( k x 2 k o ) , k o = constant ,
I ( x ) = 1 2 k o ϕ ( x ) + 1 2 .
| O 1 ( x ) | 2 d x = 1 2 k o | ( k x + k o ) 1 / 2 Õ ( k x ) | 2 d k x ,
[ ( k x + k o ) Õ ( k x ) ] [ Õ ( k x ) ] * ,
| O 1 ( x ) | 2 d x = 1 2 k o [ ( k x + k o ) Õ ( k x ) ] [ Õ ( k x ) ] * d k .
| O 1 ( x ) | 2 d x = 1 2 k o [ i d d α O ( α ) + k o O ( α ) ] [ O ( α ) ] * d α ,
O ( α ) = exp [ i ϕ ( α ) ] , d d α O ( α ) = i ϕ ( α ) O ( α ) , and O ( α ) O * ( α ) = 1 ,
| O 1 ( x ) | 2 d x = 1 2 k o [ ϕ ( α ) + k o ] d α .
x o x o | O 1 ( x ) | 2 d x = 1 2 k o x o x o [ ϕ ( α ) + k o ] d α .
x o x o + δ | O 1 ( x ) | 2 d x = 1 2 k o x o x o + δ [ ϕ ( α ) + k o ] d α .
I ( x o ) | O 1 ( x o ) | 2 = 1 2 k o ϕ ( x o ) + 1 2 .
ϕ max k max k o
d ϕ d x max 2 π Δ x .
I ( x , x o ) = [ 1 2 k o ϕ ( x ) + 1 2 + π x o f λ k o ] S ( x o ) ,
I ( x ) = I ( x , x o ) d x o ,
I ( x ) = [ 1 2 k o ϕ ( x ) + 1 2 ] S ( x o ) d x o + π f λ k o x o S ( x o ) d x o .
I ( x ) = 1 2 k o ϕ ( x ) + ( 1 2 + K ) ,
h ( x ) = 1 ( 2 k o ) 1 / 2 k o k 0 ( k x + k o ) 1 / 2 exp ( i k x x ) d k x .
h ( x ) = 1 ( 2 k o ) 1 / 2 exp ( i k o x ) x 3 / 2 0 α o ( α i ) exp ( i α 2 ) i 2 α d α ; α o ( 2 k o x ) 1 / 2 .
h ( x ) = exp ( i k o x ) ( 2 k o ) 1 / 2 x 3 / 2 [ i α o exp ( i α o 2 ) + i 0 α o exp ( i α 2 ) d α ] ,
h ( x ) = exp ( i k o x ) ( 2 k o ) 1 / 2 x 3 / 2 [ i α o exp ( i α o 2 ) + i 0 α o cos α 2 d α 0 α o sin α 2 d α ] .
h ( x ) = i x exp ( i k o x ) + π 1 / 2 exp ( i k o x ) ( 4 k o ) 1 / 2 x 3 / 2 { i C 1 [ ( 2 k o x ) 1 / 2 ] S 1 [ ( 2 k o x ) 1 / 2 ] } .
h 1 ( x ) = 2 k o sinc ( k o x ) ,

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