Abstract

The influence of additional filters upon the output of an optical correlator is studied for the case of a spatial filter measuring the size of rectangles. Addtional filters leading to increased sensitivity are derived by means of a series expansion. One type of additional filter, consisting of several transparent slits, yields a linear relation between output signal and the deviation from the desired size. The other additional filter, effecting a quadratic output characteristic, is a properly dimensioned bandpass. Experimental results demonstrate the enhancement of sensitivity achieved with additional filters. As this method can easily be extended to a correlator measuring the diameter of circular apertures, additional filters may be of interest in the field of automatic production control.

© 1973 Optical Society of America

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References

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  1. A. Vander Lugt, IEEE IT-10, 139 (1964).
  2. S. Lowenthal, Y. Belvaux, Optica Acta 14, 245 (1967).
    [CrossRef]
  3. D. Huhn et al., Phys. Lett. 27A, 51 (1968).
  4. R. A. Binns et al., Appl. Opt. 7, 1047 (1968).
    [CrossRef] [PubMed]
  5. J. Bulabois et al., Opt. Technol. 1, 191 (1969).
    [CrossRef]
  6. U. Wagner, Opt. Commun. 3, 130 (1971).
    [CrossRef]
  7. W. T. Cathey, J. Opt. Soc. Am. 61, 478 (1971).
    [CrossRef]
  8. J. E. Wasielewski, Appl. Opt. 10, 2439 (1971).
    [CrossRef] [PubMed]
  9. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York1968), p. 175.
  10. M. Abramowitz, I. Stegun, Handoook of Mathematical Functions (Dover Publications, Inc., New York, 1968), p. 231.
  11. W. Lukosz, J. Opt. Soc. Am. 56, 1463 (1966).
    [CrossRef]
  12. W. Lukosz, J. Opt. Soc. Am. 57, 932 (1967).
    [CrossRef]

1971

1969

J. Bulabois et al., Opt. Technol. 1, 191 (1969).
[CrossRef]

1968

D. Huhn et al., Phys. Lett. 27A, 51 (1968).

R. A. Binns et al., Appl. Opt. 7, 1047 (1968).
[CrossRef] [PubMed]

1967

W. Lukosz, J. Opt. Soc. Am. 57, 932 (1967).
[CrossRef]

S. Lowenthal, Y. Belvaux, Optica Acta 14, 245 (1967).
[CrossRef]

1966

1964

A. Vander Lugt, IEEE IT-10, 139 (1964).

Abramowitz, M.

M. Abramowitz, I. Stegun, Handoook of Mathematical Functions (Dover Publications, Inc., New York, 1968), p. 231.

Belvaux, Y.

S. Lowenthal, Y. Belvaux, Optica Acta 14, 245 (1967).
[CrossRef]

Binns, R. A.

Bulabois, J.

J. Bulabois et al., Opt. Technol. 1, 191 (1969).
[CrossRef]

Cathey, W. T.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York1968), p. 175.

Huhn, D.

D. Huhn et al., Phys. Lett. 27A, 51 (1968).

Lowenthal, S.

S. Lowenthal, Y. Belvaux, Optica Acta 14, 245 (1967).
[CrossRef]

Lukosz, W.

Stegun, I.

M. Abramowitz, I. Stegun, Handoook of Mathematical Functions (Dover Publications, Inc., New York, 1968), p. 231.

Vander Lugt, A.

A. Vander Lugt, IEEE IT-10, 139 (1964).

Wagner, U.

U. Wagner, Opt. Commun. 3, 130 (1971).
[CrossRef]

Wasielewski, J. E.

Appl. Opt.

IEEE

A. Vander Lugt, IEEE IT-10, 139 (1964).

J. Opt. Soc. Am.

Opt. Commun.

U. Wagner, Opt. Commun. 3, 130 (1971).
[CrossRef]

Opt. Technol.

J. Bulabois et al., Opt. Technol. 1, 191 (1969).
[CrossRef]

Optica Acta

S. Lowenthal, Y. Belvaux, Optica Acta 14, 245 (1967).
[CrossRef]

Phys. Lett.

D. Huhn et al., Phys. Lett. 27A, 51 (1968).

Other

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York1968), p. 175.

M. Abramowitz, I. Stegun, Handoook of Mathematical Functions (Dover Publications, Inc., New York, 1968), p. 231.

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Figures (8)

Fig. 1
Fig. 1

Optical processing system.

Fig. 2
Fig. 2

Essential integrands of the zero-order term (——), linear term (- - - -) and quadratic term (·····). The lower graphs show the amplitude transmittance tLP of the additional phase reversal filter FLP yielding linear output characteristic and the amplitude transmittance tLA of a similar additional amplitude filter FLA.

Fig. 3
Fig. 3

Numerical results for the output characteristic sAB() of the correlator using additional filters FLA: (a) SAB() vs the deviation for N = 1,2,3, and 4 if the slit width w = 0.5π. (b) SAB() vs for slit width reductions δw/π = 0., 0.025, …, 0.125 if N = 4. As the graph shows, an increase of δw brings about a sensitivity increase as well as an enlargement of the range of linearity. (c) Intensity level b0 as, function of δw indicating that any sensitivity increase must be paid with a decrease of the output signal.

Fig. 4
Fig. 4

Numerical results for output characteristics sAB() using additional filters FQ of bandwidths w/π = 1.5, 2.5, 3.5.

Fig. 5
Fig. 5

Experimental setup where g(x,y) is the input signal, H(p,q) the master filter, and FLA an additional filter.

Fig. 6
Fig. 6

Correlation signals in the output plane using different additional filters: (a) FQ, A = 0.75π, B = 2.25π; (b) FQ, A = 0.75π, B = 3.25π; (c) FQ, A = 0.75π, B = 4.25π; (d) FLA, N = 4, w = 0.4π; and (e) master filter H(p,q) alone. As can be seen, a detector pinhole of 15-μ diameter only transmits the central part of the main maximum.

Fig. 7
Fig. 7

Comparison of sensitivity characteristics using different additional filters: H(p,q) alone (Δ), H(p,q) combined with amplitude filter FLA described in Sec. III (○), and H(p,q) combined with FQ (A = 0.75π, B = 4.25π (□).

Fig. 8
Fig. 8

Experimental and numerical results illustrating the influence of position mismatch. (a) Using the additional amplitude filter FLA if the position mismatch is δp = −0.25π. (b) Using the additional filter FQ, A = 0.75π, B = 3.25π if position mismatches are δp = −0.25 π and p = 0.25π. To demonstrate the effect more clearly, output characteristics produced by correctly positioned (δp = 0) additional filters are plotted also.

Equations (17)

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U ( γ , δ ) = C 1 - + d p d q H * ( p , q ) G ( p , q ) exp [ j ( p γ + q δ ) ] ,
U ( γ , ) = C 0 - + d p sin p p sin p ( 1 + ) p exp ( j p γ ) .
U A B ( ) = C 0 A B d p sin p p sin p ( 1 + ) p .
U A B ( ) = ( C 0 / 2 ) { - [ 2 sin p sin p ( 1 + ) / p ] + ( 2 + ) S i ( p ( 2 + ) ) - S i ( p ) } B A ,
U A B ( ) = C 0 n a n ( A , B ) n = C 0 A B d p [ sin 2 p p 2 n ( - 1 ) n ( p ) 2 n ( 2 n ) ! + sin p cos p p n ( - 1 ) n ( p ) 2 n + 1 ( 2 n + 1 ) ! ] ,
U A B ( ) = C 0 n ( - 1 ) n { [ A B d p p 2 n - 2 sin 2 p / ( 2 n ) ! ] 2 n + [ A B d p p 2 n - 1 sin p cos p / ( 2 n + 1 ) ! ] 2 n + 1 }
S A B ( ) = U A B ( ) 2 = C 0 2 n b n ( A , B ) n ,
b n ( A , B ) = l = 0 n a l ( A , B ) a n - l ( A , B ) .
s A B ( ) = 1 + ( b 1 / b 0 ) + ( b 2 / b 0 ) 2 + ( b 3 / b 0 ) 3 ,
b 0 ( A , B ) = ( A B d p sin 2 p p 2 ) 2 ,
b 1 ( A , B ) = 2 A B d p sin 2 p p 2 A B d p sin 2 p 2 p ,
b 2 ( A , B ) = - A B d p sin 2 p p 2 A B d p sin 2 p + ( A B d p sin 2 p 2 p ) 2 ,
b 3 ( A , B ) = - 1 3 A B d p sin 2 p p 2 A B d p p sin 2 p 2 - A B d p sin 2 p 2 p A B d p sin 2 p .
P A B ( ) = Pinhole S A B ( γ , ) d γ .
γ max π / 2 ( B - A )
a n ( A , B ) = { ( - 1 ) n / 2 1 + ( - 1 ) n 2 ( A B p n - 2 sin 2 p d p / n ! ) + ( - 1 ) ( n - 1 ) / 2 1 - ( - 1 ) n 2 ( A B p n - 2 sin p cos p d p / n ! ) } .
b n ( A , B ) = l = 0 n { ( - 1 ) n / 2 1 + ( - 1 ) l + ( - 1 ) n - l + ( - 1 ) n 4 ( A B p l - 2 sin 2 p d p / l ! ) [ A B p n - l - 2 sin 2 p d p / ( n - l ) ! ] + ( - 1 ) ( n - 1 ) / 2 1 + ( - 1 ) l - ( - 1 ) n - l - ( - 1 ) n 4 ( A B p l - 2 sin 2 p d p / l ! ) [ A B p n - l - 2 sin p cos p d p / ( n - l ) ! ] + ( - 1 ) ( n - 1 ) / 2 1 - ( - 1 ) l + ( - 1 ) n - l - ( - 1 ) n 4 ( A B p l - 2 sin p cos p d p / l ! ) [ A B p n - l - 2 sin 2 p d p / ( n - l ) ! ] + ( - 1 ) ( n - 2 ) / 2 1 - ( - 1 ) l - ( - 1 ) n - l + ( - 1 ) n 4 ( A B p l - 2 sin p cos p d p / l ! ) [ A B p n - l - 2 sin p cos p d p / ( n - l ) ! ] }

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