Abstract

In correlation filtering a spatial light modulator is traditionally modeled as affecting only the phase or only the amplitude of light. Usually, however, a single operating parameter affects both phase and amplitude. An integral constraint is developed that is a necessary condition for optimizing a correlation filter having single parameter coupling between phase and amplitude. The phase-only filter is shown to be a special case.

© 1989 Optical Society of America

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References

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  1. J. L. Horner, “SLM Induced Phase Distortion and Its Effect on Optical Correlation” Proc. Soc. Photo-Opt. Instrum. Eng. 754, 157–161 (1987).
  2. R. D. Juday, S. E. Monroe, D. A. Gregory, “Optical Correlation with Phase Encoding and Phase Filtering” Proc. Soc. Photo-Opt. Instrum. Eng. 825, 149–155 (1987).
  3. R. D. Juday, B. J. Daiuto, “Relaxation Method of Compensation in an Optical Correlator” Opt. Eng. 26, 1094–1101 (1987).
    [CrossRef]
  4. L. E. Franks, Signal Theory (Dowden & Culver, Stroudsburg, PA, 1981).
  5. G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1985).

1987 (3)

J. L. Horner, “SLM Induced Phase Distortion and Its Effect on Optical Correlation” Proc. Soc. Photo-Opt. Instrum. Eng. 754, 157–161 (1987).

R. D. Juday, S. E. Monroe, D. A. Gregory, “Optical Correlation with Phase Encoding and Phase Filtering” Proc. Soc. Photo-Opt. Instrum. Eng. 825, 149–155 (1987).

R. D. Juday, B. J. Daiuto, “Relaxation Method of Compensation in an Optical Correlator” Opt. Eng. 26, 1094–1101 (1987).
[CrossRef]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1985).

Daiuto, B. J.

R. D. Juday, B. J. Daiuto, “Relaxation Method of Compensation in an Optical Correlator” Opt. Eng. 26, 1094–1101 (1987).
[CrossRef]

Franks, L. E.

L. E. Franks, Signal Theory (Dowden & Culver, Stroudsburg, PA, 1981).

Gregory, D. A.

R. D. Juday, S. E. Monroe, D. A. Gregory, “Optical Correlation with Phase Encoding and Phase Filtering” Proc. Soc. Photo-Opt. Instrum. Eng. 825, 149–155 (1987).

Horner, J. L.

J. L. Horner, “SLM Induced Phase Distortion and Its Effect on Optical Correlation” Proc. Soc. Photo-Opt. Instrum. Eng. 754, 157–161 (1987).

Juday, R. D.

R. D. Juday, S. E. Monroe, D. A. Gregory, “Optical Correlation with Phase Encoding and Phase Filtering” Proc. Soc. Photo-Opt. Instrum. Eng. 825, 149–155 (1987).

R. D. Juday, B. J. Daiuto, “Relaxation Method of Compensation in an Optical Correlator” Opt. Eng. 26, 1094–1101 (1987).
[CrossRef]

Monroe, S. E.

R. D. Juday, S. E. Monroe, D. A. Gregory, “Optical Correlation with Phase Encoding and Phase Filtering” Proc. Soc. Photo-Opt. Instrum. Eng. 825, 149–155 (1987).

Opt. Eng. (1)

R. D. Juday, B. J. Daiuto, “Relaxation Method of Compensation in an Optical Correlator” Opt. Eng. 26, 1094–1101 (1987).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

J. L. Horner, “SLM Induced Phase Distortion and Its Effect on Optical Correlation” Proc. Soc. Photo-Opt. Instrum. Eng. 754, 157–161 (1987).

R. D. Juday, S. E. Monroe, D. A. Gregory, “Optical Correlation with Phase Encoding and Phase Filtering” Proc. Soc. Photo-Opt. Instrum. Eng. 825, 149–155 (1987).

Other (2)

L. E. Franks, Signal Theory (Dowden & Culver, Stroudsburg, PA, 1981).

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1985).

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Figures (4)

Fig. 1
Fig. 1

Representation of the complex value of a filter value by a point lying on the operating curve of a spatial light modulator. The solid dot is a desired value such as the classical matched filter. The square matches the phase of the desired value. The open circle nearly matches the phase and, by having a larger amplitude than the square, transmits more energy into the correlation intensity. The optimum representation of any one value depends on the ensemble of desired values.

Fig. 2
Fig. 2

Hypothetical curves showing the change in the complex value of Er as α is changed for some choice of μ(·): (a) |Er| is not stationary at α = 0 for this combination of s(·) and μ(·). (b) Although |Er| is stationary at α = 0, it is at a minimum, (c) To maximize the correlation strength by choice of s(·) we require that all choices for μ(·) will produce a maximum in |Er| for α = 0.

Fig. 3
Fig. 3

Variation in Es, δEs is a function of μ(·) If |Er| is an extremum at α = 0, δEs is perpendicular to Es for any choice of μ(·).

Fig. 4
Fig. 4

Physical interpretation of an augmented phase; an SLM operating curve is drawn in the complex plane. For a control value s the SLM phase value is g(s), and the augmented phase is a(s). Vector t is the tangent to the operating curve. Since the operating curve of a phase-only SLM is a circle, its augmented phase would be g(s) + π/2.

Equations (27)

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X ( ω ) = exp ( j ω t ) x ( t ) d t
x ( t ) X ( ω ) = A ( ω ) exp [ j ϕ ( ω ) ]
H ( ω ) = f [ s ( ω ) ] exp { j g [ s ( ω ) ] } ,
E s ( t ) = exp ( j ω t ) A ( ω ) exp [ j ϕ ( ω ) ] f [ s ( ω ) ] × exp { j g [ s ( ω ) ] } d ω = exp ( j ω t ) X ( ω ) H ( ω ) d ω .
X ( ω ) H [ s ( ω ) ] d ω
I s : = E s * ( t = 0 ) E s ( t = 0 ) .
I = E * E = [ D exp ( j q ) * ] [ D exp ( j q ) ] = D 2 .
r ( , α ) = s ( ) + α μ ( ) ,
E r ( α ) = A ( ω ) exp [ j ϕ ( ω ) ] f [ r ( ω ) ] exp { j g [ r ( ω ) ] } d ω .
r α = μ .
δ E s : = α { α A exp ( j ϕ ) f ( r ) exp [ j g ( r ) ] d ω } | α = 0 = α { A exp ( j ϕ ) α { f ( r ) exp [ j g ( r ) ] } d ω } | α = 0 = α A exp { j [ ϕ + g ( s ) ] } μ [ f ( s ) + j f ( s ) g ( s ) ] d ω ,
E s + δ E s exp ( j k μ α ) E s ,
δ E s E s exp ( j k μ α ) 1 j k μ α .
exp { j [ ϕ + g ( s ) ] } [ f ( s ) + j f ( s ) g ( s ) ] E s
E s = j D exp ( j θ )
arg { y exp ( j x ) } = x + arg { y } ,
ϕ ( ω ) + g ( s ) + arg { f ( s ) + j f ( s ) g ( s ) } = θ .
g ( s ) + arg { f ( s ) + j f ( s ) g ( s ) } = : a ( s ) ,
ϕ ( ω ) + a [ s ( ω ) ] = θ ,
s ( ω ) = p [ θ ϕ ( ω ) ] .
A ( ω ) exp [ j ( ϕ ( ω ) + g { p [ θ ϕ ( ω ) ] } ) ] × f { p [ θ ϕ ( ω ) ] } d ω = j D exp ( j θ ) ,
arg { f ( s ) + j f ( s ) g ( s ) } = π / 2 ,
ϕ + g ( s ) + π / 2 = θ ,
g ( s ) = θ ϕ π / 2 .
E s = A exp ( j ϕ ) exp [ j ( θ ϕ π / 2 ) ] d ω = 1 j exp ( j θ ) A d ω .
ϕ ( ω ) + g ( s ) + arg { f ( s ) + j f ( s ) g ( s ) } = θ 1 .
A exp ( j ϕ ) exp [ j g ( s ) ] f ( s ) d ω = 1 j D exp ( j θ 2 ) .

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