Abstract

The changes in the interference pattern in Young’s interference experiment, produced by placing two identical narrow-band filters in front of the pinholes, are analyzed. It is shown theoretically that, in general, the fringes will not become sharp (i.e., their maximum visibility will not tend to unity) even when the filters have arbitrarily narrow passbands. The analysis brings out a relationship between the complex degree of coherence in the space–time and the space–frequency domains. When the passbands of the filters are narrow enough, the filtered light is found to be cross-spectrally pure.

© 1983 Optical Society of America

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References

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  1. For explanation of the concepts of coherence theory used in this Letter, see M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. X, or L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
    [Crossref]
  2. L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
    [Crossref]
  3. L. Mandel, “Concept of cross-spectral purity in coherence theory,” J. Opt. Soc. Am. 51, 1342–1350 (1961).
    [Crossref]
  4. W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958), pp. 42 and 67.
  5. E. Wolf, “A new description of second-order coherence phenomena in the space–frequency domain,” AIP Conf. Proc. 65, 42–48 (1981); “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981); New theory of partial coherence in the space–frequency domain. Part I: Spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
    [Crossref]
  6. It is, of course, assumed that the measurements are made on a time scale that involves averaging over a time interval that is long compared to the reciprocal bandwidth of the filters and that the detector is sensitive enough to measure the reduced intensities of the filtered light.

1981 (1)

E. Wolf, “A new description of second-order coherence phenomena in the space–frequency domain,” AIP Conf. Proc. 65, 42–48 (1981); “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981); New theory of partial coherence in the space–frequency domain. Part I: Spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
[Crossref]

1976 (1)

1961 (1)

Born, M.

For explanation of the concepts of coherence theory used in this Letter, see M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. X, or L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[Crossref]

Davenport, W. B.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958), pp. 42 and 67.

Mandel, L.

Root, W. L.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958), pp. 42 and 67.

Wolf, E.

E. Wolf, “A new description of second-order coherence phenomena in the space–frequency domain,” AIP Conf. Proc. 65, 42–48 (1981); “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981); New theory of partial coherence in the space–frequency domain. Part I: Spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
[Crossref]

L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
[Crossref]

For explanation of the concepts of coherence theory used in this Letter, see M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. X, or L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[Crossref]

AIP Conf. Proc. (1)

E. Wolf, “A new description of second-order coherence phenomena in the space–frequency domain,” AIP Conf. Proc. 65, 42–48 (1981); “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981); New theory of partial coherence in the space–frequency domain. Part I: Spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
[Crossref]

J. Opt. Soc. Am. (2)

Other (3)

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958), pp. 42 and 67.

It is, of course, assumed that the measurements are made on a time scale that involves averaging over a time interval that is long compared to the reciprocal bandwidth of the filters and that the detector is sensitive enough to measure the reduced intensities of the filtered light.

For explanation of the concepts of coherence theory used in this Letter, see M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. X, or L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[Crossref]

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Figures (2)

Fig. 1
Fig. 1

Illustrating the notation used in the analysis of Young’s interference experiment. The parameter τ that occurs in the text is equal to (s2s1)/c, where c is the speed of light in vacuo.

Fig. 2
Fig. 2

Schematic illustration of the relative behavior of the modulus of the complex amplitude transmission function T(ω) of the filters placed in front of the pinholes of Young’s interference experiments and of the absolute value of the cross-spectral density W(P1, P2, ω) of the unfiltered light at the pinholes. The effective passbands ω0 − Δω/2 ≤ ωω0 + Δω/2 of the filters are assumed to be so narrow that both the modulus and the phase (not shown) of W(P1, P2, ω) and also the spectral densities W(P1, P1, ω) and W(P2, P2, ω) (with P1 and P2 fixed) are substantially constant across it.

Equations (20)

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Γ ( P 1 , P 2 , τ ) = V * ( P 1 , t ) V ( P 2 , t + τ ) ,
γ ( P 1 , P 2 , τ ) = Γ ( P 1 , P 2 , τ ) [ Γ ( P 1 , P 1 , 0 ) ] 1 / 2 [ Γ ( P 2 , P 2 , 0 ) ] 1 / 2 ,
V ( Q ) = γ ( P 1 , P 2 , τ ) .
Γ ( P 1 , P 2 , τ ) = 0 W ( P 1 , P 2 , ω ) e - i ω τ d ω .
W ( P 1 , P 2 , ω ) = U * ( P 1 , ω ) U ( P 2 , ω ) ω ,
W ( + ) ( P 1 , P 2 , ω ) = T * ( ω ) U * ( P 1 , ω ) T ( ω ) U ( P 2 , ω ) ω ;
W ( + ) ( P 1 , P 2 , ω ) = T ( ω ) W 2 ( P 1 , P 2 , ω ) .
Γ ( + ) ( P 1 , P 2 , τ ) = 0 T ( ω ) W 2 ( P 1 , P 2 , ω ) e - i ω τ d ω ;
γ ( + ) ( P 1 , P 2 , τ ) = Γ ( + ) ( P 1 , P 2 , τ ) [ Γ ( + ) ( P 1 , P 1 , 0 ) ] 1 / 2 [ Γ ( + ) ( P 2 , P 2 , 0 ) ] 1 / 2 .
Γ ( + ) ( P 1 , P 2 , τ ) = W ( P 1 , P 2 , ω 0 ) 0 T ( ω ) e 2 - i ω τ d ω .
Γ ( + ) ( P j , P j , τ ) = W ( P j , P j , ω 0 ) 0 T ( ω ) e 2 - i ω τ d ω ,             ( j = 1 , 2 ) .
γ ( + ) ( P 1 , P 2 , τ ) = μ ( P 1 , P 2 , ω 0 ) θ ( τ ) ,
μ ( P 1 , P 2 , ω 0 ) = W ( P 1 , P 2 ω 0 ) [ W ( P 1 , P 1 , ω 0 ) ] 1 / 2 [ W ( P 2 , P 2 , ω 0 ) ] 1 / 2
θ ( τ ) = 0 T ( ω ) e 2 xp ( - i ω τ ) d ω 0 T ( ω ) d 2 ω .
μ ( + ) ( P 1 , P 2 , ω 0 ) = W ( + ) ( P 1 , P 2 , ω 0 ) [ W ( + ) ( P 1 , P 1 , ω 0 ) ] 1 / 2 [ W ( + ) ( P 2 , P 2 , ω 0 ) ] 1 / 2 ,
γ ( + ) ( P 1 , P 2 , 0 ) = μ ( P 1 , P 2 , ω 0 ) .
γ ( + ) ( P 1 , P 1 , τ ) Γ ( + ) ( P 1 , P 1 , τ ) Γ ( + ) ( P 1 , P 1 , 0 ) = θ ( τ )
γ ( + ) ( P 2 , P 2 , τ ) Γ ( + ) ( P 2 , P 2 , τ ) Γ ( + ) ( P 2 , P 2 , 0 ) = θ ( τ ) .
γ ( + ) ( P 1 , P 2 , τ ) = γ ( + ) ( P 1 , P 2 , 0 ) γ ( + ) ( P 1 , P 1 , τ ) .
γ ( + ) ( P 1 , P 1 , τ ) = γ ( + ) ( P 2 , P 2 , τ ) .

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