Abstract

A new general interference law is derived for the superposition of two random electromagnetic beams of any state of coherence and of any state of polarization when the beams are transmitted through polarizers and rotators. It includes, as special cases, a variety of interference laws that apply to particular situations. Some of them have a close bearing on the classic interference experiments of Fresnel and Arago that have played a basic role in elucidating the concept of polarization of light.

© 2004 Optical Society of America

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References

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  1. D. F. J. Arago, A. J. Fresnel, “On the action of rays of polarized light upon each other,” Ann. Chimie Physique, 288 (1819).
  2. English translation of Ref. 1is published in The Wave Theory of Light. Memories of Huygens, Young and Fresnel, H. Crew, ed. (American Book Co., New York, 1900), pp. 145–157.
  3. An excellent account of the historical background relating to Young’s explanation of the Fresnel–Arago experiments is given in E. Whittaker, A History of the Theories of Aether and Electricity: The Classical Theories, rev. enlarged ed. (Nelson, London, 1951).
  4. R. Hanau, “Interference of linearly polarized light with perpendicular polarizations,” Am. J. Phys. 31, 303–304 (1963).
    [CrossRef]
  5. E. Collett, “Mathematical formulation of the interference laws of Fresnel and Arago,” Am. J. Phys. 39, 1483–1495 (1971).
    [CrossRef]
  6. E. Collett, Polarized Light, Fundamentals and Applications (Marcel Dekker, New York, 1993).
  7. C. Brosseau, Fundamentals of Polarized Light, A Statistical Optics Approach (Wiley, New York, 1998).
  8. R. Barakat, “Analytic proofs of the Arago–Fresnel laws for the interference of polarized light,” J. Opt. Soc. Am. A 10, 180–185 (1993).
    [CrossRef]
  9. M. Henry, “Fresnel–Arago laws for interference in polarized light: a demonstration experiment,” Am. J. Phys. 49, 690–691 (1981).
    [CrossRef]
  10. J. L. Ferguson, “A simple, bright demonstration of the interference of polarized light,” Am. J. Phys. 52, 1141–1142 (1984).
    [CrossRef]
  11. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
    [CrossRef]
  12. E. Wolf, “Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation,” Opt. Lett. 28, 1078–1080 (2003).
    [CrossRef] [PubMed]
  13. H. Roychowdhury, E. Wolf, “Determination of the electric cross-spectral density matrix of a random electromagnetic beam,” Opt. Commun. 226, 57–60 (2003).
    [CrossRef]
  14. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1999).
  15. Since only the difference between the angles of rotation introduced by the rotators is relevant, we could employ only one rotator placed behind one of the pinholes. However, introduction of two rotators, one behind each pinhole, makes the analysis more symmetric.
  16. G. Parrent, P. Roman, “On the matrix formulation of the theory of partial polarization in terms of observables,” Nuovo Cimento 10, 370–388 (1960).
    [CrossRef]
  17. E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).

2003 (3)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
[CrossRef]

E. Wolf, “Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation,” Opt. Lett. 28, 1078–1080 (2003).
[CrossRef] [PubMed]

H. Roychowdhury, E. Wolf, “Determination of the electric cross-spectral density matrix of a random electromagnetic beam,” Opt. Commun. 226, 57–60 (2003).
[CrossRef]

1993 (1)

1984 (1)

J. L. Ferguson, “A simple, bright demonstration of the interference of polarized light,” Am. J. Phys. 52, 1141–1142 (1984).
[CrossRef]

1981 (1)

M. Henry, “Fresnel–Arago laws for interference in polarized light: a demonstration experiment,” Am. J. Phys. 49, 690–691 (1981).
[CrossRef]

1971 (1)

E. Collett, “Mathematical formulation of the interference laws of Fresnel and Arago,” Am. J. Phys. 39, 1483–1495 (1971).
[CrossRef]

1963 (1)

R. Hanau, “Interference of linearly polarized light with perpendicular polarizations,” Am. J. Phys. 31, 303–304 (1963).
[CrossRef]

1960 (1)

G. Parrent, P. Roman, “On the matrix formulation of the theory of partial polarization in terms of observables,” Nuovo Cimento 10, 370–388 (1960).
[CrossRef]

1819 (1)

D. F. J. Arago, A. J. Fresnel, “On the action of rays of polarized light upon each other,” Ann. Chimie Physique, 288 (1819).

Arago, D. F. J.

D. F. J. Arago, A. J. Fresnel, “On the action of rays of polarized light upon each other,” Ann. Chimie Physique, 288 (1819).

Barakat, R.

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light, A Statistical Optics Approach (Wiley, New York, 1998).

Collett, E.

E. Collett, “Mathematical formulation of the interference laws of Fresnel and Arago,” Am. J. Phys. 39, 1483–1495 (1971).
[CrossRef]

E. Collett, Polarized Light, Fundamentals and Applications (Marcel Dekker, New York, 1993).

Ferguson, J. L.

J. L. Ferguson, “A simple, bright demonstration of the interference of polarized light,” Am. J. Phys. 52, 1141–1142 (1984).
[CrossRef]

Fresnel, A. J.

D. F. J. Arago, A. J. Fresnel, “On the action of rays of polarized light upon each other,” Ann. Chimie Physique, 288 (1819).

Hanau, R.

R. Hanau, “Interference of linearly polarized light with perpendicular polarizations,” Am. J. Phys. 31, 303–304 (1963).
[CrossRef]

Henry, M.

M. Henry, “Fresnel–Arago laws for interference in polarized light: a demonstration experiment,” Am. J. Phys. 49, 690–691 (1981).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1999).

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).

Parrent, G.

G. Parrent, P. Roman, “On the matrix formulation of the theory of partial polarization in terms of observables,” Nuovo Cimento 10, 370–388 (1960).
[CrossRef]

Roman, P.

G. Parrent, P. Roman, “On the matrix formulation of the theory of partial polarization in terms of observables,” Nuovo Cimento 10, 370–388 (1960).
[CrossRef]

Roychowdhury, H.

H. Roychowdhury, E. Wolf, “Determination of the electric cross-spectral density matrix of a random electromagnetic beam,” Opt. Commun. 226, 57–60 (2003).
[CrossRef]

Whittaker, E.

An excellent account of the historical background relating to Young’s explanation of the Fresnel–Arago experiments is given in E. Whittaker, A History of the Theories of Aether and Electricity: The Classical Theories, rev. enlarged ed. (Nelson, London, 1951).

Wolf, E.

H. Roychowdhury, E. Wolf, “Determination of the electric cross-spectral density matrix of a random electromagnetic beam,” Opt. Commun. 226, 57–60 (2003).
[CrossRef]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
[CrossRef]

E. Wolf, “Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation,” Opt. Lett. 28, 1078–1080 (2003).
[CrossRef] [PubMed]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1999).

Am. J. Phys. (4)

R. Hanau, “Interference of linearly polarized light with perpendicular polarizations,” Am. J. Phys. 31, 303–304 (1963).
[CrossRef]

E. Collett, “Mathematical formulation of the interference laws of Fresnel and Arago,” Am. J. Phys. 39, 1483–1495 (1971).
[CrossRef]

M. Henry, “Fresnel–Arago laws for interference in polarized light: a demonstration experiment,” Am. J. Phys. 49, 690–691 (1981).
[CrossRef]

J. L. Ferguson, “A simple, bright demonstration of the interference of polarized light,” Am. J. Phys. 52, 1141–1142 (1984).
[CrossRef]

Ann. Chimie Physique (1)

D. F. J. Arago, A. J. Fresnel, “On the action of rays of polarized light upon each other,” Ann. Chimie Physique, 288 (1819).

J. Opt. Soc. Am. A (1)

Nuovo Cimento (1)

G. Parrent, P. Roman, “On the matrix formulation of the theory of partial polarization in terms of observables,” Nuovo Cimento 10, 370–388 (1960).
[CrossRef]

Opt. Commun. (1)

H. Roychowdhury, E. Wolf, “Determination of the electric cross-spectral density matrix of a random electromagnetic beam,” Opt. Commun. 226, 57–60 (2003).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. A (1)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
[CrossRef]

Other (7)

English translation of Ref. 1is published in The Wave Theory of Light. Memories of Huygens, Young and Fresnel, H. Crew, ed. (American Book Co., New York, 1900), pp. 145–157.

An excellent account of the historical background relating to Young’s explanation of the Fresnel–Arago experiments is given in E. Whittaker, A History of the Theories of Aether and Electricity: The Classical Theories, rev. enlarged ed. (Nelson, London, 1951).

E. Collett, Polarized Light, Fundamentals and Applications (Marcel Dekker, New York, 1993).

C. Brosseau, Fundamentals of Polarized Light, A Statistical Optics Approach (Wiley, New York, 1998).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1999).

Since only the difference between the angles of rotation introduced by the rotators is relevant, we could employ only one rotator placed behind one of the pinholes. However, introduction of two rotators, one behind each pinhole, makes the analysis more symmetric.

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).

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Figures (1)

Fig. 1
Fig. 1

Illustration of the notation: Π1, Π2 are polarizers, P1, P2 are rotators.

Equations (25)

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Wij(r1, r2, ω)=Ei*(r1, ω)Ej(r2, ω)
(i=x, y; j=x, y),
η(r1, r2, ω)=Tr W(r1, r2, ω)[Tr W(r1, r1, ω)]1/2[Tr W(r2, r2, ω)]1/2,
P(r, ω)=1-4 Det W(r, r, ω)[Tr W(r, r, ω)]21/2.
E(r, ω)=K1E(r1, ω) exp(ikR1)R1+K2E(r2, ω) exp(ikR2)R2,
Kn=-ikdAn(n=1, 2),
Ex(r, ω)Ey(r, ω)=Ex(1)(r, ω)Ey(1)(r, ω)+Ex(2)(r, ω)Ey(2)(r, ω),
Ex(n)(r, ω)Ey(n)(r, ω)=Knexp(ikRn)Rn R(αn)P(θn)Ex(rn, x)Ey(rn, x),
Ex(n)(r, ω)Ey(n)(r, ω)=Knexp(ikRn)Rn [Ex(rn, ω)cos θn+Ey(rn, ω)sin θn]cos(θn-αn)sin(θn-αn),
S(r, ω)=Tr W(r, r, ω)=E*(r, ω)E(r, ω=[E(1)*(r, ω)+E(2)*(r, ω)]×[E(1)(r, ω)+E(2)(r, ω)].
S(r, ω)=S0(r, ω)+2 cos[(θ1-θ2)-(α1-α2)]S12(r, ω).
S0(r, ω)=S0(1)(r, ω)+S0(2)(r, ω)
S0(n)(r, ω)|Kn|2Rn2 {Sx(rn, ω)cos2 θn+Sy(rn, ω)sin2 θn+2 cos θnsin θnRe[Wxy(rn, rn, ω)]},
Sx(rn, ω)Wxx(rn, rn, ω)
Sy(rn, ω)Wyy(rn, rn, ω)
S12(r, ω)=S12(1)(r, ω)+S12(2)(r, ω)+S12(3)(r, ω)+S12(4)(r, ω),
S12(1)(r, ω)=K1*K2R1R2cos θ1cos θ2×Re[Wxx(r1, r2, ω)exp(iΔ)],
S12(2)(r, ω)=K1*K2R1R2sin θ1sin θ2×Re[Wyy(r1, r2, ω)exp(iΔ)],
S12(3)(r, ω)=K1*K2R1R2cos θ1sin θ2×Re[Wxy(r1, r2, ω)exp(iΔ)],
S12(4)(r, ω)=K1*K2R1R2sin θ1cos θ2×Re[Wyx(r1, r2, ω)exp(iΔ)],
Δ=k(R2-R1)=2π (R2-R1)λ.
S(r, ω)=S0(r, ω),
S12(r, ω)=K1*K2R1R2Re[Wxx(r1, r2, ω)exp(iΔ)].
S0(r, ω)|K1|2R12 Sx(r1, ω)+|K2|2R22 Sy(r2, ω),
S12(r, ω)=K1*K2R1R2Re[Wxy(r1, r2, ω)exp(iΔ)].

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