Abstract

The complex degree of coherence and the resulting van Cittert–Zernike theorem are employed to analyze the exit of an arbitrary amplitude-division interferometer with two-beam interferences. Considering that the source is a periodic array of spatially incoherent slits and assuming negligible equivalent aberrations and no vignetting, an expression for the complex degree of coherence as a function of the position of an exit point is derived. Formulas for the location, fringe spacing, and fringe localization depth of the multilocalized fringes are given.

© 2000 Optical Society of America

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References

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  1. J W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  2. M. Born, B. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1987).
  3. P. Hariharan, “Partial coherence and localization of fringes in two-beam interference,” J. Opt. Soc. Am. 59, 1384–1386 (1969).
    [CrossRef]
  4. P. Hariharan, Optical Interferometry (Academic, Sydney, Australia, 1985).
  5. W. H. Steel, Interferometry (Cambridge U. Press, London, 1985).
  6. J. C. Wyant, “Fringe localization,” Appl. Opt. 17, 1853–1853 (1978).
    [CrossRef] [PubMed]
  7. J. M. Simon, S. A. Comastri, “Localization of interference fringes,” Am. J. Phys. 48, 665–668 (1980).
    [CrossRef]
  8. J. M. Simon, S. A. Comastri, “Fringe localization depth,” Appl. Opt. 26, 5125–5129 (1987).
    [CrossRef] [PubMed]
  9. P. Hariharan, W. H. Steel, “Fringe localization depth: a comment,” Appl. Opt. 28, 29–29 (1989).
    [CrossRef] [PubMed]
  10. J. M. Simon, S. A. Comastri, “Interferometers: equivalent sine condition,” Appl. Opt. 27, 4725–4730 (1988).
    [CrossRef] [PubMed]
  11. J. M. Simon, M. C. Simon, R. M. Echarri, M. T. Garea, “Fringe localization in interferometers illuminated by a succession of incoherent line sources,” J. Mod. Opt. 45, 2245–2254 (1998).
    [CrossRef]
  12. J. M. Simon, R. M. Echarri, M. C. Simon, M. T. Garea, “Fringe localization in wave-front division interferometers,” Cuaderno de Optica 66E, Ediciones Previas, Laboratorio de Optica, FCEN-UBA (1999), available from the authors at the address on the title page.
  13. J. M. Simon, R. M. Echarri, P. A. Walsh, “Multilocalization of interference fringes in the Mach–Zehnder interferometer,” Cuaderno de Optica 67E, Ediciones Previas, Laboratorio de Optica, FCEN-UBA (1999), available from the authors at the address on the title page.
  14. J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
    [CrossRef]
  15. K. Patorski, “Incoherent superposition of multiple self-imaging: Lau effect and moiré fringe explanation,” Opt. Acta 30, 745–758 (1983).
    [CrossRef]
  16. F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
    [CrossRef]
  17. R. Sudol, B. J. Thompson, “Lau effect: theory and experiment,” Appl. Opt. 20, 1107–1116 (1981).
    [CrossRef] [PubMed]
  18. A. W. Lohmann, J. Ojeda-Castaneda, “Spatial periodicities in partially coherent fields,” Opt. Acta 30, 475–479 (1983).
    [CrossRef]
  19. S. A. Comastri, J. M. Simon, R. Blendowske, “Generalized sine condition for image-forming systems with centering errors,” J. Opt. Soc. Am. A 16, 602–612 (1999).
    [CrossRef]
  20. J. M. Simon, S. A. Comastri, “Image-forming systems: matrix formulation of the optical invariant via Fourier optics,” J. Mod. Opt. 43, 2533–2541 (1996).
    [CrossRef]
  21. H. H. Hopkins, “Image formation by a general optical system. 1: General theory,” Appl. Opt. 24, 2491–2505 (1985).
    [CrossRef] [PubMed]
  22. J. M. Simon, S. A. Comastri, C. Tardin, “Multilocalization and the van Cittert–Zernike theorem. 2. Application to the Wollaston prism,” J. Opt. Soc. Am. A 17, 1277–1283 (2000).
    [CrossRef]

2000 (1)

1999 (1)

1998 (1)

J. M. Simon, M. C. Simon, R. M. Echarri, M. T. Garea, “Fringe localization in interferometers illuminated by a succession of incoherent line sources,” J. Mod. Opt. 45, 2245–2254 (1998).
[CrossRef]

1996 (1)

J. M. Simon, S. A. Comastri, “Image-forming systems: matrix formulation of the optical invariant via Fourier optics,” J. Mod. Opt. 43, 2533–2541 (1996).
[CrossRef]

1989 (1)

1988 (1)

1987 (1)

1985 (1)

1983 (2)

K. Patorski, “Incoherent superposition of multiple self-imaging: Lau effect and moiré fringe explanation,” Opt. Acta 30, 745–758 (1983).
[CrossRef]

A. W. Lohmann, J. Ojeda-Castaneda, “Spatial periodicities in partially coherent fields,” Opt. Acta 30, 475–479 (1983).
[CrossRef]

1981 (1)

1980 (1)

J. M. Simon, S. A. Comastri, “Localization of interference fringes,” Am. J. Phys. 48, 665–668 (1980).
[CrossRef]

1979 (2)

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

1978 (1)

1969 (1)

Blendowske, R.

Born, M.

M. Born, B. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1987).

Comastri, S. A.

Echarri, R. M.

J. M. Simon, M. C. Simon, R. M. Echarri, M. T. Garea, “Fringe localization in interferometers illuminated by a succession of incoherent line sources,” J. Mod. Opt. 45, 2245–2254 (1998).
[CrossRef]

J. M. Simon, R. M. Echarri, M. C. Simon, M. T. Garea, “Fringe localization in wave-front division interferometers,” Cuaderno de Optica 66E, Ediciones Previas, Laboratorio de Optica, FCEN-UBA (1999), available from the authors at the address on the title page.

J. M. Simon, R. M. Echarri, P. A. Walsh, “Multilocalization of interference fringes in the Mach–Zehnder interferometer,” Cuaderno de Optica 67E, Ediciones Previas, Laboratorio de Optica, FCEN-UBA (1999), available from the authors at the address on the title page.

Garea, M. T.

J. M. Simon, M. C. Simon, R. M. Echarri, M. T. Garea, “Fringe localization in interferometers illuminated by a succession of incoherent line sources,” J. Mod. Opt. 45, 2245–2254 (1998).
[CrossRef]

J. M. Simon, R. M. Echarri, M. C. Simon, M. T. Garea, “Fringe localization in wave-front division interferometers,” Cuaderno de Optica 66E, Ediciones Previas, Laboratorio de Optica, FCEN-UBA (1999), available from the authors at the address on the title page.

Goodman, J W.

J W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Gori, F.

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

Hariharan, P.

Hopkins, H. H.

Jahns, J.

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Lohmann, A. W.

A. W. Lohmann, J. Ojeda-Castaneda, “Spatial periodicities in partially coherent fields,” Opt. Acta 30, 475–479 (1983).
[CrossRef]

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Ojeda-Castaneda, J.

A. W. Lohmann, J. Ojeda-Castaneda, “Spatial periodicities in partially coherent fields,” Opt. Acta 30, 475–479 (1983).
[CrossRef]

Patorski, K.

K. Patorski, “Incoherent superposition of multiple self-imaging: Lau effect and moiré fringe explanation,” Opt. Acta 30, 745–758 (1983).
[CrossRef]

Simon, J. M.

J. M. Simon, S. A. Comastri, C. Tardin, “Multilocalization and the van Cittert–Zernike theorem. 2. Application to the Wollaston prism,” J. Opt. Soc. Am. A 17, 1277–1283 (2000).
[CrossRef]

S. A. Comastri, J. M. Simon, R. Blendowske, “Generalized sine condition for image-forming systems with centering errors,” J. Opt. Soc. Am. A 16, 602–612 (1999).
[CrossRef]

J. M. Simon, M. C. Simon, R. M. Echarri, M. T. Garea, “Fringe localization in interferometers illuminated by a succession of incoherent line sources,” J. Mod. Opt. 45, 2245–2254 (1998).
[CrossRef]

J. M. Simon, S. A. Comastri, “Image-forming systems: matrix formulation of the optical invariant via Fourier optics,” J. Mod. Opt. 43, 2533–2541 (1996).
[CrossRef]

J. M. Simon, S. A. Comastri, “Interferometers: equivalent sine condition,” Appl. Opt. 27, 4725–4730 (1988).
[CrossRef] [PubMed]

J. M. Simon, S. A. Comastri, “Fringe localization depth,” Appl. Opt. 26, 5125–5129 (1987).
[CrossRef] [PubMed]

J. M. Simon, S. A. Comastri, “Localization of interference fringes,” Am. J. Phys. 48, 665–668 (1980).
[CrossRef]

J. M. Simon, R. M. Echarri, M. C. Simon, M. T. Garea, “Fringe localization in wave-front division interferometers,” Cuaderno de Optica 66E, Ediciones Previas, Laboratorio de Optica, FCEN-UBA (1999), available from the authors at the address on the title page.

J. M. Simon, R. M. Echarri, P. A. Walsh, “Multilocalization of interference fringes in the Mach–Zehnder interferometer,” Cuaderno de Optica 67E, Ediciones Previas, Laboratorio de Optica, FCEN-UBA (1999), available from the authors at the address on the title page.

Simon, M. C.

J. M. Simon, M. C. Simon, R. M. Echarri, M. T. Garea, “Fringe localization in interferometers illuminated by a succession of incoherent line sources,” J. Mod. Opt. 45, 2245–2254 (1998).
[CrossRef]

J. M. Simon, R. M. Echarri, M. C. Simon, M. T. Garea, “Fringe localization in wave-front division interferometers,” Cuaderno de Optica 66E, Ediciones Previas, Laboratorio de Optica, FCEN-UBA (1999), available from the authors at the address on the title page.

Steel, W. H.

Sudol, R.

Tardin, C.

Thompson, B. J.

Walsh, P. A.

J. M. Simon, R. M. Echarri, P. A. Walsh, “Multilocalization of interference fringes in the Mach–Zehnder interferometer,” Cuaderno de Optica 67E, Ediciones Previas, Laboratorio de Optica, FCEN-UBA (1999), available from the authors at the address on the title page.

Wolf, B.

M. Born, B. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1987).

Wyant, J. C.

Am. J. Phys. (1)

J. M. Simon, S. A. Comastri, “Localization of interference fringes,” Am. J. Phys. 48, 665–668 (1980).
[CrossRef]

Appl. Opt. (6)

J. Mod. Opt. (2)

J. M. Simon, S. A. Comastri, “Image-forming systems: matrix formulation of the optical invariant via Fourier optics,” J. Mod. Opt. 43, 2533–2541 (1996).
[CrossRef]

J. M. Simon, M. C. Simon, R. M. Echarri, M. T. Garea, “Fringe localization in interferometers illuminated by a succession of incoherent line sources,” J. Mod. Opt. 45, 2245–2254 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Acta (2)

A. W. Lohmann, J. Ojeda-Castaneda, “Spatial periodicities in partially coherent fields,” Opt. Acta 30, 475–479 (1983).
[CrossRef]

K. Patorski, “Incoherent superposition of multiple self-imaging: Lau effect and moiré fringe explanation,” Opt. Acta 30, 745–758 (1983).
[CrossRef]

Opt. Commun. (2)

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Other (6)

P. Hariharan, Optical Interferometry (Academic, Sydney, Australia, 1985).

W. H. Steel, Interferometry (Cambridge U. Press, London, 1985).

J W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

M. Born, B. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1987).

J. M. Simon, R. M. Echarri, M. C. Simon, M. T. Garea, “Fringe localization in wave-front division interferometers,” Cuaderno de Optica 66E, Ediciones Previas, Laboratorio de Optica, FCEN-UBA (1999), available from the authors at the address on the title page.

J. M. Simon, R. M. Echarri, P. A. Walsh, “Multilocalization of interference fringes in the Mach–Zehnder interferometer,” Cuaderno de Optica 67E, Ediciones Previas, Laboratorio de Optica, FCEN-UBA (1999), available from the authors at the address on the title page.

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Figures (3)

Fig. 1
Fig. 1

General amplitude-division interferometer. (a) Entrance space: O, central point of the source; Sd and Su, points on the lower and upper borders of the source; (x, y, z), orthogonal coordinates (z normal to the source); βII and βII+Δβ, angles from the z axis to the rays OPII,loc and OPI respectively; PI and PII, images of the exit point P by inverse ray tracing through branches I and II; n, refraction index; TI, point on the ray OPII,loc such that δξ=TIPI is parallel to x; Ω˜, angle subtended by OSu at TI. (b) Exit space: O and O, images of O through branches I and II; Ploc, point at the localization surface of order zero; P, exit point at a distance Z=Ploc P; (x, y, z), coordinates at the source image through branch I (z normal to SdSu); βII, angle from the z axis to the ray OPloc; n, refractive index; T, image of TI; δξ=TP, relative coordinate; F=OPloc; F=OPloc; Λ, bifurcation angle; Ω and Ω˜, angles subtended by OSu at Ploc and T, respectively.

Fig. 2
Fig. 2

(a) Coherence pattern, (b) fringe spacing E (m) versus m, (c) localization depth LD+|m versus m, (d) localization depth LD-|m versus m.

Fig. 3
Fig. 3

Variation of parameters (distances in millimeters): (a) original parameters, (b) double δx, (c) double b, (d) double F, (e) halfN, (f) double E (0).

Equations (72)

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VOPD=LI(P)-LII(P)-LIo(P)+LIIo(P).
ω¯τ(x, y, P)=2πλ¯ [LII(P)-LI(P)]=2πλ¯ [LIIo(P)-LIo(P)]-2πλ¯ VOPD.
LI(P)=LIo(P)+L1(P)xx=y=0x+LI(P)yx=y=0y+φI(x, y, P),
LII(P)=LIIo(P)+LII(P)xx=y=0x+LII(P)yx=y=0y+φII(x, y, P),
ω¯τ(x, y, P)=a00(P)+a10(P)x+a01(P)y+Φ(x, y, P),
a00(P)=2πλ¯ [LIIo(P)-LIo(P)],
a10(P)=2πλ¯ [LII(P)-LI(P)]xx=y=0,
a01(P)=2πλ¯[LII(P)-LI(P)]yx=y=0,
Φ(x, y, P)=2πλ¯ (φII-φI)=-2πλ¯ VOPD-(a10 x+a01 y).
VIj(t-tIj)|CI|exp(iγI)LIo(P) Aj(t)exp[-iω¯(t-tIj)],
VI(t)=jVIj(t-tIj).
I(P)=[VI(t)+VII(t)][VI(t)+VII(t)]*=I(I)(P)+I(II)(P)+ΠI,II,
I(I)=VI(t)VI*(t),I(II)=VII(t)VII*(t)
ΠI,II=VI(t)VII*(t)+VII(t)VI*(t),
I(I)(P)=VI(t)VI*(t)=jVI j(t-tIj)nVIn*(t-tIn)=jVI j(t)VI j*(t),
ΠI,II=j[VI j(t-tI j)VIIj*(t-tIIj)+VI j*(t-tI j)VIIj(t-tIIj)]=2 RejVI j(t+τj)VII j*(t),
I(I)(P)=|CI|2LIo2(P) jAj(t)Aj*(t)=|CI|2LIo2(P) Itot,
ΠI,II=2|CI||CII|LIo(P)LIIo(P) Re{exp[i(γI-γII)]×jAj(t)Aj*(t)exp(-iωτ¯j)}
=2[I(I)(P)I(II)(P)]1/2 1ItotReexp(iγI,II)×σI(x, y)exp[-iω¯τ(x, y, P)]dxdy,
μI,II(P)=exp[ia00(P)]Itot σ I(x, y)×exp(-iω¯τ(x, y, P))dxdy=1Itot σ I(x, y)exp[-iΦ(x, y, P)]×exp{-i[a10(P)x+a01(P)y]}dxdy,
I(P)=I(I)(P)+I(II)(P)+2[I(I)(P)I(II)(P)]1/2×Re(μI,II(P)exp{i[γI,II-a00(P)]})=I(I)(P)+I(II)(P)+2[I(I)(P)I(II)(P)]1/2×|μI,II(P)|cos{arg[μI,II(P)]+γI,II-a00(P)}.
U(ξ, η)=EPUo(X, Y)exp-i 2πλ¯ W(X, Y)×expi n2πλ¯R (ξX+ηY)dXdY.
χ(P)=(Imax-Imin)(Imax+Imin)=2[I(I)(P)I(II)(P)]1/2[I(I)(P)+I(II)(P)] |μI,II(P)|.
sin ϕII-sin ϕII,0=sin ϕI-sin ΦI,0,
E (m)D˜λ¯d0n,
μI,II(P)|Φ=0=gItot σI(x)exp[-ia10(P)x]dx.
LI(P)=[SjP]I=pI+[SjPI],
LIo(P)=[OP]I=pI+[OPI],
LI(P)=pI+nLIo-pIn2+x2-2x (LIo-pI)n sin βI1/2
a10(P)=2πnλ¯ (sin βI-sin βII)=2πnλ¯ [sin(βII+Δβ)-sin βII].
VOPD=[SjPI]-[SjPII]-[OPI]+[OPII],
Z=PlocP,F=OPloc.
E (0)=Fλ¯nd0=-λ¯nΛ.
nδξ[sin(βII+Ω˜)-sin βII]
=nδξ[sin(βII+Ω˜)-sin βII],
nδξ[sin(βII+Ω˜)-sin βII]-Hλ¯2π a10(P),
nδξ[sin(βII+Ω˜)-sin βII]=-nΛFΩ ZF+Z.
a10(P)=2πH cos βIIHE (0) ZF+Z,
Z=F2π cos βIIH/(HE (0)a10(P))-1.
q1=H cos βIIH=-FΩH,q˜=q1δxE (0).
a10(P)=2πq1E (0) Zη(F+Z).
μI,II(P)|Φ=0=gItot j=0N-1XjXj+δxI(x)exp(-ia10(P)x)dx.
μI,II(P)|Φ=0=1N sin ζζ sin(ϑN)sin(ϑ),
ζ=ba10(P)2=nπbλ¯ (sin βI-sin βII)=πq1bE(0) Z(F+Z).
ϑ=-πλ¯ VOPD=a10(P)δx2=nπδxλ¯ (sin βI-sin βII)=πq1δxE(0) Z(F+Z).
a10(Pm)=m2π(δx).
(sin βIm-sin βIIm)=mλ¯nδx,
Zm=Fq˜/m-1=FmE(0)q1δx-mE(0).
μI,II(Pm)|Φ=0 = ±sin(πmb/δx)πmb/δx,
E (m)(F+Zm)λ¯nd0=E (0)1+ZmF=E(0)1-mE(0)/(q1δx)=E (0)1-m/q˜.
Zm±1/N=Fq˜/(m±1/N)-1.
LD+|m=Zm+1/N-Zm=Fq˜(q˜-m)(N(q˜-m)-1),
LD-|m=Zm-Zm-1/N=Fq˜(q˜-m)(N(q˜-m)+1),
LD|m=Zm+1/N-Zm-1/N=LD+|m+LD-|m=2FNq˜(1-m/q˜)2-1/(Nq˜),
LD+|(m=0)=λ¯2nΛΩ-λ¯/F,
|m|<(δx)/b.
nδξ[sin(βII+Ω˜)-sin βII]
=nδξ[sin(βII+Ω˜)-sin βII].
a10(P)=n 2πλ¯ [sin(βII+Δβ)-sin βII]n 2πλ¯ Δβ cos βII.
Ω˜-H cos βII|OTI| 1+H|OTI| sin βII,
sin(βII+Ω˜)-sin βII
Ω˜ cos βII-H cos2 βII|OTI| 1+H|OTI| sin βII.
δξ=|OTI|Δβcos βII(1-Δβ tan βII).
nδξ[sin(βII+Ω˜)-sin βII]-HnΔβ cos βII-Hλ¯2π a10(P).
Z=δξ cos βII-Λ.
|Ω||tan Ω|=|QSu|F-H sin βII.
|Ω˜||tan Ω˜|=|QSu|F-H sin βII+Z-δξ sin βII.
Ω˜=ΩFF+Z=Ω1-(δξ cos βII)/(ΛF).
nδξ[sin(βII+Ω˜)-sin βII]
nδξΩ˜ cos βII=-nΛΩF ZF+Z.
FΩ=-H cos βII,nΛFΩλ¯H cos βIIE(0).
a10(P)=ZF+Z nΛFΩ 2πλ¯H=2πH cos βIIHE(0) ZF+Z.

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