Abstract

A time-domain equivalent of the van Cittert–Zernike theorem is formulated. The two-time correlation function of an incoherent source after temporal gating and chromatic dispersion is derived. Significant spectral dispersion establishes partial temporal coherence, in a way similar to the establishment of partial spatial coherence in the far field for a spatially incoherent source after propagation through an aperture and diffraction. It is shown theoretically and experimentally that the temporal degree of coherence of the source after temporal gating and dispersive propagation is related to the modulus of the Fourier transform of the temporal transmission of the gate. The derivation of the two-time correlation function of such a source is applied to an interferometric measurement of chromatic dispersion that uses a time-gated incoherent source.

© 2004 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, 1980).
  2. P. Tournois, J.-L. Vernet, and G. Bienvenu, “Sur l’analogie optique de certains montages électroniques: formation d’images temporelles de signaux électriques,” C. R. Hebd. Seances Acad. Sci. 267, 375–378 (1968).
  3. B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
    [CrossRef]
  4. C. V. Bennett and B. H. Kolner, “Upconversion time microscope demonstrating 103× magnification of femtosecond waveforms,” Opt. Lett. 24, 783–786 (1999).
    [CrossRef]
  5. B. H. Kolner, “The pinhole time camera,” J. Opt. Soc. Am. A 14, 3349–3357 (1997).
    [CrossRef]
  6. L. Lepetit, G. Chériaux, and M. Joffre, “Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectroscopy,” J. Opt. Soc. Am. B 12, 2467–2474 (1995).
    [CrossRef]
  7. V. Wong and I. A. Walmsley, “Analysis of ultrashort pulse-shape measurement using linear interferometer,” Opt. Lett. 19, 287–289 (1994).
    [CrossRef]
  8. M. Françon, Optical Interferometry (Academic, New York, 1966).
  9. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).
  10. I. Kaminov and T. Li, eds., Optical Fiber Telecommunications (Academic, San Diego, Calif., 2002).
  11. M. G. Raymer, J. Cooper, H. J. Carmichael, M. Beck, and D. T. Smithey, “Ultrafast measurement of optical-field statistics by dc-balanced homodyne detection,” J. Opt. Soc. Am. B 12, 1801–1812 (1995).
    [CrossRef]
  12. C. Dorrer, “Measurement of chromatic dispersion using direct instantaneous frequency measurement,” Opt. Lett. 29, 204–206 (2004).
    [CrossRef] [PubMed]
  13. C. Dorrer and I. Kang, “Simultaneous characterization of telecommunication optical pulses and modulators,” Opt. Lett. 27, 1315–1317 (2002).
    [CrossRef]
  14. R. M. Fortenberry and W. V. Sorin, “Apparatus for characterizing short optical pulses,” U.S. patent 5, 684, 586 (November 4, 1997).
  15. A. Weling and D. H. Auston, “Novel sources and detectors for coherent tunable narrow-band terahertz radiation in free space,” J. Opt. Soc. Am. B 13, 2783–2791 (1996).
    [CrossRef]

2004 (1)

2002 (1)

1999 (1)

1997 (1)

1996 (1)

1995 (2)

1994 (2)

V. Wong and I. A. Walmsley, “Analysis of ultrashort pulse-shape measurement using linear interferometer,” Opt. Lett. 19, 287–289 (1994).
[CrossRef]

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
[CrossRef]

1968 (1)

P. Tournois, J.-L. Vernet, and G. Bienvenu, “Sur l’analogie optique de certains montages électroniques: formation d’images temporelles de signaux électriques,” C. R. Hebd. Seances Acad. Sci. 267, 375–378 (1968).

Auston, D. H.

Beck, M.

Bennett, C. V.

Bienvenu, G.

P. Tournois, J.-L. Vernet, and G. Bienvenu, “Sur l’analogie optique de certains montages électroniques: formation d’images temporelles de signaux électriques,” C. R. Hebd. Seances Acad. Sci. 267, 375–378 (1968).

Carmichael, H. J.

Chériaux, G.

Cooper, J.

Dorrer, C.

Joffre, M.

Kang, I.

Kolner, B. H.

Lepetit, L.

Raymer, M. G.

Smithey, D. T.

Tournois, P.

P. Tournois, J.-L. Vernet, and G. Bienvenu, “Sur l’analogie optique de certains montages électroniques: formation d’images temporelles de signaux électriques,” C. R. Hebd. Seances Acad. Sci. 267, 375–378 (1968).

Vernet, J.-L.

P. Tournois, J.-L. Vernet, and G. Bienvenu, “Sur l’analogie optique de certains montages électroniques: formation d’images temporelles de signaux électriques,” C. R. Hebd. Seances Acad. Sci. 267, 375–378 (1968).

Walmsley, I. A.

Weling, A.

Wong, V.

C. R. Hebd. Seances Acad. Sci. (1)

P. Tournois, J.-L. Vernet, and G. Bienvenu, “Sur l’analogie optique de certains montages électroniques: formation d’images temporelles de signaux électriques,” C. R. Hebd. Seances Acad. Sci. 267, 375–378 (1968).

IEEE J. Quantum Electron. (1)

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
[CrossRef]

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

J. Opt. Soc. Am. B (3)

Opt. Lett. (4)

Other (5)

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, 1980).

M. Françon, Optical Interferometry (Academic, New York, 1966).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).

I. Kaminov and T. Li, eds., Optical Fiber Telecommunications (Academic, San Diego, Calif., 2002).

R. M. Fortenberry and W. V. Sorin, “Apparatus for characterizing short optical pulses,” U.S. patent 5, 684, 586 (November 4, 1997).

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Figures (7)

Fig. 1
Fig. 1

Illustration of the temporal van Cittert–Zernike theorem. An incoherent source with field EINC is temporally gated by gate R(t), leading to field E1, which then propagates into a medium with spectral dispersion φ(ω).

Fig. 2
Fig. 2

Setup for the experimental verification of the temporal van Cittert–Zernike theorem. An ASE source is gated by an electroabsorption modulator driven by a voltage pulse after proper biasing. The temporal intensity is measured after propagation in a spool of SSMF and a free-space Michelson interferometer.

Fig. 3
Fig. 3

Spectrum of the ASE source.

Fig. 4
Fig. 4

Temporal intensities measured for (a) τ=2.2 ps, (b) τ=7.3 ps, (c) τ=12.3 ps, and (d) τ=17.2 ps.

Fig. 5
Fig. 5

Magnitude of the temporal degree of coherence as a function of the relative delay between the two fields (filled circles) compared with the amplitude of the Fourier transform of the transmission of the time aperture (continuous curve).

Fig. 6
Fig. 6

Implementation of the direct instantaneous frequency measurement with (a) a short-pulse source and (b) a gated incoherent source. In both cases the source propagates into the device under test and an interferometer, and the temporal and spectral intensities of the output light are measured.

Fig. 7
Fig. 7

Group delay measured on (a) a 4-km SSMF and (b) a DCM (b) with the gated incoherent source (continuous curves) and the short-pulse source (filled circles). In each case the difference between the two values is plotted with a dashed curve.

Equations (28)

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C(t, t)=E(t)E*(t),
C(ω, ω)=E˜(ω)E˜*(ω),
IINC(ω)=ΓINC(τ)exp(iωτ)dτ.
E1(t)=R(t)EINC(t).
E˜2(ω)=E˜1(ω)exp[iφ(ω)].
E2(t)=E1(t)K(t, t)dt.
C2(t1, t2)=C1(t1, t2)×K(t1, t1)K*(t2, t2)dt1dt2.
K(t, t)=[2πiΩ(t-t)]1/2 exp-i 0t-tΩ(t)dt.
E2(t)=[2πiΩ(t)]1/2 exp-i0tΩ(t)dt×E1(t)exp[iΩ(t)t]dt=[2iπΩ(t)]1/2E˜1[Ω(t)]exp-i0tΩ(t)dt.
I2(t)=C2(t, t)=2πΩ(t)I1[Ω(t)].
C1(ω1, ω2)=R˜(ω1)E˜INC(ω1-ω1)×R˜*(ω2)E˜INC*(ω2-ω2)dω1dω.
C1(ω1, ω2)=IINCω1+ω22R(ω1-ω2),
C2(t1, t2)=2π[Ω(t1)Ω(t2)]1/2IINC×Ω(t1)+Ω(t2)2R[Ω(t1)-Ω(t2)]×exp-it2t1Ω(t)dt.
μ2(t1, t2)=C2(t1, t2)[I2(t1)I2(t2)]1/2.
μ2(t1, t2)=R[Ω(t1)-Ω(t2)]exp-it2t1Ω(t)dt.
|μ2(t1, t2)|=R[Ω(t1)-Ω(t2)].
|μ2(t1, t2)|=|R(t)|2 expi t1-t22φ(2)tdt.
|E2(t)+E2(t-τ)|2=|I2(t)|2+|I2(t-τ)|2+2 Re[C2(t, t-τ)].
E2(t)=[2iπΩ(t)]1/2 E˜1[Ω(t)]exp-i0tΩ(t)dt.
I3(t)=|E2(t)+E2(t-τ)|2=I2(t)+I2(t-τ)+E2(t)E2*(t-τ)+c.c.
E2(t)E2*(t-τ)=2πΩ(t-τ/2)I1[Ω(t-τ/2)]×exp[iτΩ(t-τ/2)],
I3(ω)=|E˜2(ω)+E˜2(ω)exp(iωτ)|2=I2(ω)[1+cos(ωτ)].
I3(t)=|E2(t)+E2(t-τ)|2=I2(t)+I2(t-τ)+2 Re[C2(t, t-τ)],
C2(t, t-τ)=2π[Ω(t)Ω(t-τ)]1/2×IINCΩ(t)+Ω(t-τ)2R[Ω(t)-Ω(t-τ)]exp-itt-τΩ(t)dt.
C2(t, t-τ)=2πΩ(t-τ/2)IINC[Ω(t-τ/2)]×R[τΩ(t-τ/2)]exp[iτΩ(t-τ/2)].
τΩt-τ2+arg{R[τΩ(t-τ/2)]},
I3(ω)=|E˜2(ω)+E˜2(ω)exp(iωτ)|2=I2(ω)[1+cos(ωτ)].
I3(ω)=IINC(ω)[1+cos(ωτ)].

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