Abstract

The modulus of the complex degree of coherence is directly measured at the output of a step-index multimode optical fiber using lateral-sheering, delay-dithering Mach-Zehnder interferometer. Pumping the multimode fiber with monochromatic light always results in spatially-coherent output, whereas for the broadband pumping the modal dispersion of the fiber leads to a partially coherent output. While the coherence radius is a function of the numerical aperture only, the residual coherence outside the main peak is an interesting function of two dimensionless parameters: the number of non-degenerate modes and the ratio of the modal dispersion to the coherence time of the source. We develop a simple model describing this residual coherence and verify its predictions experimentally.

© 2014 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Cambridge University1999).
    [Crossref]
  2. J. Kim, D. T. Miller, E. Kim, S. Oh, J. Oh, and T. E. Milner, “Optical coherence tomography speckle reduction by a partially spatially coherent source,” J. Biomed. Opt. 10, 064034 (2005).
    [Crossref]
  3. A. Dhalla, J. V. Migacz, and J. A. Izatt, “Crosstalk rejection in parallel optical coherence tomography using spatially incoherent illumination with partially coherent sources,” Opt. Lett. 35, 2305–2307 (2010).
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  13. I. A. Deryugin, S. S. Abdullaev, and A. G. Mirzaev, “Coherence of the electromagnetic field in dielectric waveguides,” Sov. J. Quantum Electron. 7, 1243–1248 (1977).
    [Crossref]
  14. M. I. Dzhibladze, B. S. Lezhava, and T. Ya. Chelidze, “Coherence of laser radiation traveling along an optical fiber,” Sov. J. Quantum Electron. 4, 1181–1183 (1975).
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  22. We will often use the shortened notation for the equal-time complex degree of coherence γ ≡ γ12(τ = 0), where indices 1 and 2 represent the two spatial points between which the coherence is measured, as in [1].
  23. J. W. Goodman, Statistical Optics (Wiley1985).
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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2014 (1)

A. Efimov, K. Velizhanin, and G. Gelikonov, “Simultaneous scintillation measurements of coherent and partially coherent beams in an open atmosphere experiment,” Proc. SPIE 8971, 897105 (2014).
[Crossref]

2013 (1)

2012 (1)

2011 (2)

2010 (1)

2005 (1)

J. Kim, D. T. Miller, E. Kim, S. Oh, J. Oh, and T. E. Milner, “Optical coherence tomography speckle reduction by a partially spatially coherent source,” J. Biomed. Opt. 10, 064034 (2005).
[Crossref]

2003 (1)

1996 (1)

1993 (1)

H. Nakano, N. Miyanaga, K. Yagi, K. Tsubakimoto, T. Kanabe, M. Nakatsuka, and S. Nakai, “Partially coherent light generated by using single and multimode fibers in a high-power Nd:glass laser system,” Appl. Phys. Lett. 63, 580–582 (1993).
[Crossref]

1988 (1)

D. Veron, H. Ayral, C. Gouedard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaing, and C. Sauteret, “Optical spatial smoothing of Nd-glass laser beam,” Opt. Commun. 65, 42–46 (1988).
[Crossref]

1986 (1)

1983 (1)

M. Imai and Y. Ohtsuka, “The influence of mode-coupling on the degree of coherence in an optical fiber,” Opt. Commun. 45, 331–335 (1983).
[Crossref]

1982 (1)

M. Imai and Y. Ohtsuka, “Spatial coherence of laser light propagating in an optical fibre,” Opt. Quantum Electron. 14, 515–523 (1982).
[Crossref]

1981 (1)

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Strong flux scintillations of incoherent radiation in turbulent atmosphere,” Izvestia VUZ, Radiofizika 24, 703–708 (1981).

1980 (2)

P. Spano, “Connection between spatial coherence and modal structure in optical fibers and semiconductor lasers,” Opt. Commun. 33, 265–270 (1980).
[Crossref]

E. G. Rawson, J. W. Goodman, and R. E. Norton, “Frequency dependence of modal noise in multimode optical fibers,” J. Opt. Soc. Am. 70, 968–976 (1980).
[Crossref]

1977 (1)

I. A. Deryugin, S. S. Abdullaev, and A. G. Mirzaev, “Coherence of the electromagnetic field in dielectric waveguides,” Sov. J. Quantum Electron. 7, 1243–1248 (1977).
[Crossref]

1975 (1)

M. I. Dzhibladze, B. S. Lezhava, and T. Ya. Chelidze, “Coherence of laser radiation traveling along an optical fiber,” Sov. J. Quantum Electron. 4, 1181–1183 (1975).
[Crossref]

1974 (1)

B. Crosignani, B. Diano, and P. Di Porto, “Interference of mode patterns in optical fibers,” Opt. Commun. 11, 178–179 (1974).
[Crossref]

1973 (2)

C. Pask and A. W. Snyder, “The Van-Citter Zernike theorem for optical fibers,” Opt. Commun. 9, 95–97 (1973).
[Crossref]

B. Crosignani and P. Di Porto, “Coherence of an electromagnetic field propagating in a weakly guiding fiber,” J. Appl. Phys. 44, 4616–4617 (1973).
[Crossref]

1971 (1)

1964 (1)

Abdullaev, S. S.

I. A. Deryugin, S. S. Abdullaev, and A. G. Mirzaev, “Coherence of the electromagnetic field in dielectric waveguides,” Sov. J. Quantum Electron. 7, 1243–1248 (1977).
[Crossref]

Ayral, H.

D. Veron, H. Ayral, C. Gouedard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaing, and C. Sauteret, “Optical spatial smoothing of Nd-glass laser beam,” Opt. Commun. 65, 42–46 (1988).
[Crossref]

Banakh, V. A.

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Strong flux scintillations of incoherent radiation in turbulent atmosphere,” Izvestia VUZ, Radiofizika 24, 703–708 (1981).

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University1999).
[Crossref]

Buldakov, V. M.

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Strong flux scintillations of incoherent radiation in turbulent atmosphere,” Izvestia VUZ, Radiofizika 24, 703–708 (1981).

Chou, J.

Crosignani, B.

B. Crosignani, B. Diano, and P. Di Porto, “Interference of mode patterns in optical fibers,” Opt. Commun. 11, 178–179 (1974).
[Crossref]

B. Crosignani and P. Di Porto, “Coherence of an electromagnetic field propagating in a weakly guiding fiber,” J. Appl. Phys. 44, 4616–4617 (1973).
[Crossref]

Davidson, F. M.

Deryugin, I. A.

I. A. Deryugin, S. S. Abdullaev, and A. G. Mirzaev, “Coherence of the electromagnetic field in dielectric waveguides,” Sov. J. Quantum Electron. 7, 1243–1248 (1977).
[Crossref]

Dhalla, A.

Di Porto, P.

B. Crosignani, B. Diano, and P. Di Porto, “Interference of mode patterns in optical fibers,” Opt. Commun. 11, 178–179 (1974).
[Crossref]

B. Crosignani and P. Di Porto, “Coherence of an electromagnetic field propagating in a weakly guiding fiber,” J. Appl. Phys. 44, 4616–4617 (1973).
[Crossref]

Diano, B.

B. Crosignani, B. Diano, and P. Di Porto, “Interference of mode patterns in optical fibers,” Opt. Commun. 11, 178–179 (1974).
[Crossref]

Diebold, E. D.

Dzhibladze, M. I.

M. I. Dzhibladze, B. S. Lezhava, and T. Ya. Chelidze, “Coherence of laser radiation traveling along an optical fiber,” Sov. J. Quantum Electron. 4, 1181–1183 (1975).
[Crossref]

Efimov, A.

A. Efimov, K. Velizhanin, and G. Gelikonov, “Simultaneous scintillation measurements of coherent and partially coherent beams in an open atmosphere experiment,” Proc. SPIE 8971, 897105 (2014).
[Crossref]

A. Efimov, “Lateral-sheering, delay-dithering Mach-Zehnder interferometer for spatial coherence measurement,” Opt. Lett. 384522–4525 (2013).
[Crossref] [PubMed]

Feuermann, D.

Gelikonov, G.

A. Efimov, K. Velizhanin, and G. Gelikonov, “Simultaneous scintillation measurements of coherent and partially coherent beams in an open atmosphere experiment,” Proc. SPIE 8971, 897105 (2014).
[Crossref]

Gloge, D.

Goldstein, A.

Goodman, J. W.

Gordon, J. M.

Gouedard, C.

D. Veron, H. Ayral, C. Gouedard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaing, and C. Sauteret, “Optical spatial smoothing of Nd-glass laser beam,” Opt. Commun. 65, 42–46 (1988).
[Crossref]

Hon, N. K.

Husson, D.

D. Veron, H. Ayral, C. Gouedard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaing, and C. Sauteret, “Optical spatial smoothing of Nd-glass laser beam,” Opt. Commun. 65, 42–46 (1988).
[Crossref]

Iaconis, C.

Imai, M.

M. Imai, S. Satoh, and Y. Ohtsuka, “Complex degree of spatial coherence in an optical fiber: theory and experiment,” J. Opt. Soc. Am. A 3, 86–93 (1986).
[Crossref]

M. Imai and Y. Ohtsuka, “The influence of mode-coupling on the degree of coherence in an optical fiber,” Opt. Commun. 45, 331–335 (1983).
[Crossref]

M. Imai and Y. Ohtsuka, “Spatial coherence of laser light propagating in an optical fibre,” Opt. Quantum Electron. 14, 515–523 (1982).
[Crossref]

Izatt, J. A.

Jalali, B.

Kanabe, T.

H. Nakano, N. Miyanaga, K. Yagi, K. Tsubakimoto, T. Kanabe, M. Nakatsuka, and S. Nakai, “Partially coherent light generated by using single and multimode fibers in a high-power Nd:glass laser system,” Appl. Phys. Lett. 63, 580–582 (1993).
[Crossref]

Kim, E.

J. Kim, D. T. Miller, E. Kim, S. Oh, J. Oh, and T. E. Milner, “Optical coherence tomography speckle reduction by a partially spatially coherent source,” J. Biomed. Opt. 10, 064034 (2005).
[Crossref]

Kim, J.

J. Kim, D. T. Miller, E. Kim, S. Oh, J. Oh, and T. E. Milner, “Optical coherence tomography speckle reduction by a partially spatially coherent source,” J. Biomed. Opt. 10, 064034 (2005).
[Crossref]

Kumar, A.

Lauriou, J.

D. Veron, H. Ayral, C. Gouedard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaing, and C. Sauteret, “Optical spatial smoothing of Nd-glass laser beam,” Opt. Commun. 65, 42–46 (1988).
[Crossref]

Lezhava, B. S.

M. I. Dzhibladze, B. S. Lezhava, and T. Ya. Chelidze, “Coherence of laser radiation traveling along an optical fiber,” Sov. J. Quantum Electron. 4, 1181–1183 (1975).
[Crossref]

Martin, O.

D. Veron, H. Ayral, C. Gouedard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaing, and C. Sauteret, “Optical spatial smoothing of Nd-glass laser beam,” Opt. Commun. 65, 42–46 (1988).
[Crossref]

Mashaal, H.

Meyer, B.

D. Veron, H. Ayral, C. Gouedard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaing, and C. Sauteret, “Optical spatial smoothing of Nd-glass laser beam,” Opt. Commun. 65, 42–46 (1988).
[Crossref]

Migacz, J. V.

Miller, D. T.

J. Kim, D. T. Miller, E. Kim, S. Oh, J. Oh, and T. E. Milner, “Optical coherence tomography speckle reduction by a partially spatially coherent source,” J. Biomed. Opt. 10, 064034 (2005).
[Crossref]

Milner, T. E.

J. Kim, D. T. Miller, E. Kim, S. Oh, J. Oh, and T. E. Milner, “Optical coherence tomography speckle reduction by a partially spatially coherent source,” J. Biomed. Opt. 10, 064034 (2005).
[Crossref]

Mironov, V. L.

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Strong flux scintillations of incoherent radiation in turbulent atmosphere,” Izvestia VUZ, Radiofizika 24, 703–708 (1981).

Mirzaev, A. G.

I. A. Deryugin, S. S. Abdullaev, and A. G. Mirzaev, “Coherence of the electromagnetic field in dielectric waveguides,” Sov. J. Quantum Electron. 7, 1243–1248 (1977).
[Crossref]

Miyanaga, N.

H. Nakano, N. Miyanaga, K. Yagi, K. Tsubakimoto, T. Kanabe, M. Nakatsuka, and S. Nakai, “Partially coherent light generated by using single and multimode fibers in a high-power Nd:glass laser system,” Appl. Phys. Lett. 63, 580–582 (1993).
[Crossref]

Murty, M. V. R. K.

Nakai, S.

H. Nakano, N. Miyanaga, K. Yagi, K. Tsubakimoto, T. Kanabe, M. Nakatsuka, and S. Nakai, “Partially coherent light generated by using single and multimode fibers in a high-power Nd:glass laser system,” Appl. Phys. Lett. 63, 580–582 (1993).
[Crossref]

Nakano, H.

H. Nakano, N. Miyanaga, K. Yagi, K. Tsubakimoto, T. Kanabe, M. Nakatsuka, and S. Nakai, “Partially coherent light generated by using single and multimode fibers in a high-power Nd:glass laser system,” Appl. Phys. Lett. 63, 580–582 (1993).
[Crossref]

Nakatsuka, M.

H. Nakano, N. Miyanaga, K. Yagi, K. Tsubakimoto, T. Kanabe, M. Nakatsuka, and S. Nakai, “Partially coherent light generated by using single and multimode fibers in a high-power Nd:glass laser system,” Appl. Phys. Lett. 63, 580–582 (1993).
[Crossref]

Norton, R. E.

Oh, J.

J. Kim, D. T. Miller, E. Kim, S. Oh, J. Oh, and T. E. Milner, “Optical coherence tomography speckle reduction by a partially spatially coherent source,” J. Biomed. Opt. 10, 064034 (2005).
[Crossref]

Oh, S.

J. Kim, D. T. Miller, E. Kim, S. Oh, J. Oh, and T. E. Milner, “Optical coherence tomography speckle reduction by a partially spatially coherent source,” J. Biomed. Opt. 10, 064034 (2005).
[Crossref]

Ohtsuka, Y.

M. Imai, S. Satoh, and Y. Ohtsuka, “Complex degree of spatial coherence in an optical fiber: theory and experiment,” J. Opt. Soc. Am. A 3, 86–93 (1986).
[Crossref]

M. Imai and Y. Ohtsuka, “The influence of mode-coupling on the degree of coherence in an optical fiber,” Opt. Commun. 45, 331–335 (1983).
[Crossref]

M. Imai and Y. Ohtsuka, “Spatial coherence of laser light propagating in an optical fibre,” Opt. Quantum Electron. 14, 515–523 (1982).
[Crossref]

Pask, C.

C. Pask and A. W. Snyder, “The Van-Citter Zernike theorem for optical fibers,” Opt. Commun. 9, 95–97 (1973).
[Crossref]

Prabhakar, S.

Rawson, E. G.

Ricklin, J. C.

Rostaing, M.

D. Veron, H. Ayral, C. Gouedard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaing, and C. Sauteret, “Optical spatial smoothing of Nd-glass laser beam,” Opt. Commun. 65, 42–46 (1988).
[Crossref]

Satoh, S.

Sauteret, C.

D. Veron, H. Ayral, C. Gouedard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaing, and C. Sauteret, “Optical spatial smoothing of Nd-glass laser beam,” Opt. Commun. 65, 42–46 (1988).
[Crossref]

Sienicki, T.

Singh, R. P.

Snyder, A. W.

C. Pask and A. W. Snyder, “The Van-Citter Zernike theorem for optical fibers,” Opt. Commun. 9, 95–97 (1973).
[Crossref]

Spano, P.

P. Spano, “Connection between spatial coherence and modal structure in optical fibers and semiconductor lasers,” Opt. Commun. 33, 265–270 (1980).
[Crossref]

Tan, Z.

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

Tsubakimoto, K.

H. Nakano, N. Miyanaga, K. Yagi, K. Tsubakimoto, T. Kanabe, M. Nakatsuka, and S. Nakai, “Partially coherent light generated by using single and multimode fibers in a high-power Nd:glass laser system,” Appl. Phys. Lett. 63, 580–582 (1993).
[Crossref]

Vaity, P.

Velizhanin, K.

A. Efimov, K. Velizhanin, and G. Gelikonov, “Simultaneous scintillation measurements of coherent and partially coherent beams in an open atmosphere experiment,” Proc. SPIE 8971, 897105 (2014).
[Crossref]

Veron, D.

D. Veron, H. Ayral, C. Gouedard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaing, and C. Sauteret, “Optical spatial smoothing of Nd-glass laser beam,” Opt. Commun. 65, 42–46 (1988).
[Crossref]

Walmsley, I. A.

Wang, C.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University1999).
[Crossref]

Ya. Chelidze, T.

M. I. Dzhibladze, B. S. Lezhava, and T. Ya. Chelidze, “Coherence of laser radiation traveling along an optical fiber,” Sov. J. Quantum Electron. 4, 1181–1183 (1975).
[Crossref]

Yagi, K.

H. Nakano, N. Miyanaga, K. Yagi, K. Tsubakimoto, T. Kanabe, M. Nakatsuka, and S. Nakai, “Partially coherent light generated by using single and multimode fibers in a high-power Nd:glass laser system,” Appl. Phys. Lett. 63, 580–582 (1993).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

H. Nakano, N. Miyanaga, K. Yagi, K. Tsubakimoto, T. Kanabe, M. Nakatsuka, and S. Nakai, “Partially coherent light generated by using single and multimode fibers in a high-power Nd:glass laser system,” Appl. Phys. Lett. 63, 580–582 (1993).
[Crossref]

Izvestia VUZ, Radiofizika (1)

V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Strong flux scintillations of incoherent radiation in turbulent atmosphere,” Izvestia VUZ, Radiofizika 24, 703–708 (1981).

J. Appl. Phys. (1)

B. Crosignani and P. Di Porto, “Coherence of an electromagnetic field propagating in a weakly guiding fiber,” J. Appl. Phys. 44, 4616–4617 (1973).
[Crossref]

J. Biomed. Opt. (1)

J. Kim, D. T. Miller, E. Kim, S. Oh, J. Oh, and T. E. Milner, “Optical coherence tomography speckle reduction by a partially spatially coherent source,” J. Biomed. Opt. 10, 064034 (2005).
[Crossref]

J. Opt. Soc. Am. (2)

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

Opt. Commun. (5)

M. Imai and Y. Ohtsuka, “The influence of mode-coupling on the degree of coherence in an optical fiber,” Opt. Commun. 45, 331–335 (1983).
[Crossref]

D. Veron, H. Ayral, C. Gouedard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaing, and C. Sauteret, “Optical spatial smoothing of Nd-glass laser beam,” Opt. Commun. 65, 42–46 (1988).
[Crossref]

C. Pask and A. W. Snyder, “The Van-Citter Zernike theorem for optical fibers,” Opt. Commun. 9, 95–97 (1973).
[Crossref]

P. Spano, “Connection between spatial coherence and modal structure in optical fibers and semiconductor lasers,” Opt. Commun. 33, 265–270 (1980).
[Crossref]

B. Crosignani, B. Diano, and P. Di Porto, “Interference of mode patterns in optical fibers,” Opt. Commun. 11, 178–179 (1974).
[Crossref]

Opt. Express (1)

Opt. Lett. (5)

Opt. Quantum Electron. (1)

M. Imai and Y. Ohtsuka, “Spatial coherence of laser light propagating in an optical fibre,” Opt. Quantum Electron. 14, 515–523 (1982).
[Crossref]

Proc. SPIE (1)

A. Efimov, K. Velizhanin, and G. Gelikonov, “Simultaneous scintillation measurements of coherent and partially coherent beams in an open atmosphere experiment,” Proc. SPIE 8971, 897105 (2014).
[Crossref]

Sov. J. Quantum Electron. (2)

I. A. Deryugin, S. S. Abdullaev, and A. G. Mirzaev, “Coherence of the electromagnetic field in dielectric waveguides,” Sov. J. Quantum Electron. 7, 1243–1248 (1977).
[Crossref]

M. I. Dzhibladze, B. S. Lezhava, and T. Ya. Chelidze, “Coherence of laser radiation traveling along an optical fiber,” Sov. J. Quantum Electron. 4, 1181–1183 (1975).
[Crossref]

Other (4)

M. Born and E. Wolf, Principles of Optics (Cambridge University1999).
[Crossref]

We will often use the shortened notation for the equal-time complex degree of coherence γ ≡ γ12(τ = 0), where indices 1 and 2 represent the two spatial points between which the coherence is measured, as in [1].

J. W. Goodman, Statistical Optics (Wiley1985).

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

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Figures (7)

Fig. 1
Fig. 1 Schematic of the lateral-sheering, time-dithering MZ interferometer. Principal rays from object points P1 (red) and P2 (green) are shown to split at the first surface of the beamsplitter into horizontal (solid) and vertical (dashed) arms thus creating two displaced images on the Detector Plane for each point. The image of P2 through the vertical arm and the image of P1 through the horizontal arm overlap and interfere as shown on the right.
Fig. 2
Fig. 2 Modulus of the complex degree of coherence |γ| at the output of a step-index MMF pumped with a monochromatic source (black crosses) and a broadband source with different optical bandwidths: red triangles—0.75 nm; green circles—7 nm; blue squares—39 nm.
Fig. 3
Fig. 3 Modulus of the complex degree of coherence |γ| for a pair of widely separated points at the output of a MMF on log-log scale. Left column—Gaussian spectrum, right column—Lorentzian spectrum. Number of non-degenerate modes is listed for each panel. Color represents the probability of a particular value of |γ| from 0 (deep blue) to 1 (red).
Fig. 4
Fig. 4 Average residual coherence from the density plots of Fig. 3 for Gaussian (left) and Lorentzian (right) spectra. Number of modes used to compute each curve is listed on the right.
Fig. 5
Fig. 5 Modulus of the complex degree of coherence |γ| for a pair of widely separated points at the output of a MMF as a function of the number of modes on a log-log scale. Left column—Gaussian spectrum, right column—Lorentzian spectrum. The value of the parameter Δt/tc is listed for each panel. Color represents the probability of a particular value of |γ| from 0 (deep blue) to 1 (red).
Fig. 6
Fig. 6 Mean value of the complex degree of coherence |γ| from the density plots of Fig. 5 for Gaussian (left) and Lorentzian (right) spectra. Δt/tc values are shown on the right. Dashed line corresponds to Δt/tc = 1000.
Fig. 7
Fig. 7 Residual coherence from experiments (circles) and the model (solid and dashed lines) as a function of Δt/tc.

Equations (6)

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γ 12 γ ( R 1 , R 2 ) = E ( R 1 , t ) E * ( R 2 , t ) | E ( R 1 , t ) | 2 | E ( R 2 , t ) | 2 ,
γ 12 γ ( R 1 , R 2 ) = k , l u k 1 u l 2 * k , l u k 1 u l 1 * k , l u k 2 u l 2 *
u k s u l p * = a k s a l p A ( ω ) A * ( ω ) e i [ β k ( ω ) β l ( ω ) ] L i [ ω ω ] t d ω d ω
u k s u l p * = a k s a l p W ( ω ) e i [ β k ( ω ) β l ( ω ) ] L d ω
u k s u l p * = a k s a l p e i [ β k ( ω 0 ) β l ( ω 0 ) ] L i [ τ k τ l ] ω 0 L W ( ω ) e i [ τ k τ l ] ω L d ω
γ 12 = k , l a k 1 a l 2 B ( Δ τ k l L ) [ k , l a k 1 a l 1 B ( Δ τ k l L ) ] [ k , l a k 2 a l 2 B ( Δ τ k l L ) ]

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