Abstract

We report significant speckle reduction in a laser illumination system using a vibrating multimode optical fiber bundle. The optical fiber bundle was illuminated by two independent lasers simultaneously. The beams from both lasers were first expanded and collimated and were further divided into multiple beams to illuminate the fiber optic bundle with normal and oblique incidence. Static diffusers were also placed at the input and output faces of the fiber bundle, thus introducing the spatial as well as angular diversity of illumination. Experiments were carried out both in free space and in imaging geometry configuration. Standard deviation, speckle contrast and signal-to-noise ratio of the images were computed, and the results were compared with those of white light illumination. Speckle contrast close to that of white light was obtained using a vibrating fiber bundle with combined temporal, spatial, and angular diversities of the illumination.

© 2012 Optical Society of America

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    [CrossRef]
  13. S. An, A. Lapchuk, V. Yurlov, J. Song, H. Park, J. Jang, W. Shin, S. Karagpoltsev, and S. Yun, “Speckle suppression in scanning laser display using several partially coherent beams,” Opt. Express 17, 92–103 (2009).
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  14. V. Yurlov, A. Lapchuk, S. Yun, J. Song, and H. Yang, “Speckle suppression in scanning laser display,” Appl. Opt. 47, 179–187 (2008).
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  15. V. Yurlov, A. Lapchuk, S. Yun, J. Song, I. Yeo, H. Yang, and S. An, “Speckle suppression in scanning laser displays: aberration and defocusing of the projection system,” Appl. Opt. 48, 80–90 (2009).
    [CrossRef]
  16. M. N. Akram, Z. Tong, G. Ouyang, X. Chen, and V. Kartashov, “Laser speckle reduction due to spatial and angular diversity introduced by fast scanning micromirror,” Appl. Opt. 49, 3297–3304 (2010).
    [CrossRef]
  17. I. Peled, M. Zinou, B. Greenberg, and Z. Kotler, “MEMS based speckle reduction obtained by angle diversity for fast imaging,” in Conference on Lasers and Electro-Optics/International Quantum Electronics Conference (Optical Society of America, 2009), paper JTuD44.
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    [CrossRef]
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    [CrossRef]
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  23. B. Dingel, S. Kawata, and S. Minami, “Speckle reduction with virtual incoherent laser illumination using a modified fiber array,” Optik 94, 132–136 (1993).
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    [CrossRef]
  30. H. Ambar, Y. Aoki, N. Takai, and T. Asakura, “Mechanism of speckle reduction in laser-microscope images using a rotating optical fiber,” Appl. Phys. B 38, 71–78 (1985).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2010 (4)

2009 (3)

2008 (2)

2006 (2)

J. P. Parry, J. D. Shephard, J. D. C. Jones, and D. P. Hand, “Speckle contrast reduction in a large-core fiber delivering Q-switched pulses for fluid flow measurements,” Appl. Opt. 45, 4209–4218 (2006).
[CrossRef]

O. I. Kotov, A. V. Khlybov, L. B. Liokumovich, S. I. Markov, A. V. Medvedev, V. A. Rukavishnikov, A. I. Borovkov, and D. V. Shevchenko, “Polarization modulation of light in an optical waveguide under lateral compression,” Tech. Phys. 51, 1494–1499 (2006).
[CrossRef]

2004 (3)

2000 (1)

L. Wang, T. Tschudi, M. Boeddinghaus, A. Elbert, T. Halldorsson, and P. R. Petursson, “Speckle reduction in laser projections with ultrasonic waves,” Opt. Eng. 39, 1659–1664 (2000).
[CrossRef]

1998 (1)

1996 (1)

T. Iwai and T. Asakura, “Speckle reduction in coherent information processing,” Proc. IEEE 84, 765–781 (1996).
[CrossRef]

1993 (2)

B. Dingel and S. Kawata, “Speckle-free image in a laser-diode microscope by using the optical feedback effect,” Opt. Lett. 18, 549–551 (1993).
[CrossRef]

B. Dingel, S. Kawata, and S. Minami, “Speckle reduction with virtual incoherent laser illumination using a modified fiber array,” Optik 94, 132–136 (1993).

1992 (1)

P. J. Kajenski, P. L. Fuhr, and D. R. Huston, “Mode coupling and phase modulation in vibrating waveguides,” J. Lightwave Technol. 10, 1297–1301 (1992).
[CrossRef]

1986 (1)

H. Ambar, Y. Aoki, N. Takai, and T. Asakura, “Fringe contrast improvement in speckle photograph by means of speckle reduction using vibrating optical fiber,” Optik 74, 60–64(1986).

1985 (2)

H. Ambar, Y. Aoki, N. Takai, and T. Asakura, “Mechanism of speckle reduction in laser-microscope images using a rotating optical fiber,” Appl. Phys. B 38, 71–78 (1985).
[CrossRef]

S. Jutamulia, T. Asakura, and H. Ambar, “Reduction of coherent noise using various artificial incoherent sources,” Optik 70, 52–57 (1985).

1980 (3)

1979 (1)

B. Daino, G. de Marchis, and S. Piazzolla, “Analysis and measurement of modal noise in an optical fibre,” Electron. Lett. 15, 755–756 (1979).
[CrossRef]

1978 (1)

Y. Imai and Y. Ohtsuka, “Laser speckle reduction by ultrasonic modulation,” Opt. Commun. 27, 18–22 (1978).
[CrossRef]

1976 (2)

1975 (1)

T. McKechnie, “Reduction of speckle by a moving aperture-first order statistics,” Opt. Commun. 13, 35–39 (1975).
[CrossRef]

1971 (1)

Akram, M. N.

Ambar, H.

H. Ambar, Y. Aoki, N. Takai, and T. Asakura, “Fringe contrast improvement in speckle photograph by means of speckle reduction using vibrating optical fiber,” Optik 74, 60–64(1986).

S. Jutamulia, T. Asakura, and H. Ambar, “Reduction of coherent noise using various artificial incoherent sources,” Optik 70, 52–57 (1985).

H. Ambar, Y. Aoki, N. Takai, and T. Asakura, “Mechanism of speckle reduction in laser-microscope images using a rotating optical fiber,” Appl. Phys. B 38, 71–78 (1985).
[CrossRef]

An, S.

Aoki, Y.

H. Ambar, Y. Aoki, N. Takai, and T. Asakura, “Fringe contrast improvement in speckle photograph by means of speckle reduction using vibrating optical fiber,” Optik 74, 60–64(1986).

H. Ambar, Y. Aoki, N. Takai, and T. Asakura, “Mechanism of speckle reduction in laser-microscope images using a rotating optical fiber,” Appl. Phys. B 38, 71–78 (1985).
[CrossRef]

Asakura, T.

T. Iwai and T. Asakura, “Speckle reduction in coherent information processing,” Proc. IEEE 84, 765–781 (1996).
[CrossRef]

H. Ambar, Y. Aoki, N. Takai, and T. Asakura, “Fringe contrast improvement in speckle photograph by means of speckle reduction using vibrating optical fiber,” Optik 74, 60–64(1986).

H. Ambar, Y. Aoki, N. Takai, and T. Asakura, “Mechanism of speckle reduction in laser-microscope images using a rotating optical fiber,” Appl. Phys. B 38, 71–78 (1985).
[CrossRef]

S. Jutamulia, T. Asakura, and H. Ambar, “Reduction of coherent noise using various artificial incoherent sources,” Optik 70, 52–57 (1985).

Boeddinghaus, M.

L. Wang, T. Tschudi, M. Boeddinghaus, A. Elbert, T. Halldorsson, and P. R. Petursson, “Speckle reduction in laser projections with ultrasonic waves,” Opt. Eng. 39, 1659–1664 (2000).
[CrossRef]

Borovkov, A. I.

O. I. Kotov, A. V. Khlybov, L. B. Liokumovich, S. I. Markov, A. V. Medvedev, V. A. Rukavishnikov, A. I. Borovkov, and D. V. Shevchenko, “Polarization modulation of light in an optical waveguide under lateral compression,” Tech. Phys. 51, 1494–1499 (2006).
[CrossRef]

Chen, X.

Cleven, E.

Daino, B.

B. Daino, G. Demarchis, and S. Piazzola, “Speckle and modal noise in optical fibers, theory and experiment,” Opt. Acta 27, 1151–1159 (1980).
[CrossRef]

B. Daino, G. de Marchis, and S. Piazzolla, “Analysis and measurement of modal noise in an optical fibre,” Electron. Lett. 15, 755–756 (1979).
[CrossRef]

de Marchis, G.

B. Daino, G. de Marchis, and S. Piazzolla, “Analysis and measurement of modal noise in an optical fibre,” Electron. Lett. 15, 755–756 (1979).
[CrossRef]

de Mul, F. F. M.

Demarchis, G.

B. Daino, G. Demarchis, and S. Piazzola, “Speckle and modal noise in optical fibers, theory and experiment,” Opt. Acta 27, 1151–1159 (1980).
[CrossRef]

Dingel, B.

B. Dingel, S. Kawata, and S. Minami, “Speckle reduction with virtual incoherent laser illumination using a modified fiber array,” Optik 94, 132–136 (1993).

B. Dingel and S. Kawata, “Speckle-free image in a laser-diode microscope by using the optical feedback effect,” Opt. Lett. 18, 549–551 (1993).
[CrossRef]

Elbert, A.

L. Wang, T. Tschudi, M. Boeddinghaus, A. Elbert, T. Halldorsson, and P. R. Petursson, “Speckle reduction in laser projections with ultrasonic waves,” Opt. Eng. 39, 1659–1664 (2000).
[CrossRef]

Fuhr, P. L.

P. J. Kajenski, P. L. Fuhr, and D. R. Huston, “Mode coupling and phase modulation in vibrating waveguides,” J. Lightwave Technol. 10, 1297–1301 (1992).
[CrossRef]

Goodman, J. W.

Greenberg, B.

I. Peled, M. Zinou, B. Greenberg, and Z. Kotler, “MEMS based speckle reduction obtained by angle diversity for fast imaging,” in Conference on Lasers and Electro-Optics/International Quantum Electronics Conference (Optical Society of America, 2009), paper JTuD44.

Ha, W.

Halldorsson, T.

L. Wang, T. Tschudi, M. Boeddinghaus, A. Elbert, T. Halldorsson, and P. R. Petursson, “Speckle reduction in laser projections with ultrasonic waves,” Opt. Eng. 39, 1659–1664 (2000).
[CrossRef]

L. Wang, T. Tschudi, T. Halldorsson, and P. R. Petursson, “Speckle reduction in laser projection systems by diffractive optical elements,” Appl. Opt. 37, 1770–1775 (1998).
[CrossRef]

Hand, D. P.

Huston, D. R.

P. J. Kajenski, P. L. Fuhr, and D. R. Huston, “Mode coupling and phase modulation in vibrating waveguides,” J. Lightwave Technol. 10, 1297–1301 (1992).
[CrossRef]

Imai, M.

Imai, Y.

Iwai, T.

T. Iwai and T. Asakura, “Speckle reduction in coherent information processing,” Proc. IEEE 84, 765–781 (1996).
[CrossRef]

Jang, J.

Jones, J. D. C.

Joyeux, D.

Jung, Y.

Jutamulia, S.

S. Jutamulia, T. Asakura, and H. Ambar, “Reduction of coherent noise using various artificial incoherent sources,” Optik 70, 52–57 (1985).

Kajenski, P. J.

P. J. Kajenski, P. L. Fuhr, and D. R. Huston, “Mode coupling and phase modulation in vibrating waveguides,” J. Lightwave Technol. 10, 1297–1301 (1992).
[CrossRef]

Karagpoltsev, S.

Kartashov, V.

Kasazumi, K.

K. Kasazumi, Y. Kitaoka, K. Mizuuchi, and K. Yamamoto, “A practical laser projector with new illumination optics for reduction of speckle noise,” Jpn. J. Appl. Phys. 43, 5904–5906 (2004).
[CrossRef]

Katagiri, B.

Kawakami, T.

Kawata, S.

B. Dingel and S. Kawata, “Speckle-free image in a laser-diode microscope by using the optical feedback effect,” Opt. Lett. 18, 549–551 (1993).
[CrossRef]

B. Dingel, S. Kawata, and S. Minami, “Speckle reduction with virtual incoherent laser illumination using a modified fiber array,” Optik 94, 132–136 (1993).

Khlybov, A. V.

O. I. Kotov, A. V. Khlybov, L. B. Liokumovich, S. I. Markov, A. V. Medvedev, V. A. Rukavishnikov, A. I. Borovkov, and D. V. Shevchenko, “Polarization modulation of light in an optical waveguide under lateral compression,” Tech. Phys. 51, 1494–1499 (2006).
[CrossRef]

Kim, J. K.

Kitaoka, Y.

K. Kasazumi, Y. Kitaoka, K. Mizuuchi, and K. Yamamoto, “A practical laser projector with new illumination optics for reduction of speckle noise,” Jpn. J. Appl. Phys. 43, 5904–5906 (2004).
[CrossRef]

Kotler, Z.

I. Peled, M. Zinou, B. Greenberg, and Z. Kotler, “MEMS based speckle reduction obtained by angle diversity for fast imaging,” in Conference on Lasers and Electro-Optics/International Quantum Electronics Conference (Optical Society of America, 2009), paper JTuD44.

Kotov, O. I.

O. I. Kotov, A. V. Khlybov, L. B. Liokumovich, S. I. Markov, A. V. Medvedev, V. A. Rukavishnikov, A. I. Borovkov, and D. V. Shevchenko, “Polarization modulation of light in an optical waveguide under lateral compression,” Tech. Phys. 51, 1494–1499 (2006).
[CrossRef]

Kubota, S.

Kuratomi, Y.

Lapchuk, A.

Lee, S.

Liokumovich, L. B.

O. I. Kotov, A. V. Khlybov, L. B. Liokumovich, S. I. Markov, A. V. Medvedev, V. A. Rukavishnikov, A. I. Borovkov, and D. V. Shevchenko, “Polarization modulation of light in an optical waveguide under lateral compression,” Tech. Phys. 51, 1494–1499 (2006).
[CrossRef]

Lowenthal, S.

Markov, S. I.

O. I. Kotov, A. V. Khlybov, L. B. Liokumovich, S. I. Markov, A. V. Medvedev, V. A. Rukavishnikov, A. I. Borovkov, and D. V. Shevchenko, “Polarization modulation of light in an optical waveguide under lateral compression,” Tech. Phys. 51, 1494–1499 (2006).
[CrossRef]

McKechnie, T.

T. McKechnie, “Reduction of speckle by a moving aperture-first order statistics,” Opt. Commun. 13, 35–39 (1975).
[CrossRef]

Medvedev, A. V.

O. I. Kotov, A. V. Khlybov, L. B. Liokumovich, S. I. Markov, A. V. Medvedev, V. A. Rukavishnikov, A. I. Borovkov, and D. V. Shevchenko, “Polarization modulation of light in an optical waveguide under lateral compression,” Tech. Phys. 51, 1494–1499 (2006).
[CrossRef]

Minami, S.

B. Dingel, S. Kawata, and S. Minami, “Speckle reduction with virtual incoherent laser illumination using a modified fiber array,” Optik 94, 132–136 (1993).

Mizuuchi, K.

K. Kasazumi, Y. Kitaoka, K. Mizuuchi, and K. Yamamoto, “A practical laser projector with new illumination optics for reduction of speckle noise,” Jpn. J. Appl. Phys. 43, 5904–5906 (2004).
[CrossRef]

Nafarrate, A. B.

Norton, R. E.

Oh, K.

Ohtsuka, Y.

Ouyang, G.

Park, H.

Parry, J. P.

Peled, I.

I. Peled, M. Zinou, B. Greenberg, and Z. Kotler, “MEMS based speckle reduction obtained by angle diversity for fast imaging,” in Conference on Lasers and Electro-Optics/International Quantum Electronics Conference (Optical Society of America, 2009), paper JTuD44.

Petoukhova, A. L.

Petursson, P. R.

L. Wang, T. Tschudi, M. Boeddinghaus, A. Elbert, T. Halldorsson, and P. R. Petursson, “Speckle reduction in laser projections with ultrasonic waves,” Opt. Eng. 39, 1659–1664 (2000).
[CrossRef]

L. Wang, T. Tschudi, T. Halldorsson, and P. R. Petursson, “Speckle reduction in laser projection systems by diffractive optical elements,” Appl. Opt. 37, 1770–1775 (1998).
[CrossRef]

Piazzola, S.

B. Daino, G. Demarchis, and S. Piazzola, “Speckle and modal noise in optical fibers, theory and experiment,” Opt. Acta 27, 1151–1159 (1980).
[CrossRef]

Piazzolla, S.

B. Daino, G. de Marchis, and S. Piazzolla, “Analysis and measurement of modal noise in an optical fibre,” Electron. Lett. 15, 755–756 (1979).
[CrossRef]

Rawson, E. G.

Rukavishnikov, V. A.

O. I. Kotov, A. V. Khlybov, L. B. Liokumovich, S. I. Markov, A. V. Medvedev, V. A. Rukavishnikov, A. I. Borovkov, and D. V. Shevchenko, “Polarization modulation of light in an optical waveguide under lateral compression,” Tech. Phys. 51, 1494–1499 (2006).
[CrossRef]

Satoh, H.

Sekiya, K.

Shephard, J. D.

Shevchenko, D. V.

O. I. Kotov, A. V. Khlybov, L. B. Liokumovich, S. I. Markov, A. V. Medvedev, V. A. Rukavishnikov, A. I. Borovkov, and D. V. Shevchenko, “Polarization modulation of light in an optical waveguide under lateral compression,” Tech. Phys. 51, 1494–1499 (2006).
[CrossRef]

Shin, W.

Song, J.

Steenbergen, W.

Suzuki, Y.

Takai, N.

H. Ambar, Y. Aoki, N. Takai, and T. Asakura, “Fringe contrast improvement in speckle photograph by means of speckle reduction using vibrating optical fiber,” Optik 74, 60–64(1986).

H. Ambar, Y. Aoki, N. Takai, and T. Asakura, “Mechanism of speckle reduction in laser-microscope images using a rotating optical fiber,” Appl. Phys. B 38, 71–78 (1985).
[CrossRef]

Tomiyama, T.

Tong, Z.

Trisnadi, J. I.

Tschudi, T.

L. Wang, T. Tschudi, M. Boeddinghaus, A. Elbert, T. Halldorsson, and P. R. Petursson, “Speckle reduction in laser projections with ultrasonic waves,” Opt. Eng. 39, 1659–1664 (2000).
[CrossRef]

L. Wang, T. Tschudi, T. Halldorsson, and P. R. Petursson, “Speckle reduction in laser projection systems by diffractive optical elements,” Appl. Opt. 37, 1770–1775 (1998).
[CrossRef]

Uchida, T.

Wang, L.

L. Wang, T. Tschudi, M. Boeddinghaus, A. Elbert, T. Halldorsson, and P. R. Petursson, “Speckle reduction in laser projections with ultrasonic waves,” Opt. Eng. 39, 1659–1664 (2000).
[CrossRef]

L. Wang, T. Tschudi, T. Halldorsson, and P. R. Petursson, “Speckle reduction in laser projection systems by diffractive optical elements,” Appl. Opt. 37, 1770–1775 (1998).
[CrossRef]

Yamamoto, K.

K. Kasazumi, Y. Kitaoka, K. Mizuuchi, and K. Yamamoto, “A practical laser projector with new illumination optics for reduction of speckle noise,” Jpn. J. Appl. Phys. 43, 5904–5906 (2004).
[CrossRef]

Yang, H.

Yeo, I.

Yun, S.

Yurlov, V.

Zinou, M.

I. Peled, M. Zinou, B. Greenberg, and Z. Kotler, “MEMS based speckle reduction obtained by angle diversity for fast imaging,” in Conference on Lasers and Electro-Optics/International Quantum Electronics Conference (Optical Society of America, 2009), paper JTuD44.

Appl. Opt. (8)

S. Kubota and J. W. Goodman, “Very efficient speckle contrast reduction realized by a moving diffuser device,” Appl. Opt. 49, 4385–4391 (2010).
[CrossRef]

L. Wang, T. Tschudi, T. Halldorsson, and P. R. Petursson, “Speckle reduction in laser projection systems by diffractive optical elements,” Appl. Opt. 37, 1770–1775 (1998).
[CrossRef]

V. Yurlov, A. Lapchuk, S. Yun, J. Song, and H. Yang, “Speckle suppression in scanning laser display,” Appl. Opt. 47, 179–187 (2008).
[CrossRef]

V. Yurlov, A. Lapchuk, S. Yun, J. Song, I. Yeo, H. Yang, and S. An, “Speckle suppression in scanning laser displays: aberration and defocusing of the projection system,” Appl. Opt. 48, 80–90 (2009).
[CrossRef]

M. N. Akram, Z. Tong, G. Ouyang, X. Chen, and V. Kartashov, “Laser speckle reduction due to spatial and angular diversity introduced by fast scanning micromirror,” Appl. Opt. 49, 3297–3304 (2010).
[CrossRef]

Y. Imai, M. Imai, and Y. Ohtsuka, “Optical coherence modulation by ultrasonic waves: application to speckle reduction,” Appl. Opt. 19, 3541–3544 (1980).
[CrossRef]

J. P. Parry, J. D. Shephard, J. D. C. Jones, and D. P. Hand, “Speckle contrast reduction in a large-core fiber delivering Q-switched pulses for fluid flow measurements,” Appl. Opt. 45, 4209–4218 (2006).
[CrossRef]

A. L. Petoukhova, E. Cleven, F. F. M. de Mul, and W. Steenbergen, “Suppression of dynamic laser speckle signals in multimode fibers of various lengths,” Appl. Opt. 43, 2059–2065 (2004).
[CrossRef]

Appl. Phys. B (1)

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Figures (9)

Fig. 1.
Fig. 1.

(a) Schematic diagram illustrating the speckle reductions using two independent lasers illuminating the diffuser at an angle. (b) Experimental setup for speckle reduction using multimode fiber bundle with combined effect of temporal, spatial, and angular diversity.

Fig. 2.
Fig. 2.

(a) Image of speckle pattern generated by single multimode fiber. (b) Image of the speckle pattern generated by MMFB illuminated with a laser of beam size 1.5 mm. (c) Image of speckle pattern generated by MMFB with beam size 10 mm.

Fig. 3.
Fig. 3.

Images of the speckle patterns generated by MMFB illuminated with beam size >10mm with (a) a single laser switched on, (b) two lasers switched on without vibrating fiber bundle, and (c) a single laser and (d) two lasers with vibrating fiber bundle.

Fig. 4.
Fig. 4.

Intensity distributions of speckle images shown in Fig. 3; (a) single laser switched on and (b) two lasers switched on without vibrating fiber bundle; (c) a single laser and (d) two lasers with vibrating fiber bundle, respectively.

Fig. 5.
Fig. 5.

Images of speckle patterns with angular, spatial, and temporal diversity with one laser switched on and static MMFB illuminated with (a) single beam at normal incidence; (b) two beams at angular incidence (left and right); (c) all three beams made incidence; and with vibrating MMFB; (d) single beam at normal incidence; (e) two beams at angular incidence (left and right); and (f) all three beams made incidence.

Fig. 6.
Fig. 6.

Corresponding intensity distributions of speckle patterns shown in Fig. 5, (a) single beam, (b) two beams, and (c) three beams made incidence with static MMFB; and (d) single beam, (e) two beams, and (f) three beams made incidence with vibrating MMFB.

Fig. 7.
Fig. 7.

Images of speckle patterns with two lasers switched on simultaneously and static MMFB illuminated with (a) single beam at normal incidence, (b) two beams at angular incidence (left and right), and (c) three beams made incidence and with vibrating MMFB; (d) single beam at normal incidence, (e) two beams at angular incidence (left and right), and (f) three beams made incidence.

Fig. 8.
Fig. 8.

Corresponding intensity distribution of speckle patterns shown in Fig. 7 with two lasers switched on simultaneously and static MMFB illuminated with (a) single beam at normal incidence, (b) two beams at angular incidence, and (c) all three beams made incidence; and with vibrating MMFB (d) single beam at normal incidence, (e) two beams at angular incidence, and (f) all three beams made incidence.

Fig. 9.
Fig. 9.

Images of speckle patterns obtained with (a) vibrating MMFB illuminated with three beams and two lasers switched on, (b) MMFB illuminated with white light, and (c) and (d) are the respective intensity distribution patterns.

Tables (2)

Tables Icon

Table 1. Variation of Standard Deviation, Speckle Contrast, and Signal-to-Noise ratio of the Speckle Patterns Generated by Static MMFB with Different Illuminating Beams

Tables Icon

Table 2. Variation of Standard Deviation, Speckle Contrast, and Signal-to-Noise Ratio of the Speckle Patterns Generated by Vibrating MMFB with Different Illuminating Beams

Equations (4)

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pI(I)=1Iexp(II),
C=σII=I2I2I,
SN=1C=IσI.
C=M+K+1KM,

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