Abstract

A fiber bundle arrangement containing a distribution of fiber lengths has been proposed in the literature to produce a partially spatially coherent beam. Light input to the bundle with limited temporal coherence is translated into limited spatial coherence. Expressions are developed for the bundle pupil autocorrelation function and far-field irradiance pattern. A numerical simulation approach is implemented and results are compared with a speckle-free result. The fiber bundle approach tends to create an irradiance pattern whose average shape matches the pattern produced by a single fiber. A “smoothed” far-field pattern is obtained if the fiber length difference is much greater than the source temporal coherence length.

© 2013 Optical Society of America

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References

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    [CrossRef]
  3. J. Pu and X. Liu, “Beam shaping of partially coherent beams by use of the spatial coherence effect,” Proc. SPIE 5876, 58760Z (2005).
    [CrossRef]
  4. T. Shirai and E. Wolf, “Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space,” J. Opt. Soc. Am. 21, 1907–1916 (2004).
    [CrossRef]
  5. X. Xiao and D. Voelz, “Wave optics simulation for partial spatially coherent beams,” Opt. Express 14, 6986–6992 (2006).
    [CrossRef]
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    [CrossRef]
  7. W. Martienssen and E. Spiller, “Coherence and fluctuations in light beams,” Am. J. Phys. 32, 919–926 (1964).
    [CrossRef]
  8. H. Arsenault and S. Lowenthal, “Partial coherence of an object illuminated with laser light through a moving diffuser,” Opt. Commun. 1, 451–453 (1970).
    [CrossRef]
  9. H. Nakano, N. Miyanaga, K. Yagi, K. Tsubakimoto, T. Kanabe, and M. Nakatsuka, “Partially coherent light generated by using single and multimode optical fiber in a high-power Nd-glass laser system,” Appl. Phys. Lett. 63, 580–582 (1993).
    [CrossRef]
  10. A. Mussot, E. Lantz, H. Maillotte, T. Sylvestre, C. Finot, and S. Pitois, “Spectral broadening of a partially coherent CW laser beam in single-mode optical fibers,” Opt. Express 12, 2838–2843 (2004).
    [CrossRef]
  11. 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]
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  14. J. W. Goodman, Statistical Optics (Wiley, 1985).
  15. D. Voelz, Computational Fourier Optics, a MATLAB Tutorial (SPIE, 2011).
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2008 (1)

2006 (2)

X. Xiao and D. Voelz, “Wave optics simulation for partial spatially coherent beams,” Opt. Express 14, 6986–6992 (2006).
[CrossRef]

O. Korotkova, “Changes in the intensity fluctuations of a class of random electromagnetic beams on propagation,” Pure Appl. Opt. 8, 30–37 (2006).
[CrossRef]

2005 (3)

J. Pu and X. Liu, “Beam shaping of partially coherent beams by use of the spatial coherence effect,” Proc. SPIE 5876, 58760Z (2005).
[CrossRef]

Q. Wang and M. K. Giles, “Coherence reduction using optical fibers,” Proc. SPIE 5892, 58920N (2005).
[CrossRef]

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]

2004 (3)

A. Mussot, E. Lantz, H. Maillotte, T. Sylvestre, C. Finot, and S. Pitois, “Spectral broadening of a partially coherent CW laser beam in single-mode optical fibers,” Opt. Express 12, 2838–2843 (2004).
[CrossRef]

T. Shirai and E. Wolf, “Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space,” J. Opt. Soc. Am. 21, 1907–1916 (2004).
[CrossRef]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “The effect of partially coherent quasi-monochromatic Gaussian-beam on the probability of fade,” Proc. SPIE 5160, 68–77 (2004).
[CrossRef]

1993 (1)

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

1992 (1)

1988 (1)

1970 (1)

H. Arsenault and S. Lowenthal, “Partial coherence of an object illuminated with laser light through a moving diffuser,” Opt. Commun. 1, 451–453 (1970).
[CrossRef]

1964 (1)

W. Martienssen and E. Spiller, “Coherence and fluctuations in light beams,” Am. J. Phys. 32, 919–926 (1964).
[CrossRef]

Andrews, L. C.

O. Korotkova, L. C. Andrews, and R. L. Phillips, “The effect of partially coherent quasi-monochromatic Gaussian-beam on the probability of fade,” Proc. SPIE 5160, 68–77 (2004).
[CrossRef]

Arsenault, H.

H. Arsenault and S. Lowenthal, “Partial coherence of an object illuminated with laser light through a moving diffuser,” Opt. Commun. 1, 451–453 (1970).
[CrossRef]

Cai, Y.

Finot, C.

Friberg, A. T.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978).

Giles, M. K.

Q. Wang and M. K. Giles, “Coherence reduction using optical fibers,” Proc. SPIE 5892, 58920N (2005).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985).

Kanabe, T.

H. Nakano, N. Miyanaga, K. Yagi, K. Tsubakimoto, T. Kanabe, and M. Nakatsuka, “Partially coherent light generated by using single and multimode optical fiber 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]

Korotkova, O.

O. Korotkova, “Changes in the intensity fluctuations of a class of random electromagnetic beams on propagation,” Pure Appl. Opt. 8, 30–37 (2006).
[CrossRef]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “The effect of partially coherent quasi-monochromatic Gaussian-beam on the probability of fade,” Proc. SPIE 5160, 68–77 (2004).
[CrossRef]

Lantz, E.

Liu, X.

J. Pu and X. Liu, “Beam shaping of partially coherent beams by use of the spatial coherence effect,” Proc. SPIE 5876, 58760Z (2005).
[CrossRef]

Lowenthal, S.

H. Arsenault and S. Lowenthal, “Partial coherence of an object illuminated with laser light through a moving diffuser,” Opt. Commun. 1, 451–453 (1970).
[CrossRef]

Lu, Z.

Maillotte, H.

Mandel, L.

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

Martienssen, W.

W. Martienssen and E. Spiller, “Coherence and fluctuations in light beams,” Am. J. Phys. 32, 919–926 (1964).
[CrossRef]

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]

Miyanaga, N.

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

Mussot, A.

Nakano, H.

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

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]

Phillips, R. L.

O. Korotkova, L. C. Andrews, and R. L. Phillips, “The effect of partially coherent quasi-monochromatic Gaussian-beam on the probability of fade,” Proc. SPIE 5160, 68–77 (2004).
[CrossRef]

Pitois, S.

Pu, J.

J. Pu and X. Liu, “Beam shaping of partially coherent beams by use of the spatial coherence effect,” Proc. SPIE 5876, 58760Z (2005).
[CrossRef]

Shirai, T.

T. Shirai and E. Wolf, “Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space,” J. Opt. Soc. Am. 21, 1907–1916 (2004).
[CrossRef]

Spiller, E.

W. Martienssen and E. Spiller, “Coherence and fluctuations in light beams,” Am. J. Phys. 32, 919–926 (1964).
[CrossRef]

Sylvestre, T.

Tervonen, E.

Tsubakimoto, K.

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

Turunen, J.

Voelz, D.

Wang, F.

Wang, Q.

Q. Wang and M. K. Giles, “Coherence reduction using optical fibers,” Proc. SPIE 5892, 58920N (2005).
[CrossRef]

Wang, Y.

Wolf, E.

T. Shirai and E. Wolf, “Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space,” J. Opt. Soc. Am. 21, 1907–1916 (2004).
[CrossRef]

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

Xiao, X.

Yagi, K.

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

Zhao, C.

Am. J. Phys. (1)

W. Martienssen and E. Spiller, “Coherence and fluctuations in light beams,” Am. J. Phys. 32, 919–926 (1964).
[CrossRef]

Appl. Phys. Lett. (1)

H. Nakano, N. Miyanaga, K. Yagi, K. Tsubakimoto, T. Kanabe, and M. Nakatsuka, “Partially coherent light generated by using single and multimode optical fiber in a high-power Nd-glass laser system,” Appl. Phys. Lett. 63, 580–582 (1993).
[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. (1)

T. Shirai and E. Wolf, “Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space,” J. Opt. Soc. Am. 21, 1907–1916 (2004).
[CrossRef]

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

Opt. Commun. (1)

H. Arsenault and S. Lowenthal, “Partial coherence of an object illuminated with laser light through a moving diffuser,” Opt. Commun. 1, 451–453 (1970).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Proc. SPIE (3)

Q. Wang and M. K. Giles, “Coherence reduction using optical fibers,” Proc. SPIE 5892, 58920N (2005).
[CrossRef]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “The effect of partially coherent quasi-monochromatic Gaussian-beam on the probability of fade,” Proc. SPIE 5160, 68–77 (2004).
[CrossRef]

J. Pu and X. Liu, “Beam shaping of partially coherent beams by use of the spatial coherence effect,” Proc. SPIE 5876, 58760Z (2005).
[CrossRef]

Pure Appl. Opt. (1)

O. Korotkova, “Changes in the intensity fluctuations of a class of random electromagnetic beams on propagation,” Pure Appl. Opt. 8, 30–37 (2006).
[CrossRef]

Other (4)

J. W. Goodman, Statistical Optics (Wiley, 1985).

D. Voelz, Computational Fourier Optics, a MATLAB Tutorial (SPIE, 2011).

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

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978).

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Figures (5)

Fig. 1.
Fig. 1.

Fiber bundle concept where partial temporal coherence is exploited to create partial spatial coherence. The bundle is composed of fibers of different lengths.

Fig. 2.
Fig. 2.

Fiber bundle profiles with wp=32wf and maximum phase difference of (a) π, (b) 2π, and (c) 5π. Panels (d)–(f) are the autocorrelation results (○) of (a)–(c) and the corresponding speckle-free results (—).

Fig. 3.
Fig. 3.

(a)–(c) Far-field irradiance (log-amplitude) at a distance z for the fiber bundles illustrated in Figs. 1(a)1(c). Simulation results (○) and speckle-free results (—).

Fig. 4.
Fig. 4.

(a) Pupil autocorrelation functions for a source with temporal coherence length lc=0.05mm (λ0=1μm and Δλ=20nm) and fibers defined by n=1.5, where Dm=0.3, 1.0, 3.0, and 10.0 mm, and the speckle-free case (black). (b) Corresponding far-field patterns. The insets in (a) are the enlargements of the denoted sections.

Fig. 5.
Fig. 5.

Same as Fig. 4 except with source temporal coherence length lc=0.02mm (λ0=1μm and Δλ=50nm).

Equations (37)

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tf(x,y;λ0)=P(x,y)exp[jξ(x,y;λ0)],
P(x,y)=circ[x2+y2/wp],
ξ(x,y;λ0)=(1+a)k0nD(x,y),
D(x,y)=[d(x,y)comb(x2wf,y2wf)]rect(x2wf,y2wf),
[k0nD(x,y)]λ0=k0nD(x,y)λ0.
R(Δx,Δy;λ0)=tf(x,y;λ0)tf*(x+Δx,y+Δy;λ0)dxdy∫∫tf(x,y;λ0)tf*(x,y;λ0)dxdy,
R(Δx,Δy;λ0)=1πwp2exp[j(1+a)k0nD(x,y;Δx,Δy)]dxdy,
D(x,y;Δx,Δy)=D(x,y)P(x,y)D(x+Δx,y+Δy)P(x+Δx,y+Δy).
exp[j(1+a)k0nD]dxdy=exp(jk0nD)dxdyexp(jak0nD)dxdy.
exp(jak0nD)dxdy=[λ0ΔλΔλ/(2λ0)Δλ/(2λ0)exp(jak0nD)da]dxdy.
R(Δx,Δy;λ0)=1πwp2(exp(jk0nD)dxdy)×(sinc(nD/lc)dxdy),
I(x,y;λ0;z)=(λ0z)2I[R(Δx,Δy;λ0)],
RS,λ(Δx,Δy;λ)=I1{|I{tf(x,y;λ)}|2},
RS(Δx,Δy)=1Ni=(N1)/2(N1)/2RS,λi(Δx,Δy;λi).
u(x,y;λ)=tf(x,y;λ).
IS,λ(x,y;λ;z)=(λz)2|I{u(x,y;λ)}|2,
IS(x,y;λ;z)=1Ni=(N1)/2(N1)/2IS,λi(x,y;λi;z).
Iπ4(λ0z)2[wp2π+wf2(14π2)],
I2π4wf2(λ0z)2,
I5π4(λ0z)2[wp225π+wf2(1425π2)].
I0πwp2(λ0z)2.
ξ(x,y)=k0n[d(x,y)comb(x2wf,y2wf)]rect(x2wf,y2wf),
R(x,y;x+Δx,y+Δy)=P(x,y)P(x+Δx,y+Δy)exp[jξ(x,y)]exp[jξ(x+Δx,y+Δy)]×P(x,y)exp[jξ(x,y)]P(x,y)exp[jξ(x,y)]1.
R(Δx,Δy)=1πwp2ΓP(Δx,Δy)Γξ(Δx,Δy),
ΓP(Δx,Δy)=P(x,y)P*(x+Δx,y+Δy)dxdy.
ΓP(Δx,Δy)=2wp2[cos1(Δx2+Δy22wp)Δx2+Δy22wp(1Δx2+Δy24wp2)1/2]×circ(Δx2+Δy22wp).
Γξ(Δx,Δy)=ejξejξ×Pr[(x,y)and(x+Δx,y+Δy)in the same fiber]+ejξ2×Pr[(x,y)and(x+Δx,y+Δy)in different fibers],
Pr[(x,y)and(x+Δx,y+Δy)in the same fiber]=tri(Δx/2wf,Δy/2wf),
Pr[(x,y)and(x+Δx,y+Δy)in different fibers]=1tri(Δx/2wf,Δy/2wf),
ejξ=1k0nDmk0nDm/2k0nDm/2ejϕdϕ=sinc(nDm/λ0),
Γξ(Δx,Δy)=tri(Δx/2wf,Δy/2wf)+sinc2(nDm/λ0)[1tri(Δx/2wf,Δy/2wf)].
R(Δx,Δy)=ΓP(Δx,Δy)πwp2{sinc2(nDmλ0)+[1sinc2(nDmλ0)][tri(Δx2wf,Δy2wf)]}.
I(x,y;λ0;z)=(λ0z)2I{R(Δx,Δy)}.
I(x,y;λ0;z)πwp2(λ0z)2{sinc2(nDmλ0)×somb2(2wpx2+y2λ0z)+4wf2(λ0z)2[1sinc2(nDmλ0)]×somb2(2wpx2+y2λ0z)[sinc2(2wfxλ0z)sinc2(2wfyλ0z)]},
somb(r)=2J1(πr)/πr,
I(x,y;λ0;z)=4wf2(λ0z)2×[πwp2(λ0z)2somb2(2wpx2+y2λ0z)][sinc2(2wfxλ0z)sinc2(2wfyλ0z)].
I(x,y;λ0;z)4wf2(λ0z)2sinc2(2wfxλ0z,2wfyλ0z).

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