Abstract

We investigate the occurrence of self-images, or Talbot images, in a spatially multimode field that propagates along an optical fiber whose core has an annular-shaped cross section. By use of full-vectorial modal analysis, we study the effect of the transverse fiber dimensions on the self-imaging properties. According to our analysis, good self-images can be expected when the fiber core is thin and the modes are far from their cutoffs. However, as the core diameter is made larger to increase the number of modes available in the imaging, the general self-imaging properties tend to deteriorate.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. Patorski, "The self-imaging phenomenon and its applications," in Progress in Optics, E.Wolf, ed. (Elsevier, 1989), Vol. 27, pp. 1-108.
    [CrossRef]
  2. S. W. Allison and G. T. Gillies, "Observations of and applications for self-imaging in optical fibers," Appl. Opt. 33, 1802-1805 (1994).
    [CrossRef] [PubMed]
  3. A. Mehta, W. Mohammed, and E. G. Johnson, "Multimode interference-based fiber-optic displacement sensor," IEEE Photon. Technol. Lett. 15, 1129-1131 (2003).
    [CrossRef]
  4. C. Y. H. Tsao, D. N. Payne, and W. A. Gambling, "Modal characteristics of three-layered optical fiber waveguides: a modified approach," J. Opt. Soc. Am. A 6, 555-563 (1989).
    [CrossRef]
  5. N. B. Baranova and B. Ya. Zel'dovich, "Talbot effect for whispering gallery modes and modes of tubular waveguides," in Technical Digest of the International Quantum Electronics Conference, IQEC'98 (1998), pp. 184-185.
  6. T. Niemeier and R. Ulrich, "Self-imaging by ring-core fibers," in Digest of Topical Meeting on Optical Fiber Communication, I.D.Aggarwal, ed. (Optical Society of America, 1985), pp. 122-124.
  7. A. Zoubir, C. Lopez, M. Richardson, and K. Richardson, "Femtosecond laser fabrication of tubular waveguides in poly(methyl methacrylate)," Opt. Lett. 29, 1840-1842 (2004).
    [CrossRef] [PubMed]
  8. M. Wrage, P. Glas, D. Fischer, M. Leitner, N. N. Elkin, D. V. Vysotsky, A. P. Napartovich, and V. N. Troshchieva, "Phase-locking of a multicore fiber laser by wave propagation through an annular waveguide," Opt. Commun. 205, 367-375 (2002).
    [CrossRef]
  9. A. P. Napartovich and D. V. Vysotsky, "Phase-locking of multicore fibre laser due to Talbot self-reproduction," J. Mod. Opt. 50, 2715-2725 (2003).
    [CrossRef]
  10. M. Hautakorpi and M. Kaivola, "Modal analysis of M-type-dielectric-profile optical fibers in the weakly guiding approximation," J. Opt. Soc. Am. A 22, 1163-1169 (2005).
    [CrossRef]
  11. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).
  12. A. W. Snyder and J. D. Love, Optical Waveguide Theory, 1st ed. (Chapman & Hall, 1983).
  13. G. Keiser, Optical Fiber Communications, 3rd ed. (McGraw-Hill, 2000).
  14. H. Ito, K. Sakaki, T. Nakata, W. Jhe, and M. Ohtsu, "Optical potential for atom guidance in a cylindrical-core hollow fiber," Opt. Commun. 115, 57-64 (1995).
    [CrossRef]
  15. P. R. Chaudhuri, C. Lu, and W. Xiaoyan, "Scalar model and exact vectorial description for the design analysis of hollow optical fiber components," Opt. Commun. 228, 285-293 (2003).
    [CrossRef]
  16. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

2005 (1)

2004 (1)

2003 (3)

A. Mehta, W. Mohammed, and E. G. Johnson, "Multimode interference-based fiber-optic displacement sensor," IEEE Photon. Technol. Lett. 15, 1129-1131 (2003).
[CrossRef]

A. P. Napartovich and D. V. Vysotsky, "Phase-locking of multicore fibre laser due to Talbot self-reproduction," J. Mod. Opt. 50, 2715-2725 (2003).
[CrossRef]

P. R. Chaudhuri, C. Lu, and W. Xiaoyan, "Scalar model and exact vectorial description for the design analysis of hollow optical fiber components," Opt. Commun. 228, 285-293 (2003).
[CrossRef]

2002 (1)

M. Wrage, P. Glas, D. Fischer, M. Leitner, N. N. Elkin, D. V. Vysotsky, A. P. Napartovich, and V. N. Troshchieva, "Phase-locking of a multicore fiber laser by wave propagation through an annular waveguide," Opt. Commun. 205, 367-375 (2002).
[CrossRef]

1995 (1)

H. Ito, K. Sakaki, T. Nakata, W. Jhe, and M. Ohtsu, "Optical potential for atom guidance in a cylindrical-core hollow fiber," Opt. Commun. 115, 57-64 (1995).
[CrossRef]

1994 (1)

1989 (1)

Allison, S. W.

Baranova, N. B.

N. B. Baranova and B. Ya. Zel'dovich, "Talbot effect for whispering gallery modes and modes of tubular waveguides," in Technical Digest of the International Quantum Electronics Conference, IQEC'98 (1998), pp. 184-185.

Chaudhuri, P. R.

P. R. Chaudhuri, C. Lu, and W. Xiaoyan, "Scalar model and exact vectorial description for the design analysis of hollow optical fiber components," Opt. Commun. 228, 285-293 (2003).
[CrossRef]

Elkin, N. N.

M. Wrage, P. Glas, D. Fischer, M. Leitner, N. N. Elkin, D. V. Vysotsky, A. P. Napartovich, and V. N. Troshchieva, "Phase-locking of a multicore fiber laser by wave propagation through an annular waveguide," Opt. Commun. 205, 367-375 (2002).
[CrossRef]

Fischer, D.

M. Wrage, P. Glas, D. Fischer, M. Leitner, N. N. Elkin, D. V. Vysotsky, A. P. Napartovich, and V. N. Troshchieva, "Phase-locking of a multicore fiber laser by wave propagation through an annular waveguide," Opt. Commun. 205, 367-375 (2002).
[CrossRef]

Gambling, W. A.

Gillies, G. T.

Glas, P.

M. Wrage, P. Glas, D. Fischer, M. Leitner, N. N. Elkin, D. V. Vysotsky, A. P. Napartovich, and V. N. Troshchieva, "Phase-locking of a multicore fiber laser by wave propagation through an annular waveguide," Opt. Commun. 205, 367-375 (2002).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Hautakorpi, M.

Ito, H.

H. Ito, K. Sakaki, T. Nakata, W. Jhe, and M. Ohtsu, "Optical potential for atom guidance in a cylindrical-core hollow fiber," Opt. Commun. 115, 57-64 (1995).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).

Jhe, W.

H. Ito, K. Sakaki, T. Nakata, W. Jhe, and M. Ohtsu, "Optical potential for atom guidance in a cylindrical-core hollow fiber," Opt. Commun. 115, 57-64 (1995).
[CrossRef]

Johnson, E. G.

A. Mehta, W. Mohammed, and E. G. Johnson, "Multimode interference-based fiber-optic displacement sensor," IEEE Photon. Technol. Lett. 15, 1129-1131 (2003).
[CrossRef]

Kaivola, M.

Keiser, G.

G. Keiser, Optical Fiber Communications, 3rd ed. (McGraw-Hill, 2000).

Leitner, M.

M. Wrage, P. Glas, D. Fischer, M. Leitner, N. N. Elkin, D. V. Vysotsky, A. P. Napartovich, and V. N. Troshchieva, "Phase-locking of a multicore fiber laser by wave propagation through an annular waveguide," Opt. Commun. 205, 367-375 (2002).
[CrossRef]

Lopez, C.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory, 1st ed. (Chapman & Hall, 1983).

Lu, C.

P. R. Chaudhuri, C. Lu, and W. Xiaoyan, "Scalar model and exact vectorial description for the design analysis of hollow optical fiber components," Opt. Commun. 228, 285-293 (2003).
[CrossRef]

Mehta, A.

A. Mehta, W. Mohammed, and E. G. Johnson, "Multimode interference-based fiber-optic displacement sensor," IEEE Photon. Technol. Lett. 15, 1129-1131 (2003).
[CrossRef]

Mohammed, W.

A. Mehta, W. Mohammed, and E. G. Johnson, "Multimode interference-based fiber-optic displacement sensor," IEEE Photon. Technol. Lett. 15, 1129-1131 (2003).
[CrossRef]

Nakata, T.

H. Ito, K. Sakaki, T. Nakata, W. Jhe, and M. Ohtsu, "Optical potential for atom guidance in a cylindrical-core hollow fiber," Opt. Commun. 115, 57-64 (1995).
[CrossRef]

Napartovich, A. P.

A. P. Napartovich and D. V. Vysotsky, "Phase-locking of multicore fibre laser due to Talbot self-reproduction," J. Mod. Opt. 50, 2715-2725 (2003).
[CrossRef]

M. Wrage, P. Glas, D. Fischer, M. Leitner, N. N. Elkin, D. V. Vysotsky, A. P. Napartovich, and V. N. Troshchieva, "Phase-locking of a multicore fiber laser by wave propagation through an annular waveguide," Opt. Commun. 205, 367-375 (2002).
[CrossRef]

Niemeier, T.

T. Niemeier and R. Ulrich, "Self-imaging by ring-core fibers," in Digest of Topical Meeting on Optical Fiber Communication, I.D.Aggarwal, ed. (Optical Society of America, 1985), pp. 122-124.

Ohtsu, M.

H. Ito, K. Sakaki, T. Nakata, W. Jhe, and M. Ohtsu, "Optical potential for atom guidance in a cylindrical-core hollow fiber," Opt. Commun. 115, 57-64 (1995).
[CrossRef]

Patorski, K.

K. Patorski, "The self-imaging phenomenon and its applications," in Progress in Optics, E.Wolf, ed. (Elsevier, 1989), Vol. 27, pp. 1-108.
[CrossRef]

Payne, D. N.

Richardson, K.

Richardson, M.

Sakaki, K.

H. Ito, K. Sakaki, T. Nakata, W. Jhe, and M. Ohtsu, "Optical potential for atom guidance in a cylindrical-core hollow fiber," Opt. Commun. 115, 57-64 (1995).
[CrossRef]

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory, 1st ed. (Chapman & Hall, 1983).

Troshchieva, V. N.

M. Wrage, P. Glas, D. Fischer, M. Leitner, N. N. Elkin, D. V. Vysotsky, A. P. Napartovich, and V. N. Troshchieva, "Phase-locking of a multicore fiber laser by wave propagation through an annular waveguide," Opt. Commun. 205, 367-375 (2002).
[CrossRef]

Tsao, C. Y. H.

Ulrich, R.

T. Niemeier and R. Ulrich, "Self-imaging by ring-core fibers," in Digest of Topical Meeting on Optical Fiber Communication, I.D.Aggarwal, ed. (Optical Society of America, 1985), pp. 122-124.

Vysotsky, D. V.

A. P. Napartovich and D. V. Vysotsky, "Phase-locking of multicore fibre laser due to Talbot self-reproduction," J. Mod. Opt. 50, 2715-2725 (2003).
[CrossRef]

M. Wrage, P. Glas, D. Fischer, M. Leitner, N. N. Elkin, D. V. Vysotsky, A. P. Napartovich, and V. N. Troshchieva, "Phase-locking of a multicore fiber laser by wave propagation through an annular waveguide," Opt. Commun. 205, 367-375 (2002).
[CrossRef]

Wrage, M.

M. Wrage, P. Glas, D. Fischer, M. Leitner, N. N. Elkin, D. V. Vysotsky, A. P. Napartovich, and V. N. Troshchieva, "Phase-locking of a multicore fiber laser by wave propagation through an annular waveguide," Opt. Commun. 205, 367-375 (2002).
[CrossRef]

Xiaoyan, W.

P. R. Chaudhuri, C. Lu, and W. Xiaoyan, "Scalar model and exact vectorial description for the design analysis of hollow optical fiber components," Opt. Commun. 228, 285-293 (2003).
[CrossRef]

Zel'dovich, B. Ya.

N. B. Baranova and B. Ya. Zel'dovich, "Talbot effect for whispering gallery modes and modes of tubular waveguides," in Technical Digest of the International Quantum Electronics Conference, IQEC'98 (1998), pp. 184-185.

Zoubir, A.

Appl. Opt. (1)

IEEE Photon. Technol. Lett. (1)

A. Mehta, W. Mohammed, and E. G. Johnson, "Multimode interference-based fiber-optic displacement sensor," IEEE Photon. Technol. Lett. 15, 1129-1131 (2003).
[CrossRef]

J. Mod. Opt. (1)

A. P. Napartovich and D. V. Vysotsky, "Phase-locking of multicore fibre laser due to Talbot self-reproduction," J. Mod. Opt. 50, 2715-2725 (2003).
[CrossRef]

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

Opt. Commun. (3)

H. Ito, K. Sakaki, T. Nakata, W. Jhe, and M. Ohtsu, "Optical potential for atom guidance in a cylindrical-core hollow fiber," Opt. Commun. 115, 57-64 (1995).
[CrossRef]

P. R. Chaudhuri, C. Lu, and W. Xiaoyan, "Scalar model and exact vectorial description for the design analysis of hollow optical fiber components," Opt. Commun. 228, 285-293 (2003).
[CrossRef]

M. Wrage, P. Glas, D. Fischer, M. Leitner, N. N. Elkin, D. V. Vysotsky, A. P. Napartovich, and V. N. Troshchieva, "Phase-locking of a multicore fiber laser by wave propagation through an annular waveguide," Opt. Commun. 205, 367-375 (2002).
[CrossRef]

Opt. Lett. (1)

Other (7)

K. Patorski, "The self-imaging phenomenon and its applications," in Progress in Optics, E.Wolf, ed. (Elsevier, 1989), Vol. 27, pp. 1-108.
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

N. B. Baranova and B. Ya. Zel'dovich, "Talbot effect for whispering gallery modes and modes of tubular waveguides," in Technical Digest of the International Quantum Electronics Conference, IQEC'98 (1998), pp. 184-185.

T. Niemeier and R. Ulrich, "Self-imaging by ring-core fibers," in Digest of Topical Meeting on Optical Fiber Communication, I.D.Aggarwal, ed. (Optical Society of America, 1985), pp. 122-124.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).

A. W. Snyder and J. D. Love, Optical Waveguide Theory, 1st ed. (Chapman & Hall, 1983).

G. Keiser, Optical Fiber Communications, 3rd ed. (McGraw-Hill, 2000).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

(Color online) (a) Cross section of an annular-core optical fiber with inner and outer core radii denoted by a and b, respectively, and with a core thickness of d = b a . The refractive-index n 1 of the core is slightly higher than that of the inner and outer claddings, n 2 . (b) Transverse intensity patterns of the LP m , p modes of a fiber with a = 15 λ , d = 3 λ , n 1 = 1.457 , and n 2 = 1.455 , with λ denoting the vacuum wavelength.

Fig. 2
Fig. 2

(Color online) Normalized propagation constants β / k of the vector modes arranged according to index m of the LP m ,1 modes (dots). The insets illustrate that for m > 1 there are two modes (three modes for m = 1 ) with almost degenerate propagation constants (triangles). Dashed lines are least-squares fits with σ denoting their standard deviation. The core thickness is d = 5.8 λ (upper data) and d = 3.5 λ (lower data), and the other fiber parameters are a = 5 λ , n 1 = 1.457 , n 2 = 1.455 , with λ denoting the vacuum wavelength.

Fig. 3
Fig. 3

(Color online) Normalized Talbot distance z T k (dots connected with gray curve) and the beat length z B k (dashed curve) of the modes TE 0,1 and TM 0,1 as a function of the normalized core thickness d / λ for four different values of the inner radius of core a. The relative sizes of the fiber cores are depicted by gray circles for d / λ = 3 .

Fig. 4
Fig. 4

(Color online) Parameter s = σ z T as a function of the normalized core thickness d / λ . The gray curves connect the data points corresponding to a certain value of the core radius a. The points labeled with the uppercase letters correspond to the cases of Fig. 5.

Fig. 5
Fig. 5

(Color online) Intensity profiles of the fields in annular-core fibers at distances z shown above each column. The annular core is for clarity sketched with the gray dashed circle for z = 0 in (a) and (c), and the uppercase letters correspond to the points in Fig. 4. The squared modulus of the overlap integral, | γ | 2 , between the fields at planes z = 0 and z = z T is shown at each row. The example plots are chosen to illustrate (a) some general features of the Talbot effect, (b) the deterioration of the self-image near modal cutoffs, and (c) the lowering of the image quality as the fiber core is made thicker.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

( 2 + k 2 n j 2 ) { E ( r ) H ( r ) } = 0 ,
E z ( r , θ ) = { C 1 I l ( v r ) sin ( l θ + ϕ ) , [ C 2 J l ( u r ) + C 3 N l ( u r ) ] sin ( l θ + ϕ ) , C 4 K l ( v r ) sin ( l θ + ϕ ) , r a , a < r < b , b r ,
H z ( r , θ ) = { C 5 I l ( v r ) cos ( l θ + ϕ ) , [ C 6 J l ( u r ) + C 7 N l ( u r ) ] cos ( l θ + ϕ ) , C 8 K l ( v r ) cos ( l θ + ϕ ) , r a , a < r < b , b r .
z B = 2 π / | β β + | ,
β = β 0 β 1 m 2 ,
z T = 2 π / β 1 ,
γ = N 1 F * ( r , θ , 0 ) · F ( r , θ , z T ) r d r d θ ,

Metrics