Abstract

We propose to determine the optical field in multimode circular fibers by using a one-step method that measures the Wigner distribution function of a section of the field in the fiber. This method allows an estimation not only of the power carried by each mode but also of the relative phases of different modes in the fiber. An additional measurement with the same setup can even determine the propagation constants of different modes. An example is provided, and the connection of this method of field recovery to the coupling coefficient between fibers and light sources is also discussed.

© 2001 Optical Society of America

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References

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  1. S. R. Dean, The Radon Transform and Some of Its Applications (Wiley, New York, 1993).
  2. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  3. T. E. Gureyev, K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13, 1670–1682 (1996).
    [CrossRef]
  4. N. Nakajima, “Phase retrieval from Fresnel zone intensity measurements by use of Gaussian filtering,” Appl. Opt. 37, 6219–6226 (1998).
    [CrossRef]
  5. D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Prog. Opt. 37, 1–56 (1997).
    [CrossRef]
  6. D. Dragoman, “Can the Wigner transform of a 2D rotationally symmetric beam be fully recovered from the Wigner transform of its 1D approximation,” Opt. Lett. 25, 281–283 (2000).
    [CrossRef]
  7. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  8. H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
    [CrossRef]
  9. K.-H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310–314 (1982).
    [CrossRef]
  10. R. Bamler, H. Glünder, “The Wigner distribution function of two-dimensional signals coherent-optical generation and display,” Opt. Acta 30, 1789–1803 (1983).
    [CrossRef]
  11. M. Conner, Y. Li, “Optical generation of the Wigner distribution of 2-D real signals,” Appl. Opt. 24, 3825–3829 (1985).
    [CrossRef] [PubMed]
  12. T. Iwai, A. K. Gupta, T. Asakura, “Simultaneous optical production of the sectional Wigner distribution function for a two-dimensional object,” Opt. Commun. 58, 15–19 (1986).
    [CrossRef]
  13. D. Dragoman, J. P. Meunier, “Recovery of translationally variant refractive index profile from the measurement of the Wigner transform,” Opt. Commun. 153, 360–367 (1998).
    [CrossRef]
  14. H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543–559 (1992).
    [CrossRef]
  15. D. Dragoman, M. Dragoman, J.-P. Meunier, “Implementation of the spatial and temporal cross ambiguity function for waveguide fields and optical pulses,” Appl. Opt. 38, 822–827 (1999).
    [CrossRef]

2000 (1)

1999 (1)

1998 (2)

D. Dragoman, J. P. Meunier, “Recovery of translationally variant refractive index profile from the measurement of the Wigner transform,” Opt. Commun. 153, 360–367 (1998).
[CrossRef]

N. Nakajima, “Phase retrieval from Fresnel zone intensity measurements by use of Gaussian filtering,” Appl. Opt. 37, 6219–6226 (1998).
[CrossRef]

1997 (1)

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Prog. Opt. 37, 1–56 (1997).
[CrossRef]

1996 (1)

1992 (1)

H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543–559 (1992).
[CrossRef]

1986 (1)

T. Iwai, A. K. Gupta, T. Asakura, “Simultaneous optical production of the sectional Wigner distribution function for a two-dimensional object,” Opt. Commun. 58, 15–19 (1986).
[CrossRef]

1985 (1)

1983 (1)

R. Bamler, H. Glünder, “The Wigner distribution function of two-dimensional signals coherent-optical generation and display,” Opt. Acta 30, 1789–1803 (1983).
[CrossRef]

1982 (1)

K.-H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310–314 (1982).
[CrossRef]

1980 (1)

H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Asakura, T.

T. Iwai, A. K. Gupta, T. Asakura, “Simultaneous optical production of the sectional Wigner distribution function for a two-dimensional object,” Opt. Commun. 58, 15–19 (1986).
[CrossRef]

Bamler, R.

R. Bamler, H. Glünder, “The Wigner distribution function of two-dimensional signals coherent-optical generation and display,” Opt. Acta 30, 1789–1803 (1983).
[CrossRef]

Bartelt, H. O.

H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Brenner, K.-H.

K.-H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310–314 (1982).
[CrossRef]

H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Conner, M.

Dean, S. R.

S. R. Dean, The Radon Transform and Some of Its Applications (Wiley, New York, 1993).

Dragoman, D.

D. Dragoman, “Can the Wigner transform of a 2D rotationally symmetric beam be fully recovered from the Wigner transform of its 1D approximation,” Opt. Lett. 25, 281–283 (2000).
[CrossRef]

D. Dragoman, M. Dragoman, J.-P. Meunier, “Implementation of the spatial and temporal cross ambiguity function for waveguide fields and optical pulses,” Appl. Opt. 38, 822–827 (1999).
[CrossRef]

D. Dragoman, J. P. Meunier, “Recovery of translationally variant refractive index profile from the measurement of the Wigner transform,” Opt. Commun. 153, 360–367 (1998).
[CrossRef]

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Prog. Opt. 37, 1–56 (1997).
[CrossRef]

Dragoman, M.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Glünder, H.

R. Bamler, H. Glünder, “The Wigner distribution function of two-dimensional signals coherent-optical generation and display,” Opt. Acta 30, 1789–1803 (1983).
[CrossRef]

Gupta, A. K.

T. Iwai, A. K. Gupta, T. Asakura, “Simultaneous optical production of the sectional Wigner distribution function for a two-dimensional object,” Opt. Commun. 58, 15–19 (1986).
[CrossRef]

Gureyev, T. E.

Iwai, T.

T. Iwai, A. K. Gupta, T. Asakura, “Simultaneous optical production of the sectional Wigner distribution function for a two-dimensional object,” Opt. Commun. 58, 15–19 (1986).
[CrossRef]

Li, Y.

Lohmann, A. W.

K.-H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310–314 (1982).
[CrossRef]

H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Meunier, J. P.

D. Dragoman, J. P. Meunier, “Recovery of translationally variant refractive index profile from the measurement of the Wigner transform,” Opt. Commun. 153, 360–367 (1998).
[CrossRef]

Meunier, J.-P.

Nakajima, N.

Nugent, K. A.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Weber, H.

H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543–559 (1992).
[CrossRef]

Appl. Opt. (3)

J. Mod. Opt. (1)

H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543–559 (1992).
[CrossRef]

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

Opt. Acta (1)

R. Bamler, H. Glünder, “The Wigner distribution function of two-dimensional signals coherent-optical generation and display,” Opt. Acta 30, 1789–1803 (1983).
[CrossRef]

Opt. Commun. (4)

T. Iwai, A. K. Gupta, T. Asakura, “Simultaneous optical production of the sectional Wigner distribution function for a two-dimensional object,” Opt. Commun. 58, 15–19 (1986).
[CrossRef]

D. Dragoman, J. P. Meunier, “Recovery of translationally variant refractive index profile from the measurement of the Wigner transform,” Opt. Commun. 153, 360–367 (1998).
[CrossRef]

H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

K.-H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1D signals,” Opt. Commun. 42, 310–314 (1982).
[CrossRef]

Opt. Lett. (1)

Optik (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Prog. Opt. (1)

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Prog. Opt. 37, 1–56 (1997).
[CrossRef]

Other (2)

S. R. Dean, The Radon Transform and Some of Its Applications (Wiley, New York, 1993).

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

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Figures (4)

Fig. 1
Fig. 1

Setup for generating the Wigner distribution function of a section of the incident 2D field.

Fig. 2
Fig. 2

WDF of the total field distribution consisting of a superposition of the fundamental and the first higher-order modes in a fiber with a parabolic refractive-index profile, for two fiber lengths.

Fig. 3
Fig. 3

WDF of the (a) fundamental and (b) the first higher-order modes in a fiber with parabolic refractive-index profile.

Fig. 4
Fig. 4

(a) Real (solid curve) and imaginary (dotted curve) parts of the total field in the fiber if the WDF is that as in Fig. 2(b). (b) The fundamental F 01 (solid curve), and the higher-order F 02 (dotted curve) and F 03 (dashed curve) modes in a fiber with a parabolic refractive-index profile.

Equations (11)

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ϕr, θ; L=Σl,malm expiβlmLFlmrexpilθ=Σl,mblmFlmrexpilθ,
d2dr2+1rddr+k2n2r-l2r2-βlm2Flmr=0,
Wr, p; L= ϕr+r2, 0; Lϕ*×r-r2, 0; Lexpiprdr
ϕr, 0; L=12πϕ*0, 0; L  Wr/2, p; L×exp-irpdp.
F0mr=Lm-1Vr2/d2exp-Vr2/2d2,
ηdx, dy, px, py= ϕ1x+dx2, y+dy2ϕ2*×x-dx2, y-dy2×expipxx+pyydxdy2,
x=r cos θ, y=r sin θ,
dx=d cos ϕ, dy=d sin ϕ,
px=p cos ϑ,  py=p sin ϑ,
ηd, ϕ, p, ϑ= ϕ1r+, θ+ϕ2*r-, θ-×expirp cosθ-ϑrdrdθ2,
r±=r2+d2/4±rd cosθ-ϕ1/2,θ±=arctanr sin θ±d/2sin ϕr cos θ±d/2cos ϕ.

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