Abstract

Self-imaging of a periodic space-variant polarized field is demonstrated. The field is created by use of space-variant subwavelength dielectric gratings. Our observations include self-imaging of the fields at the Talbot planes as well as the translation of incident polarization variation into intensity modulation at certain planes. We demonstrate the formation of a one-dimensional nondiffracting beam with uniform intensity and a nontrivial polarization structure.

© 2002 Optical Society of America

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References

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  1. See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996), p. 87.
  2. Ch. Siegel, F. Lowenthal, J. E. Balmer, “A wavefront sensor based on the fractional Talbot effect,” Opt. Commun. 194, 265–275 (2001).
    [Crossref]
  3. H. L. Kung, A. Bhatnagar, D. A. B. Miller, “Transform spectrometer based on measuring the periodicity of Talbot self-images,” Opt. Lett. 26, 1645–1647 (2001).
    [Crossref]
  4. M. Wrage, P. Glas, D. Fischer, M. Leitner, D. Vysotsky, A. P. Napartovich, “Phase locking in a multicore fiber laser by means of a Talbot resonator,” Opt. Lett. 25, 1436–1438 (2000).
    [Crossref]
  5. V. A. Arrizón, E. Tepchin, M. Ortiz-Gutierrez, A. W. Lohmann, “Fresnel diffraction at ¼ of the Talbot distance of an anistropic grating,” Opt. Commun. 127, 171–175 (1996).
    [Crossref]
  6. H. J. Rabal, W. D. Furlan, E. E. Sicre, “Talbot interferometry with anistropic gratings,” Opt. Commun. 57, 81–83 (1986).
    [Crossref]
  7. Z. Bomzon, G. Biener, V. Kleiner, E. Hasman, “Real time analysis of partially polarized light with a space-variant subwavelength dielectric grating,” Opt. Lett. 27, 285–287 (2002).
    [Crossref]
  8. Z. Bomzon, V. Kleiner, E. Hasman, “Pancharatnam–Berry phase in space-variant polarization-manipulations with subwavelength gratings,” Opt. Lett. 26, 1424–1426 (2001).
    [Crossref]
  9. Z. Bomzon, G. Biener, V. Kleiner, E. Hasman, “Space-variant Pancharatnam–Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett.1141–1143 (2002).
    [Crossref]
  10. The Fresnel approximation for the propagation of a scalar wave is defined as E(x, y, z) = F-1HF[E(x, y, z = 0)], where E(x, y, z) is a scalar wave function, F denotes a spatial Fourier transform, H(fx, z) = exp(i2πz/λ)exp(-iπλzfx2) is the Fresnel transfer function, and fx denotes spatial frequency. See, for example, Ref. 1.
  11. Z. Bomzon, V. Kleiner, E. Hasman, “Space-variant polarization state manipulation with computer-generated subwavelength metal-stripe gratings,” Opt. Commun. 192, 169–181 (2001).
    [Crossref]
  12. Stokes parameters are used to define the polarization state. They are S0 = |Ex|2 + |Ey|2, S1 = |Ex|2 - |Ey|2, S2 = ExEy* + EyEx*, and S3 = i(ExEy* - EyEx*), where Ex and Ey are the Cartesian components of the electromagnetic field. S0 is the intensity of the field, whereas S1 … S3 define the polarization ellipse. See, for example, C. Brosseau, Polarized Light, A Statistical Optics Approach (Wiley, New York, 1998).
  13. E. Collett, Polarized Light (Marcel Dekker, New York, 1993).

2002 (2)

Z. Bomzon, G. Biener, V. Kleiner, E. Hasman, “Real time analysis of partially polarized light with a space-variant subwavelength dielectric grating,” Opt. Lett. 27, 285–287 (2002).
[Crossref]

Z. Bomzon, G. Biener, V. Kleiner, E. Hasman, “Space-variant Pancharatnam–Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett.1141–1143 (2002).
[Crossref]

2001 (4)

Z. Bomzon, V. Kleiner, E. Hasman, “Space-variant polarization state manipulation with computer-generated subwavelength metal-stripe gratings,” Opt. Commun. 192, 169–181 (2001).
[Crossref]

Z. Bomzon, V. Kleiner, E. Hasman, “Pancharatnam–Berry phase in space-variant polarization-manipulations with subwavelength gratings,” Opt. Lett. 26, 1424–1426 (2001).
[Crossref]

Ch. Siegel, F. Lowenthal, J. E. Balmer, “A wavefront sensor based on the fractional Talbot effect,” Opt. Commun. 194, 265–275 (2001).
[Crossref]

H. L. Kung, A. Bhatnagar, D. A. B. Miller, “Transform spectrometer based on measuring the periodicity of Talbot self-images,” Opt. Lett. 26, 1645–1647 (2001).
[Crossref]

2000 (1)

1996 (1)

V. A. Arrizón, E. Tepchin, M. Ortiz-Gutierrez, A. W. Lohmann, “Fresnel diffraction at ¼ of the Talbot distance of an anistropic grating,” Opt. Commun. 127, 171–175 (1996).
[Crossref]

1986 (1)

H. J. Rabal, W. D. Furlan, E. E. Sicre, “Talbot interferometry with anistropic gratings,” Opt. Commun. 57, 81–83 (1986).
[Crossref]

Arrizón, V. A.

V. A. Arrizón, E. Tepchin, M. Ortiz-Gutierrez, A. W. Lohmann, “Fresnel diffraction at ¼ of the Talbot distance of an anistropic grating,” Opt. Commun. 127, 171–175 (1996).
[Crossref]

Balmer, J. E.

Ch. Siegel, F. Lowenthal, J. E. Balmer, “A wavefront sensor based on the fractional Talbot effect,” Opt. Commun. 194, 265–275 (2001).
[Crossref]

Bhatnagar, A.

Biener, G.

Z. Bomzon, G. Biener, V. Kleiner, E. Hasman, “Real time analysis of partially polarized light with a space-variant subwavelength dielectric grating,” Opt. Lett. 27, 285–287 (2002).
[Crossref]

Z. Bomzon, G. Biener, V. Kleiner, E. Hasman, “Space-variant Pancharatnam–Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett.1141–1143 (2002).
[Crossref]

Bomzon, Z.

Z. Bomzon, G. Biener, V. Kleiner, E. Hasman, “Space-variant Pancharatnam–Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett.1141–1143 (2002).
[Crossref]

Z. Bomzon, G. Biener, V. Kleiner, E. Hasman, “Real time analysis of partially polarized light with a space-variant subwavelength dielectric grating,” Opt. Lett. 27, 285–287 (2002).
[Crossref]

Z. Bomzon, V. Kleiner, E. Hasman, “Pancharatnam–Berry phase in space-variant polarization-manipulations with subwavelength gratings,” Opt. Lett. 26, 1424–1426 (2001).
[Crossref]

Z. Bomzon, V. Kleiner, E. Hasman, “Space-variant polarization state manipulation with computer-generated subwavelength metal-stripe gratings,” Opt. Commun. 192, 169–181 (2001).
[Crossref]

Brosseau, C.

Stokes parameters are used to define the polarization state. They are S0 = |Ex|2 + |Ey|2, S1 = |Ex|2 - |Ey|2, S2 = ExEy* + EyEx*, and S3 = i(ExEy* - EyEx*), where Ex and Ey are the Cartesian components of the electromagnetic field. S0 is the intensity of the field, whereas S1 … S3 define the polarization ellipse. See, for example, C. Brosseau, Polarized Light, A Statistical Optics Approach (Wiley, New York, 1998).

Collett, E.

E. Collett, Polarized Light (Marcel Dekker, New York, 1993).

Fischer, D.

Furlan, W. D.

H. J. Rabal, W. D. Furlan, E. E. Sicre, “Talbot interferometry with anistropic gratings,” Opt. Commun. 57, 81–83 (1986).
[Crossref]

Glas, P.

Goodman, J. W.

See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996), p. 87.

Hasman, E.

Z. Bomzon, G. Biener, V. Kleiner, E. Hasman, “Real time analysis of partially polarized light with a space-variant subwavelength dielectric grating,” Opt. Lett. 27, 285–287 (2002).
[Crossref]

Z. Bomzon, G. Biener, V. Kleiner, E. Hasman, “Space-variant Pancharatnam–Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett.1141–1143 (2002).
[Crossref]

Z. Bomzon, V. Kleiner, E. Hasman, “Space-variant polarization state manipulation with computer-generated subwavelength metal-stripe gratings,” Opt. Commun. 192, 169–181 (2001).
[Crossref]

Z. Bomzon, V. Kleiner, E. Hasman, “Pancharatnam–Berry phase in space-variant polarization-manipulations with subwavelength gratings,” Opt. Lett. 26, 1424–1426 (2001).
[Crossref]

Kleiner, V.

Z. Bomzon, G. Biener, V. Kleiner, E. Hasman, “Space-variant Pancharatnam–Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett.1141–1143 (2002).
[Crossref]

Z. Bomzon, G. Biener, V. Kleiner, E. Hasman, “Real time analysis of partially polarized light with a space-variant subwavelength dielectric grating,” Opt. Lett. 27, 285–287 (2002).
[Crossref]

Z. Bomzon, V. Kleiner, E. Hasman, “Pancharatnam–Berry phase in space-variant polarization-manipulations with subwavelength gratings,” Opt. Lett. 26, 1424–1426 (2001).
[Crossref]

Z. Bomzon, V. Kleiner, E. Hasman, “Space-variant polarization state manipulation with computer-generated subwavelength metal-stripe gratings,” Opt. Commun. 192, 169–181 (2001).
[Crossref]

Kung, H. L.

Leitner, M.

Lohmann, A. W.

V. A. Arrizón, E. Tepchin, M. Ortiz-Gutierrez, A. W. Lohmann, “Fresnel diffraction at ¼ of the Talbot distance of an anistropic grating,” Opt. Commun. 127, 171–175 (1996).
[Crossref]

Lowenthal, F.

Ch. Siegel, F. Lowenthal, J. E. Balmer, “A wavefront sensor based on the fractional Talbot effect,” Opt. Commun. 194, 265–275 (2001).
[Crossref]

Miller, D. A. B.

Napartovich, A. P.

Ortiz-Gutierrez, M.

V. A. Arrizón, E. Tepchin, M. Ortiz-Gutierrez, A. W. Lohmann, “Fresnel diffraction at ¼ of the Talbot distance of an anistropic grating,” Opt. Commun. 127, 171–175 (1996).
[Crossref]

Rabal, H. J.

H. J. Rabal, W. D. Furlan, E. E. Sicre, “Talbot interferometry with anistropic gratings,” Opt. Commun. 57, 81–83 (1986).
[Crossref]

Sicre, E. E.

H. J. Rabal, W. D. Furlan, E. E. Sicre, “Talbot interferometry with anistropic gratings,” Opt. Commun. 57, 81–83 (1986).
[Crossref]

Siegel, Ch.

Ch. Siegel, F. Lowenthal, J. E. Balmer, “A wavefront sensor based on the fractional Talbot effect,” Opt. Commun. 194, 265–275 (2001).
[Crossref]

Tepchin, E.

V. A. Arrizón, E. Tepchin, M. Ortiz-Gutierrez, A. W. Lohmann, “Fresnel diffraction at ¼ of the Talbot distance of an anistropic grating,” Opt. Commun. 127, 171–175 (1996).
[Crossref]

Vysotsky, D.

Wrage, M.

Opt. Commun. (4)

V. A. Arrizón, E. Tepchin, M. Ortiz-Gutierrez, A. W. Lohmann, “Fresnel diffraction at ¼ of the Talbot distance of an anistropic grating,” Opt. Commun. 127, 171–175 (1996).
[Crossref]

H. J. Rabal, W. D. Furlan, E. E. Sicre, “Talbot interferometry with anistropic gratings,” Opt. Commun. 57, 81–83 (1986).
[Crossref]

Ch. Siegel, F. Lowenthal, J. E. Balmer, “A wavefront sensor based on the fractional Talbot effect,” Opt. Commun. 194, 265–275 (2001).
[Crossref]

Z. Bomzon, V. Kleiner, E. Hasman, “Space-variant polarization state manipulation with computer-generated subwavelength metal-stripe gratings,” Opt. Commun. 192, 169–181 (2001).
[Crossref]

Opt. Lett. (5)

Other (4)

The Fresnel approximation for the propagation of a scalar wave is defined as E(x, y, z) = F-1HF[E(x, y, z = 0)], where E(x, y, z) is a scalar wave function, F denotes a spatial Fourier transform, H(fx, z) = exp(i2πz/λ)exp(-iπλzfx2) is the Fresnel transfer function, and fx denotes spatial frequency. See, for example, Ref. 1.

Stokes parameters are used to define the polarization state. They are S0 = |Ex|2 + |Ey|2, S1 = |Ex|2 - |Ey|2, S2 = ExEy* + EyEx*, and S3 = i(ExEy* - EyEx*), where Ex and Ey are the Cartesian components of the electromagnetic field. S0 is the intensity of the field, whereas S1 … S3 define the polarization ellipse. See, for example, C. Brosseau, Polarized Light, A Statistical Optics Approach (Wiley, New York, 1998).

E. Collett, Polarized Light (Marcel Dekker, New York, 1993).

See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996), p. 87.

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Figures (4)

Fig. 1
Fig. 1

(a) Space-varying polarization diffraction grating consisting of a periodic wave plate with a spatially rotating fast axis in the x direction. The period of the diffraction grating is d. Filled arrows show the local fast axis, whereas open arrows show the local slow axis. (b) Geometry of a space-variant subwavelength dielectric structure designed to act as the space-variant wave plate depicted in (a). Bottom left, scanning-electron microscope image of the subwavelength structure; bottom right, magnification of a region on the grating, demonstrating local subwavelength period Λ(x, y), local grating orientation θ(x, y), and local subwavelength grating vector K g (x, y).

Fig. 2
Fig. 2

Diffraction from the structure in Fig. 1. The Talbot effect occurs in the region where the diffracted orders overlap (the striped region).

Fig. 3
Fig. 3

(a) Measured intensity at planes z = 0, z = Z T /4, z = Z T /2, and z = Z T for linear incident polarization. (b) Measured and predicted Stokes parameters and (c) illustration of the space-variant polarization ellipse at these planes.

Fig. 4
Fig. 4

Measured and predicted azimuthal angles when circular polarization is incident upon the polarization diffraction grating of Fig. 1 at planes z = 0, z = Z T /4, z = Z T /2, and z = Z T .

Equations (13)

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J=100eiϕ,
TCx, y=Mθx, yJM-1θx, y,
Mθ=cos θ-sin θsin θcos θ
|R=10, |L=01
U=121i1-i
Tx, y=cosϕ21001 -i sinϕ20expi2θx, yexpi2θx, y0.
Tx, y=cosϕ21001 -i sinϕ20expi2πx/dexpi2πx/d0,
|Eoutx, z=0=cosϕ2|Ein-i sinϕ2ηL|Lexp-i2πx/d+ηR|Rexpi2πx/d,
|Eoutx, z=cosϕ2|Ein-i sinϕ2ηL|Lexp-i2πxd-iπλzd2+ηR|Rexpi2πxd-iπλzd2×expi2πzλ,
|Eout=cosϕ2|R-i sinϕ2|Lexp-i2πx/d.
S0x, z=1,S1x, z=-sin ϕ sin2πx/d+πλz/d2,S2x, z=sin ϕ cos2πx/d+πλz/d2,S3=cos ϕ.
χ=½ sin-1S3/S0=ϕ/2+π/4,ψ=½ tan-1S2/S1=πx/d+πλz/2d2+π/4.
|Eoutx,z=-cos2πx/d+π/4xˆ+sin2πx/d+π/4yˆexpi2πz/λ+π/4,

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