Abstract

In our proposal a light intensity distribution generated by an incoherently illuminated planar amplitude grating is projected into a photorefractive crystal. This 3D distribution is mapped as an index refractive perturbation via the photorefractive effect thereby generating a volume phase grating. The self-imaging phenomenon in the Fresnel field of this volume phase grating coherently illuminated is theoretically and experimentally analyzed. A model to simulate this volume grating that considers the 3D light intensity distribution formed in the crystal combined with the photorefractive grating formation theory is proposed. A path-integral approach to calculate the self-image patterns which account for the inhomogeneous propagation through the photorefractive grating is employed. The experimental and theoretical results show that the self-images location coincides with that of the self-images generated by planar phase grating of the same period. Moreover, the self-images visibility depends on three parameters: the exit pupil diameter of the incoherent recording optical system, the external electric field applied on the crystal, and the crystal thickness. To study the visibility behavior, a phase parameter which includes the three mentioned parameters is proposed. The self-images visibility shows the typical sinusoidal dependence found in planar phase grating. A good agreement between theoretical and experimental results is observed.

© 2012 Optical Society of America

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References

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    [CrossRef]
  4. K. Patorski, “Optical testing of ultrasonic phase gratings using a Fresnel diffraction method,” Ultrasonics 19, 169–172(1981).
    [CrossRef]
  5. K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 1–108 (1989).
    [CrossRef]
  6. T. Jinhong, “The diffraction near fields and Lau effect of a square-wave modulated phase grating,” J. Mod. Opt. 35, 1399–1408 (1988).
    [CrossRef]
  7. M. C. Lasprilla, A. A. Amorin, M. Tebaldi, and N. Bolognini, “Self-imaging through incoherent to coherent conversion,” Opt. Eng. 35, 1440–1445 (1996).
    [CrossRef]
  8. M. Tebaldi, P. J. Rueda, and N. Bolognini, “Talbot interferometer based on a birefringence grating,” Opt. Commun. 185, 65–76 (2000).
    [CrossRef]
  9. G. Forte, A. Lencina, M. Tebaldi, and N. Bolognini, “Self-imaging by a volume grating,” Opt. Commun. 284, 2494–2499 (2011).
    [CrossRef]
  10. P. Yeh, Introduction to Photorefractive Nonlinear Optics(Wiley, 1993), pp. 89–98.
  11. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), pp. 148–151.
  12. J. W. Yu, D. Psaltis, A. Marrakchi, A. Tanguay, and R. V. Johnson, “The photorefractive incoherent-to-coherent optical converter,” in Photorefractive Materials and their Applications II, P. Gunter and J. P. Huignard, eds. (Springer-Verlag, 1989), pp. 275–323.

2011

G. Forte, A. Lencina, M. Tebaldi, and N. Bolognini, “Self-imaging by a volume grating,” Opt. Commun. 284, 2494–2499 (2011).
[CrossRef]

2000

M. Tebaldi, P. J. Rueda, and N. Bolognini, “Talbot interferometer based on a birefringence grating,” Opt. Commun. 185, 65–76 (2000).
[CrossRef]

1996

M. C. Lasprilla, A. A. Amorin, M. Tebaldi, and N. Bolognini, “Self-imaging through incoherent to coherent conversion,” Opt. Eng. 35, 1440–1445 (1996).
[CrossRef]

1989

K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 1–108 (1989).
[CrossRef]

1988

T. Jinhong, “The diffraction near fields and Lau effect of a square-wave modulated phase grating,” J. Mod. Opt. 35, 1399–1408 (1988).
[CrossRef]

1981

K. Patorski, “Optical testing of ultrasonic phase gratings using a Fresnel diffraction method,” Ultrasonics 19, 169–172(1981).
[CrossRef]

1969

G. L. Rogers, “Fourier images in electron microscopy and their possible misinterpretation,” J. Microsc. 89, 121–124 (1969).
[CrossRef]

1965

1836

H. F. Talbot, “Facts relating to Optical Science. No. IV,” Philos. Mag. 9, 401–407 (1836).

Amorin, A. A.

M. C. Lasprilla, A. A. Amorin, M. Tebaldi, and N. Bolognini, “Self-imaging through incoherent to coherent conversion,” Opt. Eng. 35, 1440–1445 (1996).
[CrossRef]

Bolognini, N.

G. Forte, A. Lencina, M. Tebaldi, and N. Bolognini, “Self-imaging by a volume grating,” Opt. Commun. 284, 2494–2499 (2011).
[CrossRef]

M. Tebaldi, P. J. Rueda, and N. Bolognini, “Talbot interferometer based on a birefringence grating,” Opt. Commun. 185, 65–76 (2000).
[CrossRef]

M. C. Lasprilla, A. A. Amorin, M. Tebaldi, and N. Bolognini, “Self-imaging through incoherent to coherent conversion,” Opt. Eng. 35, 1440–1445 (1996).
[CrossRef]

Forte, G.

G. Forte, A. Lencina, M. Tebaldi, and N. Bolognini, “Self-imaging by a volume grating,” Opt. Commun. 284, 2494–2499 (2011).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), pp. 148–151.

Jinhong, T.

T. Jinhong, “The diffraction near fields and Lau effect of a square-wave modulated phase grating,” J. Mod. Opt. 35, 1399–1408 (1988).
[CrossRef]

Johnson, R. V.

J. W. Yu, D. Psaltis, A. Marrakchi, A. Tanguay, and R. V. Johnson, “The photorefractive incoherent-to-coherent optical converter,” in Photorefractive Materials and their Applications II, P. Gunter and J. P. Huignard, eds. (Springer-Verlag, 1989), pp. 275–323.

Lasprilla, M. C.

M. C. Lasprilla, A. A. Amorin, M. Tebaldi, and N. Bolognini, “Self-imaging through incoherent to coherent conversion,” Opt. Eng. 35, 1440–1445 (1996).
[CrossRef]

Lencina, A.

G. Forte, A. Lencina, M. Tebaldi, and N. Bolognini, “Self-imaging by a volume grating,” Opt. Commun. 284, 2494–2499 (2011).
[CrossRef]

Marrakchi, A.

J. W. Yu, D. Psaltis, A. Marrakchi, A. Tanguay, and R. V. Johnson, “The photorefractive incoherent-to-coherent optical converter,” in Photorefractive Materials and their Applications II, P. Gunter and J. P. Huignard, eds. (Springer-Verlag, 1989), pp. 275–323.

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 1–108 (1989).
[CrossRef]

K. Patorski, “Optical testing of ultrasonic phase gratings using a Fresnel diffraction method,” Ultrasonics 19, 169–172(1981).
[CrossRef]

Psaltis, D.

J. W. Yu, D. Psaltis, A. Marrakchi, A. Tanguay, and R. V. Johnson, “The photorefractive incoherent-to-coherent optical converter,” in Photorefractive Materials and their Applications II, P. Gunter and J. P. Huignard, eds. (Springer-Verlag, 1989), pp. 275–323.

Rogers, G. L.

G. L. Rogers, “Fourier images in electron microscopy and their possible misinterpretation,” J. Microsc. 89, 121–124 (1969).
[CrossRef]

Rueda, P. J.

M. Tebaldi, P. J. Rueda, and N. Bolognini, “Talbot interferometer based on a birefringence grating,” Opt. Commun. 185, 65–76 (2000).
[CrossRef]

Talbot, H. F.

H. F. Talbot, “Facts relating to Optical Science. No. IV,” Philos. Mag. 9, 401–407 (1836).

Tanguay, A.

J. W. Yu, D. Psaltis, A. Marrakchi, A. Tanguay, and R. V. Johnson, “The photorefractive incoherent-to-coherent optical converter,” in Photorefractive Materials and their Applications II, P. Gunter and J. P. Huignard, eds. (Springer-Verlag, 1989), pp. 275–323.

Tebaldi, M.

G. Forte, A. Lencina, M. Tebaldi, and N. Bolognini, “Self-imaging by a volume grating,” Opt. Commun. 284, 2494–2499 (2011).
[CrossRef]

M. Tebaldi, P. J. Rueda, and N. Bolognini, “Talbot interferometer based on a birefringence grating,” Opt. Commun. 185, 65–76 (2000).
[CrossRef]

M. C. Lasprilla, A. A. Amorin, M. Tebaldi, and N. Bolognini, “Self-imaging through incoherent to coherent conversion,” Opt. Eng. 35, 1440–1445 (1996).
[CrossRef]

Winthrop, T.

Worthington, C. R.

Yeh, P.

P. Yeh, Introduction to Photorefractive Nonlinear Optics(Wiley, 1993), pp. 89–98.

Yu, J. W.

J. W. Yu, D. Psaltis, A. Marrakchi, A. Tanguay, and R. V. Johnson, “The photorefractive incoherent-to-coherent optical converter,” in Photorefractive Materials and their Applications II, P. Gunter and J. P. Huignard, eds. (Springer-Verlag, 1989), pp. 275–323.

J. Microsc.

G. L. Rogers, “Fourier images in electron microscopy and their possible misinterpretation,” J. Microsc. 89, 121–124 (1969).
[CrossRef]

J. Mod. Opt.

T. Jinhong, “The diffraction near fields and Lau effect of a square-wave modulated phase grating,” J. Mod. Opt. 35, 1399–1408 (1988).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

M. Tebaldi, P. J. Rueda, and N. Bolognini, “Talbot interferometer based on a birefringence grating,” Opt. Commun. 185, 65–76 (2000).
[CrossRef]

G. Forte, A. Lencina, M. Tebaldi, and N. Bolognini, “Self-imaging by a volume grating,” Opt. Commun. 284, 2494–2499 (2011).
[CrossRef]

Opt. Eng.

M. C. Lasprilla, A. A. Amorin, M. Tebaldi, and N. Bolognini, “Self-imaging through incoherent to coherent conversion,” Opt. Eng. 35, 1440–1445 (1996).
[CrossRef]

Philos. Mag.

H. F. Talbot, “Facts relating to Optical Science. No. IV,” Philos. Mag. 9, 401–407 (1836).

Prog. Opt.

K. Patorski, “The self-imaging phenomenon and its applications,” Prog. Opt. 27, 1–108 (1989).
[CrossRef]

Ultrasonics

K. Patorski, “Optical testing of ultrasonic phase gratings using a Fresnel diffraction method,” Ultrasonics 19, 169–172(1981).
[CrossRef]

Other

P. Yeh, Introduction to Photorefractive Nonlinear Optics(Wiley, 1993), pp. 89–98.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), pp. 148–151.

J. W. Yu, D. Psaltis, A. Marrakchi, A. Tanguay, and R. V. Johnson, “The photorefractive incoherent-to-coherent optical converter,” in Photorefractive Materials and their Applications II, P. Gunter and J. P. Huignard, eds. (Springer-Verlag, 1989), pp. 275–323.

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Figures (5)

Fig. 1.
Fig. 1.

Experimental setup: S1: white light source; FG: green interference filter; FR: red interference filter; G: Ronchi grating; L1 and L2: lenses; D: Diaphragm; BS: beam splitter; CS: collimator system.

Fig. 2.
Fig. 2.

Simulated index grating modulation for: (a), (b), and (c) different crystal thickness with D=3mm and E0=7kV/cm;(d), (e), and (f) different DC external field with D=50mm and LZ=6mm; (g), (h), and (i) different output pupil diameter with E0=7kV/cm and LZ=6mm. (normalized to the amplitude at E0=7kV/cm).

Fig. 3.
Fig. 3.

Experimental self-image patterns generated by photorefractive gratings with LZ=6mm, E0=7kV/cm, and (a) D=50mm and (b) D=3mm.

Fig. 4.
Fig. 4.

Self-image intensity profiles generated by different photorefractive grating modulations: (a) calculated and (b) experimental patterns using the gratings depicted in Figs. 2(a), 2(b), and 2(c); (c) calculated and (d) experimental patterns using gratings with increasing E0, D=3mm, and LZ=3mm; (e) calculated and (f) experimental patterns using the gratings depicted in Figs. 2(g) and 2(i).

Fig. 5.
Fig. 5.

Self-images visibility versus phase parameter βmod for: (a) theoretical patterns: ■ LZ=1, 2, 3, 4, 5, and 6 mm with fixed D=3mm and E0=7kV/cm; ● E0=5, 6, 7, 8, and 9kV/cm with fixed D=3mm and LZ=3mm; ▲ D=50, 25, 10, 5, and 3 mm with fixed LZ=6mm and E0=7×102kV/m; (b) experimental patterns: ■ LZ=1, 2, 3, 4, and 6 mm with fixed D=3mm and E0=7kV/cm; ● E0=5, 6, 7, 8, and 9kV/cm with fixed D=3mm and LZ=3mm; ▲ D=50 and 3 mm with fixed LZ=6mm and E0=7×102kV/m.

Equations (10)

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w=(1za+1zG1f)D28,
H(u)=Λ(u2u0)sinc[(8wλw)(u2u0)(1|u|2u0)],
I(z0,x0;D)=F1{t˜(u).H(u)},
t(x)=12+2m=1sin(πm/2)πmcos(2πmxd),
Δn(z0,x0;E0,D,LZ)=Δn0(E0)IN(z0,x0,D)rect(z0/LZ),
n(z0,x0;E0,D,LZ)=n0+Δn(z0,x0;E0,D,LZ),
z,x,y|0,x0,y0=kr2πi(zzt)eikr[(zzt)+(xx0)2+(yy0)22(zzt)]eikr0ztn(z,x0,y0)dz,
z,x,y|Ψ=tFkr2πi(zzt)eikr[(zzt)+(xx0)2+(yy0)22(zzt)]eikr0ztn(z,x0,y0)dz0,x0,y0|Ψdx0dy0,
V=||zT,0|Ψ|2|zT,d/2|Ψ|2||zT,0|Ψ|2+|zT,d/2|Ψ|2
βmod=kr(0LZ(Δn(z,0;E0,D,LZ)Δn(z,d/2;E0,D,LZ))dz).

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