Abstract

Formulas are given for the calculation of diffraction efficiency of reflection-type gratings recorded in a photorefractive medium. The analysis incorporates the coupled-wave theory that was developed for photorefractive hologram gratings. This analysis takes into account grating slant with respect to the medium surface, light absorption during reconstruction, any incident angle of the reference beam, and any photorefractive phase shift. General solutions for signal and reference wave functions are given in a closed-form expression by use of a hypergeometric function. The optimum media parameters and recording conditions for high diffraction efficiency are obtained by the derived formulas. The diffraction properties for off-Bragg conditions are also discussed.

© 1998 Optical Society of America

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References

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  1. T. Kume, K. Nonaka, M. Yamamoto, “Wavelength-multiplexed holographic recording in cerium-doped strontium barium niobate by using tunable laser diode,” Jpn. J. Appl. Phys. 35, 448–453 (1996).
    [CrossRef]
  2. J. M. Heaton, P. A. Mills, E. G. S. Paige, L. Solymer, T. Wilson, “Diffraction efficiency and angular selectivity of volume phase holograms recorded in photorefractive materials,” Opt. Acta 31, 885–901 (1984).
    [CrossRef]
  3. R. Hofmeister, A. Yariv, S. Yagi, “Spectral response of fixed photorefractive grating interference filters,” J. Opt. Soc. Am. A 11, 1342–1351 (1994).
    [CrossRef]
  4. P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
    [CrossRef]
  5. K. Nonaka, “Diffraction efficiency analysis in hologram gratings recorded by counterpropagating-type geometry,” J. Appl. Phys. 78, 4345–4352 (1995).
    [CrossRef]
  6. K. Nonaka, “Off-Bragg analysis of the diffraction efficiency of transmission photorefractive holograms,” Appl. Opt. 36, 4792–4800 (1997).
    [CrossRef] [PubMed]
  7. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
  8. H. Zhou, F. Zhao, F. T. S. Yu, “Effects of recording-erasure dynamics of storage capacity of a wavelength-multiplexed reflection-type photorefractive hologram,” Appl. Opt. 33, 4339–4334 (1994).
    [CrossRef] [PubMed]
  9. J. V. Alvarez-Bravo, N. Bolognini, L. Arizmendi, “Experimental study of the angular selectivity of volume phase holograms stored in LiNbO3,” Appl. Phys. B 62, 159–164 (1996).
    [CrossRef]
  10. J. V. Alvarez-Bravo, M. Carrascosa, L. Arizmendi, “Experimental effects of light intensity modulation on the recording and erasure of holographic gratings in BSO crystals,” Opt. Commun. 103, 22–28 (1993).
    [CrossRef]
  11. E. Ochoa, F. Vachss, L. Hesselink, “Higher-order analysis of the photorefractive effect for large modulation depths,” J. Opt. Soc. Am. A 3, 181–187 (1986).
    [CrossRef]
  12. B. I. Sturman, D. J. Webb, R. Kowarschik, E. Shamonina, K. H. Ringhofer, “Exact solution of the Bragg-diffraction problem in sillenites,” J. Opt. Soc. Am. B 11, 1813–1819 (1994).
    [CrossRef]
  13. S. Mallick, D. Rouède, A. G. Apostolidis, “Efficiency and polarization characteristics of photorefractive diffraction in a Bi12SiO20 crystal,” J. Opt. Soc. Am. B 4, 1247–1259 (1987).
    [CrossRef]
  14. Y. H. Ja, “Energy transfer between two beams in writing a reflection volume hologram in a dynamic medium,” Opt. Quantum Electron. 14, 547–556 (1982).
    [CrossRef]

1997 (1)

1996 (2)

T. Kume, K. Nonaka, M. Yamamoto, “Wavelength-multiplexed holographic recording in cerium-doped strontium barium niobate by using tunable laser diode,” Jpn. J. Appl. Phys. 35, 448–453 (1996).
[CrossRef]

J. V. Alvarez-Bravo, N. Bolognini, L. Arizmendi, “Experimental study of the angular selectivity of volume phase holograms stored in LiNbO3,” Appl. Phys. B 62, 159–164 (1996).
[CrossRef]

1995 (1)

K. Nonaka, “Diffraction efficiency analysis in hologram gratings recorded by counterpropagating-type geometry,” J. Appl. Phys. 78, 4345–4352 (1995).
[CrossRef]

1994 (3)

1993 (1)

J. V. Alvarez-Bravo, M. Carrascosa, L. Arizmendi, “Experimental effects of light intensity modulation on the recording and erasure of holographic gratings in BSO crystals,” Opt. Commun. 103, 22–28 (1993).
[CrossRef]

1989 (1)

P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[CrossRef]

1987 (1)

1986 (1)

1984 (1)

J. M. Heaton, P. A. Mills, E. G. S. Paige, L. Solymer, T. Wilson, “Diffraction efficiency and angular selectivity of volume phase holograms recorded in photorefractive materials,” Opt. Acta 31, 885–901 (1984).
[CrossRef]

1982 (1)

Y. H. Ja, “Energy transfer between two beams in writing a reflection volume hologram in a dynamic medium,” Opt. Quantum Electron. 14, 547–556 (1982).
[CrossRef]

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Alvarez-Bravo, J. V.

J. V. Alvarez-Bravo, N. Bolognini, L. Arizmendi, “Experimental study of the angular selectivity of volume phase holograms stored in LiNbO3,” Appl. Phys. B 62, 159–164 (1996).
[CrossRef]

J. V. Alvarez-Bravo, M. Carrascosa, L. Arizmendi, “Experimental effects of light intensity modulation on the recording and erasure of holographic gratings in BSO crystals,” Opt. Commun. 103, 22–28 (1993).
[CrossRef]

Apostolidis, A. G.

Arizmendi, L.

J. V. Alvarez-Bravo, N. Bolognini, L. Arizmendi, “Experimental study of the angular selectivity of volume phase holograms stored in LiNbO3,” Appl. Phys. B 62, 159–164 (1996).
[CrossRef]

J. V. Alvarez-Bravo, M. Carrascosa, L. Arizmendi, “Experimental effects of light intensity modulation on the recording and erasure of holographic gratings in BSO crystals,” Opt. Commun. 103, 22–28 (1993).
[CrossRef]

Bolognini, N.

J. V. Alvarez-Bravo, N. Bolognini, L. Arizmendi, “Experimental study of the angular selectivity of volume phase holograms stored in LiNbO3,” Appl. Phys. B 62, 159–164 (1996).
[CrossRef]

Carrascosa, M.

J. V. Alvarez-Bravo, M. Carrascosa, L. Arizmendi, “Experimental effects of light intensity modulation on the recording and erasure of holographic gratings in BSO crystals,” Opt. Commun. 103, 22–28 (1993).
[CrossRef]

Heaton, J. M.

J. M. Heaton, P. A. Mills, E. G. S. Paige, L. Solymer, T. Wilson, “Diffraction efficiency and angular selectivity of volume phase holograms recorded in photorefractive materials,” Opt. Acta 31, 885–901 (1984).
[CrossRef]

Hesselink, L.

Hofmeister, R.

Ja, Y. H.

Y. H. Ja, “Energy transfer between two beams in writing a reflection volume hologram in a dynamic medium,” Opt. Quantum Electron. 14, 547–556 (1982).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Kowarschik, R.

Kume, T.

T. Kume, K. Nonaka, M. Yamamoto, “Wavelength-multiplexed holographic recording in cerium-doped strontium barium niobate by using tunable laser diode,” Jpn. J. Appl. Phys. 35, 448–453 (1996).
[CrossRef]

Mallick, S.

Mills, P. A.

J. M. Heaton, P. A. Mills, E. G. S. Paige, L. Solymer, T. Wilson, “Diffraction efficiency and angular selectivity of volume phase holograms recorded in photorefractive materials,” Opt. Acta 31, 885–901 (1984).
[CrossRef]

Nonaka, K.

K. Nonaka, “Off-Bragg analysis of the diffraction efficiency of transmission photorefractive holograms,” Appl. Opt. 36, 4792–4800 (1997).
[CrossRef] [PubMed]

T. Kume, K. Nonaka, M. Yamamoto, “Wavelength-multiplexed holographic recording in cerium-doped strontium barium niobate by using tunable laser diode,” Jpn. J. Appl. Phys. 35, 448–453 (1996).
[CrossRef]

K. Nonaka, “Diffraction efficiency analysis in hologram gratings recorded by counterpropagating-type geometry,” J. Appl. Phys. 78, 4345–4352 (1995).
[CrossRef]

Ochoa, E.

Paige, E. G. S.

J. M. Heaton, P. A. Mills, E. G. S. Paige, L. Solymer, T. Wilson, “Diffraction efficiency and angular selectivity of volume phase holograms recorded in photorefractive materials,” Opt. Acta 31, 885–901 (1984).
[CrossRef]

Ringhofer, K. H.

Rouède, D.

Shamonina, E.

Solymer, L.

J. M. Heaton, P. A. Mills, E. G. S. Paige, L. Solymer, T. Wilson, “Diffraction efficiency and angular selectivity of volume phase holograms recorded in photorefractive materials,” Opt. Acta 31, 885–901 (1984).
[CrossRef]

Sturman, B. I.

Vachss, F.

Webb, D. J.

Wilson, T.

J. M. Heaton, P. A. Mills, E. G. S. Paige, L. Solymer, T. Wilson, “Diffraction efficiency and angular selectivity of volume phase holograms recorded in photorefractive materials,” Opt. Acta 31, 885–901 (1984).
[CrossRef]

Yagi, S.

Yamamoto, M.

T. Kume, K. Nonaka, M. Yamamoto, “Wavelength-multiplexed holographic recording in cerium-doped strontium barium niobate by using tunable laser diode,” Jpn. J. Appl. Phys. 35, 448–453 (1996).
[CrossRef]

Yariv, A.

Yeh, P.

P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[CrossRef]

Yu, F. T. S.

Zhao, F.

Zhou, H.

Appl. Opt. (2)

Appl. Phys. B (1)

J. V. Alvarez-Bravo, N. Bolognini, L. Arizmendi, “Experimental study of the angular selectivity of volume phase holograms stored in LiNbO3,” Appl. Phys. B 62, 159–164 (1996).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

IEEE J. Quantum Electron. (1)

P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[CrossRef]

J. Appl. Phys. (1)

K. Nonaka, “Diffraction efficiency analysis in hologram gratings recorded by counterpropagating-type geometry,” J. Appl. Phys. 78, 4345–4352 (1995).
[CrossRef]

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

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

Jpn. J. Appl. Phys. (1)

T. Kume, K. Nonaka, M. Yamamoto, “Wavelength-multiplexed holographic recording in cerium-doped strontium barium niobate by using tunable laser diode,” Jpn. J. Appl. Phys. 35, 448–453 (1996).
[CrossRef]

Opt. Acta (1)

J. M. Heaton, P. A. Mills, E. G. S. Paige, L. Solymer, T. Wilson, “Diffraction efficiency and angular selectivity of volume phase holograms recorded in photorefractive materials,” Opt. Acta 31, 885–901 (1984).
[CrossRef]

Opt. Commun. (1)

J. V. Alvarez-Bravo, M. Carrascosa, L. Arizmendi, “Experimental effects of light intensity modulation on the recording and erasure of holographic gratings in BSO crystals,” Opt. Commun. 103, 22–28 (1993).
[CrossRef]

Opt. Quantum Electron. (1)

Y. H. Ja, “Energy transfer between two beams in writing a reflection volume hologram in a dynamic medium,” Opt. Quantum Electron. 14, 547–556 (1982).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Schematic illustration of reflection-type recording geometry: (a) recording geometry of type I, (b) recording geometry of type II, (c) intensity distributions of reference (pump) and object (probe) beams, (d) vector diagram for the recording geometry of type I showing the relationship to grating vector K, the wave vector of reference beam kRW, and the wave vector of object beam kSW. In (c) R w denotes the reference beam and S w the object beam; S w0 shows the intensity of the object beam at the medium surface and R w0 shows that of the reference beam.

Fig. 2
Fig. 2

Model of a hologram grating during reconstruction.

Fig. 3
Fig. 3

Diffraction efficiency η as a function of input intensity ratio of recording beam r with variable parameter φ g . The calculation conditions are as follows: The medium parameters have a medium thickness of T = 4 mm, a refractive index of n 0 = 2.4, an index modulation depth of Δn = 80 × 10-6 (=Δn S = Δn L /2), an absorption coefficient of α = 150(1/m), a slant angle of grating φ = 0°; the recording conditions are an incident angle of the recording beam of θ0 = 10°; the reconstruction conditions are a reference beam wavelength of λ = 0.532 μm, an incident angle of the reference beam of θ0 = 10°, and a mismatch from the Bragg condition of ϑ = 0.

Fig. 4
Fig. 4

Dependence of diffraction efficiency η on index amplitude Δn (=Δn S = Δn L /2) for various values of φ g with r = 100 and T = 2 mm. The parameters used are the same as those in Fig. 3, except for Δn, r, and T.

Fig. 5
Fig. 5

Dependence of diffraction efficiency η on medium thickness T for various values of φ g with r = 100 and Δn = 80 × 10-6. The parameters used are the same as those in Fig. 4, except for T.

Fig. 6
Fig. 6

Diffraction efficiency η as a function of dephasing parameter ϑ from the Bragg condition for various values of φ g with r = 100 and T = 5 mm. The parameters used are the same as those in Fig. 4, except for T and ϑ.

Fig. 7
Fig. 7

Mismatch curves for large Δn (=200 × 10-6). The parameters used are the same as those in Fig. 6, except for Δn.

Equations (81)

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m z = 1 / 1 + M   exp γ z 1 / 2 ,
M = r 2 exp - γ T - 1 2 4 r r + 1 r   exp - γ T + 1 ,
r = R w 0 S w 0 ,
γ = 2 π Δ n S λ   cos   θ 0 sin   φ g ,
φ z = β z ,
β = π Δ n S λ   cos   θ 0 cos   φ g .
n x ,   z = n 0 + 1 / 2 ) Δ n L m z exp i φ g × exp i K · r + φ z + c . c . ) ,
c r d R d z + α R = 1 2   i   exp i φ g Γ m z × exp i φ z exp - i ϑ z S ,
c s d S d z + α S = 1 2   i   exp - i φ g Γ m z × exp - i φ z exp i ϑ z R ,
Γ = π Δ n L / λ ,
c r = cos   θ ,
c s = cos   θ - 2   cos φ - θ cos   φ ,
α = α / 2 ,
ϑ = Δ θ K   sin φ - θ 0 - Δ λ K 2 / 4 π n 0 .
d 2 S d t 2 + 1 t t - 1 3 2   t + u s - 1 2 d S d t + 1 t 2 t - 1 2 × b 2 - 1 2   h s t + h s h r + 1 2 - i Δ S = 0 .
d 2 R d t 2 + 1 t t - 1 3 2   t + u r - 1 2 d R d t + 1 t 2 t - 1 2 × b 2 - 1 2   h r t + h r h s + 1 2 + i Δ R = 0 ,
t = 1 / 1 + M   exp γ z ,
h r = α c r γ ,
h s = α c s γ ,
u s = h s + h r - i Δ ,
u r = h r + h s + i Δ ,
Δ = ϑ - β γ ,
b 2 = Γ 2 4 c r c s γ 2 .
S = C 1 S 1 + C 2 S 2 ,
R = C 3 R 1 + C 4 R 2 ,
S 1 = t h S 1 - t - u S + N / 2 2 F 1 - w + N / 2 , - w - 1 + N / 2 ,   1 / 2 - w ,   t ,
S 2 = t 1 / 2 + h S + w 1 - t - u S + N / 2 2 F 1 w + 1 - N / 2 , w + 2 - N / 2 ,   3 / 2 + w ,   t ,
R 1 = t h r 1 - t - u r + N / 2 2 F 1 w - N / 2 , w + 1 - N / 2 ,   1 / 2 + w ,   t ,
R 2 = t 1 / 2 + h r - w 1 - t - u r + N / 2 2 F 1 - w + 1 - N / 2 , - w + 2 - N / 2 ,   3 / 2 - w ,   t ,
w = h r - h s - i Δ ,
N = w 2 - 4 b 2 ,
R 0 = 1 ,     S T = 0 ,
S 0 = t 0 h S 1 - t 0 - u S + N / 2 C 1   2 F 1 - w + N / 2 ,   - w - 1 + N / 2 ,   1 / 2 - w ,   t 0 + C 2 t 0 1 / 2 + w 2 F 1 w + 1 - N / 2 , w + 2 - N / 2 ,   3 / 2 + w ,   t 0 ,
t 0 = 1 / 1 + M ,
η = | c s | c r   S 0 S * 0 ,
c r d R d z + α R = i   Γ 2 1 + M 0 1 / 2 exp i β - ϑ z S ,
c s d S d z + α S = i   Γ 2 1 + M 0 1 / 2 exp - i β - ϑ z R ,
M 0 = r - 1 2 4 r .
d 2 S d z 2 + a r + a s + i δ d S d z + a r a s + ia s δ + B 2 S = 0 ,
d 2 R d z 2 + a r + a s - i δ d R d z + a r a s - ia r δ + B 2 R = 0 ,
a r = α c r ,
a s = α c s ,
δ = β - ϑ ,
B 2 = Γ 2 4 1 + M 0 c r c s .
S z = i c r c s 1 / 2 B   exp - i δ T × q 1 - q 2 p 1 - p 2 exp p 1 z + p 2 T - exp p 2 z + p 1 T q 2 + a r exp q 2 T - q 1 + a r exp q 1 T ,
R z = q 2 + a r exp q 1 z + q 2 T - q 1 + a r exp q 2 z + q 1 T q 2 + a r exp q 2 T - q 1 + a r exp q 1 T ,
p 1 = 1 2 - a r + a s + i δ + a r - a s + i δ 2 - 4 B 2 1 / 2 ,
p 2 = 1 2 - a r + a s + i δ - a r - a s + i δ 2 - 4 B 2 1 / 2 ,
q 1 = 1 2 - a r + a s - i δ + a s - a r - i δ 2 - 4 B 2 1 / 2 ,
q 2 = 1 2 - a r + a s - i δ - a s - a r - i δ 2 - 4 B 2 1 / 2 .
S z = i c r c s 1 / 2 B × 2   exp - i 2 δ T exp - i δ z / 2 sinh Q T - z / 2 Q   cosh QT / 2 + i δ   sinh QT / 2 ,
R z = exp i δ z / 2 × Q   cosh Q T - z / 2 + i δ   sinh Q T - z / 2 Q   cosh QT / 2 + i δ   sinh QT / 2 ,
Q = - δ 2 - 4 B 2 1 / 2 .
η = 4 B 2 sinh 2 QT / 2 δ 2 + 4 B 2 cosh 2 QT / 2 .
S z = i c r c s 1 / 2 sinh | B | T - z cosh | B | T ,
R z = cosh | B | T - z cosh | B | T ,
η = tanh 2 | B | T .
c   d R d z = - Γ 2 1 + M   exp γ z 1 / 2   S ,
c   d S d z = - Γ 2 1 + M   exp γ z 1 / 2   R .
S z = sinh Γ 2 c ν T - ν cosh Γ 2 c ν T - ν 0 ,
R z = cosh Γ 2 c ν T - ν cosh Γ 2 c ν T - ν 0 ,
ν = 1 γ ln 1 + M   exp γ z 1 / 2 - 1 1 + M   exp γ z 1 / 2 + 1 ,
ν T = 1 γ ln 1 + M   exp γ T 1 / 2 - 1 1 + M   exp γ T 1 / 2 + 1 ,
ν 0 = 1 γ ln 1 + M 1 / 2 - 1 1 + M 1 / 2 + 1 .
C 1 = - i   c r c s 1 / 2 1 - 2 w 1 + 2 w × b   exp i ϑ - β T - φ g t T 1 / 2 t T 1 - t T i 2 Δ S 2 T D ,
C 2 = + i   c r c s 1 / 2 1 - 2 w 1 + 2 w × b   exp i ϑ - β T - φ g t T 1 / 2 t T 1 - t T i 2 Δ S 1 T D ,
C 3 = D 2 / D ,
C 4 = - D 1 / D ,
D = R 10 D 2 - R 20 D 1 ,
D 1 = 1 2   u r + 1 2 N - h r t T R 1 T + t T 1 - t T α 3 β 3 γ 3   X 3 T F 3 T ,
D 2 = 1 2 - w - 1 2 - w + h r - 1 2   u r - 1 2 N t T × R 2 T + t T 1 - t T α 4 β 4 γ 4   X 4 T F 4 T ,
S nT = S n z = T ,
R nT = R n z = T ,
R n 0 = R n z = 0 .
X 3 T = t T h r 1 - t T - u r + N / 2 ,
X 4 T = t T 1 / 2 + h r - w 1 - t T - u r + N / 2 ,
F 3 T = 2 F 1 w - N + 2 / 2 ,   w + 3 - N / 2 , 3 / 2 + w ,   t T ,
F 4 T = 2 F 1 - w + 3 - N / 2 ,   - w + 4 - N / 2 , 5 / 2 - w ,   t T ,
t T = 1 / 1 + M   exp γ T ,
α 3 = w - N / 2 ,   β 3 = w + 1 - N / 2 , γ 3 = 1 / 2 + w ,
α 4 = - w + 1 - N / 2 ,   β 4 = - w + 2 - N / 2 , γ 4 = 3 / 2 - w .

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