Abstract

We investigated the diffraction properties of dynamic holograms recorded in porphyrin:Zn doped nematic liquid crystals (NLCs) under the influence of an applied dc electric field for various conditions of the grating period, the writing beam intensity and the applied electric field. We also derived an analytic expression for diffraction efficiency from NLCs material equations and torque balance equations and compared the experimental results with the theory, revealing excellent agreement.

© 2008 Optical Society of America

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References

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  1. P. Günter and J. P. Huignard, Photorefractive Materials and Their Applications (Springer, Berlin, 1989), Vols. I and II.
  2. P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993).
  3. E. V. Rudenko and A. V. Sukhov, "Optically induced spatial charge separation in a nematic and the resultant orientational nonlinearity," JEPT.  78, 875-882 (1994).
  4. I. C. Khoo, B. D. Guenther, M. V. Wood, P. Chen, and M.-Y. Shin, "Coherent beam amplification with a photorefractive liquid crystal," Opt. Lett. 22, 1229-1231 (1997).
    [CrossRef] [PubMed]
  5. L. Marrucci, D. Paparo, P. Maddalena, E. Massera, E. Prudnikova, and E. Santamato, "Role of guest-host intermolecular forces in photoinduced reorientation of liquid crystals," J. Chem. Phys. 107, 9783-9793 (1997).
    [CrossRef]
  6. H. Ono and N. Kawatsuki, "High-performance photorefractivity in high- and low-molar-mass liquid crystal mixtures," J. Appl. Phys. 85, 2482-2487 (1999)
    [CrossRef]
  7. K. H. Kim, E. J. Kim, S. J. Lee, J. H. Lee, J. E. Kim, and C. H. Kwak, "Effects of applied field on orientational photorefraction in porphyrin:Zn-doped nematic liquid crystals," Appl. Phys. Lett. 85, 366-368 (2004).
    [CrossRef]
  8. I. C. Khoo, "Holographic grating formation in dye and fullerene C60-doped nematic liquid crystal film," Opt. Lett. 20, 2137-2139 (1995).
    [CrossRef] [PubMed]
  9. I. C. Khoo, "Orientational photorefractive effects in nematic liquid crystal films," IEEE J. Quantum. Electron. 32, 525-534 (1996).
    [CrossRef]
  10. I. Janossy, A. D. Lloyd, and B. S. Wherrer, "Anomalous optical Freedericksz transition in an absorbing liquid crystal," Mol. Cryst. Liq. Cryst. 179, 1-12 (1990).
  11. Y. -P. Huang, T. -Y. Tsai, W. Lee, W. -K. Chin, Y. -M. Chang, and H. -Y. Chen, "Photorefractive effect in nematic-clay nanocomposites," Opt. Express 13, 2058-2063 (2005).
    [CrossRef] [PubMed]
  12. M. Kaczmarek, M. -Y. Shin, R. S. Cudney, and I. C. Khoo, "Electrically tunable, optically induced dynamic and permanent gratings in dye-doped liquid crystals," IEEE J. Quantum. Electron. 38, 451-457 (2002).
    [CrossRef]
  13. H. Ono and N. Kawatsuki, "Orientational photorefractive effects observed in polymer-dispersed liquid crystals," Opt. Lett. 22, 1144-1146 (1997).
    [CrossRef] [PubMed]
  14. C. H. Kwak, J. Takacs, and L. Solymar, "Spatial subharmonic instabilities," Opt. Commun. 96, 278-282 (1993).
    [CrossRef]
  15. C. H. Kwak, M. Sharmonin, J. Takacs, and L. Solymar, "Spatial subharmonics in photorefractive Bi12SiO20 crystal with a square wave applied field," Appl. Phys. Lett. 62, 328-330 (1993).
    [CrossRef]
  16. I. C. Khoo and S. H. Wu, Optics and Nonlinear Optics of Liquid Crystals (World Scientific, Singapore, 1993).
  17. S. -T. Wu and C. -S. Wu, "Experimental confirmation of the Osipov-Terentjev theory on the viscosity of nematic liquid crystals," Phys. Rev. 42, 2219-2227 (1990).
    [CrossRef]

2005 (1)

2004 (1)

K. H. Kim, E. J. Kim, S. J. Lee, J. H. Lee, J. E. Kim, and C. H. Kwak, "Effects of applied field on orientational photorefraction in porphyrin:Zn-doped nematic liquid crystals," Appl. Phys. Lett. 85, 366-368 (2004).
[CrossRef]

2002 (1)

M. Kaczmarek, M. -Y. Shin, R. S. Cudney, and I. C. Khoo, "Electrically tunable, optically induced dynamic and permanent gratings in dye-doped liquid crystals," IEEE J. Quantum. Electron. 38, 451-457 (2002).
[CrossRef]

1999 (1)

H. Ono and N. Kawatsuki, "High-performance photorefractivity in high- and low-molar-mass liquid crystal mixtures," J. Appl. Phys. 85, 2482-2487 (1999)
[CrossRef]

1997 (3)

1996 (1)

I. C. Khoo, "Orientational photorefractive effects in nematic liquid crystal films," IEEE J. Quantum. Electron. 32, 525-534 (1996).
[CrossRef]

1995 (1)

1994 (1)

E. V. Rudenko and A. V. Sukhov, "Optically induced spatial charge separation in a nematic and the resultant orientational nonlinearity," JEPT.  78, 875-882 (1994).

1993 (2)

C. H. Kwak, J. Takacs, and L. Solymar, "Spatial subharmonic instabilities," Opt. Commun. 96, 278-282 (1993).
[CrossRef]

C. H. Kwak, M. Sharmonin, J. Takacs, and L. Solymar, "Spatial subharmonics in photorefractive Bi12SiO20 crystal with a square wave applied field," Appl. Phys. Lett. 62, 328-330 (1993).
[CrossRef]

1990 (2)

S. -T. Wu and C. -S. Wu, "Experimental confirmation of the Osipov-Terentjev theory on the viscosity of nematic liquid crystals," Phys. Rev. 42, 2219-2227 (1990).
[CrossRef]

I. Janossy, A. D. Lloyd, and B. S. Wherrer, "Anomalous optical Freedericksz transition in an absorbing liquid crystal," Mol. Cryst. Liq. Cryst. 179, 1-12 (1990).

Chang, Y. -M.

Chen, H. -Y.

Chen, P.

Chin, W. -K.

Cudney, R. S.

M. Kaczmarek, M. -Y. Shin, R. S. Cudney, and I. C. Khoo, "Electrically tunable, optically induced dynamic and permanent gratings in dye-doped liquid crystals," IEEE J. Quantum. Electron. 38, 451-457 (2002).
[CrossRef]

Guenther, B. D.

Huang, Y. -P.

Janossy, I.

I. Janossy, A. D. Lloyd, and B. S. Wherrer, "Anomalous optical Freedericksz transition in an absorbing liquid crystal," Mol. Cryst. Liq. Cryst. 179, 1-12 (1990).

Kaczmarek, M.

M. Kaczmarek, M. -Y. Shin, R. S. Cudney, and I. C. Khoo, "Electrically tunable, optically induced dynamic and permanent gratings in dye-doped liquid crystals," IEEE J. Quantum. Electron. 38, 451-457 (2002).
[CrossRef]

Kawatsuki, N.

H. Ono and N. Kawatsuki, "High-performance photorefractivity in high- and low-molar-mass liquid crystal mixtures," J. Appl. Phys. 85, 2482-2487 (1999)
[CrossRef]

H. Ono and N. Kawatsuki, "Orientational photorefractive effects observed in polymer-dispersed liquid crystals," Opt. Lett. 22, 1144-1146 (1997).
[CrossRef] [PubMed]

Khoo, I. C.

M. Kaczmarek, M. -Y. Shin, R. S. Cudney, and I. C. Khoo, "Electrically tunable, optically induced dynamic and permanent gratings in dye-doped liquid crystals," IEEE J. Quantum. Electron. 38, 451-457 (2002).
[CrossRef]

I. C. Khoo, B. D. Guenther, M. V. Wood, P. Chen, and M.-Y. Shin, "Coherent beam amplification with a photorefractive liquid crystal," Opt. Lett. 22, 1229-1231 (1997).
[CrossRef] [PubMed]

I. C. Khoo, "Orientational photorefractive effects in nematic liquid crystal films," IEEE J. Quantum. Electron. 32, 525-534 (1996).
[CrossRef]

I. C. Khoo, "Holographic grating formation in dye and fullerene C60-doped nematic liquid crystal film," Opt. Lett. 20, 2137-2139 (1995).
[CrossRef] [PubMed]

Kim, E. J.

K. H. Kim, E. J. Kim, S. J. Lee, J. H. Lee, J. E. Kim, and C. H. Kwak, "Effects of applied field on orientational photorefraction in porphyrin:Zn-doped nematic liquid crystals," Appl. Phys. Lett. 85, 366-368 (2004).
[CrossRef]

Kim, J. E.

K. H. Kim, E. J. Kim, S. J. Lee, J. H. Lee, J. E. Kim, and C. H. Kwak, "Effects of applied field on orientational photorefraction in porphyrin:Zn-doped nematic liquid crystals," Appl. Phys. Lett. 85, 366-368 (2004).
[CrossRef]

Kim, K. H.

K. H. Kim, E. J. Kim, S. J. Lee, J. H. Lee, J. E. Kim, and C. H. Kwak, "Effects of applied field on orientational photorefraction in porphyrin:Zn-doped nematic liquid crystals," Appl. Phys. Lett. 85, 366-368 (2004).
[CrossRef]

Kwak, C. H.

K. H. Kim, E. J. Kim, S. J. Lee, J. H. Lee, J. E. Kim, and C. H. Kwak, "Effects of applied field on orientational photorefraction in porphyrin:Zn-doped nematic liquid crystals," Appl. Phys. Lett. 85, 366-368 (2004).
[CrossRef]

C. H. Kwak, M. Sharmonin, J. Takacs, and L. Solymar, "Spatial subharmonics in photorefractive Bi12SiO20 crystal with a square wave applied field," Appl. Phys. Lett. 62, 328-330 (1993).
[CrossRef]

C. H. Kwak, J. Takacs, and L. Solymar, "Spatial subharmonic instabilities," Opt. Commun. 96, 278-282 (1993).
[CrossRef]

Lee, J. H.

K. H. Kim, E. J. Kim, S. J. Lee, J. H. Lee, J. E. Kim, and C. H. Kwak, "Effects of applied field on orientational photorefraction in porphyrin:Zn-doped nematic liquid crystals," Appl. Phys. Lett. 85, 366-368 (2004).
[CrossRef]

Lee, S. J.

K. H. Kim, E. J. Kim, S. J. Lee, J. H. Lee, J. E. Kim, and C. H. Kwak, "Effects of applied field on orientational photorefraction in porphyrin:Zn-doped nematic liquid crystals," Appl. Phys. Lett. 85, 366-368 (2004).
[CrossRef]

Lee, W.

Lloyd, A. D.

I. Janossy, A. D. Lloyd, and B. S. Wherrer, "Anomalous optical Freedericksz transition in an absorbing liquid crystal," Mol. Cryst. Liq. Cryst. 179, 1-12 (1990).

Maddalena, P.

L. Marrucci, D. Paparo, P. Maddalena, E. Massera, E. Prudnikova, and E. Santamato, "Role of guest-host intermolecular forces in photoinduced reorientation of liquid crystals," J. Chem. Phys. 107, 9783-9793 (1997).
[CrossRef]

Marrucci, L.

L. Marrucci, D. Paparo, P. Maddalena, E. Massera, E. Prudnikova, and E. Santamato, "Role of guest-host intermolecular forces in photoinduced reorientation of liquid crystals," J. Chem. Phys. 107, 9783-9793 (1997).
[CrossRef]

Massera, E.

L. Marrucci, D. Paparo, P. Maddalena, E. Massera, E. Prudnikova, and E. Santamato, "Role of guest-host intermolecular forces in photoinduced reorientation of liquid crystals," J. Chem. Phys. 107, 9783-9793 (1997).
[CrossRef]

Ono, H.

H. Ono and N. Kawatsuki, "High-performance photorefractivity in high- and low-molar-mass liquid crystal mixtures," J. Appl. Phys. 85, 2482-2487 (1999)
[CrossRef]

H. Ono and N. Kawatsuki, "Orientational photorefractive effects observed in polymer-dispersed liquid crystals," Opt. Lett. 22, 1144-1146 (1997).
[CrossRef] [PubMed]

Paparo, D.

L. Marrucci, D. Paparo, P. Maddalena, E. Massera, E. Prudnikova, and E. Santamato, "Role of guest-host intermolecular forces in photoinduced reorientation of liquid crystals," J. Chem. Phys. 107, 9783-9793 (1997).
[CrossRef]

Prudnikova, E.

L. Marrucci, D. Paparo, P. Maddalena, E. Massera, E. Prudnikova, and E. Santamato, "Role of guest-host intermolecular forces in photoinduced reorientation of liquid crystals," J. Chem. Phys. 107, 9783-9793 (1997).
[CrossRef]

Rudenko, E. V.

E. V. Rudenko and A. V. Sukhov, "Optically induced spatial charge separation in a nematic and the resultant orientational nonlinearity," JEPT.  78, 875-882 (1994).

Santamato, E.

L. Marrucci, D. Paparo, P. Maddalena, E. Massera, E. Prudnikova, and E. Santamato, "Role of guest-host intermolecular forces in photoinduced reorientation of liquid crystals," J. Chem. Phys. 107, 9783-9793 (1997).
[CrossRef]

Sharmonin, M.

C. H. Kwak, M. Sharmonin, J. Takacs, and L. Solymar, "Spatial subharmonics in photorefractive Bi12SiO20 crystal with a square wave applied field," Appl. Phys. Lett. 62, 328-330 (1993).
[CrossRef]

Shin, M. -Y.

M. Kaczmarek, M. -Y. Shin, R. S. Cudney, and I. C. Khoo, "Electrically tunable, optically induced dynamic and permanent gratings in dye-doped liquid crystals," IEEE J. Quantum. Electron. 38, 451-457 (2002).
[CrossRef]

Shin, M.-Y.

Solymar, L.

C. H. Kwak, J. Takacs, and L. Solymar, "Spatial subharmonic instabilities," Opt. Commun. 96, 278-282 (1993).
[CrossRef]

C. H. Kwak, M. Sharmonin, J. Takacs, and L. Solymar, "Spatial subharmonics in photorefractive Bi12SiO20 crystal with a square wave applied field," Appl. Phys. Lett. 62, 328-330 (1993).
[CrossRef]

Sukhov, A. V.

E. V. Rudenko and A. V. Sukhov, "Optically induced spatial charge separation in a nematic and the resultant orientational nonlinearity," JEPT.  78, 875-882 (1994).

Takacs, J.

C. H. Kwak, J. Takacs, and L. Solymar, "Spatial subharmonic instabilities," Opt. Commun. 96, 278-282 (1993).
[CrossRef]

C. H. Kwak, M. Sharmonin, J. Takacs, and L. Solymar, "Spatial subharmonics in photorefractive Bi12SiO20 crystal with a square wave applied field," Appl. Phys. Lett. 62, 328-330 (1993).
[CrossRef]

Tsai, T. -Y.

Wherrer, B. S.

I. Janossy, A. D. Lloyd, and B. S. Wherrer, "Anomalous optical Freedericksz transition in an absorbing liquid crystal," Mol. Cryst. Liq. Cryst. 179, 1-12 (1990).

Wood, M. V.

Wu, C. -S.

S. -T. Wu and C. -S. Wu, "Experimental confirmation of the Osipov-Terentjev theory on the viscosity of nematic liquid crystals," Phys. Rev. 42, 2219-2227 (1990).
[CrossRef]

Wu, S. -T.

S. -T. Wu and C. -S. Wu, "Experimental confirmation of the Osipov-Terentjev theory on the viscosity of nematic liquid crystals," Phys. Rev. 42, 2219-2227 (1990).
[CrossRef]

Appl. Phys. Lett. (2)

K. H. Kim, E. J. Kim, S. J. Lee, J. H. Lee, J. E. Kim, and C. H. Kwak, "Effects of applied field on orientational photorefraction in porphyrin:Zn-doped nematic liquid crystals," Appl. Phys. Lett. 85, 366-368 (2004).
[CrossRef]

C. H. Kwak, M. Sharmonin, J. Takacs, and L. Solymar, "Spatial subharmonics in photorefractive Bi12SiO20 crystal with a square wave applied field," Appl. Phys. Lett. 62, 328-330 (1993).
[CrossRef]

IEEE J. Quantum. Electron. (2)

I. C. Khoo, "Orientational photorefractive effects in nematic liquid crystal films," IEEE J. Quantum. Electron. 32, 525-534 (1996).
[CrossRef]

M. Kaczmarek, M. -Y. Shin, R. S. Cudney, and I. C. Khoo, "Electrically tunable, optically induced dynamic and permanent gratings in dye-doped liquid crystals," IEEE J. Quantum. Electron. 38, 451-457 (2002).
[CrossRef]

J. Appl. Phys. (1)

H. Ono and N. Kawatsuki, "High-performance photorefractivity in high- and low-molar-mass liquid crystal mixtures," J. Appl. Phys. 85, 2482-2487 (1999)
[CrossRef]

J. Chem. Phys. (1)

L. Marrucci, D. Paparo, P. Maddalena, E. Massera, E. Prudnikova, and E. Santamato, "Role of guest-host intermolecular forces in photoinduced reorientation of liquid crystals," J. Chem. Phys. 107, 9783-9793 (1997).
[CrossRef]

JEPT (1)

E. V. Rudenko and A. V. Sukhov, "Optically induced spatial charge separation in a nematic and the resultant orientational nonlinearity," JEPT.  78, 875-882 (1994).

Mol. Cryst. Liq. Cryst. (1)

I. Janossy, A. D. Lloyd, and B. S. Wherrer, "Anomalous optical Freedericksz transition in an absorbing liquid crystal," Mol. Cryst. Liq. Cryst. 179, 1-12 (1990).

Opt. Commun. (1)

C. H. Kwak, J. Takacs, and L. Solymar, "Spatial subharmonic instabilities," Opt. Commun. 96, 278-282 (1993).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. (1)

S. -T. Wu and C. -S. Wu, "Experimental confirmation of the Osipov-Terentjev theory on the viscosity of nematic liquid crystals," Phys. Rev. 42, 2219-2227 (1990).
[CrossRef]

Other (3)

I. C. Khoo and S. H. Wu, Optics and Nonlinear Optics of Liquid Crystals (World Scientific, Singapore, 1993).

P. Günter and J. P. Huignard, Photorefractive Materials and Their Applications (Springer, Berlin, 1989), Vols. I and II.

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993).

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Figures (11)

Fig. 1.
Fig. 1.

Geometry for writing orientational photorefractive hologram in porphyrin:Zn-doped NLCs sample. Ia and Ib are intensities of writing beams, θ inc is the incident half-angle between two incident beams, β is the tilt angle, q is the grating vector, and E 0 is the applied electric field, parallel to the z-axis.

Fig. 2.
Fig. 2.

(a) Complex representation of the steady state space charge field E 1(∞) as positively (E o) and negatively (-E o) increasing the applied dc field and (b) the phase shift variation ϕ against the applied dc field for various grating periods at I 0=220mW/cm2.

Fig. 3.
Fig. 3.

(a) Complex representation of the steady state space charge field E 1(∞) as positively (E o) and negatively (-E o) increasing the applied dc field and (b) the phase shift variation ϕ against the applied dc field for various input intensities at Λg=1.48µm.

Fig. 4.
Fig. 4.

Experimental setup for measuring real-time diffraction efficiency and two beam coupling gain of NLC sample (BS : beam splitter, M1~M4 : mirrors, D1~D4 : detectors).

Fig. 5.
Fig. 5.

Typical experimental data for two beam coupling experiment at E 0=1.2V/µm.

Fig. 6.
Fig. 6.

Real-time diffraction efficiencies for a grating period of Λg=1.0µm. The solid lines are the theoretical curves. The arrows (↓) represent the moment one of writing beams turned off.

Fig. 7.
Fig. 7.

Real-time diffraction efficiencies for a grating period of Λg=1.48µm. The solid lines are the theoretical curves. The arrows (↓) represent the moment one of writing beams turned off.

Fig. 8.
Fig. 8.

Dependence of 1/τ g on applied dc electric field at various grating periods. The solid lines are the theoretical predictions of Eq.(18).

Fig. 9.
Fig. 9.

Semilog plot of diffraction efficiencies as a function of total writing beam intensity at E 0=1.4V/µm and Λg=1.48µm. The solid line is the theoretical curve.

Fig. 10.
Fig. 10.

Diffraction efficiencies against applied dc field for several writing beam intensities at Λg=1.48µm. The solid lines are the theoretical curves of Eq.(25) with Eq.(24).

Fig. 11.
Fig. 11.

Diffraction efficiencies against applied dc field for several grating periods at I 0=224mW/cm2. The solid lines are the theoretical curves of Eq.(25) with Eq.(24).

Equations (34)

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

n ± t + γ R n + n ± 1 e · J ± = α I ,
J ± = e μ ± n ± E k B T μ ± n ± ,
· E = e ε 0 ε ( n + n ) ,
I ( r , t ) = I 0 ( t ) ( 1 + m cos q · r ) = I 0 ( t ) + 1 2 I 1 ( t ) exp ( i q · r ) + c . c .
Y ( r , t ) = Y 0 ( t ) ( 1 + m cos q · r ) = Y 0 ( t ) + 1 2 Y 1 ( t ) exp ( i q · r ) + c . c . ,
n 0 t + γ R n 0 2 = α I 0 ,
J 0 ± = e μ ± n 0 ± E 0 ,
n 0 + = n 0 n 0 ,
n 1 + t + ( γ R n 0 + i μ + q · E 0 + D + q · q ) n 1 + + γ R n 0 n 1 + i μ + n 0 q · E 1 = α I 1
n 1 t + ( γ R n 0 i μ q · E 0 + D q · q ) n 1 + γ R n 0 n 1 + i μ n 0 q · E 1 = α I 1
E 1 · q = i e ε ε 0 Δ n ,
q · [ d 2 E 1 d t 2 + ( a + b ) d E 1 dt + ( ab c 2 ) E 1 ]
= q . { i em γ R n 0 2 ε ε 0 [ ( μ + + μ ) E 0 i + k B T e ( μ + μ ) q ] }
q · { e ε ε 0 [ 2 n 0 2 γ R ( μ + + μ ) + q 2 n 0 2 k B T e μ + μ ] E 1 } q · [ e n 0 ε ε 0 ( μ + + μ ) d E 1 dt ] ,
q · E 0 = ( q q ̂ ) · [ ( E 0 sin β ) q ̂ + ( E 0 cos β ) q ̂ ] = q E 0 sin β ,
d 2 E 1 dt 2 + A d E 1 dt + B E 1 = m C ,
d E 1 d t + g E 1 = mh ,
E 1 ( t ) = mh g [ 1 exp ( gt ) ] = E 1 ( ) [ 1 exp ( gt ) ] .
E 1 ( ) = m 2 1 X 2 + Y 2 [ ( E D ν Y E 0 X sin β ) + i ( E D ν X + E 0 Y sin β ) ] ,
E 1 ( ) = m 2 ( E D 2 ν 2 + E 0 2 sin 2 β X 2 + Y 2 ) 1 2 ,
ϕ = tan 1 ( E D ν X + E 0 Y sin β E D ν Y E 0 X sin β ) .
γ vis θ t = K ( 2 θ z 2 + 2 θ x 2 ) + Γ E
n = ( sin θ , 0 , cos θ ) ,
E = E 0 + E 1 ( ) cos ( q · r + ϕ ) = ( E 1 ( ) cos β cos ( q · r + ϕ ) , 0 , E 1 ( ) sin β cos ( q · r + ϕ ) + E 0 ) .
Γ E Δ ε ε 0 2 [ E 1 ( ) 2 ( 2 θ cos 2 β + sin 2 β ) cos 2 ( q · r + ϕ ) 2 E 0 2 θ + 2 E 0 E 1 ( ) ( cos β 2 θ cos β ) cos ( q · r + ϕ ) ]
θ ( r , t ) = θ 1 ( t ) cos ( q · r + ϕ ) .
d θ 1 dt + θ 1 τ g = θ 1 ( ) τ g ,
θ 1 ( t ) = θ 1 ( ) [ 1 exp ( t τ g ) ] .
θ 1 ( t ) = θ 1 ( ) exp ( t τ g ) .
n e ( β ) = n n n 2 cos 2 β + n 2 sin 2 β ,
Δ n ( t ) = n n ( n n ) sin ( 2 β ) θ 1 ( t ) cos ( q · r + ϕ ) δ n 1 ( t ) cos ( q · r + ϕ ) ,
δ n 1 ( t ) = { δ n 1 ( ) [ 1 exp ( t τ g ) ] for OPR grating formation , δ n 1 ( ) exp ( t τ g ) for OPR grating erasing .
δ n 1 ( ) = n n ( n n ) E 0 E 1 ( ) E C 2 + E 0 2 cos β sin ( 2 β ) .
η ( t ) = sin 2 ( π δ n 1 ( t ) d λ r cos θ B ) ,

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