## Numerical analysis of vectorial two-beam coupling in photorefractive materials

Optics Express, Vol. 10, Issue 24, pp. 1384-1390 (2002)

http://dx.doi.org/10.1364/OE.10.001384

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### Abstract

A numerical model is presented for the analysis of vectorial two-beam coupling in photorefractive materials. A powerful software tool has been developed for design purposes. For Fe,Ce-doped lithium niobate crystals, conversion efficiency and signal gain dependences on experimental parameters and beam polarization have been found. The good accuracy of the model has been demonstrated by comparisons with experimentally measured parameters.

© 2002 Optical Society of America

## 1. Introduction

## 2. Mathematical model

1. B. Fischer, M. Cronin-Golomb, J. O. White, and A. Yariv, “Nonlinear vectorial two-beam coupling and forward four-wave mixing in photorefractive materials,” Opt. Letters **11**, 239–241 (1986). [CrossRef]

*d*). Their vectorial field amplitudes are represented by two orthogonal components, (A

_{1}, A

_{2}) and (A

_{3}, A

_{4}) respectively. We note that A

_{1}and A

_{4}are not parallel (with arbitrary mismatch angle

*ϕ*

_{14}) and observe that the angles formed by the beam propagation directions with the optic axis are in general different, i.e.,

*ϑ*

_{1}and

*ϑ*

_{2}. Without any lack in generality, the optic axis coincident with the z axis is assumed in Fig. 1 as a particular case of our model.

1. B. Fischer, M. Cronin-Golomb, J. O. White, and A. Yariv, “Nonlinear vectorial two-beam coupling and forward four-wave mixing in photorefractive materials,” Opt. Letters **11**, 239–241 (1986). [CrossRef]

*α*is linear absorption coefficient,

*g*= (

*A*

_{1}

*A*

_{2}

*I*

_{0}and

*γ*

_{I}and

*γ*

_{II}are the coupling coefficients between homologous field components (i.e. A

_{1}and A

_{4}or A

_{2}and A

_{3}) and not homologous ones (i.e. A

_{1}and A

_{3}or A

_{2}and A

_{4}), respectively. The above equations are normalized with respect to the total intensity,

*I*

_{0}. Since the field components are complex (the asterisk designating the complex conjugate), we can write them as:

*A*

_{1}}, Re{

*A*

_{2}}, Re{

*A*

_{3}}, Re{

*A*

_{4}}, Im{

*A*

_{1}}, Im{

*A*

_{2}}, Im{

*A*

_{4}}, Im{

*A*

_{4}}. The linear absorption coefficient

*α*depends on the incident wavelength, material and doping. Now, since the coupling coefficients depend on the geometry parameters, the grating vector amplitude, the dielectric

*γ*

_{I},

*γ*

_{II}the general expression as in [2]:

*k*

_{B}is the Boltzmann constant,

*q*is the electron charge,

*n*is the substrate refractive index, ∣

*K̄*∣ is the grating vector amplitude,

*e*

_{1}and

*e*

_{2}are the polarization vectors of the generic interacting components. Moreover, two important factors are present in Eq. (2), i.e. the Debye length

*k*

_{o}, and the effective electro-optic coefficient

*r*

_{eff}. The former has been evaluated by the following expression [3

3. C. Yang, Y. Zhao, R. Wang, and M. Li, “Studies of photorefractive crystals of double-doped Ce,Fe:LiNbO_{3},” Opt. Commun. **175**, 247–252 (2000). [CrossRef]

*ε*

_{0}= 8.85 10

^{-12}[F/m] and

*K̄*).

*N*

_{eff}represents the effective density of photorefractive traps and its value, experimentally evaluated, depends on the doping process and wavelength. Moreover, the generic expression for

*r*

_{eff}is [2]:

## 3. Numerical results

*I*

_{0},

*α*,

*λ*,

*T*,

*n*,

*ϑ*

_{1},

*ϑ*

_{2},

*ϕ*

_{14},

*d*,

*N*

_{eff},

_{3}crystal at room temperature (T=300 K), having width

*d*= 1

*mm*,

*N*

_{eff}= 0.1∙10

^{-15}cm

^{-3}and

*α*= 0.70 cm

^{-1}at the wavelength of 632.8 nm. This test case was experimentally investigated in Ref. [3

3. C. Yang, Y. Zhao, R. Wang, and M. Li, “Studies of photorefractive crystals of double-doped Ce,Fe:LiNbO_{3},” Opt. Commun. **175**, 247–252 (2000). [CrossRef]

*ϕ*

_{14}= 10°. The conversion efficiency

*η*(defined as the power ratio between the object beam at the output and the reference beam at the input) has been numerically evaluated for different polarizations (i.e. both extraordinary beams with

*n*=

*n*

_{e}, or both ordinary beams with

*n*=

*n*

_{o}) as a function of the crossing angle

*ϑ*

_{C}, assuming in particular

*ϑ*

_{1}=

*ϑ*

_{2}=

*ϑ*

_{C}/2 (see Fig. 2). In the routine, the grating period Λ (and so the grating vector amplitude, ∣

*K̄*∣ = 2

*π*/Λ) has been evaluated by the following expression [5

5. P. Gunter and J. P. Huignard, “Photorefractive effects and materials,” in *Photorefractive Materials and their applications I*, 7–74 (Springer-Verlag, Berlin, 1988). [CrossRef]

*γ*

_{II}= 0 , green line in Fig. 3). We can note that the approximated method gives under-estimated results.

*ϕ*

_{14}), in order to better match a real experimental condition. Figure 5 clearly shows the gain peaks around 80.3° and how the signal gain does not significantly change up to an o.d.m.a. of 45°, but it reduces from a factor around 3.5 to a factor around 2.5 at an o.d.m.a. of 90°. Thus, we have shown how critically the signal gain depends both on crossing angle and on the oblique directions mismatch angle.

_{num}, and then compared the results with respect to those experimentally found in Ref. [5

5. P. Gunter and J. P. Huignard, “Photorefractive effects and materials,” in *Photorefractive Materials and their applications I*, 7–74 (Springer-Verlag, Berlin, 1988). [CrossRef]

_{exp}), where a number of LiNbO

_{3}crystals with different Fe and Ce doping and process treatment were considered. The two-beam coupling gain coefficient has been defined as Γ = (

*I*

_{S})/

*L*[3

3. C. Yang, Y. Zhao, R. Wang, and M. Li, “Studies of photorefractive crystals of double-doped Ce,Fe:LiNbO_{3},” Opt. Commun. **175**, 247–252 (2000). [CrossRef]

*L*is the beam interaction length (i.e. the thickness of the sample,

*d*=1 mm),

*I*

_{S}is the transmitted signal beam intensity without any interaction, i.e. when only the linear absorption effect occurs.

^{2}. The results of simulations and comparisons are summarized in Table I.

## 4. Conclusions

## References and links

1. | B. Fischer, M. Cronin-Golomb, J. O. White, and A. Yariv, “Nonlinear vectorial two-beam coupling and forward four-wave mixing in photorefractive materials,” Opt. Letters |

2. | J. Feinberg and K. R. MacDonald, “Phase-conjugate mirrors and resonators with photorefractive materials,” in |

3. | C. Yang, Y. Zhao, R. Wang, and M. Li, “Studies of photorefractive crystals of double-doped Ce,Fe:LiNbO |

4. | J. O. White, S. K. Kwong, M. Cronin-Golomb, B. Fischer, and A. Yariv, “Wave propagation in photorefractive media,” in |

5. | P. Gunter and J. P. Huignard, “Photorefractive effects and materials,” in |

**OCIS Codes**

(090.2900) Holography : Optical storage materials

(130.3730) Integrated optics : Lithium niobate

(160.5320) Materials : Photorefractive materials

(210.2860) Optical data storage : Holographic and volume memories

**ToC Category:**

Research Papers

**History**

Original Manuscript: October 21, 2002

Revised Manuscript: November 12, 2002

Published: December 2, 2002

**Citation**

Vittorio Passaro and Daniele Marseglia, "Numerical analysis of vectorial two-beam coupling in photorefractive materials," Opt. Express **10**, 1384-1390 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-24-1384

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### References

- B. Fischer, M. Cronin-Golomb, J. O. White, A. Yariv, �??Nonlinear vectorial two-beam coupling and forward four-wave mixing in photorefractive materials,�?? Opt. Lett. 11, 239-241 (1986). [CrossRef]
- J. Feinberg, K. R. MacDonald, �??Phase-conjugate mirrors and resonators with photorefractive materials,�?? in Photorefractive Materials and their applications II, 151-203 (Springer-Verlag, Berlin, 1988).
- C. Yang, Y. Zhao, R. Wang, M. Li, �??Studies of photorefractive crystals of double-doped Ce,Fe:LiNbO3,�?? Opt. Commun. 175, 247-252 (2000). [CrossRef]
- J. O. White, S. K. Kwong, M. Cronin-Golomb, B. Fischer, A. Yariv, �??Wave propagation in photorefractive media,�?? in Photorefractive Materials and their applications II, 101-150 (Springer-Verlag, Berlin, 1988).
- P. Gunter, J. P. Huignard, �??Photorefractive effects and materials,�?? in Photorefractive Materials and their applications I, 7-74 (Springer-Verlag, Berlin, 1988). [CrossRef]

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