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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 10, Iss. 24 — Dec. 2, 2002
  • pp: 1384–1390
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Numerical analysis of vectorial two-beam coupling in photorefractive materials

Vittorio M. N. Passaro and Daniele Marseglia  »View Author Affiliations


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

Photorefractive crystals, making possible an efficient steady-state energy transfer from a reference beam into an object beam (phase-conjugate), have attracted particular attention as media for dynamic recording and transformation of light beams. Therefore, photorefractive crystals are considered as excellent non-linear media for implementing phase conjugate mirrors, wave-front correction and real-time interferometry. They are also viewed as parametric gain media for a new type of optical oscillator. Moreover, the relevant light wave amplitude changes induced by them result in energy exchange that can be used for image amplification and real-time processing. Therefore, it is important to develop a full vectorial model of two-beam coupling in these photorefractive crystals.

2. Mathematical model

Two laser beams writing a holographic grating in a photorefractive medium are coupled nonlinearly to each other. The nonlinear coupled-wave equations which describe this physical mechanism have been analytically solved in literature either for a scalar case, i.e. two beams with the same polarization, or for a vectorial case, i.e. two beams with arbitrary polarization, but under the heavy approximation of neglecting the cross-coupling effect [1

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]

].

Figure 1 shows the vectorial two-beam coupling geometry analysed in our model. Two beams (the signal beam and reference one) propagate in the crystal (width d). Their vectorial field amplitudes are represented by two orthogonal components, (A1, A2) and (A3, A4) respectively. We note that A1 and A4 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.

Fig. 1. Vectorial two-beam coupling geometry.

If we assume the same wavelength (i.e. the same source) for both incident waves and the same coupling constants (real) for both the polarizations [1

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]

], under the hypothesis of moderate power levels [2

2. J. Feinberg and K. R. MacDonald, “Phase-conjugate mirrors and resonators with photorefractive materials,” in Photorefractive Materials and their applications II, 151–203 (Springer-Verlag, Berlin, 1988).

] the coupled-wave equation system describing the amplitude changes inside the grating can be written as follows:

{dA1dz=γIgA4γIIgA3αA1dA2dz=γIgA3γIIgA4αA2dA3*dz=γIgA2*γIIgA1*αA3*dA4*dz=γIgA1*γIIgA2*αA4*
(1)

where the α is linear absorption coefficient, g = (A 1 A4* +A 2 A3*)/I 0 and I0=iIi=iAi2=iAiAi*.

In Eq. (1), γI and γII are the coupling coefficients between homologous field components (i.e. A1 and A4 or A2 and A3) and not homologous ones (i.e. A1 and A3 or A2 and A4), 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:

Ai=Re{Ai}+jIm{Ai}i=1,2,3,4
g=Re{g}+jIm{g}=
=(Re{A1}Re{A4}+Im{A1}Im{A4}+Re{A2}Re{A3}+Im{A2}Im{A3})+
+j(Re{A4}Im{A1}Re{A1}Im{A4}+Re{A3}Im{A2}Re{A2}Im{A3})

and, then, explicitly write the system (1) in terms of eight equations as a function of eight real unknowns, i.e. Re{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 ε00 and the electro-optic r00 tensors of the crystal, we use for γI , γII the general expression as in [2

2. J. Feinberg and K. R. MacDonald, “Phase-conjugate mirrors and resonators with photorefractive materials,” in Photorefractive Materials and their applications II, 151–203 (Springer-Verlag, Berlin, 1988).

]:

γp=4πλkBTqreffncos[ϑ1+ϑ22]·K¯1+(K¯ko)2e1ge2*p=I,II
(2)

where kB is the Boltzmann constant, q is the electron charge, n is the substrate refractive index, ∣∣ 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 ko , and the effective electro-optic coefficient reff . 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:LiNbO3,” Opt. Commun. 175, 247–252 (2000). [CrossRef]

]:

k0=4πNeffq2ε0εrKkBT
(3)

where

ε 0 = 8.85 10-12 [F/m] and εrK=K¯·ε00K¯K¯2 (dimensionless dielectric constant along ).

The parameter Neff represents the effective density of photorefractive traps and its value, experimentally evaluated, depends on the doping process and wavelength. Moreover, the generic expression for reff is [2

2. J. Feinberg and K. R. MacDonald, “Phase-conjugate mirrors and resonators with photorefractive materials,” in Photorefractive Materials and their applications II, 151–203 (Springer-Verlag, Berlin, 1988).

]:

reff=1n4e1*·{ε00[r00(K¯K¯)]·ε00}·e2
(4)

3. Numerical results

The developed software tool is a powerful instrument for design purposes due to its generality and versatility. It is based on the solution of the modified system (1) by a numerical routine, being the coupling constants evaluated as in Eq. (2). In the code, a large number of geometry and physical parameters can be selected, i.e. I 0, α,λ, T, n, ϑ 1, ϑ 2, ϕ 14, d , Neff , ε00 and r00, giving to our model the potential to predict the behaviour of the two-beam coupling mechanism occurring in several different experimental set-ups.

To point out some useful results, let us consider the real case of a Ce,Fe:LiNbO3 crystal at room temperature (T=300 K), having width d = 1mm, Neff = 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:LiNbO3,” Opt. Commun. 175, 247–252 (2000). [CrossRef]

]. We have assumed a mismatch angle between oblique directions 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 = ne , or both ordinary beams with n = no ) as a function of the crossing angle ϑC , assuming in particular ϑ 1 = ϑ 2 = ϑC /2 (see Fig. 2). In the routine, the grating period Λ (and so the grating vector amplitude, ∣∣ = 2π/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]

]:

Λ=λ2nsin(ϑC2)
(5)

since we are under the hypothesis of considering the same polarization for both the incident beams.

Fig. 2. Conversion efficiency dependence on crossing angle.

It is evident that the conversion efficiency remain around 26% for ordinary polarization (blue line), only one peak reaching 40% at 80.3°, while two high-efficiency main peaks can be detected for extraordinary polarization (red line), approaching 100% at 18.5° and 80.3°, respectively. The number of peaks in Fig. 2 are obtained when the Bragg condition is perfectly satisfied and the coupling coefficients are maximized, depending on the coupling geometry and beam polarization. For comparison purposes with more approximated approaches, the same simulation for extraordinary polarization has been carried out by neglecting the crossing coupling effect (i.e. γII = 0 , green line in Fig. 3). We can note that the approximated method gives under-estimated results.

Fig. 3. Conversion efficiency dependence on crossing angle.
Fig. 4. Signal gain dependence on crossing angle.

In the above conditions, we have also evaluated the relevant signal gain, as in Fig. 4, only for extraordinary polarization. It is clear that we can obtain a signal gain close to 3.5 at the crossing angles of 18.2° and 80.3°. These results can be useful for the design of photorefractivity-based optical amplifiers.

With the aim to show our software features, we consider now to be under the same experimental conditions as above and to change the oblique direction mismatch angle (o.d.m.a., or 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.

Fig. 5. Signal gain dependence on crossing angle.

To demonstrate the accuracy of our results and the model versatility, we have also carried out a number of simulations by assuming different experimental conditions. We have numerically evaluated the two-beam coupling gain coefficient, Γnum, 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]

] (i.e. Γexp), where a number of LiNbO3 crystals with different Fe and Ce doping and process treatment were considered. The two-beam coupling gain coefficient has been defined as Γ = (IStwm /IS )/L [3

3. C. Yang, Y. Zhao, R. Wang, and M. Li, “Studies of photorefractive crystals of double-doped Ce,Fe:LiNbO3,” Opt. Commun. 175, 247–252 (2000). [CrossRef]

], where L is the beam interaction length (i.e. the thickness of the sample, d=1 mm), IStwm is the transmitted signal beam intensity when the coupling effect occurs and IS is the transmitted signal beam intensity without any interaction, i.e. when only the linear absorption effect occurs.

The above approximated expression is valid when the pump beam intensity is much larger than the signal beam one. In our case this condition is satisfied if we consider a pump/signal intensity ratio of 750 for a normalised total intensity of 1mW/cm2. The results of simulations and comparisons are summarized in Table I.

It is clear that a good agreement between the experimental and numerical values can be obtained by our model, showing a percentage relative error lower than 5% (see the last red column) under all the experimental conditions.

Table 1. Comparisons among experimental [3] and numerical values of two-beam coupling gain coefficient Γ.

table-icon
View This Table

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 11, 239–241 (1986). [CrossRef]

2.

J. Feinberg and K. R. MacDonald, “Phase-conjugate mirrors and resonators with photorefractive materials,” in Photorefractive Materials and their applications II, 151–203 (Springer-Verlag, Berlin, 1988).

3.

C. Yang, Y. Zhao, R. Wang, and M. Li, “Studies of photorefractive crystals of double-doped Ce,Fe:LiNbO3,” Opt. Commun. 175, 247–252 (2000). [CrossRef]

4.

J. O. White, S. K. Kwong, M. Cronin-Golomb, B. Fischer, and A. Yariv, “Wave propagation in photorefractive media,” in Photorefractive Materials and their applications II, 101–150 (Springer-Verlag, Berlin, 1988).

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]

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

  1. 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]
  2. 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).
  3. 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]
  4. 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).
  5. 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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