Ph Delaye

L A De Montmorillon

I Biaggio

J C Launay

G Roosen

WAVELENGTH DEPENDENT EFFECTIVE TRAP DENSITY IN CdTe : EVIDENCE FOR THE PRESENCE OF TWO PHOTOREFRACTIVE SPECIES

An accurate determination of the effective trap density in CdTe:V is performed using both counterpropagating (small grating spacing) and copropagating (large grating spacing) two wave mixing. This enhanced precision allows to show a variation of the effective trap density with wavelength. We show that such a change is explained when two traps both coupled to the conduction and the valence bands are present. New theoretical expressions for the electron hole competition coefficient and Debye screening length are derived and used.

Photorefractive semiconductors like CdTe are characterized by a low dielectric constant compared to other photorefractive materials like sillenites or ferroelectrics. This point added to the fact that these materials are sensitive in the infrared region of the spectra leads to the point that even with a low effective trap density N eff (around 10 15 cm -3 ), the optimum grating vector (corresponding to k=k 0 , with k 0 the inverse of the Debye screening length) can not be obtained with copropagating beams. In other words, the photorefractive gain is mostly dominated by the diffusion field in most of the experiments. Then the use of curves of variation of the photorefractive gain with the grating spacing leads to a great uncertainty on the exact value of the effective trap density (compared to the good accuracy of the electron hole competition coefficient determined in these experiments). In order to decrease this uncertainty, we include in our measurement a value of two wave miwing gain obtained in a counterpropagating geometry. With this new experiment we not only deduce a more accurate value of the effective trap density, but also find that N eff is different for 1.06µm and 1.55µm. In order to explain this wavelength dependence we use the photorefractive model with two photoactive traps developed by Bashaw [1], and show that this model can perfectly explain this variation of N eff with wavelength.

Experimental study :

The determination of the photorefractive parameters (electron-hole competition coefficient ξ 0 and effective trap density N eff ) can be easily performed through a measurement of the photorefractive two beam coupling energy transfer gain Γ as a function of the grating spacing Λ (or as a function of the grating wave number k=2π/Λ) [2]. For s-polarized beams the expression of the gain according to the model with one defect coupled to both bands, when considering that the electron-hole competition factor is independent of the grating spacing, as usually admitted in semiconductors (diffusion length much longer than the grating spacing), is [3,4]:

! = 2" n 0 3 r eff # cos $ k B T e % 0 k 1 + k 2 k 0 2 (1) with k 0 2 = e 2 ! k B T
N eff . θ is the half angle between the beams inside the crystal. The linear refractive index of the material is n 0 , r eff is the effective electrooptic coefficient that depends on the crystal and the beam polarization orientations [5] and ε is the dielectric constant. All these parameters are known from the literature.

In most of the cases, the experiments are performed with two beams which are copropagating with a small angle between them, leading to values of k that are smaller than k 0 . We can then very often work with a simplified expression :

! = 2" n 0 3 r eff # cos $ k B T e % 0 k (2)
where the influence of k 0 has disappeared. We can thus see that the experiments generally allow an accurate determination of the electron-hole coefficient ξ 0 , whereas the k 0 influence appears only through perturbations to the linear variation of Γ with k. Unless a great number of experimental points are used, the study of the variation of the photorefractive gain with the grating spacing gives only a poor estimation of the effective trap density present in the photorefractive material.

A solution is to have a measurement of the gain with a large value of k. This is only possible in photorefractive semiconductors like CdTe by performing measurements with counterpropagating beams. These counterpropagating points generally lead to values of k such that we have k greater than k 0 , which allows to pass the maximum of the gain as a function of k, giving a better determination of this maximum.

It is possible to get a feeling of the parameter accuracy that can be obtained from equation (1) and a given data set using a simple and general approach [START_REF] Press | Numerical Recipes -the Art of Scientific Computing[END_REF]. It consists of simulating a data set by using the theoretical function and a random number generator which imitates the known error distribution of each data point. Simulation of many data sets and many fits gives a statistically valid way to estimate the error of the parameters. Using function (1) with ξ 0 =1 and k 0 2 =200, a set of data points at k=1,1.5,2,2.5,3,4,5, and assuming a gaussian 5% error distribution for each data point, one can generate a large number of data sets and fit Eq.(1) every time. From the distribution of ξ 0 , k 0 2 values obtained over all the fits, one can estimate the confidence intervals for the parameters ξ 0 and k 0 2 . The result for 90% confidence intervals is ξ 0 =[0.95,1.05] and k 0 2 =[100,900]. Using a data set k=1,2,3,4,5,22, which contains less data points but one point at large k, the same procedure delivers, for the 90% confidence intervals, ξ 0 =[0.96,1.04] and k 0 2 =[174,230]. In this example, the expected accuracy increase on the value of k 0 2 by the addition of a data point measured with counterpropagating beams, is evident.

The above estimation, although mathematical, is very close to the experimental practice, as we will see now on an experimental example.

The CdTe:V sample we used, named B23-3, was already characterized through beam coupling measurements in the copropagating geometry [START_REF] De Montmorillon | [END_REF]. The main results obtained in this geometry where : at λ=1.06 µm, 100 < k 0 2 < 300 µm -2 (with a mean value of k 0 2 = 166 µm -2 ), and 0.78 < ξ 0 < 0.93 (with a mean value of ! 0 = 0.84), at λ=1.55 µm, 50 µm -2 < k 0 2 (with a mean value of k 0 2 = 109 µm -2 ), and -0.94 < ξ 0 < -0.71 (with a mean value of ! 0 = -0.82). at λ=1.32 µm the gain is zero which means that the electron-hole competition is strong, we have then

ξ 0 ≈0 and k 0 2 is indeterminate.
The measured sign of the electron hole competition says that the dominant carriers are electrons at 1.06µm and holes at 1.55µm [START_REF] De Montmorillon | [END_REF]. We see on these results that the precision on the determination of ξ 0 is good whereas k 0 2 is quite indeterminate.

To these results we can add transmission measurements that gives us the absorption of the crystal (Fig. 1). Crystal absorption spectrum exhibits classical features with two absorption bands centered at 1µm and 1.5µm (Fig. 1). As for some of the other samples we studied, we see a background absorption [START_REF] De Montmorillon | [END_REF]8]. This background absorption is assumed to be due to scattering centers and is supposed not to participate in the photorefractive effect. In the following, when saying that results are corrected from absorption background, we means that we consider the crystal absorption to be zero at λ=2µm.

These corrected values of absorption are : α = 1.6 cm -1 at 1.06 µm, α = 0.85 cm -1 at 1.32 µm and α = 0.58 cm -1 at 1.55 µm One interest of the B23-3 crystal is its cut. Indeed, it is cut with a [111] polished face, (111) direction along the beam propagation direction whereas the other faces are [1 1 0] and [112 ], with the [1 1 0] plane as plane of incidence. In this configuration the effective electrooptic coefficient is non zero in both the co and counterpropagating geometries which facilitates the measurement. The effective electrooptic coefficient is r eff = 2 3 r 4 1 in the copropagating geometry and r eff = 1 3 r 4 1 in the counterpropagating geometry. With this configuration we can then easily add to our copropagating points, already presented, a counterpropagating point. Nevertheless if the absolute value of the gain is immediately obtained, care has to be taken in order to determine the sign of the gain compared to the copropagating case, i.e. to see if the majority carriers is the same at large and small grating spacings.

In order to measure the sign of the photorefractive gain we change the set-up to allow to go directly from one geometry to the other one changing the minimum of parameters (Fig. 2). We choose an angle between the two beams of 90° outside the crystal. A simple rotation of 90° of the crystal allows us to go from a copropagating to a counterpropagating geometry. The crystal orientation is chosen with still [111] as the polished face, but with now the [112 ] plane as the plane of incidence. The value of the effective electrooptic coefficient is presented in Figure 3 for s polarizations. With spolarized beams and copropagating beams with θ c =0, we have a gain that is zero. When turning a little the crystal, from this position, a small photorefractive gain with a certain sign appears (as r eff becomes different from zero), if we now continue to rotate the crystal until θ c =90° (i.e. counterpropagating geometry) the sign of the effective electrooptic coefficient stays the same and so does the gain if the majority carrier for the photorefractive effect remains alike. Using this experiment at both wavelengths, at which gain was measured, we confirm that the sign of the majority carriers is the same whatever the grating spacing. We can now place the counterpropagating points on the experimental curves of variation of the photorefractive gain as a function of k (Fig. 4), after correction of the different value of the effective electrooptic coefficient in the two geometries.

We fit the data with the theoretical expression (1). These adjustments are conducted with the same value of r 41 =5.5 pm.V -1 [9,10] for the two wavelengths and a refractive index n 0 =2.82 at 1.06µm

and n 0 = 2.74 at 1.55µm [11]. Using r eff = 2 3 r 4 1 , we determine ξ o =0.80±0.03 at 1.06 µm and ξ o =-0.81±0.03 at 1.55 µm and k 0 2 =262±30µm -2 at 1.06µm and k 0 2 =96±13µm -2 at 1.55µm. We see easily that the accuracy for k 0 2 is much better than the one determined with the copropagating points only. With the dielectric constant ε = 10.3, one obtains an effective trap density N eff =(3.8±0.5)×10 15 cm -3 at 1.06 µm and N eff =(1.4±0.2)×10 15 cm -3 at 1.55 µm.

Due to the fact that the counterpropagating point brings us an higher precision in the determination of the effective trap density we can note on these results that the effective trap density varies with the wavelength. Although this fact had already been observed in BaTiO 3 crystals [12] it has never been really explained. We will now give indications for a possible origin of this variation of the effective trap density with wavelength.

The two traps model

The model for the photorefractive effect had several evolutions since the model developed by Kuhktarev [3] that takes into account a single defect coupled to only one of the transport bands. It had rapidly been necessary to take into account two types of carriers [4] to explain some of the experimental results. Again, new results that does not fit the theory force to add to the model with one defect coupled with the two bands another defect coupled to one of the bands [13,14].

One major characteristics of all these models is that with the assumption that the diffusion length of the carriers is much longer than the grating spacing (what is generally true in semiconductors) the expression of the gain can always be reduced to an expression like equation ( 1) with only the expression of ξ 0 and k 0 2 that varies from one model to another [3,4,13,14]. Moreover, if we look at the different models in detail we can see that in all these models (in the limit of low illuminations, which corresponds to the regime generally used in experiments in semiconductors) the different concentrations of the defect species are constant with the wavelength and, more important, the value of k 0 2 is the same whatever the wavelength used. It means that none of these photorefractive models can explain our experimental results that show clearly such a variation.

Recently a new evolution of the photorefractive model appears [1], it takes into account two deep defects both coupled to both conduction and valence bands. The model established analytical expressions for the space charge field and showed that wavelength dependent charge redistribution between the two defects was possible. Such a charge redistribution can explain the variation of the effective trap density we observed.

Before looking at this, we simplify the expressions derived in Ref. [1]. First we limit ourselves to the case of the variation of the steady state space charge field with the grating spacing in the case where no external electric field is applied. Secondly we rewrite the expression found in Ref. [1], using the approximation of large diffusion lengths classically used in semiconductors, to show that the expression of the gain can be reduced to an expression of the form of equation ( 1) with new expressions for ξ 0 and k 0 2 that we derive. In every step of the calculations we compare the numerical results obtained with our expressions to the ones obtained without approximation with the general expressions of reference [1] (after correction of a typographical errors in formula (45) of Ref. [1]) to validate our expressions and our results.

Derivation of the expressions

We consider two traps (Fig. 5), with total concentrations N T and M T , present under two states of charges with ionized parts N and M and neutral parts N T -N and M T -M respectively. At each state we associate parameters for optical (S) and thermal (ß) generation of carriers and recombination (γ) in the deep traps. The superscript (N and M) deals with the two different traps, whereas the subscript (n and p) deals with the two kinds of carriers (for example, S p N represents the hole photoionization cross-section from the ionized state of the N trap). The different material equations that allow the calculation of the space charge field are [1]:

!N !t = " n N + S n N I ( ) N T # N ( ) # $ n N n N # " p N + S p N I ( ) N + $ p N p N T # N ( ) (3-a) !M !t = " n M + S n M I ( ) M T # M ( ) # $ n M n M # " p M + S p M I ( ) M + $ p M p M T # M ( ) (3-b)
j n = e nµ n E + µ n k B T !n !x (3-c) j p = e pµ p E ! µ p k B T "p "x (3-d) !n !t = 1 e !j n !x + " n N + S n N I ( ) N T # N ( ) # $ n N n N + " n M + S n M I ( ) M T # M ( ) # $ n M n M (3-e) !p !t = " 1 e !j p !x + # p N + S p N I ( )N " $ p N p N T " N ( ) + # p M + S p M I ( )M " $ p M p M T " M ( ) (3-f) !E !x = " e # N T " N ( ) + M T " M ( ) + N D " N A ( ) + n " p ( ) (3-g)
Where n and p are the free carriers concentrations, j n and j p are the electron and hole current densities and E is the space charge field. N D and N A are the donnors and acceptors, which compensation gives the semi insulating nature of the material. We limit ourselves to a one dimension problem with an illumination I given by a sinusoidal illumination pattern We solve these equations in steady state, without applied electric field, in a classical way by linearizing the different quantities [1,3,4], with a non modulated part (with subscript 0) and a space modulated component with the same grating spacing than the illumination (with subscript 1). The zeroth order is given in Appendix A and allows to calculate the non modulated concentration of the different species that will be used in the first order expressions. The resolution of the first order system equation in order to obtain the modulated space charge field is long and tedious but does not present any mathematical difficulties. Under the assumption that the free carrier grating concentrations are much smaller than the one of gratings in the deep traps (n 1 , p 1 << N 1 ,M 1 ), we obtain the expression for modulated space charge field E 1 :

I = I 0 1+ Re m e ikz
E 1 = ! i m k k B T e " # $ % NUM DENOM ( 4
)
where :

NUM = k 2 ! n N " ! p N ( ) Z M + ! n M " ! p M ( ) Z N [ ] + Z M " Z N ( ) ! n N # n M 2 " ! n M # n N 2 ( ) " ! p N # p M 2 " ! p M # p N 2 ( ) [ ] " # n 2 ! p N Z M + ! p M Z N ( ) + # p 2 ! n N Z M + ! n M Z N ( ) + $ k B T e 2 I 0 X n M " X n N ( ) # n M 2 ! n N " ! p N ( ) " # n N 2 ! n M " ! p M ( ) [ ] + $ k B T e 2 I 0 X p M " X p N ( ) # p M 2 ! n N " ! p N ( ) " # p N 2 ! n M " ! p M ( ) [ ] DENOM = k 2 + ! n 2 ( ) k 2 + ! p 2 ( )Z M Z N + k 2 + ! p 2 ( ) " k B T e 2 I 0 # $ % & ' ( Z N ! n M 2 X n M + Z M ! n N 2 X n N [ ] + k 2 + ! n 2 ( ) " k B T e 2 I 0 # $ % & ' ( Z N ! p M 2 X p M + Z M ! p N 2 X p N [ ] + " k B T e 2 I 0 # $ % & ' ( 2 X n N X p M ) X p N X n M ( ) ! n N 2 ! p M 2 ) ! p N 2 ! n M 2 ( )
With the following notations that are used classically in the space charge field expressions [4]:

! n N = S n N N T " N 0 ( ) and ! p N = S p N N 0 k 0N 2 = e 2 ! k B T N 0 N T " N 0 ( ) N T ! n N 2 = e k B T " n N N 0 µ n and ! p N 2 = e k B T " p N N T # N 0 ( ) µ p A n N = ! n N + S n N I 0 ( ) S n N I 0 and A p N = ! p N + S p N I 0 ( ) S p N I 0
and the same expressions for the M defect.

The model also uses specific parameters :

X n N = n 0 eµ n ! " # n N + S n N I 0 ( ) " $ n N n 0 and X p N = p 0 eµ p ! " # p N + S p N I 0 ( ) " $ p N p 0
To these definitions we can add some relations between the parameters, relations that are given in Appendix B.

Even if not in a factorized form, we can nevertheless note that the space charge field (Eq.4) has the form found in previous models, i.e. the diffusion field E d = k k B T e multiplied by a rational function with a numerator that is a polynomial of degree 1 in k 2 and a denominator that is a polynomial of degree 2 in k 2 .

In order to simplify this expression we will now make further approximations. The first one which is classical in semiconductor materials is that the Debye screening length is much smaller than the diffusion length, according to our notations it will correspond to

k 0N 2 , k 0M 2 >> ! n 2 ,! p 2 .
Under this approximation we have :

X n N = X n M = e 2 I 0 ! k B T Y n and X p N = X p M = e 2 I 0 ! k B T Y p (5)
The second approximation is linked to the experimental set-up we use and to the range of grating spacings on which we make the experiments, we thus have k 2 >> ! n 2 ,! p 2 , i.e. a grating spacing much smaller than the diffusion length of carriers.

Under all these approximations we can rewrite the photorefractive space charge field as :

E 1 = ! i m k B T e " # $ % & 0 k 1+ k 2 k 0 2 (6)
with

k 0 2 = k 0 N 2 Y n ! n N 2 + Y p ! p N 2 A n N " n N + Y p ! p N 2 # $ % & ' ( + k 0M 2 Y n ! n M 2 + Y p ! p M 2 A n M " n M + Y p ! p M 2 # $ % & ' ( (6a)
and

! 0 = 1 k 0 2 k 0 N 2 " n N # " p N A n N " n N + Y p $ p N 2 % & ' ( ) * + k 0 M 2 " n M # " p M A n M " n M + Y p $ p M 2 % & ' ( ) * + , - . / 0 (6b)
The photorefractive gain is then written in the form of the equation ( 1) with new expressions of ξ 0 and k 0 2 . This explains why we could adjust our experimental results with equation (1) but found different values for N eff when changing wavelength (see experimental part). We can always define an apparent "electron-hole" competition coefficient and an apparent Debye screening length, that will characterize the strength of the photorefractive effect. It can be easily shown from equations ( 6) that the results of the previous models [4,13,14] can be deduced from this model taking the right values of generation and recombination parameters.

Adjustment to the experimental results

To see whether this two photoactive traps model explains our experimental results, we look for a set of parameters that will give us theoretical values of ξ 0 and k 0 2 equal to the ones we found experimentally. To these two photorefractive parameters, we add the absorption of the crystal

! = ! n N + ! p N + ! n M + ! p M .
So we have six experimental parameters, corresponding to the two values of the wavelengths 1.06µm and 1.55µm (as well as for 1.32µm for ξ 0 and α). The free parameters of the calculations are all the coefficients for photoionisation, thermal emission, recombination of carriers and concentration of the defects, with the condition that between 1.06µm and 1.55µm we only change the wavelength dependent coefficients, i.e. the photoionization cross-sections. Moreover as in our experiments we did not see, for our sample, any illumination dependent effect in the conditions of illumination used, i.e. from 1 to 100mW.cm -2 , we will also verify that the parameters we find will not vary with illumination.

The set of parameters we find is shown in Table 1, it allows us to calculate exactly the experimental parameters, without any peculiar problem. The set of parameters we determined is certainly not unique as the adjustment is performed with eleven free parameters (we are in illumination conditions where the thermal emission is negligible and thermal emission coefficients are only taken small and non zero) which is a lot. Complementary experiments should be performed in order to have other informations on the characteristics of the two defects, this is not the goal of this study.

Here again we note that a complete calculation with the model of Bashaw [1] using the parameters we found, gives the same theoretical curves (in the range where the approximations we have done are justified, i.e. at large values of k).

The two defects we have taken have thermal emission coefficients that are low, this implies that the defects are both close to the mid-gap where the Fermi level is located [14]. Thus these two defects are different from the one found responsible for the diminution of the photorefractive gain under AC field [15,[START_REF] Biaggio | Influence of shallow traps on the enhancement of the photorefractive grating amplitude by a high frequency alternating electric field : a probabilistic analysis[END_REF]. Indeed it was shown that the reduction of the gain under AC field is due to a defect with a thermal emission coefficient around 10 3 -10 4 s -1 , very different from the one we found.

Conclusion

The measurements we performed on our CdTe:V sample show that the effective trap density was a wavelength dependent parameter in this sample. This new observation cannot be explained by the usual models of the photorefractive effect. Only a model that takes into account two traps coupled to the two bands allows charge redistribution that can explain this variation of the effective trap density with wavelength. Using the model developped by Bashaw [1], we derived an expression for the variation of the gain with grating spacing that has the same form than the one given by previous models. We establish the new expressions for the electron hole competition ξ 0 and the Debye screening length (inverse square root of k 0 2 ) with the two traps. Using these expressions we show that it is possible to explain our experimental results with this two defect model using a set of parameters that is plausible.

At this point of the study it is impossible to identify these defects and even to determine their parameters. Other experiments should be performed to have a set of parameters that can be determined unambiguously.

These experiments, as well as other studies [START_REF] Shcherbin | Relaxation of photorefractive gratings in Cadmium Telluride crystals[END_REF], shows the importance of the inclusion of a counter propagating point in the measurement of the photorefractive gain as a function of the grating spacing, as it brings a considerable improvement of the accuracy in the determination of the Debye screening length. This is not only true on the point of view of the material characterization, but also for the application of photorefractive crystals as this parameter is important for the value of the gain in the counterpropagating geometry or for experiments under applied electric field.

Table Caption :

Parameter used for the adjustment of the experimental results with the theoretical expression. N 0 and M 0 are calculated with the zeroth order equations.

  wave number k and modulation m.

Figure 1 :Figure 2 :

 12 Figure 1 : Room temperature absorption spectra of sample B23

Figure 3 :

 3 Figure 3 : Value of the effective electrooptic coefficient as a function of the angle of rotation of the crystal. The beams are s-polarised.

Figure 4 :

 4 Figure 4 : Experimental photorefractive gain as a function of the grating spacing at wavelengths λ=1.06µm (•) and 1.55µm (▲) for CdTe sample B23. Solid lines are the best theoretical adjustments.The dashed lines represent the adjustment with σ=2 σ min . The photorefractive gain at 1.32µm is nearly zero.

Figure 5 :

 5 Figure 5 : Energy diagram with two defects in total concentrations N T and M T both coupled to both the valence and the conduction band.

Figure 1 Figure 2 Figure 3 Figure 4

 1234 Figure 1

Figure 5

 5 Figure 5

Table λ

 λ

	(µm)	1.064	1.32	1.55
	N T (cm -3 )		10 16
	M T (cm -3 )		1.2×10 16
	N D -N A (cm -3 )		1.2×10 16
	N 0 (cm -3 )	7.65×10 15	8.23×10 15	9.19×10 15
	M 0 (cm -3 )	2.35×10 15	1.67×10 15	8.12×10 14
	S n N (cm 2 )	1.11×10 -16	2.55×10 -17	1.03×10 -18
	S n M (cm 2 )	1.34×10 -16	2.26×10 -17	8.77×10 -19
	S p N (cm 2 )	3.70×10 -18	6.33×10 -17	5.91×10 -17
	S p M (cm 2 )	1.85×10 -17	2.58×10 -17	3.46×10 -17
	! n N (cm 3 s -1 )		1.79×10 -9
	! p N (cm 3 s -1 )		2.39×10 -8
	! n M (cm 3 s -1 )		3.92×10 -7
	! p M (cm 3 s -1 )		1.25×10 -9
	! n N (s -1 )		0.00248
	! p N (s -1 )		0.000182
	! n M (s -1 )		0.00458
	! p M (s -1 )		0.000144
	k 0 2 (µm -2 )	260	NA	98
	α (cm -1 )	1.64	0.85	0.58
	ξ 0	0.79	0	-0.81

Acknowledgment

Support for this work by DRET (grant #94-063) is gratefully acknowleged.

Appendix A :

Under the condition of small concentration of free carriers (n 0 , p 0 << N D ! N A , N T ! N 0 ,M T ! M 0 ) the different non spatially modulated concentrations are solution of a four equations system :

The system can easily be reduced to third order equation that gives N 0 , which value gives then n 0 , p 0 and M 0 . We have 3 solutions, but only one satisfies the conditions 0 ! N 0 ! N T , 0 ! M 0 ! M T , n 0 ! 0 and p 0 ! 0 . We numerically solve the system to have the non modulated concentrations that will be used for the first order solutions.

Appendix B :

With the zeroth order equations we write relations between the different parameters that appears in the expressions of the space charge field. We have :

We can also rewrite :

With the same relation for the defect M.

These relations are used to show that the expressions for ξ 0 and k 0 2 are symmetrical when exchanging N and M on one hand and n and p on the other hand, a point that is not evidently seen on the expressions.


Louis-Anne De Montmorillon

Ivan Biaggio

Philippe Delaye

Jean-Claude Launay

Gérald Roosen
email: gerald.roosen@iota.u-psud.fr.

EYE SAFE LARGE FIELD OF VIEW HOMODYNE DETECTION USING A PHOTOREFRACTIVE CdTe:V CRYSTAL

Keywords: A8170C, A0760L, A4240K, B4340 Photorefractive effect, Homodyne detection, Non destructive testing

We demonstrate a self-adapting homodyne detection system working at a wavelength of 1.55 µm.

The system uses a photorefractive CdTe:V crystal as the element that combines local oscillator and signal beams. The device spontaneously adapts to slow phase and direction changes in the incoming light. Using a pump beam illumination of 66 mW•cm -2 and a signal power of 25 µW, we detect phase modulations corresponding to an optical path variation of a few tenths of nanometers at modulation frequencies higher than a cut-off frequency of 15 Hz. The performance of the system is not affected when the signal light consists of scattered light with an "étendue" of up to 0.1 mm 2 •sr, limited only by the lenses we are currently using. We obtain a detection limit of (6.2±0.4)×10 -7 nm• W/Hz . This is only about 20 times above the theoretical limit of an ideal interferometric system. We show how to scale these results to the optimized crystals and higher laser intensities necessary to obtain higher sensitivity (< 10 -7 nm• W/Hz ) and higher cut-off frequencies (> 1 kHz).

Introduction

Real time holographic materials are ideally suited for use as adaptive beam-splitters to coherently superpose a phase-modulated signal beam and a local oscillator beam in a homodyne detection scheme.

Photorefractive crystals are among the most sensitive materials for real-time holography and were among the first to be proposed for this kind of application [1].

Detection of phase modulated signals using two wave mixing in a photorefractive crystal is possible whenever the amplitude of the phase modulation is much lower than π/2 and its frequency is higher than a cut-off frequency characterizing the response time of the photorefractive effect. The crystal acts as the beam-combiner in a classical interferometric system : the diffracted part of the pump beam interferes with the signal beam.

The main advantage of such a device is the adaptativity of the photorefractive effect. The system adapts to distorted wavefronts and to low frequency mechanical disturbances. It allows the use of speckled beams without loss of signal power, thus decidedly improving the limit given by the antenna's theorem [2]. A homodyne detector based on a photorefractive beam-combiner is a very interesting candidate for quality testing of production line parts in an unfriendly industrial environment. The system would detect the ultrasound waves induced by strong laser pulses in mechanical parts, while at the same time adapting to slowly changing environmental conditions. Ultrasound waves arriving on the object to be tested create displacements that are measured by detecting the phase variations of a laser beam reflected by the rough surface of this object. We will show that photorefractive techniques potentially give a better detection limit and a wider phase modulation bandwidth than the best device presently used, i.e. the confocal Fabry-Perot interferometer [3]. Detection of phase [4] and amplitude [5] changes using two wave mixing has already been demonstrated using insulating [6,[START_REF] Stepanov | Proc. of II Intern. Conf. on Holographic Systems, Components and Applications[END_REF][START_REF] Ing | [END_REF] and semi-insulating [9,10] photorefractive crystals. This technique has been used for detection of ultrasonic signals [START_REF] Ing | [END_REF]10] as well as for the stabilization of holographic recording [11]. In this work we investigate the use of Vanadium doped CdTe crystals to build a self-adaptive homodyne detector working at the eye-safe wavelength of 1.55 µm [START_REF] De Montmorillon | Digest of Topical Meeting on Photorefractive Materials, Effects and Devices[END_REF]. We also present a comparison between the theoretical [START_REF] Ph | [END_REF] and experimental detection limit and frequency response of the system.

CdTe presents the advantages of a high n 3 r 41 value of ≈ 113 pm•V -1 [14], a sensitivity in the 1-1.5 µm wavelength range [15] and a low grating formation fluence of ≈ 100 µJ•cm -2 [16].

Theoretical background

We consider a two beam coupling experiment with a strong pump beam and a weak signal beam of parallel polarizations. The signal beam is phase modulated, E s (x = 0, t) = E s (0, 0)e i!( t) . We denote with E s the complex amplitude of the signal beam. We assume a phase modulation faster than the response time of the photorefractive effect and with a modulation amplitude much smaller than π/2. In this limit, the grating written in the crystal corresponds to the one that would be recorded without any phase modulation. In the non depleted pump approximation, the amplitude of the diffracted part of the pump beam only depends on this grating and is E d (x, t) = (e !x " 1)E s (x, 0), where ! = " ! + i " " ! is the amplitude photorefractive gain [START_REF] Ph | [END_REF] and x is the crystal thickness. The amplitude of the detected signal is then given by the sum of the signal beam and of the diffracted portion of the pump beam. The detected intensity is :

I s (x,t) = e i! (t ) + (e " x # 1) 2 e #$ x I s (0,0) = e 2 % " x + 2(e % " x cos( % % " x) # 1)(cos! (t) # 1) + 2e % " x sin( % % " x)sin !(t) [ ] e #$ x I s (0,0) , ( 1
)
where α is the crystal absorption.

This shows that the detected signal will be linearly proportional to ϕ(t) as soon as ! ! " # 0 . In this case one has 2(e ! " x cos( ! ! " x) # 1)(cos$(t) # 1) << 2e ! " x sin( ! ! " x)sin $(t) (as long as ϕ << π/2) and the portion of the signal which is linearly proportional to the phase change becomes dominant, even if the real part γ ʹ′ remains appreciable.

To obtain γ ʹ′ʹ′ ≠ 0 we apply a DC electric field to the crystal. The signal intensity is then given by :

I s (x,t) = e ! "x I s (0,0) e 2 # $ x + 2e # $ x sin( # # $ x) % & (t) ] [ (2)
We use a sinusoidal phase modulation ϕ(t)= 2 •Δϕ•cos(2πf t). One obtains the signal to noise ratio (SNR) by dividing the root mean square of the signal and the noise, respectively. In the case where photon noise is the limiting noise source we have :

SNR = signal variation photon noise = 2!I s (0,0) h"#f e $ %x / 2 sin( & & ' x) 4() * , ( 3
)
where η is the quantum efficiency of the detection scheme and Δf is its bandwidth. ! = " 4# $% is the root mean square displacement corresponding to the root mean square phase modulation amplitude Δϕ. λ is the wavelength of the light, ν = c/λ is the light frequency and h is Planck's constant.

In order to be able to compare the performances of different systems, we introduce a normalized detection limit, δ lim , expressed in nm• W/Hz . It corresponds to the minimum detectable displacement using a 1 W power incident signal beam and a 1 Hz detector bandwidth :

! lim = !(SNR = 1) I s (0, 0) "f = # 4$ h% 2& e +' x / 2 sin( ( ( ) x) . ( 4
)
For plane waves, we find that the normalized detection limit in a classical homodyne

interferometer is ! lim classic = " 4# h$ 2%
. At λ = 1.55 µm and for η = 0.76, which corresponds to the quantum efficiency of our InGaAs detector, one obtains

! lim classic = 3.6 " 10 #8 nm $ W Hz .
With a photorefractive beam-combiner, we obtain the same detection limit in the ideal case where α = 0 and γ ʹ′ʹ′x = π/2.

Fig. 1 shows the normalized detection limit. We observe that to obtain a detection limit lower than 10 -7 nm• W/Hz at the 1.55 µm wavelength, the α/γ ʹ′ʹ′ ratio must be less than 2.

Experimental realization

A scheme of the experimental set-up is shown in Fig. 2. The displacement of the vibrating material is simulated with an electro-optic phase modulator (Pockels cell). The scattering of the material is simulated with a diffuser used in transmission. A first optical system images the surface of the diffuser onto the photorefractive crystal and a second system of lenses images the plane of the crystal on the detector. Some experiments were made without diffuser. The grating spacing obtained for a collimated signal beam traveling along the optical axis of the system is Λ = 3.5 µm. With the diffuser, we obtain a range of grating spacings from 2.8 µm to 4.8 µm. We use a distributed feed-back laser diode emitting 10 mW of infrared light at a wavelength λ = 1.55 µm.

We use two photorefractive CdTe:V samples, B23 and B25. Their characteristics parameters are presented in Table 1. Sample B23 was characterized in a previous work [16]. The electro-optic coefficient of CdTe is r 41 = 5.5 pm•V -1 [14,15]. Sample B23 is oriented in such a way that the effective electro-optic coefficient is r eff = 2/3 r 41 . Clamped (strain-free) and unclamped (stress-free) [START_REF] Nye | Physical Properties of Crystals[END_REF] electro-optic coefficients differ less than by 2% [START_REF] Strauss | [END_REF]19] in CdTe and the piezoelectric contribution [20,21,22] can be safely neglected.

We first studied the effectiveness of a DC applied field with the B23 sample. We measured the intensity modulation of the signal without using any diffuser in the path of the signal beam. The signal beam was phase modulated at 1 kHz. Due to a residual intensity modulation generated by the Pockels cell, the detected signal is only clearly measurable for an applied voltage above 0.4 kV (applied electric field ≈ 1 kV•cm -1 ). Above this value the signal varies linearly with the applied electric field (Fig. 3, insert).

We applied a voltage of 3.5 kV, to obtain V/d = 8 kV•cm -1 , for the measurement of the detection limit of our device (sample B23). We note d the length of the crystal along the direction of the applied field, corresponding to the electrode separation. We used an InGaAs detector with a quantum efficiency η = 0.76. The detection bandwidth given by the electronics was Δf = 1.33 MHz. The incident signal illumination was 25 µW. We verified that the intensity modulation amplitude of the detected signal was proportional to the phase modulation amplitude given by the electro-optic modulator. For an equivalent root mean square displacement δ = 2.1 nm we measured a signal to noise ratio of 16. From this we extrapolate to a displacement of 0.13 nm for a signal to noise ratio of 1 and we obtain a detection limit of 5.8×10 -7 nm• W/Hz . From Eq. ( 4) we then deduce γ ʹ′ʹ′ = 0.41 cm -1 .

From the experimental data one can also obtain the imaginary part of the amplitude photorefractive gain γ ʹ′ʹ′ in a different way by measuring the ratio between the signal modulation amplitude and its DC part. From Eq. ( 2), this ratio theoretically equals

8!" # sin $ $ % x e $ % x
. From classical energy transfer measurement by two wave mixing, we deduce γ ʹ′ ≈ 0.1 cm -1 . Since γ ʹ′x is small, e ! " x # 1 , and an exact knowledge of γ ʹ′ is not required. We derive γ ʹ′ʹ′ = 0.36 cm -1 . Inserting this value in Eq. ( 4) we obtain a detection limit of 6.6×10 -7 nm• W/Hz . Both approaches give the same result, showing that our signal to noise ratio is indeed limited by photon noise.

We finally obtain γ ʹ′ʹ′ = 0.39±0.03 cm -1 and δ lim = (6.2±0.4)×10 - From the electron-hole competition model [23] and the B23 crystal parameters given in Table 1, we can compute γ th ʹ′ = 0.72 cm -1 and γ th ʹ′ʹ′ = 0.97 cm -1 for an applied electric field E 0 = 8kV•cm -1 and a grating spacing of Λ = 3.5 µm. Crystal parameters such as effective trap concentration (N eff ) and electronhole competition factor (ξ 0 ) were determined by studying the two wave mixing gain as a function of grating spacing (no applied field). ξ 0 is positive when the electrons are the majority charge carriers. Using α/γ ʹ′ʹ′ = 1.9 (α = 1.85 cm -1 ), one predicts a detection limit δ lim = 2.4×10 -7 nm• W/Hz . This is almost 3 times better than our experimental value. It is to be expected that the electric field in the center of the crystal does not correspond to V/d = 8 kV•cm -1 , as the electric field can be reduced by screening effects caused by trapped space-charge [24]. A parallel electro-optic measurement of the Pockels effect, i.e a determination of the crystal transmission when placed between crossed polarizers and in the same experimental conditions (illumination and applied voltage) showed that the screening effects lead to a decrease of the internal field in the middle of the crystal by a factor of about 0.7. Thus the real applied DC electric field is approximately 5 kV•cm -1 which gives γ th ʹ′ = 0.45 cm -1 and γ th ʹ′ʹ′ = 0.80 cm - 1 . Screening effects are not the only explanation for the smaller experimental gain value. This gain reduction could come from crystal imperfections like polycrystallinity or the existence of multiple trapping centers.

With the same experimental process in the B25 sample, we obtain a detection limit of (8.2±0.5)×10 -7 nm• W/Hz which corresponds to γ ʹ′ʹ′ = 0.22 cm -1 . We note that with this sample we can not apply an electric field higher than 5 kV•cm -1 to prevent electrical breakdown. Using the parameters of Table 1 for B25 and ignoring electric field screening effects leads to γ th ʹ′ʹ′ = 1.06 cm -1 . Similarly to what we observed in B23, the measurement gives a reduced value of the gain.

To measure the adaptability of our system to deformations of the wavefront of the signal beam, we placed the diffuser in the signal path (Fig. 2). The étendue of the signal beam is defined as G = πSsin 2 ρ, where S and ρ are the surface of the image and the half angle of the conical signal beam in front of the crystal, respectively. The étendue of a diffraction limited gaussian beam is given by G = λ 2 and is equal to 2.4×10 -6 mm 2 •sr. In our experiment with the diffuser in place, G was limited by the diameters of the lenses (f-number = 4) and was equal to G = 0.1 mm 2 •sr. This corresponds to an étendue 4×10 4 times higher than a gaussian beam. This value of 0.1 mm 2 •sr is large enough for the multimode optical fibers generally employed in this kind of system. Within the measurement accuracy of 5%, the signal beam intensity modulation observed after the crystal for a given phase modulation was the same for a gaussian signal beam and for the speckled beam produced by the diffuser in the signal beam path.

We have also experimentally studied the frequency response of the system, including at modulation periods larger than the photorefractive response time. The experiment was performed without the diffuser in the signal beam path. The results for both crystal samples are shown in Fig. 3. Using a 66 mW•cm -2 illumination intensity we obtained a cut-off frequency between 3 and 4 Hz for B23 and of 15 Hz for the B25. A slight overshoot was observed at about 15 Hz and 200 Hz for the B23 and B25 samples, respectively (Fig. 3). Theoretical description will be given in the following discussion.

Discussion.

Under CW illumination, the photorefractive response time is expected to be linear with the illumination intensity, the cut-off frequency should be around kHz at an illumination intensity of 19 W•cm -2 or 4.4 W•cm -2 for B23 and B25, respectively. Our samples have a much too high absorption to be used at these power densities. However, as shown in Ref. [16], this absorption contains a large residual part which can be eliminated with a new process of crystal growth.

It is clear from the curves in Fig. 1 that the best results will be obtained when the crystal absorption is very small. However, as the photorefractive effect relies on charge generation from light absorption, absorption can not be reduced too much. It is thus important to consider its influence on the detected signal when increasing the crystal length. As shown by Fig. 1, there is an optimal crystal thickness x opt for which the signal to noise ratio is minimal. From Eq. ( 4), we obtain

x opt = 1 ! ! " Arctan 2 ! ! " # .
When the crystal thickness exceeds this limit, the signal reduction due to absorption becomes more important than the signal increase due to amplification by the diffracted part of the pump beam. The detection limit also depends on crystal geometry. Its minimal value, and the maximum value of γ ʹ′ʹ′, is obtained when the crystal orientation is such that r eff = 2r 41 / 3 .

If the B23 crystal sample had such an optimized orientation, we would obtain γ ʹ′ʹ′ = 0.54 cm -1 .

Given the crystal absorption α = 1.85 cm -1 , the minimum of δ lim in Eq. ( 4) is reached for x opt = 1 cm. One thus expects a detection limit δ lim = 1.8×10 -7 nm• W/Hz that is more than three times better than our current experimental value. The parameter α/γ ʹ′ʹ′ of a crystal is the only constraint on the detection limit.

The large background absorption of B23 makes α/γ ʹ′ʹ′ equal to 3.4, which is too high for going beyond the threshold of 10 -7 nm• W/Hz . This is the second motivation for growing crystals with less absorption.

The crystal absorption also an effect on the frequency response of the crystal. This can be calculated by considering the full expression for the response of the system, valid whatever the relative amplitude between the period of the modulation and the photorefractive response time. Using the same nomenclature as in Eqs. (1-2), the amplitude of the signal beam after the crystal is given [START_REF] Ph | [END_REF] :

E s x, t ( ) = e ! "x 2 e #x E s 0, 0 ( ) + $ 0 %E s %t 0, T ( ) + E s 0, T ( ) ! E s 0, 0 ( ) & ' ( ) H x, t ! T ( )dT 0 t * + , - . / 0 , (5)
and

H x,t ( ) = e ! "x # 0 e ! t # 0 1 F 1 " $ ,1, e $x ! 1 e $x % & ' ( ) * t # 0 % & ' ( ) * , (6)
where τ o is the photorefractive time constant at the entrance of the crystal (x = 0), and 1 F 1 (a,b,z) is the confluent hypergeometric function. The signal beam is phase modulated at a frequency f : ϕ(t) = ϕ 0 cos(2πf t) with ϕ 0 <<π/2. From Eqs. (5-6), we compute the intensity modulation of the signal beam as a function of the frequency f. The theoretical response is calculated for B23 and B25 and for a 66 mW•cm -2 illumination using the parameters of Table 1. The modulus and phase of the response time of the crystal under DC field are experimentally determined by measuring the kinetics of energy transfer when the DC field amplitude is reversed with a periodicity slower than the response time of the crystal.

Adjustment of experimental results with theoretical expressions (Eq. ( 5) in Ref. [25]) allows to determine both modulus and phase of the crystal response time τ (Table 1). Since αx < 1, τ 0 is obtained from

τ 0 = τ•e -αx/2 [13].
For both samples, the theoretical curves are drawn using the measured gain and time constant, without any adjustments to the frequency response data. The photorefractive beam-combiner acts as a high pass filter with an overshoot. An increase of absorption produces a decrease of the cut-off frequency and a reduction of the height of the overshoot. This theoretically expected slight overshoot in the frequency response was also observed experimentally. For B23, we present two other theoretical curves where we use a phase of the response time of 0.4 rad and 1 rad instead of 0.7 rad. These two values correspond to our measurement uncertainty. We see that a small variation of the phase of the response time produces a large variation of the height of the overshoot. It allows us to tell that frequency response measurements give a more accurate determination of the phase of the response time which was otherwise quite inaccurate. An examination of Eq. ( 5) shows that a change in the modulus of the response time leads to a lateral shift of the whole curve without modifying shape.

Conclusion

We characterized a homodyne detection system using a CdTe crystal as the element which combines the local oscillator beam with the signal beam. We obtained a detection limit approaching the theoretical limit and demonstrated the insensitivity of the system to deformations in the signal beam wavefront up to an étendue of 0.1 mm 2 •sr. We studied the response of the system as a function of the phase modulation frequency in the signal beam and demonstrated the expected high band pass response.

We now plan to test the system for a signal beam with a larger étendue of the order of some mm 2 •sr. We will also approach the ideal interferometric detection limit (with δlim ≤ 10 -7 nm• W/Hz

) by using less absorbing crystals (α/γʹ′ʹ′ ≤ 1) (Fig. 1). In such low-absorbing crystals we also expect to obtain a cut-off detection frequency of 1 kHz at illumination densities of the order of some W•cm -2 .

Table Caption

Table 1 : Measured characteristics of the photorefractive crystals. The response time and the photorefractive gain were measured with a DC applied electric field of 8 kV•cm -1 for B23 and 5 kV•cm -1 for B25. The absorption at λ = 2 µm corresponds to a residual background absorption [16].

Figure Caption

Fig. 1 : Theoretical detection limit as a function of the γ ʹ′ʹ′x factor. The left vertical scale gives the detection limit resulting from Eq. ( 4) using a quantum efficiency of 0.76. The right vertical scale gives the relative detection limit that is normalized to the plane wave interferometric detection limit. The different curves correspond to different α/γ ʹ′ʹ′ values. The dashed line corresponds to the detection limit of the confocal Fabry-Perot used in transmission [26]. The circle corresponds to the experimental measurement with B23. The square corresponds to the typical sensitivity we expect to obtain with an optimized crystal. 5) and the parameters in Table 1. The dashed line and the dashdotted line give the result for the B23 parameters in Table 1, but with 0.4 rad and 1 rad for the phase of the response time, respectively.

The insert shows the detected signal amplitude as a function of the DC applied electric field.

B23 B25

α (1.55 µm) in cm -1

Fig. 2 :

 2 Fig. 2 :Experimental set-up used for the study of the homodyne detection system.

Fig. 3 :

 3 Fig. 3 :Normalized signal, i.e. amplitude of the intensity modulation normalized to the high frequency value, versus frequency. The empty circles and full triangles give the experimental results in samples B23 and B25, respectively. The thick lines give for both crystals the prediction obtained from Eq. (5) and the parameters in Table1. The dashed line and the dash-

Fig. 1 -Fig. 2 -Fig. 3 -

 123 Fig. 1 -Louis-Anne de Montmorillon

Table 1 -

 1 Louis-Anne de Montmorillon

	1.85	2.31

nm• W/Hz (experimental point in Fig.1) .


Ph Delaye

L A De Montmorillon

G Roosen

TRANSMISSION OF TIME MODULATED OPTICAL SIGNALS THROUGH AN ABSORBING PHOTOREFRACTIVE CRYSTAL

We solve the problem of the transmission of time modulated signals in a two beam coupling photorefractive set-up using an absorbing material. We obtain an analytical expression that describes this phenomenon in absence of pump beam depletion due to beam coupling. We use this expression to analyse the influence of absorption on the photorefractive build-up. We also describe, using this model, the photorefractive beamsplitter response. It is also showed that the π/2 phase shift between illumination and index gratings is detrimental to a linear detection of the phase variation of the signal beam.

I. Introduction

Coherent detection is a fundamental technique for the detection of phase modulated signals in which the coherent combination of the signal beam with a local oscillator requires a perfect adequacy of the wavefronts of the two beams. This is easily performed with guided waves in fiber optic communications, but, when considering free space communications, the problem is much more complex. The required phase matching of the signal and the local oscillator leads to a reduction of the field of view of the system, according to the antenna theorem [1]. However the use of photorefractive two-beam coupling allows to develop coherent detection schemes with large fields of view [2][3][4]. In this scheme, two beams, a pump beam and a time modulated signal beam, interfere in a photorefractive crystal. They create an index grating that is π/2-shifted relative to the illumination grating. The beams then diffract on this grating, leading to an energy transfer from one beam to the other. Then, at the exit of the crystal, in the direction of the modulated signal beam, we have generation of a local oscillator which is the non modulated diffracted pump beam, having the same wavefront structure than the signal beam. This two beam coupling scheme is equivalent to the beam splitter in the classical homodyne detection scheme, but without the antenna theorem restriction as the signal and the local oscillator have the same phase structure whatever this structure. The photorefractive two beam coupling set-up then acts as a photorefractive beam splitter for homodyne detection [2,3]. An additional property of this component is its adaptability as the photorefractive effect is a dynamic effect. If the phase structure of the signal beam is slowly varying, the device will automatically adapt to this variation creating a new index structure. So the photorefractive beam splitter adapts to slow variations of the signal whereas it remains totally transparent to the rapidly modulated information imprinted on it. This photorefractive beam splitter and its properties have been widely studied and used in the literature. Theories of operation of this component have been developed previously [2,[START_REF] Cronin-Golomb | Digest of Topical Meeting on Photorefractive Materials, Effects and Devices I[END_REF][START_REF] Boutsikaris | [END_REF][7][8]. Some of these models have taken into account the depletion of the pump beam due to the photorefractive energy transfer induced by high photorefractive gains [7,9]. However most of these models have considered that the influence of the crystal absorption on the kinetics was negligible, which is a strong and generally wrong assumption. Applications of this component in an industrial environment will require two important characteristics to be satisfied. First it should operate in the eye-safe part of the spectra, as for example the 1.5µm wavelength. Second, thanks to its adaptive properties, the component should be insensitive to environmental perturbations and should present a high cut-off frequency. A photorefractive response time of the order of 10µs is a good estimation of the required temporal performances [10]. First and second constraints impose the use of highly sensitive photorefractive semiconductors, like CdTe or GaAs [11,12]. These materials have low value electrooptic coefficients, which leads to low values of the photorefractive gain. The gains in these crystals are often of the order of magnitude or lower than the absorption. From these considerations, it is therefore not reasonable to consider that absorption is negligible, and we can not apply existing models without taking a lot of care. We then must solve the photorefractive two beam coupling model with a time varying signal taking absorption into account.

II. Situation of the problem

We send two plane waves on a photorefractive crystal, a strong unmodulated pump beam and a time varying signal beam (Fig. 1). The signal can be either phase or amplitude modulated and has a much lower intensity than the pump beam. The aim of the study is to see how this modulation is transmitted through the crystal.

The photorefractive gain is supposed to be low, which means that the energy transfer is not high enough to induce depletion of the pump beam. We will be thus in the undepleted pump beam approximation. On the other hand, as we take into account absorption, the pump beam will be attenuated when traversing the crystal. Consequently the photorefractive time constant, that is determined by the pump beam intensity, will vary in the crystal thickness.

Pump beam and signal beam interfere in the crystal creating a spatially modulated space charge field. This space charge field induces, through the electrooptic effect, an index grating on which the beams diffract. This system is governed by three differential equations that link amplitude E 1 of the space charge field, amplitude E i of the pump beam and amplitude E d of the probe beam [START_REF] Cronin-Golomb | Digest of Topical Meeting on Photorefractive Materials, Effects and Devices I[END_REF].

The first differential equation gives the kinetics of the space charge field [13] :

! " k B T I e 2 # $ % & ' (E 1 (t = E 1 A n ) n * n 2 k(k ! iV ) k o 2 + 1 k(k ! iV ) * n 2 + 1 # $ % % & ' + + + A p ) p * p 2 k(k + iV ) k o 2 + 1 k(k + iV ) * p 2 + 1 # $ % % % & ' + + , - . . . / 0 1 1 1 +im k B T e ) n * n 2 k ! iV k(k ! iV ) * n 2 + 1 # $ % % & ' + + ! ) p * p 2 k + iV k(k + iV ) * p 2 + 1 # $ % % % & ' + + , - . . . / 0 1 1 1 (1)
Equation ( 1) can be rewritten as :

!E 1 !t + E 1 " = G " E i * E d I (2)
with time constant τ of the space charge field (also labelled as photorefractive time constant)

given by :

1 ! = I e 2 " k B T A n # n $ n 2 k(k % iV ) k o 2 + 1 k(k % iV ) $ n 2 + 1 & ' ( (
) * + + + A p # p $ p 2 k(k + iV ) k o 2 + 1 k(k + iV ) $ p 2 + 1 & ' ( ( ( )
and The two other differential equations are the propagation equations for the pump and signal beams [14] :

G ! = "2i I e # $ n % n 2 k " iV k(k " iV ) % n 2 + 1 & ' ( ( ) * + + " $ p % p 2 k + iV k(k + iV ) % p 2 + 1 & ' ( ( ( )
!E i !x = " n 0 3 r eff m * E sc * 2i# cos$ E d (3a) !E d !x = " n 0 3 r eff m E sc 2i# cos$ E i (3b) % & ' ( '
r eff is the effective electrooptic coefficient that depends on the orientation of the photorefractive crystal and on the directions of polarization of the beams; n 0 is the linear index of refraction of the material, λ is the wavelength in the material, θ is the half-angle between the beams inside the material and the beams propagate along the x direction.

We set

E 1 = m E sc and ! G = " n 0 3 r eff 2i# cos$ which gives ! = " n 0 3 r eff 2i# cos$ G .
Here we can note that in the diffusion regime (without any applied electric field) γ is purely real and equals ! = " 2 with Γ the two beam coupling photorefractive intensity gain (Γ in cm -1 ) [13,14]. Taking into account all these notations and including the absorption (α absorption coefficient for intensity in cm -1 ) equations (3) become :

!E i !x = " G E 1 * E d # $ 2 E i (4a) !E d !x = " G E 1 E i # $ 2 E d (4b) % & ' ( ' (2)
and ( 4) correspond to a system of three coupled differential equations.

In this problem we are interested in the transmission through the crystal of the time modulated signal beam E d . We will rewrite these three equations in the form of one partial differential equation that governs the propagation of E d through the crystal. For this resolution we will make some assumptions and approximations, we will now precise.

1.-The pump beam is not time modulated (an eventual phase modulation present on the pump beam can be easily transferred on the signal beam as it is the phase difference between the two beams that is important).

2.-We neglect the pump beam depletion due to the beam coupling, it corresponds to a regime of small photorefractive gain and high pump to probe beam ratio.

3.-The pump beam attenuation due to the absorption is taken into account.

4.-At time t=0 (before the probe beam is modulated) we are in a steady-state regime with a grating written in the crystal. It means that before t=0, the two beams are present in the crystal and have written a saturated index grating on which they now diffracts (if no grating is present before t=0, the system can be solved as well, as it will be seen later). Points 2 and 3 allow to simplify equation (4a) that becomes :

!E i !x = " # 2 E i (5)
It is solved immediately as :

E i x ( ) = E i0 e ! " 2 x
independent of time. It finally gives the total illumination in the crystal :

I x ( ) = E i E i * + E d E d * = E i E i * = I 0 e ! "
x with I 0 = E i 0 2 the pump beam intensity at the entrance of the crystal.

We see from this first equation that the illumination decreases during propagation through the crystal. Then the time constant τ increases during this same propagation (the photorefractive effect is slower at the exit of the crystal than at its entrance). As 1 ! is proportional to I, we can write 1 ! = e "# x ! 0 with τ 0 the photorefractive time constant at the entrance of the crystal (at x=0).

Deriving equation (4b) relative to time t and replacing !E 1 !t by its expression given by equation ( 2) we obtain :

! 2 E d !t!x = " # E i E i * I E d $ " G# E 1 E i $ % 2 !E d !t (6)
We then take E 1 from equation (4b) which finally gives the partial derivative equation that governs E d x,t ( ) :

! 2 E d !t!x = E d e "# x $ 0 % " # 2 & ' ( ) " e "# x $ 0 !E d !x " # 2 !E d !t (7)
Point 4 gives us the initial condition of the problem. At t=0, a grating is written in the crystal and is in its stationary state, so we have :

E d x,0 ( ) = E d0 e ! " 2 x e #x
. At time t=0 + , we put a known time modulation (in amplitude and/or in phase) on the signal beam E d 0,t ( ) at the entrance of the crystal and we are looking for the expression of E d x,t ( ) after propagation of a distance x inside the crystal.

In order to simplify the equation we will change the function by setting

E d x,t ( ) = F x,t ( )e !x e " #x
2 . Equation ( 7) becomes :

! 2 F !t!x = "# !F !t " e " $x % 0 !F !x (8)
with the initial condition at t=0 : F x,0

( ) = E d 0,0 ( ) constant, which means that !F !x x,0 ( ) = 0.
The boundary condition F 0,t

( ) = E d 0,t ( ) is already known.

III Resolution of the transfer equation

We will solve equation ( 8) using the Laplace transform method. The Laplace transform method is prefered over the Fourier transform method [15,16] as it deals more easily with the causal signals we have in our study. We call ˜ F x,s ( ) the Laplace transform of F x,t ( ), with the initial conditions F x,0 ( ) = E d 0,0 ( ),

!F !x x,0
( ) = 0 and the boundary condition that becomes ˜ F 0,s ( ) (Laplace transform of F 0,t ( )) known. We take the Laplace transform of equation ( 8) which gives (following functions with tilde are functions of s while functions without tilde are functions of t) :

s + 1 ! 0 e "x # $ % & ' ( ) ˜ F )x = * + s ˜ F + + E d 0,0 ( ) (9)
which is rewritten :

! ˜ F !x x,s ( ) = " # s $ 0 e %x 1+ s $ 0 e %x ˜ F x,s ( ) + # $ 0 e %x 1 + s $ 0 e %x E d 0,0 ( ) (10)
The solution of this differential equation (function of x) is :

˜ F x,s ( ) = C s ( ) 1+ s ! 0 1+ s ! 0 e "x # $ % & ' ( ) " + E d 0,0 ( ) s ( 11
)
The integration constant C(s) is given by the boundary condition : ˜ F 0,s

( ) = C s ( ) + E d 0,0 ( ) s . Going back to ˜ E d x,s ( ) the Laplace transform of E d x,t ( ) , with ˜ F 0,s ( ) = ˜ E d 0,s
( ) , we obtain finally :

˜ E d x,s ( ) = ˜ E d 0,s ( ) ! E d 0, 0 ( ) s " # $ % & ' e ! (x 2 e
)x

1

+ s * 0 1 + s * 0 e (x + , - . / 0 ) ( + E d 0,0 ( ) s e ! (x 2 e )x (12)
The solution E d x,t ( ) is given by the inverse Laplace transform of expression (12).

In order to find E d x,t ( ) , we set ˜ H x,s

( ) = 1 1 + s ! 0 " # $ % & ' 1+ s ! 0 1+ s ! 0 e (x " # $ % & ' ) (
, which inverse Laplace transform is H x,t ( ). Equation ( 12) becomes :

˜ E d x,s ( ) = e ! "x 2 e #x ˜ E d 0,s ( ) ! E d 0,0 ( ) s $ % & ' ( ) 1 + s * 0 ( ) ˜ H x,s ( ) + E d 0,0 ( ) s $ % & ' ( ) (13)
which gives :

˜ E d x,s ( ) = e ! "x 2 e #x ˜ E d 0,s ( ) ! E d 0,0 ( ) s $ % & ' ( ) ˜ H x,s ( ) + * 0 s ˜ E d 0, s ( ) ! E d 0,0 ( ) [ ] ˜ H x,s ( ) + E d 0,0 ( ) s $ % & ' ( ) (14)
We then take the inverse Laplace transform of expression ( 14), using the convolution theorem, and we obtain the final expression for E d x,t ( ) :

E d x,t ( ) = e ! "
x

2 e #x E d 0,0 ( ) + $ 0 %E d %t 0,T ( ) + E d 0,T ( ) ! E d 0,0 ( ) & ' ( ) H x,t ! T ( )dT 0 t * + , - . / 0 (15)
We thus have an analytical expression for the transmission of the modulated signal through a photorefractive crystal in a two beam coupling configuration. The problem is now to find the expression of H x,t ( ).

In a general way, we are looking for the inverse Laplace transform of function

˜ H s ( ) = 1+ as ( ) ! "1 1+ b s ( ) ! .
We rewrite this function as ˜ H s

( ) = a ! "1 b ! ˜ F s + 1 a # $ % & with ˜ F s ( ) = s ! "1 s " b " a ab # $ % & ! function which
inverse Laplace transform is known from tables [START_REF]Formulaire pour le calcul operationnel[END_REF]. This function is

1 F 1 ! ,1, b " a ab # $ % & t # $ % &
with the confluent hypergeometric function 1 F 1 a,b, z ( ) [18]. We then deduce, knowing the properties of the Laplace transform, that the inverse Laplace transform of ˜ H s

( ) is H t ( ) = a ! "1 b ! e " t a 1 F 1 ! ,1, b " a ab # $ % & t # $ % &
. In our case of interest, i.e. a = ! 0 , b = ! 0 e "x and ! = " # ,

we finally obtain :

H x,t ( ) = e !" x # 0 e ! t # 0 1 F 1 " $ ,1, e $x ! 1 e $x % & ' ( ) t # 0 % & ' ( ) * (16)
So this value of function H x,t ( ) completely determines the value of E d x,t ( ) given by relation (15).

We will now show that the expression we find, reduces, in the case of no absorption, to the expression previously derived in the literature [START_REF] Cronin-Golomb | Digest of Topical Meeting on Photorefractive Materials, Effects and Devices I[END_REF].

For negligible absorption α goes to 0, and then ! = " # goes to infinity. Setting

z = ! e "x ! 1 e "x # $ % & ' !t( ") 0 # $ % & ' * "+ 0 , + , , , ! ( x t ) 0 , we rewrite 1 F 1 ! " ,1, e "x # 1 e "x $ % & ' ( t ) 0 $ % & ' ( * = 1 F 1 ! " ,1,# z " ! $ % & ' ( .
As we know that lim a! + "

1 F 1 a,1,# z a $ % & ' = J 0 2 z
( ) [18], we finally deduce that lim

! "0 1 F 1 # ! ,1, e !x $ 1 e !x % & ' ( ) t * 0 % & ' ( ) + = J 0 2 $ # x t * 0 % & ' ( ) +
and then :

lim ! "0 H x,t ( ) = e # $x % 0 e # t % 0 J 0 2 # $ x t % 0 & ' ( ) * + (17)
To be exactly in the case established by Cronin-Golomb [START_REF] Cronin-Golomb | Digest of Topical Meeting on Photorefractive Materials, Effects and Devices I[END_REF], the initial conditions have to be changed a little and we have to consider that no grating is present at time t=0. In that case we would have to slightly modify function F x,t ( ), defining it as

E d x,t ( ) = F x,t ( )e ! "x
2 with the initial conditions F x,0

( ) = E d 0,0 ( ) and !F !x x,0
( ) = 0. The equation would become :

! 2 F !t!x = " e # $x % 0 F # e #$ x % 0 !F !x (18)
The same resolution as for the previous equations would then have given :

E d x, t ( ) = e ! "x 2 e #x $ 0 E d 0, 0 ( ) H x, t ( ) + $ 0 %E d %t 0, T ( ) + E d 0, T ( ) & ' ( ) H x,t ! T ( )dT 0 t * + , - . / 0 (19)
with the same function H(x,t). So when α goes to 0 considering expression [START_REF]Formulaire pour le calcul operationnel[END_REF], equation ( 19) reduces to :

E d x, t ( ) = E d 0, 0 ( )K t ( ) + !E d !t 0, T ( ) + E d 0, T ( ) " 0 # $ % & ' ( K t ) T ( )dT 0 t * (20) with K t ( ) = e ! t " 0 J 0 2 ! # x t " 0 $ % & ' (
) . This expression is the one derived in Ref. [START_REF] Cronin-Golomb | Digest of Topical Meeting on Photorefractive Materials, Effects and Devices I[END_REF] after correction of a typographical error [START_REF] Boutsikaris | [END_REF].

So our model is a generalization of the model of Cronin-Golomb to the case where absorption is not negligible, which is the real case.

We have established an analytical expression (expressions ( 15) and ( 16)) of the transmission of a time varying signal through an absorbing photorefractive crystal in a two beam coupling scheme. The final expression has the same feature than in the case without absorption, the only difference being in the transfer function which is a less common function than the Bessel function found by Cronin-Golomb. We have transformed the problem from solving a system of coupled partial derivative equations to the one of solving an expression with still a convolution but with one variable t only. The expression is still complicated even if easy to solve numerically. In fact, in some special cases this expression can be simplified. We will now present two of these cases.

IV Two-beam coupling build-up

An example of the use of the expression we have established, is for the determination of the build-up kinetics of the two beam coupling effect in presence of absorption. Absorption creates a continuum of time constants through out the crystal as the illumination decreases due to absorption. The question is then, what is the actual kinetics of energy transfer ?

To solve the problem we use expression [START_REF] Heaton | [END_REF] with a step function for the initial value : E d 0,t ( ) = 0 for t ! 0 and E d 0,t ( ) = 1 for t > 0 . The derivative of this function is a Dirac function centered at 0. We have then :

E d x,t ( ) = e ! "
x

2 e #x $ 0 H x,t ( ) + H x,t ! T ( )dT 0 t % & ' ( ) * + = e ! "
x

2 e #x $ 0 H x,t ( ) + H x,T ( )dT 0 t % & ' ( ) * + ( 21
)
When absorption is negligible we replace H(x,t) by function K(t) and we obtain the expression derived in Ref. [START_REF] Heaton | [END_REF].

From the analytical expression ( 21), we have numerically calculated the temporal dependency of the output intensity, for a crystal with an intensity gain of Γ=0.3cm -1 , a thickness x=1cm and a time constant τ 0 =1s, for different values of absorption between 0 and 1cm -1 . Results are shown in Fig. 2 where the different curves were normalized to their steady state value :

S = E d x,t ( ) e ! "
x

2 e #x 2
. We see on these curves that the more absorbent the crystal, the slower the two beam coupling build-up. Moreover these kinetics curves are very close to exponential curves with two beam coupling build-up time constants that follows the law

! TBC " ( ) = e " x
2 ! TBC " = 0 ( ) which corresponds to a time constant value taken in the middle of the crystal. We here have to note that, in our simulations, we have neglected a contribution of the absorption on the photorefractive effect. Indeed space charge field time constant τ 0 depends directly on the absorption of the crystal (through parameters α n and α p in relation (2a)). In the calculations we only wanted to show the influence of absorption on two beam coupling, so we take the same value of time constant τ 0 for all curves, despite the fact that, in real cases, changing absorption changes the response time of the space charge field build-up.

We also note on the curve at α=0 that the exponential time constant is greater than τ 0 as already noticed in Ref. [9]. This means that the build-up of the space charge field takes a longer time in presence of beam coupling than when beam coupling is absent (when for example r eff =0). This slowing down of the kinetics with increasing beam coupling gain can be seen more clearly in Fig. 3 where we have calculated the normalized transmitted probe beam intensity for different values of the photorefractive gain between 0 and 2cm -1 . The grating build-up slowers as the photorefractive gain increases.

V Detection of rapidly phase modulated signals : Photorefractive beamsplitter

One use of the photorefractive beamsplitter is the detection of rapidly phase modulated signals [4,20]. We want to measured a signal E d 0,t ( ) which is phase modulated, on a time scale very short compared to the response time of the photorefractive effect τ 0 . We are then in experimental conditions where t<<τ 0 . As we have lim

z !0 1 F 1 a, b, z ( ) = 1, equation (15)
becomes :

E d x,t ( ) = e ! "x 2 e #x E d 0,0 ( ) + $ 0 %E d %t 0,T ( ) + E d 0,T ( ) ! E d 0,0 ( ) & '
( ) 22) which is rewritten as :

e ! #x $ 0 e ! t !T $ 0 dT 0 t * + , - . / 0 (
E d x,t ( ) = e ! "x 2 E d 0,0 ( )e #x + e ! t $ 0 % %T e T $ 0 E d 0,T ( ) ! E d 0,0 ( ) ( ) & ' ( ) * +
We have finally :

E d x,t ( ) = e ! "x 2 E d 0,0 ( )e #x + E d 0,t ( ) ! E d 0,0 ( ) ( ) [ ] (24)
Knowing that E d x,0

( ) = E d 0,0 ( )e !x e " #x
2 , we obtain the simple expression that gives the transmission of the modulated signal through the crystal (when a stationary regime is reached) :

E d x,t ( ) ! E d x,0 ( ) ( ) = e ! "x 2 E d 0,t ( ) ! E d 0,0 ( ) ( ) (25)
The modulated input signal E d 0,t ( ) can be written as

E d 0,t ( ) = E d 0, 0 ( ) ! "E d t ( ) with !E d 0 ( ) = 1.
It then gives for relation ( 24) :

E d x,t ( ) = e ! "x 2 E d 0,0 ( ) e #x ! 1 ( ) + $E d t ( ) [ ] (26)
Experimentally, we are interested in the intensity of the signal beam at the output of the crystal. It is given by :

I d x,t ( ) = e !" x I d 0,0 ( ) e #x ! 1 2 + $E d t ( ) 2 + 2 Re e #x ! 1 ( ) * $E d t ( ) ( ) [ ] (27)
The signal beam is now purely phase modulated, so we have !E d t ( ) = e i" t ( ) with ! 0

( ) = 0 , what gives !E d t ( ) 2 = 1.
We write the photorefractive gain as ! = ! ' +i!" . The gain is purely real, ! "= 0 , when the phase shift between index and illumination gratings is π/2 (photorefractive effect in the diffusion regime) and is purely imaginary, ! ' = 0 , when the two gratings are in phase. According to these notations, the rapidly modulated part of the transmitted pump intensity !I d x,t ( ) is due to the third term of expression (27) only and we finally have : ) In this expression, the most interesting term is the first term as it is, in the limit of a low amplitude of the phase modulation (! t ( ) << " ), proportional to the phase shift ! t ( ). This term only exists if its coefficient is significant, i.e. if ! "" 0 . This means that in presence of a photorefractive grating in a pure diffusion regime, this term disappears and the only remaining term is the term in cos ! t ( ) ( ). In this case we have for the modulated part :

!I d x,t ( ) = 2 e "# x I d 0, 0 ( ) e $ ' x sin $ " x ( )sin % t ( ) ( ) + e $ ' x cos $ " x ( ) " 1 ( ) cos % t ( ) ( ) [ ] (28
!I d x,t ( ) = e "# x I d 0,0 ( ) e $ ' x " 1 ( )% 2 t ( ) (29)
We thus see that the photorefractive coupling is not optimum for the photorefractive beamsplitter as it does not allow for the coupling in quadrature that is best suited for a linear phase detection. This is easy to understand as a zero phase shift between illumination and space charge field gratings corresponds to no energy transfer between the beams. A phase shift introduced on one beam shifts the illumination grating towards the space charge field grating giving possibility for energy transfer with an amplitude proportional to the phase shift. Nevertheless a quadrature detection is not always necessary as shown in ref [21] for the case of high amplitude binary phase shift modulation scheme used in coherent communication.

One way to remove this drawback is to use a polarization sensitive set-up in order to mix the components in quadrature as proposed in Ref. [4]. Another way is to introduce on the signal beam, during the time of the measurement that is supposed to be short, an additional π/2 phase shift (with for example an electrooptic modulator) that allows to mix the components in quadrature. We then have :

!I d x,t ( ) = 2 e "# x I d 0, 0 ( ) e $ ' x " 1 ( )cos % t ( ) + & 2 ' ( ) * + , - . / 0 = 2e " #x I d 0,0 ( ) 1 " e $ ' x ( )sin % t ( ) ( ) [ ] (30)
Finally, a last possibility is to find a material for which the index grating is not π/2shifted with the illumination grating. This is for example the case for photorefractive semiconductors with an externally applied electric field. In this case, we have ! "" 0 and the modulated part of the signal is dominated by sin ! t ( ) ( ) (as we suppose ! t ( ) << " we have cos ! t ( ) ( ) " 1) and we have (from (27)):

I d x,t ( ) = e !" x I d 0, 0 ( ) e 2# ' x + 2e # ' x sin # "x ( )sin $ t ( ) ( ) [ ] (31)
An illustration is given in Fig. 4 for the parameters of a CdTe:V sample [22]. With an applied electric field of 2kV.cm -1 , a grating vector k = 1µm -1 and an illumination of 100mW.cm -2 (material parameters are given in Ref. [22]). We calculate an amplitude photorefractive gain ! = 0.11" 0.64i ( )cm "1 and a complex time constant ! 0 = ! r " i ! i with ! r = 0.33ms and ! i = 0.14ms. We introduce these parameters (with an absorption α = 1cm -1 and a crystal thickness of 1cm) in expressions ( 15) and ( 16), and we calculate the transmission of a phase modulated signal (with a sinusoidal phase modulation ! t ( ) = 0.1 " sin 2# $ t ( ) of frequency ν = 32MHz). We see, in Fig. 4, that the fast phase modulation is perfectly transformed into an intensity modulation of the transmitted beam as predicted by (31) for the regime of 1/ν<<τ 0 .

Coming back to the general case, we calculate the frequency response of the component for a sinusoidal phase shift ! t ( ) = 0.1 " sin 2# $ t ( ) of frequency ν. We see in Fig. 5 the high band pass filtering properties of the photorefractive beamsplitter. The cut-off frequency is about 150Hz (cut-off at 3dB).

VI Conclusion

We have established an analytical model for two beam coupling energy transfer in presence of modulated beams for a crystal presenting absorption. We solved the two beam coupling differential equation using the Laplace transform method. The found solution allows to calculate the kinetics of the photorefractive effect in presence of absorption. It shows that in the low gain and low absorption regime, the response of the photorefractive effect is exponential with a time constant which is the space charge field time constant taken in the middle of the crystal. With this model, we can simulate the photorefractive beam splitter in crystals with low gain and an absorption even greater than the photorefractive gain, a situation which is common in systems developed in the infrared region with semiconductors.

We show that, taking into account the absorption or not, the photorefractive effect is not the best adapted for a linear detection of phase modulated signals. The π/2 shift between illumination and index gratings has to be compensated, either by an external system, like adding an additional π/2 phase shift on one of the beams, or by applying an external electric field in order to have a photorefractive effect governed by the drift regime. 2 : Photorefractive build-up in the diffusion regime for different value of absorption of the crystal between 0 and 1 cm -1 (see text for calculation parameters). The values of the intensity at the crystal output are normalized to their steady state value. The signal beam intensity at the entrance of the crystal is the same for all the curves. Figure 3 : Photorefractive build-up in the diffusion regime for different values of the photorefractive gain between 0 and 2 cm -1 . The value of the intensity at the crystal output are normalized to their steady state value. The absorption is α = 1 cm -1 (other parameters are the same as in Fig. 2). The probe intensity at the entrance of the crystal is the same for all the curves. ( ) = 0.1 " sin 2# $ t ( ) (same calculation parameters as for Fig. 4).

Pump

x Signal E i E d (0,t) E d (x,t) Frequency (Hz)

!=0cm -1 !=0.05cm -1 !=0.1cm -1 !=0.25cm -1 !=0.5cm -1 !=0.75cm -1 !=1cm -1

Figure 1 :

 1 Figure 1 : Photorefractive two beam coupling set-up. A pump beam E i and a time modulated signal beam E d propagates along x direction and interacts through a two beam coupling scheme, thus modifying the transmission of the signal beam.

Figure

  Figure2: Photorefractive build-up in the diffusion regime for different value of absorption of the crystal between 0 and 1 cm -1 (see text for calculation parameters). The values of the intensity at the crystal output are normalized to their steady state value. The signal beam intensity at the entrance of the crystal is the same for all the curves.

Figure 4 :

 4 Figure 4 : Transmitted signal (straight line) through a photorefractive beamsplitter, as a function of time. The input signal beam is sinusoidally phase modulated (! t ( ) = 0.1 " sin 2# $ t ( ) , dashed line) with an amplitude of 0.1rad and a frequency ν=32 MHz (see text for calculation parameters). The probe intensity at the entrance of the crystal is I d 0, 0 ( ) = 1.

Figure 5 :

 5 Figure 5 : Frequency response of the photorefractive beamsplitter for a phase modulation ! t

  Figure 1

  Figure 2

  is the dielectric constant of the material, α n and α p are the parts of absorption that generate electrons and holes respectively, κ n and κ p are the inverse of the diffusion lengths of electrons and holes respectively, k 0 is the inverse of the Debye screening

			,			/
			. .			+ +	1 1	(2b)
			-.			*	0 1
	with	m = 2	E i * E d I	and	I = E i * E i + E d * E d
	m is the modulation of the interference pattern and I is the total illumination (as we set
	E d <<E i , we have I = E i * E i ). k is the grating number (k =	2! "	with Λ the grating spacing). In
	these expressions ε length, A n p ( ) = S n p ( ) I + ß n p ( ) S n p ( ) I	with S n p ( ) the photoionisation cross-section of electrons (holes)
	and ß n p ( ) the thermal emission coefficient of electrons (holes), V =	e E 0 k B T	with E 0 the applied
	external electric field.
		If we neglect the influence of A n and A p (i.e. photoconductivity much larger than dark
	conductivity) we have independent of I. In the general case, G and 1 ! proportional to I and photorefractive effect governed by the diffusion regime (E 0 = 0), G proportional to I, which means that G is ! 1 are complex. In the particular case of a ! 1 ! is real and G is purely
	imaginary.


L A De Montmorillon

Ph Delaye

J C Launay

G Roosen

COMPARATIVE STUDY OF CdTe AND GaAs PHOTOREFRACTIVE PERFORMANCES FROM 1µm TO 1.55µm

We present a photorefractive investigation of the Vanadium doped CdTe at different wavelengths from 1.06µm to 1.55µm. The sensitivity and performances of different samples grown with different conditions are deduced. We compare these performances with the ones of the undoped GaAs in the same wavelength range.

Introduction. The development of the optical telecommunication networks and metrology

systems at the eye safe wavelength of 1.5µm, increases the interest for the extension of the photorefractive effect towards the wavelength region of 1.3-1.5µm. CdTe appears to be a good candidate for such applications as its sensitivity extends beyond 1.5µm and a photorefractive effect can be obtained with the low excitation energies delivered by laser diodes [1,2]. Previous studies have shown that, in general, semiconductors are very sensitive to the electron-hole competition leading to a strong reduction of the photorefractive gain that varies with optical wavelength [3]. We here study different CdTe samples which present variable electron-hole competitions in the wavelength range 1 -1.55µm. We compare the performances of these CdTe crystals to the ones of a GaAs sample for which we show a sensitivity at 1.55µm with photorefractive gains, in the diffusion regime, comparable to the absorption losses. Experimental set-up and samples. Photorefractive performances of our samples are analysed in a two beam coupling experiment. The measurement of the strength of the photorefractive energy transfer gain from a pump beam toward a probe beam (from which we deduce the photorefractive gain Γ) as a function of the grating spacing allows to obtain two important parameters of the photorefractive effect : the electron-hole competition coefficient ξ 0 and the effective trap density N eff [4]. The expression of the gain with the grating spacing Λ (or grating wavenumber k=2π/Λ), when considering that the electron-hole competition factor is independent of the grating spacing, as usually admitted in semiconductors, is [5,6]:

! = 2" n 0 3 r eff # cos$ k B T e % 0 k 1 + k 2 k 0 2 with k 0 2 = e 2 ! k B T
N eff . θ is the half angle between the beams inside the crystal. n 0 the linear refractive index of the material, r eff the effective electrooptic coefficient that depends on the crystal and the beam polarization orientations [7] and ε is the dielectric constant. All these parameters are known from the literature. The measurements are performed at the three wavelengths of 1.06µm, 1.32µm and 1.55µm given by two diode pumped Nd:YAG lasers and a laser diode respectively. The photorefractive gain is verified to be saturated with respect to the illumination for all the samples and wavelengths. The energy transfer direction and then the sign of ξ 0 is measured for the different wavelengths relatively to its sign at 1.06µm. We consider that, in all the cases, we have electrons as majority carriers (ξ 0 >0) at 1.06µm as already measured on two CdTe samples in previous studies [8].

The possible existence of absorption gratings is checked by using the symmetry properties of photorefractive gain. All the measurements presented here are corrected, when necessary, from the absorption component.

We determine the exponential time constant of the build-up of the photorefractive effect in our samples by monitoring the energy transfer kinetics when the pump beam is turned on.

The CdTe samples were grown using the modified Bridgman technique. They were doped with Vanadium at concentrations between 10 20 cm -3 and 5x10 18 cm -3 . The crystals were generally cut in the classical photorefractive orientation with a beam propagation along (110) direction. However sample (B23-3) was cut with a propagation direction along (111). The absorption spectra of the different samples exhibits similar features with two absorption bands centered at 1µm and 1.5µm (Figure 1), only the absolute value of the absorption changes with the growth conditions.

The undoped GaAs crystal was grown by the Liquid Encapsulated Czochralski technique. Conversely to CdTe, the deep defect responsible for the photorefractive effect between 1 and 1.55µm is identified [9], it is the EL2 native defect, which optical characteristics are known [10]. Concentrations of the different states of charge of this defect are determined from absorption and Electron Paramagnetic Resonance (EPR) measurements. Absorption at λ=1.19µm (Figure 2) gives a total concentration of the EL2 defect [11,9] :

[EL2]=1.3x10 16 cm -3 . [EL2 + ]=6x10 15 cm -3 is given by EPR measurements [9]. Experimental results in CdTe. We study four CdTe samples (called H1, B6, B9-1, B23-3).

Typical curves of variation of the gain with the grating spacing are obtained in these samples as it is shown in Figure 3 for sample B23-3. The adjustment of the theoretical curve to the experimental points gives us the two parameters ξ 0 and N eff . For these adjustments we take for the n 0 3 r 4 1 coefficient the same value for the three wavelengths : 120pm.V -1 [1, 12]. The parameters deduced from these experiments are summed up in Table 1. All CdTe samples present high photorefractive gain at 1.06µm, at least as strong as in GaAs (as will be seen below). According to the results in Table 1, it appears that all samples present a strong electron-hole competition at 1.32µm or 1.55µm. Photorefractive gains of different signs are observed at 1.32µm. This change in majority carrier with wavelength is very different from what was previously observed in others studies [13] where the same dominant carriers (electrons) were observed at the three wavelengths. The effective trap density is of the same order of magnitude for the different samples (Table 1). The photorefractive gain we measure is always lower than the absorption coefficient, i.e. no net gain is observed in these two beam coupling experiments without applied electric field. Experimental results in GaAs. In GaAs, conversely to CdTe, the knowledge of photoionization cross-sections (S n , S p ) [10] and concentrations allows to calculate the expected photorefractive gain for each wavelength. As

! 0 = S n EL2 0 [ ] " S p EL2 + [ ] S n EL2 0 [ ] + S p EL2 + [ ]
, we calculate ξ 0 =0.49 at λ=1.06µm, ξ 0 =-0.48 at λ=1.32µm and ξ 0 =-0.55 at λ=1.55µm with an effective trap density N eff =3.2x10 15 cm -3 . The accordance between theoretical calculations and experimental results is good (Figure 4). It shows that the photorefractive model of one defect coupled with both valence and conduction bands well explains the photorefractive behavior of undoped GaAs. The photorefractive gain changes its sign between 1.06µm and 1.32µm going from electrons to holes as majority carriers. As for CdTe, the sensitivity range of GaAs extends to 1.55µm with a photorefractive gain of the order of magnitude of the absorption (Γ≈α≈ 0.1cm -1 ).

Comparison between CdTe and GaAs. If we compare the photorefractive gain obtained in

CdTe and GaAs, we see that higher gains are obtained in CdTe for each of the three wavelengths. This is mainly due to the higher value of the electrooptic coefficient of CdTe that leads to a higher n 0 3 r 4 1 (Table 2). Nevertheless only GaAs currently exhibits a net gain at 1.55µm without applied electric field. Work has to be done to decrease the background absorption of CdTe samples in order to observe this net gain.

In order to compare the different samples, we use a figure of merit which is the index change per absorbed energy per unit volume. This sensitivity is defined as S = !n F " . Δn is the index change at saturation which is deduced from the photorefractive gain Γ with expression

! = 4" # cos$
%n. F is the grating formation fluence F=I.τ(I) (τ(I) is the time constant of the photorefractive effect at the incident illumination I), α is the sample absorption. For example, for sample B9-1 at 1.06µm, we measure a time constant τ = 580µs at an illumination of 24 mW.cm -2 , which gives F = 14 µJ.cm -2 . At the same grating spacing (Λ = 1.44 µm), the gain Γ is 0.4 cm -1 for an absorption α = 6 cm -1 . We have then for sample B9-1 a sensitivity S = 0.04 cm 3 .J -1 . The same calculation conducted for sample B23-3 gives S = 0.005 cm 3 .J -1 (see Table 2).

For the GaAs sample at 1.06µm, we measure, at a grating spacing of 1.44µm, a gain Γ = 0.2 cm -1 with an absorption α = 1 cm -1 . The response time is τ = 1 ms at an illumination of 110 mW.cm -2 . We have then a sensitivity S = 0.015 cm 3 .J -1 . Figure 2 : Absorption spectrum of GaAs sample D2. The sligthly negative value is due to a small overestimation of the refractive index [9], in all the paper we consider the crystal absorption to be zero at λ=2µm.

Figure 1 :

 1 Figure 1 : Absorption spectrum of CdTe sample B6.

Figure 3 :

 3 Figure 3 : Photorefractive gain as a function of the grating spacing at wavelegths λ=1.06µm and 1.55µm for CdTe sample B23-3. Lines are the theoretical adjustments. The photorefractive gain 1.32µm is nearly zero.

Figure 4 :

 4 Figure 4 : Photorefractive gain as a function of the grating spacing at λ=1.06µm, 1.32µm and 1.55µm for the GaAs sample D2. Lines represent the calculated photorefractive gain.

Figure 1 Figure 2 Figure 3

 123 Figure 1

Figure 4

 4 Figure 4

Table 1 :

 1 Photorefractive parameters of CdTe crystals at different wavelengths

		CdTe		GaAs
		B9-1	B23-3
	n 0 3 r 4 1 (pm.V -1 )	120		72
	F (µJ.cm -2 )	14	214	110
	α (cm --1 )	6	2.9	1
	Δn (at Λ≈1.5µm)	3.4x10 -6	2.9x10 -6	1.7x10 -6
	S (cm 3 .J -1 )	0.04	0.005	0.015

Table 2 :

 2 Sensitivity of CdTe samples and GaAs at 1.06µm

We see on these results that, regarding the sensitivity, CdTe and GaAs are comparable. The sensitivity of CdTe is very dependant on the samples with a variation of a factor of 10 between samples B9-1 and B23-3.

Conclusion.

We have presented a photorefractive study of two of the most important photorefractive materials for the wavelength range between 1 and 1.55µm : GaAs and CdTe. Both of these crystals present an electron-hole competition that can lead to an annulation of the gain at 1.32µm or 1.55µm, depending on the sample. If CdTe presents the highest gain due to its higher electrooptic coefficient, only GaAs presents net gain at 1.55µm due to its smaller absorption. The sensitivity of these two crystals are similar with a strong variation from sample to sample for CdTe.