S De Rossi

Ph Delaye

J C Launay

G Roosen

IMPLEMENTATION AND COMPARATIVE EVALUATION OF VARIOUS ARCHITECTURES OF ULTRASONIC PHOTOREFRACTIVE SENSORS

Keywords: Photorefractive effect, ultrasonic detection, screening field. 1. Presentation of various architectures

A novel optical sensor for the detection of ultrasonic motion has been recently developed. It allows to reach 25 pm displacement sensitivity with a large frequency range bandwidth 100 MHz. We will present a comparative study between various architectures of this ultrasonic detection system we have implemented. It uses InP:Fe or CdZnTe:V holographic crystals and operates with CW laser at 1.06 µm. Two configurations called the direct detection and the anisotropic diffraction configurations work with either plane or speckled waves. A third discussed configuration works with depolarized speckled waves. We measured their relative detection limits as a function of the applied electric field that governs photorefractive efficiency of the materials. Experimental results are well described by theory, using photorefractive models relevant to the used crystals.

Phase demodulation is based on a two wave mixing (TWM) mechanism. A rapidly and weakly phase modulated signal beam and a pump beam interfere in a photorefractive crystal. As the period of the phase modulation is much smaller than the response time of the crystal, both waves write an average hologram. The pump beam diffracts on it to give an unmodulated local oscillator that has the same direction and the same wavefront as the transmitted signal beam. These two beams interfere on a photodiode what allows the phase demodulation. An external applied electric field allows to control the amplitude and the phase of the index grating compared to the illumination grating given by the complex nature of the photorefractive gain in amplitude γ=γ'+iγ'', where γ' is the real part and γ'' the imaginary part of γ. In all the experiments, the pump to signal beam intensity ratio is much greater than one, what allows to be in the undepleted pump approximation. With polarized beams, two configurations are possible, the direct detection and the anisotropic diffraction ones (fig. 1). In the direct detection [1,2] both waves are polarized along the Pockels eigenaxes of the crystal (here at ±45°). Local oscillator and signal beam have the same polarization. We detect directly the demodulation with a photodiode. Linear detection is possible only if the imaginary part γ'' of photorefractive gain γ is not null, what is obtained with an external field. In the anisotropic diffraction [1] both waves are polarized along (001). The local oscillator is horizontally polarized. If we insert a 45° polarizer after the crystal, each detector receives the eigenwaves. If we make the difference between the signal of each detector, we have here also a linear detection for γ"≠0.

A third solution consists in working with depolarized beams [3]. Both waves record two decorrelated holograms of 90° polarized images along the eigenaxes (here at ±45°). We analyze them independently with a polarizer beam splitter oriented along the eigenaxes. Each wave has a local oscillator and a transmitted signal in phase quadrature thanks to the external field. Contrary to the anistropic diffraction the polarized beams along the different eigenaxes have not the same wavefront and can not interfere together. We thus have two decorrelated wave mixing process in a direct detection configuration with a gain reduced by a factor 2. By making the difference of the two photodiode signals, we obtain again a linear detection for γ"≠0.

To compare our results we use the relative detection limit δ rel . It is the ratio between the smallest displacement amplitude δ measured by our sensor (signal to noise ratio of one with an 1W power polarized wave incident on the crystal and a bandwidth of 1Hz) and the optimal detection limit of an optical technique δ opt :

! opt = " 4#
h$ 2% λ is the wavelength of the laser, hν its energy and η the quantum efficiency of the detector. For our three architectures we obtain :

! r e l d i r = e "x 2 sin( # # $ x) ; ! r e l ani = e "x 2 1+ # 2 x 2 $ $ # x ; ! r e l dep = e "x 2 2 1+ th 2 ( # $ x 2 ) sin( # # $ x 2 )
Where x is the thickness of the crystal along the wave propagation direction, α the absorption.

Photorefractive crystals and measurement of the screening factor

The photorefractive effect in semiconductors as CdZnTe:V and InP:Fe is now well-known. For the CdZnTe we use one trap model with an electron-hole competition [4]. For InP the photoconduction is governed by holes with an electron-hole competition thermally induced from the excited state of Fe 2+ [5]. From photorefractive gain measurements as a function of the grating spacing in diffusion regime, we know all the essential parameters for InP and CdTe photorefractive models. Nevertheless one problem subsists because of inhomogeneities of the crystal and nonuniformity of the pump beam illumination. The applied electric field is screened and the real field E o seen by the signal beam is not known. We define the screening factor c 1 as :

E o = E app .c 1 Where Eapp=V o /d with V o the applied voltage on crystal and d the inter-electrode spacing.

Determination of c 1 is obtained by a Pockels effect measurement (see fig. 2). We measure the signal beam transmission between crossed polarizers as a function of the applied field. The experiments have been performed with the same configuration of illumination than that of the ultrasonic detection system. To avoid having a parasite grating, pump and signal polarizations are crossed. We show easily that in our configuration, the transmitted signal I t is :

I t = I o sin 2 ( !.x " n o 3 . r 41 .E app .c 1 )
An example of transmission curve is presented in fig. 2.

Experimental setup for detection sensitivity measurement

Beams are obtained from a 600 mW CW single mode diode pumped Nd:YAG laser emitting at 1.06 µm. The signal beam is phase modulated with a Pockels cell. The phase modulation is sinusoidal with amplitude of 20 mrad and frequency 80 kHz. A multimode fiber module is inserted on each arm to have speckle depolarized beams. Furthermore the pump beam is expanded to cover fully the crystal to minimize the screening of the field and allow a good covering of the signal beam. Illumination is about 150 mW/cm 2 . A polarization module polarizes again the beams after the fiber for the direct detection or the anisotropic diffraction configurations. The typical grating spacing is between 3 and 4.5 µm. The external field is applied to silver painted electrodes during 50 ms. Detectors are InGaAs photodiodes with quantum efficiency η=0.76.

Detection sensitivity : experimental results

Measurement presented here (see fig. 3) are obtained from a speckled signal beam with similar results obtained from plane signal beam.

For the InP crystal, the screening factor is 0.64 with a plane pump beam. The agreement with the theory is good in the direct detection and anistropic diffraction. We have not been able to work with a speckle pump beam. This is explained as follow. The dark conductivity of InP is greater than the photoconductivity in the some place in the crystal because of the granular nature of the pump beam and the gain hardly saturates regarding the incident intensity. This problem disappears in the pulsed regime.

For CdZnTe we can work with a speckled pump beam without any problem. The curves have been realized with a speckled pump and signal beams. Results are identical for both planes waves or one plane wave and the other speckled. The screening factor is 0.6. We obtain a good agreement with theory if we add a supplementary screening factor c 2 =0.5. This complementary screening factor c 2 is identical whatever the architecture and the wavefront structure of the beams (plane or speckle wave). It has been observed in other tested CdZnTe crystals. Work is currently in progress to identify the origin of this supplementary screening factor.

Conclusion

The study of various architectures of an ultrasonic sensor allows us to valid theoretical TWM model in presence of a phase modulated signal in photorefractive crystals as InP:Fe and CdZnTe:V. We can now predict the detection sensitivity of each configuration and choose the optimal one. Problems have been emphasized. For CdZnTe the results are less good than expected due to the presence of a supplementary screening factor. We have to understand this drop in performances to operate the sensor at best with CdZnTe.

Fig. 1 :

 1 Fig. 1 : Implemented architectures, (A) direct detection, (B) anisotropic diffraction; (C) depolarized configuration. Both beams enter by the (110) faces and the grating vector is along (1 10 ) .

Fig. 2 :

 2 Experimental setup and example of curve for screening measurement.

Fig. 3 :

 3 Fig. 3 : Evolution of the relative detection limits as a function of the applied field (markers : experimental measures / lines : theoretical curves).


Philippe Delaye
email: philippe.delaye@iota.u-psud.fr

Gérald Roosen

EVALUATION OF A PHOTOREFRACTIVE TWO BEAM COUPLING NOVELTY FILTER

Theoretical and experimental study of a novelty filter is presented. We show that best performances are obtained when the images are presented to the set-up with a spatial light modulator operating as a linear phase modulator. We obtained a good accordance between the experimental curves and the calculated ones, what validates the model we develop for this novelty filter.

A novelty filter is an image pre-processing element that is sensitive only to the new features in a scene. It is thus perfectly adapted to the detection of an objet appearing and moving in a complex scene. Different architectures were proposed to implement a novelty filter with a photorefractive crystal [START_REF] Cronin-Golomb | Photorefractive time differentiation of coherent optical images[END_REF][START_REF] Anderson | Optical novelty filters[END_REF][START_REF] Sedlatschek | Demonstrator concepts and performances of a photorefractive optical novelty filter[END_REF][START_REF] Mathey | Performance evaluation of a photorefractive novelty filter for motion tracking and edge enhancement[END_REF]. They use phase conjugation [START_REF] Anderson | Optical novelty filters[END_REF], ring geometries [START_REF] Anderson | Optical novelty filters[END_REF], two beam coupling energy transfer [START_REF] Cronin-Golomb | Photorefractive time differentiation of coherent optical images[END_REF][START_REF] Anderson | Optical novelty filters[END_REF] or even beam fanning [START_REF] Cronin-Golomb | Photorefractive time differentiation of coherent optical images[END_REF]. Among these different setups the last two are very simple to implement. Nevertheless, the beam fanning novelty filter is not adapted to our problem as it is not a true differentiating set-up [START_REF] Cronin-Golomb | Photorefractive time differentiation of coherent optical images[END_REF] (for example it would not be sensitive to a uniform phase change of the image). That's why we choose the two beam coupling novelty filter. A signal beam carrying the image (i.e. the complex scene) interferes with a pump beam in a high gain photorefractive crystal. Energy transfer occurs through two beam coupling that causes a strong attenuation of the stationary signal beam. As soon as an objet appears or moves in the scene the attenuation condition is no more fulfilled and light passes immediately at the location of the object, and only there, until a new grating is written what restores attenuation. Such a system allows thus to detect very small moving objects, with very small contrasts compared to the textured background.

The aim of this paper is to describe both theoretically and experimentally the operation of a two beam coupling novelty filter, with the objective to detect the smallest possible grey level variation (∆n=1) on an arbitrary background. In a first part we will model this device, showing that the best operating conditions are obtained with an image coded on the beam as a phase modulation. In the second part we will describe the experimental set-up and its characterization. Last part will compare experimental results with theory showing an excellent agreement that enables to predict the performances of future systems using optimized components.

I. THEORETICAL BACKGROUND

The principle of the two beam coupling novelty filter is based on a classical two beam coupling set-up (Fig. 1). A signal beam E d carrying the information of an image imprinted on it is sent on a photorefractive crystal, with a coherent pump beam E i . Both beams interfere and write through the photorefractive effect an hologram of the signal beam wavefront. The crystal is used in such a way that the signal beam transfers its energy toward the pump beam leading to a strong depletion of it. This attenuation is due to a quasi perfect destructive interference between the transmitted signal beam and the diffracted pump beam [START_REF] Cronin-Golomb | Photorefractive time differentiation of coherent optical images[END_REF][START_REF] Anderson | Optical novelty filters[END_REF]. At a given time, the imprinted image is locally changed much rapidly than the response time of the photorefractive crystal. The transmitted image accordingly changes its wavefront whereas the hologram and thus the diffracted pump beam does not have the time to adapt to this change. The condition of destructive interference is no more fulfilled in all the parts of the image that have changed. There is no more attenuation in these locations that reappears suddenly as bright spots in the attenuated image field. After some time, the crystal adapts itself to the new wavefront by writing a new hologram leading again to the attenuation of the whole signal beam.

If we look at this description more closely, we see that we have to consider the transmission of a time modulated signal beam in a two beam coupling set-up, with a pump beam that is stationary and not depleted (it just acquires a small quantity of energy from the much weaker signal beam through beam coupling process). We can thus apply the theoretical model we previously developed [START_REF] Ph | Transmission of time modulated optical signals through an absorbing photorefractive crystal[END_REF] and we already successfully used for the description of the photorefractive two beam coupling ultrasonic detector [START_REF] Ph | Detection of ultrasonic motion of a scattering surface using photorefractive InP:Fe under applied DC field[END_REF][START_REF] De Montmorillon | Novel theoretical aspects on photorefractive ultrasonic detection and implementation of a sensor with an optimum sensitivity[END_REF]. This model is derived from the one used in Ref. [START_REF] Cronin-Golomb | Photorefractive time differentiation of coherent optical images[END_REF] (it takes into account the absorption of the photorefractive crystal that causes changes of the photorefractive time constant during the propagation through the crystal). It is also equivalent to the Laplace transform formalism used in Ref. [START_REF] Anderson | Optical novelty filters[END_REF], except in the final evaluation of the inverse Laplace transform, for which we prefer the convolution calculation, we estimated more adapted to our problem and easier to use.

The time modulated signal beam E d (0,t) is sent on a photorefractive crystal having a photorefractive gain in amplitude γ, an absorption coefficient α and a thickness x. The photorefractive time constant at the entrance of the crystal is τ 0 . In the condition in which the crystal is used, i.e. a photorefractive crystal in the diffusion regime, gain γ is real and negative (to account for an energy transfer from the signal towards the pump) and the photorefractive time constant τ 0 is real. Finally we suppose that a photorefractive grating is written in its steady state regime when the change of the image occurs at time t=0.

Taking into account all these considerations, the transmission of the time modulated signal beam through the crystal is governed by the following equation [START_REF] Ph | Transmission of time modulated optical signals through an absorbing photorefractive crystal[END_REF]:

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 (1)
with H x, t

( ) = e ! x " 0 e # t " 0 1 F 1 ! $ ,1, e $ x # 1 e $ x % & ' ( ) t " 0 % & ' ( ) * and 1 F 1 a, b, z ( ) the confluent hypergeometric function [8].
This expression is complex but perfectly usable as the hypergeometric function is known and tabulated. Nevertheless, the fact that the expression uses the derivative of E d (0,t) might bring a difficulty in the interpretation. This is why, using an integration by parts and

defining function G x, t ( ) = e ! x H x, t ( ) + " 0 #H #t x, t ( ) $ % & '
, we rewrite equation (1) [START_REF] Hermanns | The transfer function and impulse response of photorefractive two-beam coupling[END_REF][START_REF] De Montmorillon | Novel theoretical aspects on photorefractive ultrasonic detection and implementation of a sensor with an optimum sensitivity[END_REF] taking into account the properties of hypergeometric function [START_REF] Abramowitz | Handbook of mathematical functions[END_REF] : ) with G x, t

E d x, t ( ) = e ! " x 2 E d 0, t ( ) ! E d 0, 0 ( ) ( ) + E d 0, 0 ( )e # x + E d 0, T ( ) ! E d 0, 0 ( ) ( )G x, t ! T ( )dT 0 t $ [ ] (2
( ) = e !x " 1 # 0 e $ % & ' ( t # 0 $ % & ' ( ) .
Thus using equation ( 2) and knowing the expression of E d (0,t) at the entrance of the crystal, we now deduce its expression at the output.

Signal E d (0,t) presented at the entrance can be split into two parts : a constant part E d0 (0,t)=E d (0,0) and a varying part ∆E d (0,t) that is different from zero only for t>0. Expression (2) being linear, the response of the system is equal to :

E d x, t ( ) = E d 0 x, t ( ) + !E d x, t ( ) (3)
The response of each part being given by equation [START_REF] Anderson | Optical novelty filters[END_REF].

We thus have :

E d 0 x, t ( ) = E d 0, 0 ( )e ! " x 2 e # x = E 0 e ! " x 2 e # x (4)
and :

!E d x, t ( ) = e " #x 2 !E d 0, t ( ) + !E d 0, T ( )G x, t " T ( )dT 0 t $ [ ] (5)
Which gives :

E d x, t ( ) = e ! " x 2 E 0 e # x + $E d 0, t ( ) + $E d 0, T ( )G x, t ! T ( )dT 0 t % [ ] (6)
The response will thus depend on the temporal variation of the input signal. To modelize the working of the novelty filter, we consider a perfect impulse variation of the amplitude of the entrance signal beam having a duration ∆t and amplitude ∆E, given by : !E d 0, t

( ) = 0 !E d 0, t ( ) = !E !E d 0, t ( ) = 0 " # $ % $ t & 0 0 < t & !t t > !t (7)
that we can rewrite, using the Heavyside function U(t) (U(t)=1 for t > 0, U(t)=0 elsewhere) :

!E d 0, t ( ) = !E U t ( ) " !E U t " !t ( ) = !E U t ( ) " U t " !t ( ) ( ) (8)
We introduce now ∆E d (0,t) in equation ( 5), what gives : ) Taking into account the properties of the Heavyside function, both integrals can be rewritten :

!E d x, t ( ) = !E e " # x 2 U t ( ) " U t " !t ( ) + U t " T ( )G x, T ( )dT + U t " T " !t ( )G x, T ( )dT 0 t $ 0 t $ [ ] (9
!E d x, t ( ) = !E e " # x 2 U t ( ) " U t " !t ( ) + G x, T ( )dT + G x, T ( )dT 0 t " !t $ 0 t $ % & ' ( (10)
What gives in final :

!E d x, t ( ) = 0 t " 0 !E d x, t ( ) = !E e # $ x 2 1 + G x, T ( )dT 0 t % [ ] 0 < t " !t !E d x, t ( ) = !E e # $ x 2 G x, T ( )dT t > !t t # !t t % & ' ( ( ) ( ( (11)
We define a function R t,!t

( ) = G x, T ( )dT t " !t t # what allows to write R t,t ( ) = G x, T ( )dT 0 t ! .
We can now come back to the total amplitude of the signal E d (x,t) :

E d x, t ( ) = E 0 e ! " x 2 e # x t $ 0 E d x, t ( ) = E 0 e ! " x 2 e # x + %E e ! " x 2 1 + R t,t ( ) [ ] 0 < t $ %t E d x, t ( ) = E 0 e ! " x 2 e # x + %E e ! " x 2 R t, %t ( ) t > %t & ' ( ( ) ( ( (12)
The response of the novelty filter will thus depend on the parameters of the crystal, and on the characteristics of the amplitude variation.

The normal application of the novelty filter is the detection of very rapid fluctuations of the signal beam, compared to the response time of the photorefractive crystal. In order to consider the performances of the novelty filter, we will thus be in these normal conditions of operation of the novelty filter. We consequently suppose that ∆t << τ 0 (and thus that we make a measurement on a time period much smaller than the response time of the crystal t << τ 0 ). This means that the function G(x,t) that appears in the definition of the functions R(t,∆t) does not vary and stays equal to its value at time t=0 :

G x, t ( ) ! G x, 0 ( ) = e " x # 1 $ 0 e " x % & ' ( ) * + " (13)
as

1 F 1 ! + " " , 2, 0 # $ % & = 1
It means that we have :

R t,t ( ) = G x, T ( )dT 0 t ! " G x, 0 ( )dT 0 t ! = e # x $ 1 e # x % & ' ( ) * # t + 0 " 0 (14)
since t << τ 0 .

In the same manner, we have R t,!t

( ) " 0

This means that expression [START_REF] Soutar | Measurement of the complex transmittance of the Epson liquid crystal television[END_REF] becomes :

E d x, t ( ) = E 0 e ! " x 2 e # x t $ 0 E d x, t ( ) = E 0 e ! " x 2 e # x + %E e ! " x 2 0 < t $ %t E d x, t ( ) = E 0 e ! " x 2 e # x t > %t & ' ( ( ) ( ( (15)
We see on this expression the response of the novelty filter to a variation ∆E of the amplitude of the signal beam. In the normal operation of the filter we choose the photorefractive gain such that we have ∆E>>E 0 e γx . In this condition the output exactly reproduces the signal variation (except for the reduction due to the absorption of the crystal), and we have :

E d x, t ( ) ! 0 t " 0 E d x, t ( ) = #E e $ %x 2 0 < t " #t E d x, t ( ) ! 0 t > #t & ' ( ( ) ( ( (16)
This result is obtained in perfect conditions of a photorefractive crystal having a very strong photorefractive gain γ and a very slow response time compared to the duration of the novelty. The reality is somewhat different. To analyse the effect of the material response time, we have to go back to equation [START_REF] Soutar | Measurement of the complex transmittance of the Epson liquid crystal television[END_REF] to know the temporal response of the novelty filter and the rate at which the filter will readapt to the signal change, when this change is not short compared to the response time of the photorefractive crystal. For the effect of a limited value of the photorefractive gain on the response and its choice in order to have an attenuation sufficient to obtain the desired response, the analysis is less direct. In fact looking further to what happens in the novelty filter, we will have to take into account the exact nature of the amplitude change ∆E. Indeed the image can be imprinted on the signal beam in two different ways : first it can be imprinted as an intensity modulation where the image levels correspond to different intensity levels of the beam, second it can be coded as a phase modulation where the image levels correspond to local changes of the phase of the beam. Performances of the novelty filter will be analyzed in both cases.

I.1. Intensity modulation case

The image is coded as an intensity variation of the beam. Thus a novelty will be seen as a local variation of the intensity of the beam from an intensity level I 0 towards a new level I 0 +∆I. Then according to the previous notations we have :

!E = I 0 + !I " I 0 (17)
In order to have an optimum response of the novelty filter, we need to have (equation (15)) :

E 0 e ! x << "E (18)
and thus :

e ! x << I 0 + "I I 0 # 1 = 1+ "I I 0 # 1 (19)
we see that the requirement on the photorefractive gain γ depends on the initial intensity I 0 .

We thus have to take into account the coding of the image. We here take the example of an image linearly coded with N grey levels. Thus I 0 = n I G , with 0≤n<N, and I G the intensity variation corresponding to a variation of one grey level. Taking into account inequality (19), the most difficult signal to measure will thus correspond to a measurement of a variation of one grey level (smallest ∆I, i.e. ∆I = I G ), on a high level (i.e. I 0 ≈ N I G ). Inequality (19) then becomes :

e ! x << 1 + 1 N " 1 (20)
Practically, there is a large number of grey level such as N>>1 and then :

e ! x << 1 2 N (21)
To give an idea of the constraint on the photorefractive gain we take the example of an image coded on 256 level (8 bit images), then a gain γ such that e ! x " 1 5000 should be sufficient to verify relation ( 21). This corresponds to an attenuation (given by e ! 2 " x ) of the initial signal beam of the order of 2.5x10 7 . In order to have such a high exponential gain with a 0.5cm thick crystal, an intensity photorefractive gain Γ=2γ of the order of 35cm -1 is needed.

This is high but accessible with a baryum titanate crystal [START_REF] Mager | High quality selfpumped phase conjugation of nanosecond pulses at 532nm using photorefractive BaTiO 3[END_REF]. Nevertheless such an attenuation would be very difficult to obtain experimentally because of the imperfections of the set-up (residual phase modulation due to vibrations, scattering on the crystal defects, ...). Considering this optimum conditions fulfilled, the response of the novelty filter is given by equation ( 16). Then the intensity change at the output in response to the entrance intensity change is equal to :

!E 2 = I 0 + !I " I 0 2 = 2I 0 + !I " 2 I 0 I 0 + !I ( ) (22)
In the normal utilization case, we need to measure a small intensity variation on a moderate background, we thus have ∆I << I 0 . Then the response becomes :

!E 2 = !I 2 4I 0 ( 23
)
This response is not at all linear and it depends both on the incident intensity variation ∆I and on the initial level I 0 on which the variation is applied. Detection of variation of 1 grey level will be easier on a black level than on a white one as the response depends on the initial level I 0 .

To characterize the response of the novelty filter we define the novelty factor (sometimes called with the ambiguous term contrast [START_REF] Anderson | Optical novelty filters[END_REF]) as the ratio of the intensity of the transmitted signal beam during the novelty to the intensity of the transmitted signal just before the change :

C = I d x, 0 < t ! "t ( ) I d x, t ! 0 ( ) = "E 2 I 0 e 2# x = e $ 2 # x "E I 0 2 = e $ 2# x I 0 + "I I 0 $ 1 2 (24)
As ∆I << I 0 , this novelty factor becomes

C = e ! 2" x #I 2I 0 $ % & ' ( ) 2 .
Without the novelty filter the same ratio can be defined. Its value is then :

C' = I 0 + !I I 0 (25)
which gives, as ∆I << I 0 , C' = 1.

We see that the novelty filter brings a gain in the novelty factor of the order of the attenuation rate e ! 2 " x in the measurement of the intensity variation of the signal beam. Such a gain will be very strong if we can obtain the expected attenuation of 2.5x10 7 required for an optimum operation of the novelty filter with an intensity modulated beam.

I.2. Phase modulation case

We have just seen that with an intensity modulated signal beam, the response of the novelty filter depends on the initial level from which the variation is made. This important drawback does not exist in the case of a phase modulated beam. Indeed the grating written by the interference of the pump beam and the signal beam, carries the information on the spatial variation of the phase ϕ 0 of the signal beam. Diffraction on the grating will give a wave that will carry this phase information, to which will be added a constant phase shift of π (independantly of the initial phase ϕ 0 ) that will be responsible of the attenuation of the signal beam. The local value of the phase ϕ 0 can thus vary without any influence on the transmitted beam. The novelty filter only uses the fact that the diffracted beam is π phase shifted compared to the transmitted beam and that whatever the initial phase of the transmitted beam. The system reacts to a phase variation ∆ϕ from an initial phase ϕ 0 . As the transmitted beam as no reference concerning the exact value of the phase ϕ 0 , the system can not distinguish a phase variation from a level ϕ 0 , from that from a level ϕ' 0 . We will thus measure in the same manner a phase variation on a "black" level or on a "white" level, what is very interesting for an actual application of the novelty filter.

Going back to the previous expressions, the initial amplitude is E 0 = I 0 e i! 0 , whereas the amplitude after the change is

E 1 = I 0 e i! 1 = I 0 e i ! 0 + "! (
) . The amplitude variation is then

: !E = I 0 e i" 0 e i!" # 1 ( ) (26)
Then between 0 and ∆t, we have (equation (15)) :

E d x, t ( ) = I 0 e i! 0 e " # x 2 e $ x + e i%! " 1 ( ) (27)
which gives an intensity :

I d x, t ( ) = I 0 e ! " x e 2 # x ! 4 e # x sin 2 $% 2 + 4sin 2 $% 2 & ' ( ) (28)
The photorefractive gain γ has to be chosen such that the first two terms of expression (28) are negligible compared to the last one. This gives two conditions :

e 2 ! x << 4sin 2 "# 2 4 e ! x sin 2 "# 2 << 4sin 2 "# 2 $ % & ' & ( e 2 ! x << 4sin 2 "# 2 e ! x <<1 $ % ' (29)
The second condition is generally automatically verified when the first one is. In the case of the detection of small phase shift variations (∆ϕ<<π), this first condition becomes :

e 2 ! x << "# 2 ( 30
)
where ∆ϕ is the smallest phase shift variation we want to measure. For a modulator giving a linear phase modulation with a maximum value of kπ radians with N grey levels, we have ∆ϕ=kπ/N, and the condition becomes :

e 2 ! x << k" N # $ % & 2 (31)
for a maximum induced phase shift of π (k=1) and N=256 grey levels, this gives :

e 2 ! x << 1 6600 (32)
Thus an attenuation of the order of 6x10 4 should be sufficient to have an optimum response of the novelty filter. This is almost three order of magnitude smaller than in the case of the intensity modulation. This difference is due to the fact that here the condition is directly on the attenuation e 2 ! x rather than on its square root as in the case of the intensity modulation. This attenuation corresponds to a gain of 22cm -1 for a 0.5cm thick crystal, which is readily available experimentally with baryum titanate crystals [START_REF] Mager | High quality selfpumped phase conjugation of nanosecond pulses at 532nm using photorefractive BaTiO 3[END_REF].

Considering now that we have this optimum value of the gain, the intensity variation giving the response of the system is given by the last term of equation ( 28) and equals :

!E 2 = 4I 0 sin 2 !" 2 (33)
This response depends linearly with I 0 as the signal received by the detector is proportional to the constant power sent to the crystal. The independence compared to the initial level is here given by the independence of equation ( 33) with respect to the initial phase ϕ 0 . In the case where !" 2 << # 2 , we have :

!E 2 = I 0 !" 2 (34)
From this we obtain the novelty factor C which value is :

C = I d x, 0 < t ! "t ( ) I d x, t ! 0 ( ) = "E 2 I 0 e 2# x = 4e $ 2# x sin 2 "% 2 & e $ 2# x "% 2 (35)
If we compare this response to the one obtained in the case of intensity modulation, we still have a quadratic response of the novelty filter but now the response is completely independent of the initial state of the image. Moreover this optimum operation of the novelty filter is obtained with a less tough, and easier to obtain, condition on the required attenuation. Consequently a novelty filter which aim is to detect small changes in a complexe image, whatever its exact structure, will be much better and easier to implement if the image is imprinted on the beam as a phase modulation. I.3. Problems linked to the use of a phase modulated beam As we have seen, taking into account the response of the novelty filter, it is extremely advantageous to work with images coded as a phase modulation. This response will be all the more efficient as the phase variation induced by a variation of one grey level will be important. We thus see that it can be advantageous to boost the response of the Spatial Light Modulator (SLM) to have it giving a maximum phase equal to kπ with k greater than one. Nevertheless such a modulation range would bring some drawbacks, essentially due to the sinusoidal response of the novelty filter. In such a case (taking k>1) the system is very sensitive to small grey level variations but can be completely insensitive to very high variations of grey level (for example if we have a variation such that ∆ϕ=2π we then have ∆E=0). This point has to be taken into account in the design of the novelty filter, i.e. the choice of the SLM will depend on the final use of the novelty filter.

This problem of the cancellation of the response at high novelty phase shifts (around Δϕ≈2π) has nevertheless to be moderated. Indeed when establishing the expressions for the response of the novelty filter, we supposed that the phase change was instantaneous (illustrated by the use of a heavyside function). In reality this change will not be as sudden. Indeed the SLM needs a certain time to commute from one value ϕ 0 to another ϕ 1 . During this time all the phase values will be seen, giving a non zero response of the novelty filter. If we come back to the case of a phase variation between 0 and 2π, we will thus have a transient response that gives a signal even if the steady state response is zero. To simulate this effect we can not use the previous expressions [START_REF] Soutar | Measurement of the complex transmittance of the Epson liquid crystal television[END_REF] as they correspond to a sudden phase shift change. We thus have to come back to the initial expression (2) using E d 0, t ( ) = E 0 e i ! t ( ) with ϕ(t) reproducing the exact response of the SLM. In our case we use an exponential response of the SLM with a time constant τ R , which gives for the phase modulation :

! 0, t ( ) = ! 0 ! 0, t ( ) = ! 1 " ! 0 ( ) 1" e " t # R $ % & ' ( ) + ! 0 ! 0, t ( ) = ! 1 " ! 0 ( ) e " t "* t ( ) # R $ % & ' ( ) + ! 0 + , - - - . - - - t / 0 0 < t / *t *t < t (36)
The result of such a calculation is showed in Fig. 2 (only the first transition is shown, an equivalent response would be given for the second one). We see that if the phase transition is important we observe a transient response which width is characteristics of the response time of the SLM, but which height is rather constant. We thus have the possibility to detect a signal even if the phase modulation is equal to 2π. The exact interpretation of the response can be complex but because of the non instantaneous response time of the SLM we have the possibility to enhance the response of the novelty filter to small variation of the grey level, without missing any higher variation.

The finite response time of the SLM has another effect that is seen in Fig. 2.A, giving the amplitude of the response at small phase variations. The height of the peak will be all the more high as the time constant of the SLM will be short compared to the response time of the novelty filter. This comes from a partial adaptation of the hologram during the phase change when the phase is changed "slowly", giving a reduced response of the novelty filter.

With this model we can simulate the response of the novelty filter. For this we just have to introduce the exact expression of the novelty at the entrance of the crystal, in equation ( 2) and calculate the response of the novelty filter. Peculiar characteristics of the SLM (noise, mixed amplitude-phase response or response time of the SLM) can be also introduced and taken into account to describe the response of the system.

I.4. Discussion

We have shown in this theoretical part that it is very advantageous for the operation of a photorefractive novelty filter to imprint the images in the set-up as a phase modulation of the signal beam. Indeed, in such a case, we have a response to the grey level variation that is completely independent of the initial grey level. It is thus as easy to detect a one grey level change on a "black" image as on a "white" level. Moreover, the experimental conditions necessary to obtain an "ideal" condition of use of the novelty filter are less stringent than for an intensity modulated image. A less strong attenuation is required and then a smaller photorefractive gain is necessary for the crystal.

Before presenting the comparison of the experimental results with the calculated curves obtained from the theoretical model, we will describe the experimental set-up of the two beam coupling photorefractive novelty filter we implemented.

II. EXPERIMENTAL IMPLEMENTATION OF THE PHOTOREFRACTIVE NOVELTY FILTER

II.1. Experimental set-up

The basis of the experimental set-up is a two beam coupling experiment (Fig. 3). The laser source is a single mode Argon laser emitting at λ=514nm. The beam is split in two beams by a polarizing beam splitter, the respective power between the two beams being adjusted by a λ/2 plate. The signal beam is filtered and expanded to illuminate uniformly the spatial light modulator (SLM). Two λ/2 plate before and after the SLM adjust the polarization and a dichroic polarizer just before the crystal is used to accurately select an extraordinary polarization. An afocal system made with achromats makes the image of the SLM in the vicinity of the photorefractive crystal, image that is again taken by another afocal to form a final image on a CCD camera. A pinhole in the Fourier plane of the second afocal allows to transmit only the zeroth order of the diffraction pattern due to the pixelated structure of the SLM. This pinhole also eliminates some of the scattered light issued from the pump beam that can reach the CCD camera. When temporal response is to be measured, the CCD camera is replaced by a Si detector. The pump beam is also extraordinary polarized and expanded to uniformly cover the image of the SLM, which was about 3 x 3 mm 2 . Most of the energy of the laser was sent in the pump beam which power at the level of the photorefractive crystal was typically around 3W.cm -2 . The power of the signal beam was small enough to have a pump to signal power ratio at the entrance of the crystal greater than 10. This assure that we really are in the conditions of undepleted pump approximation in which the theoretical model was established. Due to the large sensitivity of the CCD camera this condition was easy to satisfy.

II.2. The photorefractive crystal

The photorefractive crystal is an antireflection-coated iron doped BaTiO 3 crystal in a classical geometry, i.e. with the C-axis perpendicular to the entrance face. The external angle between the beams is of the order of 20° and the crystal is tilted in order to take advantage of the high r 42 electrooptic coefficient (the tilt is such that the pump beam is incident quasi normally on the crystal). For each experiments the crystal is fully characterized with respect to the photorefractive parameters. A typical set of the experimental parameters is presented in Table 1. They are the ones used in the theoretical calculations which also corresponds to the experiments presented here.

The photorefractive gain is evaluated through a measurement of the attenuation of the signal beam. For the determination of the response time of the photorefractive crystal, we write a grating that gives an attenuation of the signal beam. Then the signal beam is turned off for a short period of time and the response is recorded on a Silicon detector positioned in place of the CCD camera (Fig. 3). This signal corresponds to the response of the beam coupling set-up to an intensity novelty. We thus use the theoretical model (eq. ( 12)) to adjust the experimental results. The only unknown parameter being the response time we use this adjustment to determine this response time. We measure τ 0 =750ms, with an excellent accordance between the experimental and the theoretical curves (Fig. 4).

The photorefractive gain -interaction length product (γ x = 2.83) of our crystal is clearly not sufficient to obtain an optimized implementation of the photorefractive novelty filter. Much higher gain could have been obtained with higher tilting angle of the crystal, but we have another constraint on the choice of the angle, which is the obligation to transmit the whole image of the SLM without diaphragmation and to cover this image with the pump beam. This imposes moderate incidence and tilting angles and thus a moderate photorefractive gain, nevertheless the gain is sufficient for a good operation of the novelty filter. As we will show, the operation is in fact not limited by this rather moderate gain but by the flicker of the SLM that reduces the apparent attenuation of the signal beam and thus the performances of the novelty filter.

II.3. The spatial light modulator

The SLM is issued from an Epson video projector. It is used as a phase modulator and was characterized in this regime. The method and results of this characterization of the SLM are presented elsewhere [START_REF] Ph | Simple technique for the determination of the complex transmittance of spatial light modulator[END_REF] but the important point is that quasi pure linear phase modulation has been obtained with a very small known residual intensity modulation. Two regimes of operation of the SLM are used (Fig. 5). One where a linear phase modulation between 0 and 1.3π is obtained, it will be called in the following the "0" contrast regime (by reference to the adjustment of the SLM electronics). The second regime gives a quasi linear phase modulation between 0 and 2.5π, this will be the "10" contrast regime. Curves in Fig. 5 give a perfect knowledge of the value of the amplitude (in phase and intensity) of the signal beam for any value of the grey level. These values will be used for the calculation of the theoretical curves. This SLM presents a flicker due to the leakage of the transistors that are supposed to apply the voltage to the liquid crystal cells. The phase modulation is thus modulated at a frequency corresponding to the refreshing period of the transistors (i.e. 25Hz in our case). This phase variation is seen by the novelty filter as a permanent novelty always present on the signal and it will be of course detected as a novelty. Practically it is responsible for a reduction of the apparent attenuation of the signal beam and thus for reduced performances of the novelty filter (due to the presence of a high amplitude noise). This is clearly visible in Fig. 6, where we see the attenuation of the signal beam with the SLM off, and with the SLM on with different uniform grey level applied on it.

First the flicker is at twice the period of the refreshing of the SLM (due to the quadratic response of the photorefractive crystal in the diffusion regime to a periodic phase modulation [START_REF] Ph | Transmission of time modulated optical signals through an absorbing photorefractive crystal[END_REF][START_REF] Ph | Detection of ultrasonic motion of a scattering surface using photorefractive InP:Fe under applied DC field[END_REF][START_REF] De Montmorillon | Novel theoretical aspects on photorefractive ultrasonic detection and implementation of a sensor with an optimum sensitivity[END_REF]). Second there is a reduction of the apparent attenuation of the signal beam (given by the average value of the detected signal). This reduction is all the more important as the grey level is important as the flicker noise increases with the grey level. It should be noted that the model we developed enables to take this flicker noise into account (as will be seen in the following). It can even be used to deduce the value of the photorefractive gain from the data of Fig. 6. From the "noisy" curves of Fig. 6 we measure γ=(7.8±0.2) cm -1 , and estimate the amplitude of the flicker noise phase modulation (0.5rad for the 255 level and 0.25rad for the 100 level, both case corresponds to the worst case of a "10" contrast regime).

Looking now at the attenuation of the image we goes from a 175 times attenuation without the SLM to a 8 times attenuation with the SLM on (255 level uniform image with a "10" contrast). This is clearly seen on the images in Fig. 7. For this experiment we insert a slide with a chessboard image just in front of the SLM to simulate the image and put an uniform image on the SLM (255 grey level with "10" contrast). The initial image is seen in Fig. 7.A and the attenuated image in Fig. 7.B, without changing anything to the set-up. The attenuation seems good but an enhancement of the contrast of the attenuated image, with a simple image processing computer program clearly reveals the image (Fig. 7.C). The same experiment, performed with the SLM off (SLM driver turned off), shows that in order to begin to guess the image after contrast enhancement (Fig. 7.E), the initial image (without attenuation) has to be greatly saturated (Fig. 7.D). The loss of the apparent attenuation due to the flicker is thus clearly seen in these images. These images also show the good quality of our imaging optics. The residual distortion is mostly due to dust on the optics, that have not been totally eliminated and that gives fringes structures, and to defects inside the photorefractive crystal, giving scattering points on the attenuated image.

We perform some measurements of the temporal response of the SLM using the residual amplitude modulation observed in the phase modulation regime. We measure an exponential response of the SLM, with a time constant roughly estimated around 80ms. We just make a rough estimation of the time constant because a precise determination of this response time is very difficult due to the flicker of the SLM, which varies on time scale close to the response time of the SLM. The estimated response time corresponds to what was previously observed, i.e. a build-up of the phase shift until its maximum in about 2 images [START_REF] Soutar | Measurement of the complex transmittance of the Epson liquid crystal television[END_REF].

II.4. Discussion

The set-up used as a two beam coupling photorefractive novelty filter has been presented and characterized. All the parameters of the set-up, essentially the photorefractive crystal parameters, are known as well as the image parameters, essentially the response of the SLM. The defects of the SLM might even be taken into account as we know that the SLM presents flicker and it can be introduced into the modelization, together with the response time constant of the SLM (even if the found value has a great uncertainty). All these parameters will be introduced in the theoretical model to calculate the response of the novelty filter. The comparison with the experimental response will be thus performed without having any free parameters (they are all determined more or less precisely by independent experiment).

III. EXPERIMENTAL RESULTS

To determine the performances of the photorefractive novelty filter, we send to the SLM computer generated series of images using a frame grabber connected to the SLM. The sequence of images is the following : first an uniform image with a level n 0 followed at time t=0 by a second image constituted of a 50pixels x 50pixels square with level n 1 = n 0 + ∆n.

Then after about 1s once again the initial background image is sent. In the experimental setup the CCD camera is replaced by Si detector which surface is fully covered by the image of the square. The experimental study concentrates on the response of the novelty filter to the amplitude of the novelty ∆n and on the influence of initial level n 0 .

III.1. Response to the amplitude of the novelty

We here fix the initial level n 0 = 0 and vary ∆n. In the first experiment the contrast of the SLM is set to "0", which means that we have a linear response of the SLM with a maximum amplitude of the phase variation of about 1.3π. The experimental results are presented in Fig. 8 (all the curves here and in the following are single shot curves). The temporal structure of the curves corresponds to the theory. At the appearance of the square (at t=0) the novelty filter detects it and, then it adapts to the new image, what corresponds to the disappearance of the novelty peak. Then the square disappears which gives a new peak as this disappearance is also a novelty. Then this peak disappears as the novelty filter readapts to the new image. The amplitude of the novelty peaks varies as the square root of the novelty amplitude (Fig. 8.F) as predicted by the theory (in the case considered here of linear dependence of the phase shift with grey level). The saturation (and the departure from the straight line) at high grey level change is due to the high value of the phase change, with such high value (around π/2) the influence of the sinus function (given by relation (33)) begins to appear. Moreover, we see in Fig. 8 the good accordance of the experimental curves with the calculated ones (reminding that the theoretical curves are calculated one and that no adjustment is performed).

The same measurement is performed with the SLM contrast at level "10" (Fig. 9) where the maximum phase variation is greater than 2π. Here again we have a good accordance between the experimental curves and the calculated ones. We see the peculiar behaviour predicted by the theory (Fig. 2). For example, for the curve with n 1 = 150, corresponding to a phase variation of 2π, we observe the transient response due to the finite response time of the SLM. The accordance between calculated curves is good even if a more precise adjustment could have been made, just playing with the value of τ R (as already seen in Fig. 2), and confirmed by curve at n 1 =200 calculated with a smaller value of τ R (30ms instead of the roughly estimated value of 80ms ( §II.4.)) and showing a better accordance with experiment. Nevertheless, whatever the exact value of this response time the feature of the curve is always the same. One can also note that the height of the transient peaks at high modulation overpasses the initial intensity of the signal beam (obtained without pump beam) as predicted by the theory (even if it is reduced compared to the 4 fold amplification predicted by the theory due to the finite response time of the SLM). These curves also show that it is possible to detect a novelty amplitude as low as 10 grey levels, despite the high level of fluctuations (due in fact to the flicker of the SLM as will be seen later). III.2. Independence of the response towards the initial level A peculiarity the use of a phase modulation to introduce the image into the novelty filter, is the independence of the response towards the initial level. Thus we conduct measurements where we keep a constant novelty amplitude (of opposite sign compared to previous measurements) and vary the initial level n 0 . The contrast is set at "0" to have a linear response of the SLM. The curves (Fig. 10) show that the amplitude of the novelty peaks stays constant whatever the initial level (the smaller signal at low initial level (n 0 =50) being certainly due to the small sublinearity of the response of the SLM at small grey levels (Fig. 5)).

To confirm this advantage of phase modulation against intensity modulation operation of the novelty filter, we perform the same experiment with an intensity modulated beam. The experimental conditions were slightly different as we use plane waves and an acoustooptic modulator to perform the experiment (and the parameters of the crystal (γ and τ 0 ) were slightly different). The ratio of the amplitude of the novelty compared to the maximum accessible amplitude is similar to the one in the previous measurements for a good comparison between both experiments. We change the initial level from a low level, towards a medium and a high level. We see (Fig. 11) that the amplitude of the peaks is greatly reduced when the initial level increases, indicating that, as predicted by the theory, it is easier to detect a novelty on a black level than on a white level. Again, the accordance between experimental curves and theoretical ones is good.

III.3. Comparison with theory

As seen previously for all the experimental curves we can simulate exactly the response of the novelty filter, and compare it to the experimental results. As we have seen the accordance is rather good whatever the experimental conditions, especially in the case of an intensity modulated signal beam (Fig. 4 and Fig. 11), what validates our theoretical model. However our model allows to go even farther. Indeed we notice that the curves in the phase modulation regime are characterized (especially at low novelty amplitude) by a strong "noise" that limits the performances. Thanks to the model we can now explain this "noise" that is not a real noise but is rather due to the fluctuations of the SLM.

This flicker of the SLM is known, it can be modelized as a sinusoidal phase modulation with period 40ms (Fig. 6) and with an amplitude that varies with the grey level. We will take it equal to 0.2rad for our simulation. We add this additional phase modulation to the initial variation of the incident signal beam E d (0,t) in our simulations in order to calculate the real response of the SLM and to compare it to the experiment. For this comparison (Fig. 12) we take as experimental data the curve in Fig. 8.C. The accordance between calculated and measured curves is very good, the exact feature of the response of the novelty filter being modelized. We can also see that the previous calculated curve (that does not take into account the flicker) gives nevertheless a good idea of the averaged amplitude of the novelty signal, confirming the quality of the previous simulations.

III.4. Discussion

Our results are important because they confirm the validity of our theoretical model of the photorefractive two beam coupling novelty filter. It is now possible to modelize the behaviour of the novelty filter when changing the performances of the different components of the set-up and to optimize the response. Moreover new features have been discovered that allow to operate with phase shifts greater than 2π, by using the finite response time of the SLM. This new working regime will have to be explored in more details in order to implement future versions of the system.

The results are also important because they show that the "noise" that limits the performances of the novelty filter for the detection of small grey level variation, is in fact due to the SLM. This means that the use of better SLM that does not exhibit such a flicker, would increase the performances of the system. Such spatial light modulators commercially exist [START_REF] Cho | Characteristics of a 128x128 liquid crystal spatial light modulator for wavefront generation[END_REF] and will be chosen for future implementation of such novelty filters.

IV. CONCLUSION

We have performed both theoretical and experimental evaluations of the performances of a photorefractive two beam coupling novelty filter. The theoretical model allows to deduce that the operation of this kind of filter is greatly improved if the image is imprinted on the signal beam as a phase modulation rather than as an intensity modulation. In such a case, the requirements on the photorefractive crystal (essentially the photorefractive gain) are greatly reduced, but also the response of the novelty filter to small changes in the image becomes totally independent of the initial level of the pixel. In other words, it is as easy to detect a one grey level variation on a "white" background as on a "black" background. This is of importance as the main application of this kind of system is the detection of small objets appearing or moving in a highly textured scene. These theoretical predictions are confirmed by the experiments we performed, all the predicted features being perfectly observed experimentally.

Apart this result, the model gives perfect simulations of the operation of the novelty filter. The calculated curves are indeed in excellent accordance with the experimental ones. This agreement allows to show that the main limitation of our set-up was in fact due to the spatial light modulator we used, and particularly its flicker. This limitation is important and an operating system will have be built with new generation SLMs that do not present this flicker. Taking into account the model we developed, we are now able to simulate the response of the system when changing one element or the other and to predict the performances of the novelty filter with these new elements. The steady state phase shift change is 0.5π (curves A), 2π (curves B) and 2.5π (curves C). The transmitted signal when the novelty filter does not work is 100. The crystal parameters for the calculation are x=3.63mm, α=0.7cm -1 , γ=-7.8cm 1 and τ 0 =750ms. is an half wave plate, P a dichroic polarizer, PBS a polarizing beam splitter, OI an optical isolator, and SLM is the spatial light modulator.

Figure 4 : Signal given by the novelty filter, when the signal beam is turned off during 181ms. The signal beam power in the absence of the pump beam is 96mV. The dotted line represents the theoretical curve. The offset between the curves is for clarity of presentation. For this experiment the SLM was turned off and has no action on the beam. corresponds to a calculated theoretical curves using a response time of the SLM τ R =30ms. The signal power without the pump beam is 86mV.

Figure 1 :

 1 Figure 1 : Schematic of two beam coupling in a photorefractive crystal.

Figure 2 :

 2 Figure 2 : Response to a phase change that establishes with an exponential law of increasing response time (τ R =3ms plain line, τ R =30ms dotted line, τ R =80ms dashed line).

Figure 3 :

 3 Figure 3 : Experimental set-up of the photorefractive two beam coupling novelty filter. λ/2

Figure 5 :

 5 Figure 5 : Experimental values of the phase shift (black markers) and transmission in amplitude (grey markers) of the SLM as a function of the grey level, for "0" contrast (+) and "10" contrast (∑).

Figure 6 :

 6 Figure 6 : Experimental curve of the attenuation of the signal beam with the SLM off (A)and the SLM on (B). On (A) the insert shows temporal response of the attenuated signal. (B) shows the temporal response of the attenuated signal for different levels n of the uniform image sent on the SLM (contrast "10").

Figure 7 :

 7 Figure 7 : Attenuation of an image imprinted on the signal beam with the SLM on (uniform image with level n=255, with contrast "10") (A, B, C) and with the SLM off (D, E). Images A and D are taken before attenuation, and images B, C and E are taken after attenuation and after an identical contrast enhancement for images C and E.

Figure 8 :

 8 Figure 8 : Experimental curves (grey curves) of response of the novelty filter to an increasing novelty amplitude between a level n 0 =0 and a level n 1 = n 0 +∆n (A :

Figure 9 :

 9 Figure 9 : Experimental curves (grey curves) of response of the novelty filter to anincreasing novelty amplitude between a level n 0 =0 and a level n 1 = n 0 +∆n, the contrast of the SLM is "10". The black curves are calculated theoretical curves using a response time of the SLM τ R =80ms. The dashed line for n 1 =200

Figure 10 :

 10 Figure 10 : Experimental curves of response of the novelty filter to a constant novelty amplitude (∆n=50) between a varying level n 0 and a level n 1 = n 0 -∆n. The constrast of the SLM is "0".

Figure 11 :

 11 Figure 11 : Experimental curves (grey curves) of response of the novelty filter to a quasi constant intensity variation on different initial levels. The incident signal is shown on the curves and has been horizontally shifted for clarity. The black curves are calculated theoretical curves. The intensity modulation is obtained with an acoustooptic modulator that replaced the liquid crystal SLM.

Figure 12 :

 12 Figure 12 : Comparison between the experimental curve of Fig. 8.C and the theoretical curve (dotted line), taking into account the exponential response time of the SLM τ R =80ms and the additional phase modulation due to flicker. The theoretical curves presented in Fig.8.C is also given (dashed line), it takes into account the exponential response time of the SLM but not the flicker. Without the pump beam the signal power is 86mV. The offset between the curves is for clarity of presentation.

Figure 7 :

 7 Figure 3

Table x

 x

	(mm)	3.63
	α (cm -1 )	0.7
	γ (cm -1 )	-7.8
	τ 0 (ms)	750

Table 1 :

 1 Typical set of the parameters of the photorefractive crystal used in the experiment.

Acknowledgments :

Support for this research by Direction des Systèmes de Forces et de la Prospective of DGA is greatly acknowledged.


Ph Delaye

L Frey

A Mugnier

G Roosen

EXPERIMENTAL INVESTIGATION OF A PHOTOREFRACTIVE RING PHASE MODULATION AMPLIFIER

We present the experimental investigation of a photorefractive ring phase modulation amplifier, that is not resonant and that can work with distorted beams. We show that with this geometry an amplification by a factor 5 of a phase modulation signal is possible. We also show that this strong amplification is alleviated by the presence of losses in the ring. These losses lead to a real gain in signal to noise ratio of a factor 3, that nevertheless expresses that with such a system the photon noise limit of laser ultrasonics system can be overpassed. The losses are also responsible for the strong sensitivity of the system to the vibrations, which puts a strong limitation of the ring amplifier.

Introduction

The development of the laser ultrasonic technique has increased the demand for detection scheme that are both sensitive and can work with highly deformed speckled beams. Different systems based on dynamic wavefront adaptation in photorefractive crystals have been studied and characterized [1 -6]. All these systems meet the necessary requirements concerning the étendue and the possibility to work with speckled beams. With these speckle structures, they reach the sensitivity obtained by other optical techniques (homodyne or heterodyne detection) with plane waves [START_REF] De Montmorillon | Conference on Lasers and Electro-Optics Europe[END_REF]. Nevertheless, all these optical techniques suffer from reduced sensitivity if compared to piezoelectric transducers : they are all limited by the photon noise. A solution to overcome this limit is to amplify the phase modulation by multiple pass on the phase shifting structure. For example, in the case of ultrasonic signals, the tested surface can be the back mirror of a Fabry-Perot interferometer. At resonance, the photons are stocked in the cavity which results in multiple passing on the vibrating surface, enhancing the total phase shift seen by the wave. This technique works well but requires very tough conditions on the quality and the positioning of the surface to be tested. It should be a highly reflective and flat (or eventually spherical) surface to have a high finesse of the Fabry Perot. It should be very exactly positioned for the cavity to be at resonance. Under these conditions high amplifications of the signal can be obtained. Unfortunately, most of the time, the surfaces tested with the laser ultrasonic techniques are rough low reflectivity surfaces, with a position that can vary in the range of some millimeters.

The use of such Fabry Perot cavity is thus unrealistic.

Recently, a new kind of cavity was proposed [8]. In this cavity, based on a photorefractive crystal, some of the requirements on the quality of the surface are alleviated. This is because the cavity being no more resonant, its length is no more a condition for the amplification of the phase modulation. As a consequence such an amplification system could increase the sensitivity of laser ultrasonic systems, keeping their advantages such as the possibility to have a non contact measurement system.

In this paper we will describe the experimental implementation of such a photorefractive ring amplifier. Its performances will be evaluated and compared to the theoretical expectations. Some of its limits will be finally discussed.

The photorefractive ring amplifier: Principle

The ring geometry used for the amplification of the phase modulation [8] was first used as a real time programmable ring for image storage [9]. The principle of this ring can be seen in figure 1. A beam, with power density I i , is sent on a photorefractive crystal. After transmission through the crystal, it is returned back to the crystal, as beam I d with a set of mirrors in a ring geometry. The crystal orientation is such that there is energy transfer through two beam coupling from I d towards I i . So when sent on the crystal, I i generates I d and writes with it a photorefractive grating on which I d diffracts. Through energy transfer the power of I d is increased, leading to a reinforcement of the grating, until a steady state regime is reached. In this steady state regime, we have increased the power of I d through beam coupling. We have then an accumulation of energy in the ring, all the more that the photorefractive gain is high. This accumulation of energy corresponds to photons that are trapped in the ring and thus make several turns in the ring. Then if a phase modulation ϕ(t) is introduced in the ring, as illustrated by the electrooptic modulator (EOM) in figure 1, the photons see several time this phase modulation leading to an amplification of the phase modulation. The exit beam carries an amplified phase modulation that can be detected through classical techniques like a photorefractive beam mixer [1 -3, 5, 6]. Moreover, except during a short transient behavior, all the energy put at the entrance is transferred to the exit of the ring if there is no loss in the cavity.

One problem with this principle is that, as the photons make several turns in the cavity, they see several times the different cavity losses (absorption, diffusion, ...). The energy transmitted through the cavity will decrease all the more that the photorefractive gain (i.e. the number of turns) will be high. This will cause a reduction of the efficiency of the system, as the device that will demodulate the phase modulation has its sensitivity that depends on the energy of the carrying beam. So a very low loss crystal and cavity will be necessary for the system to work. These losses will also make the system extremely sensitive to low frequency vibrations as will be seen in the following.

The energy accumulation in the ring : Comparison of experiment and theory

The first point that we will conduct in the experiment is to underscore the energy accumulation in the ring and compare it to the theory. This point is important as the whole phenomenon of phase modulation amplification is based on the capacity of the ring to trap the photons and to make them go several times through the phase modulation element. We are here only interested in the steady state regime.

The source is a single longitudinal mode frequency doubled Nd:YAG laser emitting at λ=532nm. Its total power is about 80mW. For the experiment, we use a BaTiO 3 crystal of 0.36cm thickness, with antireflection (AR) coatings, and with a crystallographic cut ( i.e. the c axis in the plane of incidence and perpendicular to the normal of the coated faces). The beams I i and I d were incident on the crystal with angles of 32° and 57° respectively, in order to maximize the photorefractive gain [10], that we measure to be around Γ= -9cm -1 (the minus sign comes from the sign of the energy transfer and indicates energy transfer from the probe beam I d towards the pump beam I i ). With these large angles the efficiency of the AR coating is reduced especially for I d . We measure a reflection coefficient of 2.7% for I d and 0.3% for I i with extraordinary polarizations (with close values for ordinary polarizations). We also measure an absorption coefficient α=0.18cm -1 for this crystal at this wavelength.

For the theoretical treatment we consider a beam I 0 incident on the crystal. This beam gives the beam I i (0) at the entrance of the crystal but inside it (we have I i (0)=T i I 0 , with T i the transmission factor of the first interface). This beam is transmitted through the crystal to give I i (d), just before the exit face, and is sent back on the crystal by the ring as beam I d (0), just after the entrance face, with an intensity transmission T. Transmission T thus includes the transmissions on the exit and entrance interfaces and the transmission of the different elements of the ring. Its value is T= 0.88. I d (0) is transmitted through the crystal to form I d (d). This beam exits from the crystal to form I F (we have I F =T d I d (d), with T d the transmission of the exit interface), and is finally sent on the phase demodulation device. All the beams I i (x) and I d (x) are taken inside the crystal.

A simple calculation [9] gives the energy accumulated in the ring as well as the energy at the output of the cavity. We have (supposing small angles inside the crystal) :

I d 0 ( ) = T I i d ( ) = I i 0 ( ) K 2 e !d and I d d ( ) = I i 0 ( ) K 2 4T e !d (1) with K = ! 1 ! T e !" d ( ) + 1 ! T e !" d ( ) 2 + 4 T e !" d e #d $ % & '
In the lossless case ( T e

! "d = 1), we have K = 2e !d 2 which gives I d 0 ( ) = I i 0 ( )e ! " d 2
and

I d d ( ) = I i 0
( ), the energy accumulation is simply given by the factor e ! " d

2 . The presence of losses reduces the energy accumulated in the ring, but also decreases the energy that goes out of the ring.

This last point will be the main limitation of the performances of the ring for the amplification of the phase modulation as, as it is well known, the signal to noise ratio of phase demodulation systems depends on intensity of the demodulated beam.

Experimentally we measure both the energy accumulation in the cavity (i.e. I d (0)) and the decrease of the output intensity (i.e. I d (d)). For I d (d) we put the detector at the output of the ring.

For I d (0) we used the reflection of the beam at the first interface of the crystal (that can be easily separated from the reflection of the same beam on the second interface). So we can monitor I d (0)

without introducing new losses in the ring. For both beams we monitor the variation of intensity with time when the beams are put on in the ring. The results are shown in figure 2. We can see the temporal evolution of both beam intensities until the steady state regime. Before the beams are put on, there is no grating in the crystal and thus Γ=0, we can thus measure I d (0) Γ=0 , whereas, at steady state, we measure I d (0) Γ≠0 (with the same measure for I d (d)). From these curves we deduce that I d (0) is amplified by a factor

I d 0 ( ) ! "0 I d 0 ( ) ! =0 = 3.2 ± 0.3, whereas I d (d) is decreased by a factor I d d ( ) ! = 0 I d d ( ) ! " 0 = 1.9 ± 0.2 .
Once noted that when Γ=0, we have K = 2T e ! "d , we can easily compare the experimental values with the theoretical ones. We have :

I d 0 ( ) ! "0 I d 0 ( ) ! =0 = # 1 # T e #$d ( ) + 1 # T e #$ d ( ) 2 + 4 T e #$d e ! d % & ' ( 2 T e #$ d e !d (2a)
and So we observe an accumulation of energy in the ring that corresponds to the expected value.

I d d ( ) ! = 0 I d d ( ) ! " 0 = 1 4e 2# d e !d $ 1 $ T e $# d ( ) + 1 $ T e $ #d ( ) 2 + 4T e $# d e !d % & ' ( 2 (
This energy accumulation is accompanied with a decrease of the energy that goes out of the cavity. This loss of energy will limit the performances of the ring as will be seen in the following. We will now study the phase modulation amplification brought by this energy accumulation.

The phase modulation amplification : Comparison of experiment and theory

The ring acts as a phase modulation amplifier. We will now analyze its amplification ability by measuring the phase modulation of the beam I d (d) at the output of the ring. For this phase demodulation we use a photorefractive beam mixer, made with a photorefractive crystal with an applied DC electric field, as previously implemented [3, 5 -7]. Because of the wavelength sensitivity range of the BaTiO 3 crystal used in the ring, the photorefractive beam mixer has to operate in the visible (here at 532nm). Then a Bi 12 SiO 20 crystal is used (Fig. 3). In this study, we will compare the signal given by the photorefractive beam mixer with the ring either working or not. For simulating a not working ring, we put in the ring, just before the crystal, an half wave plate (HW2 in Fig. 3), that turns the polarization of I d (0). When I d (0) is extraordinary polarized, the ring works. When I d (0) is ordinary polarized, it is crossed polarized with I i (0), and no photorefractive grating can be written. A simple rotation of the wave plate allows the measurement of the phase modulation amplification without changing anything in the set-up.

The phase modulation is given by a Pockels cell (EOM) introduced in the cavity. This Pockels cell gives a sinusoidal phase modulation with a RMS amplitude of 1.7mrad at frequency 100kHz.This phase modulation amplitude would correspond to a displacement of one of the mirror of the cavity of about 0.1nm.

Theoretical treatment

A. Phase modulation amplification

Once a steady state regime is obtained, the system can be seen as a steady state volume grating on which beams diffract in the Bragg regime. Using the method described in Ref. [8] but taking into account the crystal absorption and the losses of the ring, we obtain the optical field amplitudes E i and E d of the different beams at the exit of the crystal :

E i d ( ) = e ! "d 2 E i 0 ( ) 1 ! # + e ! "d 2 E d 0 ( )e ! i$ 0 # E d d ( ) = !e ! "d 2 E i 0 ( )e i$ 0 # + e ! "d 2 E d 0 ( ) 1! # % & ' ( ' (3)
with

! = T e " #d 1" e $d 2 1 + K 2 % & ' ' ( ) * * and K = ! 1 ! T e !" d ( ) + 1 ! T e !" d ( ) 2 + 4 T e !" d e #d $ % & '
where the absorption losses in the crystal are taken into account. ϕ 0 is the phase shift introduced by the transfer through the ring between E i (d) and E d (0).

Once the ring is in its steady state regime, a low amplitude phase modulation ϕ(t)

(ϕ(t)<<π/2) is introduced in the ring. This phase modulation is introduced in the transfer expression of the ring :

E d 0 ( ) = T e i! 0 e i! t ( ) E i d ( ) (4)
This transfer relation introduced in the system allows to calculate the amplitude E d (d) at the output of the crystal. We have :

E d d ( ) = E i 0 ( )e i! 0 e " #d 2 " $ + T e " #d 2 1 " $ ( )e i! t ( ) 1" T $ e " #d 2 e i! t ( ) % & ' ' ( ) * * (5)
If now, we use the fact that the phase modulation is very small (ϕ(t)<<π/2), i.e.

e i! t ( ) = 1+ i ! t ( ) and cos ! t ( ) ( ) = 1, the previous equation can be rewritten :

E d d ( ) = E i 0 ( )e i! 0 Y 1 + i X ! t ( ) [ ] (6)
with

X = e ! "d 2 and Y = K e ! "d 2

T

If we now suppose that the amplification is low enough to assure that (Xϕ(t)<<π/2) we will have :

E d d ( ) = E i 0 ( )e i! 0 Y e i X ! t ( ) (7)
indicating the amplification of the phase modulation by a factor X, whereas the intensity of the output beam intensity decreases by a factor Y 2 as seen previously (eq.1). One remarkable point is that the amplification of the phase modulation is not at all influenced by the losses in the cavity and depends on the photorefractive gain only.

B. Signal to noise ratio amplification

Now to estimate the true performances of the system we have to include the phase detection system and to remind that this is the initial intensity I 0 that has to be taken into account. The phase detection system receives a beam of amplitude :

E F = E 0 T i T d e i! 0 Y e i X ! t ( ) (8)
The detection system we use here, is a photorefractive beam mixer, but the following treatment is independent of the type of detection scheme used. Consider that the photorefractive crystal of the beam mixer has an amplitude gain ! = " ! + i " " ! , a thickness x and an absorption α'.

When a beam of amplitude E s 0, t

( ) = E s 0, 0 ( )e i ! t
( ) (with φ(t)<<π/2) is sent on such a crystal, we measure at the output [11]:

I s x, t ( ) = I s 0, 0 ( )e ! "
# x e 2 " $ x + 2 e " $ x sin " " $ x

( )% t ( ) [ ] (9)
With our previous notations, we have E s 0, 0 ( ) = E 0 T i T d e i! 0 Y and ! t ( ) = X " t ( ) . Thus, on the detector, we will have :

I s x, t ( ) = I 0 T i T d Y 2 e ! " # x e 2 " $ x + 2e " $ x sin " " $ x ( ) X % t ( ) [ ] (10)
From this, we can establish the signal to noise ratio for the whole system (phase amplifier and demodulator) [START_REF] De Montmorillon | [END_REF]:

SNR Ring = 2! d I 0 h" #f T i T d X Y $ t ( )e % & ' x 2 sin & & ( x ( ) (11)
in which η d is the quantum efficiency of the detector, hν is the photon energy, and Δf is the measurement frequency bandwidth.

This result has to be compared to the signal to noise ratio obtained with the same phase demodulation system used with the same target without going through the amplification ring but with the same incident intensity. This corresponds to a reference signal to noise ratio :

SNR Re f = 2 ! d I 0 h" #f $ t ( )e % & ' x 2 sin & & ( x ( ) (12)
Comparing equations ( 11) and ( 12), we define the gain in signal to noise ratio that can be written :

G R = X Y T i T d (13)
which gives :

G R = e ! "d 2 T T i T d ! 1 ! T e !#d ( ) + 1 ! T e !#d ( ) 2 + 4 T e !#d e " d $ % & ' (14)
This gain in signal to noise ratio, as expected, depends only on the parameters of the ring and is independent of the characteristics of the phase demodulation system.

Experimental study

A. Phase modulation amplification

The demodulation photorefractive beam mixer uses a Bi 12 SiO 20 crystal of thickness 7mm having an absorption α' = 0.86cm -1 . We applied on this crystal a DC voltage of 3kV (which corresponds to a field of about 4kV.cm -1 ). Under these conditions we measure, using the demodulation of the known phase modulation signal and equation ( 9) [START_REF] De Montmorillon | [END_REF]11], the amplitude of the amplitude gain ! = " ! + i " " ! with γ'≈0.04cm -1 and γ "≈0.3cm -1 .

To compare more easily the signal with and without the ring active, a neutral filter is put at the output, when the ring is inactive (crossed polarization and Γ=0), to compensate for the energy loss that occurs when the ring becomes active. The half wave plate at the output of the ring (HW3, Fig. 3) is here to turn the polarization when we go from the crossed polarization geometry (Γ=0) to the parallel polarization geometry (Γ≠0), i.e. to maintain a constant state of polarization for the beam entering the photorefractive beam mixer.

The measured signal has a DC part and an AC part that is the demodulated phase modulation signal (eq.10). With an inactive ring, (Γ=0) we measure a DC signal V 0 =8.2±0.1V and an AC modulated part δV RMS =8.3±0.2mV. With the active ring (Γ≠0), we measure a DC signal V 0 =8.2±0.5V and an AC modulated part δV RMS =42±2.7mV. These results are shown in figure 4.

As the DC level is the same for the two cases, the curves show directly the amplification X of the phase modulation due to the amplification ring.

As the phase modulation Δϕ is proportional to the ratio δV RMS /V 0 (eq.10) with the same proportionality coefficient in both cases (Γ=0 and Γ≠0), we have :

X exp = !" # $ 0 ( ) !" # = 0 ( ) = %V RMS V 0 ( ) # $0 ( ) %V RMS V 0 ( ) # =0 ( ) = 5 ± 0.8
This value corresponds well to the theoretical value

X th = e ! " d 2 = 5 .
We also measure the influence of the losses in the ring on this amplification of the phase modulation. For this, we introduce in the ring a variable neutral filter of transmission T NF .

For T NF =0.91, we measure with the inactive ring (Γ=0) : V 0 =8.06±0.08V and an AC modulated part δV RMS =8.3±0.17mV. With the active ring (Γ≠0) we get : V 0 =6.7±0.4V and an AC modulated part δV RMS =33±2.1mV. From this, we deduce X exp = 4.7 ± 0.8 .

For T NF =0.84, we measure with the inactive ring (Γ=0) : V 0 =7.5±0.1V and an AC modulated part δV RMS =8±0.2mV while with the active ring (Γ≠0) we obtain : V 0 =5.2±0.3V and an AC modulated part δV RMS =25±1.6mV. We deduce X exp = 4.5 ± 0.7.

From these results, we see that the phase modulation amplification does not depend, within the experimental errors, on the losses in the ring. We also see that the decrease of the DC signal is much stronger when the ring is active, which confirms the theoretical results on the influence of losses on the ring performances.

B. Gain in signal to noise ratio

We have now all the experimental parameters to evaluate the performances of the ring and to calculate the gain in signal to noise ratio G R brought by the phase amplification ring. If we remember that the transmissions of the faces are close to one we have T i T d = 1 and the value of G R is given by the product XY. We have measured in the first part of the study a loss at the output of the ring : So, from these results, we deduce the gain in signal to noise ratio provided by the ring G R =2.9±0.6. It is the amplification of the signal to noise ratio that is obtained in comparison with the one that would be obtained with the same incident power sent directly on the same target but without passing through the ring and its different losses. This gain in signal to noise ratio corresponds to the one that can be expected from the theoretical model (eq.14) with the experimental parameters : G R =3.1. This value is also not optimum due to the low value of the photorefractive gain in the crystal used (Fig. 5). With a better crystal we will reach a high gain limit

I d d ( ) ! " 0 I d d ( ) ! = 0 = 1
value of G R lim = T e !" d 1! Te ! "d ( )
, which value is 4.95. So we have the confirmation that the ring can be used to amplify the phase modulation. The gain in signal to noise ratio is modest and very dependent on the losses in the ring but it exists and is noticeable. Nevertheless, one point does not appear in this study, and we will discuss it now, this is the sensitivity of the system to external vibrations that is brought by the presence of the losses.

C. Vibration sensitivity of the ring

This vibration sensitivity is directly induced by the working principle of the amplification ring. Indeed, in its normal working mode the induced grating is π/2 phase shifted from the illumination grating which causes the energy transfer responsible for the energy accumulation in the ring. Now a low frequency vibration (but high enough frequency compared to the frequency response of the photorefractive crystal, i.e. some tenths of Hertz), will be characterized by a high amplitude phase shift that will displace the illumination grating relatively to the index grating, that remains stationary. The optimum condition for energy transfer will be no more fulfilled and some energy will be expelled from the ring, leading to an increase of the energy at the output, increase that will be totally random as the vibration that induces it. In the same time the phase modulation signal will be linearly demodulated by the photorefractive crystal of the ring that will act as a photorefractive beam mixer (as the phase shift between the illumination and the index gratings is no more π/2, it is equivalent to a complex photorefractive gain γ). The main problem with this phenomenon, which is not detrimental by essence, is the fact that the vibrations are generally completely random and uncontrollable, and thus unexploitable.

This influence of the vibrations can be easily described theoretically. For this we go back to equation [START_REF] Ph | Proc. SPIE[END_REF], that describes the beam amplitude at the output of the crystal. We can rewrite it as : allow for the amplification of a phase modulated signal, such as an ultrasonic vibration signal, by a factor that can be as high as a factor 5. We also show that this amplification should be tempered by the increased losses that occur due to the amplification of the losses that occurs in the ring. These losses strongly reduce the amplification of the signal to noise ratio brought by the ring. By limiting at maximum these losses we obtain an amplification of this signal to noise ratio by a factor 3. This amplification is sufficient to allow the break of the photon noise limit of laser ultrasonic system, and can make them more competitive compared to classical piezoelectric systems. The properties of the system keep all the advantages of the optical techniques, which are the possibility to be non contact with a good depth of field. Due to the properties of the photorefractive effect it is possible

The possibility to work with rough surfaces, if used with detection systems like the photorefractive beam mixer. This will be a great amelioration if compared to passive interferometers that are used in the same way, like Fabry Perot interferometers. Nevertheless we have shown that this result is acquired only in the case where losses remain low, i.e. on highly reflecting surfaces and if all the scattered light can be recovered and reinjected in the ring. The study also show a strong limitation of the system which is its extreme sensitivity to vibrations, because of the cavity losses.

Nevertheless this sensitivity is also partly due to the slowness of the crystal used in this demonstration. The use of faster photorefractive crystals can allow to overpass this limitations, if this enhanced rapidity is not obtained at the detriment of the efficiency or the absorption. Another solution to prevent this vibration sensitivity could be the use of an active stabilization of the ring to compensate for the vibrations.

This study also shows a good adequacy of the experimental results to the theoretical model.

The work will be continued towards the study of new geometry, like the reflection grating geometry, and to the development of dynamical models that can describe the frequency response of such a system.

0 (= 2 . 1 ,

 021 2b)With the experimental values T=0.88, e ! "d =0.935 and e ! "d =26, we calculate I d which corresponds to the experimental results.

=

  T e "2#d , we deduce Y=0.58±0.03. In the same time we have measure a phase modulation amplification X=5±0.8.

Figure 1 :

 1 Figure 1 : Schematic representation of the phase modulation amplification ring. The phase modulation is simulated by an electrooptic modulator (EOM).

Figure 2 :

 2 Figure 2 : Temporal evolution of the intensity of the beam inside the ring I d (0) and at its output I d (d). When the beam is put on there is no grating inside the crystal. The intensity are normalized to their values without grating.

Figure 3 :

 3 Figure 3 : Experimental set-up with both the amplification ring and the photorefractive beam mixer for phase demodulation. HW i are half wave plates. L i are lenses. PBS is a polarization beamsplitter.NF is a neutral filter used to compensate for the increased signal when the amplification ring doesn't work.

Figure 4 :

 4 Figure 4 : AC part of the demodulated signal with the ring operating (solid line) and with the non working ring (dashed line), showing the amplification brought by the amplification ring. Both curves corresponds to the same value of the DC part of the signal.

Figure 5 :

 5 Figure 5 : Calculated gain in signal to noise ratio of the ring G R as a function of the photorefractive gain (-Γ), showing the saturation of the amplification at high gain due to the losses of the ring. We use for the calculation the experimental values α=0.18cm -1 , T=0.88 and a crystal thickness of 0.36cm. The experimental data ( ■ ) is placed on the curve for comparison.

Figure 1 Figure 2 Figure 3 Figure 4

 1234 Figure 1

Figure 5

 5 Figure 5

( ) = ! s + "! t ( ), ! s being the phase shift given by the vibration (that is supposed to be rapid enough not to erase the index grating) and !" t ( ) the phase modulation to be measured with !" t ( ) << # / 2. In these conditions we have :

And, if we calculate the energy at the output of the ring, we can write it as :

with A = e !" d + T e ! "d ! 1

( )

We find on these expressions the different interesting points we discussed previously. First A increases when ! s changes from 0, indicating the energy increases at the output of the ring.

Second there is a phase demodulation as B is different from zero when ! s departs from 0. Another remarkable point, that also appears on the expressions, is that this effect totally disappears when there is no losses in the ring : we have A=1 and B=0 when T e ! "d = 1.

We think that this sensitivity to the low frequency vibrations is the main limitation of the use of the amplification ring. The solutions would be to increase the rapidity of the crystal used in the ring without losing in amplification, i.e. keeping a low absorption value and a high two wave mixing photorefractive gain. Thus the optimization of such a system seems to be quite difficult.

Conclusion

We have presented the experimental investigation of a photorefractive ring phase modulation amplifier proposed in Ref. [8]. The main results of this study is that such a system can