Martino Lobue

Vincent Loyau

Frédéric Mazaleyrat

F Mazaleyrat Satie

Analysis of volume distribution of power loss in ferrite cores

Keywords: numbers: 75, 50, -y, 75, 50, Gg,

We present a technique to estimate the inhomogeneities of magnetic loss across the section of ferrite cores under AC excitation. The technique is based on two distinct calorimetric methods that we presented elsewhere. Both the method are based on the measurement of the rate of increase of the sample temperature under adiabatic condition. The temperature ramp is recorded either measuring the sample bulk resistivity, or using a platinum probe pasted on the sample surface. As an example we apply the procedure to an industrial sample of Mn-Zn ferrite under controlled sinusoidal excitation with a peak induction of 50 mT in the range between 100kHz and 2MHz. The results are discussed by comparison with simulations of the dissipation field profile through the sample, calculated using a FEM code.

I. INTRODUCTION

Soft sintered Mn-Zn ferrites are widely used in high frequency applications based on switching circuits. Their relatively high resistivity and permeability make them optimal for the 30kHz-1MHz bandwidth. Miniaturization and the increase of working frequencies make thermal and loss managements the keys for future design of electronic switching devices. It is thus necessary to establish reliable methods to measure, understand and predict the losses in soft ferrites at medium and high frequencies.

Losses are commonly measured using the flux-metric method [1][2][3] . However, when excitation frequencies are above 10 kHz, measurements can be dramatically affected by spurious phase contributions due to stray capacitances and inductances of the circuits and probes. A possible alternative is the calorimetric method; in adiabatic conditions, the power dissipated in the magnetic material is directly related to the heating rate, reducing the measurement to that of a temperature ramp. We discussed in detail this technique in a recent paper 4 where an experimental set-up using a platinum probe as temperature sensor was presented.

A further improvement of the technique has been proposed 5 by measuring the temperature from the bulk DC resistivity of the ferrite core 5 . Whereas the temperature deduced from bulk resistivity represents a measurement averaged over the whole sample volume (as in the case of flux-metric measurements), the platinum probe measurement gives information on dissipation taking place in a thin strip of the material just near its surface. Consequently the comparison between the two measurements can be used to estimate the loss inhomogeneity, and thus, the relevance of skin effect for a given sample geometry and excitation frequency. It will be shown that, global low level electromagnetic parameters, obtained by impedance spectroscopy, can be used as local input parameters for the FEM computation of magnetic loss in linear regime. In this way we shall calculate the volume or surface averaged loss values, corresponding to the two types of calorimetric measurements and compare them with experimental results.

II. EXPERIMENTAL AND COMPUTATION DETAILS

Power losses in magnetic materials under AC excitation are associated with all the irreversible phenomena taking place during the core magnetisation (i.e. Barkhausen jumps, eddy currents, domain wall resonance, spin dumping, magnetoacoustic emission, etc.). Under adiabatic conditions 4 , all the dissipated power is converted into heat and induces a temperature increase of the sample. Thus the temperature field through the core T (x, y, z, t) can be described by means of the linearised heat equation giving:

∂T (x, y, z, t) ∂t = 1 ρ m c p p s (x, y, z) (1)
where p s (x, y, z) is the local heat source density (in Wm -3 ), c p is the specific heat (in

JK -1 kg -1
) of the sample and ρ m its density (in kgm -3 ). Now, if V 0 is the volume of the sample, the magnetic loss per cycle measured using the flux-metric method, at a given excitation frequency f , corresponds to the average loss over the sample volume, namely:

W = c p f V 0 V ∂T (x, y, z, t) ∂t dV. (2)
Whereas a measure of the average temperature of the whole sample will result in a correct extrapolation of W , any local measurement will be related with dissipations taking place just in a small volume near the position of the temperature probe. In the latter case W can be deduced properly only when volume losses are uniformly distributed through the sample.

Two different techniques where used to measure the temperature ramp: the first using a platinum probe pasted (Pt500) on the sample 4 and therefore sensible to the temperature change in a narrow strip just near the surface, and a second technique based on the measurement of previously calibrated bulk resistivity of the ferrite core 5 which returns a global average of the temperature ramp. The latter leads W notwithstanding the inhomogeneities of the loss field due to the geometry of the sample and to the skin effect. In both cases, the sample was a square cross section bar (I rod: 0.64 × 0.64 × 2.54 cm 3 ) made of 3E27

Ferroxcube Mn-Zn ferrite. Complex impedance spectra of the permeability µ = µ ′ + iµ ′′ and of the electrical conductivity σ = σ ′ + iσ ′′ = σ ′ + iǫω (where ǫ is the electrical permittivity and ω = 2πf is the angular frequency) where performed with a HP 4195 impedance analyser between 100 kHz and 10 MHz. The losses where measured in a vacuum chamber between 100 kHz and 2 MHz as described in ref. 4,5 from 10 second long temperature recordings.

In order to validate the interpretation of the measurements performed with the two techniques, an approach based on the solution of Maxwell equations was used. For this purpose, the code FEMM 6 was used taking measurements of complex magnetic permeability and conductivity as intrinsic material parameters. All the following analysis is based on three strong assumptions: (i ) the peak induction values are sufficiently low to consider the magnetic behaviour as linear. Thus the concept of real and imaginary magnetic permeability and their relationship with energy losses is valid. This assumption is confirmed by the fact that all the loss measurements presented in this paper scale roughly as the square of the peak induction. This means that the term µ ′′ / | µ | 2 remains constant in the investigated induction range (from 3mT to 50mT); (ii ) all the measurements of complex magnetic permeability are performed on a toroidal ferrite sample with section small enough to make eddy current loss and skin effect negligible; (iii ) all the measurements of complex conductivity are performed on bar shaped samples which section is sufficiently reduced to neglect skin effect.

From (ii ) the measured µ is a parameter related to the local phase lag between H and B due to all dissipative phenomena but eddy current losses. On the other hand, the complex electrical conductivity σ results from the mixed electrical properties of the grain and of the grain boundaries [START_REF] Goodenough | [END_REF] . Its real part σ ′ , is related to ohmic and dielectric loss effect whereas the imaginary part σ ′′ , is due to the effect of mixed permittivity. Given the geometry of the problem, with the magnetic field always perpendicular (along z axis) to the core section, the calculation of the loss field reduces to a two dimensional problem in the xy plane. Now, under sinusoidal excitation when H z (x, y, t) = H 0 (x, y)e iωt , substituting the measured σ and µ in the diffusion equation for the magnetic field we have,

1 µ ▽ 2 H z (x, y, t) = iωσH 0 (x, y). (3)
In Fig. 2 comparison between the bulk measurements of loss and the calculated average total loss W v is shown for peak inductions ranging from 3mT to 50mT. The agreement between measurements and calculations is particularly good for frequencies above 300kHz. The over estimation of computed losses may be due to the compensation error in the measurement of µ ′′ performed with the impedance analyser. Indeed, at low frequency the resistive part of the impedance is nearly zero, so even a small absolute error can yield a strong systematic error. Fig. 3 shows the comparison between the surface measured loss and corresponding computation W s . Calculations and measurements are in good agreement. With the sample geometry studied here, as the average magnetic path is not much different from the extremal paths in the sample, all the loss inhomogeneities can be ascribed to the skin effect.

So, either measurements and finite elements calculations, show that skin effect is negligible in a frequency regime below 600-700kHz. Above these frequencies skin effect becomes relevant, and a large part of the dissipation phenomena is concentrated near the surface of the sample.

In conclusion, it has been shown that the calorimetric loss measurement method, exploiting bulk and surface temperature recordings, is able to give indirect information on the eddy current losses in Mn-Zn ferrites. Using FEM computation -for which material parameters input where obtained by impedance spectroscopy-it is possible to compute the loss fields and to obtain very good agreement with measurements without introducing any fitting parameter. Hence, this model could be integrated in magnetic components design software in order take into account the magnetic losses including eddy currents for the dimensioning.

FIG. 1 .

 1 FIG.1. Total loss field w tot (x, y) calculated from Eq. (4) and (5) for f = 100kHz (a) and f=2MHz (b). The dashed rectangle indicates the region used to calculate the surface loss as measured by the platinum probe. The thickness of the rectangle is defined as the product of the heat propagation speed in the material and the measure duration (10 seconds). Grey scales indicate the local loss in mJkg -1 where the loss varies from 0.04mJkg -1 (dark regions) to 0.05mJkg -1 (white regions) in (a) , and from 0.45mJkg -1 to 0.85mJkg -1 in (b) . The ratio between the maximum and the minimum values is 1.25 at f = 100kHz (a) and 2 at f=2MHz (b)

FIG. 2 .

 2 FIG.2. Comparison between the bulk measurements of the average total loss and the calculated ones. Peak inductions are: 3mT, 10mT, 30mT, 40mT and 50mT

From this equation and from the fact that j = ▽ × H we can calculate the eddy-current loss field across the core section,

where ρ ′ = Re(1/σ). This expression includes all the eddy current related losses. The effect of ohmic and dielectric dissipation in both, the grains and their boundaries, are described by the real part of the resistivity ρ ′ . On the other hand, the complex part of the permeability µ ′′ , measured on the very small sample, will account for all the magnetisation process related dissipation phenomena ( excess losses, spin damping, etc.). So we can write the magnetic loss field as:

The total power loss will be the sum of the two terms of Eq. ( 4) and ( 5), and the loss per cycle and unit mass will be w tot (x, y) = p tot (x, y)/(f ρ m ). By averaging over the core section and multiplying by the length, we obtain

-a/2 w(x, y)dxdy corresponding to the bulk (resistance) measurement, whereas W s = ℓ a/2 0 d 0 w(x, y)dxdy yields the value corresponding to surface (PT 500) measurement. The thickness d is defined as the product of the heat diffusion velocity in the material and the duration of the measurement (10 seconds).

III. RESULTS AND DISCUSSION

In order to determine the magnetic field H z (x, y) across the section area we must specify the induction conditions under which losses have been both calculated and measured. Here we consider the case where a sinusoidal magnetic induction of average value B 0 is forced across the core section area S. This implies that the magnetic field verifies:

The magnetic field H z (x, y) has been determined solving Eq.( 3) coupled with Eq.( 6) with a two dimensional finite element code in a frequency range between 100kHz and 2MHz on the sample geometry. Calculation results are shown in Fig. 1 where the loss field w tot (x, y) is plotted for f = 100kHz in Fig. 1(a) and f = 2MHz in Fig. 1(b).


F Mazaleyrat

) A Pasko

A Bartok

M Lobue

Giant coercivity of dense nanostructured spark plasma sintered barium hexaferrite

Keywords: numbers: 75.50.Tt, 75.50.Ww, 75.50.Gg, 75.75.Cd

Enhencement of the coercivity of ferrite magnets in order to make them true hard magnets -i.e. to get a coercive field higher than the residual magnetization-is still a very important issue due to the limited ressource in rare-earth. Thus, an alternative can be found in making very fine grain ferrite magnets but it is usually impossible to get small grains and dense material together. In this paper, it is shown that the spark plasma sintering method (SPS) is able to produce close to 80% dense material with crystallites smaller than 100 nm. The as prepared bulk sintered anisotropic magnets exibits coercive field of 0.5 T which is close to 60% of the theoretical limit and only few % below that of loose nano-powders. As a result, the magnets behave nearly ideally (-1.18 slope in the BH plane second quandrant) and the energy product reaches 8.8 kJm -3 , the highest value achieved in isotropic ferrite magnet to our knowledge.

I. INTRODUCTION

In the last years, much efforts have been devoted to the enhancement of the coercivity of hexagonal M-ferrites. The most fructuous results have been obtained by substitution of La and Co in strontium ferrite according to the formula Sr 1-x La x Fe 12-y Co y O 19 1 . A very good compromize have been found for equi-molar La and Co compound with x = y = 0.2. Under optimal processing and sintering conditions, anisotropic magnets have been obtained with a very good rectangularity and an intrinsic coercivity as high as 334 kAm -1 (viz. µ 0 H CJ = 0.42 T) which represents a 20% improvement compared to regular anisotropic Sr magnets. This very good result, however, has to be compared with the anisotropy field µ 0 H K = 2.27 T of this compound, showing that, alltogether, the coercivity is only a fifth of the anisotropy field. This effect, known as Brown's paradox, is related to magnetic domain structure. On the other hand, a strong enhencement of the coercivity can be obtained by reducing the grain size below the single domain limit as it was proposed by Néel in 1942 (published ony after World War 2 2,3 ). According to this theory -usually referred under the name of Stoner & Wohlfarth 4 -the upper limit of the coercivity is H K for perfectly oriented and 0.48 × H K for an assembly of non-interacting uniaxial particles with isotropic distribution of easy axes. Several authors have succeed to produce BaFe 12 O 19 nanoparticles by soft chemical route and they have found coercivities as high as 0.58 T for a mean grain size about 100 nm 5,6 . As the anisotropy field is 1.7 T, the theoretical value for an isotropic magnet is 0.82 T, so that they a) Electronic mail: mazaleyrat@satie.ens-cachan.fr; http://www. satie.ens-cachan.fr b) Presently at LGEP, Universite Paris Sud, Orsay. reached 70% of the upper limit, showing how promising should be this route. There is however a big drawback with this approach: these results have been obtained with loose powders only, whereas in practice, it is necessary to supply solid magnets. On the one hand, embeding in resin can keep this high coercivity but reduce dramatically the residual induction (due to dilution effect) in such a way that the energy product is not improved. On the other hand, sintering produce grain growth and the coercivity drops down to about 0.2 T. In this paper, we are showing that, starting from hexaferrite nanopowder, Spark Plasma Sintering (SPS) allows to produce nonostructured Ba-ferrite with a density close to 90% with almost no reduction in coercivity by opposition to SPS synthesis of nanosize Ba-ferrite in a single step 7,8 .

II. EXPERIMENTAL DETAILS

Barium hexaferrite powders have been prepared by a sol-gel citrate precursor method [9][10][11] . High purity iron(III) nitrate, barium hydroxide and citric acid were used as starting materials with the molar ratio of citrate to metal ions 2:1. Iron(III) nitrate was dissolved in deionized water and quantitatively precipitated with excess of ammonia solution as iron(III) hydroxide. The precipitate was filtered and washed with water until neutrality. Then the obtained iron(III) hydroxide was dissolved in a vigorously stirred citric acid solution at 60-70°C. Barium hydroxide was added according to the desired composition. Polycondensation reaction with ethylene glycol to prevent segregation 9 was not used. Instead, pH value of the solution was adjusted to 6 for better chelation of metal ions 11 . Water was slowly evaporated at 80-90°C with continuing stirring until a highly viscous residue is formed. The gel was dried at 150-170°C and heat treated at 450°C for 2 hours for total elimination of organic matter. Finally, the inorganic precursor with homogeneous The samples have been sintered in a Sumitomo Dr Sinter spark plasma sintering (SPS) machine under a pressure of 50 MPa and neutral atmosphere in a graphite die. The heating rate was 160 Kmin -1 , up to 800°C. This temperature was kept for a duration between 0 and 20 min and then cooled at a rate of 200 K.min -1 .

XRD diffractograms have been recorded using a PANalytical X'Pert Pro equiped with a linear detector and the analysis of spectra was conducted with the Rietveld based software MAUD [START_REF] Luterotti | Materails analysis using diffraction[END_REF] . The morphology of sinterd sample was obsvered with a Hitachi SEM. Density was mesured using standart Archimedes method and the hysteresis loops where recorded at room temperature using a LakeShore vibrating sample magnetometer with a maximum field of 2 T.

III. RESULTS AND DISCUSSION

X-ray diffractogram of the powder after calcination shows all characteristic peaks of M-type hexaferrite (magnetoplumbite) with relatively broad shape. The lattice parameters are very close to those of literature reference samples and the average grain size was found to be about 60 nm (see Table I). This value obtained by fitting diffractograms with MAUD is confirmed by TEM (see Fig. 1). Although some particles are bigger than 100 nm, most of then remains close to 50 nm. A slight decrease in the a axis paramer has been observed together with a increase of the c axis parameter upon sintering but it doesn't seem to be significant as the unit cell volume is almost constant. If this variation could be considered as significant, one would attribute it to internal stresses produced by the sintering process under uniaxial pressure. As expected with the SPS process, the average crystallite size is not much enhanced even after 20 min compared to the calcinated powder. This effect is related to the high speed of the process which allows atomic short range diffusion (involved in sintering process) but not long range diffusion (involved in grain growth process). In contrast, the density changes appreciably with time as it has been observed with most of materials sintered by SPS. A non negligible porosity remains after sintering as it is also seen in the SEM picture in Fig. 2. Only the sample SPS4 is shown but others exhibits very similar features. From these pictures, it is concluded that the material is composed of approximatively 1 µm grains composed of 70-80 nm crystallites with ∼500 nm node pores.

The hysteresis loops of the different samples are shown in Fig. 3. Comparison with conventional coarse-grain sample (commercial) immediatly shows the interest of reducing the grain size down to nm range: the sample measured after calcination in the form of loose powder exhibits a coercive field of 0.56 T, which is twice that of the coarse grain counterpart. In comparison with the theoretical field for an anisotropic magnet, this value is 70%, very close to results obtained by others 5,6 . The rectangularity of the loop is very close to 50% as expected from Néel calculations for a uniform distribution of easy axes. After SPS sintering, the density is already high, even for a very short heating-cooling cycle. A marked increase in saturation magnetization is observed, probably due to uncomplete transformation of the precursors during calcination. However, samples sintered for 0 to 13 min, still have sensitively higher value than the coarse grain sample, and drop to the same value for longer treatment time. This effect is probably due to an excess in oxygen gradually released due to the slightly reducing conditions.

The most stricking effect of SPS processing is the high value of the coercivity. Indeed, sintering reduces the coercive field, by only several tens of mT, so that coercivity values remain very close to 0.5 T for all SPS samples. This feature is due to the fact the crystallite size remains unchanged clearly under 100 nm, much below the single domain limit, D SD = 36µ 0 √ AK 1 /J 2 S = 235 nm given after Kittel formula, where K 1 = 338 kJm -3 , A = 5 pJm -1 and J S = 0.5 T. As a consequence, nucleation of domain walls is impossible and the magnetization reversal should be processed by rotation. Although it is still far from the theoretical limit for isotropic magnets 0.48H K ≈ 0.85 T the coercivity is doubled compared to regular isotropic ferrite magnets. This feature is very important in applications, since it considerably improves the resistance upon demagnetization. From pratical point of view the characteristic in the BH plane is more relevant. Computation of the BH loop B = µ 0 (σρδ + H), with σ the specific magnetization, ρ the bulk specific mass and δ the relative density, yields for the extrinsic coercivity µ 0 H CB = 0.2 T, the remnant induction B R = 0.23 T and the energy product (BH) max = 8.9 kJm 3 compared with 0.15 T, 0.22 T and 7 kJm 3 respectively for regualar sample.

IV. CONCLUSION

It has been demonstrated that SPS technique is an efficient and powerful tool for the sintering of nanostructured isotropic ferrite magnets as the coercivity of the powder can be obtained in dense samples. The energy product have been improved by 30% and the hardness against demagnetization by a factor 2. In addition low temperature sintering meets the requirement of LTCC technology with no need of glass addition 13 .

FIG. 1 .

 1 FIG. 1. Typical hexaferrite particle as received after clacination

FIG. 2 .

 2 FIG. 2. SEM picture of sample SPS4

FIG. 3 .

 3 FIG.3. Hysteresis (specic magnetization as a function of applied induction field) loops of commercial, as-calcinated and SPS sintered samples.

ACKNOWLEDGMENTS

This work was partly supported by the EC FP7 project SSEEC under grant number NMP-SL-2008-214864. A.B. greatly aknowledge ENS Cachan for a 6 month scolarship in the frame of international scolarship program 2008/2009. P. Audebert, Pr at ENS Cachan Chemical department PPSM-CNRS is greatly acknowledged for his advices in


Ayman Alfalou
email: ayman.al-falou@isen.fr

Ali Mansour

A new double random phase encryption scheme to multiplex & simultaneous encode multiple images

In this manuscript, a new approach of multiplexing & simultaneous encoding of target images is presented. Our approach can enhance the encryption level of a classical "DRP" system, by adding a supplementary security layer. The new approach can be divided into two security layers. The first layer is called the multiplexing level which consists in using iterative Fourier transformations along with several encryption key-images. These latter can be a set of biometrical images. At the second layer, we are using a classical DRP system. The two layers enable us to encoding several target images (multi-encryption) and to reduce, at the same time, the requested decoded information (transmitted or storage information).

Introduction

In the last two decades, the development of secure transmission systems becomes the priority of many research and engineering institutions. In fact, many approaches can be found in the literature. Among these methods, cryptography methods are very attractive ones which consist in transforming original information to an illegible format (encrypted information). To retrieve the original information, encryption keys and encryption methods are required. To reduce the processing time, an increase attention has been recently paid to optical encoding methods. In addition, multi-encryption and a very good encoding level including a biometric key (as digital fingerprints) are required in many recent applications. The study proposed in this manuscript deals with the latest mentioned problems.

In [START_REF] Refregier | Optical image encryption based on input plane and Fourier plane random encoding[END_REF][START_REF] Goudail | Influence of a perturbation in a double phase-encoding system[END_REF][START_REF] Unnikrishnan | Optical encryption by double-random phase encoding in the fractional Fourier domain[END_REF][START_REF] Kishk | Information Hiding Technique with Double Phase Encoding[END_REF][START_REF] Neto | Optical implementation of image encryption using random phase encoding[END_REF][START_REF] Towghi | Fully phase encrypted image processor[END_REF][START_REF] Unnikrishnan | Optical encryption using quadratic phase systems[END_REF], methods using a double random phase have been proposed. Different other methods can be found in the literature. The latter methods are based on: Digital holography [START_REF] Javidi | Securing information by use of digital holography[END_REF][START_REF] Tajahuerce | Encrypting three-dimensional information with digital holography[END_REF], Fresnel domain [START_REF] Situ | Double random-phase encoding in the Fresnel domain[END_REF][START_REF] Matoba | Encrypted optical memory system using three-dimensional keys in the Fresnel domain[END_REF], multiplexing [START_REF] Barrera | Multiple image encryption using an aperture-modulated optical system[END_REF][START_REF] Matoba | Encrypted Optical Storage with Angular Multiplexing[END_REF], polarized light [START_REF] Barrera | Multiplexing encrypted data by using polarized light[END_REF], and interferometer [START_REF] Cai | Digital Image Encryption and Watermarking by Phase-Shifting Interferometry[END_REF][START_REF] He | Phase-only encryption and watermarking based on phase-shifting interferometry[END_REF]. We should mention here that Fractional Fourier transform is essential to perform some optical encryption algorithms, further details can be found in [START_REF] Xin | Affine cryptosystem of double-random-phase encryption based on the fractional Fourier transform[END_REF][START_REF] Hennelly | Image encryption techniques based on fractional Fourier transform[END_REF][START_REF] Liu | Double image encryption based on iterative fractional Fourier transform[END_REF]. To support modern applications, multiple image encryption methods have been recently proposed [START_REF] He | Multiple image encryption and watermarking by random phase matching[END_REF]. The main idea of the multi-encryption methods consists in encoding the phase and the amplitude of two original images and combines them into a single image.

To meet the requirements of modern applications with a high encoding level, a modified and an improved version of a Double Random Phase encryption system "DRP" is proposed in this manuscript. DRP systems are very attractive thanks to their simple implementation, robustness and their easily application on several different image formats (B&W, gray level or colored images). To increase the security level of a classic DRP system, we firstly use several and independently keys. In fact, we are using two different kinds of image-keys: random and structural images. Secondly, we are using different multiplexing algorithms. We should mention that at least one image-key can be a biometric image. Our modified method can be summarized as following:

• The first level consists in multiplexing several target images using a set of iterative Fourier transformations. In this stage, several encoding-keys, including known biometric images, should be used. This level can be done off-line (numerically) to avoid the complexity of an all optical implementation.

• The second level, more traditional, is based on the use of double random keys.

We should mention that the DRP and the IFTA are two techniques well know in the literature. Should we mention here that, to the best of our knowledge, these two techniques have been never employed in tandem. Besides that, our new algorithm has shown very good results in various scenarios: using mono-dimensional signals (as speech, music and other acoustic signals)

as well as bi-dimensional data (as black and white, grayscale or colored images). Many experiments have been conducted to corroborate the performances and the convergence of our approach.

The principal idea of our approach consists in using two encryption levels to avoid implementing a simple phase mask as the input of a classical DRP. In fact, by using IFAT and a known image (such as the fingerprint), we can encrypt at same time the amplitude and the phase of target image or mixed images. By doing so, mainly three advantages can be obtained:

1-Cracking and hacking the encrypted images become harder. Let us imagine the case when a hacker may crack the DRP key, i.e. the second mask which is the random phasis, he could still not obtain the target image as it is protected by the first mask and the other auxiliary keys (i.e the mixed images).

2-All acts of piracy on the encrypted image could affect the first mask, i.e. the hidden fingerprint. In this case, we can easily notice if the received image has been intercepted and modified.

3-This approach could be also used as a water-marketing technique. Some useful information (copyright, version, etc) can be easily hidden in the encrypted image. This manuscript is organized as follows: Section 2 describes classical DRP systems. In section 3, our modified algorithm is presented. To corroborate our new approach, simulations are shown in that section. Section 4 presents the multiplexing and simultaneous encryption of several target images. Finally, conclusions and future works are given in section 5.

Classical Double Random Phase "DRP": Mono-encryption system

In the last two decades, various optical encoding approaches have been proposed. These works can be traced back to the original work proposed initially by Ph. Refergier and B. Javidi [START_REF] Refregier | Optical image encryption based on input plane and Fourier plane random encoding[END_REF]. In their approach, Ph. Refergier and B. Javidi proposed an image encryption scheme based on the modification of the spectral distribution. Without any prior information about the spectral modification or the target image, the image decoding cannot be done. The main idea of their approach, see figure 1, consists in inserting two encoding-keys (random phase) in a setup called "4f". The proposed approach consists mainly of three stages:

• The first key, i.e. the first Random Phase Mask RP 1 , is multiplied by a target image (the image which should be encrypted). The latest image should be displayed in the input plane of a "4f'' setup and lighted with a parallel coherent light resulting from a Laser generator. This procedure introduces a first modification to the spectrum of the target image.

• The second key (i.e. RP 2 ) is directly inserted into the image spectrum in the Fourier plane. The multiplication of the RP 2 by the spectrum obtained at the first stage can introduce a second modification into the spectrum of the target image.

• Finally, a second optical Fourier transformation is carried out using a second lens to obtain the encoded image in original 2D space images. This approach can generate a high-noise encrypted image. We should mention that the simplicity and the easy implementation of this approach make it very attractive. On the other hand, its drawbacks are emphasized in the literature [START_REF] Frauel | Resistance of the double random phase encryption against various attacks[END_REF][START_REF] Carnicer | Vulnerability to chosencyphertext attacks of optical encryption schemes based on double random phase keys[END_REF][START_REF] Peng | Known-plaintext attack on optical encryption based on double random phase keys[END_REF]. Recently, the authors of [START_REF] Frauel | Resistance of the double random phase encryption against various attacks[END_REF] meticulously analyzed this approach and they proposed a large number of possible attacks. In the same work, they suggested few propositions to increase the encoding rate of DRP systems: One should use a huge number of encoding keys and we should very often change them. In this manuscript, a new scheme satisfying their propositions is proposed. In addition, another security layer has been added. Further details are provided in the following.

Multiplexing and simultaneous images encryption: multiencryption DRP system

In specific applications (such as military applications, police identification procedures, online banking systems, etc.), a robust encoding systems are highly required. Here, we reinforce the encoding rate of a classical "mono-DRP" system by adding another security layer. The main idea of our approach (multi-encryption system), consists in multiplexing target images (I 0 , I 1 , I 2 etc which are images to be encrypted) with at least a key-image. The latest procedure realizes our first security level, see figure 2-a. At our second encoding level, we introduce the outcomes of our first encoding level as the input of a classical DRP system. The result of the first layer is a complex image " i i i e I ϕ " which is completely different from the original target image (figure 2b), where "I i " is a known image used in our DRP stage. This image could be a bio-metric image such as a digital fingerprint. Finally, as shown in figure 2, the output of our system is a two layer encrypted image " c i c e I ϕ " obtained by two independent sub-systems.

To simplify the principle of our new scheme and without any loss of generality, only two images (I 0 and I 1 ) will be used in the first encoding level: I 0 is a grayscale digital image and I 1 is a known white-black image of a fingerprint or a photo. Before applying the iterative Fourier transform algorithm, the target image I 0 should be multiplied by a pure random phase function [0,π].

Encryption

The synoptic diagram of our approach with two encoding independently levels is illustrated in figure 3. The target image "I 0 " is introduced at the input of the first level (figure 3).

In the first stage, the image "I 0 " is multiplied by a selected phase function: " (amplitude of the target image spectrum I 0 modulated with a function phase) converges towards "I 1 " (with I 1 could be a known black and white image as fingerprint (its own fingerprint for example) or grayscale image as a photo, etc). After "N-iterations", we obtain at the output of our first level "

n i e I 1 1
ϕ " a digital fingerprint modulated with a function phase (Equation 1):

( )

n n i n i e I FT e I 0 1 0 1 ϕ ϕ = (1) With ( n 0 ϕ & n 1 ϕ
) are two pure phase functions necessary to tend the amplitude of the target image spectrum towards the desired digital fingerprint. The algorithm, used to find these two functions, will be detailed in the following paragraph. Then, this complex image "

n i e I 1 1
ϕ " is introduced at the input of the classical DRP (second level). Thus, this complex image is multiplied by a first random phase key " ( )

1 1 1 A n i e I ϕ ϕ +
". After a Fourier transform, we multiply this spectrum with a second random phase key " ( )

Mathematical Concepts and Convergence

In [START_REF] Liu | Double image encryption based on iterative fractional Fourier transform[END_REF] Z. Liu and Sh. Liu proposed a double encryption based on iterative fractional Fourier transform. This approach is similar to the first layer of our encryption system. In fact, in this manuscript, an iterative Fourier transform is considered. To get benefit of the high security level offered by these methods and the simplicity of the Fourier transform which can be done optically using a single convergent lens. The convergence of the iterative Fourier transform could indeed be similar to the convergence of the algorithm proposed by [START_REF] Liu | Double image encryption based on iterative fractional Fourier transform[END_REF]. We should mention that the convergence analysis has not been considered in [START_REF] Liu | Double image encryption based on iterative fractional Fourier transform[END_REF]. Indeed, the authors showed the convergence of their algorithm through the outcomes of their conducted simulations.

In this section, this analysis of the convergence will be discussed. Without loss of generality and for seek simplicity, the convergence of the iterative 1D Fourier transform is considered instead of the 2D Discrete Fourier transform. The case of 2D Fourier transform required in our algorithm is just straightforward [START_REF] Folland | Fourier Analysis and Its Applications[END_REF][START_REF] Pinsky | Introduction to Fourier Analysis and Wavelets[END_REF][START_REF] Ioana | Digital signal processing using matlab[END_REF]. Fig. 4 shows the results of applying our algorithm on speech signals. We should just mention that the absolute value of the signals should only be considered instead of the original real zero-mean signals. Fig. 5 shows the convergence of the Mean Square Error (MSE). Let x(t) and Y(f) respectively be the target signal and the key signal. The main loop of the iterative Fourier transform can be simplified as following; further details can be founded in the Matlab code provided in appendix A:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) end t y angle t f j f Y TF t y f X angle f t j t x TF f X T i for S S = = = = = - 1 1 1 1 1 exp exp : 1 ϕ ϕ ϕ ϕ (2)
By considering the above equations, we can roughly consider that at each iteration the amount of information in " ( )

t 1 ϕ
" and related to Y(f) will be increased [START_REF] Yamaguchi | Image reconstruction only by phase in phase-shifting digital holography[END_REF]. At the same time, ( )

t 1 ϕ
will also contain more information about x(t). In the following, we will give some hints to prove the previous basic reasoning.

The mean square error (MSE) is considered as the convergence criterion of our algorithm. In this case, one should prove that the nonlinear "angle" function of the Fourier transform of target signal is an optimal solution of MSE:

( ) ( ) ( ) ( ) ( ) ( ) 2 1 exp t x f j f Y TF E Min t - - ϕ ϕ (3)
It is obvious that the minimisation of equation ( 3) is a minimization of energy functional [START_REF] Rubio | The global control of nonlinear diffusion equations[END_REF][START_REF] Langhaar | Energy Methods in Applied Mechanics[END_REF][START_REF] Asaki | Abel inversion using total variation regularization: applications[END_REF]. We should mention here that there is no general or unique solution for such problem. However, few approximation numerical methods can be founded in the literature [START_REF] Van Kan | Numerical Methods in Scientific Computing[END_REF][START_REF] Atkinson | Theoretical Numerical Analysis[END_REF][START_REF] Alghofari | Problems in Analysis Related to Satellites[END_REF]. The main idea of the latter methods consists in approximating the solution using a projection in a set of kernel functions. These numerical methods can't be used to prove that the nonlinear "angle" function (i.e. the phasis of the Fourier transform) is a solution that belongs to the set of optimal solutions of equation ( 3). However, by using Taylor series and the previous methods, we can approximate the solution. The proof using the minimisation of energy functional is beyond the scope of this manuscript.

We mention that intensive simulations have been conducted to corroborate the performance and the convergence of the proposed algorithm. In following sections, some simulations are shown and discussed. ρ " and the key-image "I 1 " (which is a black and white known image or it could be a grayscale image, as its own digital fingerprint or a photo). The Mean square error "MSE" is widely used as a convergence criterion. Therefore in our algorithm, we evaluated the MSE between the amplitude of the spectrum " 1 ρ " obtained at the first stage and the image-key "I 1 " as that is shown in equation ( 4), with (M, N) are the pixel size of the images.

Algorithm

( ) ( )

2 1 1 , , 1 ∑∑ - × = N i M j j i I j i N M MSE ρ ( 4
)
The encryption is considered achieved when the value of obtained MSE becomes lower than a preselected threshold "C". The value of the threshold should be wisely adjusted. In fact, a very small value of "C" means a better performance but more computing efforts and processing time are needed. If the convergence is achieved, the first level output image can be written as:

"

n n i i e I e 1 1 1 1 ϕ ϕ ρ = ". Otherwise, if ( 1 1 I ≠ ρ
), we change " 1 ρ " by "I 1 " and we apply an inverse Fourier transform. This Inverse Fourier transform enables us to return our obtained image in the image domain. In a similar way, we should compare the amplitude " 0 ρ " of the inverse Fourier transform with the target image "I 0 ". For that, we estimated another MSE between " 0 ρ " and "I 0 " and we compared its value with a second threshold "CC". If " 0 0 I ≠ ρ ", we replaced " 0 ρ " by "I 0 " and another iteration should be done. The convergence of our algorithm is achieved when a measurement error applied to image I 0 and respectively to I 1 becomes less than a fixed threshold "є 0 = C" and respectively "є 1 = CC".

Effect of the iteration number

We should mention here that in all conducted simulations, good results have been obtained in less than 50 iterations. For this reason, we implemented a simplified version of the proposed algorithm mainly to decrease the processing time. The simplified version is given in appendix A. in that version a fixed number of iterations is used instead of the output of the MSE.

However, the first version can obviously be implemented. In order to fix the needed number of iterations to obtain optimal performance, a heuristic analysis is conducted. In this section, the effect of the iteration number on the performance of our proposed algorithm is emphasized.

The following example shows the effect of the iteration number on the quality of obtained images using the algorithm described in Fig. 6. Let us consider two images: A black-white image and a gray level image; see Fig. 7. Figure 8 shows the MSE values between the origin image and the image obtained after "500" iterations and "3000" iterations. The previous equation is the solution of equation [START_REF] Unnikrishnan | Optical encryption by double-random phase encoding in the fractional Fourier domain[END_REF]. As mentioned before, in many simulations, 50 iterations were enough to obtain satisfactory or good results. Hereinafter, we select 500 as the maximum number of iterations.

Validation by numerical simulations

Many simulations were conducted to show the effectiveness and the performance of our algorithm. Figure [START_REF] Tajahuerce | Encrypting three-dimensional information with digital holography[END_REF] presents one target image and one known image-key, both of them are grayscale images: The target image is "Florent" and "Lena" is the known key-image. After N = 25 iterations, our algorithm converges and we obtain the image shown in figure (9-c) with a "MSE = 0.0014". After "500" iterations, a slightly improved image was obtained, with "MSE = 1.2869 10 -4 " (figure 9-d). In the second stage of our encryption approach, the outcome of the first stage, i.e. the image shown in The final result presented in Fig. 10 shows the high quality of encryption that could be obtained by two stage encryption scheme. The iterative Fourier transform could reinforce the encryption done by a classical DRP system. In the following, the influence of algorithm parameters will be investigated.

Decryption

The main advantage of our proposed algorithm is its ability to be implemented using an all-optical system. In fact, once the convergence is attained, our system can be considered as a simple Fourier transform applied to the product by phase functions of input images. Besides, the classical DRP system could be realized using an all-optical system. The decryption system could also be implemented by mainly using an all-optical system. The latter system is described in the next section. We should mention here that all-optical systems have many advantages comparing to classical digital systems. The main advantages are:

1-High speed (almost the light speed) processing.

2-A Fourier transform or an inverse Fourier transform can be realized using a simple convergent lens 3-Compact systems: While classical digital systems require digital cameras, processing unites, memories, power supplies, etc.; optical systems can be made very compact. A powerful encryption system becomes useless without the existence of a corresponding decryption system which allows us to retrieve the hidden transmitted information. Hereinafter, we will describe a decryption system that could retrieve the target image from the final two level encrypted image ( ) • Then, we multiply the obtained spectrum by the second phase random key conjugate " ( )

2 A i e ϕ - ".
• Later on, we should carry out a second inverse FT.

• The outcome of the inverse Fourier transform should be multiplied by a second phase random key conjugate " ( ) By introducing a known key-image (as a digital fingerprint) at the first stage of our algorithm, two main benefits could be observed:

• One could perform the multiplex of various target images to reinforce the encryption result.

• Any unauthorized modification on the transmitted encrypted image ( ) Fig. 12 presents the outcomes of our encryption-decryption systems. These results were obtained after "25" iterations. We should mention that we succeeded in decrypting the encrypted image with a very low MSE =0.0173. The low value of MSE proves the good decryption performances obtained by our approach. This low value makes possible the encryptiondecryption of multiple target images.

Hacking

Previous sections show good experimental results which are obtained using our proposed algorithm with two layers of encryption security. In this section, the robustness of our approach to cracking or hacking procedure is considered. Several modifications of classic DRP can be found in the literature [START_REF] Alfalou | Optical Image Compression and Encryption Methods[END_REF]. Most of them can be straightforward included in our system to improve the robustness of our DRP stage. We should mention that the major weakness of a classic DRP is the possibility for a hacker to crack the second key, i.e. the random phase mask.

In this case, the hacker can easily retrieve the decrypted image. This problem could be solved using our approach. In fact, at the first stage of our algorithm, the principal image has been intentionally modified. Therefore, without priori information, the hacker can't obtain the other keys (the mixed images) by simply intercepting and processing the encrypted image. On the other hand, without the knowledge of all used keys, the original image can't be clearly retrieved.

In order to prove the latter statement, many simulations results have been conducted.

In the following simulation, the target image is a real greyscale image shown in figure (13-a).

Let us consider "Lena's" photo as the known image, i.e. the first key shown in fig ( 13-b). The encrypted image is shown in figure (13-c). If the hacker has successfully found the key of the classical DRP, he will be able to find the known image up to a phase function as shown in figure 13-d. By applying a Fourier transform on the retrieved image, one can obtain the image shown in figure 13-e. This result clearly shows that the original image can't be obtained without knowing the second key of the DRP. The task of hacking becomes more and more difficult when the number of mixed images is increased.

Multiplexing and simultaneous encryption of multiple target images.

In previous sections, the case of two target images has been discussed. Here, a most general case, with a set of target images, is considered. Hereinafter, the extension of our approach to encrypt simultaneously several target images is described. The encryption system is based on DRP-Multi-encryption. To illustrate the power of our proposed approach comparing to the classical Mono-DRP system, we compared the outcomes of both algorithms applied to encrypt same real target images (I 1 , I 2 ,…., I m ).

Mono-DRP system

From a practical point of view, the encoding of multiple target images can be simply obtained by successively encrypt various target images using the classical mono-DRP system. Two cases can be distinguished, see figure [START_REF] Barrera | Multiplexing encrypted data by using polarized light[END_REF]:

1. The first case illustrated in figure (14-a) consists in the encryption of all target images (I 1 , I 2 ,…., I m ) using the same phase random keys " ( ) ( )

1 2 & A A i i e e ϕ ϕ
". In the decryption system, all encrypted images However according to [START_REF] Frauel | Resistance of the double random phase encryption against various attacks[END_REF] this case does not offer a high encoding rate because the same encryption keys are used in the whole process.

2. The second case, shown in figure (14-b), consists in using different encryption keys for every target image. This case offers a higher encoding rate. However, it seriously increases the required transmitted information. In fact, the decryption stage necessities all encrypted images as well as all keys ( ) ( ) ( )

2 22 12
, , ,

Am A A i i i e e e ϕ ϕ ϕ L .

The proposed Multi-encryption-DRP system

Using classic DRP to encrypt a set of N t target images has two major drawbacks:

• 2 N t random phase keys should be used.

• N t encrypted images should be processed.

It is clear that this procedure involves huge amount of data which requires huge memories, enormous transmission time and computing efforts. These problems can be solved by considering our approach. In fact, the target images could be mixed together to form one mixed image using the first layer of our approach. Later on, the mixed image should be considered as the input of standard DRP. The mixing procedure can be achieved using various methods. One of them is the method proposed in [START_REF] Soualmi | Optical image compression based on segmentation of the Fourier plane: new approaches and critical analysis[END_REF] and it is based on the fusion of relevant information in the Fourier plane. However, the latter approach doesn't achieve good performances. In fact, this approach is mainly based on the extraction of critical spectrum information from each target image. Therefore, the spectrum space should be shared by all these spectrum information. When the number of target images increases the reserved spectrum information for every target image will decrease. This can result in important deterioration of reconstructed images [START_REF] Alfalou | Optical Image Compression and Encryption Methods[END_REF].

Hereinafter, an alternative method to mix multiple target images is proposed. A multiencryption is performed by multiplexing various target images together. This multiplexing is realized using our algorithm which is based on iterative Fourier transformations. We called this approach the Multi-encryption-DRP system, see Figure [START_REF] Cai | Digital Image Encryption and Watermarking by Phase-Shifting Interferometry[END_REF]. Our encryption system consists of two stages:

1. At first, various independent target images should be jointly equalized to a single image " m i m e I ϕ ". This multiplexing can be done using adequate phase functions which are making it possible to write:

( )

1 1 - - = j j i j i j e I FT e I ϕ ϕ
. The algorithm is discussed in section 4.3.

2.

A second encryption level is obtained as the output of a traditional DRP-system.

In our approach, the mixing of various target images can result in a unique image which contains all necessary information for the decryption process. The obtained mixed image should be used as the input of a classical DRP. In this case, we are getting double encryption layers. Our simulation results show very good performances of our approach. Other aspects of our algorithm, such as the convergence and the compression ability, are also studied.

Multi-Layer Encryption-Decryption Algorithm

Hereinafter, an extend version of our algorithm proposed in figure 6 is presented, see Fig. ( )

1 ' 2 1 2 ϕ ϕ i i e I FT e I = .
• Another iterative Fourier transforms should be carried out to find two new phase functions ( ) , corresponding to the second image I 2 , is evaluated.

• Using similar way, the other phase functions ( )

Convergence of Multi-Layer -Encryption-Decryption Algorithm.

Various simulations have been conducted to show the performance and the convergence of our algorithm as well as the impact of an increasing number of target images on the quality of decrypted images. It is obvious that an increasing number of target images is the most important issue to the stability, the convergence of our algorithm and the quality of reconstructed images.

In this section, we emphasize the impact of this number. Using three images (two target images and one key image), encryption and decryption parts of our algorithm have shown very good results: very weak values of MSE, 1.014*10 -4 and 5.15*10 -5 , are obtained. When the number of target images is increased to three, once again, the convergence is reached and good results are observed; see the second row of table 1. The encrypted image is shown in the third row of the same table. By increasing further the number of target images, we notice a small deterioration of decrypted images. However, the convergence has been always obtained and the quality of the reconstructed images could be improved by increasing the number of iterations.

Other simulations have proved that our algorithm is more sensitive to the bit number. To better compress the involved images, the number of bits should be decreased. On the other hand, the reduction of this number has a big impact of the quality of reconstructed images. This effect has been emphasized in our recent study presented in SPIE conference Europe Security and Defence / September 2009 in Berlin. A resume of that study is given in the following. 1): The outcomes of our multi-encryption approach: multiplexed, encrypted, decrypted and demultiplexed images.

Comparison between two encryption techniques: Successive classical DRP and the proposed multi-encryption DRP.

It is clear that in our approach, we considerably reduce the size of transmitted information necessary to carry out the decryption of multiple target images. To highlight this fact, we compared the two techniques described in this section to encrypt and decrypt a set of real target images. Many simulations have been conducted. Table [START_REF] Goudail | Influence of a perturbation in a double phase-encoding system[END_REF] shows one of the obtained results. In Table [START_REF] Goudail | Influence of a perturbation in a double phase-encoding system[END_REF], four target images (256,256) pixels have been used.

The results presented in table [START_REF] Goudail | Influence of a perturbation in a double phase-encoding system[END_REF], show that we considerably reduced the size of encrypted information (storage information and/or transmitted information), needed to encrypt a multiple target images. Indeed, a rate factor of 2 has been observed. Finally, we should notice that the compression ratio (between the two previously described techniques) increases according to the number of images. We are conducting studies to evaluate the reconstruction quality of the different target images according to the compression ratio.

Reduction of the bit number

In this section, we emphasize the impact of reducing the bit number on necessary information required to reconstruct target images. To clarify our idea and simplify the study, we only consider two grayscale images (I 0 (1) and I 0 (2)) with (256x256) pixels, see table [START_REF] Unnikrishnan | Optical encryption by double-random phase encoding in the fractional Fourier domain[END_REF]. These images should be quantized using firstly a huge quantization number (up to 64 as usually used in Matlab, the quantization number is given in the first column of table 3). Then, lower quantization numbers are considered till reaching a binary encoding (with a single bit). The second column of table 2 contents the compression ratio achieved for every phase function. The compression rate is defined as the ratio between the size of the final phase function (with respect to the quantization number) and the original size coded on 8-bit (equation 6).

Conclusions and prospects

In this manuscript, a new all-optical and two level encryption/decryption system is proposed.

The new proposed system has several advantages compared to the classical system. Indeed, while the new algorithm is faster, it can achieve two level encryption securities, enables us to obtain better compression / transmission information rate, realizes a joint encryption of various target images and enables us to use biometrical key images. Many simulations have been conduct to corroborate the effectiveness and the good performance of the proposed algorithm.

Our future works will emphasize the relationship between the reconstruction qualities of different target images according to the increase of compression ratio.

  out, a first Fourier Transform "FT", we obtain " 11 1 ϕ ρ i e ". Then, we carry out successively several Fourier Transformations (FT( ) n ) by modifying for each iteration the phase function until " 1 ρ "

  . Finally, a last Fourier transform gives, in the output of our system, the doubly encrypted image with two different and independently layers "

Figure ( 6 )

 6 Figure[START_REF] Towghi | Fully phase encrypted image processor[END_REF] shows the flowchart of our algorithm which can mix multiple target images

Figure 8

 8 Figure 8 emphasizes the strong convergence of our algorithm. Once the MSE becomes less

  Fig 9-c or Fig 9-d, is used as the input of a classical DRP. In the DRP part, two random phase masks are used. The final encrypted image is shown in figure (10).

  ϕ . At the receiver, we assume that various encryption keys: known. Fig.11presents a synoptic diagram of our decryption system:• At first, an inverse Fourier Transform of the encrypted image ( )

•

  this step, the digital fingerprint, modulated with a phase function " ( ) Finally, the decrypted image I 0 could be obtained as the modulus of another inverse Fourier transform "

  ) could be easily detected at the reception stage. Indeed, any vandalism attempt (attacks) results in a modification of the encrypted image ϕ . The latter modification could be detected at the output of the first decryption stage as the digital fingerprint is well known to all authorized users.

16 ...I•ϕ

 16 Let us considered m target images (I 1 , I 2 ,…., I m ). At first only two target images (I 1 , I 2 ) is considered as the input of the first layer. At the convergence, we obtained two other images The second step is similar to the first one except for the two considered images The convergence of the second step generates two new images ( ) be continued until the last image, i.e I m , has been taken into consideration.The final obtained image ( ) ϕ is used as the input of the DRP algorithm. The outcome of the DRP is the doubled encrypted image. The multi-encryption can be resumed using the following steps: At first, two phase functions ( ) have been adjusted to satisfy the following relationship:

•ϕ:

  After the fourth step, a new image is obtained. This latter is used as the input of our second level (classical DRP) after multiplying it with a first phase random mask. The output of the DRP, i.e. the final encrypted image final encrypted image contains different keys and necessary information to decrypt target images.The decryption system necessities the knowledge of: ϕ : encrypted image at output of our multi-encryption system, • 2 A ϕ : the second random encryption key using a traditional DRP system, • 1 A ϕ : the first random key of traditional DRP system, Various phase functions which have been used in the multiplexing of various target images.

  Hereinafter, a comparison between a standard DRP and the proposed algorithm is given. This comparison takes into consideration the required information, the image quality among other performance indexes to encrypt-decrypt multi target images. First target image Second target image Third target image Key-image Convergence between I1 & I2 Convergence between I2 & I3 Convergence between

  4 of table[START_REF] Unnikrishnan | Optical encryption by double-random phase encoding in the fractional Fourier domain[END_REF] present different reconstructed images with their corresponding MSEs values. We should mention that an optimal bit number should be used. This number should be resulted as a compromised solution to achieve good compression ratio and good encryption-decryption quality.

FIG. 1 :FIG. 2 :FIG. 4 :FIG. 5 :FIG. 6 :FIG. 7 :FIG. 8 :FIG. 9 :FIG. 11 :FIG. 12 :FIG. 15 :FIG. 16 :FIG

 1245678911121516 FIG. 1: Synoptic diagram of classical Double Random Phase encrypted system: "Mono-

Table ( 3

 ( ): simulation results

FIG. 1: Alfalou and Mansour

FIG. 2: Alfalou and Mansour

FIG. 3: Alfalou and Mansour

FIG. 5: Alfalou and Mansour

FIG. 7: Alfalou and Mansour

FIG. 10 Alfalou and Mansour

FIG. 13 Alfalou and Mansour