Self-imaging through incoherent to coherent conversion

Maria del Carmen Lasprilla A.* Alexandra Agra Amorim* Myrian Cristina Tebaldi Néstor A. Bolognini _ Centro de Investigaciones Opticas, ClOp (CONICET, CIC) Casilla de Correo 124 1900 La Plata, Argentina Eherent to coherent converter. For this purpose, a photorefractive BSO crystal that becomes uniformly birefringent due to an external applied grating that locally modulates the induced birefringence is registered. In -Ne laser is utilized. The ellipticity of the light exiting the crystal will be determined izer, the coherent output will reproduce the incoherent input. In this way, under free propagation, coherent replicas of the grating will be obtained. © 1996 Society of Photo-Optical Instrumentation Engineers.


Introduction 2 Theoretical Analysis
When a periodic transparency (for instance, a Ronchi grat ing) is illuminated with a monochromatic collimated beam, the grating amplitude transmittance is periodically repro duced along the propagation direction.These replicas under free propagation are called self-images. 1 The spatial fre quencies of self-i located on the so-called Montgomery rings.2A Ronchi grating of spatial period d fulfills the men tioned condition and therefore generates self-images.It should be mentioned that lateral periodicity of an object is sufficient but not necessary for self-imaging to occur.When the grating transparency is illuminated with a plane cated at distances z T= = 1,2,3,...) from the grat ing reproduce the input grating.2,3Furthermore, the diffraction patterns located between the transparency and the first self-image are repeated between any two successive self-images.That is, the diffracted field is longitudinally periodic.
Photorefractive crystals such as the sillenites combine real-time response, reversibility, and high sensitivity in the visible portion of the spectrum that make them very attrac tive for many optical information processing and related applications.4In this work, an incoherent to coherent con verter arrangement that utilizes a sillenite photorefractive crystal (BSO) to encode a periodic transparency is pro posed.To this end, the write-in step is detailed in the fol lowing section.Afterwards, we discuss the coherent read out process that allows the replicas of the input to be produced.
Let us consider the experimental setup schematized in Fig. 1.A transparency O, which consists of a grating of binary amplitude with an opening ratio equal to 0.5, is illuminated with a white light source S through the condenser lens L j .
tained in the crystal by means of the lens L2.The directions (110), (001), and (110) of the crystal coincide with the X, Y, and Z axes, respectively (laboratory axes).
The crystal exhibits the linear electro-optic effect.Thus, if a voltage V is applied between (110) faces separated a distance Lx , it becomes uniformly birefringent due to the = optical axes are at ±45 deg with respect to the (110) direc tion.The crystal has a strong photoconductivity in the range of wavelength between 400 and 550 nm.A filter centered at 530 nm (AX ±3 nm) is placed in front of the whitereceived by the crystal, that is, the Ronchi grating projected by the lens L2, will create photocharges that will drift due to the external field Ea into the dark regions, where they are trapped.These charges develop a space-charge field that partially compensates the external field in the illuminated areas.In this way, the induced birefringence will be locally modulated according to the light pattern, producing a local change in the ellipticity of the transmitted light.
In order to compute the resulting total internal field E in -distribution, a single-spatial-variable approach to these equations is possible: Jt =s(x)------7----+ 7 " 77" (2) where x and t are spatial and temporal variables, respec tively, J is the current density, and N is the total freecarrier concentration, which includes the contribution due to the light pattern and the contribution ND of the free charges in the dark.Furthermore, e is the electron charge, ¡x is the mobility, r is the free-carrier lifetime, and D is the diffusion coefficient.The usual assumption is made that the generation rate g(x) is proportional to the light pattern I{(x) received by =, normalized light intensity and g0 is proportional to the highest intensity value / 0. Also, the photoinduced space charge field Esc must fulfill The gradient contribution ôN/àx in the diffusion term of Eq. ( 1) can be neglected because of the smooth variation of -= concern.
Under these conditions, from Eqs. ( 1), (2), and (3), the following total internal field results: where the expression for N(x) is and G= rg0/ND.The value of K is obtained as For this configuration, the calculated expression for the modulated birefringence is (7) where r 41 is the electro-optic coefficient and n0 is the re fractive index without induced field.
In the readout or reconstruction process, a collimated Y--Ne laser is utilized.In order to avoid destructive readout of the stored information, a wavelength that is outside of the spec tral sensitivity range of the crystal is employed.Then, let us consider the light amplitude transmitted by the crystal in the readout process.It will be necessary to compute the produced changes in the light ellipticity.
The BSO crystal exhibits rotatory power per unit length p (\) (where X is the wavelength employed for readout), which combines with the induced birefringence modulation ôn{x).This means that when light of a given ellipticity enters the crystal, in order to evaluate the change in the polarization state, both effects must be computed.
-dinate system of the two induced axes results:

(ID
In the experimental setup that is schematized in Fig. 1, the polarizer P j is placed with its pass plane parallel to the Y axis, and the polarizer P2 is oriented with its pass plane making an angle ¡3 with respect to the X axis.The Jones matrix of P2 is Finally, the incoherent input I¡(x) has been converted to a coherent output Ic(x,/3), where the parameter governs the contrast, as will be detailed in the next section.It is clear that Sn(x) has the same spatial period as 7,00.Also, by using Eqs.( 10 Thus, when the linearly polarized collimated readout beam propagates through the crystal, it will emerge periodically modulated in its ellipticity.Those parts of the readout beam = will emerge linearly polarized with the plane o f polarization ro tated an angle A2=pLz with respect to that of the entering light.The remaining part of the emerging wave will be elliptically polarized, because it has traveled through the periodic stripes where Sn is not negligible (write-in dark regions).
Returning to Eq. ( 14), we have This expression analyzes the transmitted intensity of polar izer P2 into two parts: a linearly polarized part of the read out wavefront exiting the crystal, l c2 (yS), and an elliptically polarized part, Ic\{0).Figure 2 shows 7cl(y3) and Ic2{0).
The phase shift between them depends on the value of Sn x.
Then Ic(x,/3) can be rewritten as Consider the case /3-(3x=A2I =0), the system is in extinction and no light passes through the analyzer P2 ■ But ing emerging readout intensity results: is displayed in Fig. 3, where it can be observed that the = 1, is reached, because expression (23), this implies equality of the linear and the fills this constraint is = l i W , we obtain so that the output of the polarizer P2 is uniform.Another important value of ft is that for which Ic\{0) is Then ( 22) That is, the emerging wave of the readout beam will be alternately polarized linearly and elliptically with period d.
The analyzer P2 will block out the linearly polarized parts and will select a component of the elliptically modulated portion o f the wavefront.Alternate dark and bright stripes will exit the polarizer.In this way, a coherent contrastreversed replica of the input is obtained, converting the -out beam is high and that free propagation takes place.In this case the Fresnel intensity distribution along the Z axis will be the same as if the original transparency, shifted a half period, had been located at the position o f the polar ized P 2.
The visibility of the readout coherent image Ic(x,P), defined as By considering Fig. 2 it is clear that Ic\(p 3) < ^IC 2(Pi) and Eq. ( 27) can be written as: In this case, a direct-contrast output is obtained.
As p is rotated from p x to /?3, more and more of the tal will be transmitted through the analyzer P2, and less and less of the elliptically polarized parts will be transmit ted.
izer P 2 was oriented at P I -gation.In the following, all distances are measured from the crystal plane.Figure 4 The pictures displayed in Fig. 5 were registered at the distance z T for three values of the angle y3. Figure 5(a) -i =-=self-= i By comparison of Fig. 4(a), 4(b), 4(c) with Fig. 5(a), dinal displacement could be comparable to a certain value of y8 at a fixed plane.
The sequence of Fig. 6 was obtained for values of (3=i the crystal.This would be comparable to obtaining three -= f3y.

Discussion and Conclusions
By observing the visibility curve it can be concluded that -= contrast version at y3=fcertain applications.The converter itself can be planned as a subsystem of an optical device in which control of the visibility is a key point for subsequent processing.In order to achieve this situation, the condition established by Eq. ( 28) must be fulfilled.This implies a high eccentricity of the elliptical light that originates the component / cl(y3) [see Eq. ( 20)], implying a low value for Snl .As Snx increases, ÁyS=/?3| /?= 's ability to =yS].Let us suppose that the appropriate conditions are fulfilled.Then, for instance, let us suppose that a reference Ronchi grating is placed beyond the polarizer P2.Hence, the moiré pattern that arises between a readout self grating can be employed to detect phase objects.9Further more, the contrast control of the self-images could contrib ute to the easy visualization of those objects.Also, if a birefringent material is located between the analyzer and the reference grating, the shift in the moiré fringes can be used for quantitative determination of the birefringence.
In some circumstances, for subsequent processing, it could be useful to have a Talbot object in the form of an incoherent input, that is, a TV picture, 2-D array, etc.In this case, an incoherent-to-coherent converter as was pro posed in this paper could be appropriate.Some constraints arise from the nature of the converter.It is limited by the crystal response time, which depends on the intensity of the input signal.Furthermore, geometric tern.Also, a compromise must be considered between the increasing of the crystal thickness to improve the sensitivity and the corresponding constraints on the spatial resolution tailed at the beginning of the previous section, the available proved several fold by reducing the crystal thickness (Zz) and increasing the / number of lens L 2 ■ On the other hand, it should be mentioned that because of the finite crystal dimension Lx, the diffracted high harmonics in the readout " walk o f f not take part in the image formation.This inherent draw -tion of the input.
Finally, we should emphasize the nonholographic nature tage.A device such as a photorefractive incoherent-tocoherent optical converter (PICOC) could be employed as well, but it has a holographic support10 that considerably complicates the experimental arrangement.Besides, it lacks flexibility for controlling the contrast of the output.

3
Experim ental R esults = Ly-10 mm and Lz= 3 I tween their (110) faces.The intensity of the incoherent -mW He-Ne laser (\= 6 3 3 nm) was employed for the readout =22 deg/mm.=2.54, r 4I = I the input transparency.Lens L2 (ft6) images the grating in the crystal with unit magnification.The intensity distribu tion received in the crystal plane is and the normalized light intensity is (15a) (15b) According to Eq. (7), the received intensity distribution I¡{x) is encoded as a birefringence modulation.In this case, by using Eqs.(4) and (5), the total internal field becomes E, and the expression for E(x) reduces to if /,(jc) .(15a).The periodically striped illuminated = striped areas the total internal field drops to a negligible value in comparison with the magnitude of the field in the plied value Ea .As a consequence, in view of Eq. (7), the gions, while maintaining a nonnegligible value in the dark stripes, that is, Sn(x) = Snx = r^nl{2Ea) Sn2 = 0 if 7,00 = 0, ¡f /,w=/".