G. Balzer a, M. Goulkov a b, S. Matamontero a & T. Tschudi a a Institut für Angewandte Physik, Licht- und Teilchenoptik,

Transcription

1 This article was downloaded by: [Moskow State Univ Bibliote] On: 12 February 2014, At: 07:14 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Journal of Modern Optics Publication details, including instructions for authors and subscription information: Spatial Resolution and Channel Coupling in a Phase-conjugate Ringresonator G. Balzer a, M. Goulkov a b, S. Matamontero a & T. Tschudi a a Institut für Angewandte Physik, Licht- und Teilchenoptik, Hochschulstraße 6, 6100, Darmstadt, Germany b Institute of Physics, Ukrainian Academy of Sciences, 46 Science ave., , Kiev 22, Ukraine Published online: 01 Mar To cite this article: G. Balzer, M. Goulkov, S. Matamontero & T. Tschudi (1993) Spatial Resolution and Channel Coupling in a Phase-conjugate Ring-resonator, Journal of Modern Optics, 40:11, , DOI: / To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views epressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, epenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is epressly forbidden. Terms & Conditions of access and use can be found at page/terms-and-conditions

2 JOURNAL OF MODERN OPTICS, 1993, VOL. 40, NO. 11, Spatial resolution and channel coupling in a phase-conjugate ring-resonator G. BALZER, M. GOULKOVt, S. MATAMONTERO and T. TSCHUDI 1. Introduction Institut fur Angewandte Physik, Licht- and Teilchenoptik, Hochschulstral3e 6, 6100 Darmstadt, Germany j' Institute of Physics, Ukrainian Academy of Sciences, 46 Science ave Kiev 22, Ukraine (Received 30 April 1993 ; revision received 5 July 1993) Abstract. We describe our study of the spatial resolution of a phaseconjugating ring-resonator (PCR) consisting of a Sagnac interferometer and a phase-conjugating mirror based on four-wave miing in photorefractive BaTiO 3. The use of optical image processing systems depends on the amount of information channels and the nonlinear coupling between the channels. The spatial resolution or the number of independent piels is important to describe any image processing system. If the gain of the phase-conjugating mirror is higher than the losses in the passive part of the system, then the system eceeds the threshold, and without any eternally incident signal the resonator shows the effect of self-oscillation. From the analysis of this self-oscillation pattern we have calculated the spatial resolution of the PCR. Additionally, we try to investigate the piel coupling of the system by defining channels in the resonator. By changing the geometrical set-up of the system it is possible to observe a controlled coupling between neighbouring channels. A coherent optical oscillator with a Sagnac interferometer as one end mirror and a phase conjugating other mirror was proposed and studied in [1-3]. This optical arrangement is particularly attractive as it can be used for optical image processing. Below the threshold of oscillation the input image can be introduced into the cavity through one mirror, while the output can be taken from the other mirror of the interferometer allowing specially designed filters to be introduced between the two mirrors [2]. Above threshold such a cavity with a large Fresnel number and a transparency inside can produce an image bearing output beam without any eternal input signal. In this case the arrangement may be used for intensity thresholding (like e.g. in [4]) or other operations with images by using eternal input beams. One of the main characteristics of any optical arrangement for image processing is its spatial resolution or the number of independent piels in the image to be processed. The purpose of the present work is to study the spatial resolution and the coupling between the adjacent channels of this arrangement. 2. Eperimental set-up The considered cavity is shown schematically in figure 1. It consists of a Sagnac interferometer and the phase-conjugate mirror (PCM) realized by a photorefractive BaTiO 3 crystal pumped by two counterpropagating pump waves. The conjugating mirror may have a reflectivity larger than unity thus assuring the gain to be sufficient /93 $ Taylor & Francis Ltd.

3 G. Balzer et al. r IMIL CCD-Line Figure 1. Schematic eperimental set-up. Mi are the mirrors, BS, are the beam splitters, Di are the amplitude transparencies as eplained in the tet, BaTiO 3 is the photorefractive crystal, CCD is the charge-coupled device. to start self-oscillation (see e.g. [5, 6]). The lens L is placed into the cavity in such a way, that the BaTiO 3 crystal is just in its focal plane. The distances between the optical elements are as follows : MI -M2 = 100 mm, M2-BS' =BSI-M, = 70 mm, M l-l =150 mm, L-PCM= 180 mm. The rectangular aperture mm 2 of the beam splitter BSI, which is AR-coated to avoid internal reflections, limits the transverse dimensions of the beam propagating in the Sagnac interferometer. All eperiments were using a TEM oo single-mode argon ion laser at a wavelength of rim. The stabilized output was approimately 150 mw. The reflectivity of the resonator was RRes =0-86 ; therefore an amplification of the PCM of R PCM >, 1. 2 compensates for resonator losses. In the eperiments the amplification for small signals was up to 10. A sample of BaTiO 3 measuring mm 3 (with the longest edge parallel to the c ais) is immersed in an inde matching oil to increase the angle between the pump waves and the c ais. Two pump waves are directed to the sample at an angle OP =52 to the optical ais of the cavity (in air). The intensity ratio of two pump waves is adjusted with the help of the variable beam splitter placed between the sample and mirror M 4 (not shown in figure 1). The pump waves are divided from the central, uniform intensity part of the unepanded Ar + laser beam. The horizontal slit D l is placed before the beam splitter BS2 to vary the vertical dimension of both pump waves on the photorefractive sample from 0. 3 to 1-6 mm. This orientation of the slit assures the same (constant) interaction length for the oscillating wave and pump waves when vertical dimension of pump waves is changed. For idealized (aberration free) optical systems the size of a smallest resolvable element A is limited by diffraction and depends on the angular aperture O lim : A A (1) OHM The angular aperture of the considered optical cavity (figure 1) is defined by the transverse size r of the PCM and the focal length L of the lens placed before the PCM : _ r O lim ^' F (2)

4 Phase-conjugate ring-resonator 2145 Any light diffracted to the angle O > O, im inside the Sagnac interferometer will not find the PCM and therefore will not return back into the cavity. Thus the divergence of an oscillating beam can not eceed O, im by definition and imposes the ultimate limit on the smallest resolvable element :.1F r (3) This estimate is well justified for the passive (not oscillating) arrangement with the small number of round-trip propagating cycles in the cavity. Above the threshold of self-oscillation this estimate may become incorrect if only a certain limited number of cavity modes oscillate. For eample, for the aial mode (i.e. for the mode with zero transverse indices) the size of uniformly illuminated area in the near field is just comparable to the cavity aperture. Thus depending on the type of oscillation the speckle size of an oscillation pattern may be much larger than that limited by simple diffraction. 3. Eperiments In the first eperiment the pump intensity ratio p=ip,/ip, is adjusted to optimize the phase conjugate reflectivity of the PCM and to assure the onset of oscillation. We are measuring the size of the speckles in near field pattern of oscillation as a function of vertical pump dimension r. The intensity distribution is photographed at the output plane just behind the mirror M2. Figures 2 (a) and 3 (a) show eamples of near field distributions for two values of slit width. The growth of the vertical size of the speckles for smaller width is obvious. A set of quantitative measurements are taken with a charge-coupled device (CCD) line camera, which is placed vertically in the output plane. For any intensity distribution recorded by the CCD line the Fourier spectrum is calculated (figure 2 (b), 3 (b)) and the cut-off spatial frequency O cut _ off corresponding to an amplitude b slit orientation Lines/mm Figure 2. (a) Self-oscillation pattern with the maimum pump beam diameter of 1.6 mm ; the size of the pattern is mm 3. ( b) Powers spectrum of the CCD line shot of this pattern.

5 2146 G. Balzer et al. Slit onentation 1 Lines/mm Figure 3. (a) Self-oscillation pattern with the reduced pump beam diameter of 0. 5 mm ; the size of the pattern is mm 3. (b) Powers spectrum of the CCD line shot of this pattern. reaching 1% 0 of zero-frequency peak is determined. This cut-off frequency just defines the grain size of the speckles in the near-field pattern : Qcut-off and thus imposes the limit to minimum resolvable elements in the image plane. Figure 4 represents the dependence of the cut-off frequency on the slit width. The dashed line is the best least square fit to linear dependence showing a good qualitative agreement with equation (3). The solid line corresponds to the dependence Q=A - '= f(r) calculated from equation (3) for 2 = nm and 20 W Z ::j 15 i i i - 11 (4) i i 11 X W, I I PCM APERTURE/ M M Figure 4. Cut-off spatial frequency in near-field intensity distribution as a function of pump width. Dashed line is the least-square fit to linear dependence, solid line gives the diffraction limit according to equation (3). 2.0

6 Phase-conjugate ring-resonator 2147 f = 180 mm. It can be easily seen that the grain size in the near field speckle pattern is very close to the diffraction limit estimated from the angular aperture of the cavity. This result is consistent with the conclusions of previous works. It has been shown in [2, 3] that in a cavity with a large Fresnel number (N F 2 =r / tf>, 10) many high-order transverse modes with random phases oscillate simultaneously. The speckle-type intensity distribution in near-field is irregular and continously varying in time. The application of the Grassberger-Procaccia technique for analysis of the spatial distribution reveals chaotic behaviour with the correlation dimension eceeding 3. 4, thus providing the lack of correlation between particular grains in a speckle pattern. In the second eperiment we study the possible coupling of individual channels in the resonator maintained in the vicinity of the oscillation threshold. The discrete channels are imposed to the cavity by placing a Ronchi ruling D 2 between the mirrors M 1 and M2 (see figure 1) The ruling has a spatial frequency S2 R = 2 lines mm -1 and its grating vector is normal to the plane of drawing of figure 1. The oscillation pattern with Ronchi ruling inside the cavity simply repeats its transparency. By changing the pump intensity ratio p, we reduce the PCM reflectivity thus diminishing the oscillation intensity down to its complete disappearance. The whole arrangement is kept just in this state, very close but below threshold of oscillation. Now a probe signal beam is formed by a narrow horizontal slit D3 placed in the input plane near the mirror M 1. The beam width of this slit A P is equal to one spacing of the Ronchi ruling. The intensity distribution of the output signal is photographed and measured by the CCD line behind the mirror M2. For a completely open slit D 1 of the pump waves (r=1. 6 mm) we observe in the output plane the narrow bright stripe just ecited (figure 5 (a, b)). There eists a certain noise intensity in the other channels which do not correspond to the input signal. This noise is randomly distributed in other channels and the maimum peak intensity in the noise speckles does not eceed 5% of the maimum intensity in the main channel. The decrease of the width of slit D 1 up to certain value does not affect the noise in the channels that are not ecited but starting from r=0. 8 mm we can observe the beggining of the regular filling of adjacent channels with light. For further decrease of r < 0. 8 mm the obvious growth of intensity in adjacent channels is detected also by the CCD line (figure 5 (b, d)). Figure 6 represents the peak intensity in adjacent channels normalized to the peak intensity in the main channel as a function of pump wave dimension r. One can see that adjacent channels become coupled to each other for r < 0. 8 mm. From equation (3) we can evaluate the critical vertical dimension of the pump waves rcr below which two adjacent channels will be coupled by diffraction : ~F r~r = ~t mm. R (5) Referring to the eperiment (figure 6) we can say that the complete independence of two adjacent channels is established for r >, 4r cr. There may be many different reasons for channel coupling in coherent oscillators. For solid state lasers for eample, the crucial reason is the scattering from optical imperfections of the active laser rods. For the optical oscillators considered

7 i G. Balzer et al. D1! 1.6mm a) 0 7 O 300a } 200 z 100 z --^^ A I A - D1 0.5 mm b) DISTANCE / cm. Figure 5. Output intensity distribution for cavity with Ronchi ruling inside when input signal corresponds to one luminous stripe. Pump wave width r =1. 6 mm (a, b) and 0.5 mm (c, d), respectively. 1.5 X I z 3f PCM APERTURE/ MM Figure 6. Normalized intensity of the adjacent channel as a function of pump width. 1.8

8 Phase-conjugate ring-resonator 2149 here the imperfect alignment of the counterpropagating pump waves or the lateral misalignment of the Sagnac interferometer were shown to cause such a coupling [2, 3]. We believe that the most important reason for the increased sensitivity to the pump aperture is connected also with the strongly nonlinear behaviour of the system near the oscillation threshold. A part of radiation from the main channel diffracted to adjacent channels may seed the weakly damped long-lived waves producing a detectable signal in output plane. Whatever the reason for this limitation, our eperiment proves the possibility of using the described arrangement for optical processing of images with a minimum spatial resolution of 2 lines mm -1 and a pump wave width of not less than 1 mm. Therefore we have a usable area in the resonator of about mm 2 ; it means that we are able to transfer about 2500 independent channels within our system. Acknowledgments This work has been done in the framework of the agreement on scientific cooperation between Institut fur Angewandte Physik, Technische Hochschule Darmstadt and the Institute of Physics of Ukrainian Academy of Sciences. Financial support of Deutsche Forschungsgemeinschaft is gratefully acknowledged. We are grateful to Dr S. Odulov for the initial idea of this work and for helpful discussions. References [1] KLUMB, H., HERDEN, A., KOBIALKA, T., LAERI, F., TSCHUDI, T., and ALBERS, J., 1988, J. opt. Soc. Am. B, 5, [2] KOBIALKA, T., HERDEN, A., and TSCHUDI, T., 1989, Ferroelectrics, 92, 189. [3] BLUMRICH, R., KOBIALKA, T., and TSCHUDI, T., 1990, J. opt. Soc. Am. B, 7, [4] KLEIN, M., DUNNING, G. J., VALLEY, G. C., LIND, R. C., and O'MEARA, T. R., 1986, Optics Lett., 11, 575. [5] FEINBERG, J., and HELLWARTH, R. W., 1980, Optics Lett., 5. [6] ODULOV, S., SOSKIN, M., and KHYZNIAK, A., 1991, Optics Oscillators with Degenerate Four-Wave Miing (London : Harwood).