ELECTRONICS, VOL. 19, NO., DECEMBER 015 105 Some examples of photorefractive oscillators Zoran Ljuboje bstract The photorefractive effect presents a perioical change of the refractive inex of an optical environment uner the influence of a coherent light. n interesting phenomenon which occurs at this effect is optical phase conjugation (PC). Photorefractive oscillators, that is photorefractive mirrors present important ecies in photorefractive optics an their function is base on photorefractive effect. In these oscillators, a phase-conjugate light beam occurs. The basic characteristics of photorefractive oscillators, such as reflectivity, the existance of the oscillation threshol an the threshol of the coupling strength are explaine by the so-calle grating-action metho. This is analyse on a ring oscillator, semilinear mirror an linear mirror. Inex terms fotorefractive oscillators, grating-action metho, ring oscillator, semilinear mirror, linear mirror. Original Research Paper DOI: 10.751/ELS1519105L T I. INTRODUCTION HE paper will analyse photorefractive oscillators [1] whose operation is base on the photorefractive effect. The photorefractive effect is base on the four-wave mixing of laser light beams (4TM) in some crystals []. In this effect, a perioical change of refraction inex of an optic environment occurs uner the activity of light. During the illumination of some crystals by coherent light, carriers of free charge are create by the transfer of the onor atom electrons to the conuction zone. The number of electron transfers is proportional to the number of onor atoms an the light intensity. Carriers iffuse to the place with lower light intensity, an the opposite sign charges remain in their positions, so-calle holes. s the consequence of this rearrangement of charge, an internal electric fiel occurs, i.e. the phenomenon of change in local refraction inex occurs, that is, a iffraction grate is forme in the crystal at which aitional incient beams can be scattere (Figure 1). Manuscript receive 1 September 015. Receive in revise form 1 December 015. Zoran Ljuboje is with the Faculty of Electrical Engineering, University of East Sarajevo, Bosnia an Herzegovina e-mail: zoran.ljuboje@etf.unssa.rs.ba). n interesting effect which causes light in a photorefractive environment is optical phase conjugation (PC). In this situation a simultaneous rotation of the phase an the irection of the light beam wave propagation. So, the crystal is illuminate with three laser beams: two oppositely irecte pumps whose amplitues are 1 an an an amplitue signal 4. s the result, the fourth wave occurs, with the amplitue 3 which presents a phase-conjugate copy of the beam 4. Unlike the known law of reflection of light in geometrical optics, the reflecte beam 3 returns by the same path of the incient beam 4. PC beam is interesting because of great opportunities for its practical application. Fig. 1. Four-wave mixing in photorefractive crystals. The basic equations which escribe the process are3 : 1 Q 4 1 x x, Q, 3 x 4 Q, (1) 3 x 1 4 Q. () 3 In the name equations, enotes the amplitues of waves, is the absorption coefficient, Q presents the so-calle amplitue of the transmission iffraction grating, an the asterisk enotes the conjugate-complex units. The amplitue of the transmission iffraction grating satisfies the following equation:
106 ELECTRONICS, VOL. 19, NO., DECEMBER 015 Q Q 1 4 3, (3) t I where is the relaxation time, I is the total intensity of waves I i, while an are the parameters which epen on the electric fiels forme in the crystal [4]. The solving of these equations is performe numerically ans may lea to numerical instabilities, but that is not going to be iscusse in this paper [5], [6]. where a is the constant calculate from bounary conitions, is the coupling strength of the crystal, is the thickness of the crystal, an I enotes the intensities of light beams. On the basis of this metho, it is possible to construct an aequate photorefractive circuit which, in a general case, correspons to the 4TM process (Figure ). II. PHOTOREFRCTIVE OSCILLTORS The most important evices in photorefractive optics are photorefractive oscillators, that is, mirrors. Their operation is base on the photorefractive effect. Photorefractive oscillators are forms when photorefractive crystals are illuminate with one or two laser beams. In that, a iffraction grating is forme in the crystal, on which the iffraction of laser beams occurs. The most interesting ones are the configurations of oscillators which create the phase-conjugate beams of incient signals. Photorefractive oscillators with one incient beam are: linear, semilinear, ring an cat oscillators. Cat oscillator has two internal interference region, an each of the other oscillators has one. In the others, external mirrors are use, an cat oscillators use internal total reflection. Photorefractive oscillators with two incient beams are: ouble PC mirror, mutually incoherent beam-coupler, an the so-calle bir an frog oscillators. In this paper, the oscillators with one incient beam will be analyze: ring, semilinear an linear photorefractive oscillator. n oscillator operation analysis can be explaine efficiently by the grating-action metho [7]. On the basis of this metho, the relations of laser beam connections at the entrance (0) an exit of a crystal () are given by the expressions: Fig.. Optical processor representation of the 4TM process. III. RING OSCILLTOR ring oscillator has the possibility of self-starting at the oscillation threshol. This oscillator has two orinary mirrors outsie the crystal an one interaction region in the crystal (Figure 3.) where the matrix 1 10 u 4 40 u : 30 3, u 0, (4) u cos u sin u sin u cosu (5) Fig. 3. Photorefractive ring mirror. where u is the total grating action an calculate from the expression: * * 10 40 3 tan u ai coth( a / ) I I I I 40 3 10, (6) The analysis of operation of this oscillator can be efficiently explaine by the mentione grating-action metho. Figure 4 shows an aequate photorefractive circuit of this oscillator. By applying the expressions (4) an (5), for a ring oscillator is P 10 0 so it follows that the test signal 40 S others, generates the output signal 30, i.e., among the
ELECTRONICS, VOL. 19, NO., DECEMBER 015 107 M sin u 30 40, (7) where M is the prouct of reflectivity of external orinary mirrors. Threshol of the coupling strength is calculate from (9) ( for u 0 ) an from that it follows: M 1 tanh( a th / ) 3M 1 th, (10) what is equivalent to the results in [8] but obtaine in a simpler way. It is obvious that the coupling threshol has the value 1, when M 1 (Figure 6). th Fig. 4. Photorefractive circuits relate to the ring mirror. The reflectivity of beam in this oscillator is given by the expression R I / I /, so, from (7) it follows: 30 40 30 40 R Msin u, (8) while the action u is calculate from (6) an given by the expression: M 1 1 M acoth a / cos u 4M (9) Figure 5 presents the solutions which follow from (8) an (9), efining the epenence of the oscillator refection in the function of the coupling parameter, where M is given as a parameter, an 1. Fig. 6. Threshol of the coupling strength in the function of reflectivity M. t the oscillation threshol, it is a 1 M / 1 M an from that, on the basis of (10) it follows that the relation between the threshol of the coupling strength an the parameter a given by the expression; 1 th ln(1 a ), (11) a what is presente in Figure 7 (for 1). Fig. 5. Oscillator reflection in the function of coupling constant.
108 ELECTRONICS, VOL. 19, NO., DECEMBER 015 Fig. 7. Constant a in the function of the threshol of the coupling strength. Ring oscillators can have an interesting application. For example, two couples ring oscillators whose operation is base on 4TM present a combination which is analogous to an electronic flip-flop circuit [9], [10]. Such a flip-flop oscillator consists of two ring oscillators which are from the outsie pumpe by the light intensities which may be ifferent. In that, only one oscillator oscillates an the other shuts own an vice versa. IV. SEMILINER MIRROR Semilinear mirror is, also, an oscillator with one incient light beam an one interaction region. It has one external mirror an oes not have the possibility of self-restarting (Figure 8 an Figure 9). We will repeat the proceures as in the previous case [11]. The incient signal P 40 generates the beams an on the basis of (4) an (5) we get the beams at the exit: an, in that, 10 0. u, cos 4 40 u sin 1 40, (1) The beam 1 via the external mirror with reflection coefficient M presents the entrance M 1, while 3 0. By applying the relations (4) an (5) it follows: M u, cos 0 M 1 u sin 30 1, (13) The constant a is calculate from the expression a T / I, where T I I I I I 4 1 4 ( 1 4 ), I 1, I are I 4 light beam intensities for z an I is their sum. On the basis of the calculation for the constant a, an also from (1) an (13) for the reflection of this mirror R I / I it is obtaine: 30 40 R M 1 a a M 1 1 M(1 a ) 4 1. (14) By the arranging from the expression (6) for this case, it follows: 1 1 a ln a 1 a. (15) On the basis of (14) an (15) in Figure 10 the epenence R f( ) is presente, where M is the parameter, an 1. Fig. 8. Photorefractive semilinear mirror. Fig. 10. Oscillator reflection in the function of the coupling constant. Minimal reflectivity value follows from the equation (14) for the value of the constant a: Figure 9. Photorefractive circuits relate to the semilinear mirror. a M 1. (16) 1
ELECTRONICS, VOL. 19, NO., DECEMBER 015 109 The threshol of the coupling strength follows from (15) an (16): 1M 1 th 1M ln 1M 1. (17) V. LINER MIRROR We will mention the linear mirror, too. This oscillator is, also, a mirror with one incient beam, one interaction region an has the possibility of self-starting at the threshol. This mirror also has two orinary external mirrors (Figure 13). This result is ientical to the result from the paper [8] where it has been obtaine in a completely ifferent, but a more complicate way. For the case of maximal reflectivity M 1,.49. (Figure 11) th lso, the solutions given in Figure 1 follow from the expression (15). Fig. 13. Photorefractive linear mirror. Fig. 11. Threshol of the coupling strength in the function of reflexivity M. Fig. 14. Photorefractive circuits in relation to the linear mirror. We will repeat the same proceures as for the previous examples. The incient signal P 40 an beam 10 generate the beams at the exit. The beam 1 via the external reflection coefficient M presents the entrance 1 M,while 3 0. From (4) an (5) among the other, it follows: 30 M sin, 1 u 0 M cos 1 u (18) Fig. 1. Constant a in the function of the threshol of the coupling strength. The beam 0 returns via the external mirror of reflection M 1, i.e. 10 M1 0 an thus the process is repeate. It follows from the calculation that the minimal value of the threshol of coupling is:
110 ELECTRONICS, VOL. 19, NO., DECEMBER 015 ln M M (19) th 1 It is obvious that the threshol of the coupling strength tens to zero 0, when MM 1 1. th For the case when the reflectivities M1 M it follows from (19) that ln M, (0) th an that is presente in Figure 15. VI. CONCLUSION The paper analyses the examples of photorefractive oscillators with one incient beam. The operation of this oscillators is base on the photorefractive effect, an in that a phase-conjugate light beam also appears, so they are also calle phase conjugate mirrors. So-calle grating-action metho has been applie, by means of which are efficiently explaine the reflectivity of oscillator, then the so-calle oscillation threshol an the threshol of coupling strength. This is analyse on a ring oscillator, semilenear an linear mirror. nalytical calculations have been performe an characteristic units, such as reflectivity an characteristic units at the oscillation threshol have been presente graphically. Fig.15. Threshol of coupling strength in the function of reflectivity M. REFERENCES [1] Z. Ljuboje Fotorefraktivni oscilatori sa jenim upanim zrakom, INFOTEH-JHORIN, Vol. 14, March 015. [] P. Gunter an J.P. Huignar (es), Photorefractive Materials an Their pplications, I an II (Springer, Berlin, 1988, 1989) [3] W. Krolikowski, K. D. Shaw, M.Cronin-Golomb an. Bleowski, J. Opt.Soc.m.B 6 (1989); W. Krolikowski, M.R. Belić, M.Cronin- Golomb an. Bleowski, J. Opt.Soc.m.B 7 (1990). [4] N. V. Kukhtarev, V. Markov an S. Oulov, Transient energy transfer uring hologram formation in LiNbO3 in external electric fiel, Opt. Commun. 3 338 (1977). [5] M. Belić an Z. Ljuboje, Chaos in phase conjugation: physical vs numerical instabilities, Opt. Quant. Electron. 4 745 (199) [6] Z. Ljuboje, O. Bjelica Numeričke nestabilnosti pri rješavanju nekih problema u fizici, INFOTEH-JHORIN, Vol. 13, March 014. [7] M. S. Petrović, M. R. Belić, an F. Kaiser, Photorefractive circuitry an optical transistors, Opt. Commun. 11 (1995). [8] M. C. Golomb, B. Fischer, J.O. White an. Yariv, Theori an aplication of four-wave mixing in photorefractive meia IEEE J. Qant. Electron. QE-0, 1 (1984). [9] M. Petrovic, M.R. Belic, M.V. Jaric an F. Kaiser, Optical photorefractive flip-flop oscilator, Opt. Commun. 138, 349 (1997) [10] Z. Ljuboje, M. Petrović, Optički fotorefraktivni flip-flop oscilator INFOTEH-JHORIN, Vol. 3, March 003. [11] Z. Ljuboje, Numerički haos u fotorefraktivnoj optici. neobjavljeno.