Abstract • Although the sine-Gordon equation was originally obtained for the description of four wave-mixing in transmission geometry, it describes self-diffraction of the wave from shifted gratings as well. The sine-Gordon equation governs the soliton propagation. • The photoinduced amplitude of the refractive-index grating exhibits also a soliton shape in the crystal volume. The origin of this effect is the change of the contrast of light due to energy transfer between coupled waves during their propagation, which occurs in bulk crystals with strong photorefractive gain. • The theoretical description shows the possibility to control the soliton properties by changing the input intensity ratio and/or input phase difference of the wave. • The effect can lead to diffraction efficiency management, auto-oscillations and bistability of the output waves due to wave-mixing in ferroelectrics. • Results on the first experimental observation of non-uniform distribution of the grating amplitude profile and its changes versus input intensity ratio are presented.
Band-transport model of recording of non-local phase gratings The photorefractive mechanism of recording refractive index grating includes photo-excitation of free carriers by light interference pattern; their movement due to diffusion and drift with systematic trapping by defects; creation of the internal non-uniform distributed space charge field, which modulate the refractive index. In the case of diffusion dominant mechanism the grating is shifted relative to light interference pattern bya quarter of the period. Single level band transport model Photo-excitation of free carriers. I is the light interference pattern Trapping of free carriers. Creation of internal space charge distribution. E is the space-charge field that modulates the refractive index. Non-local and local gratings recorded in the crystal
The theory of thetransmission four-wave mixing withnon-local gratings The system that describes the changes of coupled wave amplitudes inside the crystal has the following form: where is the photoinduced changes of complex grating amplitude: For the value we can write the kinetic equation that describes the process of grating recording due to photorefractive mechanism and the grating relaxation with the time constant T0: In the case of diffusion dominant mechanism of the photorefraction the change of the grating amplitude is proportional to the light intensity: where is the photorefractive gain for the non-local grating and 1 is the maximum grating amplitude for local gratings. In case of only non-local gratings the kinetic equation has the following form:
Solution for non-local gratings: the soliton Introducing the new real variable u: we have found the solutions for wave intensities from the equation set of coupled waves: ud=u(z=d) where n is the nth wave phase and is the grating phase. Substituting these solutions in the kinetic equation, we obtained a damped sine-Gordon equation: where =t/T0. R and are constants defined from input intensities I10, I20, I3d, I4d and input phase differences of coupled waves. The steady state solution is: where both and may be found from the input conditions. The grating amplitude has a soliton shape. The solution describes a single stationary soliton in the coordinate system (, z). The wave-mixing diffraction efficiency is defined by the integral under the grating amplitude shape: ~sin(2ud).
Two- and four-wave mixing.Formation of the grating amplitude profile along axis z. Two-wave mixing. The energy transfer between two coupled waves (I1 and I2) is started from the input surface of the crystal. The grating amplitude maximum is located near the input boundary. Four-wave mixing. In the case of equal input intensity ratios at the crystal boundaries (I10/I20=I4d/I3d) and a strong photorefractive response, the energy transfer takes place only in the center of the crystal. The maximum grating amplitude is located in the middle of the crystal. Distribution of the grating amplitude and the intensities of coupled waves along the crystal. d=10.
Optical control of the grating amplitude shape.Diffraction efficiency management. The stationary soliton shape is unequivocally defined by the input intensity ratio and input wave phase difference. The grating amplitude maximum is located near a crystal boundary in the case of I10/I20I4d/I3d, or two-wave mixing, or usual four-wave mixing with three input waves. Alteration of the single wave-mixing soliton by changing input intensity ratio: 1) I4d/I3d=0.1 ; 2) I4d/I3d=0.5, I20=I3d=I4d ; 3) I10/I20=I4d/I3d=1 ; 4) I10/I20=I4d/I3d=0.01. (Itotal=1; d=10). The localized soliton can be moved through the medium from one side to another. The grating amplitude distributions for some input intensity ratios. The numbers on the top indicate lg((I10+I20)/(I3d+I4d)). The dashed curves correspond to disphasedinput coupled waves by at the crystal boundaries (10-20=0; 4d-3d=) or (10-20=; 4d-3d=0). Location of the grating amplitude maximum of steady state gratings in the case of FWM with four input waves. J1=J10+J20, J2=J3d+J4d. J10/J20=J4d/J3d=0.1, d=10.
Experimental set-up Four writing beams from Ar-ion laser are expanded and converged in the crystal to record a transmission grating. The expanded probe beam from He-Ne laser is entered at the Bragg angle relative to the recorded grating. We use specially grown congruently melted LiNbO3:Fe crystal with input faces ac=35 mm and the largest dimension as a thickness b=12 mm. The crystallographic dimensions a, b and c are connected with axes y, z and x, respectively. Axis x coincides with the polar crystallographic axis c. The diffracted pattern of the probe beam displays the grating amplitude distribution along the crystal. The space area of thesoliton formation inside the crystal volume. The optical scheme of four-wave mixing with four input waves.
Experimental observation of the grating amplitude distribution We change the input intensity ratios of writing beams. The left pictures show the photos of the space distribution of light induced refractive index. The right pictures show the measured intensity distribution of the diffracted probe beam along axis z. (a) - I4d/I3d=0.08, I20=I3d=I4d ; (b) – I4d/I3d=0.14, I20=I3d=I4d ; (c) - I10=I20=I3d=I4d. In case of equal input intensities of all waves the maximum of the grating amplitude is located in the center of the crystal. The obtained experimental results of the soliton shape changes versus input intensity ratio are in good qualitative agreement with the theoretically calculated curves.
Optical control of localized gratings.Diffraction efficiency management. I10/I20=I4d/I2d=1; d=10 I10/I20=I4d/I2d=0.1; d=10 By changing the input intensity ratio or/and the phase difference of the input waves, the localization degree of the grating amplitude profile changes. Thus the wave-mixing diffraction efficiency defined as the integral under the grating amplitude shape changes as well. All optical space switching I10/ I20>I4d/I3d or I4d=0; d=10. I10/ I20<I4d/I3d or I10=0; d=10. Braking the equal intensity ratio I10/I20I4d/I3d on crystal boundaries leads to a movement of the grating amplitude maximum from one boundary to the other one. The scheme of optical space switching. The waves 1-4 record the grating with different amplitude profile that determine the angle of space diffraction for the probe beam 5-6.
Breather solutions of bond soliton states The sine-Gordon equation admits multi-soliton solutions that describe the interaction of several solitons as well as bond soliton states (pulsing, or so called breather solutions). We obtained the breather solutions numerically for the case of usual four-wave mixing with three input waves and for certain area of input intensity ratios. The breather solutions. I10/I20=3, I3d=0,87, I4d=0, d=15, (Itotal=1). The oscillation of the grating amplitude causes the auto-oscillations of the diffraction efficiency and by this way the auto-oscillations of every output intensities. The reason of the bond-soliton behavior is the emergence of a local component of the grating that causes the changes of wave phases during their propagation. In this way, the light contrast changes with time, and the grating is repeatedly erased and rerecorded. Hence the auto-oscillations can be observed only in optically reversible media.
Conclusion The wave self-diffraction from non-local phase gratings in a photorefractive medium can be described by a sine-Gordon equation with a damped term in the case of transmission geometry. The sine-Gordon equation reveals the changes of the grating amplitude induced by light beam interaction in the medium. The grating amplitude distribution has a soliton shape in the direction of wave propagation. In steady state the soliton is motionless and its parameters, i.e. the amplitude, the half-with and the amplitude maximum position, are unequivocal defined by input intensity ratio and input wave phase difference. The crucial parameter of the soliton is the energy transfer gain on a given distance z in the medium. The photorefractive gain determines the change of the light contrast during the wave propagation and this way the soliton localization degree. Alteration of the soliton shape of the grating amplitude opens the ways to control the parameters of output waves.Theone of them is the all-optical management of the diffraction efficiency, as the wave-mixing diffraction efficiency is determined by the integral under the soliton shape. Multisoliton and bond-soliton behaviors lead to bistability and auto-oscillations of output intensities. For the first time, we observed experimentally the non-uniform distribution of the grating amplitude and the control of its profile by means of changes of input intensity ratio. Research was supported by OTKA contract Nos. T23092, T26088, T35044, by the Austrian-Hungarian Intergovernmental ST Program A-8/2001, and by the Centre of Excellence Program ICA1-1999-75002.