On conformity of solutions for onedimensional pho

Authors: Marek Wichtowski

Publish Date: 2015/07/03

Volume: 120, Issue: 3, Pages: 527-538

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Abstract

In the present paper the problem of onedimensional screening photorefractive solitons is reconsidered in the context of the accordance of soliton solutions with the Kukhtarev–Vinetskii model In all theoretical and experimental works dealing with the analysis of such type solitons one assumes that under the slowly varying approximation for the optical field amplitude the reduced form of photorefractive rate equations can be employed In this work we point out that the crucial and commonly accepted approximation within this scheme has a limited range of applicability as regards dark solitons This author proposes a relatively simple modification of the standard saturable photorefractive response formula to obtain the plausible selfconsistent solutions The improved solutions for screening black solitons have been derived and discussed by comparison with standard solutionsTwo decades ago it was predicted that photorefractive crystals biased with a DC electric field could support the formation and propagation of the steadystate screening spatial solitons that could be generated at very low optical power of the order of μW The theory of 1 + 1D onetransverse and onepropagation dimension of such type of solitons was presented in works 1 2 and shortly after experimentally confirmed 3 4 5 6 7 8 9 10 Photorefractive PR crystals are photoconductive specific doped electrooptics materials When an optical beam propagates in a PR crystal charge carriers are excited from photosensitive impurities into the conduction band where they move due to drift and diffusion and eventually are trapped in darker regions of the light intensity pattern The resulting space–charge redistribution produces the spatially nonuniform internal electric field which in turn modulates the material refractive index through the linear electrooptic effect Under an appropriate polarity of an external voltage a local variation in the index of refraction leads to selffocusing or selfdefocusing effect of an optical beam This allows forming bright or dark soliton states when the nonlinearity compensates exactly the diffraction spreading of a light beam As a result the beam propagates in a PR medium without changing its transverse profile For onedimensional bright spatial solitons a polarized optical beam in the form of a narrow stripe is launched at the entrance of a crystal together with orthogonally polarized background illumination In the case of dark soliton a black notch is superimposed on an otherwise uniform background illumination Both types of solitons can be generated in the same crystal by reversing the bias voltage directionDue to a possibility of soliton creation at very low laser power levels and their potential applications in all optical switching devices PR screening solitons have been subjected to intensive theoretical and experimental works for the last two decades So far the formation of 1 + 1D solitary waves has been demonstrated experimentally in various PR materials such as ferroelectrics SBN 3 4 5 LiNbO3 11 12 sillenities BTO 6 7 BSO 13 centrosymmetric paraelectrics KLTN 8 semiconductors InPFe 9 14 and CdZnTe 15 both in geometry with bulk PR crystals and in planar waveguides 10 16Typically an analysis of optical beam propagation in PR media involves the transport equations of the socalled Kukhtarev–Vinetskii K–V model 17 to find the material response then it consists in solving the paraxial wave equation with a nonlinearity term derived from the K–V modelIn original papers 1 2 the theory of 1 + 1D screening solitons was developed under an assumption of the slowly varying approximation for the optical field amplitude which corresponds to typical experimental conditions In both mentioned papers the same approximate relation referring to the space–charge field was adopted Using this approximation we can derive the simple relationship representing the local saturable nonlinearity of a PR medium As a result we obtain the specific form of the paraxial wave equation describing both bright and dark solitary state solutions Such form of the envelope wave equation has been commonly accepted by other researchers and to date it has been treated as a standard in all theoretical and experimental works concerning the analysis of screening 1 + 1D solitons 1 2 3 4 5 6 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33The purpose of this paper is to indicate that standard soliton solutions of the wave equation are not generally selfconsistent with the K–V model as far as dark beams are concerned More specifically it is shown that the approximation made within the K–V model to determine the space–charge field induced by an optical beam has a limited range of applicability Conditions for the validity of this approximation are discussed and the standard relationship has been modified to ensure much better conformity with the equations of the band transport model The modified intensity profiles of dark solitons are calculated and compared with profiles resulting from the standard theory In particular it has been found that a full width at half maximum FWHM for both profiles has very similar value The FWHM of a beam is one of relevant experimentally accessible parameters employed to plot the soliton existence curve which constitutes the basic test in comparing experimental data with theoretical predictions The similarity of existence curves derived from the standard theory and from the herein given revised theory explains why no discrepancies between experiments and standard soliton solutions have been reportedThe expression 8 is taken in literature to analyze bright solitons as well as dark solitons It is worth noting that the above approximation is used in the analysis of solitary beams in conventional bulk materials and in the analysis of nonlinear propagation in PR centrosymmetric materials exhibiting the quadratic electrooptic effect 34 35In this study calculations are carried out for ceriumdoped strontium–barium–niobate SBN crystal—the material widely used in soliton experiments Photorefractive transitions in SBN can be discussed in terms of a onelevel model with electron conductivity 36 in accord to Eqs 1a–1c Photoactive centers occur at two valence states which act as donors ions Cr3+ and acceptors ions Cr4+ The values of the material parameters adopted for SBN61CeO2 crystal are as follows 36 N D = 4 × 1018 cm−3 N A = 2 × 1016 cm−3 the photoionization cross section s = 26 × 10−19 cm2 which at a light wavelength λ 0 = 500 nm corresponds to S = 065 cm2/J and the recombination coefficient γ = 1 × 10−10 cm3/s In soliton experiments with SBN crystals the optical beam is linearly polarized in the xdirection which coincides with the optical caxis and direction of the external bias electric field In that case referring to Eq 1f coefficients n b and r eff denote respectively the extraordinary index of refraction n e = 233 and the electrooptic tensor element r 33 = 235 pm/V 20 21 36 37 Because SBN has a very large dielectric constant ε r = ε 33 = 880 screening solitons can be formed at a rather small applied electric field E a of the order of 1 kV/cm

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