- Email: [email protected]
Build up mechanisms of (1+1)-dimensional photorefractive bright spatial quasi-steady-state and screening solitons
1 January 1998
Optics Communications 145 Ž1998. 393–400
Full length article
Build up mechanisms of ž 1 q 1/ -dimensional photorefractive bright spatial quasi-steady-state and screening solitons N. Fressengeas a
a,b,)
, D. Wolfersberger
a,b
, J. Maufoy b, G. Kugel
a
MOPS, CLOES, UniÕersite´ de Metz-Supelec, ´ 2 rue Edouard Belin, 57078 Metz Cedex 3, France b SEL, Supelec, ´ 2 rue Edouard Belin, 57078 Metz Cedex 3, France Received 14 November 1996; revised 11 April 1997; accepted 4 August 1997
Abstract A Ž1 q 1.-dimensional model is studied numerically to evidence the build up mechanisms of photorefractive solitons, from the characteristic carrier recombination time, through the quasi-steady-state soliton, to the screening soliton. Three different build up regimes are evidenced and their domain of existence are computed. The transient quasi-steady-state soliton is shown to be characterized by two constants: its normalized width and its normalized build up time response multiplied by its peak intensity over dark irradiance. This latter assertion allows us to predict the photorefractive soliton response time for various optical powers. It is thus compared to existing experimental results. q 1998 Elsevier Science B.V. PACS: 42.65.Sf; 42.65.Tg; 42.65.Hw; 42.65.Jx Keywords: Bright spatial optical soliton; Self-focusing; Space-charge field; Transient state; Photorefractive effect
1. Introduction The propagation of a single light beam in a photorefractive crystal has been the focus of growing interest in the past few years. In particular, quasi-steady-state bright spatial solitons have been predicted w1,2x and observed w3x. Steady-state self-focusing features have been observed then w4x, leading to the prediction w5x and the observation w6x of bright steady-state screening solitons trapped in one dimension or in both w7–9x. These phenomena occur only if the photorefractive crystal is properly biased by an external electric field or by an adequate photovoltaic nonlinearity. Three types of photorefractive bright spatial soliton have been predicted and observed. The quasi-steady-state soliton w1,2,10,11x has a limited lifetime; it depends neither on absolute light intensity nor on its ratio to dark irradi) C o rre s p o n d in g [email protected].
a u th o r.
E -m a il:
ance, as long as the latter is larger than unity. The screening soliton w6,5x is a steady-state phenomenon due to the partial screening of the electric field induced by an external source. The photovoltaic soliton w12,13x is a steady-state phenomenon as well but can be obtained without any external source, the photorefractive material being biased by the photovoltaic effect. Both the screening and the photovoltaic soliton depend only on the ratio of the absolute light intensity over the dark irradiance. Considering a beam that is allowed to diffract only in one direction Ža Ž1 q 1.-dimensional model., the three types of soliton can be thought of as three particular aspects of a more unique phenomenon which is the slow screening of the electric field induced either by an external source or by the photovoltaic effect or by both w14x. In this paper, a numerical investigation of the Ž1 q 1.dimensional model developed in Ref. w14x is provided in order to evidence the consequences of the electric field screening process on steady state and quasi-steady-state photorefractive bright spatial solitons. Three different soli-
0030-4018r98r$17.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 7 . 0 0 4 5 5 - 0
N. Fressengeas et al.r Optics Communications 145 (1998) 393–400
394
ton generation regimes are found, and their existence domains exhibited. Numerical results show that, provided its peak intensity normalized to dark irradiance is high enough, the quasi-steady-state soliton possesses two characteristic constants depending only on the crystal physical parameters: Its normalized width and its build-up time multiplied by its normalized peak intensity, as defined above. A comparison with the build up times of other photorefractive phenomena, such as two-wave mixing, is then provided.
2. Theoretical elements
The Ž1 q 1.-dimensional model developed in Ref. w14x is based on the Kukhtarev–Vinetskii model w15x reduced to one dimension. It describes the one level charge transport mechanism in a generic photorefractive medium. Several approximations, detailed in Ref. w14x, are used in this model to analyze the process of electric field screening: the beam is assumed to be much narrower than the crystal width but wide enough to validate the slow varying approximation, and the crystal Debye wave number is assumed to be large enough w14x Žor the Debye screening length small enough.. In typical photorefractive media, the electron life time is much shorter than the dielectric relaxation time, as confirmed by Yeh Žw16x, pp. 106, 107. and by the quantitative analysis of Ref. w14x, based on experimental values available in the literature. Under these assumptions, the Kukhtarev–Vinetskii model can be reduced to Eq. Ž1., which links the space-charge field and the light intensity w14x, E Ž IEt . Ex
qk BTm
E2I Ex
2
If the light intensity is assumed to vary slowly enough with time so that its last term can be neglected, Eq. Ž1. can be analytically solved and leads to the expression of the generalized space charge field Et as a function of the generalized light intensity I w14x,
ž
Et s E0 exp y
=
2.1. Light intensity and space charge field
em
2.2. Space charge field expression
q´ 0 ´ r
I0 E 2 Et n 0 Et E x
EI ye Et
s 0.
Ž1. Here, t and x are the time and transverse spatial coordinates. e is the elementary charge, m the electron mobility, k B the Boltzmann constant, T the temperature, ´ 0 the electric permeability of vacuum, ´ r the static dielectric tensor. n 0 is the free electron density generated by an arbitrary uniform illumination I0 , so that n 0 r I0 s sŽ ND y NA . r j NA , s being the photoexcitation coefficient, j the recombination constant, ND and NA the densities of donors and acceptors. I is the sum of the light intensity and of the dark irradiance Id , which is the modelling of the thermal generation of free carriers by an equivalent optical intensity. The generalized electric field Et is the sum of the space charge field and of the photovoltaic field Eph s bph j NA r e m s, bph being the component of the photovoltaic tensor along the ferroelectric c-axis, the other components being neglected.
ž
em n0
´ 0 ´ r I0
Ž Eext q Eph .
Id I
/
ž
It q 1 y exp y
y
k BT IX e
I
/
.
em n0
´ 0 ´ r I0
It
/
Ž2.
In Eq. Ž2., E0 is the electric field initial state and is a function of the transverse spatial coordinate. Eext is the externally applied field and is a constant in the crystal. I is the generalized light intensity and thus depends on the spatial coordinate x. I X is therefore its spatial derivative. Eq. Ž2. seems to show that the applied electric field, figured as Eext , and the photovoltaic effect, figured as Eph , have similar and additive influence on the space-charge field and thus on the wave propagation. However, this assertion has to be tampered by introducing two further ideas: Ø The addition is to be understood in an algebraic way: While the sign of Eext can be reversed experimentally, the direction of Eph is a function of the very nature of the photorefractive crystal. Ø It is important to understand the scope of the addition that appears in Eq. Ž2. between Eext and Eph . We have integrated Eq. Ž1. into Eq. Ž2. using boundary conditions valid only if the photovoltaic effect is small in effect with respect to the applied field. Therefore, this addition is valid only locally and should not be understood as a generally valid addition. One other limitation to the addition of the influence of the photovoltaic effect and of the applied electric field lies in the experimental setups used, for instance, in Refs. w4,14x. In these experiments, a background illumination is used to artificially raise the dark irradiance. Via the photovoltaic effect, this illumination adds a constant component to the current density that is not accounted for in the model and used here. However, the crystal used in these experiments was Bi 12TiO 20 which is not photovoltaic, and thus does not raise that problem.
2.3. WaÕe propagation equation The space charge field induces, via the Pockels effect, a refractive index variation in the photorefractive medium. Therefore, knowing the propagation of a beam in a medium with a low index modulation, Eq. Ž2. can lead to the wave propagation equation Ž3., describing the propagation of an
N. Fressengeas et al.r Optics Communications 145 (1998) 393–400
arbitrary light beam in a generic photorefractive medium w14x, 1 E2U
EU i
q EZ
2 EX 2
ž
= N 2yD
Id Ž 1 q < U < 2 . t
ž
y 1 y exp y E
/
ž
S
1 q
S
/
U s 0.
Ž3.
EN Ž X, Z . is the space-charge field initial state, defined by the crystal history. U s SCEr Id is the beam electric field SCE normalized to the dark irradiance. X s x r X 0 and Z s z r kX 02 , where X 0 is an arbitrary length, are the transverse and propagation-wise coordinates. Eq. Ž3. is not normalized with respect to time in order to clearly show the influence of the dark irradiance, and of the energy density S defined in the paragraph below. However, in analysing the temporal evolution of soliton solutions in Section 3, we shall normalize the time to the dielectric relaxation time in the dark Te s S r Id . The quantity S s Ž ´ 0 ´ rre m .Ž I0rn 0 ., which has already implicitly appeared in Ž2., is homogeneous to an energy density. Although the above definition of S uses an arbitrary illumination I0 and the electron density n 0 associated to it, it is completely determined by the crystal physical parameters and is thus a very intimate characteristic of the crystal related to the relaxation of the self focusing process. However, since I0 is arbitrary, we can give it particular values in order to show the physical meaning of S . For instance, if I0 s Id , S is the dielectric relaxation time in the dark multiplied by the dark irradiance. We can emphasize that S is, at each point, the product of the total local optical intensity Ždark irradiance q effective illumination. multiplied by the local dielectric relaxation time. N 2 is characteristic of the quasi-local mechanisms of the nonlinearity and is due to drift and photovoltaic mechanisms of transport:
'
N 2s
k 2 n 2 reff X 02 Ž Eext q Eph . 2
,
Ž4.
k is the beam wave vector, n is the medium base refractive index, and reff is the effective electro-optic coefficient. The drift mechanism of transport is due to the externally applied electric field Eext , while the photovoltaic effect is represented in Eq. Ž4. by Eph , as defined in Section 2.1. The quantity D is characteristic of the diffusion mechanism of transport and is expressed by 2 2
Ds
k n reff X 0 k BT 2e
.
essentially dependent on the electro-optic coefficient, since the diffusion process is nearly the same in all crystals. 2.4. Soliton solutions
/
U
y EN Ž X , Z . exp y
395
Ž5.
The effects of the diffusion mechanism of transport are
In the following section, we will focus our study on a particular set of solutions of the wave propagation equation Ž3.: The solutions which give an unchanging beam profile along the propagation direction z. They are called ‘soliton solutions’. These are evidently not the only solutions to Ž3.: obtaining all the solutions of Ž3. would require a complete and very difficult mathematical analysis, which is out of the scope of this paper. However, they are an interesting class of solutions to study if one aims at understanding the temporal behaviour of photorefractive solitons. For instance, such an approach has already been done, with another wave propagation equation, by Segev et al. w17x, by Valley et al. w13x and by Christodoulides and Carvalho w18x. Nevertheless, there are some other interesting solutions, such as oscillatory ones that are not included in this class. Furthermore, this study will not provide any analysis of the soliton spatial stability. We will discuss these limitations later in Section 6.1. Feinberg w19x and Singh and Christodoulides w20x have shown that the diffusion process, accounted for by the quantity D, induces a self deflection of the beam. Therefore, strict soliton beams Žwhich do not bend. can be achieved only if this term is neglected. Using this latter approximation, we assume the existence of a soliton solution UŽ X, Z, t . s 'r g Ž X, t .e i n Z in which Ø r is the ratio of beam peak intensity over dark irradiance. Ø g is the soliton profile normalized to 1: g Ž0, t . s 1 and g Ž"`, t . s 0. Ø n is the soliton propagation constant. Assuming the above form for UŽ X, Z, t ., basic algebra allows to derive from Ž3. an equation describing the normalized soliton profile g w14x, along with the propagation constant n Ž7.: Id t y2 ng q g Y y 1 y exp y Ž 1 q rg 2 . S g Id t 2N2 y 2g exp y Ž 1 q rg 2 . s 0. 2 S 1 q rg
ž
ž
/
ž
/
/
Ž6. In Eq. Ž6., the primes stand for the derivatives of g along the transverse spatial coordinate. The soliton propagation constant n is defined by
nsy
N2 r N2
q
ln Ž 1 q r .
ž
Ei y
Id t
S Id t q exp y Ž1 q r Id tr S r S
ž
Id t
/ ž / ./ ž /
Ž 1 q r . y Ei y
S
y exp y
Id t
S
,
Ž7.
396
N. Fressengeas et al.r Optics Communications 145 (1998) 393–400
where Ei is an exponential integral function defined by ` Ž yt EiŽ z . s yHyz e r t . d t. Eq. Ž6., along with Eq. Ž7., describes analytically the behaviour of the soliton profile in time and space, whereas Eq. Ž7. describes on its own the behaviour of the propagation constant. Although it is an interesting matter, it is not the purpose of this manuscript to analyze in detail the propagation constant. However, it should be kept in mind that the numerical analysis of Ž6. that we shall make in the following section, requires evaluating Ž7. beforehand because the soliton propagation constant appears in the first term of Ž6.. Therefore, both Eqs. Ž6. and Ž7. are the basis of the following numerical study.
3. Three different temporal regimes 3.1. Introduction A simple numerical integration of Ž6. along the space coordinate, for given values of r and t, leads, through an intermediate evaluation of the propagation constant n , to a numerical determination of the soliton profile. These calculations allow in turn to compute the time evolution of the soliton profile for a given value of r. Fig. 1 shows three typical time evolutions of the soliton half width at half maximum ŽHWHM. for N s 5, for r equal to unity, 300 and 1000 for the curves 1, 2 and 3, respectively. For the sake of simplicity, we normalized the time to the dielectric relaxation time in the dark Te s S r Id . The general shape of these curves is confirmed by the computations made by Zozulya and Anderson w21x, through a direct numerical study of the space charge field equation. The curves 1, 2 and 3 of Fig. 1 evidence three different HWHM time evolution regimes. Curve 1 Ž r s 1. shows a monotonically decreasing HWHM. Curve 2 Ž r s 300. shows a HWHM which presents a local minimum but relaxes to a more focused state. Curve 3 Ž r s 1000. shows
Fig. 1. Typical half width at half maximum time evolution for N s 5 and, respectively, for r equal to unity, 300 and 1000 for the curves 1, 2 and 3. The time is normalized to the dielectric relaxation time in the dark Te s S r Id .
Fig. 2. Domains of existence for each regime of soliton temporal behaviour. Each domain is labelled with the number of the regime it represents. White dots are the computed points which show the frontiers between the domains. Their height is the error margin yielded by the computation. The straight lines are mere guides for the eye.
a HWHM that presents an absolute minimum and relaxes to a less focused state. In other words, curve 1 shows an electric field screening process that yields a steady-state screening soliton, without generating on the way a quasisteady-state soliton. On the other hand, curves 2 and 3 evidence a quasi-steady-state soliton. The former shows a quasi-steady-state soliton larger than the screening soliton whereas the latter shows a quasi-steady-state soliton narrower than the screening one. This allows us to distinguish between three temporal behaviour regimes into which each of the HWHM versus time curve we computed can be classified. In the following, regime 1 is the class of the curves which do not show a quasi-steady soliton, regime 2 is the class of the curves that show wider quasi-steady state soliton, regime 3 being the remaining class. It is important to understand that, as r grows, the HWHM versus time curve evolves continuously from regime 1, through regime 2, to regime 3. The frontiers between these regimes can, however, be mathematically defined in a precise way, using the above definition, even though they represent a smooth transition from one regime to another. In addition, owing to the assumption that lead to Ž6., thus restricting our present study to soliton solutions of Ž3., all the solutions displayed in Fig. 1, as well as in Figs. 2–4, are soliton solutions. For obvious reasons, we call the steady-state solution ‘steady-state solitons’. We call the local minimum ‘quasi-steady-state soliton’ because the properties of this minimum with respect to the variation of the power ratio r are similar to the properties of quasi-steady-state solitons, as defined in the literature w1,2,10,11x. 3.2. Domains Eq. Ž6. is the equation that governs the shape of the HWHM versus time curves. Besides the space variable, it
N. Fressengeas et al.r Optics Communications 145 (1998) 393–400
depends on two independent parameters, namely N, characteristic of nonlinearity strength, and r, the beam peak intensity to dark irradiance ratio. Fig. 2 shows, for N ranging from 1 to 10, the domains on the Ž N, 'r . plane representing each regime, as defined previously. Each domain is labelled with the regime number. The white dots are the computed points which show the frontiers between the domains. Their height shows the maximum error in the result that the numerical analysis yields. Fig. 2 suggests that, on the Ž N, 'r . plane, the frontiers between these domains are straight lines passing by the point N s 1, r s 1, thus allowing to extrapolate these results to higher values of r and N.
4. Soliton HWHM Fig. 3 shows the behaviours, as a function of r, of the steady-state soliton HWHM Ždotted curve. and of the minimum HWHM reached during transient state Žsolid curve., for the integer values of N ranging from 1 to 10 Žfrom top to bottom.. These have been computed using the numerical analysis leading to the HWHM versus time curves, like those in Fig. 1. For the sake of clarity, Fig. 3 is displayed on a bi-logarithmic scale. For small r, i.e., for the regimes 1 and 2, the minimum achieved diameter is that of the screening soliton. This is the reason why the solid and dotted curve are the same for small r. The shape of the screening soliton HWHM curve displayed in Fig. 3 is consistent with the previous theoretical results found by Christodoulides and Carvalho w18x by other means, as well as with experimental results concerning steady-state screening solitons w22x. Furthermore, Fig. 3 suggests that, if r is large enough, then, the quasi-steady state soliton HWHM does not depend on r anymore, as confirmed by Refs. w1,2,11x. However, Fig. 3 shows that, if r is large enough, then the quasi-steady-state soliton
Fig. 3. Half width at half maximum as a function of the peak power to dark irradiance ratio r, for steady-state screening solitons Ždotted curves. and for quasi-steady-state solitons Žsolid curves., for N ranging from 1 to 10, from top to bottom.
397
Fig. 4. Quasi-steady-state normalized build time multiplied by the peak power to dark irradiance ratio r as a function of N, characteristic of the nonlinearity strength. Each data point is represented with its error margin as a straight vertical line bounded by two dots. The horizontal line is a mere guide to the eye.
HWHM does not depend on N either. In other words, whatever the strength of the nonlinearity N, the normalized quasi-steady-state soliton width is a constant, provided r is large enough. The numerical analysis made here seems to show that this constant is not far from pr2, within a 5% error margin. Only a thorough mathematical analysis of Eq. Ž6., which is not in the scope of this paper, would allow to confirm that the quasi-steady state soliton normalized HWHM is indeed pr2.
5. Quasi-steady-state soliton build up time The numerical computations that lead to Figs. 1–3 allow as well to compute the time needed to reach the minimum transient HWHM. This time can be called the quasi-steady-state soliton build up time. Fig. 4 shows the computed quasi-steady-state soliton build up time Žnormalized to Te s S r Id . multiplied by r along with the error margin yielded by the computation method. The wider error margin and the different average value for N from 6 to 10 are explained by the need to change the computation method owing to computer memory problems. It is important to notice that this quasisteady-state soliton build up time has been computed only for the soliton generation regime 3, for it is the only case where the quasi-steady-state soliton is narrower than the steady-state screening soliton. Therefore, given the numerical results and the allowed error margin, Fig. 4 suggests that, whatever the values of N and r within the range of our computations, r Ž t r Te . appears as a constant roughly equal to 2. Once again, only a precise mathematical analysis of Eq. Ž6. would be able to confirm this assertion and to yield a precise analytical value for this constant. In order to extrapolate this result to other values of N and r, we need to compare it to existing experimental
398
N. Fressengeas et al.r Optics Communications 145 (1998) 393–400
results. The literature does not yield experimental results concerning the quasi-steady-state soliton generation time. We thus chose to compare the order of magnitude of the time values we can predict, to other photorefractive phenomena.
6. Experimental validation 6.1. Limits of the model It is important to understand the limits of the comparisons that are presented in the following section. The very nature of the mathematical model makes the comparison to experiments difficult. Indeed, the wave propagation equation is assumed, in Section 2.4, to possess soliton solution, whose profile we investigate numerically. Therefore, the solutions of the wave propagation equation Ž3. that we obtain are only those with unvarying profile in the propagation direction. The spatial stability of these solutions cannot be investigated this way, although previous studies have shown the photorefractive screening solitons to be stable. In addition, there are certainly other solution classes that are interesting to study, such as oscillatory w9,21x or relaxing ones. The study of these solutions needs the development of a whole new study starting from the wave propagation equation Ž3.. This study would certainly have to be based on heavy numerical computation aiming to solve Ž3. numerically. This heavy computation is out of the scope of this paper but will be the focus of near-future studies. The HWHM versus time curves of Fig. 1 thus carry the assumption that the photorefractive medium supports a soliton at every moment. However, the only way this can be physically achieved is to launch in the photorefractive crystal a beam with the correct profile at every moment: that would require the input beam profile to change continuously with time, which is impossible, to our knowledge, with technology available at present. In this spirit, we will propose, on one hand, a comparison of the build up time predicted by our model to that which has been measured for other photorefractive phenomena such as two wave mixing ŽSection 6.2.. On the other hand, we shall suggest experiments that may allow the testing of our model ŽSection 6.3.. 6.2. Comparison with other photorefractiÕe phenomena The difference between the physical process of phenomena such as TWM or FWM and the photorefractive soliton mechanism needs to be emphasized. The simple fact that the time constant of TWM-like phenomena depends on the applied electric field whereas it does not as far as quasi-steady-state solitons are concerned shows the limits of the comparison, which is thus reduced to the comparison of orders of magnitude. Further evidence is
provided if one considers that this time constant depends on the grating wave vector for TWM-like phenomena, whereas this quantity does not exist, strictly speaking, in the physics of photorefractive solitons. In spite of these limits, it is still interesting to carry out these comparisons because of the simple fact that they are the only ones that can be made today with existing results. However, in spite of its transient nature, the quasi-steadystate soliton has one important characteristic in common with TWM-like phenomena: They both occur when the photorefractive effect is not saturated, provided the saturation is understood as a local independence of the spacecharge field upon the light intensity. This latter common characteristic shows why the following comparisons can and need to be made. If t is the quasi-steady state soliton build up time and Is the beam peak intensity, then, our model predicts that r Ž t r Te . ( 2, or tIs
e m s Ž ND y NA .
´ 0 ´r j NA
( 2.
Ž8.
It is of importance to notice that Ž8. does not depend on the dark irradiance Id anymore, which implies that the build up time predicted by Ž8. is independent of Id as well. In order to compare our model’s prediction to existing experimental results, we calculated the left-hand term of Eq. Ž8., for continuous wave and pulsed lasers. One has to understand that few measured values of all the parameters in Eq. Ž8. are available in the literature, that is the reason why we will compare our model to only four previously published experiments. In order to validate the assertion that t could be as small as possible, provided Is is high enough, we tested our model with pulsed lasers experiments. We used the results of TWM experiments from Refs. w23–25x, carried out with a BSO crystal. The build up time of the TWM process reported in Ref. w24x leads to r Ž t r Te . ( 2 = 10 2 whereas Ref. w25x leads to r Ž t r Te . ( 6 = 10 2. This large discrepancy between our prediction and these experimental results may be explained by the fact that the characteristic carrier recombination time Ž6 ms. is large with respect to the pulse duration Ž10 ns.: this case is not taken into account by the model developed in Ref. w14x, where the generation and recombination process is assumed to be steady-state ŽEq. Ž15a. of Ref. w14x.. Nevertheless, numerous other reasons may provide an explanation, such as fringe spacing Ža few micrometers. too small to validate the slow varying approximation of our model. On the other hand, TWM experiments were carried out on KNbO 3 w26x, which allows a carrier recombination time of 1 ms, only five times longer than the 210 ns laser pulse. These results lead to r Ž trTe . ( 4.5, which is in agreement with our predictions within the error margin yielded by Fig. 4, in spite of the short laser pulse, slightly outside the frame of our model.
N. Fressengeas et al.r Optics Communications 145 (1998) 393–400
Furthermore, Ref. w27x reports on FWM experiments carried out on a BaTiO 3 crystal in the continuous wave regime. The experimental conditions of Ref. w27x are exactly those required by our model. Using the crystal parameters given in Ref. w28x, the results of these experiments lead to r Ž t r Te . ( 2.1, remarkably consistent with our prediction. 6.3. Experimental Õalidation of the model We consider, in Section 2.4, a particular set of solutions of the nonlinear wave propagation equation: The solutions of unchanged beam profile that can be called ‘soliton solutions’. Doing this, we assumed theoretically that the photorefractive crystal supports a spatial soliton at every moment. Of course, this is a purely mathematical assumption which cannot, obviously, be physically true for the whole length of the crystal, since it would require the input beam to continuously match the soliton profile and the beam to be stable during its propagation in the crystal. Therefore, if spatial solitons are stable – as Refs. w9,17x tend to confirm – then, if the crystal is long enough and if the input beam profile is not too far from the soliton profile at every moment, there is a possibility that the output beam profile might converge in space towards the predicted soliton profile. With these conditions, it can be thought that the output beam profile might evolve like the predicted spatial soliton. If an experiment should indeed show an output beam profile evolving like the predicted soliton, not only would it validate our model but it would also prove that spatial solitons are stable. Such experimental results, recently obtained, will be published elsewhere. We must emphasize that, to our knowledge, this is the only way to strictly test our model experimentally. For instance, measurements of self-focusing powers Že.g., zscan experiments. may qualitatively look like our prediction but are not expected to match them quantitatively, except if the crystal is long enough to allow spatial convergence of the self focusing process towards spatial soliton. The experimental conditions expressed here in order to validate our model are restrictive but necessary. For instance, if the crystal is too thin in the direction of propagation, the beam profile cannot evolve enough in space and thus cannot be expected to match the soliton profile: the measurements of the output beam diameter on a too thin crystal would probably result in a diameter larger than that of the soliton Žif the input beam diameter is itself larger than the one possible soliton.. Furthermore, if the input beam profile is too different from the predicted soliton, it cannot be expected to converge towards a spatial soliton. In that case, the break-up of the beam into several selffocused filaments may be observed w29x. The theoretical explanation of theses phenomena cannot be obtained directly with our model but requires a numerical analysis of the photorefractive processes w29x.
399
7. Conclusion We have studied numerically the theoretical model developed in Ref. w14x. Three regimes of soliton build up have been evidenced. Their domain of existence have been numerically evaluated and our results can be reasonably extrapolated to higher values. The quasi-steady-state soliton has been shown to possess a characteristic normalized half width not far from pr2 and it is generated in a time t s 2Te r r, depending only on the crystal physical parameters, aside from the beam peak intensity to which it is inversely proportional. The predictions we thus make for the build up time have been successfully matched to previous experimental TWM and FWM build up times, provided the experimental conditions roughly match the assumptions of our model.
References w1x M. Segev, B. Crosignani, A. Yariv, Phys. Rev. Lett. 68 Ž1992. 923. w2x B. Crosignani, M. Segev, D. Engin, P. Di Porto, A. Yariv, G. Salamo, J. Opt. Soc. Am. B 10 Ž1993. 446. w3x G.C. Duree, J.L. Shultz, G.J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E.J. Sharp, R.R. Neurgaonkar, Phys. Rev. Lett. 71 Ž1993. 533. w4x M.D. Iturbe Castillo, P.A. Marquez Aguilar, J.J. Sanchez Mandragon, S. Stepanov, V. Vysloukh, Appl. Phys. Lett. 64 Ž1994. 408. w5x M. Segev, G.C. Valley, B. Crosignani, P. Di Porto, A. Yariv, Phys. Rev. Lett. 73 Ž1994. 3211. w6x M.F. Shih, M. Segev, G.C. Valley, G. Salamo, B. Crosignani, P. Di Porto, Electron. Lett. 31 Ž1995. 826. w7x G.C. Duree, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. Sharp, Optics Lett. 19 Ž1994. 1195. w8x M.F. Shih, P. Leach, M. Segev, M.H. Garett, G.J. Salamo, G.C. Valley, Optics Lett. 21 Ž1996. 324. w9x A.A. Zozulya, D.Z. Anderson, Phys. Rev. A 51 Ž1995. 1520. w10x M. Morin, G. Duree, G. Salamo, M. Segev, Optics Lett. 20 Ž1995. 2066. w11x G.C. Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani, P. Di Porto, E. Sharp, A. Yariv, Phys. Rev. Lett. 74 Ž1995. 1978. w12x M.T. Taya, M.C. Bashaw, M. Segev, G.C. Valley, Phys. Rev. A 52 Ž1995. 3095. w13x G.C. Valley, M. Segev, B. Crosignani, A. Yariv, M.M. Fejer, M.C. Bashaw, Phys. Rev. A 50 Ž1994. 4457. w14x N. Fressengeas, J. Maufoy, G. Kugel, Phys. Rev. E 54 Ž1996. 6866. w15x N.V. Kukhtarev, V.B. Markov, S.G. Odulov, M.S. Soskin, V.L. Vinetskii, Ferroelectrics 22 Ž1979. 949. w16x P. Yeh, Introduction to Photorefractive Nonlinear Optics, Wiley-Interscience, New York, 1993. w17x M. Segev, B. Crosignani, P. Di Porto, G.C. Duree, G. Salamo, E. Sharp, Optics Lett. 19 Ž1994. 1296. w18x D.N. Christodoulides, M.I. Carvalho, J. Opt. Soc. Am. B 12 Ž1995. 1628.
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w19x J. Feinberg, J. Opt. Soc. Am. 72 Ž1982. 46. w20x S.R. Singh, D.N. Christodoulides, Optics Comm. 118 Ž1995. 569. w21x A.A. Zozulya, D.Z. Anderson, Optics Lett. 20 Ž1995. 837. w22x K. Kos, H. Meng, G. Salamo, M. Shih, M. Segev, G.C. Valley, Phys. Rev. E 53 Ž1996. R4330. w23x G. Le Saux, G. Roosen, A. Brun, Optics Comm. 56 Ž1986. 374. w24x G. Roosen, G. Le Saux, G. Pauliat, M. Allain, J.M.C. Jonathan, A. Brun, Revue Phys. Appl. 22 Ž1987. 1253.
w25x D.C. Jones, L. Solymar, Optics Comm. 85 Ž1991. 372. w26x M. Zgonik, I. Biaggio, P. Amrhein, P. Gunter, Ferroelectrics ¨ 101 Ž1990. 15. w27x A. Krause, G. Notni, L. Wenke, Opt. Mater. 4 Ž1995. 386. w28x G.C. Valley, IEEE J. Quantum Electron. 19 Ž1983. 1637. w29x A.V. Mamaev, M. Saffman, D.Z. Anderson, A.A. Zozulya, Phys. Rev. A 54 Ž1996. 870.
Optics Communications 145 Ž1998. 393–400
Full length article
Build up mechanisms of ž 1 q 1/ -dimensional photorefractive bright spatial quasi-steady-state and screening solitons N. Fressengeas a
a,b,)
, D. Wolfersberger
a,b
, J. Maufoy b, G. Kugel
a
MOPS, CLOES, UniÕersite´ de Metz-Supelec, ´ 2 rue Edouard Belin, 57078 Metz Cedex 3, France b SEL, Supelec, ´ 2 rue Edouard Belin, 57078 Metz Cedex 3, France Received 14 November 1996; revised 11 April 1997; accepted 4 August 1997
Abstract A Ž1 q 1.-dimensional model is studied numerically to evidence the build up mechanisms of photorefractive solitons, from the characteristic carrier recombination time, through the quasi-steady-state soliton, to the screening soliton. Three different build up regimes are evidenced and their domain of existence are computed. The transient quasi-steady-state soliton is shown to be characterized by two constants: its normalized width and its normalized build up time response multiplied by its peak intensity over dark irradiance. This latter assertion allows us to predict the photorefractive soliton response time for various optical powers. It is thus compared to existing experimental results. q 1998 Elsevier Science B.V. PACS: 42.65.Sf; 42.65.Tg; 42.65.Hw; 42.65.Jx Keywords: Bright spatial optical soliton; Self-focusing; Space-charge field; Transient state; Photorefractive effect
1. Introduction The propagation of a single light beam in a photorefractive crystal has been the focus of growing interest in the past few years. In particular, quasi-steady-state bright spatial solitons have been predicted w1,2x and observed w3x. Steady-state self-focusing features have been observed then w4x, leading to the prediction w5x and the observation w6x of bright steady-state screening solitons trapped in one dimension or in both w7–9x. These phenomena occur only if the photorefractive crystal is properly biased by an external electric field or by an adequate photovoltaic nonlinearity. Three types of photorefractive bright spatial soliton have been predicted and observed. The quasi-steady-state soliton w1,2,10,11x has a limited lifetime; it depends neither on absolute light intensity nor on its ratio to dark irradi) C o rre s p o n d in g [email protected].
a u th o r.
E -m a il:
ance, as long as the latter is larger than unity. The screening soliton w6,5x is a steady-state phenomenon due to the partial screening of the electric field induced by an external source. The photovoltaic soliton w12,13x is a steady-state phenomenon as well but can be obtained without any external source, the photorefractive material being biased by the photovoltaic effect. Both the screening and the photovoltaic soliton depend only on the ratio of the absolute light intensity over the dark irradiance. Considering a beam that is allowed to diffract only in one direction Ža Ž1 q 1.-dimensional model., the three types of soliton can be thought of as three particular aspects of a more unique phenomenon which is the slow screening of the electric field induced either by an external source or by the photovoltaic effect or by both w14x. In this paper, a numerical investigation of the Ž1 q 1.dimensional model developed in Ref. w14x is provided in order to evidence the consequences of the electric field screening process on steady state and quasi-steady-state photorefractive bright spatial solitons. Three different soli-
0030-4018r98r$17.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 7 . 0 0 4 5 5 - 0
N. Fressengeas et al.r Optics Communications 145 (1998) 393–400
394
ton generation regimes are found, and their existence domains exhibited. Numerical results show that, provided its peak intensity normalized to dark irradiance is high enough, the quasi-steady-state soliton possesses two characteristic constants depending only on the crystal physical parameters: Its normalized width and its build-up time multiplied by its normalized peak intensity, as defined above. A comparison with the build up times of other photorefractive phenomena, such as two-wave mixing, is then provided.
2. Theoretical elements
The Ž1 q 1.-dimensional model developed in Ref. w14x is based on the Kukhtarev–Vinetskii model w15x reduced to one dimension. It describes the one level charge transport mechanism in a generic photorefractive medium. Several approximations, detailed in Ref. w14x, are used in this model to analyze the process of electric field screening: the beam is assumed to be much narrower than the crystal width but wide enough to validate the slow varying approximation, and the crystal Debye wave number is assumed to be large enough w14x Žor the Debye screening length small enough.. In typical photorefractive media, the electron life time is much shorter than the dielectric relaxation time, as confirmed by Yeh Žw16x, pp. 106, 107. and by the quantitative analysis of Ref. w14x, based on experimental values available in the literature. Under these assumptions, the Kukhtarev–Vinetskii model can be reduced to Eq. Ž1., which links the space-charge field and the light intensity w14x, E Ž IEt . Ex
qk BTm
E2I Ex
2
If the light intensity is assumed to vary slowly enough with time so that its last term can be neglected, Eq. Ž1. can be analytically solved and leads to the expression of the generalized space charge field Et as a function of the generalized light intensity I w14x,
ž
Et s E0 exp y
=
2.1. Light intensity and space charge field
em
2.2. Space charge field expression
q´ 0 ´ r
I0 E 2 Et n 0 Et E x
EI ye Et
s 0.
Ž1. Here, t and x are the time and transverse spatial coordinates. e is the elementary charge, m the electron mobility, k B the Boltzmann constant, T the temperature, ´ 0 the electric permeability of vacuum, ´ r the static dielectric tensor. n 0 is the free electron density generated by an arbitrary uniform illumination I0 , so that n 0 r I0 s sŽ ND y NA . r j NA , s being the photoexcitation coefficient, j the recombination constant, ND and NA the densities of donors and acceptors. I is the sum of the light intensity and of the dark irradiance Id , which is the modelling of the thermal generation of free carriers by an equivalent optical intensity. The generalized electric field Et is the sum of the space charge field and of the photovoltaic field Eph s bph j NA r e m s, bph being the component of the photovoltaic tensor along the ferroelectric c-axis, the other components being neglected.
ž
em n0
´ 0 ´ r I0
Ž Eext q Eph .
Id I
/
ž
It q 1 y exp y
y
k BT IX e
I
/
.
em n0
´ 0 ´ r I0
It
/
Ž2.
In Eq. Ž2., E0 is the electric field initial state and is a function of the transverse spatial coordinate. Eext is the externally applied field and is a constant in the crystal. I is the generalized light intensity and thus depends on the spatial coordinate x. I X is therefore its spatial derivative. Eq. Ž2. seems to show that the applied electric field, figured as Eext , and the photovoltaic effect, figured as Eph , have similar and additive influence on the space-charge field and thus on the wave propagation. However, this assertion has to be tampered by introducing two further ideas: Ø The addition is to be understood in an algebraic way: While the sign of Eext can be reversed experimentally, the direction of Eph is a function of the very nature of the photorefractive crystal. Ø It is important to understand the scope of the addition that appears in Eq. Ž2. between Eext and Eph . We have integrated Eq. Ž1. into Eq. Ž2. using boundary conditions valid only if the photovoltaic effect is small in effect with respect to the applied field. Therefore, this addition is valid only locally and should not be understood as a generally valid addition. One other limitation to the addition of the influence of the photovoltaic effect and of the applied electric field lies in the experimental setups used, for instance, in Refs. w4,14x. In these experiments, a background illumination is used to artificially raise the dark irradiance. Via the photovoltaic effect, this illumination adds a constant component to the current density that is not accounted for in the model and used here. However, the crystal used in these experiments was Bi 12TiO 20 which is not photovoltaic, and thus does not raise that problem.
2.3. WaÕe propagation equation The space charge field induces, via the Pockels effect, a refractive index variation in the photorefractive medium. Therefore, knowing the propagation of a beam in a medium with a low index modulation, Eq. Ž2. can lead to the wave propagation equation Ž3., describing the propagation of an
N. Fressengeas et al.r Optics Communications 145 (1998) 393–400
arbitrary light beam in a generic photorefractive medium w14x, 1 E2U
EU i
q EZ
2 EX 2
ž
= N 2yD
Id Ž 1 q < U < 2 . t
ž
y 1 y exp y E
/
ž
S
1 q
S
/
U s 0.
Ž3.
EN Ž X, Z . is the space-charge field initial state, defined by the crystal history. U s SCEr Id is the beam electric field SCE normalized to the dark irradiance. X s x r X 0 and Z s z r kX 02 , where X 0 is an arbitrary length, are the transverse and propagation-wise coordinates. Eq. Ž3. is not normalized with respect to time in order to clearly show the influence of the dark irradiance, and of the energy density S defined in the paragraph below. However, in analysing the temporal evolution of soliton solutions in Section 3, we shall normalize the time to the dielectric relaxation time in the dark Te s S r Id . The quantity S s Ž ´ 0 ´ rre m .Ž I0rn 0 ., which has already implicitly appeared in Ž2., is homogeneous to an energy density. Although the above definition of S uses an arbitrary illumination I0 and the electron density n 0 associated to it, it is completely determined by the crystal physical parameters and is thus a very intimate characteristic of the crystal related to the relaxation of the self focusing process. However, since I0 is arbitrary, we can give it particular values in order to show the physical meaning of S . For instance, if I0 s Id , S is the dielectric relaxation time in the dark multiplied by the dark irradiance. We can emphasize that S is, at each point, the product of the total local optical intensity Ždark irradiance q effective illumination. multiplied by the local dielectric relaxation time. N 2 is characteristic of the quasi-local mechanisms of the nonlinearity and is due to drift and photovoltaic mechanisms of transport:
'
N 2s
k 2 n 2 reff X 02 Ž Eext q Eph . 2
,
Ž4.
k is the beam wave vector, n is the medium base refractive index, and reff is the effective electro-optic coefficient. The drift mechanism of transport is due to the externally applied electric field Eext , while the photovoltaic effect is represented in Eq. Ž4. by Eph , as defined in Section 2.1. The quantity D is characteristic of the diffusion mechanism of transport and is expressed by 2 2
Ds
k n reff X 0 k BT 2e
.
essentially dependent on the electro-optic coefficient, since the diffusion process is nearly the same in all crystals. 2.4. Soliton solutions
/
U
y EN Ž X , Z . exp y
395
Ž5.
The effects of the diffusion mechanism of transport are
In the following section, we will focus our study on a particular set of solutions of the wave propagation equation Ž3.: The solutions which give an unchanging beam profile along the propagation direction z. They are called ‘soliton solutions’. These are evidently not the only solutions to Ž3.: obtaining all the solutions of Ž3. would require a complete and very difficult mathematical analysis, which is out of the scope of this paper. However, they are an interesting class of solutions to study if one aims at understanding the temporal behaviour of photorefractive solitons. For instance, such an approach has already been done, with another wave propagation equation, by Segev et al. w17x, by Valley et al. w13x and by Christodoulides and Carvalho w18x. Nevertheless, there are some other interesting solutions, such as oscillatory ones that are not included in this class. Furthermore, this study will not provide any analysis of the soliton spatial stability. We will discuss these limitations later in Section 6.1. Feinberg w19x and Singh and Christodoulides w20x have shown that the diffusion process, accounted for by the quantity D, induces a self deflection of the beam. Therefore, strict soliton beams Žwhich do not bend. can be achieved only if this term is neglected. Using this latter approximation, we assume the existence of a soliton solution UŽ X, Z, t . s 'r g Ž X, t .e i n Z in which Ø r is the ratio of beam peak intensity over dark irradiance. Ø g is the soliton profile normalized to 1: g Ž0, t . s 1 and g Ž"`, t . s 0. Ø n is the soliton propagation constant. Assuming the above form for UŽ X, Z, t ., basic algebra allows to derive from Ž3. an equation describing the normalized soliton profile g w14x, along with the propagation constant n Ž7.: Id t y2 ng q g Y y 1 y exp y Ž 1 q rg 2 . S g Id t 2N2 y 2g exp y Ž 1 q rg 2 . s 0. 2 S 1 q rg
ž
ž
/
ž
/
/
Ž6. In Eq. Ž6., the primes stand for the derivatives of g along the transverse spatial coordinate. The soliton propagation constant n is defined by
nsy
N2 r N2
q
ln Ž 1 q r .
ž
Ei y
Id t
S Id t q exp y Ž1 q r Id tr S r S
ž
Id t
/ ž / ./ ž /
Ž 1 q r . y Ei y
S
y exp y
Id t
S
,
Ž7.
396
N. Fressengeas et al.r Optics Communications 145 (1998) 393–400
where Ei is an exponential integral function defined by ` Ž yt EiŽ z . s yHyz e r t . d t. Eq. Ž6., along with Eq. Ž7., describes analytically the behaviour of the soliton profile in time and space, whereas Eq. Ž7. describes on its own the behaviour of the propagation constant. Although it is an interesting matter, it is not the purpose of this manuscript to analyze in detail the propagation constant. However, it should be kept in mind that the numerical analysis of Ž6. that we shall make in the following section, requires evaluating Ž7. beforehand because the soliton propagation constant appears in the first term of Ž6.. Therefore, both Eqs. Ž6. and Ž7. are the basis of the following numerical study.
3. Three different temporal regimes 3.1. Introduction A simple numerical integration of Ž6. along the space coordinate, for given values of r and t, leads, through an intermediate evaluation of the propagation constant n , to a numerical determination of the soliton profile. These calculations allow in turn to compute the time evolution of the soliton profile for a given value of r. Fig. 1 shows three typical time evolutions of the soliton half width at half maximum ŽHWHM. for N s 5, for r equal to unity, 300 and 1000 for the curves 1, 2 and 3, respectively. For the sake of simplicity, we normalized the time to the dielectric relaxation time in the dark Te s S r Id . The general shape of these curves is confirmed by the computations made by Zozulya and Anderson w21x, through a direct numerical study of the space charge field equation. The curves 1, 2 and 3 of Fig. 1 evidence three different HWHM time evolution regimes. Curve 1 Ž r s 1. shows a monotonically decreasing HWHM. Curve 2 Ž r s 300. shows a HWHM which presents a local minimum but relaxes to a more focused state. Curve 3 Ž r s 1000. shows
Fig. 1. Typical half width at half maximum time evolution for N s 5 and, respectively, for r equal to unity, 300 and 1000 for the curves 1, 2 and 3. The time is normalized to the dielectric relaxation time in the dark Te s S r Id .
Fig. 2. Domains of existence for each regime of soliton temporal behaviour. Each domain is labelled with the number of the regime it represents. White dots are the computed points which show the frontiers between the domains. Their height is the error margin yielded by the computation. The straight lines are mere guides for the eye.
a HWHM that presents an absolute minimum and relaxes to a less focused state. In other words, curve 1 shows an electric field screening process that yields a steady-state screening soliton, without generating on the way a quasisteady-state soliton. On the other hand, curves 2 and 3 evidence a quasi-steady-state soliton. The former shows a quasi-steady-state soliton larger than the screening soliton whereas the latter shows a quasi-steady-state soliton narrower than the screening one. This allows us to distinguish between three temporal behaviour regimes into which each of the HWHM versus time curve we computed can be classified. In the following, regime 1 is the class of the curves which do not show a quasi-steady soliton, regime 2 is the class of the curves that show wider quasi-steady state soliton, regime 3 being the remaining class. It is important to understand that, as r grows, the HWHM versus time curve evolves continuously from regime 1, through regime 2, to regime 3. The frontiers between these regimes can, however, be mathematically defined in a precise way, using the above definition, even though they represent a smooth transition from one regime to another. In addition, owing to the assumption that lead to Ž6., thus restricting our present study to soliton solutions of Ž3., all the solutions displayed in Fig. 1, as well as in Figs. 2–4, are soliton solutions. For obvious reasons, we call the steady-state solution ‘steady-state solitons’. We call the local minimum ‘quasi-steady-state soliton’ because the properties of this minimum with respect to the variation of the power ratio r are similar to the properties of quasi-steady-state solitons, as defined in the literature w1,2,10,11x. 3.2. Domains Eq. Ž6. is the equation that governs the shape of the HWHM versus time curves. Besides the space variable, it
N. Fressengeas et al.r Optics Communications 145 (1998) 393–400
depends on two independent parameters, namely N, characteristic of nonlinearity strength, and r, the beam peak intensity to dark irradiance ratio. Fig. 2 shows, for N ranging from 1 to 10, the domains on the Ž N, 'r . plane representing each regime, as defined previously. Each domain is labelled with the regime number. The white dots are the computed points which show the frontiers between the domains. Their height shows the maximum error in the result that the numerical analysis yields. Fig. 2 suggests that, on the Ž N, 'r . plane, the frontiers between these domains are straight lines passing by the point N s 1, r s 1, thus allowing to extrapolate these results to higher values of r and N.
4. Soliton HWHM Fig. 3 shows the behaviours, as a function of r, of the steady-state soliton HWHM Ždotted curve. and of the minimum HWHM reached during transient state Žsolid curve., for the integer values of N ranging from 1 to 10 Žfrom top to bottom.. These have been computed using the numerical analysis leading to the HWHM versus time curves, like those in Fig. 1. For the sake of clarity, Fig. 3 is displayed on a bi-logarithmic scale. For small r, i.e., for the regimes 1 and 2, the minimum achieved diameter is that of the screening soliton. This is the reason why the solid and dotted curve are the same for small r. The shape of the screening soliton HWHM curve displayed in Fig. 3 is consistent with the previous theoretical results found by Christodoulides and Carvalho w18x by other means, as well as with experimental results concerning steady-state screening solitons w22x. Furthermore, Fig. 3 suggests that, if r is large enough, then, the quasi-steady state soliton HWHM does not depend on r anymore, as confirmed by Refs. w1,2,11x. However, Fig. 3 shows that, if r is large enough, then the quasi-steady-state soliton
Fig. 3. Half width at half maximum as a function of the peak power to dark irradiance ratio r, for steady-state screening solitons Ždotted curves. and for quasi-steady-state solitons Žsolid curves., for N ranging from 1 to 10, from top to bottom.
397
Fig. 4. Quasi-steady-state normalized build time multiplied by the peak power to dark irradiance ratio r as a function of N, characteristic of the nonlinearity strength. Each data point is represented with its error margin as a straight vertical line bounded by two dots. The horizontal line is a mere guide to the eye.
HWHM does not depend on N either. In other words, whatever the strength of the nonlinearity N, the normalized quasi-steady-state soliton width is a constant, provided r is large enough. The numerical analysis made here seems to show that this constant is not far from pr2, within a 5% error margin. Only a thorough mathematical analysis of Eq. Ž6., which is not in the scope of this paper, would allow to confirm that the quasi-steady state soliton normalized HWHM is indeed pr2.
5. Quasi-steady-state soliton build up time The numerical computations that lead to Figs. 1–3 allow as well to compute the time needed to reach the minimum transient HWHM. This time can be called the quasi-steady-state soliton build up time. Fig. 4 shows the computed quasi-steady-state soliton build up time Žnormalized to Te s S r Id . multiplied by r along with the error margin yielded by the computation method. The wider error margin and the different average value for N from 6 to 10 are explained by the need to change the computation method owing to computer memory problems. It is important to notice that this quasisteady-state soliton build up time has been computed only for the soliton generation regime 3, for it is the only case where the quasi-steady-state soliton is narrower than the steady-state screening soliton. Therefore, given the numerical results and the allowed error margin, Fig. 4 suggests that, whatever the values of N and r within the range of our computations, r Ž t r Te . appears as a constant roughly equal to 2. Once again, only a precise mathematical analysis of Eq. Ž6. would be able to confirm this assertion and to yield a precise analytical value for this constant. In order to extrapolate this result to other values of N and r, we need to compare it to existing experimental
398
N. Fressengeas et al.r Optics Communications 145 (1998) 393–400
results. The literature does not yield experimental results concerning the quasi-steady-state soliton generation time. We thus chose to compare the order of magnitude of the time values we can predict, to other photorefractive phenomena.
6. Experimental validation 6.1. Limits of the model It is important to understand the limits of the comparisons that are presented in the following section. The very nature of the mathematical model makes the comparison to experiments difficult. Indeed, the wave propagation equation is assumed, in Section 2.4, to possess soliton solution, whose profile we investigate numerically. Therefore, the solutions of the wave propagation equation Ž3. that we obtain are only those with unvarying profile in the propagation direction. The spatial stability of these solutions cannot be investigated this way, although previous studies have shown the photorefractive screening solitons to be stable. In addition, there are certainly other solution classes that are interesting to study, such as oscillatory w9,21x or relaxing ones. The study of these solutions needs the development of a whole new study starting from the wave propagation equation Ž3.. This study would certainly have to be based on heavy numerical computation aiming to solve Ž3. numerically. This heavy computation is out of the scope of this paper but will be the focus of near-future studies. The HWHM versus time curves of Fig. 1 thus carry the assumption that the photorefractive medium supports a soliton at every moment. However, the only way this can be physically achieved is to launch in the photorefractive crystal a beam with the correct profile at every moment: that would require the input beam profile to change continuously with time, which is impossible, to our knowledge, with technology available at present. In this spirit, we will propose, on one hand, a comparison of the build up time predicted by our model to that which has been measured for other photorefractive phenomena such as two wave mixing ŽSection 6.2.. On the other hand, we shall suggest experiments that may allow the testing of our model ŽSection 6.3.. 6.2. Comparison with other photorefractiÕe phenomena The difference between the physical process of phenomena such as TWM or FWM and the photorefractive soliton mechanism needs to be emphasized. The simple fact that the time constant of TWM-like phenomena depends on the applied electric field whereas it does not as far as quasi-steady-state solitons are concerned shows the limits of the comparison, which is thus reduced to the comparison of orders of magnitude. Further evidence is
provided if one considers that this time constant depends on the grating wave vector for TWM-like phenomena, whereas this quantity does not exist, strictly speaking, in the physics of photorefractive solitons. In spite of these limits, it is still interesting to carry out these comparisons because of the simple fact that they are the only ones that can be made today with existing results. However, in spite of its transient nature, the quasi-steadystate soliton has one important characteristic in common with TWM-like phenomena: They both occur when the photorefractive effect is not saturated, provided the saturation is understood as a local independence of the spacecharge field upon the light intensity. This latter common characteristic shows why the following comparisons can and need to be made. If t is the quasi-steady state soliton build up time and Is the beam peak intensity, then, our model predicts that r Ž t r Te . ( 2, or tIs
e m s Ž ND y NA .
´ 0 ´r j NA
( 2.
Ž8.
It is of importance to notice that Ž8. does not depend on the dark irradiance Id anymore, which implies that the build up time predicted by Ž8. is independent of Id as well. In order to compare our model’s prediction to existing experimental results, we calculated the left-hand term of Eq. Ž8., for continuous wave and pulsed lasers. One has to understand that few measured values of all the parameters in Eq. Ž8. are available in the literature, that is the reason why we will compare our model to only four previously published experiments. In order to validate the assertion that t could be as small as possible, provided Is is high enough, we tested our model with pulsed lasers experiments. We used the results of TWM experiments from Refs. w23–25x, carried out with a BSO crystal. The build up time of the TWM process reported in Ref. w24x leads to r Ž t r Te . ( 2 = 10 2 whereas Ref. w25x leads to r Ž t r Te . ( 6 = 10 2. This large discrepancy between our prediction and these experimental results may be explained by the fact that the characteristic carrier recombination time Ž6 ms. is large with respect to the pulse duration Ž10 ns.: this case is not taken into account by the model developed in Ref. w14x, where the generation and recombination process is assumed to be steady-state ŽEq. Ž15a. of Ref. w14x.. Nevertheless, numerous other reasons may provide an explanation, such as fringe spacing Ža few micrometers. too small to validate the slow varying approximation of our model. On the other hand, TWM experiments were carried out on KNbO 3 w26x, which allows a carrier recombination time of 1 ms, only five times longer than the 210 ns laser pulse. These results lead to r Ž trTe . ( 4.5, which is in agreement with our predictions within the error margin yielded by Fig. 4, in spite of the short laser pulse, slightly outside the frame of our model.
N. Fressengeas et al.r Optics Communications 145 (1998) 393–400
Furthermore, Ref. w27x reports on FWM experiments carried out on a BaTiO 3 crystal in the continuous wave regime. The experimental conditions of Ref. w27x are exactly those required by our model. Using the crystal parameters given in Ref. w28x, the results of these experiments lead to r Ž t r Te . ( 2.1, remarkably consistent with our prediction. 6.3. Experimental Õalidation of the model We consider, in Section 2.4, a particular set of solutions of the nonlinear wave propagation equation: The solutions of unchanged beam profile that can be called ‘soliton solutions’. Doing this, we assumed theoretically that the photorefractive crystal supports a spatial soliton at every moment. Of course, this is a purely mathematical assumption which cannot, obviously, be physically true for the whole length of the crystal, since it would require the input beam to continuously match the soliton profile and the beam to be stable during its propagation in the crystal. Therefore, if spatial solitons are stable – as Refs. w9,17x tend to confirm – then, if the crystal is long enough and if the input beam profile is not too far from the soliton profile at every moment, there is a possibility that the output beam profile might converge in space towards the predicted soliton profile. With these conditions, it can be thought that the output beam profile might evolve like the predicted spatial soliton. If an experiment should indeed show an output beam profile evolving like the predicted soliton, not only would it validate our model but it would also prove that spatial solitons are stable. Such experimental results, recently obtained, will be published elsewhere. We must emphasize that, to our knowledge, this is the only way to strictly test our model experimentally. For instance, measurements of self-focusing powers Že.g., zscan experiments. may qualitatively look like our prediction but are not expected to match them quantitatively, except if the crystal is long enough to allow spatial convergence of the self focusing process towards spatial soliton. The experimental conditions expressed here in order to validate our model are restrictive but necessary. For instance, if the crystal is too thin in the direction of propagation, the beam profile cannot evolve enough in space and thus cannot be expected to match the soliton profile: the measurements of the output beam diameter on a too thin crystal would probably result in a diameter larger than that of the soliton Žif the input beam diameter is itself larger than the one possible soliton.. Furthermore, if the input beam profile is too different from the predicted soliton, it cannot be expected to converge towards a spatial soliton. In that case, the break-up of the beam into several selffocused filaments may be observed w29x. The theoretical explanation of theses phenomena cannot be obtained directly with our model but requires a numerical analysis of the photorefractive processes w29x.
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7. Conclusion We have studied numerically the theoretical model developed in Ref. w14x. Three regimes of soliton build up have been evidenced. Their domain of existence have been numerically evaluated and our results can be reasonably extrapolated to higher values. The quasi-steady-state soliton has been shown to possess a characteristic normalized half width not far from pr2 and it is generated in a time t s 2Te r r, depending only on the crystal physical parameters, aside from the beam peak intensity to which it is inversely proportional. The predictions we thus make for the build up time have been successfully matched to previous experimental TWM and FWM build up times, provided the experimental conditions roughly match the assumptions of our model.
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