The Dynamics of the System

We have used an iterative evolving grating approach (for examples see [31, 33, 50, 51, 70]) to model the dynamics of this system, the equations for which are shown below in 5.2a-h. With this approach the optical fields are assumed to respond instantaneously to

any grating change. In a photorefractive material like BaTiC^, this assumption is valid because of the very slow response time of the material, particularly when illuminated with milliwatt cw laser fields. This assumption allows the spatial integration of the fields to be performed as the grating is assumed to be static at any point in time. These calculated fields can then be used to determine the grating at the next point in time. This process is then repeated until steady state is reached.

fiAp = " gm A2m ' gsA 2s " a A p dz = gm A p “ a A 2m dz M 2s = gs Ap - a A 2s dz ^A3m = gm A r + a A 3m dz d£3s = Ss A r + a A 3s dz 5.2e fiAr = " gm A 3m " gsA 3s + ^ A r dz 5 .2 f ^ = X( ApA2m + ArA3m ) " gm dt Io 5.2g T dgs = X ( ApA2s + ArA 3s ) " gs dt Iq 5.2h

As mentioned previously there is no direct coupling between the multi-mode and single mode fields, however, the two are linked via the total intensity, Ip. The boundary conditions for this system are:

5.2a

5.2b

5.2c

Ap(0) = 1 ; A2m(0) = EmAptO); A2s(0) = esAp( 0 ) ; Ar(d) = 0 ;

= rm ^2m^) » ^ 3s(d) = rs^2s(^)+ P(rs " rm) ^2m(^) 5.2i

The last boundary condition describes bootstrapping of the single mode reflectivity to the multi-mode mode reflectivity. This bootstrapping arises because the multi-mode grating requires all of its constituent modes to be reflected by a mirror of reflectivity rm to sustain itself. However, the single mode component of the multi-mode radiation is unattenuated and is reflected by a m irror of reflectivity rs, thereby creating a residual field. This residual field then enhances the single mode reflected beam. The quantity p describes the fraction of the multi-mode radiation that is single mode.

This system of equations is solved iteratively to yield the evolution of the single mode and multi-mode fields in the oscillator. Figures 5.9a-d show the evolution of these fields for various parameter values. An estimate for the parameter p, was obtained by measuring the fraction of radiation that is transmitted through the transparency ( see figure 5.7) that also passes through the pin hole, before oscillations begin in the SLPCM. However in order to obtain better agreement between experiment and model a value approximately twice this estimate was used. The model, like the experiments, demonstrates the trend that increasing the differential loss causes a more rapid turn on of the single mode radiation. As well as predicting the transient behaviour of the system the model also gives reasonable agreement with the observed steady-state behaviour, particularly in showing the trend that increasing the differential loss increases the fraction of the power in the single mode component in the steady-state.

Im portantly the model agrees with the observed behaviour in the case where the differential loss is zero (i.e. when there is no mode selector). For this case the model predicts that all modes receive an equal share of the diffracted pump power, and as the single mode is only one o f a large number of modes that comprise the multi-mode radiation, it will receive a negligible amount of the total power. Another important area of

agreement between the model and the observed behaviour, occurs for the case when the differential loss was removed after the steady-state had been reached (figure 5.9d). This result deserves particular mention as an alternative model, that is based on the interaction of one pump beam with two separated gratings, was proposed [51] to explain switching of power between two physically separated SLPCMs. In that particular model, once switching had occurred there was no way to switch the power back again. However, as figure 5.9d indicates, in the system described in this paper, the power can be switched at will between the multi-mode and single mode SLPCMs, simply by adding or removing the mode selector. This reversibility is a consequence of the fact that, in the present case, the two gratings exist in one region, i.e. they are not physically separated.

The quantity x in equations 5.2g and 5.2h, is an intensity dependent time constant characteristic of the photorefractive response of the material (Feinberg et al 1980 [20]). For all the cases shown in figures 5.9a-d the same input power was used, so it was appropriate to use the same time constant in the various simulations. This gave satisfactory results when comparing simulations of the growth of single mode power in the presence of different differential loss (figures 5.9a-c ). However, to obtain good agreement between the theoretical and observed behaviour for the case where the transparency was removed after steady-state had been reached (figure 5.9d), it was necessary to decrease the time constant by a factor of two below its previous value.

The effect of bootstrapping the single mode mirror reflectivity to the multi-mode reflectivity is not critical to the predicted mode selecting behaviour. Similar behaviour occurs without bootstrapping, except the turn on of the single mode field is delayed and the time taken to reach the steady state is increased. Figure 5.11 shows a typical behaviour of the system with and without bootstrapping.

Steady state solutions of equations 5.2a-h are needed to determine how much single mode power can be obtained as a function of differential loss. Unfortunately the complexity of

the differential equations makes it unlikely that exact solutions can be found directly. However by linking four wave mixing (FWM) and two-wave mixing (TWM) processes and making use of the solutions of an "equivalent" TWM system, it has proved possible to find approximate solutions for the system described here.

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