(z) where C = AiJrfX]). Although an approximation, this is reasonable and commonly used for diffused strip waveguides, e.g., the optical field was found in Keil and Auracher [23] to be approximately Gaussian in width and Hermite-Gaussian in depth. Since the expression for the propagation constant R is known [24], and using 23 neff = RlkQ, nejy(x) can be obtained from: P [n2(^)4)(|)*-(V4))-(V(j)*)]& \u00bb > ) = \u2014 Tm (3-7) J \u2014CO given

nejg(x)), whereas the n'^x) for the weakly guiding arm (n\\^x) < ne^x)) is calculated using the described method. In Figure 3.5, neff'(Xj) = n^Xj) + An^Xj) giving An^Xj) = n'ejff(Xj) - n^Xj), where n'^Xj) can be obtained from the E I M and nejff(Xj) is calculated for 0 V applied, i.e., the case depicted in Figure 3.4. Then at x = -Xj, n'^-Xj) = n^Xj) -An^Xj) since n^Xj) = n^-Xj). The unmodulated profile is also shown in Figure 3.5 for reference, (dashed line). 3.3 Input and Output The input to the Y-branch modulator is the fundamental TM-like mode with unit power, here called the eigenfunction UJx). UJx) is the lowest order mode which can propagate through the straight unperturbed waveguide with no loss in power. To calculate the eigenfunction UJx), a commonly used procedure described in reference [25] is employed. This procedure works well with any B P M and is described briefly here. Firstly, an arbitrary electric field distribution E(x,0), such as a Gaussian profile with a beam waist of 4 pm, is allowed to propagate down the straight unperturbed waveguide for a length Y by repeatedly applying equation 3.4. Then the propagation constants of the modes comprising E(x,0) can be determined by computing the correlation function P(y) = fE*(x,0)E(x,y)dy (3.11) If P(y) is multiplied by a Hanning window and then Fourier transformed, the propagation 25 13 20 x (urn) Figure 3.4 Unmodulated effective index neff(x) Figure 3.5 Modulated effective index n'eff(x) during voltage application constant 151 of each mode for the profile E(x,0) can be found by using a line-fitting procedure [26] and by taking into account that njc0 < B < njc0. Ut(x) can subsequently be evaluated by integrating E(x,y) with the corresponding Bt as follows [25]: where w(y) is an appropriate window function such as a Hanning window. Therefore, for our case, Y = 2048 jum for 2 1 1 steps at 1 pm intervals. Once the UJx) is determined for the particular maximum refractive index ns being used, the Y-branch modulator can be excited with the modal eigenfunction and its propagation behaviour can then be examined. The output power of the modulator at a distance y can be evaluated by taking the inner product of the propagating field with the eigenfunction UJx). In the straight section of the Y-branch (before the horn), the power is For the power output at the branching section, the eigenfunction UJx) would have to be shifted in (3.14) to overlap the two branch arms properly. In the simulations, we have decided to compute the power output when the two arms are 40 \/zm apart centre to centre so that the optical fields no longer interact with each other. The power is then (3.12) (3.13) P(y) = JE(x,y)Ul{x)dx (3.14) P(y) = f E(x,y)U*(x-20) dx\\ + f E(x,y)U*o(x + 20)dx 2 (3.15) 27 The on\/off ratio of the modulator with voltage application is taken as the ratio of the power in the strongly guiding arm to the power in the barely guiding arm. 3.4 Simulation Procedure To prevent sudden changes in the refractive index profile and to minimize reflections when the Y-branch is spreading out, the Y-j unction and the branching arms of the modulator are made to step out at 0.1 jitm (which is equal to the step size possible in a electron-beam fabricated mask). The longitudinal step length Ay varies for each branch angle so that the number of steps required to reach the 40 pm separation point can be kept approximately constant for all the branch angles. The step length and its corresponding angle are shown in Table III. The horn length is divided into 20 sections, and each of those consists of 20-30 sections of Ay. Table HI Step length Ay branch angle 6 step length Ay (\/xm) 0.5\u00b0 1.0\u00b0 1.5\u00b0 2.0\u00b0 2.5\u00b0 3.0\u00b0 1.0 0.5 0.3 0.19 0.23 0.19 28 The entire simulation procedure can be summarized as follows: 1. Calculate the eigenfunction UJx). 2. Evaluate the effective index nejJx). 3. Evaluate the modulated effective index n'ejg(x) when a voltage is applied. 4. Propagate UJx) through the Y-branch modulator. 5. Calculate the power guided and the on\/off ratio. The complete simulation program is written in Pascal and Turbo Pascal version 4.01 by Borland is used to compile and run the program. Running on a 33 M H z IBM-386 compatible machine with a math co-processor, approximately 1.5 hours are required for each simulation on average if the eigenfunction UJx) is predetermined. 29 Chapter I V Simulation Results 4.1 n s - Maximum Refractive Index at the Surface The first investigation performed using the numerical simulations was to determine a value for ns (giving the required Ti thickness) for the Y-branch waveguides. A value for ns resulting in weakly guided light which can be easily steered by applying a voltage is desirable. If the waveguides are too strongly guiding, then the voltage-induced refractive index changes will not be sufficient to make the necessary difference to the index profile. However, if ns is too small, no waveguide is created. We begin by using ns = 2.205, i.e., the maximum refractive index change AHs = ns - nb = 0.0050, which corresponds to approximately 0.667% T i concentration by weight at the surface for z-cut L i N b 0 3 at X = 632.8 nm [27]. The eigenfunction UJx) is found first, then it is used to excite a 0.5\u00b0 Y-branch modulator with a two-horn-length electrode at 80 V . The on\/off ratio evaluated is 4.6:1 and the power guided is 80% (100% power guided being for a straight waveguide). Since the on\/off ratio is very low and the power guided is high, the waveguides are too strongly guiding for the intended purpose. The next AJIS we attempt uses AHS = 0.0042, which corresponds to 0.565% T i 1 . The simulation results showed an improvement with a branch angle 6 = 0.5\u00b0. The 1 This is equivalent to indiffusing a layer of Ti approximately 480A thick (see appendix B for calculation), which past fabrication experience at UBC solid state laboratory has shown will produce waveguides at A.G = 632.8 nm. 30 on\/off ratio at 80 V is 10:1, and the power guided is approximately 70%. Then we set ATIS = 0.0035, which corresponds to 0.467% T i . With this low AUS, the guide is found to be incapable of guiding the fundamental TM-like mode. Our results showed that for a 6 hour diffusion at 1050X, AJIS = 0.0035 is too low to produce waveguides, AHS = 0.0050 is too high for the intended application, whereas Atis = 0.0042 could both be useful and was consistent with currently used fabrication procedures. Figure 4.1 shows the on\/off ratios at various voltages obtained from the various AHS values. With AHS = 0.0042, a modulator with a 2\u00b0 branch angle at 80 V has an on\/off ratio of 84:1 with about 50% of the power guided, while with AHS = 0.0050 the on\/off ratio under the same conditions is 43:1 with 53% of the power guided. Since AHS = 0.0042 provides much higher on\/off ratios while losing only a few percent of guided power, it is selected for our subsequent simulations. It should be pointed out that a value other than AHS = 0.0042 could lead to improved performance. However, as mentioned, this value is very realistic due to its correspondence to a known fabrication procedure. Figure 4.2 shows the optical power of the eigenfunction UJx) which corresponds to AUS \u2014 0.0042, the spot size (defined as the full width at half the maximum power) is 4 u-m. 4.2 Electrode Length As mentioned previously, the electrodes should be as short as possible to minimize capacitance. However, poor modulation and low on\/off ratio result i f the electrodes are too short. This is because there is not enough change in the refractive index to steer the 31 Figure 4.1 On\/off ratios for a 2.0\u00b0 branch with a two-horn-length electrode at various voltages for An, = 0.0050 and An, = 0.0042 32 X o CD 0.225 q 0.200 z_ 0.175 : 0.150 0.125 : O 0.100 H CL \u2014 0.075 H U O 0.050 0.025 H 0.000 i\u2014i\u2014i\u2014i\u2014i\u2014i\u2014\\\u2014r\u2014i\u2014i\u2014i\u2014i\u2014i\u2014\\\u2014\\\u2014r\u2014i\u2014i\u2014i\u2014i 50 - 4 0 - 3 0 - 2 0 - 1 0 0 10 20 30 40 50 x ( u m ) Figure 4.2 Eigenfunction U0(x) for An, = 0.0042 33 light into the ON branch. To investigate the effect of electrode length on the performance of the modulator, a 2\u00b0 Y-branch is simulated with three different electrode lengths: one-horn-length, two-horn-length, and three-horn-length long electrodes. A l l the electrodes originate from the same location, i.e., at the beginning of the horn. The one-horn-length electrode fails to cause substantial modulation of the optical fields in the two arms. The on\/off ratio is very low because of the small amount of steering. Due to the poor results obtained we abandoned the single-horn-length electrode simulations. Comparing the results for the two-horn-length electrode with those for the three-horn-length electrode, the two-horn-length electrode is found to give higher on\/off ratios (see Figure 4.3). This may seem surprising at first because longer electrode lengths are generally associated with higher on\/off ratios. The fact that the on\/off ratios from the three-horn-length electrode are actually lower than those from the shorter electrode may indicate cross coupling effects. Although we are dealing with large branch angles for Y -branch modulation, the angles are still fairly small, i.e., the two branch arms are still nearly parallel, such that some coupling will result. In any case, the three-horn-length electrode shows, in no instance, any notable improvement in performance over the two-horn-length electrode but obviously increases the capacitance of the modulator. Since one of the criteria in electrode selection is to minimize capacitance a two-horn-length electrode is chosen. 34 0 f i I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I M 1 I I I I I 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 appl ied vol tage (V) Figure 4.3 On\/off ratios for a 2\u00b0 Y-branch at two different electrode lengths 35 4.3 Effects of Branch Angle To investigate the effect branch angle 6 has on the performance of the modulator, devices having branch angles of 0.5\u00b0, 1.0\u00b0, 1.5\u00b0, 2.0\u00b0, 2 .5\u00b0 , 3.0\u00b0 are simulated. The AHS is 0.0042 and the electrode length used is two-horn-length long. The theoretical amount of guided power for the various 0 with 0 V , 50 V , and 75 V applied is shown in Figure 4.4. The on\/off ratios for 50 V and 75 V are shown in Figure 4.5. The general trend is that the guided power decreases with increasing branch angle (above 1 \u00b0), while the on\/off ratios increase with increasing branch angle [28]. The guided power is expected to decrease with increasing angle 0 because of the increased radiation loss, regardless of voltage application. When a voltage is applied, however, light is channelled into the ON branch but not into the other branch. As the branch angle increases, the greater this discrepancy becomes and the higher the on\/off ratios can be obtained. Referring to Figure 4.6, the operation of the modulator can be classified into three regions according to angle 0. Region I is for 0 less than 0.8\u00b0; region II is for those with 0 between 0.8\u00b0 and 2.0\u00b0; region III is for 0 greater than 2.0\u00b0. 4.3.1 Region I In this region, the on\/off ratios are low while the guided power is high. The high guided power is due to the shallow branch angle, i.e., low radiation loss. With no voltage applied, the percentage guided power is over 90%. With voltage application, however, the optical power in the ON branch increases moderately but is greatly 36 100 0 | i i i i i i i i i i i i i i i i i i i i i i i i i ' I ' ' I 0.0 0.5 1.0 1.5 2.0 2.5 3.0 branch angle Figure 4.4 Percentage guided power vs. branch angle 70 60 H 50 O D 40H O 30 20 H 10 75 V 50 V - i I i i 1 i i 1 i | \u2014 i \u2014 i \u2014 i \u2014 i \u2014 | \u2014 i \u2014 I \u2014 i \u2014 i \u2014 | \u2014 i \u2014 r ~ i \u2014 i \u2014 | \u2014 i \u2014 i \u2014 i \u2014 i \u2014 j 0.0 0.5 1.0 1.5 2.0 2.5 3.0 branch angle Figure 4.5 On\/off ratio vs. branch angle 6 100