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By W.T. Tsang (Eds.)

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4a, n(x) is given by + for x > 0 (17) n(x) = n b A n exp(-x/L) with x increasing from the surface into the bulk. The quantity n b is the refractive index in the bulk and L is the diffusion length. This dependence was used in the analysis by Conwell ( 1973), which follows. In the following analysis we assume nbto be isotropic. Since many dielectrics used in guided-wave devices are uniaxial, two indices of refraction are required in treating such dielectric guides. , Nye, 1957). Therefore, the results given below must be properly modified (Conwell, 1974)when applied to actual devices.

1. INTEGRATABLE ACTIVE AND PASSIVE OPTICAL DEVICES 27 with an abrupt index change can support more lateral modes than a guide with a corresponding but gradual index change because of a tighter confinement in the former. In both cases the number of modes increases with w/A. Either Eq. (66) or (67) should be useful in estimating the number of lateral modes. In concluding this section, we review the procedure for finding Anzff. From Eq. ( 5 I), we see that Wff An& = -Ak$/ki = AfI2/ki. (68) Therefore, the value of Anzff can be found from the difference Apz = /I2@) - P2(II) between the two regions (I) and (11) in Figs.

INTEGRATABLE ACTIVE AND PASSIVE OPTICAL DEVICES 25 The eigenvalue of p for the pth lateral mode is - ( 2 k o / w ) W f f ( 2 +p 1). I; is k$[n2fl(0)- Anfff], the maximum nump; = k;n;ff(O) ber of lateral modes is related to An& by p + 15(koW/4)&&++. 10. 1 pm, and w = 10 fim. [After Reinhart et al. 1 pm, and w = 10 pm. Figure 1 1 shows the structure and the measured mode pattern at d = l. 30. 3 for the inverted ridge guide. Actually, the rib guide had a very abrupt transition to the rib region. In other words, the change in n& should be represented by a step function and the field distribution should be represented by a sinusoidal function in the rib region.

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