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As q(xt ) = 0, the approximation given by Eq. (30) fails at the turning point. To find the solution at x = xt , the function q2 (x) in the neighborhood of the turning point is approximated by a linear function: −∞ < x < 0, x 0 Figure 6. Refractive-index profile of a graded-index planar waveguide. |q| dx)), xt < x < + ∞, xt (37) (38) and the dispersion relation where q(x) = [n2 (x)k2 − β2 ]1/2 . (31) For the TE mode, ψ and dψ/dx are continuous at x = 0, while for the TM mode, ψ and [1/n2 (x)]dψ/dx are continuous at x = 0.

21. Y. Silberberg, P. Perlmutter, and J. E. Baran, Digital optical switch, Appl. Phys. , 51 (16): 1230–1232, 1987. 22. L. M. Johnson and H. V. Rousell, Reduction of intermodulation distortion in interferometric optical modulators, Opt. , 13 (10): 928–930, 1988. 23. E. M. Zolotov and R. F. Tavlykaev, Integrated optical Mach-Zehnder modulator with a linearized modulation characteristic, Sov. J. , 18 (3): 401–403, 1988. 24. S. K. Korotky and R. M. Ridder, Dual parallel modulation scheme for low-distortion analog optical transmission, IEEE J.

The b − M relation applies to all guided ∇t n2 (x, y) ], n2 (x, y) (47) where ∇t is the transverse part of ∇. In general, the waveguide supports hybrid modes or vector modes, which contain all six field components. Once the transverse electric-field is found, the z component Ez and the magnetic field H can be calculated from Ez = 1 [∇t · Et + Et · ∇t lnn2 (x, y)], jβ (48) 1 ∇ × E. jωµ0 (49) and H =− Alternatively, one can first solve the transverse magnetic field from a wave equation similar to Eq. (47) and then find the other field components from Maxwell’s equations.

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