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987 µm, only the TE0 mode 4 Waveguides, Optical is supported and a single-mode waveguide results. As the wavelength becomes shorter and shorter, more and more modes are allowed to propagate. The number of guided modes supported by the waveguide can be determined from mc , the value of m evaluated at β/k = n2 : mc = √ 4t 2 1 (n − n22 )1/2 − tan−1 (r3 a). λ 1 π (9) The next integer larger than mc gives the number of modes. 21 and for the TE and TM modes, respectively. The waveguide therefore supports only the TE0 and TM0 modes.
18. C. S. Tsai, B. Kim, and F. R. El-Akkari, Optical channel waveguide switch and coupler using total internal reflection, IEEE J. , QE-14 (7): 513–517, 1978. 39. R. C. Alferness, Polarization-independent optical directional coupler switch using weighted coupling, Appl. Phys. , 35: 748– 750, 1979. ELECTROPHOTOGRAPHY 40. P. Granestrand and L. Thylen, Polarization independent switch and polarization splitter employing ⌬ͱ and ⌬ modulation, Electron. , 24: 1142–1143, 1988. 41. P. Granestrand, Four-sectioned polarization-independent directional coupler with extremely relaxed fabrication tolerances, IEEE Photon.
While light within the forbidden bands cannot propagate in the perfect PBG structure, it can propagate through the defects with a low loss. Defect control in a PBG structure thus provides a highly effective means for the manipulation of light.