By Keigo Iizuka; John Wiley & Sons, Inc
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Extra info for Elements of Photonics, Volume II : For Fiber and Integrated Optics
As already OTHER METHODS FOR OBTAINING THE CHARACTERISTIC EQUATIONS 629 stated, in the two outermost layers, the wave is necessarily an evanescent wave, and the first and last layers are set aside for now. The matrix that applies to the ith layer in the geometry shown in Fig. 8 is, from Eq. 93 In the event that a particular mode that has an evanescent wave in the ith layer is desired, the transmission matrix Ti xi xi 1 of the ith layer has to be derived in a similar manner as Eq. 93) was derived from Eqs.
8. xi 1 Ä x 0 , x Ä xi where xi 1 is the value of x at the lower boundary of the ith layer and xi is the value of x at the upper boundary of the ith layer. 8 Geometry of a layered medium. solving the simultaneous Eqs. 78) and back into Eqs. 79). 80) is the transmission matrix that relates the fields at x D x 0 with those at x D x. The x 0 and x can be set at any point as long as they belong to the same layer. 80) is the only type of matrix that is needed for the transmission matrix method. The transmission matrix method will first be illustrated using the example of a threelayer guide with the geometry shown in Fig.
This is illustrated in Fig. ) As Â increases, so does the effective refractive index N. For the mode-index lens shown in Fig. 7, the advance in constant phase lines is slowed down the most where N is the greatest, and the spacing between the constant phase lines narrows. The result is that the phasefronts of the transmitted wave converge toward the point F. 6 TE MODES The wave optics approach applied earlier to the TM modes is applied in a similar manner to the TE modes. Since the approach is so similar, only the key formulas are repeated for the benefit of summarizing the approach.
Elements of Photonics, Volume II : For Fiber and Integrated Optics by Keigo Iizuka; John Wiley & Sons, Inc