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Unformatted text preview: 118 Components where I is the identity matrix. Note that this relation follows merely from conserva tion of energy and can be readily generalized to a device with an arbitrary number of inputs and outputs. For a 2 2 directional coupler, by the symmetry of the device, we can set s 21 = s 12 = a and s 22 = s 11 = b . Applying (3.4) to this simplified scattering matrix, we get  a  2 +  b  2 = 1 (3.5) and ab + ba = . (3.6) From (3.5), we can write  a  = cos (x) and  b  = sin (x). (3.7) If we write a = cos (x)e i a and b = sin (x)e i b , (3.6) yields cos ( a b ) = . (3.8) Thus a and b must differ by an odd multiple of / 2 . The general form of (3.1) now follows from (3.7) and (3.8). The conservation of energy has some important consequences for the kinds of optical components that we can build. First, note that for a 3 dB coupler, though the electric fields at the two outputs have the same magnitude, they have a relative phase shift of / 2 . This relative phase shift, which follows from the conservation of energy as we just saw, plays a crucial role in the design of devices such as the MachZehnder interferometer that we will study in Section 3.3.7. Another consequence of the conservation of energy is that lossless combining is not possible. Thus we cannot design a device with three ports where the power input at two of the ports is completely delivered to the third port. This result is demonstrated in Problem 3.2....
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This note was uploaded on 01/15/2011 for the course ECE 6543 taught by Professor Boussert during the Spring '09 term at Georgia Institute of Technology.
 Spring '09
 Boussert

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