Notes 20 Spring 2005

Notes 20 Spring 2005 - Notes 20 Spring 2005 1 Propagation...

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Unformatted text preview: Notes 20 Spring 2005 1 Propagation in Dispersive Media In dealing with lossless dielectric media ( , eff = = = ) thus far weve said that the phase constant is a linear function of frequency, i.e., = , where and have been constants of the media. Under these conditions the phase velocity of the waves are independent of frequency, i.e., = = = 1 p v . In general, however, the permittivity of lossless dielectrics usually are slightly dependent upon the frequency of the wave making the index of refraction frequency dependent as well; . When this is taken in to account the medium is said to be dispersive and as a result the phase velocity is no longer independent of frequency: ( ) n n ( ) ( ) ( ) = = = n c 1 1 1 r p v ( ) assuming = In the case of sending a coherent signal down a fiber optic line, dispersion cannot be ignored as it results in signal distortion. Namely, any signal (a carrier of information) is composed of many time- domain components contained in a certain frequency band (Fourier transforms). The signal shape is determined by the relative amplitudes and phases of the time-harmonic components within this band i . If the phase velocities of the time-harmonic components are not the same due to dispersion, their relative positions, which means their relative phases, change as the signal propagates causing the overall signal shape change. The signal itself (the sum of the components) propagates at a velocity known as the group velocity . It is this velocity at which power and information are transported. The group velocity is always smaller than c. To determine the group velocity, consider a simple signal in a weakly dispersive medium. Let the signal be obtained as a superposition of two uniform plane waves propagating in the same direction, but with slightly different angular frequencies, b a and , and slightly different propagation constants (due to dispersion). We will assume that the amplitudes are the same (for...
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Notes 20 Spring 2005 - Notes 20 Spring 2005 1 Propagation...

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