By G. Landwehr

ISBN-10: 3540119965

ISBN-13: 9783540119968

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52) we already wrote the same φ0 and m. The conditions for smooth transition are NK (r = a) = NM (r = a), ∂ ∂ NK (r = a) = NM (r = a). 6. 55) r=a . 58) a a (the prime ( ) denotes the derivative with respect to the argument). 59) CK CM Jm (u) Km (w) − CK CM Km (w) Jm (u) = 0. a a From this, CK CM /a can be eliminated immediately. 60) = mKm (w) − wKm+1 (w). 61) With this the derivatives can be eliminated: Jm (u) mKm (w) − wKm+1 (w) wJm (u)Km+1 (w) = Km (w) mJm (u) − uJm+1 (u) = uKm (w)Jm+1 (u) or Km (w) Jm (u) = .

13) We rearrange the RHS using Eqs. 5) and obtain ∇(∇ · E) − ∇2 E = − ∂ ∂t μ0 ∂D ∂t ∂2 D ∂t2 ∂2 ∂2 = −μ0 0 2 E − μ0 2 P . 16) We thus ﬁnd the wave equation −∇(∇ · E) + ∇2 E = 1 ∂2 ∂2 E + μ0 2 P . 17) A fully analogous equation can be derived for the magnetic ﬁeld. Now we must make some statement about the relation between the polarization P and the ﬁeld strength E. This involves properties of the material. , quicker than any other relevant time scale involved (Approximation 4). Then we can write the polarization as P = 0 χ(1) E + χ(2) E 2 + χ(3) E 3 + · · · .

The maximum rate is given by the inverse of the maximum scatter: in our example we obtain about 70 MHz. That is not a very high rate, and 1 km is not a very long distance, either. We therefore realize that the mechanism of modal dispersion can severely hamper the usefulness of ﬁbers for practical applications. Fortunately, there are ways to avoid the problem. One can either use the so-called gradient index ﬁbers or, for the highest demands, single-mode ﬁbers. We will take a closer look at both.

### Application of High Magnetic Fields in Semiconductor Physics by G. Landwehr

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