By M. J. N. Sibley (auth.)

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5. 76). Two plots of the right-hand side have been drawn- the upper plot is for a Bessel function order of 0, while the lower plot is for v = 1. ) An interesting feature of these plots is that there are no eigenvalues for ua > V. Thus V is known as the normalised cut-off frequency. 11, we can see that, provided the argument ua is less than V, the number of eigenvalues, that is, the number of modes, is one greater than the number of zeros for the particular Bessel function order. 11 Eigenvalue graphs for the zero- and first order modes in a cylindrical waveguide Thus for the zero-order function, the number of zeros with ua < V is 4, and so the number of modes is 5.

40) H 1 = ( -axcos81 . 44). 45) reveals that re is unity if the term under the square root is zero, that is, if sin20i = (nzfn 1)2. Under these conditions, the reflected E field will equal the incident E field, and this is total internal reflection. 46) nl Substitution of this result into Snell's Law gives the angle of refraction to be 90°, and so the transmitted ray travels along the interface. If the angle of incidence is greater than Oc (that is, sin Oi > n 2/n 1) then re will be complex, but lrel will be unity.

If the angle of incidence is greater than Oc (that is, sin Oi > n 2/n 1) then re will be complex, but lrel will be unity. This implies that total internal reflection takes place. However, there will still be a transmitted wave. 47) To evaluate Et we require to find sinOt and cosOt. If Oi > Oc, then sinOi > n 2/n 1• If we substitute this into Snell's Law, then we find that sinOt > 1, which is physically impossible. 50) So, although total internal reflection takes place, an E field exists in the lower refractive index material.