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Even though these mode functions diverge at infinity, it is still possible to expand the fields in the whole of space in terms of them, provided the initial excitation conditions are explicitly taken into account (see Ref. [36]). However, such an approach is not quite suitable for quantum electrodynamics.
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note
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Because g̃ are purely incoming, they have to increase in time rather than decrease.
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90
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note
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In high-Q cavities the leakage appears to be delocalized throughout the cavity. Then we can adopt mode functions that do not reproduce the fields faithfully at the place where the outcoupling occurs [15], but this is only an approximation that is violated when Q increases and the true localized nature of the leakage becomes evident.
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n vanishes.
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The discontinuity of the refractive index at the origin is just a way of modeling the very abrupt variation of the index at an interface. It is this (implicit) fast spatial dependence of the refractive index that is the physical reason behind the breakdown, at the origin, of the notion that we can separate forward and backward propagating components. See F. W. Sluijter, J. Opt. Soc. Am. 60, 8 (1970).
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