THE CASIMIR FORCE in .NET

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Let the z axis be perpendicular to the plates, with one of the plates located at z = 0 and the other at z = l. We will assume that the plate at z = 0 is a perfect reflector but only for the moment, we will think of the other plate as having a nonvanishing transmissivity. In the end of our mode calculation, we will take the limit of a vanishing transmissivity (Le., the perfect reflector limit). This way of determining the cavity modes is a concrete example of a suggestion by Michael Berry40 [50]. He suggested that every confined mode would correspond to the continuation to the interior of the cavity of an external superposition of plane waves for which the cavity is effectively transparent (Le., for which there is no reflected wave). This way of thinking about cavity modes will tum out to be very useful when the cavity is no longer perfect, and we will come back to it in a later chapter. For now, just notice the key idea of the scattering method. In the absence of the plates, we have seen that the free-space field can be written in the form of a plane-wave expansion. Then if we determine how each plane-wave component is modified as it is scattered by the plates, we just have to sum the modified components to obtain an expression for the total quantized field. Due to the nature of the boundary <:onditions (see Appendix B), it is convenient to decompose the electric field of each plane wave in two parts: a part perpendicular to the plane of incidence, Q; = ..1, and another lying on the plane of incidence, Q; = II. SO we write the electric field of such a plane-wave component as
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where VI,a gives the direction ofthis i:lectric field component and E~,a its amplitude. Just as in the usual theory of the Fabry-Perot interferometer [69], we can obtain the field inside the cavity due to the plane-wave component EI,a simply by adding the multiple reflections depicted in Figure 2.9. Plane waves with their electric fields perpendicular to the plane of incidence are multiplied by m 1. = -1 upon reflection by the plate at z = 0 (see Appendix B). Plane waves whose electric field is on the plane of incidence get multiplied by mil = 1 upon reflection by the plate at z = O. The reason that a plane wave with an electric field on the plane of incidence does not get multiplied by -Ion reflection is best understood by looking at Figure 2.10. As we have just mentioned, we do not assume yet that the plate at z = 1 is a perfect reflector. So for the moment, we take its amplitude reflectivities and transmissivities to be the following generic functions of the incidence angle B: T1. (B), Til (B), t1. (B), and til (B), where T stands for reflectivity, t for transmissivity, ..1 for being perpendicular to the plane of incidence, and II for being on the plane of incidence. Using this notation, between the plates, the multiple scattering makes the electric field of the incident plane wave (2.172) change into
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E~,ata(B) (VI,a {~ [e2iOmaTa(B)r }
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40See also [154, 162, 163].
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Figure 2.9 Multiple reflections in the planar Fabry-Perot cavity with the variables used in the calculations indicated.
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(2.173) where vR,o gives the direction of the electric field of the reflected wave and
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is the extra phase gained on each trip inside the cavity. In free space (i.e., above the plate at z = l). the field is the sum of E[,o, its reflection on the plate at z = l, and the partial transmissions to the outside of each multiple scattered wave between the plates:
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