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Taking the vector product of (2.49) with k from the left and using (2.47) and (2.50), we find that (2.51) Proceeding in an analogous way but starting with (2.50) rather than (2.49), we find an identical equation for B: (2.52) Equations (2.51) and (2.52) are harmonic oscillator equations, and they certainly have nonvanishing solutions as long as the electric and magnetic fields do not both vanish initially. This amazing result is also the reason why we can study light on its own in this chapter.
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In this chapter we are concerned only with closed cavities, such as the one shown in Figure 2.1. There are also open cavities. These are discussed in detail in 9. Here we only introduce the main idea of cavity. For a detailed calculation of the modes of a perfect cavity, see Appendix A.
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Figure 2.1 This is a closed superconducting microwave cavity of the type used in the first (two-photon) micromaser experiments. 13 There is actually a tiny hole across this cavity through which the atoms go, but as it is smaller than the wavelength ofthe mode, its effect is negligible. 14 This cavity belongs to Serge Haroche's cavity QED group of the Laboratoire Kastler Brossel at the Ecole Normale Superieure in Paris, France, where the first two-photon micromaser was realized.
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13For the curious. the hand belongs to Dr. Paulo A. Nussenzveig. 14This is true for superconducting microwave cavities [51] but curiously enough not in the visible region of the electromagnetic spectrum [14. 432].
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In quantum optics, the word cavity means a resonator for electromagnetic radiation. Acoustic resonators have been known since antiquity. They are the basis of most musical instruments. In ancient China, for example, vessels filled with water up to certain levels were used as resonators to give standard tones [527]. But what exactly is a resonator Broadly speaking, a resonator is a device that is resonant only to certain frequencies, whose vibrating fields at these frequencies form spatial patterns called resonator modes. A stretched string with both ends fixed is a closed onedimensional resonator. The body of a violin is an open three-dimensional acoustic resonator. The vibrating field in the string is the vertical displacement of the string at each point along it. String modes can be seen with the naked eye. Figure 2.2 shows the mode corresponding to the second lowest resonance frequency. In the violin, the
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Figure 2.2 The output of a function generator fed through an audio amplifier controls an electromechanical device that drives one end of a string with a precise frequency. The other end of the string is kept fixed. In this figure, the frequency was set in such a way that the wavelength ofthe standing-wave pattern approximately coincides with the length ofthe string. This is a good way of measuring v, the speed of wave propagation along the string, as the frequency here should be equal to 21rv/l, with l being the length of the string, which is also known. This picture was taken by Dr. Gary Chottiner from Case Western Reserve University, where this experiment is part of their introductory laboratory courses for undergraduates.
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vibrating field is the small variation in air pressure at each point inside the violin. A two-dimensional picture of a mode can be obtained by spreading a fine powder at the back of the violin and then sounding one of its resonances. The powder will settle at the nodes of the mode on the two-dimensional surface of the back of the violin because these are the places where the surface will not vibrate. Figure 2.3 shows the resulting picture, and Figure 2.4 shows the actual standing-wave patterns in a classic guitar. In this section we use simple models to explain the essence of what a resonator is, its modes, and how they arise. But to understand resonators we must first understand the phenomenon of resonance. This is the topic of the next subsection. Readers familiar with that might wish to skip it and move to the second and last subsection of this section, where we address resonators directly.
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