Chapter 4
Mode Formation in Optical Cavities 1
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Kenneth W. Busch, Aurélie Hennequin , and Marianna A. Busch Department of Chemistry, Baylor University, Waco, TX 76798-7348
Longitudinal mode formation in optical cavities is introduced with a discussion of the Fabry-Perot cavity. Transverse mode formation is discussed in terms of Hermite-Gaussian waves for cavities with rectangular cross-sectional symmetry and in terms of Laguerre-Gaussian waves for cavities with circular cross-sectional symmetry. Cavity resonance frequencies for both types of waves are discussed. Finally, the effect of the g-parameters on mode spacing is discussed.
We have seen in the previous chapter that an optical cavity, in the most general sense, is a volume of space with boundary conditions that depend upon the placement and characteristics of the optical elements that make up the cavity. Such a cavity may be stable or unstable depending on the values of the g-parameters (7). In the previous discussion (7), cavity behavior was discussed entirely from the point of view of geometrical optics, without regard for the wave character of the radiation. In this chapter, we will examine the development of resonant modes in optical cavities. To understand the development of resonant modes in optical cavities, we must consider the oscillating character of the light beam as it passes back and forth within the cavity. In our discussion, we will limit our attention to stable cavities where the energy is confined near the optical axis (paraxial conditions). For a mode to develop in such an optical cavity, the electric vector of the oscillating light wave must be zero at the boundaries of the cavity. The standing waves that develop within the cavity are known as cavity modes. The role of mode formation in cavity-ringdown spectroscopy has been discussed extensively in the literature (2-6). When a pulsed laser with a very short pulse duration is used, the coherent wavetrain emitted during the pulse can be less than the length of the optical cavity. If this is true, mode formation is considered to be unimportant since the returning wave can never overlap the initial wave because the wavetrain is too short. Under these experimental conditions, the pulse is believed to behave like a "photon bullet" (6,7). While the photon bullet model is clearly applicable to the case where short duration pulses (-15 ns) from a pulsed laser source are used, experiments where continuous-wave lasers are used in cavity-ringdown spectroscopy may involve mode
1
Current address: École Supérieure d'Optique, Institut d'Optique Théorique et Appliquée, Centre Universitaire Bât 503-BP147-91403, Orsay Cedex, France
34
© 1999 American Chemical Society In Cavity-Ringdown Spectroscopy; Busch, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1999.
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35 formation. Moreover, since some authors (3,5) discuss the cavity-ringdown experiment in terms of mode formation, this chapter will attempt to provide the background material needed for understanding mode formation in optical cavities. Modes in cavities are classified into two basic types. Longitudinal (or axial) modes determine the resonant oscillation frequencies that satisfy the wavelength requirements of the cavity along a given optical path (i.e., the electric vector is zero at the reflecting surface). Tranverse modes, which travel over slightly different optical paths, determine the intensity pattern and divergence of the propagating beam. In other words, transverse modes determine the cross-sectional geometry of the beam. Each transverse mode, propagating along a unique direction, can have various allowed longitudinal modes associated with it. Fabry-Perot Cavity Since one of the later contributions to this volume deals with the use of a FabryPerot (FP) cavity, we will begin our discussion of cavity modes with a brief introduction to the principles behind this type of cavity. In the previous chapter (7), we saw that a Fabry-Perot cavity consists of two plane mirrors facing each other and separated by some distance. Figure 1 shows a schematic diagram of an FP-cavity where the distance between the reflecting surfaces is L and the index of refraction of the medium between the mirrors is n. Outside the cavity, the index of refraction of the medium is taken as ri. Let us consider a hypothetical plane wave of infinite extent that strikes the cavity at an angle 9' to the normal. The complex amplitude of the wave can be expressed as (1) where j is V=T, co is the angular frequency of the light wave, k is the propagation number given by k = 2n/X, X is the vacuum wavelength, and < > | is a phase angle. A(z) is an amplitude factor that depends on distance z along the optical axis. As shown in Figure 1, a series of waves is reflected and transmitted by the cavity. To determine the transmission of the cavity, we must sum the amplitudes of the transmitted waves. Transmission of a Fabry-Perot Cavity. We begin our discussion by considering the path difference between two successive partial waves. As shown in Figure 1, the path difference between two successive partial waves is given by (2a)
where AB = BC = L /cos 9, AC = 2L tan 9, and AD = AC sin 9'. Substituting Snell's law (n sin 9 = n'sin 9') into eq 2b gives 8 = np U _ „ f J L ) \cos 0 / \cos9;
2 s i n
2
0
= „ U U i -sin e) \,cos6;v )
8 = 2nL cos 6
In Cavity-Ringdown Spectroscopy; Busch, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1999.
(3a) (3b)
36 n'
n 6
2
r r wexp (3jty)
L
) B
2 2
Downloaded by UNIV MASSACHUSETTS AMHERST on October 11, 2012 | http://pubs.acs.org Publication Date: April 8, 1999 | doi: 10.1021/bk-1999-0720.ch004
r r wexp (/) 2
ft
Figure 1. Propagation of an incident wave in a Fabry-Perot cavity Since the corresponding phase delay is given by = 2rc8A, the path difference in eq 3b becomes x _ 4nnL cos 8 _ 4nnLv cos 8 • X " c
(4)
where v is the frequency of the light and c is the speed of light. If we define r and t as the amplitude reflection and transmission coefficients, respectively, from n to n' or from n' to n, the corresponding reflectance and transmittance (in terms of intensity) will be given by R = r2 and T = ft, respectively (recall that the intensity is the square of the amplitude). Figure 1 shows the situation for the transmitted amplitudes that result from an incident wave whose complex amplitude is given by U = u exp(-j ), where is the phase delay between the incident wave and the first transmitted wave. The total transmitted amplitude for the wave, At, will be given as the sum of the individual amplitudes 0
0
2
0
2 2
4 2
A = t u + r t u exp (;
