Abstract
This chapter deals with the theory of passive optical resonators i.e. where no active medium is present within the cavity. The most widely used laser resonators have either plane or spherical mirrors of rectangular (or, more often, circular) shape, separated by some distance L. Typically, L may range from a few centimeters to a few tens of centimeters, while the mirror dimensions range from a fraction of a centimeter to a few centimeters. Laser resonators thus differ from those used in the microwave field (see e.g. Sect. 2.2.1) in two main respects: (1) The resonator dimensions are much greater than the laser wavelength. (2) Resonators are usually open, i.e. no lateral surfaces are used. The resonator length is usually much greater than the laser wavelength because this wavelength usually ranges from a fraction of a micrometer to a few tens of micrometers. A laser cavity with length comparable to the wavelength would then generally have too low a gain to allow laser oscillation. Laser resonators are usually open because this drastically reduces the number of modes which can oscillate with low loss. In fact, with reference to example 5.1 to be considered below, it is seen that even a narrow linewidth laser such as a He-Ne laser would have a very large number of modes ( ≈ 109) if the resonator were closed. By contrast, on removing the lateral surfaces, the number of low-loss modes reduces to just a few ( ≈ 6 in the example). In these open resonators, in fact, only the very few modes corresponding to a superposition of waves traveling nearly parallel to the resonator axis will have low enough losses to allow laser oscillation.
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- 1.
The fact that we are considering here an equivalent lens of focal length f 1 = R 1 while the focal length of the equivalent lens-guide structure was f 1 = R 1 ∕ 2 (see Fig. 5.5b) may generate some confusion. One should note, however, that, due to the reflection at the plane mirror of Fig. 5.8c, the lens f 1 in the figure is traversed twice by the beam and its effect is thus equivalent to a single lens of overall focal length f 1 ∕ 2.
- 2.
The usage of the terms “longitudinal mode” and “transverse mode” in the laser literature has sometimes been rather confusing, and can convey the (mistaken) impression that there are two distinct types of modes, viz., longitudinal modes (sometimes called axial modes) and transverse modes. In fact any mode is specified by three numbers, e.g., n, m, l of (5.5.24). The electric and magnetic fields of the modes are nearly perpendicular to the resonator axis. The variation of these fields in a transverse direction is specified by l, m while field variation in a longitudinal (i.e., axial) direction is specified by n. When one refers, rather loosely, to a (given) transverse mode, it means that one is considering a mode with given values for the transverse indices (l, m), regardless of the value of n. Accordingly a single transverse mode means a mode with a single value of the transverse indexes (l, m). A similar interpretation can be applied to the “longitudinal modes”. Thus two consecutive longitudinal modes mean two modes with consecutive values of the longitudinal index n [i.e., n and (n + 1) or (n − 1).
- 3.
The two modes still differ with respect to the total round trip phase shift, i.e., they still differ in the field variation along the longitudinal z axis and thus in their resonance frequencies.
References
A.E. Siegman, Lasers (University Science Books, Mill Valley, California, 1986) sect. 14.2
M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon Press, London 1980) sect. 1.6.5
H. Kogelnik and T. Li, Laser Beams and Resonators, Appl. Opt. 5, 1550 (1966)
Reference [1], Chap. 19
A.G. Fox and T. Li, Resonant Modes in a Maser Interferometer, Bell. Syst. Tech. J. 40, 453 (1961)
T. Li, Diffraction Loss and Selection of Modes in Maser Resonators with Circular Mirrors, Bell. Syst.Tech. J. 44, 917 (1965)
V. Magni, Resonators for Solid-State Lasers with Large-Volume Fundamental Mode and High Alignment Stability, Appl. Opt., 25, 107–117 (1986). See also erratum Appl. Opt., 25, 2039 (1986)
Reference [1], p. 666
W. Koechner, Solid-State Laser Engineering, Vol. 1, Springer Series in Optical Sciences, fourth edition (Springer, New York, 1996)
Reference [1], Chap. 22
A.E. Siegman, Unstable Optical Resonators for Laser Applications, Proc. IEEE 53, 277–287 (1965)
D.B. Rensch and A.N. Chester, Iterative Diffraction Calculations of Transverse Mode Distributions in Confocal Unstable Laser Resonators, Appl. Opt. 12, 997 (1973)
A.E. Siegman, Stabilizing Output with Unstable Resonators, Laser Focus 7, 42 (1971)
H. Zucker, Optical Resonators with Variable Reflectivity Mirrors, Bell. Syst. Tech. J. 49, 2349 (1970)
A.N. Chester, Mode Selectivity and Mirror Misalignment Effects in Unstable Laser Resonators, Appl. Opt. 11, 2584 (1972)
Ref. [1], sect. 23.3
S. De Silvestri, P. Laporta, V. Magni and O. Svelto, Solid-State Unstable Resonators with Tapered Reflectivity Mirrors: The super-Gaussian Approach, IEEE J. Quant. Electr. QE-24, 1172 (1988)
G. Cerullo et al., Diffraction-Limited Solid State Lasers with Supergaussian Mirror, In: OSA Proc. on Tunable Solid-State Lasers, Vol. 5 ed. by M. Shand and H. Jenssen (Optical Society of America, Washington 1989) pp. 378–384
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Svelto, O. (2010). Passive Optical Resonators. In: Principles of Lasers. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-1302-9_5
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