Resonator Modes

Resonator modes are essentially self-consistent field distributions of light – more precisely, electric field distributions which are self-reproducing (apart from a possible loss of power) in each resonator round trip.
The properties of resonator modes depend very much on various details:

• In waveguide resonators, the transverse mode structure is determined by the waveguide properties only, and is constant everywhere in the resonator, if the waveguide properties are constant. Only a single transverse mode shape may exist, e.g. if single-mode fibers are used.
• In bulk-optical resonators, the mode properties depend on the overall optical setup, and differ very much between stable and unstable resonators (see above). For stable resonators, the transverse mode patterns can be described e.g. with Hermite–Gaussian functions. The lowest-order mode (axial mode, fundamental mode) has the simplest profile; more complicated shapes correspond to higher-order modes. The mode properties can be calculated using an ABCD matrix algorithm. Unstable resonators exhibit a much more complicated mode structure, which can be calculated only with numerical means. Generally, the transverse mode size varies along the resonator axis.

For each of the transverse mode patterns, there are only certain optical frequencies for which the optical phase is self-consistently reproduced after each round trip (i.e. the round-trip phase shift is an integer multiple of 2π). These are called the mode frequencies or resonance frequencies and are approximately equidistant (but not exactly equidistant due to chromatic dispersion). The frequency spacing of the resonator modes, also called free spectral range (FSR), is the inverse round-trip time, or more precisely the inverse round-trip group delay. This means that the FSR becomes smaller as the resonator length is increased. The ratio of the frequency spacing to the width of the resonances (resonator bandwidth) is called the finesse and is determined by the power losses per resonator round trip. A related measure is the Q factor, which is the ratio of resonance frequency and bandwidth.
The article on resonator modes gives more details.

Resonant Enhancement

If e.g. an end mirror is partially transparent, light can be fed into the resonator from outside. The highest internal optical power (and the maximum transmission through a resonator) can be achieved when the (monochromatic) input light has a frequency matching that of one of the modes, and the spatial shapes are also matched (→ mode matching). Particularly for low-loss resonators, the circulating intracavity power can then greatly exceed the input power by means of resonant enhancement (→ enhancement cavities).
Resonant enhancement is also possible for a regular train of light pulses, when the frequencies of the pulse train match the optical resonances. In the time domain, this means that the pulse period matches the resonator's round-trip time, or an integer fraction of it.

Subtle Properties of Bulk-optical Resonators

The physics of bulk-optical resonators is surprisingly rich in nature. Some interesting aspects are:

• The modes of a resonator with a transverse variation of optical gain or loss in general do not form an orthogonal set of functions. These non-normal modes have some peculiar properties. For example, the total power in a superposition of such modes is not simply the sum of the power in the different modes. Under some conditions, resonators with nonnormal modes can be treated with complex Gaussian beam analysis, where e.g. the elements of the ABCD matrix and the Gaussian beam radius can be complex numbers.
• In situations with general astigmatism (such as in some nonplanar ring laser resonators), there are interesting effects such as image rotation, polarization rotation, and so-called twisted beams.
• There are technically interesting methods (often of numerical nature) for designing a resonator with given properties.
• The design of a laser resonator has important influences on various aspects of laser operation, e.g. on the alignment sensitivity and the beam quality.

Application of Optical Resonators

Optical resonators are used for, e.g., the following purposes:

• as laser resonators, where the resonator losses are compensated by a gain medium to maintain or build up optical power
• as etalons for filtering the frequency content of optical radiation
• for filtering the transverse shape of optical radiation (→ mode cleaner cavities)
• as short-term optical frequency standards (when e.g. the frequency of a laser is locked to a resonance frequency of a stable reference cavity)
• for precise length measurements, e.g. exploiting the periodically occurring resonances when the resonator length is changed
• for exploiting the resonant enhancement of intracavity power (→ enhancement cavities), e.g. in order to achieve efficient frequency doubling of light from a low-power single-frequency laser
• for precisely measuring low-level losses by recording the decay of the power of intracavity radiation (cavity ring-down spectroscopy)
• for generating chromatic dispersion effects, e.g. with a Gires–Tournois interferometer

Ricardo Monroy      C.I. 17646658

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