Optical cavity
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An optical cavity or optical resonator is an arrangement of mirrors that forms a standing wave cavity resonator for light waves. Optical cavities are a major component of lasers, surrounding the gain medium and providing feedback of the laser light. They are also used in optical parametric oscillators and some interferometers.
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Resonator modes
Light confined in a resonator will reflect multiple times from the mirrors, and due to the effects of interference, only certain patterns and frequencies of radiation will be sustained by the resonator, with the others being suppressed by destructive interference. In general, radiation patterns which are reproduced on every round-trip of the light through the resonator are the most stable, and these are known as the modes of the resonator.
Resonator modes can be divided into two types: longitudinal modes, which differ in frequency from each other; and transverse modes, which may differ in both frequency and the intensity pattern of the light. The basic, or fundamental transverse mode of a resonator is a Gaussian beam.
Resonator types
The most common types of optical cavities consist of two facing plane (flat) or spherical mirrors. The simplest of these is the plane-parallel or Fabry-Perot cavity, consisting of two opposing flat mirrors. While simple, this arrangement is rarely used in large-scale lasers due the difficulty of alignment; the mirrors must be aligned parallel within a few seconds of arc, or "walkoff" of the intracavity beam will result in it spilling out of the sides of the cavity. However, this problem is much reduced for very short cavities with a small mirror separation distance (L < 1 cm). Plane-parallel resonators are therefore commonly used in microchip and microcavity lasers and semiconductor lasers. In these cases, rather than using separate mirrors, a reflective optical coating may be directly applied to the laser medium itself. The plane-parallel resonator is also the basis of the Fabry-Perot interferometer.
For a resonator with two mirrors with radii of curvature R1 and R2, there are a number of common cavity configurations. If the two curvatures are equal to half the cavity length (R1 = R2 = L / 2), a concentric or spherical resonator results. This type of cavity produces a diffraction-limited beam waist in the centre of the cavity, with large beam diameters at the mirrors, filling the whole mirror aperture. Similar to this is the hemispherical cavity, with one plane mirror and one mirror of curvature equal to the cavity length.
A common and important design is the confocal resonator, with equal curvature mirrors equal to the cavity length (R1 = R2 = L). This design produces the smallest possible beam diameter at the cavity mirrors for a given cavity length, and is often used in lasers where the purity of the transverse mode pattern is important.
A concave-convex cavity has one convex mirror with a negative radius of curvature. This design produces no intracavity focus of the beam, and is thus useful in very high-power lasers where the intensity of the intracavity light might be damaging to the intracavity medium if brought to a focus.
Stability
Only a narrow range of possible values for R1, R2, and L produce stable resonators in which periodic refocussing of the intracavity beam is produced. If the cavity is unstable, the beam size will grow without limit, eventually growing larger than the size of the cavity mirrors and being lost. By using methods such as ray transfer matrix analysis, it is possible to calculate a stability criterion:
- <math> 0 < \left( 1 - \frac{L}{R_1} \right) \left( 1 - \frac{L}{R_2} \right) < 1 </math>.
Values which satisfy the inequality correspond to stable resonators.
The stability can be shown graphically by defining a stability parameter, g for each mirror:
- <math> g_1 = 1 - \frac{L}{R_1} ,\qquad g_2 = 1 - \frac{L}{R_2}</math>,
and plotting g1 against g2 as shown. Areas bounded by the line g1 g2 = 1 and the axes are stable. Cavities at points exactly on the line are marginally stable; small variations in cavity length can cause the resonator to become unstable, and so lasers using these cavities are in practice often operated just inside the stability line.
Practical resonators
If the optical cavity is not empty (e.g., a laser cavity which contains the gain medium), the value of L used is not the physical mirror separation but the optical path length between the mirrors. Lenses placed in the cavity alter the stability and mode size. In addition, for most gain media, thermal and other inhomogeneities create a lensing effect in the medium.
Practical laser resonators may contain more than two mirrors; three- and four-mirror arrangements are common. This is called folding. These designs allow compensation of the cavity beam's astigmatism, which is produced by brewster cut elements in the cavity. A 'z' arrangement of the cavity also compensates for third order Aberrations while the 'delta' cavity does not. Intracavity acousto-optic modulators work faster if placed at a beam waist. Curved mirrors can be used to create a second waist in the cavity for this purpose. Commonly, a pair of curved mirrors form one or more confocal sections, with the rest of the cavity being quasi-collimated and using plane mirrors. Filters, prisms and gratings often need large quasi-collimated beams.
Optical Delay Line
Optical cavities can be used as multipass optical delay lines. When flat mirrors are used, the light travels in a flat zigzag path. Flat mirrors are very sensitve to mechanical disturbances. When curved mirrors are used in a nearly confocal configuration, the beam travels on a circular zigzag path. The latter are called Herriott-type delay line. A fixed insertion mirror is placed off-axis near one of the curved mirrors, and a mobile pickup mirror is similarly placed near the other curved mirror. A flat linear stage with one pickup mirror is used in case of flat mirrors and a rotational stage with two mirrors is used for the Herriott-type delay line. The rotation disturbs the polarization. As only full roundtrips can be added, a single pass delay line is also needed, made of either a three or two mirrors in a 3d respective 2d retro-reflection configuration on top of a linear stage. To adjust for beam divergence a second car on the linear stage with two lenses can be used. The two lenses act as a telescope producing a flat phase front of a Gaussian beam on a virtual end mirror.
References
- Koechner, William. Solid-state laser engineering, 2nd ed. Springer-Verlag (1988).de:Optischer Resonator