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. Light confined in the cavity reflect multiple times producing standing waves for certain resonance frequencies. The standing wave patterns produced are called modes; longitudinal modes differ only in frequency while transverse modes differ for different frequencies and have different intensity patterns across the cross section of the beam.[1]

Different resonator types are distinguished by the focal lengths of the two mirrors and the distance between them. (Flat mirrors are not often used because of the difficulty of aligning them to the needed precision.) The geometry (resonator type) must be chosen so that the beam remains stable (that the size of the beam does not continually grow with multiple reflections. Resonator types are also designed to meet other criteria such as minimum beam waist or having no focal point (and therefore intense light at that point) inside the cavity.[2]

Certain devices when excited through a gunn oscillator doesnot produce satisfactory oscillations. There are certain jitters in whole range of frequencies. For this we use a resonator in which the beam of just a particular frequency is choosen to be reflected multiple times in order to produce oscillations. All the other frequencies are not reflected and hence no oscillations are generated at those frequencies. The advantage of the open cavity is that it is non reflecting hence the beam of frequencies which are not focussed at the mirrors move out of the cavity.
We have constructive interference at resonance and a destructive interference at all others. So a sharp peak is detected.[1]

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 the eigenmodes, 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:
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.

• It works on the principle of optics and probability(statistics).
• 2 walls are made reflecting to an extent that they behave as mirrors.
• Only 2 walls are contributing to receive the dominant mode at the coupling hole at resonance.

a=22.86 mm b=10.16 mm
Formulas used to calculate d:

d=(l π)/[k²-(π/a)]^0.5
d=16.99021661 mm



• Hemispherical resonator
• Resonant frequency(theoretical)=11 GHz
• Resonant frequency(practical)=11.5 GHz
• Length of the resonator=d=16.99021661 mm
• Height of the resonator=2a=45.72 mm
• Mode used: TE



Only certain ranges of 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 it is possible to calculate a stability criterion:
0 ≤ ≤ 1.
Values which satisfy the inequality correspond to stable resonators.
The stability can be shown graphically by defining a stability parameter, g for each mirror:
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.
A simple geometric statement describes the regions of stability: A cavity is stable if the line segments between the mirrors and their centers of curvature overlap, but one does not lie entirely within the other.

Putting our values into the inequality we get:
(since our R1=35mm>L=16.99021661mm and R2= ∞)
Thereby proving that our resonator is stable!

The Q factor (quality factor) of a resonator is a measure of the strength of the damping of its oscillations, or for the relative linewidth.
• Definition via resonance bandwidth: the Q factor is the ratio of the resonance frequency ν0 and the full width at half-maximum (FWHM) bandwidth δν of the resonance:

Loaded and unloaded Q factors are important characteristics of a resonator used in high-power mm-wave sources[3]
The resonator is composed of parallel curved-plate waveguide sections [4], shorted at the end, with entrance and exit holes for transporting an electron beam through the resonator. In order to study the resonator’s characteristics, it is excited from an external source, such as a network analyzer. The polarizing grid coupler is illuminated by a free-space Gaussian beam, formed by means of a mode exciter and a mirror, as described in [5]

 Q Factor of an Optical Resonator
The Q factor of a resonator depends on the optical frequency ν0, the fractional power loss l per round trip, and the round-trip time Trt:

(assuming that l
<<1). For a resonator consisting of two mirrors with air (or vacuum) in between, the Q factor rises as the resonator length is increased, because this decreases the energy loss per optical cycle. However, extremely high Q values (see below) are often achieved not by using very long resonators, but rather by strongly reducing the losses per round trip.

 High-Q Resonators
One possibility for achieving very high Q values is to use super mirrors with extremely low losses, suitable for ultra-high Q factors of the order of 1011. Also, there are toroidal silica microcavities with dimensions of the order of 100 μm and Q factors well above 108, and silica microspheres with whispering gallery resonator modes exhibiting Q factors around 1010.
High-Q optical resonators have various applications in fundamental research (e.g. in quantum optics) and also in telecommunications (e.g. as optical filters for separating WDM channels). Also, high-Q reference cavities are used in frequency metrology, e.g. for optical frequency standards. The Q factor then influences the precision with which the optical frequency of a laser can be stabilized to a cavity resonance. [6]

3. Phase Noise Reduction
Q (unloaded) = 1097
Bandwidth (practical) = 300Mhz
Q (loaded) = 38.33
1/QL = 1/Qext + 1/Qo
Qe = 39.71
PNR=20 log (1+ Qo/Qext)
PNR= 28.82 dB
Narrow bandwidth given that input is in GHz
A load with high Q factor will improve it even further.

[2] A STUDY OF INTER-INJECTION-LOCKED PHASED ARRAYS Final Report by Karl D. Stephan October 10, 1989 Department of Electrical and Computer Engineering University of Massachusetts.
[3] Paschotta, Rüdiger. "Q Factor". Encyclopedia of Laser Physics and Technology. RP Photonics.
[4] M.E. Hill, W.R. Fowlex, X.E. Lin, and D.H. Whittum, Beam-cavity interaction circuit at W-band, IEEE Trans MTT 49 (2001), 998–1000.

[5] M.A. Shapiro and S.N. Vlasov, Study of a combined millimeter-wave resonator, IEEE Trans MTT 45 (1997), 1000–1002.

[6] "Quasioptical microwave and millimeter-wave power combining," by K.D. Stephan
and S.L. Young, SPIE (vol. 947) Conf. on Interconnection of High Speed and High Frequency Devices and Systems, Newport Beach, CA, March "1988.

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