where r is the radius of the ring resonator and is the effective index of refraction of the waveguide material. Due to the total internal reflection requirement, must be greater than the index of refraction of the surrounding fluid in which the resonator is placed (e.g. air). For resonance to take place, the following resonant condition must be satisfied:
where is the resonant wavelength and m is the mode number of the ring resonator. This equation means that in order for light to interfere construcitvely inside the ring resonator, the circumference of the ring must be an integer multiple of the wavelength of the light. As such, the mode number must be a positive integer for resonance to take place. As a result, when the incident light contains multiple wavelengths (such as white light), only the resonant wavelengths will be able to pass through the ring resonator fully.
The quality factor of an optical ring resonator can be quantitatively described using the following formula:
where is the finesse of the ring resonator, is the free spectral range, and is the full-width half-max of the transmission spectra. The quality factor is useful in determining the spectral range of the resonance condition for any given ring resonator. The quality factor is also useful for quantifying the amount of losses in the resonator as a low Q factor is usually due to large losses.
Read more about this topic: Optical Ring Resonators
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