Editing Optical ring resonators (section)

== Theory ==
To understand how optical ring resonators work, we must first understand the optical path length difference (OPD) of a ring resonator. This is given as follows for a single-ring ring resonator:

: <math>\mathbf{OPD} = 2 \pi r n_\text{eff}</math>

where ''r'' is the radius of the ring resonator and ''<math>n_\text{eff}</math>'' is the effective [[index of refraction]] of the waveguide material. Due to the total internal reflection requirement, <math>n_\text{eff}</math> 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:

: <math>\mathbf{OPD} = m \lambda_{m}</math>

where ''<math>\lambda_{m}</math>'' is the resonant wavelength and ''m'' is the mode number of the ring resonator. This equation means that in order for light to interfere constructively 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 [[Q factor|quality factor]] and the finesse of an optical ring resonator can be quantitatively described using the following formulas (see: eq: 2.37 in,<ref>{{Cite journal|last=Rabus|first=Dominik Gerhard|date=2002-07-16|title=Realization of optical filters using ring resonators with integrated semiconductor optical amplifiers in GaInAsP/InP|url=https://depositonce.tu-berlin.de/handle/11303/862|language=en|doi=10.14279/depositonce-565}}</ref> or eq:19+20 in,<ref>{{Cite journal|last1=Hammer|first1=Manfred|last2=Hiremath|first2=Kirankumar R.|last3=Stoffer|first3=Remco|date=2004-05-10|title=Analytical Approaches to the Description of Optical Microresonator Devices|url=https://aip.scitation.org/doi/abs/10.1063/1.1764013|journal=AIP Conference Proceedings|volume=709|issue=1|pages=48–71|doi=10.1063/1.1764013|bibcode=2004AIPC..709...48H |issn=0094-243X}}</ref> or eq:12+19 in <ref>{{Cite journal|last1=Yang|first1=Quankui |date=2023-08-22|title= Finesse of ring resonators |url= https://pubs.aip.org/aip/adv/article/13/8/085225/2907706/Finesse-of-ring-resonators |journal=AIP Advances |volume=13|issue=8 |pages=085225|doi=10.1063/5.0157450|doi-access=free|bibcode=2023AIPA...13h5225Y }}</ref>):

: <math>\mathbf{Q} = \frac{\nu}{\delta\nu}
</math>
:<math>\mathcal{F} = \frac{\nu_{f}}{\delta\nu}</math>

where ''<math>\mathcal{F}</math>'' is the finesse of the ring resonator, <math>\nu</math> is the operation frequency, <math>\nu_{f}</math> is the [[free spectral range]] and <math>\delta\nu</math> is the [[Full width at half maximum|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 <math>Q</math> factor is usually due to large losses.

[[File:MultipleResonances.png|600px|thumb|center|A transmission spectra depicting multiple resonant modes (<math>m=1,m=2,m=3,\dots,m=n</math>) and the [[free spectral range]].]]