Editing Optical ring resonators (section)

== Double ring resonators ==
[[File:Double Optical Ring Resonator.png|thumb|225px|A double ring resonator with rings of varying radii in series showing the relative intensities of light passing through on the first cycle. Note that the light passing through a double ring resonator would more often travel in multiple loops around each ring rather than as pictured.]] In a double ring resonator, two ring waveguides are used instead of one. They may be arranged in series (as shown on the right) or in parallel. When using two ring waveguides in series, the output of the double ring resonator will be in the same direction as the input (albeit with a lateral shift). When the input light meets the resonance condition of the first ring, it will couple into the ring and travel around inside of it. As subsequent loops around the first ring bring the light to the resonance condition of the second ring, the two rings will be coupled together and the light will be passed into the second ring. By the same method, the light will then eventually be transferred into the bus output waveguide. Therefore, in order to transmit light through a double ring resonator system, we will need to satisfy the resonant condition for both rings as follows:

: <math>\ 2 \pi n_{1} R_{1} = m_{1} \lambda_{1}</math>

: <math>\ 2 \pi n_{2} R_{2} = m_{2} \lambda_{2}</math>

where <math>m_{1}</math> and <math>m_{2}</math> are the mode numbers of the first and second ring respectively and they must remain as positive integer numbers. For the light to exit the ring resonator to the output bus waveguide, the wavelength of the light in each ring must be same. That is, <math>\lambda_{1} = \lambda_{2}</math> for resonance to occur. As such, we get the following equation governing resonance:

: <math>\ \frac{n_{1} R_{1}}{m_{1}} = \frac{n_{2} R_{2}}{m_{2}} </math>
Note that both <math>m_{1}</math> and <math>m_{2}</math> need to remain integers.
[[File:DoubleRingReflector.jpg|thumb|Optical mirror (reflector) made of a double ring system coupled to a single waveguide. Forward propagating waves in the waveguide (green) excite anti-clockwise traveling waves in both rings (green). Due to the inter-resonator coupling, these waves generate clockwise rotating waves (red) in both rings which in turn excite backward propagating (reflected) waves (red) in the waveguide. The reflected wave exists only in the part of the waveguide to the left of the coupling point to the right ring.<ref name=":0" />]]
A system of two ring resonators coupled to a single waveguide has also been shown to work as a tunable reflective filter (or an  optical mirror).<ref name=":0">{{Cite journal|last1=Chremmos|first1=I.|last2=Uzunoglu|first2=N.|date=2010|title=Reflective properties of double-ring resonator system coupled to a waveguide|journal=IEEE Photonics Technology Letters|volume=17|issue=10|pages=2110–2112|doi=10.1109/LPT.2005.854346|s2cid=23338710 |issn=1041-1135}}</ref> Forward propagating waves in the waveguide excite anti-clockwise rotating waves in both rings. Due to the inter-resonator coupling, these waves generate clockwise rotating waves in both rings which are in turn coupled to backward propagating (reflected) waves in the waveguide. 
In this context, the utilization of nested ring resonator cavities has been demonstrated in recent studies.<ref name = "Nested1">{{cite journal |last1=Selim|first1=M. A.|last2=Anwar|first2=M.|title=Enhanced Q-factor and effective length silicon photonics filter utilizing nested ring resonators|journal=Journal of Optics|date=12 September 2023|volume=25|number=11|pages=115801|doi=10.1088/2040-8986/acf5fd |doi-access=free|arxiv=2309.02775|bibcode=2023JOpt...25k5801S }}</ref><ref name = "Nested2">{{cite book|last1=Shalaby|first1=R. A.|last2=Selim|first2=M. A.|last3=Adib|first3=G. A.|last4=Sabry|first4=Yasser|last5=Khalil|first5=Diaa|title=Silicon Photonics XIV |chapter=Silicon photonics dual-coupler nested coupled cavities |editor-first1=Graham T. |editor-first2=Andrew P. |editor-last1=Reed |editor-last2=Knights |series=Proceedings of the SPIE|date=2019 |volume=10923|pages=187–193|doi=10.1117/12.2509661|bibcode=2019SPIE10923E..1PS |isbn=978-1-5106-2488-7 }}</ref> These nested ring resonators are designed to enhance the quality factor (Q-factor) and extend the effective light-matter interaction length. These nested cavity configurations enable light to traverse the nested cavity multiple times, a number equal to the round trips of the main cavity multiplied by the round trips of the nested cavity, as depicted in Figure below.

[[File:Nested Cavity.png|thumb|Nested Cavity Configuration: Light undergoes multiple round trips within the nested cavity, the number of which is approximately determined by the product of the round trips within the main cavity and the nested cavity.<ref name = "Nested1"/><ref name = "Nested2"/>]]