Editing Optical ring resonators (section)

=== Optical coupling ===
[[File:CouplingCoefficients.png|thumb|225px|A pictorial representation of the coupling coefficients]]
[[File:Wave guiding.gif|300px|thumb|right|Visualization of: how the light from a point source is guided by a waveguide, how the waveguide is coupled to a ring resonator, and how the ring resonator is in turn coupled to another waveguide.]]
Important for understanding how an optical ring resonator works, is the concept of how the linear waveguides are coupled to the ring waveguide. When a beam of light passes through a wave guide as shown in the graph on the right, part of light will be coupled into the optical ring resonator. The reason for this is the [[Evanescent field#Evanescent-wave coupling|phenomenon of the evanescent field]], which extends outside of the waveguide mode in an exponentially decreasing radial profile. In other words, if the ring and the waveguide are brought closely together, some light from the waveguide can couple into the ring. There are three aspects that affect the optical coupling: the distance, the coupling length and the refractive indices between the waveguide and the optical ring resonator. In order to optimize the coupling, it is usually the case to narrow the distance between the ring resonator and the waveguide. The closer the distance, the easier the optical coupling happens. In addition, the coupling length affects the coupling as well. The coupling length represents the effective curve length of the ring resonator for the coupling phenomenon to happen with the waveguide. It has been studied that as the optical coupling length increases, the difficulty for the coupling to happen decreases.{{Citation needed|reason=Reliable study needed to prove this sentence|date=October 2016}} Furthermore, the refractive index of the waveguide material, the ring resonator material and the medium material in between the waveguide and the ring resonator also affect the optical coupling. The medium material is usually the most important feature under study since it has a great effect on the transmission of the light wave. The refractive index of the medium can be either large or small according to various applications and purposes.

One more feature about optical coupling is the critical coupling. The critical coupling shows that no light is passing through the waveguide after the light beam is coupled into the optical ring resonator. The light will be stored and lost inside the resonator thereafter. 
<ref name = "Xiaoetal">
{{cite journal
 |author=Xiao, Min
 |author2=Jiang, Dong
 |author3=Yang
 |name-list-style=amp
 |title=Coupling Whispering-Gallery-Mode Microcavities With Modal Coupling Mechanism
 |journal=IEEE Journal of Quantum Electronics
 |date=November 2008
|volume=44
 |issue=11
 |page=1065
 |doi=10.1109/JQE.2008.2002088
 |bibcode=2008IJQE...44.1065X
 }}</ref> Lossless coupling is when no light is transmitted all the way through the input waveguide to its own output; instead, all of the light is coupled into the ring waveguide (such as what is depicted in the image at the top of this page).<ref name = "Caietal">
{{cite journal
 |author1=Cai
 |author2=Painter
 |author3=Vahala
 |name-list-style=amp
 |title=Observation of Critical Coupling in a Fiber Taper to a Silica-Microsphere Whispering-Gallery Mode System
 |journal=Physical Review Letters
 |date=July 2000
|volume=85
 |issue=1
 |pages=74–77
 |doi=10.1103/PhysRevLett.85.74
 |pmid=10991162
 |bibcode=2000PhRvL..85...74C
 |url=https://resolver.caltech.edu/CaltechAUTHORS:CAIprl00
 }}</ref> For lossless coupling to occur, the following equation must be satisfied:

: <math>|\Kappa|^2 + |t|^2 = \mathbf{1}</math>

where t is the transmission coefficient through the coupler and <math>\Kappa</math> is the taper-sphere mode coupling amplitude, also referred to as the coupling coefficient.