Two recipes for ensuring critical coupling between a single-mode optical fiber and a whispering-gallery-mode (WGM) optical resonator have been devised. The recipes provide for phase matching and aperture matching, both of which are necessary for efficient coupling. There is also a provision for suppressing intermodal coupling, which is detrimental because it drains energy from desired modes into undesired ones.

A Typical Whispering-Gallery-Mode Resonator is characterized by, among other parameters, radii of curvature r and R that appear in the equations that describe the conditions for critical coupling.

According to one recipe, the tip of the single-mode optical fiber is either tapered in diameter or tapered in effective diameter by virtue of being cleaved at an oblique angle. The effective index of refraction and the phase velocity at a given position along the taper depend on the diameter (or effective diameter) and the index of refraction of the bulk fiber material. As the diameter (or effective diameter) decreases with decreasing distance from the tip, the effective index of refraction also decreases. Critical coupling and phase matching can be achieved by placing the optical fiber and the resonator in contact at the proper point along the taper. This recipe is subject to the limitation that the attainable effective index of refraction lies between the indices of refraction of the bulk fiber material and the atmosphere or vacuum to which the resonator and fiber are exposed.

The other recipe involves a refinement of the previously developed technique of prism coupling, in which the light beam from the optical fiber is collimated and focused onto one surface of a prism that has an index of refraction greater than that of the resonator. Another surface of the prism is placed in contact with the resonator. The various components are arranged so that the collimated beam is focused at the prism/resonator contact spot. The recipe includes the following additional provisions:

  • In fabricating the resonator, one strives to obtain

    r = R[1–(nd/np)2],

    where r is the vertical radius of curvature at the contact spot as defined in the figure; R is the horizontal radius of curvature, also as defined in the figure; nd is the effective index of refraction of the desired mode in the resonator; and np is the index of refraction of the prism.

  • The reason for this choice of r and R is that it ensures aperture matching with a Gaussian beam cross section at the contact spot.
  • The numerical aperture (NA) of the collimated beam must be chosen to have the following value:

    NA = sin(λ/h),

    where λ is the vacuum wavelength of the light that one seeks to couple into and out of the resonator, and h is a magnitude of the evanescent electromagnetic field of the resonator, given by

    h ≈ λ/[2π(nd2–np2)1/2].

  • In practice, the fabrication process does not yield precisely the desired radius r : instead, it yields a slightly different value, r′. Therefore, after fabrication, in order to ensure phase matching, one must select a new desired mode for which the effective index of refraction is given by

    nd = np(1–r′/R)1/2.

  • Intermodal coupling is suppressed by use of what, at the time of writing this article, was reported to be a “single mode technique” but not otherwise described. The technique was reported to be described in “Morphology-dependent photonic circuit elements,” Optics Letters Vol. 31, Issue 9, page 1313.

This work was done by Andrey Matsko, Lute Maleki, Vladimir Iltchenko, and Anatoliy Savchenkov of Caltech for NASA’s Jet Propulsion Laboratory.

This invention is owned by NASA, and a patent application has been filed. Inquiries concerning nonexclusive or exclusive license for its commercial development should be addressed to

the Patent Counsel
NASA Management Office–JPL.

Refer to NPO-45462.

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Critical Coupling Between Optical Fibers and WGM Resonators

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This article first appeared in the November, 2009 issue of Photonics Tech Briefs Magazine.

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