# US 10,310,186 B2 · Method of engineering the dispersion of whispering gallery mode resonators and the resonators with dispersion engineered by the method

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- Number: US 10,310,186 B2
- Document ID: US10310186B2
- Status: Granted
- Kind: B2
- Application number: 15/069,725
- Filed: 2016-03-14
- Published: 2016-09-15
- Granted: 2019-06-04
- Earliest claimed: 2015-03-13
- Assignees: California Institute of Technology
- Applicants: California Institute of Technology
- Inventors: Ivan S. Grudinin; Nan Yu; Lukas M. Baumgartel
- Web page: https://patentgrep.com/patent/US10310186B2

## CPC

- G02B 6/29341: {Loop resonators operating in a whispering gallery mode evanescently coupled to a light guide, e.g. sphere or disk or cylinder (evanescent coupling for sensing fluorescence G01N21/648)}
- G02B 27/0012: {Optical design, e.g. procedures, algorithms, optimisation routines}
- G06F 30/00: Computer-aided design [CAD]
- G06F 30/23: using finite element methods [FEM] or finite difference methods [FDM]
- G02B 6/138: by using polymerisation
- G06F 2119/10: Noise analysis or noise optimisation
- G02B: OPTICAL ELEMENTS, SYSTEMS OR APPARATUS
- G06F: ELECTRIC DIGITAL DATA PROCESSING (computer systems based on specific computational models G06N)

## Abstract

A method for fabricating a resonator structure, comprising engineering dispersion of electromagnetic radiation guided along a boundary of an axially symmetric substrate, the engineering comprising micro-structuring a geometry of the boundary, wherein the structure defines a waveguide for electromagnetic radiation.

## Claims

### Claim 1

A resonator, comprising:
a protrusion formed on an axially symmetric boundary of a substrate, the protrusion:
guiding a mode of the electromagnetic radiation in the resonator,
having a rectangular cross-section having a height and a width and an aspect ratio A (the width divided by the height) wherein A≤2, and
wherein the aspect ratio is capable of significantly altering a shape of the dispersion curve of the electromagnetic radiation in the resonator, rather than merely shifting the dispersion curve, resulting in flattening of the wavelength dependence of dispersion of the electromagnetic radiation guided within the resonator.

### Claim 2 (depends on 1)

The resonator of claim 1, wherein the protrusion comprises a photonic belt on the substrate.

### Claim 3 (depends on 1)

The resonator of claim 1, wherein the boundary is on a surface of revolution about an axis of the resonator .

### Claim 4 (depends on 1)

The resonator of claim 1, wherein the dimensions and the aspect ratio of the protrusion:
flatten the wavelength dependence of dispersion so that the dispersion is flat or nearly zero over a range of wavelengths of the electromagnetic radiation,
support a single mode family, or more than one single mode family of the electromagnetic radiation, and
obtain a controlled Q-factor for the resonator.

### Claim 5 (depends on 1)

The resonator of claim 1, wherein:
the protrusion on the boundary comprises the height in a range of 0.1-100 micrometers, the width in a range of 0.1-100 micrometers that is smaller than a height of the resonator, and a shape that is distinct from a shape of the resonator,
the mode comprises a whispering gallery mode, and
the resonator comprises a whispering gallery mode resonator.

### Claim 6 (depends on 1)

The resonator of claim 1, wherein the boundary is on a vertical or angled sidewall of the substrate.

### Claim 7 (depends on 1)

The resonator of claim 1, wherein the resonator comprises a cylindrical surface and the protrusion comprises a photonic belt formed on a middle, top, and/or bottom of the cylindrical surface.

### Claim 8 (depends on 1)

The resonator of claim 1, wherein the electromagnetic radiation has a wavelength in a range of 0.1-400 micrometers.

### Claim 9 (depends on 1)

The resonator of claim 1, wherein the resonator has a diameter of more than 0.005 mm.

### Claim 10 (depends on 1)

The resonator of claim 1, wherein the substrate comprises a cylindrical substrate on a chip.

### Claim 11 (depends on 1)

The resonator of claim 1, wherein a shape and dimensions of the protrusion form the resonator generating a comb of frequencies having a frequency span including one octave.

### Claim 12 (depends on 1)

The resonator of claim 1, wherein a shape and dimensions of the protrusion form the resonator generating a comb of repetition rate below 100 GHz.

### Claim 13 (depends on 1)

The resonator of claim 1, wherein:
the protrusion is structured using a computer-controlled process according to a pattern that is determined by numerically solving a three dimensional vectorial Maxwell equation.

### Claim 14 (depends on 13)

The resonator of claim 13, wherein the numerical solving calculates eigenvalues with a precision achieved using a finite element method having meshes containing up to 0.5 million elements.

### Claim 15 (depends on 1)

The resonator of claim 1, wherein the protrusion includes a cascaded cross-section.

### Claim 16 (depends on 1)

The resonator of claim 1, wherein the resonator comprises a whispering gallery mode resonator and the height and the width are in a range of 0.1-100 micrometers.

### Claim 17

A method for fabricating a resonator structure, comprising:
forming a protrusion on an axially symmetric boundary of a substrate, the protrusion:
guiding a mode of electromagnetic radiation along the boundary,
having a rectangular cross-section having a height and a width and an aspect ratio A (the width divided by the height) wherein A≤2, and
wherein the aspect ratio significantly alters and flattens the wavelength dependence of dispersion of the electromagnetic radiation guided within the resonator.

### Claim 18 (depends on 17)

The method of claim 17, further comprising:
calculating the dispersion by numerically solving a three dimensional vectorial Maxwell equation using a finite element method (FEM), wherein:
the FEM utilizes meshes, and
the solving includes calculating eigenvalues with a precision achieved using one or more of the meshes containing up to 0.5 million elements.

### Claim 19 (depends on 18)

The method of claim 18, wherein the dispersion is calculated for the geometry having dimensions measured using the method comprising:
making a cast of the resonator;
filling the cast with a glue having different refractive index from the cast to form a replica;
after hardening the glue, cutting and polishing the replica to reproduce the resonator's geometry;
using an optical microscope to obtain an image of the resonator's geometry; and
measuring the dimensions using the image.

### Claim 20 (depends on 17)

The method of claim 17, further comprising:
solving a Maxwell equation numerically and using a finite element method to find each of three adjacent eigenfrequencies representing mode frequencies of the resonator structure, including:
(a) solving the Maxwell equation to compute one of the eigenfrequencies using the finite element method using a rough mesh,
(b) refining the rough mesh by increasing the number of mesh vertices within the rough mesh, to form a refined mesh,
(c) solving the Maxwell equation to compute the one of the eigenfrequencies using the finite element method using the refined mesh,
(d) repeating steps (a)-(c) using the refined mesh obtained in step (b) as the rough mesh in the next computing step (a), thereby generating a plurality of refined meshes, until the one of the eigenfrequencies converges with sufficient precision,
(e) solving the Maxwell equation to compute the remaining two eigenfrequencies using the rough mesh,
(f) solving the Maxwell equation to compute the remaining two eigenfrequencies using the refined meshes obtained in steps (b),(d) and using the same number of repeating steps as in step (d), while adjusting the refractive index only such that errors introduced by the meshes are the same for all three eigenfrequencies; and
calculating the dispersion from the three eigenfrequencies.

### Claim 21 (depends on 17)

The method of claim 17, further comprising:
solving a Maxwell equation numerically and using a finite element method to find each of three adjacent eigenfrequencies fl , fl+1, and fl−1 representing mode frequencies of the resonator structure, including:
(a) for each eigenfrequency, building a rough finite element mesh using a set of input approximations,
(b) solving the Maxwell equation to compute each of the three eigenfrequencies using the rough finite element mesh,
(c) refining each of the rough finite element meshes by increasing the number of mesh vertices within the rough finite element meshes, to form a refined mesh for each of the eigenfrequencies,
(d) solving the Maxwell equation to compute the eigenfrequencies using the finite element method using the refined meshes,
(e) repeating steps (b)-(d) using the refined meshes obtained in step (c) as the finite element rough mesh in the next solving step (a), until the eigenfrequencies converge with sufficient precision; and
calculating the dispersion from the three eigenfrequencies using
D
=
cD
2
2
⁢
πλ
2
⁢
RF
3
×
10
6
⁡
[
ps
nm
·
km
]
where F=0.5 (fl+1 −fl−1) is free spectral range near the wavelength)λ, D2 =fl+1 −2f1−fl−1, fl is the optical mode frequency corresponding to l field nodes along the light-guiding boundary of the resonator, and l is an integer.