US 11,255,746 B2

7

and known temperature values associated with those mea-
surements are provided as input in regression model where
regularization is used to auto select the most import features
and a linear non-parametric model, or non-linear model

(deep neural network) is trained on the data to generate as

model representation of the photonic resonator 200. The
model can be used to infer temperature of the resonator from
its spectra. The use of long short term memory algorithms in
constructing the model can allow the use of future spectral
inputs of known temperature to adjust the model’s param-
eters to compensate for drift.

Photonic resonator analyzer 220 and processes herein
have numerous advantageous and unexpected benefits and
uses. In an embodiment, a process for characterizing a
photonic resonator includes providing photonic resonator
200; communicating probe light 201 from light source 202
by a fiber optic; receiving, by photonic resonator 200, probe
light 201 by evanescent coupling between a coupling wave-
guide and photonic resonator 200; producing, by photonic
resonator 200, product light 203 from probe light 201 by
evanescent coupling standing modes of photonic resonator
200 into a coupling waveguide in communication with a
fiber optic; communicating product light 203 from photonic
resonator 200 to optical detector 204 by evanescent coupling
from photonic resonator 200 to a coupling waveguide;
receiving, by optical detector 204, product light 203 by a
fiber optic; producing, by optical detector 204, product
signal 205 by converting optical power into an electrical
signal; communicating product signal 205 from optical

detector 204 to mode analyzer 206 by a coaxial or serial 3

cable; receiving, by mode analyzer 206, product signal 205
by a coaxial or serial cable; producing, by mode analyzer
206, resonator spectrum 207 from product signal 205 by
collating wavelength and photocurrent data into a dictionary,

resonator spectrum 207 including: a plurality of mode 3

features 208, each mode feature 208 individually including:
primary spline 211 including primary slope 215; secondary
spline 212 including secondary slope 216; and mode peak
210 spectrally interposed between primary spline 211 and
secondary spline 212; and modal width 213 that is a full
width at half maximum of mode feature 208 from primary
spline 211 to secondary spline 212. such that mode peaks
210 of adjacent mode features 208 are spectrally separated
by mode distance 214 and spectrum valley 209: analyzing
resonator spectrum 207 by: producing modal kurtosis by
normalizing primary slope 215 to secondary slope 216 by
peak fitting to identify regions of interest, fitting a cubic
spline over the region, solving for roots to determine width,
using the half-width to determine peak wavelength, using
peak wavelength to determine height and ratio of area under
the primary and secondary spline to determine kurtosis;
producing fringe visibility by normalizing the product of
height squared and mode peak 210 to modal width 213 by
setting up a new data array where pairwise computed values
are stored; and performing regression by: fitting non-para-
metric model 221 to resonator spectrum 207 by using
machine learning algorithm, wherein spectrum valley 209,
mode peak 210, primary spline 211, secondary spline 212,
modal width 213, mode distance 214, primary slope 215,
secondary slope 216, and modal kurtosis are fitting param-
eters of non-parametric model 221; and determining thermal
response function 222 of photonic resonator 200 from fitting
non-parametric model 221 to resonator spectrum 207 to
characterize photonic resonator 200 by using gradient
descent to train and validate the 221

Additional steps can be performed in characterizing a
photonic resonator. Such steps can include using regulariza-

S

8

tion to auto-select parameters fitting parameters and mea-
suring changes in fitting parameters due to change in polar-
ization of light 201.

Photonic resonator analyzer 220 and characterizing a
photonic resonator use a mode frequency (v), mode-to-mode
separation (FSRmn), and their respective dependence on a
thermo-physical perturbation (e.g., temperature) to uniquely
identify each mode of photonic resonator 200. Photonic
resonator 200 can be a nanophotonic thermometer such as a
whispering gallery mode resonator that can be a disk reso-
nator or ring resonator. Resonator spectrum 207 from pho-
tonic resonator 200 can be described by the equation:

hy) ~2 Mtg,

wherein n, is a group index; 2, is a resonance wavelength
of m” mode; and r is resonator radius; and m is a particular
mode of photonic resonator 200. Group index n, is given by:

a
aa aol te )

wherein nyy is the effective index of the mode; 4,,, is the
wavelength of the mth mode, n is refractive index; and on/3h.
is a change in refractive index with photon wavelength. The
temperature dependence of photonic resonator 200 is given
by:

an neg ar
a _Iy oaneg( on
oF p»( ar +2ma( 5)

wherein OW/8T is temperature sensitivity of the mode fre-
quency; r is 8n,9/@T is thermo-optic coefficient and can be
modeled as

1 n-1 E,G. aT
2A-|-3ap x" - > JL | me - 2
prerrsh me BE (« T+B

or as empirically fitted polynomial equation; and ar/oT is a
thermal linear expansion of the resonator.

The first term on the right-hand side independently can be
considered since that term can be 100 times greater than the
second term and can be orders of magnitude greater than the
third term. In certain measurements, the second and third
terms can contribute a non-negligible quantity. A calibration
of non-parametric model 221 for photonic resonator 200
resonator spectrum 207 can be fit to measured data of
resonator spectrum 207 can provide these smaller contribu-
tions. If all modes of photonic resonator 200 occupy the
same volume, then the temperature dependence of all such

5 modes are similar. Higher-order modes can be located closer

to an edge of photonic resonator 200 than a lower-order
mode. Without wishing to be bound by theory, it is believed
that if modes sample different amounts of different materials
of photonic resonator 200 (e.g., if mode m+1 samples a
slightly greater volume of silicon oxide cladding than an
m-th mode), modes have proportional differences in n,yand
dispersion. Respective dependencies of mode frequency and
FSR on temperature vary systematically from mode to mode
and are used to identify modes. Measurements of product
light 203 from of a silicon ring resonator as photonic
resonator 200 had each successive mode with higher tem-
perature sensitivity as shown in FIG. 3 and FIG. 4.


U.S. Patent

Temperature sensitivity

0.079
9.0788
0.0786
0.0784
0.0782

0.078
0.0778
0.0776

0.0774

Feb. 22, 2022 Sheet 3 of 8 US 11,255,746 B2
m mth mt2 m+3 m4 mS m+6
Mode

FIG. 3


U.S. Patent Feb. 22, 2022 Sheet 8 of 8 US 11,255,746 B2

(A)

_ xy pK ;
v 1s «Rx Xx,
= 10
©
3 asd
ny,
wo 064
o
084
“16
1551.50 1951.75 1852.00 1552.25 1952.50 1592.75 1853.00
Wavelength/ nm
te! ' . oy
UT be g989%°
= be #@ Ut of aT } al jit
3 a0 af
2 t aay :
G oe a
ad oy
000 abesegnhdzotoetfeogeese gang disetg
&

¥ ¢ r ¥ t F

ogo 825 650) O75) 100) 12S) SGOLTS

Scaled Wavelength/ AU
FIG. 8


U.S. Patent Feb. 22, 2022 Sheet 5 of 8 US 11,255,746 B2

Temperature dependence of FSR

5.926 4

$.924

5.922 4
5.920 4
5.918 4

5.916
5.914

5.912 4

5.910

Temperature (°C)

FIG. 5


U.S. Patent Feb. 22, 2022 Sheet 4 of 8 US 11,255,746 B2

Temperature dependence of FSR

§.8825

5.8800

5.BT75 4

5.8750 4

5.8725 4

5.8700 4

5.8675

5.8650 4

Temperature °C)

FIG. 4


U.S. Patent Feb. 22, 2022 Sheet 7 of 8 US 11,255,746 B2

ne
o
be

st
al
oe

79.04

ws
ba
%

736
78.44

78.2 4

‘Temperature Sensitivity {pra/k}

78.0

77.84

a a 2 3 4 5 6
mTM mode number {AU}

a
&.
g
zg
® 77.00
&
@ 76.75
é
76.50
6 i 2 3 4 8 6
Ta
c
T72

wd
Da
&

x
>
&

x
>
&

Temperature Sensitivity (pm/K}
x x
> a
NO co)

x
Sd
&

ng
“i
&

0 i 2 3 4 5
mi mode number (AU)

FIG. 7


a2 United States Patent
Ahmed

US011255746B2

US 11,255,746 B2
Feb. 22, 2022

(10) Patent No.:
(45) Date of Patent:

(54) PHOTONIC RESONATOR ANALYZER AND
CHARACTERIZING A PHOTONIC
RESONATOR

(71) Applicant: Government of the United States of
America, as represented by the
Secretary of Commerce, Gaithersburg,
MD (US)

(72) Inventor: Zeeshan Ahmed, Washington, DC (US)

(73) Assignee: GOVERNMENT OF THE UNITED
STATES OF AMERICA, AS
REPRESENTED BY THE
SECRETARY OF COMMERCE,
Gaithersburg, MD (US)

(*) Notice: Subject to any disclaimer, the term of this,

patent is extended or adjusted under 35
U.S.C. 154(b) by 0 days.

(21) Appl. No.: 17/113,222

(22) Filed: Dee. 7, 2020

(65) Prior Publication Data
US 2021/0172833 Al Jun. 10, 2021

Related U.S. Application Data
(60) Provisional application No. 62/944,463, filed on Dec.

6, 2019
(51) Int. CL
GOIM 11/00 (2006.01)
G02B 1/00 (2006.01)
(Continued)

(52) US. CL
svseeee GOIM 11/33 (2013.01); GOK 11/32

(2013.01); G02B 1/005 (2013.01); HOIP 7/00

(2013.01); GO2B 6/29338 (2013.01)

20

(58) Field of Classification Search
CPC .... GOIM 11/08; GO1M 11/088; GO1M 11/30;
GOIM 11/33; GOIM 11/331;

(Continued)
(56) References Cited
U.S. PATENT DOCUMENTS

7,352,466 B2* 4/2008 Cao B82Y 20/00

356/437

9,726,553 B2* 8/2017 Ahmed GOIK 11/00
(Continued)

FOREIGN PATENT DOCUMENTS,

AU 2019100679 A4 * 8/2019

OTHER PUBLICATIONS

Lee, J-M., et al., “Temperature Dependence of Silicon Nanophotonic
Ring Resonator With a Polymeric Overlayer”, Journal of Lightwave
Technology, 2007, p. 2236-2243, vol. 25 No 8.

Primary Examiner — Gordon J Stock, Jr.
(74) Attorney, Agent, or Firm — Otlice of Chief Counsel
for National Institute of Standards and Technology

(57) ABSTRACT

‘A photonic resonator analyzer characterizes a photonic
resonator and incudes a light source that provides a probe
light; a photonic resonator that receives the probe light and
produces product light; an optical detector that receives the
product light and produces a product signal; a mode analyzer
that receives the product signal and produces a resonator
spectrum; and a spectral analyzer that receives the resonator
spectrum, performs regression by fitting a non-parametric
model to the resonator spectrum, and produces a thermal
response function of the photonic resonator from fitting the
non-parametric model to the resonator spectrum to charac-
terize the photonic resonator.

2 Claims, 8 Drawing Sheets

20



US 11,255,746 B2

3

FIG. 8 shows (A) a time series temperature response of a
FBG is fitted to a simple linear regression model that
calculates the relationship between peak center and tem-
perature, wherein the Y-axis is a difference is measured
temperature and predicted temperature from the linear
model; (B) shows a time series temperature response of a
FBG that is fitted to a simple linear regression model that
calculates the relationship between peak center, state nor-
malized (width, kurtosis and amplitude) and temperature,
wherein the Y-axis is a difference is measured temperature
and predicted temperature from the linear model

DETAILED DESCRIPTION

A detailed description of one or more embodiments is
presented herein by way of exemplification and not limita-
tion.

It has been discovered that a photonic resonator analyzer
220 and process for characterizing a photonic resonator
distinguish optical modes of a resonator based on charac-
terization of a temperature sensitivity of a resonance fre-
quency and dispersion characteristics of a mode of the
resonator. The photonic resonator analyzer 220 and charac-
terizing a photonic resonator can include an interrogator that
can include a tunable laser, broadband light source, fre-
quency comb, or a combination thereof. The interrogator
measures a resonator response that can span a free spectral
range (FSR). Photonic resonator analyzer 220 acquires spec-
tra spanning multiple FSRs over multiple temperatures,

pressure, or stimuli to characterize the stimuli-specific (e.g., 3

temperature-specific) sensitivity of individual modes and a
mode spacing (FSR) dependence on the stimuli. The stimuli
(e.g., temperature) dependence of the mode frequency and
dispersion uniquely identifies a mode number of the

observed mode, providing matching calibration coefficients 3

to the mode under observation.

Photonic resonator analyzer 220 characterizes photonic
resonator 200. In an embodiment, with reference to FIG. 1
and FIG. 2, photonic resonator analyzer 220 includes: light
source 202 that provides probe light 201; photonic resonator
200 in optical communication with light source 202 and that
receives probe light 201 from light source 202 and produces
product light 203 from probe light 201: optical detector 204
in optical communication with photonic resonator 200 and
that receives product light 203 from photonic resonator 200
and that produces product signal 205 from product light 203;
mode analyzer 206 in communication with optical detector
204 and that receives product signal 205 from optical
detector 204 and produces resonator spectrum 207 from
product signal 205, resonator spectrum 207 including: a
plurality of mode features 208, each mode feature 208
individually including: primary spline 211 including pri-
mary slope 215; secondary spline 212 including secondary
slope 216; and mode peak 210 spectrally interposed between
primary spline 211 and secondary spline 212; and modal
width 213 that is a full width at half maximum of mode
feature 208 from primary spline 211 to secondary spline 212,
such that mode peaks 210 of adjacent mode features 208 are
spectrally separated by mode distance 214 and spectrum
valley 209; and spectral analyzer 219 in communication
with mode analyzer 206 and that: receives resonator spec-
trum 207 from mode analyzer 206; analyzes resonator
spectrum 207 by: producing modal kurtosis by normalizing
primary slope 215 to secondary slope 216; and producing
fringe visibility by normalizing mode peak 210 to modal
width 213; and performs regression by: fitting non-paramet-
ric model 221 to resonator spectrum 207, wherein spectrum

S

3

40

&

4

valley 209, mode peak 210, primary spline 211, secondary
spline 212, modal width 213, mode distance 214, primary
slope 215, secondary slope 216, and modal kurtosis are
fitting parameters of non-parametric model 221; and pro-
duces thermal response function 222 of photonic resonator
200 from fitting non-parametric model 221 to resonator
spectrum 207 to characterize the photonic resonator 200.

Components of photonic resonator analyzer 220 can be
made from and include various materials. Photonic resona-
tor 200 can includes a ring resonator, a concave disk
resonator, a photonic waveguide, disk resonator, a micro-
sphere resonator, a photonic crystal cavity, a photonic array,
or a combination thereof. As such, photonic resonator 200
can be a fiber (e.g., fiber Bragg grating), strip, ring, disk,
spherical structure, and the like that is disposed at a selected
distance from a waveguide or other medium that commu-
nicates probe light 201 to photonic resonator 200. For a
curved photonic resonator 200, a radius of photonic reso-
nator 200 is selected so an effective length of a circumfer-
ence is an integer of a resonant wavelength of photonic
resonator 200. Various oscillators are described in U.S. Pat.
No. 10,782,421, the disclosures of which is incorporated by
reference in its entirety. Probe light 201 can have a fre-
quency from 20 kilohertz (kHz) to 700 terahertz (THz),
specifically from 1 gigahertz (GHz) to 500 THz, and more
specifically from 500 GHz to 400 THz. Accordingly, pho-
tonic resonator 200 produces product light 203 that can have
a frequency from 20 kilohertz (kHz) to 700 terahertz (THz),
specifically from 1 gigahertz (GHz) to 500 THz, and more
specifically from 500 GHz to 400 THz. Photonic resonator
200 has an arbitrary number m of modes by which reso-
nances of photonic resonator 200 sustain standing electro-
magnetic modes in photonic resonator 200. Photonic reso-
nator 200 can be made from and include various materials
including optically transmissive medium such as quartz,
silicon, and the like.

Optical detector 204 can include a photodiode, photomul-
tiplier tube, and the like that receives product light 203 and
transduces product signal 205 from product light 203. Prod-
uct signal 205 can have a magnitude that provides an optical
amount of product light 203. Moreover, optical detector 204
can be calibrated to provide a linear response or an absolute
amount of product light 203.

Mode analyzer 206 receives product signal 205 from
optical detector 204 and analyzes product signal 205 to
produce resonator spectrum 207. Here, mode analyzer 206
can include hardware or software components that subject
product signal 205 to analysis. Such components include
analog to digital converter, microprocessor, data array tabu-
lation software to create a lookup dictionary, and the like.
Exemplary mode analyzers 206 includes a process such as a
computer equipped with digitizer card and data tabulation
software.

With reference to FIG. 2, resonator spectrum 207 can be

5 an optical amount of product light 203 as a function of

wavelength such that resonator spectrum 207 includes a
plurality of mode features 208. Each mode feature 208
corresponds to a mode of photonic resonator 200 and
individually includes: primary spline 211 including primary
slope 215; secondary spline 212 including secondary slope
216; and mode peak 210 spectrally interposed between
primary spline 211 and secondary spline 212; and modal
width 213 that is a fall width at half maximum of mode
feature 208 from primary spline 211 to secondary spline 212.

5 Mode peaks 210 (e.g., mode peak 210.1, mode peak 210.2)

of adjacent mode features 208 are spectrally separated by
mode distance 214 and spectrum valley 209. It should be


US 11,255,746 B2

1
PHOTONIC RESONATOR ANALYZER AND
CHARACTERIZING A PHOTONIC
RESONATOR

CROSS REFERENCE

This application claims priority to U.S. Provisional Patent
Application Ser. No. 62/944,463, filed Dec. 6, 2019, the
disclosure of which is incorporated herein by reference in its
entirety.

STATEMENT REGARDING FEDERALLY
SPONSORED RESEARCH

This invention was made with United States Government
support from the National Institute of Standards and Tech-
nology (NIST), an agency of the United States Department
of Commerce and The Government has certain rights in the
invention. Licensing inquiries may be directed to the Tech-
nology Partnerships Office, NIST, Gaithersburg, Md.,
20899; voice (301) 975-2573; email tpo@nist.gov.

BRIEF DESCRIPTION

Disclosed is a photonic resonator analyzer for character-
izing a photonic resonator, the photonic resonator analyzer
comprising: a light source that provides a probe light; a
photonic resonator in optical communication with the light
source and that receives the probe light from the light source

and produces product light from the probe light; an optical 3

detector in optical communication with the photonic reso-
nator and that receives the product light from the photonic
resonator and that produces a product signal from the
product light; a mode analyzer in communication with the

optical detector and that receives the product signal from the 3

optical detector and produces a resonator spectrum from the
product signal, the resonator spectrum comprising: a plural-
ity of mode features, each mode feature individually com-
prising: a primary spline comprising a primary slope: a
secondary spline comprising a secondary slope; and a mode
peak spectrally interposed between the primary spline and
the secondary spline; and a modal width that is a full width
at half maximum of the mode feature from the primary
spline to the secondary spline. such that mode peaks of
adjacent mode features are spectrally separated by a mode
distance and a spectrum valley; or a mode height that can be
twice a distance from the mode peak to mode width; and a
spectral analyzer in communication with the mode analyzer
and that: receives the resonator spectrum from the mode
analyzer; analyzes the resonator spectrum by: producing a
modal kurtosis by normalizing the primary slope to the
secondary slope; and producing a fringe visibility by nor-
malizing the product square of mode height and mode peak
to the modal width; producing time-state referenced param-
eters by normalizing all measurements of peak by measure-
ment of peak at the first measurement of 200, creating
time-state referenced parameters by normalizing all mea-
surements of kurtosis by measurement of kurtosis at the first
measurement of 200; creating time referenced parameters by
normalizing all measurements of mode width by measure-
ment of mode width at the first measurement of 200; and
performs regression by: fitting a non-parametric model to
the resonator spectrum, wherein the spectrum valley, the
mode peak, primary spline, secondary spline, modal width,
mode distance, primary slope, secondary slope, and modal
kurtosis, state referenced peak, state referenced kurtosis,
state referenced height, and time reference width are fitting

2

40

45

2

parameters of the non-parametric model; and produces a
thermal response function of the photonic resonator from
fitting the non-parametric model to the resonator spectrum to
characterize the photonic resonator.

Disclosed is a process for characterizing a photonic
resonator, the process comprising: providing a photonic
resonator; communicating a probe light from a light source,
receiving, by the photonic resonator, the probe light; pro-
ducing, by the photonic resonator, a product light from the
probe light; communicating the product light from the
photonic resonator to an optical detector; receiving, by the
optical detector, the product light; producing, by the optical
detector, a product signal; communicating the product signal
from the optical detector to a mode analyzer; receiving, by
the mode analyzer, the product signal; producing, by the
mode analyzer, a resonator spectrum from the product
signal, the resonator spectrum comprising: a plurality of
mode features, each mode feature individually comprising:
a primary spline comprising a primary slope; a secondary
spline comprising a secondary slope; and a mode peak
spectrally interposed between the primary spline and the
secondary spline; and a modal width that is a full width at
half maximum of the mode feature from the primary spline
to the secondary spline, such that mode peaks of adjacent
mode features are spectrally separated by a mode distance
and a spectrum valley; analyzing the resonator spectrum by:
producing a modal kurtosis by normalizing the primary
slope to the secondary slope; producing a fringe visibility by
normalizing the product of square of mode height and mode
peak to the modal width; and performing regression by:
fitting a non-parametric model to the resonator spectrum,
wherein the spectrum valley, the mode peak, primary spline,
secondary spline, modal width, mode distance, primary
slope, secondary slope, and modal kurtosis are fitting param-
eters of the non-parametric model; and determining a ther-
mal response function of the photonic resonator from fitting
the non-parametric model to the resonator spectrum to
characterize the photonic resonator.

BRIEF DESCRIPTION OF THE DRAWINGS.

The following description should not be considered lim-
iting in any way. With reference to the accompanying
drawings, like elements are numbered alike

FIG. 1 shows a photonic resonator analyzer 220;

FIG. 2 shows a resonator spectrum 207;

FIG. 3 shows a graph of temperature sensitivity versus
mode for mode m to (m+4);

FIG. 4 shows a graph of temperature sensitivity versus
mode for a temperature dependence of FSR between mode
m and (m+4) (0.0001 nnvV/K);

FIG. 5 shows a graph of temperature sensitivity versus
mode for a temperature dependence of FSR between mode

5m and (m+6) (0.00016 nm/K);

FIG. 6 shows (a) temperature dependent spectra of device
Aat 20° C., 50° C. and 100° C. for eight modes from 1520
nm to 1570 nm; (b) a shift in mode frequency relative to
mode frequency at 20° C. for dependence on temperature
with differences between higher-order and lower-order
mode; mode 5 shows a shoulder on the blue side that is
labeled as mode m+5a;

FIG. 7 shows a temperature sensitivity of each mode for
three different photonic resonators in panels A, B, and C,

5 respectively, wherein temperature sensitivities have statisti-

cally significant variances between different modes for the
three photonic resonators; and


U.S. Patent Feb. 22, 2022 Sheet 6 of 8 US 11,255,746 B2

——~ 25°C a
50°C
—— 100 °C

Transmission (AU}

1520 1530 1540 1550 1560 1570
Wavelength (nm)

b *
&
2 %
Ps +”
Dp
5 a som
2 oe mad
= de * - m2
3 ~* + om+3
3 a? ema
€ * + m+t5a
§ *
2 & m+5b
* co * m+
i? m+7
20 40 60 80 100 120 140

Temperature (°C)

FIG. 6


U.S. Patent Feb. 22, 2022 Sheet 2 of 8 US 11,255,746 B2

b>
a

;

Optical amount

Li L2

Wavelength

FIG. 2


US 11,255,746 B2

9

In photonic resonator 200, a single mode is identified by
a selected characteristic, such as high Q or a mode fre-
quency. The single mode’s frequency is measured as a
function of measurand to create a calibration plot. Charac-
terizing a photonic resonator can include, at some known
state such as temperature or pressure, locating the mode and
following the mode as the measurand is slowly changed. The
measurand can include a temperature, pressure, radiation,
humidity and the like. Further, the process can include a
broadband light source. For a change that is continuous,
mode dispersion can be linear or about zero, wherein a mode
can be selected by progressively moving up or down modes,
counting modes to track frequency change to measurand.
Advantageously, the initial state can be independently
known. Photonic resonator 200 and characterizing a pho-
tonic resonator provides identification of the modes avail-
able for photonic resonator 200 and application of calibra-
tion coeflicient in an absence of a broadband light source or
an independent measurement device. Moreover, photonic
resonator 200 and characterizing a photonic resonator allows
use of lower size, weight and power (SWAP) value lasers in
lieu of expensive lasers that provides a reduction in instru-
ment cost.

It should be appreciated that photonic resonator analyzer
220 and characterizing a photonic resonator can be used to
measure the spectra of photonic resonator 200 in response to
a wide variety stimuli. Different stimuli, e.g., temperature or
pressure interact with photonic resonator 200 in unique ways
that are in part dependent on the exact geometry, material
composition and defects of both in photonic resonator 200.
The interaction of device geometry, material composition
and defects with stimuli alter the underlying physics of
photonic resonator 200 in ways that leave tell-tale signatures
in the spectra, e.g., changes in the cladding layer can exert

stress on photonic resonator 200, resulting in slight geomet- 3:

ric distortion that manifest themselves as changes in FSR
and peak. These small but significant changes in the spectral
characteristics include changes to the composition of mode
features. These changes can be reliably ascertained by
measuring the spectra spanning multiple FSR, allowing for
determination of relative changes between modes providing
a useful internal standard. Using such self-normalized mea-
surements provides multi-dimensional data set with each
feature representing a different aspect related to the physics
of the device e.g. the width is a proxy for coupling coefli-
cient between the coupling waveguide and photonic reso-
nator 200. It should be appreciated that calibration models of
sensors based on conventional interrogators correlate peak
to an amount of stimuli. Using photonic resonator 200,
multiple, non-colinear parameters were extracted that show
negative and positive correlations on the amount of stimuli.
Using a detailed dictionary, assembled using an exemplary
set of data acquired under known amounts of stimuli,
non-parametric model 221 identified spectral characteristic
unique to each mode.

Photonic resonator analyzer 220 and processes disclosed
herein have numerous beneficial uses, including the ability
to uniquely identify a mode number of a resonance of
photonic resonator 200 at any condition. The ability to
identify mode number under arbitrary conditions has the
benefit of eliminating the need for one or both of a check
thermometer and a temperature control bath for recreating a
known condition which could be compared to previously
measured standard condition to identify the mode. Advan-
tageously, photonic resonator analyzer 220 and characteriz-
ing a photonic resonator overcome limitations of technical
deficiencies of conventional compositions that can require

0

30

40

&

0

3!

10

users to identify the mode number by taking measurements
under known conditions, allowing the user to identify the
mode for which a calibration model was developed using its
resonance wavelength (4). Using photonic resonator ana-
lyzer 220, one analyzes response of photonic resonator 200
over a small arbitrary range of stimuli and uses the non-
parametric model 221 to identify the mode number and
chose appropriate parameters of non-parametric model 221
for use in the calibration model. As such, the user employs
small, inexpensive lasers with narrower tuning ranges, e.g.
distributed feedback lasers in lieu of widely tunable lasers,
and can mode hop across different modes whilst using
appropriate model parameters. FIG. 8 shows exemplary data
and fitting.

Photonic resonator analyzer 220 and processes herein
unexpectedly reduce measurement error due to long-term
drift in photonic resonators. Conventional devices allow the
measurement of a single parameter such as a peak. Changes
in peak are measured as a function of changes in stimuli and
a calibration model is developed using standard linear
regression techniques. Photonic resonator analyzer 220 pro-
vides multiple parameters, many of whom are non-colinear
with each other, and representative of different aspects of the
device physics, as the calibration models built using the
complete set of parameters allows us to capture reversible
and irreversible changes in device physics and chemistry.
The resulting models allow for corrections of small errors,
eg. apparent shift in peak due to width broadening.
Accounting for such changes leads to a 2-10 fold reduction
in validation error when inference is made with photonic
resonator 200.

The articles and processes herein are illustrated further by
the following Example, which is non-limiting,

EXAMPLE

Mode Inter-Changeability in Photonic Thermometer

Nano-photonics can provide an ultra-stable photonic tem-
perature sensor that replace resistance thermometry. A tem-
perature dependence of a photonic resonator’s mode fre-
quency is used to measure temperature. Following the same
mode over a large temperature range can involve a tunable
laser. Adjacent modes of a silicon ring resonator can be
treated as being inter-changeable because temperature sen-
sitivities of nearby modes, though very close in magnitude,
are statistically different. For certain resonators, the differ-
ence in temperature sensitivity between the mode at position
m and m+4 amounts to an error of 2.4 mK to 17.7 mK at
room temperature. Over a 325 K interval, this error can
result in a temperature measurement error of 0.77 K to 5 K.

Temperature measurements are used in manufacturing
process control, physiological monitoring, environmental
monitoring, and the like. Conventional thermometer tech-
nology includes resistance measurement of a thin metal film
or wire. Resistance thermometers can measure temperature
with uncertainties of 100 mK to 10 mK and are sensitive to
humidity, chemical oxidation, and mechanical shock that
causes resistance of the conventional thermometer to change
over time that can be subject to frequent off-line, expensive,
and time-consuming calibration that adds to the cost of
sensor maintenance. The photonic resonator described here
overcomes these technical deficiencies of conventional
resistance thermometry.

The photonic resonator is a photonic temperature sensor
that exploits temperature-dependent change in material
properties of the photonic resonator. These properties
include a thermo-optic effect, thermal expansion, and the


US 11,255,746 B2

13

a light source that provides a probe light;

a photonic resonator in optical communication with the
light source and that receives the probe light from the
light source and produces product light from the probe
light;

an optical detector in optical communication with the
photonic resonator and that receives the product light
from the photonic resonator and that produces a prod-
uct signal from the product light;

a mode analyzer in communication with the optical detec-

tor and that receives the product signal from the optical

detector and produces a resonator spectrum from the
product signal, the resonator spectrum comprising:

a plurality of mode features, each mode feature indi-
vidually comprising:

a primary spline comprising a primary slope;

a secondary spline comprising a secondary slope;
and

a mode peak spectrally interposed between the pri-
mary spline and the secondary spline; and

a modal width that is a full width at half maximum
of the mode feature from the primary spline to the
secondary spline,

such that mode peaks of adjacent mode features are
spectrally separated by a mode distance and a
spectrum valley; and

spectral analyzer in communication with the mode

analyzer and that:

2

receives the resonator spectrum from the mode ana- ,

lyzer,
analyzes the resonator spectrum by:
producing a modal kurtosis by normalizing the pri-
mary slope to the secondary slope; and

producing a fringe visibility by normalizing the ,

mode peak to the modal width; and
performs regression by:

fitting a non-parametric model to the resonator spec-
trum, wherein the spectrum valley, the mode peak,
primary spline, secondary spline, modal width,
mode distance, primary slope, secondary slope,
and modal kurtosis are fitting parameters of the
non-parametric model; and

produces a thermal response function of the photonic
resonator from fitting the non-parametric model to
the resonator spectrum to characterize the pho-
tonic resonator.

14

2. A process for characterizing a photonic resonator, the
process comprising:
providing a photonic resonator;
communicating a probe light from a light source;
receiving, by the photonic resonator, the probe light;
producing, by the photonic resonator, a product light from.
the probe light;
communicating the product light from the photonic reso-
nator to an optical detector;
receiving, by the optical detector, the product light;
producing, by the optical detector, a product signal;
communicating the product signal from the optical detec-
tor to a mode analyzer;
receiving, by the mode analyzer, the product signal;
producing, by the mode analyzer, a resonator spectrum.
from the product signal, the resonator spectrum com-
prising:
a plurality of mode features, each mode feature indi-
vidually comprising:
a primary spline comprising a primary slope:
a secondary spline comprising a secondary slope;
and
a mode peak spectrally interposed between the pri-
mary spline and the secondary spline; and
a modal width that is a full width at half maximum
of the mode feature from the primary spline to the
secondary spline,
such that mode peaks of adjacent mode features are
spectrally separated by a mode distance and a
spectrum valley;
analyzing the resonator spectrum by:
producing a modal kurtosis by normalizing the primary
slope to the secondary slope;
producing a fringe visibility by normalizing the mode
peak to the modal width; and
performing regression by:
fitting a non-parametric model to the resonator spec-
trum, wherein the spectrum valley, the mode peak,
primary spline, secondary spline, modal width, mode
distance, primary slope, secondary slope, and modal
kurtosis are fitting parameters of the non-parametric
model; and
determining a thermal response function of the pho-
tonic resonator from fitting the non-parametric
model to the resonator spectrum to characterize the
photonic resonator.

+ ke ee


U.S. Patent Feb. 22, 2022 Sheet 1 of 8 US 11,255,746 B2

A
220 222
221

—\_»! 219
207

206
205

204

FIG. 1


US 11,255,746 B2

Page 2
(51) Int. CL

GOIK 1132 (2021.01)

HOIP 7/00 (2006.01)

GO2B 6/293 (2006.01)

(58) Field of Classification Search

CPC weeeeeeeee GOIM 11/332; GOIM 11/333; GOIM.
11/334; GOIM 11/335; GO1M 11/336;
GOIM 11/337; GOIM 11/338; GO1K
11/32; GOIB 11/16; GO1B 11/18; G02B
6/1225; G02B 6/4266; G02B 6/29338;
G02B 6/2934; G02B 6/29341; GO2B
6/29343;, G02B 1/005; HO1P 7/00; HOIP
7/005; HO1P 7/02; HO1P 7/04; HO1P.
7/06; HO1P 7/065; HO1P 7/08; HO1P
7/082; HO1P 7/084; HO1P 7/086; HO1P.
7/088; HO1P 7/10; HO1P 7/105

See application file for complete search history.

(56) References Cited
U.S. PATENT DOCUMENTS

10,718,872 B2* 7/2020 Tosh GOIT 1/12
10,782,421 B2* 9/2020 Tosh GOIT 7/00
10,816,325 B2* 10/2020 Ahmed GOLB 9/02008
10,935,370 B2* 3/2021 Ahmed GOLB LL/16
10,955,617 B2* 3/2021 Klimov G02B 6/30
2006/0227331 Al* 10/2006 Vollmer GOIN 21/39
356/483

2012/0044970 Al* 2/2012 Arsenault GOIK 11/125
374/159

2017/0331550 AL* 11/2017 Liu GOLI 3/0256

* cited by examiner


US 11,255,746 B2

5

appreciated that there can be m number of mode peaks 210
in resonator spectrum 207. Further resonator spectrum 207
can be an absorption spectrum or transmission spectrum.
Further resonator spectrum 207 can be a time-varying
inverse Fourier transform signal containing the absorption or
transmission spectrum.

Spectral analyzer 219 receives resonator spectrum 207
from mode analyzer 206 and analyzes resonator spectrum
207 to produce thermal response function 222 from non-
parametric model 221 and resonator spectrum 207. Here,
spectral analyzer 219 can include hardware or software
components that subject resonator spectrum 207 to analysis
based on non-parametric model 221. Such components
include a microprocessor, data tabulation software, machine
learning algorithm, and the like. Exemplary spectral analyz-
ers 219 includes a computer running software configured to
compute linear regression, ridge regression, lasso regres-
sion, Gaussian processes, deep neural networks, and the like.
The analysis performed by spectral analyzer 219 on reso-
nator spectrum 207 can include various operations including
background subtraction, background correction, peak iden-
tification, or fitting, e.g., regression using an arbitrary func-
tion that includes selected fitting parameters such as non-
parametric model 221. Spectral analyzer 219 uses labeled
output, e.g., a spectrum produced by a known amount of
stimuli, to determine 221. Subsequent 207 inputs for which
amount of stimuli is unknown, 219 may use be used by 221

to modify the parameters of 221 or identify the mode _

number or infer the amount of stimuli,

Non-parametric model 221 is a fitting function that
includes adjustable fitting parameters. The fitting parameters
can include spectrum valley 209, the mode peak 210,
primary spline 211, secondary spline 212, modal width 213,

mode distance 214, primary slope 215, secondary slope 216, ~

or modal kurtosis. Exemplary non-parametric model 221
includes {(T.m)=a,,*peak+b,,*peak*peak+c,,*(time refer
enced kurtosis)+d,,*(time referenced width), where m is the
mode number, a,,, b,,, C,,. d,, are fitting parameters for m”
mode.

‘A device specific model can be used, ¢.g., a model for a

Bragg grating such as:

S

40

wherein o is electron-phonon interaction term; is related
to a Debye temperature of the material; ) is Bragg wave-
length at measurement condition, e.g., an unknown tempera-
ture; A, is a Bragg wavelength at the reference temperature,
e.g. 20° C.; A is grating period; a, is the thermal expansion
coefficient; E, is the energy of the j” state, G, is a dispersion
parameter associated with the j” state; T is the temperature;
nis the refractive index; E is the energy of the photon; and
jis the number of states considered and is an adjustable real
number greater than zero and typically less than ten
Thermal response function 222 is a relation between
temperature, wavelength, and mode number for photonic
resonator 200. The relation can be data represented as a
graph, table, order tuple, and the like or a mathematical
function with functional inputs that include temperature or

6

wavelength and has functional output of mode, or mode can
be provided to thermal response function 222 to obtain
wavelength or temperature.

Photonic resonator analyzer 220 can be made in various
ways. Photonic resonator analyzer 220 includes a number of
optical and electrical components, wherein such components
can be interconnected and placed in communication (e.g.,
optical communication, electrical communication, and the
like) by physical or free-space interconnects. The compo-
nents can be disposed on mounts that can be disposed on a
bulkhead for alignment and physical compartmentalization.
As a result, photonic resonator analyzer 220 can be disposed
in a terrestrial environment, space environment, biological
environment, manufacturing environment, and the like.

In an embodiment, a process for making photonic reso-
nator analyzer 220 includes: providing light source 202;
connecting photonic resonator 200 to light source 202 by pig
tailing a fiber optic to a coupling waveguide using either butt
coupling, grating couplers or evanescent coupling. A cou-
pling waveguide can be disposed within 100 nm to 100
micrometer of the resonator. In an embodiment, the resona-
tor is fabricated within the coupling waveguide. The product
light from the resonator is coupled back into a coupling
waveguide which communicates the product light to a fiber

5 optic. The process can include connecting optical detector

204 to photonic resonator 200 by a fiber optic; connecting
mode analyzer 206 to optical detector 204 by a coaxial or
serial cable; and connecting spectral analyzer 219 to mode
analyzer 206 by an electrical communication waveguide,
e.g.,a cable. The light source 202 can include a tunable laser,
and a wavelength of light can be swept over a range of
wavelength equaling multiple FSR. A small amount of 201
can be diverted to a frequency reference e.g. wavemeter by
using an optical splitter. The wavemeter provide accurate
measurements of laser wavelength which is communicated
via a serial or USB connection to 206,

The process for making photonic resonator analyzer 220
also can include interposing an optical splitter between 202
and 200. The optical splitter delivers a part of probe light
201 to optical detector 204, wherein product light 203 and
probe light 201 are beat against each other on a fast
photoreceiver to generate a beat note. Light source 202 may
bea frequency comb. In an embodiment, heterodyne readout
techniques laser locking, e.g., Pound-Drever-Hall or dither
locking, are used. For this purpose, a laser is locked to a
known frequency source such a molecular gas (acetylene).
The output of frequency stabilized laser can be communi-
cated to a phase adjustment apparatus e.g. delay line or
acousto-optic modulator followed by communication to
optical detector 204. A second laser, in communication with
photonic resonator 200 is swept over a wide range of
wavelengths spanning two or more FSR. In an embodiment,
a wide bandwidth light source such a super luminescent
diode (SLD) may be used. Here, light output from a SLD is
coupled to photonic resonator 200 and product light 203 is
communicated to optical detector 204 where it beats against
the reference light generated. Reference light may be a
frequency comb or a tunable laser. Using mode analyzer
206, output signal 205 is plotted against laser wavelength
measurement to construct a resonator spectra. Standard peak
finding algorithms are then used to find regions where
spectral features are located. The resonator spectra is fitted
to a cubic spline or gaussian function and mode character-
istics such as peak, kurtosis, width and FSR are calculated.
From these parameters, state-referenced features are calcu-
lated by taking the values of each feature at 20° C. recorded
at the first instance of 200 is measured. All of these features


US 11,255,746 B2

11

like. Fiber Bragg gratings (FBG) exhibit temperature-de-
pendent shifts in resonant Bragg wavelength of10 pm/K in
dry, strain-free environments and can measure temperature
with uncertainties of 500 mK (k=2) from 233 K to 293 K,

Silicon-on-insulator devices such as Bragg waveguides, ring
resonators, and photonic crystal cavities have a superior
thermo-optic coeflicient that provide temperature sensitivity
of 60 pm/K to 80 pm/K. Humidity effect on silicon photonic
devices can be mitigated by passivating silicon dioxide layer
on top of the device. Silicon photonic thermometers can
measure temperature with sub-mK resolution.

This Example provides information for temperature-de-
pendence of up to seven adjacent modes of a silicon ring
resonator across three different chips drawn from the same
fabrication run. The device include a Si straight-probe
waveguide with cross-sections of 510 nm by 220 nm,
evanescently coupled to a 15 jum diameter ring over an air
gap of 130 nm and covered with a 1.5 jum thick layer of
plasma-enhanced chemical vapor deposition (PECVD)
SiO,. The silicon photonic sensors were made using CMOS
manufacturing technology.

The device is probed using a C-band laser that was swept
cover the sensor resonance. Ten percent of laser power was
picked up from the laser output for wavelength monitoring,
and the rest of the laser power, after passing through
polarization paddles, was injected into the photonic device
and detected by a power meter. Light from single mode fiber
was coupled into and out of the waveguide by grating
couplers. Some experiments were performed with~100 nW
of incident laser power.

As shown in FIG. 6a, at 20° C., over the range of 1520 nm
to 1570 nm the spectra shows seven resonances, located at
1522.30828 nm, 1528.10100 nm, 1533.95890 nm,
1539.84587 nm, 1545.75360 nm, 1551.76479 nm,

1557.75820 nm and 1561.4100 nm separated by FSR 3

of=5.79 nm. As temperature increased, the highest wave-
length mode, mode 7, shifted out of the laser scanning
window, and new modes (m-1, m-2) appeared. Mode m47,
m-1, and m-2 over limited temperatures were considered in
analysis.

Temperature-dependent shifts in the mode resonances
have similar profiles with linear fit slopes of78 jun/K (FIG.
6b). Temperature-dependence of silicon ring resonators”
different modes is weakly quadratic. Temperature-depen-
dence of modes were subjected to linear fits of each mode,
As shown in FIG. 7, the temperature sensitivities trended
towards increasing temperature sensitivity with mode num-
ber. Uncertainty of fits were determined by a leave-one-out
bootstrap analysis with fit uncertainty of approximately
0.12%. The first two modes (mode m and m+1) had statis-
tically similar slopes. Higher-order modes, e.g. m+4 or
greater, had statistical differences from the m” mode. The
three samples had from 0.2% to 2% change in temperature
sensitivity per mode hop. Between modes m and m+4 the
percentage corresponded to a measurement error of 2.4 mK,
8.93 mK, and 17.7 mK for individual devices. When apply-
ing calibration of mode m to m+4, the change in temperature
sensitivity had measurement uncertainty. Over the tuning
range of the laser, equivalent to a 325 K temperature range,
the measurement error was 770 mK (device A, FIG. 7a) and
up to 5 K (device C, FIG. 7c).

For certain measurements, the combined uncertainty
should be 10 mK or lower. Resistance thermometers such as
platinum resistance thermometers (PRT) are interchangeable
to within 100 mK-400 mK over a 300 K range. Results here
show that, for photonic devices, mode inter-changeability
can provide measurement performance of a Grade B PRT.

S

g

12

‘The mode number dependence of the temperature sensitivity
can be due to cross-term interaction between dispersion and
thermo-optic coefficient, inter-mode mixing or other mode
structure deviations from ideal transverse electric (TE) mode
due to fabrication errors and limitations such as sidewall
roughness and sloping. There is a need for in-depth analysis
modal dispersion and mode structure. The impact of fabri-
cation errors and waveguide geometry on mode structure
and its evolution with temperature should be understood.
Precise characterization of propagating mode families and
their interactions across the temperature range provide
design, fabrication, and selection of devices where mode
inter-changeability can be employed as low-cost laser inter-
rogators.

While one or more embodiments have been shown and
described, modifications and substitutions may be made
thereto without departing from the spirit and scope of the
invention. Accordingly, it is to be understood that the present
invention has been described by way of illustrations and not
limitation. Embodiments herein can be used independently
or can be combined.

All ranges disclosed herein are inclusive of the endpoints,
and the endpoints are independently combinable with each
other. The ranges are continuous and thus contain every
value and subset thereof in the range. Unless otherwise
stated or contextually inapplicable, all percentages, when
expressing a quantity, are weight percentages. The
suflix “(s)” as used herein is intended to include both the
singular and the plural of the term that it modifies, thereby
including at least one of that term (e.g., the colorant(s)
includes at least one colorants). “Optional” or “optionally”
means that the subsequently described event or circumstance
can or cannot occur, and that the description includes
instances where the event occurs and instances where it does
not. As used herein, “combination” is inclusive of blends,
mixtures, alloys, reaction products, and the like.

As used herein, “a combination thereof” refers to a
combination comprising at least one of the named constitu-
ents, components, compounds, or elements, optionally
together with one or more of the same class of constituents,
components, compounds, or elements.

All references are incorporated herein by reference.

The use of the terms “a” and “an” and “the” and similar
referents in the context of describing the invention (espe-
cially in the context of the following claims) are to be
construed to cover both the singular and the plural, unless
otherwise indicated herein or clearly contradicted by con-
text. “Or” means “and/or.” It should further be noted that the
terms “first,” “second,” “primary,” “secondary,” and the like
herein do not denote any order, quantity, or importance, but
rather are used to distinguish one element from another. The
modifier “about” used in connection with a quantity is
inclusive of the stated value and has the meaning dictated by
the context (e.g., it includes the degree of error associated
with measurement of the particular quantity). The conjunc-
tion “or” is used to link objects of a list or alternatives and
is not disjunctive; rather the elements can be used separately
or can be combined together under appropriate circum-
stances.

What is claimed is:

1. A photonic resonator analyzer for characterizing a
photonic resonator, the photonic resonator analyzer com-
prising: