WO2021156731A1

WO2021156731A1 - Temperature sensor and temperature-measurement method

Info

Publication number: WO2021156731A1
Application number: PCT/IB2021/050795
Authority: WO
Inventors: Bernard P. Masterson; Andrew D. Sappey
Original assignee: Onpoint Technologies, Llc
Priority date: 2020-02-03
Filing date: 2021-02-01
Publication date: 2021-08-12

Abstract

A temperature-measurement method includes a cavity-signal step, a free-spectral-range step, and a temperature-determination step. The cavity-signal step includes generating a cavity signal by measuring optical power output by an optical cavity in response to interrogation by a laser being tuned through a plurality of resonance wavelengths of the optical cavity. The free-spectral-range step includes determining, from the cavity signal, a free-spectral range of the optical cavity at a first resonance wavelength of the plurality of resonance wavelengths. The temperature-determination step includes determining a temperature of the optical cavity from the free-spectral range, the first resonance wavelength, a temperature-dependent refractive index of the optical cavity, and a temperature-dependent length of the optical cavity.

Description

TEMPERATURE SENSOR AND TEMPERATURE-MEASUREMENT METHOD

CROSS-REFERENCE TO RELATED APPLICATION

[0001] This application claims priority to U.S. Provisional Patent Application No. 62/969,219, filed February 3, 2020, which is incorporated by reference as if fully set forth herein.

BACKGROUND

[0002] Furnaces used for cracking and methane reforming include heated process tubes. Optimal operation of such furnaces requires holding the process tubes at a prescribed temperature. Current means for determining tube temperature lack robustness and/or accuracy. These methods include use of IR cameras and thermocouples.

SUMMARY OF THE EMBODIMENTS

[0003] One aspect of the present embodiments includes the realization that variations in operator methodology result in IR thermometers producing inconsistent and hence unreliable temperature measurements. The present embodiments solve this problem by leveraging the temperature sensitivity of optical cavities, which can be attached to a process tube and remotely interrogated to yield operator-independent temperature measurements.

[0004] In a first aspect, a temperature-measurement method includes a cavity-signal step, a free-spectral-range step, and a temperature-determination step. The cavity-signal step includes generating a cavity signal by measuring optical power output by an optical cavity in response to interrogation by a laser being tuned through a plurality of resonance wavelengths of the optical cavity. The free-spectral-range step includes determining, from the cavity signal, a free-spectral range of the optical cavity at a first resonance wavelength of the plurality of resonance wavelengths. The temperature-determination step includes determining a temperature of the optical cavity from the free-spectral range, the first resonance wavelength, a temperature- dependent refractive index of the optical cavity, and a temperature-dependent length of the optical cavity.

[0005] In a second aspect, a temperature measurement method includes a first detecting step, a first recovering step, a second detecting step, a second recovering step, and a temperature- determining step. The first detecting step (a) includes detecting a first cavity signal output by an optical cavity in response to interrogation by a temporally -tuned laser operating at a first resonance wavelength. The optical cavity has a first free-spectral range at the first resonance wavelength and a second free-spectral range at a second resonance wavelength. The first recovering step includes recovering a first spectral response of the optical cavity from the first cavity signal. The first spectral response includes a first plurality of resonances spectrally separated by a first free-spectral range. The second detecting step includes detecting a second cavity signal output by the optical cavity in response to interrogation a temporally -tuned laser operating at the second resonance wavelength. The second recovering step includes recovering a second spectral response of the optical cavity from the second cavity signal. The second spectral response includes a second plurality of resonances spectrally separated by the second free- spectral range. The temperature-determining step includes determining a temperature of the optical cavity from the first spectral response and the second spectral response.

[0006] In a third aspect, a temperature sensor includes an optical device and a retroreflector. The optical device includes a first port, a second port, and a third port. The optical device is configured to: receive a first input optical signal at the first port, output the first input optical signal at the second port, receive a second input optical signal at the second port, and output the second input optical signal at the third port. The retroreflector includes a planar front- face configured to reflect the first input optical signal output from the second port as the second input optical signal along an optical path between the second port and the retroreflector.

BRIEF DESCRIPTION OF THE FIGURES

[0007] FIG. 1 is a schematic of a temperature sensor that includes collimator optics and a compound optical cavity, in an embodiment.

[0008] FIGs. 2 and 3 are respective examples of collimator optics of FIG. 1.

[0009] FIG. 4 is a graphical representation of cavity signals of the compound optical cavity of FIG. 1 in response to interrogation by temporally-tuned lasers, in an embodiment.

[0010] FIG. 5 is a graphical representation of spectral responses obtained by applying a frequency-domain filter to the cavity signals of FIG. 4, in an embodiment.

[0011] FIG. 6 is a graphical representation a beat-note signal obtained from the spectral responses of FIG. 4, in an embodiment. [0012] FIG. 7 is a flowchart illustrating a temperature-measurement method implementable by the temperature sensor of FIG. 1, in embodiments.

[0013] FIG. 8 is a schematic of a temperature sensor, which is an embodiment of the temperature sensor of FIG. 1.

[0014] FIG. 9 is a time-series plot of example tap signals and cavity signal detected by an embodiment of the temperature sensor of FIG. 8.

[0015] FIG. 10 is a plot of a normalized cavity signal derived from the cavity signal of FIG. 9, in an embodiment.

[0016] FIG. 11 is a plot of a filtered cavity signal derived from the normalized cavity signal of FIG. 10.

[0017] FIG. 12 is a plot of temperature-dependent free-spectral range of an optical cavity of the temperature sensor of FIG. 8, in an embodiment.

[0018] FIG. 13 is a flowchart illustrating a temperature-measurement method implementable with the temperature sensor of FIG. 8, in embodiments.

DETAILED DESCRIPTION OF THE EMBODIMENTS

[0019] FIG. 1 is a schematic of a temperature sensor 100 in an example use scenario involving a process tube 194 of a furnace 190. Temperature sensor 100 monitors a process-tube temperature 195 of process tube 194. Furnace 190 also includes a burner 192 proximate process tube 194.

[0020] Temperature sensor 100 includes an optical device 120 and an optical cavity 140. In embodiments, temperature sensor 100 also includes at least one of optoelectronics 110, collimator optics 130, and a temperature-sensing processor 160. Optoelectronics 110 includes at least one of a light source 111 and an optical sensor 116. Light source 111 includes at least one laser 114, In embodiments, laser 114 is a scannable single-frequency laser, such as a diode laser, examples of which include a vertical-cavity surface-emitting laser (VCSEL), and a distributed feedback laser. In embodiments, optoelectronics 110 includes a demultiplexer 118.

[0021] When optoelectronics 110 includes both optical sensor 116 and demultiplexer 118, optical sensor 116 may include a respective optical sensor 116 (k) corresponding to each demultiplexed port k = {1, 2, ... } of demultiplexer 118. For example, FIG. 1 illustrates demultiplexer 118 with two demultiplexed ports 119(1) and 119(2), which are optically coupled to optical sensors 116(1) and 116(2), respectively.

[0022] Temperature-sensing processor 160 includes electronics 152, a processor 154, and a memory 156. Processor 154 is communicatively coupled to memory 156 and may be communicatively coupled to electronics 152. Electronics 152 is communicatively coupled to memory 156, and may include one or more of an operational amplifier and a microcontroller, and may be a data acquisition device. Electronics 152 may include an optical spectrum analyzer.

[0023] Memory 156 may be transitory and/or non-transitory and may include one or both of volatile memory (e.g., SRAM, DRAM, computational RAM, other volatile memory, or any combination thereof) and non-volatile memory (e.g., FLASH, ROM, magnetic media, optical media, other non-volatile memory, or any combination thereof). Part or all of memory 156 may be integrated into processor 154.

[0024] Memory 156 stores at least one of software 161, a first spectral response 174, a second spectral response 184, and a cavity temperature 188. In embodiments, process-tube temperature 195 is derived from cavity temperature 188 via previously determined calibration data that maps cavity temperature 188 to process-tube temperature 195. In embodiments, cavity temperature 188 equals, or is an effective proxy for, process-tube temperature 195.

[0025] First spectral response 174 includes cavity resonance frequencies 176 and a free- spectral range (FSR) 177. Second spectral response 184 includes cavity resonance frequencies 186 and an FSR 187. In embodiments, software 161 includes a wavelength mapper 162 and a solver 168 in the form of computer-readable instructions that when executed by at least one processor implement the functionality of the wavelength mapper 162 and solver 168 discussed herein. In embodiments, solver 168 determines cavity temperature 188 from spectral responses 174 and 184. In embodiments, solver 168 implements one or more numerical methods for finding an extremum of an objective function. The extremum may be local or global, and be either a minimum or maximum. Examples of numerical methods include, but are not limited to, direct search methods such as Nelder-Mead, probabilistic methods such as simulated annealing, and combinations thereof.

[0026] Optical device 120 includes ports 121-123. Port 121 is configured to receive an optical signal. For example, light source 111 generates an input optical signal 115 received by port 121. Port 122 is configured to output input optical signal 115, as an optical signal 125, along an optical path traversed by optical signal 125, between port 122 and optical cavity 140. In embodiments, optical device 120 includes one of a l-by-2 optical fiber coupler, a 2-by-2 optical fiber coupler, and an optical circulator.

[0027] Optical cavity 140 includes a retroreflector 144. Retroreflector 144 has a refractive index n₄ and includes a planar front-face 145. Retroreflector 144 is configured to retroreflect optical signal 125, as a temperature-sensitive signal 147, back to port 122 along the optical path, such that optical signals 125 and 147 counterpropagate along the optical path. Port 123 is configured to output temperature-sensitive signal 147.

[0028] In embodiments, retroreflector 144 is a retroreflector prism, such as a corner-cube prism. In embodiments, retroreflector 144 is formed of one or more materials that can withstand high temperatures of furnace 190. These materials include, but are not limited to, silicon, sapphire, magnesium aluminate spinel, aluminum oxynitride, and fused silica. Retroreflector 144 may be formed of a material that has a melting point exceeding 700 °C. Retroreflector 144 has a thickness 148, which in embodiments is between five millimeters and seven millimeters.

[0029] Temperature sensor 100 may include a cavity mount 149 that secures optical cavity 140 to process tube 194. In embodiments, cavity mount 149 is in physical contact with optical cavity 140 such that respective temperatures of optical cavity 140 and process tube 194 are correlated. In embodiments, cavity mount 149 attaches optical cavity 140 to process tube 194 by means of one of welding, spring-loaded hinges, and a combination thereof.

[0030] In embodiments, optical cavity 140 includes an etalon 142 between retroreflector 144 and port 122 along the optical path of optical signal 125, in which case optical cavity 140 is referred to herein as a compound optical cavity. Unless otherwise stated, statements about optical cavity 140 apply to instances of optical cavity 140 that either include or do not include etalon 142.

[0031] Etalon 142 includes a rear planar surface 143 adjoining planar front-face 145. Etalon 142 has a front planar surface 141 opposite rear planar surface 143. Rear planar surface 143 and planar front-face 145 may be held together via one of a spring-loaded mechanism, optical contacting, a glass fritting material, and a combination thereof.

[0032] Etalon 142 has a refractive index n₂. In embodiments, \n₂ — n₄| > 1, which results in the reflection coefficient at the interface between etalon 142 and retroreflector 144 being sufficiently high to enable etalon 142 to function as a high-finesse optical cavity. In embodiments, the ambient medium adjacent to front planar surface 141 has refractive index ⁿ _amb < ⁿ4_> such that \n₂ — n_amb | also exceeds one such that the reflection coefficient at the interface between etalon 142 and the ambient medium is also sufficiently high to enable etalon 142 to function as a high- finesse optical cavity. In embodiments, the ambient medium is air, and may include combustion gasses, e.g., during furnace operation. Etalon 142 may be a Fabry-Perot etalon and, in embodiments, is formed of silicon. In embodiments, a thickness of etalon 142 between surfaces 141 and 143 is between one millimeter and three millimeters, e.g., two millimeters.

[0033] In embodiments, optoelectronics 110 includes a light source monitor 101, which includes an optical tap 102 and a photodetector 104. Optical tap 102 directs a tap signal 103 to photodetector 104 which monitors the output power of optical signal 115.

[0034] In embodiments, light source monitor 101 also includes an optical tap 106 and a wavemeter 108. Optical tap 106 directs a tap signal 107 to wavemeter 108, which monitors a center wavelength of optical signal 115 and outputs a time-series of measured wavelengths 109. Wavemeter 108 includes a photodetector and at least one of an optical etalon and an interferometer. Each of tap signals 103 and 107 is a fraction of optical signal 115.

[0035] FIG. 2 is a schematic of optical device 120 coupled to optical cavity 140 by collimator optics 230, which is an example of collimator optics 130. Collimator optics 230 includes an optical fiber 232 and an off-axis collimator 234. In embodiments, collimator optics 230 lacks a refractive optical lens, as its additional surfaces might introduce additional long-path (high frequency) etalons, resonances of which appear on the desired signal as high-frequency noise.

[0036] In operation, optical signal 125 exits optical fiber 232 and is collimated upon being reflected by off-axis collimator 234. Optical signal 125 exits optical cavity 140 as temperature-sensitive signal 147. Temperature-sensitive signal 147 is incident on collimator optics 230, which focuses temperature-sensitive signal 147 such that temperature-sensitive signal 147 is coupled into port 122 via optical fiber 232.

[0037] FIG. 3 is a schematic of optical device 120 coupled to optical cavity 140 by collimator optics 330, which is an example of collimator optics 130. Collimator optics 330 includes optical fiber 332 and a lens 334. Optical fiber 332 is directly or indirectly optically coupled to port 122. In embodiments, collimator optics 330 is a fiber-optic collimator. [0038] In operation, optical signal 125 exits optical fiber 332 and is collimated by collimator optics 330. Optical signal 125 exits optical cavity 140 as temperature-sensitive signal 147. Temperature-sensitive signal 147 is incident on collimator optics 330, which couples temperature-sensitive signal 147 to port 122 via optical fiber 332.

[0039] When optical cavity 140 includes etalon 142, optical cavity 140 includes two optical cavities with respective spectral responses each characterized by a respective plurality of resonance frequencies. The first optical cavity is within etalon 142 between its surfaces 141 and 143, illustrated in FIG. 1. The second cavity is within retroreflector 144, where reflecting surfaces include planar front- face 145 and retro-reflecting surfaces of retroreflector 144 surfaces function as the reflecting surfaces. The dashed-dot line within retroreflector 144 illustrates a U- shaped optical path 146 of the second cavity (retroreflector 144), where U-shaped optical path 146 begins and ends at respective locations on planar front-face 145. Accordingly, retroreflector 144 will also be referred to as retroreflector optical cavity 144. While surfaces of etalon 142 and retroreflector 144 combine to form additional optical cavities, these cavities do not significantly contribute to the observed temperature-sensitive signal 147.

[0040] In embodiments, temperature sensor 100 determines process-tube temperature 195 of process tube 194 according to spectral responses of one of etalon 142 and retroreflector optical cavity 144, which in part determine the optical spectrum of temperature-sensitive signal 147. In such embodiments, temperature sensor 100 includes optical sensor 116, which receives temperature-sensitive signal 147. When light source 111 is temporally-tuned, e.g., periodically tuned, through its wavelength range at a particular chip temperature the time-dependent amplitude of temperature-sensitive signal 147 exhibits beat frequencies resulting from the differences in resonance frequencies of etalon 142 and retroreflector optical cavity 144. Herein, tuning of light source 111 refers to tuning of at least one laser 114 thereof and the time- dependent amplitude of temperature-sensitive signal 147 is referred to as a cavity signal. Temperature-sensing processor 160 stores first and second cavity signals 170 and 180. In embodiments, light source 111 is temporally -tuned through its wavelength range at a frequency between 100 Hz and 10 kHz. In embodiments, optoelectronics 110 includes a laser driver 112 coupled to laser 114 that operates to temporally tuner laser 114 via a drive signal 113, which may be a current or voltage that is time- varying, e.g., a periodic waveform. [0041] The temperature dependence of cavity signals 170 and 180 depends in part on the temperature dependence of refractive index n₄ of retroreflector 144 and, in embodiments, also refractive index n₂ of etalon 142. Herein D_Tn₄ and D_Tn₂ denote the temperature dependence of n₄ and n₂ respectively. Since both etalon 142 and optical cavity 144 contribute to resonance frequencies of temperature-sensitive signal 147, tracking the temperature of process tube 194 is complicated when temperature dependencies D_Tn₄ and D_Tn₂ are similar, e.g., differing by less than a factor of two. In embodiments, one of D_Tn₄/D_Tn₂ and D_Tn₂/D_Tn₄ is greater than eight. For example, when D_Tn₄/D_Tn₂ exceeds eight, temperature sensor 100 tracks process-tube temperature 195 according to temperature-dependent resonance frequencies of etalon 142.

[0042] The temperature dependence of cavity signals 170 and 180 also depends in part on respective thermal expansion coefficients of etalon 142 and retroreflector 144, denoted herein as a₄ and a₂ respectively. In embodiments, D_Tn₄ exceeds a₄ by at least a factor of ten, and D_Tn₂ exceeds a₂ by at least a factor of ten, such that thermal expansion has a negligible effect on cavity signals 170 and 180, which simplifies calibration of temperature sensor 100. In embodiments, etalon 142 and retroreflector 144 are formed of silicon and silicon dioxide, respectively. In such embodiments, a₂ = 2.6 x 10^-6 K ¹, a₄ = 5.7 x 10^-7 K ¹, and at near- infrared wavelengths, D_Tn₂ = 1.9 x 10^-4 K^~ and D_Tn₄ = 1.3 x 10^-5 K ¹. In embodiments, optical cavity 140 does not include etalon 142, and includes retroflector 144, which is formed of silicon.

[0043] In embodiments, light source 111 is optically coupled to port 121 and generates input optical signal 115. Light source 111 is configured to generate at least one of (i) a first optical signal 115(1) having a free-space wavelength l₁ and (ii) a second optical signal having a free-space wavelength l₂. Input optical signal 115 is one of first optical signal 115(1) and second optical signal 115(2). Etalon 142 has a first free-spectral range at free-space wavelength l and a second free-spectral range at free-space wavelength l₂. In embodiments, each of free-space wavelength l and free-space wavelength l₂ are between 1.0 pm and 1.6 pm. In an embodiment, l₄ = 1.35 ± 0.05 pm and l₂ = 1.56 ± 0.05 pm.

[0044] In embodiments, each of first and second cavity signals 170 and 180 is a respective time-series of signal amplitudes that include resonant responses of optical cavity 140 at free-space wavelength l and free-space wavelength l₂ respectively. Memory 156 may store at least one of a wavelength time-series 172 and 182. Temperature-sensing processor 160 may receive wavelength time-series 172 and 182 from wavemeter 108 as measured wavelengths 109.

[0045] In embodiments of temperature sensor 100 that do not include tap 106 and wavemeter 108, laser 114 and laser driver 112 have been previously calibrated, via use of a wavemeter to generate a wavelength map 153 that maps values of drive signal 113 to respective center wavelengths of laser 115. Hence, at a time t, wavelength A(t) of laser 114 may be derived from the value of drive signal 113 and wavelength map 153. In such embodiments, wavelength time-series 172 and 182 are mapped wavelengths. In embodiments, software 161, e.g., wavelength mapper 162, determines wavelength time-series 172 and 182 from drive signal 113 and wavelength map 153. Memory 156 may store wavelength map 153.

[0046] Wavelength time-series 172 includes a plurality of timestamps each paired with a respective wavelength of input optical signal 115(1), and each respective wavelength is in a narrow spectral band Dl that includes free-space wavelength A₁. Wavelength time-series 182 includes a plurality of timestamps each paired with a respective wavelength of input optical signal 115(2), and each respective wavelength is in a narrow spectral band Dl₂ that includes free-space wavelength l₂. In embodiments, a width of each narrow spectral bands Dl and Dl₂ is less than one nanometer. In embodiments, wavelength mapper 162 converts cavity signals 170, 180 to respective spectral responses 174 and 184 by mapping each signal amplitude of cavity signals 170, 180 to a wavelength of respective wavelength time-series 172 and 182.

[0047] FIG. 4 is a graphical representation of measured cavity signals 470 and 480 of a compound optical cavity interrogated by light source 111 while being temporally -tuned about respective laser center wavelengths A and A₂. The compound optical cavity is an example of optical cavity 140, where etalon 142 is a 3 -mm thick silicon Fabry -Perot etalon. In this example, measured cavity signals 470 and 480 correspond to A_± = 1.349 pm and l = 1.559 pm, respectively, and the maximum tuning range is approximately DA = 0.5 nm. Measured cavity signals 470 and 480 are respective examples of unprocessed cavity signals for laser wavelengths A and A₂, and are examples of cavity signals 170 and 180 generated by electronics 152, FIG. 1.

[0048] FIG. 5 is a graphical representation of a computed spectral response 672 of a silicon etalon and a computed spectral response 682 of a silicon dioxide retroreflector. The silicon etalon and the silicon dioxide retroreflector are examples of etalon 142 and retroreflector 144 respectively. Spectral responses 572 and 582 are computed independently, that is, not with the silicon etalon and silicon dioxide retroreflector configured as in optical cavity 140. Spectral responses 572 and 582 correspond to temporal-tuning about wavelengths l and l₂, respectively. Spectral responses 572 and 582 are plotted as function of tuning magnitude 502, which ranges from zero to 0.6 nm. Spectral response 672 includes a plurality of resonances 573 and a free- spectral range 574, which are respective examples of resonance frequencies 176 and free-spectral range 177. Spectral response 682 includes a plurality of resonances 583 and a free-spectral range 584, which are respective examples of resonance frequencies 186 and free-spectral range 187.

[0049] Tuning magnitude 502 has a tuning range 506 equal to 0.6 nm. Within tuning range 506 of tuning magnitude 502, spectral response 672 (A_x = 1.349) includes seven resonances 573 while spectral response 682 (A₂ = 1.559 pm) includes five resonances 573. The beat frequency can be derived from this difference in the number of resonances within tuning range 506.

[0050] FIG. 6 is a graphical representation of spectral responses 672 and 682 and a beat- note signal 689, each as a function of a temperature 602. Beat-note signal 689 equals spectral response 672 divided by spectral response 682. In embodiments, temperature 602 is derived from tuning magnitude 502, FIG. 5, based on calibration of etalon 142 by setting etalon 142 to a plurality of known temperatures and tracking its resonant wavelengths. Spectral responses 672 and 682 are spectral responses 572 and 582 with tuning magnitude 502 mapped to a temperature 602.

[0051] FIG. 7 is a flowchart illustrating a temperature-measurement method 700. Method 700 includes steps 721, 722, 731, 732, and 740. In embodiments, method 700 also includes at least one of steps 711, 712, and 742. Method 700 may be implemented within one or more aspects of temperature sensor 100. In embodiments, method 700 is implemented by processor 154 executing computer-readable instructions of software 161.

[0052] Step 711 includes interrogating a compound optical cavity with a temporally - tuned laser operating at a first resonance wavelength. The compound optical cavity includes a retroreflecting optical cavity in series with an etalon having a first free-spectral range at the first resonance wavelength and a second free-spectral range at a second resonance wavelength. In an example of step 711, light source 111 operating at free-space wavelength A₁ interrogates optical cavity 140. The temporal tuning of light source 111 modulates free-space wavelength l by a tuning amplitude Dl, which in embodiments ranges from -0.6 nm to +0.6 nm during interrogation of optical cavity 140.

[0053] Step 712 includes interrogating the compound optical cavity with a temporally- tuned laser operating at a second resonance wavelength. The temporally -tuned laser of step 712 may be the same laser of step 711. In an example of step 712, light source 111 operating at free- space wavelength l₂ interrogates optical cavity 140. The temporal tuning of light source 111 modulates free-space wavelength l₂ by a tuning amplitude Dl, which in embodiments ranges from -0.6 nm to +0.6 nm during interrogation of optical cavity 140. In the following description, temperature-sensitive signals 147(1) and 147(2) refers to temperature-sensitive signals 147 generated in response to steps 711 and 712, respectively.

[0054] Step 721 includes detecting a first cavity signal output by the compound optical cavity in response to the interrogation of step 711. In an example of step 721, optical sensor 116 detects temperature-sensitive signal 147(1), which electronics 152 receives and therefrom produces first cavity signal 170 stored in memory 156.

[0055] Step 722 includes detecting a second cavity signal output by the compound optical cavity in response to the interrogation of step 712. In an example of step 722, optical sensor 116 detects temperature-sensitive signal 147(2), which electronics 152 receives and therefrom produces second cavity signal 180 stored in memory 156.

[0056] In embodiments, a difference Dh between the refractive index of the optical cavity at the first resonance wavelength and the second resonance wavelength is at least 0.01, such that the first and second cavity signals are non-trivially different. This difference facilitates determination of cavity temperature 188 described in subsequent steps of method 700.

[0057] Step 731 includes recovering a first spectral response of the etalon from the first cavity signal. The first spectral response includes a first plurality of resonances spectrally separated by a first free-spectral range. In an example of step 731, wavelength mapper 162 recovers spectral response 672, FIG. 6, from cavity signal 470, FIG. 4.

[0058] Step 732 includes recovering a second spectral response of the etalon from the second cavity signal. The second spectral response includes a second plurality of resonances spectrally separated by a second free-spectral range. In an example of step 732, wavelength mapper 162 recovers spectral response 682, FIG. 6, from cavity signal 480, FIG. 4. [0059] Step 740 includes determining a temperature T of the optical cavity from the first spectral response and the second spectral response. In an example of step 740, solver 168 determines cavity temperature 188 of optical cavity 140 from spectral responses 174 and 184.

[0060] Step 740 may include step 742, in which the optical cavity has temperature- dependent refractive indices n ( T ) and n₂ ( T ) at the first and second resonance wavelengths respectively. Step 742 includes determining the temperature T as a temperature that minimizes a merit function that is an increasing function of both (i) a difference between the first spectral response and a first mathematical expression /_x (n₁(T)) and (ii) a difference between the second spectral response and a second mathematical expression ^(^( ))·

[0061] In embodiments, the first mathematical expression includes an Airy distribution in which an argument to a first periodic function includes n (T). In such embodiments, the second mathematical expression includes an Airy distribution in which an argument to a second periodic function includes n₂(T). Equation (3) is an example of an Airy distribution where the following terms are attributes of optical cavity 140: finesse T, geometric length , temperature-dependent refractive index n_k(T) (at A_fc), and resonance frequency spacing v_F = c/(2n_k(T)L).

[0062] When optical cavity 140 includes etalon 142, the mathematical expression used in step 742 is a product of Airy distributions corresponding to the aforementioned first cavity within etalon 142 and the aforementioned second cavity within retroreflector 144.

[0063] The embodiments of systems and methods described herein may be implemented in hardware, software, firmware, and any combination thereof. As such, it should be appreciated that the functionality disclosed herein, unless otherwise indicated, may be implemented via computer-readable instructions that, when executed by one or more processors, operate to cause the functionality disclosed herein. The hardware, software, and/or firmware may operate to utilize and analyze data captured by, or produced using, the components of the embodiments of the systems and methods disclosed herein.

[0064] FIG. 8 is a schematic of a temperature sensor 800, which is an embodiment of temperature sensor 100, FIG. 1. Temperature sensor 800 includes optical device 120 and optical cavity 140. In embodiments, temperature sensor 100 also includes at least one of optoelectronics 810, collimator optics 130, and a temperature-sensing processor 860. Optoelectronics 810 is an example of optoelectronics 110 and includes a light source monitor 801, which is an example of light source monitor 101. Optical sensor 116 receives a cavity signal 847 and therefrom produces a detected cavity signal 857. Cavity signal 847 is an example of temperature-sensitive signal 147. Optical cavity 840 is an example of optical cavity 140 that includes retroreflector 144 and etalon 142.

[0065] Temperature-sensing processor 860 includes electronics 152, processor 154, and a memory 856, which is an example of memory 156. Processor 154 is communicatively coupled to memory 856 and may be communicatively coupled to electronics 152. Electronics 152 is communicatively coupled to memory 856. Memory 856 stores input values 850, software 861, and intermediate outputs 870.

[0066] Input values 850 include tap signal 103, drive signal 113, and detected cavity signal 857, each of which is received by electronics 152. Input values 850 also include wavelength map 153 and, for the material that forms etalon 142, temperature-dependent refractive index data 854 and temperature-dependent length data 856. FIG. 8 represents data 854 and 856 as n(T) and L(T) respectively. In embodiments, temperature-dependent refractive index data 854 includes one of a look-up table of temperature-dependent refractive index data and coefficients of an equation for temperature-dependent refractive index, such as a temperature- dependent Sellmeier equation. In embodiments, temperature-dependent length data 856 includes a linear thermal expansion coefficient.

[0067] Software 861 includes a wavelength mapper 865, an FSR extractor 866 and a solver 868. Intermediate outputs 870 include a normalized cavity signal 872, a filtered cavity signal 874, mapped wavelengths 875, and a free-spectral range 876. Processor 154 executes machine-readable instructions of software 861 to generate intermediate outputs 870, and ultimately a cavity temperature 878, from input values 850.

[0068] In embodiments, input values 850, software 861, and intermediate outputs 870 respectively include tap signal 103, a normalizer 862, and a normalized cavity signal 872 output by normalizer 862. Input values 850 may also include an insertion loss 105 of optical tap 102. In embodiments, normalizer 862 computes normalized cavity signal 872 as cavity signal 847 divided by the product of tap signal 103 and insertion loss 105.

[0069] In embodiments, input values 850, software 861, and intermediate outputs 870 include a nominal cavity length 852, filter 864, and filtered cavity signal 874, respectively. Nominal cavity length 852 is the geometric thickness of etalon 142 at a known temperature, such as an ambient temperature when burner 192 is off. Filter 864 computes filtered cavity signal 874 from normalized cavity signal 872 and nominal cavity length 852. Filtered cavity signal 874 is a low-pass filtered version of normalized cavity signal 872, where the cutoff frequency of the low- pass filter is higher than an expected free-spectral range of the etalon having nominal cavity length 852, and lower than smaller free-spectral ranges, e.g., of etalons with shorter cavity lengths.

[0070] Mapper 865 computes mapped wavelengths 875 from drive signal 113 and wavelength map 153. FSR extractor 866 computes free-spectral range 876 from filtered cavity signal 874 and mapped wavelengths 875. In embodiments, FSR extractor 866 computes free- spectral range 876 via one or more of the following methods as applied to filtered cavity signal 874: fringe-counting, Fourier analysis, a least-squared fit to a sinusoid, and other methods known in the art.

[0071] In embodiments, FSR extractor 866 maps each data point of filtered cavity signal 874 to a wavelength of mapped wavelengths 875. In such embodiments, filtered cavity signal 874 includes a plurality of signal magnitudes each paired with a respective wavelength value of mapped wavelengths 875. For example, each of mapped wavelengths 875 and filtered cavity signal 874 may be a time series of mapped wavelengths and cavity signals, respectively, such that values of filtered cavity signal 874 associated with a measurement times {t , t₂, t₃, ... } can be mapped to a respective wavelength value of mapped wavelengths 875 measured at the respective measurement times {t , t₂, t₃, ... }.

[0072] Expressed in units of length, the free-spectral range (FSR) of an optical cavity is FSR = A_Q/2II(T)L(T), hereinafter equation (1). In equation (1), A₀ is a free-space wavelength, n(T) is the temperature-dependent refractive index of the medium between the high-reflectors of the optical cavity, and L(T) is the temperature-dependent length of the optical cavity. Inverting equation (1) yields n(T)L(T ) = A_Q/(2 FSR), hereinafter equation (2). For candidate optical materials of etalon 142 and retroreflector 144, such as fused silica and silicon, n(T) and L(T) are monotonic and may be expressible in closed form. In embodiments, at least one of (i) n(T) is a temperature-dependent Sellmeier equation and (ii) L(T) includes a linear thermal expansion coefficient, where each is applied to the material that forms retroreflector 144.

[0073] Solver 868 determines a cavity temperature 878 of optical cavity 840 from free- spectral range 876, temperature-dependent refractive index data 854, and temperature-dependent length data 856. In embodiments, solver 868 determines cavity temperature 878 from equation (2) by the following steps (a) extracting measured FSR at free-space wavelength A₀ of plot 1100 as described above, (b) in equation (2), expressing n(T ) and L(T) by their respective functions, (c) solving equation (2) for temperature T, e.g., either analytically or numerically.

[0074] FIG. 9 is a time-series plot 900 of a tap signal 903, a tap signal 907, and a detected cavity signal 957 as measured in an embodiment of temperature sensor 100. In this embodiment, optical cavity 840 includes an etalon 942 a corner-cube retroreflector 944 and , which are respective examples of etalon 142 and retroreflector 144. Corner-cube retroreflector has two corner-adjacent surfaces 943 and a corner-opposite surface 945, which is an example of planar front-face 145 of retroreflector 144. Etalon 942 and corner-cube retroreflector 944 have respective (room temperature) thicknesses of 2.0 millimeters and 6.1 millimeters in a direction perpendicular to surface 945. Etalon 942 and corner-cube retroreflector 944 are formed of silicon and fused silica respectively.

[0075] Signals 903, 907, and 957 were measured over a single scanning cycle through a wavelength range of light source 111. Signals 903, 907, and 957 are respective examples of tap signal 103, tap signal 107, and detected cavity signal 857. For each signal 903, 907, and 957, plot 900 includes four thousand data points each acquired at a respective one of four thousand sample times 912 that span a time-interval 910 equal to two milliseconds.

[0076]

[0077] FIG. 10 is a plot 1000 of a normalized cavity signal 1072 as a function of a detuning magnitude 1010. Each detuning magnitude 1010 is an example of a measured wavelength of mapped wavelengths 875. The vertical axis of plot 1000 is reflectivity.

Normalized cavity signal 1072 is an example of normalized cavity signal 872, FIG. 8, and was obtained from detected cavity signal 957 by (a) dividing detected cavity signal 957 by tap signal 903, and (b) mapping the horizontal axis of plot 900 to mapped wavelengths 875. That is, wavelength mapper 865 tracked the center wavelength (or equivalently detuning magnitude 1010, FIG. 10) of light source 111 at which each data point of signals 903, 907, and 957 were measured.

[0078] The FSR of etalon 942 can be derived from plot 1000 by filtering out high- frequency interference fringes of normalized cavity signal 1072, which arise from a U-shaped optical path 946 within retroreflector 944, illustrated in FIG. 9, that has a first end at corner- opposite surface 945, changes directions at each of surfaces 943 , and has a second and at corner- opposite surface 945. U-shaped optical path 946 is an example of U-shaped optical path 146. The desired FSR (or primary FSR) is that of etalon 942.

[0079] FIG. 11 is a plot 1100 of a filtered cavity signal 1174 as a function of detuning magnitude 1010. As in plot 1000, the vertical axis of plot 1000 is reflectivity. Filtered cavity signal 1174, an example of filtered cavity signal 874, is a low-pass filtered version of normalized cavity signal 1072, where the cutoff frequency is higher than an expected free-spectral range of etalon 942, and lower than the smaller free-spectral range corresponding to U-shaped optical path 946.

[0080] FIG. 11 denotes a one-nanometer wide detuning range 1110 between detuning values of 1.0 nm and 2.0 nm. Filtered cavity signal 1174 has an FSR 1176 within detuning range 1110 equal to 0.1 nm, and l₀ is the center wavelength of light source 111 with zero detuning plus 1.5 nanometers - the center of wide detuning range 1110. FSR 1176 may be derived by Fourier analysis of filtered cavity signal 1174 or a least-squared fit of filtered cavity signal 1174 to a sinusoid.

[0081] FIG. 12 is a plot 1200 of free-spectral ranges of corner-cube reflector 944 as a function of its temperature. Plot 1200 includes FSR values 1210 and FSR values 1220. FSR values 1210 are extracted directly from filtered cavity signal 1174 via a means described above, for example. FSR values 1220 are computed using equation (1).

[0082] FIG. 13 is a flowchart illustrating a temperature-measurement method 1300. In embodiments, method 1300 is implemented within one or more aspects of temperature sensor 800. In embodiments, method 1300 is implemented by processor 154 executing computer- readable instructions of software 861. Method 1300 includes steps 1310, 1340, and 1350. In embodiments, method 1300 also includes at least one of steps 1320 and 1330. Without departing from the scope of the embodiments, steps 1320 and 1330 may be performed in a single step, and/or in either order.

[0083] Step 1310 includes generating a cavity signal by measuring optical power output by an optical cavity in response to interrogation by a laser being tuned through a plurality of resonance wavelengths of the optical cavity. In an example of step 1310, optical sensor 116 generates detected cavity signal 957, FIG. 9. [0084] Step 1320 includes normalizing the cavity signal by a wavelength dependent power of the laser to yield a normalized cavity signal. In an example of step 1320, normalizer 862 divides cavity signal 957 by a product of tap signal 103 and insertion loss 105 to yield normalized cavity signal 1072, FIG. 10.

[0085] Step 1330 includes low-pass filtering the cavity signal to yield an optical-cavity spectral response that includes a plurality of spectral-response values each derived from a respective one of the plurality of cavity-signal values. In an example of step 1330, filter 864 low- pass filters normalized cavity signal 1072 to yield filtered cavity signal 1174, FIG. 11.

[0086] Step 1340 includes determining, from the cavity signal, a free-spectral range of the optical cavity at a first resonance wavelength of the plurality of resonance wavelengths. In a first example of step 1340, FSR extractor 866 determines, from detected cavity signal 957, FSR 1176 within detuning range 1110, FIG. 11.

[0087] In embodiments, step 1310 includes step 1312 and 1314, and step 1340 includes step 1342. Step 1312 includes measuring, at each of plurality of sample times during a sampling interval, a respective one of a plurality of cavity-signal values of the cavity signal. In an example of step 1313, temperature sensor 800 measures, at each sample time 912 in time-interval 910, a respective cavity-signal value of detected cavity signal 957, FIG. 9.

[0088] Step 1314 includes determining, from a drive signal applied to the laser, a wavelength of the laser at each of the plurality of sample times to yield a plurality of mapped wavelengths. In an example of step 1314, mapper 865 determines the wavelength of laser 114 at each sample time 912 from wavelength map 153 and drive signal 113 to yield detuning magnitudes 1010, FIG. 10.

[0089] Step 1342 includes determining the free-spectral range from the plurality of measured wavelengths and the plurality of cavity-signal values. In example of step 1342, FSR extractor 866 determines free-spectral range 876 from detuning magnitudes 1010 and detected cavity signal 957.

[0090] When method 1300 includes steps 1312, 1314, and 1330, step 1340 may include step 1344. Step 1344 includes determining the free-spectral range from the plurality of measured wavelengths and the plurality of spectral-response values. In example of step 1342, FSR extractor 866 determines free-spectral range 1176 from detuning magnitudes 1010 and filtered cavity signal 1174. [0091] Step 1350 includes determining a temperature of the optical cavity from the free- spectral range, the first resonance wavelength, a temperature-dependent refractive index of the optical cavity, and a temperature-dependent length of the optical cavity. In an example of step 1350, solver 868 determines cavity temperature 878 from free-spectral range 1176, temperature- dependent refractive index data 854, and temperature-dependent length data 856. In this example, temperature-dependent refractive index data 854 is that of silicon, and temperature- dependent length data 856 is that of corner-cube retroreflector 944.

[0092] Combinations of Features

[0093] Features described above as well as those claimed below may be combined in various ways without departing from the scope hereof. The following enumerated examples illustrate some possible, non-limiting combinations.

[0094] (Al) A temperature-measurement method includes a cavity-signal step, a free- spectral-range step, and a temperature-determination step. The cavity-signal step includes generating a cavity signal by measuring optical power output by an optical cavity in response to interrogation by a laser being tuned through a plurality of resonance wavelengths of the optical cavity. The free-spectral-range step includes determining, from the cavity signal, a free-spectral range of the optical cavity at a first resonance wavelength of the plurality of resonance wavelengths. The temperature-determination step includes determining a temperature of the optical cavity from the free-spectral range, the first resonance wavelength, a temperature- dependent refractive index of the optical cavity, and a temperature-dependent length of the optical cavity.

[0095] (A2) In embodiments of method (Al), generating a cavity signal includes (i) measuring, at each of plurality of sample times during a sampling interval, a respective one of a plurality of cavity-signal values of the cavity signal; and (ii) determining, from a drive signal applied to the laser, a wavelength of the laser at each of the plurality of sample times to yield a plurality of mapped wavelengths. In such embodiments, determining the free-spectral range includes determining the free-spectral range from the plurality of measured wavelengths and the plurality of cavity-signal values.

[0096] (A3) Embodiments of method (A2) further include low-pass filtering the cavity signal to yield an optical-cavity spectral response that includes a plurality of spectral-response values each derived from a respective one of the plurality of cavity-signal values. Such embodiments also include determining the free-spectral range comprising determining the free- spectral range from the plurality of measured wavelengths and the plurality of spectral-response values.

[0097] (Bl) A temperature measurement method includes a first detecting step, a first recovering step, a second detecting step, a second recovering step, and a temperature determining step. The first detecting step (a) includes detecting a first cavity signal output by an optical cavity in response to interrogation by a temporally -tuned laser operating at a first resonance wavelength. The optical cavity has a first free-spectral range at the first resonance wavelength and a second free-spectral range at a second resonance wavelength. The first recovering step includes recovering a first spectral response of the optical cavity from the first cavity signal. The first spectral response includes a first plurality of resonances spectrally separated by a first free-spectral range. The second detecting step includes detecting a second cavity signal output by the optical cavity in response to interrogation a temporally -tuned laser operating at the second resonance wavelength. The second recovering step includes recovering a second spectral response of the optical cavity from the second cavity signal. The second spectral response includes a second plurality of resonances spectrally separated by the second free- spectral range. The temperature-determining step includes determining a temperature T of the optical cavity from the first spectral response and the second spectral response.

[0098] (B2) In embodiments of method (Bl), the optical cavity has temperature- dependent refractive indices n ( T ) and n₂ ( T ) at the first and second resonance wavelengths respectively, and step (a) includes determining the temperature T as a temperature that minimizes a merit function that is an increasing function of both (i) a difference between the first spectral response and a first mathematical expression /_x (u₁(T)) and (ii) a difference between the second spectral response and a second mathematical expression ^(^(O)·

[0099] (B3) In embodiments of method (B2), the first mathematical expression including an Airy distribution in which an argument to a first periodic function includes n (T), the second mathematical expression including an Airy distribution in which an argument to a second periodic function includes n₂(T).

[0100] (B4) Embodiments of any one of methods (Bl) - (B3) further include interrogating the optical cavity with the temporally -tuned laser operating at the first resonance wavelength; and interrogating the optical cavity with the temporally -tuned laser operating at the second resonance wavelength.

[0101] (Cl) A temperature sensor includes an optical device and a retroreflector. The optical device includes a first port, a second port, and a third port. The optical device is configured to: receive a first input optical signal at the first port, output the first input optical signal at the second port, receive a second input optical signal at the second port, and output the second input optical signal at the third port. The retroreflector includes a planar front-face configured to reflect the first input optical signal output from the second port as the second input optical signal along an optical path between the second port and the retroreflector.

[0102] (C2) Embodiments of temperature sensor (Cl) further include an off-axis collimator along the optical path and configured to optically couple the second input optical signal to the second port.

[0103] (C3) Embodiments of either of one temperature sensors (Cl) and (C2), further include an etalon between the retroreflector and the second port along the optical path, including a second planar surface adjoining the planar front-face, and having a refractive index n₂,

\n₂ — ^hi I > 1_> where nl is the refractive index of the retroreflector.

[0104] (C4) In embodiments of temperature sensor (C3), the etalon is a Fabry-Perot etalon.

[0105] (C5) In embodiments of any one of temperature sensors (C3) - (C4), the refractive index n₂ exceeds (n₁ + 1).

[0106] (C6) In embodiments of any of temperature sensor (C3) - (C5), the refractive indices n and n₂ have respective temperature dependencies D_Tn and D_Tn₂, wherein one of D_rn₁/D_rn₂ > 8 and D_rn₂/£>_7’n₁ > 8.

[0107] (C7) In embodiments of temperature sensor (C3), D_Tn₂/D_Tn exceeds 8.

[0108] (C8) In embodiments of any one of temperature sensors (Cl) - (C7), the optical device being one of an optical circulator, a l-by-2 optical fiber coupler, and a 2-by-2 optical fiber coupler.

[0109] (C9) Embodiments of any one of temperature sensors (Cl) - (C8) further include: at least one laser optically coupled to the first port and configured to generate (i) the first input optical signal, (ii) a first optical signal having a first free-space wavelength A₁, (iii) a second optical signal having a second free-space wavelength l_2. The first input optical signal is one of the first optical signal and the second optical signal.

[0110] (CIO) Embodiments of any one of temperature sensors (Cl) - (C9) further include an optical sensor optically coupled to the third port configured to quantify the second input optical signal received from the third port.

[0111] (Cl 1) In embodiments of any one of temperature sensors (Cl) - (CIO), the retroreflector is formed of one of silicon, sapphire, magnesium aluminate spinel, aluminum oxynitride, fused silica, and a combination thereof.

[0112] (C12) In embodiments of any one of temperature sensors (Cl) - (Cl 1), the retroreflector is formed of a material having a melting point exceeding 700 °C.

[0113] (C13) Embodiments of any one of temperature sensors (Cl) - (C12) further include an optical cavity that includes the retroreflector; a processor; and a memory. The memory includes non-transitory machine-readable instructions that, when executed by the processor, control the processor to execute any one of methods (Al) - (A4)

[0114] (C14) Embodiments of any one of temperature sensors (Cl) - (C12) further include an optical cavity that includes the retroreflector; a processor; and a memory. The optical cavity includes the retroreflector and has a first free-spectral range at a first resonance wavelength and a second free-spectral range at a second resonance wavelength. The memory includes non-transitory machine-readable instructions that, when executed by the processor, control the processor to execute any one of methods (Bl) - (B4).

[0115] (Dl) A temperature sensor includes an optical cavity, a processor, and a memory. The optical cavity includes a retroreflector and having a first free-spectral range at a first resonance wavelength and a second free-spectral range at a second resonance wavelength. The memory includes non-transitory machine-readable instructions that, when executed by the processor, control the processor to execute any one of methods (Bl) - (B4).

[0116] (D2) In embodiments of temperature sensor (Dl), the optical cavity includes an etalon in series with the retroreflector, the first free-spectral range the second free-spectral range being respective free-spectral ranges of the etalon.

[0117] (D3) In embodiments of temperature sensor of (D2), retroreflector has a refractive index n , the etalon having a refractive index n₂, \n₂ — n₁\ > \ . [0118] (D4) Embodiments of any one of temperature sensors (Dl) - (D3) further include an optical device including a first port, a second port, and a third port. The optical device is configured to: receive a first input optical signal at the first port, output the first input optical signal at the second port, receive a second input optical signal at the second port, and output the second input optical signal at the third port. The retroreflector includes a planar front-face configured to reflect the first optical signal output by the second port as the second input optical signal along an optical path between the second port and the retroreflector.

[0119] (D5) Embodiments of temperature sensor (D4) further includes an etalon located between the retroreflector and the second port along the optical path. The etalon includes a second planar surface adjoining the planar front-face.

[0120] Changes may be made in the above methods and systems without departing from the scope of the present embodiments. It should thus be noted that the matter contained in the above description or shown in the accompanying drawings should be interpreted as illustrative and not in a limiting sense. Herein, and unless otherwise indicated the phrase “in embodiments” is equivalent to the phrase “in certain embodiments,” and does not refer to all embodiments. The following claims are intended to cover all generic and specific features described herein, as well as all statements of the scope of the present method and system, which, as a matter of language, might be said to fall therebetween.

Claims

CLAIMS What is claimed is:

1. A temperature sensor comprising: an optical device including a first port, a second port, and a third port; and being configured to: receive a first input optical signal at the first port, output the first input optical signal at the second port, receive a second input optical signal at the second port, and outputting the second input optical signal at the third port; and a retroreflector including a planar front-face configured to reflect the first input optical signal output from the second port as the second input optical signal along an optical path between the second port and the retroreflector.

2. The temperature sensor of claim 1, the retroreflector having a refractive index n , and further comprising: an etalon between the retroreflector and the second port along the optical path, including a second planar surface adjoining the planar front- face, and having a refractive index n₂, I n₂ - n > 1.

3. The temperature sensor of claim 2, the etalon being a Fabry-Perot etalon.

4. The temperature sensor of claim 2, the refractive index n₂ exceeding (n + 1).

5. The temperature sensor of claim 2, the refractive indices n and n₂ having respective temperature dependencies D_Tn and D_Tn₂, wherein one of D_Tn /D_Tn₂ > 8 and

6. The temperature sensor of claim 5, wherein D_Tn₂/D_Tn > 8.

7. The temperature sensor of claim 1, further comprising: at least one laser optically coupled to the first port and configured to generate (i) the first input optical signal, (ii) a first optical signal having a first free-space wavelength l_c, (iii) a second optical signal having a second free-space wavelength l₂, the first input optical signal being one of the first optical signal and the second optical signal.

8. The temperature sensor of claim 1, further comprising an optical sensor optically coupled to the third port configured to quantify the second input optical signal received from the third port.

9. The temperature sensor of claim 1, the retroreflector being formed of one of silicon, sapphire, magnesium aluminate spinel, aluminum oxynitride, fused silica, and a combination thereof.

10. A temperature sensor of claim 1 , the retroreflector being formed of a material having a melting point exceeding 700 °C.

11. A temperature sensor of claim 1 , the optical device being one of an optical circulator, a 1 - by-2 optical fiber coupler, and a 2-by-2 optical fiber coupler.

12. The temperature sensor of claim 1, further comprising: an optical cavity that includes the retroreflector; a processor; and a memory including non-transitory machine-readable instructions that, when executed by the processor, control the processor to: generate a cavity signal by measuring optical power output by the optical cavity in response to interrogation by a laser being tuned through a plurality of resonance wavelengths of the optical cavity; determine, from the cavity signal, a free-spectral range of the optical cavity at a first resonance wavelength of the plurality of resonance wavelengths; and determine a temperature of the optical cavity from the free-spectral range, the first resonance wavelength, a temperature-dependent refractive index of the optical cavity, and a temperature-dependent length of the optical cavity.

13. The temperature sensor of claim 1, further comprising: an optical cavity that includes the retroreflector and having a first free-spectral range at a first resonance wavelength and a second free-spectral range at a second resonance wavelength; a processor; and a memory including non-transitory machine-readable instructions that, when executed by the processor, control the processor to: detect a first cavity signal output by an optical cavity in response to interrogation by a temporally-tuned laser operating at a first resonance wavelength, the optical cavity having a first free-spectral range at the first resonance wavelength and a second free-spectral range at a second resonance wavelength; recover a first spectral response of the optical cavity from the first cavity signal, the first spectral response including a first plurality of resonances spectrally separated by a first free-spectral range; detect a second cavity signal output by the optical cavity in response to interrogation a temporally-tuned laser operating at the second resonance wavelength; recover a second spectral response of the optical cavity from the second cavity signal, the second spectral response including a second plurality of resonances spectrally separated by the second free-spectral range; and determine a temperature T of the optical cavity from the first spectral response and the second spectral response.

14. The temperature sensor of claim 13, the optical cavity including an etalon in series with the retroreflector, the first free-spectral range the second free-spectral range being respective free-spectral ranges of the etalon.

15. A temperature-measurement method comprising: generating a cavity signal by measuring optical power output by an optical cavity in response to interrogation by a laser being tuned through a plurality of resonance wavelengths of the optical cavity; determining, from the cavity signal, a free-spectral range of the optical cavity at a first resonance wavelength of the plurality of resonance wavelengths; and determining a temperature of the optical cavity from the free-spectral range, the first resonance wavelength, a temperature-dependent refractive index of the optical cavity, and a temperature-dependent length of the optical cavity.

16. The temperature- measurement method of claim 15, generating comprising: (i) measuring, at each of plurality of sample times during a sampling interval, a respective one of a plurality of cavity-signal values of the cavity signal; and (ii) determining, from a drive signal applied to the laser, a wavelength of the laser at each of the plurality of sample times to yield a plurality of mapped wavelengths; and determining the free-spectral range comprising: determining the free-spectral range from the plurality of measured wavelengths and the plurality of cavity-signal values.

17. The temperature-measurement method of claim 16, further comprising: low-pass filtering the cavity signal to yield an optical-cavity spectral response that includes a plurality of spectral-response values each derived from a respective one of the plurality of cavity-signal values; and determining the free-spectral range comprising determining the free-spectral range from the plurality of measured wavelengths and the plurality of spectral-response values.

18. A temperature-measurement method comprising: detecting a first cavity signal output by an optical cavity in response to interrogation by a temporally-tuned laser operating at a first resonance wavelength, the optical cavity having a first free-spectral range at the first resonance wavelength and a second free- spectral range at a second resonance wavelength; recovering a first spectral response of the optical cavity from the first cavity signal, the first spectral response including a first plurality of resonances spectrally separated by a first free-spectral range; detecting a second cavity signal output by the optical cavity in response to interrogation a temporally-tuned laser operating at the second resonance wavelength; recovering a second spectral response of the optical cavity from the second cavity signal, the second spectral response including a second plurality of resonances spectrally separated by the second free-spectral range; and determining a temperature T of the optical cavity from the first spectral response and the second spectral response.

19. The method of claim 18, the optical cavity having temperature-dependent refractive indices n ( T ) and n₂ ( T ) at the first and second resonance wavelengths respectively, determining comprising: determining the temperature T as a temperature that minimizes a merit function that is an increasing function of both (i) a difference between the first spectral response and a first mathematical expression /_x (n₁(T)) and (ii) a difference between the second spectral response and a second mathematical expression (712(D)·

20. The method of claim 19, the first mathematical expression including an Airy distribution in which an argument to a first periodic function includes n (T), the second mathematical expression including an Airy distribution in which an argument to a second periodic function includes n₂(T).