A method and system for detecting the low-temperature thermal expansion coefficient of sapphire based on weak measurement

Abstract

This invention discloses a method and system for detecting the low-temperature thermal expansion coefficient of sapphire based on weak measurement, belonging to the field of precision measurement technology of material thermal properties. It applies weak measurement technology to the low-temperature thermal expansion measurement of bulk sapphire materials. By constructing an interference optical path containing the sapphire sample as the optical sensing element, a weak amplification mechanism is used to convert the slight optical path difference (or phase change) caused by temperature changes in sapphire into a significant observable shift in the spectral domain. Then, by measuring the change of the characteristic spectral wavelength with temperature, combined with the known temperature coefficient of refractive index of sapphire, its thermal expansion coefficient can be accurately determined. This invention enables non-contact, high-sensitivity measurement of the thermal expansion coefficient and has strong interference resistance, making it suitable for precise measurement of the thermal expansion coefficient of sapphire and other transparent brittle materials across the entire temperature range from extremely low temperatures to room temperature.

Description

Technical Field

[0001] This invention relates to the field of precision measurement technology of material thermal properties. More specifically, this invention relates to an optical method for non-contact, ultra-sensitive measurement of the coefficient of thermal expansion of transparent optical materials such as sapphire (Al2O3) at low temperatures, specifically to a method and system for detecting the low-temperature coefficient of thermal expansion of sapphire based on weak measurement. Background Technology

[0002] Sapphire, due to its excellent high-temperature stability, high hardness, wide spectral transmittance, and good dielectric properties, is widely used in extreme environments such as aerospace windows, semiconductor substrates, high-power laser components, and low-temperature physics experimental devices. Its coefficient of thermal expansion is a key parameter involved in thermo-structural design, especially in low-temperature regions (such as liquid helium and liquid nitrogen temperatures). Accurate thermal expansion data is crucial for ensuring the stability of optical systems, avoiding thermal stress damage, and achieving precision assembly. Currently, methods for measuring the low-temperature thermal expansion coefficient of materials mainly include interferometric dilatation, capacitive sensor methods, and strain gauge methods. Interferometry (such as Fizeau interferometers and laser interferometers) offers high accuracy, but at low temperatures, environmental vibrations, thermal drift, and the difficulty in interpreting interference fringes limit its sensitivity and stability. Capacitive methods require the sample to be part of the capacitor plates, which is a contact measurement, potentially introducing additional stress and requiring specific sample shapes. Strain gauge methods are also contact-based, and the bonding process and the behavior of adhesives at low temperatures can introduce systematic errors. Therefore, existing low-temperature thermal expansion measurement technologies suffer from technical problems such as contact interference, insufficient sensitivity, and weak resistance to environmental vibrations.

[0003] Weak measurement technology is an ultrasensitive optical measurement technique based on the concepts of pre-selection and post-selection in quantum mechanics. By weakly coupling the measuring instrument to the system under test and then performing near-orthogonal post-selection on the system state, it can amplify minute physical quantities (such as phase and displacement) by tens to hundreds of times, thereby significantly improving the signal-to-noise ratio and measurement sensitivity. This technology has been successfully applied to the precision measurement of physical quantities such as beam deflection, phase measurement, and frequency estimation.

[0004] Therefore, in principle, applying weak measurement techniques to the measurement of the thermal expansion coefficient of materials, especially for transparent materials like sapphire where optical displacement changes are minimal at low temperatures, has unique advantages. It eliminates the need to fabricate sensitive microstructures on the sample, avoiding processing damage; it utilizes non-contact optical measurement, eliminating the influence of contact stress; and its weak value amplification characteristics effectively suppress low-frequency noise, making it particularly suitable for extracting weak signals in low-temperature environments.

[0005] However, there is currently no publicly available scheme that specifically uses weak measurement technology for the precise measurement of the low-temperature thermal expansion coefficient of sapphire bulk materials, which means that there is still a technical bottleneck in the precise measurement of the low-temperature thermal expansion coefficient. Summary of the Invention

[0006] One object of the present invention is to solve at least the above-mentioned problems and / or defects, and to provide at least the advantages described below.

[0007] To achieve these objectives and other advantages of the present invention, a method for detecting the low-temperature thermal expansion coefficient of sapphire based on weak measurement is provided, comprising:

[0008] S1. Construct a weak measurement optical system, and place the sapphire sample to be measured in the measurement optical path corresponding to the weak measurement optical system.

[0009] S2, Based on sample temperature change The thickness variation will affect the optical path length of light propagating in the sample. If the time delay of light propagation changes, then... It is characterized by the following formula:

[0010]

[0011] In the above formula, c The speed of light in a vacuum. n For sapphire l The refractive index at 0, α The coefficient of thermal expansion is... d 0 represents the initial thickness of the sapphire sample;

[0012] S3. Turn on the superluminescent diode light source, adjust the optical path, and establish the following center wavelength offset. With time delay Relationship:

[0013]

[0014] In the above formula, l 0 represents the initial center wavelength of the light source. s Indicates spectral width, A w For the weak value of a weak measurement optical system, Im[ A w [] indicates taking the imaginary part of the weak value;

[0015] S4. Based on the effect of temperature on refractive index, using the following formula... α Characterizing the coefficient of thermal expansion and center wavelength shift Relationship:

[0016]

[0017] In the above formula, The change in temperature T Indicates temperature in Celsius. Indicates the thermoluminescence coefficient of sapphire;

[0018] S5. Start the low-temperature thermostat and cool down in a step-by-step manner. Record the temperature T at each set point after the temperature has fully stabilized. i corresponding l i ;

[0019] S6. Calculate the wavelength offset Δ at each temperature point relative to the initial temperature. l i Based on the relational formulas in S5, the sapphire's properties are calculated. The average thermal expansion coefficient α within the range T ).

[0020] Preferably, in S3, A w Weak values ​​in weak measurement optical systems A w It is characterized by the following formula:

[0021]

[0022] In the above formula, For system pointers, Let I be the wave function after pre-selection processing, and It is characterized by the following formula:

[0023]

[0024] In the above formula, , Let represent the two components of the incident light after preselective decomposition, and let ω represent the angular frequency. This represents the spectrum of a light source with a Gaussian distribution. e Indicates the pre-selection angle, i Represents the imaginary unit. The wave function II is the result of post-selection processing, and It is characterized by the following formula:

[0025]

[0026] In the above formula, t Let d represent the total time delay of the system, and d represent the derivative.

[0027] Preferably, in S3, the initial center wavelength of the light source is... l The method to obtain 0 is as follows:

[0028] At an initial temperature T0, the phase compensator of the weak measurement optical system is adjusted to make the spectrometer output a symmetrical double peak of interference spectrum, thereby obtaining the initial center wavelength. l 0.

[0029] Preferably, in S5, the stepped cooling method refers to gradually changing the temperature of the sapphire sample from T0 to T0. n .

[0030] Preferably, in S6, Δ l i It is characterized by the following formula:

[0031] Δ l i = l i - l 0.

[0032] A sapphire low-temperature thermal expansion coefficient testing system includes:

[0033] A light source that outputs a smooth spectrum;

[0034] A sapphire sample to be tested is placed inside a cryostat, and the sapphire sample to be tested has a parallel plane.

[0035] A weak measurement module is constructed using a pre-selective polarizer, a Soleil-Babinet phase compensator, and a post-selective polarizer, wherein the transmission axis of the post-selective polarizer is set to be approximately orthogonal to the transmission axis of the pre-selective polarizer.

[0036] A high-resolution spectrometer is set downstream of the post-selective polarizer to receive interference light;

[0037] The output light from the light source passes through the optical window and is perpendicularly incident into the vacuum chamber. After being perpendicularly reflected by mirror I, it enters the plane on one side of the sapphire sample.

[0038] Light transmitted through the other side of the sapphire sample is vertically reflected by mirror II and output to the post-selective polarizer.

[0039] The preselective polarizer and Soleil-Babinet phase compensator are positioned between the light source and the optical glass.

[0040] Preferably, the light source is a superluminescent diode;

[0041] The sapphire sample to be tested is a cylindrical sapphire made of c-cut or a-cut material;

[0042] The sapphire sample to be tested has a diameter of approximately 10 mm and a height of approximately 50 mm.

[0043] The parallel surfaces of the sapphire sample to be tested are optically polished to ensure that the parallelism of the parallel planes is better than 10 arcseconds.

[0044] The present invention has at least the following beneficial effects:

[0045] Firstly, the detection method of this invention has extremely high sensitivity. When applied, the detection system can be used to amplify the small optical path changes caused by temperature, and the theoretical sensitivity can be 2-3 orders of magnitude higher than that of traditional interferometry. It is particularly suitable for sapphire materials with extremely small thermal expansion coefficients at low temperatures.

[0046] Secondly, the detection method of this invention is a non-contact, non-destructive measurement method. That is, this invention is entirely based on optical detection, does not contact the sample surface, and does not introduce additional stress or damage, thus ensuring the authenticity of the measurement and the usability of the sample.

[0047] Third, the present invention has excellent anti-interference ability. That is, the post-selection process in the weak measurement module of the present invention has a filtering effect, which can effectively suppress common-mode noise (such as light source intensity fluctuations and low-frequency vibrations) and improve the measurement stability in low-temperature complex environments.

[0048] Fourth, the sapphire sample preparation of the present invention is simple, that is, only the sapphire needs to be processed into a cylinder, without the need for complex micro-nano processing or electrode preparation, which reduces the difficulty and cost of sample preparation.

[0049] Fifth, the detection method of the present invention is applicable to a wide temperature range, that is, the principle of the method of the present invention is applicable to a wide temperature range from extremely low temperature at the mK level to room temperature, providing a powerful tool for the establishment of a full-temperature-range thermal expansion database for sapphire.

[0050] Other advantages, objectives and features of the present invention will become apparent in part from the following description, and in part from those skilled in the art through study and practice of the invention. Attached Figure Description

[0051] Figure 1 This is a block diagram of the sapphire low-temperature thermal expansion coefficient detection system based on weak sapphire according to the present invention.

[0052] Figure 2 This is a calibration diagram of the sapphire low-temperature thermal expansion coefficient detection system of the present invention;

[0053] Figure 3 This is a center wavelength shift diagram on the spectrum analyzer of the present invention;

[0054] Among them, there are superluminescent light-emitting diode light source-1, front selective polarizer-2, quarter-wave plate-3, phase compensator-4, reflector I-7, sample stage-8, second reflector II-9, rear selective polarizer-10, and high-resolution spectrometer-11. Detailed Implementation

[0055] The present invention will now be described in further detail with reference to the accompanying drawings, so that those skilled in the art can implement it based on the description.

[0056] This invention discloses a method for detecting the low-temperature thermal expansion coefficient of sapphire based on weak measurement. It mainly utilizes the optical path change caused by the sapphire sample itself in the weak measurement device as a sensing signal, and achieves high-precision detection of nanometer-level or even sub-nanometer-level length changes through a weak value amplification mechanism, thereby retrieving the high-precision thermal expansion coefficient.

[0057] Specifically, the sapphire low-temperature thermal expansion coefficient detection method of this invention uses the sapphire bulk material itself as the optical sensing element. It applies the weak-value amplification mechanism from weak measurement techniques to detect the optical phase change of the sapphire sample caused by temperature, thereby achieving non-contact, high-sensitivity measurement of its thermal expansion coefficient. This is achieved by constructing a weak measurement optical system and placing the sapphire sample to be tested within the measurement optical path of the weak measurement optical system, i.e., the entire sapphire sample to be tested is installed inside a low-temperature cryostat. Light emitted from a broadband light source enters the low-temperature cryostat after pre-polarization selection. After the light beam passes through the sapphire sample to be tested, its optical path is affected by the sample thickness (i.e., length). d 0) and refractive index n The combined effect. When the sample temperature changes by ∆T, its length changes by Δ. d = α× d 0×ΔT, while the refractive index changes Δ n = (d n The sum of ΔT and ΔT (dT / dT) leads to a slight phase change. This phase change is polarized and selected (the analyzer direction is nearly orthogonal to the pre-selected direction). Utilizing the weak-value amplification effect, this slight phase change is converted into a large shift Δλ in the center wavelength of the interference spectrum that can be clearly detected by the spectrometer. By accurately measuring Δλ at different temperatures and separating the contribution of refractive index changes, the length change purely caused by thermal expansion can be calculated, thus yielding the coefficient of thermal expansion. α (T).

[0058] Furthermore, the method for detecting the low-temperature thermal expansion coefficient of sapphire in this invention mainly includes the following processing steps:

[0059] (1) System initialization: At room temperature T0 (e.g., 300 K), the sapphire sample is placed into the thermostat.

[0060] A sapphire sample with parallel planes is placed on a sample stage and installed in a cryostat. When the superluminescent diode light source is turned on, the light source spectrum exhibits a Gaussian distribution. It is expressed as follows:

[0061] (1)

[0062] in, s Indicates spectral width, and , c The speed of light in a vacuum h This represents the full width at half maximum (FWHM) of the light source. l 0 is the initial center wavelength of the light source, and l 0=830 nm ω represents the angular frequency, ω0 represents the center angular frequency of the light source, and .

[0063] Adjust the optical path so that the light beam is incident on the sample from the right side. This is done by adjusting mirrors I and II to ensure the light is incident on the sample and that the outgoing light is parallel to the incident light. Adjust the linear polarizer so that its polarization direction makes an angle with the horizontal direction of [value missing]. ,in e =0.04° represents the size of the preselection angle. Adjust the quarter-wave plate 3 so that the angle between its fast axis direction and the horizontal direction is... The incident light is decomposed into mutually perpendicular components. , The preselective state preparation is completed, as shown below:

[0064] (2)

[0065] After the initial selection is completed, the wave function I of the system can be expressed as follows:

[0066] (3)

[0067] Then, the phase compensator is finely adjusted so that the angle between the polarization direction of the selected polarizer and the horizontal direction is... After completion, the wave function II after selection can be expressed as:

[0068] (4)

[0069] System Weakness A w Defined as:

[0070] (5)

[0071] Substituting into the above expression, we can obtain the following relationship between the center wavelength shift and the time delay:

[0072] (6)

[0073] Furthermore, establish the coefficient of thermal expansion. α The relationship with the shift of the center wavelength.

[0074] Sample temperature change Below, thickness change for:

[0075] (7)

[0076] In the above formula, d 0 represents the initial thickness of the sample.

[0077] This change in thickness directly alters the optical path difference of light propagating within the sample. :

[0078] (8)

[0079] Time delay for:

[0080] (9)

[0081] Substituting formula (9) into formula (6) yields the center wavelength offset. :

[0082] (10)

[0083] The coefficient of thermal expansion can be obtained by simplification. α With center wavelength shift The relationship is as follows:

[0084] (11)

[0085] If we consider the effect of temperature on the refractive index, the total time delay can be expressed as follows:

[0086] (12)

[0087] in, This indicates the thermoluminescence coefficient of sapphire.

[0088] Substituting formula (12) into formula (6) yields:

[0089] (13)

[0090] Considering the effect of temperature on the refractive index of sapphire, the final relationship between the coefficient of thermal expansion and the shift in the center wavelength can be obtained as follows:

[0091] (14)

[0092] in, When the temperature remained unchanged, the spectrometer observed the following: Figure 2The left-right symmetrical double-peak interference spectrum shown is obtained by adjusting the phase compensator at the initial temperature T0 to make the spectrometer output a symmetrical double-peak interference spectrum, and recording the center wavelength of the double peak at this time. l 0. When the temperature changes, the center wavelength shifts, and asymmetrical double peaks can be observed on the spectrometer, such as... Figure 3 As shown.

[0093] (2) Cooling and Data Acquisition: Start the cryostat and cool down in a step-by-step manner (even if the sapphire sample temperature changes from T0 to T in stages). n For example, 300K -> 200K - 100K -> 77K -> 4.2K). At each temperature setpoint, after the temperature has fully stabilized (fluctuation < 0.1 K), the interference spectrum on spectrometer 11 is recorded. Because temperature changes cause changes in the thickness and refractive index of the sapphire sample, they result in a system optical path difference. The change, after weak amplification, manifests as a shift in the center wavelength of the interference spectrum to... l 0. Record each temperature T i corresponding for .

[0094] (3) Data processing and coefficient calculation: Calculate the wavelength offset ∆ of each temperature point relative to the initial temperature. l i = l i - l 0. Then change Δ l i Substituting into formula (14) to calculate the coefficient of thermal expansion, and then by measuring continuous temperature points, the average coefficient of thermal expansion can be fitted. α The curve showing how the temperature changes with temperature T.

[0095] A sapphire low-temperature thermal expansion coefficient measurement system is proposed, which aims to apply weak measurement technology to the low-temperature thermal expansion measurement of sapphire bulk materials. By constructing an interference optical path containing the sapphire sample, the weak optical path difference (or phase change) caused by temperature change in sapphire is converted into a significant shift observable in the spectral domain using a weak value amplification mechanism.

[0096] In applications, the sapphire low-temperature thermal expansion coefficient measurement system uses the sapphire sample itself as the sensing element. This eliminates the need for internal microstructure fabrication, avoiding stress and defects introduced during processing. Therefore, from an optical path perspective, as... Figure 1As shown, the system mainly includes: a superluminescent diode light source 1, a front-selective polarizer 2, a quarter-wave plate 3, a phase compensator 4, a reflector I 7, a sample stage 8, a second reflector II 9, a back-selective polarizer (also known as a back-selective polarizer) 10, and a high-resolution spectrometer 11. The sapphire sample to be tested is placed on the sample stage inside a cryostat, which includes: an optical window 5 and a vacuum chamber 6.

[0097] In practical applications, a sapphire sample with parallel planes is placed on a sample stage and installed in a cryostat. The optical path is adjusted so that the beam is incident on the sample from the right. The sapphire sample is a C-cut or A-cut sapphire, optically polished on both sides, with a parallelism better than 10 arcseconds, and is cylindrical in shape with a diameter of 10 mm and a height of 50 mm. The weak measurement module in this scheme sets the pre-selective polarizer and the post-selective polarizer to be nearly orthogonal and finely adjusts the phase compensator angle to make the system operate in the weak value amplification region, thereby amplifying small optical phase changes into significant spectral shifts.

[0098] This system measures the change in spectral characteristic wavelengths with temperature and, combined with the known temperature coefficient of refractive index of sapphire, can accurately deduce its coefficient of thermal expansion. This invention offers advantages such as non-contact operation, high sensitivity, and strong anti-interference capabilities, making it suitable for precise measurement of the coefficient of thermal expansion of sapphire and other transparent, brittle materials across the entire temperature range from extremely low temperatures to room temperature. It is expected to have significant applications in fields such as superconducting materials, aerospace cryogenic structures, quantum devices, and precision optical systems.

[0099] The above solution is merely an illustration of a preferred example and is not limited thereto. When implementing this invention, appropriate substitutions and / or modifications can be made according to the user's needs.

[0100] Although embodiments of the present invention have been disclosed above, they are not limited to the applications listed in the specification and embodiments. It can be applied to various fields suitable for the present invention. Other modifications can be readily made by those skilled in the art. Therefore, without departing from the general concept defined by the claims and their equivalents, the present invention is not limited to the specific details and examples shown and described herein.

Claims

1. A method for detecting the low-temperature thermal expansion coefficient of sapphire based on weak measurement, characterized in that, include: S1. Construct a weak measurement optical system, and place the sapphire sample to be measured in the measurement optical path corresponding to the weak measurement optical system. S2, Based on sample temperature change The thickness variation will affect the optical path length of light propagating in the sample. If the time delay of light propagation changes, then... It is characterized by the following formula: In the above formula, c The speed of light in a vacuum. n For sapphire λ The refractive index at 0, α The coefficient of thermal expansion is... d 0 represents the initial thickness of the sapphire sample; S3. Turn on the superluminescent diode light source, adjust the optical path, and establish the following center wavelength offset. With time delay Relationship: In the above formula, λ 0 represents the initial center wavelength of the light source. σ Indicates spectral width, A w For the weak value of a weak measurement optical system, Im[ A w [] indicates taking the imaginary part of the weak value; S4. Based on the effect of temperature on refractive index, using the following formula... α Characterizing the coefficient of thermal expansion and center wavelength shift Relationship: In the above formula, The change in temperature T Indicates temperature in Celsius. Indicates the thermoluminescence coefficient of sapphire; S5. Start the low-temperature thermostat and cool down in a step-by-step manner. Record the temperature T at each set point after the temperature has fully stabilized. i corresponding λ i ; S6. Calculate the wavelength offset Δ at each temperature point relative to the initial temperature. λ i Based on the relational formula in S4, the sapphire's properties are calculated. The average thermal expansion coefficient α within the range T ).

2. The method for detecting the low-temperature thermal expansion coefficient of sapphire based on weak measurement as described in claim 1, characterized in that, In S3, A w Weak values ​​in weak measurement optical systems A w It is characterized by the following formula: In the above formula, For system pointers, Let I be the wave function after pre-selection processing, and It is characterized by the following formula: In the above formula, , Let represent the two components of the incident light after preselective decomposition, and let ω represent the angular frequency. This represents the spectrum of a light source with a Gaussian distribution. ε Indicates the pre-selection angle, i Represents the imaginary unit. The wave function II is the result of post-selection processing, and It is characterized by the following formula: In the above formula, τ Let d represent the total time delay of the system, and d represent the derivative.

3. The method for detecting the low-temperature thermal expansion coefficient of sapphire based on weak measurement as described in claim 1, characterized in that, In S3, the initial center wavelength of the light source λ The method to obtain 0 is as follows: At an initial temperature T0, the phase compensator of the weak measurement optical system is adjusted to make the spectrometer output a symmetrical double peak of interference spectrum, thereby obtaining the initial center wavelength. λ 0.

4. The method for detecting the low-temperature thermal expansion coefficient of sapphire based on weak measurement as described in claim 1, characterized in that, In S5, the stepped cooling method refers to gradually changing the temperature of the sapphire sample from T0 to T0. n .

5. The method for detecting the low-temperature thermal expansion coefficient of sapphire based on weak measurement as described in claim 1, characterized in that, In S6, Δ λ i It is characterized by the following formula: D λ i = λ i - λ 0。 6. A sapphire low-temperature thermal expansion coefficient testing system, applied in the sapphire low-temperature thermal expansion coefficient testing method based on weak measurement as described in any one of claims 1-5, characterized in that, include: A light source that outputs a smooth spectrum; A sapphire sample to be tested is placed inside a cryostat, and the sapphire sample to be tested has a parallel plane. A weak measurement module is constructed by a pre-selective polarizer, a Soleil-Babinet phase compensator, and a post-selective polarizer, wherein the transmission axis of the post-selective polarizer is set to be orthogonal to the transmission axis of the pre-selective polarizer. A high-resolution spectrometer is set downstream of the post-selective polarizer to receive interference light; The output light from the light source passes through the optical window and is perpendicularly incident into the vacuum chamber. After being perpendicularly reflected by mirror I, it enters the plane on one side of the sapphire sample. Light transmitted through the other side of the sapphire sample is vertically reflected by mirror II and output to the post-selective polarizer. The preselective polarizer and Soleil-Babinet phase compensator are positioned between the light source and the optical glass.

7. The sapphire low-temperature thermal expansion coefficient testing system as described in claim 6, characterized in that, The light source is a superluminescent diode; The sapphire sample to be tested is a cylindrical sapphire made of c-cut or a-cut material; The sapphire sample to be tested has a diameter of 10 mm and a height of 50 mm. The parallel surfaces of the sapphire sample to be tested are optically polished to ensure that the parallelism of the parallel planes is better than 10 arcseconds.

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