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Abstract
Non-contact temperature measurement for a solar telescope mirror is critical for improving the mirror seeing and thermal deformation of solar telescopes, a long-standing challenge in astronomy. This challenge arises from the telescope mirror’s inherent weak thermal radiation, often overwhelmed by reflected background radiations due to its high reflectivity. In this work, an infrared mirror thermometer (IMT) is equipped with a thermally-modulated reflector, and a measurement method based on an equation for extracting mirror radiation (EEMR) has been developed for probing the accurate radiation and temperature of the telescope mirror. Using this approach, we can extract the mirror radiation from the instrumental background radiation via the EEMR. This reflector has been designed to amplify the mirror radiation signal incident on the infrared sensor of IMT, while inhibiting the radiation noise from the ambient environment. In addition, we also propose a set of evaluation methods for IMT performance based on EEMR. The results reveal that the temperature measurement accuracy of IMT to the solar telescope mirror using this measurement method can be achieved better than ±0.15°C.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
During the operation of the solar telescope, the solar radiation incident on the telescope mirror is partially absorbed by its metal coating of the mirror surface, leading to a rise in the mirror surface temperature. The surface temperature exceeds that of both the mirror blanks and the surrounding environment [1,2]. This increase in temperature can induce thermal deformation in the telescope mirror and exacerbate hot airflow disturbances above the mirror surface. These disturbances degrade the telescope’s image quality, a phenomenon known as mirror seeing. In order to mitigate thermal deformation and improve mirror seeing, it is crucial to maintain the mirror temperature as close to the ambient temperature and uniform as possible [3–5]. Therefore, accurately measuring the mirror surface temperature becomes the foundational step towards improving mirror seeing and reducing thermal deformation. Contact temperature measurement methods would damage the mirror coating, and the temperature of the sensor exposed to solar radiation often does not equivalent to the mirror temperature. Meanwhile, the heat exchange between the mirror blanks, the supporting structure, and the surrounding environment is sophisticated and challenging to measure as well. Finite element simulation methods often yield inaccurate mirror surface temperatures.
Non-contact temperature measurement methods do not damage the mirror coating. Among them, infrared temperature measurement is almost the only feasible choice. However, due to the high reflectivity and low emissivity of the mirror surface, the mirror radiation is often overwhelmed by the massive ambient and instrumental radiations reflected from the mirror surface, leading to erroneous capture of the mirror radiation and temperature. Though the cryogenic coolers and vacuum optical path design of radiation thermometry systems could eliminate the interference of ambient radiations on mirror temperature measurement [6,7], they are unsuitable for telescope mirrors due to limited usage conditions. Radiation shields could reduce ambient radiation [8], yet the shield’s radiation reflected from the mirror is still significantly larger than the mirror radiation. External radiation source thermometry can estimate the metal surface’s temperature by measuring the heat source’s temperature and its apparent temperature after reflection on the metal surface [9]. However, factors such as the extremely small emissivity of the mirror surface, the reflectivity change with the incident angle, and the non-uniform temperature field of the mirror surface can cause severe temperature measurement errors. Bicolor pyroreflectometry measures the metal’s reflectivity to obtain the diffusion factor, which is then used to determine the temperature of hot metal, but it is not applicable for telescope mirrors with feeble diffuse reflection [10–12]. The ratio between the S and P polarized radiance, and the emissivity are adopted to measure the temperature of hot metal and silicon wafers [13,14]. However, it is difficult to measure the polarized radiances of the telescope mirror. Presently, various non-contact temperature measurement methods based on reflectivity and polarized radiation are not suitable for telescope mirrors. Due to the interference of the reflector radiation, the infrared temperature measurement device equipped with the reflector cannot really measure the radiation and temperature of the telescope mirror, but can only be used for metals whose temperature is much higher than that of the reflector [15–25]. Other existing methods for measuring metal temperature are also not applicable to telescope mirrors [26–30]. Non-contact temperature measurement of telescope mirrors remains an unsolved problem in the field of temperature measurement and astronomy. As a result, it has been impossible to establish a closed-loop mirror surface temperature control system for solar telescopes, as the mirror surface temperature cannot be used as feedback signals.
As the critical obstacles for truly identifying the mirror temperatures, the extremely weak mirror radiation submerged by ambient and instrumental radiations via reflection in the measurement process is the primary problem to solve. Here in this work, the equation for extracting mirror radiation (EEMR) is proposed and it can separate the mirror radiation from the instrumental radiation, so that the mirror radiation can be measured accurately. In addition, we introduce an infrared mirror thermometer (IMT), which uses a temperature-controlled reflector to significantly increase the mirror radiation incident on the infrared sensor and eliminate the interference of ambient radiation, simultaneously.
2. Design of IMT and temperature-controlled mirror
2.1 IMT design
As illustrated in Fig. 1, the IMT consists of a reflector, an infrared sensor, and a temperature control system. This study proposes two types of reflector designs: a parabolic reflector (a) and a conical reflector (b). The IMT proposed in this study is an infrared spot thermometer, where the diameter of the measurement area is approximately equal to the reflector diameter. The diameter of the measurement area determines the spatial resolution of the mirror temperature, the number of temperature measurements, and the total measurement time. An appropriately sized measurement area must be selected for a discrete sampling of the telescope mirror to obtain its temperature distribution. The primary mirror diameter of the solar telescope is usually on the order of meters. Balancing between spatial resolution and measurement efficiency, we have chosen the diameter of the measurement area to be 100 millimeters. Consequently, the diameter of the designed reflectors is uniformly set to 100 mm.
Fig. 1. Schematic diagrams of the proposed IMT designs: (a) parabolic IMT and (b) conical IMT. Both reflector types aim to amplify the mirror radiation power incident on the photosurface. The integrated temperature control system is designed to modulate the reflector temperature. The three differently colored lines within each reflector represent the potential pathways for mirror radiation to reach the photosurface: direct radiation, single reflection, and double reflection.
The infrared sensor is an uncooled IR focal plane array (FPA: 640 × 512) with a detection cut-off wavelength range of 8∼14 µm and a 14-bit AD converter. The infrared sensor type is the LA6110 of IRay Technology. Without a cold shield and Dewar packages, the uncooled infrared detector can collect mirror radiation at large angles from the normal to the photosurface. As a result, it has a far greater ability to collect mirror radiation than cooled infrared detectors. The infrared sensor is equipped with a thermoelectric cooler (TEC), and the measurement frequency of the infrared sensor is 25 Hz. The size of the FPA is 10.88mm × 8.7 mm, and the surface of the FPA is the photosurface. To facilitate numerical calculation, we simplify the photosurface to a circle with a diameter of 10.88 mm. The infrared sensor is located at the detection pinhole on the reflector top. The opening of the parabolic reflector is located directly in the middle of the parabolic vertex and the focus. The opening and the detection pinhole radii are 50 mm and 5.44 mm, respectively. The parabolic curve of the parabolic reflector is illustrated as shown in Fig. 2:
The equation of the reflector’s parabola is in form y^2 = 2px, with its focal point at the coordinate (p/2, 0). By substituting the coordinate of one endpoint of the opening (p/4, 50) into the parabolic equation, the p-value is calculated to be 70.71 mm. The reflector depth, calculated by substituting the coordinate of one endpoint of the detection hole (p/4 - Depth, 5.44) into the equation, is 17.47 mm. This design allows the mirror radiation near the normal directly incident on the photosurface, when the radiation is reflected by the reflector and mirror surface, as presented by the purple light in Fig. 1(a). The conical reflector’s opening and detection hole radii are the same as those of the parabolic reflector. The mirror radiation power incident on the photosurface at different cone depths is simulated, indicating that the mirror radiation power incident on the photosurface is maximized when the depth of the conical reflector is 26 mm. Detailed simulation results will be presented in Section 3.1. Based on the above calculation results, we fabricated a copper parabolic reflector and a conical reflector, each with the inner surface coated with gold.
During the temperature measurement using IMT, the reflector is placed very close to the mirror, most of the ambient radiations will be shielded from the reflector, while the ambient radiation leakage through the small gap will be absorbed after multiple rounds of reflections. In addition to the mirror radiation directly incident on the photosurface, considerable mirror radiation would also cast on the photosurface by reflection (the blue line and purple line in Fig. 1). Compared to the conventional infrared devices, the equipped reflector would remarkably increase the mirror radiation incident on the photosurface by multiple reflections, which effectively improves the IMT’s capability to collect the mirror radiation. When IMT measures the mirror temperature, the reflector would also remarkably increase its internal radiation incident on the photosurface via multiple reflections. The emissivity of the gold coating and the telescope mirror is fairly close, while the reflector’s inner surface area is larger than that of the measured mirror, which causes the mirror radiation incident on the photosurface to be submerged by the reflector internal radiation. For extracting mirror radiation from reflector radiation, an mirror temperature measurement method based on EEMR have been proposed and it can extract the mirror radiation from the instrumental radiation, so that the mirror radiation can be measured. To measure the EEMR, we have designed a temperature control system to modulate the reflector temperature. Measuring the IMT’s EEMR is equivalent to performing a calibration. During the measurement of the telescope mirror temperature, temperature modulation of the reflector is not required. Temperature control system of reflector mainly comprises a water jacket and a water chiller. The schematic diagram is shown in Fig. 3(a).
Fig. 3. (a) Schematic representation of the IMT and its associated temperature control system. (b) Photograph of the IMT and the temperature-controlled mirror. The temperature-controlled mirror is coated with aluminum, mimicking the typical coating on solar telescope’s primary mirrors.
The water jacket forms a water-cooled annular cavity on the outer surface of the reflector. The circulating water flows into the annular cavity through the inlet at the bottom and the outlet at the top. Due to gravity, the annular cavity is filled with circulating water. The wall thickness of the reflector is approximately 1 mm. Given the extremely high thermal conductivity of copper, the temperature of the circulating water can be considered as the temperature of the inner surface of the reflector. The circulating water is supplied by a KD-3AS chiller which maintains a temperature control accuracy of ±0.1°C. Moreover, a temperature sensing device, model TLTP-TEC2415-3, is installed near the inlet. TLTP-TEC2415-3, equipped with both temperature measurement and control capabilities, is only used in this study to monitor the temperature fluctuations of the circulating water near the inlet. The host of the TLTP-TEC2415-3 and its temperature monitoring and control module are shown in Fig. 4.
Fig. 4. Photographs of the TLTP-TEC2415-3 host (a) and its associated temperature monitoring and control module (b), which are only utilized to monitor temperature fluctuations in the experiment.
The temperature measurement results indicate that during the experiment, the temperature fluctuations of the circulating water near the inlet are less than ±0.1°C.
2.2 Temperature-controlled mirror design
Figure 3(b) presents a photograph of the actual parabolic IMT and temperature-controlled mirror. The coating of the temperature-controlled mirror is identical to the typical mirror coating of the primary mirror of a solar telescope. To precisely control the temperature of the temperature-controlled mirror, a copper cylindrical water block is employed as the mirror blank for the temperature-controlled mirror. The water block is designed with internal channels that have approximately the same cross-sectional area for water flow, as depicted in Fig. 5.
Fig. 5. (a) Internal structure of the water block, and (b) the temperature-controlled mirror featuring the inlet and outlet.
Figure 5. (a) showcases a photograph of the water block’s internal structure. The circulating water enters the water block from the inlet, then flows along the internal channels, as indicated by the blue curve. The internal channels have approximately the same cross-sectional area and flow rate. Additionally, numerous protrusions on the internal channels are designed to enhance the convective heat exchange area between the circulating water and the inner wall of the water block, thereby increasing their convective heat exchange capacity. Due to the uniform roughness of the inner wall of the water block, the convective heat transfer coefficient in regions with equal flow velocity is also approximately equal, ensuring a uniform surface temperature on the temperature-controlled mirror.
Figure 5. (b) shows the temperature-controlled mirror mounted on an adjustable mirror mount, with the inlet and outlet at the back of the mirror. The temperature control device (DC-0530) regulates the temperature of the circulating water entering the temperature-controlled mirror with an accuracy of ±0.05°C. Given the 1 mm wall thickness of the water block and the extremely high thermal conductivity of copper, when the internal channels of the water block are filled with circulating water, the surface temperature of the temperature-controlled mirror can be considered equivalent to the temperature of the circulating water. A TLTP-TEC2415-3 is installed near the inlet to monitor the temperature fluctuations of the circulating water.
3. Numerical calculations of mirror radiation collecting capability of IMT
3.1 Numerical calculation method
When employing the IMT for mirror temperature measurement, a substantial amount of mirror radiation incident on the photosurface, directly or after multiple reflections, can be observed. This multiple reflected radiation effect makes obtaining analytical solutions for mirror radiation power incidents on the photosurface difficult. This study defines the mirror radiation power incident on the photosurface as the infrared device’s Mirror Radiation Collecting Capability (MRCC), assuming a condition of full light absorption. The mirror surface is a typical non-Lambertian surface, so the emissivity from different angles varies significantly [31]. At present, no established method exists for calculating the IMT’s MRCC. Consequently, we employed the non-sequential ray tracing method of Zemax software to simulate the IMT’s MRCC. Key settings of the non-sequential ray tracing simulations are presented in Table 1.
Table 1. Key settings of non-sequential ray tracing simulation
The mirror is modeled as a circular source radial (object type: source radial) with a circular speculum (material: mirror) on the back. The radiation and reflection characteristics of the mirror are simulated by using the source radial and circular speculum options in Zemax. The light emitted by the source radial is adopted to simulate the infrared radiation characteristic for the mirror. The emissivity of the mirror is determined according to an aluminum coating [31]. The total radiation power of the mirror is denoted as $S \times {M_{\Delta \lambda }}$, where S denotes the mirror area, and ${M_{\Delta \lambda }}$ is the spectral radiant emittance of the mirror in the wavelength range of 8∼14µm. The angular luminous power of the source radial is distributed symmetrically with respect to the normal. To accurately simulate the infrared radiation characteristics of the mirror, we configured the angular luminous power to reflect the spatial distribution of the infrared radiation intensity. A relative angular luminous power is set every 5 degrees in the range of 0 to 90o with the normal, which is equal to the product of the mirror’s directional emissivity (${\varepsilon _\theta }$) and the cosine of the included angle ($\theta $). The inner surface of the reflector is defined as a mirror with a reflectivity of 0.97. By applying the abovementioned method, we calculated the MRCC of the conical IMT measuring a temperature-controlled mirror at 20°C under varying reflector depths. The results are depicted in Fig. 6:
The relationship between the depth of the conical reflector and the MRCC is not straightforward. The mirror radiation power incident on the photosurface gradually increases in an oscillating manner as the depth of the reflector increases, peaking at a value of $4.0125 \times {10^{ - 3}}$W at a reflector depth of 26 mm. As the reflector depth increases beyond this point, the MRCC oscillates and gradually decreases.
3.2 Measurement distance design
The MRCC of the IMT will therefore change with the measurement distance due to multiple reflections between the reflector and mirror. In addition, for the smaller gaps, the environmental radiation interference would be less, while the risk of damaging the mirror coating become greater. Therefore, the design of the measurement distance needs to take into account in the aspect of the MRCC and safety. Figure 7 presents the schematic diagram of MRCC for both ITMs calculated by non-sequential ray tracing method. The blue lines in the cavity between the mirror and the reflector are the simulated mirror radiations.
Fig. 7. Schematic diagrams of MRCC numerical calculation for both ITMs. The measurement distance of IMT is the gap between the reflector and the mirror, denoted as L.
The primary mirror diameter of the solar telescope is usually on the order of meters. The large-sized thermally-modulated mirrors used for IMT measurement experiments are difficult to manufacture and very costly, and it is necessary to choose a suitable mirror size. We used the method as described in Section 3.1 to calculate the MRCCs for two IMTs with different mirror sizes at varying measurement distances, as shown in Fig. 8.
The measurement distances are set from 0 to 20 mm. The mirror temperature is set at 20°C, and the mirror emissivity is referenced to aluminum coating [31]. The calculated MRCC of IMTs for mirrors with three sizes varies similarly with the measurement distances, indicating that the mirror radiation incident on the photosurface mainly originates from the mirror area with a diameter of 100 mm covered by the reflector. When the measurement distance is less than 7 mm, the MRCCs of the two IMTs for mirrors with diameters of 150 mm and 1000 mm are almost completely equal. Therefore, we set the diameter of the temperature-controlled mirror used in the temperature measurement experiment as 150 mm.
When the measurement distance is zero, the MRCC of the conical IMT is greater than that of the parabolic IMT, and the MRCCs of the two IMTs are equal when the measurement distance is close to 2 mm. As the measurement distance increases, the MRCC of the conical IMT tends to decrease exponentially. At the same time, the MRCC curve for the parabolic IMT is not as smooth as that of the conical IMT, exhibiting three distinct valleys and three peaks. The measurement distances for these three valleys are approximately 0, 4, and 7 mm, respectively. The large MRCC of the IMT, compared to traditional infrared devices, can be attributed to the multiple reflection effect of the mirror radiation between the mirror and the reflector. The focus plays a significant role in affecting the reflection effect of the parabolic IMT. The focal length of the parabolic reflector is 35.36 mm (p/2), and the distances from the mirror to the photosurface at the three valleys are approximately 0.5, 0.6, and 0.7 times the focal length, each being an integer multiple of one-tenth of the focal length. The measurement distance at 0.8 times the focal length is approximately 11 mm, which is also a weaker valley. The three peaks are local maxima located between the four valleys. While it is challenging to prove directly, these peaks/valleys likely originate from the focal point of the parabolic reflector. As the measurement distance increases, a significant amount of mirror radiation enters the external environment after multiple reflections between the mirror and the reflector. This results in an increase in the proportion of mirror radiation that is directly incident on the photosurface and mirror radiation that is reflected once by the inner surface of the reflector before incident on the photosurface. These two types of radiation are hardly affected by the focus. The impact of the focus diminishes, resulting in a gradually smoother response of the MRCC curve of parabolic IMT as the measurement distance increases.
The parabolic IMT consistently exhibits a higher MRCC than the conical IMT, and the deviation in measurements induced by identical changes in measurement distance is smaller for the former. In general terms, the MRCC of the parabolic IMT surpasses that of the conical IMT. Considering both the MRCC and the deviation in measurements prompted by variations in the measurement distance, 2 mm, and 5 mm emerge as the most suitable measurement distances for the parabolic IMT. For mirror coating safety, a measurement distance of 5 mm is selected. The accuracy of a typical vernier caliper is approximately 0.05 mm, and the change in MRCC brought about by a distance deviation of 0.05 mm is a mere 0.25%. When the measurement distance stands at 5 mm, the discrepancy in MRCC of the parabolic IMT for mirrors with a diameter of 150 mm and 1000 mm is only 0.137%, thereby indicating that mirrors with a diameter of 150 mm can serve as adequate representatives for the measurement performance of the IMT for solar telescope mirrors in practical applications.
3.3 MRCC of IMT and conventional infrared device
Ignoring the absorption of the mirror radiation by the lens of conventional infrared device, all the mirror radiation from the mirror in the field-of-view (FOV) incident on the lens can be taken as the signals. The conventional infrared device in this study is composed of the same infrared sensor as the IMT and an infrared lens with a diameter of 50 mm and an F-number of 1. The MRCC of the conventional infrared device is obtained using the radiative angle factors between the mirror and the lens, which is denoted as ${P_W}$:
where ${M_{\Delta \lambda }}$ represents the spectral radiant emittance of the mirror;${R_F}$ denotes the radius of the mirror in the FOV ; ${X_{ms}}$ expresses the angle factor of the mirror in the FOV to the lens, i.e., the ratio of the mirror radiation incident to the lens in the total radiation emitted by the mirror in the FOV. Equation (1) is used to calculate the radiation collecting capability of the telescope mirror and the blackbody measured by the conventional infrared device at a measurement distance of 80 mm. When the measurement distance is 5 mm, the MRCC of two IMTs to a mirror with a diameter of 150 mm are listed in Table 2.Table 2. Object radiation power incident on the photosurface
The calculation results indicate that the MRCC of the parabolic and conical types of IMTs are 22.4 times and 16.04 times, respectively, that of the traditional infrared device. It is shown that the two IMTs have excellent MRCC, and the parabolic IMT is better than the conical IMT. The mirror radiation power incident on the photosurface of the two kinds of IMTs is equivalent to a gray body measured by the traditional infrared device with emissivity values of 0.896 and 0.642, respectively. These results show that the IMT’s reflector makes the mirror radiation power fluctuation incident on the photosurface caused by the same mirror temperature change much larger than that of the conventional infrared device. Therefore, when the same infrared sensor is used, the mirror temperature measurement accuracy of ITM will be expected higher than that of conventional infrared device.
4. Temperature measurement method and performance evaluation method
As shown in the above calculations, the reflectors are capable of increasing the radiation power incident on the photosurface from the mirror significantly. However, the reflector radiation incident on the photosurface is also amplified by multiple reflection effects. The mirror radiation is still submerged by the reflector radiation. To truly measure the mirror temperature, in addition to enhancing the collection of signals with a designed reflector, we have further developed an EEMR and a temperature measurement method capable of separating the mirror radiation from the reflector radiation background.
4.1 EEMR
The average output of all FPA pixels of infrared sensor is termed the sensor reading unit, denoted as D. The temperature of a solar telescope mirror is generally slightly higher than the ambient temperature. The mirror radiation power collected by the two types of IMTs when measuring the mirror is equivalent to a gray body measured by a traditional infrared device with emissivity values of 0.896 and 0.642, respectively. When IMTs measure a solar telescope mirror, the mirror radiation power incident on the photosurface is sufficient, ensuring that the sensor readings of the two types of IMTs are within the linear zone of the infrared detector’s response. When the integration time is properly selected, the sensor reading follows an overall linear relationship with the infrared radiation power incident on the photosurface. Since there is only the reflector in front of the photosurface, the IR radiation incident on the photosurface is mainly composed of the mirror radiation and the reflector radiation. Obviously, the reflector radiation power incident on the photosurface follows a linear relationship with the radiant exitance of the gold coating on the reflector’s inner surface, the mirror radiation power incident on the photosurface is also proportionate to the mirror radiant exitance. Therefore, the sensor reading (D) is expressed as:
where $MO$ denotes the spectral radiant exitance at 8∼14 µm waveband of an ideal blackbody(emissivity = 1) with the temperature equal to the mirror; When measuring an actual blackbody, $MO$ denotes the spectral radiant exitance at 8∼14 µm waveband of an ideal blackbody with the temperature equal to the actual blackbody used in this study. $ML$ represents the spectral radiant exitance at 8∼14 µm waveband of a ideal blackbody with the temperature equal to the reflector; ${\varepsilon _O}$ and ${\varepsilon _L}$ expresses the emissivity of the mirror and the gold coating in the 8∼14 µm waveband, respectively.$\textrm{u}$ denotes the response coefficient of the IMT to the mirror radiation, which is expressed as follows: ${C_O}$ represents the relative quantity of IMT’s MRCC, while $R$ denotes the average pixel responsivity. Once the integration time is fixed, $R$ maintains a constant value. Following the determination of distance, ${C_O}$ also becomes constant, so $\textrm{u}$ would be a constant and exhibits a linear relationship with ${C_O}$. Likewise, $v$ represents the product of the relative quantity of IMT’s reflector radiation collecting capability (${\textrm{C}_L}$) and $R$. The $\textrm{u} \times {\varepsilon _O} \times MO$ represents the sensor reading contributed by the mirror radiation and $v \times {\varepsilon _L} \times ML$ denotes the sensor reading contributed by the reflector radiation. $W$ denotes the intercept term, which represents the readout circuit bias caused by taking out the photoelectric signal and correcting the sensor reading to the linear regime of infrared sensor response. $W$ also includes the sensor reading generated by the dark current accumulation and the infrared sensor’s thermal radiation accumulation. Given that u and $v$ are constants, the sensor reading equation can be simplified as follows: $U$ denotes the response coefficient of the IMT to the unit radiant exitance of the mirror, $U = {C_O} \times R \times {\varepsilon _O}$. $V$ represents the response coefficient of the IMT to the unit radiant exitance of the reflector, $V = {C_L} \times R \times {\varepsilon _L}$. The correlation between radiant exitance and temperature follows the Stefan-Boltzmann law, and the sensor reading equation establishes the correlation between the mirror radiation, the reflector radiation, and the sensor reading. When the distance between the IMT and the mirror is fixed, both $U$ and $V$ are constants. The control variates method is adopted to establish an overdetermined system of equations by measuring the sensor readings of different temperatures of the mirror and reflector, and then the values of U, V, and $W$ can be calculated. And then, the mirror radiation can be extracted from the reflector radiation and intercept term. Therefore, the sensor reading equation is referred to as the EEMR.The measurement distance between the reflector of the IMT and the mirror surface is 5 mm. The DC-0530 is used to control the mirror temperature, denoted as $TO$, and the KD-3AS to control the reflector temperature, denoted as $TL$. For each combination of temperature parameters, which are expressed as $[{T{O_i},T{L_j}} ]$(i = 1, 2, 3, …, M; j = 1, 2, 3, …, N), the corresponding sensor reading can be obtained, denoted as ${D_{ij}}$. The spectral radiant exitance of the ideal blackbody with temperature $T{O_i}$ and $T{L_j}$ in the 8∼14 µm waveband is denoted as $M{O_i}$ and $M{L_j}$, respectively. The spectral radiant exitance combination $[{M{O_i},M{L_j}} ]$ corresponds the ${D_{ij}}$ elements. The above radiant exitance combination and reading are brought into Eq. (4) to obtain K (K = M × N, K$> $3) three-variable linear equations. The K equations form an overdetermined system of equations, which can be written in the form of a matrix as:
The least squares solutions for the overdetermined system of equations are obtained using ordinary least squares. These solutions are then substituted into Eq. (4) to establish the EEMR.
When measuring ultra-low temperature optical mirrors with IMT, the sensor reading may be in the non-linear region, requiring a non-linear correction model to correct the non-linear effects of the infrared detector: First, measure a series of radiation illuminance (Ri) covering the dynamic range of the infrared detector corresponding to the sensor reading (Di). These data are used to fit a non-linear correction model, denoted as D = f (R). Subsequently, calculate the inverse function of the non-linear correction model, denoted as R = f-1(D). Each term in Eq. (4) essentially represents an sensor reading. By substituting inverse function into each sensor reading in Eq. (4), which is expressed as follows:
Finally, controlling the temperature of the mirror and the reflector and recording the corresponding sensor readings can obtain a nonlinear overdetermined system of equations about U, V, and W. The nonlinear least squares solutions of the nonlinear overdetermined system of equations can be obtained using nonlinear least squares. These solutions can then be substituted into Eq. (6) to establish the nonlinear EEMR.
4.2 Mirror temperature measurement method
When measuring the mirror temperature, we set the measurement distance to 5 mm, then start the IMT to obtain the sensor reading. The reflector temperature is recorded firstly, then the spectral radiant emittance is given at 8∼14 µm waveband of the ideal blackbody at a temperature equal to the reflector temperature, which is denoted as $ML$. The sensor reading and $ML$ are substituted into the EEMR or the nonlinear EEMR to obtain the mirror spectral radiant emittance, which is expressed as $MO$. It means that the temperature of the ideal blackbody with the spectral radiant emittance in 8∼14µm equal to $MO$ can be calculated, which is the measured mirror temperature. In this case, it is not necessary to measure the emissivity of the mirror.
The proposed method uses the reflector temperature to obtain the sensor reading contributed by the reflector, and subsequently separates the sensor reading contributed by the mirror radiation from the total sensor reading. It is equivalent to separating the mirror radiation from the total radiation on the photosurface, by which the measurement of the mirror radiation and mirror temperature can be achieved. The IMT’s reflector is equivalent to the lens of the conventional infrared device. Such that, combined to the lens temperature control design, the conventional infrared device can also use this method to true measure the mirror radiation and mirror temperature.
4.3 Evaluation method of MRCC
In both IMTs and conventional infrared device cases, the same infrared sensor is used for a fair comparison. When the three devices are used to measure the same mirror with the identical integration time, $R$ and ${\varepsilon _O}$ will be equal. The ratio of the coefficient $U$ in the different EEMR is the relative ratio of the MRCC for the three devices. The MRCC of both IMT can be evaluated using the MRCC of the conventional infrared device as the reference. In addition, when the measured object is full of FOV, the conventional infrared device would exhibit the same radiation collection capability for the mirror and the blackbody, simultaneously. On that basis, the ratio of the $U$ coefficient is the emissivity ratio of the mirror and the blackbody.
4.4 Evaluation method of mirror radiation measuring capability
Based on EEMR, in the case of a larger $U$ and a smaller V, a higher signal-to-noise ratio can be obtained when measuring the mirror radiation. The measurement capability of IMT to the mirror radiant exitance, denoted as $P$, is dependent on the $U/V$ ratio and $R$, and the correlation is expressed as follows:
The $R$ should be the same for the three devices, implying that the mirror radiant exitance’s measurement capability is proportional to the $U/V$ ratio. In accordance with Stefan Boltzmann law, the measurement capability of the mirror radiant exitance also represents the measuring capability of the mirror temperature to a large extent.
4.5 Evaluation method of the mirror temperature measurement accuracy
Mirror temperature measurement accuracy is a vital indicator of IMT. Temperature measurement accuracy is the degree of closeness of temperature measurements to that temperature’s true value and comprises random and systematic temperature measurement errors. The magnitude of the random temperature measurement error is represented by the temperature measurement precision, while the temperature measurement trueness represents the magnitude of the systematic temperature measurement error. Temperature measurement accuracy is the combined contribution of temperature measurement precision and temperature measurement trueness. The random temperature measurement error depends on the random error in measuring the mirror radiant exitance, which can be converted between each other via the Stefan-Boltzmann law. The random error in measuring the mirror radiant exitance depends on the random error of the sensor reading contributed by the mirror radiation incident on the photosurface. The EEMR establishes the relationship between the mirror radiant exitance and the sensor readings. The sensor reading contributed by the reflector radiation incident on the photosurface is a fixed value calculated from the reflector temperature (set by the water chiller), the radiant exitance of the inner surface of the reflector is considered a constant during temperature measurement, and the intercept term is also constant. Therefore, the random error in measuring the mirror radiant exitance can be directly represented by the random error of the sensor reading. The ratio between them is the response coefficient of the IMT to the unit radiant exitance of the mirror ($U$). The sources of random error in sensor reading include the intrinsic noise of the infrared detector, the temperature control error of the mirror, the temperature-controlled mirror, and the infrared detector. These factors together lead to the random error in the IMT measurement of mirror temperature. The probability distribution of the random measurement errors of the sensor readings can be considered to conform to the Gaussian distribution [32]. The precision of the sensor readings, denoted ${\sigma _r}$, is the standard deviation of the Gaussian distribution function of considerable sensor readings obtained by repeated measurements. ${\sigma _r}$ means that under the same measurement conditions, the sensor reading has a 68.3% probability of falling within the range of $mu \pm {\sigma _r}$, where $mu$ denotes the expected value of the Gaussian distribution of the sensor reading. The random errors of sensor readings are converted into random errors in measuring mirror radiant exitance, which is expressed by the precision of the measurement of mirror radiant exitance, denoted as ${\sigma _{\textrm{m}p}}$. Based on the EEMR, the correlation between ${\sigma _r}$ and ${\sigma _{\textrm{m}p}}$ is expressed as follows:
According to the Stefan-Boltzmann law, ${\sigma _{\textrm{m}p}}$ is converted into the temperature measurement precision, denoted as ${\sigma _{tp}}$, and the conversion calculation is expressed as follows:
The systematic error in mirror temperature measurement is a type of error that emerges due to inherent flaws or biases in mirror temperature measurement. Essentially, the EEMR acts as the ruler for mirror temperature measurement. Errors in the EEMR measurement process include the nonlinear response of the infrared sensor, the error induced by intrinsic sensor noise, and the temperature control error of the mirror and reflector. These errors cause a deviation of the sensor readings from their true values when measuring EEMR, collectively leading to fitting residuals of the EEMR. The fitting residuals represent the systematic error in subsequent mirror temperature measurements based on the EEMR. The magnitude of fitting residuals is quantified using the root mean square error (RMSE), which will be elaborated upon in Section 5.1. The trueness for the mirror radiant exitance measurement can be expressed by the ratio of the RMSE of the fitting residual to the coefficient U, which is denoted as ${\sigma _{\textrm{mt}}}$:
It is assumed that the measurement to obtain the EEMR is independent of the mirror temperature measurement. The accuracy of mirror radiant exitance measurement, is recorded as ${\sigma _{\textrm{ma}}}$. In accordance with the propagation of uncertainty, ${\sigma _{\textrm{ma}}}$ is expressed as follows:
Equation (9) is adopted to convert ${\sigma _{\textrm{ma}}}$ into temperature measurement accuracy, which is recorded as ${\sigma _{\textrm{ta}}}$.
5. Results and discussion
5.1 EEMR
The EEMR’s coefficients of conventional infrared device and two IMTs are obtained using the method as described in Section 4.1. The reflector/lens temperatures are set at 10, 12, 15, and 17°C, while the mirror temperatures for obtaining the IMT’s EEMR are set at 8, 12, 16, and 20°C. The mirror temperatures for obtaining the conventional infrared device’s EEMR are set at 10, 18, 26, and 34°C. Here, the large interval for the mirror temperatures is taken because the conventional infrared device exhibits a weaker MRCC, so that the variation in the sensor readings caused by a 4°C temperature difference would be significantly affected by random errors. The measurement process is carried out in a walk-in environmental test chamber and the ambient temperature fluctuates in the range of 15 ± 2°C. The EEMR’s coefficients of 4 cases as presented in the Table 2 are listed in Table 3:
Table 3. EEMR’s coefficients
The $U$, $V$, and $W$ values in Table 3 are the point estimates for the coefficients of the EEMR, and the interval range is the interval estimate of the 95% confidence interval. RMSE is the root mean square value of the error between the 16 measured sensor readings and the 16 fitted readings obtained by bringing the point estimate into the overdetermined system of equations. R-square is the coefficient of determination. The W coefficients are negative due to the readout circuit bias. A silicone tube connected to the inlet and outlet of the KD-3AS chiller is densely wrapped around the lens of the conventional infrared device to control its temperature (see, Fig. 9).
Fig. 9. (a) Schematic diagram illustrating temperature control of the lens in the conventional infrared device. (b) Image showing the conventional infrared device measuring a temperature-controlled mirror with a diameter of 150 mm. (c) The blackbody from CI Systems.
Since there is no heat source inside the lens, the temperature difference between the thermally stabilized lens and the cool running water should be slight. Thus, the temperature of the cooling water can be taken as the lens temperature. The blackbody used is model SR800N-12D-LT from CI Systems, with an emission surface area of 300 mm × 300 mm. The blackbody control accuracy over the temperature is ± 0.007°C, and its emissivity is 0.97 ± 0.02.
The $V$ coefficient of the EEMR of conventional infrared device (blackbody) represents the relative radiation intensity of the lens internal radiation directly incident on the photosurface, while the relative mirror radiation intensity incident on the photosurface is expressed by the $U$ coefficients of EEMR of conventional infrared device (mirror) . As indicated in conventional infrared device (mirror) of Table 3, the lens internal radiation incident on the photosurface is significantly larger than that of the mirror radiation. The lens external radiation that is incident on the photosurface through mirror surface reflection and the lens internal radiation that is directly incident on the photosurface constitute the $V$ coefficient of conventional infrared device (mirror). The lens external radiation incident on the photosurface through mirror reflection can be expressed by the difference between the $V$ coefficient of conventional infrared device (mirror) and conventional infrared device (blackbody), which is 17.2 times larger than that of the mirror radiation. At present, various conventional infrared devices consider the lens external radiation incident on the photosurface as mirror radiation, which makes it impossible to measure the true mirror radiation. The mirror radiation is separated from the reflector/lens radiation through EEMR. The EEMR proposed in this study enables both IMTs and various conventional infrared devices to true measure mirror radiation. The MRCC of the conventional infrared device is significantly smaller than that of IMT, and its temperature measurement accuracy is inevitably lower than IMT.
5.2 Temperature-controlled test mirror and emissivity error
The mirror temperature measurement method proposed in this study involves measuring the EEMR of a temperature-controlled mirror with the same coating as the solar telescope, and then using the EEMR to measure the solar telescope mirror. The emissivity difference between the temperature-controlled mirror and the solar telescope mirror will cause temperature measurement error when applying EEMR to the solar telescope mirror. The mirror’s emissivity is included in the U coefficient of the EEMR. Assuming we can measure the difference in emissivity between the two mirrors, we can then adjust the U coefficient of EEMR to reduce the temperature measurement error caused by this difference. This adjustment involves multiplying the ratio of the emissivity of the solar telescope mirror to that of the temperature-controlled mirror by the EEMR’s U coefficient derived from IMT’s measurement of the temperature-controlled mirror, to obtain the EEMR’s U coefficient for measuring the solar telescope mirror with the IMT. However, this correction method primarily applies to instances where the differences in emissivity are minor. In scenarios where significant differences in emissivity occur, inducing notable changes in reflectance would result in further modifications to the U coefficient, due to alterations in the multiple reflections between the reflector and the mirror. In fact, there have been few research reports on the measurement of telescope mirror emissivity. The emissivity differences between the temperature-controlled mirror and the solar telescope mirror are challenging to measure, and the errors in temperature measurement induced by these emissivity differences are also difficult to evaluate. Reducing the emissivity difference between the temperature-controlled mirror and the solar telescope can improve the temperature measurement accuracy. To minimize the emissivity discrepancy between the temperature-controlled mirror and the solar telescope mirror under operation, the companion coating mirror of the primary mirror of the solar telescope can be employed as the temperature-controlled mirror. This temperature-controlled mirror can be mounted on the tube of the solar telescope, ensuring that the temperature-controlled mirror and the solar telescope mirror are under identical conditions. Nevertheless, the emissivity of the companion coating mirror mounted on the solar telescope tube might still have minuscule differences compared to the solar telescope mirror. It should be noted that such minuscule emissivity differences are nearly impossible to measure.
Considering that we cannot control the temperature of the solar telescope mirror and that there are no other temperature measurement techniques available for telescope mirrors, it is not feasible to use the solar telescope mirror to validate the temperature measurement accuracy of the IMT. This study used a test mirror with a minuscule emissivity difference from the temperature-controlled mirror as the measurement object. The mirror blank, mirror coating, and temperature control method of the test mirror are identical to those of the temperature-controlled mirror, which we refer to as the temperature-controlled test mirror. To simulate the impact of minuscule emissivity differences on temperature measurement, the mirror coating of the temperature-controlled test mirror has been in use for three months longer than that of the temperature-controlled mirror. Figure 10 presents photographs of two types of IMTs measuring the temperature-controlled test mirror.
The temperature-controlled test mirror employs a circulating water system identical to the temperature-controlled mirror, maintaining a temperature control accuracy of ±0.05°C. We hypothesize that the difference in emissivity between the companion coating mirror and the solar telescope mirror is comparable to the difference in emissivity between the temperature-controlled mirror and the temperature-controlled test mirror. To a certain extent, the results of our temperature measurements in this experiment can be used to evaluate the IMT’s accuracy in measuring the solar telescope mirror under operation.
5.3 Mirror temperature measurement of IMT
The proposed mirror temperature measurement method is demonstrated by parabolic IMT as an example. As depicted in Table 3, the EEMR is written as follows:
The measurement distance is 5 mm. The temperature measurement starts when the reflector’s temperature is stabilized. The measurement takes 1 sec, during which 25 sensor readings are obtained. The average of the 25 sensor readings is recorded as Dm, and the measurement is repeated thrice. $ML$ obtained according to the reflector temperature and Dm are substituted into the EEMR, from which $MO$ can be calculated, the temperature of temperature-controlled test mirror is obtained from $MO$. Table 4 shows the readings and temperature measurement results from the mirror at 28°C and 32°C, respectively.
Table 4. Sensor readings and temperature measurement results
Clearly, it verifies that repeated measurements of mirror temperature achieve a low measurement error of less than 0.15°C, suggesting that the IMT proposed in this study exhibits a high measurement accuracy. Here, we should point out that the IMT is an experimental prototype. The program for calculating the temperature also needs to be integrated into the final product, allowing for rapid determination of the mirror temperature during the measurement.
5.4 MRCC of IMT
The $U$ coefficients listed in Table 3 suggest that the MRCC of the parabolic IMT is 21.03 times higher than that of the conventional infrared device. Likewise, for the conical IMT, the MRCC would be enhanced by 16.89 times. Compared to conical IMT, the parabolic IMT has a better MRCC. As shown in Table 2 and Table 3, the measured results for the relative MRCC values are consistent with the simulation results. The conventional infrared device has the same radiation collecting capability for the mirror and the blackbody. Consequently, the ratio of the two U coefficients, those of the conventional infrared device (mirror) and conventional infrared device (blackbody), is equivalent to the ratio of the emissivity of the mirror to that of the blackbody. The blackbody emissivity is 0.97, and then the mirror emissivity can be calculated to be 0.0394.
5.5 Mirror radiation measuring capability of IMT
As depicted in Table 3, the $U/V$ ratio derived from the EEMR’s coefficients of the conventional infrared device (blackbody) is 0.206, while for the mirror, this ratio falls to just 0.00732. This suggests that the conventional infrared device’s mirror radiant exitance measurement accuracy would be very low. The mirror radiant exitance measurement accuracy by the conical IMT and parabolic IMT is 22.043 and 29.892 times higher than that of the conventional infrared device. This suggests a significant improvement in their temperature measurement accuracies compared to the conventional infrared device. The significant improvement in the IMT’s measurement accuracy of mirror radiant exitance can be attributed to the high MRCC of the reflector designed in this study, coupled with its reduced interference of the reflector radiation compared to the conventional lens.
5.6 Mirror temperature measurement accuracy of IMT
Figure 11(a) displays the 63,000 sensor readings obtained by the parabolic IMT from repeated measurements of the mirror under identical conditions. In general, the readings exhibit significant high-frequency random noisy behaviors, accompanied by large-period fluctuations in the counts.
Statistically, the overall 63,000 readings follow a Gaussian distribution, as shown in Fig. 6(b). The expected value ($mu$) is 8659, and the measurement precision of sensor reading (${\sigma _r}$) is 5.005. Based on Eq. (7∼10), by combining the corresponding $U$ coefficient and RMSE of parabolic IMT in Table 3, the mirror temperature measurement accuracy of parabolic IMT is calculated. Using the same method, we obtain three other temperature measurement accuracies, as listed in Table 5.
Table 5. Mirror temperature accuracies achieved using different measurement setups
Empowered with the EEMR, both the IMT and conventional infrared devices gain the ability to truly measure mirror radiation and temperature for the first time. We achieved true measurement of mirror temperature with the traditional infrared device, with a low accuracy of only ±2.921°C. Notably, the accuracy using the IMTs can be dramatically increased to ± 0.134°C (21.8 times) and ±0.112°C (26.1 times), respectively. The IMT has been designed to amplify the mirror radiation incident on the photosurface through the reflector, and greatly improves the measurement accuracy of mirror radiation and mirror temperature. The primary reason for this improvement in measurement accuracy is that the reflector’s MRCC is significantly higher than that of a conventional lens. This leads to significantly larger mirror radiation power fluctuations on the photosurface caused by the same mirror temperature fluctuation.
5.7 Application of IMT in the primary mirror of solar telescope
As shown in Fig. 12, when measuring the temperature of a solar telescope’s primary mirror, the IMT can be mounted on a motorized rotation arm above the mirror, with the IMT directed towards the mirror surface at a measurement distance of 5 mm.
Fig. 12. Schematic representation of the application of the IMT to the primary mirror of a solar telescope.
The IMT operates at a measurement frequency of 25 Hz, capable of ascertaining the average temperature of a 100 mm diameter measurement area within 40 ms. We consider this average temperature as the temperature at the center point of the measurement area, denoted as Ti. When the motorized rotation arm moves, multiple continuous measurement areas can form an annulus sector, thereby obtaining the discrete temperature distribution of the annulus sector. The motorized linear stage can adjust the radial position of the IMT, measuring the temperature distribution of the annulus sector at different radial positions and thus capturing the temperature distribution of the entire mirror surface.
By utilizing a stepper motor to precisely control the rotation of the motorized rotation arm, the solar radiation shielding time for the measurement area can be approximately 40 ms. If an infrared detector with a measurement frequency of several hundred Hz is used, the solar radiation shielding time for the measurement area can be reduced to the millisecond level. Considering the large volume and enormous thermal inertia of the typical solar telescope primary mirror, temperature changes of mirror surface within a few to tens of milliseconds are extremely minimal. To reduce the impact on solar observations, measuring the temperature of a particular annulus sector once per hour is possible, with a measurement duration on the order of tens to hundreds of milliseconds. The average temperature of the annulus sector can be taken as the primary mirror temperature. Another measurement method involves fitting an empirical formula for the mirror temperature based on the mirror temperature, air temperature, solar irradiance, wind speed, and telescope altitude angle measured during non-formal observation periods. During formal observations, the mirror temperature can be calculated using the measured solar irradiance, air temperature, wind speed, and telescope altitude angle, thus eliminating the impact of the IMT on formal solar observations. Of course, during non-formal observation periods, the complete mirror temperature distribution at different times can be measured to study the primary mirror’s temperature behavior.
Solar telescopes are primarily cooled by an air-cooling system installed at the rear of the primary mirror. Based on the obtained mirror surface temperature, the air-cooling system can establish a closed-loop control system using real-time mirror surface temperature and air temperature as feedback signals. This closed-loop control can significantly improve the precision of mirror surface temperature control, ensuring that the temperature difference between the mirror and the air remains minimal. Reducing the temperature difference between the mirror and the air will improve the mirror seeing and minimize the primary mirror’s thermal deformation.
6. Conclusions
A mirror temperature measurement method based equation for extracting mirror radiation and two reflectors have been proposed in this study. This measurement method can extract weak mirror radiation from various radiation incidents on the photosurface, and enable the infrared mirror thermometer and conventional infrared device to have the ability to truly measure the mirror radiation and mirror temperature. Compared with traditional lenses, the reflector has been designed to amplify the mirror radiation collecting capability of the infrared mirror thermometer. The combination of the measurement method and the reflector realizes the precise temperature measurement of the telescope mirror for the first time. Furthermore, the performance of infrared mirror thermometer has been evaluated comprehensively by using the proposed evaluation method. Compared to conventional infrared devices, the temperature measurement accuracies of the parabolic and conical infrared mirror thermometers can be improved by over 26.1 times and 21.8 times, respectively, reaching an accuracy better than ±0.15°C. A method of using IMT to measure the primary mirror temperature of a solar telescope under operation is designed. The accurate temperature measurement of the optical mirror can be used to improve the mirror seeing and thermal deformation of solar telescopes and high-power laser mirrors. In addition, the equation for extracting mirror radiation and the design method of infrared mirror thermometer can also provide theoretical guidance in infrared temperature measurement of non-high temperature metals, smooth coatings and silicon wafers.
Funding
National Natural Science Foundation of China (11673064, U1931124, U2031148, U2241226).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper may be obtained from the authors upon reasonable request.
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