This paper investigates a simple noncontact optical thermometry technique based on the laser interferometric measurement of the thermal expansion and refractive index change of a thin transparent substrate or temperature sensor. The technique is shown to be extendible from room temperature to at least 900°C with the proper choice of a thermally stable sensor. Sensor materials investigated included c-axis Al2O3, MgO, MgAl2O4 (spinel), Y2O3–ZrO2 (yttria stabilized zirconia), and fused silica. Calibration data were taken at 633 nm by measuring the sensor response to known temperature changes. These data provided (1) the information needed for quantitative thermometry (i.e., the functional relationship between interference fringes and temperature for samples of known thickness) and (2) the thermal coefficient of refractive index for those materials with known thermal expansion coefficients.
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Experimentally Determined Fitting Parameters Relating Fringes to Temperature (25–870°C) via Eq. (5) or (6) for λ = 633 nm; the Quantity (ΔT/fringe)|25°C is the Room Temperature Degrees per Fringe Calibration for a 0.1-cm Thick Sensor
Parameter
Al2O3 (c-axis) (Corundum)
MgO < 100 > (Periclase)
MgAl2O4 < 111 > (Spinel)
Y2O3–ZrO2 (YSZ; Yttria (9%) stabilized Zirconia)
SiO2 (Suprasil Fused Silica)
(ΔT/fringe)|25°C (°C)
13.5
10.2
14.7
13.1
33
T0 (°C)
23
29
31
26
36
a1 (cm−1°C−1)
7.3727 × 10−1
9.8584 × 10−1
6.8741 × 10−1
7.6689 × 10−1
3.0794 × 10−1
a2 (cm−1°C−2)
3.8938 × 10−4
5.8646 × 10−4
4.3193 × 10−4
3.4055 × 10−4
2.2806 × 10−4
a3 (cm−1°C−3)
−0.8759 × 10−7
−2.5928 × 10−7
−0.9743 × 10−7
−0.4747 × 10−7
−0.8395 × 10−7
b1 (cm − °C)
1.3412
0.99906
1.4256
1.2956
3.1727
b2 (cm − °C)
−7.6892 × 10−4
−4.4815 × 10−4
−9.5170 × 10−4
−6.4390 × 10−4
−5.2699 × 10−3
b3 (cm − °C)
6.2409 × 10−7
3.1410 × 10−7
8.0880 × 10−7
4.6318 × 10−7
1.1397 × 10−5
b4 (cm − °C)
−2.3323 × 10−10
−8.6779 × 10−11
−3.0736 × 10−10
−1.6439 × 10−10
−1.0137 × 10−8
Sample Thickness, L0 (cm)
0.0318
0.0635
0.0762
0.0495
0.3231
Table II
Thermal Expansion Fitting ParametersaAm and Room Temperature Refractive Indices at 633 nm for Some Optically Transparent Sensor Materials
Ref. 10.
Ref. 11.
Ref. 12.
Ref. 13, p. 6-46, linear interpolation from indices measured at 589.3 and 656.3 nm.
Ref 14.
Table III
Measured and Literature Values of Thermal Coefficients, α = (1/L)(dL/dT) and β ≡ (1/n)(dn/dT) at Room Temperature (25°C) and 800°C for Several Materials; the Wavelength is 633 nm
Calculated via Eq. (8) with Am parameters from Ref. 10 (listed in Table II).
Ref. 11; average value over the visible region from differences in indices measured at 17, 21 and 31 °C.
Ref. 12; from differences in indices measured at 20 and 30 °C interpolated for 633 nm from values at 589.3 and 656.3 nm.
Ref. 15; from differences in indices measured at 0 and 23 °C for 656.3 nm.
Ref. 14; from differences in indices measured at 20 and 30 °C interpolated for 633 nm from a plot of dn/dT vs. wavelength.
Ref. 13 and 16; from differences in indices measured at 26 and 828 °C for 578 nm. This value is better compared to the derived β found for the midpoint of the quoted temperature range, i.e. β(427°C) = 9.2 × 10−6.
Tables (3)
Table I
Experimentally Determined Fitting Parameters Relating Fringes to Temperature (25–870°C) via Eq. (5) or (6) for λ = 633 nm; the Quantity (ΔT/fringe)|25°C is the Room Temperature Degrees per Fringe Calibration for a 0.1-cm Thick Sensor
Parameter
Al2O3 (c-axis) (Corundum)
MgO < 100 > (Periclase)
MgAl2O4 < 111 > (Spinel)
Y2O3–ZrO2 (YSZ; Yttria (9%) stabilized Zirconia)
SiO2 (Suprasil Fused Silica)
(ΔT/fringe)|25°C (°C)
13.5
10.2
14.7
13.1
33
T0 (°C)
23
29
31
26
36
a1 (cm−1°C−1)
7.3727 × 10−1
9.8584 × 10−1
6.8741 × 10−1
7.6689 × 10−1
3.0794 × 10−1
a2 (cm−1°C−2)
3.8938 × 10−4
5.8646 × 10−4
4.3193 × 10−4
3.4055 × 10−4
2.2806 × 10−4
a3 (cm−1°C−3)
−0.8759 × 10−7
−2.5928 × 10−7
−0.9743 × 10−7
−0.4747 × 10−7
−0.8395 × 10−7
b1 (cm − °C)
1.3412
0.99906
1.4256
1.2956
3.1727
b2 (cm − °C)
−7.6892 × 10−4
−4.4815 × 10−4
−9.5170 × 10−4
−6.4390 × 10−4
−5.2699 × 10−3
b3 (cm − °C)
6.2409 × 10−7
3.1410 × 10−7
8.0880 × 10−7
4.6318 × 10−7
1.1397 × 10−5
b4 (cm − °C)
−2.3323 × 10−10
−8.6779 × 10−11
−3.0736 × 10−10
−1.6439 × 10−10
−1.0137 × 10−8
Sample Thickness, L0 (cm)
0.0318
0.0635
0.0762
0.0495
0.3231
Table II
Thermal Expansion Fitting ParametersaAm and Room Temperature Refractive Indices at 633 nm for Some Optically Transparent Sensor Materials
Ref. 10.
Ref. 11.
Ref. 12.
Ref. 13, p. 6-46, linear interpolation from indices measured at 589.3 and 656.3 nm.
Ref 14.
Table III
Measured and Literature Values of Thermal Coefficients, α = (1/L)(dL/dT) and β ≡ (1/n)(dn/dT) at Room Temperature (25°C) and 800°C for Several Materials; the Wavelength is 633 nm
Calculated via Eq. (8) with Am parameters from Ref. 10 (listed in Table II).
Ref. 11; average value over the visible region from differences in indices measured at 17, 21 and 31 °C.
Ref. 12; from differences in indices measured at 20 and 30 °C interpolated for 633 nm from values at 589.3 and 656.3 nm.
Ref. 15; from differences in indices measured at 0 and 23 °C for 656.3 nm.
Ref. 14; from differences in indices measured at 20 and 30 °C interpolated for 633 nm from a plot of dn/dT vs. wavelength.
Ref. 13 and 16; from differences in indices measured at 26 and 828 °C for 578 nm. This value is better compared to the derived β found for the midpoint of the quoted temperature range, i.e. β(427°C) = 9.2 × 10−6.