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When electric rotation stage is controlled to rotate, the optical path changes in the material with rotating. As shown in Figure 3, when the angle between the normal line of the material surface and the light is θ, the optical distance L in the material is
where d is the thickness of material, n denotes the refractive index to be measured, i denotes the refraction angle in the material, n0 denotes the air refractive index, and x is geometric length from optical position incident on the material to the measurement mirror. According to Snell equation [15]
derive as
Assuming that the angle between the surface normal of material and the beam before rotation is θ0, and the angle after rotation is θ, the optical path change ΔL is
where the optical path change Δϕr of reference signal comes from the air disturbance between ML and MR and the thermal effect of the acousto-optic modulators, and the optical path change Δϕm of the measurement signal comes from the air disturbance between ML and ME, thermal effect and the optical path change. When the material is rotated, θ and ΔL selected at multiple angles are substituted into equation (5) to solve this over-determined equation expressed by equation (6). It excludes the effect of thickness measurement uncertainty on refractive index accuracy.
In the experiment, the material S is polished thin wafer which is Sapphire Crystal or GaAs. Because sapphire is an anisotropic uniaxial crystal, it has two refractive indices. Choose ordinary ray of W as the measurement light B2. The thickness of Sapphire Crystal and GaAs are 0.302 mm and 0.686 mm respectively measured by a micrometer. The air sensor is used to detect the air in the experiment. The indoor temperature of Sapphire Crystal is 23.34 ℃, the air pressure is 101.93 kPa, the relative humidity is 18.5%, and the air refractive index n0 is calculated as 1.0002678 by the modified Edlén equation[16]. For GaAs, they are 23.60 ℃, 101.89 kPa, 16.5%, and 1.0002675. Considering the transmittance of the materials and reducing the measurement uncertainty requirement, decide that Sapphire Crystal is rotated from 0° to 30° with pausing every two degrees, and GaAs from 0° to 20° with pausing every angle. The optical path difference of rotating material at each angle is measured 20 times within 20 seconds. The average optical path difference is shown in Table 1 and Table 2.
Rotation angle/(°) Optical path change/nm Rotation angle/(°) Optical path change/nm 1 145 11 8924 2 354 12 10693 3 689 13 12594 4 1186 14 14502 5 1827 15 16679 6 2598 16 19042 7 3558 17 21524 8 4674 18 24134 9 6015 19 26870 10 7482 20 30037 Table 1. The average optical path difference in Sapphire Crystal with angle
Rotation angle/(°) Optical path change/nm Rotation angle/(°) Optical path change/nm 2 69 18 6550 4 270 20 8047 6 700 22 9811 8 1275 24 11703 10 2036 26 13771 12 2896 28 16006 14 3937 30 18365 16 5158 Table 2. The average optical path difference in GaAs with angle
Substitute the experiment data in Table 1 and Table 2 and n0 into equation (5) by fitting refractive index and thickness. The theoretical curve and experimental data points are drawn in Fig. 4(a). It is shown that, consistent with the experimental data, the solution of the Sapphire Crystal n is 1.7551, d is 0.3017 mm, the GaAs n is 3.4653, and d is 0.6862 mm. The difference between theoretical fit data and experimental data of optical path change at every angle is presented in Fig. 4(b), and the difference range is from −80-80 nm.
Figure 4. (a) Theoretical and experimental data of optical path change of two materials with angle of rotation; (b) Difference between theoretical fit data and experimental data
There are five times of the measurement of Sapphire Crystal and GaAs, and the measurement results are presented in Table 3. Calculate the measurement average value (refractive index and thickness) and type A evaluation of uncertainty[17](that is, the standard deviation calculated by the Bessel equation[18]). From Table 3, the refractive index of Sapphire Crystal is 1.7550±0.0005. The reference value is based on the value calculated by Sellmeier equation[2], which is 1.7545, so the deviation of the experimental data with the reference data is 0.0005. The measurement accuracy of Sapphire Crystal reaches 10−4 and the material does not need to be processed into prismatic shape. However, the accuracy of ellipsometry method and m-lines method are 10−2 and 10−3 respectively, and the common reflective gem refractometer is 10−3. The refractive index of GaAs is 3.4719±0.0039. The reference refractive index value is 3.4727 which is calculated by Sellmeier equation[19], so the deviation of the experimental data with the reference data is 0.0008. The measurement accuracy range of GaAs is between 10−3-10−4. However, the ellipsometry method accuracy is 10−2 and GaAs is beyond the measurement range of Abbé refractometer. From Table 3, the thickness of Sapphire Crystal is (0.3022±0.0004) mm, and the thickness measured by micrometer is (0.302±0.001) mm. The thickness of GaAs is (0.6861±0.0001) mm, and the thickness measured by micrometer is (0.686±0.001) mm. Therefore, the results are satisfactory and the accuracy has been improved.
Number Sapphire Crystal GaAs n d/mm n d/mm 1 1.7553 0.3028 3.4653 0.6862 2 1.7551 0.3026 3.4759 0.6861 3 1.7539 0.3019 3.4739 0.6860 4 1.7554 0.3018 3.4696 0.6859 5 1.7551 0.3017 3.4749 0.6861 Table 3. Measurement results
There are several reasons for refractive index measurement error Δn. First of all, the material composition, processing technology and surface morphology are the main reasons for the measurement error Δn. Among them, the main component of Sapphire Crystal is Al2O3, which contains the trace elements titanium (Ti4+) or iron (Fe2+), and different doping concentrations lead to different refraction effects. Due to the different crystal growth process, the melting temperature, the number of pulling times and other steps are also different. However, the temperature, pulling number and other steps will affect refraction results. Secondly, both materials are polished wafers, so the parallelism, roughness and uniformity of their surfaces will affect the measurement accuracy and the corresponding Δn cannot be accurately calculated. In addition, the air refractive index and the laser wavelength will vary with the ambient temperature. In a normal laboratory environment, the refractive index of air changes less than 10−5, and Δn is less than 7×10−6. The frequency-stabilized microchip laser has the wavelength drift Δλ less than 2.6×10−4 nm, and Δn is only 1.94×10−7. Finally, GaAs is a semiconductor material, and refractive index of semiconductor material is complex number because its conductivity is not zero. This article is measuring the real part of the complex refractive index. However, when light is incident on the GaAs sheet obliquely at a certain angle, the extinction coefficient of the semiconductor material (that is, the imaginary part of the complex refractive index) will affect the real part, thereby affecting the measurement data of refractive index.
Improving the measurement accuracy of refractive index of GaAs and Sapphire Crystal by laser feedback interferometry
doi: 10.3788/IRLA20210400
- Received Date: 2021-06-15
- Rev Recd Date: 2021-08-16
- Publish Date: 2022-04-07
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Key words:
- GaAs /
- Sapphire Crystal /
- feedback interferometry /
- refractive index /
- accuracy
Abstract: GaAs and Sapphire Crystal has been widely used in infrared region, optoelectronics field and military equipment, so the measurement of refractive index of two materials is of great significance to optical design, metrological inspection and industrial application. To improve the measurement accuracy of refractive index of two materials, microchip laser feedback interferometer technology was used to simultaneously measure refractive index and thickness. The system combined heterodyne modulation and quasi-common path to compensate for airflow and vibration, so it has the characteristics of high sensitivity, high precision and high stability, especially the simultaneous measurement and only the material needs to be processed into flake rather than prism shape. The experimental results demonstrate that the measurement accuracy of refractive index of GaAs and Sapphire Crystal (under ordinary light) has been enhanced to 10−3 and 10−4 respectively and thickness is 10−4 mm.