# Measurement of the radius of curvature and decentering of the backings of laser mirrors on an interferometer computer profilometer

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DOI: 10.1007/s11018-012-0058-0

- Cite this article as:
- Vishnyakov, G.N. & Tselmina, I.Y. Meas Tech (2012) 55: 904. doi:10.1007/s11018-012-0058-0

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A method of measuring the radius of curvature and the decentering of the backings of laser mirrors using the PIK-30 interferometer computer profilometer is considered.

### Keywords

Michelson interferometerphase steps methodlaser mirrorradius of curvatureOptical monitoring is an important part of the production of optical components [1]. Measurements of the deviations of the measured optical surface from planar or spherical are most essential. For example, the development of laser gyroscopes involves an increase in the requirements on the accuracy with which the resonator mirrors of a ring laser are manufactured. A specific feature of such mirrors is the considerable radius of curvature (2–10 m) over small dimensions (a diameter of 10–30 mm). The important monitored parameters of the mirrors are the radius of curvature of the reflecting surface (usually a sphere) and decentering, i.e., the value of the deviation of the center of curvature of the reflecting surface with respect to the center of the mirror itself.

Visual-type interference instruments have previously been used for optical monitoring, for example, the IT-40, IT-70, IT-100, and IT-200 series interferometers, based on the Fizeau scheme [2]. These interferometers were not automated, and hence the accuracy of the measurements did not exceed λ/10, where λ is the wavelength of the radiation employed. Recently interferometers have appeared with digital recording of the interferometers – instruments with which the topography of the measured surface can be obtained automatically in one or several interferograms. Thus, the IKD-110 automated interferometer (manufactured by the MTPK-LOMO Company, St Petersburg) is being used for the production monitoring of plane and spherical articles. However, for automatic deciphering of the interferograms, this instrument uses the frame method, i.e., it searches for the center of the fringes with subsequent interpolation, which is limited in the accuracy with which the phase is calculated to a value of λ/50. In another interferometer (the FTI-100, made by the Difraktsiya Company, Novosibirsk) for automatic deciphering of the interferograms, the more modern phase-shift method is used [3]. As a result, the accuracy with which the phase can be reconstructed increases to λ/100. Modern imported interference instruments, for example, those made by the ZYGO Company, are all provided with automatic systems of deciphering the interferograms, and hence possess higher measurement accuracy, but, unfortunately, are fairly expensive.

All this serves as a reason for developing a new interferometer, which satisfies the following requirements: high accuracy (greater than λ/100) in measuring the parameters of optical components with a diameter of up to 30 mm, automation of the measurements, low sensitivity to vibrations, simple and easy to use software, and small dimensions.

After analyzing possible arrangements, we decided on the Michelson interferometer with a mechanical phase shift for automatic deciphering of the interferograms. This instrument – the PIK-30 inteferometric computer profilometer – was developed at the All-Russia Research Institute of Optophysical Measurements (VNIIOFI) and is included in the State Register of Measuring Instruments under the number 30003-08.

In this paper, we describe a method of measuring the radius of curvature and the decentering of the backings of laser mirrors using the PIK-30. We present the order of the operations which enable the radius of curvature of an optical surface and its decentering with respect to the center of the backing itself to be measured by an interference method using the PIK-30. The diameter of the measured backings of the laser mirrors does not exceed 30 mm, and the reflection coefficient is not less than 4%.

The method is applicable to spheres with a radius exceeding several meters. In this case, a plane mirror can serve as the standard surface, and the interference pattern for correct adjustment is a set of concentric rings.

*x*,

*y*) between the object and reference waves is established. The required difference in heights

*z*(

*x*,

*y*) of the profile of the surface being investigated (the topogram) is calculated from the formula

The measured topogram of the laser mirror backing surface is approximated by a spherical surface by minimizing the root mean square deviation (RMSD). The radius of curvature *R* of the spherical surface obtained is calculated, and the center of curvature is the coordinates of the point with maximum value of the height *z*. The coordinates of the geometrical center of the backing are obtained from the video image of the laser mirror backing. The decentering *d* is calculated as the distance between the geometrical center of the laser mirror backing and the center of curvature of its spherical surface.

We also calculate, as additional parameters, the limit deviation of the measured topogram from a spherical surface Δ*Z*_{max} and the RMSD of the spherical surface σ.

We take as the characteristics of the measurement error the boundaries of the limits of the acceptable error in measuring a radius of curvature of 50 mm and a decentering of 0.1 mm; the range of measurements of the radius of curvature is 2000–7000 mm and of decentering is 0.1–5.0 mm.

The main means of measurement employed are the PIK-30, auxiliary equipment and the materials expended: a plane-parallel plate from the PM-15 set [4], bedding, a rinsing system, a thermometer, immersion liquid, an artist’s brush, wadding, nephras, and deionized water.

Before making measurements, we prepared the object being investigated. This is because the measured objects are made of transparent optical material (glass and glass ceramic) and have polished surfaces. Hence, when such an object is illuminated with a plane beam of laser radiation, additional reflections occur from the lower surface, which leads to parasitic interference fringes. To eliminate these fringes, it is necessary to deposit uniform thin layers of immersion liquid on the flat parts of the laser backing being measured (the object), opposite the surfaces being measured.

To calibrate the profilometer, we used a Zygo 1776-666-03 precision supersmooth flat plate made of silicon carbide, having high planarity and a roughness in the nanorange (*PV* = 0.5 nm, RMSD = 0.083 nm) or PM-15 plane-parallel glass plates with a nonplanarity of the measured surfaces of 0.07 μm. For this purpose, we initially recorded the surface profile (the error file) of the plate, carrying the phase aberrations of the optical system, which is then calculated from the results of the measurements.

The results of the measurements are processed as described in [5, 6] and include the following operations.

The determination of the limit permissible absolute error in measuring *R* and *d*. Since this operation is similar for both parameters of the backings, we will henceforth use the single notation *R*.

*R*

_{i}from the formula

*i*= 1, 2, …,

*N*is the number of the measurement, and

*N*= 10 is the number of measurements.

_{PIK}is the limit permissible absolute error of measurements with the PIK-30, given in the certification.

*S*is calculated:

if θ/

*S*< 0.8, the uneliminated systematic error is neglected and one can use as the error the result of a measurement of the confidence limit of the random error ε;if θ/

*S*> 8, the uneliminated systematic error can be neglected and one can use as the error of the result of a measurement the confidence limit of the uneliminated systematic error θ;- if 0.8 ≤θ/
*S*≤ 8, the confidence limit of the error of the result of measurements Δ for a confidence coefficient of 0.95 can be calculated from the formula$$ \varDelta =0.76\left( {\theta +\varepsilon } \right). $$

The result of a measurement of the parameters of the laser mirror backings is presented in the form *R* ± Δ for a confidence coefficient *P* = 0.95.

This research was supported by the Ministry of Education and Science of the Russian Federation (State Contract No. 16.552.11.7049 dated from July 29, 2011).