Use of Michelson and Fabry-Perot interferometry for independent determination of the refractive index and physical thickness of wafers

Abstract

We present a method to independently measure the refractive index and the thickness of materials having flat and parallel sides by using a combination of Michelson and Fabry-Perot interferometry techniques. The method has been used to determine refractive-index values in the infrared with uncertainties in the third decimal place and thicknesses accurate to within Ϯ5 m for materials at room and cryogenic temperatures. © 2005 Optical Society of America OCIS codes: 120.2230, 120.3180, 120.4290, 160.4760. The refractive index, n, and the thermo-optic coefficient, dn͞dT, of materials are often determined by one's interferometrically measuring the phase change that light undergoes in passing through a plane-parallel slab of the material. Because the phase change depends on the value of n as well as the slab thickness, d, to obtain accurate values of n and dn͞dT, it is important to know d accurately. FabryPerot etalon interferometry has been used to optically measure d, 5 but the precision of thickness measurements with this method is limited by the precision of the known refractive-index value. Recent research by Coppala et al. 6 demonstrated that independent values for n and d can be obtained with interferometry and a continuously tunable laser source. In this paper we demonstrate that the Michelson and the Fabry-Perot interferometric methods can be used sequentially to determine independent and absolute values of both the material's thickness and the material's refractive index over a wide range of temperatures of practical interest. The method does not require that either quantity be initially well known. With this method, both n and d can be determined by use of a fixed-wavelength laser source. First, by use of data from both experiments, the material's physical thickness is determined. Then the thickness value is used to determine the material's refractive index (and thermo-optic coefficient) with either of the interferometric methods. We present experimental verification of this method by measuring n and d for a range of common infrared materials at both room temperature and cryogenic temperatures. The intensity of a coherent collimated beam of light transmitted by a plane-parallel transparent plate is given by the Airy formula 7 : where I o is the incident intensity, r is the reflection coefficient for the electric field, and f is the phase difference accumulated by the light beam in a double traversal through the plate. As the sample is rotated in the path of the laser light, the net transmitted intensity will modulate owing to the changing phase, f . The angle-dependent phase difference between subsequent transmitted light paths through the sample is given by 7 f () ϭ 4nd cos t ϭ 4d ͙n 2 Ϫ sin 2 , where d is the sample thickness, is the laser wavelength, t is the angle of refraction, and is the angle of incidence of the laser path with respect to the normal of the sample surface