The complex ordinary (No) and extraordinary (Ne) refractive indices of an absorbing uniaxial crystal can be determined using reflection ellipsometry. The measurements are taken with the optic axis parallel and perpendicular to the crystal’s surface. The equations obtained are solved without resort to iterative methods; No and Ne are determined separately. Sixteen solution sets (No, Ne) are obtained and the correct solution can be easily identified. We present an optimum angle of incidence that minimizes the relative errors in No and Ne

Azzam, R. M.A.

Elshazly-Zaghloul, M.

English

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Inversion of the nonlinear equations of reflection ellipsometry for uniaxial crystals in symmetrical orientations

University of New Orleans

University of New Orleans ScholarWorks@UNO Electrical Engineering Faculty Publications Department of Electrical Engineering 7-1980 Inversion of the nonlinear equations of reflection ellipsometry for uniaxial crystals in symmetrical orientations M. Elshazly-Zaghloul R. M.A. Azzam University of New Orleans, razzam@uno.edu Follow this and additional works at: https://scholarworks.uno.edu/ee_facpubs  Part of the Electrical and Electronics Commons, and the Physics Commons Recommended Citation M. Elshazly-Zaghloul and R. M. A. Azzam, "Inversion of the nonlinear equations of reflection ellipsometry for uniaxial crystals in symmetrical orientations," J. Opt. Soc. Am. 70, 880-883 (1980) This Article is brought to you for free and open access by the Department of Electrical Engineering at ScholarWorks@UNO. It has been accepted for inclusion in Electrical Engineering Faculty Publications by an authorized administrator of ScholarWorks@UNO. For more information, please contact scholarworks@uno.edu. 880 J. Opt. Soc. Am., Vol. 70, No. 7, July 1980 0030-3941/80/070880-04$00.50 © 1980 Optical Society of America 880 Inversion of the nonlinear equations of reflection ellipsometry for uniaxial crystals in symmetrical orientations M. Elshazly-Zaghloul Electrical Engineering Department, Faculty of Engineering, Cairo University, Cairo, Egypt R. M. A. Azzam Department of Electrical Engineering, School of Engineering, University of New Orleans, Lakefront, New Orleans, Louisiana 70122 (Received 15 June 1979; revised 7 January 1980) The complex ordinary (No) and extraordinary (Ne) refractive indices of an absorbing uniaxial crystal can be determined using reflection ellipsometry. The measurements are taken with the optic axis parallel and perpendicular to the crystal's surface. The equations obtained are solved without resort to iterative methods; Nn and Nc are determined separately. Sixteen solution sets (Not Ne) are obtained and the correct solution can be easily identified. We present an optimum angle of incidence that minimizes the relative errors in No and Ne. INTRODUCTION With the increased interest in uniaxial crystals and their applications, e.g., in electro-optics, the need arises for accurate determination of their optical properties. One of the powerful tools for this purpose is ellipsometry. By studying the change of the polarization state of light upon reflection from the surface of a crystal, the optical properties of the crystal can be determined. Until now the equations relating the optical properties of a uniaxial crystal and the quantities measured by a conven­tional ellipsometer were solved using iterative numerical methods. In this paper we present a new inversion method for determining separately the ordinary (N0) and extraordi­nary (Ne) refractive indices. An eighth-degree polynomial in N0 is first obtained whose coefficients are determined from the measured data. This polynomial is solved using a new numerical method that is not iterative in nature.1 Knowing the solution for N0, we obtain Ne by direct substitution.2 A detailed example in which this method is applied to a GaSe crystal is presented. Also, using a simulated error analysis, we determine the range of angles of incidence φ with the least percent error in N0 and Ne caused by an error of φ for ice, calcite, and GaSe crystals. I. METHOD For a uniaxially symmetric crystal, two relatively simple cases can be distinguished depending on whether the optic axis (axis of symmetry) is in the plane of incidence or per­pendicular to it.3-5 With respect to the first case, consider only the configuration where the optic axis is perpendicular to the reflecting crystal surface. We take the optic axis par­allel to the z axis of a Cartesian xyz coordinate system. The optical constants N0 (Nx) and Ne (Nz) of the crystal may then be written as When the z axis is perpendicular to the crystal surface, the Fresnel reflection coefficients take the form where φ is the angle of incidence, n is the refractive index of the ambient, and TABLE I. Different solution values for Eq. (11). where If the 2 axis (optic axis) is in the crystal surface and is per­pendicular to the plane of incidence, the Fresnel reflection coefficients become where n, φ, X0, and Xe have the same meaning as before. From Eqs. (2), we can write ' This equation can be rewritten as where and Similarly, from Eqs. (4), we have which can be rewritten as where Substituting the value of Xe from Eq. (8a) into Eq. (6a) and squaring both sides, we obtain From Eqs. (3), we can write Substituting Eqs. (10) and the value of Xe from Eq. (8a) into Eq. (9) and rearranging, we obtain an eighth-degree polyno­mial in X0 in the form; 881 J. Opt. Soc. Am., Vol. 70, No. 7, July 1980 JOSA Letters 881 FIG. 1. Percent error of Nθ for ice and calcite crystals as a function of φ (°) caused by an angle-of-incidence error δφ = +0.1°. FIG. 3. Percent error of the real part of Nσ and Nθ in the case of a GaSe crystal as a function of φ (°) caused by an angle-of-incidence error δφ = +0.1°. Note, from Eqs. (12) and (13), that the polynomial coefficients Mn of Eq. (11) are all functions of the known (n, φ) and mea­sured (K1, K2) quantities. Using a readily available computer program, Eq. (11) can easily be solved for Xo.l Accordingly, we obtain eight values for Xo. For each value of Xo, we obtain one value for Xe by direct substitution in Eq. (8a). Now we have eight pairs of solutions (X0,Xe). From Eqs. (10) we obtain four (N0,Ne) solution pairs for each (XO,Xe) pair. In general, most of these 32 solution pairs for (N0,Ne) can be readily rejected by applying the physical constraint of a complex refractive index with a positive real part and a neg­ative imaginary one. The unrejected solution pairs can be further screened on the basis of a reflectance measurement.6 The correct solution pair can easily be distinguished using such a method. The eight solutions of the above obtained polynomial are thus I I . NUMERICAL EXAMPLE We have checked the above scheme by using the published values of No and Ne for several uniaxial crystals7,8 in forward calculations, Eqs. (5)-(7), to get the ratios of reflection coef­ficients ρ1 and ρ2 at selected values of φ. The inversion scheme of Sec. I, Eqs. (8)-(11), was applied using ρ1, ρ2, and φ (in backward calculations) to determine the solution sets (No,Ne). We take as a detailed example the case of a GaSe crystal at a wavelength of 0.3 μ at φ = 70° and assuming n = 1 (i.e., an air ambient). The coefficients of the eighth-degree polyno­mial, Eq. (11), obtained from direct calculations are given by Table I gives the solutions obtained for No and Ne. From this FIG. 2. Percent error of N0 for ice and calcite crystals as a function of φ (°) caused by an angle-of-incidence error δφ = +0.1°. FIG. 4. Percent error of the Imaginary part of N0 and Nθ In the case of a GaSe crystal as a function of φ (°) caused by an angle-of-incidence error áφ = +0.1°. 882 J. Opt. Soc. Am., Vol. 70, No. 7, July 1980 JOSA Letters 882 t ab le , it is clear t h a t t h e correct solution is easily identified, solut ion 8; N0 = (3.766, -0 .906 ) and Ne = (2.797, - 0 . 4 6 9 ) . III. ERROR ANALYSIS Forward calculations combined with the inversion me thod expla ined in Sec. I have been used to de te rmine the effect of errors of t he angle of incidence φ on the derived ordinary and ex t raord ina ry refractive indices as ob ta ined from the con­vent ional el l ipsometric measu remen t s of ρ1 a n d ρ2. Figures 1-4 show t h e pe rcen t error of N0 and Ne as a function of φ caused by 0.1° error of φ in t h e cases of ice, calcite, arid GaSe crystals. F rom these figures the conclusion can be drawn tha t t he choice of φ in the range from 50-70° leads to the min imum pe rcen t errors of t h e derived optical proper t ies (N0 and Ne) arising from given errors of angles of incidence.9 1A computer program based on the Automated Taylor Series method of Dr. Chang of the Computer Department of the University of Nebraska-Lincoln was used. It yields highly accurate results in a very short time (better than 12 significant figures). The program is in FORTRAN IV and can be obtained by writing to Dr. Chang or to the authors. The closeness of N0 and Ne should cause no problem because they are separated in two polynomials. 2The suggested inversion method is superior to the previously used numerical methods in several aspects. First, it does not require previous knowledge of the optical properties to be measured. Of course this is essential whenever we have an unknown crystal to be characterized. Second, its accuracy does not depend on the starting values of N0 and Ne (it needs no starting values). Third, it sepa­rates the determination of N0 and Ne. 3R. A. W. Graves, "Determination of the optical constants of aniso­tropic crystals," J. Opt. Soc. Am. 59, 1225-1228 (1969). 4A. Wünsche, "Neue formeln für die reflexion und brechung des lichtes an anisotropen medien," Ann. Phys. (Leipzig) 25, 201-214 (1970). 5D. den Engelsen, "Transmission ellipsometry and polarization spectrometry of thin layers," J. Phys. Chem. 76, 3390-3397 (1972). 6M. Elshazly-Zaghloul, R. M. A. Azzam, and N. M. Bashara, "Explicit solution for the optical properties of a uniaxial crystal in generalized ellipsometry," in Proceedings of the Third International Confer-ence on Ellipsometry, edited by N. M. Bashara and R. M. A. Azzam (North-Holland, Amsterdam, 1977), pp. 281-292. 7F. Meyer, E. E. de Kluizenaar, and D. den Engelsen, "Ellipsometric determination of the optical anisotropy of gallium selenide," J. Opt. Soc. Am. 63, 529-552 (1973). 8J. M. Bennett and H. E. Bennett, "Polarization," in Handbook of Optics, edited by W. G. Driscoll and W. Vaughan (McGraw-Hill, New York, 1978), pp. 10-1 to 10-164. 9This range of φ for minimum percent error of No and Ne (caused by 0.1° error of φ) is the same as we have previously found6 in the case of generalized ellipsometry applied to uniaxial crystals with the optic axis in the plane of the surface but is neither parallel nor perpendicular to the plane of incidence. JOSA Letters 883 883 J. Opt. Soc. Am., Vol. 70, No. 7, July 1980 

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Inversion of the nonlinear equations of reflection ellipsometry for uniaxial crystals in symmetrical orientations

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