Refractive index dispersion measurement of absorbing layers from transmittance spectra

Abstract

Abstract. We describe a fast and accurate method for the measurement of refractive index spectra of absorbing layers from transmittance spectra at two angles of incidence. The method is less sensitive to surface conditions than other photometric techniques. © Photometric techniques1-4 used to measure the optical constants of materials are easily adapted to obtain spectra of the optical constants and are suitable to be used with thin films. The accuracy is typically on the second or third significant digit. With adequate instrumentation the measurement of the reflectance R and transmittance T of a layered sample can be measured with relatively high precision ͑commonly of about 0.1%͒. Also, robust inversion algorithms have been devised to obtain the optical constants and thickness of thin absorbing films from a few optical transmittance measurements. 5 However, in all reported photometric techniques, the inversion of the experimental data relies on the use of Fresnel reflection coefficients; and surface roughness, scratches, or small particles on the surface may introduce large errors. In fact, in many cases the surfaces of a sample material do not have optical quality.In this letter, we describe a simple and accurate method to determine the refractive index spectra of an absorbing layer that is less sensitive to the surface conditions than other techniques.Consider a slab with a complex refractive index n + i and an optical beam incident at an angle i . If the slab absorbs most of the incident light in a single pass, we can ignore multiple reflections within the slab. It is not difficult to show that in this case the intensity of the transmitted beam is given bywhere I 0 is the intensity of the incident beam, d is the thickness of the slab, k 0 is the wave number in a vacuum, and T 1 and T 2 are the transmittance coefficients of the interfaces of the slab,where a = n 2 − 2 − sin 2 i , b =2n, i is the angle of incidence, and we have assumed that the refractive index of the incidence medium is one. If we suppose Ӷ n, and expand Eq. ͑2͒ in a power series of and neglect terms of order 2 we have Х / cos m where cos m = ͑1 − sin 2 i / n 2 ͒ 1/2 . Thus, m is given by the usual Snell's law with the real part of the refractive index of the slab.The method proposed in this letter consists of measuring the exponential decay factor of the transmittance, exp͑−2k 0 d / cos m ͒, at two angles of incidence, 1 and 2 , and then solving for the real part of the refractive index of the slab as n = ͩ where e j = exp͓2k 0 d / ͑1 − sin 2 j / n 2 ͒ 1/2 ͔. Note that the determination of n from Eq. ͑3͒ does not require knowledge of the film thickness, nor does it depend on the validity of Fresnel relations at the interfaces of the layer system. Now, to measure only the exponential factor we must normalize the transmittance measurements given by Eq. ͑1͒ by the factor I 0 T 1 T 2 . This factor must be estimated by additional measurements at angles of incidence, 1 and 2 .The error in determining n will come from the errors in estimating t...