1 Refractive index measurements of Ge John H. Burnett*a, Simon G. Kaplana, Erik Stoverb, and Adam Phenisc aNational Institute of Standards and Technology, Physical Measurement Laboratory, 100 Bureau Dr., Gaithersburg, MD USA 20899; bM3 Measurement Solutions, Inc., 938 S. Andreasen Dr., Suite I, Escondido, CA USA 92029; cAMP Optics, 13308 Midland Rd., #1304, Poway, CA USA 92074 ABSTRACT A program has been started at NIST to make high-accuracy measurements of the infrared (IR) index properties of technologically important IR materials, in order to provide the IR optics community with updated values for the highest- quality materials now available. For this purpose, we designed and built a minimum-deviation-angle refractometry system enabling diffraction-limited index measurements for wavelengths from 0.12 µm to 14 µm. We discuss the apparatus and procedures that we use for IR measurements. First results are presented for germanium for the wavelength range from 2 µm to 14 µm, with standard uncertainties ranging from 2 × 10-5 near 2 µm to 8 × 10-5 near 14 µm. This is an improvement by about an order of magnitude of the uncertainty level for index data of germanium generally used for optic design. A Sellmeier formula fitting our data for this range is provided. An analysis of the uncertainty is presented in detail. These measurements are compared to previous measurements of Ge. Keywords: index, index of refraction, refractive index, dispersion, refractometry, germanium, Ge, infrared 1. INTRODUCTION Since germanium (Ge) first began to be developed for transistor and infrared (IR) optics applications in the late 1940s, the material quality has continually improved to meet increasingly stringent demands. For optics applications, this has driven the need for more accurate optical properties measurements, especially the index of refraction in the high IR transmission wavelength region (2 µm to 14 µm for dry air), to meet the requirements of optical design. Ge measurements by various methods in the 1950s reported absolute index measurement accuracies ranging from high 10-3 to a few parts in 10-4, with results between labs differing by as much as 1 percent.1,2,3,4 These materials had various doping and defect levels. The measurements by Salzberg and Villa3,4 in 1957 and 1958 for the range 2 µm to 16 µm have been used extensively for IR optics design for two decades since their publication, and to some extent even today. By the mid-1970s, driven by the importance of Ge for IR applications, material quality and index measurement methods had improved to the point where reported index measurements for the range of 2 µm to 14 µm generally had stated uncertainty estimates in the mid to low 10-4 range.5,6 In particular, Icenogle et al.5 reported measurements for the refractive indexes of Ge over the range 2.5 µm to 12 µm and their variations with temperature over the ranges 95 K to 298 K, with conservative estimates of the their standard uncertainties of ±6 × 10-4. These differed from previous measurements, e.g., those of Salzberg and Villa4, by as much as 5 × 10-3, when account is taken of the different measurement wavelengths and temperatures used. Icenogle et al.5 suggested that the difference may be due to different sample impurities. Edwin et al.6 reported measurements of a Ge sample at two laboratories over a subrange 8 µm to 14 µm by different deviation-angle methods, with a discrepancy of 3 × 10-4 between labs. These results differed from those of Icenogle et al. by ≤ 10 × 10-4 over the range of overlap (with the exception of longest wavelength index). In 1980, Li7 reviewed and tabulated all the measured index data on Ge and Si since 1949. Based on an analysis of all the data, he recommended index values for Ge for wavelength and temperature in the ranges 1.0 µm to 18 µm and 100 K to 550 K, in the form of an expression with the wavelength dispersion given by a single 1/λ2 term. The stated standard uncertainty for the index was ±2 × 10-3 over the wavelength and temperature range, consistent with the uncertainties claimed for the 1970s measurements discussed above. _______________________________________________ *[email protected]; phone 301-975-2679 2 Since the late 1970s, the results of Icenogle et al.5, and to some extent the previous results of Salzberg and Villa3,4, have been widely used in IR optics design, including incorporation in commercial optics design software. These values have been recommended in well-established index data reference books, such as the Handbook of Optical Constants of Solids8 and the Handbook of Infrared Optical Materials.9 With no substantial additions to these index data over the following 30 years, they remain the values IR optical designers must rely on, accepting a measurement uncertainty for each sample of at best (1 to 2) × 10-3 and sample-to-sample variation up to an order of magnitude higher. There have been a number of different methods used to measure the index of bulk materials, including techniques based on refraction angle, interference in slabs, surface reflection, and ellipsometry. For the transparent region of IR materials, the methods reporting the highest accuracy are variations of the minimum-deviation-angle refractometry method, based on measuring the angle of refraction through a prism of the material set at the angle of minimum deviation, discussed below. The accuracy of this approached is limited by the diffraction of the beam through the prism. For a prism about 3 cm on a side, it is straightforward to calculate that the diffraction-limited index uncertainty is approximately 5 × 10-7 in the ultraviolet, which has been verified.10 This diffraction-limited uncertainty should scale with wavelength approximately two orders of magnitude to the mid 10-5 range in the mid-IR. This is an order of magnitude smaller than the uncertainties reported for Ge discussed above. Thus, with an optimized refractometer design, an order of magnitude improvement in index accuracies should be achievable. Besides the inherent value of this higher accuracy for more precise IR optical design, this level of accuracy would be of value to sort out index variations in samples of commercial material, feeding back to material fabricators to improve material quality. Over the last several years a program has been established at NIST to pursue these goals for materials of importance to the IR optics community. In this paper we discuss the minimum-deviation-angle refractometry system developed at NIST and the procedures designed to achieve diffraction-limited index measurements in the IR out to 14 µm. Achieving this performance requires substantially reducing numerous sources of uncertainty compared to previous measurements, in some cases by an order of magnitude or more, as will be discussed. We will present our index measurements of a sample of high-quality Ge for wavelengths from 2 µm to 14 µm at the temperature of 22.000 °C. We fit these data to a Sellmeier formula and compare the values to previous measurements. 2. MEASUREMENT The well-established minimum-deviation-angle refractometry method is based on determining the angle δ (λ) that a collimated beam of known wavelength λ refracts through a prism of known apex angle α.11,12 If the prism is oriented so that the beam travels symmetrically through the prism with respect to the apex angle, then the refraction angle (deviation angle) is at a minimum and the refractive index of the prism material nmat(λ) relative to the index of the gas surrounding the prism ngas(λ) is given by the following formula: (λ) = (λ) (λ) = sin + (λ) 2 sin( 2). (1) ⁄ �� �⁄ �⁄ ⁄ A substantial benefit of measuring the deviation angle δ (λ) at the angle of minimum deviation is that the measurement is at an extremum condition, making it fairly insensitive to the alignment of the prism, a substantial source of error. Due to the conceptual simplicity of the index determination, not depending on any models other than Snell’s law, it is relatively secure from subtle sources of systematic errors. The accuracy of the determination of nmat(λ) simply depends on the accuracy of the of the determination of the parameters, λ, ngas(λ), α, and δ (λ). The uncertainties in α and δ (λ) depend in part on the geometric and surface properties of the prism, along with the prism index homogeneity and isotropy. We give some discussion on those aspects of our implementation of the minimum deviation angle technique relevant to the uncertainties of these various components. 2.1 NIST Minimum-Deviation-Angle Refractometer A custom minimum deviation angle refractometer system used for high-accuracy index measurements in the visible and ultraviolet, was modified with sources, detectors, and-all reflective optics appropriate for operation in the IR. A schematic diagram of the system is shown in Figure 1. The narrow-band IR light for the δ (λ) measurements was generated by filtering radiation from a broadband 1200 °C blackbody source by a grating monochromator and from the a 3.39 µm HeNe 3 laser. The grating orders were separated by the prism sample. The entire path of the radiation from the source to the detector was enclosed in sealed housings purged with dry N2 gas, to enable measurements in the low transmission regions of air with substantial H2O and CO2 content. θ/2θ Stacked Prism Goniometers Temperature Entrance Slit Control Unit Purge Box Sample Exit Slit Prism M5 M6 M3,M4 Monochromator Focusing Optics Collimating Optics 1 m Focal Length M8 Purge Housing Entrance Slit Exit Slit Gas Temperature Purge Box Controlled M9,M10 M2 Blackbody M7 HgCdTe Source Detector 1200 °C Chopper L-N2 Cooled M1 Figure 1.