Infrared Refractive Index and Thermo-optic Coefficient ...

1 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 19, NUMBER 3 (1998)293 Infrared Refractive Index AND Thermo-optic COEFFICIENTDI nfrared Refractive Index and Thermo-optic CoefficientMeasurement at APLW illiam J. Tropf and Michael E. Thomasevelopment of Infrared optical systems requires knowledge of the refractiveindex of the transmissive optical elements. Many Laboratory applications require theknowledge of the Refractive Index over a wide temperature range. The simplestexpression of temperature dependence is the derivative of Refractive Index withrespect to temperature, , the Thermo-optic Coefficient . A convenient method fordetermining Refractive Index in the Infrared is measuring interference between surfacesof a thick lamina.

2 Such transmittance measurements, made at several temperatures,provide a comprehensive picture of the temperature-dependent Refractive Index .(Keywords: Diamond, Infrared materials, KRS-5, Refractive Index , Silicon carbide, Thermo-optic Coefficient .)INTRODUCTIONFor many years, APL has measured and modeled theoptical properties of materials. Articles on this topicin past issues of the Technical Digest include the mea-surement of scatter,1 emissivity,2 reflectance,3 and thevisible Thermo-optic Coefficient ,4 as well as many ofthe optical properties of natural diamond5 and In this article, we discuss interferometricmeasurements to determine Refractive Index over abroad spectral range in the Infrared .

3 Temperaturedependence is obtained by taking measurements atseveral temperatures. Such temperature-dependentdata are rarely available in the literature. Results havebeen applied to several applications, particularly guid-ed missile INDEXThe Refractive Index arises from the molecularpolarizability a according to the Lorentz Lorenzformula:nnTT220123 +=rare()(, ),(1)where r is molecular density, T is temperature, and e0is the permittivity of free space. Differentiating Eq. 1with respect to temperature gives the temperaturedependence of Refractive Index (or Thermo-optic coef-ficient). Decreasing density with temperature (positivethermal expansion Coefficient , the usual case) decreas-es Refractive Index , whereas a positive change in po-larizability with temperature (the usual case) increasesrefractive Index .

4 Therefore, high thermal expansionmaterials such as the alkali and thallium halides havenegative Thermo-optic coefficients, whereas low ther-mal expansion materials such as diamond and siliconcarbide have positive Thermo-optic MEASUREMENTSThe highest accuracy Refractive Index measurementis made with the prism or minimum deviation method,APPLIED RESEARCH294 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 19, NUMBER 3 (1998)W. J. TROPFANDM. E. THOMAS which requires the material to be made into a prismwith a precisely known vertex. Some polycrystallinematerials, including those made by chemical vapordeposition, are difficult to make sufficiently thick foraccurate Index determination using this method.

5 Onthe other hand, interferometric measurements areeasily made on thin plates and provide a convenientway to determine Refractive Index , especially for ex-perimental Refractive Index n of a laminar sample withhighly parallel surfaces, separated by a distance L, canbe determined by measuring interference betweendirectly transmitted and internally reflected a sample acts as a solid etalon. Interferenceproduces intensity modulation with maxima occurringwhen the optical path length difference 2nL betweenthe two paths is exactly an integer number m ofwavelengths l, hencemLn mLnln==22or,(2)where n is the wavenumber (reciprocal of the wave-length, usually measured in inverse centimeters).

6 Thedifference between maxima (or minima) is called thefree spectral range D and is related to the refractiveindex (assuming small D) by12 LndndD().nnn=+(3)High spectral resolution measurements as in Fig. 1 areused to determine peak locations and hence free spec-tral range. Also shown in Fig. 1 is an ideal transmissioncalculation. The smaller range of the transmissionmodulation is due to effects such as nonparallel sur-faces, absorption and scatter, and imperfectly collimat-ed sample of Refractive Index 2 has thenominal free spectral range of cm21; therefore, aspectral resolution of cm21 or better is needed toaccurately determine the free spectral range.

7 We usea Bomem DA3 Fourier transform spectrometer resolution to make these Refractive INDEXE quation 3 gives the relationship between refrac-tive Index and free spectral range. We have developedtwo methods to extract n(n) from D(n) data:1. Use a model for n(n) and fit the free spectralrange data to this model. This method was firstapplied to the material KRS-57 and since then tosilicon carbide, diamond, and other Integrate Eq. 3 to directly determine n(n).Although method 1 is often used to model directlymeasured Refractive Index data, we believe we are thefirst to apply this method to determine Refractive indexfrom interferometric free spectral range 2 is also believed to be new and is a powerfultechnique for extending Refractive Index using inter-ferometric measurements.

8 The two methods comple-ment each other; in particular, method 1 can providea starting point for method to a Refractive Index ModelPutting n(n) in a suitable analytic form allows us tofit free spectral range data (in the form of 21/2LD)using a nonlinear, least-squares routine to derive themodel parameters. Long experience has shown theSellmeier equation to accurately model Refractive in-dex in transparent regions:8nSiiii22221(),nnnn=+ (4)where the parameters Si and ni of each term in thesummation are the strength and location, respectively,of resonances (absorptions) in the material. Experi-ence has also shown that, at most, three terms (cm 1)Transmission Figure of the room-temperature Infrared transmission ofa slab of polycrystalline diamond.

9 The modula-tion of transmission is due to interference between directly trans-mitted and internally reflected light. The black curve is a measure-ment and the red curve is a theoretical prediction for a perfectsample. The free spectral range D is the distance betweenmaxima (or, alternatively, minima).JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 19, NUMBER 3 (1998)295 Infrared Refractive Index AND Thermo-optic Coefficient adequate to model the most precise Refractive indexdata over the entire range of transparency: two repre-senting short-wavelength electronic absorption andone representing the Infrared lattice vibration absorp-tion. Often, however, interferometric data do notcover the entire range of transparency and all of themodel parameters of Eq.

10 4 cannot be determined. Insuch a case, terms of Eq. 4 are approximated by ex-panding them in a power series and selecting low-orderterms that can be determined using the data. Sellmeiermodel terms representing high-frequency (ultravioletor electronic) resonances are expanded as9 SSSS iiiiiiiiinnnnnnnnn2222244 =+++<L,(5a)which is also known as the Cauchy model. Thelow-frequency ( Infrared or vibrational) terms are ex-panded asSSSiiiiiiiinnnnnnnnn2222244 = >L.(5b)Once a suitable model is selected, the accuracy of thederived Refractive Index is dependent only on accuratesample thickness L and accurate location of the inter-ferometric maxima (accurate free spectral range).