Method for calculating optical properties of a thermal control material in the mid-infrared range

Abstract

The application relates to a method for calculating the optical properties of a thermal control material in the mid-infrared waveband, and belongs to the field of thermal control material design of a spacecraft. The application aims to solve the problem that the existing mainstream calculation ideas are mainly focused on the related calculation in the sunlight waveband, and the calculation and research on the optical properties of the material in the mid-infrared waveband are relatively less. The method comprises the following steps: establishing a unit cell model, optimizing the structure model with high precision through a step-by-step optimization method, calculating a second-order force constant matrix, a BORN effective charge, a dielectric constant tensor, a third-order force constant matrix and a three-phonon scattering rate, determining an optical phonon vibration mode of a Brillouin zone center with infrared activity, calculating the dielectric function of the material according to a Lorentz oscillator model, and further calculating the refractive index and extinction coefficient of the material. The application significantly improves the calculation precision and efficiency of the optical properties of the thermal control material in the mid-infrared waveband, provides a reliable theoretical basis for the research and development of new materials, and can effectively reduce the experimental cost and shorten the research and development period.

Description

Technical Field

[0001] This invention belongs to the field of spacecraft thermal control material design technology, specifically, it relates to a method for calculating the infrared optical properties of thermal control materials. Background Technology

[0002] Aerospace thermal control coatings, as a special functional material, can achieve precise temperature control by adjusting the infrared thermal radiation characteristics of spacecraft, and are widely used on the surfaces of spacecraft such as satellites, manned spacecraft, and space stations. Traditional thermal control materials include ZnO, TiO2, ZrO2, and SiO2, while newer intelligent thermal control materials include VO2 and WO2. x La 1-x Sr x Materials such as MnO3. The intrinsic infrared optical properties of these thermal control materials have a significant impact on the performance of their thermal control coatings. Therefore, developing new thermal control materials with excellent infrared optical properties is of great importance to meeting the thermal control requirements of spacecraft.

[0003] The core parameters for designing novel thermal control materials include their absorptivity in the solar band and their emissivity in the infrared band, which reflect the material's ability to regulate thermal radiation in space. The intrinsic optical constants of a material (refractive index n and extinction coefficient k) are fundamental to calculating its optical properties and are crucial for the development of novel thermal control materials and the design of thermal control coatings. Typically, the complex permittivity (ε) of a new material can be obtained through theoretical calculations, and then the optical constants can be derived using the relationship between the complex refractive index and the complex permittivity. However, current mainstream methods for calculating optical properties primarily focus on the analysis of the material's electronic behavior: based on Fermi's golden rule, the imaginary part of the permittivity is calculated by describing the photon absorption caused by electronic transitions; then, the real part of the permittivity is obtained through the Kramers-Kronig transformation. This method, because the light absorption process caused by electronic transitions is mainly concentrated in the higher-energy, shorter-wavelength solar band, cannot accurately calculate the optical properties of thermal control materials in the longer-wavelength mid-infrared band.

[0004] Due to the wavelength limitations of traditional calculation methods, the optical properties in the mid-infrared band cannot be accurately obtained when designing new thermal control materials. This makes it difficult to accurately predict the thermal control performance of materials, thus affecting the accuracy and reliability of spacecraft thermal control functions. Therefore, there is an urgent need to develop a calculation method for the mid-infrared optical properties of thermal control materials to accurately describe their infrared thermal radiation characteristics. This would provide a theoretical foundation and technical support for the research and development of new mid-infrared optical materials, and strongly promote the design of thermal control materials and the development of thermal control coatings. Summary of the Invention

[0005] Compared with the optical property calculation only for the electronic behavior, the present application provides a method for calculating the infrared optical property of dielectric material based on the phonon behavior. For the infrared waveband optical property of thermal control material, the present application is based on the density functional theory, adopts the VASP, Phonopy, THIRDORDER.PY and ShengBTE software package to perform phonon spectrum calculation, and fits the dielectric function in the infrared waveband according to the Lorentz oscillator model.

[0006] In order to achieve the above technical problems, the present application adopts the following technical solutions:

[0007] The method for calculating the optical property of dielectric material in the mid-infrared waveband comprises the following steps:

[0008] Step one, a unit cell model is established, and the unit cell model is optimized with high precision by a step-by-step optimization method;

[0009] Step two, the second-order force constant matrix, the BORN effective charge, the dielectric constant tensor, the third-order force constant matrix and the three-phonon scattering rate are calculated;

[0010] Step three, the optical phonon vibration mode of the Brillouin zone center with infrared activity is determined;

[0011] Step four, the dielectric function of the material is calculated according to the Lorentz oscillator model; the refractive index and the extinction coefficient of the material are calculated by using the relationship between the complex refractive index of the material and the dielectric function; that is, the calculation is completed.

[0012] Further limitation, the thermal control material is inorganic material, such as silicon dioxide (SiO2), aluminum oxide (Al2O3), vanadium dioxide (VO2), lanthanum strontium manganese oxide (La 1-x Sr x MnO3, LSMO), silicon nitride (Si3N4), silicon carbide (SiC), calcium fluoride (CaF2) and other inorganic materials.

[0013] Further limitation, in step one, the unit cell model is established by retrieving the unit cell information through the Materials Studio software or The Materials Project crystal website, and exporting the unit cell model cif file.

[0014] Further limitation, in step one, the unit cell model is optimized with high precision by a step-by-step optimization method through the following steps:

[0015] Step 1.1: convert the cif file to POSCAR file through VESTA software; generate POTCAR file according to unit cell elements and functional; generate KPOINTS and INCAR files through K-point and convergence test of cutoff energy ENCUT;

[0016] Step 1.2: Set the force convergence parameters in INCAR so that IBRION = 1, ISIF = 3, EDIFFG = -1E-2, and perform the first structure optimization; then copy the calculated CONTCAR to POSCAR, set EDIFFG to -1E-3, and perform the second optimization.

[0017] Step 1.3: Further improve the force convergence accuracy, set ISIF = 2, EDIFFG = -1E-4, and copy the CONTCAR calculated in the previous step to POSCAR, and perform the third optimization; complete the high-precision optimization.

[0018] Further limitation, in step two, the second-order force constant matrix, BORN effective charge, and dielectric constant tensor are calculated by the following steps: call the Phonopy software package, set IBRION = 8, LEPSILON =.TRUE. in INCAR, and calculate using density functional perturbation theory (DFPT); extract the high-frequency dielectric constant tensor and BORN effective charge from the calculated OUTCAR, and generate the BORN file; extract the FORCE_CONSTANTS second-order force constant matrix file from the calculated vasprun.xml file.

[0019] Further limitation, in step two, the third-order force constant matrix is calculated by the following steps: call the THIRDORDER.PY software package, appropriately expand the primitive cell, and consider a suitable number of nearest neighbors as the cutoff value, generate a number of POSCAR-00* files using the finite displacement method; copy the POSCAR-00* files to the job-00* folder and rename them as POSCAR, prepare the POTCAR, INCAR, KPOINTS files, set NSW = 0, IBRION = -1 in INCAR, and perform single-point energy calculations in sequence; after all calculations are completed, collect the FORCE_CONSTANTS_3RD third-order force constant matrix file from the vasprun.xml file obtained in each calculation.

[0020] Further limitation, in step two, the three-phonon scattering rate is calculated by the following steps: generate the CONTROL file according to the element type, lattice information, etc., and place it in the same folder as the FORCE_CONSTANTS_2ND second-order force constant matrix file calculated in step 2.1 and the FORCE_CONSTANTS_3RD third-order force constant matrix file calculated in step 2.2; call the ShengBTE software package for calculation; extract the three-phonon scattering rate from the calculated BTE.w_anharmonic file.

[0021] Further limited, the optical phonon vibration mode with the Brillouin zone center of infrared activity is determined by the following steps:

[0022] Step 3.1: The tolerance is set to 0.001, the symm.conf configuration file is generated, and the command phonopy--readfc--dim="111" symm.conf is executed, wherein phonopy represents calling the phonopy program, --readfc represents reading the force constant, --dim="111" represents setting the supercell size of the crystal, "111" represents only considering the initial primitive cell, and symm.conf represents the symm.conf configuration file for assisting phonon calculation; the frequency of the optical phonon vibration mode at the Gamma point (0, 0, 0) and the irreducible representation are extracted from the calculation result irreps.yaml file;

[0023] Step 3.2: Upload the crystal cif file to the Bilbao Crystallographic Server website (https: / / www.cryst.ehu.es / rep / sam.html) to obtain the irreducible representation of the infrared active optical phonon mode of the crystal; by comparing the results of step 3.1, the frequency corresponding to the optical phonon mode with infrared activity can be obtained;

[0024] Step 3.3: According to the high symmetry point path of the unit cell, the band.conf configuration file is generated, and the command phonopy-cPOSCAR-p-s band.conf is executed, wherein phonopy represents calling the phonopy program, -c POSCAR represents reading the crystal structure file POSCAR, -p represents calculating the phonon band structure diagram, and -s band.conf represents specifying the band.conf file for calculation; the phonon spectrum is obtained after the calculation is completed;

[0025] Step 3.4: Considering the infrared active mode longitudinal wave-transverse wave (LO-TO) splitting, enabling non-analytical correction, setting NAC=.TRUE., and specifying the non-analytical correction direction Q_DIRECTION, generating a new symm.conf configuration file and a band.conf configuration file; the commands of steps 3.1 and 3.3 are executed respectively to obtain the frequency of the modified optical phonon vibration mode at the Gamma point (0, 0, 0) and the phonon spectrum;

[0026] Step 3.5: Comparing the frequencies of the optical phonon vibration modes before and after considering the LO-TO splitting and the phonon spectrum, at the Gamma point, the higher frequency after the LO-TO splitting is the LO frequency, and the lower frequency is the TO frequency.

[0027] Further limited, in step four, the dielectric function of the material ε (ω) is calculated by the Lorentz oscillator model shown in the following formula:

[0028]

[0029] Wherein ε (ω) is the dielectric function, ε ∞ is the high-frequency dielectric constant, ω j is the resonance frequency, j represents all infrared active modes, ω is the frequency of the mid-infrared band, S j is the oscillator strength of the jth phonon mode, γ j is the three-phonon scattering rate of the jth phonon mode, LO and TO represent longitudinal and transverse optical phonon modes, respectively;

[0030] The refractive index n (ω) and extinction coefficient k (ω) of the material are calculated by the following relationship:

[0031]

[0032] Wherein n (ω) is the refractive index, k (ω) is the extinction coefficient, ε1 (ω) is the real part of the dielectric function, and ε2 (ω) is the imaginary part of the dielectric function.

[0033] The present application can accurately describe the optical properties of thermal control materials in the infrared band by using the influence of phonon action on the dielectric function in the infrared band. By calculating the phonon mode with infrared activity and using the Lorentz oscillator model to describe it, the present application effectively captures the contribution of the phonon mode to the dielectric function of the material. The Lorentz oscillator model provides accurate frequency dependence for the calculation of the dielectric function by simulating the dynamic behavior of the interaction between the phonon mode and the photon, thereby enhancing the prediction ability of the optical response of the material in the mid-infrared band. The method of the present application provides a more detailed theoretical tool for optimizing infrared optical performance and designing new materials.

[0034] Compared with the prior art, the present application has the following beneficial effects:

[0035] The present application provides a new method for calculating the optical properties of thermal control materials in the mid-infrared band, which can more comprehensively consider the optical properties of the material in the band, and makes up for the shortcomings of the prior art in the application of the mid-infrared band. Based on the density functional theory, the VASP, Phonopy, THIRDORDER.PY and ShengBTE software packages are used to fit the dielectric function by the Lorentz oscillator model, considering the influence of phonon action on the optical properties, which significantly improves the calculation accuracy and efficiency of the optical properties of thermal control materials in the mid-infrared band.

[0036] The method provides a reliable theoretical basis for research and development of new materials in a mid-infrared wave band, promotes development of intelligent thermal control and other technologies, reduces material consumption in experiments, thereby reducing cost and shortening a research and development period, and has a wide application prospect.

[0037] For further understanding of the features and technical contents of the present application, please refer to the following detailed description of the present application and the accompanying drawings, which are provided for reference and illustration only, and do not limit the present application. BRIEF DESCRIPTION OF DRAWINGS

[0038] Figure 1 The calculation flowchart of embodiment 1 of the present application. DETAILED DESCRIPTION

[0039] The present application will be described in detail below with reference to specific embodiments. The following embodiments will help those skilled in the art to further understand the present application, and do not limit the present application in any form. It should be pointed out that, for those skilled in the art, without departing from the concept of the present application, a number of modifications and improvements can be made. These all belong to the protection scope of the present application. DETAILED DESCRIPTION 1

[0041] The present embodiment provides a calculation method for optical properties of a thermal control material in a mid-infrared wave band. The calculation uses VASP, Phonopy, THIRDORDER.PY, ShengBTE software packages, and fits a dielectric function according to a Lorentz oscillator model, as shown in the following formula: Figure 1 The specific calculation of the dielectric function of α-SiO2 in the mid-infrared wave band is as follows:

[0042] Step 1. Establish a unit cell model, and optimize the structure model with high precision;

[0043] The step 1 is specifically as follows:

[0044] Step 1.1: Retrieve α-SiO2 unit cell information through The Materials Project crystal website (https: / / legacy.materialsproject.org), and export a cif file;

[0045] Step 1.2: Convert the cif file into a POSCAR file through VESTA software; generate a POTCAR file by using a PBE (Perdew-Burke-Ernzerhof) exchange correlation functional; set a Gamma k-point network of 9x9x5 for K points, and set a cutoff energy ENCUT of 520eV, to generate KPOINTS and INCAR files;

[0046] Step 1.3: First structure optimization is performed by using VASP software package, setting IBRION = 1, ISIF = 3, EDIFFG = -1E-2 in INCAR; then copying the calculated CONTCAR as POSCAR, setting EDIFFG as -1E-3, and performing second optimization;

[0047] Step 1.4: Further improving the force convergence accuracy, setting ISIF = 2, EDIFFG = -1E-4, and copying the CONTCAR calculated in the previous step as POSCAR, and performing third optimization; high-precision optimization is completed.

[0048] Step 2: Calculating the second-order force constant matrix, BORN effective charge, dielectric constant tensor, third-order force constant matrix, and three-phonon scattering rate;

[0049] The step 2 is specifically:

[0050] Step 2.1: Calling Phonopy software package, setting IBRION = 8, LEPSILON =.TRUE. in INCAR, and performing calculation by using density functional perturbation theory (DFPT); extracting high-frequency dielectric constant tensor and BORN effective charge in the calculated OUTCAR, and generating BORN file, and extracting FORCE_CONSTANTS second-order force constant matrix file from the calculated vasprun.xml file;

[0051] Step 2.2: Calling THIRDORDER.PY software package, performing 2x2x2 supercell expansion on the primitive cell, selecting the fifth nearest neighbor atom as the cutoff value, and generating 264 POSCAR-00* files by using finite displacement method; copying the POSCAR-00* files to the job-00* folder and renaming them as POSCAR, preparing POTCAR, INCAR, KPOINTS files, setting NSW = 0, IBRION = -1 in INCAR, and sequentially performing single-point energy calculation; after all the calculations are completed, collecting and generating FORCE_CONSTANTS_3RD third-order force constant matrix file from the vasprun.xml file obtained in each calculation;

[0052] Step 2.3: Generating CONTROL file according to element type and lattice information, and placing the FORCE_CONSTANTS_2ND second-order force constant matrix file calculated in step 2.1 and the FORCE_CONSTANTS_3RD third-order force constant matrix file calculated in step 2.2 in the same folder; calling ShengBTE software package for calculation; extracting three-phonon scattering rate from the calculated BTE.w_anharmonic file.

[0053] Step 3: determining the optical phonon vibration mode with infrared activity in the Brillouin zone center;

[0054] The step 3 is specifically:

[0055] Step 3.1: setting the tolerance to 0.001, generating a symm.conf configuration file, and executing the command phonopy--readfc--dim="222" symm.conf, wherein phonopy represents calling the phonopy program, --readfc represents reading the force constant, --dim="222" represents setting the supercell size of the crystal to 2x2x2 times the primitive cell, and symm.conf represents the symm.conf configuration file used to assist phonon calculation; the frequency and irreducible representation of the optical phonon vibration mode at the Gamma point (0,0,0) are extracted from the calculation result irreps.yaml file;

[0056] Step 3.2: uploading the cif file of the α-SiO2 crystal on the Bilbao Crystallographic Server website (https: / / www.cryst.ehu.es / rep / sam.html) to obtain the irreducible representation of the infrared active optical phonon mode of the crystal; by comparing the results of step 3.1, the frequency corresponding to the infrared active optical phonon mode can be obtained;

[0057] Step 3.3: generating a band.conf configuration file according to the high-symmetry point path of the unit cell, and executing the command phonopy-cPOSCAR-p-s band.conf, wherein phonopy represents calling the phonopy program, -c POSCAR represents reading the crystal structure file POSCAR, -p represents calculating the phonon band structure diagram, and -s band.conf represents specifying the band.conf file used for calculation; the phonon spectrum diagram is obtained after the calculation is completed;

[0058] Step 3.4: considering the infrared active mode longitudinal wave-transverse wave (LO-TO) splitting, enabling non-analytical correction, setting NAC=.TRUE., and specifying the non-analytical correction direction Q_DIRECTION, generating a new symm.conf configuration file and a band.conf configuration file; the commands of steps 3.1 and 3.3 are executed respectively to obtain the frequency of the optical phonon vibration mode at the Gamma point (0,0,0) and the phonon spectrum diagram after the correction;

[0059] Step 3.5: Comparing the frequencies of the optical phonon vibration modes before and after the LO-TO splitting with the phonon spectrum, at the Gamma point, the higher frequency after the LO-TO splitting is the LO frequency, and the lower frequency is the TO frequency.

[0060] Step 4: Calculate the dielectric function of the material according to the Lorentz oscillator model; calculate the refractive index and extinction coefficient of the material by using the relationship between the complex refractive index and the dielectric function of the material;

[0061] The step 4 is specifically:

[0062] The dielectric function ε(ω) of the material is calculated by the Lorentz oscillator model as shown in the following formula:

[0063]

[0064] Wherein, ε(ω) is the dielectric function, ε ∞ is the high-frequency dielectric constant, ω j is the resonance frequency, j represents all infrared active modes, ω is the frequency in the mid-infrared band, S j is the oscillator strength of the jth phonon mode, γ j is the three-phonon scattering rate of the jth phonon mode. LO and TO represent longitudinal and transverse optical phonon modes, respectively; the refractive index n(ω) and extinction coefficient k(ω) of the material are calculated by the following relationship:

[0065]

[0066] Wherein, n(ω) is the refractive index, k(ω) is the extinction coefficient, ε1(ω) is the real part of the dielectric function, and ε2(ω) is the imaginary part of the dielectric function.

[0067] The method of the embodiment significantly improves the calculation accuracy and efficiency.

[0068] The above is only a preferred embodiment of the method for calculating the optical properties of a thermal control material in the mid-infrared band, and the protection scope of the method for calculating the optical properties of a thermal control material in the mid-infrared band is not limited to the above-mentioned embodiments. Any technical solution under this idea shall be within the protection scope of the present application. It should be noted that for those skilled in the art, some improvements and changes without departing from the principles of the present application shall be considered as the protection scope of the present application.

Claims

1. A method for calculating optical properties in the mid-infrared range of a thermal control material, characterized in that, Comprising the following steps: Step one, establish the unit cell model, through the method of step-by-step optimization of the unit cell model for high-precision optimization; Step two, calculate the second order force constant matrix, BORN effective charge, dielectric constant tensor, third order force constant matrix, three phonon scattering rate; Step three, determine the optical phonon vibration mode with the center of the Brillouin zone of infrared activity; Step four, according to the Lorentz oscillator model to calculate the dielectric function of the material; namely, the calculation is completed; In step one, high-precision optimization is realized by the following steps: Step 1.1: convert cif file to POSCAR file by VESTA software, then generate POTCAR file, and then test the convergence of K points and cutoff energy ENCUT to generate KPOINTS and INCAR file; Step 1.2: set the force convergence parameter in INCAR, so that IBRION=1, ISIF=3, EDIFFG=-1E-2, and perform the first structure optimization, then copy the calculated CONTCAR to POSCAR, and set EDIFFG to-1E-3 for the second optimization; Step 1.3: further improve the force convergence accuracy, so that ISIF=2, EDIFFG=-1E-4, and copy the CONTCAR calculated in the previous step to POSCAR for the third optimization; complete high-precision optimization; In step two, the second order force constant matrix, BORN effective charge and dielectric constant tensor are calculated by the following steps: call the Phonopy software package, set IBRION=8 and LEPSILON=.TRUE. in INCAR, and calculate using density functional perturbation theory (DFPT); extract the high-frequency dielectric constant tensor and BORN effective charge from the calculated OUTCAR, and generate the BORN file; extract the FORCE_CONSTANTS second order force constant matrix file from the calculated vasprun.xml file; In step two, the third order force constant matrix is calculated by the following steps: call the THIRDORDER.PY software package, appropriately expand the primitive cell, and consider a suitable number of adjacent atoms as the cutoff value, generate a number of POSCAR-00* files using the finite displacement method; copy the POSCAR-00* file to the job-00* folder and rename it as POSCAR, prepare POTCAR, INCAR, KPOINTS files, set NSW=0 and IBRION=-1 in INCAR, and perform single-point energy calculation in turn; after all calculations are completed, collect the FORCE_CONSTANTS_3RD third order force constant matrix file from the vasprun.xml file obtained in each calculation; In step two, the three-phonon scattering rate is calculated by the following steps: generating a CONTROL file according to the element type, lattice information, etc., and placing it in the same folder as the FORCE_CONSTANTS_2ND second-order force constant matrix file and the FORCE_CONSTANTS_3RD third-order force constant matrix file; The ShengBTE software package is called for calculation; the three-phonon scattering rate is extracted from the BTE.w_anharmonic file obtained by calculation.

2. The method of claim 1, wherein, The thermal control material is an inorganic material.

3. The method of claim 1, wherein, In step one, the unit cell model is established by retrieving the unit cell information through the Materials Studio software or The Materials Project crystal website, and exporting the unit cell model cif file.

4. The method of claim 1, wherein, The POTCAR file is generated according to the unit cell elements and the functional.

5. The method of claim 1, wherein, In step three, the optical phonon vibration mode with the center of the Brillouin zone having infrared activity is determined by the following steps Step 3.1: The tolerance is set to 0.001, a symm.conf configuration file is generated, and the command phonopy --readfc --dim="1 1 1" symm.conf is executed, where phonopy represents calling the phonopy program, --readfc represents reading the force constant, --dim="1 1 1" represents setting the supercell size of the crystal, "1 1 1" is only considering the initial unit cell, symm.conf represents the symm.conf configuration file, which is used to assist phonon calculation; the frequency and irreducible representation of the optical phonon vibration mode at the Gamma point (0, 0, 0) are extracted from the calculation result irreps.yaml file; Step 3.2: Upload the crystal cif file to the Bilbao Crystallographic Server website to obtain the irreducible representation of the infrared active optical phonon mode of the crystal; compare the results of step 3.1 to obtain the frequency corresponding to the infrared active optical phonon mode; Step 3.3: According to the high-symmetry point path of the unit cell, a band.conf configuration file is generated, and the command phonopy -c POSCAR -p -s band.conf is executed, where phonopy represents calling the phonopy program, -c POSCAR represents reading the crystal structure file POSCAR, -p represents calculating the phonon band structure diagram, and -s band.conf represents specifying the band.conf file for calculation; The phonon spectrum is obtained after calculation; Step 3.4: Considering the LO-TO splitting of the infrared active mode longitudinal wave-transverse wave, enabling non-analytic correction, setting NAC=.TRUE., and specifying the non-analytic correction direction Q_DIRECTION, a new symm.conf configuration file and a band.conf configuration file are generated; The commands of steps 3.1 and 3.3 are executed respectively to obtain the frequency of the modified optical phonon vibration mode at the Gamma point (0, 0, 0) and the phonon spectrum; Step 3.5: Comparing the frequencies of the optical phonon vibration modes before and after the LO-TO splitting with the phonon spectrum, at the Gamma point, the higher frequency is the LO frequency and the lower frequency is the TO frequency after the LO-TO splitting occurs.

6. The method of claim 1, wherein, In step four, the dielectric function of the material is calculated by the following equation e ( w ) where e w is a dielectric function, e ∞ is the high-frequency dielectric constant, w j is the resonance frequency, j denotes all infrared active modes, e is the frequency of the mid-infrared band, S j is the oscillator strength of the j th phonon mode, w j is the three-phonon scattering rate of the j th phonon mode, LO and TO denote longitudinal and transverse optical phonon modes, respectively;​ The refractive index of the material is calculated by the following relation n ( w ) and the extinction coefficient k ( g ) : wherein n w n is the refractive index, k w k is the extinction coefficient, w 1 w ε1 is the real part of the dielectric function, w 2 e w e w ε2 is the imaginary part of the dielectric function.​​

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