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Abstract
Linear optical constants of thermally evaporated Zinc Selenide thin films are computed on the account of only transmittance experimental data. The refractive index and absorption coefficient dispersions of Zinc Selenide layers of small to close-to-bulk (50 nm ÷ 800 nm) thicknesses deposited on transparent quartz substrates are reckoned over a broad wavelength spectrum (300 nm ÷ 2500 nm). To distinct optical thickness classes, distinct analytical approach methods are implemented to compute the refractive index dispersions (Sellmeier equations) and absorption coefficient values, with the substrate’s influence accounted for. The implemented methods are exploited to infer the linear optical constants for practically any thin film thickness, with the premise that all considered films are deposited by a same deposition method. Although different deposition methods may lead to different thickness dependencies of the linear optical constants (thin films of the same material having to be regarded accordingly to the deposition process), the implemented approach stays. The study substantiates the possibility and uplifts the importance of prior knowledge on thickness dependent linear optical constants to design thin film structures with particular, custom features for linear and nonlinear photonics applications.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Chalcogenide glasses have been proved as suitable materials to be used in integrated optics, optical imaging, optical data storage, etc. [1–6]. Their transparency over a broadband spectrum and their high refractive index are important for optical designs that require high refractive index contrast.
Broad wavelength transmission spectrum up to 22 µm, high linear refractive index, n > 2, due to higher densities and weaker inter-atomic bonds than those characteristic to oxides glasses, non-hygroscopic, low crystallization tendency and low phonon energies suitable for rare-earth doping, qualify ZnSe as a particularly meaningful chalcogenide material for linear and nonlinear photonic applications [1–9].
Facing continuous increase in the number and in the complexity of particular requests from photonic applications, single and multilayered 1-dimensional, as well as 2-dimensional structures of thin films are more and more required to meet custom demands [10]. The optical properties of materials of sub-wavelength dimensionality may significantly differ from those of their bulk form. That raises the requisiteness for an accurate weight of the film thickness influence on refractive index and absorption coefficient over a broad spectral range, addressed to custom thin film designs with distinctive features. Therefore, the knowledge on the thickness dependent optical constants, prior to physical attainment of the thin films is a sought after data of importance for rigorous thin film structures. Besides, thin films obtained by different deposition methods, even using the same bulk material as source, may have different physical properties -grain size, crystallite size, crystal orientation, microstrain- and, as a consequence, different thickness dependent linear optical properties [11–19]. The actual limited knowledge on thickness and wavelength dependent thin films’ refractive index and absorption coefficient commonly leads to the employment of those of the bulk in order to design systems for linear and nonlinear photonic applications. That adds to the real time errors in determining the thin film thickness during the deposition process, especially for high deposition rates.
There is a constant interest to a straightforward way to compute the thin film linear optical constants’ wavelength dependencies over a broadband spectrum.
In this paper is studied the thickness influence on the refractive index and absorption coefficient for thermally evaporated Zinc Selenide (ZnSe) thin films with the substrate influence, in terms of its optical constants, being analyzed and accounted for. Two independent formalisms developed by Swanepoel [20], and Barybin and Shapovalov [21], respectively, have been considered in our approach for an accurate computation of the optical constants of ZnSe thin films with different thicknesses. The dependencies over a broad wavelength spectrum for film thicknesses ranging from ∼ 1 µm to tens of nanometers are computed only from the transmittance data. This study aims also to a general applicability to any transparent material in thin film form of any uniform thickness within the considered range, for after and, by numerical inter- and extrapolation, for prior thin film deposition process, independent of the deposition method used. For distinct optical thickness classes, distinct analytical approaches have been considered in computing the optical constants of the thin films.
A first thin film optical thickness class consists of thin films with a significant number of oscillation extremes in the transmission spectrum, meaningful for a mathematical built dispersion curve over the considered spectrum. For this class of thin film optical thicknesses, the Swanepoel method [20] is used to settle the corresponding refractive index and film thickness values, while the Barybin and Shapovalov method [21] is used to determine the absorption coefficient, for after and, subsequently from that, by numerical interpolation, for prior to deposition process of thin films in this class.
A second thin film optical thickness class consists of thin films of meaningless or null number of oscillation extremes in the transmission spectrum, for which the Swanepoel approach can no longer be applied. The results on optical constant dispersions of thin films with optical thicknesses in the first class are subsequently exploited, by numerical extrapolation, to estimate the linear optical constants of thin films with optical thicknesses in this second class. The refractive index and absorption coefficient of the thin films in the second class are subsequently settled by the numerical synthesis of their respective transmittance data.
The two thin film optical thickness classes cover any possible transmission optical response of transparent thin films over the considered transparency spectrum.
A validity check for the computed thicknesses of the thin films in the first class is performed by atomic force microscopy (AFM). As the thicknesses of the thin films in the second class cannot be computed, they have been calibrated by AFM measurements, too.
The results of the study are the computed thicknesses, the thickness dependent spectral dispersions of the refractive index (Sellmeier equations) and of the absorption coefficient for the ZnSe thin films in the first class, and also the estimated spectral dispersions of the same optical constants for the ZnSe thin films in the second class, all obtained from transmittance data only. The considered approach can be used to determine the linear optical parameters of any transparent thin film with uniform thickness, independent of the deposition method.
2. Experimental details
ZnSe thin films of different thicknesses have been obtained by thermal evaporation deposition technique using a Balzers BA 510 coating equipment. This deposition method is well known for thin film excellent homogeneity and constant layer thickness over significant areas [7,11,22].
The films have been deposited from high purity (99.99%) ZnSe pellets, from a Balzers’ genuine Tantalum boat, on optical quality plane parallel faces Synthetic Quartz Glass HPFS Standard Grade substrates (hereinafter referred to as quartz) of thickness ${d_s}$ = 1.31 mm and approximately 5 cm2 area in 1 × 10−6 Torr vacuum, with a deposition rate of ∼ 2 nm/s. In order to avoid the thin film contamination, high attention has been paid to a proper cleaning of the quartz substrate: multiple rinses in high purity ethyl alcohol, followed by 12 min acetone ultrasonic cleanse and subsequently dryness in high purity Ar flow.
The film thickness, ${d_f}$, was optically controlled by the equipment’s built-in optical monitoring system by means of Balzers interferential filters. The central wavelengths of the filters used are ${\lambda _1}$ = 397 nm, ${\lambda _2}$ = 553 nm, ${\lambda _3}$ = 615 nm, ${\lambda _4}$ = 992 nm, respectively.
Thin films of five different thicknesses, ${d_{fi}}({i = 1,\ldots ,5} )$ have been deposited. The wavelength dependent optical thickness, $({{d_f}} )(\lambda ),$ is defined as the wavelength dependent refractive index of the film, ${n_f}(\lambda ),$ multiplied by its geometrical thickness, $({{d_f}} )(\lambda )= {n_f}(\lambda )\cdot {d_f}.$ The optical thicknesses of the five deposited films in ascending order are $({{d_{f1}}} )= {\lambda _1}/4,$ $({{d_{f2}}} )= {\lambda _1}/2,$ $({{d_{f3}}} )= 3{\lambda _2}/4,$ $({{d_{f4}}} )= 3{\lambda _3}/2,$ $({{d_{f5}}} )= 7{\lambda _4}/4.$ For the real time estimation (i.e. during deposition process) of the thin film thickness, the unknown ZnSe thin film refractive index values, ${n_f}(\lambda ),$ have been substituted with known ZnSe bulk refractive index values, ${n_{bulk}}(\lambda ),$ at the same wavelengths.
Several Sellmeier type refractive index dispersion curves for ZnSe bulk can be found in literature, e.g. Connolly’s and Marple’s data [23,24], as shown in Fig. 1. For the real time estimation of the film thicknesses we used Marple’s data, with the refractive index values ${n_{bulk}}({{\lambda_1}} )$ = 3.27306, ${n_{bulk}}({{\lambda_2}} )$ = 2.64830, ${n_{bulk}}({{\lambda_3}} )$ = 2.59027, ${n_{bulk}}({{\lambda_4}} )$ = 2.47920, at the aforementioned filter wavelengths. The estimated thin film thicknesses are ${d_{f1}}$ = 30.3 nm, ${d_{f2}}$ = 60.6 nm, ${d_{f3}}$ = 156.6 nm, ${d_{f4}}$ = 356.1 nm, ${d_{f5}}$ = 700.2 nm, respectively.
The optical thicknesses of the deposited ZnSe thin films are specifically minded to cover both classes of optical responses in terms of transmittance within the considered transmission spectrum (i.e. 300 nm ÷ 2500 nm), gradually increasing from null to a meaningful number of oscillation extremes.
As the refractive index dispersion, with the thin film thickness as parameter, is unknown prior to the deposition process -an issue that the present paper addresses to-, the thin film thickness is only estimated in real time due to the obvious error of using the bulk refractive index and to the errors introduced by the in-situ optical thickness monitoring system of the coating equipment. These errors have a significant impact to the subsequent determination of the linear optical constants unless a calibration of the thickness, based on its direct measurement, is performed. Therefore, for each deposition process on the quartz substrate, corresponding to each intended thin film thickness, a twin, pre-cleaned glass substrate disk (2.5 cm diameter) was used as witness substrate on which a witness thin film is deposited, too. The witness glass substrate was placed in the immediate vicinity to the quartz substrate. Prior deposition, a narrow band (∼ 3 mm) of self-stick, vacuum proof Kapton tape strip was placed all along the central zone of the witness substrate. The corresponding tape covered area is, after the tape strip removal, a film-free zone. This zone delimits two opposite side zones of the deposited film, with step-like level profiles that lay along the witness’ diameter, subsequently allowing a direct measurement of the thin film thickness by means of atomic force microscopy (AFM).
3. Results and discussion
To compute the dependence of the refractive index of thermally evaporated ZnSe thin films on wavelength, with the film thickness as parameter, on the account of only the transmittance experimental data, we consider five ZnSe thin films of different thicknesses. The thickness of each of the five layers is constant over the entire 5 cm2 area of the quartz substrate. The thin film optical thicknesses, $({{d_{f1}}} )$ to $({{d_{f5}}} ),$ cover both classes of optical response in terms of transmission: oscillating type with meaningful number of maxima and minima, meaningless and oscillation free type, all for wavelengths within the considered spectrum, respectively.
Their transmission curves, together with the one of the quartz substrate, obtained using a Perkin-Elmer UV-Vis-NIR Lambda 900 spectrophotometer, are shown in Fig. 2. The surrounding medium for the system is air, with the corresponding refractive index ${n_{air}}$ = 1. The quartz substrate has the thickness ${d_s}$ (orders of magnitude larger than the film thickness, ${d_f}$), and its absorption is quasi negligible (${\alpha _s}$≅ 0).
Apart the critical requirement for the substrate’s high transmittance over the whole considered broadband spectrum, an accurate and complete prior knowledge on substrate’s refractive index dispersion is mandatory for the proper determination of the thin film optical properties, as it can have a significant contribution to the subsequent results of the employed numerical methods.
Numerous studies consider the compulsory for calculations substrate’s refractive index, ${n_s},$ determined from its transmittance data. According to the Swanepoel formalism, the substrate’s refractive index expression in terms of its transmittance data is [20,25–27]
where ${T_s}$ is the normalized transmittance of the substrate. According to Eq. (1), to quasi-constant transmittance values over the spectrum correspond quasi-constant refractive index values as well, which does not describe exactly the refractive properties for a transparent substrate. Moreover, apart from the spectral domains for which the Synthetic Quartz Glass HPFS has quasi-constant optical transmittance values, there are two narrow spectral regions for which sharp losses in its transmission spectrum can be observed (Fig. 3). In this case, the optical losses are due to the optical absorption of the material to the respective wavelengths and is not determined by a sudden, sharp, increase of the refractive index as Eq. (1) infers (Fig. 4, blue line).
Fig. 4. Refractive index of Synthetic Quartz Glass HPFS: 
Let us now consider the general expression of the Sellmeier equation:
An ellipsometric measurement for the quartz substrate describes its refractive index dispersion by the following Cauchy equation:
where $\lambda $ is the wavelength, in µm (Fig. 4, green curve).In the case of the quartz substrate, there are significant discrepancies between the refractive index values obtained from Eq. (1) and the producer’s catalogue data over the broadband (300 nm ÷ 2500 nm) considered spectrum (Fig. 4, blue and red curves). The discrepancies between the refractive index values obtained by ellipsometry and the producer’s catalogue data over the broadband (300 nm ÷ 2500 nm) considered spectrum are less significant (Fig. 4, green and red curves).
Therefore, according to Eq. (1), to the two sharp losses in the transmission curve of the substrate, positioned at wavelengths in the middle and towards the end of the considered spectrum, two sharp peaks in the substrate’s refractive index dispersion are supposed to correspond. These particular refractive index values, as well as the other refractive index values computed from Eq. (1) are in flagrant disagreement with the ones computed from Eq. (2) with the producer’s Sellmeier coefficients, thus representing an important source of errors for subsequent calculations. The errors in setting the ${n_s}$ values from Eq. (1) are greater for wavelengths in the near infrared domain of the considered broadband spectrum than for wavelengths in the ultraviolet and visible spectral domains. The refractive index values computed from Eq. (3), derived from ellipsometric measurements, are in a significant better agreement with those from Eq. (2) with the producer’s Sellmeier coefficients: for $\lambda $ = 0.5 µm, the refractive index value ${n_s}$ from Eq. (3) is 0.6% less than the value from Eq. (2), while for the same wavelength, the refractive index value ${n_s}$ from Eq. (1) is 2% greater. Thus, for transparent materials, which prevalently have a constant transmission over a broad spectrum, but with localized, sharp variations in the transmission curve, Eq. (1) must be precautionarily used.
For the first class of thin films, for which the transmission response is of oscillating type with a meaningful number of maxima and minima over the considered spectrum, we have used the Swanepoel formalism [20] to determine the Sellmeier equation for the refractive index dispersion as well as the thin film thickness. The thin film transmittance, $T,$ is a function of the light wavelength, of the film and substrate’s refractive indices $({{n_f},{n_s}} ),$ absorption coefficients $({{\alpha_f},{\alpha_s}} )$ and thicknesses $({{d_f},{d_s}} ),$ $T = T({\lambda ,{n_f},{n_s},{\alpha_f},{\alpha_s},{d_f},{d_s}} ),$ respectively. In the Swanepoel formalism, the contributions of ${\alpha _s}$ and ${d_s}$ are considered negligible and $T = T({\lambda ,{n_f},{n_s},{\alpha_f},{d_f}} ),$ as in Eq. (4), detailed in Appendix by Eqs. (16)–(19),
The Swanepoel formalism has been employed for ZnSe thin films of $({{d_{f3}}} )= 3{\lambda _2}/4,$ $({{d_{f4}}} )= 3{\lambda _3}/2,$ and $({{d_{f5}}} )= 7{\lambda _4}/4$ optical thicknesses. Two envelope functions, one connecting the maxima, ${T_M}(\lambda ),$ and one connecting the minima, ${T_m}(\lambda ),$ of the experimental transmittance are computed for each film in this class (Fig. 5).Fig. 5. Experimental transmittance of ZnSe thin film of $({{d_{f5}}} )$ optical thickness and the two envelopes, ${T_M}(\lambda )$ for maxima and ${T_m}(\lambda )$ for the minima.
For each value of a maximum ${T_M}({{\lambda_M}} )$ on ${T_M}(\lambda )$ envelope, the ${T_m}({{\lambda_M}} )$ value corresponding to the same wavelength, ${\lambda _M},$ on the opposed ${T_m}(\lambda )$ envelope is determined. In the same way, for each value of a minimum ${T_m}({{\lambda_m}} )$ on ${T_m}(\lambda )$ envelope, the corresponding ${T_M}({{\lambda_m}} )$ value from the ${T_M}(\lambda )$ envelope is determined at the wavelength ${\lambda _m}.$ As a consequence, for each film in this class there is a number of pairs of correlated points on the two envelopes, ${T_M}(\lambda )$ and ${T_m}(\lambda ),$ for an equal number of wavelengths, $\lambda .$ The thinnest film in this class, of optical thickness $({{d_{f3}}} ),$ has 3 pairs of correlated points on the two envelopes, while the thickest one, of optical thickness $({{d_{f5}}} ),$ has 11. The ${T_M}({{\lambda_M}} ),$ ${T_m}({{\lambda_M}} ),$ and, respectively, ${T_m}({{\lambda_m}} ),$ ${T_M}({{\lambda_m}} )$ pairs are used to compute, in a first approximation, the thin film refractive index using the expression [20,28,29]
where $N = {N_{m - w}}$ for the medium and weak absorption region and $N = {N_t}$ for the transparent region: The values for ${n_f}$ obtained, in the first approximation, using Eqs. (5) and (7), $N = {N_t},$ at wavelengths in the near infrared region of the considered spectrum, significantly differ from the values for ${n_f}$ obtained, in the first approximation, using Eqs. (5) and (6), $N = {N_{m - w}}.$ In the same time, the same values for ${n_f}$ computed in the first approximation using Eqs. (5) and (7), $N = {N_t},$ significantly differ, being much larger, than the ones from the Marple’s Sellmeier equation for the bulk ZnSe [24]. This is a clear indication that the visible and near infrared regions ($\lambda $ < 2.5 µm) correspond to the weak absorption region, $N = {N_{m - w}},$ for ZnSe thin films obtained by thermal evaporation. Consequently, only the Eqs. (5) and (6) are used to compute the ${n_f}$ values.In all analytical expressions, the substrate’s refractive index ${n_s},$ follows the Sellmeier equation provided by the producer of the quartz substrate, thus eliminating the errors that come from considering it as having the values derived by the use of Eq. (1).
Let us consider the values $n_f^1$ and $n_f^2$ of the linear refractive index computed, in the first approximation, at the wavelengths ${\lambda _1}$ and ${\lambda _2},$ corresponding to two adjacent maxima (or minima) of the transmittance for a thin film in the first class, as previously described. The thickness of the thin film, ${d_f},$ is given by [20,28,29]:
Thus, thickness values $d_f^j({j = 1,\ldots ,n} ),$ where n is the total number of adjacent maxima and minima pairs, are determined for each pair of adjacent maxima and each pair of adjacent minima of the experimental transmission curve. The average ${\bar{d}_f}$ of all $d_f^j$ values is the first approximation of the film thickness.Considering all the refractive index values determined in the first approximation, $n_f^k({k = 1,\ldots ,p} )$, where p is the total number of pairs of correlated points on the ${T_M}(\lambda )$ and ${T_m}(\lambda )$ envelopes, and the average thickness ${\bar{d}_f},$ we have determined the order numbers of the transmission curve extremes, ${M_k},$ as the closest integers (for maxima) or half integers (for minima) to the values ${m_k}$ satisfying [20,29,30]:
The determined order numbers, ${M_k},$ together with the values, $n_f^k,$ of the refractive index for the wavelengths corresponding to each pair of correlated points on ${T_M}(\lambda )$ and ${T_m}(\lambda )$ are introduced in and lead to new, more accurate, values for the thin film thickness, $d_f^{^{\prime}k}.$ A new, more accurate, value of the film thickness, $\bar{d}^{\prime}_f,$ is obtained by averaging all $d_f^{^{\prime}k}$ values. The values of ${M_k}$ and the new average value $\bar{d}^{\prime}_f$ can now be used to compute with increased accuracy, from an equation similar to Eq. (9), in which $d_f^{^{\prime}k}$ is replaced by $\bar{d}^{\prime}_f,$ the refractive index values, $n_f^{^{\prime}k},$ for the wavelengths ${\lambda _k}$ corresponding to the transmission extremes.The Sellmeier equations that describe the refractive index dispersions over the considered spectrum are obtained by numerically fitting the $n_f^{^{\prime}k}$ values for each ZnSe thin film, $n_f^{^{\prime}k} \equiv n_{fi}^{^{\prime}k}({i = 3,4,5} ),$ in the first class. Figure 6 is a graphical representation of the computed Sellmeier equations for each of the three ZnSe thin films in this class, with the numerically determined film thickness values $\bar{d}^{\prime}_{fi} \equiv {d_{fi}}$ as parameter, where $({i = 3,4,5} )$ is the identification index for the films in this class, together with the Marple’s Sellmeier equation for bulk ZnSe.
Fig. 6. Graphic representations of the Sellmeier equations for the refractive index of ZnSe bulk and ZnSe thin films in the first class.
The ${B_1},$ ${B_2},$ ${B_3}$ and ${C_1},$ ${C_2},$ ${C_3}$ coefficients of the Sellmeier equations in Fig. 6 are shown in Table 1.
Table 1. The coefficients of the Sellmeier equations for the ZnSe bulk and ZnSe thin films in the first class
In order to validate the computed Sellmeier equations and film thicknesses, as well as to test the thickness uniformity of ZnSe thin films in the first class, direct AFM measurements were performed using an XE-100 Park Systems microscope. The AFM measurements of step-like profiles were made on both sides of the film-free zone of the witness glass substrate surface, following several path lines orthogonal to the film-free zone borders.
For statistical reasons, 256 successive, noncontact mode scans have been carried out to create images of 40 µm x 40 µm size areas. The height (thickness) scale is translated into an unequivocally associated colour scale in which the darkest areas correspond to lowest heights and the lightest areas to the highest heights. In Fig. 7 is shown the colour-coded AFM image obtained for the thin film with the thickness ${d_{f5}}.$ Two profiles, obtained along the marked red and green paths, orthogonal to the film-free zone borders, and the histograms of the film thickness and of the film-free zone reference level are visible. The colour uniformity, the constant profiles, and the narrowness of the histograms associated to the film thickness and to the reference level are the experimental confirmation of the thin film thickness uniformity. The numerically determined thin film thicknesses $\bar{d}^{\prime}_{fi} \equiv {d_{fi}}({i = 3,4,5} ),$ are within 3% error of AFM thickness measurements for each witness sample, respectively.
The measured thicknesses and the computed thicknesses of the three ZnSe thin films in the first class are shown in the Table 2.
Table 2. The computed and the AFM measured thicknesses of ZnSe thin films in the first class.
For the second class of thin films, that is the oscillating type with meaningless number of maxima and minima (< 3) and oscillation free type transmittance data over the considered spectrum, the Swanepoel method cannot longer be applied. That is the case of ZnSe thin films of $({{d_{f2}}} )= {\lambda _1}/2$ and $({{d_{f1}}} )= {\lambda _1}/4$ optical thicknesses, respectively. To estimate the refractive index dispersion over the considered spectrum for the ZnSe thin films in this class, we used the results (Sellmeier equations) already obtained for the three ZnSe thin films in the first class. A number of refractive index values, corresponding to wavelengths separated by the same interval, is determined from each of these three Sellmeier equations and plotted as function of film thickness, this time with the wavelength, within the considered spectrum, as parameter. Twelve refractive index values separated by $\Delta \lambda $ = 200 nm wavelength intervals have been considered as shown in Fig. 8.
To the best of our knowledge, there is not any available approach to compute the thickness of the films in the second class from the transmittance data only. As the thicknesses are only estimated during the film deposition process by considering ZnSe bulk refractive index as previously described, more precise values were determined by AFM measurements for the two films in this class. Their measured thicknesses are ${d_{f1}}$ = 49.5 nm and ${d_{f2}}$ = 72.5 nm, respectively.
In a first instance, each of the twelve sets of three points (refractive index values, at the same wavelength, for the three thicknesses of the films in the first class) is fitted with a decaying towards small thickness function for ${n_f}$ of the form
From the twelve fitting functions, one for each wavelength considered, we determined two sets of refractive index values, one for each of the two films in the second class. Each of these two sets is subsequently fitted with a Sellmeier type equation.
The first instance$B_1^{1st},$ $B_2^{1st},$ $B_3^{1st}$ and $C_1^{1st},$ $C_2^{1st},$ $C_3^{1st}$ coefficients of the Sellmeier equations for the two films in the second class are shown in Table 3.
Table 3. The first instance coefficients of the Sellmeier equations for the ZnSe thin films in the second class
The graphic representation of the first instance Sellmeier equations for the two ZnSe films in the second class are shown together with the Sellmeier equations for the three ZnSe films in the first class and with the Marple’s Sellmeier equation for the ZnSe bulk in Fig. 9.
Fig. 9. Graphic representation of the Sellmeier equations for the refractive index of ZnSe bulk, for the refractive index of ZnSe thin films in the first class and for the first instance estimation of the refractive index of ZnSe thin films in the second class.
Besides the thickness validation ascertained by the direct AFM measurements, the validation of the numerically determined refractive indices and absorption coefficients of ZnSe thin films and their dependencies on the film thickness is ascertained by the numerically synthesized transmission curves and their match to the respectively experimental ones. The numerical process of ascertainment can lead to possible refinements of the already obtained data for the thickness influence on the refractive index and absorption coefficient for the ZnSe thin films.
There are numerous analytical studies to determine the absorption coefficient values once the refraction index ${n_f}(\lambda )$ dispersion, with the thin film thickness as parameter, are known [8,12,31–36]. Unlike the analytical approaches to determine the Sellmeier equation for ${n_f}(\lambda ),$ that are ultimately based on numerical methods to determine fitting functions extrapolated over the considered wavelength spectrum, the thin films’ absorption coefficients, ${\alpha _f},$ are determined by an wavelength-to-wavelength step-like numerical approach from equations derived from the expression for transmission [20,21,26,28,29,37–43].
A comprehensive study to numerically determine the thickness dependent thin films’ absorption coefficient is that elaborated by A. Barybin and V. Shapovalov [21] for both semi-infinite substrate thickness (3M structure) and finite substrate thickness (4M structure). In addition to the Swanepoel formalism [20], their analytical study takes into account the substrate’s absorption coefficient, ${\alpha _s}(\lambda ),$ and its thickness, ${d_s},$ in the case of a 4M structure. Therefore, all the contributions in terms of refractive index, absorption coefficient, and thickness of both the thin film and of the substrate are considered in deriving an expression for the thin film transmittance, $T,$ in the general form $T = T({\lambda ,{n_f},{n_s},{\alpha_f},{\alpha_s},{d_f},{d_s}} )$ [21]:
In order to wavelength-to-wavelength numerically determine the values for ${\alpha _f},$ the experimental values of $T$ and all the other variables in Eq. (12), detailed in Appendix by Eqs. (20)–(23), must be known. As previously mentioned, ${n_f}(\lambda )$ computed using the Swanepoel method for the thin films in the first class, respectively estimated as a first instance extrapolation for the thin films in the second class, and ${n_s}(\lambda )$ are known. The thin film thicknesses ${d_{fi}}({i = 1,\ldots ,5} )$ and the substrate thickness ${d_s}$ are known as well. For the thin films in the first class, the ${d_{fi}}({i = 3,4,5} )$ values were numerically determined using the Swanepoel method and validated by AFM measurements. For the thin films in the second class, the ${d_{fi}}({i = 1,2} )$ values were determined only by AFM measurements. While the substrate’s refractive index dispersion ${n_s}(\lambda )$ is well known from the producer data, ${\alpha _s}(\lambda )$ had to be determined. For that we used the geometric series formula for transmittance [44],With all variables ${n_f}(\lambda ),$ ${n_s}(\lambda ),$ ${\alpha _s}(\lambda )$ and the values of ${d_f}$ and ${d_s}$ known, we used the analytical expression for $T({\lambda ,{n_f},{n_s},{\alpha_f},{\alpha_s},{d_f},{d_s}} ),$ derived by A. Barybin and V. Shapovalov for a 4M structure in the case of a slightly absorbing thin film on a transparent substrate to numerically determine the ${\alpha _f}$ values [21,40].
The wavelength dependencies of the absorption coefficient for ZnSe thin films in the first class and of the first instance estimation of the absorption coefficient for ZnSe thin films in the second class are shown in Fig. 10.
Fig. 10. Absorption coefficients for ZnSe thin films in the first class and the first instance estimation of the absorption coefficient for ZnSe thin films in the second class.
For the thin films in the first class, the values of ${\alpha _f}$ decrease with the decreasing of film thickness in the region of strong absorption (short wavelengths of the considered spectrum), while at the end of the considered spectrum, at near infrared wavelengths, the opposite is valid -the values of ${\alpha _f}$ decrease with the increasing of thin film thickness.
For near infrared wavelengths ($\lambda $ ≥ 2.1 µm), the substrate’s contribution to the overall absorption of the system is dominant in regard to the substrate to thin film thickness ratio.
For the thin films in the second class, as for the refractive index, values of ${\alpha _f}$ in a first instance have been numerically determined. In this case, the derived parametric dependencies of both ${n_f}(\lambda )$ and ${\alpha _f}(\lambda )$ on film thickness are refined by numerically synthesizing the transmission curve and subsequently matching it to the experimental one. To validate the derived parametric dependencies of both ${n_f}(\lambda )$ and ${\alpha _f}(\lambda )$ on film thickness, we used the Swanepoel’s analytical expression for the transmittance $T({\lambda ,{n_f},{n_s},{\alpha_f},{d_f}} )$ [20] to numerically synthesize the film transmittance data. By the use of the Barybin and Shapovalov formalism, a perfect reproduction of the data would have been obtained as long as the ${\alpha _f}$ values have been computed by this method. For a good interpretation of the results it must be noted that the Swanepoel’s formalism analytical expression for T does not take into account the substrate’s contribution to the overall absorption of the structure.
For the ZnSe thin films in the first class there is a good match between the experimental transmittance data and the numerically determined ones, as shown in Figs. 11,12, and 13. The maxima and minima in the transmission spectrum overlap, both in what regards the position within the considered broadband spectrum -occurring at the same wavelengths credit to the refractive index-, as well in what regards the transmittance values -same transmittance values credit to the absorption coefficient-. As the thicknesses for the thin films in this class have been determined by the Swanepoel method and also validated by the AFM measurements, with the conditions imposed by Eq. (9) for maxima and minima being met in what concerns the optical thickness $({{d_{fi}}} )= {n_{fi}} \cdot {d_{fi}}({i = 3,4,5} ),$ the validity of the refractive index dispersions ${n_{fi}}(\lambda )$ and of the absorption coefficient values, ${\alpha _{fi}}(\lambda ),$ for the thin films in this class are proven.
For the second class of thin films there are two different behaviours: the one of the thin film of thickness equal to 72.5 nm, with only one maximum and one minimum in the transmission spectrum and the one of the thin film of thickness equal to 49.5 nm, with no maximum and no minimum in the transmission spectrum.
3.1 72.5 nm thin film
The first instance ${n_{f2}}(\lambda )$ and ${\alpha _{f2}}(\lambda )$ values are used to numerically reconstruct the transmittance data for the thin film of 72.5 nm by the use of the Swanepoel formalism’s analytical expression for T (substrate’s absorption and thickness contributions neglected). The reconstructed transmittance data presented a maximum in the strong absorption region at a lower wavelength than that of the experimental one (Fig. 14, dotted line). This is an indication that the refractive index values from the first instance ${n_{f2}}(\lambda )$ Sellmeier equation are smaller than the real ones (the maximum occurs at a lower wavelength). To correct this behavior in terms of the refractive index values, the conditions from Eq. (9) have been imposed to the maximum and to the minimum in this thin film experimental transmittance data for the given, measured by AFM, thickness, with ${M_k}$ = 1 for the maximum and ${M_k}$ = 0.5 for the minimum. These two particular new refractive index values, $n_{f2}^{\prime}(\lambda ),$ together with the first instance values of ${n_{f2}}(\lambda )$ for wavelengths in the near infrared domain determined a new Sellmeier equation for the 72.5 nm thin film and new $\alpha _{f2}^{\prime}(\lambda )$ values numerically computed using the Barybin and Shapovalov approach. The new numerically reconstructed transmittance data using the same Swanepoel formalism’s analytical expression for T has a maximum that now overlaps the one of the experimental transmittance and is also shown in Fig. 14.
Fig. 14. Experimental, first instance, and computed transmittance data for ${d_{f2}}$ ZnSe thin film.
The differences in transmittance data (higher values for the theoretical reconstructed curve than the values of the experimental one, for $\lambda $ ≥ 0.7 µm) are due to the fact that the numerically reconstructed transmittance is obtained using the Swanepoel formalism’s analytical expression for T which does not take into account the absorption of the substrate and its thickness, while the Barybin and Shapovalov formalism does (a perfect reconstruction of the experimental transmittance data would have been obtained using the later as long as the ${\alpha _f}$ values are computed from it).
The oscillating behaviour in the strong absorption region (for $\lambda $ ≤ 0.5µm) is due to the fact that the refractive index values for this region, obtained by extrapolating the resulted new Sellmeier equation for lower wavelengths, are overestimated, being an indication that these values are higher than the real ones.
As the absorption coefficient of the substrate could not be determined from the Eq. (11) and was neglected even in Barybin - Shapovalov formalism (from which the absorption coefficient of the film was determined), its contribution to the overall absorbance of the film – substrate system still exists. In deriving the absorption coefficient of the thin film from the experimental transmittance data, all absorption contributions (film + substrate) appear as being of the film only and making the absorption coefficient of the film larger than the real one.
However, even with these limitations in deriving the refractive index and the absorption coefficient of the considered thin film, the reconstructed transmission curve is in qualitative agreement with the experimental one.
3.1 49.5 nm thin film
As in the case of the previous thin film, the refractive index dispersion ${n_{f1}}(\lambda )$ and the numerically ${\alpha _{f1}}(\lambda )$ values determined in a first instance are introduced in Swanepoel formalism’s analytical expression for T to synthesize the transmittance data. The reconstructed transmittance presents a maximum in the region of strong absorption (Fig. 15, dotted line). This is an indication that the values for the refractive index from the first instance ${n_{f1}}(\lambda )$ are higher than the real ones. To correct this behavior, a supplementary condition expressing the fact that for a thin film thickness ${d_{f1}} \to $ 0, the refractive index value ${n_{f1}} \to $ 1 (like if there is not a single atom of the material). As the computed ${n_{f1}}(\lambda )$ and ${\alpha _{f1}}$ values stand for the thin film itself, the value for ${n_{f1}},$ when ${d_{f1}}$ “thickness” equals precisely the value ${d_{f1}}$ = 0, is that of the surrounding medium that is air, and equals 1. Each of the twelve sets of now four points of refractive index values plotted against thin film thickness with the wavelength as parameter has been refitted with the same curve given by Eq. (10), which determines new values for the fitting constants ${a_i},$ ${b_i},$ and ${d_{fic}},$ $i = 1$. Subsequently, a new Sellmeier equation for the linear refractive index $n_{f1}^{\prime}(\lambda )$ has been determined and, using again the Barybin and Shapovalov approach, new $\alpha _{f1}^{\prime}$ values. The new numerically synthesized transmittance data using Swanepoel formalism’s analytical expression for T is shown in Fig. 15.
Fig. 15. Experimental, first instance, and computed transmittance data for ${d_{f1}}$ ZnSe thin film.
We are attributing the higher values of the reconstructed transmittance data face to the values of the experimental one to the fact that the Swanepoel formalism’s analytical expression for T does not take into account the substrate contribution to the overall absorption of the system. The substrate influence is higher for thin films in the second class than for thin films in the first class.
The new ${B_1},$ ${B_2},$ ${B_3}$ and ${C_1},$ ${C_2},$ ${C_3}$ coefficients of the Sellmeier equations for the thin films in the second class, derived as above, are given in Table 4.
Table 4. The coefficients of the Sellmeier equations for the ZnSe thin films in the second class
The wavelength dependencies of the refractive index, ${n_{fi}}(\lambda )$ and of the absorption coefficient values, ${\alpha _{fi}}(\lambda ),({i = 1,\ldots ,5} ),$ for the five thermally deposited ZnSe thin films over the broadband considered spectrum, with the film thickness as parameter, are shown in Figs. 16 and 17, respectively.
Fig. 16. Graphic representation of the Sellmeier equations for the linear refractive index of ZnSe bulk and for the linear refractive index of ZnSe thin films in the first class, and for the linear refractive index of ZnSe thin films in the second class.
Fig. 17. Wavelength dependencies of the absorption coefficients for ZnSe thin films in the first class and in the second class.
The new values of the absorption coefficient for the thin films with optical thicknesses in the second class add to the previous observation that for wavelengths at the end of the considered spectrum, ${\alpha _f}$ decrease with the increasing of thin film thickness, with the individual substrate contribution to the overall optical absorption being considered (Barybin – Shapovalov formalism). The computed band gaps for the thin films are shown in Fig. 18.
The optical band gap, ${E_g},$ has been computed using the Tauc law’s equation for amorphous materials [31,45,46]
where h is the Plank constant, $\nu $ the frequency of incident light in vacuum, and A is a characteristic constant of the material.The optical band gap values decrease with the decrease of ZnSe thin film thickness for thin films in the first class and for the thin film of null number of oscillation extremes in the transmission spectrum, while an inflection point in the thickness dependent ${E_g}$ trend occur for the thin film of meaningless number of oscillation extremes in the transmission spectrum with, in this case, an ${E_g}$ value higher than the one for the thickest thin film.
Additionally, for further validation of our results, the linear refractive index dispersions ${n_f}(\lambda )$ for two ZnSe thin films independently obtained in [42] using the same deposition technique, have been computed by numerical interpolations and compared with the experimental transmittance data reported in the same Ref. [42]. The two considered ZnSe thin films from [42], the 272.5 nm (the thinnest one) and 676.8 nm (the thickest one), are of intermediate thicknesses between the thicknesses (${d_{f3}}$ = 183.3 nm, ${d_{f4}}$ = 399.3 nm, ${d_{f5}}$ = 790.8 nm) of the three films in the first class.
The refractive index dispersions ${n_f}(\lambda )$ for the two considered films from [42] have been determined using Eq. (10) for interpolation (Fig. 19).
Fig. 19. Graphic representation of the Sellmeier equations for the linear refractive index of two intermediate ZnSe thin film thicknesses in the first class.
The reconstructed transmittance data are in qualitative agreement with the experimental transmittance data for the two ZnSe thin films in [42], as shown in Figs. 20(a) and 20(b) for the ZnSe thin film of 275.2 nm and Figs. 21(a) and 21(b) for the ZnSe thin film of 676.8 nm, respectively.
Fig. 20. Transmittance data of ZnSe thin film of 275.2 nm thickness. (a) transmittance data (Ref. [42], Fig. 8 a); (b) computed transmittance data using the derived Sellmeier equation for the refractive index of a ZnSe thin film of 275.2 nm thickness.
Fig. 21. Transmittance data of ZnSe thin film of 676.8 nm thickness. (a) transmittance data (Ref. [42], Fig. 8 d); (b) computed transmittance data using the derived Sellmeier equation for the refractive index of a ZnSe thin film of 676.8 nm thickness.
As an empirical observation, for ZnSe thin films with thicknesses for which the transmittance data present a number of maxima = 1 and/or number of minima = 1, Eq. (10) without the condition ${n_f} \to $ 1 when the film “thickness” ${d_f} \to $ 0 is better for calculating refractive index values at larger wavelengths and, subsequently, to better agreement with the experimental transmittance data, while for ZnSe thin films with thicknesses that present transmittance data of null maxima and null minima, Eq. (10) with the condition ${n_f} \to $ 1 when the film “thickness” ${d_f} \to $ 0 leads to a better agreement with the corresponding experimental transmittance data, respectively.
The differences between the experimental transmittance data in [42] and our synthesized transmittance data using the Swanepoel formalism’s analytical expression for T are due to the use of different substrates (glass in [42] versus quartz with documented refractive properties in our study) and to the fact that while the Sellmeier equations ${n_f}(\lambda )$ are reckoned for the specific thin film thicknesses of 272.5 nm and 676.8 nm, the absorption coefficient values ${\alpha _f}$ used are the ones of the 399.8 nm and 790.8 nm ZnSe thin films, respectively, due to unknown values of the transmittance data in [42].
4. Conclusion
The linear optical constants of ZnSe layers of small to close-to-bulk thicknesses (50 nm ÷ 800 nm), deposited by thermal evaporation technique on transparent quartz substrates, have been computed for a broad wavelength spectrum (300 nm ÷ 2500 nm) from transmittance data only. Two distinct thin film optical thickness classes have been defined to compute the refractive index dispersion (Sellmeier equations): a first class of thin films that exhibit a significant, meaningful -for fitting purposes-, number of oscillation extremes in the transmission spectrum, for which the Swanepoel’s method to settle the corresponding refractive index is considered an exact approach and a second class of meaningless or null oscillation extremes in the transmission spectrum for which the Sellmeier equations of the films in the first class are extrapolated to sub-wavelength thin film optical thicknesses in this later class by means of refractive index representation as function of thickness with wavelengths within the considered spectrum as parameter. The substrate influence to reckon the linear optical constants of ZnSe thin films has also been considered.
For both thin film optical thickness classes, the AFM thickness measurements stand as references. The thicknesses of the thin films in the first class, computable from the transmittance data using the Swanepoel formalism, are in good agreement to the film thicknesses determined by AFM measurements.
The Sellmeier equations for the thin films’ refractive index, computed using the Swanepoel formalism, have been subsequently used to numerically determine the thin films’ absorption coefficient values using the Barybin and Shapovalov formalism, by a wavelength-to-wavelength step-like approach that takes into account the substrate thickness and its individual contribution to the overall optical absorption. To validate the results and to avoid a self-proof of the Barybin and Shapovalov formalism, the synthesized transmittance data have been computed by means of Swanepoel formalism’s analytical expression and compared to the corresponding experimental data.
For a further, more comprehensive substantiation of the results on the linear optical constants, the refractive index values for two ZnSe thin films with intermediate thicknesses to those of our thin films in the first class, obtained independently of our study, yet by the same deposition method, have been computed. The two corresponding synthesized transmittance data are in good qualitative agreement and consistent with the independently reported experimental transmittance data.
The results are of importance for after deposition and for prior to deposition knowledge on thickness dependent linear optical constants, with impact on rigorous design of thin film structures for linear and nonlinear photonics applications.
Appendix
General expression for transmittance in Swanepoel formalism
The coefficients of Eq. (4) are
General expression for transmittance, $T,$ for a finite substrate in Swanepoel formalism [20], where ${k_f}$ is the extinction coefficient of a thin film.
General expression for transmittance in Barybin-Shapovalov formalism
The coefficients of Eq. (12) are given by the expressions
Funding
Ministry of Research and Innovation (16N/2019).
Acknowledgments
The authors thank Dr. Mihaela Filipescu for the thin film thickness measurements by Atomic Force Microscopy.
Disclosures
The authors declare no conflicts of interest.
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Swanepoel formalism, Eq. (
producer, Sellmeier equation; 
ellipsometry, Cauchy equation.