Comparison of Measurement and Electromagnetic Overall Transfer Matrix Simulation Results of MgF 2 -Nb 2 O 5 Distributed Bragg Reectors with Different Layers

In this study, glasses were used as substrates and an e-beam was used the method to deposit MgF 2 and Nb 2 O 5 single-layer films, and the optical properties, including extinction coefficients (k values) and refractive indices (n values), were measured by using the light wavelength as variable. The equation d = λ/(4n) was used to calculate the thickness (d) of 1/4 wavelength (λ) for each layer of the MgF 2 -Nb 2 O 5 bilayer films in distributed Bragg reflectors (DBRs) with a designed reflective wavelength at blue light (~450nm). Each MgF 2 -Nb 2 O 5 bilayer film was called a period, and the glass substrates were used to deposit the films with two, four, and six periods for fabricating the DBRs. The field emission scanning electron microscope equipped with a focused ion beam was used to measure the thickness of only the maximum reflective ratio at a specific wavelength. An overall transfer matrix was investigated to calculate the reflective spectra by incorporating the variable n values and thicknesses of the MgF 2 -Nb 2 O 5 films in each layer. We show that the measured results of the fabricated DBRs matched the results simulated using Sheppard’s approximate equation and the overall transfer matrix.


Introduction
Multilayer films are composite films deposited using numbers of alternating layers of different materials (for example, metal oxides), these films have their specific thicknesses as they are with nanometer scale. Multilayer films with different materials have been widely used in a large number of applications because they have outstanding optical, photoactive, and protective properties. For example, to achieve a long lifetime for semiconductive lasers with a microcavity exciton-polariton, it is necessary to use a highly reflective distributed Bragg reflector (DBR) 1,2 . A DBR is a reflector designed for a specific wavelength with a narrow bandwidth, which can be used to design a notch filter 3 . DBRs are also used in waveguides, and they can be constructed from multiple layers of  9 , and Wang et al. used a CO2 laser to fabricate SiO2-Al2O3 glasses with a thermally stable n value 10 . Dubey and Ganesan used a sol-gel spin-coating technique to fabricate DBRs based on TiO2-SiO2 stacks, but as the TiO2-SiO2 films were deposited in seven periods, only about 90% reflectance was observed 11 . Muallem et al. used a thermal evaporation method to fabricate multilayer DBRs using CaF2 as material with lower n value and ZnS as material with higher n value, and they found that the fabricated DBRs exhibited a center wavelength of 550nm with a maximum reflectance more than 99% 12 .
Willey and Shakoury used ion beam-assisted deposition to deposit an MgF2 film without additional heat or fluorine, and the deposited film exhibited low absorbance, low scattering, and a low n value 13 . Chen et al. used a sputtering method to deposit Nb2O5 films under different oxygen concentrations and found that when they introduced oxygen during the deposition process, the deposited Nb2O5 film had low absorbance and a high n value 14 . In the present study, we used the MgF2 and Nb2O5 films as low and high refractive-index films to fabricate a DBR with a centrally reflective frequency of 450nm. An electron-beam (e-beam) deposition can be a simple method to deposit the different oxide films, and the properties of the deposited films depend on the deposition parameters, such as temperature and pressure, and the used gas during deposition 5,6,14 . The e-beam deposition method makes it easy to control the thicknesses of the films, even in multilayer structures or special structures, so the e-beam technology was used for deposition the MgF2-Nb2O5 bilayer films.
In the past, Du et al. also used bi-layer MgF2 and Nb2O5 films to design a 500 nm DBR, but they only used the e-beam to deposit the bi-layer MgF2 and Nb2O5 films and investigated their properties 15 .
In this study, we would investigate a transfer matrix to find the difference between the simulated and measured results of the fabricated DBRs. The variations in the reflective ratios of multilayer MgF2-Nb2O5 DBRs versus the variations in period numbers is simulated using Sheppard's approximate equation 16 : where nL, nH, nS, and n0 are MgF2 (nL = 1.39), Nb2O5 (nH = 2.29), glass substrate (nS = 1.52), and air (n0 = 1), R represents the reflective ratio at a special wavelength of light, and P is the deposited period of the MgF2-Nb2O5 bilayer films.
If the n values of the two films differ greatly, the fabricated DBR will have a similar reflective ratio if fewer periods are deposited, or a higher reflective ratio if the designed device has the same number of periods. Knowing this saves time and money when designing DBRs, due to the n values of the MgF2 (nL= 1.39) and Nb2O5 films (nH =2.29). MgF2 and Nb2O5 films have low extinction coefficients (k values), as well as high transparent ratios in the near-infrared, visible, and ultraviolet regions. In optical applications, Nb2O5 film can be used as a low-loss and high-index material, and MgF2 film can be used as a low-loss and low-index material. There are two reasons for us to use MgF2 and Nb2O5 films to design reflective films having a multi-period L (MgF2, 1/4 λ)/H (Nb2O5, 1/4 λ) bilayer structure. The first is that the MgF2 film (n ~1.39 in light range of 400~700nm) 17 and the Nb2O5 film (n ~2.30 in visible light) 14 have very different n values. The second is that both materials have low k values 14,17 . In this paper, we show that even with only four periods of the MgF2-Nb2O5 bilayer films deposited, the fabricated DBRs had a reflective ratio of over 90%.
However, Equation (1) is only used to simulate the maximum reflective ratio at a designed wavelength, and it cannot simulate the reflective spectrum in a very wide range of light wavelength.
In this study, we investigated the transfer matrix of each layer and obtained an overall transfer matrix of the multilayer coatings, which is formed using the product of each single transfer matrix. The following steps were carried out for investigating the potential difference between the experimental measurements and simulation estimations for an MgF2-Nb2O5 bilayer blue-light Bragg reflector with different periods. First, we deposited single-layer MgF2 and Nb2O5 films on glass substrates, their k values and n values were measured, and the thicknesses to match the λ/4 MgF2 and Nb2O5 films could be obtained. Next, a characteristic matrix for simulating the multilayer films was constructed to calculate the reflective spectra of a bilayer MgF2-Nb2O5 Bragg reflector with different periods by using the optical properties of the single-layer MgF2 and Nb2O5 films. Third, we deposited bilayer MgF2-Nb2O5 DBRs according to the calculated thicknesses of the λ/4 MgF2 and Nb2O5 films, then measured the reflective spectra of the fabricated DBRs. In what follows, we provide a detailed comparison of the experimental and theoretical results for the multilayer blue-light MgF2-Nb2O5 DBRs and discuss reasons for deviations between the measured and simulated results.

Experimental Procedure
Corning 1737 glass substrates (the area was 2 × 2 cm 2 ) were cleaned, then MgF2 and Nb2O5 single-layer films were deposited on them using e-beam technology to measure their k values and n values. The thickness of the Corning 1737 glass was about 2 mm. The deposited MgF2 and Nb2O5 single-layer films and the MgF2-Nb2O5 bilayer films were much thinner than the glass substrates, to prevent the coherent condition that would be caused by multiple light reflections in the substrates.
The base chamber pressure to start the deposition was 6x10 -6 Torr, for the electron beam the applied voltage and current were 4 kV and 20 mA, and 25 o C (room temperature) was used the deposition temperatures of the MgF2 and Nb2O5 films. The thicknesses of the MgF2 and Nb2O5 films were 57.7 and 58.4nm, which were controlled by a film gauge on the e-beam and verified by field-emission scanning electron microscopy (FESEM). When the MgF2 and Nb2O5 films were deposited at higher temperatures or were annealed, the films' densities increased and/or the average crystallite sizes increased. There were two important reasons for us to deposit the MgF2 and Nb2O5 films at room temperature. First, as the films' densities increased, their thicknesses decreased and became not easy to control, causing variations in the central wavelength of the reflective band. Second, as the crystallite sizes increased, grain boundaries formed between crystallites of different sizes; this increased the chance of the light scattering, which in turn affected the reflectance and transmittance efficiencies of the multilayer films.
For the deposition of the MgF2-Nb2O5 bilayer films, at first an MgF2 film was deposited on a glass substrate, after that a Nb2O5 film was deposited on the MgF2 film, and then one period of the MgF2-Nb2O5 bilayer film was formed, as Figure 1 shows. For the investigation of the effect of period number on the properties of the designed blue-light DBRs, different periods of the MgF2-Nb2O5 bilayer films were deposited. The DBRs' center wavelength was 450nm, and the thicknesses of the and Nb2O5 films were measured along the direction of thickness using a normal light incidence on single-layer films with an n&k analyzer. The n&k analyzer was able to measure thickness (d), n value, k value, transmittance spectrum, and reflectance spectrum, and it used a nonlinear curve-fitting method, which fit the measured results using polynomial terms in linear regression, and these measured data can be retrieved directly. X-ray diffraction (XRD) pattern was used to measure the crystalline structure of the MgF2 and Nb2O5 single-layer films and the MgF2-Nb2O5 bilayer film with different periods. The surface observations of top-layer films for different periods of MgF2-Nb2O5 bilayer film were measured using FESEM. For measuring the real thicknesses of each MgF2 and Nb2O5 film for different periods, we used a focused ion beam to prepare the samples and used the FESEM to observe the cross sections of the MgF2-Nb2O5 bilayer films. The n&k analyzer was also used to measure the reflective spectra of the MgF2-Nb2O5 bilayer films with different periods in the light wavelength range of 300-800nm. Light is one kind of electromagnetic wave, for a single-layer film on a substrate (substrate-thin film-air), the interface between substrate and film and that between film and air were denoted by b and a, respectively, and the transmission (τ) and amplitude reflection (ρ) coefficients were 18,19 : Therefore, the relationship between magnetic field (Ha) and electrical field (Ea) can be written in matrix equation shown below 18,19 : where δ represents the phase factor and it can be shown below δ = 2π Nd cos(θ)/λ (4) Where N is the complex index and it is equal to (nr -ikr), d is the thickness of the film, λ is the wavelength of light, k is the extinction coefficient, and n is the refractive index. N is a modified optical admittance of the deposited film and it can be shown as below: For the multilayer films, when light is propagated from the front face of a reference plane (often is the air) to the surfaces of the multilayer films, and then to the substrate, the variation in the input optical admittance can be listed as follows for single-layer, two-layer (one period, P=1), four-layer (two periods, P=2), eight-layer (four periods, P=4), and twelve-layer (six periods, P=6) structures: where ys is the admittance of the substrate.

Results and Discussion
At first, we used XRD patterns to measure the crystalline structures of the MgF2 and Nb2O5 single-layer films and one-period MgF2-Nb2O5 bilayer films; the results are shown in Figure 2. No characteristic peaks were found in the XRD patterns, because these films deposited using the e-beam process would present in the amorphous phase. We believe that because the e-beam deposition process was carried out at room temperature, the natures of the single-layer and bilayer films remained unchanged. The XRD patterns of MgF2-Nb2O5 bilayer films with different periods were also measured (not shown here); again, only the amorphous phase was observed.  350 to 650nm, the k value of the Nb2O5 single-layer film decreased quickly from ~3.6% to ~1.6%, and the k value was less than 1% when the light wavelength was longer than 1200nm. Previously, Chen et al. used a sputtering method to deposit Nb2O5 films in different atmospheres 14 . They found that when pure argon was used as the deposition atmosphere, the Nb2O5 film had a higher k value; when oxygen was injected with the argon during the deposition, the Nb2O5 film had a lower k value, which dropped to zero as the light wavelength was longer than 346-370nm.
This was because when pure argon was used, a small amount of Nb2O5 decomposed to form Nb2O5-x, which had semiconducting characteristics and thus increased the k value. When oxygen was mixed with the argon, the Nb2O5 film did not decompose to form the Nb2O5-x and had a lower k value.
Because we deposited the Nb2O5 film in a high vacuum, we believe that only a small amount of the Nb2O5 decomposed to form Nb2O5-x, creating oxygen vacancies in the film. The oxygen vacancies increased the film's conductivity and absorption, so the Nb2O5 (Nb2O5-x) film had semiconducting properties and a higher k value. However, the deposited MgF2 film had a k value close to zero even when the measured wavelength was 200nm. This result suggested that the deposited MgF2 film was a stable and lossless dielectric material. The results in Figure 3  After the thicknesses of the MgF2 and Nb2O5 films matched for the λ/4 condition were found, the MgF2-Nb2O5 bilayer films with the periods of two, four, and six were deposited using the e-beam process. The surface morphologies in Figure 4 compares the surface morphologies of the bilayer films.
These FESEM images show that when the deposition was done at room temperature, a dense surface morphology was presented on the surfaces of Nb2O5 films, consisting of growing nanocrystalline particles, which were polycrystalline materials with a grain size of only a few nanometers. Although nanocrystalline particles were observed, the XRD patterns indicated that the MgF2 and Nb2O5 singlelayer films and the MgF2-Nb2O5 bilayer films presented amorphous structures. The results in Figure   4 also show that the nanocrystalline Nb2O5 grains presented the uniform particle sizes. The average crystallite sizes were 12.3, 16.7, and 22.1nm for two (Figure 4(a)), four (Figure 4(b)), and six ( Figure   4(c)) periods. An AFM with a tapping mode could be used to analyze the surfaces' roughness of the Nb2O5 films on top of the MgF2-Nb2O5 bilayer films with different period numbers. Because we deposited the MgF2-Nb2O5 bilayer films at room temperature, we believed they would have a lower activation energy, and the Nb2O5 films would have smaller particle sizes and less roughness. As two, four, and six periods were deposited, as Figure 5 shows, their average roughness (Sa) was 1.21, 1.28, and 1.35nm and their root-mean-square (rms, Sq) surface roughness of the Nb2O5 films was 1.56, 1.68, and 1.75nm. These results mean that the roughness of the Nb2O5 films on four-and six-period bilayers was greater than on two periods. The results also prove that all the Nb2O5 films' surfaces were relatively smooth, but their roughness slightly increased with the period number.    After the two, four, and six periods of MgF2-Nb2O5 bilayer films were deposited, the optimum reflective ratios of optical spectra were 66.8%, 92.5%, and 95.3% at light wavelengths of 436nm, 456nm, and 448nm. Using Equation (1), the maximum reflective ratios were calculated to be 70.0%, 95.3%, and 99.3% when the period numbers of the designed DBRs were two, four, and six. When these simulation results are compared with the measured values, they have no apparent differences in the central light wavelength. The calculated bandwidth for the reflective band of the fabricated DBRs with the MgF2-Nb2O5 bilayer films at a specific wavelength (λ) can be found using Equation (14) 18 : where nL and nH are the n values of the MgF2 and Nb2O5 films, λ is the central light wavelength for reflectance of the designed DBR, and Δλ is the reflective bandwidth of DBRs at a specific wavelength.
When the measured n values of the MgF2 (nL = 1.39) and Nb2O5 (nH = 2.29) films are used in Equation (13) to find the bandwidth at the stop-band of blue light (450nm) for the MgF2-Nb2O5 bilayer films, the calculated value is 142nm, meaning light in the light wavelength of 379-521nm can be reflected.
The measured reflective bandwidths and simulated reflective bandwidths (using an overall transfer matrix) are defined as that the measured reflective ratio is higher than or equal to 90% of the measured maximum reflective ratio. Hence, as Figure 8 shows, for the bilayer films with two, four, or six periods their measured bandwidths were 88nm (385-497nm), 113nm (409-522nm), and 116nm (395-511nm). The full width at half maximum (FWHM) values of the reflected bands with two, four, or six periods were 263nm (347-610nm), 186nm (386-572nm), and 160nm (384-544nm). These results prove that the simulated results have matched the measured ones and also prove that differences in thicknesses and the n values of the MgF2 and Nb2O5 films are the reasons for the variation in the central light wavelength and the difference in bandwidth.    matrix prove this to be the case. As the periods of the bilayer films were two, four, and six, their simulated bandwidths were 103nm (390-493nm), 140nm (400-540nm), and 104nm (398-502nm).
However, the bandwidths for these bilayer films with the same periods and simulated with an overall transfer matrix matched the measured results.   When light strikes the surface of a film, part of light can be transmitted, part will be reflected, and the remaining part can be absorbed. The first law of thermodynamics defines that the sum of the transmitted, reflected, and absorbed radiation energies is equal to the energy of incident light. In other words, the relationships between absorptivity (α), reflectivity (η), and transmissivity (τ) are: α + η + τ = 1 (15) In the past, Liu et al. used lossless Al2O3 and TiO2 films (or k values close to zero in the visible light range, defined as α = 0) to fabricate DBRs and found that the reflective ratios of the fabricated devices were higher than the simulation results using Sheppard's approximate equation. 5 For our DBRs fabricated using MgF2-Nb2O5 bilayer films, maximum α plus maximum τ was smaller than 1.
From the measurements of the transmission and reflection ratios, we can quantify that the total scattering and absorption losses at 450nm are α = 2.2%. These results prove that the higher k value of the Nb2O5 film meant the absorptivity (α) was not equal to zero, so the reflective ratios of the MgF2-Nb2O5 bilayer films were smaller than those obtained using Sheppard's approximate equation. Figure 11. Comparison of maximum reflective ratios using different simulation methods, and measurement results of fabricated DBRs with different periods.

Conclusions
When MgF2-Nb2O5 bilayer films were used to fabricate DBRs with two, four, and six periods, the total thicknesses of the calculated MgF2-Nb2O5 bilayer films were 259.8, 519.6, and 779.4nm, and the measured total thicknesses of the deposited MgF2-Nb2O5 bilayer films were 242.8, 520.0, and 780.5nm. When we used two, four, and six as the period numbers to deposit the MgF2-Nb2O5 bilayer films, the simulated maximum reflective ratios using Sheppard's approximation equation were 70.0%, 95.3%, and 99.3%; using an overall transfer matrix, the values were 68.0%, 94.3%, and 98.3%, and the measured maximum reflective ratios were 66.8% (at 436nm), 92.8% (448nm), and 95.3% (456nm). The simulated bandwidth of the designed DBRs with MgF2-Nb2O5 bilayer films at 450nm was 142nm. As the designed MgF2-Nb2O5 bilayer films were two, four, and six periods, their simulated bandwidths with an overall transfer matrix were 103nm (390-493nm), 140nm (400-540nm), and 104nm (398-502nm), and the measured bandwidths for the deposited bilayer films were 88nm (385-497nm), 113nm (409-522nm), and 116nm (395-511nm). The results in this study prove that the investigated overall transfer matrix can be successfully used as a simulation tool to calculate the reflective spectra of fabricated MgF2-Nb2O5 bilayer films.

Figure 1
Structure of designed MgF2-Nb2O5 bilayer lms with different periods.      Cross-sectional EDX mapping measurements of four-period MgF2-Nb2O5 bilayer lms.

Figure 8
Measured re ectance rates of MgF2-Nb2O5 bilayer lms as a function of period number.

Figure 9
Measured transmittance rate of MgF2-Nb2O5 bilayer lms with different period numbers.

Figure 10
Simulated and measured re ectance spectra of DBRs designed using the MgF2-Nb2O5 bilayer lms: (a) two, (b) four, and (c) six periods.

Figure 11
Comparison of maximum re ective ratios using different simulation methods, and measurement results of fabricated DBRs with different periods.