XRD patterns of the Ti, TiO2, and SiO2 thin films (with different sputtering powers) are shown in Fig. 1. Both TiO2 and SiO2 thin films show predominantly an amorphous phase, while the Ti metal thin films show crystalline phases. An amorphous phase was observed in the TiO2 and SiO2 thin films regardless of sputtering power, as shown in Fig. S1. Pulsed-DC and RF sputtering power are abbreviated as X/Y kW. When the Ti thin film was deposited using a sputtering power of 6/0 kW, a reflection is present at 81.07° 2θ coinciding with the absence of the (110) reflection (62.79°). The reflection at 81.07° 2θ can be identified as the (004) plane of the β-Ti (bcc) structure. This suggests that there is preferential film growth along the c-axis, which may be due to a lower surface energy of the (00l) than other lattice planes20–22.
Cross-sectional and surface micrographs obtained from AFM and SEM show that high sputtering power increases roughness and also a more dense film structure. Figures 2(a), (b) and (c) show the cross-sections of optimized Ti, TiO2, and SiO2 thin films prepared using sputtering power 13/1 kW, 8/1 kW and 6/0 kW, respectively. All thin films show smooth surfaces, confirmed by low RMS obtained from AFM measurements. Additional cross-sectional images of the Ti, TiO2, and SiO2 thin films are provided in Fig. S2. It is well known that pulsed-DC supply is advantageous for obtaining dense and uniform structures, because surface diffusion of the sputtered particles on the substrate promotes homogenous film growth23,24. Accordingly, the low roughness and dense structure of the TiO2, SiO2 and Ti thin films presented here suggests that pulsed-DC sputtering is beneficial for preparation of smooth and dense thin films.
Elemental composition and chemical states of the constitutive elements are important factors in the resultant microstructure, refractive index and extinction coefficient of thin films. XPS spectra of a TiO2 thin film are shown in Fig. 3(a). Two major peaks at 458.51 and 464.22 eV can be assigned to Ti 2p3/2 and 2p1/2 energy levels, respectively. The binding energy of Ti2p indicates that the oxidation state of the Ti is 4 + 25. The O1s spectra show two major peaks at 530.05 eV and at 531.67 eV, which can be attributed to OI and OII, respectively26. The OI Peak corresponds to O2− in the lattice sites of the TiO2 structure, while the OII Peak is assigned to OH− bonded to Ti3+. A dense microstructure in commonly formed in thin films when the ratio of OI to OII is higher. The ratio of O to Ti in the film is 1.97, which indicates the chemical composition of the thin film is TiO1.97. XPS spectra of the SiO2 thin film is shown in Fig. 3(b). Si2p at 103.38 eV and O1s at 532.82 eV can be assigned to Si4+ and O2−, respectively27. The ratio of O to Si is 1.89, indicating the formation of SiO1.89 thin film.
The refractive indices (n) and the extinction coefficients (k) of Ti, TiO2 and SiO2 single-layer films were measured in the wavelength range 300 nm to 1,800 nm by Ellipsometer, and a selection of the results is shown in Fig. 4 (results at other selected wavelengths are provided in Fig. S3). The refractive index at 550 nm wavelength of Ti is 2.43, TiO2 is 1.48, and SiO2 is 1.99Also, the extinction coefficients of TiO2 and SiO2 are 0.00, while the extinction coefficient of Ti is 3.05. The high refractive index and extinction coefficient of the metal film can be attributed to large absorption of the incident radiation through electronic conduction in the metal film28. It is suggested that the incorporation of metal films into multilayer dielectric films can efficiently modulate the optical properties, and thus achieve desired optical properties of the narrow bandpass filter.
Transmittance and reflectivity of the multilayer thin films may be estimated using the following relation, which is described by the Fourier Transform relationship29:
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPEAAAAoCAYAAAAv+hzuAAAJRUlEQVR4Ae3agZHUPAwF4GuBGmiBHiiBGmiBDuiADqiACmiABuiAHvafj+HNL4yzSW6TXecumskkdmRZetaTnb17upxyInAicGgEng7t/en8icCJwOUk8StLgu/fv1/evHlzeffu3eXHjx+Xb9++/b5eGQwvKtzDkvjnz5+zC/Hr169Zndem8OnTp98hf/78+fL09HT5+PHjJARLMM7gE+sgcf/74Uhs9/jw4cOi3SO6X79+vT+yz5gREeyUewtymmeKpPxAbkRfItFPgVgy5tTZDoFDkVjSvX379rK26iP9EYjMx2s7463LDrf379//Pk7n3pvPUfs5xeTLly+7+n9r/C91/DAkljQhp3ttB3wJ1yOjHbfuKhmfcdq+A2M//SPdxeuqcWzpn9iR0xUc3B2pK6ZLCgm842+7Ts8tAFvG+tpsPZzEEslOaYfNjpldQjsiuSVcEjD9ks5Y7+i4JJJ2ldiufSM9x2ek2EMcjXuFDNb12AxLP3a1AndFND+KuRvrqmty7sYtcvu3/870/ef7ZwZV3SWRJFC+q7QlSERi1Xbt94wESSBJ5bkKe7Uo1HcjPCNvr0ht5RvSBdtqU39IDLe2+NHVTw+JPaft3oo46J5yPwQeTuKEKkGQOKJdSdeSOnq5S1DjFYSezI3vjblnX04Ua+cUr7iRD2Y9oeN9u8vndJOdN4WktdGujfc9e/qnbLQ292jLl2AgZs8KCl9tAOJ9iTIMiSVidgRAa9fddI6EIfHUIs2Nnxp3r34JV4vWknklZQoXrHonFXZCrDaJYSLJs6NGr527XYuQn34rUzZavb3bsEj+IHROEnvP+wj7Q5BYEtXKniQBfqSXpPSMtZMgMRtJyIzLfXQSt0SJ39fuYl5C/BCr4pnEhktEHwxbQYBK2Jb8VT9z1b69nzPntfVH6mAlRxRN+lNXfKab3Ku61os4QemPbRh6N1VQY3fL+z8rFic4tpcjyAdEF0FC80WyKCGnfn2SqYokllABEHja7JqjCl0XsSjGsDmCSBTxr/Un8S6JwW8GiVniTe1M/JADVYwznn/wpcNGT6wZ3XsLv0Ksdm74ipdvxPqLRQzJnV4Bk0Ns0vGefnI2c8ipjE1eayfXote7B8vnrH219z9zSm+Sag8SJ4ECqGlbcMwPOIB5jgAUQBFAVR12+FxtR9fYShLPkm0J2LGx152/FnKtGCMRlggcxQoHF6x6Ak9JXiVjYWtdKo5Vz7PxS31qx9a22KauXl6KLYSsduITv1oxJrG6t3a1a5+4p9YJpvKp3TzaOds2H6ZstrpT7cnMYbgGMGVgTX9AqERcMx7QUwt1zU7I3epIThU6C9m+v1dbgq3dvYLlWt9hYW2TbLknVmsjIWvxzLu5O1vwfM7YOdtz7+HXKx6w7RGYPWNSkOQ6QkXEAKe6IVwrtu342Jm78+1Wnt2VxJKjAjUXYO+9oNfYsEhAmkqsNql7c+7VhzDmr0e9pXOFxEnCpePoSd7sHL2CmqI3hVlvLgSGc89eT3/LvhCuxeIagflbd8CWsMG3+ske3FoJXs8h4xacWExikwk0C+WuLQEFMSd06LeVf25c7z2Al+xA0ZtLRjGsKQw9n57TZ04ni1rtl9rJ91SbuEvHG3cNF+u0FBN26C5dW/r8T065PweDxCoWuVVFvtXTjTnrKc78lXTGsyOO4KIvuS3ftKuf4QJyK17eJ7bqS57Z5xM9OZc1rDajG/zDM2Ny1bjo/x15LPz5O2AN0itguxKoibQ5NCcApDuijOzbFF5JAIlxJEnOyC3Pkl5etLm2JiZYtIktJ5P0uWcO87aF23h9Ia35PfPNePeWbPrZDOl7NhIHW/T5Sj9x68v4qssXeak4uLRbveivIjGHWyImmeaOUZwIiJl8lHtimAIpfkYP8EuujNvjHl+ORmI50BJOu+6Sa/Ey3m44qqRotDH2+BTCWt8q8m3q9LmaxC0RlyYTJ1rHOLmEDFvrVHA8L42hHXdLe0lM1+yv8XnJXHvotP7bycyj8CiY7hJbgZ/bBFpbabMRm+kb7e7kykdkrmJDbIsPPNqN0hjjW93YejiJ48gj72sI8Ug/69xH9FkSSsZcNoQ139I1fs/sSPh6BG51RmjzsT19IDT/W9/1WdsqCp5+WPXkJPGKnTjEAeiSqwf4Vn3x5UjH6d5Reis8RrYjV9qjdNavnkCmThX5nm4Jn5jvRuKX8E0c0Ea4JwmOTmK7TJvgI+C7pQ9IXI/CiGt31l8lu3P7A5rit/qHrWzf7RHAxK4qOe+3E1cdz1Nn/VbvEe2RfZvCIyRuv7Om9Efoj88pPO5ybGqHGcHnLXwIb/I7AFLCQr8+BM86wkM+6s87ZL+G0d+l4I/HjBtYKwjD6fNMss3rn/txIrpxdgtwtrLB96nvja3m2NpO1mNru3vbS9GXM5I5hN573kfazy/O4RNy5kc+XKsY0M3a0vdc3/fi6JK4p7hFH4frsWILm7faGLm4XIvttX5fXsPktb67K4lTferH/COBVxHtwo42RxMkHq0gHg3Dl+LvXUkMNDsf4sx9QwdghK9HcEeL2o7e2ju7vj8eeYxWRMTTu+YK3RoM12JT9flR/ePzKWMhcHcSCx8J7SJzBPIe0SSsZLL7OJJr3yL+82XJt8Ytc8yNVczEJh5Xjsdi89zDxjdSSGXsXoKo1ocvwZ9PrpPEe6H+fLsPIfESdyWL3douUH8QSHuJjZF1kJgoJnl2rNeeEmSHBTJtcRrpzQN35EViz2m7nzImAsOSOHBJcImbpNWWZC9FEDOx2fUe/X2OvHyqAn/F85QxERiexJLKzhPRbncrJJD8jslH2jH4nYLE7xHIgsD1H+35OIJfWf/z/i8Cw5O43Z3aJPONKMl8QyL7nt+K/8J3W0++zVkZ5TNBUam7ruKo70jF8bZVOd7ooUnc250QVvLnmw158yNQ9BH7CKIg5fgcEvsdoD1p3DMWcyuE/OEbvPPNfk8/zrmWIzA0ifOLdN0FEFaihah23xBB2Ec5+iEr3xOHGMWV4rR8CbfV5EdONfypu/K2M53WtkJgaBIvCVLSu0h24vxQtGT8qXMicHQEDk/iHEOzQ9cfwY6+OKf/JwJLEDg8iQWZ7zffbvXovQSAU+dE4OgIvAgSH30RTv9PBG5B4CTxLeidY08EBkDgP+t2yF+NmpTKAAAAAElFTkSuQmCC)
(1)
where n(x) is the refractive index profile, and
is the complex function of the transmittance or reflectivity. The effective thickness, x, can be calculated by:
![](data:image/png;base64,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)
(2)
It can be shown from Maxwell’s equations that the transmittance of the thin film based optical filter can be expressed as Eq. (3).
is integrand given in Eq. (4).
![](data:image/png;base64,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)
(4)
where the complex function
is expressed as:
![](data:image/png;base64,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)
(5)
In TFNBFs, the cutoff frequency of transmittance can be determined by the complex function
depending on the refractive index n(x) and effective thickness x. Accordingly, the application of a metal film between dielectric films as well as thickness control can tune the transmittance in a specific wavelength, thereby selectively controlling the reflectance or absorption of light in a specific wavelength30 − 33.
The number of thin films and effective thickness for optimized transmittance of TFNBFs was obtained using the Essential Macleod Program (EMP). We chose 8-layered thin films for computational calculation which demonstrates the effect of Ti layer on the optical properties of the dielectric (TiO2 and SiO2) based multilayered thin films. Based on the EMP simulations, the thickness of each Ti, TiO2, and SiO2 thin film was precisely controlled in the deposition of 8-layered thin films, as shown in Fig. 5. Transmittance of 8-layered thin films was investigated in the wavelength range of 300 nm to 1,100 nm. Different thicknesses of the Ti thin film were deposited between 4F SiO2 (108.0 nm) and 5F TiO2 (20.0 nm) as shown in Fig. 5 and Fig. S4. For comparison, transmittance of 7-layered thin films consisting of dielectric (TiO2 and SiO2) thin films only is also presented. The transmittance in 7-layered thin films was 82.29 % (at 360 nm wavelength), 89.72 % (at 400 nm), 84.81 % (at 500 nm) and 79.44 % (above 750 nm). With the addition of Ti metal film with different thicknesses (8-layered thin films), transmittance significantly changed in specific wavelengths as shown in Fig. 5, which shows a transmittance of 8-layered thin films was 33.71 % (at 360 nm wavelength), 71.08 % (at 400 nm), 29.34 % (at 500 nm), and 33.58 % (above 750 nm). It appears that thicker Ti metal films result in larger ΔT. Simulated transmittance at specific wavelengths, highlighted by the dotted line, matches well with experimental data, indicating that the inclusion of metallic thin films with different thickness into dielectric thin films can effectively tailor the optical properties of the TFNBF.
Based on the EMP simulations, transmittance at 485 nm can be selectively modulated when thickness and a sequence of inserted Ti metal films is precisely controlled in the deposition of 26-layers thin films: the deposition sequence and thickness of the Ti, TiO2 and SiO2 thin films are shown in Fig. 6 and Table 1. For comparison, 23-layered TiO2/SiO2 films were also prepared as shown in Fig. S5 and Table 1. There was little difference in the total thickness between 23-layers and 26-layers thin films. Interfacial diffusion among thin films was not observed in the 26-layers thin films. Transmittance of the 23-layer and 26-layer thin films in the wavelength range from 300 nm and 1100 nm was evaluated, as presented in Fig. 7. For 23-layer films, transmittance of 90.5 % with a FWHM of 21 nm was observed at low cutoff frequency (485 nm). Transmittance of 65.98 % and 57.21 % was observed at high cutoff frequency of 400 nm and 680 nm, respectively. With the insertion of Ti metal layers, a high transmittance of 80.42 % with FWHM of 19 nm at 485 nm was observed. A lower transmittance of 10.29 % and 10.48 % was observed at 400 nm and 680 nm, respectively. That is, increased ΔT at both a low cutoff frequency (485 nm) and a high cutoff frequency (400 nm, 680 nm) is achieved in the 26-layer thin films. Experimental results matched well with simulated optical properties as shown in Fig. 7. Transmittance spectra of Ti/TiO2/SiO2 multilayer thin films with increasing the number of Ti layers is also shown in Fig. S6, which implies that Ti layer is beneficial to improve the spectral selectivity. For application of multilayer thin films to narrow bandpass filters, a square bandwidth with a steep slope of the transmittance at specific wavelength is essential, highlighted in the red squares in the inset of Fig. 7.
In the present work, a targeted approach for the design of the multilayered thin films with desired optical properties is presented. Based on computational simulation of the optical properties depending on materials with different refractive indices, thickness, and the number and sequence of layers, multilayer thin films were carefully prepared and thus, efficiently tailoring the optical properties for the possible application for narrow bandpass filters. The introduction of metal films into dielectric-based multilayer thin films open possibilities to efficiently tune the optical properties at specific wavelengths.