Figure 1a plots the normalized admittance loci of a five-layered Fabry–Perot design
Ni/
M1D2M3D4M5/
Nsub. The bottom
D4 and
M5 cause the equivalent admittance to be a complex value
N =
n +
ik with large positive
k and small
n, causing the locus of
M3 to be a large contour. To form an angle-insensitive filter, the metal and dielectric films were made of silver and silicon, respectively. For smaller index of extinction coefficient, the semiconductor of a-Si was used as a dielectric in near-infrared region. The refractive indexes of Ag and a-Si were taken from the database in the Macleod thin-film package [
6], and the dispersion of metal and dielectric films was taken into account. The refractive indexes as functions of wavelength are shown in
Appendix A. The filter was designed to pass a wavelength of 950 nm. The thicknesses of
M5 and
D4 were 20 nm and 97 nm, respectively, causing the terminal normalized admittance (NA) to be at (14.003, 19.189). The thickness of
M3 was set to 10 nm, which is feasible for an ultra-thin silver film that was formed by e-beam evaporation or sputtering evaporation. The top two layers
M1 and
D2 functioned as matching layers to bring the equivalent admittance back to the admittance of the incident medium. Therefore, the designed five-layered Fabry–Perot filter became
Ni/
M1D2M3D4M5/
Nsub = air/Ag(13 nm)/a-Si(90 nm)/Ag(10 nm)/a-Si(97 nm)/Ag(20 nm)/BK7 glass. The associated transmittance spectrum of the five-layered system in
Figure 1b indicated that the passband was not a symmetrical peak and sideband occured on one side of the design wavelength.
Figure 1c shows the NA diagram at the central wavelength of a sideband, 1326 nm. Although the wavelength was longer (shorter) than the design wavelength, all loci were shortened (lengthened), but the terminal point remains around unity on the real axis. The compensation was caused by the top two layers
M1 and
D2. The shrinkage or expansion of loci
M1 and
D2 kept the admittance of the terminal close to that of the incident medium. For the five-layered Fabry–Perot design, the oxidation of metal film on the top would affect the performance of the filter. Although a low-index, thin film with half-wave thickness can be coated on the top of a metal film to function as a protective layer, the performance of filtering only maintains at normal incidence. At oblique incidence, the bandpass spectrum is affected due to the angular dependent admittance and phase thickness of the protective layer. Therefore, it was desired to modify the design to have dielectric layer on the top to support the avoidance of oxidation.
To eliminate the sideband effect and to increase the sensitivity to wavelength, the multilayer system was changed to a four-layered
Ni/
D1M2D3M4/
Nsub = air/a-Si(67 nm)/Ag(15 nm)/a-Si(84 nm)/Ag(16 nm)/BK7 glass system. The sideband comes from the admittance matching that occurs at another wavelength in the five-layered system. A four-layered system proposed here can avoid the unnecessary admittance matching. The loci of
M2 and
D1 are sensitive to the variation of wavelength to lead to a narrow passband.
Figure 2a shows the designed loci of a passband filter. After the large locus of
M2, the locus of
D1 directly brings the admittance of the terminal back to that of the incident medium. This design yields the spectrum that is shown in
Figure 2b. The maximum transmittance at a wavelength of 950 nm is 85% and the full width at half maximum (FWHM) of the passband is 117 nm. In the wavelength range from 700 nm to 1500 nm, the only transmission peak was the designed one. To increase the resolution of the bandpass filter, the four-layered design was modified to reduce the FWHM. The thickness of
D3 was increased from 84 nm to 212 nm to take one more loop of locus in NAD. The large phase thickness of
D3 varies markedly with wavelength. As shown in
Figure 3, the maximum transmittance of the four-layered
Ni/
D1M2D3M4/
Nsub = air/a-Si(67 nm)/Ag(15 nm)/a-Si(212 nm)/Ag(16 nm)/BK7 glass at a wavelength of 950 nm is 85% and the FWHM of the passband is 83 nm.
The admittance matching design can be extended to six-layered and eight-layered structures as
Ni/
D1M2D3M4D5M6/
Nsub = air/a-Si(52 nm)/Ag(11 nm)/a-Si(40 nm)/Ag(13 nm)/a-Si(75 nm)/Ag(20 nm)/BK7 glass system and
Ni/
D1M2D3M4D5M6D7M8/
Nsub = air/a-Si(60 nm)/Ag(13 nm)/a-Si(50 nm)/Ag(11 nm)/a-Si(53 nm)/Ag(13 nm)/a-Si(50 nm)/Ag(20 nm)/BK7 glass system, respectively. The corresponding spectra are shown in
Figure 2b. The transmittance decreases with increasing number of layers. In the case of eight-layered structure, although the transmittance at wavelengths over 1050 nm is suppressed to be less than 4.7%, the peak transmittance is also reduced to be 50%.