1. Introduction

In 1899, Fabry and Perot developed an interferometer that has greatly influenced the development of thin-film optics and is well-known as the Fabry-Perot interferometer [1]. The Fabry-Perot interferometer consists of two flat, highly reflective and parallel surfaces that are separated by a distance to reflect light beams between them many times. Since the device is a multiple-beam interferometer, the interference fringes are much sharper than those of two-beam interferometers so it exhibits high accuracy and high resolution. The Fabry-Perot interferometer is utilized principally to observe directly fine spectral structures and the concept on which it is based is also used in the design of band-pass interference filters [2]. The Fabry-Perot geometry has been applied as a laser resonator [3] and the Fabry-Perot etalon can be used in astronomy observation [4]and gravitational wave detection [5].An optical filter is then developed based on Fabry-Perot interferometer. Such a filter is realized by arranging a dielectric film between two highly reflective coatings that can be thin metal films or all-dielectric highly reflective coatings; the band-pass interference filters thus formed are called metal-dielectric-metal Fabry-Perot filters and all-dielectric Fabry-Perot filters [6], respectively. The bandwidth of each transmitted band is inversely proportional to the reflectivity of the highly reflective coating, and it is related to the order of interference and the extinction of the entire layered structure. The band width of the traditional metal-dielectric-metal Fabry-Perot filter is limited because of the extinction property of metal. The all-dielectric Fabry-Perot filter must include enough films to achieve high reflection.

The other metal-dielectric structured filter is induced transmission filter [6]. By desiging dielectric multilayers on both sides of a metal film for a designated wavelength, the potential transmittance would reach a maximum value to cause a high transmission peak in the spectrum [7]. The high value of potential transmittance represents low electric field intensity distributed within the metal film. The passband is narrow to reflect the fact that the maximum potential transmittance is reached only under strict circumstance.

In this work, the traditional Fabry-Perot(FP) filter is modified to allow one or more metal films to be arranged between two metal films to form a metal-dielectric multilayered structure. The additional metal films do not reduce the transmittance peak or broaden the pass-band. Rather, the additional metal films narrow down the pass-band and keep the transmittance high. The low-loss metal-dielectric multilayer is designed by plotting normalized admittance locus of each film in the normalized admittance diagram (NAD) to be a brief narrow band pass filter. The loci of the metal films between two mirrors are designed to be huge loci in the admittance diagram to reduce the energy dissipation within the metal films. That measns the transmission is induced based on the loci of normalized admittance in the NAD rather than the multilayer design for maximum potential transmittance. The method is straight forward and easy to design a multilayered filter for any wavelength. The huge loci are sensitive to the wavelength of the incident light so a narrow band pass filter can be achieved.

When the modified FP filter is approached as a periodic subwavelength structure, the equivalent refractive index at the pass band may be negative real. Although a subwavelength structure that comprises alternating metal and dielectric films has been proposed for use as a negative index medium at a ultra-violet wavelength of 363.8nm [8], an attempt has been made to understand its negative index using concepts from thin-film optics [9]. Some efforts are devoted to investigation of the associated photonic band gaps [10,11] and equivalent negative indices of refraction [12–14] of periodic metal-dielectric multilayers in which metal and dielectric films are arranged alternatively and all of the metal (dielectric) films are the same. Scientists and engineers need a method to improve the transmission and for designing a medium with a negative index at any designated wavelength by tuning the thickness of each layer of a metal-dielectric multilayer. In this work, a metal-dielectric multilayer with dielectric layers that are composed α-Si, amorphous silicon, which is taken as a high refractive index optical material, and silver as the metal film, is designed as a highly transparent FP filter at a visible wavelength of 660 nm. Although the α-Si film unavoidably has an extinction coefficient(The Essential Macleod Package. Version 8.), 0.18, the transmission peak of the FP filter still can be improved. The transmittance spectrum varies weakly with the angle of incidence. The exact equivalent index is then evaluated using the equivalent optical parameter retrieval method. Such a metamaterial with a negative index can be applied for a wide range of angles of incidence (AOIs).

2. Loci of MDMDM in normalized admittance diagram (NAD)

The normalized admittance diagram (NAD) is used here to elucidate the design of the modified FP filter. Consider the single boundary of a medium with a refractive index of N that occupies a half-space: the wave is obliquely incident from the other half-space occupied with a cover medium with refractive index N_i. The normalized admittance (NA) η equivalent optical admittance, in the transmitted medium is defined for the p-polarized state and the s-polarized state as [6], Sec. 2.2.3.

 

Fig. 1 (a) Loci of NAD for a typical metal-dielectric-metal FP filter; dashed lines represent possible loci of metal films with refractive index n-ik; (b) NAD loci of modified five-layered FP filter with structure N_i/M₁D₂M₃D₄M₅/N_sub.

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As a metal film that is grown on any surface, the equivalent NA of the whole structure in the NAD travels along one of the loci toward the common terminal point (n, -k) of all possible loci [6], which represents the vanishing of the transmitted light, the whole coating behaves as a bulk metal. The NAD of the traditional three-layered metal-dielectric-metal FP filter, N_i/M₁D₂M₃/N_sub, presented in Fig. 1(a), is analyzed. Figure 1(b) presents the parametric design of the modified five-layered FP filter, N_i/M₁D₂M₃D₄M₅/N_sub, where M and D represent metal and dielectric films, respectively. The equivalent NA of traditional FP filter N_i/M₁D₂M₃/N_sub starts from the admittance of substrate N_sub and follow the locus of metal layer M₃ to point A. The dielectric film D₂ brings the NA to point B above the real axis. The NA of the third layer follows the locus of the metal M₁ and terminates at C, which is close to or equal to the NA of the cover medium N_i, establishing an antireflection condition at the pass-band. Once the illumination wavelength deviates from the design wavelength, the terminal admittance departs from N_i, providing the high reflectivity of the stop band.

The NA locus in an admittance diagram can be used to design a modified FP band-pass filter. Firstly, the locus of the bottom layer M₅ ends at A and the locus of the second layer D₄ must end at point B, where the imaginary part is large enough to enable the locus of the middle layer M₃ to become huge and terminate at point C. The next two layers play as important roles in bringing the end of the NA locus to N_i, so the reflection vanishes. Therefore, the locus of layer D₂ next to the top layer M₁ should be above the real axis and end at the locus of the top layer M₁, which passes through N_i. The thickness of each layer can be tuned to design a band-pass filter. The loci of the M₁D₂M₃D₄M₅ are similar to those of an ultra long-range surface-plasmon-polariton [15]. According to the definition of normalized admittance, the large NA represent large ratio of magnetic field to electric field is large so that the electric field is relative small. The huge locus of the metal film indicates the magnitude of electric field within the third layer of M₃ is small, reducing the dissipation of energy in that metal film.

An example is presented here to elucidate aforementioned configuration. The five-layered system is N_i/M₁D₂M₃D₄M₅/N_sub = Air/Ag(22.7 nm)/Ta₂O₅(100.5 nm)/Ag(17 nm)/Ta₂O₅(107 nm)/Ag(22.7 nm)/BK7 glass. The refractive indices of Ag and Ta₂O₅ are taken from the database in the Macleod thin-film package and consider the dispersion of metal and dielectric films. Figure 2(a) shows that the normalized admittance locus is consistent with the aforementioned design of a modified FP filter at a wavelength of 600 nm. The NAs at the five interfaces are A(0.75, −2.88), B(8.23, 8.62), C(3.10, −7.26), D(0.57, 2.75) and E(1.32,-0.14) The locus of the middle layer intersects the real axis at (17, 0). Figure 2(b) shows the transmittance spectrum at normal incidence. The pass band is centered at a wavelength of 600 nm and the half-width of the pass band is 41nm. The figure also shows the spectrum of the N_i/M₁D₂M₃/N_sub = Air/Ag(22.7 nm)/Ta₂O₅(84 nm)/Ag(22.7 nm)/BK7 glass system that is designed for a wavelength of 600 nm; the transmittance peak value is 0.78 and the half-width is 80 nm, which is larger than that of a five-layered filter.

 

Fig. 2 (a) N_i/M₁D₂M₃D₄M₅/N_sub = NAD loci of a Air/Ag(22.7 nm)/Ta₂O₅(100.5 nm)/Ag(17 nm)/Ta₂O₅(107 nm)/ Ag(22.7 nm)/BK7 glass system designed at a wavelengthof 600 nm, (b) Transmittance spectra of N_i/M₁D₂M₃D₄M₅/N_sub = Air/Ag(22.7 nm)/Ta₂O₅(100.5 nm)/Ag(17 nm)/Ta₂O₅(107 nm)/ Ag(22.7 nm)/BK7 glass system and Air/Ag(22.7 nm)/Ta₂O₅(84 nm)/Ag(22.7 nm)/BK7 glass system

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The additional metal layer does not reduce the transmittance peak of the filter. The huge loop in the admittance diagram indicates that the electric field is reduced in the middle metal layer, reducing the dissipation of energy. Here, the finite element method (COMSOL 3.5a) is used with triangular high-order edge elements to simulate the distribution of field intensity over the five-layered structure when an electromagnetic wave with electric field amplitude of unity is incident on the films. As shown in Fig. 3, the field intensity is minimal in the third layer and most of the electric field intensity is distributed within the dielectric layers.

 

Fig. 3 Distribution of magnitude of electric filed throughout five-layered structure

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The aforementioned modified FP filter can be extended to a seven-layered system. N_i/M₁D₂M₃D₄M₅D₆M₇/N_sub = Air/Ag(15 nm)/Ta₂O₅₍102 nm)/Ag(25 nm)/Ta₂O₅(106.5 nm)/Ag(25 nm)/Ta₂O₅(108 nm)/Ag(22.7 nm)/BK7 glass system is considered as an example. The NA loci in the diagram, Fig. 4(a), include two huge loops. The normalized admittances that are associated with the eight interfaces of the multilayered system, in order from the substrate to the top surface, are A(0.75, −2.88), B(9.86, 8.59), C(0.98, −5.33), D(0.90, 5.09), E(11.19, −6.48), F(0.60, 2.09) and G(0.94, 0.03). The admittance loci of C and E are two huge loops that intersect the real axis at 14 and 17, respectively. Figure 4(b) shows the transmittance spectrum at normal incidence. The transmittance peak value is 0.78 and the half width is 33 nm, which is smaller than that of the five-layered filter.

 

Fig. 4 (a) NAD loci of N_i/M₁D₂M₃D₄M₅D₆M₇/N_sub = Air/Ag (15 nm)/Ta₂O₅₍102 nm)/Ag(25 nm)/Ta₂O₅(106.5 nm)/Ag(25 nm)/Ta₂O₅(108 nm)/Ag(22.7 nm)/BK7 glass system designed for a wavelength of 600 nm. (b) Transmittance as a function of wavelength.

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The narrow bands of the five-layered and seven-layered FP filters that are proposed herein are based on the huge loci in NADs, which are sensitive to wavelength. The design in Fig. 4 demonstrates the sensitivity of normalized admittance on which the reduction of the half-width is based. Interestingly, the transmission peak is not symmetric and the transmittance raises with wavelength over 640 nm. A transmittance peak occurs at a wavelength of 688 nm. At a wavelength that exceeds the design wavelength of 600 nm, all loci are shortened, but at a wavelength of 688 nm, the terminal point is still around unity on the real axis. Since many multilayer arrangements involve metal layers with large loci, suitably modifying the thickness of each layer can produce designs with more symmetrical transmittance peaks.

The transmission and passband performance of a metal-dielectric FP filter are not as good as those of an all-dielectric narrow bandpass filter. The transmission-induced method proposed here can be applied to improve the loss problem when a metal-dielectric multilayer performs extraordinary optical properties such as negative refraction.

3. Negative-index FP filter

The design of metal-dielectric FP filters can be applied to enhance transmission of multilayered metamaterials. An example of MDMDM with a wide-angle negative index at a wavelength of 363.8 nm has been developed [8]. Silver films and TiO₂ films formed a five-layered structure as a nearly symmetrical film stack. This work introduces a method for inducing transmittance by finely tuning the thickness of each layer. The optical constants are obtained from previous measurements. The five-layered MDMDM is modified from N_i/M₁D₂M₃D₄M₅/N_sub = Air/Ag(33 nm)/TiO₂(28 nm)/Ag(30 nm)/TiO₂(28 nm)/Ag(33 nm)/BK7 glass in Ref. 8 to N_i/M₁D₂M₃D₄M₅/N_sub = Air/Ag(11 nm)/TiO₂(26 nm)/Ag(25 nm)/TiO₂(28 nm)/Ag(20 nm)/BK7 glass to yield the admittance loci that are presented in Fig. 5. The middle Ag film in MDMDM has a large locus that intersects the real axis at η = 4.78. The equivalent refractive index is plotted as a function of angle of incidence using transfer matrix method [16]. The complex transmission and reflection coefficients of the five-layered system are calculated firstly. Then the equivalent relative permittivity ɛ and permeability μ are derived according to the formula in ref.16. The equivalent refractive index N and impedance Z are derived via the relationships: $Z=\sqrt{\mu /\epsilon }$ and $N=\sqrt{\mu \epsilon }$. The extinction coefficient is successfully suppressed to under 0.12, which is only half of that previously obtained [8]. The associated Figure of Merit(FOM), is defined as $FOM=\left|\mathrm{Re}\left(N_{equivalent}\right)/\mathrm{Im}\left(N_{equivalent}\right)\right|$, is varied from 12.8 at 0° to 15 at 70°, which is larger than the previously obtained FOM, which is varied from 4.0 at 0° to 5.7 at 70°.

 

Fig. 5 (a) Admittance loci of N_i/M₁D₂M₃D₄M₅/N_sub = Air/Ag(11 nm)/TiO₂(26 nm)/Ag(25 nm)/TiO₂(28 nm)/Ag(20 nm)/BK7 glass system.(b) Equivalent refractive index and FOM as functions of angle of incidence.

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4. Ag-α-Si multilayered filter with negative index at a visible wavelength

The multilayered metamaterial was developed for ultra-violet wavelengths; developing one for longer wavelengths is difficult. At visible wavelengths, the structure must be on the sub-wavelength scale and the same optical path for a longer wavelength requires thicker dielectric films or dielectric films with a higher refractive index. However, a highly refractive material with an index of refraction of over 2.5 generally has an extinction coefficient at visible wavelengths that would reduce the transmittance. At a wavelength of 660 nm, the index of refraction of the dielectric film in M₁D₂M₃D₄M₅ must exceed 3.5 to satisfy the sub-wavelength scale requirement. One possible material for D_i is α-Si, but its extinction coefficient yields a very low transmittance. For a symmetrical film stack MDMDM with a negative index of refraction, the transmittance is too low to be detected. Based on the design method herein, the transmittance can be raised to enable the negative refraction to be observed.

The transmission induced metal-dielectric multilayer with negative index of refraction is designed as follows. (1) Arrange a five-layered multilayer to perform NA loci as those in Fig. 2; (2) Calculate the transmission and reflection coefficients of the system; (3) Retrieve the equivalent refractive indices (multi-solutions); (4) The refraction of light wave is simulated using a near field software COMSOL to determine the correct equivalent refractive index. If the equivalent index of refraction is not negative, then the procedure must be repeated from step (1) to tune the thickness of each film. The equivalent negative index of refraction typically requires that the metal film is not too much thinner than the dielectric film, because if metal constitutes only a small fraction of each film, then the plasmonic effect that yield negative refraction will be reduced.

4.1 Induced transmission design in NAD

In fabrication, a single layer of silver and another single layer of α-Si were both deposited by sputtering. The optical constants at the wavelength of 660nm are Ag(N = 0.24-i4.11) and α-Si(N = 3.80-i0.18). Based on the measured optical constants, a five-layered metamaterial with negative index is designed as N_i/M₁D₂M₃D₄M₅/N_sub = Air/Ag(10 nm)/α-Si(34 nm)/Ag(42 nm)/α-Si(33 nm)/Ag(13 nm)/glass. The NAD of the designed five-layered system is shown in Fig. 6(a).

 

Fig. 6 (a) Admittance loci of N_i/M₁D₂M₃D₄M₅/N_sub = Air/Ag(10 nm)/α-Si(34 nm)/Ag(42 nm)/α-Si(33nm)/Ag(13nm)/glass system. (b) Admittance loci of N_i/M₁D₂M₃D₄M₅/N_sub = Air/Ag(12.0 nm)/α-Si(33.4 nm)/Ag(32.5 nm)/α-Si(35.1 nm)/Ag(11.0 nm)/glass system.

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The five-layered structure was then deposited according to the proposed design. It is difficult to deposited an uniform silver fim with precise thickness less than 20nm. Fig. 7 shows the cross-sectional SEM image, which shows the thickness of each layer as N_i/M₁D₂M₃D₄M₅/N_sub = Air/Ag(12.0 nm)/α-Si(33.4 nm)/Ag(32.5 nm)/α-Si(35.1 nm)/Ag(11.0 nm)/glass.

 

Fig. 7 The SEM image of cross-section of fabricated sample; thicknesses of layers are N_i/M₁D₂M₃D₄M₅/N_sub = Air/Ag (12.0 nm)/α-Si (33.4 nm)/Ag(32.5 nm)/α-Si(35.1 nm)/Ag(11.0 nm)/glass.

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Figure 8 (a) presents the transmittance spectra of the designed and deposited multilayers. The transmittance peak of the deposited multilayered is shifted to 670 nm from the designated wavelength of 660 nm and the transmittance maximum is reduced to be 0.165. The refractive index of each layer was then measured using an ellipsometer (J. A. Woollam) to be N_i/M₁D₂M₃D₄M₅/N_sub = Air/Ag(N = 0.27-i3.71) / α-Si(N = 3.95-i0.07) / Ag(N = 0.77-i4.30) / α-Si(N = 3.33-i0.08) / Ag(N = 0.13-i5.02) / glass. The NAD of the deposited five-layered system is shown in Fig. 6(b).

 

Fig. 8 (a) Transmittance spectra of designed and deposited experimental multilayers, Air/Ag/α-Si/Ag/α-Si/Ag/glass. (b) Equivalent refractive index and FOM as functions of angle of incidence.

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4.2 Optical properties of the FP filter: negative index and transmittance spetra

It is necessary to discuss the phenomenon that the measured and designed transmission curves in Fig. 8(a) are so close to each other. The wavelength of the transmittance peak is associated with the optical thickness between two mirrors of FP filter. The index of refraction of D₄ in the deposited Air/M₁D₂M₃D₄M₅ /BK7 glass system, is less than the designed value but that of D₂ exceeds the designed value. Therefore, the total optical thickness of D₂M₃D₄ is 273 nm, which is very close to the designed optical thickness of 264 nm. The extinction coefficients of the deposited M₁, D₂, and D₄ are smaller than the designed values and those of other two layers, M₃ and M₅, exceeds the designed values. Therefore, the two transmission curves are close to each other. Figure 8 (b) shows the equivalent refractive index and FOM as functions of AOI. The index of refraction changes from −4.24 at normal incidence to −4.04 at AOI of 70°.The extinction coefficient decreases from 0.86 at normal incidence to 0.74 at AOI of 70°. The associated FOM increases from 4.93 at 0° to 5.43 at 70°.

Figure 9 plots the transmittance as a function of wavelength and angle of incidence. The transmittance peak in p-polarized spectra is located between 670 nm and 660 nm at AOIs from 0° to 60°, the pass band is almost angle-independent. The maximum p-polarized transmittance is 0.165 occurs at wavelength of 670 nm and AOI of 0°. The maximum of p-polarized transmittance wavelength spectrum is larger than 0.16 over a wide range of AOI from 0° to 60°. The transmittance peak in s-polarized spectrum is located in a small range from 670 nm to 668 nm at AOIs from 0° to 60°. The pass band is blue-shifted from 670 nm at 0° to 668 nm at 70°. The maximum of s-polarized transmittance wavelength spectrum decays from 0.165 at 0° to 0.073 at 70°

 

Fig. 9 Measurement of (a) P-polarized and (b) S-polarized transmittance of deposited multilayer as functions wavelength and angle of incidence.

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5. Conclusions

This paper suggests a novel design for a FP filter in which metal films and dielectric films alternate between two metal films. The design is based on NA loci in the admittance diagram to minimize the dissipation of energy in the metal films and increase the sensitivity of the pass band to wavelength. This design can be utilized for any specified wavelength. When the multilayer becomes a subwavelength structure, the equivalent refractive index at the pass band maybe a negative real index. Since the pass-band remains high transmission, the imaginary part of the refractive index is so small that negative refraction can be achieved. Extraordinary optical properties, including angle-independence of the pass-band, high transparency, and a negative index are achieved. Negative refraction and negative refractive index from a metal-dielectric multilayer can be developed for different wavelengths realized by this loss-improved method. It is expected to design an induced transmission filter according to the requirement of equivalent index and impedance in the future. This work can be extended for a two dimensional and three dimensional nanostructures to produce novel polarization-dependent optical devices.