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Abstract
The frustrated total internal reflection (FTIR) spatial-frequency thin-film filter with metamaterial inclusions operating in a visible spectral range is investigated. Influence of nanoparticles embedded into a resonator layer is analyzed taking into account the frequency dispersion. Narrow band spectral lines and highly directional transmitted light beams for s- and p- polarizations are demonstrated at different values of refractive indices and thicknesses of layers. It is shown that the dispersion causes splitting of the filter bandwidth and spatial splitting of incidence beam into three angularly separated output beams. The type of thin-film filter considered should be useful in many applications, such as high-resolution spectroscopy and color imaging systems.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Due to interference, resonant effects may occur when waves are tunneled through the barrier system. Resonant tunneling effects, which are an optical analogy of quantum mechanical tunnelling, have found application in promising areas of applied and experimental physics, including integrated and fiber optics, electron transport in solid-state super-lattices, and FTIR spectroscopy. Currently, there are various types of devices such as filters, polarizers, and beam splitters based on light interference and FTIR in thin films.
The frustrated total reflection filter was first proposed in 1947 by Leirgens and Turner [1]. This is a device that uses resonant tunneling of light through a flat dielectric optical waveguide sandwiched between two low-refractive index thin films that act as potential barriers. The theory of the frustrated total internal reflection effect is presented in [2–5]. Recent advances in manufacturing technology have led to the appearance of dielectric thin films with extreme thickness accuracy and low absorption and scattering losses. A circular polarization beam splitter based on FTIR effect was considered in [6]. In [7,8] high performance FTIR based thin-film linear polarizing beam splitter has been demonstrated. In [9] a bandpass filter based on resonant tunnelling through an air layer in the frustrated total internal reflection regime is described. In [10] the three-layer interference filter based on FTIR effect for near-IR wavelengths was analyzed. In [11] the lateral shift and photon tunneling in a FTIR structure containing a negative-zero-positive index metamaterial barrier was studied. Spatial-frequency filtering is widely used to image enhancement in color imaging systems, color display devices, etc. In [12] the FTIR based color filtering device was proposed to split white color beam into three color beams. It splits the incident white linearly polarized light into three spatially separated light beams of primary colors, allowing all three components to be projected simultaneously. In [13] a tunable bandpass filter and polarizer based on resonant tunneling through an air gap between two hemi-cylindrical prisms coated with 4-layer thin-film matching stacks is described.
It is well established that FTIR structures reveal transmission peaks at specific angles and wavelengths, which are highly sensitive to the parameters of the prisms and of the embedded layers. It was shown in [14] that when measuring the spectral characteristics of FTIR filters, the divergence of the incident light beam should not exceed a certain limit value, which does not exceed a few angular minutes. However, the effect of frequency dispersion on the FTIR process has not been previously considered.
In this paper, the effect of anomalous frequency dispersion caused by metal nanoparticles embedded in the resonator layer of the FTIR filter on the resonant transmission of light is investigated theoretically. A schematic model of a filter is presented in Fig. 1. The device consists of a three-layer structure placed between two prisms with refractive indices ${n_p}$ and ${n^{\prime}_p}$. A trilayer structure consists of a high-index central layer with refractive index ${n_2}$ and thickness d2 sandwiched between two low-index films with refractive indices n1 and ${n^{\prime}_1}$ and thicknesses d1 and ${d^{\prime}_1}$, accordingly. A chromatic and angular filtration of an incident light beam takes place due to the resonance diffraction effect at a light propagation through the inhomogeneous stratified medium (layered structure), i.e. frustrated total internal reflection effect.
2. Simulation results
The integral [3–5] as well as the differential [2,3] methods can be applied for the description of the FTIR device. The plane-wave approach can be used to consider spatially collimated wave beams passing through the structure if the beam width w0 > λ [15]. As it was shown in [2], the filter is transparent if the filter length satisfies the condition L > l0 ∝ tgφ·d2exp(2qzd1), where l0 is the characteristic diffraction length, d1 is the thickness of low-index layer, d2 is the thickness of the central layer, φ is the incidence angle and qz is the imaginary part of the wave vector in a low-index layer.
Using the differential method developed in [2,3], for the solution of Maxwell equations, we find the local coefficient of transmittance T(x) = |Bout|2/|Bin|2, where Bin and Bout are the amplitudes of incident and output fields, accordingly.
2.1 Resonance condition
The resonance condition for the s-polarization taking into account the influence of prisms is determined by the equation [12]:
Analogical expressions for the diffraction length and resonance condition can be obtained in case of p- polarization.
The spectral shape and transmittance bandwidth are determined by the wave vectors deviations from their values at exact resonance. In the first order of approximation relatively to small parameters Δω/ω0 << 1 and Δφ/φ0 <<1, the spectral shape of the transmitted light is determined as:
where l0 is the characteristic diffraction length, ${l_p} = {l_0} \cdot 2{q_z}{k_z}/(k_z^2 - q_z^2)$, ζ is the integrated amount off-resonance deviation from the resonance frequency or resonance angle of incidence.The generalized off-resonance value is defined by
For the simplicity, a symmetrical three-layer structure with ${n^{\prime}_1} = {n_1}$ is considered. Spectral bandwidth and angular divergence of transmitted light are calculated for different layer thicknesses and refractive indices. Note that the results obtained also apply when the structure is asymmetric. Calculations show that the effects under consideration occur if the refractive indices ${n_1}$ and ${n^{\prime}_1}$ are noticeably less than ${n_2}$ and ${n_p}$.
In Fig. 2 the resonance wavelength dependence on the incident angle corresponding to the resonance transmission is shown for different values of the thickness of the central layer without metamaterial inclusions.
Fig. 2. Dependence of resonance wavelength on incident angle of s- polarized (a) and p- polarized (b) beams for different values of central layer thickness. 1 - d2 = 70 nm; 2 - d2 = 80 nm; 3 - d2 = 90 nm. ${n_2} = 2.0$, ${n_p} = 1.85$, ${n_1} = 1.38$.
A decrease in the refractive indices n1 and n2 leads to a shift in the range of working angles towards smaller values. Note that the ranges of working angles for s- and p-polarized beams do not coincide. This indicates that the device is also a polarizer.
2.2 Influence of frequency dispersion
Influence of frequency dispersion can be estimated from the resonance condition. For a fixed frequency of incident radiation (angular filter), the dependence of the angle of incidence on the dispersion $\Delta {\varepsilon ^{\prime}_2}$ follows from the resonant condition (1):
At the specified angle of incidence (frequency filter), the deviation of the permittivity value $\Delta {\varepsilon ^{\prime}_2}$ will cause a deviation of the resonant frequency $\Delta \omega = \omega - {\omega _0}$:
2.2.1 Plasmon resonance excitation
It is known that plasmon modes can be subdivided into two classes: propagating surface plasmons and localized surface plasmons (LSPs). Below the localized plasmon resonances associated with metallic nanoparticles are considered. LSPs are combined oscillation of free electrons in a metallic nanoparticle and associated oscillations of the electromagnetic field. The resonance frequency depends on the size, shape, and local optical environment of the particle [16,17] and usually occurs in the visible to near-infrared part of the spectrum for noble metal (Ag, Au) nanostructures.
Let’s consider a central layer with embedded metallic nanoparticles. It is assumed that the size of nanoparticles is substantially smaller than the wavelength of light and they are randomly distributed. In the framework of the Maxwell-Garnett model, such a medium is described by the effective dielectric permittivity, which for spherical nanoparticles satisfies the relation [18]:
It should be noted that the birefringence effect exists in the case of prolate or oblate spheroids and nanorods [18], so the effective dielectric permittivity will depend on the polarization of incident radiation.
Note that the Maxwell-Garnett model of an isotropic effective medium agrees well with experimental data, provided that the particle sizes are smaller than the radiation wavelength, and the distances between the particles are greater than their radii. The advantage of this model is that to analyze the propagation of radiation in a heterogeneous medium, it is not necessary to solve the Maxwell equations at each point in space and take into account the scattering on particles and the interference of scattered waves [19].
The optical properties of metal nanoparticles will be described by the expression in the framework of the Drude model:
In Fig. 3 the dependences of the real and imaginary parts of the value $\Delta {\varepsilon _2} = {\varepsilon _{eff}} - {\varepsilon _m}$ on the wavelength are presented. It is noteworthy that the plasmon resonces occur in a visible spectral range.
Fig. 3. Dispersion $\Delta {\varepsilon _r}$ (black color) and absorption $\Delta {\varepsilon _i}$ (red color) as function of wavelength for Ag (a) and Au (b) nanoparticles. a = 30 nm, $\eta = {10^{ - 3}}$.
For a silver nanoparticle with parameters ε0 = 5, ${\omega _p} = 1.38 \cdot {10^{16}}$s−1,${\gamma _0} = 2.39 \cdot {10^{13}}$s−1 [20,22] and ${\textrm{v}_F} = 1.4 \cdot {10^6}$ m s−1 [21], the plasmon resonance in a medium with ${\varepsilon _m} = 4.0$ occurs at a wavelength ${\lambda _p} \simeq 493$nm. For a gold nanoparticle with parameters ε0 = 9.5, ${\omega _p} = 1.3 \cdot {10^{16}}$s−1, ${\gamma _0} = 1.67 \cdot {10^{13}}$ s−1 [20,22], the plasmon resonance occurs at a wavelength ${\lambda _p} \simeq 607$nm.
2.3 Angular filter
Consider an angular filter when the wavelength of the incident beam is fixed. In Fig. 4 the dependences of resonance wavelength on incident angle of s-polarized beam for different values of the central layer thickness are presented using Drude model parameters for silver and gold.
Fig. 4. Resonance wavelengths as function of incident angle of s-polarized beam for different values of the thickness of the central layer. np = 1.85, n1 = 1.25, n2 = 2.20, d1 = 300 nm, 1 - d2 = 70 nm, 2 - d2 = 80 nm, 3 - d2 = 90 nm; η = 10−3. (a) Ag; (b) Au.
It is seen that the frequency dispersion of the heterogeneous layer causes the angular splitting of the beam for a given wavelength of the incident light. One, two or three angularly separated output beams can be observed by changing the wavelength of the incident beam.
In Fig. 5 the angular intensity distributions corresponding to the incident beam with wavelength λ = 570 nm are presented for the layer with silver nanoparticles. The layer parameters are the same as in Fig. 4.
Fig. 5. Angular bandwidths of transmitted light at d1 = 300 nm (a, b) and d1 = 500 nm (c, d). d2 = 80 nm; λ = 570 nm.
It follows from the calculations that an angular divergence of transmitted light beams is sensitive to the low-index layer’s thicknesses and decreases with the increase of their thicknesses. The angular bandwidths $\delta \varphi = {0.05^ \circ }$ [Fig. 5(b)] and $\delta \varphi = {0.00036^ \circ }$ [Fig. 5(d)] were obtained for d1 = 300 nm and d1 = 500 nm, respectively.
2.4 Frequency filter
Consider now a frequency filter when the incident angle is fixed.
In Fig. 6 the resonance incidence angle dependence on the wavelength is shown for different values of the thickness of the central layer.
Fig. 6. Dependence of resonance incident angle of s-polarized beam on the wavelength. np = 1.85, n1 = 1.25, n2 = 2.20, d1 = 300 nm; 1 - d2 = 70 nm, 2 - d2 = 80 nm, 3 - d2 = 90 nm; η = 10−3 . (a) Ag; (b) Au.
It is seen that the changing the incident angle leads to the displacement of the resonance spectral lines. This means that it is possible to obtain physical information about the structure of layers from measurements of spectral lines at different incident angles. By changing the angle of incidence within certain limits, it is possible to get one, two or three resolved spectral lines at the output.
In Fig. 7 the spectral shapes of s- polarized transmitted light are shown for the central layers with silver [Figs. 7(a), 7(b)] and gold [Figs. 7(c), 7(d)] nanoparticles. The spectral bandwidths decrease with the increase of the low-index layer thickness and the spectral line widths $\delta \lambda = 0.013$ nm [Fig. 7(b)] and $\delta \lambda = 0.17$ nm [Fig. 7(d)] were obtained at d1 = 500 nm for silver and gold layers, respectively.
Fig. 7. Spectral shapes of transmitted light: np = 1.85, n1 = 1.25, n2 = 2.20, d1 = 500 nm, d2 = 70 nm, η = 10−3. (a, b) Ag: $\varphi = {61.4^ \circ }$; (c, d) Au: $\varphi = {57^ \circ }$.
It is seen that three resonance bands exist at once for a given incidence angle. This indicates that the resonance condition in the resonator is fulfilled for three wavelengths at once. This property of the device can be applied in imaging systems. Note that significant decrease in spectral line width occurs arising from both photonic and plasmonic effects.
3. Discussion and conclusions
Thus, a new type of narrowband FTIR filter has been proposed that uses a three-layer structure with metamaterial inclusions embedded between two prisms. The structure exhibits resonant transmission of three spectral lines or beams near the anomalous dispersion region at specific wavelengths or angles. This indicates that in contrast to the case discussed in [12], the resonance condition in the resonator is fulfilled for three wavelengths (frequency filter) or three incident angles (angular filter) at once. High spectral and angular sensitivities of the device are demonstrated for different values of refractive indices and thicknesses of layers.
It is shown that frequency dispersion in the layer’s substance has a significant effect on the processes in the filter. Usually, dispersion, as well as absorption, degrades the filter characteristics. However, the FTIR system can be also used as a device for determining fundamental optical characteristics, such as the permittivity of a layer material. It should be noted that the determination of optical characteristics in a wide area of intrinsic absorption of matter is one of the important problems of solid state physics and spectroscopy: their knowledge helps to understand many physical processes.
Resonant transmission of three spectral lines at once near the anomalous dispersion region is demonstrated. This indicates that “white” laser source can be created using the core-shell spherical nanoparticles formed by a semiconductor gain medium which are embedded in the resonator layer. Interest in studying the effect of frequency dispersion on the processes in the filter is also related to the fact that exciton absorption lines are observed in the optical region of the spectrum of solid states. The dispersion associated with exciton states of impurities and embedded molecules can also affect the operation of the filter.
Effects of anomalous frequency dispersion caused by metamaterial are also of practical interest for resonant tunneling structures such as liquid-crystal [23], acousto-optic [24,25] and volume-Bragg-grating filters. Note that acousto-optical filters with metamaterial inclusions, considered in [24,25], also provide ultra-narrow spectral lines due to resonant Bragg diffraction on a periodic volume grating created by ultrasound in a crystal. However, these devices require a much larger thickness of the crystalline material. Liquid crystals or acousto-optic materials are the most important due to their advantages of continuous tuning and wide dynamic range. Tunable optical filters are becoming increasingly important due to the growing demand for hyperspectral imaging, optical coherence tomography, etc. [23]. Resonant structures based on inhomogeneous inclusions opens up opportunities to create devices of infrared (IR) and terahertz technology, inaccessible to ordinary materials. In [26] a refractive index sensor for fluid media at terahertz frequencies using plasmonic enhancement of fields contained within a one-dimensional photonic bandgap medium was proposed. In [27] the resonance tunneling and the Goos-Hanchen (GH) shift in a FTIR configuration coated by a graphene sheet were studied. Tunable resonant Goos–Hänchen and Imbert–Fedorov shifts of terahertz beams from graphene plasmonic metasurfaces were investigated in [28]. In [29] large GH shifts near the surface plasmon resonance in subwavelength gratings were demonstrated.
In conclusion, a new FTIR filter was proposed based on a combination of photonic and plasmonic effects that lead to narrowing of the bandwidth and conversion of the transmission spectrum. It is shown that the incident beam of a given wavelength is split into three angularly separated beams. Splitting of the filter bandwidth into three narrowband spectral lines for the specific incidence angle is demonstrated. This type of a thin-film beam splitter should be useful in many application areas, including the general field of spectroscopy, sensors, with spectral regions extending from the ultraviolet to the far infrared. Note that the considered resonant tunnelling effects can occur for waves of any nature, including electromagnetic, acoustic, etc.
Funding
Ministry of Science and Higher Education of the Russian Federation (0069-2019-0006); Russian Foundation for Fundamental Investigations (19-29-11026).
Disclosures
The authors declare no conflicts of interest.
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