1. Introduction

Resonant optical structures reveal transmission, reflection or absorption peaks at specific wavelengths or angles, which are highly sensitive to the surrounding media or to the parameters of the structure itself. Examples include Fabry-Perot cavities, resonant gratings, grating coupled waveguide structures, whispering gallery modes, surface plasmon resonances, Bloch surface waves, Tamm surface waves and more [1–3]. Coupling between two or more resonators is another emerging methodology for obtaining high transparency window through what is called Fano resonances, electromagnetically induced transparency or plasmon induced transparency [4]. With the rising metamaterials field and the progress in nanofabrication of different structures in large areas this whole topic of resonance structures design and manufacturability is becoming increasingly active due to its technological importance as well as the ability it provides to demonstrate fundamental physical phenomena [5]. The phenomenon of the appearance of almost perfect transparency in a narrow spectral window in an otherwise absorbing system is a phenomenon that appears in several fields such as atomic physics, plasmonic materials and even classical systems of coupled oscillators [5]. Together with the intriguing phenomenon there appears the slowing down of light and even possible stopping of light pulses. However, the narrow resonance which might act as an optical filter is usually limited in its dynamic range and its manufacturability requires very precise materials parameters and alignment processes which increase the cost and usually limit its size. In electromagnetic induced transparency, since the absorption band is already narrow, the transparency window tunability range is limited [5]. Even in non-absorbing resonant structures such as the guided mode resonance explained as Fano type resonance, the tuning range is limited to few tens of nm up until a maximum of 100nm for optimized structures [6]. In an attempt to increase the tunability, I have searched for materials and surface waves which exhibit high absorption in a wide spectral range. The broadband lossy modes (LMs) [7–10] have been shown to exist in absorbing ultrathin films and they are more pronounced when the imaginary part of the dielectric constant is much larger than the real part, for example to demonstrate perfect absorption in ultrathin layer [11–13]. On the nature of the LM and whether it can be defined as a Fabry-Perot, surface electromagnetic wave, or lossy resonance, the reader is referred to the relevant literature [1,3,7–13]. The main purpose here is to use the fact that it is a broadband lossy mode with strong damping and therefore when coupled to a waveguide it is expected to reveal a transparency window similar to Fano resonance which can then be tuned to act as tunable filter or light modulator. The structure proposed is planar, and hence easy to fabricate, exhibiting controllable bandwidth resonances over a wide dynamic range covering over 500nm in the visible and the near infrared ranges and in principle can be designed for any spectral range. Its tunability can be achieved by tuning a sub-micron electrooptic, magnetooptic, thermooptic or piezoelectric layer with relatively small external fields or by simple angular scanning. Therefore, the proposed structure has a great potential for spectral imaging, Raman or infrared spectroscopy, sensing and the optical telecommunications industry. Such a narrow tunable filter can be used in many fields such as telescopes, to discover life in space or in miniature biosensors, to detect pollutants in the environment, food, agriculture, water quality, blood analytes, or with an endoscope inside the human body to diagnose diseases or to boost the capacity and speed of the telecommunication channels using the wavelength division multiplexing approach. Using fast modulation materials, it can be used for generating intense laser pulses through Q-switching or mode locking of lasers.

2. Geometry of the planar structure

The geometry of the structure is shown in Fig. 1. It consists of a three-layer planar structure deposited on a prism. The first layer is the ultrathin lossy layer $l$ necessary to generate the lossy modes. This layer does not need to be metallic as the only requirement for lossy wave in thin films is to have non-zero, and preferably large imaginary part of the dielectric constant ${\epsilon }_i$ without condition on the negative real part ${\epsilon }_r$.

 

Fig. 1 Schematic drawing of the resonating planar multilayered structure.

Download Full Size | PDF

When the material has negative real part it becomes plasmonic, although we shall show that even with metals having ${\epsilon }_i>>\left|{\epsilon }_r\right|$ such as Cr, it is possible to excite the LMs in ultrathin films down to 6nm thickness. The optimum thickness for maximum absorption in the prism coupling geometry can be easily estimated for TE wave using the Airy formulae for reflection and transmission of a single layer, writing the absorption as: $A=1-R-T$ and finding the conditions for maximum absorption. Considering the absorptive layer on the prism under total internal reflection (T = 0) and ignoring ${\epsilon }_r$, the result for the optimum thickness is:

For Cr, in the visible range we have ${\epsilon }_i\approx 20-22$, giving $d_{l-op}\approx 6-7nm$which agrees with the optimum thickness found to give the best contrast as will be seen below. An exact calculation of the TE reflectivity above the critical angle shows less than 20% reflection over the entire spectral range 400-1300nm, meaning over 80% absorbance (T = 0). This is the essence of the broadband lossy mode (LM). The 2nd layer acts as a clad, semi-clad or coupling layer $c$ to the 3rd waveguide layer $w$. The substrate $s$ is a low index semi-infinite substrate, which for biosensing application can be the environment medium such as water or air via the evanescent wave in the substrate. The existence of the lossy layer is crucial to generate the lossy mode (LM) which can then allow the excitation of the guided mode (GM). Its thickness preferably be very small in order to minimize the losses as we are seeking a high reflectance peak, and here we will present a design in which 6nm of Cr is an optimum thickness for the visible and near infrared ranges. It should be noted that without the lossy layer modes cannot propagate inside the waveguide and will be reflected back through the prism. The thickness of the clad or semi-clad layer controls the coupling between the LM and the waveguide, hence the width of any resonance that may emerge is expected to be controlled by modulating this layer. When it is very thick, the escape of the radiation from the waveguide becomes more difficult and the resonance will be satisfied only at very narrow range (spectral or angular). At moderate thicknesses comparable to the wavelength or less, more radiation can escape the waveguide at the resonance. Hence one expects to control the width of the resonance by varying the optical thickness of this layer, which is why we refer to it also as the coupling layer.

3. Results and discussions

To simulate the reflection properties of this structure the 2x2 transfer matrix method of Abeles is used. Figure 2 shows results for TM (a) and (TE) reflectivity spectra. The TM reflection spectrum shows dips because of the lossy wave excitation, while the TE reflection spectrum shows narrow peaks, both are tuned by varying the refractive index of the waveguide layer over a range of 500nm. Then 2nd order peak for the 1.68 case corresponds to the coupling between the lossy mode and next waveguide mode TE2.

 

Fig. 2 Reflectivity spectra from the stack of Fig. 1 using the following layers: 6nm Cr/575nm MgF₂/603nm variable index layer/MgF₂ substrate. The incidence angle in the SF11 prism is ${\theta }_p={52}^{\circ }$. (a) TM, (b) TE.

Download Full Size | PDF

Note that the refractive indices of the variable index layer are comparable to those of liquid crystals and the refractive index change required is nearly 0.152 which is easily achievable with ordinary nematic liquid crystal such as E7 or 5CB. Obtaining a narrow peak over such large tuning range is superior over many resonant structures and it will be shown later that the peak width can be made narrower by changing the optical thickness of the clad layer. As a comparison to other resonant devices, to achieve 500nm free spectral range (FSR) of a Fabry-Perot cavity the thickness of the cavity should be around $d={\lambda }^2/2nFSR\approx 350nm$by taking the value for wavelength of 750nm near the middle of the range in Fig. 2(b). In this case, the finesse required to get to full width at half maximum of FWHM = 2nm will be $F=FSR/FWHM\approx 175$which imposes high reflectivity condition on the cavity mirrors of more than 98%, making it more difficult to manufacture as well as requiring liquid crystal birefringence of $\Delta n=2d/FSR\approx 0.71nm$, which is very high and difficult to achieve. Hence the proposed tunable filter is superior over cavity and other resonant tunable structures. To vary the FWHM of the filter it is enough to increase the thickness of the cladding layer as shown in Fig. 3. Note that the dip in TM and the peak in TE shift towards higher wavelengths as the clad layer thickness increases. This is due to the increase of the phase accumulated in this layer as well as because the effective index of the GM is affected by the thickness of the clad layer when it is comparable to the wavelength.

 

Fig. 3 Calculated reflectivity spectra for the same structure of Fig. 2 at fixed refractive index of the waveguide layer of 1.56 but variable thickness of the clad layer. (a) TM, (b) TE.

Download Full Size | PDF

To explain the shift and the dependence on the cladding layer thickness let us write the phase difference between the transmitted and reflected waves which may be written as:

The fact that the reflectivity in TM consists of a dip-only without any signs of a peak indicates that the TM wave is not interacting efficiently with the waveguide or weakly interacting. This will be verified later based on the field distribution calculations. As one expects, the cutoff condition for TE guided mode excitation is smaller than that for TM. One can increase the interaction of the TM LM with the waveguide by decreasing the wavelength or increasing the angle. In fact, Fig. 2(a) shows some signs of a weak peak at the shortest wavelength spectrum around 500nm superimposed to the dip. To enhance this further we performed simulations for the same structure of Fig. 1 but at larger angle of incidence ${\theta }_p={55}^{°}$ (see Fig. 4). In Fig. 4(a) the TM dip becomes clearly similar to Fano type resonance shape indicating the coupling between two resonators. This is expected in TM waves similar to the case of plasmon coupled waveguide modes observed recently [14]. To strengthen further the fact that the emerging transparency windows are due to coupling between LM and waveguide mode we calculated for variety of materials shown in Fig. 6 including metals.

 

Fig. 4 TM (a) and TE (b) spectra calculated for the same structure of Fig. 1 at an angle of ${\theta }_p={55}^{°}$to increase the interaction of the TM LM with the waveguide.

Download Full Size | PDF

Note that when Ag, Cu are used an SP wave is excited in TM polarization and the transparency window appears with high contrast at the center of the SPR dip. This is well known now and investigated by several authors both using extended plasmons geometry [14] as well as other plasmonic and metamaterial configurations [15–18], but we would like to highlight the fact that Cu and Al are not suitable plasmonic materials for the near infrared range showing very broad peak, wherein here our original thought was to use this fact in order to increase the tunability range of the transparency window. Besides this, note that for VO₂ at 20 °C, which is an insulator, only lossy mode is excited with low contrast while VO₂ at the metallic phase (100 °C) shows beginning of very broad minimum with superimposed peak, although it becomes plasmonic (${\epsilon }_r<0$) only for wavelengths above 1200nm. The reason for the broad minimum is due to its larger ratio of $\left|{\epsilon }_i/{\epsilon }_r\right|$. Note that for Al, only 7nm is required, while for thin Al layer on a prism there is no symmetric SPR dip in the visible range and usually the minimum reflectance is not close to zero, wherein here we get near zero dip and high contrast peak, hence we conclude that even for Al in this case the wave is a lossy mode.

For TE polarization VO₂ is showing pronounced peaks, although with low contrast for the non-metallic phase at 20 °C, it demonstrates the possibility of modulating the peak thermooptically as this material is well known to have strong thermooptic property and undergoes insulator to metal transition at around 68 °C. Although being metallic, the real part of the dielectric constant becomes negative only at the infrared wavelength starting from nearly 1200nm, which is why it is used as smart window [19]. The better contrast obtained for the metallic phase in Fig. 5(b) is because the ratio $\left|{\epsilon }_i/{\epsilon }_r\right|$ is larger (~4) in the metallic phase compared to the insulating phase (~1/3). At the relatively high thicknesses of Ag and Cu, the structure is mainly reflecting at TE polarization and no coupling to the waveguide occurs. The peaks can also be observed at specific angles, which is important to show as the modulation at a single wavelength is very important for generating laser pulses and precision measurement or sensing applications. Figure 6 shows calculated reflectance versus angle inside the prism medium at different thicknesses of the waveguide layer made of AlN as this material is known to be piezoelectric. Almost 100% modulation can be reached with few nm modulation of the thickness at the optical telecommunication window wavelength of 1550nm.

 

Fig. 5 Calculated reflectance spectra for variety of materials including metals and non-metals to demonstrate the generality of the observed phenomenon to lossy materials. Parameters of the calculations for each material are given in Table 1 (a) TM, (b) TE.

Download Full Size | PDF

Table 1. Parameters of the layers used in the calculations of Fig. 5. The clad layer is MgF₂ while the lossy material is variable and the refractive indices and thicknesses fixed to give the optimum contrast.

View Table

 

Fig. 6 (a) and (b): Reflectance versus angle in the prism medium at different thicknesses of the AlN waveguide layer. The structure is: TiO₂ prism/15nm Cr/800nm MgF₂/AlN with variable thickness/MgF₂ substrate. (a) TM, (b) TE. (c) and (d) show the TE reflectivity spectra at different incidence angles (c) or different substrate refractive indices (d). In (c) and (d) the prism is SF11, 6nm Cr, 400nm MgF₂, 603 waveguide layer at refractive index 1.68. In (d) the incidence angle is 52 degrees while in (c) the substrate is MgF₂.

Download Full Size | PDF

Another interesting property highlighted in Fig. 6(c) and that is the large angular sensitivity of the resonance wavelength showing that one can scan nearly 500nm over 7.5 degrees and with further optimization it is possible even within 1 degree. The blue shift with the incidence angle can be explained based on the phase matching condition for exciting guided wave expressed in Eq. (6). As the wave vectors $k_c\propto \cos {\theta }_c/\lambda$, in the coupling layer are proportional to the cosine of the propagation angle ${\theta }_c$and inversely to the wavelength. Increasing the incidence angle ${\theta }_p$ causes an increase in ${\theta }_c$, hence for the phase matching condition to be satisfied it must be accompanied by a decrease in the resonance wavelength. However, for a full explanation the dispersion relation needs to be derived from which the resonance wavelengths can be found as a function of the different parameters. The dispersion relation can found from the condition of having poles in the reflectivity function. Such an equation for four layers’ system is not a simple expression and again numerical solution is necessary. Further discussion is elaborated below when the waveguide thickness becomes large and the refractive index of the waveguide layer becomes close to that of the coupling layer.

As a refractive index sensor the refractive index of the substrate should be variable and in Fig. 6(d) the shift of the resonance peaks with the analyte index is shown for refractive indices from 1 till 1.38. Note that the sensitivity is higher in the high index side reaching around 800nm/RIU while it drops to around 200nm/RIU in the low index side. This is because the wave becomes more confined in the waveguide layer as the substrate index decreases. However, the peak becomes narrower by the same factor, meaning the resolution will not change. In order to increase the sensitivity further as a sensor the waveguide layer refractive index may be chosen close to that of the analyte in which case the electromagnetic field will penetrate further in the analyte thus enhancing the sensitivity. One can think also of a porous waveguide layer in which a fluid can penetrate the pores and modify the effective refractive index of the mode, thus causing shift in the resonance wavelength at high sensitivity. As a porous waveguide layer its refractive index will be easier to make closer to that of the coupling layer (relatively small such as for MgF₂), in which case the sensitivity increases. To demonstrate this, we present in Fig. 7 simulations for lower waveguide layer refractive index. In Figs. 7(a) and 7(b) it is clearly seen that the resonance shifts by more than 500nm upon changing the WG RI by 0.06, almost three times less from the case of Fig. 2, meaning the sensitivity increased by around factor of x3. The sensitivity has strong dependence on the different parameters of the structure, similar to the case of grating coupled waveguide sensors [20,21], however here it is more complicated because the number of layers is larger. A more detailed study is planned for the sensitivity, its optimum conditions dependence on the structure parameters as well as from a practical point of view because when the sensitivity increases the tolerances on the thicknesses and the RIs of the different layers become more tight. When the spectral sensitivity increases, the angular one increases drastically and we verified that in this case tuning within more than 300nm range occurs within less than a degree change of the incidence angle. This is one of the reasons why we started our design in Fig. 2 with moderate but still high enough sensitivity using standard nematic liquid crystal material.

 

Fig. 7 TE and TM Reflectance spectra for different values of the waveguide refractive index closer to that of the coupling layer for the same parameters as in Fig. 2 except that $d_w=1000nm$for (a) and (c) while $d_w=2000nm$for (b) and (d).

Download Full Size | PDF

However, two important phenomena occur as the index mismatch $\delta n=n_w-n_c$ decreases: (i) the WG thickness has to increase in order to bring the peak back to the VIS-NIR range, which is easily understood from the WG mode Eq. (3); (ii) splitting of the peak into two with increasing gap as $\delta n$ increases similar to Rabi type splitting known in other systems such as when coupling occurs between extended and localized plasmons [22]. The new emerging peaks (Fig. 7(b)) shift in opposite manner to the original ones, meaning they blue shift as the WG RI increases and as the WG thickness increases or the incidence angle decreases. Amazingly even when $\delta n=n_w-n_c$ is not so large the splitting is over more than 500nm, so one can use the newly emerging peak tunability for practical applications over broadband. Figures 7(c) and 7(d) show for the case of $d_w=2000nm$and $n_w=1.417-1.384$ tunability over 700nm range. As it can be seen this new structure is rich in interesting and applicable phenomena which certainly triggers further detailed investigations and applications.

To verify that the waveguide mode is indeed excited at the resonance, we have used both our analytic algorithm based on the 2x2 Abeles matrices as well as COMSOL multiphysics to simulate the field distribution. Figure 8 shows the intensity distribution of the y-component of the electric field. From left to right, at 730nm far away from the resonance corresponding to refractive index 1.59 of Fig. 2(b) (resonance at 794.5nm), the field is mainly in the prism medium, that is no interaction between the lossy mode and the waveguide mode. The reflection at 730nm is only 5% according to Fig. 2(b) and since there is no transmission (incidence angle above the critical angle), the conclusion is that most of the energy at this wavelength and all wavelengths around the peaks get absorbed. This is the essence of the broadband lossy mode and our proposition to use this fact for wide tuning range. As the wavelength becomes closer to the resonance wavelength the energy becomes more confined in the waveguide and at the resonance it is totally in the waveguide. Hence this is the proof that the resonance corresponds to the guided mode, but due to the interaction with the lossy mode through the thin clad layer it escapes the waveguide in a Fano type resonance shape. To verify that the TM is not a waveguide mode we calculated the field distribution for the TM wave around the corresponding dip in Fig. 2(a) at 752nm as presented in Fig. 9. It clearly shows that most of the energy is outside the waveguide, hence the mode is mainly the lossy mode which gives dip corresponding to absorption in the lossy layer.

 

Fig. 8 TE polarization electric field intensity distribution ${\left|E_y\right|}^2$ for the structure of Fig. 2(b) at different wavelengths around and at the resonance of 794.6nm.

Download Full Size | PDF

 

Fig. 9 TM polarization electric field intensity distribution ${\left|E_{tot}\right|}^2={\left|E_x\right|}^2+{\left|E_z\right|}^2$for the structure of Fig. 2(a) at different wavelengths around and at the dip of 752nm.

Download Full Size | PDF

To understand the analogy between the observed phenomenon, the Fano resonance and the coupled classical oscillators, in Fig. 10 we present a diagram for illustration. In Fig. 10(a) a cross section side view of the structure is shown with the LM generated at the lossy layer-clad interface and the GM wave is in the waveguide. The quantum mechanical analog is drawn with the zero level representing the incident light from the continuum (prism medium), the 1st energy level with some width determined by the damping ${\gamma }_1$ represents the frequency of the LM ($|1〉\equiv |LM〉$) while the 2nd level represents the GM frequency ($|2〉\equiv |GM〉$) with damping ${\gamma }_2$. According to Fano [23] there are two possible transition paths that interfere together to give the Fano shape resonance: the direct transition $|0〉\to |1〉\equiv |LM〉$ and the indirect one $|0〉\to |1〉\equiv |LM〉\to |2〉\equiv |GM〉\to |1〉\equiv |LM〉$. The classical analog is depicted in Fig. 10(c) which shows two coupled oscillators having coupling constant $\kappa$ provided by the clad layer which is acting as a coupling medium. The coupling constant can be estimated by calculating the overlap integral between the GW and the LM field distributions as it is known from the coupled mode waveguide theory. Without the clad layer, using such an ultrathin lossy layer the coupling to the GM will be over a broadband and weak but as the clad layer thickness increases the coupling increases and becomes very strong at the resonances with increasing quality factor as it was shown in Fig. 3. The damping constant ${\gamma }_2$ is very small as the waveguide material is transparent which fits the requirement that one should have ${\gamma }_2<<{\gamma }_1$. The normalized shape of Fano resonance can be expressed by [23]:

 

Fig. 10 Schematic representation of the structure giving the two interacting waves LM and GM (a), their quantum mechanical analog (b), and their classical analog (c). (d) and (e) are plots of Fano shapes for different S values and signs for FWHM of 6nm and zero background reflectivity.

Download Full Size | PDF

At S = 0 a pure dip is observed while as S becomes very large in its absolute value a pure peak is obtained and if we compare to our results in Fig. 2(b) they correspond to large S around 8-10 while the TM case corresponds to small S values. Hence we may conclude that to get a peak with large contrast, a high S value is needed. For intermediate S values the shape has a peak superimposed on a dip. The interference between the two transition paths is actually expressed when S has intermediate values. To explain this, we write Eq. (7) in the following form:

4. Conclusions

To conclude, a narrowband tunable filter with high contrast and wide tuning range is proposed based on a planar multilayered structure that exhibits transparency window in reflection with a controllable FWHM. Examples are given showing tunability over wide spectral range (>500nm in the visible and near infrared) using a modulation layer of submicron thickness or small angular scan. The tuning sensitivity and range depend strongly on the waveguide layer thickness and the refractive index difference between the waveguide and the coupling layer. Under small index mismatch new peaks can appear via Rabi type splitting with gaps as high as 700nm or more exhibiting ultrahigh tuning with negative sensitivity. The phenomenon is observed in TE and TM polarizations with much higher contrast peak in TE. Analysis confirmed by field distribution calculations show that coupling between waveguide modes and lossy modes through a thin film coupling layer explains the observed phenomenon followed by Fano resonance analysis. The wide tuning range with high contrast is mainly achieved using an absorptive layer with high ratio between the imaginary to real parts of the dielectric constant that enables excitation of lossy modes known to exist over a wide spectral band in thin films. To avoid large losses, it is found that the lossy layer should be ultrathin (6nm Cr layer in TE or 7nm Al layer in TM for example). The tuning is achieved either by small angular scan of few degrees or by modulating the refractive index or thickness of the submicron thick waveguide layer from the visible till the near infrared range and in principle it can be designed to operate in any spectral range. Such a thin variable index or variable thickness layer can allow tuning at ultrahigh speed using conventional electrooptic, magnetooptic, piezoelectric or thermooptic materials at relatively low external fields. Scanning over such small angular range can also be achieved in high rate these days using microelectromechanical (MEMS) or liquid crystal scanners. Obtaining such a narrowband fast tunable filter over wide range can boost the field of hyperspectral imaging, Raman imaging, infrared sensing, diagnostics and many other applications. It should also be noted that since many materials have large ratio of the imaginary to real part of the dielectric function in the UV, infrared, THz and microwave regions, this concept should allow designing tunable filters and modulators for these regions as well. The combination of the proposed multilayered structure with anisotropic layers such as nanocolumns prepared by the oblique angle deposition technique, porous materials, graphene layer, near zero index materials, 2D materials, subwavelength gratings and other metamaterials is expected to reveal to additional interesting optical phenomena with applications of high potential. It should be mentioned that the intellectual property related to the concept proposed here and its potential applications are protected in a provisional patent application and a PCT [24]. Attempts to manufacture it and demonstrating its functionality are on-going and will be reported shortly.