1. Introduction

The past two decades gave birth to several new branches of optics expanding the classical understanding of light-matter interaction processes. Photonic crystals, materials with negative refraction, cavities with extremely high Q-factors and subwavelength mode volume – these are only few cases of experimentally realized novel optical systems. Another example which demanded new theories to explain an experimentally observed effect is the so-called extraordinary optical transmission (EOT) [1–12]. In the pioneer work of Ebbesen et. al. [1] an unexpectedly high transmittance through a square lattice of holes in a metal plate was observed at the wavelengths about 10 times larger than the diameter of the holes. The measured transmission was called “extraordinary” since the transmission efficiency, which is defined as transmittance normalized to the total area of the holes, exceeded unity. Currently, the EOT effect is confirmed for a large variety of metal hole arrays (MHAs) including different shapes of the holes [1–4], square [1, 4, 5] and triangular [6, 7] lattices, perfect electric conductors [8], various real metals [1, 4, 5] and even doped Si [9].

The EOT is caused by the excitation of surface plasmon-like electromagnetic modes in the vicinity of metal. Existence of periodic holes can lead to the excitation of (evanescent) surface plasmon modes by propagating incident wave if the difference between the wavevectors of surface plasmon and incident wave fits the reciprocal lattice vector of the hole array [7, 10–12].

A combination of a MHA with another optical element exhibiting resonance can give rise to additional transmission peaks or to enhancement of the extraordinary transmission if resonances from different constituents fit each other. This provides additional tools to control and adjust the transmission of light. Among such hybrid structures a MHA sandwiched between two dielectric slabs [13], a dielectric slab sandwiched between two MHAs [14], a MHA lying on top of a 1D PhC [15], and a MHA placed on top of an active semiconductor whose optical properties can be controlled by an applied voltage [16] should be mentioned.

In Refs [17, 18] a metal layer was deposited on top of a GaAs/AlGaAs quantum well infrared photodetector (QWIP) structure. Then a triangular lattice of air holes was etched through the metal and quantum well stack. It was shown that photocurrent peaks appear at the wavelengths corresponding to the eigenmodes of the photonic crystal (PhC) formed by the air holes and the GaAs/AlGaAs layers. However, transmittance/reflectance properties of such structures as well as physical mechanisms of light coupling remained unexplored. Besides, there was no explanation given why the amplitudes of some photocurrent peaks are orders of magnitude larger than the amplitude of others.

In this paper we consider structures which can be regarded as a MHA lying on top of a 2D photonic crystal slab. The holes of a MHA are aligned with the pores of a 2D PhC. By means of 3D finite-difference time-domain (FDTD) simulations we will show that such MHA-PhC structures exhibit EOT occurring due to the resonant coupling of the wave, incident normally onto the MHA, to the specific eigenmodes of the PhC. In contrast to “classical” EOT effect [1–12] the spectral positions of the transmission peaks and corresponding reflection dips in our hybrid structures are defined by the spectral positions of the corresponding eigenmodes of a 2D PhC.

2. Model description

The schematic illustration of the investigated structures as well as their vertical composition are shown in Fig. 1 and Table 1 , respectively. Such a structure simulates the experimentally fabricated quantum well infrared photodetectors investigated in [17, 18]. In these papers the active region of the QWIPs consisted of 50 periods of GaAs/AlGaAs quantum wells. In our simulations we treat the active layer as a homogeneous one. The metal layer in our model has the optical constants which are typical for “good” metals in the mid-infrared region, namely, huge and negative real part of the dielectric constant and high extinction coefficient [19].

 

Fig. 1 Schematic illustration of the investigated structure.

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Table 1. Vertical composition of the structure considered. “Re(n)” and “Im(n)” are the real and imaginary parts of the refractive index, respectively.

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The plane wave is incident normally to the surface, i.e. parallel to the pores. The transmittance is recorded inside the GaAs substrate at the distance of 4.5 μm below the metal layer; the reflectance is detected at the distance of 3.5 μm above the metal layer (in air). The simulations are performed using the 3D FDTD method provided by the FullWAVE [20] commercial package. Periodic boundary conditions are applied for the four facets of the computational domain which are parallel to the pores axes. The other two facets have the perfectly matched layer absorbing boundary conditions.

3. Simulation results

Figure 2 shows the transmittance and reflectance of the structure with the following parameters of the triangular-lattice PhC: a = 3 μm, r = 0.3a, h = 4 μm, where a is the lattice constant, r is the radius of the pores, and h is the depth of the pores. We observe at least three pronounced dips in reflectance (and corresponding peaks in transmittance) which will be referred to as reflection dips 1, 2, and 3 corresponding to the labels in Fig. 2. In the calculation shown in Fig. 2 the source was linearly polarized with non-zero E_z and H_x components, in this case we will refer to the source as E_z-polarized. The same calculation performed for E_x-polarized source (E_x≠0, H_z≠0) shows no distinguishable differences in transmittance and refractance.

 

Fig. 2 Normal incidence reflectance (solid) and transmittance (dotted) spectra of the structure with the vertical parameters shown in Table 1 and for a 2D PhC with a = 3 μm, r = 0.9 μm, and h = 4 μm.

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It is important to note, that, for the given parameters, 67% of the surface is covered by an opaque metal layer and only 33% is occupied by the pores. However, the reflectance at λ = 6.84 μm (dip 3) is only 0.01 and the corresponding transmittance is 0.6 which means that the transmission efficiency, defined as the transmitted power per unit area of the free surface, is as high as 180%.

In Fig. 3 the xy cross sections of the near-field distributions of the amplitudes of E_y and E_z field components are shown for the minima of dip 1 (a), dip 2 (b), and dip 3 (c). In the cases (a) and (c) the amplitude of E_y component below the MHA is much higher than of E_z component. Thus, the interaction of the linearly polarized incident wave (having only E_z and H_x non-zero field components) with the MHA-PhC structure results in the appearance of a strong E_y-polarized mode directly underneath the metal layer. Dip 2 (Fig. 3(b)) corresponds to a transmitted wave which retains the initial polarization. It is also important to note that the E_y-polarized modes (Figs. 3(a) and 3(c)) have their maximal amplitude just below the metal layer while in the case of dip 2 the maximum of E_z field is in the middle of the pore.

 

Fig. 3 The xy cross sections of the near-field distributions of the amplitudes of E_y and E_z field components for the minima of dip 1 (a), dip 2 (b), and dip 3 (c) from Fig. 2. The grey scale is normalized to the amplitude of E_z component in the incident wave. The white color corresponds to the electric field amplitudes which are 4 and more times higher than in the incident wave. Two horizontal arrows on the left show the position of the metal layer. Field distributions for the entire spectral range are given in a movie (Media 1).

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In order to interpret the diagrams of Fig. 3 let us consider the eigenmodes of a perfect infinite 2D PhC consisting of a triangular lattice of air holes with radii r = 0.3a, embedded in a dielectric with refractive index of 3.2. These parameters correspond to the PhC formed by the air pores within the “active region” (see Fig. 1 and Table 1). Figures 4(a) and 4(c) show the electric field distributions for doubly degenerate dipole eigenmodes [21] at the Γ-point of the TM band structure (the E-field is parallel to the pores) at reduced frequencies a/λ = 0.692 and a/λ = 0.451, respectively. These pictures were obtained by using the 2D plane-wave expansion method (PWEM).

 

Fig. 4 The distribution of E_y component of the electric field. Cases (a) and (c) are calculated by 2D PWEM and correspond to doubly degenerate dipole mode at the Γ-point of the TM band structure at a/λ = 0.692 and a/λ = 0.451, respectively. Cases (b) and (d) show instantaneous distributions calculated by 3D FDTD which correspond to the dips 1 and 3 from Fig. 2, respectively. The volume shown corresponds to that of the “active region”. The upper figures correspond to E_x-polarized source, the lower figures are for E_z-polarized source. The black horizontal lines mark the position of the cross-sections shown in Fig. 3.

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Figure 4(b) shows the top view of the E_y component distribution within the “active region” at the wavelength corresponding to dip 1 for E_x-polarized source (top) and E_z-polarized source (bottom). The same but for the dip 3 is shown in Fig. 4(d).

From the comparison of Figs. 4(a) with 4(b) as well as Figs. 4(c) with 4(d) it is evident that the modes excited under the metal layer by an incident plane wave are TM-polarized eigenmodes of the 2D PhC. Since these eigenmodes are doubly degenerate and have dipole symmetry, two different patterns are observed in 3D simulations: if the source is E_x-polarized then the “dipoles” are oriented along the x-direction (top in Figs. 4(b) and 4(d)), while for an E_z-polarized incident wave the “dipoles” are oriented in z-direction (bottom in Figs. 4(b) and 4(d)).

The spectral positions of the TM dipole eigenmodes are also in good agreement with the spectral positions of the reflection dips: the discrepancy is about 1% for dip 1 and about 3% for dip 3. We want to emphasize that the spectral positions of the PhC eigenmodes were calculated by idealized 2D PWEM which implies that a PhC is infinite in all directions, with periodic boundary conditions in the plane of periodicity (xz-plane) and infinitesimal translation invariance in y-direction.

We attribute dip 2 to the excitation of a TE-polarized PhC eigenmode. Repeating the procedure which was performed for the dips 1 and 3 we plot first the 3D distributions of E_x and E_z fields within the structure for the two corresponding polarizations of the source (Fig. 5(a) and 5(b)). By examining the eigenmodes of the TE band structure at the Γ-point it is easy to find the corresponding two eigenmodes (Figs. 5(c) and 5(d)). The discrepancy between the spectral position of dip 2 and the corresponding PhC eigenmodes is about 5%.

 

Fig. 5 (a), (b): Instantaneous 3D distributions of the strength of the E_x and E_z components of the electric field, respectively; both are for λ = 5.01 μm (dip 2). The polarization of the source is E_x for (a) and E_z for (b). The volume shown is the volume of the “active region”. (c), (d): the distributions of the amplitude of the E_x and E_z components, respectively, calculated by the 2D PWEM for the TE eigenmode located at reduced frequency a/λ = 0.5745. The parameters of the structure are the same as in Fig. 2.

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One has to consider that even though we are not in the effective medium range (i.e. λ>>2a) a PhC slab can act as a Fabry-Perot cavity due to the reflections at the interface to the air (n = 1) region and at the interface to the substrate (where the holes end) providing additional resonances for normally incident light. The most direct way to check whether a resonance has a Fabry-Perot nature is to fix all parameters and to change the thickness of the slab. The results of such simulations are shown in Fig. 6 . We found that dip 2 is very sensitive to the depth of the pores. The decrease of the depth of the pores to 2.5 μm results in a blue-shift, broadening and weakening of dip 2 while dips 1 and 3 exhibit only a small broadening keeping their spectral positions unchanged. If the pores are only 1.5 μm deep, the dip 2 disappears while dips 1 and 3 are still present being, however, broader and shallower. Additionally, a weak reflection dip located near 4.5 μm disappears already for the pores depth of 2.5 μm. Therefore we argue that dip 2 appears due to a combination of Fabry-Perot and PhC modes since it demonstrates the properties of both: field distributions similar to PhC eigenmodes (Fig. 5) and redshift of the spectral position for decreasing pore depth.

 

Fig. 6 Reflectance spectra for three different depths of the pores: 4 μm (black), 2.5 μm (red), and 1.5 μm (blue). The other parameters of the structure are the same as in Fig. 2.

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In contrast to dip 2, dips 1 and 3 represent coupling to pure PhC modes since the thickness of the slab does not change their spectral positions. Of course, the coupling to dips 1 and 3 becomes weak if the depth of the pores is significantly lower than the excitation wavelength, e.g. as in the case of 1.5 μm deep pores.

Thus, from the point of view of potential applications dips 1 and 3 are of the most interest since they are well-pronounced, not sensitive to the depth of the pores and provide high field intensities.

To reveal the influence of different parameters of MHA-PhC structures the reflectance curves for different radii of the pores (a-c) and for a higher refractive index of the PhC background material (d) are shown in Fig. 7 . Both, the decrease of pore radius and the increase of the refractive index of the background material result in an increase of the average refractive index and should cause a red-shift of the band structure [22, 23] and consequently of the spectral positions of the reflection dips. That is exactly what is observed for dips 1 and 3. The vertical dotted lines show the spectral positions of the corresponding PhC eigenmodes calculated by the 2D PWEM. Please note that the period of the structure is the same in all cases.

 

Fig. 7 Normal incidence reflectance spectra for the MHA-PhC structures with different parameters. The period of the PhC is a = 3 μm and the depth of the pores is h = 4 μm in all the cases. (a), (b), (c): the QWIP-like structures with vertical composition shown in Table 1 and r = 1.095 μm, r = 0.9 μm, r = 0.75 μm, respectively. (d): the metal layer is lying on a dielectric with n = 3.5; r = 0.9 μm. The vertical dotted lines show the positions of corresponding eigenmodes calculated by the 2D PWEM.

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For a quantitative characterization of the polarization of the excited TM eigenmodes the densities of the field components averaged over the volume of the “active region” were calculated according to the formula:$〈|E_{\alpha }{|}^2〉=\frac{1}{N_V}{\displaystyle \sum _{i=1}^{N_V}|E_{\alpha }^{(i)}{|}^2},$where α denotes one the components (x, y, or z) of the electric field E, N_V is the number of the computational grid points within the volume, and $E_{\alpha }^{(i)}$ is the electric field amplitude in a given grid point within the volume. For the case shown in Fig. 2 we have obtained $〈|E_y{|}^2〉/〈|E_z{|}^2〉=13.4$ for the dip 3 and $〈|E_y{|}^2〉/〈|E_z{|}^2〉=13.8$ for the dip 1. Thus, within the “active region”, the energy concentrated in E_y component is more than an order of magnitude higher than the energy carried by E_z component.

4. Discussion and conclusions

It is necessary to highlight the differences between the “classical” EOT which appears in a single MHA (in air or on a homogeneous substrate) and the EOT observed in our MHA-PhC structures. First of all, in our model the medium below the MHA is not homogeneous but a PhC exhibiting a complicated (in comparison to a homogeneous substrate) dispersion. Secondly, an approximate condition for the spectral position of the first-order extraordinary transmission peak, which is given by [7, 24]$\lambda =\frac{\sqrt{3}}{2}a\sqrt{{\epsilon }_d}$ (ε_d is the dielectric constant of a substrate), is no more applicable. In a MHA-PhC hybrid structure the positions of the reflection dips due to excitation of the TM eigenmodes (dips 1 and 3) are defined by the positions of the corresponding PhC eigenmodes. As a consequence, the dips shift significantly if the radius of the pores changes even if the lattice constant is kept constant (see Fig. 7). Dip 2 is associated with mixing between Fabry-Perot and TE PhC modes since the spectral position of this dip depends on the depth of the pores.

Nevertheless, we believe that the physical explanation of the observed effects is very similar to the explanation of the “classical” EOT and that the excitation of the plasmon-like surface standing mode on the MHA plays a crucial role. Our analysis of the time-dependent transmittance, reflectance and electric field distribution shows that the formation of dips 1 and 3 can be described by three concurrent processes: (i) the incident wave excites a plasmon-like surface standing mode in the vicinity of the MHA, (ii) plasmon-like surface standing mode couples to the corresponding PhC eigenmode at the Γ-point (i.e. a standing PhC eigenmode is excited), and (iii) the coupling from a standing PhC mode to a propagating outgoing wave occurs.

It is necessary to note that excitation of a strong E_y polarized mode was also observed in photocurrent measurement on QWIP structures in [18]. The position of a strong photocurrent peak at reduced frequency a/λ = 0.454 is in good correspondence with the position of our dip 3 in Fig. 2 (a/λ = 0.439), in both cases the lattice constant was a = 3 μm and the pore radius r = 0.3a.

According to the results of [19] high conductivity metals like Au, Cu, or Ag, have similar optical properties in the frequency range (far below plasma frequency) studied here. In particular the real part of the dielectric constant is negative and extremely large. Consequently, the skin depth is very small. Variation of these properties has only a minor influence on the field distributions outside the metal. Therefore we argue that the results using a typical value for the refractive index of the metal (as in Table 1) are generally valid for these metals. This argument is further supported by the results of simulations in the vicinities of dips 1, 2 and 3 using gold as a metal including the dispersion relation of [19]. In these simulations only a small blueshift (<1%) of dips 1 and 3 (Fig. 2) was observed, although they are associated with strong fields directly under the metal layer. Furthermore, a comparison of the simulations of dip 2 performed with and without dispersion showed even no change in the wavelength position of Fig. 2.

In conclusion, it was shown that the extraordinary transmission peaks and the corresponding reflection dips in MHA-PhC hybrid structures appear due to the resonant excitation of the eigenmodes of the 2D PhC. In the case of TM eigenmodes (the electric field is parallel to the pores) the spectral positions of associated reflection dips do not depend on the thickness of the PhC slab and are defined by the planar geometrical parameters of the PhC. Moreover, the excited TM PhC modes are localized within a subwavelength distance below the MHA. In contrast to the EOT observed in MHAs without PhC the spectral positions of the reflection dips can be tuned by changing the radius of the pores only. The presented results offer new possibilities to control light on subwavelength scale and can be utilized in integrated photonics, quantum well photodetectors and lasers.