A review and analysis is performed of various resonance effects associated with subwavelength one-dimensional (1-D) metal gratings for transverse electric (TE) and transverse magnetic (TM) polarized incident radiation. It is shown that by tuning the structural geometry (especially the groove width) and material composition of the 1-D gratings, polarization independent enhanced optical transmission (EOT) can be achieved. Three different cases of EOT have been studied for 1-D metal gratings: a) EOT for TM-polarized incident radiation b) EOT for TE-polarized incident radiation, and most importantly c) EOT for un-polarized incident light. Potential uses of these results in the design and improvement of various optoelectronic devices, such as polarizers, photodetectors and wavelength filters are discussed.
©2007 Optical Society of America
One-dimensional (1-D) reflection and transmission gratings have been researched for decades because of their interesting and “anomalous” optical characteristics and potential applications in various fields (reference  and references therein), including beam splitting polarizers, photodetectors, Raman scattering and super lenses [1–21]. These anomalous optical characteristics of 1-D gratings are explained by individual optical and electromagnetic resonance modes (and combinations or hybrids of these modes) that exist in these structures. Figure 1 shows the electromagnetic fields of three different types of resonances that are usually associated with the grating anomalies in 1-D gratings: 1) Wood-Rayleigh anomalies (WRs): WRs occur when one diffraction order grazes the surface of the grating as it changes from an evanescently decaying surface-confined mode to a radiating diffracted mode as the wavelength of the incident beam is decreased. 2) Surface plasmons (SPs): SPs are surface charge oscillations and their associated electromagnetic fields at a metal/dielectric interface. In one of the earliest papers addressing optical anomalies in 1-D gratings, Fano proposed that SPs were responsible for some of these optical anomalies . 3) Cavity modes (CM): CMs are resonantly excited modes within the groove of the grating. Only a TM-polarized incident beam (i.e., a beam with the magnetic field oriented parallel to the metal wires (see Appendix)) has a nonzero electric field component in the same direction as an electric field component of an SP mode in a lamellar grating and therefore only TM-polarized light can couple with and excite SPs [1,6]. In contrast, WRs and CMs occur for both TE-polarized and TM-polarized incident light.
Recently enhanced optical transmission (EOT) for TM-polarized incident light has received a lot of attention because of its potential use in various optoelectronic applications [13–19]. Initially, it has been suggested that SPs and CMs play significant roles in enhanced transmission, but there has been considerable disagreement in their relative roles and the physics underlying EOT remains in dispute. There are several works that address the issue of which resonance modes are responsible for EOT with a majority of them concluding that EOT is related to the excitation of SPs. In one of the earliest papers, Porto et. al showed that EOT for TM-polarized light can be achieved in two possible ways: 1) coupled SPs on the top and bottom interface and 2) CMs inside the slits . On the other hand Cao et. al have shown that SPs play a negative role in EOT and it is the CMs that are primarily responsible for EOT . Recently, Treacy et. al proposed an explanation in terms of coherent dynamical diffraction and asserted that it inherently includes surface plasmons . To clear the dispute of the physics of EOT, in a recent article, we have shown that it is the CM component of CM/SP hybrid mode that produces EOT in subwavelength metal gratings . Ultimately, we showed that SPs play a positive or negative role in EOT depending on the vortex of energy flow around the groove. The result demonstrating that CMs play a primary role in EOT for TM-polarized light is not only important by itself, but also suggests that similar EOT can be achieved for TE-polarized light, which is known to exhibit CM resonances in 1-D gratings. This is confirmed by Borisov et al., in a recent paper where they studied TE polarization EOT produced by quasi-stationary trapped modes (i.e., CMs) in a 1-D metal gratings with a high dielectric material (ε = 11.9) in the grooves .
It is the purpose of this paper to describe the optical properties of subwavelength metal gratings and the various optical and electromagnetic resonance effects associated with them, determine the geometrical and compositional dependencies of (and ability to independently tune) TE-polarization EOT and TM-polarization EOT, and determine if a practical grating structure can be designed that exhibits simultaneous EOT for TE-polarized light and TM-polarized light (i.e., EOT for both polarizations occurring at the same wavelength and angle of incidence). For the sake of completeness, we describe three different cases of EOT in subwavelength metal transmission gratings (see Fig. 2): (a) Case 1 - EOT of TM-polarized incident radiation, (b) Case 2 - EOT of TE-polarized incident radiation, (c) Case 3 - EOT of un-polarized (TE+TM) incident radiation. Case 1 has been studied extensively and it has been shown that subwavelength metal gratings can achieve EOT for TM-polarized radiation, with CMs playing a significant role in EOT. A more detailed review of Case 1 can be found in reference . On the other hand, Case 2 has received very little attention, primarily due to the inability of TE-polarized light to excite SPs, which were initially considered to be the modes primarily responsible for EOT. Case 3, to our knowledge, has never been studied. It will be shown in this paper that EOT can be achieved simultaneously for TE-polarized and TM-polarized incident light for a simple and realizable classical 1-D metal grating (i.e., a device with realistic structural dimensions and material parameters). The result is of great importance from both a theoretical point of view and for its use in optoelectronic devices. In fact, our present study was motivated by the need to improve the performance of metal-semiconductor-metal photodetectors (MSM-PDs), a structure that is essentially a 1-D metal grating on top of a semiconductor substrate. We have previously demonstrated that the performance of MSM-PDs can be greatly improved for TM-polarized incident light by tapping into a specific combination of resonance effects associated with EOT [14–17]. However, in order for the idea to gain practical significance it was necessary to design a 1-D grating structure that exhibits EOT for both TE-polarized and TM-polarized light. With the results of polarization independent EOT described in this work, along with the use of the light localization techniques described in references [14–17], it is expected that high performance (i.e., high responsivity and high bandwidth) MSM-PDs can be developed. More generally, the ability to tune and achieve enhanced or suppressed transmission for both TE-polarized and TM-polarized light at a particular wavelength by changing a grating’s structural geometry and/or material composition can be used in developing novel, or improving the performance of existing, optoelectronic devices such as TM selective polarizers (Case 1), TE selective polarizers (Case 2), and wavelength filters (Case 3).
2. Polarization independent enhanced optical transmission
Figure 2 shows the geometry of the 1-D metal grating studied in this paper and the three different cases of achieving EOT. In this paper, we consider metal gratings made of gold (dielectric constant from reference ) and all the calculations shown are for normal incident light. The electromagnetic fields in the top and bottom layer are expressed in pseudo-Fourier expansion (Floquet’s theorem) and the fields inside the grooves are expressed as a linear combination of orthonormal modes. By applying electromagnetic boundary conditions and a surface impedance boundary condition (SIBC) as described in [14, 20], a set of coupled equations is obtained that that is cast into a matrix equation that is solved to obtain the electromagnetic field expansion coefficients. The SIBC technique [1,8,9,13–17] has become increasingly popular for modeling metal grating structures in the infrared wavelength range because of its experimentally observed accuracy and because it is computationally less demanding than other techniques, such as finite difference time domain techniques  and rigorous coupled wave analysis . The SIBC technique is extensively discussed in references [14, 20] and a summary of the calculations capable of modeling both TM-polarized and TE-polarized light with arbitrary dielectrics in the groove and substrate are summarized in the Appendix. In order to test the accuracy of the calculations, we have analyzed the optical properties of the structures studied in reference  for TM radiation and found good agreement. For the sake of comparison, we have studied the optical response of the structure studied in reference  for TE-polarized light and found similar results. However, it should be noted that in reference  the dielectric constant of the metal was modeled using the Drude model and we have used the dielectric constant from the data published in reference  and hence the location, amount of the peak of transmission and linewidth of the peak is slightly different. The imaginary part of the dielectric constant used in reference  is larger than in reference  and hence transmission peaks are broader and the absorption losses are higher (high damping coefficient) when compared to the peaks modeled using the dielectric constant from reference .
Because of the fact that CMs are primarily responsible for EOT, an analysis of these modes, as well as their dependencies on groove width, groove height and dielectric constant of the material inside the groove, are important. CMs are essentially resonance modes that satisfy a Fabry-Perot condition inside the grooves, thereby producing high fields in these regions. It was found that for both TE and TM polarizations, the peaks of transmission due to CMs move to lower energies as the groove height or dielectric constant of the groove is increased, a result that is consistent with previous findings by other authors [1, 9, 12]. However, it was found that the most important design parameter that facilitates the tuning of TE-polarized and TM-polarized CMs are their dependencies on the groove width. It is interesting that this dependence has received very little attention compared to the groove height dependence. For a given polarization and fixed groove height and period, changes in the groove width alters the number of groove modes, energy at which EOT occurs, and the electromagnetic field distribution inside the grooves. For TM-polarized light CMs produced in very narrow groove openings, the resonantly enhanced electromagnetic fields is relatively uniform throughout the groove and as one increases in the groove width, the field redistributes with high intensity electromagnetic fields remaining close to the groove walls for wide openings. On the other hand, for TE polarization, the electromagnetic fields inside the grooves are concentrated more at the center of the groove, with very little fields on the side walls. As the groove width is increased, more resonance modes start occurring, redistributing the fields into lobes of high field intensities.
Figure 3 shows the location of the peak of transmission for TM-polarized and TE-polarized light as a function of groove width (with groove width varying from 0.35μm to 0.66μm) for a structure with period d = 1.75μm, groove height h = 1μm, and silicon inside the groove with a ε = 11.9. As can be seen from Fig. 3, the peak at which EOT occurs moves to higher energies for TM-polarized light and lower energies for TE-polarized light and that for an energy of 0.5eV (λ = 2.5 μm) and the point of intersection of the two curves produces a device design where simultaneous EOT can be achieved for TE and TM polarization. Figure 4(a) and 4(b) shows the reflectance and transmittance as a function of energy for TE-polarized and TM-polarized light for the above structure with the groove widths being 0.45μm for 0.615μm respectively. From Fig. 4(b), it can be concluded that for unpolarized incident light with an equal contribution from both polarization states (50% TM, 50% TE), as high as 94% of the incident light can be transmitted into the substrate. This result will have the potential to effect significant design improvements in a variety of optoelectronic devices, which are typically operated using polarization-independent radiation.
By plotting the peaks of transmission for TM-polarized light and the dips of transmission for TE-polarized light as a function of groove width, we can find the optimum groove width (point of intersection of the two curves ) for a design for a TM polarization transmitter and TE polarization reflector (Case 1). Case 1 is a well-known result and is used in designing polarizers, which separate the TE and TM polarization states. On the other hand, Case 2, in which TE-polarized light is transmitted and TM-polarized light is reflected, is equally important and has not been studied before. Using the same analysis, namely plotting the peaks of transmission of TE-polarized light and dips of transmission of TM-polarized light as a function of groove width, we can find the optimum groove width for the design of a TE-polarization transmitter and a TM polarization reflector.
Figure 4a shows one possible structure, which acts both as a TM polarization transmitter (TE reflector) and a TE polarization transmitter (TM reflector) at λ = 3.729μm (ħω = 0.333eV) and λ = 2.992μm (ħω = 0.415eV) respectively. Even though the line-widths of the peak transmission are different for Case 1 and Case 2 for the structure shown, it is possible to design narrow or broad peaks by changing the groove aspect ratio (i.e., groove height divided by groove width) depending on the application of interest. For example, photodetectors generally require a broad transmission peak, while wavelength filters may require narrow or broad transmission peaks depending on if they are being used as wavelength selectors or band-pass filters. Also, the phase of the reflected and transmitted beams have not been explicitly stated in this work, and while they may not be important for polarizers or optical transmission gratings, it can be important in other sensor applications.
To further understand the light channeling characteristics of these subwavelength gratings that exhibit simultaneous TE and TM polarization EOT, the electric and magnetic fields are plotted in Fig. 5 and the Poynting vector plotted in Fig. 6 for the above structure, which achieves EOT for 0.5eV TE-polarized and TM-polarized light. As mentioned before, the electromagnetic fields for TM-polarized light are more concentrated on the groove walls (this will be more prominent for wider grooves) compared to TE-polarized light in which the electromagnetic fields are positioned towards the center of the groove. Also, it can be seen that the resonant excitation of CMs is associated with significantly more field enhancement for TM as compared to TE, with a field enhancement for TM reaching as high as ~11 and a field enhancement for TE at ~4 (corresponding to intensity enhancements of ~121 and ~16 respectively) for this device geometry and specific CMs. A similar behavior has been previously reported for reflection gratings, where in it was shown that anomalous absorption of reflection gratings for TE is not accompanied by significant field enhancement, in contradistinction to anomalous absorption in TM polarization . The Poynting vector shows that the grooves act as light funnels, collecting and channeling the incident light through the grooves. However, the channeling is distinctly different for TE-polarized and TM-polarized light. For TM-polarized light, the channeling of energy occurs close to metal contacts and for TE-polarized light, channeling occurs more towards the center of the groove.
Two additional differences between the characteristics of TE-polarized and TM-polarized light incident on these lamellar grating structures are worth noting. First, the two polarization states have different cut-off frequencies at which a CM, and therefore EOT, occurs due to the different electromagnetic boundary conditions of the two polarization states at metal/dielectric interfaces. More specifically, there is no cut-off frequency for TM-polarization but there is for TE-polarization. Second, for TM-polarized light the energy of the peak of transmission and amount of transmission may be strongly dependent on the metal wire width and period due to the interactions between CMs and SPs if these two resonances have similar energies . For TE-polarized light however, no SPs are present and the location of peak transmission and the amount of transmission is almost independent of the metal wire width and hence independent of the period for a fixed groove opening.
3. Device applications
The ability to independently control TE polarization and TM polarization EOT, as well as align EOT for both polarization states, can be applied to many optoelectronic devices, both active devices and passive devices. We have shown previously that the light channeling described in this paper, and light localization techniques described in references [14–17], for TM polarization can be used to improve the performance of metal-semiconductor-metal photodetectors (MSM-PDs). The results of this paper will allow for the performance of MSM-PDs for a TE polarized incident beam, as well as an un-polarized incident beam, to be optimized as well. The use of the independent control of TE polarization EOT and TM polarization EOT in the development of two types of passive devices is described below.
Polarizers are essential optical devices for most optical systems and networks and can be classified into two general categories: absorptive polarizers and beam splitting polarizers. In absorptive polarizers, the unwanted polarization state is absorbed, whereas in beam splitting polarizers an un-polarized beam is split into two beams with the different polarization states traveling in different directions. Beam splitting polarizers are typically made of metal wire gratings, but suffer from internal metal absorption and scattering losses [25, 26]. Also, a typical beam splitter is designed to allow TM-polarized light to be transmitted and TE-polarized light to be reflected. This is based on the idea that subwavelength-metal gratings are transparent for TM-polarized light and act as perfect reflectors for the TE-polarized light. In this paper, we showed that by selectively tuning the resonance peaks of subwavelength gratings for TE and TM, we can design a polarizer, which can be used both as a TM transmitter (Case 1) and a TE transmitter (Case 2), with total transmission as high as 87% and 97% respectively (see Fig. 4) . The ability to selectively transmit a particular polarization state will be of great use in integrated optoelectronic systems, which require either TE-polarized or TM-polarized light depending on the application of interest. Also, the losses incurred due to absorption in the metal and scattering are significantly reduced by tapping into these resonance modes. It is interesting to note that, in fact, Case 2 would serve as a better polarizer than Case 1 with less absorption in the metal. This can be explained from our previous analysis of the electromagnetic field distribution for the two polarization states. In Case 1, large fields are concentrated on the metal walls leading to high absorption in the lossy metal as compared to Case 2, which has negligible fields on the metal walls and hence low absorption.
3.2 Wavelength Filters
Optical wavelength filters are important components for optical communication systems where they serve, for example, as wavelength-selective elements in optical receivers or as noise filters in optical amplifiers [27, 28]. In particular, there is great interest in filters that are fabricated on semiconductor materials, since they may be directly integrated with semiconductor optical amplifiers and detectors. For applications in fiber-optic networks, however, it is essential that these filters be independent of the state of polarization of the incident light. Typically, optical wavelength filters are designed with waveguides with Bragg gratings containing grating lines perpendicular to the propagation direction. In this paper, we show that by carefully designing the structural geometry of a classical 1-D metal grating, polarization independent wavelength filters can be achieved. A very important design consideration for wavelength filters is the line-width of transmission peak. For a given geometry, the linewidth of the peak of transmission for TM-polarized light can be changed by changing the groove aspect ratio, with high aspect ratios producing broader peaks and vice-versa. On the other hand, for TE-polarized light, the peaks are typically very narrow. Using the tuning techniques described in this paper, we have designed a 1-D metal grating (on top of SiO2) to act as a polarization independent wavelength filter at the communication wavelength. Figure 7 shows the optical response of this 1-D grating filter, which has peak transmission of 86% at 1550nm and FWHM of the transmission equal to 41nm. The present design is not only highly efficient in transmitting light for a particular wavelength, but also can be easily integrated with various other optical components.
We have shown that EOT can be achieved for both TM-polarized and TE-polarized light, both separately and simultaneously (in terms of wavelength and angle of incidence). It has been shown that the groove width is a very important design parameter in tuning the peak of EOT in TE and TM polarized light. The results of EOT are discussed in relation to improvements in, or the development of, novel optoelectronic devices including MSM-PDs, metal grating polarizers that can transmit TE and TM radiation and optical wavelength filter at 1550nm.
The optical and electromagnetic characteristics of lamellar gratings are modeled in this work using a coupled mode algorithm that uses the surface impedance boundary condition (SIBC) approximation. This method is described in detail in references [14, 20] and only summarized below. This method uses the following approximation relating the tangential components of the electric and magnetic fields at a dielectric/metal interface: 
where Z = 1/nmetal, with nmetal being the complex index of refraction of the metal. This approximation is valid if the dielectric constant of the metal is much larger than the neighboring dielectric (which is largely true in the infrared and visible spectral regions).
The electromagnetic fields are expressed as a linear combination of orthogonal modes as follows:
where fi(x, y) is the ẑ component of the magnetic field or the ẑ component of the electric field depending on if the TM polarization or TE polarization is being modeled respectively. The other electric and magnetic field components can be obtained using relations derived from Maxwell’s equations. Also, αn = kosinθincident+nK, K = 2π/d, , with n is an integer, d being the period of the structure θincident the angle of incidence, λ the wavelength, and εi the dielectric constant of the i th region. In Eqs. (A1) and (A3), the orthogonal modes used in the modal expansion are plane waves in the air and substrate layers and the following orthogonal modes Φ n(x, y) are used in the grooves:
where the terms μn and υn obey the relation:
Applying the SIBC condition to the left-hand and right hand sides of the grooves results in the following equations (respectively):
where c is the width of the groove and ηgroove = koεgrooveZ/i for TM polarization and ηgroove = ko/iZ for TE polarization. An important step in the above method is the solution to Eq. (A10). We have used the technique described by Tayeb and Petit , which is shown to be very effective and computationally less demanding. In this method the roots of Eq. (A10) are found by integration starting from an initial value. We have performed the integration using the Runge-Kutta method.
Applying boundary conditions equating the tangential field components and the SIBC conditions at the metal/dielectric interfaces at y = h/2 and y = -h/2 yields the following equations.
where γair = εair = 1, γgroove = εgroove, γsubstrate = εsubstrate, ηair = koZ/i and ηsubstrate = koεsubstrateZ/i for the TM polarization and γair = γgroove = γsubstrate = 1, ηair = ηsubstrate = ko/iZ for the TE polarization and φm = e iυmh/2.
Then multiplying Eqs. (A11) and (A13) by Xm(x) and integrating over the region 0≤x≤c and multiplying Eqs. (A12) and (A14) by e iαqx/d and integrating over the region 0≤x≤d yields the following matrix equations that are used to determine the unknown coefficients Rn, Tn, an and bn:
where the matrices φ, β, υ are square matrices with nonzero components along the main diagonal given by φm, βn, υm that have been previously defined; G, N, J, K are matrices with components given by:
The number of modes used in the electromagnetic field expansions were large and the solutions were convergent. The results obtained using the above approach were checked using another method that assumes that the walls of the grooves are perfectly conducting. These results yield practically identical results indicating that even though the convergence of TE polarization solutions using the SIBC approximation is worse than the convergence of TM polarization solutions, the main results showing EOT for both TM and TE polarizations will hold true when more accurate methods are used for the calculations.
Once Eq. (A15) is used to find all of the unknown coefficients, the reflectance (i = air in Eq. A23), transmittance and diffraction efficiencies (i = substrate in Eq. A23) can be calculated as the ratio of the ŷ -component of the Poynting vector for an outward propagating mode and the ŷ -component of the incident beam (assuming a normalized incident beam and a top layer being air):
where Ψ outward,n is either Rn or Tn and θoutput,n is the angle of the outward propagating mode.
This material is based upon work supported by the National Science Foundation under Grant No. 0539541. Also part of the work is performed at the Cornell NanoScale Facility, a member of the National Nanotechnology Infrastructure Network, which is supported by the National Science Foundation.
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