## Abstract

The properties of silver and aluminium wire-grid polarizers are examined in the volume plasmon frequency region where the transmittances of field with polarizations parallel and perpendicular to the grid lines are reversed with respect to their behavior outside the plasma region. Analysis of the behavior is conducted with effective approximate refractive index formulae and by simulations with rigorous Fourier modal method. The parallel polarization behaves as in a homogenous thin metal film while the perpendicular field is absorbed in the plasma region and transmitted otherwise. We further explain the performance by viewing the distribution of the field intensities inside the grating.

© 2009 Optical Society of America

## 1. Introduction

Wire-grid gratings are known to be able to transmit light that has an electric field perpendicular to the metallic wires and to reflect light with an electric field parallel to the wires. Thus they are mainly utilized as polarizers or polarizing beam splitters in applications such as projection displays. Wire-grid polarizers have been available for infrared region since 1960’s [1,2], however, it was not until last few years that they were fabricated also for visible light [3,4] and started to be commercialized.

The term plasmon was introduced in 1956 [5], although plasmons were known from the beginning of the 20th century. Plasmons are collective electron oscillations that can be categorized into two main groups; volume plasmons and surface plasmons. Surface plasmons, that is the confined electron oscillations at metal/insulator surfaces, in connection with gratings have been a subject in various papers, see for example Refs. [6–8]. Both theoretical and experimental research has been made and many applications, such as nanoconfined light sources [9] and biological sensors [10] have been proposed based on them [11,12].

In this paper, however, we investigate wire-grid gratings in the vicinity of the volume plasmon frequency, the fundamental frequency of a free oscillation of the electron sea in the metal. For ideal metals, the volume plasma frequency can be defined from the Drude model by setting the complex permittivity to zero. Therefore, the spectral position of the resonance is characteristic to the bulk material properties and always independent on the surface parameters, in contrast to the surface plasmons.

In the following, we introduce a switching wire-grid grating that works as a traditional polarizer at off-resonance wavelengths but as an inversed polarizer at the volume plasma wavelength. A similar inverse behavior has also been reported earlier for the Wood’s anomaly [13] when the grating period is slightly smaller than the wavelength of light [14]. The first theoretical analysis of the plasmon-type Wood’s anomaly in metallic gratings was given by Fano [15]. There are also some works about nanowires where increase of the extinction for perpendicular polarization has been obtained by particle plasmons, the localized surface plasmons [16,17].

In the next section we specify the conditions for volume plasma resonance and provide effective medium theory to describe the grating. In Section 3 we introduce wire-grid polarizers with the inverse behavior at the plasma wavelength and analyze the behavior with the field intensities and the effective medium theory. Finally we end our work with the conclusions.

## 2. Theory

There are two conditions for the volume plasmon resonance of real metals. Firstly, the real part of the permittivity has to be zero

and secondly, the imaginary part has to be much smaller than unity [18,19]

Normally this would not happen at all or it would happen only in the extreme ultraviolet region for metals such as lithium and aluminium, whereas for silver the conditions are satisfied in near UV at 330 nm as we can see in Fig. 1. This longer wavelength is more convenient because the period has to be much smaller than the operating wavelength for wire-grid polarizers, which sets certain boundaries for the possible fabrication process.

Let us now define the perpendicular field/polarization to indicate light with electric field perpendicular to the grating lines, and the parallel field/polarization to indicate light with electric field parallel to the lines. The behavior of subwavelength grating is a consequence of material properties as well as structure parameters. The grating can be considered as a birefringent homogenous thin film with different approximate effective refractive indexes for the perpendicular and parallel polarizations. The relationship between complex permittivity *ε* and complex refractive index *n* for non-magnetic medium is given by

We can derive the approximate effective refractive indexes from the theory of periodic waveguides by retaining only the zeroth order terms in the eigenvalue equations for the perpendicular and parallel fields [21]. This procedure gives the following equations:

for the perpendicular polarization and

for the parallel polarization. Refractive indexes of the modulated grating area are denoted by *n*
_{1} and *n*
_{2}, and *f* stands for the fill factor of medium with *n*
_{1}. In this paper we are considering gratings for which *n*
_{1} is metal and *n*
_{2} is air. The grating geometry is given in Fig. 2.

Equations (4) and (5) hold sufficiently well in the quasi-static limit *d/λ* → 0. Here *d* is the period of the grating and *λ* is the wavelength of light. In other situation the effective refractive indexes have to be calculated rigorously. For that purpose we have used Fourier modal method (FMM) for which the effective refractive index is obtained by dividing the zeroth order

propagation constant, *γ*
_{0}, by the wave number in vacuum, *k*,

where *k* = 2*π*/*λ*.

## 3. Results and analysis

Transmittance and absorbance at the plasma region for silver grating are illustrated in Figs. 3(a) and (b), respectively. The results have been calculated using FMM that is based on Refs. [22, 23], for a grating with the period 30 nm, the depth 100 nm, and the line width 5 nm. The substrate material is silica and the angle of incidence is normal to the grating surface. Solid line indicates the electric field parallel to the grating lines and the dashed line represents the perpendicular field.

The transmittance spectra are typical for a wire-grid polarizer below and above the plasmon region. The perpendicular field is transmitted and the parallel field is absorbed below and mostly reflected above the plasma wavelength. But as we reach the plasmon region the situation changes rapidly to the opposite. The grating becomes an inverse polarizer letting mainly the parallel field pass through the grating.

If we want to understand the underlying physics of this unusual behavior, we should examine the field intensities inside the grating. The intensities of the perpendicular and parallel fields at the wavelength 630 nm and at the plasma wavelength 330 nm are illustrated in Figs. 4(a)–(d).

Let us first consider the parallel field. From Fig. 3(b) we notice that most part of the field is absorbed below the plasma wavelength. This is because the field penetrates into the metal. And why part of the field does not slip trough the air grooves to the other side of the grating, is that the parallel field has to be continuous in the lateral direction according to the boundary conditions. Moreover, because of the subwavelength structure, the parallel field is almost constant in lateral direction as seen in Fig 4(c). Thus, the field in the air grooves behaves the same as in the metal wires, it attenuates along the propagation.

At the plasma region, the metal is more like dielectric with a small extinction coefficient. Therefore, the parallel field penetrates again into the metal. However, now the absorption of silver is small and the area of the metal wires is small compared to the air gaps between them. Thus, the small absorption of silver does not effect on the field and the field is transmitted, as seen in Fig. 4(d). In other words, in this case the transmission is solely defined by the proportion of the transparent area, air.

Above the plasma region silver is a good conductor. Therefore the parallel field cannot penetrate into the metal and is reflected, as seen in Fig. 4(c). Overall, it can be concluded that the parallel field acts the same as it would in the case of thin silver film.

The situation at the plasma wavelength can be also approached from the viewpoint of the effective refractive index and to consider the grating as an effectively homogenous thin film. For ideal metals at the plasma wavelength it holds that ℜ{*n*} = ℑ {*n*} ~ 0. Thus, we can expand Eq. (5) in Taylor series and take only the first term into account, which gives us an approximation for the effective refractive index

It can be seen from Eq. (7) that the grating indeed acts as dielectric with refractive index defined by the fill factor at the plasma wavelength. The corresponding values for silver at this wavelength are ℜ{*n*} = ℑ{*n*} ≈ 0.48. Therefore we would have to take more terms from the Taylor expansion which would lead to an appearance of a small imaginary part for the effective refractive index. This is consistent with the fact that in the case of silver, a small part of the field is absorbed.

The behavior of the perpendicular field can be explained by the boundary conditions according to which the normal component of the electric displacement density *D* has to be continuous across a boundary. Thus, in the modulated grating region it applies that

where *E* is the electric field. For off-resonance wavelengths it holds that | *ε*
_{Ag} | ≫ |*ε*
_{air}|. Then the intensity of the field in air must be much higher than in the metal

This means that the field is squeezed mostly in air and since air does not posses any extinction the field is transmitted, as shown in Fig. 3 and Fig. 4(a).

At the plasma wavelength we have the opposite situation: |*ε*
_{Ag}| ≪ |*ε*
_{air}|. From this it follows that

Thus, the field is highly confined in the metal and gets damped, due to the absorption of silver, when the depth of the wires is more than the skin depth. However, there still remains a small part of the perpendicular field that travels in the air grooves, as seen in Fig. 4(b). That part is transmitted and therefore the transmittance is not completely zero.

Again, we can expand the effective refractive index of the perpendicular field given by Eq. (4) in Taylor series. Assuming the ideal case at the plasma wavelength ℜ{*n*} = ℑ{*n*} ~ 0, the first term approximation gives us

Thus, the behavior of the grating is determined by the nature of silver, which is absorptive in the plasma region. Also now this approach is in good correlation with the results given above.

Many metals, such as gold, have too high imaginary part of the permittivity when *ε*
_{1} = 0. This would mean a considerable absorption for the both polarizations and no similar inverse behavior would occur.

The effective refractive indexes calculated with Eqs. (4) and (5) are presented in Fig. 5. The imaginary part is responsible for the extinction of the electric field and the real part the reduction of the speed of light. It is seen that the imaginary part stays low for the perpendicular field, but in the plasmon region it becomes significant and exceeds the corresponding value of the parallel field. In contrast, the extinction of the parallel field is high everywhere else but in the plasma region. These results correlate well with the transmittance values. A change occurs also in the real part of the refractive indexes, especially for the perpendicular field that has a dip at the plasma resonance. In conclusion, at off-resonance the effective medium behaves like dielectric and at the resonance like absorptive metal for the perpendicular polarization. For the parallel field the medium is almost like dielectric at the plasma wavelength, absorptive metal below it, and good conductor above it.

Unfortunately, inverse silver polarizer even with plasma resonance near the visible spectrum is not likely to be fabricated with the period of 30 nm and the line width 5 nm. More reasonable feature size could be 50 nm that can be achieved with electron beam lithography. Therefore, we calculated transmittances for silver grating with more practical parameters. The results are shown in Fig. 6(a). The period of the grating is 100 nm, line width 50 nm and height 100 nm. The approximate formulae Eqs. (4) and (5) are no longer valid because the period is close to the wavelength. We have therefore used Eq. (6) to determine the effective refractive indexes in Fig. 6(b).

The interaction of light with the grating is now more complicated than in the previous case because of the structure dimensions so large that it can not be analyzed as in the quasi-static limit. It is seen that the parallel field behaves characteristically the same as before but the perpendicular field has now higher extinction coefficient and lower transmittance below the resonance wavelength. Furthermore, the transmittance peak and the extinction minimum for the parallel field have remained their position but for the perpendicular field the transmission minimum is now shifted to 339 nm.

The performance of the silver wire-grid grating could be further optimized so that the transmittance contrast of the two polarizations at the plasma wavelength goes over hundred. In such a case, the transmittance would be near zero for the parallel field at off-resonance and for the perpendicular field at the resonance. Then the contrast would also be high at off-resonance. The drawback is that below the resonance the transmittance of the perpendicular field would drop down to near 10 %. Herein, it must be reminded that the bulk resonance induced inverse behavior is a consequence of the material property and therefore its position cannot be changed by structure optimization. This is of course a limiting factor, but on the other hand, because of the structure independence, the performance of the grating is not so sensitive to manufactural defects.

As mentioned earlier in Section 2, aluminium has a plasma wavelength in extreme UV, more precisely at 82 nm. Again, the quasi-static approximation does not apply and the performance of the grating as a polarizer falls short of perfection. Regardless, the performance can be enhanced by replacing the substrate layer with air and thus preventing the propagation of other orders than zeroth in the substrate. This is seen from the grating equation that defines the maximum wavelength that generates higher diffraction orders as *λ*
_{max} = *n _{s}d*, where

*n*is the refractive index of the substrate and

_{s}*d*the grating period.

Transmittances and effective refractive indexes around the plasma wavelength for a self-supporting aluminium wire-grid polarizer are given in Fig. 7(a) and (b), respectively. For this grating the period is 40 nm, line width 20 nm, and height 100 nm. The grating does not act as a polarizer in the domain where the wavelength is smaller than the grating period because many higher orders are propagating. But as we get closer to the plasma wavelength, we obtain the switched transmittances and switched imaginary parts of effective refractive indexes given by Eq. (6). In the wavelength range of 80–105 nm the transmission for the perpendicular field drops to zero while the transmission of the parallel field is around 80 %. This is exactly the opposite to the behavior at wavelengths above 250 nm. Therefore, the aluminium wire-grid can be regarded as a 2D photonic crystal with inversed band gap structures. Above the wavelength 200 nm the performance is characteristic of a wire-grid polarizer with a contrast of 10 000 and higher.

## 4. Conclusions

We have introduced a switching wire-grid polarizer for which the transmittances of the polarization parallel and perpendicular to the grating lines are reversed at the volume plasma wavelength. The polarizer transmits the perpendicular field in the off-resonance region and absorbs it near the resonance, while the parallel field acts as in a thin metal film. The inverse behavior has been investigated and explained with the analysis of the boundary conditions, field intensity distribution, and the effective refractive indexes, that correlated well with each other.

It was shown that the change from normal polarizer into inverse one was most evident when the grating period was much smaller than the plasma resonance wavelength. Although this would set challenges for the present-day fabrication methods due to the tiny dimensions, it does not prevent its potential application as a functional device when new nanofabrication techniques become available.

## Acknowledgment

This work was supported by the Finnish Graduate School of Modern Optics and Photonics. The EU Network of Excellence on Micro-Optics (NEMO) and the Research and Development Project on Nanophotonics funded by the Ministry of Education are acknowledged as well. We would also like to thank Prof. Martti Kauranen for the fruitful discussions.

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