## Abstract

It is shown from numerical results deduced from a rigorous theory of diffraction that diffraction gratings made with two-dimensional dielectric photonic crystals may present blazing effects. Since these structures are lossless, efficiencies of 100% in the -1st order can be obtained in polarized light. Efficiency curves in Littrow mount are shown.

©2001 Optical Society of America

## 1. Introduction

Diffraction gratings [1–5] have played a vital role in many fields of science and technology, including spectroscopy, astronomy, beam-sampling for high power lasers, filtering, etc. The most commonly used grating is the metallic reflection grating. One of the main requirements for this kind of grating is the so-called “blazing effect”, viz. the concentration of the reflected light in one scattered beam only, at a given wavelength [1–4]. It is well known that ruled and holographic gratings present blaze effects, however, in the visible or near infrared regions, the power in the blazed scattered beam is always significantly lower than the incident power, due to losses in the metal coating. In order to improve the scattered power, many techniques have been proposed, namely the deposit of a stack of reflective dielectric layers on top of the metal. Unfortunately, dielectric coated gratings may present strong Wood anomalies [1–4,6], i.e. strong absorption phenomena caused by the excitation of guided waves in the stack of dielectric layers.

The aim of the paper is to show that the use of dielectric photonic crystals [7–9] could initiate the development of a new kind of diffraction grating, the photonic crystal diffraction grating. Such a grating is made by carving adequately a dielectric two-dimensional photonic crystal (for instance a crystal made of circular dielectric cylinders with hexagonal symmetry). It is well known that photonic crystals allow a total control of light propagation: inside some ranges of wavelengths (transmission gaps), such a structure does not permit light propagation. Thus, it is a natural idea to use reflection properties of photonic crystals inside the gaps to realize lossless diffraction gratings.

Numerical data is obtained from a generalization of a rigorous integral formalism devoted to the calculation of the field reflected and transmitted by a stack of dielectric periodic grids [10] to the case where the grids have not the same periods (however, the periods must remain a multiple of each other). This theory reduces the scattering problem to the solution of a set of integral equations and leads to the inversion of linear systems of equations.

After a brief presentation of this theory, it will be shown from rigorous numerical results that a photonic crystal grating can be blazed inside the transmission gap of the photonic crystal, as metallic gratings. However, its remarkable peculiarity lies in the fact that it can concentrate in the blazed scattered order the totality of the incident power, at least for polarized light. This amazing property could lead to a break-through in grating technology, provided the construction of such gratings can be achieved. Efficiency curves in Littrow mount will be compared with those of classical metallic gratings for both polarizations.

## 2. Presentation of a photonic crystal grating.

Figure 1 shows the structure of two photonic crystal gratings. At the left-hand side, the grating is made of dielectric rods of radius R and index ν placed in air while the one at the right-hand side is made by a dielectric substrate pierced by air holes (inverted contrast). It is worth noting that these gratings are composed by two different parts:

• The lowest part is a two-dimensional photonic crystal with hexagonal symmetry and period d. This photonic crystal has an infinite extension in x, while the extension in y is limited to N_{g} grids separated by a distance d√3/2 (N_{g}=6 in the figure)

• The upper part is a grid of period d_{g}=Md (M positive integer) placed at a distance h from the upper grid of the photonic crystal, the rods having a radius R′ and the same index ν as the rods of the photonic crystal.

In the case of the inverted contrast (right hand side), the top of the dielectric substrate is placed at a distance ht of the top grid whilst the bottom of the substrate is located at the distance h_{b} of the lowest grid. Throughout the paper, the period d of the photonic crystal is equal to 1 and the index of the dielectric material to 3.

In order to make efficient diffraction gratings with photonic crystals, two wavelength constraints must be simultaneously satisfied: one that characterizes the blaze configuration and another that ensures the existence of a photonic band gap, enabling the back plane to act as a mirror supporting only the single, specular, propagating order. The presence of the top grid is justified by scattering requirements. In general, photonic crystals present gaps in a range of wavelengths greater than the period d of the crystals, the center of the gap being placed typically at a value equal to 2d or 3d [9,10]. As a consequence, a grating having the same period d cannot be used in -1^{st} order Littrow mount, at least in the region where blazing effects are expected. Let us recall that in the - 1^{st} order Littrow mount, the -1st diffracted order and the incident wave are propagating in opposite directions. From the classical grating formula, it can be shown easily that such a mount requires the following relation between the grating parameters to be satisfied:

where d_{g} is the grating period, θ the incidence angle and λ the wavelength.

The most interesting range of wavelength is the “blazing region”, viz. the interval (2 d_{g}/3, 2 d_{g}), since in that range, two orders only are diffracted, the -1^{st} and 0^{th} orders. It is in this mount and in this range of wavelength that blazing effects are obtained even though, due to their particular geometry, echelette gratings allow one to get blazing effects outside that range as well. As a consequence, when the period d_{g} of the grating coincides with the period d of the photonic crystal, the blazing region is placed below the forbidden band gaps of the photonic crystal. In other words, when λ/d_{g} is greater than 2, the only reflected order is the zeroth order which has no interest in spectroscopy. This conclusion does not hold for the gratings of fig.1, where the period d_{g} of the grating is a multiple of the period d of the photonic crystal. In that case, the Littrow relation becomes:

in such a way that λ/d can be greater than 2.

## 3. The theory in outline.

We have used an integral theory of gratings. This theory has been developed in our laboratory since the 70’s [1,3,5] and its last version permitted us to deal with rod gratings as well as relief gratings, of arbitrary shapes and in arbitrary ranges of wavelengths [10]. The peculiarity of this code lies in the fact that it can deal with classical gratings used in spectroscopy as well as two-dimensional photonic crystals. Thus, it constitutes a valuable tool for the analysis of diffraction gratings made of photonic crystals.

The theory is based on the use of the notion of scattering matrix. This matrix expresses the linear relations between the plane waves propagating upwards and downwards on both sides of an horizontal grid. Computing successively the scattering matrix of each grid, from the bottom to the top by solving an integral equation, then taking into account the radiation conditions for the scattered field leads to the solution of a linear system of equations of infinite size. The validity of the numerical truncation of the matrix is checked by the convergence of the results as the size of the truncated matrix is increased.

This integral theory has been widely checked by extensive comparisons with experimental data [1–5]. From classical tests (energy balance, reciprocity,…), it appears that the results have in general a relative precision better than 1% but for photonic crystal diffraction gratings studied in this paper, the relative precision is of the order of 10^{-4}, thanks to the circular shape of the rods.

It is worth noting that the particular shape of photonic crystal diffraction gratings required a generalization of the theory described in ref. 10. Indeed, in that case, the field in the region of the photonic crystal has not the period d of the grids but the period d_{g} of the top grid. Consequently, the discretization points used in the solution of the integral equations must be placed on M rods, and not on 1 rod only as usual. Of course, this feature increases the size of the matrices to handle in the calculations by a multiplicative factor of M. Nevertheless, the computation time remains moderate since it does not exceed 30 seconds on a PC (processor Intel 450MHz) for a single wavelength calculation on the diffraction grating represented in figure 1 with M=3 and n_{g}=5.

## 4. Numerical results.

#### 4.1 Efficiency for s-polarized light

Fig. 2 shows the efficiency curves of the photonic crystal represented at the left of figure 1, made with dielectric rods, for some values.of the radius R′ of the upper rods. Indeed, previous studies have shown that this crystal was much better than that with inverted contrast for getting large and deep gaps of photonic crystals for s-polarization [9,10]. With an index of 3, large and deep gaps are obtained for R=0.2 and this value has been adopted in our calculations. Since the gap extends from λ=2. to λ=3.2, M has been taken equal to 2, in order to have the gap of the photonic crystal inside the blazing region of the grating. The number N_{g} of grids in the photonic crystal is equal to 5. The distance h between the photonic crystal and the top grid has been taken equal to √3/2 (i.e. the same value as the distance between two grids of the photonic crystal). The red curves in figure 2 give the energy reflected by the photonic crystal (without the top grid) for the same incidence as in Littrow mount. One can see that the main gap extends from λ=2.0 to λ=3.6. Inside this gap, the reflectivity always exceeds 99% and culminates at 99.98%, but higher values could be reached by increasing N_{g}.

The main conclusion that can be drawn from figure 2 is the existence of a blazing effect as soon as R′/d reaches 0.15. It is worth noting that the width of the efficiency curves is limited by the width of the gap of the photonic crystal. The maximum efficiency culminates at values greater than 99.96% in fig. 2c, 2d, 2e and 2f. Since these values are nothing but the reflectivity of the photonic crystal, it can be deduced that the maximum efficiency will tend to unity as N_{g} is increased. The relative width of the best efficiency curve (fig. 2f) is comparable to the widths of the best efficiency curves obtained for echelette, sinusoidal or lamellar gratings in s-polarized light. However, it must be noticed that the shape of the efficiency curve in fig. 2f is very peculiar, with two blaze wavelengths (λ=2.2 and λ=2.5) separated by a small interval where the drop of efficiency does not exceed 15%. In the same curve, a third blazing effect is obtained at a wavelength of 1.5, but that one is separated from the other ones by a strong anomaly where the efficiency drops to a value less than 5%.

Since the efficiency curves of figure 2 have been obtained without any optimization process, it can be expected that further improvements of the efficiency curves could be obtained, by varying h for instance, even though some calculations have shown that this parameter is not a crucial one for this polarization.

#### 4.2 Efficiency for p-polarized light

For this polarization, it has been shown that photonic crystals with inverted contrast are the best [9,10], thus we will consider photonic crystals represented at the right hand side of figure 1. With an index of 3, interesting gaps are obtained for radii R greater than 0.325, and we have adopted this value in order to keep a moderate size of the air holes. The distance h_{t} between the upper interface and the top grid is equal to 0.433 in all the calculations, in order to permit the radius R of the top grid to reach large values. With these parameters, the main gap is located between 3d and 4.2d, thus we have been led to adopt M=3, in order to get the gap of the photonic crystal and the blazing region of the grating close to each other.

The efficiency curves obtained in figure 3 show that, in contrast with the results obtained for s-polarization, the efficiency obtained without any optimization of h is quite small. No blaze effect is obtained and, even though the efficiency obtained in fig.3c culminates at a value close to 90%, the width of the peak is small. In order to improve the efficiency, three changes have been introduced in the parameters:

• Keeping M=3, we have introduced two rods per period (instead of 1) in the top grid. It emerges that such a change does not improve the efficiency, thus it has been abandoned.

• Keeping M=3 and adopting R′=0.4 as in figure 3c, h has been varied. It appears that this parameter is a crucial one and that the efficiency is improved as h is decreased. Taking into account the constraints (two rods cannot interpenetrate…), h=0.41 (instead of h=0.866 in figure 3) has been chosen. That way, a blazing effect is obtained, as it will be seen, but a further improvement cannot be reached since the radius R′ cannot be increased notably.

• Adopting M=3 and h=0.41, the size of the rods has been increased by replacing the circular shape by an elliptic one. The length a of the semi-axis a parallel to the y axis remains equal to 0.4 but the length b of the semi-axis parallel the x axis takes greater values, from b=0.4 (circular shape) to b=1.2.

Fig. 4 shows the result of this last optimization. Obviously, blaze effects are obtained, despite a strong anomaly which moves away from the right to the left of the gap as the horizontal semi-axis is increased. As a consequence, the best efficiency curves are obtained for values of b less than 0.8. For this polarization, the relative width of the efficiency curve around the blaze wavelength is significantly lower than that obtained with metallic gratings. The efficiency in fig.4 a, b, c, d, e culminates at values exceeding 99.5%, i.e. the reflectivity of the photonic crystal. Larger peak efficiencies could be obtained by increasing N_{g}.

## 5. Conclusion.

Thanks to a generalization of our integral theory of gratings, it has been shown that photonic crystal diffraction gratings can provide blazing effects with absolute efficiencies very close to 100% for both polarizations. The efficiency curves of this new kind of grating have a shape which is similar to those obtained with classical gratings for s-polarized light, but for p-polarized light, it appears that the widths of the efficiency curves cannot challenge those of the classical gratings. The explanation of this strange behavior could be the excitation of guided modes at the top of the grating with inverted contrast, between the top grid of the photonic crystal and the upper plane interface. It is well known that the propagation of surface or guided waves leads to “Wood anomalies” which can deteriorate the blaze effect of classical gratings [11]. Efficiency curves of figure 4 show efficiency drops which look like anomalies of classical metallic diffraction gratings and the explanation should be the same. This problem will be investigated in the next months. It could be interesting to continue this work by considering other kinds of photonic crystals in order to improve the efficiency curves for polarized or unpolarized light. Obtaining interesting efficiency curves for unpolarized light would require large complete gaps of the photonic crystal. It seems that photonic crystals with graphite symmetry could reach this goal [12]. Another peculiarity of this new kind of grating could be the possibility to introduce short drops in the efficiency curve by removing rods inside the photonic crystal. This property could be interesting for filtering of light.

## References and links

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