1. Introduction

Dielectric gratings are one of the most important components in various optical systems recently. Since dielectric materials have high resistance against laser induced damage, the dielectric gratings are important especially for high power laser systems. Various dielectric reflection gratings have been proposed and developed up to now. A standard dielectric reflection grating has grooves on a dielectric multilayer mirror [1]. A total internal reflection grating is also one of the dielectric reflection gratings [2, 3]. The dielectric reflection gratings have high efficiency. However, the high efficiency is obtained only when the incident angle is near the Littrow angle and the angular bandwidth with the high efficiency is not so wide. The reflection gratings cannot be used at the exact Littrow angle, because the diffraction wave overlaps on the incident wave. Dielectric transmission gratings have been also developed [4, 5], which can be used at the exact Littrow angles and therefore are useful for various setups. However, the diffraction efficiency is limited by an amount of reflection loss. In order to suppress the reflection loss and to obtain higher diffraction efficiency, a buried grating was proposed, where the grating was immersed in a fused silica medium [6]. However, it seems that the practical fabrication of the buried grating is not so easy.

In this paper, we propose a new practical approach for increasing the diffraction efficiency of dielectric transmission gratings. The reflection loss is suppressed by using a multilayer coating under the dielectric grating. Therefore, the multilayer coating works as an anti-reflection (AR) coating for the grating.

2. Design of grating

An excellent explanation about dielectric transmission gratings can be found in Ref.[7], which is based on the modal method [8, 9]. Figure 1(a) shows a simple transmission grating with rectangular grooves. The incident wave is partly reflected on the air-grating boundary and partly transmitted into the grating region. Only waves with a discrete set of modes can propagate in the grating region [7]. For a fused silica grating with deep rectangular grooves, there are only two propagating modes and the other modes are evanescent. After propagation through the grating region, the modes are partly reflected on the grating-substrate boundary and partly transmitted into the substrate. The diffraction efficiency is determined by the coupling between the modes and the diffraction orders. The maximum diffraction efficiency is obtained for the Littrow configuration, in which −1^st and 0^th orders are symmetric. However, the diffraction efficiency is limited by the amount of reflection, which increases with the incident angle. The behaviors of reflection on transmission gratings are similar to those of the Fresnel reflection [7].

 

Fig. 1 Dielectric gratings with rectangular grooves. A standard transmission grating (a), and a grating with dielectric multilayers between the grating region and the substrate (b).

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For a plane air-substrate boundary, it is possible to reduce the reflection by a technique of dielectric AR coating. A physical mechanism of this standard AR coating is explained by interference between the original reflection wave (the wave reflected on the surface) and the reflection waves from the additional boundaries. One of the simplest AR coating can be made by using a pair of dielectric layers, which have a low index and a high index. The reflection wave can be suppressed by adjusting the thicknesses of these two layers. This technique of AR coating must be effective for transmission gratings. Here, we propose a method of making AR coatings under dielectric gratings. Figure 1(b) shows a grating with an AR coating, which consists of two additional layers. The returned waves are generated at the additional boundaries and they influence the reflection waves through the backward wave in the grating region. Main parameters of the AR coating are a number of layers, their materials and thicknesses. These parameters should be determined in order to minimize the amount of reflection.

Here, we design a transmission grating with 1740 grooves/mm for a TE-polarized wave at a wavelength of 1030 nm, which is the wavelength of Yb:YAG lasers. The grating pitch is 575 nm. This pitch is one of the standard gratings, which are used for pulse compressors of picosecond or sub-picosecond pulse laser systems. Firstly, we examine a standard transmission grating, which has rectangular grooves on a surface of fused silica substrate. The refraction index of fused silica is 1.45 and the Littrow angle is 63.59°. The Fresnel reflection at the plane air-substrate boundary is 19.2%. Only two diffraction waves with −1^st and 0^th orders exist in this grating. Figure 2 shows a contour plot of the diffraction efficiency for the −1^st order transmission wave. The two parameters of the contour are a filling factor f and a depth of grooves h. The filling factor is defined as a ratio of ridge width over pitch length. The diffraction efficiency was calculated by the modal method including multilayer reflection [10, 11, 12]. In the figure, a maximum efficiency is 92.3% for the parameters with f = 0.48 and h = 1.23 μm. The total loss is 7.7%, which is smaller than the Fresnel reflection. The main part of the loss is due to the reflection with −1^st order.

 

Fig. 2 Contour plot of diffraction efficiency for a standard fused silica transmission grating. The two parameters, f and h, are the filling factor and the depth of grooves.

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The diffraction efficiency can be improved by using an AR coating. One of the simplest AR coating is a pair of low and high index layers, as shown in Fig. 1(b). This AR coating can be made by using fused silica as the first layer and a material with higher refractive index as the second layer. One of the typical high index materials is Ta₂O₅, of which index value is 2.15. We found that the diffraction efficiency was improved by this simple AR coating. If the thicknesses of the two layers are given, a contour plot of the diffraction efficiency can be calculated as shown in Fig. 2. We found that the diffraction efficiency had several peaks in the contour plot and a peak with high efficiency was located at a position of f = 0.3 and h = 1.1–1.2 μm. This peak value depended on the thicknesses of the two layers. The optimal thicknesses were determined by scanning the two thicknesses independently. The improvement of the diffraction efficiency was found in cases using the other materials instead of Ta₂O₅. For example, the same quantitative improvement was obtained by using HfO₂, of which index value is 2.0. Dielectric layers are made by an evaporation process and rectangular grooves are fabricated by an etching process. Generally, controllability of thickness by the evaporation process is better than that by the etching process. In the case using two layers of SiO₂ and Ta₂O₅, the grating and the first layer are both fused silica. Therefore, the control of the depth is difficult. We should use a material as the first layer, which is not etched by the etching process for fused silica. Typical one of the materials is alumina. When two layers of Al₂O₃ and Ta₂O₅ are used as a AR coating, improvement of the diffraction efficiency is lower than that in the case using SiO₂ and Ta₂O₅. Therefore, we designed an AR coating using three layers, of which the first layer is alumina. In this case, we can control the depth of grooves by the evaporation process for the fused silica top layer. Figure 3 shows a AR coating with the three layers.

 

Fig. 3 Design of a dielectric transmission grating with the AR coating, which consists of three layers.

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We searched optimal thicknesses of these layers in order to maximize the diffraction efficiency. The optimal thicknesses were easily determined by scanning these thicknesses independently. A set of the optimal thicknesses is shown in Fig. 3. Figure 4 shows a contour plot of the diffraction efficiency using the AR coating with three layers. It is found that there are three areas with high efficiencies in the contour. The first, second and third areas locates at f = 0.2–0.4, 0.6 and 0.7, respectively. The first area spreads with a bow shape. These areas are characterized by an aspect ratio, which is defined as a ratio of depth over width of the grooves. Values of the aspect ratio are 2.9, 6.2 and 11 for the first, second and third areas, respectively. A low aspect ratio is better for fabricating the grooves by the etching process. Therefore, the first area should be selected as a target of the grating. The maximum value of the diffraction efficiency is 99.3%, which is obtained with the parameters of f = 0.31 and h = 1.14 μm. Figure 5 shows a profile of the square of relative field amplitude |E/E₀|² in the grating. The peak value is 1.69 at the center in the grating region. The Ta₂O₅ layer is the weakest in the AR coating for laser-induced damage. The value of |E/E₀|² is 0.3–0.4 in the Ta₂O₅ layer and the AR coating does not affect tolerance for laser-induced damage. For the standard transmission grating without the AR coating, the peak value of |E/E₀|² is 1.42 in the grating region. Therefore, the tolerances for damage are not so different between the gratings with and witout the AR coating.

 

Fig. 4 Contour plot of diffraction efficiency using the AR coating with three layers.

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Fig. 5 Contour profile of the square of relative field amplitude in the grating. The materials of layers are shown in the right hand side of the contour and the upper layer is the substrate.

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3. Discussions

As mentioned in the previous section, a standard AR coating makes additional boundaries and generates reflection waves at the boundaries. These reflection waves return to the surface and interfere with the original reflection wave (the reflection wave generated at the surface). As a result of the interference, the reflection waves are suppressed. This interference can be controlled by changing the thicknesses of the dielectric layers. In a case of the grating with the AR coating, the physical aspect is similar to that of the standard AR coating. The reflection waves generated by the AR coating propagate in the grating region as backward propagation modes. Each backward mode couples with two reflection waves with −1^st and 0^th orders. The coupling between the reflection waves and the propagation modes can be controlled by the two grating parameters f and h. So, the two reflection waves can be suppressed by choosing the grating parameters appropriately. The optimal grating parameters are different between the gratings with and without the AR coating. It is confirmed by that the peak positions in the contour plots (f and h) are quite different between Fig. 2 and Fig. 4. The physical mechanism to suppress the reflection is essentially equal to that of the standard AR coating. Therefore, this technique is useful for any incident angles and any polarizations.

In general, gratings are required to have high diffraction efficiency and wide angular bandwidth. So, we examined how the AR coating affects the angular bandwidth. Figure 6 shows diffraction efficiencies as a function of the incident angle. The two plots are values with and without the AR coating. From these plots, we can see that the reflection waves are suppressed by the AR coating. The angular bandwidth with high efficiency over 95% is 8.4° for the grating with the AR coating. The bandwidth is slightly narrower than that without the AR coating, but it is enough for practical applications. It is found that there is a turning point at 52.2°. The reflection wave with −1^st order does not exist below this angle.

 

Fig. 6 Plots of the diffraction efficiency as a function of the incident angle. A standard transmission grating (a), and a grating with the AR coating (b). The circles and triangles represent the values of transmission waves and reflection waves, respectively.

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4. Conclusions

In this paper we proposed a new approach for increasing diffraction efficiency of dielectric transmission gratings. The efficiency of standard transmission gratings is limited by reflection. In our approach, reflection waves are suppressed by the AR coating, which is made between the grating region and the substrate. We designed a grating with 1740 grooves/mm using the AR coating, and succeeded in increasing the efficiency from 92% to 99%. This technique using the AR coating is useful for many applications.