## Abstract

A new approach for the realization of highly dispersive dielectric transmission gratings is presented, which enables the suppression of any reflection losses and, thus, 100% diffraction efficiency. By applying a simple two-mode-model a comprehensible explanation as well as a theoretical design of such a reflection-free transmission grating is presented.

© 2008 Optical Society of America

## 1. Introduction

Dielectric transmission gratings have emerged as the key components in many present-day optical setups. One reason for this is their high resistance against laser induced damage, but also the possibility to realize various optical functions simply by optimizing their profile parameters. Many applications such as high-power femtosecond laser setups require a highly-efficient diffraction into the dispersive −1^{st} order. Usually, the highest diffraction efficiency can be achieved, if the grating is illuminated in the Littrow configuration at an angle *φ _{in}* which fulfills the condition

where *λ* is the vacuum wavelength and p the grating period [1]. In this configuration, the −1^{st} transmitted order propagates symmetrically to the 0^{th} order (Fig. 1).

If, furthermore, the grating period *p* fulfills the condition

(*n* is the refractive index of the substrate material) it is theoretically possible to deflect 100% of the transmitted light to the −1^{st} order. In an earlier paper [2] we gave a physical explanation for this effect for the case of a rectangular transmission grating. However, this consideration neglected any reflection effects, which is why the diffraction efficiency of a real grating is always less than the predicted 100%. These reflection losses of rectangular transmission gratings increase rapidly as the dispersion increases, or rather as the ratio between the period and the wavelength is decreased. Figure 2 shows a numerical simulation of the reflection losses as a function of the grating period for a rectangular fused silica transmission grating. Here and in the following, a 1.064µm wavelength and a 1.45 refractive index of the substrate and the ridges is assumed. Furthermore, TE-polarized light is considered, where the electric field vector oscillates parallel to the grating lines. Some remarks on TM-polarized light are made at the end of this paper. The numerical simulations were made using the commercial software *Gsolver*, which is based upon the rigorous Fourier modal method [3]. Depending on the groove depth and the fill factor (ratio between grating ridge width and period) the reflection of a rectangular surface relief grating can vary within the values that are illustrated by the grey area. For comparison, the dashed blue line shows the Fresnel-reflection at a plane interface by illuminating it under an angle that corresponds to the Littrow angle of a particular grating period. The reflection at the grating is mostly less than the reflection at a plane interface, but similar to the Fresnel-reflection, it increases significantly if the grating period approaches λ/2. As an example, the reflectivity of a rectangular fused silica transmission grating with a period of 600nm cannot be less than 7% for a wavelength of 1064nm (Littrow angle 62.45°). The diffraction efficiency of the −1^{st} transmitted order can, thus, theoretically not exceed 93%.

One route to overcome this limitation is to use triangular instead of rectangular grooves. By optimizing such a profile, a diffraction efficiency exceeding 99% is theoretically possible even for a grating period which is nearly half of the wavelength. Unfortunately, such profiles can hardly be fabricated with the required profile accuracy. Another approach, which is the subject of this paper, is to embed a rectangular grating into the dielectric substrate material. By optimizing the profile parameters (groove depth and width) of the buried grating a complete suppression of any reflection and, thus, a diffraction efficiency of 100% is theoretically possible. Previously, it was proposed that a dielectric diffraction grating could be protected against mechanical damage and dust by burying it [4]. It was demonstrated that the diffraction efficiency barely changes by applying an appropriate coating procedure. In the following, it will be shown how this approach can even be exploited in order to significantly increase the diffraction efficiency.

In Ref. 2 the diffraction of a rectangular surface relief transmission grating was explained on the basis of a simple two-mode-model. In the present paper this model will be expanded by reflection effects in order to investigate the origin of the reflection losses. This consideration easily explains why the reflection losses can completely be suppressed in case of a buried grating.

## 2. Investigation of the grating reflection

One possible rigorous description of the diffraction of a binary grating is based on a modal expansion of the electromagnetic field inside the grating [5]. This approach can not only be used to numerically simulate the optical response of such a grating, but it also has much potential for a very understandable imagination of the diffraction process. It can be shown that in many cases only a few propagating grating modes need to be considered to describe the diffraction of such gratings [2, 6, 7]. These modes are similar to those within a well-known slab waveguide.

According to Ref. 1, the diffraction of a deep dielectric transmission grating can be described by only two modes, when it is illuminated in the Littrow-configuration and its period fulfills the condition (2). These two modes, namely the 0^{th} and the 1^{st}, are equally excited by the incident wave. They propagate through the grating and couple to the diffraction orders at the grating bottom (Fig. 3). If the reflection at the air-grating and grating-substrate interface is neglected, the intensities of the two transmitted diffraction orders (the 0^{th} and the −1^{st}) are a result of a two-beam-interference mechanism. Their intensities are determined by the phase differences between the two grating modes at the grating bottom. The propagation of the two modes through the grating (in *z*-direction) is described by their propagation constants *k ^{m}_{z}*=

*k*·

_{0}*n*, or rather their effective refractive indices

^{m}_{eff}*n*, where

^{m}_{eff}*m*is the number of the mode. According to this simple two-mode-model, the efficiencies

*η*and

_{0T}*η*-

_{1T}of the 0

^{th}and the −1

^{st}transmitted diffraction order can be calculated by

and

with

Here, *h _{max}* describes the groove depth, where the two modes exhibit a phase difference of π and a maximum diffraction to the −1

^{st}transmitted order is achieved. In this model the groove depth

*h*just represents a propagation distance and the diffraction is determined by the effective indices of the two modes. These

*n*in turn depend on the grating period, the fill factor, the wavelength and the polarization of light incident upon the grating, and so does the diffraction behavior. For the following investigation a 600nm grating period will be assumed, which is pretty close to λ/2 in the case of illumination with a 1064nm wavelength. Figure 4 illustrates the dependence of both effective indices on the fill factor for this grating. From this dependency, which can easily be found by applying the Modal Method (see e.g. [8]), the diffraction efficiencies can be calculated for every fill factor according to Eqs. (3) and (4). In this model the diffraction efficiency of the −1

_{eff}^{st}order would be 100%, if the two modes accumulated a phase difference of π (and odd-numbered multiples of π) at the grating bottom. Exact numerical calculations for a conventional surface relief grating show a maximum diffraction efficiency of 93%. The missing 7% are due to reflection effects that have been neglected in this simple model.

However, the two-mode-model can easily be expanded to investigate the reflection at a grating. In Fig. 4 the effective indices approach the one of the incident wave in air *n _{eff}^{air}*=

*cosφ*or the refracted wave within the substrate material

_{in}*n*=

_{eff}^{sub}*ncosφ’*for fill factors

_{in}*f*→0 or

*f*→1, respectively. This is obvious since the grating represents a plane interface in this case. The reflectivity of such a “grating” with a fill factor of 0 or 1 is well known; it corresponds to the Fresnel-reflection at a plane interface. This reflectivity depends on how much the propagation direction changes. In other words: the more the effective index is changed at the interface, the higher is the reflectivity. This description of the reflection at a plane interface can also be used for the reflection of grating modes. At the air-grating interface the effective index of the incident wave is changed to those of the two grating modes. Furthermore, at the grating bottom, both modes couple to the two transmitted diffraction orders, where the effective index is changed again. Due to the Littrow configuration (symmetrical propagation of 0

^{th}and −1

^{st}order) both transmitted and reflected diffraction orders possess the same effective index

*n*in air or

_{eff}^{air}*n*within the substrate, respectively. Therefore, in analogy to the Fresnel equations, the reflection of the grating modes at the interface to the air superstrate

_{eff}^{sub}*R*or to the substrate

_{m}^{air}*R*, can be calculated by:

_{m}^{sub}where *m* is the index of the respective mode. Figure 5 shows the calculated reflectivity of both propagating grating modes at the grating-air (blue lines) and the grating-substrate interface (green lines). The reflection of the 0^{th} mode is higher at the interface to air, while the 1^{st} mode shows a higher reflection at the substrate interface. The reflection of the whole grating device is a result of the complex interplay of all contributions from both modes and both interfaces. This fact makes it impossible to suppress the reflection losses.

A more detailed analysis of Fig. 5 reveals that for fill factors larger than 0.5 only the 1^{st} mode exhibits a significant reflection at the grating-substrate interface. Therefore, the entire reflection process would be much easier to understand and to control, if there was only this kind of interface. More precisely, the complex reflection process can be reduced to a very simple problem by embedding the fused silica – air grating into its substrate material.

## 3. The new approach: a buried grating

The considerations above showed that the reflection behavior of a grating with a fill factor larger than 0.5, which is embedded into fused silica, is almost completely determined by the 1^{st} mode. If the fill factor of the grating is less than 0.5 the reflectivity of the 0^{th} mode will increase, and this assumption becomes less precise. However, even for a fill factor of 0.2 the reflectivity of the 1^{st} mode is much higher than the one of the 0^{th} mode. Even there, the reflection behavior is mainly dominated by the 1^{st} mode. Since the buried grating adjoins to homogeneous fused silica at its top as well as at its bottom, the 1^{st} mode exhibits the same reflectivity at both interfaces. Therefore, the reflection of the grating can be described by a symmetrical Fabry-Perot-Resonator which is filled with a medium of refractive index *n _{eff}^{1}*. If the length of this resonator or rather the groove depth is a whole-number multiple of

the reflection will be completely suppressed. In general, the reflectivity *R* can be described by

where

*R _{1}^{sub}* is the reflectivity of the 1

^{st}mode at the grating-substrate interface. The multiplier ½, in contrast to a classical Fabry-Perot-Resonator, is due to the undisturbed transmission of the 0

^{th}mode, which carries half of the energy and does not take part in the reflection. The reflectivity is

*R*=0 for groove depths of

*h*=

*k*·

*h*(where

_{R}*k*is a positive whole number). On the contrary, the maximum is

for *h*=(*m*+*1/2*)·*h _{R}*, where

*F*and, thus,

*R*depend on the fill factor of the grating. In Fig. 6 the reflectivity according to Eq. (7) is plotted as a function of the groove depth and the fill factor of the rectangular grating profile. The groove depths

_{max}*h*and their whole-numbered multiples, which effect a complete suppression of reflection, are illustrated by dashed white lines. The plot reveals a decrease of the reflectivity

_{R}*R*with increased grating fill factor. Thus, the tolerance of the grating parameters to obtain a grating with a transmission near 100% is much larger for large fill factors.

_{max}In order to realize a buried transmission grating with 100% diffraction efficiency it is necessary to fulfill two conditions. On the one hand, the two propagating grating modes have to accumulate a phase difference of π in order to deflect all transmitted light to the −1^{st} order. This is achieved for a groove depth according to Eq. (4) and its whole-number multiples. On the other hand, the mode that is reflected, has to accumulate a phase of 2π by propagating twice through the grating. This is achieved for groove depths according to Eq. (6) and whole-number multiples. The diffraction efficiency η_{-1} of a buried grating can be expressed using Eqs. (7) and (4)

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}={\mathrm{sin}}^{2}\left(\frac{\pi}{2}\frac{h}{{h}_{max}\left(f\right)}\right)\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\xb7\phantom{\rule{.2em}{0ex}}\frac{1+\frac{1}{2}F\left(f\right)\xb7{\mathrm{sin}}^{2}\left(\pi \frac{h}{{h}_{R}\left(f\right)}\right)}{1+F\left(f\right)\xb7{\mathrm{sin}}^{2}\left(\pi \frac{h}{{h}_{R}\left(f\right)}\right)}.$$

Based on Eq. (9) the diffraction efficiency, comprising the reflection as well as the deflection of the transmitted light, is calculated as a function of the two profile parameters for a buried grating with a 600nm period and a 1064nm wavelength (Fig. 7(a)). In order to check the validity of the simple modal approach the diffraction efficiency is numerically simulated in Fig. 7(b). The images demonstrate that the diffraction behavior of the buried grating is very well described by the simple modal approach.

In Figs. 7(a) and (b) diffraction efficiencies of higher than 98% are illustrated by the white areas. The groove depths for which Eq. (4) and (6) are fulfilled are represented by dashed lines in Fig. 7(a). The intersections of these lines represent the grating profile parameters, which are necessary to achieve 100% diffraction efficiency. Exemplarily, a grating with a fill factor of 0.57 (corresponding to a groove width of 260nm) and a groove depth of 1.44µm theoretically exhibits a 99.9% diffraction efficiency. An efficiency higher than 98% would still be achieved, if the fill factor varies by ±0.03 (groove width variation of ±20nm). For fill factors larger than 0.68 the parameter tolerance increases significantly, but the groove parameters become more challenging for the fabrication. For a grating period of 600nm a fill factor of 0.7 corresponds to a groove width of 180nm and the necessary groove depth would be 1.9µm. Even though it is possible to fabricate gratings with comparable parameters [9], a lower fill factor is much more convenient for the fabrication process.

In Fig. 8 the numerically simulated diffraction efficiency is illustrated for a corresponding surface relief grating. It has a maximum of 93%, which is not sufficient for many optical setups.

In the discussion above TE-polarized illumination was assumed, which naturally exhibits higher reflection losses than TM-polarized light. One could ask if the problem of high reflectivity of highly-dispersive gratings could be solved by changing the polarization. However, it can be shown that for a small period to wavelength ratio, the TM-polarization requires a very large groove depth to obtain a significant diffraction to the −1^{st} order. A grating with a period as discussed above even shows almost no diffraction for a fill factor near 0.5. This is caused by the very small difference of the effective indices of the two TM-modes for small periods, which has been discussed in more detail, e.g. in Ref. [10]. Therefore, it is hardly possible to deflect the transmitted light to the −1^{st} order, even though the reflectivity could be reduced.

## 4. Discussion and conclusion

In this paper a new approach for the realization of transmission gratings that overcome reflection losses was presented. These gratings are embedded into their substrate material (fused silica), resulting in a symmetrical layout, and, thus, enabling the suppression of any reflection. As an example a grating period of 600nm (~1670 grooves/mm) was discussed for illumination at a 1064nm wavelength. In order to understand the origin of reflection losses, a two-mode-model has been applied, offering a very comprehensible explanation of the diffraction process. This physical investigation gives rise to the idea of embedded gratings. Based on the analogy to a Fabry-Perot-Resonator it explains why and how this approach physically works.

The concept of buried gratings is still valid if the gratings are embedded into air. In this case not the 1^{st} but the 0^{th} mode has to be considered, according to Fig. 5. Small fill factors instead of large ones would be necessary. This concept, however, appears to be much more difficult to fabricate, which is why it has not been discussed here in detail.

A promising route to fabricate a grating embedded in fused silica is to etch a rectangular profile with the parameters selected from Fig. 7 into the surface of a plane fused silica substrate and afterwards bond a second plane substrate on top of the first one. The fabrication and characterization of samples is currently in progress.

## Acknowledgment

This work was funded by the *Deutsche Forschungsgemeinschaft* within the Sonderforschungsbereich “Gravitational wave astronomy” and the Gottfried-Wilhelm-Leibniz-Program.

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