1. Introduction

With the enhancement of lithographic fabrication techniques, dielectric diffraction gratings with a period in the submicron range became interesting and applicable in a large number of different applications, e.g. in high-power laser systems. The optimization for a specific application requires a precise design of the grating parameters. The optical properties of a grating, whose period is very large with respect to the groove depth (low aspect ratio grating), can be simulated by approximate analytical methods like the Thin Element Approach [1]. This analytical description of the diffraction process can be inverted to design the geometrical parameters of the grating profile, thus achieving a desired optical function (the “inverse diffraction problem”). However, for some applications, higher dispersion and therefore gratings with smaller period and higher aspect ratio are needed. If the grating depth becomes comparable with the period, the aforementioned approximate methods are not valid. Rigorous methods that directly solve Maxwell’s equations have to be applied. Within the past decades, countless different methods have been developed that numerically solve the Maxwell equations in their time-dependent (e.g. Finite Difference Time Domain Method [2]), integral (e.g. the Generalized Source Method [3]) or differential form (e.g. Fourier Modal Method [4], Rigorous Coupled Wave Analysis [5]). Though these theories are capable of simulating the optical properties of the gratings, they cannot be easily inverted to solve the inverse diffraction problem. Furthermore, their numerical treatment does not give much insight into the processes that take place inside the grating region. However, in the case of dielectric transmission gratings with deep rectangular grooves, the modal method, developed originally by Collin [6] and Rytov [7] and later applied to the diffraction problem by Botten et.al.[8], opens up the possibility of reducing the difficult diffraction process to an easy and intelligible modal interference mechanism. This modal description, which is discussed in greater detail in [9], helps to understand the propagation of light through the grating region and the processes that determine the diffraction efficiencies. A currently interesting example is highly efficient dielectric gratings. Recently it was demonstrated that a grating etched directly into a fused silica substrate, can exhibit theoretically a diffraction efficiency of higher than 97% for TE-[10] or even unpolarized light [11]. In this paper the modal description is applied to these high-efficiency gratings. We show how the efficiency is influenced by the grating parameters and which conditions have to be fulfilled for high diffraction efficiency.

 

Fig. 1. Geometry of the diffraction problem

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2. Essentials of the modal method

If a dielectric transmission grating is illuminated by a plane wave as it is shown in Fig. 1, the reflected and the transmitted field can be described as a sum of plane waves: the diffraction orders. The propagation directions of these orders are given by the well-known grating equation

(λ wavelength in air, φ_in angle of incidence, φ_m diffraction angle of the m^th diffraction order in air). They are only affected by the period of the grating, but not by the geometrical shape of the grooves. However, the amplitudes or diffraction efficiencies (intensity in one order divided by the intensity of the incident wave) of the orders cannot be calculated by such a simple equation. For their calculation, the electromagnetic field inside the grating region and therefore the exact groove shape has to be considered. According to the modal method, the propagation of light in the z-direction through the grating region is very similar to a simple slab waveguide. Such a waveguide is able to guide a discrete set of modes. The propagation constants of these modes ${k_z}^m$ = k₀ ·${n_{\mathit{\text{eff}}}}^m$ , or rather their effective indices ${n_{\mathit{\text{eff}}}}^m$ , are characteristics of the waveguide geometry. Modes with ${n_{\mathit{\text{eff}}}}^2$ > 0 propagate through the waveguide, while those with ${n_{\mathit{\text{eff}}}}^2$ < 0 are evanescent. The more similar the field distributions of the exciting wave and the excited mode are at the matching plane, the higher the excitation efficiency of these modes by the incident field is. This “similarity” is expressed by an overlap integral. Furthermore, the difference of the propagation constant of the incident wave ${k_{\text{z}}}^{\mathrm{in}}$ = k₀ ·cosφ_in relative to ${k_z}^m$ results in a reflection similar to the Fresnel-reflection at the interface of two homogeneous media. Both effects, the overlap and the matching of effective indices, determine how much energy of the incident wave is coupled to a specific mode.

A wave incident upon a grating, or periodic waveguide, also excites discrete modes comparable to the simple case of a slab waveguide. The efficiency of this excitation, the propagation of the modes through the grating region, and how they couple to the diffraction orders of the grating determine the optical properties of the grating. The propagation constants or effective indices of these modes can be derived according to the following procedure (for a more detailed description see [12]):

Let us assume a TE-polarized incident wave with the electric field vector perpendicular to the plane of incidence. The y-invariant electric field ${E_{\mathit{\text{in}}}}^y$ writes:

with the vacuum wave number k₀ = 2π/λ. The geometry of the problem permits the separation of the sole electric field component $E_y^{\mathit{\text{gr}}}$ (x, z) in the grating region into two parts

The field inside the grating grooves and the ridges fulfills the Helmholtz-equation for homogeneous media, which is for the z-part:

and for the x-part:

where ${k_{i,x}}^2=k_i^2-k_z^2\mathrm{and}i=\{\begin{array}{c}b\mathit{in}\mathit{the}\mathit{ridge}\\ g\mathit{in}\mathit{the}\mathit{groove}\end{array}$ k_i = n_ik₀ is the wave number in the grooves (in case of air: n_g = 1) and ridges, respectively. The propagation constant k_z = k ₀ n_eff is equal in both media. Eq. (4b) can be solved in grooves and ridges separately by

The field continuity conditions at the boundaries between the ridges and grooves lead to a transcendent equation for k_z or rather the effective index n_eff , which can be written in the case of TE-polarized illumination as

with

where

b and g are the ridge and groove widths, respectively, ε _g and ε _b are their dielectric permittivities. The left part of Eq. (6) contains the angle of incidence and the period to wavelength ratio. It thus represents the incidence conditions. The cosine is 1 in the case of normal incidence and -1 in the case that one diffraction order propagates symmetrically to the 0^th order (“Littrow-mounting”). The function F(${n_{\text{eff}}}^2$) depends on the fill factor f of the grating, which will in the following be defined as the ratio between the ridge width b and the period d, and the refractive indices. An example for this function is shown in Fig. 2 (details will be discussed in the next section). The intersections between F(${n_{\text{eff}}}^2$) and cos(αd) lead to discrete values of the effective index n _eff, which characterize discrete grating modes. A solution of Eq. (4a) is therefore a field composed of upward and downward propagating modes, whose z-dependent part v_m (z) is given by

If the reflectivity of mode m at the grating-substrate and the grating-air interface is low, the upward propagating part and every effect caused by multiple reflections can be neglected, then v_m(z) can be expressed by

where C describes the part of the power of the incident wave that has been coupled to the mode. Modes with ${n_{\text{eff}}}^2$ < 0 are evanescent, and those with ${n_{\text{eff}}}^2$ > 0 propagate along the z-direction. The effective index of the evanescent modes determines how fast their amplitude decreases with increasing groove depth (z-dependence). In cases of shallow gratings, these modes cannot be neglected, whereas they will play a minor role for deep grooves. The diffraction properties of deep gratings are therefore mainly determined by the propagating modes. The x-dependent amplitude u_m (x) of the m^th mode can be calculated by inserting the respective effective index into Eq. (5) and matching the amplitude at the groove-ridge boundaries. According to the Floquet-Bloch Theorem every u_m (x) fulfills the condition of pseudo-periodicity

where d is the grating period and $\alpha =\frac{2\pi }{\lambda }\sin {\phi }_{\mathit{in}}.$ In the case of normal incidence (α = 0), u_m(x) repeats itself every period, and for Littrow-mounting (α= π/d), every two periods. The efficiency of excitation of these modes by the incident wave is, analogous to the waveguide theory, determined by the overlap integral between the field of the incident wave and mode m at the air-grating boundary [13]

as well as by their Fresnel-like transmission coefficient at this interface, which is determined by the change of effective index (“impedance matching”). After propagation through the grating region, the modes are partly reflected at the grating-substrate interface and partly transmitted into the substrate. By doing this, every mode distributes its energy to all possible diffraction orders specified in Eq. (1). The coupling efficiency is again defined by the overlap of the fields and the transmission at this interface caused by the change of propagation constants. The contributions of all grating modes interfere and determine the intensity of one diffraction order.

 

Fig. 2. Eigenvalue-relation F(${n_{\mathit{\text{eff}}}}^2$) for d = 800nm

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3. Modal Description applied to highly efficient gratings in fused silica

Ref. [10] describes the optimization of a fused silica transmission grating with a period of 800nm at a wavelength of 1060nm. It was shown by rigorous calculations using the Fourier modal method that a diffraction efficiency of 97% can be theoretically achieved under -1^st order Littrow-mounting and TE-polarization for a groove depth of 1.54μm and a fill factor of 0.45. It was also demonstrated that a slight deviation from the optimal fill factor can be compensated by adapting the groove depth. This amazing behaviour can be explained without complicated numerical calculations, but by an analysis of Eq. (6) or rather by considering the effective indices that characterize the modal field propagation inside the grating. Figure 2 shows F(${n_{\mathit{\text{eff}}}}^2$) from Eq. (7) for 800nm grating period and a fill factor of 0.4, 0.5 and 0.6 assuming a refractive index of 1.45 for the grating ridges. Since the grating is illuminated under Littrow-mounting, the intersection of the illustrated functions F(${n_{\mathit{\text{eff}}}}^2$) with cos(αd) = -1 gives the effective indices of all modes that might be excited by the incident wave. There are only two propagating modes with real effective indices; all higher order modes are evanescent, since their n_eff is imaginary. Figure 2 illustrates that the function F(${n_{\mathit{\text{eff}}}}^{\mathit{2}}$) depends on the fill factor of the grating, so the effective indices do as well. Figure 3 shows the effective indices of the first two modes as a function of the fill factor. The horizontal lines illustrate the effective indices upon illumination from the air side

and the orders transmitted into the substrate

where $\tilde{\phi }$ is the refracted angle in the substrate and n its refractive index. For small fill factors, the modes approach the effective index of air. For large fill factors, they approach the effective index of the substrate material. If the fill factor was 0.45, the two modes would propagate with ${n_{\mathit{\text{eff}}}}^{\mathit{0}}$ = 1.210 and ${n_{\mathit{\text{eff}}}}^{\mathit{1}}$ = 0.864.

 

Fig. 3. Effective indices of the first two modes as a function of the fill factor

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As mentioned above, the excitation efficiency of these modes is determined by the field overlap between the incident wave and the field of the specific mode at the boundary between air and the grating region. According to Eq. (2), the field of the incident wave is given by

which for Littrow-mounting (α= π/d) is

It consists of a “sine-part” and a “cosine-part” with a periodicity of 2d. The field amplitudes of the first four grating modes are illustrated in Fig. 4. Since the x-dependent part of the Helmholtz-operator defined by Eq. (4b) is linear and self-adjoint, its eigenfunctions and therefore the fields of the modes are either odd or even functions. The periodicity of the modes is determined by Eq. (10), assuming that α is π/d. The modes repeat their amplitudes after two grating periods and whole-numbered multiples of 2d. The field of the 0^th grating mode in Fig. 4(a) looks very similar to a cosine function with respect to the definition of the coordinate system in Fig. 1. On the other hand, the amplitude of the first mode (Fig. 4(b)) nearly follows a sine-function. Their periodicity is the same as that of the incident wave, so it is obvious that the incident wave excites these two propagating modes very well. Higher order modes, though they exist, are hardly excited. The exact values of the overlap are 0.4905 and 0.4994 for the propagating modes and 0.0054 and 0.0046 for the two illustrated evanescent modes (Fig. 4(c), 4(d)).

 

Fig. 4. Amplitudes of the first four TE modal fields in the grating region

(a) mode 0 (propagating) dashed line: cosine function

(b) mode 1 (propagating) dashed line: sine function

(c) mode 2 (evanescent)

(d) mode 3 (evanescent)

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These two excited modes, each of which carries nearly half of the energy of the incident wave, propagate along the z-direction. Since their effective indices are different, they accumulate a phase difference. In the case of a fused silica grating, the reflection of the two modes at the grating-substrate interface is low, so most of their energy is coupled out to the transmitted diffraction orders defined in Eq. (1). For Littrow-mounting, the two propagating diffraction orders (the 0^th and the -1^st order) have symmetrical field distributions. So the coupling efficiency between a grating mode and each one of the two diffraction orders is the same as the coupling between the incident wave and the modes. The intensity in one of the two orders is determined by the interference of the corresponding parts of both modes.

 

Fig. 5. Analogy to a Mach-Zehnder-Interferometer. The two grating modes propagate through different optical paths; the intensity in port 1 or 2 are determined by their phase difference.

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Fig. 6. Numerically calculated diffraction efficiency of the 0^th and the -1^st order as a function of the groove depth.

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This diffraction process is very similar to a Mach-Zehnder Interferometer (Fig. 5). In such an interferometer the incident wave is split into two equal parts that afterwards propagate through different media or different optical paths. After passing the second beam splitter, the light leaves the interferometer either through port 1 or 2 or partly through both of them. The intensity in one port is determined by the interference of the two beams, so it changes sinusoidally with the path difference. This is analogous to the propagation of the two grating modes with different effective indices. If the difference of the optical paths of the two modes is zero, for example if the groove depth is zero, then all light propagates in the 0^th transmitted order (the light does not see a grating). For increasing path differences, the intensity of the 0^th order decreases continuously until the two modes have accumulated a phase difference of π (resp. a path length difference of λ/2). In this case, all light is diffracted into the -1^st order. Since the diffraction is based on a two-beam interference mechanism the diffraction efficiency changes sinusoidally when the groove depth is increased. Fig. 6 shows the numerical calculation of this correlation for a fill factor of 0.5. Both graphs indeed show a nearly sinusoidal trend. Small deviations are caused by the reflection effects, which have been neglected here.

The complex diffraction process is therefore reduced to a two-beam interference of two grating modes that propagate along the z-direction with different effective indices. For the given grating period, fused silica grating ridges and Littrow-mounting, the values of the effective indices are only affected by the fill factor of the grating. The accumulated phase difference and therefore the diffraction efficiency of the 0^th or -1^st order is determined by the propagation distance, and therefore the groove depth of the grating. A grating is highly efficient if the two modes carry nearly the same energy and interfere at the grating-substrate interface with a phase difference of an odd-numbered multiple of π. The groove depth h, which is necessary to accumulate this phase shift, depends on the difference of the effective index of both modes. According to Eq. (9), a phase shift of π is achieved at:

Figure 7(a) shows this groove depth and its odd-numbered multiples as a function of the fill factor. For a fill factor of 0.45 a phase shift of π is reached at a groove depth of 1.55μm, which is pretty near to the value found in [10]. Figure 7(a) furthermore illustrates, that it is possible to assign a groove depth to every fill factor. So if the fill factor of a fabricated grating deviates from the desired value, it can be compensated by the proper groove depth. Figure7(b) shows the rigorously calculated diffraction efficiency as a function of the profile parameters, as it was shown in [10]. The light areas illustrate for every fill factor, which groove depth is necessary for a high diffraction efficiency. The comparison of Fig. 7(a) and 7(b) show that the model presented here makes impressively good predictions of the performance of a fused silica grating. However, the calculation of the groove depth using the effective indices does not regard the amplitude condition explained above and the reflection, which also depends on the fill factor. So to find the global maximum of the diffraction efficiency, these effects also have to be considered.

 

Fig. 7(a). Groove depth to accumulate a phase difference of π, 3π and 5π

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Fig. 7(b). Rigorously calculated diffraction efficiency from ref. [10]

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4. Some remarks on unpolarized light

The formulas given in section 2 are only valid for TE-polarized light. However, the derivation for TM-polarized light can be made analogous, by using the sole magnetic component H_y . The eigenvalue equation is then in the form

The simple model, which explains the diffraction as a modal interference mechanism, can be applied here, too. In order to achieve high diffraction efficiency for both, TE and TM polarized light, it is necessary to fulfil the phase condition Eq. (14) at the same groove depth. This does not mean that the propagating modes have to posses the same effective index for both polarizations, but their difference |${n_{\mathit{\text{eff}}}}^{\mathit{0}}$ - ${n_{\mathit{\text{eff}}}}^{\mathit{1}}$ | should be equal. As the effective indices depend on the period of the grating, as well as its fill factor, these parameters can be used to fulfil this condition.

5. Summary

In this paper it has been shown that the diffraction properties of deep dielectric rectangular gratings with a period in the range of the wavelength, which are illuminated under Littrow mounting, is mainly determined by two modes that are excited within the grating region. These two modes propagate through the grating with different effective indices and couple out at the grating-substrate interface. A high diffraction efficiency is achieved if the two modes carry nearly the same energy and interfere at the grating substrate interface with a phase difference of π (or a whole numbered multiple of it). The groove parameters estimated with these simple considerations basically correspond to those that have been previously calculated numerically. The example described in the presented paper vividly shows how physically meaningful a modal approach of diffraction can be and how easily it leads to the design of new optical functions in the application field of deep periodical structures of small feature size.