Analytic theory of soft x-ray diffraction by lamellar multilayer gratings

An analytic theory describing soft x-ray diffraction by Lamellar Multilayer Gratings (LMG) has been developed. The theory is derived from a coupled waves approach for LMGs operating in the single-order regime, where an incident plane wave can only excite a single diffraction order. The results from calculations based on these very simple analytic expressions are demonstrated to be in excellent agreement with those obtained using the rigorous coupled-waves approach. The conditions for maximum reflectivity and diffraction efficiency are deduced and discussed. A brief investigation into p-polarized radiation diffraction is also performed. ©2011 Optical Society of America OCIS codes: (050.1950) Diffraction gratings; (230.4170) Multilayers; (340.0340) X-ray optics; (340.7480) X-rays, soft x-rays, extreme ultraviolet (EUV); (230.1480) Bragg reflectors. References and links 1. I. V. Kozhevnikov and A. V. Vinogradov, “Basic formulae of XUV multilayer optics,” Phys. Scr. T17, 137–145 (1987). 2. A. Sammar, J.-M. André, and B. Pardo, “Diffraction and scattering by lamellar amplitude multilayer gratings in the X-UV region,” Opt. Commun. 86(2), 245–254 (1991). 3. A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. A 333(2-3), 599–606 (1993). 4. A. Sammar, M. Ouahabi, R. Barchewitz, J.-M. Andre, R. Rivoira, C. K. Malek, F. R. Ladan, and P. Guerin, “Theoretical and experimental study of soft X-ray diffraction by a lamellar multilayer amplitude grating,” J. Opt. 24(1), 37–41 (1993). 5. R. Benbalagh, J.-M. André, R. Barchewitz, P. Jonnard, G. Julié, L. Mollard, G. Rolland, C. Rémond, P. Troussel, R. Marmoret, and E. O. Filatova, “Lamellar multilayer amplitude grating as soft-X-ray Bragg monochromator,” Nucl. Instrum. Meth. Res. A 541(3), 590–597 (2005). 6. I. V. Kozhevnikov, R. van der Meer, H. M. J. Bastiaens, K.-J. Boller, and F. Bijkerk, “High-resolution, highreflectivity operation of lamellar multilayer amplitude gratings: identification of the single-order regime,” Opt. Express 18(15), 16234–16242 (2010). 7. R. Benbalagh, “Monochromateurs Multicouches à bande passante étroite et à faible fond continu pour le rayonnement X-UV,” PhD Thesis, University of Paris VI, Paris, 2003. 8. L. I. Goray, “Numerical analysis of the efficiency of multilayer-coated gratings using integral method,” Nucl. Instrum. Meth. Res. A 536(1-2), 211–221 (2005). 9. R. Petit, Electromagnetic Theory of Gratings, (Springer-Verlag, 1980. 10. A. V. Vinogradov and B. Ya. Zeldovich, “X-ray and far uv multilayer mirrors: principles and possibilities,” Appl. Opt. 16(1), 89–93 (1977). 11. A. V. Vinogradov and B. Ya. Zeldovich, “Multilayer mirrors for x-ray and far-ultraviolet radiation,” Opt. Spektrosk. 42, 404–407 (1977). 12. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980.


Introduction
Lamellar Multilayer Gratings (LMG) offer increased spectral and angular resolution in the soft x-ray (SXR) region compared to conventional multilayer mirrors (MM). The resolution of MMs in the SXR region is inherently limited due to absorption [1]. By fabricating a grating structure into a MM, an LMG structure is obtained that allows the SXR to penetrate deeper into the multilayer stack. Effectively, more bi-layers contribute to the reflection, improving the spectral and angular resolution of the MM. A schematic representation of an LMG is shown in Fig. 1 and more detailed descriptions of the working principles of LMGs can be found in Refs. [2][3][4][5][6][7].
Till now, diffraction of SXR from LMGs has been analyzed by time-consuming numerical simulations using complex rigorous theories [2,7,8]. Generally, one would not expect that computation time would be a significant limitation when addressing a physical problem. However, the optimization of LMG performance, in terms of resolution and reflectivity, is very difficult, as many of these time-consuming simulations need to be performed. It is even more difficult for the inverse problem of obtaining multilayer structural data, such as intermixing or roughness, from reflectivity measurements. Here, a multi-dimensional fitting procedure needs to be performed requiring hundreds or even thousands of simulations. Computation time is, thus, an important aspect when trying to understand the optical properties of LMGs. The lack of easy optimization and difficulties in understanding the LMG physical structure hampers the implementation of these dispersive elements in the SXR region.
Some progress in reducing computation time has recently been made in Ref. [6]. where we used a coupled-waves approach (CWA). Additionally, we identified a single-order operating regime. In this regime, the incident wave only excites one diffraction order and, hence, the problem of diffraction is simplified as a two-wave approximation can be used. Nevertheless, the approach described in this work still requires numerical calculations [6].
In this paper, we will derive analytic expressions for the diffraction efficiency of SXR radiation by LMGs operating in the single-order regime. We will demonstrate that these expressions are in excellent agreement with results from rigorous diffraction theories. This analytical description of the optical properties of LMGs makes earlier, complex, theories redundant. Furthermore, this theory can be used to improve our understanding of the optical properties of LMGs, simplify their design and optimization, and, most importantly, reveal their ultimate potential performance.
In section 2, we will first describe the differential equations of the CWA that can be rigorously used to calculate LMG SXR diffraction efficiency with numerical methods. Next, in section 3, we will derive the analytical solution to these differential equations for the zeroorder diffraction efficiency (reflectivity) for LMGs operating in single-order regime. We will compare the reflectivity obtained using our analytic solution to that obtained from rigorous diffraction theory and demonstrate excellent agreement between them. In section 4, the analytical solution for the diffraction efficiency of higher diffraction orders is derived and discussed. Again, we will show an excellent agreement between rigorous diffraction theory and our simple analytical expressions. Finally, in section 5, a concise discussion of the diffraction of p-polarized radiation by LMGs will be presented.

Differential equations of the coupled waves approach
A comprehensive description of a CWA used to calculate diffraction and reflection by LMGs is given in Ref. [6]. Certain parts of that description are repeated here, as they are required for our derivation of analytical expressions. We begin with a schematic representation of an LMG as shown in Fig. 1(a). Here, a two-component (absorber A and spacer S) periodic multilayer structure with bi-layer period d and thickness ratio γ is depicted. The Z-axis is defined as directed into the depth of the substrate and L is the total thickness of the multilayer structure. The spatial distribution of the dielectric permittivity ε and the piece-wise periodic function U can be written as ( , 0) 1;   is the polarizability of matter. The piece-wise periodic function U, shown in Fig. 1(b), describes the lamellar profile in the X-direction and can be expanded into a Fourier series: The solution to the 2D-wave equation 22   then has the following form (chapter 1, Ref. [9].): where 0  is the grazing angle of the incident monochromatic plane wave, n q is the Xcomponent of the wave vector for the n th diffraction order and k is the wave number in vacuum.
Substituting Eqs. (1)-(3) into the 2D wave equation (s-polarized radiation case) we obtain a system of coupled equations for waves of different diffraction orders: where the prime symbol in A' n and C' n indicates a first derivative with respect to z. Cz vary slowly with z compared with the quickly oscillating exponential multipliers as long as the angle of incidence does not satisfy total external reflection.
By substituting Eqs. (5) and (6) into Eqs. (4) we obtain a system of first order differential equations for the amplitudes () n Az and () with the following boundary conditions ,0 The dielectric permeability of the substrate is set to unity in Eq. (8), which is quite reasonable for soft x-rays as the polarizability is very small. We can, therefore, neglect the effect of reflection and diffraction of the incident wave from the substrate. Evidently, diffraction and reflection from an LMG can be determined by numerically solving Eqs. (7) and (8) and calculating the amplitudes of the waves diffracted into the vacuum ( (0) nn rC  ) and substrate ). In Ref. [6], we have already compared numerical simulations using Eqs. (7) and (8) to calculations performed by other authors [7] and found excellent agreement provided sufficient diffraction orders are taken into account. In that work, we also identified a single-order operating regime for LMGs in which, an incident plane wave can only excite a single diffraction order. It was shown that, in this regime, the LMG zeroth-order diffraction efficiency (reflectivity) can be calculated by replacing the LMG by a conventional multilayer mirror (MM) with its material densities reduced by a factor of Γ (the ratio of the lamel width to grating period). The angular width of the zeroth-order diffraction peak of an LMG then simply reduces by a factor of 1/Γ without loss of peak reflectivity compared to a conventional MM. The necessary condition for LMG operation in the single-order regime is quite evident: the angular width of the LMG Bragg peak conventional MM) should be small compared with the angular distance between neighboring diffraction peaks d/D, such that [6]:

Analytical solution for the zeroth-order diffraction efficiency (reflectivity)
In the previous section, we briefly discussed the most important aspects of the differential equations used in the coupled-waves approach as applicable to the LMG single-order regime [6]. From these differential equations, we will now derive analytical expressions for the diffraction efficiency of s-polarized SXR radiation by LMGs in single-order operation. In the single-order regime, all higher diffraction orders in Eqs. (7) can be neglected, leaving only the incident and specularly reflected waves 0 0 2 2 0 00 0 where 0 (0) 1 A  and 0 ( ) 0 CL . One can see that Eqs. (10) coincide with those describing the reflectivity of a conventional MM, except for the factor Γ that has appeared as a multiplier in front of the polarizability. As the polarizability in the SXR region is directly proportional to the material density, we can conclude that Eqs. (10) describe the reflection of an SXR wave from a conventional multilayer structure with the material density scaled by Γ, as was also found in Ref. [6].
Let us now consider a periodic multilayer structure having abrupt interfaces and consisting of two materials, namely a spacer and an absorber, with polarizabilities S  and A  , respectively. Then describes the modulation of the multilayer and the piece-wise function u is similar to the function U that describes the lamellar profile:    (12) We limit ourselves to the most important case of a wave incident onto the multilayer structure within or near the Bragg resonance of the j th order, i.e. we will suppose that  is the mean polarizability of a multilayer structure. The left-hand side of Eqs. (13) contain all terms that vary slowly with z. The functions A  and C  on the right-hand side denotes all other terms that oscillate quickly with z. These only weakly influence the amplitudes A 0 and C 0 and, therefore, can be neglected. Formally, this can be expressed by averaging Eqs. (13) over an interval, z  , that is substantially larger than the period of oscillations of the functions A  and C  , but much smaller than the typical length scale over which the functions A 0 and C 0 vary.
A system of coupled differential equations with constant coefficients can be obtained by introducing 0 with the same boundary conditions 0 (0) 1 a  and 0 ( ) 0 cL as for Eqs. (10). By solving Eqs. (14) we obtain an analytical expression for the zeroth-order diffraction efficiency where the Bragg parameter, b, characterizes a deviation from the Bragg resonance, the parameters B  describe the modulation of the structure, and the parameter S characterizes the variation of the amplitudes A 0 and C 0 with z. The number N is the total number of bi-layers in the multilayer structure.
Vinogradov and Zeldovich have previously derived Eq. (15) in Refs. [10,11]. However, they used a somewhat different mathematical technique and, in contrast to our approach, neglected the second derivatives with respect to z of the slowly varying amplitudes. As a result, there is a small difference in the expression for the Bragg parameter b.
To demonstrate the effect of the different Bragg parameters, b, Fig. 2 shows calculated reflectivities versus the grazing incidence angle using 3 different approaches. These calculations were performed for a conventional Mo/B 4 C multilayer mirror (i.e. for Γ = 1) at E = 183.4eV, which is the characteristic boron K α -line. Curve 1 is the result of exact calculations using a recurrent algorithm [12], while curve 2 was calculated via Eqs. (15) and (16). Curve 3 was calculated using the Bragg parameter derived by Vinogradov and Zeldovich. It can easily be seen that curves 1 and 2 are in better agreement outside the Bragg peak than curves 1 and 3. Figure (2), thus, demonstrates that the Bragg parameter, b, deduced above (Eq. (16)) is better suited for the calculation of MM reflectivity than the parameter deduced by Vinogradov and Zeldovich.
We will now continue our investigation by deriving the generalized Bragg condition for LMGs. For simplicity, we begin by assuming a semi-infinite multilayer structure ( N ). Equation (15) then reduces to its simplest form with tanh( ) 1 SNd  . The Bragg peak is very narrow, because of the small polarizability of matter in the SXR wavelength region. Therefore, we can neglect the wavelength dependence of the dielectric constant inside the peak. The reflectivity then only depends on the incidence angle and the radiation wavelength through the Bragg parameter b.   (15) and (16) deduced in the present paper (curve 2, blue), and formulas obtained in Ref. [10,11]. (curve 3, green).
If the Bragg condition (Eq. (17)) is fulfilled, the reflectivity achieves a peak value, which can be written in a very simple manner [1]  ; From Eq. (18), it can be seen that the peak reflectivity is independent of the grating parameters and corresponds to that of a conventional multilayer mirror [1]. This is because the parameters f and y determine the peak reflectivity entirely and these parameters are not changed if the density of both materials is scaled by the same factor, Γ. In contrast, the penetration depth of the radiation into the multilayer structure L MS , and therefore also the spectral and angular resolution of an LMG, is inversely proportional to Г: LMGs can, thus, offer improved resolution without loss of peak reflectivity. Note that the peak reflectivity, (Eq. (18)), chieves its maximum possible value when the parameter y is maximal and, hence, the thickness ratio, γ, of a multilayer structure obeys the equation which is well-known in the theory of conventional SXR multilayer mirrors [10,11]. Figure 3 demonstrates the accuracy of the analytic expressions (15) and (16) for the reflectivity of LMGs operating in the single-order regime. For these calculations, the lamellar width ГD = 70 nm was fixed, while the parameter Г and the grating period D were varied. The number of bi-layers N was chosen to be large enough to provide the maximum possible peak reflectivity. The colored curves were calculated using rigorous diffraction theory, as described by Eqs. (7), where 5 diffraction orders were taken into account. The black dashed curves were calculated using the analytic Eq. (15). Figure 3 shows that the agreement between the curves is excellent with a deviation in reflectivity peaks of less than 0.6%. In addition, it can be seen that the width of the Bragg peak decreases proportionally to Γ and that the peak reflectivity of the LMG corresponds to that of a conventional MM. The shift in the peak position is caused by the dependence on the Γ-ratio of the Bragg condition (Eq. (17)) as the "effective" polarizabilities of both materials are scaled by this factor. Here, the "effective" polarizability refers to the average polarizabiliy of an individual layer.

Analytic solution for higher-order diffraction efficiencies
In the previous section, we derived analytical expressions for the specular reflection of an LMG where it is, essentially, used as a mirror. This is because the diffraction angle of all orders except the zeroth order falls into an angular range where the multilayer shows no noticeable reflection. A more correct term would, thus, actually be Lamellar Multilayer Mirror (LMM). However, an LMG can also be used as a conventional diffraction grating, decomposing incident radiation of a single direction into diffracted light where the emission angle depends on the wavelength. Let us now consider the m th order diffraction efficiency, where we limit the analysis to angles and wavelengths close to the Bragg resonance (quasi-Bragg resonance), i.e. assuming 0 (sin sin ) Here, the index, j, is the order of the Bragg reflection from a multilayer structure, and the index, m, is the order of diffraction from the grating surface. Quasi-Bragg resonance means that there is constructive interference of waves diffracted from different interfaces of the multilayer structure and, hence, a high diffraction efficiency. In addition, the LMG will be limited to the single order regime.
One can check directly from Eqs. CWA system (Eqs. (7)). , and obtain a system of differential equations with constant coefficients: Please note that the diffraction angle, The peak value of the diffraction efficiency then has the same form as in Eq. (18), but the parameter y is more complex: As a quick check of these equations, one would expect that, for specular reflectivity [i.e. inserting m = 0 into Eqs. (21)-(24)], we should obtain Eqs. (14)-(17), which is indeed the case.
To investigate the accuracy of the expressions (15) and (22), we compared results deduced from these equations with those obtained using rigorous diffraction theory (Eqs. (7)). As an example of such a comparison, Fig. 4 shows the zeroth to 5th order diffraction peaks as a function of the grazing angle of the incident beam. The colored curves were numerically calculated using rigorous diffraction theory, where 9 diffraction orders, from + 2nd to 6th order, were taken into account. The black dashed curves were calculated using the analytic expressions (15) and (22). The parameters of the multilayer structure were the same as for Fig. 3. The grating period D was 210 nm, the lamellar width ГD was 70 nm and the number of bi-layers N was 300. As can be seen, the agreement between the curves is excellent with less than 0.6% deviation in diffraction peaks, except for the very small 3rd order diffraction peak. A closer look at the 3rd order diffraction peak shows that our parameter choices (specifically  21)). From these observations it can be concluded that these extremely small peaks can be neglected in practical applications. Hence, as was found for the reflectivity calculations, complicated rigorous diffraction theories are not required for the calculation of the diffraction efficiencies of LMGs operating in the single order regime. While the expressions for the reflectivity and the higher-order diffraction efficiencies are very similar, they differ in certain details. The main difference is the dependence of the maximum diffraction efficiency on the Γ-ratio in Eq. (24) for the parameter y. The diffraction efficiency becomes higher for larger values of y. However, the Γ-ratio is always smaller than unity for a grating, so each multiplier in Eq. (24) is also less than unity, and the parameter y for the higher order diffraction efficiencies is always less than that of the zeroth order diffraction efficiency (Eq. (18)). Hence, the peak value of the higher order diffraction efficiencies, R n , will increase with decreasing Г, but not exceed the reflectivity peak value R 0 . Note that the parameter γ providing the maximum of diffraction efficiency is the same as for the reflectivity and is determined by Eq. (20).
In Fig. 5, the Г-dependence of higher order diffraction efficiencies is demonstrated by showing the 1st order diffraction efficiencies for different values of Г. The parameters of the multilayer structure are the same as for Figs. 3 and 4. The lamellar width ГD = 70 nm is fixed, while Г and the grating period, D, are varied. The number of bi-layers N is again chosen large enough (100/Г) to obtain the maximum possible diffraction efficiency. The colored curves were calculated on the basis of rigorous diffraction theory (where 5 diffraction orders were taken into account) and black curves were calculated with the use of the analytical Eqs. (15) and (22). The agreement between the curves is excellent for all values of Г. By comparing Figs. 4 and 5, it can be seen that the 1st order diffraction efficiency is almost as high as the reflectivity, with a relative difference of only 0.85% for Г = 1/20.

Diffraction of p-polarized radiation by LMG
In the previous sections we have considered diffraction and reflection of s-polarized radiation by LMGs. In this section, we will discuss diffraction and reflection of p-polarized radiation. For p-polarized radiation, a slightly different expression for the dielectric constant distribution in the LMG is required. In accordance with Eqs. (1) and (12), the distribution of the dielectric constant inside an LMG (0 < z < L) can be written as the 2D Fourier series