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

Optics Express, Vol. 19, Issue 10, pp. 9172-9184 (2011)

http://dx.doi.org/10.1364/OE.19.009172

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### Abstract

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 OSA

## 1. Introduction

1. I. V. Kozhevnikov and A. V. Vinogradov, “Basic formulae of XUV multilayer optics,” Phys. Scr. **T17**, 137–145 (1987). [CrossRef]

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). [CrossRef]

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). [CrossRef]

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). [CrossRef]

6. I. V. Kozhevnikov, R. van der Meer, H. M. J. Bastiaens, K.-J. Boller, and F. Bijkerk, “High-resolution, high-reflectivity operation of lamellar multilayer amplitude gratings: identification of the single-order regime,” Opt. Express **18**(15), 16234–16242 (2010). [CrossRef] [PubMed]

6. I. V. Kozhevnikov, R. van der Meer, H. M. J. Bastiaens, K.-J. Boller, and F. Bijkerk, “High-resolution, high-reflectivity operation of lamellar multilayer amplitude gratings: identification of the single-order regime,” Opt. Express **18**(15), 16234–16242 (2010). [CrossRef] [PubMed]

## 2. Differential equations of the coupled waves approach

6. I. V. Kozhevnikov, R. van der Meer, H. M. J. Bastiaens, K.-J. Boller, and F. Bijkerk, “High-resolution, high-reflectivity operation of lamellar multilayer amplitude gratings: identification of the single-order regime,” Opt. Express **18**(15), 16234–16242 (2010). [CrossRef] [PubMed]

*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 aswhere

*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

*X*-component of the wave vector for the

*n*

^{th}diffraction order and

*k*is the wave number in vacuum.

*n*

^{th}diffraction order in vacuum and in the substrate, respectively. The boundary conditions in Eqs. (4) signify that only a single plane wave is incident on the LMG at the grazing angle

**18**(15), 16234–16242 (2010). [CrossRef] [PubMed]

*n*

^{th}order diffracted wave

_{n}and C'

_{n}indicates a first derivative with respect to

*z*.

*z*compared with the quickly oscillating exponential multipliers as long as the angle of incidence does not satisfy total external reflection.

## 3. Analytical solution for the zeroth-order diffraction efficiency (reflectivity)

**18**(15), 16234–16242 (2010). [CrossRef] [PubMed]

**18**(15), 16234–16242 (2010). [CrossRef] [PubMed]

*u*is similar to the function

*U*that describes the lamellar profile: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

*z*. The functions

*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,

*A*

_{0}and

*C*

_{0}vary.

*b*, characterizes a deviation from the Bragg resonance, the parameters

*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.

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). [CrossRef] [PubMed]

*z*of the slowly varying amplitudes. As a result, there is a small difference in the expression for the Bragg parameter

*b.*

*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.

*b*. At maximum reflectivity, we have the condition

1. I. V. Kozhevnikov and A. V. Vinogradov, “Basic formulae of XUV multilayer optics,” Phys. Scr. **T17**, 137–145 (1987). [CrossRef]

1. I. V. Kozhevnikov and A. V. Vinogradov, “Basic formulae of XUV multilayer optics,” Phys. Scr. **T17**, 137–145 (1987). [CrossRef]

*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*, 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

_{MS}*y*is maximal and, hence, the thickness ratio, γ, of a multilayer structure obeys the equationwhich is well-known in the theory of conventional SXR multilayer mirrors [10

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). [CrossRef] [PubMed]

*D*= 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.

## 4. Analytic solution for higher-order diffraction efficiencies

*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

*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.

*b*, and

*S*are somewhat different to those in Eqs. (16):

*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.

*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).

*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.

## 5. Diffraction of p-polarized radiation by LMG

*z*<

*L*) can be written as the 2D Fourier seriesThe single-order regime is characterized by a suitable choice of parameters such that one only needs to keep the term with

*m*= 0 in the first sum and only one harmonic with fixed

*m*and

*j*in the second sum. The first term,

*H*is the nonzero Y-component of the magnetic field perpendicular to the plane of Fig. 1 and where

*m*

^{th}order diffraction efficiency goes to zero if the diffracted beam propagates perpendicular to the incident one, i.e. at

## 6. Conclusions

**18**(15), 16234–16242 (2010). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | I. V. Kozhevnikov and A. V. Vinogradov, “Basic formulae of XUV multilayer optics,” Phys. Scr. |

2. | A. Sammar, J.-M. André, and B. Pardo, “Diffraction and scattering by lamellar amplitude multilayer gratings in the X-UV region,” Opt. Commun. |

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 |

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. |

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 |

6. | I. V. Kozhevnikov, R. van der Meer, H. M. J. Bastiaens, K.-J. Boller, and F. Bijkerk, “High-resolution, high-reflectivity operation of lamellar multilayer amplitude gratings: identification of the single-order regime,” Opt. Express |

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 |

9. | R. Petit, |

10. | A. V. Vinogradov and B. Ya. Zeldovich, “X-ray and far uv multilayer mirrors: principles and possibilities,” Appl. Opt. |

11. | A. V. Vinogradov and B. Ya. Zeldovich, “Multilayer mirrors for x-ray and far-ultraviolet radiation,” Opt. Spektrosk. |

12. | M. Born and E. Wolf, |

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(230.1480) Optical devices : Bragg reflectors

(230.4170) Optical devices : Multilayers

(340.0340) X-ray optics : X-ray optics

(340.7480) X-ray optics : X-rays, soft x-rays, extreme ultraviolet (EUV)

**ToC Category:**

X-ray Optics

**History**

Original Manuscript: February 23, 2011

Revised Manuscript: April 21, 2011

Manuscript Accepted: April 21, 2011

Published: April 26, 2011

**Citation**

I. V. Kozhevnikov, R. van der Meer, H. M. J. Bastiaens, K.-J. Boller, and F. Bijkerk, "Analytic theory of soft x-ray diffraction by lamellar multilayer gratings," Opt. Express **19**, 9172-9184 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-10-9172

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### References

- I. V. Kozhevnikov and A. V. Vinogradov, “Basic formulae of XUV multilayer optics,” Phys. Scr. T17, 137–145 (1987). [CrossRef]
- 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). [CrossRef]
- 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). [CrossRef]
- 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). [CrossRef]
- 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). [CrossRef]
- I. V. Kozhevnikov, R. van der Meer, H. M. J. Bastiaens, K.-J. Boller, and F. Bijkerk, “High-resolution, high-reflectivity operation of lamellar multilayer amplitude gratings: identification of the single-order regime,” Opt. Express 18(15), 16234–16242 (2010). [CrossRef] [PubMed]
- 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.
- 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). [CrossRef]
- R. Petit, Electromagnetic Theory of Gratings, (Springer-Verlag, 1980.
- A. V. Vinogradov and B. Ya. Zeldovich, “X-ray and far uv multilayer mirrors: principles and possibilities,” Appl. Opt. 16(1), 89–93 (1977). [CrossRef] [PubMed]
- A. V. Vinogradov and B. Ya. Zeldovich, “Multilayer mirrors for x-ray and far-ultraviolet radiation,” Opt. Spektrosk. 42, 404–407 (1977).
- M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980.

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