## High-resolution, high-reflectivity operation of lamellar multilayer amplitude gratings: identification of the single-order regime |

Optics Express, Vol. 18, Issue 15, pp. 16234-16242 (2010)

http://dx.doi.org/10.1364/OE.18.016234

Acrobat PDF (966 KB)

### Abstract

High resolution while maintaining high peak reflectivities can be achieved for Lamellar Multilayer Amplitude Gratings (LMAG) in the soft-x-ray (SXR) region. Using the coupled waves approach (CWA), it is derived that for small lamellar widths only the zeroth diffraction order needs to be considered for LMAG performance calculations, referred to as the single-order regime. In this regime, LMAG performance can be calculated by assuming a conventional multilayer mirror with decreased density, which significantly simplifies the calculations. Novel analytic criteria for the design of LMAGs are derived from the CWA and it is shown, for the first time, that the resolution of an LMAG operating in the single-order regime is not limited by absorption as in conventional multilayer mirrors. It is also shown that the peak reflectivity of an LMAG can then still be as high as that of a conventional multilayer mirror (MM). The performance of LMAGs operating in the single-order regime are thus only limited by technological factors.

© 2010 OSA

## 1. Introduction

2. R. A. M. Keski-Kuha and A. M. Ritva, “Layered synthetic microstructure technology considerations for the extreme ultraviolet,” Appl. Opt. **23**(20), 3534 (1984). [CrossRef] [PubMed]

6. T. Peng, “Rigorous formulation of scattering and guidance by dielectric grating waveguides: general case of oblique incidence,” J. Opt. Soc. Am. **6**(12), 1869–1883 (1989). [CrossRef]

9. 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. Methods **536**(1–2), 211–221 (2005). [CrossRef]

8. L. I. Goray, “Numerical analysis of the efficiency of multilayer-coated gratings using integral method,” Nucl. Instrum. Methods **536**(1–2), 211–221 (2005). [CrossRef]

10. K. Krastev, J.-M. André, and R. Barchewitz, “Further applications of a recursive modal method for calculating the efficiencies of X-UV multilayer gratings,” J. Opt. Soc. Am. A **13**(10), 2027 (1996). [CrossRef]

13. V. V. Martynov, B. Vidal, P. Vincent, M. Brunel, D. V. Roschupkin, A. Yu. Agafonov, A.I. Erko, and A. Yakshin, “Comparison of modal and differential methods for multilayer gratings,” Nucl. Instrum. Methods Phys. Res. **339**(3), 617–625 (1994). [CrossRef]

4. 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. Methods **541**(3), 590–597 (2005). [CrossRef]

15. I. V. Kozhevnikov and A. V. Vinogradov, “Basic formulae of XUV multilayer optics,” Phys. Scr. T137–145 17, (1987). [CrossRef]

## 2. Basic equations of the coupled waves approach

*d*and thickness ratio

*γ*).

*L*is the total thickness of the multilayer structure. The piece-wise periodic function

*U*, shown in Fig. 1b, describes the lamellar profile in the

*X*-direction normalized to

*L*. Other functions can also be used to describe different lamellar profiles, for instance trapezoidal. The spatial distribution of the dielectric permittivity

*ε*is then written as follows:

*χ*(

*z*) is simply the complex polarizability, which varies with depth in the multilayer structure. Although Fig. 1a, for simplicity, displays a polarizability

*χ*(

*z*) that varies between two values associated with materials A and S, we note that also arbitrary depth distributions of the polarizability can be used. The lamellar-profile function

*U*(

*z*) can be expanded into the Fourier series:

^{2}

*E*(

*x*,

*z*) +

*k*

^{2}

*ε*(

*x*,

*z*)

*E*(

*x*,

*z*) = 0, where the dielectric permittivity is a periodic function of

*x*, as defined in Eqs. (1) and (2). The general solution then has the following form (chapter 1, Ref [16].):

*θ*

_{0}is the grazing angle of the incident monochromatic plane wave,

*q*is the

_{n}*X*-component of the wave vector for the

*n*diffraction order and

^{th}*k*is the wave number in vacuum. The boundary conditions for our problem constitute that the wave field in vacuum and in the substrate should represent a superposition of plane waves propagating at different angles to the

*X*-axis.

*κ*= (

_{n}*k*

^{2}−

*q*

^{2}

_{n})

^{1/2}and

*κ*= (

_{n}*k*

^{2}

*ε*−

_{sub}*q*

^{2}

_{n})

^{1/2}are the Z-components of the wave vector for the

*n*

^{th}diffraction order in vacuum and in the substrate, respectively, and

*δ*

_{n,0}is the Kronecker symbol. The boundary conditions (Eqs. (5)) signify that plane waves are only incident onto the LMAG from the vacuum at a single, grazing angle

*θ*

_{0}. The results discussed in the following sections were obtained by direct numerical integration of the system (4)–(5) without imposing any restriction on LMAG parameters.

*U*then turn to zero except the coefficient

_{n−m}*U*

_{0}, which equals 1. Equation (4) is then reduced to the simplest equation:

## 3. Calculation of LMAG diffraction efficiency

4. 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. Methods **541**(3), 590–597 (2005). [CrossRef]

_{4}C multilayer structure and operating at a SXR energy

*E*of 183.4 eV. The parameters of the LMAG are:

*D*= 2 µm, Γ = 0.3,

*N*= 150,

*d*= 6 nm, and γ = 0.33. Using Eq. (4) we numerically calculated the diffraction efficiency of the zeroth order (reflectivity) |

*r*

_{0}|

^{2}, which is shown in Fig. 2 as a function of the grazing angle of the incident wave

*θ*

_{0}. The grazing incidence angle at which the highest reflectivity is obtained corresponds to the Bragg angle and for our example amounts to about 34.5°, as shown in Fig. 2f.

^{th}diffraction order. Please note the difference in scale along the axes of the diffraction efficiency in Fig. 2. A further increase in the number of diffraction orders does neither changes the shape of the reflectivity curve nor the peak reflectivity. Such a behavior of the reflectivity curves for increasing number of diffraction orders is quite understandable from a physical point of view as incident energy must be distributed over all orders taken into account.

^{th}diffraction order into account. The figure clearly shows that the diffraction efficiency near the Bragg angle is high for the lower diffraction orders and rapidly becomes negligible for higher diffraction orders. This is a specific feature of the SXR spectral range, where the very small polarizability of materials results in very narrow reflection and diffraction peaks. We can now conclude that for an accurate calculation of the zeroth order reflectivity curve for this specific LMAG, it is sufficient to consider up to the ± 5thdiffraction order (11 orders in total).

4. 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. Methods **541**(3), 590–597 (2005). [CrossRef]

^{th}diffraction order and conclude that the shape of the curves as well as the peak reflectivity are nearly identical. As an illustration, a peak reflectivity value for the zeroth order of 0.103 (see Fig. 2f) was obtained in our calculations and a value of 0.100 by Benbalagh. In our calculations the peak reflectivity decreases with increasing number of diffraction orders, whereas the results of Benbalagh show an increase. As discussed previously, a decrease in peak reflectivity is physically more understandable as energy needs to be conserved.

## 4. LMAG single-order operating regime

*χ*(

*z*). As the polarizability in the SXR region is proportional to the material density, we can conclude that Eq. (7) describes the reflection of a wave from a conventional multilayer structure consisting of materials whose densities are effectively reduced by a factor of Γ.

*θ*) should be small compared to the angular distance, in terms of the incidence angle, between the zeroth and first order diffraction peaks. This angular distance equals Δ

_{LMAG}*θ*≈

*d*/

*D*. The angular width of the reflectivity peak of a conventional MM (Δ

*θ*) is determined by the difference in the polarizabilities of the materials in the multilayer structure [15

_{MM}15. I. V. Kozhevnikov and A. V. Vinogradov, “Basic formulae of XUV multilayer optics,” Phys. Scr. T137–145 17, (1987). [CrossRef]

*θ*≈ Γ·Δ

_{LMAG}*θ*. The final condition for operation in the single-order regime is then written as

_{MM}*D*rather than on the grating period

*D*. By comparing calculated reflectivity peaks of several LMAGs with different lamellar widths, we determined that “much less” in Eq. (8) means less by a factor of 3, at least. The peak reflectivity for calculations only considering the zeroth order then differs by less than 1% compared to calculations considering many (11) orders.

_{4}C LMAG as before, but with a smaller lamel width (Γ

*D*) of 100 nm and a reduced grating period (

*D*) of 0.3 µm (i.e. Γ = 1/3). The incident photon energy

*E*is kept at 183.4 eV, The diffraction efficiencies of the zeroth (LMAG 0) and first (LMAG ± 1) orders are shown in Fig. 4. It is clearly visible that the angular distance (

*Δθ*) between the diffraction orders increases by roughly a factor of 7 as compared to Fig. 3. As a result, the diffraction efficiency of higher orders is very low near the Bragg angle where the zeroth order reflectivity is high.

_{4}C LMAG (

*D*= 0.3 µm) reaches 0.38, which is almost 4 times more than the peak reflectivity of the long-period LMAG (

*D*= 2 µm) shown in Fig. 2f. This can be explained by the re-distribution of incident intensity into the diffracted orders. In single-order operation, the incident intensity is diffracted almost entirely into one order (Fig. 4), whereas the intensity must be distributed over several orders for LMAGs with longer periods (Fig. 2). As stated previously, Eq. (8) describes the reflection of waves from a conventional multilayer structure with reduced density. This is also demonstrated in Fig. 4, where a comparison is shown between the calculated reflectivity curve for an LMAG (LMAG 0) and that for a conventional multilayer mirror (MM) with material densities reduced by a factor of Γ = 1/3. As can be seen, the agreement between the curves is excellent. In the single-order regime, sophisticated diffraction theories are thus not necessary for proper calculation of the reflectivity of an LMAG in the SXR region.

*θ*≈ Γ·Δ

_{LMAG}*θ*, as was discussed when deriving Eq. (8), and the resolution is thus only limited by the

_{MM}*Γ*that can be obtained technology-wise. As stated previously, Γ can be interpreted as a reduction factor for the material density and a proportional variation in the density of both bi-layer materials does not change the parameters

*f*and

*g*. Hence, the peak reflectivity of an LMAG operating in the single-order regime can be the same as that of a conventional MM consisting of regular density materials. The number of bi-layers in the multilayer structure of the LMAG that is required to obtain the maximum reflectance is inversely proportional to |

*χ*−

_{A}*χ*| and so must be increased by a factor of 1/Γ as compared to a conventional multilayer mirror [15

_{S}15. I. V. Kozhevnikov and A. V. Vinogradov, “Basic formulae of XUV multilayer optics,” Phys. Scr. T137–145 17, (1987). [CrossRef]

*E*= 183 eV) of a conventional Mo/B

_{4}C multilayer mirror with multilayer parameters as before (Fig. 2) and

*N*= 100. The angular width of the Bragg peak is Δ

*θ*= 0.82°. The three other curves show the reflectivity of LMAGs based on the same Mo/B

_{MM}_{4}C multilayer structure, but with different parameters Γ and

*D*, such that the lamellar width (Γ

*D*= 70 nm) remains the same for all LMAGs. A lamellar width of 70 nm satisfies condition (8) and is quite practicable for existing fabrication technologies. From curves 2–4, it can be seen that the width of the reflectivity curve indeed decreases by a factor of 1/Γ. The angular width of curve 4 is only 0.083°, which is actually about 1.5 times less than the minimal possible angular width ((Δ

*θ*)

_{MM}_{min}= 0.13°) for this MM (Eq. (9). Yet, the peak reflectivity of the LMAGs is still the same as that of the conventional MM, although the number of bi-layers required for this is very high.

*θ*in Eq. (8) is the width of the Bragg peak for p-polarization.

_{MM}## 5. Conclusions

*N*required for maximum reflectance scale with 1/Γ in comparison to a conventional MM. This allowed us to define novel analytic design rules for LMAGs. We have also shown, for the first time, that the resolution and reflectivity of an LMAG are only limited by the number of bi-layers

*N*and the lamel-to-period ratio Γ that can be obtained technology-wise. An LMAG can thus reach much higher resolutions than a conventional MM, without loss of peak reflectivity.

## Acknowledgements

## References and links

1. | A. E. Yakshin, R. W. E. van de Kruijs, I. Nedelcu, E. Zoethout, E. Louis, F. Bijkerk, H. Enkisch, S. Müllender, and M. J. Lercel, “Enhanced reflectance of interface engineered Mo/Si multilayers produced by thermal particle deposition”, 651701:1–651701:9, Proc SPIE , |

2. | R. A. M. Keski-Kuha and A. M. Ritva, “Layered synthetic microstructure technology considerations for the extreme ultraviolet,” Appl. Opt. |

3. | B. Vidal, P. Vincent, P. Dhez, and M. Neviere, “Thin films and gratings - Theories to optimize the high reflectivity of mirrors and gratings for X-ray optics”, 142–149, Proc. SPIE , |

4. | 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. Methods |

5. | A. Sammar, M. Ouahabi, R. Barchewitz, J.-M. André, R. Rivoira, C. Khan Malek, F. R. Ladan, and P. Guérin, “Theoretical and experimental study of soft X-ray diffraction by a lamellar multilayer amplitude grating,” J. Opt. |

6. | T. Peng, “Rigorous formulation of scattering and guidance by dielectric grating waveguides: general case of oblique incidence,” J. Opt. Soc. Am. |

7. | A. Coves, B. Gimeno, J. Gil, M. V. Andres, A. A. San Blas, and V. E. Boria, “Full-wave analysis of dielectric frequency-selective surfaces using a vectorial modal method,” IEEE Trans. Antenn. Propag. |

8. | L. I. Goray, “Numerical analysis of the efficiency of multilayer-coated gratings using integral method,” Nucl. Instrum. Methods |

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

10. | K. Krastev, J.-M. André, and R. Barchewitz, “Further applications of a recursive modal method for calculating the efficiencies of X-UV multilayer gratings,” J. Opt. Soc. Am. A |

11. | L. I. Goray and J. F. Seely, “Wavelength separation of plus and minus orders of soft-x-ray-EUV multilayer-coated gratings at near-normal incidence”, 81–91, Proc. SPIE , |

12. | 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. |

13. | V. V. Martynov, B. Vidal, P. Vincent, M. Brunel, D. V. Roschupkin, A. Yu. Agafonov, A.I. Erko, and A. Yakshin, “Comparison of modal and differential methods for multilayer gratings,” Nucl. Instrum. Methods Phys. Res. |

14. | 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. |

15. | I. V. Kozhevnikov and A. V. Vinogradov, “Basic formulae of XUV multilayer optics,” Phys. Scr. T137–145 17, (1987). [CrossRef] |

16. | R. Petit, Electromagnetic Theory of Gratings, Springer-Verlag, Berlin, 1980. |

**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:**

Diffraction and Gratings

**History**

Original Manuscript: May 17, 2010

Revised Manuscript: June 30, 2010

Manuscript Accepted: July 8, 2010

Published: July 16, 2010

**Citation**

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**, 16234-16242 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-15-16234

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### References

- A. E. Yakshin, R. W. E. van de Kruijs, I. Nedelcu, E. Zoethout, E. Louis, F. Bijkerk, H. Enkisch, S. Müllender, and M. J. Lercel, “Enhanced reflectance of interface engineered Mo/Si multilayers produced by thermal particle deposition”, 651701:1–9, Proc SPIE, 6517 (2007)
- R. A. M. Keski-Kuha and A. M. Ritva, “Layered synthetic microstructure technology considerations for the extreme ultraviolet,” Appl. Opt. 23(20), 3534 (1984). [CrossRef] [PubMed]
- B. Vidal, P. Vincent, P. Dhez, and M. Neviere, “Thin films and gratings - Theories to optimize the high reflectivity of mirrors and gratings for X-ray optics ”, 142–149, Proc. SPIE, 563 (1985)
- 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. Methods 541(3), 590–597 (2005). [CrossRef]
- A. Sammar, M. Ouahabi, R. Barchewitz, J.-M. André, R. Rivoira, C. Khan Malek, F. R. Ladan, and P. Guérin, “Theoretical and experimental study of soft X-ray diffraction by a lamellar multilayer amplitude grating ,” J. Opt. 24(1), 37–41 (1993). [CrossRef]
- T. Peng, “Rigorous formulation of scattering and guidance by dielectric grating waveguides: general case of oblique incidence,” J. Opt. Soc. Am. 6(12), 1869–1883 (1989). [CrossRef]
- A. Coves, B. Gimeno, J. Gil, M. V. Andres, A. A. San Blas, and V. E. Boria, “Full-wave analysis of dielectric frequency-selective surfaces using a vectorial modal method,” IEEE Trans. Antenn. Propag. 52(8), 2091–2099 (2004). [CrossRef]
- L. I. Goray, “Numerical analysis of the efficiency of multilayer-coated gratings using integral method,” Nucl. Instrum. Methods 536(1-2), 211–221 (2005). [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]
- K. Krastev, J.-M. André, and R. Barchewitz, “Further applications of a recursive modal method for calculating the efficiencies of X-UV multilayer gratings,” J. Opt. Soc. Am. A 13(10), 2027 (1996). [CrossRef]
- L. I. Goray, and J. F. Seely, “Wavelength separation of plus and minus orders of soft-x-ray-EUV multilayer-coated gratings at near-normal incidence”, 81–91, Proc. SPIE, 5900 (2005)
- 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. 333(2-3), 599–606 (1993). [CrossRef]
- V. V. Martynov, B. Vidal, P. Vincent, M. Brunel, D. V. Roschupkin, A. Yu Agafonov, A.I. Erko, and A. Yakshin, “Comparison of modal and differential methods for multilayer gratings,” Nucl. Instrum. Methods Phys. Res. 339(3), 617–625 (1994). [CrossRef]
- 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.
- I. V. Kozhevnikov and A. V. Vinogradov, “Basic formulae of XUV multilayer optics,” Phys. Scr. T 137–14517, (1987). [CrossRef]
- R. Petit, Electromagnetic Theory of Gratings, Springer-Verlag, Berlin, 1980.

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