Using numerical optimization algorithm, non-periodic Mo/Si, Mo/Be, and Ni/C broad angular multilayer analyzers have been designed. At the wavelength of 13 nm and the angular range of 45~49°, the Mo/Si and Mo/Be multilayer can provide the plateau s-reflectivity of 65% and 45%, respectively. At 5.7 nm, the s-reflectivity of Ni/C multilayer is 16% in the 44~46° range. The non-periodic Mo/Si broad angular multilayer was also fabricated using DC magnetron sputtering, and characterized using the soft X-ray polarimeter at BESSY. The s-reflectivity is higher than 45.6% over the angular range of 45~49° at 13 nm, where, the degree of polarization is more than 99.98%.
© 2006 Optical Society of America
For polarization-sensitive studies, such as circular dichroism spectroscopy, spin-polarized photoelectron spectroscopy and spectroscopic ellipsometry [1–3], accurate evaluation of the polarization state of the radiation is clearly crucial. Traditionally, in the visible and ultraviolet regions of the spectrum, birefringence and dichroic analyzers are used. But, in the soft X-ray region and extreme ultraviolet regions, collectively known as the EUV, the closeness of the (real part of the) refractive index, coupled with high absorption, makes the realization of these kinds of analyzers impossible. Schemes utilizing multilayer interference structures to function as EUV analyzers and phase retarders have been proposed and demonstrated [1, 2, 11]. A linear analyzer based on Mo/B4C/Si multilayer that achieved almost 99.9% polarization with 70% s-component reflectivity at 13.5 nm has been realized . However, these multilayer-based analyzers are effective only at the Quasi-Brewster’s angle, and the narrowband wavelength and angular properties may be disadvantageous for some studies. To overcome this shortcoming, a non-periodic broad angular analyzer using the numerical method is developed in this letter. The broad angular multilayer analyzer can deviate the Quasi-Brewster’s angle several degrees and shows very high polarization. Moreover, the broad angular analyzer can also be used in the reflection of point light source to obtain the high reflectivity in a broad angular range. In this letter, non-periodic Mo/Si, Mo/Be, and Ni/C broad angular multilayer analyzers have been designed. The non-periodic Mo/Si broad angular multilayer was also fabricated using DC magnetron sputtering, and characterized using the high precision 8-axis ultra-high vacuum compatible soft X-ray polarimeter at BESSY facility. The reflectivity for s-polarized light was found to be 53.8% at 49° and higher than 45.6% over the angular range of 45~49° at 13 nm, where, the degree of polarization is more than 99.98%.
2. Design procedure
The broad angular multilayer analyzers design requires the choice of materials, incidence angle and layer thickness distribution to give a s-component reflectivity as high and flat as possible for the desired angular range. In the soft X-ray region, the complex refractive indices of all materials are very close to unity, so that the quasi-Brewster angles, where large differences between the reflectivities for the s- and p-components can be obtained, are close to 45° for reflective multilayers. The polarization degree and the reflectivity of the beam reflected from a multilayer are determined by usual manner,
where, Rs and Rp are the reflectivity for s- and p-polarized radiation, respectively. But, since near the quasi-Brewster angle, Rp is close to zero, then R is approximately Rs/2.
The best material pairs for multilayers are those that form smooth and compositionally abrupt interfaces with high optical contrast and low absorption . In the range of 12.5~20 nm, molybdenum and silicon are suitable materials, providing good optical performance. In this letter, the Mo/Si multilayer was used as broad angular analyzers at the wavelength ranges of 13 nm over the angular range of 45~49°. The Mo/Be multilayer was also designed just for comparison. At the wavelength of 5.7 nm, the Ni/C multilayer was chosen.
In the design of a reflective multilayer analyzer, both the s-reflectivity (Rs) and the polarization degree (P) should be maximized. From Eq. (1), when Rs is maximum, P approaches to 1. Therefore, the optimization is achieved by minimizing the Merit Function
where the summation is over a selection of discrete angles in the desired range. The layer thickness distribution is considered as independent variable. During the recursive optimization, only randomly selected layer thickness changes that decrease MF are retained, finally leading to an optimized layer thickness distribution that provides a minimum value of MF. However, in general, a merit function for the non-periodic multilayer depends on many variables, and may have a large number of local minima. The ideal solution would be the global minimum certainly, which requires a lengthy optimizing calculation process and is difficult to realize. But, in practice, it is enough to find a deep minimum of the merit function so that both the reflectivity profile and polarization degree are approximation to the desired ones. That is to say, the optimization method is the recursive calculation of the layer thicknesses, which provides the best approximation to the required profile of the reflectivity. As long as the initial thickness of each layer is given, the performance of non-periodic multilayer analyzer can be calculated numerically using the Fresnel formulas and the optical constant referred from Ref. . It is certain that the calculated results vary with the initial values of layer thickness. So, the initial thickness distribution for optimizing calculation is crucially important for obtaining a good optimization result. However, the approaches based on simplified analytic formulas lead to multilayer structures do not provide the desired reflectivity profile and usually result in a strongly oscillating reflectivity spectrum [6, 7]. Therefore, in our design, the initial period distribution was deduced from the method described in Ref.  and Ref. . The optimization of minimizing the MF was performed using the standard Levenberg-Marquardt algorithms. During the design, we will consider the multilayer with the constant s-component reflectivity in the desired angular range, i.e. Rs(θ)=R0. In addition, this design method is generally applicable for other material pairs and wavelength ranges. For different wavelengths, the suitable material combination can be selected and the broad angular analyzer can be designed. The results of optimization of the broad angular Mo/Si (at 13 nm), Mo/Be (at 13 nm) and Ni/C (at 5.2 nm) multilayer mirrors were shown in Fig. 1. Compared with the periodic Mo/Si analyzer (Fig. 1(a), the angular-width (FWHM) increases 1.6° and 4.6° for non-periodic analyzers [Figs. 1(b) and 1(c)], respectively.
As an example, the broad angular Mo/Si multilayer analyzer working at fixed wavelength of 13 nm with s-reflectivity of 65% in the 45~49° angular range was optimized and fabricated. The layer thickness distribution of Mo/Si multilayer was shown in Fig. 2. The layer thicknesses oscillate from 3 nm to 11 nm, a range which is feasible to manufacture using DC magnetron sputtering system.
3. Results and discussion
The Mo/Si multilayer were fabricated using a calibrated ultra-high vacuum DC magnetron sputtering system (JGP560C6, made in China) with targets of Mo (purity 99.95%) and Si (99.999%) in Ar (99.999%). The multilayers were deposited onto 20 mm × 30 mm silicon substrates at room temperature. The deposited multilayers were measured, for quality control, using a small angle X-ray diffractometer (D1 system, Bede Ltd, UK). The optical performances of broad angular multilayer analyzers were evaluated using the high precision 8-axes ultra-high vacuum compatible polarimeter on beamline UE56/2-PGM-1 at BESSY . The measurement results and the design ones were listed in Table 1 and shown in Fig. 3.
The reflectivity for s-polarized light was found to be higher than 45.6% over the angular range of 45~49° at 13 nm, where, the degree of polarization is more than 99.98%, and the average and the root-mean-square of experimental s-reflectivity are 50.7% and 0.0231, respectively. The s-reflectivity of multilayer analyzer is less than that calculated one. This is due to the interfacial roughness and diffusion.
The non-periodic multilayer analyzer not only exhibit broad angular width at the design wavelength, but also can be used at nearby wavelengths. Figure 4 shows the s- and p-reflectivity of the Mo/Si multilayer analyzer measured at different wavelengths: (I) 16.5 nm, (II) 15.5 nm and (III) 14.6 nm, which was designed for wavelength of 15.5 nm. At the wavelength of 15.5 nm, the s-reflectivity is flat and as high as 36.6% from 43° to 54° [(Fig. 4(II)]. At the wavelengths of 16.5 nm and 14.6 nm, the analyzer can also work in angular ranges of 48~56° [(Fig. 4(I)] and 40~48° [(Fig. 4(III)].
The method of designing the non-periodic multilayer with broad angular range used as analyzers is described in this paper. The main feature of the method is the use of non-periodic multilayer as broad angular analyzers. These broad angular analyzers will greatly simplify experimental arrangements, since the same multilayer could be used over a broad angular range, negating the necessity of translating or rotating mirrors. On the other hand, analyzer also can be used at nearby wavelengths. Furthermore, the design and manufacturing methods described are generally applicable for other material pairs, merit functions and angular ranges.
The authors are indebted to Igor V. Kozhevnikov for useful discussions. This work was supported by the National Natural Science Foundation of China (60378021 and 10435050), by the Royal Society, London (NC/China/16660) and by the European Union through the BESSY-EC-IA-SFS (BESSY-ID.05.2.165).
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