Influence of Oblique Incidence on Transmission Grating Diffraction in Soft X-Ray Region

Transcription

1 Materials Express Copyright 2012 by American Scientific Publishers All rights reserved. Printed in the United States of America /2012/2/151/006 doi: /mex Influence of Oblique Incidence on Transmission Grating Diffraction in Soft X-Ray Region Wanli Shang, Wenhai Zhang, Jiamin Yang, Tuo Zhu, Longyu Kuang, and Sanwei Li Research Center of Laser Fusion, China Academy of Engineering Physics, Mianyang , People s Republic of China Transmission grating diffraction of soft X-ray plays an important role in grating calibrations and soft X-ray measurements. In order to investigate the influence of oblique incidence on the applications of conventional transmission grating spectrometer in the soft X-ray area, the Kirchhoff diffraction theory with the Fraunhofer approximation is applied and detailed asymmetrical diffraction characteristics are demonstrated. The formation of the asymmetrical diffraction patterns is analyzed with the Fourier optics theory. Various grating structures are performed to explore the optimal structures to minimize the oblique incident influence and this can be a convenient means to control the oblique incident influence in future grating designs. Keywords: Transmission Grating, X-Ray Diffraction, X-Ray Optics, Soft X-Ray, Diffraction Efficiency. 1. INTRODUCTION The transmission grating usually consisting of fine parallel gold bars, represents a very convenient diagnostic tool in several fields including the studies of inertial confinement fusion (ICF) and astrophysics. 1 8 The design, construction, and diffraction characteristics of the transmission gratings have been investigated both experimentally and theoretically It has been reported that the grating parameters, including grating period, spatial-phase coherence, linewidth control, bar thickness, and patterned size determine the diffraction characteristics, and emphasis Author to whom correspondence should be addressed. wanlishang@gmail.com is placed on different parameters according to a particular condition. 16 The effects of photon incident angle on the grating efficiencies have been considered to indicate different characteristics toward which the grating is tilted. Asymmetrical diffraction spectra were reported experimentally in the extreme-ultraviolet spectral region with obliquely illuminated gratings. 18 Afterwards experimental evidence and coupled-wave theoretical analyses demonstrated that the nanostructured volume gratings diffract a significant proportion of the incident extreme-ultraviolet light into high orders when oblique incidence happens In recent years, a new type of grating, critical-angle transmission (CAT) grating, integrating the advantages of transmission grating with blazed reflection grating, was designed for the first time Under the non-normal incidence in the EUV and soft X-ray band, the fabrication and performance of the CAT grating were investigated Recently, a high-resolution soft X-ray grating spectrometer concept for the International X-Ray Observatory (IXO) was presented, having great effective area (>1000 cm 2 for energy <1000 ev) and spectral resolution (E/ E > The CAT grating was utilized in this grating spectrometer with the help of the non-normal incidence However, within the conventional transmission grating spectrometer applications such as the operations in SG III prototype high power laser facility (the spectrometer layout is shown in Fig. 1), the oblique incidence and asymmetrical diffraction always exist, leading to troubles for our experiments. Great uncertainties in the X-ray measurements as well as grating calibrations could be caused to the precise studies of ICF and other research areas. As an investigation towards the oblique incidence in the soft X-ray region, this paper reports the influence of the oblique incidence on the conventional transmission grating efficiencies of different diffraction orders. The formation of the asymmetrical diffraction patterns is analyzed Mater. Express, Vol. 2, No. 2,

2 Materials Express Influence of Oblique Incidence on Transmission Grating Diffraction (a) Grating slit Space resolving slit Grating Source S 2 S 1 (b) CCD 5 to 5 deg. The theoretical model applied is based on the widely used Kirchhoff diffraction theory with the Fraunhofer approximation Within the scope of conventional gold transmission grating applications, the blazed reflection on the sidewall surface is not taken into account for the reason that sub-nm microroughness sidewalls are always not achieved. m = 1 d 2 d 0 t x exp i2m x d dx 2 (1) Fig. 1. (a) Schematic layout of the space-resolving transmission grating spectrometer. (b) Scheme of grating structure with period d, bar distance a, thickness h and oblique incident angle. (a = 120 nm, d = 400 nm, h = 500 nm, 5deg 5deg. A grating period is divided into four regions 0 a htan, a h tan a, a d h tan and d h tan d according to the light path. t x =exp ikz x i 1 0 a htan ( h exp ik cos a x ) i a htan a sin = exp ik h cos i a d htan exp ik d x sin i d htan d (2) with the Fourier optics theory. In addition, different grating structures have been taken into consideration to minimize the influence of the oblique incidence and then to facilitate more effective control strategies for future transmission grating designs. 2. TRANSMISSION GRATING SPECTROMETER SETUP AND THEORETICAL DIFFRACTION EFFICIENCY MODEL The schematic layout of the conventional space-resolving transmission grating spectrometer and the related key parameters are shown in Figure 1(a). 5 To achieve high spectral resolution collimation or focusing is required. Collimation is often achieved by inserting a grating slit in front of the grating. In this case the grating was fabricated inside a slit of S 2 = 50 m. 5 A slit with width S 1, called as space-resolving slit, is set between the source and the grating for the space resolving. The space-resolved X-ray spectral image is recorded with an X-ray charge-coupled device (CCD). To bring out characteristics of soft X-ray oblique incidence transmission within that kind of transmission grating spectrometer in Figure 1(a), we consider freestanding transmission grating made up of parallel gold bars, separated by gaps. Grating structure is described in Figure 1(b). Period is 400 nm with a = 0 3 d and h = 500 nm, where d is the period, a is the bar distance, and h is the thickness of grating. The photon energy range is from 100 to 5000 ev with tiny oblique incident angle from where m is the efficiency in the mth order, d is the grating period, t x is the transmission function, and k is the wave-number (2 /. z x is the grating path-length function; at normal incidence, it is the grating bar thickness. 1 + i is the complex index of refraction cited from X-ray optics web site of Lawrence Berkeley Laboratory s Center ( and is often expressed in real and imaginary parts (parameters in the soft X-ray area are presented in Fig. 2). Because the grating linear dimension is much larger than the X-ray wavelength, the Fraunhofer approximation is applicable in our work although the rigorous solution to the classical problems of diffraction is the vector theory. 30 Fig. 2. Complex index of refraction of Au ˆn = 1 + i cited from X-ray optics web site of Lawrence Berkeley Laboratory s Center ( 152 Mater. Express, Vol. 2, 2012

3 Influence of Oblique Incidence on Transmission Grating Diffraction Materials Express 3. RESULTS AND DISCUSSION The photon energy dependent ratios of oblique to normal incident diffraction intensities of different orders are shown in Figure 3. As we can see, the general principle is that greater deviation from the normal diffraction occurs as the incident angle increases, especially for the higher orders, which is consistent with the previous works. It should be particularly noted that such an effect is quite evident in the relative low photon energy region ( ev) and not so significant in the high photon energy area ( ev) in most of the cases (see Figs. 3(b), (d), and (e)). Similar results have been obtained with the diffraction of CAT grating. The reason is that at high energy the grating bars become transparent and introduce less phase shift and amplitude decrease to the X-rays. However, the states of ±2nd diffraction orders are quite opposite in Figure 3(c) that less deviation is obtained from 100 to 2000 ev due to the considered grating structure. In addition, the diffraction deviation of positive from negative order intensities caused by the oblique incidence is significant in ±3rd and ±4th diffraction orders in Figures 3(d) and (e). On the contrary, this positive-negative order deviation becomes less significant in ±1st and ±2nd diffraction orders in Figures 3(b) and (c). It has been reported that energy is transferred into high diffraction orders when oblique incidence angle is increased, 19 and this energy redistribution lead to the diffraction deviation from the normal diffraction especially in the condition of high orders and great oblique incidence angles. Fourier optics analysis is used to explain the formation of the asymmetrical diffraction patterns owing to the oblique incidence. The comparison of the formations of the normal and oblique incident diffraction is demonstrated in Figure 4. Figures 4(a) and (b) are descriptions of diffraction by a single period of transmission grating. Figure 4(c) is the comb function of the infinite grating periods used to keep the integrals of all the delta functions at unity, 1/d comb x/d = x nd.31 As we can see, Figure (a) Figure (c) Figures (d) and (b) Figure (c) Figure (e) Figure 4(d) shows that the normal incidence leads to symmetrical diffraction patterns due to the symmetrical transmission function and comb function. When the oblique incidence with a tiny titled angle happens, light changes its transmission path length. Employing the complex index of refraction 1 + i of grating bar material gold, changes are induced in the transmission properties of the grating slits and bars owing to the variation of light path length. With the Fourier transform operation on the transmission function in a single grating period, it can be concluded that the spatial frequency has a shift ( f x compared with the normal condition. The reason is that an exponent e i x correlated to the light path variation is brought to the transmission function (e i x expresses the phase shift when the light Fig. 3. Photon energy dependent ratios of oblique to normal incident diffraction intensities of different orders. In these figures solid lines mean the positive diffraction orders and dash lines represent the negative orders. passes through the grating bar). According to the properties of the Fourier transform, the space shift will bring frequency shift and then f x is excited. Therefore, as shown in Figure 4(b), the envelope diffracted by a grating period deviate from the normal incident diffraction in Figure 4(a). Mater. Express, Vol. 2,

4 Materials Express Influence of Oblique Incidence on Transmission Grating Diffraction Fig. 4. Comparison of the formations of the normal and oblique incident diffraction patterns by Fourier optics analysis (a = 0 3d. The blue lines depict the envelope due to the individual grating period while the red areas are the diffracted field produced by a transmission grating. For simplicity, a grating contained an infinite number of slits is considered for the reason that the slit number has nothing to do with the diffraction symmetry. However, the comb function in Figure 4(c) has nothing to do with the incident angle but the grating period d while it keeps constant all along. As a result, the interaction between the single grating period transmission function and the comb function takes the asymmetrical diffraction patterns in Figure 4(e). It should be noted that there is another cause of the asymmetrical diffraction patterns. The envelope structure changes and this will help to excite the diffraction asymmetry. The influence of the oblique incidence on the transmission grating diffraction efficiencies with the selected grating structures (a/d = 0 3, d = 400 nm, h = 500 nm) has been displayed. With these diffraction characteristics the effects of oblique incidence to X-ray measurements as well as grating calibrations upon the conventional gold transmission grating could be judged to a certain extent. However, key questions remain to be explored. In particular, how about the situation with different grating structures? Is there any optimal grating structure that could minimize the influence of the oblique incidence? In order to test the influence of roughness and interdiffusion in zone plate structures on the diffraction efficiency, the root-meansquare (rms) roughness was introduced to characterize the influence of roughness.34 In this paper, we introduce an rms deviation between the oblique and normal incidence Fig of diffraction orders from 5th to 5th and oblique incident angle from 5 to 5 deg (the increment is 0.1 deg) as well as photon energy from 100 to 5000 ev (the increment is 1 ev). It is defined as: = m= 5 = 5 E=100 m E m E 0 m E 0 2 (3) where is the oblique incident angle, E is the photon energy, m is the diffraction order, and is the diffraction efficiency. Figure 5(a) is presented to indicate the normalized rms deviation with the grating period fixed as 400 nm. Nine clearly shown streaks imply the a/d values of 1/5, 2/5, 3/5, 4/5, 1/4, 2/4, 3/4, 1/3, 2/3 with which the diffraction orders vanish according to the Kirchhoff diffraction theory.13 It is also illustrated that less grating bar thickness produces less rms deviation. In Figure 5(b), a/d is set as 0.3 and it shows great deviation happens when the grating thickness nears 800 nm and the period shrinks to nearly 100 nm. It means that optimal grating structures could be obtained to resist the oblique incident deviation with relatively small grating thickness and great period. In Figure 5(c) the grating thickness is set as 500 nm. Similarly with that in Influence of the oblique incidence with different grating structures (rms deviation from normal incidence based on Eq. (3)). Mater. Express, Vol. 2, 2012

5 Influence of Oblique Incidence on Transmission Grating Diffraction Materials Express Figure 5(a), nine luminous streaks occur. Moreover, in this figure we can see that great grating period helps to reduce the oblique incident deviation. As mentioned above about Figure 5, we can conclude that relatively small grating bar thickness and great period as well as properly choosing the bar-distance-period-ratio (a/d to avoid the luminous streak area are of benefit to resist the oblique incident deviation. It has been established that the 200 nm period grating shows high sensitivity to changes in photon oblique incident angle, whereas the 400 nm period grating shows almost no sensitivity with the same grating thickness and bar-distance-period-ratio (a/d, 20 which is in agreement with our results. However, there is a disadvantage that a great period grating leads to a low spectral resolution of the transmission grating spectrometer when the grating is used as diffracting element. 35 A compromise design of the grating structure as well as increasing the photon diffraction distance and decreasing the source size can solve this problem. 4. CONCLUSION In this paper, to explore the oblique incidence diffraction characteristics within the applications of conventional transmission grating spectrometer in the soft X-ray region, the detailed investigation on the absolute diffraction of transmission grating with oblique incidence is performed with the selected grating structure. The influence of different oblique incident angles on the various diffraction orders has been demonstrated. It is found that the diffraction characteristics of photon oblique incidence change greatly when the photon energy increases. 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