Focusing single-order diffraction transmission ... - OSA Publishing

Focusing single-order diffraction transmission grating with a focusing plane perpendicular to the grating surface Quanping Fan,1,2 Yuwei Liu,2,3 Zuhua Yang,2 Lai Wei,2 Qiangqiang Zhang,2 Yong Chen,2 Feng Hu,4 Chuanke Wang,2 Yuqiu Gu,2 Weimin Zhou,2 Gang Jiang,1,5 and Leifeng Cao2,6 1 Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China Research Center of Laser Fusion, China Academy of Engineering Physics, Mianyang 621900, China 3 Joint Laboratory for Extreme Conditions Matter Properties, Southwest University of Science and Technology, Mianyang 621900, China 4 School of Mathematic and Physical Science, Xuzhou Institute of Technology, Xuzhou 221400, China 5 [email protected] 6 [email protected] 2

Abstract: By combining the single-order dispersion properties of quasisinusoidal single-order diffraction transmission gratings (QSTG) and the single-foci focusing properties of annulus-sector-shaped-element binary Gabor zone plate (ASZP), we propose a novel focusing single-order diffraction transmission grating (FSDTG). Different from the diffraction patterns of a normal transmission grating (TG), it has a focusing plane perpendicular to the grating surface. Numerical simulations are carried out to verify its diffraction patterns in the framework of Fresnel-Kirchhoff diffraction. Higher-order diffraction components of higher harmonics can be effectively suppressed by the FSDTG we designed. And we find that the focal depth and resolving power are only determined by the structure parameters of our FSDTG by theoretical estimations. ©2015 Optical Society of America OCIS codes: (050.1950) Diffraction gratings; (340.7480) X-rays, soft x-rays, extreme ultraviolet (EUV).

References and links 1.

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#234601 - $15.00 USD Received 17 Feb 2015; revised 17 May 2015; accepted 20 May 2015; published 11 Jun 2015 (C) 2015 OSA 15 Jun 2015 | Vol. 23, No. 12 | DOI:10.1364/OE.23.016281 | OPTICS EXPRESS 16281

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1. Introduction Transmission gratings (TGs) are widely used in soft x-ray spectral measurements of both laboratory and astrophysical plasmas [1–4]. Since TGs do not have focusing effect, slits or other focusing optics must be added to improve the resolving power when used in an x-ray spectrometer. As we know, zone plates (ZPs) are generally regarded as an x-ray lens to focus soft and hard x-ray [5–8]. By combining TGs’ diffraction property and ZPs’ focusing property, it may be possible to design a transmission grating with focusing effect in soft x-ray region. By combining the transmission functions of a grating and a zone plate, Chang Chang et al. [9–11] have presented a novel Fourier optical element, which is used as an interferometer by setting the first ring radius of ZPs r1 and the grating constant d in the same order of magnitude. In their work, nanoscale resolution soft x-ray differential interference contrast (DIC) imaging is achieved [9]. In addition, they have proved that the distance ( Δx ) between 0th order and 1rt order is independent with wavelength for their proposed optical element, which equals to a constant of r12 / d [10]. Though not pointed out explicitly, it implies that a focal plane exists when the novel element are served as a spectrometer by setting r1 several orders of magnitude larger than d, making Δx in the scale of several centimeters. For the novel Fourier optical element presented by Chang Chang, ordinary TGs and conventional ZPs are used. However, ordinary TGs and conventional ZPs suffer from multi order diffraction and multi order foci respectively due to their specific structures [12]. In the past decade, several binary single order diffraction gratings are proposed to suppress high diffraction orders, namely binary sinusoidal transmission grating (BSTG) [13], quantum-dotarray diffraction gratings (QDADG) [14], zigzag transmission gratings (ZZTG) [15], quasisinusoidal single-order diffraction transmission gratings (QSTG) [16], spectroscopic photon sieves (SPS) [17] and modulated groove position gratings (MGPGs) [18]. To restrain multi order foci of ZPs, some binary Gabor zone plates (BGZP) are also developed, among which the so-called annulus-sector-shaped-element binary Gabor zone plate (ASZP) is proposed by Lai Wei [19]. In this paper, we present a novel focusing single-order diffraction transmission grating (FSDTG) by combining the transmission functions of one-dimension ASZP and the QSTG proposed by Longyu Kuang [16]. The similar methods of Chang Chang et al. are used for


reference to design the novel optical element but we use it here as a spectrometer to measure spectrum and single-order diffraction grating and single-foci zone plate are used. To reduce the fabrication difficulties, we convert the circular ASZP into one dimension and onedimensional focusing is realized. To improve the diffraction efficiency, the XOR pattern technique [11] is used and the diffraction of 0th order can be eliminated completely after the consideration of that. In the following, we describe how to design the FSDTG, and the diffraction patterns are simulated and discussed in detail. 2. Design of the FSDTG

As shown in Fig. 1(a), the QSTG proposed by Longyu Kuang [16] consists of a series of rectangular holes distributed by cosinusoidal probability, in which grating constant d and the height w of rectangular holes are used to describe the structure characteristics of a QSTG. Normally, the width of each hole is setting to d/2. By converting the circular ASZP designed by Lai Wei [19] into one dimension, Fig. 1(b) gives the structure of the one-dimensional ASZP. The height of rectangular holes of the one-dimensional ASZP equals to the height of rectangular holes of the QSTG, also marked by w . rN = N r1 is the radius of every half wave zone, where r1 is the first ring radius of ZPs.

Fig. 1. The structures of (a) a QSTG and (b) a one-dimensional ASZP.

The XOR pattern technique [11] is used to improve the diffraction efficiency of our novel FSDTG by a factor of four compared with that only put a separate QSTG and a onedimensional ASZP together. The steps of adding the XOR pattern are as follows detailedly. The binary QSTG and one-dimensional ASZP are first pixelized, overlapped, and then compared pixel by pixel. At each pixel position, a logical exclusive OR (XOR) operation is performed by setting the corresponding pixel value of the resultant XOR pattern zero if the pixel values from the QSTG and one-dimensional ASZP are the same and one otherwise. The whole structures of the FSDTG without and with considering the XOR pattern are shown in Fig. 2(a) and Fig. 2(b) and the partial structures are shown in the insets of both figures after setting r1 =100d as an example, thus makes it working as a spectrometer without higher-order diffraction contamination. The absolute diffraction efficiency of Fig. 2(a) is only 0.39%, which is the product of a QSTG and a one-dimensional ASZP. After using the XOR pattern to turn the one-dimensional ASZP into a phase-type, as shown in Fig. 2(b), the efficiency increases to 1.6%, four times the case of Fig. 2(a).


Fig. 2. The whole structures of a FSDTG (a) without the XOR pattern, (b) with the XOR pattern, and the insets of (a) and (b) are the partial structures in overlapped regions.

3. Simulation of the diffraction patterns

For a transmission grating (TG), the distance Δx between 0th order and 1rt order has Δx =λ L / d , where L is the distance between the grating and screen, d is grating constant, and λ is wave length. For a zone plate (ZP), the focal length satisfies f =r12 / λ , where r1 is the first ring radius. Considering a plane wave of amplitude A0 and wavelength λ incident on our FSDTG, we have L = f and then get Δx =r12 / d , which is independent with wavelength, implying that there is a focal plane perpendicular to the surface of our FSDTG. Figure 3 shows the working pattern of a FSDTG, which is different from the working pattern of a normal TG. The distance between focal spot and grating z satisfies z=r12 / λ and we know the relationship between photon energy E and wavelength λ is E =1240 nm ⋅ eV / λ . Therefore, we get z =r12  E / 1240 nm ⋅ eV , which means that our FSDTG can realize a linear measurement of photon energy E in the focal plane rather than a linear measurement of wavelength λ as a normal TG does.

Fig. 3. The working pattern of a FSDTG.

To validate the diffraction patterns of our FSDTG, numerical simulation is carried out based on Fresnel-Kirchhoff diffraction equation, in which the complex amplitude U(P) satisfies,


    A e jkr ' cos(n, r ) − cos(n, r ') e jkr U ( P) = t (S ) [ ] dS jλ  r' 2 r S

(1)

Where A is a constant; t ( S ) is the transmission function of our FSDTG; r’ is the distance  from the point source p to a point on the FSDTG aperture surface and r ' is its unit vector; r is  the distance from the observation point q to a point on the aperture surface and r is its unit  vector; n is the unit vector normal to the aperture surface, respectively. Table 1. Simulation parameters of our FSDTG Photon energy E Wavelength λ Distance between screen centre and grating z0 Grating constant d The first ring radius of zone plate r1

122.8eV, 124.0 eV, 125.2 eV 10.1 nm, 10.00 nm, 9.90 nm 1m 1 μm 100

The distance between 1st order and 0th order Δx The periods of grating N The number of half wave zones N’ The length of screen L

10 mm 4000 400 5 cm

μm

The detailed parameters in simulations are listed in Table 1. Plane waves with photon energy 122.8 eV, 124.0 eV and 125.2 eV incident on the surface of our FSDTG, then are diffracted and focused on the focal plane. The simulated intensity distribution of these plane waves is shown in Fig. 4(a). We find that the three kinds of photons are separated linearly and high resolving power is achieved under our parameters. Figure 4(b) gives the diameter of the first null d null of the focal spot with photon energy 124.0 eV. According to Rayleigh resolution limit, the resolution along the focal plane for 124.0 eV is rnull = d null / 2 = 255 μ m . Therefore, the resolving power of 3921 is estimated by E / ΔE . However, the range of spectral measurement ΔE is narrow for our novel FSDTG. When the length of screen is L = Δz = 5 cm as shown in Table 1, the range of spectral measurement is only ΔE = 6.2 eV for Δz =r12 ΔE / 1240 nm ⋅ eV . By increasing the length of screen to L = Δz = 50 cm , the range of spectral measurement ΔE is extended to 62 eV. That means larger ΔE needs larger Δz . To obtain broad spectrum, the length of screen is too large to be detected by CCD. An x-ray film with large areas or a photodiode that moves along focal plane is an alternative way.

Fig. 4. The intensity distribution of focal spot in focal plane.

To highlight our FSDTG’s properties of effective suppression for higher-order harmonics, the diffraction intensity distribution of each harmonic (from second to fifth) with the same


amplitude A0 is calculated and compared with fundamental frequency (124.0 eV) in the same position (z = 1 m), as shown in Fig. 5. From comparisons, we find that higher-order diffraction components of higher harmonics are effectively suppressed, which means single order diffraction is realized by the FSDTG we designed.

Fig. 5. The intensity distribution in focal plane at z = 1m for higher-order diffraction components of higher harmonics.

4. Theoretical estimates of diffraction properties

Here, we estimate theoretically the diffraction properties of our FSDTG such as focal depth perpendicular to the focal plane and resolving power along the focal plane. For conventional ZPs, the focal depth along optical axis z is Δf = 4Δr 2 / λ , and the lateral spatial resolution is Δξ = 1.22Δr , where Δr is the outermost zone width of the zone plates, and satisfies Δr =r1 / (2 N ′ ) when the number of half wave zones N ′ is large enough.

Fig. 6. The propagation process of a plane wave (124.0 eV) for our FSDTG. (a) near the region of focal plane; (b) partial magnification of focal spot in the range of focal depth; (c) further partial magnification of focal spot in the range of spatial resolution.


Figure 6(a) shows the propagation process of a plane wave (124.0 eV) near the region of focal plane with the same parameters of Table 1, where exit angle θ ≤ 10 must be satisfied and θ ≈ sin θ ≈ tan θ in our situation of paraxial approximation. After magnified to focal range, as shown in Fig. 6(b), since the focal depth along exit angle θ is equal to the focal depth of a conventional ZP Δf , therefore, the focal depth perpendicular to the focal plane Δf ′ satisfies Δf ′ = 4Δr 2 / λ θ =4Δr 2 / d , where θ = λ / d . After magnified further to the range of spatial resolution, as shown in Fig. 6(c), since the spatial resolution perpendicular to the direction of θ is equal to the value of a conventional ZP Δξ = 1.22Δr , the spatial resolution along focal plane for our FSDTG has approximately Δξ ′ = 1.22Δr / θ = 1.22Δrd / λ . It can be also expressed as Δξ ′ = 1.22Δrdf N / r12 for

f N = r12 / λ . Then the resolving power η can be obtained by η = f N / Δξ ′ = r12 /1.22Δrd . From above, we find both Δf ′ and η are independent with wavelength and only determined by the structure parameters of a FSDTG. Since rN ′ = N ′r1 and D =2rN ′ , the resolving power η also can be expressed by η = N / 1.22 , where N = D / d is the number of grooves irradiated. With the same parameters in Table 1, the resolving power η is 3279 theoretically, which is a little smaller than the simulated value of 3921, but gives a proper estimate anyhow. However, the theoretical focal depth Δf ′ is only 25 μ m calculated by the same parameters in Table 1, the value of which seems too short to put a detector on the focal plane. From Δf ′ = 4Δr 2 / d and η = r12 / 1.22Δrd , we can extend focal depth Δf ′ appropriately while keeping the resolving power η invariable. For example, if we keep d invariable and extend the focal depth Δf ′ to 1 mm, Δr should be increased to 2 10 times the original, and r1

should be increased to (2 10)1/ 2

times the original. Since

Δx =r12 / d

and

z =r  E / 1240 nm ⋅ eV , both of these values should be increased to 2 10 times the originals, which implies that the extension of focal depth Δf ′ depends on increasing the length of emergent arm. A numerical simulation is performed with modified parameters, and the results of focal depth Δf ′ and the resolving power η are in a good agreement with theoretical anticipations. 2 1

5. Conclusion

In conclusion, we have proposed a novel focusing single-order diffraction transmission grating named FSDTG in this paper. Compared with normal TGs, it has a distinctive work pattern, in which a focal plane exists and is perpendicular to the surface of our FSDTG in the case of a plane wave incidence. The absolute diffraction efficiency is 1.6% after considering the XOR pattern. The length of screen along diffraction direction z should be large enough to obtain broad spectrum. The resolving power and focal depth are only determined by the structure parameters of our FSDTG. By combining QSTG and one-dimensional ASZP, higher-order diffraction components are effectively suppressed. In fact, there are two focal planes symmetrically along optical axis z due to the symmetrical structure of our FSDTG, thus produces two identical spectra that can be used for single-shot x-ray absorption finestructure spectrum (XAFS) measurements where one beam passes a sample and the other is used for normalization.


Acknowledgments

This work is supported by the National Science Instruments Major Project of China (Grant No. 2012YQ130125) and National Science Foundation of China (Grant No. 11375160, 11174213 and 11304266).
