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Grating beam splitters have been fabricated for soft X-ray Mach-Zehnder interferometer using holographic interference lithography. The grating beam splitter consists of two gratings, one works at X-ray laser wavelength of 13.9 nm with the spatial frequency of 1000 lines/mm as the operation grating, the other works at visible wavelength of 632.8 nm for pre-aligning the X-ray interferometer with the spatial frequency of 22 lines/mm as the pre-alignment grating. The two gratings lie vertically on the same substrate. The main feature of the beam splitter is the use of lowspatial-frequency beat grating of a holographic double frequency grating as the pre-alignment grating of the X-ray interferometer. The grating line parallelism between the two gratings can be judged by observing the diffraction patterns of the pre-alignment grating directly.
©2008 Optical Society of America
1. Introduction
The interferometry with an X-ray laser as probe is a powerful tool to diagnose the electron density of a high temperature and dense plasma. Soft X-ray Mach-Zehnder (M-Z) interferometer is an important experimental setup for this interferometry. In 1995, Da Silva et al. developed the first amplitude division soft X-ray M-Z interferometer based on multilayer beam splitter [1]. The beam splitters used in the interferometer are the most critical elements of the setup. However, multilayer beam splitter presents two limitations [2]. One is that multilayer beam splitter cannot be implemented at all wavelengths, owing to the lack of materials with adequate optical constants for the multilayer beam splitter in some regions of the spectrum. The other is that multilayer beam splitter is thin strained multilayer films which tends to be fragile.
Besides multilayer beam splitter, diffraction grating is another kind of beam splitter for soft X-ray M-Z interferometer, which can overcome the limitations of the multilayer beam splitter. By designing and optimising the ridge structure of gratings, the grating can split the X-ray at any selected soft X-ray wavelength. In 1999, Rocca et al. developed an amplitude division soft X-ray interferometer based on diffraction gratings as beam splitters [2]. Since then, diagnostic experiments using soft X-ray M-Z interferometer based on grating beam splitters have been performed [3–11]. The soft-X-ray diffraction grating interferometer, which consists of a skewed M–Z configuration [12] is schematically illustrated in Fig. 1. The soft X-ray laser beam incident upon the first grating beam splitter (G1) is diffracted in the zeroth and the first order as the two arms of the interferometer. These two beams are reflected at grazing incidence angle towards the second grating beam splitter (G2) by two mirrors (M1) and (M2). The two beams recombine on G2 and generate the interference pattern. If the laser plasma to be studied is introduced into one arm of the interferometer, we can diagnose the electron density of the plasma from the interference pattern.
The diffraction grating interferometer needs to be pre-aligned with a visible or an infrared semiconductor laser diode. To allow the alignment of the interferometer with the prealignment source, the grating beam splitter needs to be fabricated with two vertically separated rulings on the same substrate. Therefore, in fact, the soft X-ray laser interferometer consists of two interferometers. One is a pre-alignment interferometer as shown in the upper part of Fig. 1 the other is a plasma-probing interferometer as shown in the lower part of Fig. 1. The grating used in the pre-alignment interferometer is called the pre-alignment grating working at the wavelength of the pre-alignment source λa with the spatial frequency y of νa. And the grating used in the plasma probing interferometer is called the operation grating working at the wavelength of the soft X-ray laser λo with the spatial frequency of νo. In order to make these two gratings have the same dispersion at their own working wavelengths, their spatial frequencies and working wavelengths satisfy the relation of νoλo = νaλa. Moreover, the two gratings should keep good parallelism to make the two interferometers have parallel diffraction planes.
Ruled grating was used as the beam splitter of soft X-ray M-Z interferometer [2–11]. The X-ray grating beam splitter was ruled with two vertically separated rulings on the same substrate. One part of the beam splitter substrate is with 300 lines/mm to diffract 46.9 nm X-ray laser beam and the other is with 17.06 lines/mm to diffract the beam of the pre-alignment laser diode [2–11]. It is easy to keep the parallelism of the pre-alignment grating and the operation grating by ruling. However, the fabrication of ruled grating is time consuming. Holographic interference lithography, on the other hand, allows for the production of grating by exposure and development step. The grating pattern is produced by the interference of two wavefronts on a photoresist coated on the substrate. The difficulty of holographic interference lithography is the fabrication of the pre-alignment grating with low spatial frequency.
In this paper, we propose and demonstrate a holographic approach for the fabrication of grating beam splitters for soft X-ray M-Z interferometer. Especially, a holographic double frequency grating (HDFG) with a low-spatial-frequency is introduced as the pre-alignment grating of the soft X-ray M-Z interferometry.
2. Fabrication of HDFGs with a low-spatial-frequency beat grating
In the case of single-exposure interference lithography, the spatial period of the linear grating pattern, d, is given by, d = λ/(2nsinθ), where λ is the wavelength of the laser, n is the refractive index of the exposing medium, and 2θ is the angle of the two interfering beams. If the interference lithography is performed in the air, the refractive index of the exposing medium n = 1 and the grating period is λ/(2sinθ). Accordingly, the spatial frequency of the grating, ν, is given by, ν = 2sinθ/λ. In our experiment, the exposure was carried out using Llyod’s mirror setup with a 413.1 nm Kr+ continuous wave laser as the light source, as shown in Fig. 2. The Llyod’s mirror and grating substrate are fixed together on one precision rotation table (Huber). By rotating the table, the angle of the two interfering beams can be changed, and accordingly, the spatial period of the grating is changed.
HDFG is a kind of holographic optical element which emerged in the 1970s [13]. The HDFG is produced by exposing one photoresist substrate twice with two adjacent spatial frequencies [13, 15]. A Llyod’s mirror setup as shown in Fig. 2 was also adopted in the fabrication of the HDFG. In the first holographic exposure, the angles of the two interfering beams are 2θ 1, and the period of the grating patterns is d 1 = λ/(2sinθ 1). Accordingly, the intensity spatial distribution (see Fig. 3(a)) in the grating substrate of the interference patterns can be written as
Where I 0 is the intensity of the interference beam at x = 0 for the grating patterns of the period d 1, ν 1 is the grating spatial frequency, and the x direction is perpendicular to the grating lines. Then in the second exposure for the same grating substrate of the first exposure, the angles of the two interfering beams are 2θ 2, and the period of the grating patterns is d 2 = λ/(2sinθ 2). Because laser power is assumed equal for both exposures and the two exposure times are equal, the amplitudes of the two intensities are equal for both exposures. After the second exposure, the intensity spatial distribution (see Fig. 3(b)) in the grating substrate of the interference patterns is the sum of the intensities of the two exposures and can be written as
Where I2=I0[1+cos(2πx/d2)]=I0[1+ cos (2πν2x)] (see the red curve of Fig. 3(a)).
Fig. 3. Intensity spatial distribution in the photoresist (a) I 1(x) and I 2(x), (b) I after two exposures.
Assuming that the photoresist ablation produced by the development is proportional to the intensity of the interference patterns, the photoresist grating profile is the same as the spatial distribution of the intensity. Therefore, the photoresist grating profile h(x) after two exposures is the same as that of I as shown in Fig. 3(b), and can be written as
That is to say, Fig. 3(b) shows the photoresist pattern profile of the HDFG.
The grating produced by each exposure is called a component grating. As seen in Fig. 3(a), the spatial frequencies of the two component gratings are 1000 and 1020, respectively. From Fig. 3(b), we can deduce that the amplitude of the high-spatial-frequency component grating structures is modulated by a grating with low-spatial-frequency of 20. This phenomenon can also be understood as the Moire beat phenomenon formed by the superposition of the two high-spatial-frequency gratings of different pattern periods [16, 17]. That is to say, the two exposures will bring a beat-frequency grating. And this kind of grating is called HDFG. The spatial frequency of the beat grating (ν′) in the HDFG is the difference in the spatial frequency between the two component gratings. The less the difference in the spatial frequency of the two component gratings is, the lower the spatial frequency of the beat grating is, and the larger the spatial period of the beat grating is. If the grating spatial periods or frequencies of the two exposures have values that are close to each other, the beat grating is with large period or low spatial frequency. Therefore, we can fabricate low-spatial-frequency gratings with the HDFG technique.
Fig. 4. Micrograph images of the two HDFGs (a) grating lines of the two component gratings are parallel; (b) grating lines of the two component gratings are not parallel during two exposures.
Figure 4 shows the microscope photographs of the HDFGs, which is magnified by a factor of about 500. As shown in Fig. 4, d′ is the period of the beat grating. The photographs correspond to two cases of HDFGs. One is the two component gratings with parallel grating lines during two exposures as shown in Fig. 4(a), the other is the two component gratings whose grating lines are not parallel as shown in Fig. 4(b).
3. Diffraction characteristic of HDFGs
Let us consider a HDFG which reflectance e r(x) is given by
where n i is the refractive index of incident medium, and l is the length of grating line. If the HDFG is used in the air, the refractive index of the incident medium n i = 1. Based on the Fourier transform method, the grating spectra of a HDFG can be written as
Where α is the incident angle, β(p,q) is the diffraction angle, p and q are integers. The grating spectra are described by the two integers, p and q. If we consider one term p or q, Eq. (5) reduces to the well-known grating equation.
Then, let us compare the diffraction characteristics of the general single-spatial-frequency grating and HDFG. Figure 5 shows the experimental diffraction patterns of three cases of gratings at normal incidence. The diffraction patterns were taken separately for the limited field of view of the CCD camera. In fact, all of the diffraction angular distributions satisfied the relation of Eq. (5). Figure 5(a) shows the experimental diffraction patterns of the general single-spatial-frequency grating of ν 1. Figures 5(b) and 5(c) show the experimental diffraction patterns of the two HDFGs as shown in Figs. 4(a) and 4(b). For example, the symbol “(-1)” in Fig. 5(a) indicates the location of the diffraction angle of a general grating can be given by, sinβ(-1)=λ(-1/d)=λ(-1·ν 1). The symbol “(-1, 1)” in Fig. 5(b) indicates the location of the diffraction angle of a HDFG can be written as sinβ(-1,1)=λ(-1/d 1+1/d 2)=λ(-ν 1+ν 2).
Fig. 5. Schematic diffraction patterns of the gratings illuminated by a He-Ne laser (a) a general grating with the spatial frequency of ν 1 ; (b) a HDFG which is fabricated by the two component gratings with parallel grating lines during two exposures; (c) a HDFG in which the grating lines of the two component gratings are not parallel during two exposures.
As seen in Figs. 5(b) and 5(c), the symbols “(-1,0), (0,0), and (1,0)” indicate the diffraction plane (I) of one component grating with the spatial frequency of ν 1, its grating lines lie in the plane normal to the diffraction plane (I). Similarly, the symbols “(0,-1), (0,0), and (0,1)” indicate the diffraction plane (II) of the other component grating with the spatial frequency of 2 ν, its grating lines lie in the plane normal to the diffraction plane (II). Therefore, we can easily judge the parallelism of the two component gratings during exposures by the diffraction patterns of the HDFG. If the grating lines of the two exposures are parallel, the diffraction patterns of both component gratings are on the same line as shown in Fig. 5(b), and the diffraction planes of both component gratings are parallel. Otherwise, if the two sets of grating lines are not parallel during the exposures, the dispersion orientations of the two component gratings are not on the same orientation as shown in Fig. 5(c). An angle between the two diffraction planes is introduced.
4. Holographic grating as X-ray beam splitter
Based on the above introduction, we demonstrate an X-ray holographic grating beam splitter as shown in Fig. 6. The upper area of the substrate is a HDFG as the pre-alignment grating. The lower area of the substrate is a common grating with high spatial frequency as the operation grating. The two gratings lie vertically on the same substrate, especially the beat grating of the HDFG with low spatial frequency acts as the pre-alignment grating. As mentioned in section 1, their spatial frequencies and working wavelengths satisfy the relation of ν o λ o = νa λa. For an example, the fabrication and utilization details of a 13.9 nm soft X-ray grating beam splitter are shown in the following statement. The grating beam splitter consists of two gratings, one works at X-ray laser wavelength of 13.9 nm (λo) with the spatial frequency of 1000 lines/mm (νo) as the operation grating, the other works at visible wavelength of 632.8 nm (λa) for pre-aligning the X-ray interferometer. The spatial-frequency of the pre-alignment grating should be 22 lines/mm (νa) to meet the dispersion requirement.
The exposure process plays a key role in the fabrication of an X-ray grating beam splitter by use of holographic method. From Section 2, we can deduce that the X-ray grating beam splitter needs to be exposed twice at least. In the first exposure, the angle of the two interfering beams was set to be 2θo, and the whole grating substrate is exposed with the spatial frequency of νo = 2sinθ o/λ. A 413.1nm (λ) Kr+ laser is used in the interference lithography to produce the grating patterns. To obtain the grating patterns with spatial frequency of 1000lines/mm, the angle of the two interfering beams θo is set to be 11.9°.
If we need to produce a HDFG with a beat frequency of νa, the grating patterns spatial frequency of the second exposure should be changed to ν 2 = νo+νa or ν′2 = νo-νa by changing the angle of the two interfering beams into 2θ 2 or 2θ 2, where ν 2 = 2sinθ 2/λ or ν′2 = 2sinθ′2/λ. If νa is 22lines/mm, ν2 equals 1022 lines/mm or ν′2 equals 978 lines/mm, and accordingly, θ 2 is 12.2° or θ′2 is 11.6°.
In one case, after the first exposure, we shelter the lower part of the substrate and rotate the precision rotation table with the angle Δθ = θ 2-θ 1(0.3°) in the clockwise direction, on which the grating substrate and the Lloyd’s Mirror are fixed and rotated together, round the interference grating fringes to record the grating spatial frequency of ν 2. During the rotation, the relative position of the grating substrate and the Lloyd’s Mirror is fixed.
After the exposure and development process, the HDFG with beat frequency lies on the upper area of the beam splitter substrate as the pre-alignment grating and the general grating with high spatial frequency lies on the lower area of the substrate as the operation grating. In the use of X-ray interferometer, the operation grating and the pre-alignment grating are incident at the same angle α. The 0 and -1 diffraction orders of the operation grating act as the two arms of the X-ray interferometer. The grating equation of the operation grating can be written as,
To make these two gratings have the same dispersion at their own working wavelengths, the (0, 0) and (1,-1) diffraction orders of the pre-alignment grating act as the two arms of the prealignment interferometer. That is to say, according to Eq. (5), the grating equations of the prealignment grating can be written as,
Because of the relation of ν0λ0 = νaλa, sinβ o(-1) = sinβ a(1,1).
Similarly, in the other case, if we shelter the lower part of the substrate and rotate the precision rotation table with the angle Δθ′=θ 1-θ 2(0.3°) in the counterclockwise direction for the second exposure, the spatial frequency of the interference fringes will be ν′2, the (0, 0) and (-1, 1) diffraction orders of the pre-alignment grating are utilized.
The lateral shearing interferometer based on a HDFG was proposed and experimentally demonstrated in Ref. 13. In this paper, we demonstrated the fabrication of a HDFG referred to Ref. 13 for the low-spatial-frequency pre-alignment grating of X-ray holographic grating beam splitter. However, the uses of the HDFG in Ref.13 and this paper are different. In Ref. 13, a double frequency crossed diffraction grating is used to produce the shear for the two orthogonal directions. In this paper, the beat frequency of a HDFG is used to as a low-spatial-frequency grating. The (0, 0) and (-1, 1) (or: (0, 0) and (1, -1)) diffraction orders of the HDFG are utilized as the two arms of the pre-alignment interferometer during the pre-alignment of the soft X-ray interferometer.
Two features of this X-ray holographic beam splitter is as the following,
(1) In the pre-alignment of the X-ray interferometer, the low-spatial-frequency beat grating of a HDFG is used as the pre-alignment grating. Under the same incident angle, the pre-alignment grating and the operation grating have the same dispersion angle if they are illuminated by their own sources. Therefore, through the pre-alignment of the interferometer by the pre-alignment grating, which is a HDFG, the relative positions of the optical elements of the soft X-ray interferometer can be determined.
(2) The pre-alignment grating consists of two component gratings and one of them has the same spatial frequency as that of the operation grating. The diffraction patterns of the prealignment grating include the diffraction of the operation grating. Therefore, according to the diffraction characteristic of a HDFG as shown in Figs. 5(b) and 5(c), we can judge the grating line parallelism of the pre-alignment grating and the operation grating from the diffraction pattern of the pre-alignment grating illuminated by a He-Ne laser.
These features enable the fabrication of a HDFG for the pre-alignment of X-ray laser interferometer as a beam splitter and thus with a simple and fast fabrication process.
As mentioned above, it is the main fabrication process of the soft X-ray holographic grating beam splitter.
After the holographic lithography, the operation grating needs to be etched and deposited Au film to meet the diffraction efficiency requirement. To meet the diffraction efficiency requirement of splitting the soft X-ray laser, the operation grating with laminar profile is designed to have equal efficiency at 0 and -1 diffraction orders. The Au film with the thickness of ~40nm is deposited onto the grating beam splitter to improve its diffraction efficiency. Therefore, in the holographic interference lithography process, the photoresist grating masks of the operation grating should meet the requirements for etching, especially clean troughs and proper duty cycles.
The surface-relief profiles of the pre-alignment grating are sinuous as shown in Fig. 3(b). Because the pre-alignment grating works at the visible or IR spectral range, the pre-alignment grating can exhibit enough efficiency for the interferometer as long as the thickness of the photoresist thickness in the interference lithography is high enough. In our holographic lithography process, the thickness of the photoresist grating is usually about 300~400 nm, which is high enough to satisfy the efficiency requirement. The spatial frequency of the prealignment grating has significant effect on the optical element arrangement of the interferometer. Therefore, it is a key parameter to meet the pre-alignment of the X-ray interferometer requirement. In the holographic lithography, the spatial frequencies of the HDFG during the two exposures are the key things in its process compared with its efficiency.
An X-ray holographic grating beam splitter for X-ray interferometer is shown in Fig. 7 which is fabricated by the above process. Figure 7(a) shows the AFM image of the operation grating, whose spatial frequency is 1000 lines/mm. For the grating working at wavelength of 13.9nm, the operation grating exhibits the efficiency of ~25% at its 0 and 1 diffraction orders at grazing incidence angle of ~8°, whose duty cycle is ~0.40 and groove depth is ~14nm. The operation grating efficiencies were measured by synchrotron radiation (National Synchrotron Radiation Laboratory, Hefei, China).
Fig. 7. Digital camera photo of an X-ray grating beam splitter fabricated by holographic method (a) AFM image of groove profile of the operation grating with 1000 lines/mm.
5. Discussion
The holographic method of fabricating grating beam splitters for soft X-ray M-Z interferometers is described in this paper. The main feature of the method is the use of beat grating with low spatial frequency of a HDFG as the pre-alignment grating of the X-ray interferometer. In general, it is easier to keep the relations of the spatial frequency and parallelism between the pre-alignment grating and the operation grating.
The grating fabrication proposed has two advantages. On the one hand, it is convenient to fabricate a low-spatial-frequency grating by using holographic double frequency grating technique. On the other hand, the grating line parallelism between the pre-alignment grating and the operation grating can be judged by observing the diffraction patterns of the pre-alignment grating directly. Furthermore, the grating tiling process of the pre-alignment grating and the operation grating can be omitted.
Acknowledgments
The research is supported by the National Nature Science Associate Foundation of China (NSAF No. 10676032) and National 863 Program of China. The authors would like to thank Prof. Wang Zhanshan for his helpful suggestions and discussions. We are grateful to Dr. Hongjun Zhou and Tonglin Huo for their help with the operation grating efficiency measurements on National Synchrotron Radiation Laboratory beam line U27.
References and links
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