1. Introduction

Plane varied-line-spacing gratings have applications in monochromators of third-generation synchrotron radiation and in high-resolution tunable spectrographs for x-ray laser linewidth measurements.[1–3] Traditionally, the plane varied-line-spacing grating is fabricated mainly by holographic lithography and mechanical ruling.[1] The advantage of holographic lithography is high efficiency of fabrication; its disadvantage is that it cannot generate some computer-designed patterns or gratings with special line-spacing variation. Therefore interference lithography was introduced to generate plane varied-line-spacing gratings in the MIT Microsystems Technology Laboratories.[4] However, the line spacing produced by this method varies with segment or the variation is not continuous. Mechanical ruling also produces the same segment variation. Low fabrication efficiency and errors introduced by the environment are also disadvantages compared with holographic lithography and interference lithography. We also have tried to get varied-line-spacing grating on plane surfaces by laser writer, but the process control is very difficult. [5–8].

Recently elastic optics or soft lithography have appeared very promising in the fabrication of optical elements.[9–11] Wilbur et al. use elastic membrane to generate gratings with variable uniform line spacing.[10] Whitehead and Clark generate variable-spacing diffraction grating by employing elastomeric surface waves.[11] To obtain continuous diversity variation, in this letter we demonstrate a method that combines interference lithography with soft lithography to fabricate plane varied-line-spacing grating by employing elastic medium with variable section. The advantages of this technique include continuous diversity of variation of line spacing, high fabrication efficiency, low cost, etc.

2. Fabrication and characterization

Fabrication begins with preparation of the compound of polydimethylsiloxane(PDMS), which comes in two parts, a base and a curing agent, and montmorillonite (which is the mineral NaMMT, available on the open market). To improve the physical characteristics of the PDMS, montmorillonite is introduced into the PDMS, which makes it more conducive to producing varied line spacing in fine patterns on gratings. In our experiment, 8% of montmorillonite is used. When the prepolymer of PDMS and the montmorillonite are mixed, they should be agitated adequately. Then they are put in a vacuum container for ≈10 min to reduce any gas bubbles in the compound.

 

Fig. 1. Schematic of the fabrication of variable-line-spacing grating.

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A grating of silicon can be produced by interference lithography or by electron beam direct writing with ion beam etching. After the grating of silicon is steamed by CF ₃(CF ₂) (CH ₂)₂ SiCl ₃ for ≈20 min [Fig. 1(a)], the compound of PDMS and montmorillonite is poured on the grating. The assembly is then put in the oven at 60°C for ≈1h [Fig. 1(b)]. After curing, the PDMS is peeled from the silicon grating, transferring the pattern on the silicon substrate to the PDMS [Fig. 1(c)]. The PDMS is then cut so its vertical section varies according to the desired line-spacing variation of the grating. Equal tension on each end of the PDMS blank will now produce a grating with line spacing varying according to the vertical section variation [Fig. 1(d)].

The formation of the varied line spacing can be explained physically as follows:

 

Fig.2 . implified model of the physics of the deformation.

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Consider the simplified model of the deformation of the elastic blank as shown in Fig. 2. The stress on the vertical section b-b will be

$\overrightarrow{\sigma }=\frac{\overrightarrow{F}}{S}.$

Here F⃗ is the force on the vertical section, S is the area of the vertical section, and σ⃗ is the stress in the vertical section. Now consider the variable area vertical section:

${\overrightarrow{\sigma }}_x=\frac{\overrightarrow{F}}{S_x}$

Here F⃗ is again the force on the vertical section, but S_x is now the variable area of the vertical section, which is a function of the length coordinate x. σ⃗_x is now the variable stress in the vertical section, which is also a function of the length coordinate x.

The line space of period of n will be

$d_n={\int }_{x_{n1}}^{x_{n2}}\frac{{\sigma }_x}{E}dx,$,

where x_n1 is the length coordinate of the start point of the period of n, x_{n 2} is the length coordinate of the end point of the period of n, E is the strain coefficient of the substance, and σ_x is the variable stress of the vertical section in the horizontal direction.

It is obvious that the line space is a function of the area of the vertical section. Therefore the variation of line-spacing goes according to the variation of the area of the vertical section of the substance.

Figure 3 is a microscopic view of the variable-line-spacing grating fabricated in part by the method described here. In this picture, the line spacing varies from 2 µm to 7 µm in the length of 130 µm. Such variation is obvious, and it is very difficult or impractical to produce by traditional interference or holographic lithography. We must note that in the groove direction, or in the non-VLS direction, there may be distortions in the uniformity of the groove spacing due to inhomogeneities in the medium. This variation can be avoided by fine control of processes, including appropriate distribution of the tension.

With the wavelength of 1328 nm, the reflective diffraction efficiency is about 20% for the PDMS, and it is more than 95% after an aluminum film is deposited on the surface of the PDMS. As a mold, the pattern can also be transferred to a traditional silicon substrate. For the significant variation of the line space, this grating can be used in optical position sensors in aviation because it can improve the precision of the sensor.

 

Fig. 3. Microscopic view of the varied-line-spacing grating.

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3. Conclusion

Here we demonstrate a new method of fabricating plane varied-line-spacing grating by employing elastic medium with variable vertical section. The advantages of this technique include continuous variation of line spacing, high fabrication efficiency, and low cost. The variation of line spacing can be obtained according to the variation of area of the vertical section of the grating blank. The variation of the line spacing is obvious, and it is very difficult or impractical to produce it by traditional interference lithography. Through the tensioning of the elastic medium we can in effect transfer the plane grating pattern to the curved surface to form a varied-line-spacing grating. This grating will be used as an optical position sensor in planned aviation projects.