Abstract

A two-axis Lloyd’s mirrors interferometer based optical fabrication system was theoretically investigated and constructed for patterning high-uniformity nanoscale crossed grating structures over a large area with a high throughput. The current interferometer was configured with two reflected mirrors and a grating holder, which are placed edge by edge and orthogonal with each other. In such a manner, the two beams reflected from the two mirrors interfere with the incident beam, respectively, forming a crossed grating patterns with only one exposure. Differing from the conventional solution for elimination of unexpected interference between the two reflected beams, a systematical analysis, that is based on the proposed index indicating the non-orthogonality between the two beams at different incident angles, was conducted by using a spatial full polarization tracing method. Without polarization modulation to eliminate the additional interference, an optimal exposure condition with small non-orthogonality between reflected beams was found at a certain incident angle range, while the two required interferences to construct cross grating still remain high. A pattern period of ∼1 µm-level crossed grating structure could be obtained through balancing the structure area and the non-orthogonality. Finally, the exposure setup with orthogonal two-axis Lloyd’s mirrors interferometer is established, and the crossed grating structure with the periods of 1076 nm along X-direction and 1091 nm along Y-direction was successfully fabricated on a silicon substrate via microfabrication technology over a large area of 400 mm2. The uniformity of crossed grating array over the whole area was evaluated by an atomic force microscope, and the standard deviations of structure periods along X- and Y-directions smaller than 0.3% are achieved. It is demonstrated that the orthogonal two-axis Lloyd’s mirrors interferometer based on single-beam single-exposure scheme with non-orthogonality systematic analysis is an effective approach to fabricate crossed grating patterns of 1 µm-level period with high uniformity over a large area.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Periodically patterned structure with micro- and nano-scale in a large area catches burgeoning attention, because this configuration itself with particular material could possess certain functions [13], and also could provide a platform to further grow the functional materials and to realize the powerful functions [46]. Recently, micro- and nano-pillar array structures are widely used in the fields of biological detections of surface modifications by enlarging the contact area with targets [7,8] and nano-photonics like photonic crystals [9], photodetectors [10] and so on. Silicon (Si)-based micro- and nano-pillar structures with high uniformity in a large scale, easily transferred from photoresist (PR) patterning via standard dry etching process, are appealing and essential media components, such as master mask in nano-imprint and two-dimensional (2D) diffraction grating in multi-degree surface encoder [1114] to realize precise positioning and displacement measurement.

Normally, there are three typical technologies to fabricate 2D-grating, including mechanically ruling [15], projection lithography [16], and laser interference lithography (LIL) [1719]. The ruling technique has difficulty in fabricating 2D-graing with sub-micro-pitch due to its limitation in tool size. Also, it is rather time-consuming to fabricate the grating structure over an area larger than 100 mm2. For photolithography, though it is not a big challenge to fabricate 2D crossed grating patterns, it always requires expensive facilities when fabricating a sub-micron pattern feature and simultaneously over a large area. Compared with these two methods abovementioned, a maskless lithography, called holographic interference lithography (HIL), also known in some literatures as holographic exposure, has many characteristics for this requirement, such as flexible adjustment in grating pitch on a sub-micron order, compact optical system with high efficiency, low-cost, high uniformity, and so forth. Thanks to these abovementioned advantages, the HIL technique has been widely employed for fabrication of 2D crossed grating patterns.

Generally speaking, there are mainly two methods of HIL for 2D crossed grating patterns fabrication, classified as double-exposure and single-exposure [13,2029]. Double-beam double-exposure type, as the most popular one, requires two steps for exposure, i.e. rotating the grating substrate by 90 degree for second exposure after finishing the first exposure [13,2022]. It is obvious that optical system of double-exposure with double-beam is vulnerable to the disturbance of environment. Furthermore, asymmetric groove structures of 2D-grating always occur because it is rather difficult to eliminate the influence of first exposure on the second one. Thus, for solving this problem, many researchers recently focused on single-exposure, in which additional mechanical operations like rotating the substrate are not necessary [2328]. Among these single-exposure methods, there are two typical optical systems, named as amplitude-division and wavefront-division. For the amplitude-division single-exposure system, a two-axis diffraction combined with some reflectors forms two double-beam sub-systems, and these two sub-systems share the single light source, so that the 2D pattern can be achieved with only one exposure [24]. But it should be noted that, the amplitude-division type is also based on double-beam system essentially, and it is still vulnerable to the environmental disturbance and additionally needs complicated supplementary optics, such as phase locking loop to adjust the light phase.

In the contrast, the wavefront-division type propagates towards an interferometer in terms of a single beam, which makes the system compact and stable. The single beam is divided into multiple sub-beams by the interferometer that is generally composed of two mirrors (X-mirror and Y-mirror) and a grating holder. The interferometer can be classified into two types, orthogonal type and non-orthogonal type. For the non-orthogonal interferometer based single exposure method, high uniformity 2D grating patterns can be easily fabricated by using a stable polarization modulation technique. However, an inclination occurred in the structure along the depth direction, due to asymmetric projection of the two interference beams, further resulting in a challenge for the pattern transformation [26]. At the same time, orthogonal type was also proposed [29]. In this type, all the interference beams are symmetrically incident into the substrate, ensuring the structure symmetric along the depth direction. However, due to the existing additional interference between reflected beams, the generated interference patterns are also imposed on the substrate, dramatically reducing the pattern uniformity over a large area. What is more, the spatial polarization of multiple divided beams is required to be precisely modeled, so that the unnecessary patterns can be effectively eliminated [30,31]. In our previous research [32], to fabricate high-uniformity crossed grating with ∼1 µm period, an orthogonal two-axis Lloyd’s mirrors interference processes with and without polarization modulation were performed at an incident angle of 71.8 degree, which of the fabricated results show nearly the same morphologies, revealing that there is a unopened story to find the interference relationship between the reflected beams at a certain incident angle.

For solving these problems mentioned above, to fabricate a 2D crossed grating structure with high uniformity over a relatively large area, we thoroughly analyzed the orthogonal type interferometer based optical system with single-beam single-exposure scheme and modeled the spatial state of these beams for precisely tracing the spatial polarization states. Based on the systematical analysis, we found that there is an optimal condition with a certain incident angle that can simultaneously achieve all the above-mentioned targets. In such a condition, the unnecessary patterns can be automatically eliminated without additional optics, which ensures the grating area and no any reduction of the grating uniformity. Finally, based on this point, a fabrication system was constructed, and fabrication experiment and patterns evaluation were carried out to demonstrate the proposal.

2. Principle

The exposure optical path system including a laser source module, a beam shaping module, and an orthogonal two-axis Lloyd’s mirrors module is simply schematic, as shown in Fig. 1(a). Normally, the light source module is composed of a laser source with suitable wavelength, a shutter to control the exposure time and a half-wave plate to adjust the polarization direction of outgoing beam. In the beam shaping module, firstly, the incident beam is enlarged with a certain divergent angle and filtered to cut the edge part of beam by a spatial filter consisting of a focusing objective lens and a pinhole, then the filtered beam is collimated by a collimating lens to adjust the divergent spherical wave to a plane wave. A fan-shaped diaphragm between spatial filter and collimating lens is utilized to block the undesired secondary-deflected sub-beams located in upper side of the beam. Finally, the collimated beam is projected on the orthogonal two-axis Lloyd’s mirrors to make the interference and achieve the crossed patterns on grating substrate.

 figure: Fig. 1.

Fig. 1. Schematics of exposure system (a) and orthogonal two-axis Lloyd’s mirrors (b).

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The orthogonal two-axis Lloyd’s mirrors module consists of X-mirror, Y-mirror and a grating holder, which are perpendicular with each other. A Cartesian coordinate system is established on the orthogonal two-axis Lloyd’s mirrors, and the direction vector of incident beam is located in the plane of OABZ with an angle φ of 45 degree to X-mirror, as shown in Fig. 1(b). Here, the symbol θ represents the angle of incident beam to grating substrate. When the polarized, shaped and collimated beam reaches to the two-axis Lloyd’s mirrors, the projected beam is divided into three sub-beams, including sub-beam 1 as direct beam on grating substrate, sub-beams 2 and 3 as reflected beams (corresponding to X-beam and Y-beam) on grating substrate, respectively.

Normally, to fabricate the crossed grating pattern in the orthogonal two-axis Lloyd’s mirrors interferometer, the direct sub-beam 1 interferes with the reflected X-beam and Y-beam, respectively. However, the reflected X-beam and Y-beam also interfere with each other, resulting in additional pattern on the grating substrate. In order to eliminate this influence to achieve high-uniformity pattern, a three-dimension (3D) polarized light tracing theory is systematically analyzed to establish the transformation model of polarization vector. Because only the electric vector in the light wave plays a part in patterning the PR, in this paper, the magnetic vector in the light wave is ignored. The electric vector e of a light wave is expressed as

$${\boldsymbol e}({\boldsymbol r},t) = A \cdot {\boldsymbol E} \cdot {\mathop{\rm Re}\nolimits} \{ \exp [i(k{\boldsymbol k} \cdot {\boldsymbol r}) - \omega t + \delta ]\},$$
where A is the amplitude, E is the unit of polarization vector, i is the imaginary unit, k is the wave-number, k is the unit of wave vector, r is the position vector in the coordinate, ω is the angular frequency, t is the time value and δ is the phase, respectively. When the light waves of three sub-beams are superimposed, the equation is presented as
$$\begin{array}{l} {\boldsymbol e}({\boldsymbol r},t) = \sum\limits_{m\textrm{ = 1}}^3 {{A_m}{{\boldsymbol E}_m} \cdot {\mathop{\rm Re}\nolimits} \{ \exp [i(k{{\boldsymbol k}_m} \cdot {\boldsymbol r}) - \omega t + {\delta _m}]} \} ,\end{array}$$
where m is the positive integer of 1, 2 and 3. Thereby, the intensity distribution I of three sub-beams superposition can be obtained as
$$I({\boldsymbol r}) = \sum\limits_{m = 1}^3 {A_m^2} + 2\sum\limits_{m = \textrm{1}}^\textrm{2} {\sum\limits_{m < n} {{\mathop{\rm Re}\nolimits} \{{{{\boldsymbol e}_m}({\boldsymbol r},t) \cdot {{[{{\boldsymbol e}_n}({\boldsymbol r},t)]}^\ast }} \}} },$$
and then is expanded as
$$I({\boldsymbol r}) = \sum\limits_{m = 1}^3 {A_m^2} + 2\sum\limits_{m = \textrm{1}}^\textrm{2} {\sum\limits_{m < n} {{A_m}{A_n}{{\boldsymbol E}_m} \cdot {{\boldsymbol E}_n}\cos [({{\boldsymbol k}_m} - {{\boldsymbol k}_n}) \cdot {\boldsymbol r} + ({\delta _m} - {\delta _n})]} },$$
where * means the conjugate function and n is the positive integer of 2 and 3. From the intensity distribution expression, it can be seen that there are three sets (e1 (r, t)e2(r, t)*, e1 (r, t)e3(r, t)*, e2 (r, t)e3(r, t)*) of interference fringes in the interference field, formed by interfering sub-beam 1 with X-beam in a direction parallel to X-axis, sub-beam 1 with Y-beam in a direction parallel to Y-axis, and X-beam with Y-beam in a direction of 45 degree to X-axis, respectively. Among them, the first two sets of interference fringes with mutual orthogonality are able to fabricate the required crossed grating pattern, however, the third set of interference fringes, as additional interference abovementioned, would be superimposed on the first two sets of fringes, causing distortion of interference field and un-uniformity of crossed grating structure. To qualitatively analyze interference intensity between two light waves, the non-orthogonality Dnm (D12, D13 and D23) of polarization vector is introduced, expressed as
$${D_{nm}} = |{{{\boldsymbol E}_n}{\boldsymbol E}_m^ \ast } |,$$
where ${\boldsymbol E}_m^ \ast $ is the conjugate matrix of Em. If the non-orthogonality of polarization vector between two beams meets the following condition
$${D_{nm}} = |{{{\boldsymbol E}_n}{\boldsymbol E}_m^\ast } |= |{{{\boldsymbol E}_m}{\boldsymbol E}_n^\ast } |= 0,$$
the polarization state of the two beams means orthogonal polarization. To meet our requirements, it is desirable that the reflected X-beam and Y-beam are close to or even with an orthogonal polarization state, while the states of the direct sub-beam 1 with reflected X-beam and Y-beam maintain non-orthogonality polarization simultaneously. Therefore, this study adopts a 3D polarized light tracing method to solve the previous problem, and a polarization tracing model of the orthogonal two-axis Lloyd’s mirrors is established, as shown in Fig. 2.

To integrate the polarization vectors of three beams including directed sub-beam 1 and reflected X beam and Y beam in the same coordinate possibly, two coordinates are built as local coordinate (p, s, k) and global coordinate (X, Y, Z). Because the sub-beams 2 and 3 are reflected by X-mirror and Y-mirror, respectively, the transforming relations of polarization vector are expressed as

$${{\boldsymbol s}^{\prime}}_m\textrm{ = }{{\boldsymbol s}_m}\textrm{ = }\frac{{{{\boldsymbol k}_1} \times {{\boldsymbol k}_m}}}{{|{{{\boldsymbol k}_1} \times {{\boldsymbol k}_m}} |}},$$
$${{\boldsymbol p}_m} = {{\boldsymbol s}_m} \times {{\boldsymbol k}_1},$$
$${{\boldsymbol p}^{\prime}}_m = {{\boldsymbol s}_m} \times {{\boldsymbol k}_m}.$$

 figure: Fig. 2.

Fig. 2. Polarization tracing model of orthogonal two-axis Lloyd’s mirrors.

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According to the transformation relationship of coordinate system, the units of wave vectors of the direct sub-beam 1, reflected X-beam and reflected Y-beam can be obtained as follows

$${{\boldsymbol k}_1}\textrm{ = }\left( {\begin{array}{c} { - \cos \theta \cos \varphi }\\ { - \cos \theta \sin \varphi }\\ { - \sin \theta } \end{array}} \right),$$
$${{\boldsymbol k}_2}\textrm{ = }\left( {\begin{array}{c} {\cos \theta \cos \varphi }\\ { - \cos \theta \sin \varphi }\\ { - \sin \theta } \end{array}} \right),$$
$${{\boldsymbol k}_3}\textrm{ = }\left( {\begin{array}{c} { - \cos \theta \cos \varphi }\\ {\cos \theta \sin \varphi }\\ { - \sin \theta } \end{array}} \right).$$
By combining Eqs. (4) and (12), the equation of periods of crossed grating along X- and Y- directions can be derived as
$${g_{_{12}}} = {g_{_{13}}} = \frac{{2\pi }}{{|{{{\boldsymbol k}_1} - {{\boldsymbol k}_2}} |}} = \frac{{2\pi }}{{|{{{\boldsymbol k}_1} - {{\boldsymbol k}_\textrm{3}}} |}}\textrm{ = }\frac{\lambda }{{\sqrt 2 \cos \theta }},$$
where λ is the wavelength of incident beam, g12 and g13 represent the interference periods of sub-beam 1 with X-beam and Y-beam, respectively. For sub-beam 1, the coordinate components s and p in its local coordinate are expressed as
$${{\boldsymbol s}_1} = \left( {\begin{array}{c} {\textrm{ - }\sin \phi }\\ {\cos \phi }\\ 0 \end{array}} \right),$$
$${p_1} = \left( {\begin{array}{c} { - \sin \theta \cos \phi }\\ { - \sin \theta \sin \phi }\\ {\cos \theta } \end{array}} \right).$$
To calculate the non-orthogonality of polarization vectors between three sub-beams, firstly, the original polarization vectors of three sub-beams need to be standardized in the global coordinate with a transform matrix O1-out, expressed as
$${O_{\textrm{1 - }out}} = \left( {\begin{array}{ccc} {{{\boldsymbol p}_1}}&{{{\boldsymbol s}_1}}&{{{\boldsymbol k}_1}} \end{array}} \right) = \left( {\begin{array}{ccc} {{p_{\textrm{1 - }x}}}&{{s_{\textrm{1 - }x}}}&{{k_{\textrm{1 - }x}}}\\ {{p_{\textrm{1 - }y}}}&{{s_{\textrm{1 - }y}}}&{{k_{\textrm{1 - }y}}}\\ {{p_{\textrm{1 - }z}}}&{{s_{\textrm{1 - }z}}}&{{k_{\textrm{1 - }z}}} \end{array}} \right) = \left( {\begin{array}{ccc} { - \sin \theta \cos \phi }&{ - \sin \phi }&{ - \cos \theta \cos \varphi }\\ { - \sin \theta \sin \phi }&{\cos \phi }&{ - \cos \theta \sin \varphi }\\ {\cos \theta }&0&{ - \sin \theta } \end{array}} \right).$$
Because the sub-beam 1 is directly projected on the grating substrate, the original polarization vector E1-in of sub-beam 1 can be transformed into the global coordinate system only by the transform matrix O1-out with a polarization vector E1, expressed as
$${{\boldsymbol E}_1}\textrm{ = }{O_{\textrm{1 - }out}} \cdot {{\boldsymbol E}_{1\textrm{ - }in}}.$$
However, due to sub-beams 2 and 3 reflected by X-mirror and Y-mirror, respectively, the polarization vectors of sub-beams 2 and 3 need to be transformed from standardized state to its local coordinate state with an input transformation matrix $O_{\textrm{2 - }in}^{ - 1}$ , and then from local coordinate to global coordinate with an output transformation matrix ${O_{\textrm{2 - }out}}$. With an example of sub-beam 2, the input and output transformation matrixes are respectively expressed as
$$\begin{array}{l} O_{\textrm{2 - }in}^{ - 1} = {\left( {\begin{array}{ccc} {{{\boldsymbol p}_2}}&{{{\boldsymbol s}_2}}&{{{\boldsymbol k}_1}} \end{array}} \right)^T}\\ = \left( {\begin{array}{ccc} {\frac{{ - {{\cos }^2}\theta \sin \varphi \cos \varphi }}{{\sqrt {{{\sin }^2}\theta \textrm{ + }{{\cos }^2}\theta {{\cos }^2}\varphi } }}}&{\sqrt {{{\sin }^2}\theta \textrm{ + }{{\cos }^2}\theta {{\cos }^2}\varphi } }&{\frac{{ - \sin \theta \cos \theta \sin \varphi }}{{\sqrt {{{\sin }^2}\theta \textrm{ + }{{\cos }^2}\theta {{\cos }^2}\varphi } }}}\\ {\frac{{\sin \theta }}{{\sqrt {{{\sin }^2}\theta \textrm{ + }{{\cos }^2}\theta {{\cos }^2}\varphi } }}}&0&{\frac{{ - \cos \theta \cos \varphi }}{{\sqrt {{{\sin }^2}\theta \textrm{ + }{{\cos }^2}\theta {{\cos }^2}\varphi } }}}\\ { - \cos \theta \cos \varphi }&{ - \cos \theta \sin \varphi }&{ - \sin \theta } \end{array}} \right) \end{array},$$
$$\begin{array}{l} {O_{\textrm{2 - }out}} = \left( {\begin{array}{ccc} {{{\boldsymbol p}^{\prime}}_2}&{{{\boldsymbol s}^{\prime}}_2}&{{{\boldsymbol k}_2}} \end{array}} \right)\\ \textrm{ = }\left( {\begin{array}{ccc} {\frac{{{{\cos }^2}\theta \sin \varphi \cos \varphi }}{{\sqrt {{{\sin }^2}\theta \textrm{ + }{{\cos }^2}\theta {{\cos }^2}\varphi } }}}&{\frac{{\sin \theta }}{{\sqrt {{{\sin }^2}\theta \textrm{ + }{{\cos }^2}\theta {{\cos }^2}\varphi } }}}&{ - \cos \theta \cos \varphi }\\ {\sqrt {{{\sin }^2}\theta \textrm{ + }{{\cos }^2}\theta {{\cos }^2}\varphi } }&0&{\cos \theta \sin \varphi }\\ {\frac{{\sin \theta \cos \theta \sin \varphi }}{{\sqrt {{{\sin }^2}\theta \textrm{ + }{{\cos }^2}\theta {{\cos }^2}\varphi } }}}&{\frac{{ - \cos \theta \cos \varphi }}{{\sqrt {{{\sin }^2}\theta \textrm{ + }{{\cos }^2}\theta {{\cos }^2}\varphi } }}}&{ - \sin \theta } \end{array}} \right) \end{array}.$$
Finally, the polarization vector of sub-beam 2 in the global coordinate system can be expressed as
$${{\boldsymbol E}_2} = {O_{2\textrm{ - }out}} \cdot {J_{r\textrm{ - }X}} \cdot O_{\textrm{2 - }in}^{ - 1} \cdot {O_{\textrm{1 - }out}} \cdot {{\boldsymbol E}_{\textrm{2 - }in}}.$$
The Jones matrix Jr-X of X-mirror appeared in Eq. (18) is expressed by
$${J_{r\textrm{ - }X}} = \left( {\begin{array}{ccc} {{r_{p\textrm{ - }X}}}&0&0\\ 0&{{r_{s\textrm{ - }X}}}&0\\ 0&0&1 \end{array}} \right),$$
where the amplitude reflection coefficients rp-X and rs-X of the X-mirror are calculated according to Fresnel's law. And so on, the polarization vector of sub-beam 3 in the global coordinate system can be represented as
$${{\boldsymbol E}_3} = {O_{3\textrm{ - }out}} \cdot {J_{r\textrm{ - }Y}} \cdot O_{\textrm{3 - }in}^{ - 1} \cdot {O_{\textrm{1 - }out}} \cdot {{\boldsymbol E}_{\textrm{3 - }in}},$$
where Jr-Y is the Jones matrix of Y-mirror. Finally, the non-orthogonality of polarization vector between reflected X-beam and Y-beam can be derived as
$${D_{23}} = |{{{\boldsymbol E}_2}{\boldsymbol E}_3^\ast } |.$$
Based on the derived Eq. (23), the non-orthogonality (D12, D13 and D23) of polarization vector between the directed sub-beam 1, reflected X-beam and reflected Y-beam is curved in Fig. 3.

It can be seen that the non-orthogonality of polarization vector between reflected X-beam and Y-beam could get the smallest value of ∼3.0×10−5 at an incident angle of 74.4 degree, meanwhile, the non-orthogonality of polarization vector between the directed sub-beam 1 with reflected X-beam and Y-beam is still with a high value. Thus, at this particular incident angle, the polarization vectors of X-beam and Y-beam are closest to the orthogonal polarization state, meaning that the additional interference could be minimal and a relatively symmetric crossed grating structure could be achieved. It is also revealed that a certain range of incident angle around the optimum value is acceptable by balancing the non-orthogonality of polarization vector, grating area and grating pitch.

3. Experiments and results

To demonstrate the feasibility of the proposed orthogonal two-axis Lloyd’s mirror interferometer, the exposure system mainly including a light source, a spatial filter, diaphragms, collimating lens and a two-axis Lloyd’s mirrors unit is established on a shockproof platform, as shown in Fig. 4. Here, a commercial single-longitudinal-mode He-Cd laser (KIMMON KOHA) with a wavelength of 442 nm and a power output of 180 mW is adopted as the light source, through considering the stability and polarization of light source properties, PR sensitivity to laser wavelength and low-complexity optical alignment to laser visibility. The output laser light, continuously reflected by mirror-1, mirror-2, mirror-3 and mirror-4, is incident into a beam-expanding unit, which is composed of an objective lens with a working distance of 3.3 mm, a pinhole with a diameter of 10 µm and a collimating lens with a working distance of 400 mm. Then, the expanded beam with a diameter of 130 mm is projected onto the two-axis Lloyd’s mirror unit. An annular diaphragm and a fan-shaped diaphragm, located between the pinhole and collimating lens, are used to filter the edge part of the beam and block the unwanted secondary-reflection beam from Lloyd’s X-mirror to Y-mirror or otherwise, respectively. After collimation, the beam geometry is divided into three parts: sub-beam 1 (①), sub-beam 2 (②) and sub-beam 3 (③), which are incident to grating substrate, X-mirror and Y-mirror, respectively. The two-axis Lloyd’s mirror unit is consisted of three components including X-mirror, Y-mirror and a grating holder, which are perpendicular with each other. The two Lloyd’s mirrors coated with aluminum (Al) film have a surface flatness of λ/10 and a reflectivity of 95% in the currently used wavelength range. The rotation of the two-axis Lloyd’s mirror unit could be adjusted precisely, to perform the interference within the optimum incident angle range.

 figure: Fig. 3.

Fig. 3. Curve of non-orthogonality of polarization vector with the incident angle.

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 figure: Fig. 4.

Fig. 4. Experiment setup of exposure system with amplified two-axis Lloyd’s mirrors.

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A single-crystal (100) Si substrate with a size of 20×20 mm2, a thickness of 625 µm and a surface roughness of 1nm is employed as the grating substrate, aiming to transfer the PR-based micro-pattern into Si substrate by microfabrication technology of inductive coupled plasma-reactive ion etching (ICP-RIE). Before PR spin-coating process, the Si substrate, cleaned with a mixture of H2SO4 and H2O2 (v:v 2:1), is dehydrated on a hot plate with 140℃ for 30 min to improve the surface adhesion with PR. A positive PR of S1805, diluted by PGMEA with volume ratio 1:1, is spin-coated (1000 rpm×3 sec→6000 rpm×30 sec) on the Si substrate to form a thin thickness of ∼170 nm. After soft-baking on a hot plate with a 100℃ for 3 min, the spin-coated Si substrate is placed onto the three beams interference area of the substrate of two-axis Lloyd’s mirror unit. The two-axis Lloyd’s mirror unit is rotated with a certain angle of 71.8°, to make sure the 2D grating structure with a grating pitch of ∼1 µm-level. After an exposure dose of 117 mJ is applied to get uniform pillar structure, the development is performed by immersing the Si substrate into TMAH 2.3% for 13 sec. Then, the Si substrate is rinsed with de-ionized water and blow-dried by N2 gas. A hard-baking on a hot plate with a 110℃ for 10 min is performed to solidify the PR patterning on Si substrate.

To transfer the PR-based micro-pillar-pattern into Si substrate with vertical and smooth sidewalls, the ICP-RIE with Bosch-process is performed with detailed parameters shown in Table 1. Then, the remaining PR on Si substrate is removed by stripper. Finally, a 150 nm thickness of Al film is sputtered on the surface of Si grating substrate, to improve the reflectivity for the further application of grating ruler. Figure 5(a) shows the optical image of the finally fabricated Si substrate with a size of ∼20×20 mm2. Several small areas located in both bottom corners show black, due to un-blocked beams making the additional superposition interference on the Si substrate. Figure 5(b) shows the orientation of the fabricated 20×20 mm2 2D grating located in the two-axis Lloyd’s mirrors configuration. The 20×20 mm2 2D grating is equally divided into 16 small meshes named from (a) to (p) with a size of 5×5 mm2, where the 16 labelled positions will be scanned by the atomic force microscopy (AFM).

 figure: Fig. 5.

Fig. 5. Optical image of finally fabricated 2D grating substrate (a) and orientation of 2D grating substrate in two-axis Lloyd’s mirrors configuration with 16 identified positions measured by AFM (b).

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Tables Icon

Table 1. Parameters of ICP-RIE with Bosch-process.

To evaluate the whole surface morphology characteristics of the fabricated 2D grating substrate, an AFM (Bruker Dimension Icon) is used with a peak tapping mode. As the main setting parameters of AFM, the aspect ratio of AFM tip with Si3N4 material, the peak force setpoint and peak force amplitude are set as 1, 0.1 and 300 nm, respectively. As shown in Fig. 6, 16 positions equidistantly distributed in the whole 20×20 mm2 2D grating, corresponding to the 16 AFM images from (a) to (p), are scanned by AFM with a height sensor data density of 512×512 in an area of 10×10 µm2. From the scanned 16 AFM images, it shows that the fabricated 2D Si-grating with pillar structure possesses high pattern uniformity in an area as large as 20×20 mm2. The quantitative analysis, including 2D grating periods in X-axis and Y-axis directions, and duty cycle of pillar structure on overall consistency, will be evaluated in the following.

 figure: Fig. 6.

Fig. 6. AFM images of the fabricated 2D grating: 16 images from (a) to (p) corresponding to the scanning positions with equal distance located in the whole 20×20 mm2 2D grating.

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Based on the AFM scanning data, the uniformities of 2D grating periods are analyzed in X-axis and Y-axis directions, respectively. Table 2 shows the average values of grating periods in each scanned position, which is corresponding to the sequence of Fig. 6 from (a) to (p). The average grating periods between the wholly measured 16 positions in X-axis and Y-axis directions are calculated as 1076 nm and 1091 nm, and the corresponding standard deviations are as small as 0.23% and 0.29%, respectively. Based on the Eq. (13) of grating period, the error of incident angle plays an important role to affect the varieties of grating period, resulting in the periods larger than what expected. The initial position of two-axis Lloyd’s mirrors should be precisely calibrated to achieve the grating period with high resolution. The grating periods in X-axis and Y-axis directions show high uniformity in the whole area of 20×20 mm2, respectively. However, the grating period difference between X-axis and Y-axis directions is ∼15 nm. The possible reason is that the two Lloyd’s mirrors are not kept with orthogonality due to small loosening of the fixtures during long-term placement.

Tables Icon

Table 2. 2D grating periods analysis in X-axis and Y-axis directions.

To intuitively see the fabricated pillar shape of 2D grating, the AFM data (Fig. 6(a)) of scanning stroke in X-axis direction versus depth is graphed, as shown in Fig. 7. The fabricated pillars with a height of ∼230 nm shows a relatively smooth and vertical sidewall. Based on the good parameters in ICP-RIE Bosch-process, there is no obvious scalloping structure on the sidewall of micro-pattern pillars. Table 3 shows the pillar duty cycle analysis of scanned 16 positions in 20×20 mm2 2D grating. Because the sidewall of pillar is a little tilted, the diameter value of pillar in the middle position is selected as the evaluation data, which is expressed in Fig. 7. The average value of pillar duty cycle and standard deviation is calculated as 53.2% and 3.1%, respectively. From the measurement results, the duty cycles of pillars become larger gradually from lower-left corner to upper-right corner in the 2D grating. Because the distribution of the exposure density is uneven onto the whole 20×20 mm2 substrate, resulting in the variation of the pillar duty cycle. In the future, the uniformity of pillar duty cycle around the whole 20×20 mm2 area could be further improved by adjusting the uniformity of exposure density. Additionally, some small pitting appears on the top of Si pillars, the possible reason is that the thickness of PR on Si substrate is a little thin, resulting in nearly no remaining PR after ICP-RIE process.

 figure: Fig. 7.

Fig. 7. Scanning data graph of grating pillar-structure and schematic of pillar duty circle measurement standard.

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Tables Icon

Table 3. Pillar duty cycle analysis in 20×20 mm2 2D grating.

4. Conclusions

In this work, we theoretically and experimentally investigated an orthogonal two-axis Lloyd’s mirrors interferometer based on single-beam single-exposure scheme, to fabricate nanoscale crossed grating structures with high uniformity over a large area. The interferometer consists of two reflected mirrors and a grating holder, which are perpendicular with each other, and a single incident collimated beam can be divided into multiple sub-beams by the interferometer. A non-orthogonality analysis of polarization vectors between the divided sub-beams is introduced to precisely trace the spatial polarization states. Based on the established systematical analysis, an optimal exposure condition with small non-orthogonality between reflected beams was found at a certain incident angle range of ∼74.4 degree, which could eliminate the unnecessary additional interference patterns on the grating substrate. A pattern constant of 1 µm-level crossed grating structure could be obtained through balancing the structure area and non-orthogonality.

Then, the exposure setup with orthogonal two-axis Lloyd’s mirrors is established, based on the single-beam single-exposure with non-orthogonality consideration. An incident angle of 71.8° is selected by considering both the required grating period and the influence of additional interference. With a proper exposure dose, the 2D grating with pillar array, is firstly recorded by a PR layer coated onto a Si substrate, and then transferred into the Si substrate by ICP-RIE. From the AFM scanning data, the 2D grating with pillar array over a large area of 400 mm2 is fabricated with high grating-period consistencies of 1076 nm along X-direction and 1091 nm along Y-direction, and the standard deviations of grating periods along X- and Y-directions are less than 0.3%. The uniformity of pillar duty cycle with a standard deviation of 3.1% could be further improved by adjusting the uniformity of exposure density. It is demonstrated that the orthogonal two-axis Lloyd’s mirrors interferometer based on single-beam single-exposure scheme with non-orthogonality consideration is an effective approach to fabricate nanoscale crossed grating patterns of 1 µm-level period with high uniformity over a large area at a high throughput.

Funding

National Natural Science Foundation of China (61905129); Shenzhen Fundamental Research Funding (JCYJ20170817160808432, ,JCYJ20180508152013054); Natural Science Foundation of Guangdong Province (2018A030313748).

Acknowledgments

Part of this research work was performed at the Testing Technology Center of Materials and Devices of Tsinghua Shenzhen International Graduate School of Tsinghua University.

Disclosures

The authors declare no conflicts of interest.

References

1. X. Chen, F. Yang, C. Zhang, J. Zhou, and L. Guo, “Large-Area High Aspect Ratio Plasmonic Interference Lithography Utilizing a Single High-k Mode,” ACS Nano 10(4), 4039–4045 (2016). [CrossRef]  

2. G. Liang, X. Chen, Z. Wen, G. Chen, and L. Guo, “Super-resolution photolithography using dielectric photonic crystal,” Opt. Lett. 44(5), 1182–1185 (2019). [CrossRef]  

3. X. Yin, H. Zhu, H. Guo, M. Deng, T. Xu, Z. Gong, X. Li, Z. Hang, C. Wu, H. Li, S. Chen, L. Zhou, and L. Chen, “Hyperbolic Metamaterial Devices for Wavefront Manipulation,” Laser Photonics Rev. 13(1), 1800081 (2019). [CrossRef]  

4. G. A. Ozin and S. M. Yang, “The race for the photonic chip: Colloidal crystal assembly in silicon wafers,” Adv. Funct. Mater. 11(2), 95–104 (2001). [CrossRef]  

5. Y. Xia, Y. Yin, Y. Lu, and J. McLellan, “Template-assisted self-assembly of spherical colloids into complex and controllable structures,” Adv. Funct. Mater. 13(12), 907–918 (2003). [CrossRef]  

6. D. Xia, A. Biswas, D. Li, and S. R. J. Brueck, “Directed self-assembly of silica nanoparticles into nanometer-scale patterned surfaces using spin-coating,” Adv. Mater. 16(16), 1427–1432 (2004). [CrossRef]  

7. T. Kan, K. Matsumoto, and I. Shimoyama, “Nano-pillar structure for sensitivity enhancement of SPR sensor,” presented at the 15th International Conference on Solid-State Sensors, Actuators and Microsystems. Transducers 2009, Denver, CO, USA, 21-25 June 2009.

8. D. R. Lincoln, J. J. Charlton, N. A. Hatab, B. Skyberg, N. V. Lavrik, I. I. Kravchenko, J. A. Bradshaw, and M. J. Sepaniak, “Surface Modification of Silicon Pillar Arrays To Enhance Fluorescence Detection of Uranium and DNA,” ACS Omega 2(10), 7313–7319 (2017). [CrossRef]  

9. D. Yang, C. Li, C. Wang, Y. Ji, and Q. Quan, “High Figure of Merit Fano Resonance in 2-D Defect-Free Pillar Array Photonic Crystal for Refractive Index Sensing,” IEEE Photonics J. 8(6), 1–14 (2016). [CrossRef]  

10. P. Senanayake, C.-H. Hung, J. Shapiro, A. Scofield, A. Lin, B. S. Williams, and D. L. Huffaker, “3D Nanopillar optical antenna photodetectors,” Opt. Express 20(23), 25489–25496 (2012). [CrossRef]  

11. W. Gao, S. W. Kim, H. Bosse, H. Haitjema, Y. Chen, X. D. Lu, W. Knapp, A. Weckenmann, W. T. Estler, and H. Kunzmann, “Measurement technologies for precision positioning,” CIRP Ann. 64(2), 773–796 (2015). [CrossRef]  

12. W. Gao and A. Kimura, “A three-axis displacement sensor with nanometric resolution,” CIRP Ann. 56(1), 529–532 (2007). [CrossRef]  

13. A. Kimura, W. Gao, W. Kim, K. Hosono, Y. Shimizu, L. Shi, and L. Zeng, “A sub-nanometric three-axis surface encoder with short-period planar gratings for stage motion measurement,” Precis. Eng. 36(4), 576–585 (2012). [CrossRef]  

14. X. Li, W. Gao, H. S. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013). [CrossRef]  

15. W. Gao, T. Araki, S. Kiyono, Y. Okazaki, and M. Yamanaka, “Precision nano-fabrication and evaluation of a large area sinusoidal grid surface for a surface encoder,” Precis. Eng. 27(3), 289–298 (2003). [CrossRef]  

16. C. J. Richard, “Introduction to Microelectronic fabrication: Volume 5 of modular series on solid state devices,” Am. Scientist 77(3), 301–302 (1989).

17. H. Wolferen and L. Abelmann, Lithography: Principles, Processes and Materials (Nova Science, 2011).

18. S. R. J. Brueck, “Optical and interferometric lithography - Nanotechnology enablers,” Proc. IEEE 93(10), 1704–1721 (2005). [CrossRef]  

19. D. Xia, Z. Ku, S. C. Lee, and S. R. J. Brueck, “Nanostructures and Functional Materials Fabricated by Interferometric Lithography,” Adv. Mater. 23(2), 147–179 (2011). [CrossRef]  

20. X. Mao and L. Zeng, “Design and fabrication of crossed gratings with multiple zero-reference marks for planar encoders,” Meas. Sci. Technol. 29(2), 025204 (2018). [CrossRef]  

21. I. Byun and J. Kim, “Cost-effective laser interference lithography using a 405 nm AlInGaN semiconductor laser,” J. Micromech. Microeng. 20(5), 055024 (2010). [CrossRef]  

22. H. Korre, C. P. Fucetola, J. A. Johnson, and K. K. Berggren, “Development of a simple, compact, low-cost interference lithography system,” J. Vac. Sci. Technol., B: Nanotechnol. Microelectron.: Mater., Process., Meas., Phenom. 28(6), C6Q20–C6Q24 (2010). [CrossRef]  

23. C. Lin, S. Yan, and F. You, “Fabrication and characterization of short-period double-layer cross-grating with holographic lithography,” Opt. Commun. 383, 17–25 (2017). [CrossRef]  

24. J. K. Chua and V. M. Murukeshan, “Patterning of two-dimensional nanoscale features using grating-based multiple beams interference lithography,” Phys. Scr. 80(1), 015401 (2009). [CrossRef]  

25. H. Zhou and L. Zeng, “Method to fabricate orthogonal crossed gratings based on a dual Lloyd’s mirror interferometer,” Opt. Commun. 360, 68–72 (2016). [CrossRef]  

26. X. Li, W. Gao, Y. Shimizu, and S. Ito, “A two-axis Lloyd’s mirror interferometer for fabrication of two dimensional diffraction gratings,” CIRP Ann. 63(1), 461–464 (2014). [CrossRef]  

27. Y. Shimizu, R. Aihara, Z. Ren, Y. Chen, S. Ito, and W. Gao, “Influences of misalignment errors of optical components in an orthogonal two-axis Lloyd’s mirror interferometer,” Opt. Express 24(24), 27521–27535 (2016). [CrossRef]  

28. Y. Shimizu, R. Aihara, K. Mano, C. Chen, Y. Chen, X. Chen, and W. Gao, “Design and testing of a compact non-orthogonal two-axis Lloyd’s mirror interferometer for fabrication of large-area two-dimensional scale gratings,” Precis. Eng. 52, 138–151 (2018). [CrossRef]  

29. M. Vala and J. Homola, “Flexible method based on four-beam interference lithography for fabrication of large areas of perfectly periodic plasmonic arrays,” Opt. Express 22(15), 18778–18789 (2014). [CrossRef]  

30. X. Chen, Z. Ren, Y. Shimizu, Y. Chen, and W. Gao, “Optimal polarization modulation for orthogonal two-axis Lloyd’s mirror interference lithography,” Opt. Express 25(19), 22237–22252 (2017). [CrossRef]  

31. X. Chen, Y. Shimizu, C. Chen, Y. Chen, and W. Gao, “Generalized method for probing ideal initial polarization states in multibeam Lloyd’s mirror interference lithography of 2D scale gratings,” J. Vac. Sci. Technol., B: Nanotechnol. Microelectron.: Mater., Process., Meas., Phenom. 36(2), 021601 (2018). [CrossRef]  

32. X. Li, H. Lu, Q. Zhou, G. Wu, K. Ni, and X. Wang, “An Orthogonal Type Two-Axis Lloyd’s Mirror for Holographic Fabrication of Two-Dimensional Planar Scale Gratings with Large Area,” Appl. Sci. 8(11), 2283 (2018). [CrossRef]  

References

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  1. X. Chen, F. Yang, C. Zhang, J. Zhou, and L. Guo, “Large-Area High Aspect Ratio Plasmonic Interference Lithography Utilizing a Single High-k Mode,” ACS Nano 10(4), 4039–4045 (2016).
    [Crossref]
  2. G. Liang, X. Chen, Z. Wen, G. Chen, and L. Guo, “Super-resolution photolithography using dielectric photonic crystal,” Opt. Lett. 44(5), 1182–1185 (2019).
    [Crossref]
  3. X. Yin, H. Zhu, H. Guo, M. Deng, T. Xu, Z. Gong, X. Li, Z. Hang, C. Wu, H. Li, S. Chen, L. Zhou, and L. Chen, “Hyperbolic Metamaterial Devices for Wavefront Manipulation,” Laser Photonics Rev. 13(1), 1800081 (2019).
    [Crossref]
  4. G. A. Ozin and S. M. Yang, “The race for the photonic chip: Colloidal crystal assembly in silicon wafers,” Adv. Funct. Mater. 11(2), 95–104 (2001).
    [Crossref]
  5. Y. Xia, Y. Yin, Y. Lu, and J. McLellan, “Template-assisted self-assembly of spherical colloids into complex and controllable structures,” Adv. Funct. Mater. 13(12), 907–918 (2003).
    [Crossref]
  6. D. Xia, A. Biswas, D. Li, and S. R. J. Brueck, “Directed self-assembly of silica nanoparticles into nanometer-scale patterned surfaces using spin-coating,” Adv. Mater. 16(16), 1427–1432 (2004).
    [Crossref]
  7. T. Kan, K. Matsumoto, and I. Shimoyama, “Nano-pillar structure for sensitivity enhancement of SPR sensor,” presented at the 15th International Conference on Solid-State Sensors, Actuators and Microsystems. Transducers 2009, Denver, CO, USA, 21-25 June 2009.
  8. D. R. Lincoln, J. J. Charlton, N. A. Hatab, B. Skyberg, N. V. Lavrik, I. I. Kravchenko, J. A. Bradshaw, and M. J. Sepaniak, “Surface Modification of Silicon Pillar Arrays To Enhance Fluorescence Detection of Uranium and DNA,” ACS Omega 2(10), 7313–7319 (2017).
    [Crossref]
  9. D. Yang, C. Li, C. Wang, Y. Ji, and Q. Quan, “High Figure of Merit Fano Resonance in 2-D Defect-Free Pillar Array Photonic Crystal for Refractive Index Sensing,” IEEE Photonics J. 8(6), 1–14 (2016).
    [Crossref]
  10. P. Senanayake, C.-H. Hung, J. Shapiro, A. Scofield, A. Lin, B. S. Williams, and D. L. Huffaker, “3D Nanopillar optical antenna photodetectors,” Opt. Express 20(23), 25489–25496 (2012).
    [Crossref]
  11. W. Gao, S. W. Kim, H. Bosse, H. Haitjema, Y. Chen, X. D. Lu, W. Knapp, A. Weckenmann, W. T. Estler, and H. Kunzmann, “Measurement technologies for precision positioning,” CIRP Ann. 64(2), 773–796 (2015).
    [Crossref]
  12. W. Gao and A. Kimura, “A three-axis displacement sensor with nanometric resolution,” CIRP Ann. 56(1), 529–532 (2007).
    [Crossref]
  13. A. Kimura, W. Gao, W. Kim, K. Hosono, Y. Shimizu, L. Shi, and L. Zeng, “A sub-nanometric three-axis surface encoder with short-period planar gratings for stage motion measurement,” Precis. Eng. 36(4), 576–585 (2012).
    [Crossref]
  14. X. Li, W. Gao, H. S. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013).
    [Crossref]
  15. W. Gao, T. Araki, S. Kiyono, Y. Okazaki, and M. Yamanaka, “Precision nano-fabrication and evaluation of a large area sinusoidal grid surface for a surface encoder,” Precis. Eng. 27(3), 289–298 (2003).
    [Crossref]
  16. C. J. Richard, “Introduction to Microelectronic fabrication: Volume 5 of modular series on solid state devices,” Am. Scientist 77(3), 301–302 (1989).
  17. H. Wolferen and L. Abelmann, Lithography: Principles, Processes and Materials (Nova Science, 2011).
  18. S. R. J. Brueck, “Optical and interferometric lithography - Nanotechnology enablers,” Proc. IEEE 93(10), 1704–1721 (2005).
    [Crossref]
  19. D. Xia, Z. Ku, S. C. Lee, and S. R. J. Brueck, “Nanostructures and Functional Materials Fabricated by Interferometric Lithography,” Adv. Mater. 23(2), 147–179 (2011).
    [Crossref]
  20. X. Mao and L. Zeng, “Design and fabrication of crossed gratings with multiple zero-reference marks for planar encoders,” Meas. Sci. Technol. 29(2), 025204 (2018).
    [Crossref]
  21. I. Byun and J. Kim, “Cost-effective laser interference lithography using a 405 nm AlInGaN semiconductor laser,” J. Micromech. Microeng. 20(5), 055024 (2010).
    [Crossref]
  22. H. Korre, C. P. Fucetola, J. A. Johnson, and K. K. Berggren, “Development of a simple, compact, low-cost interference lithography system,” J. Vac. Sci. Technol., B: Nanotechnol. Microelectron.: Mater., Process., Meas., Phenom. 28(6), C6Q20–C6Q24 (2010).
    [Crossref]
  23. C. Lin, S. Yan, and F. You, “Fabrication and characterization of short-period double-layer cross-grating with holographic lithography,” Opt. Commun. 383, 17–25 (2017).
    [Crossref]
  24. J. K. Chua and V. M. Murukeshan, “Patterning of two-dimensional nanoscale features using grating-based multiple beams interference lithography,” Phys. Scr. 80(1), 015401 (2009).
    [Crossref]
  25. H. Zhou and L. Zeng, “Method to fabricate orthogonal crossed gratings based on a dual Lloyd’s mirror interferometer,” Opt. Commun. 360, 68–72 (2016).
    [Crossref]
  26. X. Li, W. Gao, Y. Shimizu, and S. Ito, “A two-axis Lloyd’s mirror interferometer for fabrication of two dimensional diffraction gratings,” CIRP Ann. 63(1), 461–464 (2014).
    [Crossref]
  27. Y. Shimizu, R. Aihara, Z. Ren, Y. Chen, S. Ito, and W. Gao, “Influences of misalignment errors of optical components in an orthogonal two-axis Lloyd’s mirror interferometer,” Opt. Express 24(24), 27521–27535 (2016).
    [Crossref]
  28. Y. Shimizu, R. Aihara, K. Mano, C. Chen, Y. Chen, X. Chen, and W. Gao, “Design and testing of a compact non-orthogonal two-axis Lloyd’s mirror interferometer for fabrication of large-area two-dimensional scale gratings,” Precis. Eng. 52, 138–151 (2018).
    [Crossref]
  29. M. Vala and J. Homola, “Flexible method based on four-beam interference lithography for fabrication of large areas of perfectly periodic plasmonic arrays,” Opt. Express 22(15), 18778–18789 (2014).
    [Crossref]
  30. X. Chen, Z. Ren, Y. Shimizu, Y. Chen, and W. Gao, “Optimal polarization modulation for orthogonal two-axis Lloyd’s mirror interference lithography,” Opt. Express 25(19), 22237–22252 (2017).
    [Crossref]
  31. X. Chen, Y. Shimizu, C. Chen, Y. Chen, and W. Gao, “Generalized method for probing ideal initial polarization states in multibeam Lloyd’s mirror interference lithography of 2D scale gratings,” J. Vac. Sci. Technol., B: Nanotechnol. Microelectron.: Mater., Process., Meas., Phenom. 36(2), 021601 (2018).
    [Crossref]
  32. X. Li, H. Lu, Q. Zhou, G. Wu, K. Ni, and X. Wang, “An Orthogonal Type Two-Axis Lloyd’s Mirror for Holographic Fabrication of Two-Dimensional Planar Scale Gratings with Large Area,” Appl. Sci. 8(11), 2283 (2018).
    [Crossref]

2019 (2)

X. Yin, H. Zhu, H. Guo, M. Deng, T. Xu, Z. Gong, X. Li, Z. Hang, C. Wu, H. Li, S. Chen, L. Zhou, and L. Chen, “Hyperbolic Metamaterial Devices for Wavefront Manipulation,” Laser Photonics Rev. 13(1), 1800081 (2019).
[Crossref]

G. Liang, X. Chen, Z. Wen, G. Chen, and L. Guo, “Super-resolution photolithography using dielectric photonic crystal,” Opt. Lett. 44(5), 1182–1185 (2019).
[Crossref]

2018 (4)

Y. Shimizu, R. Aihara, K. Mano, C. Chen, Y. Chen, X. Chen, and W. Gao, “Design and testing of a compact non-orthogonal two-axis Lloyd’s mirror interferometer for fabrication of large-area two-dimensional scale gratings,” Precis. Eng. 52, 138–151 (2018).
[Crossref]

X. Chen, Y. Shimizu, C. Chen, Y. Chen, and W. Gao, “Generalized method for probing ideal initial polarization states in multibeam Lloyd’s mirror interference lithography of 2D scale gratings,” J. Vac. Sci. Technol., B: Nanotechnol. Microelectron.: Mater., Process., Meas., Phenom. 36(2), 021601 (2018).
[Crossref]

X. Li, H. Lu, Q. Zhou, G. Wu, K. Ni, and X. Wang, “An Orthogonal Type Two-Axis Lloyd’s Mirror for Holographic Fabrication of Two-Dimensional Planar Scale Gratings with Large Area,” Appl. Sci. 8(11), 2283 (2018).
[Crossref]

X. Mao and L. Zeng, “Design and fabrication of crossed gratings with multiple zero-reference marks for planar encoders,” Meas. Sci. Technol. 29(2), 025204 (2018).
[Crossref]

2017 (3)

C. Lin, S. Yan, and F. You, “Fabrication and characterization of short-period double-layer cross-grating with holographic lithography,” Opt. Commun. 383, 17–25 (2017).
[Crossref]

D. R. Lincoln, J. J. Charlton, N. A. Hatab, B. Skyberg, N. V. Lavrik, I. I. Kravchenko, J. A. Bradshaw, and M. J. Sepaniak, “Surface Modification of Silicon Pillar Arrays To Enhance Fluorescence Detection of Uranium and DNA,” ACS Omega 2(10), 7313–7319 (2017).
[Crossref]

X. Chen, Z. Ren, Y. Shimizu, Y. Chen, and W. Gao, “Optimal polarization modulation for orthogonal two-axis Lloyd’s mirror interference lithography,” Opt. Express 25(19), 22237–22252 (2017).
[Crossref]

2016 (4)

X. Chen, F. Yang, C. Zhang, J. Zhou, and L. Guo, “Large-Area High Aspect Ratio Plasmonic Interference Lithography Utilizing a Single High-k Mode,” ACS Nano 10(4), 4039–4045 (2016).
[Crossref]

Y. Shimizu, R. Aihara, Z. Ren, Y. Chen, S. Ito, and W. Gao, “Influences of misalignment errors of optical components in an orthogonal two-axis Lloyd’s mirror interferometer,” Opt. Express 24(24), 27521–27535 (2016).
[Crossref]

H. Zhou and L. Zeng, “Method to fabricate orthogonal crossed gratings based on a dual Lloyd’s mirror interferometer,” Opt. Commun. 360, 68–72 (2016).
[Crossref]

D. Yang, C. Li, C. Wang, Y. Ji, and Q. Quan, “High Figure of Merit Fano Resonance in 2-D Defect-Free Pillar Array Photonic Crystal for Refractive Index Sensing,” IEEE Photonics J. 8(6), 1–14 (2016).
[Crossref]

2015 (1)

W. Gao, S. W. Kim, H. Bosse, H. Haitjema, Y. Chen, X. D. Lu, W. Knapp, A. Weckenmann, W. T. Estler, and H. Kunzmann, “Measurement technologies for precision positioning,” CIRP Ann. 64(2), 773–796 (2015).
[Crossref]

2014 (2)

X. Li, W. Gao, Y. Shimizu, and S. Ito, “A two-axis Lloyd’s mirror interferometer for fabrication of two dimensional diffraction gratings,” CIRP Ann. 63(1), 461–464 (2014).
[Crossref]

M. Vala and J. Homola, “Flexible method based on four-beam interference lithography for fabrication of large areas of perfectly periodic plasmonic arrays,” Opt. Express 22(15), 18778–18789 (2014).
[Crossref]

2013 (1)

X. Li, W. Gao, H. S. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013).
[Crossref]

2012 (2)

A. Kimura, W. Gao, W. Kim, K. Hosono, Y. Shimizu, L. Shi, and L. Zeng, “A sub-nanometric three-axis surface encoder with short-period planar gratings for stage motion measurement,” Precis. Eng. 36(4), 576–585 (2012).
[Crossref]

P. Senanayake, C.-H. Hung, J. Shapiro, A. Scofield, A. Lin, B. S. Williams, and D. L. Huffaker, “3D Nanopillar optical antenna photodetectors,” Opt. Express 20(23), 25489–25496 (2012).
[Crossref]

2011 (1)

D. Xia, Z. Ku, S. C. Lee, and S. R. J. Brueck, “Nanostructures and Functional Materials Fabricated by Interferometric Lithography,” Adv. Mater. 23(2), 147–179 (2011).
[Crossref]

2010 (2)

I. Byun and J. Kim, “Cost-effective laser interference lithography using a 405 nm AlInGaN semiconductor laser,” J. Micromech. Microeng. 20(5), 055024 (2010).
[Crossref]

H. Korre, C. P. Fucetola, J. A. Johnson, and K. K. Berggren, “Development of a simple, compact, low-cost interference lithography system,” J. Vac. Sci. Technol., B: Nanotechnol. Microelectron.: Mater., Process., Meas., Phenom. 28(6), C6Q20–C6Q24 (2010).
[Crossref]

2009 (1)

J. K. Chua and V. M. Murukeshan, “Patterning of two-dimensional nanoscale features using grating-based multiple beams interference lithography,” Phys. Scr. 80(1), 015401 (2009).
[Crossref]

2007 (1)

W. Gao and A. Kimura, “A three-axis displacement sensor with nanometric resolution,” CIRP Ann. 56(1), 529–532 (2007).
[Crossref]

2005 (1)

S. R. J. Brueck, “Optical and interferometric lithography - Nanotechnology enablers,” Proc. IEEE 93(10), 1704–1721 (2005).
[Crossref]

2004 (1)

D. Xia, A. Biswas, D. Li, and S. R. J. Brueck, “Directed self-assembly of silica nanoparticles into nanometer-scale patterned surfaces using spin-coating,” Adv. Mater. 16(16), 1427–1432 (2004).
[Crossref]

2003 (2)

W. Gao, T. Araki, S. Kiyono, Y. Okazaki, and M. Yamanaka, “Precision nano-fabrication and evaluation of a large area sinusoidal grid surface for a surface encoder,” Precis. Eng. 27(3), 289–298 (2003).
[Crossref]

Y. Xia, Y. Yin, Y. Lu, and J. McLellan, “Template-assisted self-assembly of spherical colloids into complex and controllable structures,” Adv. Funct. Mater. 13(12), 907–918 (2003).
[Crossref]

2001 (1)

G. A. Ozin and S. M. Yang, “The race for the photonic chip: Colloidal crystal assembly in silicon wafers,” Adv. Funct. Mater. 11(2), 95–104 (2001).
[Crossref]

1989 (1)

C. J. Richard, “Introduction to Microelectronic fabrication: Volume 5 of modular series on solid state devices,” Am. Scientist 77(3), 301–302 (1989).

Abelmann, L.

H. Wolferen and L. Abelmann, Lithography: Principles, Processes and Materials (Nova Science, 2011).

Aihara, R.

Y. Shimizu, R. Aihara, K. Mano, C. Chen, Y. Chen, X. Chen, and W. Gao, “Design and testing of a compact non-orthogonal two-axis Lloyd’s mirror interferometer for fabrication of large-area two-dimensional scale gratings,” Precis. Eng. 52, 138–151 (2018).
[Crossref]

Y. Shimizu, R. Aihara, Z. Ren, Y. Chen, S. Ito, and W. Gao, “Influences of misalignment errors of optical components in an orthogonal two-axis Lloyd’s mirror interferometer,” Opt. Express 24(24), 27521–27535 (2016).
[Crossref]

Araki, T.

W. Gao, T. Araki, S. Kiyono, Y. Okazaki, and M. Yamanaka, “Precision nano-fabrication and evaluation of a large area sinusoidal grid surface for a surface encoder,” Precis. Eng. 27(3), 289–298 (2003).
[Crossref]

Berggren, K. K.

H. Korre, C. P. Fucetola, J. A. Johnson, and K. K. Berggren, “Development of a simple, compact, low-cost interference lithography system,” J. Vac. Sci. Technol., B: Nanotechnol. Microelectron.: Mater., Process., Meas., Phenom. 28(6), C6Q20–C6Q24 (2010).
[Crossref]

Biswas, A.

D. Xia, A. Biswas, D. Li, and S. R. J. Brueck, “Directed self-assembly of silica nanoparticles into nanometer-scale patterned surfaces using spin-coating,” Adv. Mater. 16(16), 1427–1432 (2004).
[Crossref]

Bosse, H.

W. Gao, S. W. Kim, H. Bosse, H. Haitjema, Y. Chen, X. D. Lu, W. Knapp, A. Weckenmann, W. T. Estler, and H. Kunzmann, “Measurement technologies for precision positioning,” CIRP Ann. 64(2), 773–796 (2015).
[Crossref]

Bradshaw, J. A.

D. R. Lincoln, J. J. Charlton, N. A. Hatab, B. Skyberg, N. V. Lavrik, I. I. Kravchenko, J. A. Bradshaw, and M. J. Sepaniak, “Surface Modification of Silicon Pillar Arrays To Enhance Fluorescence Detection of Uranium and DNA,” ACS Omega 2(10), 7313–7319 (2017).
[Crossref]

Brueck, S. R. J.

D. Xia, Z. Ku, S. C. Lee, and S. R. J. Brueck, “Nanostructures and Functional Materials Fabricated by Interferometric Lithography,” Adv. Mater. 23(2), 147–179 (2011).
[Crossref]

S. R. J. Brueck, “Optical and interferometric lithography - Nanotechnology enablers,” Proc. IEEE 93(10), 1704–1721 (2005).
[Crossref]

D. Xia, A. Biswas, D. Li, and S. R. J. Brueck, “Directed self-assembly of silica nanoparticles into nanometer-scale patterned surfaces using spin-coating,” Adv. Mater. 16(16), 1427–1432 (2004).
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I. Byun and J. Kim, “Cost-effective laser interference lithography using a 405 nm AlInGaN semiconductor laser,” J. Micromech. Microeng. 20(5), 055024 (2010).
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D. R. Lincoln, J. J. Charlton, N. A. Hatab, B. Skyberg, N. V. Lavrik, I. I. Kravchenko, J. A. Bradshaw, and M. J. Sepaniak, “Surface Modification of Silicon Pillar Arrays To Enhance Fluorescence Detection of Uranium and DNA,” ACS Omega 2(10), 7313–7319 (2017).
[Crossref]

Chen, C.

Y. Shimizu, R. Aihara, K. Mano, C. Chen, Y. Chen, X. Chen, and W. Gao, “Design and testing of a compact non-orthogonal two-axis Lloyd’s mirror interferometer for fabrication of large-area two-dimensional scale gratings,” Precis. Eng. 52, 138–151 (2018).
[Crossref]

X. Chen, Y. Shimizu, C. Chen, Y. Chen, and W. Gao, “Generalized method for probing ideal initial polarization states in multibeam Lloyd’s mirror interference lithography of 2D scale gratings,” J. Vac. Sci. Technol., B: Nanotechnol. Microelectron.: Mater., Process., Meas., Phenom. 36(2), 021601 (2018).
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Chen, G.

Chen, L.

X. Yin, H. Zhu, H. Guo, M. Deng, T. Xu, Z. Gong, X. Li, Z. Hang, C. Wu, H. Li, S. Chen, L. Zhou, and L. Chen, “Hyperbolic Metamaterial Devices for Wavefront Manipulation,” Laser Photonics Rev. 13(1), 1800081 (2019).
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Chen, S.

X. Yin, H. Zhu, H. Guo, M. Deng, T. Xu, Z. Gong, X. Li, Z. Hang, C. Wu, H. Li, S. Chen, L. Zhou, and L. Chen, “Hyperbolic Metamaterial Devices for Wavefront Manipulation,” Laser Photonics Rev. 13(1), 1800081 (2019).
[Crossref]

Chen, X.

G. Liang, X. Chen, Z. Wen, G. Chen, and L. Guo, “Super-resolution photolithography using dielectric photonic crystal,” Opt. Lett. 44(5), 1182–1185 (2019).
[Crossref]

X. Chen, Y. Shimizu, C. Chen, Y. Chen, and W. Gao, “Generalized method for probing ideal initial polarization states in multibeam Lloyd’s mirror interference lithography of 2D scale gratings,” J. Vac. Sci. Technol., B: Nanotechnol. Microelectron.: Mater., Process., Meas., Phenom. 36(2), 021601 (2018).
[Crossref]

Y. Shimizu, R. Aihara, K. Mano, C. Chen, Y. Chen, X. Chen, and W. Gao, “Design and testing of a compact non-orthogonal two-axis Lloyd’s mirror interferometer for fabrication of large-area two-dimensional scale gratings,” Precis. Eng. 52, 138–151 (2018).
[Crossref]

X. Chen, Z. Ren, Y. Shimizu, Y. Chen, and W. Gao, “Optimal polarization modulation for orthogonal two-axis Lloyd’s mirror interference lithography,” Opt. Express 25(19), 22237–22252 (2017).
[Crossref]

X. Chen, F. Yang, C. Zhang, J. Zhou, and L. Guo, “Large-Area High Aspect Ratio Plasmonic Interference Lithography Utilizing a Single High-k Mode,” ACS Nano 10(4), 4039–4045 (2016).
[Crossref]

Chen, Y.

Y. Shimizu, R. Aihara, K. Mano, C. Chen, Y. Chen, X. Chen, and W. Gao, “Design and testing of a compact non-orthogonal two-axis Lloyd’s mirror interferometer for fabrication of large-area two-dimensional scale gratings,” Precis. Eng. 52, 138–151 (2018).
[Crossref]

X. Chen, Y. Shimizu, C. Chen, Y. Chen, and W. Gao, “Generalized method for probing ideal initial polarization states in multibeam Lloyd’s mirror interference lithography of 2D scale gratings,” J. Vac. Sci. Technol., B: Nanotechnol. Microelectron.: Mater., Process., Meas., Phenom. 36(2), 021601 (2018).
[Crossref]

X. Chen, Z. Ren, Y. Shimizu, Y. Chen, and W. Gao, “Optimal polarization modulation for orthogonal two-axis Lloyd’s mirror interference lithography,” Opt. Express 25(19), 22237–22252 (2017).
[Crossref]

Y. Shimizu, R. Aihara, Z. Ren, Y. Chen, S. Ito, and W. Gao, “Influences of misalignment errors of optical components in an orthogonal two-axis Lloyd’s mirror interferometer,” Opt. Express 24(24), 27521–27535 (2016).
[Crossref]

W. Gao, S. W. Kim, H. Bosse, H. Haitjema, Y. Chen, X. D. Lu, W. Knapp, A. Weckenmann, W. T. Estler, and H. Kunzmann, “Measurement technologies for precision positioning,” CIRP Ann. 64(2), 773–796 (2015).
[Crossref]

Chua, J. K.

J. K. Chua and V. M. Murukeshan, “Patterning of two-dimensional nanoscale features using grating-based multiple beams interference lithography,” Phys. Scr. 80(1), 015401 (2009).
[Crossref]

Deng, M.

X. Yin, H. Zhu, H. Guo, M. Deng, T. Xu, Z. Gong, X. Li, Z. Hang, C. Wu, H. Li, S. Chen, L. Zhou, and L. Chen, “Hyperbolic Metamaterial Devices for Wavefront Manipulation,” Laser Photonics Rev. 13(1), 1800081 (2019).
[Crossref]

Dian, S.

X. Li, W. Gao, H. S. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013).
[Crossref]

Estler, W. T.

W. Gao, S. W. Kim, H. Bosse, H. Haitjema, Y. Chen, X. D. Lu, W. Knapp, A. Weckenmann, W. T. Estler, and H. Kunzmann, “Measurement technologies for precision positioning,” CIRP Ann. 64(2), 773–796 (2015).
[Crossref]

Fucetola, C. P.

H. Korre, C. P. Fucetola, J. A. Johnson, and K. K. Berggren, “Development of a simple, compact, low-cost interference lithography system,” J. Vac. Sci. Technol., B: Nanotechnol. Microelectron.: Mater., Process., Meas., Phenom. 28(6), C6Q20–C6Q24 (2010).
[Crossref]

Gao, W.

Y. Shimizu, R. Aihara, K. Mano, C. Chen, Y. Chen, X. Chen, and W. Gao, “Design and testing of a compact non-orthogonal two-axis Lloyd’s mirror interferometer for fabrication of large-area two-dimensional scale gratings,” Precis. Eng. 52, 138–151 (2018).
[Crossref]

X. Chen, Y. Shimizu, C. Chen, Y. Chen, and W. Gao, “Generalized method for probing ideal initial polarization states in multibeam Lloyd’s mirror interference lithography of 2D scale gratings,” J. Vac. Sci. Technol., B: Nanotechnol. Microelectron.: Mater., Process., Meas., Phenom. 36(2), 021601 (2018).
[Crossref]

X. Chen, Z. Ren, Y. Shimizu, Y. Chen, and W. Gao, “Optimal polarization modulation for orthogonal two-axis Lloyd’s mirror interference lithography,” Opt. Express 25(19), 22237–22252 (2017).
[Crossref]

Y. Shimizu, R. Aihara, Z. Ren, Y. Chen, S. Ito, and W. Gao, “Influences of misalignment errors of optical components in an orthogonal two-axis Lloyd’s mirror interferometer,” Opt. Express 24(24), 27521–27535 (2016).
[Crossref]

W. Gao, S. W. Kim, H. Bosse, H. Haitjema, Y. Chen, X. D. Lu, W. Knapp, A. Weckenmann, W. T. Estler, and H. Kunzmann, “Measurement technologies for precision positioning,” CIRP Ann. 64(2), 773–796 (2015).
[Crossref]

X. Li, W. Gao, Y. Shimizu, and S. Ito, “A two-axis Lloyd’s mirror interferometer for fabrication of two dimensional diffraction gratings,” CIRP Ann. 63(1), 461–464 (2014).
[Crossref]

X. Li, W. Gao, H. S. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013).
[Crossref]

A. Kimura, W. Gao, W. Kim, K. Hosono, Y. Shimizu, L. Shi, and L. Zeng, “A sub-nanometric three-axis surface encoder with short-period planar gratings for stage motion measurement,” Precis. Eng. 36(4), 576–585 (2012).
[Crossref]

W. Gao and A. Kimura, “A three-axis displacement sensor with nanometric resolution,” CIRP Ann. 56(1), 529–532 (2007).
[Crossref]

W. Gao, T. Araki, S. Kiyono, Y. Okazaki, and M. Yamanaka, “Precision nano-fabrication and evaluation of a large area sinusoidal grid surface for a surface encoder,” Precis. Eng. 27(3), 289–298 (2003).
[Crossref]

Gong, Z.

X. Yin, H. Zhu, H. Guo, M. Deng, T. Xu, Z. Gong, X. Li, Z. Hang, C. Wu, H. Li, S. Chen, L. Zhou, and L. Chen, “Hyperbolic Metamaterial Devices for Wavefront Manipulation,” Laser Photonics Rev. 13(1), 1800081 (2019).
[Crossref]

Guo, H.

X. Yin, H. Zhu, H. Guo, M. Deng, T. Xu, Z. Gong, X. Li, Z. Hang, C. Wu, H. Li, S. Chen, L. Zhou, and L. Chen, “Hyperbolic Metamaterial Devices for Wavefront Manipulation,” Laser Photonics Rev. 13(1), 1800081 (2019).
[Crossref]

Guo, L.

G. Liang, X. Chen, Z. Wen, G. Chen, and L. Guo, “Super-resolution photolithography using dielectric photonic crystal,” Opt. Lett. 44(5), 1182–1185 (2019).
[Crossref]

X. Chen, F. Yang, C. Zhang, J. Zhou, and L. Guo, “Large-Area High Aspect Ratio Plasmonic Interference Lithography Utilizing a Single High-k Mode,” ACS Nano 10(4), 4039–4045 (2016).
[Crossref]

Haitjema, H.

W. Gao, S. W. Kim, H. Bosse, H. Haitjema, Y. Chen, X. D. Lu, W. Knapp, A. Weckenmann, W. T. Estler, and H. Kunzmann, “Measurement technologies for precision positioning,” CIRP Ann. 64(2), 773–796 (2015).
[Crossref]

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X. Yin, H. Zhu, H. Guo, M. Deng, T. Xu, Z. Gong, X. Li, Z. Hang, C. Wu, H. Li, S. Chen, L. Zhou, and L. Chen, “Hyperbolic Metamaterial Devices for Wavefront Manipulation,” Laser Photonics Rev. 13(1), 1800081 (2019).
[Crossref]

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D. R. Lincoln, J. J. Charlton, N. A. Hatab, B. Skyberg, N. V. Lavrik, I. I. Kravchenko, J. A. Bradshaw, and M. J. Sepaniak, “Surface Modification of Silicon Pillar Arrays To Enhance Fluorescence Detection of Uranium and DNA,” ACS Omega 2(10), 7313–7319 (2017).
[Crossref]

Homola, J.

Hosono, K.

A. Kimura, W. Gao, W. Kim, K. Hosono, Y. Shimizu, L. Shi, and L. Zeng, “A sub-nanometric three-axis surface encoder with short-period planar gratings for stage motion measurement,” Precis. Eng. 36(4), 576–585 (2012).
[Crossref]

Huffaker, D. L.

Hung, C.-H.

Ito, S.

Y. Shimizu, R. Aihara, Z. Ren, Y. Chen, S. Ito, and W. Gao, “Influences of misalignment errors of optical components in an orthogonal two-axis Lloyd’s mirror interferometer,” Opt. Express 24(24), 27521–27535 (2016).
[Crossref]

X. Li, W. Gao, Y. Shimizu, and S. Ito, “A two-axis Lloyd’s mirror interferometer for fabrication of two dimensional diffraction gratings,” CIRP Ann. 63(1), 461–464 (2014).
[Crossref]

X. Li, W. Gao, H. S. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013).
[Crossref]

Ji, Y.

D. Yang, C. Li, C. Wang, Y. Ji, and Q. Quan, “High Figure of Merit Fano Resonance in 2-D Defect-Free Pillar Array Photonic Crystal for Refractive Index Sensing,” IEEE Photonics J. 8(6), 1–14 (2016).
[Crossref]

Johnson, J. A.

H. Korre, C. P. Fucetola, J. A. Johnson, and K. K. Berggren, “Development of a simple, compact, low-cost interference lithography system,” J. Vac. Sci. Technol., B: Nanotechnol. Microelectron.: Mater., Process., Meas., Phenom. 28(6), C6Q20–C6Q24 (2010).
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T. Kan, K. Matsumoto, and I. Shimoyama, “Nano-pillar structure for sensitivity enhancement of SPR sensor,” presented at the 15th International Conference on Solid-State Sensors, Actuators and Microsystems. Transducers 2009, Denver, CO, USA, 21-25 June 2009.

Kim, J.

I. Byun and J. Kim, “Cost-effective laser interference lithography using a 405 nm AlInGaN semiconductor laser,” J. Micromech. Microeng. 20(5), 055024 (2010).
[Crossref]

Kim, S. W.

W. Gao, S. W. Kim, H. Bosse, H. Haitjema, Y. Chen, X. D. Lu, W. Knapp, A. Weckenmann, W. T. Estler, and H. Kunzmann, “Measurement technologies for precision positioning,” CIRP Ann. 64(2), 773–796 (2015).
[Crossref]

Kim, W.

A. Kimura, W. Gao, W. Kim, K. Hosono, Y. Shimizu, L. Shi, and L. Zeng, “A sub-nanometric three-axis surface encoder with short-period planar gratings for stage motion measurement,” Precis. Eng. 36(4), 576–585 (2012).
[Crossref]

Kimura, A.

A. Kimura, W. Gao, W. Kim, K. Hosono, Y. Shimizu, L. Shi, and L. Zeng, “A sub-nanometric three-axis surface encoder with short-period planar gratings for stage motion measurement,” Precis. Eng. 36(4), 576–585 (2012).
[Crossref]

W. Gao and A. Kimura, “A three-axis displacement sensor with nanometric resolution,” CIRP Ann. 56(1), 529–532 (2007).
[Crossref]

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W. Gao, T. Araki, S. Kiyono, Y. Okazaki, and M. Yamanaka, “Precision nano-fabrication and evaluation of a large area sinusoidal grid surface for a surface encoder,” Precis. Eng. 27(3), 289–298 (2003).
[Crossref]

Knapp, W.

W. Gao, S. W. Kim, H. Bosse, H. Haitjema, Y. Chen, X. D. Lu, W. Knapp, A. Weckenmann, W. T. Estler, and H. Kunzmann, “Measurement technologies for precision positioning,” CIRP Ann. 64(2), 773–796 (2015).
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H. Korre, C. P. Fucetola, J. A. Johnson, and K. K. Berggren, “Development of a simple, compact, low-cost interference lithography system,” J. Vac. Sci. Technol., B: Nanotechnol. Microelectron.: Mater., Process., Meas., Phenom. 28(6), C6Q20–C6Q24 (2010).
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Kravchenko, I. I.

D. R. Lincoln, J. J. Charlton, N. A. Hatab, B. Skyberg, N. V. Lavrik, I. I. Kravchenko, J. A. Bradshaw, and M. J. Sepaniak, “Surface Modification of Silicon Pillar Arrays To Enhance Fluorescence Detection of Uranium and DNA,” ACS Omega 2(10), 7313–7319 (2017).
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D. Xia, Z. Ku, S. C. Lee, and S. R. J. Brueck, “Nanostructures and Functional Materials Fabricated by Interferometric Lithography,” Adv. Mater. 23(2), 147–179 (2011).
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W. Gao, S. W. Kim, H. Bosse, H. Haitjema, Y. Chen, X. D. Lu, W. Knapp, A. Weckenmann, W. T. Estler, and H. Kunzmann, “Measurement technologies for precision positioning,” CIRP Ann. 64(2), 773–796 (2015).
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D. R. Lincoln, J. J. Charlton, N. A. Hatab, B. Skyberg, N. V. Lavrik, I. I. Kravchenko, J. A. Bradshaw, and M. J. Sepaniak, “Surface Modification of Silicon Pillar Arrays To Enhance Fluorescence Detection of Uranium and DNA,” ACS Omega 2(10), 7313–7319 (2017).
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D. Xia, Z. Ku, S. C. Lee, and S. R. J. Brueck, “Nanostructures and Functional Materials Fabricated by Interferometric Lithography,” Adv. Mater. 23(2), 147–179 (2011).
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D. Yang, C. Li, C. Wang, Y. Ji, and Q. Quan, “High Figure of Merit Fano Resonance in 2-D Defect-Free Pillar Array Photonic Crystal for Refractive Index Sensing,” IEEE Photonics J. 8(6), 1–14 (2016).
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D. Xia, A. Biswas, D. Li, and S. R. J. Brueck, “Directed self-assembly of silica nanoparticles into nanometer-scale patterned surfaces using spin-coating,” Adv. Mater. 16(16), 1427–1432 (2004).
[Crossref]

Li, H.

X. Yin, H. Zhu, H. Guo, M. Deng, T. Xu, Z. Gong, X. Li, Z. Hang, C. Wu, H. Li, S. Chen, L. Zhou, and L. Chen, “Hyperbolic Metamaterial Devices for Wavefront Manipulation,” Laser Photonics Rev. 13(1), 1800081 (2019).
[Crossref]

Li, X.

X. Yin, H. Zhu, H. Guo, M. Deng, T. Xu, Z. Gong, X. Li, Z. Hang, C. Wu, H. Li, S. Chen, L. Zhou, and L. Chen, “Hyperbolic Metamaterial Devices for Wavefront Manipulation,” Laser Photonics Rev. 13(1), 1800081 (2019).
[Crossref]

X. Li, H. Lu, Q. Zhou, G. Wu, K. Ni, and X. Wang, “An Orthogonal Type Two-Axis Lloyd’s Mirror for Holographic Fabrication of Two-Dimensional Planar Scale Gratings with Large Area,” Appl. Sci. 8(11), 2283 (2018).
[Crossref]

X. Li, W. Gao, Y. Shimizu, and S. Ito, “A two-axis Lloyd’s mirror interferometer for fabrication of two dimensional diffraction gratings,” CIRP Ann. 63(1), 461–464 (2014).
[Crossref]

X. Li, W. Gao, H. S. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013).
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D. R. Lincoln, J. J. Charlton, N. A. Hatab, B. Skyberg, N. V. Lavrik, I. I. Kravchenko, J. A. Bradshaw, and M. J. Sepaniak, “Surface Modification of Silicon Pillar Arrays To Enhance Fluorescence Detection of Uranium and DNA,” ACS Omega 2(10), 7313–7319 (2017).
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Lu, H.

X. Li, H. Lu, Q. Zhou, G. Wu, K. Ni, and X. Wang, “An Orthogonal Type Two-Axis Lloyd’s Mirror for Holographic Fabrication of Two-Dimensional Planar Scale Gratings with Large Area,” Appl. Sci. 8(11), 2283 (2018).
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Lu, X. D.

W. Gao, S. W. Kim, H. Bosse, H. Haitjema, Y. Chen, X. D. Lu, W. Knapp, A. Weckenmann, W. T. Estler, and H. Kunzmann, “Measurement technologies for precision positioning,” CIRP Ann. 64(2), 773–796 (2015).
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Y. Shimizu, R. Aihara, K. Mano, C. Chen, Y. Chen, X. Chen, and W. Gao, “Design and testing of a compact non-orthogonal two-axis Lloyd’s mirror interferometer for fabrication of large-area two-dimensional scale gratings,” Precis. Eng. 52, 138–151 (2018).
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T. Kan, K. Matsumoto, and I. Shimoyama, “Nano-pillar structure for sensitivity enhancement of SPR sensor,” presented at the 15th International Conference on Solid-State Sensors, Actuators and Microsystems. Transducers 2009, Denver, CO, USA, 21-25 June 2009.

McLellan, J.

Y. Xia, Y. Yin, Y. Lu, and J. McLellan, “Template-assisted self-assembly of spherical colloids into complex and controllable structures,” Adv. Funct. Mater. 13(12), 907–918 (2003).
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Murukeshan, V. M.

J. K. Chua and V. M. Murukeshan, “Patterning of two-dimensional nanoscale features using grating-based multiple beams interference lithography,” Phys. Scr. 80(1), 015401 (2009).
[Crossref]

Muto, H. S.

X. Li, W. Gao, H. S. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013).
[Crossref]

Ni, K.

X. Li, H. Lu, Q. Zhou, G. Wu, K. Ni, and X. Wang, “An Orthogonal Type Two-Axis Lloyd’s Mirror for Holographic Fabrication of Two-Dimensional Planar Scale Gratings with Large Area,” Appl. Sci. 8(11), 2283 (2018).
[Crossref]

Okazaki, Y.

W. Gao, T. Araki, S. Kiyono, Y. Okazaki, and M. Yamanaka, “Precision nano-fabrication and evaluation of a large area sinusoidal grid surface for a surface encoder,” Precis. Eng. 27(3), 289–298 (2003).
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D. Yang, C. Li, C. Wang, Y. Ji, and Q. Quan, “High Figure of Merit Fano Resonance in 2-D Defect-Free Pillar Array Photonic Crystal for Refractive Index Sensing,” IEEE Photonics J. 8(6), 1–14 (2016).
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Sepaniak, M. J.

D. R. Lincoln, J. J. Charlton, N. A. Hatab, B. Skyberg, N. V. Lavrik, I. I. Kravchenko, J. A. Bradshaw, and M. J. Sepaniak, “Surface Modification of Silicon Pillar Arrays To Enhance Fluorescence Detection of Uranium and DNA,” ACS Omega 2(10), 7313–7319 (2017).
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Shapiro, J.

Shi, L.

A. Kimura, W. Gao, W. Kim, K. Hosono, Y. Shimizu, L. Shi, and L. Zeng, “A sub-nanometric three-axis surface encoder with short-period planar gratings for stage motion measurement,” Precis. Eng. 36(4), 576–585 (2012).
[Crossref]

Shimizu, Y.

Y. Shimizu, R. Aihara, K. Mano, C. Chen, Y. Chen, X. Chen, and W. Gao, “Design and testing of a compact non-orthogonal two-axis Lloyd’s mirror interferometer for fabrication of large-area two-dimensional scale gratings,” Precis. Eng. 52, 138–151 (2018).
[Crossref]

X. Chen, Y. Shimizu, C. Chen, Y. Chen, and W. Gao, “Generalized method for probing ideal initial polarization states in multibeam Lloyd’s mirror interference lithography of 2D scale gratings,” J. Vac. Sci. Technol., B: Nanotechnol. Microelectron.: Mater., Process., Meas., Phenom. 36(2), 021601 (2018).
[Crossref]

X. Chen, Z. Ren, Y. Shimizu, Y. Chen, and W. Gao, “Optimal polarization modulation for orthogonal two-axis Lloyd’s mirror interference lithography,” Opt. Express 25(19), 22237–22252 (2017).
[Crossref]

Y. Shimizu, R. Aihara, Z. Ren, Y. Chen, S. Ito, and W. Gao, “Influences of misalignment errors of optical components in an orthogonal two-axis Lloyd’s mirror interferometer,” Opt. Express 24(24), 27521–27535 (2016).
[Crossref]

X. Li, W. Gao, Y. Shimizu, and S. Ito, “A two-axis Lloyd’s mirror interferometer for fabrication of two dimensional diffraction gratings,” CIRP Ann. 63(1), 461–464 (2014).
[Crossref]

X. Li, W. Gao, H. S. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013).
[Crossref]

A. Kimura, W. Gao, W. Kim, K. Hosono, Y. Shimizu, L. Shi, and L. Zeng, “A sub-nanometric three-axis surface encoder with short-period planar gratings for stage motion measurement,” Precis. Eng. 36(4), 576–585 (2012).
[Crossref]

Shimoyama, I.

T. Kan, K. Matsumoto, and I. Shimoyama, “Nano-pillar structure for sensitivity enhancement of SPR sensor,” presented at the 15th International Conference on Solid-State Sensors, Actuators and Microsystems. Transducers 2009, Denver, CO, USA, 21-25 June 2009.

Skyberg, B.

D. R. Lincoln, J. J. Charlton, N. A. Hatab, B. Skyberg, N. V. Lavrik, I. I. Kravchenko, J. A. Bradshaw, and M. J. Sepaniak, “Surface Modification of Silicon Pillar Arrays To Enhance Fluorescence Detection of Uranium and DNA,” ACS Omega 2(10), 7313–7319 (2017).
[Crossref]

Vala, M.

Wang, C.

D. Yang, C. Li, C. Wang, Y. Ji, and Q. Quan, “High Figure of Merit Fano Resonance in 2-D Defect-Free Pillar Array Photonic Crystal for Refractive Index Sensing,” IEEE Photonics J. 8(6), 1–14 (2016).
[Crossref]

Wang, X.

X. Li, H. Lu, Q. Zhou, G. Wu, K. Ni, and X. Wang, “An Orthogonal Type Two-Axis Lloyd’s Mirror for Holographic Fabrication of Two-Dimensional Planar Scale Gratings with Large Area,” Appl. Sci. 8(11), 2283 (2018).
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X. Yin, H. Zhu, H. Guo, M. Deng, T. Xu, Z. Gong, X. Li, Z. Hang, C. Wu, H. Li, S. Chen, L. Zhou, and L. Chen, “Hyperbolic Metamaterial Devices for Wavefront Manipulation,” Laser Photonics Rev. 13(1), 1800081 (2019).
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Wu, G.

X. Li, H. Lu, Q. Zhou, G. Wu, K. Ni, and X. Wang, “An Orthogonal Type Two-Axis Lloyd’s Mirror for Holographic Fabrication of Two-Dimensional Planar Scale Gratings with Large Area,” Appl. Sci. 8(11), 2283 (2018).
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W. Gao, T. Araki, S. Kiyono, Y. Okazaki, and M. Yamanaka, “Precision nano-fabrication and evaluation of a large area sinusoidal grid surface for a surface encoder,” Precis. Eng. 27(3), 289–298 (2003).
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C. Lin, S. Yan, and F. You, “Fabrication and characterization of short-period double-layer cross-grating with holographic lithography,” Opt. Commun. 383, 17–25 (2017).
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D. Yang, C. Li, C. Wang, Y. Ji, and Q. Quan, “High Figure of Merit Fano Resonance in 2-D Defect-Free Pillar Array Photonic Crystal for Refractive Index Sensing,” IEEE Photonics J. 8(6), 1–14 (2016).
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X. Chen, F. Yang, C. Zhang, J. Zhou, and L. Guo, “Large-Area High Aspect Ratio Plasmonic Interference Lithography Utilizing a Single High-k Mode,” ACS Nano 10(4), 4039–4045 (2016).
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G. A. Ozin and S. M. Yang, “The race for the photonic chip: Colloidal crystal assembly in silicon wafers,” Adv. Funct. Mater. 11(2), 95–104 (2001).
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X. Yin, H. Zhu, H. Guo, M. Deng, T. Xu, Z. Gong, X. Li, Z. Hang, C. Wu, H. Li, S. Chen, L. Zhou, and L. Chen, “Hyperbolic Metamaterial Devices for Wavefront Manipulation,” Laser Photonics Rev. 13(1), 1800081 (2019).
[Crossref]

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Y. Xia, Y. Yin, Y. Lu, and J. McLellan, “Template-assisted self-assembly of spherical colloids into complex and controllable structures,” Adv. Funct. Mater. 13(12), 907–918 (2003).
[Crossref]

You, F.

C. Lin, S. Yan, and F. You, “Fabrication and characterization of short-period double-layer cross-grating with holographic lithography,” Opt. Commun. 383, 17–25 (2017).
[Crossref]

Zeng, L.

X. Mao and L. Zeng, “Design and fabrication of crossed gratings with multiple zero-reference marks for planar encoders,” Meas. Sci. Technol. 29(2), 025204 (2018).
[Crossref]

H. Zhou and L. Zeng, “Method to fabricate orthogonal crossed gratings based on a dual Lloyd’s mirror interferometer,” Opt. Commun. 360, 68–72 (2016).
[Crossref]

A. Kimura, W. Gao, W. Kim, K. Hosono, Y. Shimizu, L. Shi, and L. Zeng, “A sub-nanometric three-axis surface encoder with short-period planar gratings for stage motion measurement,” Precis. Eng. 36(4), 576–585 (2012).
[Crossref]

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H. Zhou and L. Zeng, “Method to fabricate orthogonal crossed gratings based on a dual Lloyd’s mirror interferometer,” Opt. Commun. 360, 68–72 (2016).
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X. Chen, F. Yang, C. Zhang, J. Zhou, and L. Guo, “Large-Area High Aspect Ratio Plasmonic Interference Lithography Utilizing a Single High-k Mode,” ACS Nano 10(4), 4039–4045 (2016).
[Crossref]

Zhou, L.

X. Yin, H. Zhu, H. Guo, M. Deng, T. Xu, Z. Gong, X. Li, Z. Hang, C. Wu, H. Li, S. Chen, L. Zhou, and L. Chen, “Hyperbolic Metamaterial Devices for Wavefront Manipulation,” Laser Photonics Rev. 13(1), 1800081 (2019).
[Crossref]

Zhou, Q.

X. Li, H. Lu, Q. Zhou, G. Wu, K. Ni, and X. Wang, “An Orthogonal Type Two-Axis Lloyd’s Mirror for Holographic Fabrication of Two-Dimensional Planar Scale Gratings with Large Area,” Appl. Sci. 8(11), 2283 (2018).
[Crossref]

Zhu, H.

X. Yin, H. Zhu, H. Guo, M. Deng, T. Xu, Z. Gong, X. Li, Z. Hang, C. Wu, H. Li, S. Chen, L. Zhou, and L. Chen, “Hyperbolic Metamaterial Devices for Wavefront Manipulation,” Laser Photonics Rev. 13(1), 1800081 (2019).
[Crossref]

ACS Nano (1)

X. Chen, F. Yang, C. Zhang, J. Zhou, and L. Guo, “Large-Area High Aspect Ratio Plasmonic Interference Lithography Utilizing a Single High-k Mode,” ACS Nano 10(4), 4039–4045 (2016).
[Crossref]

ACS Omega (1)

D. R. Lincoln, J. J. Charlton, N. A. Hatab, B. Skyberg, N. V. Lavrik, I. I. Kravchenko, J. A. Bradshaw, and M. J. Sepaniak, “Surface Modification of Silicon Pillar Arrays To Enhance Fluorescence Detection of Uranium and DNA,” ACS Omega 2(10), 7313–7319 (2017).
[Crossref]

Adv. Funct. Mater. (2)

G. A. Ozin and S. M. Yang, “The race for the photonic chip: Colloidal crystal assembly in silicon wafers,” Adv. Funct. Mater. 11(2), 95–104 (2001).
[Crossref]

Y. Xia, Y. Yin, Y. Lu, and J. McLellan, “Template-assisted self-assembly of spherical colloids into complex and controllable structures,” Adv. Funct. Mater. 13(12), 907–918 (2003).
[Crossref]

Adv. Mater. (2)

D. Xia, A. Biswas, D. Li, and S. R. J. Brueck, “Directed self-assembly of silica nanoparticles into nanometer-scale patterned surfaces using spin-coating,” Adv. Mater. 16(16), 1427–1432 (2004).
[Crossref]

D. Xia, Z. Ku, S. C. Lee, and S. R. J. Brueck, “Nanostructures and Functional Materials Fabricated by Interferometric Lithography,” Adv. Mater. 23(2), 147–179 (2011).
[Crossref]

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C. J. Richard, “Introduction to Microelectronic fabrication: Volume 5 of modular series on solid state devices,” Am. Scientist 77(3), 301–302 (1989).

Appl. Sci. (1)

X. Li, H. Lu, Q. Zhou, G. Wu, K. Ni, and X. Wang, “An Orthogonal Type Two-Axis Lloyd’s Mirror for Holographic Fabrication of Two-Dimensional Planar Scale Gratings with Large Area,” Appl. Sci. 8(11), 2283 (2018).
[Crossref]

CIRP Ann. (3)

X. Li, W. Gao, Y. Shimizu, and S. Ito, “A two-axis Lloyd’s mirror interferometer for fabrication of two dimensional diffraction gratings,” CIRP Ann. 63(1), 461–464 (2014).
[Crossref]

W. Gao, S. W. Kim, H. Bosse, H. Haitjema, Y. Chen, X. D. Lu, W. Knapp, A. Weckenmann, W. T. Estler, and H. Kunzmann, “Measurement technologies for precision positioning,” CIRP Ann. 64(2), 773–796 (2015).
[Crossref]

W. Gao and A. Kimura, “A three-axis displacement sensor with nanometric resolution,” CIRP Ann. 56(1), 529–532 (2007).
[Crossref]

IEEE Photonics J. (1)

D. Yang, C. Li, C. Wang, Y. Ji, and Q. Quan, “High Figure of Merit Fano Resonance in 2-D Defect-Free Pillar Array Photonic Crystal for Refractive Index Sensing,” IEEE Photonics J. 8(6), 1–14 (2016).
[Crossref]

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I. Byun and J. Kim, “Cost-effective laser interference lithography using a 405 nm AlInGaN semiconductor laser,” J. Micromech. Microeng. 20(5), 055024 (2010).
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J. Vac. Sci. Technol., B: Nanotechnol. Microelectron.: Mater., Process., Meas., Phenom. (2)

H. Korre, C. P. Fucetola, J. A. Johnson, and K. K. Berggren, “Development of a simple, compact, low-cost interference lithography system,” J. Vac. Sci. Technol., B: Nanotechnol. Microelectron.: Mater., Process., Meas., Phenom. 28(6), C6Q20–C6Q24 (2010).
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X. Chen, Y. Shimizu, C. Chen, Y. Chen, and W. Gao, “Generalized method for probing ideal initial polarization states in multibeam Lloyd’s mirror interference lithography of 2D scale gratings,” J. Vac. Sci. Technol., B: Nanotechnol. Microelectron.: Mater., Process., Meas., Phenom. 36(2), 021601 (2018).
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Laser Photonics Rev. (1)

X. Yin, H. Zhu, H. Guo, M. Deng, T. Xu, Z. Gong, X. Li, Z. Hang, C. Wu, H. Li, S. Chen, L. Zhou, and L. Chen, “Hyperbolic Metamaterial Devices for Wavefront Manipulation,” Laser Photonics Rev. 13(1), 1800081 (2019).
[Crossref]

Meas. Sci. Technol. (1)

X. Mao and L. Zeng, “Design and fabrication of crossed gratings with multiple zero-reference marks for planar encoders,” Meas. Sci. Technol. 29(2), 025204 (2018).
[Crossref]

Opt. Commun. (2)

C. Lin, S. Yan, and F. You, “Fabrication and characterization of short-period double-layer cross-grating with holographic lithography,” Opt. Commun. 383, 17–25 (2017).
[Crossref]

H. Zhou and L. Zeng, “Method to fabricate orthogonal crossed gratings based on a dual Lloyd’s mirror interferometer,” Opt. Commun. 360, 68–72 (2016).
[Crossref]

Opt. Express (4)

Opt. Lett. (1)

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Precis. Eng. (4)

Y. Shimizu, R. Aihara, K. Mano, C. Chen, Y. Chen, X. Chen, and W. Gao, “Design and testing of a compact non-orthogonal two-axis Lloyd’s mirror interferometer for fabrication of large-area two-dimensional scale gratings,” Precis. Eng. 52, 138–151 (2018).
[Crossref]

A. Kimura, W. Gao, W. Kim, K. Hosono, Y. Shimizu, L. Shi, and L. Zeng, “A sub-nanometric three-axis surface encoder with short-period planar gratings for stage motion measurement,” Precis. Eng. 36(4), 576–585 (2012).
[Crossref]

X. Li, W. Gao, H. S. Muto, Y. Shimizu, S. Ito, and S. Dian, “A six-degree-of-freedom surface encoder for precision positioning of a planar motion stage,” Precis. Eng. 37(3), 771–781 (2013).
[Crossref]

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[Crossref]

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Other (2)

H. Wolferen and L. Abelmann, Lithography: Principles, Processes and Materials (Nova Science, 2011).

T. Kan, K. Matsumoto, and I. Shimoyama, “Nano-pillar structure for sensitivity enhancement of SPR sensor,” presented at the 15th International Conference on Solid-State Sensors, Actuators and Microsystems. Transducers 2009, Denver, CO, USA, 21-25 June 2009.

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Figures (7)

Fig. 1.
Fig. 1. Schematics of exposure system (a) and orthogonal two-axis Lloyd’s mirrors (b).
Fig. 2.
Fig. 2. Polarization tracing model of orthogonal two-axis Lloyd’s mirrors.
Fig. 3.
Fig. 3. Curve of non-orthogonality of polarization vector with the incident angle.
Fig. 4.
Fig. 4. Experiment setup of exposure system with amplified two-axis Lloyd’s mirrors.
Fig. 5.
Fig. 5. Optical image of finally fabricated 2D grating substrate (a) and orientation of 2D grating substrate in two-axis Lloyd’s mirrors configuration with 16 identified positions measured by AFM (b).
Fig. 6.
Fig. 6. AFM images of the fabricated 2D grating: 16 images from (a) to (p) corresponding to the scanning positions with equal distance located in the whole 20×20 mm2 2D grating.
Fig. 7.
Fig. 7. Scanning data graph of grating pillar-structure and schematic of pillar duty circle measurement standard.

Tables (3)

Tables Icon

Table 1. Parameters of ICP-RIE with Bosch-process.

Tables Icon

Table 2. 2D grating periods analysis in X-axis and Y-axis directions.

Tables Icon

Table 3. Pillar duty cycle analysis in 20×20 mm2 2D grating.

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

e ( r , t ) = A E Re { exp [ i ( k k r ) ω t + δ ] } ,
e ( r , t ) = m  = 1 3 A m E m Re { exp [ i ( k k m r ) ω t + δ m ] } ,
I ( r ) = m = 1 3 A m 2 + 2 m = 1 2 m < n Re { e m ( r , t ) [ e n ( r , t ) ] } ,
I ( r ) = m = 1 3 A m 2 + 2 m = 1 2 m < n A m A n E m E n cos [ ( k m k n ) r + ( δ m δ n ) ] ,
D n m = | E n E m | ,
D n m = | E n E m | = | E m E n | = 0 ,
s m  =  s m  =  k 1 × k m | k 1 × k m | ,
p m = s m × k 1 ,
p m = s m × k m .
k 1  =  ( cos θ cos φ cos θ sin φ sin θ ) ,
k 2  =  ( cos θ cos φ cos θ sin φ sin θ ) ,
k 3  =  ( cos θ cos φ cos θ sin φ sin θ ) .
g 12 = g 13 = 2 π | k 1 k 2 | = 2 π | k 1 k 3 |  =  λ 2 cos θ ,
s 1 = (  -  sin ϕ cos ϕ 0 ) ,
p 1 = ( sin θ cos ϕ sin θ sin ϕ cos θ ) .
O 1 -  o u t = ( p 1 s 1 k 1 ) = ( p 1 -  x s 1 -  x k 1 -  x p 1 -  y s 1 -  y k 1 -  y p 1 -  z s 1 -  z k 1 -  z ) = ( sin θ cos ϕ sin ϕ cos θ cos φ sin θ sin ϕ cos ϕ cos θ sin φ cos θ 0 sin θ ) .
E 1  =  O 1 -  o u t E 1  -  i n .
O 2 -  i n 1 = ( p 2 s 2 k 1 ) T = ( cos 2 θ sin φ cos φ sin 2 θ  +  cos 2 θ cos 2 φ sin 2 θ  +  cos 2 θ cos 2 φ sin θ cos θ sin φ sin 2 θ  +  cos 2 θ cos 2 φ sin θ sin 2 θ  +  cos 2 θ cos 2 φ 0 cos θ cos φ sin 2 θ  +  cos 2 θ cos 2 φ cos θ cos φ cos θ sin φ sin θ ) ,
O 2 -  o u t = ( p 2 s 2 k 2 )  =  ( cos 2 θ sin φ cos φ sin 2 θ  +  cos 2 θ cos 2 φ sin θ sin 2 θ  +  cos 2 θ cos 2 φ cos θ cos φ sin 2 θ  +  cos 2 θ cos 2 φ 0 cos θ sin φ sin θ cos θ sin φ sin 2 θ  +  cos 2 θ cos 2 φ cos θ cos φ sin 2 θ  +  cos 2 θ cos 2 φ sin θ ) .
E 2 = O 2  -  o u t J r  -  X O 2 -  i n 1 O 1 -  o u t E 2 -  i n .
J r  -  X = ( r p  -  X 0 0 0 r s  -  X 0 0 0 1 ) ,
E 3 = O 3  -  o u t J r  -  Y O 3 -  i n 1 O 1 -  o u t E 3 -  i n ,
D 23 = | E 2 E 3 | .

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