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In this paper, a novel optical encoder enabling the simultaneous measurement of displacement and the position of precision stages is presented. The encoder is composed of an improved single-track scale grating and a compact two-probe reading head. In the scale grating, multiple reference codes are physically superimposed onto the incremental grooves, in contrast to conventional designs, where an additional track is necessary. The distribution of the reference codes follows a specific mathematical algorithm. For the reading head, a two-probe structure is designed to identify the discrete reference codes by means of the superimposition of the codes with a stationary mask and to read the continuous incremental grooves by means of a grating interferometry, respectively. A prototype encoder was designed, constructed and evaluated, and experimental results show that the distance code precision achieved is 0.5 μm, while the linearity error of the linear displacement measurement is less than 0.06%.
© 2016 Optical Society of America
1. Introduction
Due to their competitive cost, robustness and improved resolution, linear encoders are employed for various precision positioning applications, such as precision machine tools, coordinate measuring machines, and semiconductor manufacturing [1–3]. In order to achieve increased production efficiency, the non-sensing motion of the measurement probe or cutting tool has to be minimized. Consequently, absolute-type encoders, able to provide displacement information and at the same time determine the absolute position of the object, have been gradually developed and implemented. For example, more than 60% of linear encoders used in precision machine tools are currently estimated to belong to the absolute types [4, 5].
At present, absolute position information is mainly encoded through two approaches: multiple tracks of standard binary/gray codes [6–8] and ultra-high speed camera imaging of a single absolute track structure [2]. For the multiple tracks-type absolute encoders, codes can be either binary or gray. This approach, although having the advantage of simplicity, requires as many tracks as output bits. For example, a resolution of 0.1 μm over a range of 4 m requires 26 tracks. This is impractical in a manufacturing setting, both in terms of using the scale and in the implementation of the read head. In order to minimize the number of tracks, the M-code method was developed. In this method, a serial track structure is generated on a single track, the pattern of which is unique over each segment of the length measured. This unique pattern provides the absolute position information. An additional incremental track is added to the absolute serial code structure for position interpolation and measurement of incremental displacement [9]. The main advantage of this type of encoder is that the position value is available immediately upon switch-on, and can be obtained at any time by the electronic circuits, which means there is no need to move the axes to obtain the reference position. What is more, when using a combination of these two tracks, we can achieve a resolution of 1 nm over a range of 4240 mm [9]. However, fabrication of the serial track structure over a large length with high precision requires a large and complicated fabrication system, which is only available to large manufacturing facilities. What is more, it is difficult to expand this two-track type absolute encoder to be a two-axis planar encoder for planar position determination except by using two such encoders in a stacked configuration, which will inevitably cause Abbe errors [10].
The second type of absolute encoders that has been developed uses an ultra-high speed camera as the reading head [2]. A single absolute track structure in the scale is illuminated by a light source. The ultra-high speed camera captures images of the track structures, which are then analyzed by a high-speed digital signal processor (DSP) to determine the absolute positions. A resolution of 1 nm for a movement speed of up to 100 m/s over a range of 4295 mm has been achieved. However, implementing this method includes overcoming similar challenges to the binary-type approaches, since fabrication of the serial track structure is also difficult and it is also impractical to expand the structure for planar positioning.
In order to provide alternatives to the current absolute-type encoders so that they can be applied to emerging planar motion stages [11, 12], in this study we propose some modifications to conventional M-code type absolute encoders. Firstly, in contrast to conventional M-code type encoders, the additional serial absolute track is replaced by discretely located reference codes. The reference codes are the same but the distance between every two codes is different. Secondly, the multiple reference codes are not located on an additional track but are physically superimposed onto the incremental grating track to form a compound single track. In this manner, the scale grating can be easily expanded to a biaxial configuration to form a planar encoder. A specially-designed reading head is used, in which a two-probe structure is designed to identify the discrete reference codes through superimposition of the codes with a stationary mask [13, 14] and to read the continuous incremental grooves by means of grating interferometry [15–18], respectively. A prototype encoder was designed, constructed and basic performances are evaluated.
2. Principle
Figure 1 illustrates the principle of the above-mentioned optical encoder, which can perform simultaneous and precise measurement of position and linear displacement of the moving table. The encoder is composed of an improved scale grating and a two-probe reading head. For the scale grating, apart from the incremental grooves with an equal spacing structure like the one found in conventional linear scales, reference codes are physically superimposed onto the grooves. These reference codes are specifically designed according to a mathematical algorithm and the distances between adjacent codes are slightly different.
In the reading head, a laser beam from the laser diode (LD) is divided into two beams by a beam splitter (BS1). One beam passes through BS1 and propagates towards the displacement assembly, while the other is projected onto the scale grating after passing through the reference mask. The beam passing through BS1 is then divided into another two beams by BS2 and projected onto the scale grating and a reference grating, respectively. The reference grating grooves have the same period as those of the incremental grooves of the scale grating. Diffraction beams are generated on the surfaces of the gratings, and the directions of these beams are then changed by mirrors. The beams diffracted from the scale grating pass through BS2 and coincide with those from the reference gratings. These beams interfere with each other and generate interference signals, from which both the incremental displacement along the moving axis (X-axis) and a slight straightness error along the axis vertical to the moving axis (Z-axis) can be obtained simultaneously. The other beam passing through the reference mask is also projected onto the scale grating, in which the reference codes have same pattern with the mask and are fabricated so that when these two codes coincide a pulse signal is obtained. These pulse signals are used to identify the position of the moving table.
Figure 2 shows the optical configuration of the reference reading section. Figure 2(a) shows the sectional views of the scale grating and the reference mask. The scale grating that moves with the motion table has an incremental groove area and a reference code area. In the incremental groove area, an equally spaced line structure is formed by a mechanical engine ruling method or optical interference lithography [19–21]. In the reference code area, contact lithography using a pre-made mask are employed to form a reference code with length D on the incremental grooves. The incremental grooves are not shown in this figure for clarity. The dark area of the reference mask denotes an opaque area where light cannot pass, while slits with different width are transparent. The dark area on the reference code area represents the segments where there are reflective-type incremental grooves, while in the transparent area no light can be reflected.
Figure 2(b) shows the light intensity received by the photodiode (PD), which varies and has pulse characteristics. The laser beam emitted from the LD is collimated by the lens and then directed onto the reference mask by a beam splitter (BS), and then projected onto the scale grating. When the mask has no overlap with the reference code, as shown in Fig. 2(a), the output of PD is unchanged. When the scale grating moves forwards, the light received by the PD begins to change, and only the beams passing through the transparent area of the mask and projected onto the dark area of the reference code are reflected and then pass through the mask and finally detected by the PD. When the mask totally overlaps with the reference code, all the light beams passing through the mask are projected onto the transparent area of the scale grating and there is no light reflected. Under this condition, the light intensity is minimum, which is the condition used to identify the reference position.
During the design of the reference mask and the corresponding code, the number of transparent slits in the mask is a parameter that is needs to be carefully balanced. For reference position identification, the more transparent slits, the greater contrast of the reference signal will be [13]. However, considering the continuous incremental interference signals, when there are more transparent areas, the interference signals will be lower, which will result in a low signal-noise-ratio and low position accuracy. Generally speaking, the number of transparent slits should not be greater than 20% of the width of the reference mask.
Mathematically, the mask and reference code can be described by the matrix a = [a1, a2, …, an], where n is number of the basic units in a reference code/mask, ai = 1 if a transparent slit is located at the i-th position, and ai = 0 elsewhere. The width of every slit is b. The transmittance of a reference mask code can be expressed as the following:
where function rect is the rectangle function, x is the horizontal ordinate of the reference code, while D is the width of every reference code/mask. We use the angular spectrum of plane waves to calculate the propagated beam, which is shown in Fig. 2(b) as the laser beam passing through the mask and projected onto the grating. The field amplitude on the grating is:
Function F{t(x)} is the Fourier transform of the transmittance of the reference mask, is the wavelength of the laser, v is the spatial frequency and z is the distance between the reference mask and the grating. When the laser beam is reflected by the grating, the field amplitude will be modulated by the reference code on the grating. The structure of the reference codes are the same as the reference codes on the mask. The difference is the opaque slits of the reference code reflecting the light; therefore the reflexivity of the grating can be expressed as the following equation:
When the reference code laterally shifts with respect to the mask, the field amplitude of the reflected laser beam is:
where x0 is the relative displacement between the reference mask and the scale grating. Then, the reflected laser beam will be projected onto the reference mask, in which case we use the angular spectrum to analyze the propagated beam, and the field amplitude of the laser beam reflected from the reference mask is:
Finally, the laser beam will pass through the reference mask again and finally arrive at the photodiode (PD), where the field amplitude of the reference pulse signal will be:
In grating measurement systems, the most important parameters of the reference signal are the width of the central peak and the effective signal amplitude [13]. The effective signal amplitude is the height difference between the central and second minimum. The width of the central peak is associated with the system resolution, while the effective amplitude is associated with the stability and robustness of the system. To ensure the stability of the system, the amplitude of the central peak has to be twice than that of the second peak. The resolution of the system is inversely related to the width of the central peak, which is proportional to the width of the reference mask unit. In theory, this is the full-width at half-effective-signal-amplitude closest to the width of the basic unit. Usually, the width of the basic reference mark units is about 10 to 30 times that of the incident wavelength of the laser source, which yields both a good resolution and a relatively low diffraction effect.
However, it should be noted that although the reference position can provide a peak signal, the position cannot be determined only by this single signal and at least one more reference position measurement is required. The distance between these two reference positions is determined in advance. Figure 3 shows multiple reference codes on the scale grating, located at different distances. Since the distance between adjacent reference codes is pre-determined, the reading of only two pulse signals allows us to determine the position of the scale grating. For example, if the scale grating stops at the position corresponding to point O, denoted by PO, as the scale grating moves forwards it will pass through reference code II, and then reference code III, denoted by PII and PIII, respectively, and then stop at any point O’, denoted by PO’. Let the distance between PO and PO’ be denoted by x1, and the distance between the stop point PO’ and the central position of PIII is denoted by x2, then from the incremental signals, position of PII is known and thus position O can be obtained and be expressed by:
On the other hand, with the configuration shown in Fig. 1, the displacement of the moving table can be obtained but the motion direction cannot be determined because of the cosine nature of the interference signal [15]. What is more, due to the DC component of the interference signals, the measurement results are considerably influenced by the input power. In order to overcome these shortcomings, a modification of this configuration using an expanded optical layout was employed. Figure 4 shows the expanded optical layout of displacement assembly.
In the modified optical design, a PBS is used to replace the BS in order to reduce power loss, while quarter wave plates (QWPs) are used to modulate the polarization directions of the beams for interference. In order to eliminate the influence of the DC component and allow the recognition of the direction of motion, three additional sets of interference signals are added. These four sets of interference signals have a 90° phase delay between them and are denoted as Ii(s), (i = X + 1, X-1, s = 0°, 90°, 180°, 270°), respectively.
Finally, the displacement along the direction ΔX and the error along the axis ΔZ can be calculated as follows [15, 16]:
where g is the period of the incremental grooves, θ is the first-order diffraction angle, λ is the wavelength of the incident light, and the quadrature interference signals Sα and S’α (α = X + 1, X-1) are given by
3. Experiments and results
In order to verify the validity of our approach, a prototype encoder was designed, constructed and evaluated. In the prototype encoder, a diode laser source with a central wavelength of 682 nm was used. Correspondingly, the width of the reference mask unit was determined to be 10 μm, i.e. about 15 times that of the wavelength, in order to reduce the diffraction effect. The whole width of the reference mask was 1.5 mm, and there were 150 elements, so that high resolution was achieved. According to the analysis presented in Section II, after balancing the signal contrast with the need to obtain a good incremental interference signal, 17 elements were designed to be transparent, covering 11.3% of the complete mask’s width. By using the genetic algorithm introduced in Ref [14], a contrast factor up to 2.5 was obtained, which was satisfactory for the reference position identification. Figure 5(a) shows the schematic of the designed reference mask and the simulated reference signal. The binary vector a = [1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1]. The diffraction effect is directly related to the distance z between the reference mask and grating, and a small distance will weaken the effect of diffraction, but too small a distance will be difficult to implement during installation, so a suitable choice was z = 500 μm. Figure 5(b) shows the simulation waveform of the reference pulse signal, where the signal contrast is 2.5 and the width at half-central-peak is 23 μm.
Interference lithography was employed to fabricate the incremental grooves. The surface relief photoresist grating was etched by an ion beam [21–23].The pitch of the incremental groove is 1019 nm, which was previously calibrated and is able to provide a nanometer-scale measurement resolution [16, 17].
The reference code on the scale grating and the reference mask were fabricated using a contact lithography. In the lithography, a transparent mask with a same code design was fabricated on a precisely polished glass substrate by using a conventional mask fabrication process. This process can ensure a linewidth error of less than 0.5%. The flatness of glass substrate is about 50 nm over an area of 50 mm ☓ 50 mm.
A negative type photoresist layer was coated onto a slide glass substrate. The mask was placed paralleling with the coated slide glass with a gap of about 10 μm. A parallel laser beam was projected onto the mask along the normal direction for 60 s. After development of the exposed slide glass substrate, these areas blocked by the opaque areas of the mask were removed and a new photoresist type mask was formed. The formed photoresist mask was coated by an aluminum layer of about 250 nm thickness using an evaporation process. The aluminum layer deposited onto the photoresist was then lifted off by washing it in an acetone solution and the reference mask was obtained.
Fabrication process of the reference code on the incremental grooves is slightly different from that of the reference mask. A transparent type incremental grooves pattern was preliminary fabricated by using interference lithography and ion beam etching, like that introduced in Ref [19]. The negative type photoresist was than coated onto the surface of grooves but not a flat slide glass. The following process is same with that of fabrication of the reference mask.
Fabrication results were tested before they were used for experiments. Figure 6 shows the fabricated reference mask and the scale grating with reference code. It can be seen from the optical microscope images that the fabrication results have a high consistency with the designed code. What is more, the high shape consistency between the reference mask and the reference code can ensure a high repeatability of the peak position in a single pulse signal, which is significant for improvement of positioning accuracy of the new encoder.
Fig. 6 Optical microscope image of the fabricated reference mask and its sectional images tested by an AFM (a) and picture of the scale grating with reference code, the optical microscope image and the AFM images of the reference code (b).
However, the shapes of the transparent areas of the reference mask like M1 to M4 in Fig. 6(a) and the opaque areas of the reference codes in the scale grating like G1 to G4 in Fig. 6(b) are different from the designed ones. The widths of the bottom are smaller than the designed values, they are 10 μm or 20 μm. This shape distortion is mainly caused by the imperfection of exposure process, in which an inverse bell shape exposure curve but not an ideal rectangular shape was generated. The shape imperfection is a factor influencing the signal to noise ratio (SNR) and thereby influencing the sharpness of the pulse signal. This finally affects the positioning accuracy and is experimentally demonstrated below.
Figure 7 shows the prototype encoder and the experimental setup. A compact prototype encoder was configured within a volume of 95 mm (X) ☓110 mm (Y) ☓ 30 mm (Z) by sharing a same laser source for both measurements. As shown in Fig. 7(a), the reading head was fixed onto a five degrees-of-freedom (DOF) manual stage for adjusting the position and orientation of the reading head. The scale grating was mounted onto a motorized precision stage (M-112.1DG, PI Corp.). The diameter of the collimated laser beam through the aperture was slightly larger than 1.5 mm, covering every reference code. The reference and incremental interference signals were acquainted by an I-V converter circuit and a data acquisition (DAQ) card (NI Corp.). The signal was then processed using a Labview implementation of the above-presented algorithm.
Figure 8 shows the experimental results from the reference signal. As evident from the figure, the signal contrast is 2.7, while the width at half-central-peak is 29 μm, therefore there is good consistency between the experimental result and the simulation result shown in Fig. 5(b), which proves that the proposed design provides accurate readings.
The central peak of the reference signal is used to confirm the reference position, while the width of the central peak is associated with positional accuracy. The key point is to find the vertex of the peak and set a threshold near the vertex. The sharper of the peak, the higher the position accuracy will be. As illustrated in Fig. 5(b), the pulse signal with a sharp vertex can be obtained and the value of nominalized light intensity is about 0.15 when the diffraction effect is taken into consideration. However, the measurement results shown in Fig. 8 is slightly different from the simulated one. The pulse width at half amplitude is about 29 μm, larger than the simulated value 23 μm. Diffraction effect in the experiment was enlarged because the slit width at the bottom is smaller than the designed value. But it should be noted that the pulse shape distortion is not a direct factor influencing the resolution neither the position accuracy, the position of vertex is.
For the measured reference signal, it should be noted that at the top of the pulse, it is not a vertex, but a flat range. The photodiodes used in this research (S5870, Hamamatsu Photonics) provides a photo sensitivity of about 0.45 A/W at 660 nm and a current of approximately 0.7 mA can be generated at the vertex area in the experiments. The resolve capability of the photodiodes at 0.7 mA will not influence the sharpness of the signal. The reduced sharpness of the pulse signal is mainly caused by the low SNR and the imperfection of slits shape is the main reason of the reduced SNR as mentioned above.
Therefore, in this manuscript, the threshold was determined based on the flat range. Figure 9 illustrates the details. a1 and a2 are two points at different sides of the vertex, and they have a same vertical coordinate. The line passing through a1 and a2 moves downwards until reaching the flat range. The vertical coordinate of a1 is determined to be the threshold. Then, the reference position r can be written by:
In this research, the vertical value corresponding to the flat range is 0.15. Under such a value, the distance between a1 and a2 is 0.5 μm. This 0.5 μm corresponds to the positioning accuracy of this encoder.
Figure 10 shows the measurement results of X-directional displacements and their errors. The speed of the PI motorized positioning systems is set to be 10μm/s and the sampling rate of NI DAQ Card is 800Hz. As shown in Fig. 10(a), scale grating is mounted onto the holder and the perpendicularity between the moving axis and the beam from the displacement assembly is previously calibrated. Five distances located at different areas on the scale grating are measured to evaluate the positioning accuracy of the displacement assembly. Distances 1, 2, 3, 4 are approximately 1 mm, while distance 5 that covers the reference code is about 4 mm.
Figures 10(b) and 10(c) show the results. Comparison between the inputs and the outputs verified that the positioning accuracy of the prototype encoder is better than 0.06% over a measurement range no less than 4 mm. The nonlinear component with a maximum value of ± 0.7 μm vary at different areas of the grating surface. Apart from the interpolation errors, straightness error of the moving axis and the flatness error of the grating surface formed on the slide glass also contribute to the nonlinear components. Systematical analysis of the errors and approaches to reduce them was thoroughly studied in previous works [24, 25]. Periodic error components with a pitch of about 400 μm in these results coincide with the screw pitch of the motorized stage, that is, 0.4 mm. Employment of a linear motor driven stage may eliminate these components. Also, a scale grating formed on a precisely polished optics glass substrate with a flatness on an order of tens of nm will be employed to replace the one here in the next step.
Experiments were taken to investigate the crosstalk between these two sets of codes. Figure 10(c) reveals that the special designed reference code superimposed in this proposal has no influence on displacement measurement. As mentioned above, the ratio between the transparent units and the complete reference code is a balance between the reference signal contrast and SNR of the incremental signals. According to result shown here, the transparent units’ ratio, i.e., 11.3%, can be further enlarged for a higher contrast. In such case, the reference positioning accuracy can be improved.
Position identification principle as illustrated by Eq. (7) using a scale grating with multiple reference codes is similar with the commercial encoders, like LIP 5x1C series of Heidenhain. Fabrication of such a scale grating with a sub-micron distance precision between any two reference codes by using current mask fabrication technology is available. Demonstration of this will be carried out as the future work, in which the scale grating will be expanded to be a planar type two-axis one.
4. Discussion and conclusion
In this paper, we proposed a two-probe absolute type encoder that can simultaneously measure the X-directional displacement and provide position information. An improved scale grating composed of a continuous incremental groove with a 1 μm period and multiple 150-bits reference codes was designed, simulated and fabricated. The newly proposed scale grating has the potential to be expanded to a planar scale grating due to its single-track characteristics. A reading head comprising two independent units, a reference assembly and a displacement assembly, was developed for use with this scale grating. The superimposition of reference codes and a mask in the reference assembly ensures a pulse signal with a length twice the line width. The grating interferometry principle in the displacement assembly ensures a nanometer-scale resolution and a comparable position scale interpolation for the reference assembly. The same laser source was used for both measurements to achieve a compact design volume. A prototype encoder was finally configured within a volume of 95 mm (X) ☓ 110 mm (Y) ☓ 30 mm (Z). Preliminary evaluation shows that the developed encoder can provide absolute position information with an accuracy about 0.5 μm, with a linearity error of X-directional displacement no larger than 0.06%. These results verify the feasibility of the proposed encoder design.
It should be noted that only one reference code was fabricated for the purposes of this manuscript, and more codes will be added to evaluate the entire function of this encoder. Future work also includes improvement of reference positioning accuracy and extending the one-axis single track scale grating to a two-axis one, thus forming an absolute-type planar encoder.
Funding
National Natural Science Foundation of China (NSFC) (51427805); China Postdoctoral Science Foundation Fund (2016T90089); Shenzhen Science and Technology Plan (JSGG20150512162908714); National Key Research and Development Program (2016YFF0100704); Shenzhen Fundamental Research Program (JCYJ20140417115840232, JCYJ20160301153417873).
References and links
1. Heidenhain, Sealed Linear Encoder LC (Heidenhain Co, 2015).
2. Renishaw, Absolute Optical Encoder with BiSS Serial Communications (Renishaw PLC, 2014).
3. Magnescale Co, Ltd, “Laser scale, Linear scale,” www.mgscale.com.
4. T. Oiwa, M. Katsuki, M. Karita, W. Gao, S. Makinouchi, K. Sato, and Y. Oohashi, “Report of Questionnaire Survey on Ultra-precision Positioning,” Int. J. Jpn Soc. Precis. Eng. (2015).
5. W. Gao, W. Kim, H. Bosse, H. Haitjema, Y. Chen, X. Lu, W. Knapp, A. Weckenmann, W. T. Estler, and H. Kunzmann, “Measurement technologies for precision positioning,” CIRP Ann. 64(2), 773–796 (2015). [CrossRef]
6. Y. Matsuzoe, N. Tsuji, T. Nakayama, K. Fujita, and T. Yoshizawa, “High-performance absolute rotary encoder using multitrack and m-code,” Opt. Eng. 42(1), 124–131 (2003). [CrossRef]
7. L. Wang, K. Li, Z. Liu, Y. Zhang, and J. Niu, “Rapid detection of index for encoder in rotary inertial navigation system,” Opt. Express 24(5), 5591–5599 (2016). [CrossRef]
8. S. Ishimura and K. Kikuchi, “Eight-state trellis-coded optical modulation with signal constellations of four-dimensional M-ary quadrature-amplitude modulation,” Opt. Express 23(5), 6692–6704 (2015). [CrossRef] [PubMed]
9. Heidenhain, Sealed Linear Encoders LC (Heidenhain Co, 2003).
10. H. Schwenke, W. Knapp, H. Haitjema, A. Weckenmann, R. Schmitt, and F. Delbressine, “Geometric error measurement and compensation of machines – an update,” CIRP Ann. 57(2), 660–675 (2008). [CrossRef]
11. W. Gao, S. Dejima, H. Yanai, K. Katakura, S. Kiyono, and Y. Tomita, “A surface motor driven planar motion stage integrated with an XYZ surface encoder for precision positioning,” Precis. Eng. 28(3), 329–337 (2004). [CrossRef]
12. Sumitomo Heavy Industries Co, Ltd, “Mechatronics division,” http://www.shi-mechatronics.jp.
13. J. Saez-Landete, J. Alonso, and E. Bernabeu, “Design of zero reference codes by means of a global optimization method,” Opt. Express 13(1), 195–201 (2005). [CrossRef] [PubMed]
14. J. Saez-Landete, S. Salcedo-Sanz, M. Rosa-Zurera, J. Alonso, and E. Bernabeu, “Generation of optical reference signals robust to diffractive effects,” in Proceedings of IEEE Conference on Photonics Technology Letters (IEEE, 2007) 19(15), pp. 1133–1135. [CrossRef]
15. W. Gao and A. Kimura, “A three-axis displacement sensor with nanometric resolution,” CIRP Ann. 56(1), 529–532 (2007). [CrossRef]
16. H. L. Hsieh and W. Chen, “Heterodyne wollaston laser encoder for measurement of in-plane displacement,” Opt. Express 24(8), 8693–8707 (2016). [CrossRef] [PubMed]
17. X. Li, W. Gao, H. 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]
18. X. Li, Y. Shimizu, T. Ito, Y. Cai, S. Ito, and W. Gao, “Measurement of six-degree-of-freedom planar motions by using a multi-probe surface encoder,” Opt. Eng. 53(12), 122405 (2014). [CrossRef]
19. H. Wolferen, L. Abelmann, and T. C. Hennessy, Laser interference lithography (Lithography Principles Processes and Materials, 2011), Chap. 5.
20. 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]
21. S. R. J. Brueck, “Optical and Interferometric Lithography-Nanotechnology Enablers,” in Proceedings of the IEEE (IEEE, 2005) 93(10), pp. 1704–1721.
22. X. Li, Y. Shimizu, S. Ito, and W. Gao, “Fabrication of scale gratings for surface encoders by using laser interference lithography with 405 nm laser diodes,” Int. J. Precis. Eng. Manuf. 14(11), 1979–1988 (2013). [CrossRef]
23. S. Wang and L. Zeng, “Analysis and minimization of spacing error of holographic gratings recorded with spherical collimation lenses,” Opt. Express 23(5), 5532–5546 (2015). [CrossRef] [PubMed]
24. W. Gao and A. Kimura, “A fast evaluation method for pitch deviation and out-of-flatness of a planar scale grating,” CIRP Ann. 59(1), 505–508 (2010). [CrossRef]
25. A. Kimura, W. Gao, Y. Arai, and L. Zeng, “Design and construction of a two-degree of-freedom linear encoder for nanometric measurement of stage position and straightness,” Precis. Eng. 34(1), 145–155 (2010). [CrossRef]
