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Focusing error detection based on astigmatic method with a double cylindrical lens group
Optics and Laser Technology 106 (2018) 145–151
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Optics and Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Full length article
Focusing error detection based on astigmatic method with a double cylindrical lens group Zhen Bai a,b, Jingsong Wei a,⇑ a b
Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China University of Chinese Academy of Sciences, Beijing 100049, China
a r t i c l e
i n f o
Article history: Received 30 January 2018 Received in revised form 15 March 2018 Accepted 3 April 2018
a b s t r a c t Autofocusing and autotracking are widely used in maskless laser lithography, optical imaging, and recording and readout of optical information storage. With the development of optoelectronic techniques, the accuracy of autofocusing and autotracking thus needs to be improved to the nanoscale, and the autotracking range also needs to be extended to tens of micrometers. In this work, an astigmatic method with two cylindrical lenses is proposed, where the focusing error signal is extracted to detect the focusing error. A theoretical analysis and simulation are carried out, accordingly. A focusing error detection system is established to demonstrate the theoretical analysis. The experimental results indicate that the focusing error signal curves present good ‘‘S” characteristics. The linear tracking range is up to 18 lm, and the tracking accuracy is approximately 50 nm. The theoretical and experimental results indicate that the astigmatic method with two cylindrical lenses is a good method for autofocusing and autotracking with both high accuracy and large dynamic range. This work is useful in the fields of high-resolution maskless laser lithography and optical imaging. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction Autofocusing and autotracking are widely used in maskless laser lithography, optical imaging, and recording and readout of optical information storage, when the laser beam is focused on a small spot with an objective lens having a high numerical aperture (NA) and short depth of focus [1–10]. In autofocusing and autotracking, one of the critical factors is focusing error detection [11–17]. For focusing error detection, the astigmatic method is widely used owing to its simple operation and low cost, in which only one cylindrical lens and four-quadrant detector (FQD) are usually required [18–21]. The basic principle of the astigmatic method is as follows: when the sample is out of focus, the cylindrical lens changes the distribution of the reflected laser beam, resulting in a different spot shape on the FQD. By measuring the focusing error signal on the FQD, one can calculate the defocus amount. Then, autofocusing and autotracking are realized by compensating the defocus amount, which may be conducted through servo systems. With the development of optoelectronic techniques, the objective lenses with high NA are being used in the autotracking unit of maskless laser lithography and optical imaging systems. The depth
⇑ Corresponding author. E-mail address: [email protected] (J. Wei). https://doi.org/10.1016/j.optlastec.2018.04.005 0030-3992/Ó 2018 Elsevier Ltd. All rights reserved.
of focus (DOF) of systems can be estimated as DOF ¼ 0:5k=ðNAÞ2 [11], where NA and k are numerical aperture of objective lens and laser wavelength of the system, respectively. The DOF is becoming smaller and smaller for the maskless laser lithography and optical imaging systems. For example, the DOF is about 200 nm for a system with NA of 0.95 and laser wavelength of k ¼ 405 nm. The DOF is shortened to about 100 nm when the NA of objective lens is increased to 1.40 [22,23]. The accuracy of focusing and tracking of the objective lens is required to be improved to the nanoscale, and the tracking range is also required to be extended to tens of micrometers, accordingly. However, to our knowledge, the astigmatic method with a single cylindrical lens does not meet the requirements [24–26]. This is because the high accuracy is accompanied by a short linear range. The short linear range means a very small dynamic autotracking range. In contrast, the accuracy is too low when the dynamic range is large enough. In this work, an astigmatic method is proposed in which a double cylinder lens group replaces the single cylinder lens. Focusing error detection is carried out using a combination of the double cylinder lens group and FQD. With this method, the linear dynamic tracking range can reach up to approximately 18 lm, and the focusing accuracy is improved to 50 nm theoretically and experimentally. The proposed method provides a good way of autofocusing and autotracking with both higher accuracy and larger dynamic range.
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2. Theory of astigmatic method with a double cylindrical lens group The schematic of the optical path in the proposed method is presented in Fig. 1a, where two cylindrical lenses (marked as CLx and CLy ) are used as astigmatic elements. The focal lengths of CLx and CLy are marked as f x and f y , respectively. The cylindrical lenses CLx and CLy are placed into a mutually orthogonal system. CLx tunes the x-directional optical axis of spot on the FQD and CLy changes the y-directional optical axis of spot on the FQD. Thus, CLx and CLy together form a double cylindrical lens group (DCLGÞ. The FQD, which is placed behind the DCLG, is used as the signal detection element and acquires the focusing error signal (FES). Lo is the objective lens with a focal length of f o , and is used to focus the laser beam onto a spot. Pf is the focal plane of Lo , while Pa and P p are the apofocal plane and perifocal plane of Lo , respectively. In Fig. 1a, the DCLG actually functions as an astigmatic element and generates the FES. The generation principle of FES is as follows. When the sample is placed on the focal plane P f of Lo , the reflected light passes through Lo and DCLG, then arrives at the FQD. The spot on the FQD is circular (shown in Fig. 1b-b2). The spot on the FQD becomes an upright ellipse when the sample is placed on the plane Pa (apofocus) of Lo (shown in Fig. 1b-b1). A horizontal elliptical spot can be formed if the sample is placed on the plane P p (perifocus) of Lo (shown in Fig. 1b-b3). The FES can be analyzed as follows: A1 , A2 , A3 , and A4 are the areas of the spot on each quadrant of the FQD, as shown in Fig. 2, with intensities of I1 , I2 , I3 , and I4 , respectively. r x and r y are the radii of the x-axis and y-axis of the spot on the FQD, respectively. The FES is defined as:
I1 þ I2 I3 þ I4 FES ¼ I 1 þ I2 þ I 3 þ I4
ð1Þ
Let us assume that the spot on the FQD has a uniform intensity distribution; then, one can obtain
Ii ¼ CAi
ði ¼ 1; 2; 3; 4Þ
A1 ¼ A3 ; A2 ¼ A4
ð3Þ
ð4Þ
The values of r x and ry can be obtained through the imaging principle, where the light is reflected by the sample, as shown in Fig. 3. The reflected light goes through Lo and DCLG, before arriving
1=ax þ 1=bx ¼ 1=f x
ð5Þ
Based on the homothetic triangle theory, one can obtain:
rx =r clx ¼ ðmx bx Þ=bx ;
r clx =ro ¼ ax =bo
ð6Þ
Then r x can be calculated as:
mx bx mx bx ax r clx ¼ r o bx bx bo
ð7Þ
From Eqs. (5) and (7), one can obtain:
rx ¼ ro
3
26 jr x j jr y j 7 arcsin qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi arcsin qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 2 2 rx þ ry r 2x þ r 2y
p4
1=ao þ 1=bo ¼ 1=f o ;
rx ¼
Based on Eqs. (1)–(3) and geometrical relationship of the spot, one can calculate FES as follows:
FES ¼
at the FQD. ao and bo are the object and image distances of Lo , respectively. ax and bx are the object and image distances of CLx , respectively. ay and by are object and image distances of CLy , respectively. lxy is the distance between CLx and CLy . In order to calculate the r x value, one can consider CLy as a glass plate, as is shown Fig. 3a, where ro and r clx the radii of light beam onto Lo and CLx , respectively. mx is the distance between CLx and the FQD. Based on the Gaussian imaging formula, one can obtain
ð2Þ
where C is a constant. Considering the symmetry of the spot,
2
Fig. 2. The spot profile onto the FQD.
mx mx ðao f o Þ þ 1 f x ao f o lx ao þ lx f o
1
lx ðao f o Þ ao f o
ð8Þ
Similarly, ry can be calculated by considering CLx as a glass plate, as is shown in Fig. 3b, where the rcly is the radius of light beam onto the CLy and my is the distance between CLy and the FQD. Using a series of mathematical operations, one has
" # my my ðao f o Þ ly ðao f o Þ ry ¼ ro þ 1 1 ao f o f y ao f o ly ao þ ly f o
ð9Þ
Based on Fig. 3, one can see that my ¼ mx lxy , ly ¼ lx þ lxy , hence Eq. (9) can be further written as " ry ¼ ro
mx lxy ðmx lxy Þðao f o Þ þ 1 fy ao f o ðlx þ lxy Þao þ ðlx þ lxy Þf o
# 1
ðlx þ lxy Þðao f o Þ ao f o
ð10Þ
Fig. 1. Schematic of astigmatic method with a double cylindrical lens group. (a) Optical path and (b) shape change of spot on the FQD.
Therefore, one can obtain the dependences of the FES, rx , and ry on the defocus amount of Da0 through the Eqs. (4), (8), and (10), respectively, where Da0 is defined as the change in the amount of ao . The relation between FES and Da0 is called as ‘‘S” curve. The maximum of r x and r y are marked as Rx and Ry , respectively. In practical applications, the values of Rx and Ry are also limited by the size of the FQD.
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Fig. 3. Simplified optical path for the calculation of (a) r x and (b) r y .
3. Calculated and simulated results For an autofocus and autotracking optical system, the parameters lx , lxy , and mx need to be determined using Eqs. (4), (8), and (10) when the values of Lo , CLx and CLy are fixed. The parameters are set as follows f o ¼ 2 mm and ro ¼ 5 mm for Lo ; f x ¼ 80 mm for CLx ; f y ¼ 150 mm for CLy . The sample is initially placed on the focal plane of Lo , thus ao ¼ f o ¼ 2 mm. One can optimize the parameters lx , lxy , and mx by calculating the dependence of FES on Da0 , i.e., from ‘‘S” curve with large dynamic tracking range and high tracking accuracy. For the ‘‘S” curve, there is a linear range between peak point and valley point. For example, Fig. 4 shows a typical ‘‘S” curve, where the peak point and valley point are marked as A and B, respectively. The linear response of FES on the defocus amount can be set between points C 1 and C 2 . The defocus amount L shown in Fig. 4 is thus defined as a linear range. Before optimizing the parameters, one needs to set the initial values of lx , lxy , and mx according to the experimental requirements. For example, in maskless laser lithography and optical imaging
systems, the defocus amount Da0 is approximately tens of micrometers; here, the dynamic linear range can be set as 10 lm < L < 40 lm. One can set the initial values as lx ¼ lx0 ¼ 350 mm; lxy ¼ lxy0 ¼ 20 mm; and mx ¼ mx0 ¼ 140 mm. In order to obtain a threadlike elliptical spot that also presents symmetrical change on the FQD, Rx and Ry should be chosen appropriately. On one hand, the FQD cannot receive all the light if Rx and Ry are too large; on the other hand, the change of the spot is insufficient to result in a large change in the FQD if Rx and Ry are too small. That is, improper Rx and Ry values cannot obtain a good ‘‘S” curve. Therefore, Rx and Ry should be restricted to an appropriate range to meet the experimental requirements. Simultaneously, to decrease the noise from stray lights, it is required that Rx Ry . According to the requirements, one can set 3:3 mm < Rx < 3:8 mm, 3:3 mm < Ry < 3:8 mm, and jRx Ry j < 0:1 mm. Then, the other parameters, lx , lxy , and mx , can be optimized using the flow diagram shown in Fig. 5. By parameter optimization, lx ¼ 400 mm;mx ¼ 150 mm; and lxy ¼ 30 mm are obtained, and the ‘‘S” curve thus obtained is presented in Fig. 6a; the typical spots on the FQD are also given in Fig. 6b-d. Fig. 6b-d correspond to the sample placed on the apofocal, focal, and perifocal planes, respectively. One can see from Fig. 6 that the linear range L of the ‘‘S” curve is approximately 18 lm, and the spots on the FQD are threadlike ellipses when the sample is placed on the apofocal and perifocal planes, respectively. rx ¼ 3:427 mm;r y ¼ 0:023 mm for the perifocal plane and rx ¼ 0:023 mm;r y ¼ 3:427 mm for the apofocal plane. The spot on the FQD becomes circular with r x ¼ 1:727 mm and ry ¼ 1:728 mm when the sample is placed on the focal plane.
4. Experimental result and discussion
Fig. 4. The schematic of the definition of linear range.
In order to verify the optimized results obtained through our theory, a focusing error detection system, shown in Fig. 7, was established. In this setup, a laser beam with a wavelength of 658
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Fig. 5. The flow chart of the parameter optimization process.
Fig. 6. Calculated ‘‘S” curve and spot on the FQD. (a) ‘‘S‘‘ curve. Spot shape for the sample placed onto (b) apofocal plane, (c) focal plane, and (d) perifocal plane.
nm passes through the 1/2 waveplate, polarized beam splitter (PBS), 1/4 waveplate, and an objective lens Lo in succession. It is then focused onto the sample surface. The beam reflected from the sample surface passes through the objective lens Lo and PBS, goes through the DCLG, and focuses on the FQD. The objective lens is installed on a piezoelectric tube (PZT) and can be moved up and down along the z-directional optical axis by feeding the FES into the PZT controller. Based on the theoretical calculations, the parameters are chosen as follows: f o ¼ 2 mm and ro ¼ 5 mm for f x ¼ 80 mm for CLx ; f y ¼ 150 mm for CLy ; Lo ; lx ¼ 400 mm;mx ¼ 150 mm;lxy ¼ 30 mm.
The experimental results of the real spot shape on the FQD are presented in Fig. 8. The spot is basically circular when one places the sample on the focal plane, as shown in Fig. 8c. Compared with original spot of laser beam (as shown in Fig. 8a), the spot size becomes smaller due to the converging effect of cylindrical lens, however, the spot shape is similar to the original shape of laser beam. The spot shape changes from an upright ellipse (Fig. 8b), to a circle (Fig. 8c), and to a horizontal ellipse (Fig. 8d) when one moves the sample from the apofocal plane, to the focal plane, and to the perifocal plane, respectively. By comparing Figs. 6 and 8, one can see that the experimental spots are approximately equal
Z. Bai, J. Wei / Optics and Laser Technology 106 (2018) 145–151
Fig. 7. Experimental schematics of focusing error detection system. (1/2WP-1/2 wave plate, 1/4WP-1/4 wave plate, PBS-polarized beam splitter).
to the calculated results. A slight discrepancy between the real and calculated spots may be due to the parameters of the objective lens Lo . In the calculations, Lo is considered as an ideal lens composed of a single piece of the lens. However, in our experiment, Lo is an objective lens of high NA. The NA of Lo is 0.90. Lo is actually composed of a battery of lenses. Hence, there is no focal length, but only working distance. Thus, in our calculation of the FES and spot, the working distance is used to replace the focal length of Lo . By observing the change of spot shape on the FQD with the defocus amount Da0 , the FES curve is obtained, as shown in Fig. 9. In Fig. 9a, the dependence of intensity signal of each quadrant of the FQD on Da0 is presented; one can see that signal values in quadrants A and C are the same, and those in quadrants B and D are the same, which indicates that the shape change of the spot on the FQD with Da0 is symmetrical and uniform. The spot is at the center of the FQD. Further, the ‘‘S” curve (the dependence of FES on Da0 ) is obtained through the combination of Eq. (1) and Fig. 9a. Fig. 9b is based on the experimental results and presents a good ‘‘S” curve characteristic. The FES value changes from 0.845 to 0.752. The diaphragm in the system and the gap
149
between the quadrants of the FQD restrict the FES value from reaching one. The linear range is up to 18 lm as seen in Fig. 9, and it is consistent with the theoretical value. Another experiment is carried out to test the accuracy of our results. In this experiment, if one lets the PZT move up and down with a frequency of 5 Hz, the FES can be detected. The defocus amount Da0 can be seen and analyzed through the FES signal. Fig. 10 shows the experimental results. In Fig. 10a, a square wave stimulation electric signal is sent into the PZT, and the PZT moves up and down within a distance of 500 nm, with a frequency of 5 Hz. That is, the Da0 is periodically changed in the up and down directions, with an amplitude and frequency of 500 nm and 5 Hz, respectively. The movement of PZT causes the defocus of the sample and the defocus is measured by using the established setup in Fig. 7. Fig. 10b gives the corresponding detection results. It is obvious that the FES curve presents a clear change at a frequency of 5 Hz implying that the focusing error detection system could detect a 500 nm defocus amount clearly and precisely. One can continue to decrease the defocus amount to 50 nm, as shown in Fig. 10c, where a square wave stimulation electric signal is sent into the PZT which moves up and down within a distance of 50 nm with a frequency of 5 Hz. Fig. 10d shows the detection results; although the curve is not smooth, the time response of FES still changes periodically with time. The noise of the FES value is also measured when the PZT actuator is motionless and static. Fig. 11 gives the results. One can see that the FES noise is in the range of ±0.002, and without any periodic characteristic. This can tell us that the periodical signal in Fig. 10 is from PZT actuator, not from the noise. The focusing error detection method could detect a defocus amount of 50 nm, and the detection accuracy can reach up to 50 nm. 5. Conclusion In this work, an astigmatic method with two cylindrical lenses is proposed. The theoretical analysis and simulation, to obtain the optimal system parameters, have been provided. The focusing
Fig. 8. The original shape of the laser beam (a) and experimental spot on the FQD. The sample is placed on (b) apofocal plane, (c) focal plane, and (d) perifocal plane of the objective lens.
Fig. 9. Experiment results of ‘‘S” curves. (a) Dependence of signal in every quadrant on defocus amount, (b) ‘‘S” curve.
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Fig. 10. Accuracy detection of defocus amount Da0 by moving the PZT up and down at a frequency of 5 Hz. (a) For a defocus amount Da0 of 500 nm, (b) the corresponding FES detection; (c) for a defocus amount Da0 of 50 nm, (d) the corresponding FES detection.
cooperation program in science and technology innovation (2016YFE0110600). International Science & Technology Cooperation Program of Shanghai (16520710500).
References
Fig. 11. The FES noise change with time (the PZT actuator keeps static and motionless).
error detection system has been established to demonstrate the theoretical analysis. The experimental results indicate that the obtained FES curve presents a good ‘‘S” characteristic. The linear tracking range and tracking accuracy are found to be nearly 18 lm and approximately 50 nm, respectively. The theoretical and experimental results indicate that the astigmatic method with two cylindrical lenses provides a good way to autofocusing and autotracking, while maintaining high accuracy and large dynamic range simultaneously. The proposed method can be widely applied in high-resolution maskless laser lithography system and the confocal optical microscopy. Acknowledgment National Natural Science Foundation of China (Nos. 51672292 and 61627826). International Science & Technology Cooperation Program of China: intergovernmental international
[1] T. Wei, J. Wei, K. Zhang, L. Zhang, Image lithography in telluride suboxide thin film through controlling ‘‘virtual” bandgap, Photonic. Res. 5 (1) (2017) 22–26. [2] J.C. Cheng, D.Y.C. Choi, Three-step lithography to the fabrication of vertically coupled micro-ring resonators in amorphous silicon-on-insulatorcorrigendum, Chin. Opt. Lett. 14 (4) (2016) 043501. [3] Y.H. Zhang, Z.H. Zhang, C. Geng, S. Xu, T.B. Wei, W.G. Bi, Versatile nanosphere lithography technique combining multiple-exposure nanosphere lens lithography and nanosphere template lithography, Chin. Opt. Lett. 15 (6) (2017) 062201. [4] D. Huerta-Murillo et al., Fabrication of multi-scale periodic surface structures on Ti-6Al-4V by direct laser writing and direct laser interference patterning for modified wettability applications, Opt. Lasers Eng. 98 (2017) 134–142. [5] Y. Chen, J. Qin, J.F. Chen, L. Zhang, C.F. Ma, J.R. Chu, X.F. Xu, L. Wang, 16 nmresolution lithography using ultra-small-gap bowtie apertures, Nanotechnology 28 (5) (2017) 055302. [6] N.N. Luo, Z.M. Zhang, Fabrication of a curved microlens array using double gray-scale digital maskless lithography, J. Micromechanics Microengineering 27 (3) (2017) 035015. [7] M. Deubel, G. Von Freymann, M. Wegener, S. Pereira, K. Busch, C.M. Soukoulis, Direct laser writing of three-dimensional photonic-crystal templates for telecommunications, Nat. Mater. 3 (7) (2004) 444–447. [8] M.D. Kong, C. Guo, B.C. Li, W.Y. He, M. Wei, Fabrication of multi-wavelength visible and infrared filter for solar atmosphere tomographic imaging, Chin. Opt. Lett. 15 (12) (2017) 123101. [9] R. Wang, J.S. Wei, Y.T. Fan, Chalcogenide phase-change thin films used as grayscale photolithography materials, Opt. Express 22 (5) (2014) 4973–4984. [10] D.Y. Xia, Z.Y. Ku, S.C. Lee, S.R.J. Brueck, Nanostructures and functional materials fabricated by interferometric lithography, Adv. Mater. 23 (2) (2011) 147–179. [11] H.G. Rhee, D.I. Kim, Y.W. Lee, Realization and performance evaluation of high speed autofocusing for direct laser lithography, Rev. Sci. Instrum. 80 (7) (2009) 073103. [12] J. Wei, K. Zhang, T. Wei, Y. Wang, Y. Wu, M. Xiao, High-speed maskless nanolithography with visible light based on photothermal localization, Sci. Rep. 7 (2017) 43892. [13] Z. Bai, J. Wei, X. Liang, K. Zhang, T. Wei, R. Wang, High-speed laser writing of arbitrary patterns in polar coordinate system, Rev. Sci. Instrum. 87 (12) (2016) 125118.
Z. Bai, J. Wei / Optics and Laser Technology 106 (2018) 145–151 [14] I. Stoica, A.I. Barzic, C. Hulubei, Fabrication of nanochannels on polyimide films using dynamic plowing lithography, Appl. Surf. Sci. 426 (2017) 307–314. [15] D.E. Denk, A.G. Poleshchuk, Methods of improving the accuracy of operation of an autofocus in a circular laser writing system, Optoelectronics, Instrum. Data Process. 46 (1) (2010) 87–95. [16] Y.C. Lee, S. Chao, C.C. Huang, C.T. Yang, Design of a dual-wavelength optical head for submicron-scale and nanoscale lithography, IEEE Trans. Magn. 47 (3) (2011) 696–700. [17] A.G. Poleshchuk et al., Polar coordinate laser pattern generator for fabrication of diffractive optical elements with arbitrary structure, Appl. Opt. 38 (8) (1999) 1295–1301. [18] H.G. Rhee, D. Kim, S.K. Hong, Y.W. Lee, 300 mm reference wafer fabrication by using direct laser lithography, Rev. Sci. Instrum. 79 (10) (2008) 103103. [19] H.G. Rhee, D.I. Kim, J.B. Song, H.J. Lee, Y.W. Lee, J.H. Jo, Performance evaluation of laser lithographic machine for computer-generated hologram, Int. J. Adv. Manuf. Technol. 52 (9–12) (2010) 1005–1009.
151
[20] B. Hnilicka, A. Besançon-Voda, H.-J. Schroder, G. Filardi, Modelling the focus error characteristic of a DVD player, in: Control Applications, Proceedings of the 2002 International Conference on IEEE, vol. 2, 2002, pp. 629–630. [21] C.-H. Liu, Z.-H. Li, Application of the astigmatic method to the thickness measurement of glass substrates, Appl. Opt. 47 (21) (2008) 3968–3972. [22] A. Nikolov, D. Wasan, Oil lenses on the air-water surface and the validity of Neumann’s rule, Adv. Colloid Interface Sci. 244 (2017) 174–183. [23] J. Fu, Y. Wang, P.F. Chen, Generation of sub-half-wavelength non-diffracting beams with tunable depth of focus by focusing azimuthally polarized beams, Appl. Phys. B 123 (7) (2017) 214. [24] J. Steele, H. Yuan, C. Tan, M. Keshavarz, C. Steuwe, M. Roeffaers, J. Hofkens, Direct laser writing of d- to a-phase transformation in Formamidinium lead Iodide, ACS Nano. 11 (2017) 8072–8083. [25] R. Wang, J. Wei, Y. Fan, Chalcogenide phase-change thin films used as grayscale photolithography materials, Opt. Express 22 (2014) 4973–4984. [26] M. Mansuripur, Analysis of astigmatic focusing and push–pull tracking error signals in magnetooptical disk systems, Appl. Optics 26 (1987) 3981–3986.
Contents lists available at ScienceDirect
Optics and Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Full length article
Focusing error detection based on astigmatic method with a double cylindrical lens group Zhen Bai a,b, Jingsong Wei a,⇑ a b
Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China University of Chinese Academy of Sciences, Beijing 100049, China
a r t i c l e
i n f o
Article history: Received 30 January 2018 Received in revised form 15 March 2018 Accepted 3 April 2018
a b s t r a c t Autofocusing and autotracking are widely used in maskless laser lithography, optical imaging, and recording and readout of optical information storage. With the development of optoelectronic techniques, the accuracy of autofocusing and autotracking thus needs to be improved to the nanoscale, and the autotracking range also needs to be extended to tens of micrometers. In this work, an astigmatic method with two cylindrical lenses is proposed, where the focusing error signal is extracted to detect the focusing error. A theoretical analysis and simulation are carried out, accordingly. A focusing error detection system is established to demonstrate the theoretical analysis. The experimental results indicate that the focusing error signal curves present good ‘‘S” characteristics. The linear tracking range is up to 18 lm, and the tracking accuracy is approximately 50 nm. The theoretical and experimental results indicate that the astigmatic method with two cylindrical lenses is a good method for autofocusing and autotracking with both high accuracy and large dynamic range. This work is useful in the fields of high-resolution maskless laser lithography and optical imaging. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction Autofocusing and autotracking are widely used in maskless laser lithography, optical imaging, and recording and readout of optical information storage, when the laser beam is focused on a small spot with an objective lens having a high numerical aperture (NA) and short depth of focus [1–10]. In autofocusing and autotracking, one of the critical factors is focusing error detection [11–17]. For focusing error detection, the astigmatic method is widely used owing to its simple operation and low cost, in which only one cylindrical lens and four-quadrant detector (FQD) are usually required [18–21]. The basic principle of the astigmatic method is as follows: when the sample is out of focus, the cylindrical lens changes the distribution of the reflected laser beam, resulting in a different spot shape on the FQD. By measuring the focusing error signal on the FQD, one can calculate the defocus amount. Then, autofocusing and autotracking are realized by compensating the defocus amount, which may be conducted through servo systems. With the development of optoelectronic techniques, the objective lenses with high NA are being used in the autotracking unit of maskless laser lithography and optical imaging systems. The depth
⇑ Corresponding author. E-mail address: [email protected] (J. Wei). https://doi.org/10.1016/j.optlastec.2018.04.005 0030-3992/Ó 2018 Elsevier Ltd. All rights reserved.
of focus (DOF) of systems can be estimated as DOF ¼ 0:5k=ðNAÞ2 [11], where NA and k are numerical aperture of objective lens and laser wavelength of the system, respectively. The DOF is becoming smaller and smaller for the maskless laser lithography and optical imaging systems. For example, the DOF is about 200 nm for a system with NA of 0.95 and laser wavelength of k ¼ 405 nm. The DOF is shortened to about 100 nm when the NA of objective lens is increased to 1.40 [22,23]. The accuracy of focusing and tracking of the objective lens is required to be improved to the nanoscale, and the tracking range is also required to be extended to tens of micrometers, accordingly. However, to our knowledge, the astigmatic method with a single cylindrical lens does not meet the requirements [24–26]. This is because the high accuracy is accompanied by a short linear range. The short linear range means a very small dynamic autotracking range. In contrast, the accuracy is too low when the dynamic range is large enough. In this work, an astigmatic method is proposed in which a double cylinder lens group replaces the single cylinder lens. Focusing error detection is carried out using a combination of the double cylinder lens group and FQD. With this method, the linear dynamic tracking range can reach up to approximately 18 lm, and the focusing accuracy is improved to 50 nm theoretically and experimentally. The proposed method provides a good way of autofocusing and autotracking with both higher accuracy and larger dynamic range.
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2. Theory of astigmatic method with a double cylindrical lens group The schematic of the optical path in the proposed method is presented in Fig. 1a, where two cylindrical lenses (marked as CLx and CLy ) are used as astigmatic elements. The focal lengths of CLx and CLy are marked as f x and f y , respectively. The cylindrical lenses CLx and CLy are placed into a mutually orthogonal system. CLx tunes the x-directional optical axis of spot on the FQD and CLy changes the y-directional optical axis of spot on the FQD. Thus, CLx and CLy together form a double cylindrical lens group (DCLGÞ. The FQD, which is placed behind the DCLG, is used as the signal detection element and acquires the focusing error signal (FES). Lo is the objective lens with a focal length of f o , and is used to focus the laser beam onto a spot. Pf is the focal plane of Lo , while Pa and P p are the apofocal plane and perifocal plane of Lo , respectively. In Fig. 1a, the DCLG actually functions as an astigmatic element and generates the FES. The generation principle of FES is as follows. When the sample is placed on the focal plane P f of Lo , the reflected light passes through Lo and DCLG, then arrives at the FQD. The spot on the FQD is circular (shown in Fig. 1b-b2). The spot on the FQD becomes an upright ellipse when the sample is placed on the plane Pa (apofocus) of Lo (shown in Fig. 1b-b1). A horizontal elliptical spot can be formed if the sample is placed on the plane P p (perifocus) of Lo (shown in Fig. 1b-b3). The FES can be analyzed as follows: A1 , A2 , A3 , and A4 are the areas of the spot on each quadrant of the FQD, as shown in Fig. 2, with intensities of I1 , I2 , I3 , and I4 , respectively. r x and r y are the radii of the x-axis and y-axis of the spot on the FQD, respectively. The FES is defined as:
I1 þ I2 I3 þ I4 FES ¼ I 1 þ I2 þ I 3 þ I4
ð1Þ
Let us assume that the spot on the FQD has a uniform intensity distribution; then, one can obtain
Ii ¼ CAi
ði ¼ 1; 2; 3; 4Þ
A1 ¼ A3 ; A2 ¼ A4
ð3Þ
ð4Þ
The values of r x and ry can be obtained through the imaging principle, where the light is reflected by the sample, as shown in Fig. 3. The reflected light goes through Lo and DCLG, before arriving
1=ax þ 1=bx ¼ 1=f x
ð5Þ
Based on the homothetic triangle theory, one can obtain:
rx =r clx ¼ ðmx bx Þ=bx ;
r clx =ro ¼ ax =bo
ð6Þ
Then r x can be calculated as:
mx bx mx bx ax r clx ¼ r o bx bx bo
ð7Þ
From Eqs. (5) and (7), one can obtain:
rx ¼ ro
3
26 jr x j jr y j 7 arcsin qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi arcsin qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 2 2 rx þ ry r 2x þ r 2y
p4
1=ao þ 1=bo ¼ 1=f o ;
rx ¼
Based on Eqs. (1)–(3) and geometrical relationship of the spot, one can calculate FES as follows:
FES ¼
at the FQD. ao and bo are the object and image distances of Lo , respectively. ax and bx are the object and image distances of CLx , respectively. ay and by are object and image distances of CLy , respectively. lxy is the distance between CLx and CLy . In order to calculate the r x value, one can consider CLy as a glass plate, as is shown Fig. 3a, where ro and r clx the radii of light beam onto Lo and CLx , respectively. mx is the distance between CLx and the FQD. Based on the Gaussian imaging formula, one can obtain
ð2Þ
where C is a constant. Considering the symmetry of the spot,
2
Fig. 2. The spot profile onto the FQD.
mx mx ðao f o Þ þ 1 f x ao f o lx ao þ lx f o
1
lx ðao f o Þ ao f o
ð8Þ
Similarly, ry can be calculated by considering CLx as a glass plate, as is shown in Fig. 3b, where the rcly is the radius of light beam onto the CLy and my is the distance between CLy and the FQD. Using a series of mathematical operations, one has
" # my my ðao f o Þ ly ðao f o Þ ry ¼ ro þ 1 1 ao f o f y ao f o ly ao þ ly f o
ð9Þ
Based on Fig. 3, one can see that my ¼ mx lxy , ly ¼ lx þ lxy , hence Eq. (9) can be further written as " ry ¼ ro
mx lxy ðmx lxy Þðao f o Þ þ 1 fy ao f o ðlx þ lxy Þao þ ðlx þ lxy Þf o
# 1
ðlx þ lxy Þðao f o Þ ao f o
ð10Þ
Fig. 1. Schematic of astigmatic method with a double cylindrical lens group. (a) Optical path and (b) shape change of spot on the FQD.
Therefore, one can obtain the dependences of the FES, rx , and ry on the defocus amount of Da0 through the Eqs. (4), (8), and (10), respectively, where Da0 is defined as the change in the amount of ao . The relation between FES and Da0 is called as ‘‘S” curve. The maximum of r x and r y are marked as Rx and Ry , respectively. In practical applications, the values of Rx and Ry are also limited by the size of the FQD.
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Fig. 3. Simplified optical path for the calculation of (a) r x and (b) r y .
3. Calculated and simulated results For an autofocus and autotracking optical system, the parameters lx , lxy , and mx need to be determined using Eqs. (4), (8), and (10) when the values of Lo , CLx and CLy are fixed. The parameters are set as follows f o ¼ 2 mm and ro ¼ 5 mm for Lo ; f x ¼ 80 mm for CLx ; f y ¼ 150 mm for CLy . The sample is initially placed on the focal plane of Lo , thus ao ¼ f o ¼ 2 mm. One can optimize the parameters lx , lxy , and mx by calculating the dependence of FES on Da0 , i.e., from ‘‘S” curve with large dynamic tracking range and high tracking accuracy. For the ‘‘S” curve, there is a linear range between peak point and valley point. For example, Fig. 4 shows a typical ‘‘S” curve, where the peak point and valley point are marked as A and B, respectively. The linear response of FES on the defocus amount can be set between points C 1 and C 2 . The defocus amount L shown in Fig. 4 is thus defined as a linear range. Before optimizing the parameters, one needs to set the initial values of lx , lxy , and mx according to the experimental requirements. For example, in maskless laser lithography and optical imaging
systems, the defocus amount Da0 is approximately tens of micrometers; here, the dynamic linear range can be set as 10 lm < L < 40 lm. One can set the initial values as lx ¼ lx0 ¼ 350 mm; lxy ¼ lxy0 ¼ 20 mm; and mx ¼ mx0 ¼ 140 mm. In order to obtain a threadlike elliptical spot that also presents symmetrical change on the FQD, Rx and Ry should be chosen appropriately. On one hand, the FQD cannot receive all the light if Rx and Ry are too large; on the other hand, the change of the spot is insufficient to result in a large change in the FQD if Rx and Ry are too small. That is, improper Rx and Ry values cannot obtain a good ‘‘S” curve. Therefore, Rx and Ry should be restricted to an appropriate range to meet the experimental requirements. Simultaneously, to decrease the noise from stray lights, it is required that Rx Ry . According to the requirements, one can set 3:3 mm < Rx < 3:8 mm, 3:3 mm < Ry < 3:8 mm, and jRx Ry j < 0:1 mm. Then, the other parameters, lx , lxy , and mx , can be optimized using the flow diagram shown in Fig. 5. By parameter optimization, lx ¼ 400 mm;mx ¼ 150 mm; and lxy ¼ 30 mm are obtained, and the ‘‘S” curve thus obtained is presented in Fig. 6a; the typical spots on the FQD are also given in Fig. 6b-d. Fig. 6b-d correspond to the sample placed on the apofocal, focal, and perifocal planes, respectively. One can see from Fig. 6 that the linear range L of the ‘‘S” curve is approximately 18 lm, and the spots on the FQD are threadlike ellipses when the sample is placed on the apofocal and perifocal planes, respectively. rx ¼ 3:427 mm;r y ¼ 0:023 mm for the perifocal plane and rx ¼ 0:023 mm;r y ¼ 3:427 mm for the apofocal plane. The spot on the FQD becomes circular with r x ¼ 1:727 mm and ry ¼ 1:728 mm when the sample is placed on the focal plane.
4. Experimental result and discussion
Fig. 4. The schematic of the definition of linear range.
In order to verify the optimized results obtained through our theory, a focusing error detection system, shown in Fig. 7, was established. In this setup, a laser beam with a wavelength of 658
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Fig. 5. The flow chart of the parameter optimization process.
Fig. 6. Calculated ‘‘S” curve and spot on the FQD. (a) ‘‘S‘‘ curve. Spot shape for the sample placed onto (b) apofocal plane, (c) focal plane, and (d) perifocal plane.
nm passes through the 1/2 waveplate, polarized beam splitter (PBS), 1/4 waveplate, and an objective lens Lo in succession. It is then focused onto the sample surface. The beam reflected from the sample surface passes through the objective lens Lo and PBS, goes through the DCLG, and focuses on the FQD. The objective lens is installed on a piezoelectric tube (PZT) and can be moved up and down along the z-directional optical axis by feeding the FES into the PZT controller. Based on the theoretical calculations, the parameters are chosen as follows: f o ¼ 2 mm and ro ¼ 5 mm for f x ¼ 80 mm for CLx ; f y ¼ 150 mm for CLy ; Lo ; lx ¼ 400 mm;mx ¼ 150 mm;lxy ¼ 30 mm.
The experimental results of the real spot shape on the FQD are presented in Fig. 8. The spot is basically circular when one places the sample on the focal plane, as shown in Fig. 8c. Compared with original spot of laser beam (as shown in Fig. 8a), the spot size becomes smaller due to the converging effect of cylindrical lens, however, the spot shape is similar to the original shape of laser beam. The spot shape changes from an upright ellipse (Fig. 8b), to a circle (Fig. 8c), and to a horizontal ellipse (Fig. 8d) when one moves the sample from the apofocal plane, to the focal plane, and to the perifocal plane, respectively. By comparing Figs. 6 and 8, one can see that the experimental spots are approximately equal
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Fig. 7. Experimental schematics of focusing error detection system. (1/2WP-1/2 wave plate, 1/4WP-1/4 wave plate, PBS-polarized beam splitter).
to the calculated results. A slight discrepancy between the real and calculated spots may be due to the parameters of the objective lens Lo . In the calculations, Lo is considered as an ideal lens composed of a single piece of the lens. However, in our experiment, Lo is an objective lens of high NA. The NA of Lo is 0.90. Lo is actually composed of a battery of lenses. Hence, there is no focal length, but only working distance. Thus, in our calculation of the FES and spot, the working distance is used to replace the focal length of Lo . By observing the change of spot shape on the FQD with the defocus amount Da0 , the FES curve is obtained, as shown in Fig. 9. In Fig. 9a, the dependence of intensity signal of each quadrant of the FQD on Da0 is presented; one can see that signal values in quadrants A and C are the same, and those in quadrants B and D are the same, which indicates that the shape change of the spot on the FQD with Da0 is symmetrical and uniform. The spot is at the center of the FQD. Further, the ‘‘S” curve (the dependence of FES on Da0 ) is obtained through the combination of Eq. (1) and Fig. 9a. Fig. 9b is based on the experimental results and presents a good ‘‘S” curve characteristic. The FES value changes from 0.845 to 0.752. The diaphragm in the system and the gap
149
between the quadrants of the FQD restrict the FES value from reaching one. The linear range is up to 18 lm as seen in Fig. 9, and it is consistent with the theoretical value. Another experiment is carried out to test the accuracy of our results. In this experiment, if one lets the PZT move up and down with a frequency of 5 Hz, the FES can be detected. The defocus amount Da0 can be seen and analyzed through the FES signal. Fig. 10 shows the experimental results. In Fig. 10a, a square wave stimulation electric signal is sent into the PZT, and the PZT moves up and down within a distance of 500 nm, with a frequency of 5 Hz. That is, the Da0 is periodically changed in the up and down directions, with an amplitude and frequency of 500 nm and 5 Hz, respectively. The movement of PZT causes the defocus of the sample and the defocus is measured by using the established setup in Fig. 7. Fig. 10b gives the corresponding detection results. It is obvious that the FES curve presents a clear change at a frequency of 5 Hz implying that the focusing error detection system could detect a 500 nm defocus amount clearly and precisely. One can continue to decrease the defocus amount to 50 nm, as shown in Fig. 10c, where a square wave stimulation electric signal is sent into the PZT which moves up and down within a distance of 50 nm with a frequency of 5 Hz. Fig. 10d shows the detection results; although the curve is not smooth, the time response of FES still changes periodically with time. The noise of the FES value is also measured when the PZT actuator is motionless and static. Fig. 11 gives the results. One can see that the FES noise is in the range of ±0.002, and without any periodic characteristic. This can tell us that the periodical signal in Fig. 10 is from PZT actuator, not from the noise. The focusing error detection method could detect a defocus amount of 50 nm, and the detection accuracy can reach up to 50 nm. 5. Conclusion In this work, an astigmatic method with two cylindrical lenses is proposed. The theoretical analysis and simulation, to obtain the optimal system parameters, have been provided. The focusing
Fig. 8. The original shape of the laser beam (a) and experimental spot on the FQD. The sample is placed on (b) apofocal plane, (c) focal plane, and (d) perifocal plane of the objective lens.
Fig. 9. Experiment results of ‘‘S” curves. (a) Dependence of signal in every quadrant on defocus amount, (b) ‘‘S” curve.
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Fig. 10. Accuracy detection of defocus amount Da0 by moving the PZT up and down at a frequency of 5 Hz. (a) For a defocus amount Da0 of 500 nm, (b) the corresponding FES detection; (c) for a defocus amount Da0 of 50 nm, (d) the corresponding FES detection.
cooperation program in science and technology innovation (2016YFE0110600). International Science & Technology Cooperation Program of Shanghai (16520710500).
References
Fig. 11. The FES noise change with time (the PZT actuator keeps static and motionless).
error detection system has been established to demonstrate the theoretical analysis. The experimental results indicate that the obtained FES curve presents a good ‘‘S” characteristic. The linear tracking range and tracking accuracy are found to be nearly 18 lm and approximately 50 nm, respectively. The theoretical and experimental results indicate that the astigmatic method with two cylindrical lenses provides a good way to autofocusing and autotracking, while maintaining high accuracy and large dynamic range simultaneously. The proposed method can be widely applied in high-resolution maskless laser lithography system and the confocal optical microscopy. Acknowledgment National Natural Science Foundation of China (Nos. 51672292 and 61627826). International Science & Technology Cooperation Program of China: intergovernmental international
[1] T. Wei, J. Wei, K. Zhang, L. Zhang, Image lithography in telluride suboxide thin film through controlling ‘‘virtual” bandgap, Photonic. Res. 5 (1) (2017) 22–26. [2] J.C. Cheng, D.Y.C. Choi, Three-step lithography to the fabrication of vertically coupled micro-ring resonators in amorphous silicon-on-insulatorcorrigendum, Chin. Opt. Lett. 14 (4) (2016) 043501. [3] Y.H. Zhang, Z.H. Zhang, C. Geng, S. Xu, T.B. Wei, W.G. Bi, Versatile nanosphere lithography technique combining multiple-exposure nanosphere lens lithography and nanosphere template lithography, Chin. Opt. Lett. 15 (6) (2017) 062201. [4] D. Huerta-Murillo et al., Fabrication of multi-scale periodic surface structures on Ti-6Al-4V by direct laser writing and direct laser interference patterning for modified wettability applications, Opt. Lasers Eng. 98 (2017) 134–142. [5] Y. Chen, J. Qin, J.F. Chen, L. Zhang, C.F. Ma, J.R. Chu, X.F. Xu, L. Wang, 16 nmresolution lithography using ultra-small-gap bowtie apertures, Nanotechnology 28 (5) (2017) 055302. [6] N.N. Luo, Z.M. Zhang, Fabrication of a curved microlens array using double gray-scale digital maskless lithography, J. Micromechanics Microengineering 27 (3) (2017) 035015. [7] M. Deubel, G. Von Freymann, M. Wegener, S. Pereira, K. Busch, C.M. Soukoulis, Direct laser writing of three-dimensional photonic-crystal templates for telecommunications, Nat. Mater. 3 (7) (2004) 444–447. [8] M.D. Kong, C. Guo, B.C. Li, W.Y. He, M. Wei, Fabrication of multi-wavelength visible and infrared filter for solar atmosphere tomographic imaging, Chin. Opt. Lett. 15 (12) (2017) 123101. [9] R. Wang, J.S. Wei, Y.T. Fan, Chalcogenide phase-change thin films used as grayscale photolithography materials, Opt. Express 22 (5) (2014) 4973–4984. [10] D.Y. Xia, Z.Y. Ku, S.C. Lee, S.R.J. Brueck, Nanostructures and functional materials fabricated by interferometric lithography, Adv. Mater. 23 (2) (2011) 147–179. [11] H.G. Rhee, D.I. Kim, Y.W. Lee, Realization and performance evaluation of high speed autofocusing for direct laser lithography, Rev. Sci. Instrum. 80 (7) (2009) 073103. [12] J. Wei, K. Zhang, T. Wei, Y. Wang, Y. Wu, M. Xiao, High-speed maskless nanolithography with visible light based on photothermal localization, Sci. Rep. 7 (2017) 43892. [13] Z. Bai, J. Wei, X. Liang, K. Zhang, T. Wei, R. Wang, High-speed laser writing of arbitrary patterns in polar coordinate system, Rev. Sci. Instrum. 87 (12) (2016) 125118.
Z. Bai, J. Wei / Optics and Laser Technology 106 (2018) 145–151 [14] I. Stoica, A.I. Barzic, C. Hulubei, Fabrication of nanochannels on polyimide films using dynamic plowing lithography, Appl. Surf. Sci. 426 (2017) 307–314. [15] D.E. Denk, A.G. Poleshchuk, Methods of improving the accuracy of operation of an autofocus in a circular laser writing system, Optoelectronics, Instrum. Data Process. 46 (1) (2010) 87–95. [16] Y.C. Lee, S. Chao, C.C. Huang, C.T. Yang, Design of a dual-wavelength optical head for submicron-scale and nanoscale lithography, IEEE Trans. Magn. 47 (3) (2011) 696–700. [17] A.G. Poleshchuk et al., Polar coordinate laser pattern generator for fabrication of diffractive optical elements with arbitrary structure, Appl. Opt. 38 (8) (1999) 1295–1301. [18] H.G. Rhee, D. Kim, S.K. Hong, Y.W. Lee, 300 mm reference wafer fabrication by using direct laser lithography, Rev. Sci. Instrum. 79 (10) (2008) 103103. [19] H.G. Rhee, D.I. Kim, J.B. Song, H.J. Lee, Y.W. Lee, J.H. Jo, Performance evaluation of laser lithographic machine for computer-generated hologram, Int. J. Adv. Manuf. Technol. 52 (9–12) (2010) 1005–1009.
151
[20] B. Hnilicka, A. Besançon-Voda, H.-J. Schroder, G. Filardi, Modelling the focus error characteristic of a DVD player, in: Control Applications, Proceedings of the 2002 International Conference on IEEE, vol. 2, 2002, pp. 629–630. [21] C.-H. Liu, Z.-H. Li, Application of the astigmatic method to the thickness measurement of glass substrates, Appl. Opt. 47 (21) (2008) 3968–3972. [22] A. Nikolov, D. Wasan, Oil lenses on the air-water surface and the validity of Neumann’s rule, Adv. Colloid Interface Sci. 244 (2017) 174–183. [23] J. Fu, Y. Wang, P.F. Chen, Generation of sub-half-wavelength non-diffracting beams with tunable depth of focus by focusing azimuthally polarized beams, Appl. Phys. B 123 (7) (2017) 214. [24] J. Steele, H. Yuan, C. Tan, M. Keshavarz, C. Steuwe, M. Roeffaers, J. Hofkens, Direct laser writing of d- to a-phase transformation in Formamidinium lead Iodide, ACS Nano. 11 (2017) 8072–8083. [25] R. Wang, J. Wei, Y. Fan, Chalcogenide phase-change thin films used as grayscale photolithography materials, Opt. Express 22 (2014) 4973–4984. [26] M. Mansuripur, Analysis of astigmatic focusing and push–pull tracking error signals in magnetooptical disk systems, Appl. Optics 26 (1987) 3981–3986.