In this paper, we propose a reflective two-dimensional (2D) metal-dielectric grating based on cylindrical hole nano arrays with excellent polarization-independent high diffraction efficiency. The effects of the geometrical parameters on the polarization characteristic and diffraction efficiency are studied. Optimized results show that the (-1, 0) order diffraction efficiency of transverse electric (TE) and transverse magnetic (TM) polarizations under Littrow mounting is 98.31% and 98.05% at 780 nm incident wavelength, and the diffraction efficiency equilibrium is 99.74%, which is a significant improvement over the previously reported 2D gratings. The high efficiency in both TE and TM polarizations makes it a potential candidate as planar grating rulers for high precision multi-axis displacement measurement. Moreover, the cylindrical hole-based structure performs well in manufacturing tolerances, which provides the possibility for practical applications.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
With the development of advanced manufacturing technology, the feature size of crucial integrated circuits and systems is rapidly approaching the nano-scale [1,2], which has promoted the advancement of precision displacement measurement at nanometer level. Common measuring techniques for nano-scale measurement include optical fiber displacement sensor , atomic force microscope , capacitive inductance micrometer , scanning tunneling microscope , displacement laser interferometer [7,8], and grating ruler [9,10]. The laser interferometer for nano-scale accurate displacement measurement is most widely used, which obtains the displacement by detecting the interference information to achieve high precision long range measurement. However, the laser interferometer is bulky and based on the laser wavelength, so disturbances such as temperature fluctuations and air drift will seriously damage the wavelength stability of the laser interferometer and cause considerable measurement errors.
Compared with displacement laser interferometers, grating rulers based on grating period are more insensitive to environmental disturbances. According to the latest research of PTB, National Metrology Institute of Germany, the high-precision grating ruler is superior to the laser interferometer in the repetitive positioning accuracy and long-term stability in the non-vacuum environment, since the actual optical path of the grating ruler is much smaller . Another research from semiconductor industry indicates that the signal error measured by the grating ruler is less than 0.2 nm, while the laser interferometer is less than 0.9 nm in a displacement range of 15 mm, which shows that the grating ruler has obvious advantages in the measurement stability compared with the laser interferometer .
Since two-dimensional (2D) gratings exhibit periodicity in both x and y directions, they are more suitable for multi-axis displacement measurement as grating rulers than one-dimensional (1D) gratings. In addition, the advantage of 2D gratings compared to 1D gratings is that the Abbe error is eliminated and the accuracy is improved. Based on two dimensional gratings, a variety of high-precision multi-axis measurement schemes has emerged [13–16]. Hung-Lin Hsieh et al.  proposed a 2D displacement measurement method based on a special quasi-common light path structure with a resolution of 2.52 nm. Akihide Kimura et al.  proposed a three-axis planar encoder for sub-nanometer step motion measurement, which can achieve a three-axis resolution of about 1nm. It is worth noting that 2D diffraction grating rulers are also used in semiconductor industry for nanoscale lithography, reducing the processing error from 4.8nm to 2.5nm .
Extensive studies on two dimensional gratings have been performed, ranging from novel structure design to fabrication techniques and applications [18–21]. In the case of high precision measurement, the 2D grating is required to have polarization-independent characteristics and high diffraction efficiency . Poor polarization-independent characteristics decrease the modulation depth and signal contrast when performing numerical processing. Low diffraction efficiency, on the other hand, leads to the introduction of unnecessary noise which will reduce the signal-to-noise ratio (SNR). Higher diffraction efficiency and better polarization-independent characteristics are always being pursued. In Ref. , a chessboard gold-coated 2D grating was proposed for normal incidence, the simulated maximum diffraction efficiency reached only 16% for four first orders, which is not satisfactory. Recently, Junming Chen et al.  designed a metal-dielectric grating based on conical frustum ridge structure with diffraction efficiency of 81% (TE) and 70% (TM) at 780 nm wavelength. The diffraction efficiency and polarization insensitivity are greatly improved.
In this paper, a novel cylindrical hole-based reflective 2D grating with polarization-independent high diffraction efficiency is proposed. The efficiency of (-1, 0) order for TE and TM polarizations reaches 98.31% and 98.05%, at the wavelength of 780 nm under Littrow mounting, which, to the best of our knowledge, is superior to previously reported results. The diffraction property with varied incident wavelength and angle is given. Moreover, the manufacturing tolerances are also analyzed, revealing the feasibility of its fabrication and promising prospect in the application of high precision displacement measurement as grating rulers.
2. Structure and materials
Diffraction occurs when light strikes the grating and it can be divided into different diffraction orders according to the wavelength, grating period and incident angle. The 2D grating is an extension of the 1D grating, and the diffraction equation of the 2D grating can be expressed as 
According to the diffraction grating equations, the diffracted light mainly exists in the 0th order and ±1st orders and the higher order diffracted light disappears in the form of evanescent waves on the grating surface, by reasonably selecting the incident wavelength and grating period. In this paper, we propose a reflective 2D metal-dielectric grating based on cylindrical hole nano arrays under Littrow mounting. The grating period Λ in two directions is 833.33nm (line density 1200l/mm), and the incident wavelength λ is 780nm. According to Eqs. (1)–(3), only (-1, 0) order and (0, 0) order exist.
As shown in Fig. 1, the inorganic glass of Zerodur supports an extremely low coefficient of thermal expansion for a wide range of temperatures, which is the best choice to be the substrate . On the substrate, silver (Ag) film is used as a reflective layer due to its considerable broadband high reflection in the near infrared region . A tantalum pentoxide (Ta2O5) layer is deposited on top of the film with cylindrical hole patterns to form the grating structure, of which the sidewall angle is ψ and the depth is l. And since diffraction light needs reasonable phase modulation, a buffer layer which is made of fused silicon (SiO2) is inserted between the grating layer and the Ag film. The refractive index of Ag, SiO2 and Ta2O5 film at the operating wavelength of 780 nm are 0.0905+i5.0617, 1.45 and 2.1, which are taken from Refs. [26–28].
3. Parameter optimizations
The geometric parameters of the grating, including duty cycle f (which is defined as the ratio of the hole diameter to the period), sidewall angle ψ, Ta2O5 thickness h1, SiO2 thickness h2, and cylindrical hole depth l, have significant influences on the diffraction efficiency. These parameters need to be optimized delicately to achieve high diffraction efficiency in (-1, 0) order for both TE and TM polarizations. In addition, as reflective layer, the Ag film is not involved in the grating optimization process, so a thickness (h3) of 200 nm is chosen to reflect incident light totally. Although electromagnetic field calculation based on Finite-Difference Time-Domain (FDTD) method can be performed for any structures, it requires tremendous computer memory and costs too much time . The Rigorous Coupled Wave Analysis method (RCWA, also called Fourier modal method, FMM) can be used to calculate periodic structures like 2D gratings by solving the Maxwell equations of each repeated cell based on Fourier expansion of the relative permittivity [30,31]. The speed and accuracy of calculation can be controlled by adjusting the number of Fourier modes, which has the advantages of small computer memory and fast convergence speed.
To find the highest diffraction efficiency for both TE and TM polarizations, simulated annealing (SA) algorithm is used to optimize f, ψ, h1, h2, and l. SA algorithm is an optimization technique that can find the global solution by skipping out of local minima with a certain probability . The probability is determined by Boltzmann distribution which is used to predict distribution of particle state in Thermodynamics. The cost function (CF) is defined as CF = 1 - 1/2 × (η(-1,0)TE + η(-1,0)TM), where η(-1,0)TE and η(-1,0)TM are Littrow diffraction efficiencies of TE and TM polarizations, respectively .
When the minimum cost function is achieved in the optimization process, the diffraction efficiency of η(-1,0)TE = 98.31% and η(-1,0)TM = 98.05% is obtained, which, to the best of our knowledge, is the highest in all reported studies. The optimized grating parameters are f = 0.465, ψ = 88°, h1 = l = 556 nm, h2 = 520 nm. Considering the Ag reflectance of S-(98.89%) and P-(98.57%) polarized light at 780 nm under incident angle of 27.9° (Littrow angle) , almost all of reflected light is diffracted into (-1, 0) order. In contrast with previously reported 2D gratings (no more than 81% in Ref. ), this improvement is tremendous, and thus should be attractive for applications in multi-axis high precision displacement measurement.
Figure 2 shows the influence of SiO2 thickness and Ta2O5 thickness on the polarized diffraction at duty cycle of 0.465 and sidewall angle of 88°. As can be seen in Fig. 2, in the region enclosed by the solid black line, where the thickness of SiO2 varies from 477 nm to 570 nm and the thickness of Ta2O5 changes from 543 nm to 568 nm, incident light of both TE and TM polarizations can achieve diffraction efficiency of higher than 90%. In particular, the diffraction efficiency can reach more than 95% when the thickness of SiO2 is within the range of 490nm-570 nm and the thickness of Ta2O5 is within the range of 550nm-560 nm, which is compatible with film deposition techniques.
Before we dig into the diffraction property of the cylindrical hole-based grating, it is necessary to define the extinction ratio and diffraction efficiency equilibrium, since reflected light of (0, 0) order as well as polarization-dependent diffraction will reduce the interference fringe contrast and phase demodulation accuracy in grating ruler system. Ignoring the energy absorbed by the metal, the (-1, 0) order diffraction efficiency η-1,0 divided by the (0, 0) order diffraction efficiency η0,0 is defined as the extinction ratio , which is given by34]
Figure 3 shows the diffraction efficiency and Eq versus incident wavelength under Littrow condition with the optimized grating parameters. At the central wavelength of 780 nm, the polarization-independent characteristic and high efficiency can be achieved. When the wavelength shifts away from 780 nm, the diffraction efficiency for both polarizations declines. Although the diffraction efficiency is higher than 90% only in the narrow band from 778.5 nm to 781.5 nm, it is still large enough to relieve the drift effect of the operating wavelength in grating ruler system. For Littrow mounting grating rulers, a certain incident angle bandwidth is required for actual multi-axis measurement applications. Figure 4 shows the efficiency versus incident angle for 780 nm wavelength, in which the efficiency varies little around the Littrow angle (θ-1,0=27.9°). (0, 0) order of each polarization has been sufficiently suppressed with the optimized grating parameters, while (-1, 0) order efficiency is more than 90% and diffraction efficiency equilibrium is higher than 99% for the incidence angle of 26.75°-29.05°.
4. Tolerance analysis
To fabricate the cylindrical hole-based grating, the process of two-dimensional photolithography and reactive ion etching (RIE) is usually involved [35,36]. During the process, it is inevitable that the actual shape of the grating structure may slightly deviate from the optimized value. So, it is necessary to analyze the manufacturing tolerances of etching depth, diameter (duty cycle) and sidewall angle of the cylindrical holes. Under the optimal thickness of Ta2O5 and SiO2, Fig. 5 shows the dependence of diffraction efficiency on the cylindrical hole depth, which is not equal to the thickness of Ta2O5 film any more. Both TE and TM polarized incident light can achieve diffraction efficiency of over 90% and diffraction efficiency equilibrium higher than 93% with the cylindrical hole depth ranging from 538 nm to 598 nm, which provides 60 nm production tolerance for cylindrical hole etching.
In addition to the etching depth, the variation of duty cycle (i.e. the diameter of the hole) also has great influence on the diffraction property. Although in-situ development endpoint detection method has been proposed to control the duty cycle of holographic cross grating , the control accuracy is not high enough to ignore its effect. Furthermore, in the development and RIE etching process, the nano hole grating profile obtained is not completely vertical, and as shown in Fig. 1(c) the upper diameter of the hole is usually larger than that in the bottom. Figure 6 depicts the relationship between the TE, TM polarization (-1, 0) order diffraction efficiency and the sidewall angle and duty cycle, when the hole depth is equal to the thickness of Ta2O5. In the area of interest, diffraction efficiency higher than 90% for TE and TM polarizations can be achieved, and the highest efficiency is 98%. The high diffraction efficiency can be found within the side wall angle of 85°-90° and the duty cycle of 0.45-0.5. The figures show that when the sidewall angle decreases, the duty cycle will increase to ensure high diffraction efficiency. It is obvious that the tolerance under TE polarization is not as good as that under TM polarization, but they still exhibit excellent polarization-independent characteristics as the two parameters change.
To improve the duty cycle tolerance, the hole depth is varied, since it is not necessarily equal to the thickness of the Ta2O5 film. It is found that although the maximum diffraction efficiency of the grating decreases, the grating exhibits better duty cycle compatibility when the hole depth is less than the thickness of Ta2O5. As shown in Fig. 7, when the hole depth is 530 nm, the acceptable duty cycle range is twice as large as that when the hole depth is equal to the thickness of Ta2O5(556 nm).
In summary, a novel reflective 2D metal-dielectric grating with cylindrical hole nano arrays is proposed, which can be used as grating rulers. High diffraction efficiency and perfect polarization-independent characteristic is achieved due to the specific structure. After simulation and optimization based on RCWA, (-1,0) order diffraction efficiency of TE and TM polarizations reaches 98.31% and 98.05% respectively, and the diffraction efficiency equilibrium is higher than 99%, which is superior to previously reported 2D gratings. Moreover, it also shows moderate manufacture tolerance when the etching depth, duty cycle and side wall angle vary from optimal conditions. Therefore, this 2D grating is beneficial to improve the interference fringe contrast of the grating ruler system and achieve high SNR as well as high accuracy, which provides considerable potential in high precision displacement measurement applications.
Fundamental Research Funds for the Central Universities (21619343); Guangzhou Science and Technology Program key projects (202007010001).
The authors declare no conflicts of interest.
1. R. Xie, P. Montanini, K. Akarvardar, N. Tripathi, B. Haran, S. Johnson, T. Hook, B. Hamieh, D. Corliss, and J. Wang, “A 7 nm FinFET technology featuring EUV patterning and dual strained high mobility channels,” in IEEE International Electron Devices Meeting (IEEE, 2016), pp. 2. 7.1–2. 7. 4.
2. T. Singh, S. Rangarajan, D. John, R. Schreiber, S. Oliver, R. Seahra, and A. Schaefer, “2.1 Zen 2: The AMD 7 nm Energy-Efficient High-Performance x86-64 Microprocessor Core,” in IEEE International Solid- State Circuits Conference (IEEE, 2020), pp. 42–44.
3. B. Zuanetti, T. Wang, and V. Prakash, “A compact fiber optics-based heterodyne combined normal and transverse displacement interferometer,” Rev. Sci. Instrum. 88(3), 033108 (2017). [CrossRef]
4. F. J. Giessibl, “The qPlus sensor, a powerful core for the atomic force microscope,” Rev. Sci. Instrum. 90(1), 011101 (2019). [CrossRef]
5. H. Furuichi and R. S. Fearing, “A planar capacitive micro positioning sensor,” in MHS’96 Proceedings of the Seventh International Symposium on Micro Machine and Human Science (IEEE, 1996), pp. 85-90.
6. G. Binnig, H. Rohrer, C. Gerber, and E. Weibel, “Surface Studies by Scanning Tunneling Microscopy,” Phys. Rev. Lett. 49(1), 57–61 (1982). [CrossRef]
7. J. Qi, Z. Wang, J. Huang, Q. Wang, and J. Gao, “Heterodyne interferometer with two parallel-polarized input beams for high-resolution roll angle measurement,” Opt. Express 27(10), 13820–13830 (2019). [CrossRef]
8. B. Wang, X. Wang, Z. Li, and O. Sasaki, “Sinusoidal phase-modulating laser diode interferometer insensitive to intensity modulation for real-time displacement measurement with feedback control system,” Opt. Commun. 285(18), 3827–3831 (2012). [CrossRef]
9. 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]
10. Y. Shimizu, R. Aihara, K. Mano, C. Chen, Y.-L. Chen, X. Chen, and W. J. P. E. 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]
11. J. Guan, P. Köchert, C. Weichert, and R. Tutsch, “A high performance one-dimensional homodyne encoder and the proof of principle of a novel two-dimensional homodyne encoder,” Precis. Eng. 37(4), 865–870 (2013). [CrossRef]
12. T. Castenmiller, F. van de Mast, T. de Kort, C. van de Vin, M. de Wit, R. Stegen, and S. van Cleef, “Towards ultimate optical lithography with NXT:1950i dual stage immersion platform,” Proc. SPIE 7640, 76401N (2010). [CrossRef]
13. K. Fan, B. Liao, Y. Chung, and T. Chung, “Displacement measurement of planar stage by diffraction planar encoder in nanometer resolution,” in IEEE International Instrumentation and Measurement Technology Conference Proceedings (IEEE, 2012), pp. 894–897.
14. P. Wei, X. Lu, D. Qiao, L. Zou, X. Huang, J. Tan, and Z. Lu, “Two-dimensional displacement measurement based on two parallel gratings,” Rev. Sci. Instrum. 89(6), 065105 (2018). [CrossRef]
15. H.-L. Hsieh, J.-C. Chen, G. Lerondel, and J.-Y. J. O. E. Lee, “Two-dimensional displacement measurement by quasi-common-optical-path heterodyne grating interferometer,” Opt. Express 19(10), 9770–9782 (2011). [CrossRef]
16. 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]
17. F. de Jong, B. van der Pasch, T. Castenmiller, B. Vleeming, R. Droste, and F. van de Mast, “Enabling the lithography roadmap: an immersion tool based on a novel stage positioning system,” Proc. SPIE 7274, 72741S (2009). [CrossRef]
18. Y. Ra’di, D. L. Sounas, and A. Alu, “Metagratings: Beyond the Limits of Graded Metasurfaces for Wave Front Control,” Phys. Rev. Lett. 119(6), 067404 (2017). [CrossRef]
19. P. Sun, C. Zhou, W. Jia, J. Wang, C. Xiang, Y. Xie, and D. Zhao, “Self-referenced refractive index sensor based on hybrid mode resonances in 2D metal-dielectric grating,” J. Phys. D: Appl. Phys. 53(14), 145101 (2020). [CrossRef]
20. H. Kaur and M. Kumar, “Two-way reflector based on two-dimensional sub-wavelength high-index contrast grating on SOI,” Opt. Commun. 366, 266–270 (2016). [CrossRef]
21. S. Wang and L. Zeng, “Controlling the duty cycle of holographic crossed gratings by in situ endpoint detection during development,” Appl. Opt. 55(10), 2657–2663 (2016). [CrossRef]
22. C. Lin, S. Yan, Z. Du, C. Wei, and G. Wang, “High-efficiency gold-coated cross-grating for heterodyne grating interferometer with improved signal contrast and optical subdivision,” Opt. Commun. 339, 86–93 (2015). [CrossRef]
23. J. Chen, Y. Zhang, Y. Wang, F. Kong, Y. Jin, P. Chen, J. Xu, S. Sun, and J. Shao, “Polarization-independent two-dimensional diffraction metal-dielectric grating,” Appl. Phys. Lett. 113(4), 041905 (2018). [CrossRef]
24. E. Noponen and J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11(9), 2494–2502 (1994). [CrossRef]
25. M. Balat-Pichelin, D. De Sousa Meneses, and J. Annaloro, “Behavior and optical properties of Zerodur® at high temperatures,” Infrared Phys. Technol. 101, 68–77 (2019). [CrossRef]
26. A. Ciesielski, L. Skowronski, M. Trzcinski, and T. Szoplik, “Controlling the optical parameters of self-assembled silver films with wetting layers and annealing,” Appl. Surf. Sci. 421, 349–356 (2017). [CrossRef]
27. I. H. Malitson, “Interspecimen Comparison of the Refractive Index of Fused Silica,” J. Opt. Soc. Am. 55(10), 1205–1209 (1965). [CrossRef]
28. T. J. Bright, J. I. Watjen, Z. M. Zhang, C. Muratore, A. A. Voevodin, D. I. Koukis, D. B. Tanner, and D. J. Arenas, “Infrared optical properties of amorphous and nanocrystalline Ta2O5 thin films,” J. Appl. Phys. 114(8), 083515 (2013). [CrossRef]
29. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 687–702 (2010). [CrossRef]
30. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12(5), 1068–1076 (1995). [CrossRef]
31. L. J. J. A. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14(10), 2758–2767 (1997). [CrossRef]
32. Z. Yin, J. Yu, Y. Lu, and C. Zhou, “Broadband High-Efficiency Gratings Operating at the 2nd Order Designed by Simplified Modal Method,” IEEE Photonics Technol. Lett. 32(6), 309–312 (2020). [CrossRef]
33. B. Wang, L. Chen, L. Lei, and J. Zhou, “Enhancement of the Efficiency Based on a Sandwiched Grating for Separating Polarized Beams,” IEEE Photonics Technol. Lett. 24(17), 1513–1515 (2012). [CrossRef]
34. Y. Tang, X. Chen, C. Li, R. Wang, H. Xu, and Y. Cheng, “Design of crossed planar phase grating for metrology,” Proc. SPIE 10621, 10621D (2018). [CrossRef]
35. C. Lin, S. Yan, and F. J. O. C. You, “Fabrication and characterization of short-period double-layer cross-grating with holographic lithography,” Opt. Commun. 383, 17–25 (2017). [CrossRef]
36. H. Jansen, H. Gardeniers, M. D. Boer, M. Elwenspoek, and J. Fluitman, “A survey on the reactive ion etching of silicon in microtechnology,” J. Micromech. Microeng. 6(1), 14–28 (1996). [CrossRef]