Abstract

Moiré interferometry is widely used as the precise metrology in many science and engineering fields. The schemes of moirés-based interferometry adopting diffraction gratings are presented in this paper for applications in a proximity lithographic system such as wafer-mask alignment, the in-plane twist angle adjustment, and tilts remediation. For the sake of adjustment of lateral offset as well as the tilt and in-plane twist angle, schemes of the (m,m) and (m,0) moiré interferometry are explored, respectively. Fundamental derivation of the moiré interferometry and schemes for related applications are provided. Three pairs of gratings with close periods are fabricated to form the composite grating. And experiments are performed to confirm the moiré interferometry for related applications in our proximity lithographic system. Experimental results indicate that unaligned lateral offset is detectable with resolution at the nanometer level, and the tilt and in-plane twist angle between wafer and mask could be manually decreased down to the scope of 103 and 104rad, respectively.

© 2014 Optical Society of America

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    [CrossRef]
  26. E. E. Moona and H. I. Smith, “Nanometer-precision pattern registration for scanning-probe lithographies using interferometric-spatial-phase imaging,” J. Vac. Sci. Technol. B 24, 3083–3087 (2006).
    [CrossRef]
  27. S. Zhou, Y. Fu, X. Tang, S. Hu, W. Chen, and Y. Yang, “Fourier-based analysis of moiré fringe patterns of superposed gratings in alignment of nanolithography,” Opt. Express 16, 7869–7880 (2008).
    [CrossRef]
  28. S. Zhou, C. Xie, Y. Yang, S. Hu, X. Xu, and J. Yang, “Moiré-based phase imaging for sensing and adjustment of in-plane twist angle,” Photonics Technol. Lett. IEEE 25, 1847–1850 (2013).
    [CrossRef]

2013

S. Zhou, C. Xie, Y. Yang, S. Hu, X. Xu, and J. Yang, “Moiré-based phase imaging for sensing and adjustment of in-plane twist angle,” Photonics Technol. Lett. IEEE 25, 1847–1850 (2013).
[CrossRef]

J. C. Wyant, “Computerized interferometric surface measurements,” Appl. Opt. 52, 1–8 (2013).
[CrossRef]

2012

W. Chen, R. H. W. Lam, and J. Fu, “Photolithographic surface micromachining of polydimethylsiloxane (PDMS),” Lab Chip 12, 391–395 (2012).
[CrossRef]

Y. Zhang, N. Gao, and C. Xie, “Using circular Dammann gratings to produce impulse optic vortex rings,” Appl. Phys. Lett. 100, 041107 (2012).
[CrossRef]

2011

T. Scharnweber, R. Truckenmüller, A. M. Schneider, A. Welle, M. Reinhardt, and S. Giselbrecht, “Rapid prototyping of microstructures in polydimethylsiloxane (PDMS) by direct UV-lithography,” Lab Chip 11, 1368–1371 (2011).
[CrossRef]

2008

2007

2006

E. E. Moona and H. I. Smith, “Nanometer-precision pattern registration for scanning-probe lithographies using interferometric-spatial-phase imaging,” J. Vac. Sci. Technol. B 24, 3083–3087 (2006).
[CrossRef]

2005

J.-H. Lee, C.-H. Kim, Y.-S. Kim, K.-M. Ho, K. Constant, W. Leung, and C.-H. Oh, “Diffracted moiré fringes as analysis and alignment tools for multilayer fabrication in soft lithography,” Appl. Phys. Lett. 86, 204101 (2005).
[CrossRef]

P. Yao, G. J. Schneider, B. Miao, D. W. Prather, E. D. Wetzel, and D. J. O’Brien, “Fabrication of three-dimensional photonic crystals with multilayer photolithography,” Proc. SPIE 5720, 27–35 (2005).
[CrossRef]

2004

R. Menon, E. E. Moon, M. K. Mondol, F. J. Castaño, and H. I. Smith, “Scanning-spatial-phase alignment for zone-plate-array lithography,” J. Vac. Sci. Technol. B 22, 3382–3385 (2004).
[CrossRef]

2003

E. E. Moon, L. Chen, P. N. Everett, M. K. Mondol, and H. I. Smith, “Interferometric-spatial-phase imaging for six-axis mask control,” J. Vac. Sci. Technol. B 21, 3112–3115 (2003).
[CrossRef]

2002

G. Pugh and M. Giorgi, “Evaluation of ASML ATHENA alignment system on Intel front-end processes,” Proc. SPIE 4689, 286–294 (2002).
[CrossRef]

2001

R. Navarro and S. Keij, “Extended ATHENA alignment performance and application for the 100-nm technology node,” Proc. SPIE 4344, 682–694 (2001).
[CrossRef]

J. C. Love, D. B. Wolfe, H. O. Jacobs, and G. M. Whitesides, “Microscope projection photolithography for rapid prototyping of masters with micron-scale features for use in soft lithography,” Langmuir 17, 6005–6012 (2001).
[CrossRef]

G. M. Whitesides, E. Ostuni, S. Takayama, X. Jiang, and D. E. Ingber, “Soft lithography in biology and biochemistry,” Annu. Rev. Biomed. Eng. 3, 335–373 (2001).
[CrossRef]

1999

1994

1993

1989

M. Suzuki and A. Une, “An optical-heterodyne alignment technique for quarter-micron x-ray lithography,” J. Vac. Sci. Technol. B 7, 1971–1976 (1989).
[CrossRef]

1988

C. A. Walker, “Moiré interferometry for strain analysis,” Opt. Lasers Eng. 8, 213–262 (1988).
[CrossRef]

J. Itoh, T. Kanayama, N. Atoda, and K. Hoh, “An alignment system for synchrotron radiation x‐ray lithography,” J. Vac. Sci. Technol. B 6, 409–412 (1988).
[CrossRef]

1987

Y. Uchida, S. Hattori, and T. Nomura, “An automatic mask alignment system using moire interference,” J. Vac. Sci. Technol. B 5, 244–247 (1987).
[CrossRef]

1979

G. Bouwhuis and S. Wittekoek, “Automatic alignment system for optical projection printing,” IEEE Trans. Electron Devices ED-26, 723–728 (1979).
[CrossRef]

1977

D. C. Flanders and H. I. Smith, “A new interferometric alignment technique,” Appl. Phys. Lett. 31, 426–428 (1977).
[CrossRef]

1972

Atoda, N.

J. Itoh, T. Kanayama, N. Atoda, and K. Hoh, “An alignment system for synchrotron radiation x‐ray lithography,” J. Vac. Sci. Technol. B 6, 409–412 (1988).
[CrossRef]

Bennett, V. P.

Bouwhuis, G.

G. Bouwhuis and S. Wittekoek, “Automatic alignment system for optical projection printing,” IEEE Trans. Electron Devices ED-26, 723–728 (1979).
[CrossRef]

Brueck, S. R.

Caber, P. J.

Castaño, F. J.

R. Menon, E. E. Moon, M. K. Mondol, F. J. Castaño, and H. I. Smith, “Scanning-spatial-phase alignment for zone-plate-array lithography,” J. Vac. Sci. Technol. B 22, 3382–3385 (2004).
[CrossRef]

Chen, D. J.

Chen, L.

E. E. Moon, L. Chen, P. N. Everett, M. K. Mondol, and H. I. Smith, “Interferometric-spatial-phase imaging for six-axis mask control,” J. Vac. Sci. Technol. B 21, 3112–3115 (2003).
[CrossRef]

Chen, W.

Chen, X.

Chiang, F. P.

Constant, K.

J.-H. Lee, C.-H. Kim, Y.-S. Kim, K.-M. Ho, K. Constant, W. Leung, and C.-H. Oh, “Diffracted moiré fringes as analysis and alignment tools for multilayer fabrication in soft lithography,” Appl. Phys. Lett. 86, 204101 (2005).
[CrossRef]

Everett, P. N.

E. E. Moon, L. Chen, P. N. Everett, M. K. Mondol, and H. I. Smith, “Interferometric-spatial-phase imaging for six-axis mask control,” J. Vac. Sci. Technol. B 21, 3112–3115 (2003).
[CrossRef]

Flanders, D. C.

D. C. Flanders and H. I. Smith, “A new interferometric alignment technique,” Appl. Phys. Lett. 31, 426–428 (1977).
[CrossRef]

Fu, J.

W. Chen, R. H. W. Lam, and J. Fu, “Photolithographic surface micromachining of polydimethylsiloxane (PDMS),” Lab Chip 12, 391–395 (2012).
[CrossRef]

Fu, Y.

Gao, N.

Y. Zhang, N. Gao, and C. Xie, “Using circular Dammann gratings to produce impulse optic vortex rings,” Appl. Phys. Lett. 100, 041107 (2012).
[CrossRef]

Giorgi, M.

G. Pugh and M. Giorgi, “Evaluation of ASML ATHENA alignment system on Intel front-end processes,” Proc. SPIE 4689, 286–294 (2002).
[CrossRef]

Giselbrecht, S.

T. Scharnweber, R. Truckenmüller, A. M. Schneider, A. Welle, M. Reinhardt, and S. Giselbrecht, “Rapid prototyping of microstructures in polydimethylsiloxane (PDMS) by direct UV-lithography,” Lab Chip 11, 1368–1371 (2011).
[CrossRef]

Hattori, S.

Y. Uchida, S. Hattori, and T. Nomura, “An automatic mask alignment system using moire interference,” J. Vac. Sci. Technol. B 5, 244–247 (1987).
[CrossRef]

Ho, K.-M.

J.-H. Lee, C.-H. Kim, Y.-S. Kim, K.-M. Ho, K. Constant, W. Leung, and C.-H. Oh, “Diffracted moiré fringes as analysis and alignment tools for multilayer fabrication in soft lithography,” Appl. Phys. Lett. 86, 204101 (2005).
[CrossRef]

Hoh, K.

J. Itoh, T. Kanayama, N. Atoda, and K. Hoh, “An alignment system for synchrotron radiation x‐ray lithography,” J. Vac. Sci. Technol. B 6, 409–412 (1988).
[CrossRef]

Hu, S.

S. Zhou, C. Xie, Y. Yang, S. Hu, X. Xu, and J. Yang, “Moiré-based phase imaging for sensing and adjustment of in-plane twist angle,” Photonics Technol. Lett. IEEE 25, 1847–1850 (2013).
[CrossRef]

S. Zhou, Y. Fu, X. Tang, S. Hu, W. Chen, and Y. Yang, “Fourier-based analysis of moiré fringe patterns of superposed gratings in alignment of nanolithography,” Opt. Express 16, 7869–7880 (2008).
[CrossRef]

Ingber, D. E.

G. M. Whitesides, E. Ostuni, S. Takayama, X. Jiang, and D. E. Ingber, “Soft lithography in biology and biochemistry,” Annu. Rev. Biomed. Eng. 3, 335–373 (2001).
[CrossRef]

Iskander, M.

M. Iskander, “Optical measurement of strain and stress,” in Modelling with Transparent Soils SE - 4 (Springer, 2010), pp. 27–44.

Itoh, J.

J. Itoh, T. Kanayama, N. Atoda, and K. Hoh, “An alignment system for synchrotron radiation x‐ray lithography,” J. Vac. Sci. Technol. B 6, 409–412 (1988).
[CrossRef]

Jacobs, H. O.

J. C. Love, D. B. Wolfe, H. O. Jacobs, and G. M. Whitesides, “Microscope projection photolithography for rapid prototyping of masters with micron-scale features for use in soft lithography,” Langmuir 17, 6005–6012 (2001).
[CrossRef]

Jiang, X.

G. M. Whitesides, E. Ostuni, S. Takayama, X. Jiang, and D. E. Ingber, “Soft lithography in biology and biochemistry,” Annu. Rev. Biomed. Eng. 3, 335–373 (2001).
[CrossRef]

Kanayama, T.

J. Itoh, T. Kanayama, N. Atoda, and K. Hoh, “An alignment system for synchrotron radiation x‐ray lithography,” J. Vac. Sci. Technol. B 6, 409–412 (1988).
[CrossRef]

Keij, S.

R. Navarro and S. Keij, “Extended ATHENA alignment performance and application for the 100-nm technology node,” Proc. SPIE 4344, 682–694 (2001).
[CrossRef]

Kim, C.-H.

J.-H. Lee, C.-H. Kim, Y.-S. Kim, K.-M. Ho, K. Constant, W. Leung, and C.-H. Oh, “Diffracted moiré fringes as analysis and alignment tools for multilayer fabrication in soft lithography,” Appl. Phys. Lett. 86, 204101 (2005).
[CrossRef]

Kim, Y.-S.

J.-H. Lee, C.-H. Kim, Y.-S. Kim, K.-M. Ho, K. Constant, W. Leung, and C.-H. Oh, “Diffracted moiré fringes as analysis and alignment tools for multilayer fabrication in soft lithography,” Appl. Phys. Lett. 86, 204101 (2005).
[CrossRef]

Kuznetsova, Y.

Lam, R. H. W.

W. Chen, R. H. W. Lam, and J. Fu, “Photolithographic surface micromachining of polydimethylsiloxane (PDMS),” Lab Chip 12, 391–395 (2012).
[CrossRef]

Lee, J.-H.

J.-H. Lee, C.-H. Kim, Y.-S. Kim, K.-M. Ho, K. Constant, W. Leung, and C.-H. Oh, “Diffracted moiré fringes as analysis and alignment tools for multilayer fabrication in soft lithography,” Appl. Phys. Lett. 86, 204101 (2005).
[CrossRef]

Leung, W.

J.-H. Lee, C.-H. Kim, Y.-S. Kim, K.-M. Ho, K. Constant, W. Leung, and C.-H. Oh, “Diffracted moiré fringes as analysis and alignment tools for multilayer fabrication in soft lithography,” Appl. Phys. Lett. 86, 204101 (2005).
[CrossRef]

Love, J. C.

J. C. Love, D. B. Wolfe, H. O. Jacobs, and G. M. Whitesides, “Microscope projection photolithography for rapid prototyping of masters with micron-scale features for use in soft lithography,” Langmuir 17, 6005–6012 (2001).
[CrossRef]

Menon, R.

R. Menon, E. E. Moon, M. K. Mondol, F. J. Castaño, and H. I. Smith, “Scanning-spatial-phase alignment for zone-plate-array lithography,” J. Vac. Sci. Technol. B 22, 3382–3385 (2004).
[CrossRef]

Miao, B.

P. Yao, G. J. Schneider, B. Miao, D. W. Prather, E. D. Wetzel, and D. J. O’Brien, “Fabrication of three-dimensional photonic crystals with multilayer photolithography,” Proc. SPIE 5720, 27–35 (2005).
[CrossRef]

Mondol, M. K.

R. Menon, E. E. Moon, M. K. Mondol, F. J. Castaño, and H. I. Smith, “Scanning-spatial-phase alignment for zone-plate-array lithography,” J. Vac. Sci. Technol. B 22, 3382–3385 (2004).
[CrossRef]

E. E. Moon, L. Chen, P. N. Everett, M. K. Mondol, and H. I. Smith, “Interferometric-spatial-phase imaging for six-axis mask control,” J. Vac. Sci. Technol. B 21, 3112–3115 (2003).
[CrossRef]

Moon, E. E.

R. Menon, E. E. Moon, M. K. Mondol, F. J. Castaño, and H. I. Smith, “Scanning-spatial-phase alignment for zone-plate-array lithography,” J. Vac. Sci. Technol. B 22, 3382–3385 (2004).
[CrossRef]

E. E. Moon, L. Chen, P. N. Everett, M. K. Mondol, and H. I. Smith, “Interferometric-spatial-phase imaging for six-axis mask control,” J. Vac. Sci. Technol. B 21, 3112–3115 (2003).
[CrossRef]

Moona, E. E.

E. E. Moona and H. I. Smith, “Nanometer-precision pattern registration for scanning-probe lithographies using interferometric-spatial-phase imaging,” J. Vac. Sci. Technol. B 24, 3083–3087 (2006).
[CrossRef]

Navarro, R.

R. Navarro and S. Keij, “Extended ATHENA alignment performance and application for the 100-nm technology node,” Proc. SPIE 4344, 682–694 (2001).
[CrossRef]

Neumann, A.

Nomura, T.

Y. Uchida, S. Hattori, and T. Nomura, “An automatic mask alignment system using moire interference,” J. Vac. Sci. Technol. B 5, 244–247 (1987).
[CrossRef]

O’Brien, D. J.

P. Yao, G. J. Schneider, B. Miao, D. W. Prather, E. D. Wetzel, and D. J. O’Brien, “Fabrication of three-dimensional photonic crystals with multilayer photolithography,” Proc. SPIE 5720, 27–35 (2005).
[CrossRef]

Oh, C.-H.

J.-H. Lee, C.-H. Kim, Y.-S. Kim, K.-M. Ho, K. Constant, W. Leung, and C.-H. Oh, “Diffracted moiré fringes as analysis and alignment tools for multilayer fabrication in soft lithography,” Appl. Phys. Lett. 86, 204101 (2005).
[CrossRef]

Ostuni, E.

G. M. Whitesides, E. Ostuni, S. Takayama, X. Jiang, and D. E. Ingber, “Soft lithography in biology and biochemistry,” Annu. Rev. Biomed. Eng. 3, 335–373 (2001).
[CrossRef]

Prather, D. W.

P. Yao, G. J. Schneider, B. Miao, D. W. Prather, E. D. Wetzel, and D. J. O’Brien, “Fabrication of three-dimensional photonic crystals with multilayer photolithography,” Proc. SPIE 5720, 27–35 (2005).
[CrossRef]

Pugh, G.

G. Pugh and M. Giorgi, “Evaluation of ASML ATHENA alignment system on Intel front-end processes,” Proc. SPIE 4689, 286–294 (2002).
[CrossRef]

Reinhardt, M.

T. Scharnweber, R. Truckenmüller, A. M. Schneider, A. Welle, M. Reinhardt, and S. Giselbrecht, “Rapid prototyping of microstructures in polydimethylsiloxane (PDMS) by direct UV-lithography,” Lab Chip 11, 1368–1371 (2011).
[CrossRef]

Scharnweber, T.

T. Scharnweber, R. Truckenmüller, A. M. Schneider, A. Welle, M. Reinhardt, and S. Giselbrecht, “Rapid prototyping of microstructures in polydimethylsiloxane (PDMS) by direct UV-lithography,” Lab Chip 11, 1368–1371 (2011).
[CrossRef]

Schneider, A. M.

T. Scharnweber, R. Truckenmüller, A. M. Schneider, A. Welle, M. Reinhardt, and S. Giselbrecht, “Rapid prototyping of microstructures in polydimethylsiloxane (PDMS) by direct UV-lithography,” Lab Chip 11, 1368–1371 (2011).
[CrossRef]

Schneider, G. J.

P. Yao, G. J. Schneider, B. Miao, D. W. Prather, E. D. Wetzel, and D. J. O’Brien, “Fabrication of three-dimensional photonic crystals with multilayer photolithography,” Proc. SPIE 5720, 27–35 (2005).
[CrossRef]

Schwider, J.

Smith, H. I.

E. E. Moona and H. I. Smith, “Nanometer-precision pattern registration for scanning-probe lithographies using interferometric-spatial-phase imaging,” J. Vac. Sci. Technol. B 24, 3083–3087 (2006).
[CrossRef]

R. Menon, E. E. Moon, M. K. Mondol, F. J. Castaño, and H. I. Smith, “Scanning-spatial-phase alignment for zone-plate-array lithography,” J. Vac. Sci. Technol. B 22, 3382–3385 (2004).
[CrossRef]

E. E. Moon, L. Chen, P. N. Everett, M. K. Mondol, and H. I. Smith, “Interferometric-spatial-phase imaging for six-axis mask control,” J. Vac. Sci. Technol. B 21, 3112–3115 (2003).
[CrossRef]

D. C. Flanders and H. I. Smith, “A new interferometric alignment technique,” Appl. Phys. Lett. 31, 426–428 (1977).
[CrossRef]

Suzuki, M.

M. Suzuki and A. Une, “An optical-heterodyne alignment technique for quarter-micron x-ray lithography,” J. Vac. Sci. Technol. B 7, 1971–1976 (1989).
[CrossRef]

Takayama, S.

G. M. Whitesides, E. Ostuni, S. Takayama, X. Jiang, and D. E. Ingber, “Soft lithography in biology and biochemistry,” Annu. Rev. Biomed. Eng. 3, 335–373 (2001).
[CrossRef]

Tang, X.

Truckenmüller, R.

T. Scharnweber, R. Truckenmüller, A. M. Schneider, A. Welle, M. Reinhardt, and S. Giselbrecht, “Rapid prototyping of microstructures in polydimethylsiloxane (PDMS) by direct UV-lithography,” Lab Chip 11, 1368–1371 (2011).
[CrossRef]

Uchida, Y.

Y. Uchida, S. Hattori, and T. Nomura, “An automatic mask alignment system using moire interference,” J. Vac. Sci. Technol. B 5, 244–247 (1987).
[CrossRef]

Une, A.

M. Suzuki and A. Une, “An optical-heterodyne alignment technique for quarter-micron x-ray lithography,” J. Vac. Sci. Technol. B 7, 1971–1976 (1989).
[CrossRef]

Walker, C. A.

C. A. Walker, “Moiré interferometry for strain analysis,” Opt. Lasers Eng. 8, 213–262 (1988).
[CrossRef]

Welle, A.

T. Scharnweber, R. Truckenmüller, A. M. Schneider, A. Welle, M. Reinhardt, and S. Giselbrecht, “Rapid prototyping of microstructures in polydimethylsiloxane (PDMS) by direct UV-lithography,” Lab Chip 11, 1368–1371 (2011).
[CrossRef]

Wetzel, E. D.

P. Yao, G. J. Schneider, B. Miao, D. W. Prather, E. D. Wetzel, and D. J. O’Brien, “Fabrication of three-dimensional photonic crystals with multilayer photolithography,” Proc. SPIE 5720, 27–35 (2005).
[CrossRef]

Whitesides, G. M.

G. M. Whitesides, E. Ostuni, S. Takayama, X. Jiang, and D. E. Ingber, “Soft lithography in biology and biochemistry,” Annu. Rev. Biomed. Eng. 3, 335–373 (2001).
[CrossRef]

J. C. Love, D. B. Wolfe, H. O. Jacobs, and G. M. Whitesides, “Microscope projection photolithography for rapid prototyping of masters with micron-scale features for use in soft lithography,” Langmuir 17, 6005–6012 (2001).
[CrossRef]

Wittekoek, S.

G. Bouwhuis and S. Wittekoek, “Automatic alignment system for optical projection printing,” IEEE Trans. Electron Devices ED-26, 723–728 (1979).
[CrossRef]

Wolfe, D. B.

J. C. Love, D. B. Wolfe, H. O. Jacobs, and G. M. Whitesides, “Microscope projection photolithography for rapid prototyping of masters with micron-scale features for use in soft lithography,” Langmuir 17, 6005–6012 (2001).
[CrossRef]

Wyant, J. C.

Xie, C.

S. Zhou, C. Xie, Y. Yang, S. Hu, X. Xu, and J. Yang, “Moiré-based phase imaging for sensing and adjustment of in-plane twist angle,” Photonics Technol. Lett. IEEE 25, 1847–1850 (2013).
[CrossRef]

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Figures (13)

Fig. 1.
Fig. 1.

Schematic of (m,m) moiré interferometry for wafer-mask alignment.

Fig. 2.
Fig. 2.

Schematic of the (0,m) and (m,0) moiré interferometry.

Fig. 3.
Fig. 3.

Scheme of (m,0) and (0,m) moiré interferometry for wafer-mask alignment plus tilt remediation and twist adjustment.

Fig. 4.
Fig. 4.

Schematic of beam deflection induced by wafer tilt and corresponding change of the phase distribution (fringe vector) of interference field.

Fig. 5.
Fig. 5.

Equal-thickness interference caused by tilted wafer with regard to the mask at a small gap.

Fig. 6.
Fig. 6.

Scheme of (m,m) moiré interferometry arranged in a reflective way for in-plane twist angle adjustment and subsequent wafer-mask alignment.

Fig. 7.
Fig. 7.

Composite grating with left and right part arranged as (a) G1 and (b) G2 shown in Fig. 3.

Fig. 8.
Fig. 8.

Results of fringe distribution of moiré interference and equal-thickness interference that correspond to tilts existing in (a) both sections, (b) cross section, (c) longitudinal section, respectively.

Fig. 9.
Fig. 9.

Results recorded (a) without, and (b) with participation of the (1, 0) and (0, 1) diffraction orders.

Fig. 10.
Fig. 10.

Results of the (1,1) moiré interferometry: (a) the background with (1,1) and (1,1) orders filtered. (b) Obscure fringe pattern recorded at an arbitrary gap of non-Talbot distance. (c) Clear one recorded at the gap of Talbot distance.

Fig. 11.
Fig. 11.

Results of twist angle adjustment and alignment using gratings with periods of P1=6μm, P2=8μm, where (a) certain in-plane twist angle exist, (b) twist angle remedied with certain unaligned offset, and (c) aligned with offset almost eliminated.

Fig. 12.
Fig. 12.

Process of in-plane twist angle remediation by the (0, 1) and (1, 0) interference fringes from (a) certain twist angle to (b) the state of adequately adjusted and (c) ideally adjusted.

Fig. 13.
Fig. 13.

Stepwise increased offset leads to two parts of fringes with varied phase differences of (a) 0, (b) π/2, (c) π, (d) 3π/2.

Equations (13)

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E(m,m)=A(m,m)exp[i2πsinθ(m,m)λx],
E(m,m)=A(m,m)exp[i2πsinθ(m,m)λx],
E=E(0,0)+mE(m,m)+E(m,m)=A(0,0)+2mA(m,m)cos(2πm|f1f2|x).
Im=I1m+I2m+2I1mI2mcos[2πλ(sinθ1msinθ2m)x],
Im=I1m+I2m+2I1mI2mcos[2π(f1f2)x+Δφ],
Δφ=2π(mf1Δx+1+1/cosθ1mλΔG).
I=I1+I2+2I1I2cos(F·X+φ0),
F=2πλ(tan(θ1+2δθ)1+tan2(θ1+2δθ)+tan22δφsinθ2,tan2δφ1+tan2(θ1+2δθ)+tan22δφ)
Fup=2πλ(tan(θ1+2δθ)1+tan2(θ1+2δθ)+tan22δφsinθ2,tan2δφ1+tan2(θ1+2δθ)+tan22δφ),
Fbottom=2πλ(sinθ1tan(θ2+2δθ)1+tan2(θ2+2δθ)+tan22δφ,tan2δφ1+tan2(θ2+2δθ)+tan22δφ).
Δf=2δθλ·(cosθ1+cosθ2),
Δθatan(cosθ1tan2δφsinθ1sinθ2)+atan(cosθ2tan2δφsinθ1sinθ2).
δx=δφ2π·P1P2P1+P2.

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