Abstract

Sub-aperture coherence (SAC) is a classical phase control method for high-precision beam steering using liquid crystal optical phased arrays (LCOPA). On this basis, radial sub-aperture coherence (RSAC) and symmetrical radial sub-aperture coherence (SRSAC) were proposed, which guarantee the stability of steering angles when the beam aperture and incident position fluctuate. In this article, the pre-existing one-dimensional SRSAC was firstly extended to a more universal 2D phase generation algorithm. Meanwhile, for the intractable problem of local precision defects caused by the basic two-dimensional variable period grating (2D-VPG) algorithm, we tracked their locations accurately and designed a targeted elimination method carefully. So these remarkable error peaks could be thoroughly removed by using 2D-SRSAC optimized by the local precision defect elimination method. Since then, all the excellent performance of 1D-SRSAC can be perfectly transplanted to 2D, which makes the non-mechanical beam steering technology using LCOPA more mature and competitive in the applications required ultra-high precision.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (1)

2017 (1)

X. Wang, J. Xu, Z. Huang, W. Liang, T. Zhang, S. Wu, and Q. Qi, “Theoretical model and experimental verification on the PID tracking method using liquid crystal optical phased array,” Proc. SPIE 10096, 1009618 (2017).
[Crossref]

2016 (1)

S. Davis, S. D. Rommel, D. Gann, B. Luey, J. D. Gamble, M. Ziemkiewicz, and M. Anderson, “A lightweight, rugged, solid state laser radar system enabled by non-mechanical electro-optic beam steerers,” Proc. SPIE 9832, 98320K (2016).
[Crossref]

2015 (1)

Z. Tang, X. Wang, Z. Huang, Q. Tan, Y. Duan, G. Suo, J. Du, and Q. Qiu, “Sub-aperture coherence method to realize ultra-high resolution laser beam deflection,” Opt. Commun. 335, 1–6 (2015).
[Crossref]

2014 (1)

Z. Zhang, Z. You, and D. Chu, “Fundamentals of phase-only liquid crystal on silicon (LCOS) devices,” Light Sci. Appl. 3(10), e213 (2014).
[Crossref]

2012 (1)

2010 (2)

2008 (2)

D. Engström, J. Bengtsson, E. Eriksson, and M. Goksör, “Improved beam steering accuracy of a single beam with a 1D phase-only spatial light modulator,” Opt. Express 16(22), 18275–18287 (2008).
[Crossref] [PubMed]

J. Kim, C. Oh, M. J. Escuti, L. Hosting, and S. Serati, “Wide-angle, nonmechanical beam steering using thin liquid crystal polarization gratings,” Proc. SPIE 7093, 709302 (2008).
[Crossref]

2006 (2)

N. R. Smith, D. C. Abeysinghe, J. W. Haus, and J. Heikenfeld, “Agile wide-angle beam steering with electrowetting microprisms,” Opt. Express 14(14), 6557–6563 (2006).
[Crossref] [PubMed]

N. L. Seldomridge, J. A. Shaw, and K. S. Repasky, “Dual-polarization lidar using a liquid crystal variable retarder,” Opt. Eng. 45(10), 106202 (2006).
[Crossref]

2005 (1)

Y. Lin, M. Mahajan, D. Taber, B. Wen, and B. Winker, “Compact 4 cm aperture transmissive liquid crystal optical phased array for free-space optical communications,” Proc. SPIE 5892, 58920C (2005).
[Crossref]

2003 (1)

E. Haellstig, J. Stigwall, M. Lindgren, and L. Sjoqvist, “Laser beam steering and tracking using a liquid crystal spatial light modulator,” Proc. SPIE 5087, 13–23 (2003).
[Crossref]

1998 (1)

D. Winick, B. Duewer, S. Chaudhury, J. Wilson, J. Tucker, U. Eksi, and P. Franzon, “MEMS-based diffractive optical-beam-steering technology,” Proc. SPIE 3276, 81–87 (1998).
[Crossref]

Abeysinghe, D. C.

Anderson, M.

S. Davis, S. D. Rommel, D. Gann, B. Luey, J. D. Gamble, M. Ziemkiewicz, and M. Anderson, “A lightweight, rugged, solid state laser radar system enabled by non-mechanical electro-optic beam steerers,” Proc. SPIE 9832, 98320K (2016).
[Crossref]

Bengtsson, J.

Bos, P. J.

P. F. Mcmanamon, P. J. Bos, M. J. Escuti, J. Heikenfeld, S. Serati, H. Xie, and E. A. Watson, “A review of phased array steering for narrow-band electrooptical eystems,” in Proceedings of IEEE (IEEE), 97(6), 1078–1096 (2009).

Brewer, C. D.

E. A. Watson, W. E. Whitaker, C. D. Brewer, and S. R. Harris, “Implementing optical phased array beam steering with cascaded microlens arrays,” in Proceedings of IEEE Aerospace Conference (IEEE,) 3, 1429–1436(2002).
[Crossref]

Chaudhury, S.

D. Winick, B. Duewer, S. Chaudhury, J. Wilson, J. Tucker, U. Eksi, and P. Franzon, “MEMS-based diffractive optical-beam-steering technology,” Proc. SPIE 3276, 81–87 (1998).
[Crossref]

Chen, W.

Chu, D.

Z. Zhang, Z. You, and D. Chu, “Fundamentals of phase-only liquid crystal on silicon (LCOS) devices,” Light Sci. Appl. 3(10), e213 (2014).
[Crossref]

Corkum, D. L.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, and L. J. Friedman, “Optical phased array technology,” in Proceedings of IEEE (IEEE,), 84(2), 268–298 (1996).

Davis, S.

S. Davis, S. D. Rommel, D. Gann, B. Luey, J. D. Gamble, M. Ziemkiewicz, and M. Anderson, “A lightweight, rugged, solid state laser radar system enabled by non-mechanical electro-optic beam steerers,” Proc. SPIE 9832, 98320K (2016).
[Crossref]

Dorschner, T. A.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, and L. J. Friedman, “Optical phased array technology,” in Proceedings of IEEE (IEEE,), 84(2), 268–298 (1996).

Du, J.

Z. Tang, X. Wang, Z. Huang, Q. Tan, Y. Duan, G. Suo, J. Du, and Q. Qiu, “Sub-aperture coherence method to realize ultra-high resolution laser beam deflection,” Opt. Commun. 335, 1–6 (2015).
[Crossref]

Duan, Y.

Z. Tang, X. Wang, Z. Huang, Q. Tan, Y. Duan, G. Suo, J. Du, and Q. Qiu, “Sub-aperture coherence method to realize ultra-high resolution laser beam deflection,” Opt. Commun. 335, 1–6 (2015).
[Crossref]

Duewer, B.

D. Winick, B. Duewer, S. Chaudhury, J. Wilson, J. Tucker, U. Eksi, and P. Franzon, “MEMS-based diffractive optical-beam-steering technology,” Proc. SPIE 3276, 81–87 (1998).
[Crossref]

Eksi, U.

D. Winick, B. Duewer, S. Chaudhury, J. Wilson, J. Tucker, U. Eksi, and P. Franzon, “MEMS-based diffractive optical-beam-steering technology,” Proc. SPIE 3276, 81–87 (1998).
[Crossref]

Engström, D.

Eriksson, E.

Escuti, M. J.

J. Kim, C. Oh, M. J. Escuti, L. Hosting, and S. Serati, “Wide-angle, nonmechanical beam steering using thin liquid crystal polarization gratings,” Proc. SPIE 7093, 709302 (2008).
[Crossref]

P. F. Mcmanamon, P. J. Bos, M. J. Escuti, J. Heikenfeld, S. Serati, H. Xie, and E. A. Watson, “A review of phased array steering for narrow-band electrooptical eystems,” in Proceedings of IEEE (IEEE), 97(6), 1078–1096 (2009).

Franzon, P.

D. Winick, B. Duewer, S. Chaudhury, J. Wilson, J. Tucker, U. Eksi, and P. Franzon, “MEMS-based diffractive optical-beam-steering technology,” Proc. SPIE 3276, 81–87 (1998).
[Crossref]

Friedman, L. J.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, and L. J. Friedman, “Optical phased array technology,” in Proceedings of IEEE (IEEE,), 84(2), 268–298 (1996).

Gamble, J. D.

S. Davis, S. D. Rommel, D. Gann, B. Luey, J. D. Gamble, M. Ziemkiewicz, and M. Anderson, “A lightweight, rugged, solid state laser radar system enabled by non-mechanical electro-optic beam steerers,” Proc. SPIE 9832, 98320K (2016).
[Crossref]

Gann, D.

S. Davis, S. D. Rommel, D. Gann, B. Luey, J. D. Gamble, M. Ziemkiewicz, and M. Anderson, “A lightweight, rugged, solid state laser radar system enabled by non-mechanical electro-optic beam steerers,” Proc. SPIE 9832, 98320K (2016).
[Crossref]

Goksör, M.

Haellstig, E.

E. Haellstig, J. Stigwall, M. Lindgren, and L. Sjoqvist, “Laser beam steering and tracking using a liquid crystal spatial light modulator,” Proc. SPIE 5087, 13–23 (2003).
[Crossref]

Harris, S. R.

E. A. Watson, W. E. Whitaker, C. D. Brewer, and S. R. Harris, “Implementing optical phased array beam steering with cascaded microlens arrays,” in Proceedings of IEEE Aerospace Conference (IEEE,) 3, 1429–1436(2002).
[Crossref]

Haus, J. W.

Heikenfeld, J.

N. R. Smith, D. C. Abeysinghe, J. W. Haus, and J. Heikenfeld, “Agile wide-angle beam steering with electrowetting microprisms,” Opt. Express 14(14), 6557–6563 (2006).
[Crossref] [PubMed]

P. F. Mcmanamon, P. J. Bos, M. J. Escuti, J. Heikenfeld, S. Serati, H. Xie, and E. A. Watson, “A review of phased array steering for narrow-band electrooptical eystems,” in Proceedings of IEEE (IEEE), 97(6), 1078–1096 (2009).

Hosting, L.

J. Kim, C. Oh, M. J. Escuti, L. Hosting, and S. Serati, “Wide-angle, nonmechanical beam steering using thin liquid crystal polarization gratings,” Proc. SPIE 7093, 709302 (2008).
[Crossref]

Huang, Z.

X. Wang, J. Xu, Z. Huang, W. Liang, T. Zhang, S. Wu, and Q. Qi, “Theoretical model and experimental verification on the PID tracking method using liquid crystal optical phased array,” Proc. SPIE 10096, 1009618 (2017).
[Crossref]

Z. Tang, X. Wang, Z. Huang, Q. Tan, Y. Duan, G. Suo, J. Du, and Q. Qiu, “Sub-aperture coherence method to realize ultra-high resolution laser beam deflection,” Opt. Commun. 335, 1–6 (2015).
[Crossref]

Hunwardsen, M.

B. Winker, M. Mahajan, and M. Hunwardsen, “Liquid crystal beam directors for airborne free-space optical communications,” in Proceedings of IEEE Aerospace Conference (IEEE, 2004), Vol.3, pp. 1702–1709.
[Crossref]

Kim, J.

J. Kim, C. Oh, M. J. Escuti, L. Hosting, and S. Serati, “Wide-angle, nonmechanical beam steering using thin liquid crystal polarization gratings,” Proc. SPIE 7093, 709302 (2008).
[Crossref]

Kong, L.

Lane, S. A.

Li, S.

Liang, W.

X. Wang, J. Xu, Z. Huang, W. Liang, T. Zhang, S. Wu, and Q. Qi, “Theoretical model and experimental verification on the PID tracking method using liquid crystal optical phased array,” Proc. SPIE 10096, 1009618 (2017).
[Crossref]

Lin, Y.

Y. Lin, M. Mahajan, D. Taber, B. Wen, and B. Winker, “Compact 4 cm aperture transmissive liquid crystal optical phased array for free-space optical communications,” Proc. SPIE 5892, 58920C (2005).
[Crossref]

Lindgren, M.

E. Haellstig, J. Stigwall, M. Lindgren, and L. Sjoqvist, “Laser beam steering and tracking using a liquid crystal spatial light modulator,” Proc. SPIE 5087, 13–23 (2003).
[Crossref]

Liu, Y.

Luey, B.

S. Davis, S. D. Rommel, D. Gann, B. Luey, J. D. Gamble, M. Ziemkiewicz, and M. Anderson, “A lightweight, rugged, solid state laser radar system enabled by non-mechanical electro-optic beam steerers,” Proc. SPIE 9832, 98320K (2016).
[Crossref]

Mahajan, M.

Y. Lin, M. Mahajan, D. Taber, B. Wen, and B. Winker, “Compact 4 cm aperture transmissive liquid crystal optical phased array for free-space optical communications,” Proc. SPIE 5892, 58920C (2005).
[Crossref]

B. Winker, M. Mahajan, and M. Hunwardsen, “Liquid crystal beam directors for airborne free-space optical communications,” in Proceedings of IEEE Aerospace Conference (IEEE, 2004), Vol.3, pp. 1702–1709.
[Crossref]

Mcmanamon, P. F.

P. F. Mcmanamon, P. J. Bos, M. J. Escuti, J. Heikenfeld, S. Serati, H. Xie, and E. A. Watson, “A review of phased array steering for narrow-band electrooptical eystems,” in Proceedings of IEEE (IEEE), 97(6), 1078–1096 (2009).

P. F. McManamon, T. A. Dorschner, D. L. Corkum, and L. J. Friedman, “Optical phased array technology,” in Proceedings of IEEE (IEEE,), 84(2), 268–298 (1996).

Miniscalco, W. J.

Mu, Q.

Oh, C.

J. Kim, C. Oh, M. J. Escuti, L. Hosting, and S. Serati, “Wide-angle, nonmechanical beam steering using thin liquid crystal polarization gratings,” Proc. SPIE 7093, 709302 (2008).
[Crossref]

Peng, Z.

Qi, Q.

X. Wang, J. Xu, Z. Huang, W. Liang, T. Zhang, S. Wu, and Q. Qi, “Theoretical model and experimental verification on the PID tracking method using liquid crystal optical phased array,” Proc. SPIE 10096, 1009618 (2017).
[Crossref]

Qiu, Q.

Z. Tang, X. Wang, Z. Huang, Q. Tan, Y. Duan, G. Suo, J. Du, and Q. Qiu, “Sub-aperture coherence method to realize ultra-high resolution laser beam deflection,” Opt. Commun. 335, 1–6 (2015).
[Crossref]

Repasky, K. S.

N. L. Seldomridge, J. A. Shaw, and K. S. Repasky, “Dual-polarization lidar using a liquid crystal variable retarder,” Opt. Eng. 45(10), 106202 (2006).
[Crossref]

Rommel, S. D.

S. Davis, S. D. Rommel, D. Gann, B. Luey, J. D. Gamble, M. Ziemkiewicz, and M. Anderson, “A lightweight, rugged, solid state laser radar system enabled by non-mechanical electro-optic beam steerers,” Proc. SPIE 9832, 98320K (2016).
[Crossref]

Seldomridge, N. L.

N. L. Seldomridge, J. A. Shaw, and K. S. Repasky, “Dual-polarization lidar using a liquid crystal variable retarder,” Opt. Eng. 45(10), 106202 (2006).
[Crossref]

Serati, S.

J. Kim, C. Oh, M. J. Escuti, L. Hosting, and S. Serati, “Wide-angle, nonmechanical beam steering using thin liquid crystal polarization gratings,” Proc. SPIE 7093, 709302 (2008).
[Crossref]

P. F. Mcmanamon, P. J. Bos, M. J. Escuti, J. Heikenfeld, S. Serati, H. Xie, and E. A. Watson, “A review of phased array steering for narrow-band electrooptical eystems,” in Proceedings of IEEE (IEEE), 97(6), 1078–1096 (2009).

Shaw, J. A.

N. L. Seldomridge, J. A. Shaw, and K. S. Repasky, “Dual-polarization lidar using a liquid crystal variable retarder,” Opt. Eng. 45(10), 106202 (2006).
[Crossref]

Sjoqvist, L.

E. Haellstig, J. Stigwall, M. Lindgren, and L. Sjoqvist, “Laser beam steering and tracking using a liquid crystal spatial light modulator,” Proc. SPIE 5087, 13–23 (2003).
[Crossref]

Smith, N. R.

Song, Y.

Stigwall, J.

E. Haellstig, J. Stigwall, M. Lindgren, and L. Sjoqvist, “Laser beam steering and tracking using a liquid crystal spatial light modulator,” Proc. SPIE 5087, 13–23 (2003).
[Crossref]

Suo, G.

Z. Tang, X. Wang, Z. Huang, Q. Tan, Y. Duan, G. Suo, J. Du, and Q. Qiu, “Sub-aperture coherence method to realize ultra-high resolution laser beam deflection,” Opt. Commun. 335, 1–6 (2015).
[Crossref]

Taber, D.

Y. Lin, M. Mahajan, D. Taber, B. Wen, and B. Winker, “Compact 4 cm aperture transmissive liquid crystal optical phased array for free-space optical communications,” Proc. SPIE 5892, 58920C (2005).
[Crossref]

Tan, Q.

Z. Tang, X. Wang, Z. Huang, Q. Tan, Y. Duan, G. Suo, J. Du, and Q. Qiu, “Sub-aperture coherence method to realize ultra-high resolution laser beam deflection,” Opt. Commun. 335, 1–6 (2015).
[Crossref]

Tang, Z.

Z. Tang, X. Wang, Z. Huang, Q. Tan, Y. Duan, G. Suo, J. Du, and Q. Qiu, “Sub-aperture coherence method to realize ultra-high resolution laser beam deflection,” Opt. Commun. 335, 1–6 (2015).
[Crossref]

Tucker, J.

D. Winick, B. Duewer, S. Chaudhury, J. Wilson, J. Tucker, U. Eksi, and P. Franzon, “MEMS-based diffractive optical-beam-steering technology,” Proc. SPIE 3276, 81–87 (1998).
[Crossref]

Vettese, D.

D. Vettese, “Liquid crystal on silicon,” Nat. Photonics 4(11), 752–754 (2010).
[Crossref]

Wang, C.

Wang, Q.

Wang, X.

X. Wang, J. Xu, Z. Huang, W. Liang, T. Zhang, S. Wu, and Q. Qi, “Theoretical model and experimental verification on the PID tracking method using liquid crystal optical phased array,” Proc. SPIE 10096, 1009618 (2017).
[Crossref]

Z. Tang, X. Wang, Z. Huang, Q. Tan, Y. Duan, G. Suo, J. Du, and Q. Qiu, “Sub-aperture coherence method to realize ultra-high resolution laser beam deflection,” Opt. Commun. 335, 1–6 (2015).
[Crossref]

Watson, E. A.

E. A. Watson, W. E. Whitaker, C. D. Brewer, and S. R. Harris, “Implementing optical phased array beam steering with cascaded microlens arrays,” in Proceedings of IEEE Aerospace Conference (IEEE,) 3, 1429–1436(2002).
[Crossref]

P. F. Mcmanamon, P. J. Bos, M. J. Escuti, J. Heikenfeld, S. Serati, H. Xie, and E. A. Watson, “A review of phased array steering for narrow-band electrooptical eystems,” in Proceedings of IEEE (IEEE), 97(6), 1078–1096 (2009).

Wen, B.

Y. Lin, M. Mahajan, D. Taber, B. Wen, and B. Winker, “Compact 4 cm aperture transmissive liquid crystal optical phased array for free-space optical communications,” Proc. SPIE 5892, 58920C (2005).
[Crossref]

Whitaker, W. E.

E. A. Watson, W. E. Whitaker, C. D. Brewer, and S. R. Harris, “Implementing optical phased array beam steering with cascaded microlens arrays,” in Proceedings of IEEE Aerospace Conference (IEEE,) 3, 1429–1436(2002).
[Crossref]

Wilson, J.

D. Winick, B. Duewer, S. Chaudhury, J. Wilson, J. Tucker, U. Eksi, and P. Franzon, “MEMS-based diffractive optical-beam-steering technology,” Proc. SPIE 3276, 81–87 (1998).
[Crossref]

Winick, D.

D. Winick, B. Duewer, S. Chaudhury, J. Wilson, J. Tucker, U. Eksi, and P. Franzon, “MEMS-based diffractive optical-beam-steering technology,” Proc. SPIE 3276, 81–87 (1998).
[Crossref]

Winker, B.

Y. Lin, M. Mahajan, D. Taber, B. Wen, and B. Winker, “Compact 4 cm aperture transmissive liquid crystal optical phased array for free-space optical communications,” Proc. SPIE 5892, 58920C (2005).
[Crossref]

B. Winker, M. Mahajan, and M. Hunwardsen, “Liquid crystal beam directors for airborne free-space optical communications,” in Proceedings of IEEE Aerospace Conference (IEEE, 2004), Vol.3, pp. 1702–1709.
[Crossref]

Wu, S.

X. Wang, J. Xu, Z. Huang, W. Liang, T. Zhang, S. Wu, and Q. Qi, “Theoretical model and experimental verification on the PID tracking method using liquid crystal optical phased array,” Proc. SPIE 10096, 1009618 (2017).
[Crossref]

Xie, H.

P. F. Mcmanamon, P. J. Bos, M. J. Escuti, J. Heikenfeld, S. Serati, H. Xie, and E. A. Watson, “A review of phased array steering for narrow-band electrooptical eystems,” in Proceedings of IEEE (IEEE), 97(6), 1078–1096 (2009).

Xu, J.

X. Wang, J. Xu, Z. Huang, W. Liang, T. Zhang, S. Wu, and Q. Qi, “Theoretical model and experimental verification on the PID tracking method using liquid crystal optical phased array,” Proc. SPIE 10096, 1009618 (2017).
[Crossref]

Yang, J.

You, Z.

Z. Zhang, Z. You, and D. Chu, “Fundamentals of phase-only liquid crystal on silicon (LCOS) devices,” Light Sci. Appl. 3(10), e213 (2014).
[Crossref]

Zhang, T.

X. Wang, J. Xu, Z. Huang, W. Liang, T. Zhang, S. Wu, and Q. Qi, “Theoretical model and experimental verification on the PID tracking method using liquid crystal optical phased array,” Proc. SPIE 10096, 1009618 (2017).
[Crossref]

Zhang, Z.

Z. Zhang, Z. You, and D. Chu, “Fundamentals of phase-only liquid crystal on silicon (LCOS) devices,” Light Sci. Appl. 3(10), e213 (2014).
[Crossref]

Zhao, Z.

Zhu, Y.

Ziemkiewicz, M.

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S. Davis, S. D. Rommel, D. Gann, B. Luey, J. D. Gamble, M. Ziemkiewicz, and M. Anderson, “A lightweight, rugged, solid state laser radar system enabled by non-mechanical electro-optic beam steerers,” Proc. SPIE 9832, 98320K (2016).
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[Crossref]

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Figures (16)

Fig. 1
Fig. 1 A phase diagram for 2D steering derived from that for 1D steering.
Fig. 2
Fig. 2 Simulated diagram of equidistant scanning angle generation within scanning intervals.
Fig. 3
Fig. 3 The fragments of 1D tilted optical path delays. The red, green, and blue floating ranges correspond to the error peaks around the positions that the desired steering angles equal to λ/NGd, −2λ/NGd and λ/2NGd, respectively.
Fig. 4
Fig. 4 Simulated diagram of the overall steering error based on 1D-VPG.
Fig. 5
Fig. 5 Position and relative size diagram of the error peaks in [0, 1]2 based on 2D-VPG.
Fig. 6
Fig. 6 Simulated diagram of error peaks based on 2D-VPG. (a) The main error peak for N = 1 and (mx , my ) = (1,1). (b) The minor error peak for N = 2 and (mx , my ) = (1,1/2). (c) The minor error peak for N = 3 and (mx , my ) = (0,1/3). (d) The minor error peak for N = 4 and (mx , my ) = (3/4,1/4).
Fig. 7
Fig. 7 Simulated diagram of the overall steering error based on conventional 1D-SRSAC.
Fig. 8
Fig. 8 Schematic diagram of the interpolation segment adjustment in the local error elimination process.
Fig. 9
Fig. 9 Simulated diagram of the overall steering error based on the optimized 1D-SRSAC.
Fig. 10
Fig. 10 The relationship between correlation coefficient and the interpolation segment length.
Fig. 11
Fig. 11 Simulated diagram of the correction effect of error peaks based on 2D-SRSAC optimized by the precision defect elimination method. (a) The main error peak for N = 1 and (mx , my ) = (1,1). (b) The minor error peak for N = 2 and (mx , my ) = (1,1/2). (c) The minor error peak for N = 3 and (mx , my ) = (0,1/3). (d) The minor error peak for N = 4 and (mx , my ) = (3/4,1/4).
Fig. 12
Fig. 12 Schematic diagram of the 2D steering angle measuring setup and phase distributions.
Fig. 13
Fig. 13 Physical diagram of the 2D steering angle measuring setup and the actual phase driving instructions, where the difference of stripe widths in two sub-apertures has been exaggerated in Fig. 12.
Fig. 14
Fig. 14 The measuring interface with the steering angles at the zero point and around different positions of error peaks.
Fig. 15
Fig. 15 Measured data of error peaks based on the primitive 2D-VPG algorithm. (a) The main error peak for N = 1 and (mx , my ) = (1,1). (b) The minor error peak for N = 2 and (mx , my ) = (1,1/2). (c) The minor error peak for N = 3 and (mx , my ) = (0,1/3). (d) The minor error peak for N = 4 and (mx , my ) = (3/4,1/4).
Fig. 16
Fig. 16 Measured data of error peaks with different order N based on the optimized 2D-SRSAC algorithm. (a) The main error peak for N = 1 and (mx , my ) = (1,1). (b) The minor error peak for N = 2 and (mx , my ) = (1,1/2). (c) The minor error peak for N = 3 and (mx , my ) = (0,1/3). (d) The minor error peak for N = 4 and (mx , my ) = (3/4,1/4).

Tables (1)

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Table 1 RMS of steering angle error at different position (μrad)

Equations (9)

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| θ I |= θ step floor(| θ |/ θ step ), | θ II |=| θ I |+ θ step , angle( θ I )=angle( θ II )=angle( θ )=β.
θ norm | θ || θ I | | θ II || θ I | .
η I = (1 θ norm ) 2 5 .
θ p = λ N G d .
W 1 = 2λ N G R .
W N = 2λ N N G R .
| θ I |=2 θ step floor[ ( | θ | θ step )/2 θ step ]+ θ step , | θ II |=| θ I |+2 θ step .
E( θ )= θ norm E( θ II )+(1 θ norm )E( θ I ),
var( θ )=[1(2 θ norm 2 θ norm 2 )(1c)] σ 2 , c=[E( θ I θ II )E( θ I )E( θ II )]/ σ 2 .

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