Abstract

Low-power integrated projection technology can play a key role in development of low-cost mobile devices with built-in high-resolution projectors. Low-cost 3D imaging and holography systems are also among applications of such a technology. In this paper, an integrated projection system based on a two-dimensional optical phased array with fast beam steering capability is reported. Forward biased p-i-n phase modulators with 200MHz bandwidth are used per each array element for rapid phase control. An optimization algorithm is implemented to compensate for the phase dependent attenuation of the p-i-n modulators. Using rapid vector scanning technique, images were formed and recorded within a single snapshot of the IR camera.

© 2015 Optical Society of America

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References

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    [Crossref]
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2014 (1)

2013 (3)

J. Sun, E. Timurdogan, A. Yaacobi, E. Shah Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493(7431), 195–199 (2013).
[Crossref] [PubMed]

B. W. Yoo, M. Megens, T. Chan, T. Sun, W. Yang, C. J. Chang-Hasnain, D. A. Horsley, and M. C. Wu, “Optical phased array using high contrast gratings for two dimensional beamforming and beamsteering,” Opt. Express 21(10), 12238–12248 (2013).
[Crossref] [PubMed]

A. Novack, Y. Liu, R. Ding, M. Gould, T. Baehr-Jones, Q. Li, Y. Yang, Y. Ma, Y. Zhang, K. Padmaraju, K. Bergman, A. E. J. Lim, G. Q. Lo, and M. Hochberg, “A 30 GHz silicon photonic platform,” Proc. SPIE 8781, 878107 (2013).
[Crossref]

2010 (1)

2008 (1)

N. Satyan, W. Liang, F. Aflatouni, A. Yariv, A. Kewitsch, G. Rakuljic, and H. Hashemi, “Phase-controlled apertures using heterodyne optical phase-locked loops,” IEEE Photon. Technol. Lett. 20(11), 897–899 (2008).
[Crossref]

2007 (2)

2005 (1)

A. Natarajan, A. Komijani, and A. Hajimiri, “A fully integrated 24-GHz phased-array transmitter in CMOS,” IEEE J. Solid-State Circ. 40(12), 2502–2514 (2005).
[Crossref]

1996 (1)

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84(2), 268–297 (1996).
[Crossref]

Abiri, B.

B. Abiri, F. Aflatouni, A. Rekhi, and A. Hajimiri, “Electronic two-dimensional beam steering for integrated optical phased arrays,” in Proceedings of the Optical Fiber Communication Conference (OFC), paper M2K.7 (2014).

Aflatouni, F.

N. Satyan, W. Liang, F. Aflatouni, A. Yariv, A. Kewitsch, G. Rakuljic, and H. Hashemi, “Phase-controlled apertures using heterodyne optical phase-locked loops,” IEEE Photon. Technol. Lett. 20(11), 897–899 (2008).
[Crossref]

W. Liang, N. Satyan, F. Aflatouni, A. Yariv, A. Kewitsch, G. Rakuljic, and H. Hashemi, “Coherent beam combining with multi-level optical phase locked loops,” J. Opt. Soc. Am. B 24(12), 2930–2939 (2007).
[Crossref]

B. Abiri, F. Aflatouni, A. Rekhi, and A. Hajimiri, “Electronic two-dimensional beam steering for integrated optical phased arrays,” in Proceedings of the Optical Fiber Communication Conference (OFC), paper M2K.7 (2014).

F. Aflatouni and H. Hashemi, “An electronically controlled semiconductor laser phased array,” in IEEE Microwave Symposium Digest (MTT), WEPN-6 (2012).

Baehr-Jones, T.

A. Novack, Y. Liu, R. Ding, M. Gould, T. Baehr-Jones, Q. Li, Y. Yang, Y. Ma, Y. Zhang, K. Padmaraju, K. Bergman, A. E. J. Lim, G. Q. Lo, and M. Hochberg, “A 30 GHz silicon photonic platform,” Proc. SPIE 8781, 878107 (2013).
[Crossref]

Baets, R.

Bergman, K.

A. Novack, Y. Liu, R. Ding, M. Gould, T. Baehr-Jones, Q. Li, Y. Yang, Y. Ma, Y. Zhang, K. Padmaraju, K. Bergman, A. E. J. Lim, G. Q. Lo, and M. Hochberg, “A 30 GHz silicon photonic platform,” Proc. SPIE 8781, 878107 (2013).
[Crossref]

Bovington, J. T.

J. K. Doylend, M. J. R. Heck, J. T. Bovington, J. D. Peters, L. A. Coldren, and J. E. Bowers, “Free-space beam steering in two dimensions using a silicon optical phased array,” in Proceedings of the Optical Fiber Communication Conference (OFC), OM2J (2012).

Bowers, J. E.

J. K. Doylend, M. J. R. Heck, J. T. Bovington, J. D. Peters, L. A. Coldren, and J. E. Bowers, “Free-space beam steering in two dimensions using a silicon optical phased array,” in Proceedings of the Optical Fiber Communication Conference (OFC), OM2J (2012).

Chan, T.

Chang-Hasnain, C. J.

Coldren, L. A.

J. K. Doylend, M. J. R. Heck, J. T. Bovington, J. D. Peters, L. A. Coldren, and J. E. Bowers, “Free-space beam steering in two dimensions using a silicon optical phased array,” in Proceedings of the Optical Fiber Communication Conference (OFC), OM2J (2012).

Corkum, D. L.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84(2), 268–297 (1996).
[Crossref]

Cui, J. C.

Ding, R.

A. Novack, Y. Liu, R. Ding, M. Gould, T. Baehr-Jones, Q. Li, Y. Yang, Y. Ma, Y. Zhang, K. Padmaraju, K. Bergman, A. E. J. Lim, G. Q. Lo, and M. Hochberg, “A 30 GHz silicon photonic platform,” Proc. SPIE 8781, 878107 (2013).
[Crossref]

Dorschner, T. A.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84(2), 268–297 (1996).
[Crossref]

Doylend, J. K.

J. K. Doylend, M. J. R. Heck, J. T. Bovington, J. D. Peters, L. A. Coldren, and J. E. Bowers, “Free-space beam steering in two dimensions using a silicon optical phased array,” in Proceedings of the Optical Fiber Communication Conference (OFC), OM2J (2012).

Fang, Z. L.

Friedman, L. J.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84(2), 268–297 (1996).
[Crossref]

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Gould, M.

A. Novack, Y. Liu, R. Ding, M. Gould, T. Baehr-Jones, Q. Li, Y. Yang, Y. Ma, Y. Zhang, K. Padmaraju, K. Bergman, A. E. J. Lim, G. Q. Lo, and M. Hochberg, “A 30 GHz silicon photonic platform,” Proc. SPIE 8781, 878107 (2013).
[Crossref]

Hajimiri, A.

A. Natarajan, A. Komijani, and A. Hajimiri, “A fully integrated 24-GHz phased-array transmitter in CMOS,” IEEE J. Solid-State Circ. 40(12), 2502–2514 (2005).
[Crossref]

K. Sengupta and A. Hajimiri, “A 0.28THz 4 × 4 power-generation and beam-steering array,” in Proceedings of the IEEE International Solid-State Circuits Conference, 15.3, (2012), pp. 256–257.

B. Abiri, F. Aflatouni, A. Rekhi, and A. Hajimiri, “Electronic two-dimensional beam steering for integrated optical phased arrays,” in Proceedings of the Optical Fiber Communication Conference (OFC), paper M2K.7 (2014).

Hashemi, H.

N. Satyan, W. Liang, F. Aflatouni, A. Yariv, A. Kewitsch, G. Rakuljic, and H. Hashemi, “Phase-controlled apertures using heterodyne optical phase-locked loops,” IEEE Photon. Technol. Lett. 20(11), 897–899 (2008).
[Crossref]

W. Liang, N. Satyan, F. Aflatouni, A. Yariv, A. Kewitsch, G. Rakuljic, and H. Hashemi, “Coherent beam combining with multi-level optical phase locked loops,” J. Opt. Soc. Am. B 24(12), 2930–2939 (2007).
[Crossref]

F. Aflatouni and H. Hashemi, “An electronically controlled semiconductor laser phased array,” in IEEE Microwave Symposium Digest (MTT), WEPN-6 (2012).

Heck, M. J. R.

J. K. Doylend, M. J. R. Heck, J. T. Bovington, J. D. Peters, L. A. Coldren, and J. E. Bowers, “Free-space beam steering in two dimensions using a silicon optical phased array,” in Proceedings of the Optical Fiber Communication Conference (OFC), OM2J (2012).

Hobbs, D. S.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84(2), 268–297 (1996).
[Crossref]

Hochberg, M.

A. Novack, Y. Liu, R. Ding, M. Gould, T. Baehr-Jones, Q. Li, Y. Yang, Y. Ma, Y. Zhang, K. Padmaraju, K. Bergman, A. E. J. Lim, G. Q. Lo, and M. Hochberg, “A 30 GHz silicon photonic platform,” Proc. SPIE 8781, 878107 (2013).
[Crossref]

Holz, M.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84(2), 268–297 (1996).
[Crossref]

Horsley, D. A.

Hosseini, E. Shah

J. Sun, E. Timurdogan, A. Yaacobi, E. Shah Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493(7431), 195–199 (2013).
[Crossref] [PubMed]

Huang, Y. C.

Kewitsch, A.

N. Satyan, W. Liang, F. Aflatouni, A. Yariv, A. Kewitsch, G. Rakuljic, and H. Hashemi, “Phase-controlled apertures using heterodyne optical phase-locked loops,” IEEE Photon. Technol. Lett. 20(11), 897–899 (2008).
[Crossref]

W. Liang, N. Satyan, F. Aflatouni, A. Yariv, A. Kewitsch, G. Rakuljic, and H. Hashemi, “Coherent beam combining with multi-level optical phase locked loops,” J. Opt. Soc. Am. B 24(12), 2930–2939 (2007).
[Crossref]

Komijani, A.

A. Natarajan, A. Komijani, and A. Hajimiri, “A fully integrated 24-GHz phased-array transmitter in CMOS,” IEEE J. Solid-State Circ. 40(12), 2502–2514 (2005).
[Crossref]

Landau, L.D.

L.D. Landau and E.M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Perganom Press, 1982).

Li, Q.

A. Novack, Y. Liu, R. Ding, M. Gould, T. Baehr-Jones, Q. Li, Y. Yang, Y. Ma, Y. Zhang, K. Padmaraju, K. Bergman, A. E. J. Lim, G. Q. Lo, and M. Hochberg, “A 30 GHz silicon photonic platform,” Proc. SPIE 8781, 878107 (2013).
[Crossref]

Liang, W.

N. Satyan, W. Liang, F. Aflatouni, A. Yariv, A. Kewitsch, G. Rakuljic, and H. Hashemi, “Phase-controlled apertures using heterodyne optical phase-locked loops,” IEEE Photon. Technol. Lett. 20(11), 897–899 (2008).
[Crossref]

W. Liang, N. Satyan, F. Aflatouni, A. Yariv, A. Kewitsch, G. Rakuljic, and H. Hashemi, “Coherent beam combining with multi-level optical phase locked loops,” J. Opt. Soc. Am. B 24(12), 2930–2939 (2007).
[Crossref]

Liberman, S.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84(2), 268–297 (1996).
[Crossref]

Lifshitz, E.M.

L.D. Landau and E.M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Perganom Press, 1982).

Lim, A. E. J.

A. Novack, Y. Liu, R. Ding, M. Gould, T. Baehr-Jones, Q. Li, Y. Yang, Y. Ma, Y. Zhang, K. Padmaraju, K. Bergman, A. E. J. Lim, G. Q. Lo, and M. Hochberg, “A 30 GHz silicon photonic platform,” Proc. SPIE 8781, 878107 (2013).
[Crossref]

Liu, Y.

A. Novack, Y. Liu, R. Ding, M. Gould, T. Baehr-Jones, Q. Li, Y. Yang, Y. Ma, Y. Zhang, K. Padmaraju, K. Bergman, A. E. J. Lim, G. Q. Lo, and M. Hochberg, “A 30 GHz silicon photonic platform,” Proc. SPIE 8781, 878107 (2013).
[Crossref]

Lo, G. Q.

A. Novack, Y. Liu, R. Ding, M. Gould, T. Baehr-Jones, Q. Li, Y. Yang, Y. Ma, Y. Zhang, K. Padmaraju, K. Bergman, A. E. J. Lim, G. Q. Lo, and M. Hochberg, “A 30 GHz silicon photonic platform,” Proc. SPIE 8781, 878107 (2013).
[Crossref]

Ma, Y.

A. Novack, Y. Liu, R. Ding, M. Gould, T. Baehr-Jones, Q. Li, Y. Yang, Y. Ma, Y. Zhang, K. Padmaraju, K. Bergman, A. E. J. Lim, G. Q. Lo, and M. Hochberg, “A 30 GHz silicon photonic platform,” Proc. SPIE 8781, 878107 (2013).
[Crossref]

McManamon, P. F.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84(2), 268–297 (1996).
[Crossref]

Megens, M.

Mu, G. G.

Natarajan, A.

A. Natarajan, A. Komijani, and A. Hajimiri, “A fully integrated 24-GHz phased-array transmitter in CMOS,” IEEE J. Solid-State Circ. 40(12), 2502–2514 (2005).
[Crossref]

Nguyen, H. Q.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84(2), 268–297 (1996).
[Crossref]

Novack, A.

A. Novack, Y. Liu, R. Ding, M. Gould, T. Baehr-Jones, Q. Li, Y. Yang, Y. Ma, Y. Zhang, K. Padmaraju, K. Bergman, A. E. J. Lim, G. Q. Lo, and M. Hochberg, “A 30 GHz silicon photonic platform,” Proc. SPIE 8781, 878107 (2013).
[Crossref]

Padmaraju, K.

A. Novack, Y. Liu, R. Ding, M. Gould, T. Baehr-Jones, Q. Li, Y. Yang, Y. Ma, Y. Zhang, K. Padmaraju, K. Bergman, A. E. J. Lim, G. Q. Lo, and M. Hochberg, “A 30 GHz silicon photonic platform,” Proc. SPIE 8781, 878107 (2013).
[Crossref]

Pan, J. W.

Peters, J. D.

J. K. Doylend, M. J. R. Heck, J. T. Bovington, J. D. Peters, L. A. Coldren, and J. E. Bowers, “Free-space beam steering in two dimensions using a silicon optical phased array,” in Proceedings of the Optical Fiber Communication Conference (OFC), OM2J (2012).

Rakuljic, G.

N. Satyan, W. Liang, F. Aflatouni, A. Yariv, A. Kewitsch, G. Rakuljic, and H. Hashemi, “Phase-controlled apertures using heterodyne optical phase-locked loops,” IEEE Photon. Technol. Lett. 20(11), 897–899 (2008).
[Crossref]

W. Liang, N. Satyan, F. Aflatouni, A. Yariv, A. Kewitsch, G. Rakuljic, and H. Hashemi, “Coherent beam combining with multi-level optical phase locked loops,” J. Opt. Soc. Am. B 24(12), 2930–2939 (2007).
[Crossref]

Rekhi, A.

B. Abiri, F. Aflatouni, A. Rekhi, and A. Hajimiri, “Electronic two-dimensional beam steering for integrated optical phased arrays,” in Proceedings of the Optical Fiber Communication Conference (OFC), paper M2K.7 (2014).

Resler, D. P.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84(2), 268–297 (1996).
[Crossref]

Rogier, H.

Satyan, N.

N. Satyan, W. Liang, F. Aflatouni, A. Yariv, A. Kewitsch, G. Rakuljic, and H. Hashemi, “Phase-controlled apertures using heterodyne optical phase-locked loops,” IEEE Photon. Technol. Lett. 20(11), 897–899 (2008).
[Crossref]

W. Liang, N. Satyan, F. Aflatouni, A. Yariv, A. Kewitsch, G. Rakuljic, and H. Hashemi, “Coherent beam combining with multi-level optical phase locked loops,” J. Opt. Soc. Am. B 24(12), 2930–2939 (2007).
[Crossref]

Sengupta, K.

K. Sengupta and A. Hajimiri, “A 0.28THz 4 × 4 power-generation and beam-steering array,” in Proceedings of the IEEE International Solid-State Circuits Conference, 15.3, (2012), pp. 256–257.

Sharp, R. C.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84(2), 268–297 (1996).
[Crossref]

Sun, J.

J. Sun, E. Timurdogan, A. Yaacobi, E. Shah Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493(7431), 195–199 (2013).
[Crossref] [PubMed]

Sun, T.

Timurdogan, E.

J. Sun, E. Timurdogan, A. Yaacobi, E. Shah Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493(7431), 195–199 (2013).
[Crossref] [PubMed]

Van Acoleyen, K.

Watson, E. A.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, “Optical phased array technology,” Proc. IEEE 84(2), 268–297 (1996).
[Crossref]

Watts, M. R.

J. Sun, E. Timurdogan, A. Yaacobi, E. Shah Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493(7431), 195–199 (2013).
[Crossref] [PubMed]

Wu, M. C.

Yaacobi, A.

J. Sun, E. Timurdogan, A. Yaacobi, E. Shah Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493(7431), 195–199 (2013).
[Crossref] [PubMed]

Yang, W.

Yang, Y.

A. Novack, Y. Liu, R. Ding, M. Gould, T. Baehr-Jones, Q. Li, Y. Yang, Y. Ma, Y. Zhang, K. Padmaraju, K. Bergman, A. E. J. Lim, G. Q. Lo, and M. Hochberg, “A 30 GHz silicon photonic platform,” Proc. SPIE 8781, 878107 (2013).
[Crossref]

Yariv, A.

N. Satyan, W. Liang, F. Aflatouni, A. Yariv, A. Kewitsch, G. Rakuljic, and H. Hashemi, “Phase-controlled apertures using heterodyne optical phase-locked loops,” IEEE Photon. Technol. Lett. 20(11), 897–899 (2008).
[Crossref]

W. Liang, N. Satyan, F. Aflatouni, A. Yariv, A. Kewitsch, G. Rakuljic, and H. Hashemi, “Coherent beam combining with multi-level optical phase locked loops,” J. Opt. Soc. Am. B 24(12), 2930–2939 (2007).
[Crossref]

Yoo, B. W.

Zhang, W.

W. Zhang, “LIDAR-based road and road-edge detection,” in Proceedings of the IEEE Intelligent Vehicles Symposium, WeE1.12 (2010).

Zhang, X.

Zhang, Y.

A. Novack, Y. Liu, R. Ding, M. Gould, T. Baehr-Jones, Q. Li, Y. Yang, Y. Ma, Y. Zhang, K. Padmaraju, K. Bergman, A. E. J. Lim, G. Q. Lo, and M. Hochberg, “A 30 GHz silicon photonic platform,” Proc. SPIE 8781, 878107 (2013).
[Crossref]

Zhao, X.

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Figures (9)

Fig. 1
Fig. 1 The M × N-element optical phased array.
Fig. 2
Fig. 2 The cross section of the array patterns formed at z0 = 3cm plane for an OPA with d = 50µm and λ = 1.55µm for both loss-less and lossy phase shifters. (a)M = N = 4, α = π 8 [ 1 r a d ], Δ ϕ m = Δ ϕ n = π 3 [ r a d ], and (b) M = N = 16, α = 3 π 16 [ 1 r a d ], Δ ϕ m = Δ ϕ n = π 3 [ rad ].
Fig. 3
Fig. 3 (a) The structure of the reported 4 × 4 integrated optical phased array with per-channel high speed phase control and (b) the chip micro-photograph of the fabricated OPA.
Fig. 4
Fig. 4 (a) The cross section of the p-i-n phase modulator used in the proposed OPA, (b) the propagation mode profile of the modulator, and (c) the measured excess attenuation of the p-i-n modulators in proposed OPA.
Fig. 5
Fig. 5 The flowchart of the optimization process.
Fig. 6
Fig. 6 The pattern of the OPA for zero relative phase setting (Δϕm = Δϕn = 0) captured using FJW View-R-Scope 85400A IR camera;(a) before gradient decent optimization, and (b) after gradient decent optimization.
Fig. 7
Fig. 7 Top: the OPA element radiation status (only highlighted elements are radiating), bottom: the far-field pattern. (a) The far-field pattern of the OPA when all elements except for the ones at the corners of the array are turned off, (b) the far-field pattern of the OPA when elements on every other columns are turned off, and (c) the far-field pattern of the OPA when all elements on every other rows are turned off.
Fig. 8
Fig. 8 The vertical beam steering; (a) simulation, (b) measurement, and (c) far-filed pattern cross section. The horizontal beam steering;(d) simulation, (e) measurement, and (f) far-filed pattern cross section.
Fig. 9
Fig. 9 Projected images by fast vector scan of the beam spot; (a) smiley face (simulation on left), (b)sad face (simulation on left), and (c) individual letters of CIT (California Institute of Technology).

Equations (8)

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E m n ( x , y , 0 ) = E m n , 0 e j m Δ ϕ m e j n Δ ϕ n e 4 d s 2 [ ( x m d ) 2 + ( y n d ) 2 ] ,
I ( x , y , z 0 ) = E 0 2 d s 4 π 2 16 λ 0 2 z 0 2 e π 2 d s 2 2 λ 0 2 z 0 2 ( x 2 + y 2 ) × | A F x | 2 × | A F y | 2 .
| A F x | = sin [ M ( π d λ 0 z 0 x Δ ϕ m 2 ) ] sin [ ( π d λ 0 z 0 x Δ ϕ m 2 ) ] , a n d | A F y | = sin [ N ( π d λ 0 z 0 y Δ ϕ n 2 ) ] sin [ ( π d λ 0 z 0 y Δ ϕ n 2 ) ] .
A F = 1 E 0 k = 1 M k = 1 N E m n e j ( 2 π d λ 0 z 0 x Δ ϕ m ) k e j ( 2 π d λ 0 z 0 y Δ ϕ n ) k .
E k k = E 0 ( 1 α k ϕ m ) ( 1 α k ϕ n ) ,
A F = k = 1 M ( 1 α k ϕ m ) e j ( 2 π d λ 0 z 0 x Δ ϕ m ) k A F x , u × k = 1 N ( 1 α k ϕ n ) e j ( 2 π d λ 0 z 0 y Δ ϕ n ) k A F y , u .
k = 1 M q k = q q M 1 q 1 , a n d k = 1 M k q k = q 1 ( M + 1 ) q M + M q M + 1 ( 1 q ) 2 ,
| A F x , u | 2 = | A F x | 2 × { ( 1 M + 1 2 α ϕ m ) 2 + α 2 ϕ m 2 4 [ c o t a n ( u m ) M c o t a n ( M u m ) ] 2 } , | A F y , u | 2 = | A F y | 2 × { ( 1 N + 1 2 α ϕ n ) 2 + α 2 ϕ n 2 4 [ c o t a n ( u n ) N c o t a n ( N u n ) ] 2 } ,

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