Abstract

Two-dimensional (2-D) integrated optical phased arrays (OPA) have many applications from optical imaging to LiDAR. Conventionally, 2-D beam-steering in an N × N OPA requires N2 phase shifters placed within the phased array aperture, resulting in a high per-element power consumption while limiting the minimum achievable element-to-element spacing. In this paper, we report an OPA architecture, where for 2-D beam-steering in an N × N OPA, only 2N phase shifters outside of the array aperture are used, which significantly reduces the total OPA power consumption and eliminates electrical routing within the aperture. As a proof of concept, an 8 × 8 OPA is implemented that uses 16 phase shifters to perform 2-D beam-steering without tuning the wavelength. Using the aperture size of 77 µm  × 77 µm for the implemented OPA transmitter, far-field beam-steering over a range of about 7° is demonstrated.

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

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References

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    [Crossref]
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    [Crossref]
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2019 (2)

C. V. Poulton, M. J. Byrd, P. Russo, E. Timordugan, M. Khandaker, D. Vermeulen, and M. R. Watts, “Long-range LiDAR and free-space data communication with high-performance optical phased arrays,” IEEE J. Sel. Top. Quantum Electron. 25(5), 1–8 (2019).
[Crossref]

W. Xie, T. Komljenovic, J. Huang, M. Tran, M. Davenport, A. Torres, P. Pintus, and J. Bowers, “Heterogeneous silicon photonics sensing for autonomous cars [Invited],” Opt. Express 27(3), 3642–3663 (2019).
[Crossref]

2017 (1)

2016 (1)

2015 (3)

2013 (2)

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

Y. Zhang, S. Yang, A. E. Lim, G. Q. Lo, C. Galland, T. Baeher-Jones, and M. Hochberg, “A compact and low loss Y-junction for submicron silicon waveguide,” Opt. Express 21(1), 1310–1316 (2013).
[Crossref]

2011 (2)

J. K. Doylend, M. J. R. Heck, J. T. Bovington, J. D. Peters, L. A. Coldren, and J. E. Bowers, “Two-dimensional free-space beam steering with an optical phased array on silicon-on-insulator,” Opt. Express 19(22), 21595–21604 (2011).
[Crossref]

A. Mekis, S. Gloeckner, G. Masini, A. Narasimha, T. Pinguet, S. Sahni, and P. D. Dobbelaere, “A grating-coupler-enabled CMOS photonics platform,” IEEE J. Sel. Top. Quantum Electron. 17(3), 597–608 (2011).
[Crossref]

2010 (1)

Abediasl, H.

S. Chung, H. Abediasl, and H. Hashemi, “15.4 A 1024-element scalable optical phased array in 0.18µm SOI CMOS,” in 2017IEEE International Solid-State Circuits Conference (ISSCC), San Francisco.

Abiri, B.

F. Aflatouni, B. Abiri, A. Rekhi, and A. Hajimiri, “Nanophotonic coherent imager,” Opt. Express 23(4), 5117–5125 (2015).
[Crossref]

F. Aflatouni, B. Abiri, A. Rekhi, and A. Hajimiri, “Nanophotonic projection system,” Opt. Express 23(16), 21012–21022 (2015).
[Crossref]

R. Fatemi, B. Abiri, and A. Hajimiri, “An 8×8 heterodyne lens-less OPA camera,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2017), paper JW2A.9.

B. Abiri, F. Aflatouni, A. Rekhi, and A. Hajimiri, “Electronic two-dimensional beam steering for integrated optical phased srrays,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper M2K.7.

Aflatouni, F.

F. Aflatouni, B. Abiri, A. Rekhi, and A. Hajimiri, “Nanophotonic coherent imager,” Opt. Express 23(4), 5117–5125 (2015).
[Crossref]

F. Aflatouni, B. Abiri, A. Rekhi, and A. Hajimiri, “Nanophotonic projection system,” Opt. Express 23(16), 21012–21022 (2015).
[Crossref]

B. Abiri, F. Aflatouni, A. Rekhi, and A. Hajimiri, “Electronic two-dimensional beam steering for integrated optical phased srrays,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper M2K.7.

Baeher-Jones, T.

Baets, R.

Benney, L. D.

S. J. Spector, B. F. Lane, M. R. Watts, L. D. Benney, J. G. Delva, A. E. Hare, A. F. Kelsey, J. M. Mlynarczyk, E. S. Hosseini, C. V. Poulton, and J. P. Laine, “Broadband imaging and wireless communication with an optical phased array,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), paper SM3I.7

Bovington, J. T.

Bowers, J.

Bowers, J. E.

Byrd, M. J.

C. V. Poulton, M. J. Byrd, P. Russo, E. Timordugan, M. Khandaker, D. Vermeulen, and M. R. Watts, “Long-range LiDAR and free-space data communication with high-performance optical phased arrays,” IEEE J. Sel. Top. Quantum Electron. 25(5), 1–8 (2019).
[Crossref]

C. V. Poulton, P. Russo, E. Timurdogan, M. Whitson, M. J. Byrd, E. Hosseini, B. Moss, Z. Su, D. Vermeulen, and M. R. Watts, “High-performance integrated optical phased arrays for chip-scale beam steering and LiDAR,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), paper ATu3R.2.

Chung, S.

S. Chung, H. Abediasl, and H. Hashemi, “15.4 A 1024-element scalable optical phased array in 0.18µm SOI CMOS,” in 2017IEEE International Solid-State Circuits Conference (ISSCC), San Francisco.

Coldren, L.

Coldren, L. A.

Davenport, M.

Davenport, M. L.

Delva, J. G.

S. J. Spector, B. F. Lane, M. R. Watts, L. D. Benney, J. G. Delva, A. E. Hare, A. F. Kelsey, J. M. Mlynarczyk, E. S. Hosseini, C. V. Poulton, and J. P. Laine, “Broadband imaging and wireless communication with an optical phased array,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), paper SM3I.7

Dobbelaere, P. D.

A. Mekis, S. Gloeckner, G. Masini, A. Narasimha, T. Pinguet, S. Sahni, and P. D. Dobbelaere, “A grating-coupler-enabled CMOS photonics platform,” IEEE J. Sel. Top. Quantum Electron. 17(3), 597–608 (2011).
[Crossref]

Doylend, J. K.

Fatemi, R.

R. Fatemi, B. Abiri, and A. Hajimiri, “An 8×8 heterodyne lens-less OPA camera,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2017), paper JW2A.9.

Feshali, A.

Galland, C.

Gloeckner, S.

A. Mekis, S. Gloeckner, G. Masini, A. Narasimha, T. Pinguet, S. Sahni, and P. D. Dobbelaere, “A grating-coupler-enabled CMOS photonics platform,” IEEE J. Sel. Top. Quantum Electron. 17(3), 597–608 (2011).
[Crossref]

Goodman, J.

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

Hajimiri, A.

F. Aflatouni, B. Abiri, A. Rekhi, and A. Hajimiri, “Nanophotonic coherent imager,” Opt. Express 23(4), 5117–5125 (2015).
[Crossref]

F. Aflatouni, B. Abiri, A. Rekhi, and A. Hajimiri, “Nanophotonic projection system,” Opt. Express 23(16), 21012–21022 (2015).
[Crossref]

R. Fatemi, B. Abiri, and A. Hajimiri, “An 8×8 heterodyne lens-less OPA camera,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2017), paper JW2A.9.

B. Abiri, F. Aflatouni, A. Rekhi, and A. Hajimiri, “Electronic two-dimensional beam steering for integrated optical phased srrays,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper M2K.7.

Hare, A. E.

S. J. Spector, B. F. Lane, M. R. Watts, L. D. Benney, J. G. Delva, A. E. Hare, A. F. Kelsey, J. M. Mlynarczyk, E. S. Hosseini, C. V. Poulton, and J. P. Laine, “Broadband imaging and wireless communication with an optical phased array,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), paper SM3I.7

Hashemi, H.

S. Chung, H. Abediasl, and H. Hashemi, “15.4 A 1024-element scalable optical phased array in 0.18µm SOI CMOS,” in 2017IEEE International Solid-State Circuits Conference (ISSCC), San Francisco.

Heck, J.

Heck, M. J. R.

Helkey, R.

Hochberg, M.

Hosseini, E.

C. V. Poulton, P. Russo, E. Timurdogan, M. Whitson, M. J. Byrd, E. Hosseini, B. Moss, Z. Su, D. Vermeulen, and M. R. Watts, “High-performance integrated optical phased arrays for chip-scale beam steering and LiDAR,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), paper ATu3R.2.

Hosseini, E. S.

S. J. Spector, B. F. Lane, M. R. Watts, L. D. Benney, J. G. Delva, A. E. Hare, A. F. Kelsey, J. M. Mlynarczyk, E. S. Hosseini, C. V. Poulton, and J. P. Laine, “Broadband imaging and wireless communication with an optical phased array,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), paper SM3I.7

Huang, J.

Hulme, J. C.

Hutchison, D. N.

Kelsey, A. F.

S. J. Spector, B. F. Lane, M. R. Watts, L. D. Benney, J. G. Delva, A. E. Hare, A. F. Kelsey, J. M. Mlynarczyk, E. S. Hosseini, C. V. Poulton, and J. P. Laine, “Broadband imaging and wireless communication with an optical phased array,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), paper SM3I.7

Khandaker, M.

C. V. Poulton, M. J. Byrd, P. Russo, E. Timordugan, M. Khandaker, D. Vermeulen, and M. R. Watts, “Long-range LiDAR and free-space data communication with high-performance optical phased arrays,” IEEE J. Sel. Top. Quantum Electron. 25(5), 1–8 (2019).
[Crossref]

Kim, W.

Komljenovic, T.

Kumar, R.

Laine, J. P.

S. J. Spector, B. F. Lane, M. R. Watts, L. D. Benney, J. G. Delva, A. E. Hare, A. F. Kelsey, J. M. Mlynarczyk, E. S. Hosseini, C. V. Poulton, and J. P. Laine, “Broadband imaging and wireless communication with an optical phased array,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), paper SM3I.7

Lane, B. F.

S. J. Spector, B. F. Lane, M. R. Watts, L. D. Benney, J. G. Delva, A. E. Hare, A. F. Kelsey, J. M. Mlynarczyk, E. S. Hosseini, C. V. Poulton, and J. P. Laine, “Broadband imaging and wireless communication with an optical phased array,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), paper SM3I.7

Lim, A. E.

Lo, G. Q.

Masini, G.

A. Mekis, S. Gloeckner, G. Masini, A. Narasimha, T. Pinguet, S. Sahni, and P. D. Dobbelaere, “A grating-coupler-enabled CMOS photonics platform,” IEEE J. Sel. Top. Quantum Electron. 17(3), 597–608 (2011).
[Crossref]

Mekis, A.

A. Mekis, S. Gloeckner, G. Masini, A. Narasimha, T. Pinguet, S. Sahni, and P. D. Dobbelaere, “A grating-coupler-enabled CMOS photonics platform,” IEEE J. Sel. Top. Quantum Electron. 17(3), 597–608 (2011).
[Crossref]

Mlynarczyk, J. M.

S. J. Spector, B. F. Lane, M. R. Watts, L. D. Benney, J. G. Delva, A. E. Hare, A. F. Kelsey, J. M. Mlynarczyk, E. S. Hosseini, C. V. Poulton, and J. P. Laine, “Broadband imaging and wireless communication with an optical phased array,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), paper SM3I.7

Moss, B.

C. V. Poulton, P. Russo, E. Timurdogan, M. Whitson, M. J. Byrd, E. Hosseini, B. Moss, Z. Su, D. Vermeulen, and M. R. Watts, “High-performance integrated optical phased arrays for chip-scale beam steering and LiDAR,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), paper ATu3R.2.

Narasimha, A.

A. Mekis, S. Gloeckner, G. Masini, A. Narasimha, T. Pinguet, S. Sahni, and P. D. Dobbelaere, “A grating-coupler-enabled CMOS photonics platform,” IEEE J. Sel. Top. Quantum Electron. 17(3), 597–608 (2011).
[Crossref]

Peters, J. D.

Phare, C. T.

Pinguet, T.

A. Mekis, S. Gloeckner, G. Masini, A. Narasimha, T. Pinguet, S. Sahni, and P. D. Dobbelaere, “A grating-coupler-enabled CMOS photonics platform,” IEEE J. Sel. Top. Quantum Electron. 17(3), 597–608 (2011).
[Crossref]

Pintus, P.

Poulton, C. V.

C. V. Poulton, M. J. Byrd, P. Russo, E. Timordugan, M. Khandaker, D. Vermeulen, and M. R. Watts, “Long-range LiDAR and free-space data communication with high-performance optical phased arrays,” IEEE J. Sel. Top. Quantum Electron. 25(5), 1–8 (2019).
[Crossref]

S. J. Spector, B. F. Lane, M. R. Watts, L. D. Benney, J. G. Delva, A. E. Hare, A. F. Kelsey, J. M. Mlynarczyk, E. S. Hosseini, C. V. Poulton, and J. P. Laine, “Broadband imaging and wireless communication with an optical phased array,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), paper SM3I.7

C. V. Poulton, P. Russo, E. Timurdogan, M. Whitson, M. J. Byrd, E. Hosseini, B. Moss, Z. Su, D. Vermeulen, and M. R. Watts, “High-performance integrated optical phased arrays for chip-scale beam steering and LiDAR,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), paper ATu3R.2.

Rekhi, A.

F. Aflatouni, B. Abiri, A. Rekhi, and A. Hajimiri, “Nanophotonic coherent imager,” Opt. Express 23(4), 5117–5125 (2015).
[Crossref]

F. Aflatouni, B. Abiri, A. Rekhi, and A. Hajimiri, “Nanophotonic projection system,” Opt. Express 23(16), 21012–21022 (2015).
[Crossref]

B. Abiri, F. Aflatouni, A. Rekhi, and A. Hajimiri, “Electronic two-dimensional beam steering for integrated optical phased srrays,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper M2K.7.

Rogier, H.

Rong, H.

Russo, P.

C. V. Poulton, M. J. Byrd, P. Russo, E. Timordugan, M. Khandaker, D. Vermeulen, and M. R. Watts, “Long-range LiDAR and free-space data communication with high-performance optical phased arrays,” IEEE J. Sel. Top. Quantum Electron. 25(5), 1–8 (2019).
[Crossref]

C. V. Poulton, P. Russo, E. Timurdogan, M. Whitson, M. J. Byrd, E. Hosseini, B. Moss, Z. Su, D. Vermeulen, and M. R. Watts, “High-performance integrated optical phased arrays for chip-scale beam steering and LiDAR,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), paper ATu3R.2.

Sahni, S.

A. Mekis, S. Gloeckner, G. Masini, A. Narasimha, T. Pinguet, S. Sahni, and P. D. Dobbelaere, “A grating-coupler-enabled CMOS photonics platform,” IEEE J. Sel. Top. Quantum Electron. 17(3), 597–608 (2011).
[Crossref]

Shah Hosseini, E.

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

Spector, S. J.

S. J. Spector, B. F. Lane, M. R. Watts, L. D. Benney, J. G. Delva, A. E. Hare, A. F. Kelsey, J. M. Mlynarczyk, E. S. Hosseini, C. V. Poulton, and J. P. Laine, “Broadband imaging and wireless communication with an optical phased array,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), paper SM3I.7

Su, Z.

C. V. Poulton, P. Russo, E. Timurdogan, M. Whitson, M. J. Byrd, E. Hosseini, B. Moss, Z. Su, D. Vermeulen, and M. R. Watts, “High-performance integrated optical phased arrays for chip-scale beam steering and LiDAR,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), paper ATu3R.2.

Sun, J.

D. N. Hutchison, J. Sun, J. K. Doylend, R. Kumar, J. Heck, W. Kim, C. T. Phare, A. Feshali, and H. Rong, “High-resolution aliasing-free optical beam steering,” Optica 3(8), 887–890 (2016).
[Crossref]

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

Timordugan, E.

C. V. Poulton, M. J. Byrd, P. Russo, E. Timordugan, M. Khandaker, D. Vermeulen, and M. R. Watts, “Long-range LiDAR and free-space data communication with high-performance optical phased arrays,” IEEE J. Sel. Top. Quantum Electron. 25(5), 1–8 (2019).
[Crossref]

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

Timurdogan, E.

C. V. Poulton, P. Russo, E. Timurdogan, M. Whitson, M. J. Byrd, E. Hosseini, B. Moss, Z. Su, D. Vermeulen, and M. R. Watts, “High-performance integrated optical phased arrays for chip-scale beam steering and LiDAR,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), paper ATu3R.2.

Torres, A.

Tran, M.

Van Acoleyen, K.

Vermeulen, D.

C. V. Poulton, M. J. Byrd, P. Russo, E. Timordugan, M. Khandaker, D. Vermeulen, and M. R. Watts, “Long-range LiDAR and free-space data communication with high-performance optical phased arrays,” IEEE J. Sel. Top. Quantum Electron. 25(5), 1–8 (2019).
[Crossref]

C. V. Poulton, P. Russo, E. Timurdogan, M. Whitson, M. J. Byrd, E. Hosseini, B. Moss, Z. Su, D. Vermeulen, and M. R. Watts, “High-performance integrated optical phased arrays for chip-scale beam steering and LiDAR,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), paper ATu3R.2.

Watts, M. R.

C. V. Poulton, M. J. Byrd, P. Russo, E. Timordugan, M. Khandaker, D. Vermeulen, and M. R. Watts, “Long-range LiDAR and free-space data communication with high-performance optical phased arrays,” IEEE J. Sel. Top. Quantum Electron. 25(5), 1–8 (2019).
[Crossref]

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

S. J. Spector, B. F. Lane, M. R. Watts, L. D. Benney, J. G. Delva, A. E. Hare, A. F. Kelsey, J. M. Mlynarczyk, E. S. Hosseini, C. V. Poulton, and J. P. Laine, “Broadband imaging and wireless communication with an optical phased array,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), paper SM3I.7

C. V. Poulton, P. Russo, E. Timurdogan, M. Whitson, M. J. Byrd, E. Hosseini, B. Moss, Z. Su, D. Vermeulen, and M. R. Watts, “High-performance integrated optical phased arrays for chip-scale beam steering and LiDAR,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), paper ATu3R.2.

Whitson, M.

C. V. Poulton, P. Russo, E. Timurdogan, M. Whitson, M. J. Byrd, E. Hosseini, B. Moss, Z. Su, D. Vermeulen, and M. R. Watts, “High-performance integrated optical phased arrays for chip-scale beam steering and LiDAR,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (online) (Optical Society of America, 2018), paper ATu3R.2.

Xie, W.

Yaacobi, A.

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

Yang, S.

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IEEE J. Sel. Top. Quantum Electron. (2)

C. V. Poulton, M. J. Byrd, P. Russo, E. Timordugan, M. Khandaker, D. Vermeulen, and M. R. Watts, “Long-range LiDAR and free-space data communication with high-performance optical phased arrays,” IEEE J. Sel. Top. Quantum Electron. 25(5), 1–8 (2019).
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Nature (1)

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

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Figures (5)

Fig. 1.
Fig. 1. Conceptual schematic of the proposed N × N OPA, where optical signals traveling in the row and column waveguides have (a) different phases and amplitudes and (b) different phases but the same amplitude.
Fig. 2.
Fig. 2. (a) The far-field interference pattern of an 8 × 8 OPA with 64 per-element phase control (top row) and that of an 8 × 8 OPA with identical element structure and spacing but implemented based on the proposed architecture (bottom row). The diagonal beam-steering (with same relative phase between the elements in rows and columns) is performed. (b) The peak intensity of the main lobe in the far-field pattern for the conventional and proposed 8 × 8 OPAs.
Fig. 3.
Fig. 3. (a) The structure of the implemented 8 × 8 OPA with 16 phase shifters placed outside of the array aperture. (b) The structures of the grating couplers (as OPA elements) and directional couplers with varying lengths. (c) The microphotograph of the implemented 8 × 8 OPA chip fabricated in the IME 180 nm SOI process.
Fig. 4.
Fig. 4. (a) The structure and the microphotograph of the implemented thermal phase shifter. (b) The structures of the nanophotonic waveguides, the waveguide crossing, directional couplers with varying length, the Y-junction [18], and the grating coupler serving as the OPA element. (c) Finite-Difference Time-Domain (FDTD) simulation results for the directional couplers with varying length and different silicon etch levels within the coupling region.
Fig. 5.
Fig. 5. (a) Measurement setup used to characterize the OPA chip. (b) Measured far-field interference pattern showing four grating lobes. (c) Demonstration of the 2-D far-field beam-steering. (d) Two consecutive resolvable spots while diagonal beam-steering is performed. The bottom figure shows the beam profiles.

Equations (4)

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E m n = 1 N ( a m 2 + b n 2 + 2 a m b n cos ( ϕ m θ n ) ) exp [ j tan 1 ( a m sin ϕ m + b n sin θ n a m cos ϕ m + b n cos θ n ) ] .
Δ ϕ m + 1 , m , n = tan 1 ( a m + 1 sin ϕ m + 1 + b n sin θ n a m + 1 cos ϕ m + 1 + b n cos θ n ) tan 1 ( a m sin ϕ m + b n sin θ n a m cos ϕ m + b n cos θ n ) .
Δ θ m , n + 1 , n = tan 1 ( a m sin ϕ m + b n + 1 sin θ n + 1 a m cos ϕ m + b n + 1 cos θ n + 1 ) tan 1 ( a m sin ϕ m + b n sin θ n a m cos ϕ m + b n cos θ n ) .
E m n = 2 N a 0 | cos ( ϕ m θ n 2 ) | e j ( ϕ m + θ n 2 )

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