Abstract

Programmable integrated photonics is an emerging new paradigm that aims at designing common integrated optical hardware resource configurations, capable of implementing an unconstrained variety of functionalities by suitable programming, following a parallel but not identical path to that of integrated electronics in the past two decades of the last century. Programmable integrated photonics is raising considerable interest, as it is driven by the surge of a considerable number of new applications in the fields of telecommunications, quantum information processing, sensing, and neurophotonics, calling for flexible, reconfigurable, low-cost, compact, and low-power-consuming devices that can cooperate with integrated electronic devices to overcome the limitation expected by the demise of Moore’s Law. Integrated photonic devices exploiting full programmability are expected to scale from application-specific photonic chips (featuring a relatively low number of functionalities) up to very complex application-agnostic complex subsystems much in the same way as field programmable gate arrays and microprocessors operate in electronics. Two main differences need to be considered. First, as opposed to integrated electronics, programmable integrated photonics will carry analog operations over the signals to be processed. Second, the scale of integration density will be several orders of magnitude smaller due to the physical limitations imposed by the wavelength ratio of electrons and light wave photons. The success of programmable integrated photonics will depend on leveraging the properties of integrated photonic devices and, in particular, on research into suitable interconnection hardware architectures that can offer a very high spatial regularity as well as the possibility of independently setting (with a very low power consumption) the interconnection state of each connecting element. Integrated multiport interferometers and waveguide meshes provide regular and periodic geometries, formed by replicating unit elements and cells, respectively. In the case of waveguide meshes, the cells can take the form of a square, hexagon, or triangle, among other configurations. Each side of the cell is formed by two integrated waveguides connected by means of a Mach–Zehnder interferometer or a tunable directional coupler that can be operated by means of an output control signal as a crossbar switch or as a variable coupler with independent power division ratio and phase shift. In this paper, we provide the basic foundations and principles behind the construction of these complex programmable circuits. We also review some practical aspects that limit the programming and scalability of programmable integrated photonics and provide an overview of some of the most salient applications demonstrated so far.

© 2020 Optical Society of America

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A. López, D. Pérez, P. DasMahapatra, and J. Capmany, “Auto-routing algorithm for field-programmable photonic gate arrays,” Opt. Express 28, 737–752 (2020).
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W. Zhang and J. Yao, “Photonic integrated field-programmable disk array signal processor,” Nat. Commun. 11, 406 (2020).
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2019 (6)

2018 (16)

D. Pérez, I. Gasulla, and J. Capmany, “Field-programmable photonic arrays,” Opt. Express 26, 27265–27278 (2018).
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J. G. M. Van der Tol, J. van der Tol, V. Pogoretskii, J. van Engelen, A. A. Kashi, S. Reniers, Y. Wang, X. Zhao, W. Yao, T. Liu, F. Pagliano, A. Fiore, X. Zhang, Z. Cao, R. R. Kumar, H. K. Tsang, R. van Veldhoven, T. de Vries, E.-J. Geluk, J. Bolk, H. Ambrosius, M. Smit, and K. Williams, “Indium phosphide integrated photonics in membranes,” IEEE J. Sel. Top. Quantum Electron. 24, 6100809 (2018).
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P. L. Mennea, W. R. Clements, D. H. Smith, J. C. Gates, B. J. Metcalf, R. H. S. Bannerman, R. Burgwal, J. J. Renema, W. S. Kolthammer, I. A. Walmsley, and P. G. R. Smith, “Modular linear optical circuits,” Optica 5, 1087–1090 (2018).
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D. Pérez, E. Sánchez, and J. Capmany, “Programmable true-time delay lines using integrated waveguide meshes,” J. Lightwave Technol. 36, 4591–4601 (2018).
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J. Zheng, A. Khanolkar, P. Xu, S. Colburn, S. Deshmukh, J. Myers, J. Frantz, E. Pop, J. Hendrickson, J. Doylend, N. Boechler, and A. Majumdar, “GST-on-silicon hybrid nanophotonic integrated circuits: a non-volatile quasi-continuously reprogrammable platform,” Opt. Mater. Express 8, 1551–1561 (2018).
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M. Bahadori, A. Gazman, N. Janosik, S. Rumley, Z. Zhu, R. Polster, and K. Bergman, “Thermal rectification of integrated microheaters for microring resonators in silicon photonics platform,” J. Lightwave Technol. 36, 773–788 (2018).
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J. Capmany, I. Gasulla, and D. Pérez, “Toward programmable microwave photonics processors,” J. Lightwave Technol. 36, 519–532 (2018).
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Q. Cheng, S. Rumley, M. Bahadori, and K. Bergman, “Photonic switching in high performance datacenters,” Opt. Express 26, 16022–16043 (2018).
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N. C. Harris, J. Carolan, D. Bunandar, M. Prabhu, M. Hochberg, T. Baehr-Jones, M. L. Fanto, A. M. Smith, C. C. Tison, P. M. Alsing, and D. Englund, “Linear programmable nanophonic processor,” Optica 5, 1623–1631 (2018).
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X. Qian, X. Zhou, J. Wang, C. M. Wilkes, T. Loke, S. O’Gara, L. Kling, G. D. Marshall, R. Santagati, T. C. Ralph, J. B. Wang, J. L. O’Brien, M. G. Thompson, and J. C. F. Matthews, “Large-scale silicon quantum photonics implementing arbitrary two-qubit processing,” Nat. Photonics 12, 534–539 (2018).
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G. Choo, S. Cai, B. Wang, C. K. Madsen, K. Entesari, and S. Palermo, “Automatic monitor-based tuning of reconfigurable silicon photonic APF-based pole/zero filters,” J. Lightwave Technol. 36, 1899–1911 (2018).
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Z. Guo, L. Lu, L. Zhou, L. Shen, and J. Chen, “16 × 16 silicon optical switch based on dual-ring-assisted Mach–Zehnder interferometers,” J. Lightwave Technol. 36, 225–232 (2018).
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S. Han, T. J. Seok, K. Yu, N. Quack, R. S. Muller, and M. C. Wu, “Large-scale polarization-insensitive silicon photonic MEMS switches,” J. Lightwave Technol. 36, 1824–1830 (2018).
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H. Peng, M. A. Nahmias, T. F. de Lima, A. N. Tait, and B. J. Shastri, “Neuromorphic photonic integrated circuits,” IEEE J. Sel. Top. Quantum Electron. 24, 6101715 (2018).
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A. N. Tait, H. Jayatilleka, T. Ferreira De Lima, P. Y. Ma, M. A. Nahmias, B. J. Shastri, S. Shekhar, L. Chrostowski, and P. R. Prucnal, “Feedback control for microring weight banks,” Opt. Express 26, 26422–26443 (2018).
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J. Tang, T. Hao, W. Li, D. Domenech, R. Baños, P. Muñoz, N. Zhu, J. Capmany, and M. Li, “Integrated optoelectronic oscillator,” Opt. Express 26, 12257–12265 (2018).
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2017 (11)

L. Qiao, W. Tang, and T. Chu, “32 × 32 silicon electro-optic switch with built-in monitors and balanced-status units,” Sci. Rep. 7, 42306 (2017).
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T. Ferreira de Lima, B. J. Shastri, A. N. Tait, M. A. Nahmias, and P. R. Prucnal, “Progress in neuromorphic photonics,” Nanophotonics 6, 577–599 (2017).
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H. Y. Hwang, J. S. Lee, T. J. Seok, A. Forencich, H. R. Grant, D. Knutson, N. Quack, S. Han, R. S. Muller, G. C. Papen, M. C. Wu, and P. O’brien, “Flip chip packaging of digital silicon photonics MEMS switch for cloud computing and data centre,” IEEE Photon. J. 9, 2900210 (2017).
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D. Pérez, I. Gasulla, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, “Multipurpose silicon photonics signal processor core,” Nat. Commun. 8, 636 (2017).
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D. A. B. Miller, “Silicon photonics: meshing optics with applications,” Nat. Photonics 11, 403–404 (2017).
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N. C. Harris, G. R. Steinbrecher, J. Mower, Y. Lahini, M. Prabhu, D. Bunandar, C. Chen, F. N. C. Wong, T. Baehr-Jones, M. Hochberg, S. Lloyd, and D. Englund, “Quantum transport simulations in a programmable nanophotonic processor,” Nat. Photonics 11, 447–452 (2017).
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D. Pérez, I. Gasulla, F. J. Fraile, L. Crudgington, D. J. Thomson, A. Z. Khokhar, K. Li, W. Cao, G. Z. Mashanovich, and J. Capmany, “Silicon photonics rectangular universal interferometer,” Laser Photon. Rev. 11, 1700219 (2017).
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Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljačić, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11, 441–446 (2017).
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D. Zibar, H. Wymeersch, and I. Lyubomirsky, “Machine learning under the spotlight,” Nat. Photonics 11, 749–751 (2017).
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Z. Lu, J. Jhoja, J. Klein, X. Wang, A. Liu, J. Flueckiger, J. Pond, and L. Chrostowski, “Performance prediction for silicon photonics integrated circuits with layout-dependent correlated manufacturing variability,” Opt. Express 25, 9712–9733 (2017).
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B. Corbett, R. Loi, W. Zhou, D. Liu, and Z. Ma, “Transfer print techniques for heterogeneous integration of photonic components,” Prog. Quantum Electron. 52, 1–17 (2017).
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2016 (19)

J. S. Fandiño, P. Muñoz, D. Doménech, and J. Capmany, “A monolithic integrated photonic microwave filter,” Nat. Photonics 11, 124–129 (2016).
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W. Liu, M. Li, R. S. Guzzon, E. J. Norberg, J. S. Parker, M. Lu, L. A. Coldren, and J. Yao, “A fully reconfigurable photonic integrated signal processor,” Nat. Photonics 10, 190–195 (2016).
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C. J. Oton, C. Manganelli, F. Bontempi, M. Fournier, D. Fowler, and C. Kopp, “Silicon photonic waveguide metrology using Mach-Zehnder interferometers,” Opt. Express 24, 6265–6270 (2016).
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L. Carroll, J. S. Lee, C. Scarcella, K. Gradkowski, M. Duperron, H. Lu, and S. Collins, “Photonic packaging: transforming silicon photonic integrated circuits into photonic devices,” Appl. Sci. 6, 426 (2016).
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A. Ribeiro, A. Ruocco, L. Vanacker, and W. Bogaerts, “Demonstration of a 4 × 4-port universal linear circuit,” Optica 3, 1348–1357 (2016).
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O. Graydon, “Birth of the programmable optical chip,” Nat. Photonics 10, 1 (2016).
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D. Pérez, I. Gasulla, J. Capmany, and R. A. Soref, “Reconfigurable lattice mesh designs for programmable photonic processors,” Opt. Express 24, 12093–12106 (2016).
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J. Capmany, I. Gasulla, and D. Pérez, “Microwave photonics: the programmable processor,” Nat. Photonics 10, 6–8 (2016).
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W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, and I. A. Walmsley, “Optimal design for universal multiport interferometers,” Optica 3, 1460–1465 (2016).
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T. J. Seok, N. Quack, S. Han, R. S. Muller, and M. C. Wu, “Large-scale broadband digital silicon photonic switches with vertical adiabatic couplers,” Optica 3, 64–70 (2016).
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L. Lu, S. Zhao, L. Zhou, D. Li, Z. Li, M. Wang, X. Li, and J. Chen, “16 × 16 non-blocking silicon optical switch based on electro-optic Mach–Zehnder interferometers,” Opt. Express 24, 9295–9307 (2016).
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N. Harris, D. Bunandar, M. Pant, G. R. Steinbrecher, J. Mower, M. Prabhu, T. Baehr-Jones, M. Hochberg, and D. Englund, “Large-scale quantum photonic circuits in silicon,” Nanophotonics 5, 456–468 (2016).
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J. W. Silverstone, D. Bonneau, J. L. O’Brien, and M. G. Thompson, “Silicon quantum photonics,” IEEE J. Sel. Top. Quantum Electron. 22, 390–402 (2016).
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M. Poot, C. Schuck, X.-S. Ma, X. Guo, and H. X. Tang, “Design and characterization of integrated components for SiN photonic quantum circuits,” Opt. Express 24, 6843–6860 (2016).
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P. R. Prucnal, B. J. Shastri, T. Ferreira de Lima, M. A. Nahmias, and A. N. Tait, “Recent progress in semiconductor excitable lasers for photonic spike processing,” Adv. Opt. Photon. 8, 228–299 (2016).
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A. N. Tait, T. F. de Lima, M. A. Nahmias, B. J. Shastri, and P. R. Prucnal, “Continuous calibration of microring weights for analog optical networks,” IEEE Photon. Technol. Lett. 28, 887–890 (2016).
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A. N. Tait, T. Ferreira de Lima, M. A. Nahmias, B. J. Shastri, and P. R. Prucnal, “Multichannel control for microring weight banks,” Opt. Express 24, 8895–8906 (2016).
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D. Pérez, I. Gasulla, J. Capmany, J. S. Fandiño, P. Muñoz, and H. Alavi, “Figures of merit for self-beating filtered microwave photonic systems,” Opt. Express 24, 10087–10102 (2016).
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2015 (12)

D. Perez, I. Gasulla, and J. Capmany, “Software-defined universal microwave photonics processor,” Opt. Express 23, 14640–14654 (2015).
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S. Liu, Q. Cheng, M. R. Madarbux, A. Wonfor, R. V. Penty, I. H. White, and P. M. Watts, “Low latency optical switch for high performance computing with minimized processor energy load,” J. Opt. Commun. Netw. 7, A498–A510 (2015).
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K. Tanizawa, K. Suzuki, M. Toyama, M. Ohtsuka, N. Yokoyama, K. Matsumaro, M. Seki, K. Koshino, T. Sugaya, S. Suda, G. Cong, T. Kimura, K. Ikeda, S. Namiki, and H. Kawashima, “Ultra-compact 32 × 32 strictly-non-blocking Si-wire optical switch with fan-out LGA interposer,” Opt. Express 23, 17599–17606 (2015).
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N. Dupuis, “Design and fabrication of low-insertion-loss and low-crosstalk broadband 2 × 2 Mach–Zehnder silicon photonic switches,” J. Lightwave Technol. 33, 3597–3606 (2015).
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S. Han, T. J. Seok, N. Quack, B.-W. Yoo, and M. C. Wu, “Large-scale silicon photonic switches with movable directional couplers,” Optica 2, 370–375 (2015).
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L. Zhuang, C. G. H. Roeloffzen, M. Hoekman, K.-J. Boller, and A. J. Lowery, “Programmable photonic signal processor chip for radiofrequency applications,” Optica 2, 854–859 (2015).
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J. Lyke, C. G. Christodolou, G. Vera, and A. H. Edwards, “An introduction to reconfigurable systems,” Proc. IEEE 103, 291–317 (2015).
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S. M. Trimberger, “Three ages of FPGAs: a restrospective on the first thirty years of FPGA technology,” Proc. IEEE 103, 318–331 (2015).
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J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, “Universal linear optics,” Science 349, 711–716 (2015).
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D. A. B. Miller, “Perfect optics with imperfect components,” Optica 2, 747–750 (2015).
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N. Zecevic, M. Hofbauer, and H. Zimmermann, “Integrated pulsewidth modulation control for a scalable optical switch matrix,” IEEE Photon. J. 7, 7803007 (2015).
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2014 (6)

N. Spagnolo, C. Vitelli, M. Bentivegna, D. J. Brod, A. Crespi, F. Flamini, S. Giacomini, G. Milani, R. Ramponi, P. Mataloni, R. Osellame, E. F. Galvão, and F. Sciarrino, “Experimental validation of photonic boson sampling,” Nat. Photonics 8, 615–620 (2014).
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B. Nunes, M. Mendonca, X.-N. Nguyen, K. Obraczka, and K. T. Turletti, “A survey of software-defined networking: past, present, future of programmable networks,” Proc. IEEE 16, 1617–1634 (2014).
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W. Bogaerts, M. Fiers, and P. Dumon, “Design challenges in silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 20, 8202008 (2014).
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M. K. Smit, X. J. M. Leijtens, H. P. M. M. Ambrosius, E. A. J. M. Bente, J. J. G. M. Tol van der, E. Smalbrugge, T. Vries, E. J. Geluk, J. Bolk, P. J. Veldhoven, L. M. Augustin, P. J. A. Thijs, D. D’Agostino, H. R. Haghighi, K. Lawniczuk, S. T. Stopinski, M. S. Tahvili, A. Corradi, E. Kleijn, D. O. Dzibrou, M. Felicetti, E. Bitincka, V. Moskalenko, J. Zhao, R. M. Lemos, A. Dos Santos, G. Gilardi, W. Yao, K. A. Williams, R. Stabile, P. I. Kuindersma, J. Pello, S. P. Bhat, Y. Jiao, D. Heiss, G. C. Roelkens, M. J. Wale, P. Firth, F. M. Soares, N. Grote, M. Schell, H. Debregeas, M. Achouche, J.-L. Gentner, A. Bakker, T. Korthorst, D. Gallagher, A. Dabbs, A. Melloni, F. Morichetti, D. Melati, A. Wonfor, R. V. Penty, R. G. Broeke, B. Musk, and D. J. Robbins, “An introduction to InP-based generic integration technology,” Semicond. Sci. Technol. 29, 083001 (2014).
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P. DasMahapatra, R. Stabile, A. Rohit, and K. A. Williams, “Optical crosspoint matrix using broadband resonant switches,” IEEE J. Sel. Top. Quantum Electron. 20, 5900410 (2014).
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K. Vandoorne, P. Mechet, T. Van Vaerenbergh, M. Fiers, G. Morthier, D. Verstraeten, B. Schrauwen, J. Dambre, and P. Bienstman, “Experimental demonstration of reservoir computing on a silicon photonics chip,” Nat. Commun. 5, 3451 (2014).
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2013 (13)

D. Marpaung, Y. Liu, B. Isaac, J. Kalkavage, E. Adles, and T. Clark, “Integrated microwave photonics,” Laser Photon. Rev. 7, 506–538 (2013).
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Q. Cheng, A. Wonfor, R. V. Penty, and I. H. White, “Scalable, low-energy hybrid photonic space switch,” J. Lightwave Technol. 31, 3077–3084 (2013).
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J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. Smith, and I. A. Walmsley, “Boson sampling on a photonic chip,” Science 339, 798–801 (2013).
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A. Crespi, R. Osellame, R. Ramponi, D. J. Brod, E. F. Galvao, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, and F. Sciarrino, “Integrated multimode interferometers with arbitrary designs for photonic boson sampling,” Nat. Photonics 7, 545–549 (2013).
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F. Kish, R. Nagarajan, D. Welch, P. Evans, J. Rossi, J. Pleumeekers, A. Dentai, M. Kato, S. Corzine, R. Muthiah, M. Ziari, R. Schneider, M. Reffle, T. Butrie, D. Lambert, M. Missey, V. Lal, M. Fisher, S. Murthy, R. Salvatore, S. Demars, A. James, and C. Joyner, “From visible light-emitting diodes to large-scale III–V photonic integrated circuits,” Proc. IEEE 101, 2255–2270 (2013).
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C. G. H. Roeloffzen, L. Zhuang, C. Taddei, A. Leinse, R. G. Heideman, P. W. L. van Dijk, R. M. Oldenbeuving, D. A. I. Marpaung, M. Burla, and K.-J. Boller, “Silicon nitride microwave photonic circuits,” Opt. Express 21, 22937–22961 (2013).
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M. J. R. Heck, J. F. Bauters, M. L. Davenport, J. K. Doylend, S. Jain, G. Kurczveil, S. Srinivasan, Y. Tang, and J. E. Bowers, “Hybrid silicon photonic integrated circuit technology,” IEEE J. Sel. Top. Quantum Electron. 19, 6100117 (2013).
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S. Keyvaninia, M. Muneeb, S. Stanković, P. J. Van Veldhoven, D. Van Thourhout, and G. Roelkens, “Ultra-thin DVS-BCB adhesive bonding of III-V wafers, dies and multiple dies to a patterned silicon-on-insulator substrate,” Opt. Mater. Express 3, 35–46 (2013).
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C. Papagianni, A. Leivadeas, S. Papavassiliou, V. Maglaris, C. Cervello-Pastor, and A. Monje, “On the optimal allocation of virtual resources in cloud computing networks,” IEEE Trans. Comput. 62, 1060–1071 (2013).
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B. J. Metcalf, N. Thomas-Peter, J. B. Spring, D. Kundys, M. A. Broome, P. C. Humphreys, X.-M. Jin, M. Barbieri, W. S. Kolthammer, J. C. Gates, B. J. Smith, N. K. Langford, P. G. R. Smith, and I. A. Walmsley, “Multiphoton quantum interference in a multiport integrated photonic device,” Nat. Commun. 4, 1356 (2013).
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D. A. B. Miller, “Self-aligning universal beam coupler,” Opt. Express 21, 6360–6370 (2013).
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D. A. B. Miller, “Self-configuring universal linear optical component,” Photon. Res. 1, 1–15 (2013).
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M. Rudé, J. Pello, R. E. Simpson, J. Osmond, G. C. Roelkens, J. M. van der Tol, and V. Pruneri, “Optical switching at 1.55  µm in silicon racetrack resonators using phase change materials,” Appl. Phys. Lett. 103, 141119 (2013).
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2012 (3)

D. A. B. Miller, “All linear optical devices are mode converters,” Opt. Express 20, 23985–23993 (2012).
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R. G. Heideman, M. Hoekman, and F. Schreuder, “TriPleX-based integrated optical ring resonators for lab-on-a-chip and environmental detection,” IEEE J. Sel. Top. Quantum Electron. 18, 1583–1596 (2012).
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X. Luo, J. Song, S. Feng, A. W. Poon, T.-Y. Liow, M. Yu, G.-Q. Lo, and D.-L. Kwong, “Silicon high-order coupled-microring-based electrooptical switches for on-chip optical interconnects,” IEEE Photon. Technol. Lett. 24, 821–823 (2012).
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2011 (10)

M. G. Thompson, A. Politi, J. C. Matthews, and J. L. O’Brien, “Integrated waveguide circuits for optical quantum computing,” IET Circuits Devices Syst. 5, 94–102 (2011).
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A. Wonfor, H. Wang, R. Penty, and I. White, “Large port count high-speed optical switch fabric for use within datacenters,” J. Opt. Commun. Netw. 3, A32–A39 (2011).
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R. S. Guzzon, E. J. Norberg, J. S. Parker, L. A. Johansson, and L. A. Coldren, “Integrated InP-InGaAsP tunable coupled ring optical bandpass filters with zero insertion loss,” Opt. Express 19, 7816–7826 (2011).
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L. Maleki, “Sources: the optoelectronic oscillator,” Nat. Photonics 5, 728–730 (2011).
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E. J. Norberg, R. S. Guzzon, J. S. Parker, L. A. Johansson, and L. A. Coldren, “Programmable photonic microwave filters monolithically integrated in InP–InGaAsP,” J. Lightwave Technol. 29, 1611–1619 (2011).
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N. Thomas-Peter, N. K. Langford, A. Datta, L. Zhang, B. J. Smith, J. B. Spring, B. J. Metcalf, H. B. Coldenstrodt-Ronge, M. Hu, J. Nunn, and I. A. Walmsley, “Integrated photonic sensing,” New J. Phys. 13, 055024 (2011).
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L. N. Chen, E. Hall, L. Theogarajan, and J. Bowers, “Photonic switching for data center applications,” IEEE Photon. J. 3, 834–844 (2011).
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A. Peruzzo, A. Laing, A. Politi, T. Rudolph, and J. L. O’Brien, “Multimode quantum interference of photons in multiport integrated devices,” Nat. Commun. 2, 224 (2011).
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J. F. Song, T.-Y. Liow, H. Cai, M. B. Yu, G. Q. Lo, and D.-L. Kwong, “Ultralow power silicon photonics thermo-optic switch with suspended phase arms,” IEEE Photon. Technol. Lett. 23, 525–527 (2011).
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2010 (6)

W. A. Zortman, D. C. Trotter, and M. R. Watts, “Silicon photonics manufacturing,” Opt. Express 18, 23598–23607 (2010).
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M. Hochberg and T. Baehr-Jones, “Towards fabless silicon photonics,” Nat. Photonics 4, 492–494 (2010).
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H. Chen, A. W. Fang, J. D. Peters, Z. Wang, J. Bovington, D. Liang, and J. E. Bowers, “Integrated microwave photonic filter on a hybrid silicon platform,” IEEE Trans. Microwave Theory Tech. 58, 3213–3219 (2010).
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R. Boeck, N. A. F. Jaeger, N. Rouger, and L. Chrostowski, “Series-coupled silicon racetrack resonators and the Vernier effect: theory and measurement,” Opt. Express 18, 25151–25157 (2010).
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E. J. Norberg, R. S. Guzzon, S. C. Nicholes, J. S. Parker, and L. A. Coldren, “Programmable photonic lattice filters in InGaAsP–InP,” IEEE Photon. Technol. Lett. 22, 109–111 (2010).
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M. F. Saleh, G. Di Giuseppe, B. E. A. Saleh, and M. C. Teich, “Modal and polarization qubits in Ti:LiNbO3 photonic circuits for a universal quantum logic gate,” Opt. Express 18, 20475–20490 (2010).
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2009 (5)

A. Politi, J. Matthews, M. G. Thompson, and J. L. O’Brien, “Integrated quantum photonics,” IEEE J. Sel. Top. Quantum Electron. 15, 1673–1684 (2009).
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J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3, 687–695 (2009).
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J. Van Campenhout, W. M. J. Green, S. Assefa, and Y. A. Vlasov, “Low-power, 2 × 2 silicon electro-optic switch with 110-nm bandwidth for broadband reconfigurable optical networks,” Opt. Express 17, 24020–24029 (2009).
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A. W. Poon, X. Luo, F. Xu, and H. Chen, “Cascaded microresonator-based matrix switch for silicon on-chip optical interconnection,” Proc. IEEE 97, 1216–1238 (2009).
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A. Densmore, S. Janz, R. Ma, J. H. Schmid, D. X. Xu, A. Delâge, and P. Cheben, “Compact and low power thermo-optic switch using folded silicon waveguides,” Opt. Express 17, 10457–10465 (2009).
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2008 (1)

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646–649 (2008).
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2007 (2)

P. Kok, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
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J. Yao, D. Leuenberger, M.-C. M. Lee, and M. C. Wu, “Silicon microtoroidal resonators with integrated MEMS tunable coupler,” IEEE J. Sel. Top. Quantum Electron. 13, 202–208 (2007).
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2006 (2)

M.-C. M. Lee and M. C. Wu, “Tunable coupling regimes of silicon microdisk resonators using MEMS actuators,” Opt. Express 14, 4703–4712 (2006).
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R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 1678–1687 (2006).
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2005 (2)

M. Rasras, C. Madsen, M. Cappuzzo, E. Chen, L. Gomez, E. Laskowski, A. Griffin, A. Wong-Foy, A. Gasparyan, A. Kasper, J. L. Grange, and S. Patel, “Integrated resonance-enhanced variable optical delay lines,” IEEE Photon. Technol. Lett. 17, 834–836 (2005).
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G. Qi, J. Yao, J. Seregelyi, S. Paquet, and C. Belisle, “Generation and distribution of a wide-band continuously tunable millimeter-wave signal with an optical external modulation technique,” IEEE Trans. Microwave Theory Tech. 53, 3090–3097 (2005).
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2002 (1)

S. Mookherjea and A. Yariv, “Coupled resonator optical waveguides,” IEEE J. Sel. Top. Quantum Electron. 8, 448–456 (2002).
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2001 (1)

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1998 (1)

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Figures (40)

Figure 1.
Figure 1. Classification and features of the main programmable integrated photonic families (authors’ elaboration).

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Figure 2.
Figure 2. Waveguide structures for the InP (upper row), SOI (intermediate row), and ${\text{Si}_3}{\text{N}_4} - {\text{SiO}_2}$ (lower row) platforms (authors’ elaboration).

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Figure 3.
Figure 3. (a) Top view representation of a photonic integrated waveguide of length L with input and output fields corresponding to the different propagated modes. (b) Black-box representation of the integrated waveguide action over the fundamental ${\text{TE}_{01}}$ mode. Reprinted with permission from Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984) [49].

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Figure 4.
Figure 4. (a) Schematic layout of an integrated phase shifter where an actuator placed on top of the waveguide generates a change in its refractive index ${n_{\text{TE}}}(s)$ and a subsequent phase shift by means of a control signal $s$ . (b) A two waveguide tunable phase shifter. (c) Representation of bulk beam splitter including input and output fields corresponding to its two surfaces. Reprinted with permission from Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984) [49].

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Figure 5.
Figure 5. Schematic (a) cross-section and lateral and (b) top view of a directional coupler. (c) Tunable dual drive directional coupler layout including common mode and differential control signals. (d) Tuning curve for the coupling constant value of a reconfigurable directional coupler versus the differential bias signal. Reprinted with permission from Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984) [49].

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Figure 6.
Figure 6. (a) Tunable 3 dB MZI coupler, (b) layout of the $2 \times 2$ 3 dB multimode interference (MMI) coupler used to embed the balanced interferometer. Reprinted with permission from Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984) [49].

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Figure 7.
Figure 7. (a) Schematic layout of an electronic FPGA containing a set of interconnected configurable logic elements (CLEs), input/output pads and high-performance blocks (HPBs), and examples of irreversible logic Boolean gates employed in the design of CLEs. (b) XOR gate; (c) AND gate. Reprinted with permission from Trimberger, ed., Field-Programmable Gate Array Technology (Springer, 1994) [55].

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Figure 8.
Figure 8. (a) Schematic layout for the photonic implementation of $Rx(\theta)$ based on a tunable directional coupler. (b) Schematic layout for the photonic implementation of $Ry(\theta)$ based on a tunable 3 dB MZI coupler. (c) Schematic layout for the photonic implementation of $Rz(\theta)$ based on a differential push–pull phase shifter. (d) Integrated photonic implementation of an arbitrary unitary $2 \times 2$ matrix transform using a cascade of $Rx(\theta)$ and $Rz(\theta)$ rotations. (e) Integrated photonic implementation of an arbitrary unitary $2 \times 2$ matrix transform using a cascade of $Ry(\theta)$ and $Rz(\theta)$ rotations. Reprinted from [59].

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Figure 9.
Figure 9. (a) Bulk optics implementation of a tunable beam splitter (adapted from [21]). (b) Symbol representation of the beam splitter layout. (c) Triangular decomposition implementing a general $U(4)$ ${4}\times {4}$ unitary transformation and layout (d). Colors represent the propagation of a given input port. (e)  $U(4)$ layout of Fig. 8(d) and its implementation using MZI-based TBUs. TBUs in red color operate as phase shifters.

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Figure 10.
Figure 10. Illustration of the algorithm for programming a universal multiport interferometer for the $U(5)$ interferometer. The left-hand side presents the decomposition procedure, and the right-hand side shows the decomposition for building up the corresponding interferometer. (a) The starting point is any random unitary matrix $U(5)$ and a blank interferometer. (b)–(e) Successive stages of the implementation algorithm. (f) Final expression for $U(5)$ in terms of a cascade of $U(2)$ interferometers and a diagonal matrix. Reprinted with permission from [22]. Copyright 2016 Optical Society of America.

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Figure 11.
Figure 11. Integrated rectangular interferometers. (a) Design layout for a U(9) transformer (after [22]) and (b) implementation layout suitable for integrated optics using 3 dB MZIs (after [23]). (c) Photograph of a three-module silica on silicon rectangular interferometer. (d) Detail of an individual module showing 10 parallel TBUs. (c) and (d) Reprinted with permission from [63]. Copyright 2018 Optical Society of America.

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Figure 12.
Figure 12. Different integrated waveguide mesh arrangements: (a) squared feedforward/backward and implementation in ${\text{Si}_3}{\text{N}_{4.}}$ Reprinted with permission from [17]. Copyright 2015 Optical Society of America. (b) Hexagonal feedforward/backward (reprinted from [20]) and implementation in SOI. (c) Triangular feedforward/backward and implementation in ${\text{Si}_3}{\text{N}_{4.}}$ Reprinted from [66].

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Figure 13.
Figure 13. (a) Upper, labeled schematic of a general tunable coupler acting as the basic building block of the mesh. The basic unit length (BUL) is illustrated as the sum of the tunable coupler length and the arc length of the access waveguides. Lower, particular case of an integrated balanced MZI-based tunable coupler. (b) Signal flow for the different TBU configuration states. Reprinted from [18].

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Figure 14.
Figure 14. Basic programming of medium-complex circuits in square waveguide mesh topology: delay line, unbalanced MZI, and optical ring resonator (after [18]).

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Figure 15.
Figure 15. Basic programming of medium-complex circuits in triangular waveguide mesh topology: delay line, unbalanced MZI, and optical ring resonator (after [18]).

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Figure 16.
Figure 16. Basic programming of medium-complex circuits in hexagonal waveguide mesh topology: delay line, unbalanced MZI, and optical ring resonator (after [18]).

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Figure 17.
Figure 17. FIR filter implementations (after [20]). Left, hexagonal mesh setting for, right, three different targeted UMZI filters. Note the color code that describes the programming status of each TBU in the mesh.

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Figure 18.
Figure 18. Single-cavity IIR filter implementations (after [20]). Left, hexagonal mesh setting for, right, three different targeted ORR filters. Note the color code that describes the programming status of each TBU in the mesh.

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Figure 19.
Figure 19. ${1}\times {8}$ delay line array based on discrete optical delay lines (after [65]): configuration examples for 0-, 2-, 4-, 6-, 8-, 10-, 12-, and 14-BUL path differences. Each path length is labeled at each output. The color code used for each TBU configuration is Cross State (black), Bar State (Orange), Tunable Coupler (Green).

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Figure 20.
Figure 20. Universal interferometers emulated using an hexagonal waveguide mesh (after [23]): (a) classical triangular arrangement and (b) hexagonal mesh-based implementation of a $4\times 4$ interferometer. (c) Beam splitter for the classical approach and (d) corresponding beam splitter implementation with 3 TBUs for the hexagonal waveguide mesh.

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Figure 21.
Figure 21. Schematic diagram example of the proposed FPPGA device. The zoom shows a detail of the programmable photonic analog block (PPAB) as it pertains to the left-up to right-bottom direction of propagation (after [78]).

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Figure 22.
Figure 22. (a) Two-dimensional array arrangement of reversible $G$ gates. (b) Detailed explanation of how the concatenation of a given Gate ${G_{\textit{ij}}}$ actually results in the possibility of implementing a unitary arbitrary transformation by the ${{\textit{ij}}}$ element. Reprinted from [79].

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Figure 23.
Figure 23. Physical layouts for the implementation of the 2D gate array in Figure 21 (after [78]). Each square box represents a ${2\times 2}$ tunable coupling element and a (preceding) phase shifter. The internal broken-dotted lines illustrate the possible physical connections between input and output ports. (a) and (b) show structures where the gates in adjacent columns are rotated by 180º. (c) and (d) show structures where the gates in adjacent columns are rotated by 90º.

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Figure 24.
Figure 24. Identification between the fundamental unit blocks employed to construct integrated waveguide meshes and the main physical layouts for 2D gate arrays shown in Fig. 23 (after [78]).

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Figure 25.
Figure 25. Left, main steps involved in the design flow of a FPPGA device (after [78]). Right, FPPGA soft and hard tiers and expanded layout including peripheral high-performance blocks.

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Figure 26.
Figure 26. (a) Schematic and photo of the ${\text{Si}_3}{\text{N}_4}$ waveguide technology (TriPleX) chip implementing a two square cell waveguide mesh reported in [17]. (b) Experimental results for different programmed circuit configurations obtained by varying phase-tuning elements in the chip and the measurements of their corresponding frequency responses. Reprinted with permission from [17]. Copyright 2015 Optical Society of America.

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Figure 27.
Figure 27. Experimental results for 6-BUL ring resonator IIR and FIR + IIR filters (after [20]). (a) Waveguide mesh connection diagram, (b) circuit layout, and (c) measured modulus transfer function for a IIR filter for different values of the coupling constants ${K_1}$ and ${K_2};$ (d) IIR filter along a full spectral period for different values of the optical ring resonator round trip phase shift; (e) time response for critical coupling condition.

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Figure 28.
Figure 28. Triangular feedforward/feedbackward waveguide mesh arrangement based on dual-drive directional couplers. (a) Targeted layout and TBU settings; (b) reflection response of optical circuits; (c) transmission response of optical circuits. First row, single ring resonator and add–drop. Second row, coupled ring resonator structure. Reprinted from [66].

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Figure 29.
Figure 29. Overall power consumption for waveguide meshes of 50, 100, and 150 activated TBUs for four different TBU power consumption. Reprinted from [18].

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Figure 30.
Figure 30. (a) Schematic bulk optics implementation of a programmable photonic quantum processor including all the relevant optical and electronic elements. (b) A proposed mock-up for implementation of the main parts of the processor in SOI technology (after [116]). From left to right, photon sources (magenta), pump removal filters (yellow), passive and active optics (green), single-photon detectors (cyan), and control and feedback electronics (blue). Labels indicate the following: i, pump input and splitter; ii, spiralled waveguide photon-pair source; iii, ring resonator photon-pair source; iv, Bragg reflector pump removal filter; v, coupled-resonator optical waveguide (CROW) pump removal filter; vi, asymmetric MZI wavelength division multiplexer (WDM); vii, ring resonator WDM; viii, thermal phase tuner; ix, multimode interference waveguide coupler (MMI); x, waveguide crossing; xi, superconducting nanowire single-photon detector; xii, grating-based fiber-to-chip coupler; xiii, control and logic electronics. Reprinted from [59].

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Figure 31.
Figure 31. (a) Rectangular arrangement of a $9\times 9$ interferometer as proposed in [22]. (b) Equivalent implementation using the hexagonal waveguide mesh (after [23]).

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Figure 32.
Figure 32. Layout of a probabilistic implementation of a linear combiner of operators. The linear combination is implemented when all $n$ control qubits are measured to be 0. The success probability is ${1/}k$ . Reprinted by permission from Macmillan Publishers Ltd.: Qian et al., Nat. Photonics 12, 534-539 (2018) [120].

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Figure 33.
Figure 33. Layout of the one photon quantum walker circuit (adapted from [15]). Time $(\tau)$ is defined from left to right. Space (i) is defined from top to bottom.

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Figure 34.
Figure 34. (a) Layout of a passive ${8}\times {8}$ InP switch fabric (after [140]). (b) Detail of the elementary ${2}\times {2}$ switch cell corresponding to (a). (c) Active ${2}\times {2}$ interferometric InP switch cell combining an MZI and semiconductor optical amplifiers and construction of a ${4}\times {4}$ switch fabric by dilated combination of such cells (after [142]). (d) Photograph and detail of a ${2}\times {2}$ switch cell of a silicon photonics ${32}\times {32}$ switch fabric incorporating 1024 MZIs switching cells and thermo-optic phase shifters (after [143]). (e) Layout and chip photograph of a ${32}\times {32}$ switch fabric incorporating 144 MZIs switching cells and carrier injection PIN activated phase shifters. Also shown: the basic switch cell unit with push–pull biasing and the ${8}\times {8}$ Benes building block (after [135]).

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Figure 35.
Figure 35. Schematic description of a neuromorphic processor (adapted from [147]). While two layers of electronics provide control, supervision, and reconfigurability, the programmable artificial neural network operates as the core processor.

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Figure 36.
Figure 36. (a) The schematic feedforward artificial neural network architecture implemented in [24] composed of an input layer, a number of hidden layers, and an output layer. (b) Decomposition of the general neural network into individual layers with a detail of the optical interference and nonlinearity units that compose each layer of the artificial neural network. (c) Representation of the two-layer ONN experiment reported in [24]. (d) Experimental feedback and control loop used in the experiment. Laser light is coupled to the OIU, transformed, measured on a photodiode array, and then read on a computer (drawings adapted from [24]).

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Figure 37.
Figure 37. General-purpose photonic integrated processor architecture and candidate fabrication platforms for each subsystem (after [27]).

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Figure 38.
Figure 38. General-purpose signal processor configuration for RF filtering implementation (left) based on a self-homodyne modulation/detection scheme (right). The optical filter is composed of six cascaded ORRs defined by a cavity length of 6 BULs (after [27]).

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Figure 39.
Figure 39. General-purpose signal processor configuration for microwave and mm-wave tone generation based on external modulator approach (upper) and optoelectronic oscillation approach (bottom). The right-hand figures illustrate the targeted configuration schemes (after [27]).

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Figure 40.
Figure 40. Significant milestones and evolution periods in integrated electronics and photonics (authors’ elaboration).

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Tables (6)

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Table 1. Summary of Values for the Figures of Merit of the Different Waveguide Mesh Designs

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Table 2. Experimental Demonstrators of Feedforward Meshes/ Multiport Interferometers a

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Table 3. Summary of the Main Figures of the Multipurpose Waveguide Meshes a

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Table 4. Required Minimal Number of TBU and Phase Actuators for Waveguide Meshes a

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Table 5. Maximum Number of TBUs Defining a Synthetized Path for a Mean Insertion Loss per TBU and Maximum Mesh Loss

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Table 6. Frequency Grid Associated with Four Different BULs

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Equations (49)

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E = ∑ μ e μ [ a f μ e − j β μ z + a b μ e j β μ z ] , H = ∑ μ h μ [ a f μ e − j β μ z − a b μ e j β μ z ] ,
e μ = E ( x , y ) e ^ μ , h μ = H ( x , y ) h ^ μ ,
b 2 = a f e − j β d = a 1 e − j β d , b 1 = a b e − j β d = a 2 e − j β d .
S = ( 0 e − j β d e − j β d 0 ) .
β TE = 2 π n TE d λ .
S = e − j θ ( s ) ( 0 1 1 0 ) , θ ( s ) = 2 π n TE ( s ) d λ .
S DPP ( ϕ 1 , ϕ 2 ) = ( e − j 2 π n TE ( s 1 ) d λ 0 0 e − j 2 π n TE ( s 2 ) d λ ) = e − j ( ϕ 1 + ϕ 2 ) 2 ( e − j Δ ϕ 2 0 0 e j Δ ϕ 2 ) , ϕ 1 = 2 π n TE ( s 1 ) d λ , ϕ 2 = 2 π n TE ( s 2 ) d λ .
E 1 − = r 11 E 1 + + t 12 E 2 + , E 2 − = t 21 E 1 + + r 22 E 2 + .
S = ( r 11 t 12 t 21 r 22 ) .
t 21 = t 12 = t .
| r 11 | 2 + | t | 2 = 1 , | r 22 | 2 + | t | 2 = 1 , | r 11 t | e j ( ϕ 12 − ϕ 11 ) + | r 22 t | e j ( ϕ 22 − ϕ 21 ) = 0.
| r 11 | 2 = | r 22 | 2 = | r | 2 = 1 − | t | 2 ,
ϕ 12 − ϕ 11 − ϕ 22 + ϕ 21 = ± ( 2 k + 1 ) π , k ∈ N .
2 ϕ = π ⇒ ϕ = π 2 .
S = ( r j t j t r ) ; t = 1 − r 2 .
d a 1 d z = − j β 1 a 1 + κ 12 a 2 d a 2 d z = − j β 2 a 2 + κ 21 a 1 ,
( a 1 ( z ) a 2 ( z ) ) = S TCD ( θ = f ( β 1 , β 2 , κ ) ) ( a 1 ( 0 ) a 2 ( 0 ) ) , S TCD ( θ = f ( β 1 , β 2 , κ ) ) = e − j ( β 1 + β 2 ) z 2 ( e j φ ( Δ β ) cos ( θ / 2 ) j sin ( θ / 2 ) j sin ( θ / 2 ) e − j φ ( Δ β ) cos ( θ / 2 ) ) , K = ( κ sin ⁡ β o z β o ) 2 = sin 2 ( θ / 2 ) , Δ β = β 2 − β 1 , β o = ( Δ β 2 ) 2 + κ 2 , φ ( Δ β ) = tan − 1 [ Δ β 2 β o tan ⁡ β o z ] ,
S TDC ( β , θ ) = e − j β z ( cos ( θ / 2 ) j sin ( θ / 2 ) j sin ( θ / 2 ) cos ( θ / 2 ) ) , K = sin 2 β o z = sin 2 ( θ 2 ) .
β 1 = β + γ ( s o + δ s ) , β 2 = β + γ ( s o − δ s ) ,
δ s = 3 | κ | γ .
Δ = ( β 1 + β 2 2 ) = β + γ s o .
φ ( δ s ) = tan − 1 [ 1 1 + ( | κ | γ δ s ) 2 tan ⁡ ( π γ δ s 2 | κ | 1 + ( | κ | γ δ s ) 2 ) ] .
S = j e j φ o 2 ( 1 j j 1 ) ,
Δ = 2 φ o + γ ( s 1 + s 2 ) 2 , θ = γ ( s 2 − s 1 ) + π ,
S MZI ( Δ , θ ) = − j e j Δ ( cos ( θ / 2 ) sin ( θ / 2 ) sin ( θ / 2 ) − cos ( θ / 2 ) ) .
σ o = I = ( 1 0 0 1 ) ; σ 1 = X = ( 0 1 1 0 ) , σ 2 = Y = ( 0 − j j 0 ) , σ 3 = Z = ( 1 0 0 − 1 ) .
R x ( θ ) = e − j θ 2 X , R y ( θ ) = e − j θ 2 Y , R z ( θ ) = e − j θ 2 Z .
R x ( θ ) = ( cos ( θ / 2 ) j sin ( θ / 2 ) j sin ( θ / 2 ) cos ( θ / 2 ) ) , R y ( θ ) = ( cos ( θ / 2 ) − sin ( θ / 2 ) sin ( θ / 2 ) cos ( θ / 2 ) ) , R z ( θ ) = ( e − j ( θ / 2 ) 0 0 e j ( θ / 2 ) ) .
R x ( θ ) = ( cos ( θ / 2 ) j sin ( θ / 2 ) j sin ( θ / 2 ) cos ( θ / 2 ) ) = e − j ( β 1 + β 2 ) z 2 e j ( β 1 + β 2 ) z 2 ( e j φ ( Δ β ) / 2 0 0 e − j φ ( Δ β ) / 2 ) ( e j φ ( Δ β ) cos ( θ / 2 ) j sin ⁡ ( θ / 2 ) j sin ⁡ ( θ / 2 ) e − j φ ( Δ β ) cos ( θ / 2 ) ) ( e j φ ( Δ β ) / 2 0 0 e − j φ ( Δ β ) / 2 ) = e j ( β 1 + β 2 ) z 2 e j 2 φ o S DPP ( φ o + φ ( Δ β ) , φ o − φ ( Δ β ) ) S TDC ( θ = f ( β 1 , β 2 , κ ) ) × S DPP ( φ o + φ ( Δ β ) 2 , φ o − φ ( Δ β ) 2 ) .
R x ( θ ) = S DPP ( φ o + φ ( Δ β ) , φ o − φ ( Δ β ) ) S TDC ( θ = f ( β 1 , β 2 , κ ) ) × S DPP ( φ o + φ ( Δ β ) , φ o − φ ( Δ β ) ) .
R y ( θ ) = ( cos ( θ / 2 ) − sin ( θ / 2 ) sin ( θ / 2 ) cos ( θ / 2 ) ) = j e − j Δ ( − j e j Δ ) ( cos ⁡ ( θ / 2 ) sin ⁡ ( θ / 2 ) sin ⁡ ( θ / 2 ) − cos ⁡ ( θ / 2 ) ) × ( 1 0 0 − 1 ) = j e − j Δ e j φ o S MZI ( Δ , θ ) S DPP ( φ o + π , φ o − π ) .
R y ( θ ) = S MZI ( Δ , θ ) S DPP ( φ o + π , φ o − π ) .
R z ( θ ) = ( e − j ( θ / 2 ) 0 0 e j ( θ / 2 ) ) = S DPP ( θ , − θ ) .
U = e j δ R z ( α ) R y ( β ) R z ( γ ) , U = e j δ R z ( α ) R x ( β ) R z ( γ ) ,
T mn ( φ , θ ) = ( 1 0 ⋯ ⋯ ⋯ ⋯ ⋯ 0 0 1 ⋯ ⋯ ⋯ ⋯ ⋯ 0 ⋮ ⋮ ⋱ ⋯ ⋮ ⋮ ⋮ ⋮ 0 0 ⋯ e j φ cos ⁡ θ ⋯ − sin ⁡ θ ⋯ 0 ⋮ ⋮ ⋯ ⋮ ⋱ ⋮ ⋯ 0 0 0 ⋯ e j φ sin ⁡ θ ⋯ cos ⁡ θ ⋯ 0 ⋮ ⋮ ⋯ ⋯ ⋯ ⋯ ⋱ ⋮ 0 0 ⋯ 0 ⋯ 0 ⋯ 1 ) 1 2 ⋮ m ⋮ n ⋮ N ,
T mn − 1 ( φ , θ ) = ( 1 0 ⋯ ⋯ ⋯ ⋯ ⋯ 0 0 1 ⋯ ⋯ ⋯ ⋯ ⋯ 0 ⋮ ⋮ ⋱ ⋯ ⋮ ⋮ ⋮ ⋮ 0 0 ⋯ e − j φ cos ⁡ θ ⋯ e − j φ sin ⁡ θ ⋯ 0 ⋮ ⋮ ⋯ ⋮ ⋱ ⋮ ⋯ 0 0 0 ⋯ − sin ⁡ θ ⋯ cos ⁡ θ ⋯ 0 ⋮ ⋮ ⋯ ⋯ ⋯ ⋯ ⋱ ⋮ 0 0 ⋯ 0 ⋯ 0 ⋯ 1 ) 1 2 ⋮ m ⋮ n ⋮ N
U ( N ) T N N − 1 − 1 T N N − 2 − 1 … T N 1 − 1 = ( ∗ ∗ ∗ ∗ 0 ∗ ∗ ∗ … 0 … … … … 0 ∗ ∗ ∗ ∗ 0 0 0 0 0 e j α N ) = ( U ′ ( N − 1 ) 0 0 e j α N ) .
U ( N ) T N N − 1 − 1 T N N − 2 − 1 ⋯ T N 1 − 1 ⋯ T 32 − 1 T 31 − 1 T 21 − 1 = ( e j α 1 0 0 0 0 0 e j α 2 0 ⋯ 0 ⋯ ⋯ ⋯ ⋯ 0 0 0 0 e j α N − 1 0 0 0 0 0 e j α N ) = D .
U − 1 ( N ) = T N N − 1 − 1 T N N − 2 − 1 ⋯ T N 1 − 1 ⋯ T 32 − 1 T 31 − 1 T 21 − 1 D − 1 ,
U ( N ) = D T 21 T 31 T 32 ⋯ T N − 1 N − 2 T N 1 ⋯ T N N − 2 T N N − 1 = D ∏ i = 2 N [ ∏ j = 1 i − 1 T i j ] .
T 45 T 34 T 23 T 12 T 45 T 34 U T 12 − 1 T 34 − 1 T 23 − 1 T 12 − 1 = D .
U = D ′ T 34 T 45 T 12 T 23 T 34 T 45 T 12 T 23 T 34 T 12 .
BUL = L access + L Tunable − Coupler ,
BUD = n g BUL / c ,
I L ( dB ) = 20 log 10 ( γ ) = 10 log 10 ( | a out1 | 2 + | a out2 | 2 | a in1 | 2 + | a in2 | 2 ) , C T Bar/Cross ( dB ) | | a in1 | 2 = 1 | a in2 | 2 = 0 = 10 log 10 ( | a out2,1 | 2 | a out1,2 | 2 ) .
Δ ν FSR = c n g ⋅ N ⋅ BULs ,
G = e j δ R y ( β ) R z ( γ ) , G = e j δ R x ( β ) R z ( γ ) .
U = ∑ i = 0 3 α i A i ⊗ B i = ∑ i = 0 3 α i ( P 1 σ i Q 1 ) ⊗ ( P 2 σ i Q 2 ) ,
i ( t ) ∝ s | H ( f o ) | ⋅ | H ( f o + f R F ) | ⋅ cos ⁡ ( 2 π f R F − ∠ H ( f o ) + ∠ H ( f o + f R F ) ) ,

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