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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 22 — Nov. 4, 2013
  • pp: 25619–25631
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Ultrabroadband nonreciprocal transverse energy flow of light in linear passive photonic circuits

Keyu Xia, M. Alamri, and M. Suhail Zubairy  »View Author Affiliations


Optics Express, Vol. 21, Issue 22, pp. 25619-25631 (2013)
http://dx.doi.org/10.1364/OE.21.025619


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Abstract

Using a technique, analogous to coherent population trapping in an atomic system, we propose schemes to create transverse light propagation violating left-right symmetry in a photonic circuit consisting of three coupled waveguides. The frequency windows for the symmetry breaking of the left-right energy flow span over 80 nm. Our proposed system only uses linear passive optical materials and is easy to integrate on a chip.

© 2013 Optical Society of America

1. Introduction

The integration of nonreciprocal photonic devices on Si or CMOS platforms has been challenging in the past decades. The known optical nonreciprocity can be divided into two classes: forward-backward nonrecipricity (FBNR) and left-right nonrecipricity (LRNR). The first class of nonreciprocal component is well studied [1

1. R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717–754(2004). [CrossRef]

] and can be realized by various approaches using the magneto-optic effects [2

2. J. Fujita, M. Levy, R. M. Osgood, L. Wilkens, and H. Dötsch, “Waveguide optical isolator based on machzehnder interferometer,” Appl. Phys. Lett. 76, 2158–2160(2000). [CrossRef]

7

7. L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photon. 5, 758–762(2011). [CrossRef]

], nonlinearity [8

8. L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450(2012). [CrossRef]

], a modulation media [9

9. Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94(2009). [CrossRef]

11

11. Q. Wang, F. Xu, Z. Yu, X. Qian, X. Hu, Y. Lu, and H. Wang, “A bidirectional tunable optical diode based on periodically poled linbo3,” Opt. Express 18, 7340–7346(2010). [CrossRef] [PubMed]

], optomechanics [12

12. S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. 102, 213903(2009). [CrossRef] [PubMed]

, 13

13. M. Hafezi and P. Rabl, “Optomechanically induced non-reciprocity in microring resonators,” Opt. Express 20, 7672–7684(2012). [CrossRef] [PubMed]

], or magnetized plasmonic metal [14

14. Y. Hadad and B. Z. Steinberg, “Magnetized spiral chains of plasmonic ellipsoids for one-way optical waveguides,” Phys. Rev. Lett. 105, 233904(2010). [CrossRef]

, 15

15. A. B. Khanikaev, S. H. Mousavi, G. Shvets, and Y. S. Kivshar, “One-way extraordinary optical transmission and nonreciprocal spoof plasmons,” Phys. Rev. Lett. 105, 126804(2010). [CrossRef] [PubMed]

]. It can be used for optical isolators. To date, the LRNR, indicating the nonreciprocal light flow between left and right ports of photonic circuits, has been discussed in an array of coupled waveguides by only a few research groups [16

16. C. Eüter, K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of paritytime symmetry in optics,” Nat. Phys. 6, 192–195(2010). [CrossRef]

18

18. H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Transition from discrete to continuous townes solitons in periodic media,” Phys. Rev. A 82, 043803(2010). [CrossRef]

]. If the complex optical potential causes the parity-time (PT) symmetry breaking, two coupled waveguides can show a LRNR light transfer in the transverse direction [16

16. C. Eüter, K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of paritytime symmetry in optics,” Nat. Phys. 6, 192–195(2010). [CrossRef]

,17

17. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902(2009). [CrossRef] [PubMed]

]. The realization relies on the precise control of the active medium. Although not for optical isolators, the photonic circuit with the left-right nonrecipricity may switch or route the incident light beams.

The need for integration of optical nonreciprocal elements on a Si/CMOS chip platform is a long-standing problem. The realization of the nonreciprocal light propagation in a completely linear optical medium can strongly impact on both fundamental physics, and also vast applications for integrated optics because of the compatibility with the Si material and CMOS chips. Again based on the PT symmetry breaking induced by a periodic modulation of complex optical potential, Feng et al. stated that they, for the first time, observed the nonreciprocal light propagation in a linear passive optical material [19

19. L. Feng, M. Ayache, J. Huang, Y. Xu, M. Lu, Y. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733(2011). [CrossRef] [PubMed]

]. Unfortunately, they admitted their mistake [20

20. L. Feng, M. Ayache, J. Huang, Y. Xu, M. Lu, Y. Chen, Y. Fainman, and A. Scherer, “Response to comment on “nonreciprocal light propagation in a silicon photonic circuit,” Science 335, 38–c(2012). [CrossRef]

] after Fan et al. commented on their work [21

21. S. Fan, R. Baets, A. Petrov, Z. Yu, W. F. J. D. Joannopoulos, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on “nonreciprocal light propagation in a silicon photonic circuit,” Science 335, 38–b(2012). [CrossRef]

]. Another experimental realization of on-chip optical diodes using all-dielectric, passive, and linear silicon photonic crystal structures is reported by Wang et al. [22

22. C. Wang, C. Zhou, and Z. Li, “On-chip optical diode based on silicon photonic crystal heterojunctions,” Opt. Express 19, 26948–26955(2011). [CrossRef]

, 23

23. C. Wang, X. Zhong, and Z. Li, “Linear and passive silicon optical isolator,” Sci. Rep. 2, 1–6(2012).

]. They confidentially explained why it is possible to make optical diodes using a spatial symmetry breaking geometry in a passive and linear optical medium [23

23. C. Wang, X. Zhong, and Z. Li, “Linear and passive silicon optical isolator,” Sci. Rep. 2, 1–6(2012).

].

In the LRNR, the input light is always localized in one waveguide [16

16. C. Eüter, K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of paritytime symmetry in optics,” Nat. Phys. 6, 192–195(2010). [CrossRef]

18

18. H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Transition from discrete to continuous townes solitons in periodic media,” Phys. Rev. A 82, 043803(2010). [CrossRef]

]. This is similar to the coherent population trapping in a three-level Λ–type atom [24

24. M. O. Scully and M. S. Zubairy, Quantum Optics(Cambridge University, 1997). [CrossRef]

]. While the bending waveguide array can simulate well the quantum dynamics of atoms. The classical optical analogs of coherent population transfer [25

25. S. Longhi, “Transfer of light waves in optical waveguides via a continuum,” Phys. Rev. A 78, 013815(2008). [CrossRef]

27

27. S. Longhi, “Quantum-optical analogies using photonic structures,” Laser & Photon. Rev. 3, 243–261(2009). [CrossRef]

] and population trapping in the continuum [28

28. S. Longhi, “Optical analog of population trapping in the continuum: Classical and quantum interference effects,” Phys. Rev. A 79, 023811(2009). [CrossRef]

] and Rabi oscillation [29

29. A. Crespi, S. Longhi, and R. Osellame, “Photonic realization of the quantum rabi model,” Phys. Rev. Lett. 108, 163601(2012). [CrossRef] [PubMed]

] as well has been proved by Longhi’s group. Although the trapping of equal light in two waveguides has been discussed [25

25. S. Longhi, “Transfer of light waves in optical waveguides via a continuum,” Phys. Rev. A 78, 013815(2008). [CrossRef]

], the results did not show a valid LRNR of light flow. A theory work indicated that the critical large nonlinearity is necessary to induce nonrecipricity [18

18. H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Transition from discrete to continuous townes solitons in periodic media,” Phys. Rev. A 82, 043803(2010). [CrossRef]

] in two evanescently coupled waveguides. However the LRNR in the transverse light flow has been recently observed [16

16. C. Eüter, K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of paritytime symmetry in optics,” Nat. Phys. 6, 192–195(2010). [CrossRef]

, 17

17. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902(2009). [CrossRef] [PubMed]

]. Moreover, the three/many-body systems behavior essentially different from the simple two-body system studied in [18

18. H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Transition from discrete to continuous townes solitons in periodic media,” Phys. Rev. A 82, 043803(2010). [CrossRef]

]. At least, the optical trapping in coupled three waveguides analogous to the atomic CPT can not be achieved in an optical system composing of two waveguides. It is interesting if one can realize the nonreciprocal wave propagation in evanescently coupled linear and passive waveguides. We expect to achieve the LRNR in a waveguide array. This is the motivation of our work.

Here we propose simple methods to generate the second class of optical nonreciprocity in an array of three coupled waveguides only making from linear, passive optical materials. We focus on the nonreciprocal transverse energy flow between left and right optical waveguides. This left-right nonreciprocity does not violate the Lorentz reciprocity theorem. Thanks to a small dispersion of a linear waveguide, our system can behave in a nonreciprocal manner in an ultrabroad band. The results by numerical simulation of beam propagating method (BPM) and solving the coupled mode equation (CME) demonstrate the breaking of symmetry of transverse energy flow.

2. Setup and model

Our system, shown in Fig. 1, is composed of three coupled waveguides embedded in a wafer of width Wand length L. The middle waveguide D3couples to the waveguides D1and D2. We assume that the coupling between the waveguides D1and D2is negligible. We also assume that the two side waveguides are lossless but some loss can be included in the middle one. In our photonic system, eigenmodes in the individual waveguide exchange energy via their evanescent fields when two waveguides are close. The couplings are denoted as κ13and κ23, and decrease as the distance d13,23between two waveguides increases. The coupling between D1and D2is assumed vanishing because these two waveguides are far enough from each other. The field in D3decays exponentially with a constant γthat can be controlled [16

16. C. Eüter, K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of paritytime symmetry in optics,” Nat. Phys. 6, 192–195(2010). [CrossRef]

, 17

17. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902(2009). [CrossRef] [PubMed]

, 19

19. L. Feng, M. Ayache, J. Huang, Y. Xu, M. Lu, Y. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733(2011). [CrossRef] [PubMed]

, 30

30. V. Lordi, H. B. Yuen, S. R. Bank, and J. S. Harris, “Quantum-confined stark effect of gainnas(sb) quantum wells at 1300–1600 nm,” Appl. Phys. Lett. 85, 902–904(2004). [CrossRef]

, 31

31. S. Sandhu, M. L. Povinelli, and S. Fan, “Stopping and time reversing a light pulse using dynamics loss tuning of coupled-resonators delay lines,” Opt. Lett. 32, 3333–3335(2007). [CrossRef] [PubMed]

].

Fig. 1 Light trapping in a photonic circuit consisting of three waveguides embedded in a Wwide, Llong substrate. The coupling κ13κ23. d13,23is the distance between two waveguides. The dielectric constants of substrate is εs, while εi(i∈ {1, 2, 3}) is the constant profile of corresponding individual waveguides Diwithout others. The widths of waveguides are ti, respectively.

The evolution of power of field in photonic circuits can be studied either by numerical solving the Helmholtz equation or with the derived coupled mode theory. However the later presents a clearer physic understanding. We first present the approach based on Helmholz equation.

2.1. Beam propagating method

We consider our system to be two-dimensional (2D). This is reasonable if the size of waveguide in the ydirection is much larger than that in the xdirection. The propagation of the field Ein the photonic circuits in a 2D space can be described by the Helmholtz equation, which takes the form
2Ez2+2Ex2+k02ε(x,z)E=0,
(1)
where zis the propagating direction, xis the transverse direction, and k0=2πλis the wave vector of field with wavelength λin the free space. The dielectric constant ε(x, z) plays the role of the optical potential. In Eq. (1), we assume that Et~0and it is reasonable for our interest in the steady-state behavior of the system. We can resolve the field Einto its slowly varying amplitude and a fast oscillating factor, E= ψe±jβz, where the propagation constant β= k0neff. neffis the effective index of waveguides. The sign before βindicates the propagating direction: minus (plus) for the propagation along the positive (negative) zdirection. In the paraxial approximation |2jβψz||2ψz2|, the amplitude of the field in the photonic circuit evolves according to the equation [32

32. K. Okamoto, Fundamentals of Optical Waveguides(Academic Press, 2000).

]
2jβψz2ψx2+k02[ε(x,z)neff2]ψ.
(2)

We numerically solve Eq. (2)with BPM to simulate the propagation of field with a spatial resolution δz= 1 μm and δx= 0.1 μm. A finer spatial grid gives the same results. Throughout simulation, we use the zero-order eigenmode profile ψ(x, x0, z= 0, L) of TE mode by solving the eigen equation [32

32. K. Okamoto, Fundamentals of Optical Waveguides(Academic Press, 2000).

], where x0=x0(i)(i∈ {1, 2}) is the center position of input ports PL1,L2,R1. This mode profile is very close to a Gaussian function exp((xx0)2/2wp2)with a half width of waist wp= 1 μm. The profiles ψ(x, x0(1),0) and ψ(x, x0(1), L) corresponds to the input field launching into the waveguide D1from the port PL1at z= 0 and PR1at z= L, respectively. While the profile ψ(x, x0(2), 0) means an input to the port PL2at z= 0. The intensity of the field at the peak is unity. This profile is very close to the fundamental eigenmode of D1or D2.

Our simulation focuses on a light with wavelength λ0= 1.55 μm, which is of interest in optical communications. The photonic circuit can be integrated in a wafer with a substrate dielectric constant εs= 10.56ε0, where ε0is the permittivity of free space. We consider weakly guiding waveguides with εcore= 10.76ε0in order to ensure the valid of our BPM and CME method. The imaginary part of the dielectric constant in the middle waveguide D3is ℑ[ε3] = −0.01ε0. This induces a loss of γ= 11.6 mm−1according to our numerical simulation. The propagation constant is calculated by solving the eigenvalue equation [32

32. K. Okamoto, Fundamentals of Optical Waveguides(Academic Press, 2000).

] of TE mode.

In our numerical method, the Neumann boundary condition (NBC) is used to greatly suppress the reflection field from the transverse boundary. A small reflection, which can be a practical noise from the boundary of device in experiments, is responsible for the background noise of our numerical results.

2.2. Coupled mode equation method

Before discussing the results, we present the equivalent, but physically transparent, coupled mode equation approach to explain our system.

Due to the equivalence between the Helmholtz equation in photonic circuits and the Schrödinger equation in quantum mechanics, the behavior of light propagating in a photonic circuit is similar to the dynamics of the internal atomic states of a quantum system [25

25. S. Longhi, “Transfer of light waves in optical waveguides via a continuum,” Phys. Rev. A 78, 013815(2008). [CrossRef]

27

27. S. Longhi, “Quantum-optical analogies using photonic structures,” Laser & Photon. Rev. 3, 243–261(2009). [CrossRef]

]. For example, the normalized light power trapped in optical waveguides plays the role of atomic population. Optical nonreciprocities in an array of coupled waveguides can then be considered as the trapping of input light on demand. As is well known, in a Λ–type three-level atomic systems, we can adiabatically create a target state independent of the initial state of system via the so-called coherent population trapping (CPT) [24

24. M. O. Scully and M. S. Zubairy, Quantum Optics(Cambridge University, 1997). [CrossRef]

]. Our system is analogous to such a Λ-type three-level system. Just as in CPT in the atomic system, we expect to trap most light energy in a selected optical waveguide by suitably controlling the coupling between the waveguides. This is the basis of the optical nonreciprocity studied in this paper. Note that the LRNR we propose here is substantially different from the FBNR used for optical isolator on the basis of the breaking of Lorentz reciprocity theorem [1

1. R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717–754(2004). [CrossRef]

,35

35. H. A. Haus, Waves and Fields in Solid State(Prentice-Hall Inc., 1984).

]. In the former case, both two sources input into ports in the left hand side but their responses come out from the right hand side, while the source and the response must exchange in the later. As a result, Js(1)ER(2)dS=Js(2)ER(1)dS=0for our 2D case, where the response ER(i)at port PR1is the electric field created by a source Js(j)at port PL1,L2with ijand i, j∈ {1,2}, because the source and response are separated in space, i.e. Js(1)ER(2)=Js(2)ER(1)=0. Thus the implementation of the LRNR in a linear, passive medium does not violate the Lorentz reciprocity theorem [1

1. R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717–754(2004). [CrossRef]

, 35

35. H. A. Haus, Waves and Fields in Solid State(Prentice-Hall Inc., 1984).

], which requires the exchange of the place of source and response.

2.3. Connection between two methods

The structure of photonic circuit to create the nonreciprocal transverse energy flow of light is shown in Fig. 2(a). Through our system, we assume no loss in waveguide D1and D2. The dielectric constant is εs= 10.56ε0in the substrate, while it is εcore= 10.76ε0in core of D1and D2. In waveguide D3, the dielectric constant in core is ε3= (10.76 − 0.01i)ε0. To fit the numerical results, the coupling κ13and κ23are assumed to vary corresponding to the central positions w1(z), w2(z) and w3(z) of waveguides. The other parameters for CME are given by:
γ=11.6mm1,
(6)
Δ13=Δ23=23mm1.
(7)
The mismatch of propagating constant Δ13= Δ23= −23 mm−1is obtained by solving the eigenvalue of zero-order TE mode. To simulate the varying gaps between waveguides, we assume two gradient changing coupling strength κ13and κ23in the propagating direction for Eq. (4)as shown in Fig. 2(b). Two waveguides in the same chip always couples to each other even if the coupling strength is very small. To consider this coupling, we assume small values as the distance between two waveguides are large. The intensity in waveguides given by the CMEs change very slightly if we neglect this small coupling. We note that, in the absence of loss in the waveguide w3, the system is reciprocal (not shown here). However if we include loss in the middle waveguide, we create left-right nonreciprocities.

Fig. 2 (a) The waveguide structure for left-right nonrecipricity. The straight waveguide D3is 4 μm wide around its center w3(0) = 26.7 μm. The waveguides D1and D2with width t1= t2= 2 μm are curves along their varying central positions w1(z) and w2(z) defined as w1[z][μm] = 31 for z< 1.1 mm; w1[z][μm] = 34 for z> 1.6 mm and w1[z][μm] = 31 + 3(1 + sin(2π(z− 1350)/1000))/2 for 1.1 mm ≤ z≤ 1.6 mm. w2[z] is constant 22 μm for z< 0.1 mm and 23.4 μm for z> 1.1 mm. During the transition region, w2[z][μm] = 22 + 1.4(1 + sin(2π(z− 600)/2000))/2 for 0.1 mm ≤ z≤ 1.1mm. (b) The coupling as a function of propagating distance z. Blue lines (i) for κ13, red lines (ii) for κ23. Solid lines for coupling rates are evaluated by Eq. (5), while dashed lines indicates coupling rates for fitting the numerical results. Detailedly, the coupling rates for fitting are κ13(z)[mm−1]=4 for z< 1.1 mm; κ13(z)[mm−1]=0.03 for z> 1.44 mm and κ13(z)[mm−1]=0.03+3.97(1−sin(2π(z−1270)/680))/2.0 for 1.1 mm ≤z≤ 1.44 mm. While κ23(z) is 0.6 mm−1for z< 0.24 mm and 8.5 mm−1for z> 1.1 mm, and 0.6 + 7.9(1 + sin(2π(z− 670)/1720))/2.0 for 0.24 mm ≤z≤ 1.1 mm. Here λ0= 1.55 μm.

Let us assume that a light with unit amplitude is incident on the port PL1or PL2at z= 0. If the photonic circuit is reciprocal, the transmission from port PR1or PR2exchanges as well if the incident exchange. However, in the case of left-right nonrecipricity, the light launched into port PL1and PL2always effectively transfer to the waveguide D1and comes out from port PR1. If we use constant couplings κ13and κ23, the LRNR is obtained but the transmissions are small. The energy trapped in waveguide D1also decays because part of the energy couples to the middle waveguide from which the energy is lost into the environment at a rate γ. To avoid a strong coupling of energy between D1and D3, we gradually change the distance between the two side waveguides and the middle one to guarantee an adiabatic process. In addition, large phase mismatchings Δ13and Δ23are used to suppress the energy coupling to waveguide D3. In the output side, we decouple the waveguides D1and D3by introducing a large distance to keep the light energy in D1almost constant. The profile of the mode is also kept stable after z/λ0= 1500. The couplings used in the CME for fitting the following numerical results are dashed lines shown in Fig. 2(b). The solid lines are numerically evaluated by Eq. (5). These coupling are strongly dependent on the distance d13,23. In spite the exact numerical solution of coupling rates κ13and κ23from Eq. (5)is different from the numbers we use to fit the distribution of fields below, it provide us a good guide for the fitting function.

3. Results

Now we study the left-right nonreciprocity where we have nonreciprocal light transfer in the transverse direction [16

16. C. Eüter, K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of paritytime symmetry in optics,” Nat. Phys. 6, 192–195(2010). [CrossRef]

, 36

36. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904(2008). [CrossRef] [PubMed]

]. Similar to coherent population trapping in quantum optics, we can trap most light energy in the selected waveguide D1by designing a weak coupling κ13in comparison with κ23. Our numerical results shown in Figs. 3 (a) and (c)demonstrate a left-right nonreciprocal transverse energy flow. Whatever port PL1or PL2we choose to lauch the light into, most of the light is trapped in the waveguide D1, and comes out from the same port PR1(blue lines). The transmission for light input into port PL2is about 25% (Figs. 3(b)) but it increases to 40% if the light is incident into port PL1(Figs. 3(d)). The light in the waveguide D2leaks to D3and subsequently is absorbed as it propagates. The contrast ratios of light intensities in waveguides D1and D2are higher than 29 dB in both cases.

Fig. 3 Left-right nonreciprocity corresponding to Fig. 2(a). The field propagates from left to right in photonic circuits. (a) Light incident into the waveguide D2; (c) light enters the waveguide D1; (b) and (d) Intensities of field at the middle of waveguides D1[blue lines (i)], D2[red lines (ii)] and D3[green lines (iii)]. Dashed thin lines are the corresponding plots by solving Eq. (4).

It is interesting to check whether the device displays the FBNR, a counterpart of LRNR, because the former is the basis for optical isolators. According to the Lorentz reciprocity theorem [35

35. H. A. Haus, Waves and Fields in Solid State(Prentice-Hall Inc., 1984).

] and Fan et al. [21

21. S. Fan, R. Baets, A. Petrov, Z. Yu, W. F. J. D. Joannopoulos, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on “nonreciprocal light propagation in a silicon photonic circuit,” Science 335, 38–b(2012). [CrossRef]

], the FBNR is impossible in a linear, time-independent medium. Our numerical simulation agrees with it and demonstrates the forward-backward reciprocity, as shown in Fig. 4. To check the forward-backward reciprocity, we interchange the source and the response and solve Eqs. (2)and (4). The light is incident into the port PR1and the output from ports PL1and PL2are monitored. Again, the results by solving Eq. (4)using the same parameter and coupling rates fit the numerical results by BPM well. As predicted by the Lorentz Reciprocity theorem, the transmission from PR1to PL1(PL2) in the numerical simulations equals to those from PL1(PL2) to PR1. So the Lorentz Reciprocity theorem still rules the dynamics of our system.

Fig. 4 Numerical simulation of propagation of light in Fig. 2(a). The field incident into port PR1propagates from right to left. (a) Distribution of field (intensity), (b) Intensities of field at the middle of waveguides D1[blue lines (i)], D2[red lines (ii)],D3[green lines (iii)]. Dashed lines are the fitting plots evaluated by Eq. (4)using the same parameters as in Fig. 3.

Fig. 5 Nonreciprocal transmission as a function of frequency of input light using structure as in Fig. 2(a). Thin blue lines (i) and (iii) show the light trapped in waveguide D1, thick red lines (ii) and (iv) show the light energy in D2. Solid lines for light launching to D2, dashed lines for light input into D1.

For a practical application, the performance of device need be robust against small deviation in structure. Figure 6shows how robust the nonreciprocal transfer of light is when the length, width of and gaps between waveguides change. It can be seen from Fig. 6(a)that the nonreciprocal performance varies slowly as the total length ηzLof device changes. There is more than 24% of input light is trapped in D1when the device scales in the z-direction from ηz= 1.0 to ηz= 1.3. As shown in Fig. 6(b), the light trapped in D1oscillates as a function of the width ηtt1= ηtt2but it is stable for 1.0 ⩽ ηt⩽ 1.02, which means that the width of waveguides can vary 40 nm. In contrast, the LRNR of our design is more sensitive to the distance between waveguides. The transmission is larger than 21% if the shift/offset of waveguides gis negative. It means a smaller distance between waveguides is preferable. While the nonreciprocity deteriorates rapidly as the distance increases.

Fig. 6 Nonreciprocal transmission in the structure as in Fig. 2(a)changes in the length (a), width (b) of, gap (c) between waveguide. (a) scale the device by ηzin the z-direction; (b) scale the width of waveguides D1and D2; (c) shift the center of waveguides by gas w1[z] + gand w2[z] − g. Thin blue lines (i) and (iii) show the light trapped in waveguide D1, thick red lines (ii) and (iv) show the light energy in D2. Solid lines for light launching to D2, dashed lines for light input into D1.

The deviation of dielectric constant in fabrication changes the mistmatching of propagation constants and the coupling rate as well. First, we check the performance when the global dielectric constant εcorechanges in all waveguides, as shown in Fig. 7(a). The dielectric constant εcoreneed be accurately engineered to pursue for a good LRNR behavior. Only the region 10.73 ⩽ εcore/ε0⩽ 10.77 is useful to trap light in waveguide D1. When εcore/ε0changes from 10.68 to 10.9 corresponding to Δn/ncore∼ 1%, the output from PR1is switched from “on” (“off”) to “ off” (“on”) for input to PL2(PL1). Then the output is investigated as the dielectric constant ε3of waveguide D3changes only. For ℜ[ε3]/ε0< 10.7, no LRNR displays in our system. When ℜ[ε3] is larger, the LRNR occurs and the light trapped in D1fluctuates as the refractive index increases. However, more than 25% of light can be trapped in D1over the region of 10.76 ⩽ ℜ[ε3]/ε0⩽ 10.78. In contrast, the LRNR is very robust against the loss of waveguide D3. The light in D2decreases rapidly as the loss increases. The light trapped in D1is stable for an input to PL1and decays exponentially for an input to PL2. When ℑ[ε3]/ε0< −0.005 corresponding to γ⩾ 60 cm−1, no light in D2and there is only light in D1. In the range of −0.01 ⩽ ℑ[ε3]/ε0⩽ −0.005, the system can trap more than 25% of light in D1.

Fig. 7 Nonreciprocal transmission as the structure as in Fig. 2(a)changes of the global refractive index ℜ[εcore] (a), refractive index ℜ[ε3] (b) and loss ℑ[εcore] of waveguide D3. Thin blue lines (i) and (iii) show the light trapped in waveguide D1, thick red lines (ii) and (iv) show the light energy in D2. Solid lines for light launching to D2, dashed lines for light input into D1.

Our simulations show a relative flexible parameters to the LRNR. The existing modern technology can fabricate the device in a much more accuracy. Therefore a linear, passive medium can display highly optical nonrecipricity in our design.

According to Eqs. (1)and (2), the structure of system is scalable in size to shift the frequency window of nonrecipricity, e.g. the LRNR around λ0= 800 nm. It can be seen from the coupled mode theory Eq. (4)that the LRNR occurs if all parameters are scaled in a similar structure according to the propagation constant βfor a different wavelength. In Fig. 8(a), we first scale the photonic circuit in the x-direction and then adjust the structure parameters and the width of input light for keeping the parameters mismatching and propagation constants close to those in Fig. 1(b). As a result, the distributions of field for the inputs into ports PL1and PL2are similar to Fig. 3(see Fig. 9). There is about 25% of light trapped in the waveguide D1, while the light from PL2is vanishing small. Then the intensities outcoming from ports PR1and PR2are scanned in wavelength between 700 ∼ 900 nm. It can been clearly seen from Fig. 8(b)that the second structure allow a high performance of LRNR over 40 nm from 780 nm to 820 nm, allowing to control a ultrashort laser pulse with duration 1/Δω∼ 30 fs. Thus our scheme promises an ultrabroadband LRNR at difference wavelengths. A shorter wavelength means a larger loss in the bending region due to the stronger dipole radiation. To reduce this unwanted loss, we use a finer grid δx= 50 nm and δz= 0.5 μm, and adjust wp= 375 nm according to the eigen mode profile in our simulation.

Fig. 8 (a) The waveguide structure for left-right nonreciprocity at λ= 800 nm. Similar to Fig. 1(a), the straight waveguide D3is 1 μm wide around its center w3(0) = 26.7 μm. The waveguides D1and D2with width t1= t2= 0.6 μm are curves along their varying central positions w1(z) and w2(z) defined as w1[z][μm] = 28.4 for z< 1.2 mm; w1[z][μm] = 30.4 for z> 3.1 mm and w1[z][μm] = 28.4 + 2(1 + sin(2π(z− 2150)/3800))/2 for 1.2 mm ≤ z≤ 3.1 mm. w2[z] is constant 24 μm for z< 0.2 mm and 23.4 μm for z> 1.95 mm. During the transition region, w2[z][μm] = 24 + 1.4(1 + sin(2π(z− 1075)/3500))/2 for 0.2 mm ≤ z≤ 1.95mm. (b) Nonreciprocal transmission as a function of frequency of input light in (a). Thin blue lines (i) and (iii) show the light trapped in waveguide D1, thick red lines (ii) and (iv) show the light energy in D2. Solid lines for light launching to D2, dashed lines for light input into D1. The Gaussian profile of input light is adjusted to be wp= 0.375 nmwide.
Fig. 9 Left-right nonreciprocity corresponding to Fig. 8(a). The field propagates from left to right in photonic circuits. (a) Light incident into the waveguide D2; (c) light enters the waveguide D1; (b) and (d) Intensities of field at the middle of waveguides D1[blue lines (i)], D2[red lines (ii)] and D3[green lines (iii)].

4. Discussion on two types of nonreciprocity

Two different types of nonrecipricity, FBNR and LRNR, have been discovered in optical systems [2

2. J. Fujita, M. Levy, R. M. Osgood, L. Wilkens, and H. Dötsch, “Waveguide optical isolator based on machzehnder interferometer,” Appl. Phys. Lett. 76, 2158–2160(2000). [CrossRef]

18

18. H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Transition from discrete to continuous townes solitons in periodic media,” Phys. Rev. A 82, 043803(2010). [CrossRef]

]. The FBNR requires the breaking of the Lorentz reciprocity theorem and means that the forward and backward transmissions are not equal when the sources and responses interchange [21

21. S. Fan, R. Baets, A. Petrov, Z. Yu, W. F. J. D. Joannopoulos, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on “nonreciprocal light propagation in a silicon photonic circuit,” Science 335, 38–b(2012). [CrossRef]

]. It is the basis of optical isolators. If the forward transmission Tfis much larger than the backward transmission Tb, then the device with the FBNR can allow the forward propagating light to go through but block the back scattering light. Whereas the LRNR in our photonic circuit means that the light launching into different waveguides in the left hand side comes out from the same port from right. It does not violate the Lorentz reciprocity theorem [1

1. R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717–754(2004). [CrossRef]

, 35

35. H. A. Haus, Waves and Fields in Solid State(Prentice-Hall Inc., 1984).

] and as a result can not be used to isolate a light scattered backward.

In spite of the absence of ability for optical isolator, our scheme can dynamically route the light into difference paths. Our device also provide a novel method to switch on/off the light via dynamically tuning the loss of waveguide, shown in Fig. 7(c). For ℑ[ε3]/ε0< −0.05 (γ⩾ 450 cm−1), the light input to PL2can be switched off, while it can effectively transfer to the output port PR1for −0.005 < ℑ[ε3]/ε0< −0.01. More importantly, the loss of waveguide can be tune faster (< 1 ps) and more efficiently [30

30. V. Lordi, H. B. Yuen, S. R. Bank, and J. S. Harris, “Quantum-confined stark effect of gainnas(sb) quantum wells at 1300–1600 nm,” Appl. Phys. Lett. 85, 902–904(2004). [CrossRef]

,31

31. S. Sandhu, M. L. Povinelli, and S. Fan, “Stopping and time reversing a light pulse using dynamics loss tuning of coupled-resonators delay lines,” Opt. Lett. 32, 3333–3335(2007). [CrossRef] [PubMed]

] than the refractive index modulation [37

37. V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084(2004). [CrossRef] [PubMed]

]. Moreover, the intensity of light outcoming from port PR2can be increased essentially using a shorter output length. An alternative method to route light is to dynamically tune the refractive index of waveguides. Although our nonreciprocal photonic circuit is discussed in a linear optical medium, we also can realize the setup in a nonlinear medium like fused silica [38

38. R. Keil, M. Heinrich, F. Dreisow, T. Pertsch, A. Tünnermann, S. Nolte, D. N. Christodoulides, and A. Szameit, “All-optical routing and switching for three-dimensional photonic circuitry,” Scientific Rep. 1, 94(2011).

] or silicon [37

37. V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084(2004). [CrossRef] [PubMed]

] tuned by ultrashort laser pulses. Dynamically tuning the refractive index of waveguides, see Figs. 7(a) and (b), and subsequently the coupling strength and phase mismatch using ultrashort laser pulses [38

38. R. Keil, M. Heinrich, F. Dreisow, T. Pertsch, A. Tünnermann, S. Nolte, D. N. Christodoulides, and A. Szameit, “All-optical routing and switching for three-dimensional photonic circuitry,” Scientific Rep. 1, 94(2011).

], one can switching on/off the light outcoming from port PR1(Fig. 7(a)) or route the light incident to PL2into port PR1for 10.74 ⩽ ℜ[ε3]/ε0⩽ 10.78 or port PR2for <[ε3]/ε0⩽ 10.78 (Fig. 7(b)). The dynamical tuning range is Δn< 1%, which can be obtained using the existing technology [37

37. V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084(2004). [CrossRef] [PubMed]

39

39. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126(2008). [CrossRef]

]. Therefore, our nonreciprocal design can be a ultrafast, broadband optical router.

5. Conclusion

In conclusion, using a technique analogous to the coherent population trapping in quantum optics, we broke the symmetry of transverse light propagation in the photonic circuits of three coupled waveguides. Our proposed system is made only from linear, passive materials. Our simulations indicate the possibility of asymmetric transverse energy flow in an ultrabroadband window spanning over 80 nm in frequency. Although our proposed system has a relatively large insertion loss, it opens a door to the possibility of highly efficient optical nonreciprocity in a linear, passive medium.

Acknowledgments

This research is supported by a grant from the King Abdulaziz City for Science and Technology (KACST). One of us (MSZ) is grateful for the NPRPgrant 5-102-1-071from the Qatar National Research Fund (QNRF). KX also gratefully acknowledge the hospitality at ARCCenter for Engineered Quantum Systems and Department of Physics and Astronomy, Macquarie University.

References and links

1.

R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717–754(2004). [CrossRef]

2.

J. Fujita, M. Levy, R. M. Osgood, L. Wilkens, and H. Dötsch, “Waveguide optical isolator based on machzehnder interferometer,” Appl. Phys. Lett. 76, 2158–2160(2000). [CrossRef]

3.

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904(2008). [CrossRef] [PubMed]

4.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775(2009). [CrossRef] [PubMed]

5.

Y. Shoji, M. Ito, Y. Shirato, and T. Mizumoto, “MZI optical isolator with si-wire waveguides by surface-activated direct bonding,” Opt. Express 20, 18440–18448(2012). [CrossRef] [PubMed]

6.

Y. Shen, M. Bradford, and J. T. Shen, “Single-photon diode by exploiting the photon polarization in a waveguide,” Phys. Rev. Lett. 107, 173902(2011). [CrossRef] [PubMed]

7.

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photon. 5, 758–762(2011). [CrossRef]

8.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450(2012). [CrossRef]

9.

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94(2009). [CrossRef]

10.

M. S. Kang, A. Butsch, and P. S. J. Russell, “Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre,” Nat. Photonics 5, 549–553(2011). [CrossRef]

11.

Q. Wang, F. Xu, Z. Yu, X. Qian, X. Hu, Y. Lu, and H. Wang, “A bidirectional tunable optical diode based on periodically poled linbo3,” Opt. Express 18, 7340–7346(2010). [CrossRef] [PubMed]

12.

S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. 102, 213903(2009). [CrossRef] [PubMed]

13.

M. Hafezi and P. Rabl, “Optomechanically induced non-reciprocity in microring resonators,” Opt. Express 20, 7672–7684(2012). [CrossRef] [PubMed]

14.

Y. Hadad and B. Z. Steinberg, “Magnetized spiral chains of plasmonic ellipsoids for one-way optical waveguides,” Phys. Rev. Lett. 105, 233904(2010). [CrossRef]

15.

A. B. Khanikaev, S. H. Mousavi, G. Shvets, and Y. S. Kivshar, “One-way extraordinary optical transmission and nonreciprocal spoof plasmons,” Phys. Rev. Lett. 105, 126804(2010). [CrossRef] [PubMed]

16.

C. Eüter, K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of paritytime symmetry in optics,” Nat. Phys. 6, 192–195(2010). [CrossRef]

17.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902(2009). [CrossRef] [PubMed]

18.

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Transition from discrete to continuous townes solitons in periodic media,” Phys. Rev. A 82, 043803(2010). [CrossRef]

19.

L. Feng, M. Ayache, J. Huang, Y. Xu, M. Lu, Y. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733(2011). [CrossRef] [PubMed]

20.

L. Feng, M. Ayache, J. Huang, Y. Xu, M. Lu, Y. Chen, Y. Fainman, and A. Scherer, “Response to comment on “nonreciprocal light propagation in a silicon photonic circuit,” Science 335, 38–c(2012). [CrossRef]

21.

S. Fan, R. Baets, A. Petrov, Z. Yu, W. F. J. D. Joannopoulos, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on “nonreciprocal light propagation in a silicon photonic circuit,” Science 335, 38–b(2012). [CrossRef]

22.

C. Wang, C. Zhou, and Z. Li, “On-chip optical diode based on silicon photonic crystal heterojunctions,” Opt. Express 19, 26948–26955(2011). [CrossRef]

23.

C. Wang, X. Zhong, and Z. Li, “Linear and passive silicon optical isolator,” Sci. Rep. 2, 1–6(2012).

24.

M. O. Scully and M. S. Zubairy, Quantum Optics(Cambridge University, 1997). [CrossRef]

25.

S. Longhi, “Transfer of light waves in optical waveguides via a continuum,” Phys. Rev. A 78, 013815(2008). [CrossRef]

26.

G. Della Valle, M. Ornigotti, T. Fernandez, P. Laporta, S. Longhi, A. Coppa, and V. Foglietti, “Adiabatic light transfer via dressed states in optical waveguide arrays,” Appl. Phys. Lett. 92, 011106(2008). [CrossRef]

27.

S. Longhi, “Quantum-optical analogies using photonic structures,” Laser & Photon. Rev. 3, 243–261(2009). [CrossRef]

28.

S. Longhi, “Optical analog of population trapping in the continuum: Classical and quantum interference effects,” Phys. Rev. A 79, 023811(2009). [CrossRef]

29.

A. Crespi, S. Longhi, and R. Osellame, “Photonic realization of the quantum rabi model,” Phys. Rev. Lett. 108, 163601(2012). [CrossRef] [PubMed]

30.

V. Lordi, H. B. Yuen, S. R. Bank, and J. S. Harris, “Quantum-confined stark effect of gainnas(sb) quantum wells at 1300–1600 nm,” Appl. Phys. Lett. 85, 902–904(2004). [CrossRef]

31.

S. Sandhu, M. L. Povinelli, and S. Fan, “Stopping and time reversing a light pulse using dynamics loss tuning of coupled-resonators delay lines,” Opt. Lett. 32, 3333–3335(2007). [CrossRef] [PubMed]

32.

K. Okamoto, Fundamentals of Optical Waveguides(Academic Press, 2000).

33.

H. F. Taylor and A. Yariv, “Guided wave optics,” Proc. IEEE 62, 1044–1060(1974). [CrossRef]

34.

A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Tech. 3, 1135–1146(1985). [CrossRef]

35.

H. A. Haus, Waves and Fields in Solid State(Prentice-Hall Inc., 1984).

36.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904(2008). [CrossRef] [PubMed]

37.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084(2004). [CrossRef] [PubMed]

38.

R. Keil, M. Heinrich, F. Dreisow, T. Pertsch, A. Tünnermann, S. Nolte, D. N. Christodoulides, and A. Szameit, “All-optical routing and switching for three-dimensional photonic circuitry,” Scientific Rep. 1, 94(2011).

39.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126(2008). [CrossRef]

OCIS Codes
(130.3120) Integrated optics : Integrated optics devices
(230.1150) Optical devices : All-optical devices
(270.1670) Quantum optics : Coherent optical effects
(130.2755) Integrated optics : Glass waveguides

ToC Category:
Integrated Optics

History
Original Manuscript: March 25, 2013
Revised Manuscript: August 22, 2013
Manuscript Accepted: September 20, 2013
Published: October 21, 2013

Citation
Keyu Xia, M. Alamri, and M. Suhail Zubairy, "Ultrabroadband nonreciprocal transverse energy flow of light in linear passive photonic circuits," Opt. Express 21, 25619-25631 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-22-25619


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References

  1. R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys.67, 717–754(2004). [CrossRef]
  2. J. Fujita, M. Levy, R. M. Osgood, L. Wilkens, and H. Dötsch, “Waveguide optical isolator based on machzehnder interferometer,” Appl. Phys. Lett.76, 2158–2160(2000). [CrossRef]
  3. F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett.100, 013904(2008). [CrossRef] [PubMed]
  4. Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature461, 772–775(2009). [CrossRef] [PubMed]
  5. Y. Shoji, M. Ito, Y. Shirato, and T. Mizumoto, “MZI optical isolator with si-wire waveguides by surface-activated direct bonding,” Opt. Express20, 18440–18448(2012). [CrossRef] [PubMed]
  6. Y. Shen, M. Bradford, and J. T. Shen, “Single-photon diode by exploiting the photon polarization in a waveguide,” Phys. Rev. Lett.107, 173902(2011). [CrossRef] [PubMed]
  7. L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photon.5, 758–762(2011). [CrossRef]
  8. L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science335, 447–450(2012). [CrossRef]
  9. Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics3, 91–94(2009). [CrossRef]
  10. M. S. Kang, A. Butsch, and P. S. J. Russell, “Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre,” Nat. Photonics5, 549–553(2011). [CrossRef]
  11. Q. Wang, F. Xu, Z. Yu, X. Qian, X. Hu, Y. Lu, and H. Wang, “A bidirectional tunable optical diode based on periodically poled linbo3,” Opt. Express18, 7340–7346(2010). [CrossRef] [PubMed]
  12. S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett.102, 213903(2009). [CrossRef] [PubMed]
  13. M. Hafezi and P. Rabl, “Optomechanically induced non-reciprocity in microring resonators,” Opt. Express20, 7672–7684(2012). [CrossRef] [PubMed]
  14. Y. Hadad and B. Z. Steinberg, “Magnetized spiral chains of plasmonic ellipsoids for one-way optical waveguides,” Phys. Rev. Lett.105, 233904(2010). [CrossRef]
  15. A. B. Khanikaev, S. H. Mousavi, G. Shvets, and Y. S. Kivshar, “One-way extraordinary optical transmission and nonreciprocal spoof plasmons,” Phys. Rev. Lett.105, 126804(2010). [CrossRef] [PubMed]
  16. C. Eüter, K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of paritytime symmetry in optics,” Nat. Phys.6, 192–195(2010). [CrossRef]
  17. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett.103, 093902(2009). [CrossRef] [PubMed]
  18. H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Transition from discrete to continuous townes solitons in periodic media,” Phys. Rev. A82, 043803(2010). [CrossRef]
  19. L. Feng, M. Ayache, J. Huang, Y. Xu, M. Lu, Y. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science333, 729–733(2011). [CrossRef] [PubMed]
  20. L. Feng, M. Ayache, J. Huang, Y. Xu, M. Lu, Y. Chen, Y. Fainman, and A. Scherer, “Response to comment on “nonreciprocal light propagation in a silicon photonic circuit,” Science335, 38–c(2012). [CrossRef]
  21. S. Fan, R. Baets, A. Petrov, Z. Yu, W. F. J. D. Joannopoulos, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on “nonreciprocal light propagation in a silicon photonic circuit,” Science335, 38–b(2012). [CrossRef]
  22. C. Wang, C. Zhou, and Z. Li, “On-chip optical diode based on silicon photonic crystal heterojunctions,” Opt. Express19, 26948–26955(2011). [CrossRef]
  23. C. Wang, X. Zhong, and Z. Li, “Linear and passive silicon optical isolator,” Sci. Rep.2, 1–6(2012).
  24. M. O. Scully and M. S. Zubairy, Quantum Optics(Cambridge University, 1997). [CrossRef]
  25. S. Longhi, “Transfer of light waves in optical waveguides via a continuum,” Phys. Rev. A78, 013815(2008). [CrossRef]
  26. G. Della Valle, M. Ornigotti, T. Fernandez, P. Laporta, S. Longhi, A. Coppa, and V. Foglietti, “Adiabatic light transfer via dressed states in optical waveguide arrays,” Appl. Phys. Lett.92, 011106(2008). [CrossRef]
  27. S. Longhi, “Quantum-optical analogies using photonic structures,” Laser & Photon. Rev.3, 243–261(2009). [CrossRef]
  28. S. Longhi, “Optical analog of population trapping in the continuum: Classical and quantum interference effects,” Phys. Rev. A79, 023811(2009). [CrossRef]
  29. A. Crespi, S. Longhi, and R. Osellame, “Photonic realization of the quantum rabi model,” Phys. Rev. Lett.108, 163601(2012). [CrossRef] [PubMed]
  30. V. Lordi, H. B. Yuen, S. R. Bank, and J. S. Harris, “Quantum-confined stark effect of gainnas(sb) quantum wells at 1300–1600 nm,” Appl. Phys. Lett.85, 902–904(2004). [CrossRef]
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