## Ultrabroadband nonreciprocal transverse energy flow of light in linear passive photonic circuits |

Optics Express, Vol. 21, Issue 22, pp. 25619-25631 (2013)

http://dx.doi.org/10.1364/OE.21.025619

Acrobat PDF (2299 KB)

### 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

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]

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]

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]

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]

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]

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]

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. 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]

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]

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]

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

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]

**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]

**82**, 043803(2010). [CrossRef]

## 2. Setup and model

*W*and length

*L*. The middle waveguide D

_{3}couples to the waveguides D

_{1}and D

_{2}. We assume that the coupling between the waveguides D

_{1}and D

_{2}is 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

*κ*

_{13}and

*κ*

_{23}, and decrease as the distance

*d*

_{13,23}between two waveguides increases. The coupling between D

_{1}and D

_{2}is assumed vanishing because these two waveguides are far enough from each other. The field in D

_{3}decays exponentially with a constant

*γ*that can be controlled [16

**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]

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. 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]

### 2.1. Beam propagating method

*y*direction is much larger than that in the

*x*direction. The propagation of the field

*E*in the photonic circuits in a 2D space can be described by the Helmholtz equation, which takes the form where

*z*is the propagating direction,

*x*is the transverse direction, and

*λ*in the free space. The dielectric constant

*ε*(

*x*,

*z*) plays the role of the optical potential. In Eq. (1), we assume that

*E*into its slowly varying amplitude and a fast oscillating factor,

*E*=

*ψe*

^{±jβz}, where the propagation constant

*β*=

*k*

_{0}

*n*.

_{eff}*n*is the effective index of waveguides. The sign before

_{eff}*β*indicates the propagating direction: minus (plus) for the propagation along the positive (negative)

*z*direction. In the paraxial approximation

*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*,

*x*

_{0},

*z*= 0,

*L*) of TE mode by solving the eigen equation [32], where

*i*∈ {1, 2}) is the center position of input ports

*P*

_{L1,L2,R1}. This mode profile is very close to a Gaussian function

*w*

_{p}= 1 μm. The profiles

*ψ*(

*x*,

*ψ*(

*x*,

*L*) corresponds to the input field launching into the waveguide D

_{1}from the port

*P*

_{L1}at

*z*= 0 and

*P*

_{R1}at

*z*=

*L*, respectively. While the profile

*ψ*(

*x*,

*P*

_{L2}at

*z*= 0. The intensity of the field at the peak is unity. This profile is very close to the fundamental eigenmode of D

_{1}or D

_{2}.

*λ*

_{0}= 1.55 μm, which is of interest in optical communications. The photonic circuit can be integrated in a wafer with a substrate dielectric constant

*ε*= 10.56

_{s}*ε*

_{0}, where

*ε*

_{0}is the permittivity of free space. We consider weakly guiding waveguides with

*ε*

_{core}= 10.76

*ε*

_{0}in order to ensure the valid of our BPM and CME method. The imaginary part of the dielectric constant in the middle waveguide D

_{3}is ℑ[

*ε*

_{3}] = −0.01

*ε*

_{0}. This induces a loss of

*γ*= 11.6 mm

^{−1}according to our numerical simulation. The propagation constant is calculated by solving the eigenvalue equation [32] of TE mode.

### 2.2. Coupled mode equation method

*E*in photonic circuits can be expressed as a supermode of eigenmodes

*E*of individual waveguide D

_{i}*, i.e., where the amplitude of eigenmode in the*

_{i}*i*th waveguide is denoted by

*A*(

_{i}*i*∈ {1, 2, 3}). Here

*ψ*is the slowly varying envelope of eigenmode

_{i}*E*and

_{i}*β*is the corresponding propagation constant, which can be controlled by designing the dielectric constant

_{i}*ε*of individual waveguide D

_{i}*and its width*

_{i}*t*. This parameter also depends on the wavelength of light. According to the coupled mode theory [32–34

_{i}34. A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Tech. **3**, 1135–1146(1985). [CrossRef]

_{13}=

*β*

_{1}−

*β*

_{3}and Δ

_{23}=

*β*

_{2}−

*β*

_{3}are the phase mismatch between waveguides D

_{1}(D

_{2}) and D

_{3}. The power in the

*i*th waveguide is evaluated by

*P*= ∑

*. The phase mismatching and coupling in the CME Eq. (4)can be derived from Helmholtz equation Eq. (1). These parameters are dependent on*

_{i}P_{i}*k*

_{0}and the optical potential

*ε*(

*x*,

*z*). The coupling is given by [32–34

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

*s*denotes the cross section of space and

*δε*=

_{n}*ε*(

*x*,

*z*) −

*ε*at the cut position

_{n}*z*. For simplicity’s sake, we have assumed weakly guiding waveguides and the relation

*κ*=

_{mn}*κ*. We also neglect the second-order spatial derivatives of the amplitude

_{nm}*A*and the small self phase shifts due to the perturbation of neighbor waveguides. A full study of the relation of parameter to the Helmholtz equation has been presented by Hardy et al. [33

_{i}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]

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

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

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

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

*P*

_{R1}is the electric field created by a source

*P*

_{L1,L2}with

*i*≠

*j*and

*i*,

*j*∈ {1,2}, because the source and response are separated in space, i.e.

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

### 2.3. Connection between two methods

*D*

_{1}and

*D*

_{2}. The dielectric constant is

*ε*

_{s}= 10.56

*ε*

_{0}in the substrate, while it is

*ε*

_{core}= 10.76

*ε*

_{0}in core of

*D*

_{1}and

*D*

_{2}. In waveguide

*D*

_{3}, the dielectric constant in core is

*ε*

_{3}= (10.76 − 0.01

*i*)

*ε*

_{0}. To fit the numerical results, the coupling

*κ*

_{13}and

*κ*

_{23}are assumed to vary corresponding to the central positions

*w*

_{1}(

*z*),

*w*

_{2}(

*z*) and

*w*

_{3}(

*z*) of waveguides. The other parameters for CME are given by: The mismatch of propagating constant Δ

_{13}= Δ

_{23}= −23 mm

^{−1}is obtained by solving the eigenvalue of zero-order TE mode. To simulate the varying gaps between waveguides, we assume two gradient changing coupling strength

*κ*

_{13}and

*κ*

_{23}in 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

*w*

_{3}, the system is reciprocal (not shown here). However if we include loss in the middle waveguide, we create left-right nonreciprocities.

*P*

_{L1}or

*P*

_{L2}at

*z*= 0. If the photonic circuit is reciprocal, the transmission from port

*P*

_{R1}or

*P*

_{R2}exchanges as well if the incident exchange. However, in the case of left-right nonrecipricity, the light launched into port

*P*

_{L1}and

*P*

_{L2}always effectively transfer to the waveguide D

_{1}and comes out from port

*P*

_{R1}. If we use constant couplings

*κ*

_{13}and

*κ*

_{23}, the LRNR is obtained but the transmissions are small. The energy trapped in waveguide D

_{1}also 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 D

_{1}and D

_{3}, we gradually change the distance between the two side waveguides and the middle one to guarantee an adiabatic process. In addition, large phase mismatchings Δ

_{13}and Δ

_{23}are used to suppress the energy coupling to waveguide D

_{3}. In the output side, we decouple the waveguides D

_{1}and D

_{3}by introducing a large distance to keep the light energy in D

_{1}almost constant. The profile of the mode is also kept stable after

*z*/

*λ*

_{0}= 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

*d*

_{13,23}. In spite the exact numerical solution of coupling rates

*κ*

_{13}and

*κ*

_{23}from 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

**6**, 192–195(2010). [CrossRef]

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]

_{1}by designing a weak coupling

*κ*

_{13}in comparison with

*κ*

_{23}. Our numerical results shown in Figs. 3 (a) and (c)demonstrate a left-right nonreciprocal transverse energy flow. Whatever port

*P*

_{L1}or

*P*

_{L2}we choose to lauch the light into, most of the light is trapped in the waveguide D

_{1}, and comes out from the same port

*P*

_{R1}(blue lines). The transmission for light input into port

*P*

_{L2}is about 25% (Figs. 3(b)) but it increases to 40% if the light is incident into port

*P*

_{L1}(Figs. 3(d)). The light in the waveguide D

_{2}leaks to D

_{3}and subsequently is absorbed as it propagates. The contrast ratios of light intensities in waveguides D

_{1}and D

_{2}are higher than 29 dB in both cases.

_{1}in Figs. 3(b)are slightly higher than the numerical results of BPM method. A small discrepancy is that the light in waveguide D

_{2}decays slower than the numerical results mainly because the loss in waveguide D

_{2}in the bending region is not included in the coupled mode theory Eq. (4). A full coupled mode theory [33

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

**3**, 1135–1146(1985). [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]

_{R1}and the output from ports P

_{L1}and P

_{L2}are 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 P

_{R1}to P

_{L1}(P

_{L2}) in the numerical simulations equals to those from P

_{L1}(P

_{L2}) to P

_{R1}. So the Lorentz Reciprocity theorem still rules the dynamics of our system.

_{1}and D

_{2}are the same, the propagation constants

*β*

_{1}and

*β*

_{2}are equal. Thus the phase mismatch Δ

_{13}is equal to Δ

_{23}ideally. However each propagation constant itself and the coupling rates are dependent on the wavelength of input field. As a result, the transmission to P

_{R1}from P

_{L1}(P

_{L2}) decreases (increases) gradually as the wavelength of incident light increases. Our system traps more than 24% in waveguide D

_{1}from 1.56 μm to 1.64 μm. So it has an ultrabroadband nonreciprocal window over 80 nm. In the nonreciprocal windows, the light in waveguide D

_{2}is always vanishing because it couples to the lossy channel D

_{3}. Next we concentrate our discussion in the nonreciprocal window of interest. It can be clearly seen in Fig. 5, whatever waveguide the light is incident to, more than 24% energy is trapped in D

_{1}and comes out of port

*P*

_{R1}. The frequency-dependence of transmission comes from the change of eigenmode profiles, propagation constants and their couplings

*κ*

_{13}and

*κ*

_{23}, which are also dependent on the profiles of eigenmodes and wavelength [ref. to Eq. (5)]. The deviation in fabrication may result in a small difference between Δ

_{13}and Δ

_{23}. However the transmission change slightly if Δ

_{13}≈ Δ

_{23}. A longer bending waveguide can tune the coupling between waveguides slower but is not necessary to provide a wider nonrecipricity window because it also changes the effective coupling length and the propagation constants are dependent on the wavelength as well.

*η*of device changes. There is more than 24% of input light is trapped in D

_{z}L_{1}when the device scales in the z-direction from

*η*= 1.0 to

_{z}*η*= 1.3. As shown in Fig. 6(b), the light trapped in D

_{z}_{1}oscillates as a function of the width

*η*

_{t}t_{1}=

*η*

_{t}t_{2}but it is stable for 1.0 ⩽

*η*⩽ 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

_{t}*g*is negative. It means a smaller distance between waveguides is preferable. While the nonreciprocity deteriorates rapidly as the distance increases.

*ε*changes in all waveguides, as shown in Fig. 7(a). The dielectric constant

_{core}*ε*need be accurately engineered to pursue for a good LRNR behavior. Only the region 10.73 ⩽

_{core}*ε*/

_{core}*ε*

_{0}⩽ 10.77 is useful to trap light in waveguide D

_{1}. When

*ε*/

_{core}*ε*

_{0}changes from 10.68 to 10.9 corresponding to Δ

*n*/

*n*∼ 1%, the output from P

_{core}_{R1}is switched from “on” (“off”) to “ off” (“on”) for input to P

_{L2}(P

_{L1}). Then the output is investigated as the dielectric constant

*ε*

_{3}of waveguide D

_{3}changes only. For ℜ[

*ε*

_{3}]/

*ε*

_{0}< 10.7, no LRNR displays in our system. When ℜ[

*ε*

_{3}] is larger, the LRNR occurs and the light trapped in D

_{1}fluctuates as the refractive index increases. However, more than 25% of light can be trapped in D

_{1}over the region of 10.76 ⩽ ℜ[

*ε*

_{3}]/

*ε*

_{0}⩽ 10.78. In contrast, the LRNR is very robust against the loss of waveguide D

_{3}. The light in D

_{2}decreases rapidly as the loss increases. The light trapped in D

_{1}is stable for an input to P

_{L1}and decays exponentially for an input to P

_{L2}. When ℑ[

*ε*

_{3}]/

*ε*

_{0}< −0.005 corresponding to

*γ*⩾ 60 cm

^{−1}, no light in D

_{2}and there is only light in D

_{1}. In the range of −0.01 ⩽ ℑ[

*ε*

_{3}]/

*ε*

_{0}⩽ −0.005, the system can trap more than 25% of light in D

_{1}.

*λ*

_{0}= 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

*β*for 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

*P*

_{L1}and

*P*

_{L2}are similar to Fig. 3(see Fig. 9). There is about 25% of light trapped in the waveguide D

_{1}, while the light from P

_{L2}is vanishing small. Then the intensities outcoming from ports

*P*

_{R1}and

*P*

_{R2}are 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/Δ

*ω*∼ 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

*w*= 375 nm according to the eigen mode profile in our simulation.

_{p}## 4. Discussion on two types of nonreciprocity

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]

**82**, 043803(2010). [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]

*T*is much larger than the backward transmission

_{f}*T*, 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

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

*ε*

_{3}]/

*ε*

_{0}< −0.05 (

*γ*⩾ 450 cm

^{−1}), the light input to P

_{L2}can be switched off, while it can effectively transfer to the output port P

_{R1}for −0.005 < ℑ[

*ε*

_{3}]/

*ε*

_{0}< −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. 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]

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]

_{R}_{2}can 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] 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]

_{R1}(Fig. 7(a)) or route the light incident to P

_{L2}into port P

_{R1}for 10.74 ⩽ ℜ[

*ε*

_{3}]/

*ε*

_{0}⩽ 10.78 or port P

_{R}_{2}for <[

*ε*

_{3}]/

*ε*

_{0}⩽ 10.78 (Fig. 7(b)). The dynamical tuning range is Δ

*n*< 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. 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]

## 5. Conclusion

## Acknowledgments

## References and links

1. | R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. |

2. | J. Fujita, M. Levy, R. M. Osgood, L. Wilkens, and H. Dötsch, “Waveguide optical isolator based on machzehnder interferometer,” Appl. Phys. Lett. |

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. |

4. | Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature |

5. | Y. Shoji, M. Ito, Y. Shirato, and T. Mizumoto, “MZI optical isolator with si-wire waveguides by surface-activated direct bonding,” Opt. Express |

6. | Y. Shen, M. Bradford, and J. T. Shen, “Single-photon diode by exploiting the photon polarization in a waveguide,” Phys. Rev. Lett. |

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. |

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 |

9. | Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics |

10. | M. S. Kang, A. Butsch, and P. S. J. Russell, “Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre,” Nat. Photonics |

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 |

12. | S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. |

13. | M. Hafezi and P. Rabl, “Optomechanically induced non-reciprocity in microring resonators,” Opt. Express |

14. | Y. Hadad and B. Z. Steinberg, “Magnetized spiral chains of plasmonic ellipsoids for one-way optical waveguides,” Phys. Rev. Lett. |

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. |

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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 |

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 |

22. | C. Wang, C. Zhou, and Z. Li, “On-chip optical diode based on silicon photonic crystal heterojunctions,” Opt. Express |

23. | C. Wang, X. Zhong, and Z. Li, “Linear and passive silicon optical isolator,” Sci. Rep. |

24. | M. O. Scully and M. S. Zubairy, |

25. | S. Longhi, “Transfer of light waves in optical waveguides via a continuum,” Phys. Rev. A |

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. |

27. | S. Longhi, “Quantum-optical analogies using photonic structures,” Laser & Photon. Rev. |

28. | S. Longhi, “Optical analog of population trapping in the continuum: Classical and quantum interference effects,” Phys. Rev. A |

29. | A. Crespi, S. Longhi, and R. Osellame, “Photonic realization of the quantum rabi model,” Phys. Rev. Lett. |

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. |

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. | K. Okamoto, |

33. | H. F. Taylor and A. Yariv, “Guided wave optics,” Proc. IEEE |

34. | A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Tech. |

35. | H. A. Haus, |

36. | K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. |

37. | V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature |

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. |

39. | F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. |

**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

Sort: Year | Journal | Reset

### References

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