Transmission Nonreciprocity in a Mutually Coupled Circulating Structure

Bing He, Liu Yang, Xiaoshun Jiang, and Min Xiao

Phys. Rev. Lett. 120, 203904 – Published 18 May 2018

Abstract

Breaking Lorentz reciprocity was believed to be a prerequisite for nonreciprocal transmissions of light fields, so the possibility of nonreciprocity by linear optical systems was mostly ignored. We put forward a structure of three mutually coupled microcavities or optical fiber rings to realize optical nonreciprocity. Although its couplings with the fields from two different input ports are constantly equal, such system transmits them nonreciprocally either under the saturation of an optical gain in one of the cavities or with the asymmetric couplings of the circulating fields in different cavities. The structure made up of optical fiber rings can perform nonreciprocal transmissions as a time-independent linear system without breaking Lorentz reciprocity. Optical isolation for inputs simultaneously from two different ports and even approximate optical isolator operations are implementable with the structure.

Received 25 January 2018

DOI:https://doi.org/10.1103/PhysRevLett.120.203904

Physics Subject Headings (PhySH)

Light propagation, transmission & absorption Nonlinear optics

Atomic, Molecular & Optical

Authors & Affiliations

Bing He^1,*, Liu Yang², Xiaoshun Jiang³, and Min Xiao^1,3,†

¹Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701, USA
²College of Automation, Harbin Engineering University, Harbin 150001, China
³National Laboratory of Solid State Microstructures, College of Engineering and Applied Sciences, School of Physics, Nanjing University, Nanjing 210093, China

^*binghe@uark.edu
^†mxiao@uark.edu

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Vol. 120, Iss. 20 — 18 May 2018

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Images

Figure 1
Mutually coupled circulating structure and its dynamical properties. (a) The schematic of the structure, which can be constructed either with coupled microcavities or with coupled optical fiber rings. Cavity 3 (the yellow one) carries gain medium. The concerned nonreciprocity refers to the realized phenomena that the identical $S_{i, f}$ and $S_{i, b}$ induce the unequal $S_{o, f}$ and $S_{o, b}$ . (b1) An example of the maximum real part of the eigenvalues of such a coupled system neglecting gain saturation, which varies with the couplings $J_{12}$ and $J_{23}$ , respectively. The transitions between instability (vertical axis value $> 0$ ) and stability (vertical axis value $< 0$ ) occur at the marked points. (b2) The reduction to the dynamics of a $P T$ -symmetric system when cavity 2 is detached, having the real parts of the eigenmodes to merge at an exceptional point. (c1) An example of the evolved field modes contributing to the outputs, due to a drive at $P_{1}$ . (c2) The corresponding field modes due to the same drive at $P_{4}$ . (c3) The corresponding field modes by placing the drive at $P_{3}$ . The plot for $+ (-)$ mode in (c1) and (c2) swaps the position with that for the $- (+)$ mode in (c3).
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Figure 2
Transmission nonreciprocity due to gain saturation. Here the relative parameters are chosen as $γ_{1} = γ_{2} = γ$ , $γ_{3} = 9.72 γ$ , $J_{i j} = 2.2 γ$ , which are converted from a set of experimental parameters in [32]. (a), (b) The evolutions of the forward intensity $| S_{o, f} |^{2} / (2 κ_{e}) = | a_{3, f}^{-} |^{2}$ (solid curve) and backward intensity $| S_{o, b} |^{2} / (2 κ_{e}) = | a_{1, b}^{-} |^{2}$ (dashed curve), with $g_{0} = 10.55 γ$ , $Δ = 0$ in (a) and $Δ = 5 γ$ in (b). These dimensionless intensities scale with the transmitted field powers. The dimensionless saturation intensity, which is determined by the scattering cross section and lifetime of the dopant, and the effective volume of the field inside cavity, is given as $I_{0} = 1.33 \times 10^{3}$ in (a) and $1.33 \times 10^{7}$ in (b). (Insets) Corresponding gain rates. (c),(d) The relations between isolation ratio and system parameters for the evolved time-independent steady modes by choosing $g_{0} = γ_{3}$ and the $I_{0}$ in (a), with different ratios $η = J_{23} / J_{12}$ . In (c), the detuning $Δ / γ$ is set to be $η$ . The drive amplitude in these examples is $E_{f} = E_{b} = 100 γ$ .
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Figure 3
Optical isolation for two simultaneous inputs. (a1), (a2) The induced field modes by the forward and backward drive, respectively. Here $g_{0} = 0$ , $Δ = 5 γ$ , $J_{12} = J_{13} = γ$ , $E_{f} = E_{b} = 10 γ$ , and $η = 5$ . $γ$ is the damping rate of the three cavities. (a3) The isolation ratios for two opposite drives acting together on the system, with $g_{0} = 0$ and $Δ = 20 γ$ . (b1)–(b3) The corresponding results to those in (a1)–(a3), with a difference of $g_{0} = 1.9 γ$ . (c) The relation between optical isolation ratio and $η$ , where the finally time-independent field modes are obtained under any pair of identical and simultaneously acting inputs with $Δ = 10 γ$ . (Inset) An example of unstably evolving $| a_{3, f}^{-} (t) + a_{3, b}^{-} (t) |^{2}$ (solid) and $| a_{1, f}^{-} (t) + a_{1, b}^{-} (t) |^{2}$ (dashed), with $η = 0.5$ on the left of the marked dot on the curve of $g_{0} = 1.4 γ$ . Gain saturation is neglected here.
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Figure 4
Optical nonreciprocity and optical isolation via coupling geometry. (a) A design of directional coupler. Out of a small area on the turnings of two fibers, which is removed of cladding and close to the joint of two fibers, the bending radiation from the field propagating along one direction hits the other fiber that is in a symmetric position with the first fiber. Meanwhile, the identical radiation due to the reversely propagating field mostly misses the other. (b) The consequent nonreciprocity of the individually acting fields. The gain rate $g_{0}$ with its saturation neglected is $0.9 γ$ , where $γ$ is the damping rate of the three cavities. Two pairs of CL and CCL modes in cavity 1 and cavity 3 couple at the rate $J_{13} \pm ε$ with $J_{13} = 10^{- 2} γ$ . Because of the directionality of bending radiation, the ratio $(J_{13} + ε) / (J_{13} - ε)$ can be much higher than the displayed here, though the radiation’s coupling into another fiber could have a low rate, i.e., $J_{13} \pm ε ≪ γ$ only given this type of coupling. The detuning $Δ$ is zero for the coupling between fibers. (c1),(c2) The improvement of the optical isolation with $J_{12} = J_{23} = 2.2 γ$ in (c1) to an approximate isolator operation with $J_{12} = J_{23} = 10^{- 2} γ$ in (c2). The other parameters are fixed as $γ_{1} = γ_{2} = γ$ , $γ_{3} = 9.72 γ$ , $g_{0} = 9.7 γ$ , $J_{13} \pm ε = (10^{- 2} \pm 9 \times 10^{- 3}) γ$ , and $E_{f} = E_{b} = 10 γ$ . In (c2), the isolation ratio defined in Eq. (3) is greatly enhanced together with the forward transmission proportional to $| a_{3, f}^{-} |$ .
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