Transmission Nonreciprocity in a Mutually Coupled Circulating Structure

TL;DR

This paper proposes a linear optical structure of mutually coupled microcavities or fiber rings that achieves nonreciprocal light transmission without breaking Lorentz reciprocity, enabling optical isolation and isolator functions.

Contribution

It introduces a novel linear optical system with mutual coupling that achieves nonreciprocity through gain saturation or asymmetric coupling, challenging traditional reciprocity assumptions.

Findings

Achieves nonreciprocal transmission in a linear system without breaking Lorentz reciprocity.

Enables optical isolation for inputs from two different ports.

Supports approximate optical isolator operations.

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.

Equations

$${\dot{a}}_1^{+}$$

$${\dot{a}}_1^{-}$$

$${\dot{a}}_2^{+}$$

$${\dot{a}}_2^{-}$$

$${\dot{a}}_3^{+}$$

$${\dot{a}}_3^{-}$$

$$\displaystyle\hat{S}=\left(\begin{array}[]{cccccc}S_{1+}^{1+}&S_{1-}^{1+}&S_{2+}^{1+}&S_{2-}^{1+}&S_{3+}^{1+}&S_{3-}^{1+}\\ S_{1-}^{1+}&S_{1-}^{1-}&S_{2+}^{1-}&S_{2-}^{1-}&S_{3+}^{1-}&S_{3-}^{1-}\\ S_{2+}^{1+}&S_{2+}^{1-}&S_{2+}^{2+}&S_{2-}^{2+}&S_{3+}^{2+}&S_{3-}^{2+}\\ S_{2-}^{1+}&S_{2-}^{1-}&S_{2-}^{2+}&S_{2-}^{2-}&S_{3+}^{2-}&S_{3-}^{2-}\\ S_{3+}^{1+}&S_{3+}^{1-}&S_{3+}^{2+}&S_{3+}^{2-}&S_{3+}^{3+}&S_{3-}^{3+}\\ S_{3-}^{1+}&S_{3-}^{1-}&S_{3-}^{2+}&S_{3-}^{2-}&S_{3-}^{3+}&S_{3-}^{3-}\end{array}\right),$$

$$10{\log }_{10}{\left|\frac{a_{3,f}^{-}+a_{3,b}^{-}}{a_{1,f}^{-}+a_{1,b}^{-}}\right|}^2=10{\log }_{10}{\left|\frac{S_{3-}^{1+}{E}_f+S_{3-}^{3+}{E}_b}{S_{1-}^{1+}{E}_f+S_{3+}^{1-}{E}_b}\right|}^2,$$

Full text

The availability of the approaches other than those based on Faraday effect mag ; mag1 ; mag3 is essential to realizing integrated circuits of optical nonreciprocity. “Optical nonreciprocity” here refers to the phenomena that the transmission of a field with certain frequency or bandwidth from port $P_{a}$ to port $P_{b}$ in a circuit is asymmetric with its reversed transmission from $P_{b}$ to $P_{a}$, being different from ideal “optical isolator” criteria that completely blocks the field from one of the ports, irrespective of its modal contents and polarizations. Synthetic magneto-optical effects, such as those of spatio-temporal modulations match1 ; al1 ; al2 ; al3 ; aa0 ; al4 ; aa ; lattice and optomechanics oms1 ; oms2 ; oms3 , were primarily considered as the replacement for Faraday effect. Numerous other methods b2 ; b3 ; b4 ; b5 ; b6 ; b7 ; b8 ; bragg ; wen ; jiang ; match ; atom1 ; atom2 ; o1 ; o2 ; band ; gain were also found for the purpose.

To propagate asymmetrically, light fields should be under an effect depending on their propagation directions. For example, the momenta of photons should satisfy a phase-matching condition for interband transitions match1 or in parametric processes match . Or else, nonreciprocal transmissions can occur by unequal couplings of a system to the inputs from two directions. Such examples include the ones making use of the optical gain saturated disparately under the fields unequally coupled from two waveguides gain ; wen ; jiang , as well as certain atomic systems non-identically coupled to photons from different directions atom1 ; atom2 . One question is whether there exists a nonreciprocal structure that is only made of isotropic media and can couple to the inputs from two different sides identically. It is also a widely held notion that the relation derived from Lorentz reciprocity theorem criteria ; book must be broken for achieving optical nonreciprocity. Whether this assumed restriction in designing nonreciprocal devices can be removed for certain systems, which well perform nonreciprocal transmissions, is fundamentally meaningful to understanding optical nonreciprocity.

In this Letter we provide the definite answers to these questions, using the mutually coupled circulating structure (MCCS) in Fig. 1(a). Developed from the widely studied PT-symmetric optical systems gain ; pt1 ; pt2 ; pt4 ; pt3 ; pt5 ; pt7 ; pt8 by adding one more passive component (cavity 2), such system can be constructed with three mutually coupled microresonators 3unit1 ; 3unit2 or optical fiber rings. Its relevant features are illustrated in what follows.

Inside each cavity of the MCCS, a circulating field mode couples to the reversely circulating ones in the neighboring cavities. Any external drive thus creates three pairs of circulating modes (one clockwise (CL) and one counter-clockwise (CCL) in each cavity) through the mutual couplings of the topological structure. In the frame rotating with the cavities’ resonance frequency, the dynamical equations for the coupled cavity modes read

[TABLE]

where “$+(-)$” represents a CL (CCL) mode. Besides the dissipation rate $\gamma_{k}$ ($k=1,2$ and $3$) and amplification rate $g(t)$, the main parameters are the coupling coefficients $J_{ij}$ ($i,j=1,2$ and $3$ for $i\neq j$) determined by the distances between two microcavities or by the designs of fiber couplers. The MCCS enjoys richer dynamical properties than the typical PT-symmetric systems—as in Fig. 1(b1) the system will undergo the transition from instability to stability and vice versa, as the couplings of cavity 2 to the two other cavities change. This distinguishes the structure from another topology of linking the components into a string string1 ; string2 .

Unlike many other systems for implementing nonreciprocal transmissions, the couplings of the MCCS to the forward input $S_{i,f}$ and the backward input $S_{i,b}$ are always the same with a fixed coupling constant $\kappa_{e}$. Relative to a forward drive at port $P_{1}$, we mainly consider a corresponding backward drive with the identical intensity from $P_{4}$, while the backward one can be applied on $P_{3}$ too. Under a constant optical gain, the backward input from $P_{3}$ also gives the same output amplitude $S_{o,b}=\sqrt{2\kappa_{e}}a_{1}^{-}$ at $P_{1}$ as the forward one $S_{o,f}=\sqrt{2\kappa_{e}}a_{3}^{+}$ at $P_{3}$. Such relations of the outputs, partially due to Lorentz reciprocity, are displayed in Figs. 1(c1)-1(c3).

A convenient tool to describe the relevant properties is scattering matrix, which takes the symmetric form,

[TABLE]

from Eqs. (1a)-(1f). Rewriting a single-frequency input as $\sqrt{2\kappa_{e}}S_{i,f(b)}=E_{f(b)}e^{i\Delta t}$, where $\Delta$ is the detuning with respect to cavities’ resonance frequency, one sees that the transmission reciprocity between $P_{1}$ and $P_{4}$ manifests as $S_{3-}^{1+}E=S_{3+}^{1-}E$, when two inputs with $E_{f}=E_{b}=E$ act separately on the system with a constant $g$. It is the consequence of a dynamical symmetry of the structure, which exhibits as $S_{3-}^{1+}=S_{3+}^{1-}$, the equality of the scattering matrix element $(16)$ with $(25)$ and $(61)$ with $(52)$ (the first and second number represent the matrix element’s row and column, respectively), in addition to the overall symmetry $(ij)=(ji)$ of the scattering matrix.

The transmission reciprocity can be easily lost under gain saturation. The actual gain rate g(t)=g_{0}/\big{(}1+|a_{3}^{+}(t)+a_{3}^{-}(t)|^{2}/I_{0}\big{)} becomes stable with time according to the saturation intensity $I_{0}$ of the given gain medium, as in Figs. 2(a) and 2(b). With the different gain rates $g(t)$ in Eqs. (1a)-(1f) for the inputs $S_{i,f}$ and $S_{i,b}$ that are turned on separately, the nonreciprocal transmissions between ports $P_{1}$ and $P_{4}$ exist. The trick to implement such nonreciprocity is that, through the mutual couplings, the identical $S_{i,f}$ and $S_{i,b}$ induce unequal $|a_{3,f}^{+}(t)+a_{3,f}^{-}(t)|^{2}$ and $|a_{3,b}^{+}(t)+a_{3,b}^{-}(t)|^{2}$ in the active component. The transmission nonreciprocity is measured by the log-ratio $10\log_{10}(|a_{3,f}^{-}|^{2}/|a_{1,b}^{-}|^{2})$, because the concerned forward and backward outputs are proportional to $a_{3}^{-}$ and $a_{1}^{-}$, respectively. Its relations with the system parameters are displayed in Figs. 2(c) and 2(d), where $\eta=J_{23}/J_{12}$ is an important factor adjusted by the couplings of cavity 2 with the two others. Rayleigh scattering rayleigh ; thesis that couples a CL mode with a CCL mode inside each cavity exists in realistic systems, and its effect is discussed in Sec. I of Supplementary Material supp .

To use the MCCS as an approximate optical isolator, a necessary condition is the highly asymmetric transmission contrast under the forward and backward drives together match ; nonlinear . The isolation of two simultaneous inputs is measured by the logarithmic ratio

[TABLE]

which includes the contributions from the reflected fields of the simultaneous forward input $E_{f}$ and backward input $E_{b}$. Optical isolation (a nonzero ratio of the above when $E_{f}=E_{b}$) immediately appears after breaking the symmetry of identical coupling between the cavities (having unequal distances between cavities so that $\eta\neq 1$), and it exists even without gain [Figs. 3(a1) and 3(a2)] and becomes more significant under gain [Figs. 3(b1) and 3(b2)]. Figs. 3(a3) and 3(b3) show that the associated isolation ratios defined in Eq. (3) stabilize with increased $E_{f}/E_{b}$. Moreover, they have two groups of peak values with the opposite signs, one of which is around $\eta=\Delta/\gamma$ due to a transmission resonance. In Fig. 3(c) showing this feature, the left of the marked dot on each curve is the unstable regime described in Fig. 1(b1). The outputs in the unstable regime are still asymmetric, as seen from an example of the time evolutions of their intensities.

Another symmetry breaking leads to a realization of nonreciprocity without gain saturation by the structure made up of optical fiber rings. The coupler illustrated in Fig. 4(a) capitalizes on directional macrobending loss macrobend , so that more light from one direction couples into another fiber than from the reverse direction. One choice as in Fig. 4(b) is that, through the coupler, mode $a_{1}^{+}$ in cavity 1 and mode $a_{3}^{-}$ in cavity 3 couple more strongly than the other pair $a_{1}^{-}$ and $a_{3}^{+}$, to replace the coefficient $J_{13}$ in Eqs. (1a) and (1f) [Eqs. (1b) and (1e)] by $J_{13}+\epsilon$ ($J_{13}-\epsilon$). The couplings via such bending radiation are rather weak. To implement an optical isolator operation, which requires that both the reflection $a_{1,f}^{-}=S_{1-}^{1+}E_{f}$ of the forward input and the backward transmission $a_{1,b}^{-}=S_{3+}^{1-}E_{b}$ be highly suppressed while the forward transmission $a_{3,f}^{-}=S_{3-}^{1+}E_{f}$ keep to be significant, one can lower the coupling constants $J_{12}$ and $J_{23}$ for cavity 2 to the order of $J_{13}\pm\epsilon$, as illustrated by the example in Figs. 4(c1)-4(c2). More details of this mechanism are in Sec. II of Supplementary Material supp .

The above scenario provides a better understanding of the relation between Lorentz reciprocity and transmission nonreciprocity. The scattering matrix is still symmetric after introducing the directional couplings in Fig. 4(a), but one of its additional symmetries is broken such that $S_{3-}^{1+}\neq S_{3+}^{1-}$. For two identical drives acting on $P_{1}$ and $P_{4}$ individually, such symmetry breaking results in the unequal forward output $\sqrt{2\kappa_{e}}a_{3,f}^{-}$ at $P_{4}$ and backward output $\sqrt{2\kappa_{e}}a_{1,b}^{-}$ at $P_{1}$, while the other pair of outputs, $\sqrt{2\kappa_{e}}a_{3,f}^{+}$ at $P_{3}$ and $\sqrt{2\kappa_{e}}a_{1,b}^{+}$ at $P_{2}$, constantly keep equal. The relation $E_{f}a_{1,b}^{+}=E_{b}a_{3,f}^{+}$, as the manifestation of Lorentz reciprocity for the system due to its symmetric scattering matrix, is explicitly satisfied for any pair of forward input $E_{f}$ and backward input $E_{b}$; see Sec. III in supp for more details. However, the nonreciprocal transmissions by applying the directional coupler in Fig. 4(a) to only two coupled cavities (cavity $2$ is detached) violate Lorentz reciprocity.

Such nonreciprocal transmissions by an MCCS are distinct from the asymmetric power flows in some other linear systems a1 ; l1 . For example, due to the drastically unequal transitions from one field mode to another mode with a different wave vector, asymmetric power flows between two ends exist in the system discussed in Refs. a1 ; a2 . But nonetheless, its intermodal transitions are identical, i.e. the transition from mode $1$ on the first end to mode $2$ on the second end has the same rate as that from mode $2$ on the second to mode $1$ on the first end. Corresponding to the circulating modes in MCCS, the forward transition from $a_{1}^{+}$ to $a_{3}^{-}$ is indeed symmetric with the backward one from $a_{3}^{-}$ to $a_{1}^{+}$, since the scattering matrix keeps symmetric. However, mode $a_{3}^{-}$ is from $P_{3}$ and mode $a_{1}^{+}$ leaves to $P_{2}$, not being the reversed transmission from $P_{1}$ to $P_{4}$. A field mode outputting to any port of such MCCS must be oppositely circulating with the mode that is excited by an input from the same port. At port $P_{1}$, for example, an input excites CL mode $a_{1}^{+}$, but any output comes from CCL mode $a_{1}^{-}$. The mutually reversed transmissions in Fig. 1(a), therefore, have to be the transitions from $a_{1}^{+}$ to $a_{3}^{-}$ and from $a_{3}^{+}$ to $a_{1}^{-}$, respectively. Such transitions from the CL (“$+$”) to the CCL (“$-$”) modes, unlike the intermodal ones mentioned above, will become nonidentical once the symmetry $S_{3-}^{1+}=S_{3+}^{1-}$ is broken, bringing about the transmission nonreciprocity.

In conclusion, we have demonstrated the nonreciprocal transmissions with an MCCS, illustrating such phenomena due to gain saturation and asymmetric field mode couplings. The structure exemplifies a new concept of realizing optical nonreciprocity and optical isolation with the topology of its connected components and the geometry to couple the fields in different components, bearing fundamental dissimilarity with the other approaches that rely on direction-dependent processes in the used media or nonidentical couplings to the inputs from different ports. There also exist two distinct mechanisms for the phenomena: (1) to break Lorentz reciprocity simply with gain saturation, so that the inputs from different ports follow different dynamical equations; (2) to break the additional symmetries of the scattering matrix elements, with unequal couplings between pairs of circulating modes and/or unequal distances between pairs of cavities, while preserving the Lorentz reciprocity of the system at the same time. The nonreciprocal phenomena without breaking Lorentz reciprocity were previously thought to be impossible. By choosing the proper system parameters, the structure can work as an approximate optical isolator.