1. Introduction

The development of recent nanofabrication technologies prompted the emergence of artificial structures such as photonic crystals, metamaterials, plasmonic components, and more recently so-called “PT symmetric devices” [1–35], referring to Parity-Time symmetry. Aside from fundamental research motivations, the tremendous interest for these artificial systems is also strongly driven by practical outcomes targeting light manipulation at the nanoscale and subwavelength components that shrink the miniaturization gap between photonics and electronics.

In this respect, the plasmonic approach probably stands as one of the most promising routes but the inherently high Joule losses remain a critical issue. The opportunity for loss compensation with gain was envisioned in several studies [35–48]. The interest of such systems combining gain and plasmonics was recently boosted by the demonstration of the first “plasmon lasers” or “spasers” [42], and by precise gain measurements in long-range plasmonic guides (LRSPP) [45]. The operation in this compensation regime requires however a rather high gain level, of the order of a few hundreds of cm⁻¹ to overcome metal-related losses.

While the propagation losses are generally considered as harmful in optical systems, we demonstrate in this contribution that they can play a positive role in active photonic components. We illustrate this point with PT symmetric couplers (PTSCs, obtained by coupling a lossy waveguide with another guide with gain) and demonstrate that these structures can operate as remarkably efficient switches when the proper amount of losses is included in the system.

The potential of PTSCs for switching operations has already been partly identified in the past: by concomitantly varying the gain and loss levels in the system, it was shown that the output signal could be abruptly modulated [11]. As will be recalled in the next section, this property stems from the fact that the eigenvalues of the structure coalesce when gain and loss reach a critical value. The most promising features of PTSCs are to be sought in the singular nature of this critical point (also called exceptional point) whereby a small change of gain/loss causes a quick evolution of the eigenvalues from the real to the imaginary axis.

The potential for achieving a large differential gain with a modest variation of the combined gain/loss in the vicinity of the critical gain point was further substantiated in our recent work [49–51]. Little attention, however, has been paid to the total amount of gain variation required to switch the signal. It is the aim of the present study to analyze the optimal conditions to perform a switching operation with the lowest amount of gain: we will see that, surprisingly, such optimized designs require a significant level of losses by absorption. This makes plasmonics a privileged area of application for PT symmetric systems since the implementation of gain in combination with plasmonics is increasingly mastered [45] and leads to consider more general combinations of gain and loss than the sole loss compensation.

A directional coupler with length L, coupling constant κ and gain/loss |g|/χ is the basis to which we refer to present the paper organization. In section 2, we introduce the different configurations that will be investigated in the paper and provide an analytical description of their eigenvalues. We start with an idealized PTSC and then generalize these results to two generic imperfect systems with unbalanced gain and losses: one with gain and losses that are not balanced but proportional (as one would operate with an “uncalibrated” device) and one with fixed losses (akin to operating with a “biased” device), which can be implemented, for example, with plasmonic waveguides. In section 3, we analyze the switching diagram in the (L, |g|/κ) plane for an “ideal” PT symmetry coupler where the amount of gain in one waveguide is exactly balanced with the amount of losses in the second waveguide. In section 4, we make use of a “gauge transform” tool to analyze “uncalibrated” PTSC switching on the same footing as the perfect case, substantiating the positive role of losses that reduce the amount of gain required for switching. In section 5, the “”biased” PTSC case of constant losses (plasmonics) is examined in a similar manner. In section 6, a realistic “biased” PTSC case is considered with a LRSPP as the constant loss waveguide. In Sec. 7, we report on the mirror biased PTSC configuration: a waveguide with fixed gain coupled to another one with variable losses, which proves of high interest. Arbitrary combinations (bias + uncalibrated) are left for more detailed studies.

2. Eigenvalue diagrams for perfect, unbalanced and biased switches

A sketch of a PT symmetry directional coupler is shown in Fig. 1(a). Gain in the top guide is denoted g₁ and losses in the bottom guide is denoted χ₂, although both can formally be positive or negative. Figure 1(b) plots the well-known eigenvalue evolution for the perfect case of balanced gain and losses as a function of the gain (g₁ = χ₂). Positive imaginary parts are for losses. This diagram of PTSC eigenvalues shows a characteristic transition between purely real eigenvalues to complex ones after the exceptional point. To gain more insight, we heuristically extend this perfect diagram to positive and negative values of gain and losses, resulting in a perfectly symmetric appearance [Fig. 1(c)].

 

Fig. 1 (a) Sketch of a PT symmetry directional coupler; (b) Real parts (“Re”, solid lines) and Imaginary parts (“Im”, dashed lines) of the two normalized eigenvalues β_eff1 and β_eff2 of a coupled system; (c) Same diagram presented on the whole algebraic gain axis. “GT*” refers to the gauge transform that realigns the real axis with the dotted line; (d) “Uncalibrated” case with g₁/χ₂ = 1.5 using the same gain axis; (e) “Uncalibrated case with g₁/χ₂=0.5; (f) “Biased” case with relatively small fixed losses; (g) “Biased” case with stronger fixed losses. The motion of the centre of the “Re” and “Im” patterns to the left and/or to the top is pointed out in the two last cases.

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Figures 1(d) and 1(e) show the diagram of eigenvalues for two “uncalibrated” cases where gain and losses are proportional but not equal. Here gain was chosen to be 1.5 times the losses [Fig. 1(d)] or 0.5 times the gain [Fig. 1(e)]. We see that the slope of the imaginary part is biased one way or the other depending on the dominant quantity, and that by the same token, the singularity occurs at larger or smaller gain values. Note that the biased slope also extends to the split imaginary branches. Therefore, compensating this slope by a common factor shall restore a symmetric diagram. We will see in section 4 that this operation corresponds to a “gauge transform” [11,13,14] that amounts to inserting a factor in front of all the transmission matrix quantities. Thus, this procedure will not affect the zeros of the PTSC and will help getting a simpler view of the design.

Finally, Figs. 1(f) and 1(g) presents the “biased” scenario, characterized by a variable gain and constant losses. We see that the characteristic eigenvalue pattern becomes offset and tilted, and that this distortion depends on the magnitude of the loss factor with respect to the coupling constant. As a consequence, the right-hand singular point (the more physical one in our device-oriented spirit) can be on either side of the zero imaginary value, resulting in a diagram that is not symmetrical with respect to g = 0 but symmetrical with respect to an offset centre. Such a sequence will be helpful to understand the details of PTSC modification. The situation with the right-hand exceptional point exactly on the zero-imaginary axis (intermediate between f and g) is the one underlined in [48] as most favorably reproducing the perfect switch (cases b and c).

3. PT Symmetry perfect switching

In this section, we show that it is possible to create an ideal switch, that is, a PTSC in which light can be dynamically switched from one guide to the other and delivering the same output intensity as that injected in the input waveguide. When such conditions are met, the “bar” and “cross” states denominations elaborated for conservative couplers are generalized for PTSCs. We will exploit modified “switching diagrams”, which are color maps of the coupler power transfer coefficient T^ij, in the (gain/loss, length) axis (instead of {index detuning/length} for classical couplers [52]), where lines or points for the various switching conditions are made apparent. In the spirit of [52] we discuss how the extra complexity still neatly provides both “cross” and “bar” states in a coupler that does not operate at the proper length, coupling, or wavelength, by tweaking the gain/loss parameters.

3.1 Analysis of the transmission maps

The PTSC operation can be analyzed through its transfer matrix for a given length. The detuning δ of the waveguides propagation constants β_j in the presence of the combined gain/loss is:

From Eqs. (3)-(4), it is possible to calculate the transmitted light intensity T^ij at the output of the j-th waveguide for light injected through the i-th waveguide for z = L:

Figure 2 shows the values of T^ij as a function of the gain g₁ (x-axis) and coupling length L (y-axis). The T¹¹ and T²² diagrams represent the two “through” transmissions while the T¹² and T²¹ diagrams represent the “crossover” transmissions. In the case of a conservative coupler without gain or loss (vertical cross-sections at g₁ = 0, left axis of the maps), light injected in one input sloshes with period L_c from one waveguide to the other. The coupler is in the perfect cross state ⊗ whenever a complete energy crossover between waveguides (Tⁱⁱ = 0, T^ij = 1) takes place. As is well known, this requires that:

 

Fig. 2 Color map of a transmission T^ij(L,g₁) in a dB scale. The thin black solid lines on the maps correspond to the iso-level curves T^ij = 1. The ordinate axes points corresponding to a complete power crossover are marked by a Ө, and those indicating the bar state with zero net crossover are marked by a ⊗. The green and magenta color arrows correspond to a perfect switching operation from Tⁱⁱ = 1, T^ij = 0 to Tⁱⁱ = 0, T^ij = 1, where i≠j. Note that arrows from the two top panels (injection in the “gainy” guide 1) are identical, and differ from arrows of the two bottom ones (injection in the “lossy” guide).

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The perfect bar state Ө corresponding to Tⁱⁱ = 1, T^ij = 0 is obtained when:

The perfect switching operation in a conservative directional coupler, achieved for example by refractive index tuning [52], is the bar-cross transition (Ө → ⊗). The first point of our analysis consists in determining whether the perfect switching operation using PT symmetry couplers is still possible, despite the absence of energy conservation.

According to Fig. 2, the introduction of balanced gain/loss in the system leads to the splitting of the initial lossless bar (respectively cross) states and to the genesis of two branches where Tⁱⁱ = 1 (resp. T^ij = 1 where i≠j,). These branches, corresponding to the iso-level curves on the T^ij surface map, are drawn as thin black solid lines on Fig. 2. In addition, the curves of the switching diagram shift upward and tend to crowd in the vicinity of the critical point g₁ = κ. This behavior can be understood by the fact that the detuning of the eigenvalues diminishes when approaching the singular point [Fig. 1(b)]. As a consequence, the beating phenomenon characteristic of directional couplers demands larger and eventually diverging lengths.

Before further analysis, it is important to note that the T¹² and T²¹ maps are identical. The situation is however quite different for bar channels (compare T¹¹ and T²² maps) and depends on whether there is gain or loss in the injected waveguide. Such a so-called “non-reciprocal” behavior is a characteristic feature of PT-symmetry devices [11,13,14] (the inputs are not pure modes, hence actual modal reciprocity and time-reversal symmetry holds, as expected for nonmagnetic systems). In other words, the PTSC behavior depends on which of the two input ports is excited. We must therefore analyze these two injection scenarios separately, as detailed in the next two sub-sections.

3.2 Injection into the gain waveguide

Let us first examine injection into the gain guide (T¹¹ and T¹²). It is seen that for L≥2L_c, the lower branches of the T¹¹ = 1 iso-level curves coincide with the states where T¹² = 0 (deep-blue regions on color map). Indeed, since β₁ = β₂ and (g₁-χ₂) = 0, from the transmission matrix of Eq. (1), satisfying the conditions for a perfect bar state (T¹¹ = 1, Ө, T¹² = 0, ⊗) requires that:

The system of Eqs. (8) is satisfied when ΩL is an integer multiple m of π. Note also that those arguments ΩL satisfying only the upper line of Eqs. (8), correspond to the upper branches of the T¹¹ = 1 contour. Although T¹¹ = 1 (Ө) is verified, these branches do not correspond to a perfect bar state since T¹²≠0 (not ⊗).

The cross state involves also two split branches, of which only one achieves perfect switching: for L≥L_c the upper branch of T¹² = 1 contours coincides in position with the states where T¹¹ = 0. These branches correspond to the solutions of the system of equations:

It is easy to see that whenever the lower equation, ensuring T¹¹ = 0, is satisfied, the upper one ensuring T¹² = 1 is also fulfilled thanks to Eq. (9). In contrast, the reverse is not true, since only the upper of the T¹² = 1 branches corresponds to a perfect cross state.

In summary, when operating the switch through the gain waveguide input, the PTSC displays perfect bar and cross states for L≥2L_c. Consequently, it is possible to perform a perfect switching operation by adapting the level of combined gain/loss in the system, as indicated by the generic dashed arrows running along constant length cuts on the T¹¹ and T¹² maps of Fig. 2.

The number S₁(L) of available switching configurations for gain-branch injection increases with the coupler length, with some analogy to the quantum harmonic oscillator. S₁(L) = 0 if L<2L_c. S₁(L) = 1 when 2L_c≤L<3L_c. In the interval 3L_c≤L<4L_c this configuration persists (the violet segment can be imagined to “glide up” the black curves) but it is shrunk towards the singularity, while there is an additional configuration at lower gain from the second cross state to the second bar state, thus, S₁(L) = 2, and so on when climbing up the different multiples of L_c as suggested by the examples of concatenated segments of Fig. 2.

3.3 Injection into the lossy waveguide

Let us perform a parallel analysis for injection into the lossy waveguide [maps of T²² and T²¹, number of configurations S₂(L)]. It is now seen that S₂(L) = 0 until a length L slightly lower than 3L_c. This lowest-length configuration corresponds now to the transition from the upper branches of the T²² = 1 (bar) contours, at high Δ_im or g₁, to the lower branches of the T²¹ = 1 (cross) contours, at lower Δ_im or g₁. The algebra is similar to the previous case, cf. Equations (8) and (10). The difference in the sign for the second terms of M₁₁ and M₂₂ elements explains the different selection of branches corresponding to a perfect cross and bar states. Overall, not only do we have a larger minimal length L~3L_c to get perfect switching, but also, for the same length, we nearly have S₂(L) ≈S₁(L)-1.

Therefore, this configuration appears slightly less interesting than injection into the gain waveguide, see section 3.b. Nevertheless, we not only see that perfect switching is possible for both injection scenarios, but also that it can be finely tuned by adjusting the gain level in the system. This point is of primary importance for practical applications because adjusting the gain offers the possibility to compensate for possible fabrication errors that typically spoils the degree of cancellation of the switch, e.g. the exact gap between the two waveguides. This flexibility of design is similar to that offered by the so-called Δβ couplers, as discussed by Kogelnik and Schmidt in [52].

3.d Total amount of gain required for switching

We shall close the discussion on perfect PTSCs by evaluating the total amount of gain required to perform switching. More precisely, we look for the minimal amplification level corresponding to the gain-length product g₁L, or, for this balanced case, for the minimal Δ_imL product. Also, we know from above that the shortest configuration exploits the gain input waveguide (T¹¹ and T¹²).

Constant gain-length product loci form, in Fig. 2, a familiar set of hyperbolas that have the Cartesian axes as asymptotes. Thus we look preferably for switching between close pairs of hyperbolas. When a zero-gain situation holds for the bar or cross switch states (Δ_im = 0), we seek the other state at a hyperbola with the smallest gain-length product.

In such a generic case, the coupler length has to hit multiples of 2L_c, L = 2mL_c. From Eq. (8), the detuning Δ_im needed for perfect switching, starting from Δ_im = 0, is then given by the m-dependent solution of the transcendental equation:

Let us discuss the m = 1 case. For this simpler case, it is interesting to get a compact insight by casting Eq. (9) into a trigonometric picture. Consider a circle of radius κL, and on this circle the point at angle ψ whose cosine is Δ_imL and sine is ΩL. The plot naturally enforces Eq. (9), stating Ω² = κ²-Δ_im². The condition T¹¹ = 0 reads tan(ΩL) = −Ω/Δ_im. Remarking that L = 2L_c = π/κ, we have tan(Ωπ/κ) = -Ω/Δ_im. Figure 3(a) tells us that Ω/κ = sinψ, while Ω/Δ_im = tan(ψ). We therefore have tan(πsinψ) = − tan(ψ). We change sign by taking tan(π-πsinψ) = tan(ψ), hence we get for our range of ψ values,

 

Fig. 3 (a) Trigonometric picture illustrating Eq. (9). (b) Cross and bar states intensity variation with gain g₁ (χ₂ = g₁)_.

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Thus, Eq. (12) gets a nicer form for m = 1, easy to solve graphically (sinψ = 1 − ψ/π). Standard math gives ψ = 47.43° and cosψ = Δ_im/κ = 0.6765, thus Δ_imL = 0.6765π = 2.125.

For higher m, and with more generality, as detailed in Appendix A, Eq. (13) becomes $\gamma \pi \sin \psi =m\pi -\psi$[see Eq. (20)] where:

Practically, the Δ_imL product grows typically like m^2/3 for m<~10 values.

Coming back to the m = 1 case, we insist that the fixed amount Δ_imL = πcosψ necessary to perform switching does not depend on the physical implementation of the PT symmetry coupler. For the waveguide with gain in isolation, Δ_imL = 0.6765π corresponds to an amplification level of 18.6dB, related to exp(2Δ_imL) [Fig. 3(b), dashed line and arrow, with the red and blue curves showing the overall behavior for injection in guide 1].

The different behavior for light injection into the loss waveguide, seen in Fig. 2, is made more explicit in Fig. 3(b) (green and magenta curves), evidencing the asymmetry. Despite the cross coupling to the second waveguide leading to T²¹ = 1 (0 dB), there is still light flowing in the loss waveguide (T²²≠0) with an intensity amplified almost twice. This type of asymmetric operation, different from that used in the optical memories described in [6], can nevertheless be used for the implementation of the buffer memory function by proper re-injection of output 1 to input 2, allowing to re-amplify the signal or to compensate losses and also to perform memory state monitoring.

A last aspect of our views is their relation with the “quantum brachistochrone” problem, i.e. the quest of the Hamiltonian that transfers a given initial state to a final one in the shortest time [7,26]. The bandwidth-limited time of true conservative Hamiltonians was shown in these works to be possibly arbitrarily reduced in the case of PT-symmetric pseudo-Hamiltonians owing to the modified metric of the underlying Hilbert space. Equations are rather similar, and there are certainly parallels with our quest of the shortest coupler and smallest gain provided we restate the normalized problem among the choice of constant and variable quantities. We currently think that the coupler shortening, of high interest, would be possible if going to a strong coupling regime and high gain, but we shall not treat this prospect further in the present paper.

4. The “uncalibrated” PT symmetry and the positive role of losses

Another remarkable feature, genuinely connected to the underlying PT-symmetric layout, appears when working with the so-called “uncalibrated” situation with gain/loss ratio differing from unity but constant, defined by the ratio χ₂/g₁ = ξ. We shall see that in this case, the gain level g₁ required for switching can be reduced by increasing the loss contribution χ₂. This follows from the more “compact” evolution of the eigenvalues in this configuration [Figs. 1(d) and 1(e)]. The diagrams for T¹¹ and T¹² are given in Figs. 4(a)-4(d), in a limited range of length now, and for the two ratios ξ = χ₂/g₁ = 0.5 [Figs. 4(a) and 4(b)] and ξ = 3.0 [Figs. 4(c) and 4(d)]. The solid contours again mark unity values. We can see that their shapes are rounded with respect to Fig. 2, i.e. the splitting occurs but with a different local evolution while the zero lines are fully similar in shape. This could be usefully understood with the “gauge transformation” (GT) [11] which brings back the tilted imaginary eigenvalue evolution apparent in Figs. 1(d) and 1(e) to horizontal by applying a single scalar corrective factor exp[-(g₁-χ₂)L/2] to all T_ij. The “gauge-corrected” maps for this situation need not be shown since they are topologically identical to Fig. 2. In particular the splitting of the two “unity-contour” branches of the corrected map is no more rounded. Although there is no genuine added information in these GT-corrected maps, this approach gives a synthetic argument for the switching configurations of all the situations of Fig. 1: all the T_ij = 0 contours are unchanged by the GT, so this crucial aspect of switching, i.e., the cancellation in the injection waveguide for the cross state, shall be achieved. Similarly, the identification of the whole set of “switching configurations”, e.g., those with minimal length or minimal gain-length product, shall not be modified, nor shall the scenario described in the case of exact PT symmetry.

 

Fig. 4 Color maps of switching operation with injection in the gain waveguide, for two “uncalibrated” configurations: (a), (b) ξ = χ₂/g₁ = 0.5, maps of T₁₁ and T₁₂ as indicated ; (c), (d) same for ξ = χ₂/g₁ = 3.0; (e) T^12,cross gain penalty in dB as a function of loss/gain ratio ξ (log scale).

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In the present case, we represent with horizontal arrows and small circles the switching points. We see that the price to pay is a penalty on the cross transmission level, i.e. T₁₂ now differs from unity at the point where all power has disappeared from T₁₁.

We first provide an analytical account of this penalty: The length being 2L_c = π/κ for T¹² cancellation at g₁ = 0 (bar state), the cancellation of T¹¹ for the cross state is expressed exactly as before since it involves only Ω, Δ_im, and the same κ. T^12,cross (cross state) is just given by the same quantity (κ/Ω)² sin²(ΩL) exp[(g₁-χ₂)L] as before. We thus still have ΩL = Ωπ/κ = π sinψ = π-ψ [Eq. (13)], hence sin(ΩL) = sin(π-ψ) = sin ψ. The two first factors of T^12,cross then compensate each other. With the rule that χ₂ = ξ g₁, we write Δ_imL = (1 + ξ)g₁L/2 = κL cosψ = π cosψ and get the value of g₁L. Injecting it in the last factor exp[(g₁-χ₂)L] = exp[((1−ξ) g₁L] and converting to dB, we thus find:

We quantify graphically this “imperfect switching” penalty on Fig. 4(e) for a range of gain-to-loss ratio ξ, and find it to be quite affordable: the +/− 5 dB “good” switching range, whereby T₁₂ at switching remains within + or – 5 dB gain or loss, admits a broad ratio of ξ = χ₂/g₁ from 0.574 to 1.74 .

Coming back to the fact that this situation features a counterintuitive positive role of the losses, we see that this behavior arises because only Δ_im, i.e., the sum of gain and loss, but not their individual values, is entering Eq. (4). Thus, the minimal switching gain for the m = 2 length L = 2L_c can for instance be written as g₁ = cosψ(2πL⁻¹)(1 + ξ)⁻¹ by just applying Δ_imL = πcosψ with the explicit expression Δ_im = g₁ + χ₂ = g₁(1 + ξ).

It could be noted that this result was obtained by benefiting from the condition of waveguides transparency at the initial “cold” state, then invoking only “uncalibrated” PT operation. The fulfillment of this initial condition in practical applications, even though it is relaxed vs. the exact one, is however not easily granted except in the specific case of parametric gain used by Rüter [14]. Active semiconductors have density-of-states that need to be pumped to attain transparency. It is even excluded to achieve “cold” transparency in the case of plasmonic type waveguides like PIROW [48,50] or LRSPP where metal losses are fixed [36,37,44,45]. Nevertheless, as shown in Section 4 below, we may still achieve near-perfect switching.

5. Lossless switching in plasmonic type structures with a fixed loss level

We now consider that χ₂ is a fixed loss, and that gain g₁ is still variable. This corresponds to a typical plasmonic configuration, whereby metal losses cannot be varied. The issue in this case, motivated by practical outcomes, is whether it is possible to get, if not perfect, at least lossless switching. We need also to determine whether there are some optimal conditions on the loss level leading to switching with a minimal amount of amplification g₁L in the gain waveguide.

To explain the principle of a lossless switching, we consider a generic example with a fixed loss level 0<χ₂<κ: from the eigenvalue analysis, the loss level χ₂ should not exceed the value of the coupling coefficient κ. For a convenient illustration, the loss level is fixed to the median value χ₂ = 0.5κ, giving the switching diagrams of Fig. 5. When gain equals the loss level (g₁ = χ₂) the condition for a perfect PT symmetry is locally fulfilled. As explained in section 1, it is possible in this case to get either perfect cross or bar states. These discrete points (p = 1,…) are located on the intersection of a vertical axis originating from g₁ = χ₂ with either the iso-level curves of bar (Tⁱⁱ = 1, T^ij = 0) or cross (Tⁱⁱ = 0, T^ij = 1) cases. Hence, the design procedure is now reduced to choose among the discrete loss-dependent lengths L_p(χ₂) illustrated in Fig. 5. For larger gain g₁>χ₂, switching occurs, as indicated by violet (bar→cross) or green (cross→bar) arrows. Note that for the final states, exact zero transmission in one of the waveguides is still fulfilled, either T^ij = 0, or Tⁱⁱ = 0. But the second waveguide output signal is now larger than the input signal. This results in a globally lossless switching operation.

 

Fig. 5 Color maps of switching operation corresponding to the typical plasmonic setting with fixed loss χ₂ = 0.5κ and variable gain. The thin black solid lines on the maps correspond to the iso-level curves T^ij = 1. The violet arrows indicate a switching from an initial bar state Ө to a final cross state ⊗, green arrows correspond to a switching from an initial cross state ⊗ to a final bar state Ө. The lengths for the two top T^1j diagrams are denoted L_p. The vertical dotted line corresponds to the “exact PT symmetry” operation point.

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Figure 6(a) displays the dependence of switching gain g₁ as a function of the constant losses χ₂ for the lowest switching configurations (p <7 and we denote these cases as S1…S6). The odd and even indices p correspond to a bar→cross and a cross→bar switching, respectively (solid and dashed lines). It can be observed that the required switching gain decreases with p and is always lower for a bar→cross switching than for the next cross→bar operation. In turn, as evidenced in Fig. 6(b), operation at a higher switching state requires a longer coupler length. The increased coupler length L_p has a strong negative impact on the total amplification level required for switching. As can be observed from Fig. 6(c), only the lowest switching state S1 (p = 1) seems of practical interest. The operation at a higher switching state S3 (p = 3) requires 20dB of additional amplification, which is quite substantial.

 

Fig. 6 Analysis of the six first switching configurations S1 to S6 of length L₁ to L₆ (a) Gain g₁ required for switching. (b) PTSC length L₁ to L₆; the two extremes are labeled for clarity. (c) Switching amplification level. (d) Transmitted signal amplification level. The vertical dashed line points relate to Fig. 5, χ₂ = 0.5κ case. The red dot further signals the S₁ configuration of Fig. 5.

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The minimal amplification level of 31dB obtained for S1 switching is reached for a fixed loss χ₂≈0.42κ. It means that when designing a plasmonic type PT coupler, its coupling strength and loss level can be mutually adjusted for an optimal switching operation. Fortunately for many plasmonic applications, not only is the minimum of the optimal loss level still quite large, but also, a kind of “half width” of χ₂ ranging from 0.07κ<χ₂<0.67κ holds if we consider a 3dB penalty criterion for the S1 amplification level (34dB). This makes the design quite robust with respect to the technological errors on propagation loss level. The cross and bar intensity corresponding to the optimal switching operation with χ₂≈0.42κ fixed losses are displayed in Fig. 7(a).

 

Fig. 7 (a) Cross and bar states intensity variation with gain g₁ for a fixed loss level χ₂ = 0.42κ. (b) Cross-sectional view of the hybrid plasmonic/dielectric coupler. The modes propagate along the third dimension (the z-axis). The total length L of the device along the z axis, not shown here, is 5 mm. (c) Cross and bar states intensity vs. g = Im(ε_SU8). The coupler is fed by the SU8 waveguide. To compute these curves, we assumed that all the electromagnetic power contained in the left semi-infinite xy plane is carried by the SU8 waveguide while all the electromagnetic power contained in the right semi-infinite xy plane is carried by the metallic stripe.

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It can be further noted that the 31 dB price paid to fixed loss for a lossless operation are the 13dB of additional extra amplification as compared to the 18.4dB required for a perfect PT symmetry switching. As shown in Fig. 6(d), most of this additional gain fortunately goes into the 7.8 dB amplification of the transmitted signal. The amount of amplification can be of course somewhat decreased depending on the tolerance with respect to allowed switch losses.

6. Numerical example with LRSPP and dielectric waveguides

To validate our ideas, we apply the previous conclusions to the design of an active gold-based plasmonic switch operating at the wavelength λ = 1.55 µm. Figure 7(b) shows a cross-section of the device under investigation: it consists of an SU8 polymer waveguide doped with uniform gain g (relative permittivity ε_SU8 = 1.57² + ig), in close proximity to a metal stripe with fixed losses (relative permittivity ε_Au = −132-12.65i, representative of Au at 1.55 µm). The resulting coupler is itself buried in a dielectric slab (BCB, with relative permittivity ε_BCB = 1.535²) supported by a SiO₂ substrate, ensuring that the isolated metallic waveguide supports a long-range plasmon mode. The dimensions of the different elements and their relative position within the BCB layer are shown in Fig. 7(b); they have been chosen so as to maximize the coupling from the SU8 waveguide to the plasmonic stripe, as explained in [44].

To operate the coupler as a switch, two additional geometrical parameters must be properly adjusted: the separation distance d between the two arms of the device and the propagation length L. From Appendix B, we know that these parameters are interdependent: d sets the coupling coefficient κ, which itself determines the propagation length L necessary to perform the switching operation along the lines discussed on Fig. 5. Based on these considerations, we have designed a switch with a commercial FEM eigenmode solver (Comsol Multiphysics). Since the structure is invariant along the propagation direction, the computational domain is reduced to the cross-section shown in Fig. 7(b). Once the coupled eigenmodes of the system are found, it is possible to reconstruct the propagation along the third direction by just using a complex linear combination of these solutions, as explained in [44] following the prescriptions of [53].

Figure 7(c) shows the result of the simulations for a separation distance d = 3.845 µm and propagation length L = 5 mm. In this example, the coupler is fed by the SU8 waveguide and we have plotted the power at the output of each waveguide as a function of the gain g. The plot is in excellent qualitative agreement with the predictions of the coupled mode theory shown in Fig. 7(a): in particular, we note that each curve has a pronounced minimum when the other carries approximately 100% of the signal. By varying the gain between these two states, it is therefore possible to transition from isolation to complete transmission, i.e. to use the coupler as a switch. We will discuss the quantitative discrepancies and their origin elsewhere, to avoid blurring the issues genuinely at stake in the switching and PT symmetry. Again, we emphasize that the dimensions of the structure are set by the fixed losses of the system. To reduce the length of the device (here L = 5mm), it would be necessary to consider much more confined (and therefore more absorbing) plasmonic geometries (PIROW, grooves, metal-insulator-metal structures, asymmetric stripes…).

7. Fixed gain switching

The final question that we would like to address in our study is whether it is possible to further decrease the amount of amplification required for switching in a system differing from ideal PT symmetry in a way still not considered, namely a fixed gain configuration. The case of PTSCs with fixed losses revealed the necessity to bring in the system at least 13dB of additional amplification in order to achieve a lossless switching operation. This additional amplification is needed because the gain and losses are equal in the initial state while we need to further increase the gain level to perform switching.

In contrast, when starting from a fixed gain case as proposed here, the switching is performed by decreasing the losses level. It can be expected that this would result in a lower amount of required amplification level. As detailed in Appendix C, the minimal amount of gain g₁ = 0.71κ is achieved for the lowest bar→cross switching with the gain input waveguide. The variation of the bar and cross intensities for this case is shown on Fig. 8. The required amplification level in the system is of only 20.8dB in this case, that is only 2.3dB of penalty as compared to the perfect PT symmetry case.

 

Fig. 8 Cross and bar states intensity variation with optimized fixed gain (g₁ = 0.71κ) and variable loss χ_2.

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It could be noted however, that the switching diagrams are quite different from the previous cases. As evidenced by Fig. 9, for a given amount of gain above the threshold level required for switching, only one fully perfect switching state is possible (i.e. with unit transmissions for both bar and cross cases).

 

Fig. 9 Color maps of switching operation with fixed gain g₁ = 0.8κ and variable loss (χ₂ variable negative gain). The thin black solid lines on the maps correspond to the iso-level curves T^ij = 1. The violet arrows indicate a switching from an initial bar state Ө to a final cross state ⊗. The vertical dotted line corresponds to the “exact PT symmetry” operation point. In the top maps, the switching shown is the only one with final unity transmission, whereas in the top map, there is not even a single such possibility.

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7. Summary and conclusions

Directional couplers combining loss and gain have a disruptive potential as optical switches. We showed that perfect switching from complete isolation to full transmission is possible when gain and loss are exactly balanced, i.e. when the system exhibits a perfect PT symmetry. For less perfect PT symmetry of two types, unbalanced, or with fixed loss such as in plasmonic systems, switching can still be realized on one channel and with perfect cancellation, but with typically excess transmission in one switch configuration.

Importantly, our results show that the combination of gain and plasmonics can be used for a functional structure, well beyond the simple idea of loss compensation. The positive role of losses allowing lowering the total amount of gain required for switching was particularly emphasized. For a plasmonic case with fixed loss, the required amplification level of 31dB is within the reach for most of practical applications. A practical example based on long-range plasmon waveguide and polymer substantiated our most significant prospect of a plasmonic PT-symmetry switch. This numerical example allowed to settle issues for forthcoming practical designs and also served as a validation of our ideas which were made within the tractable but simplified framework of the coupled-mode theory. This confirmation was necessary given that coupled-mode-theory can be difficult to handle properly in the vicinity of the PT-symmetry singularity [49,51].

One interesting feature, opening promising perspectives for the miniaturization of this kind of plasmonic devices, is related to the “brachistochrone” dynamics in PT systems [7,26]. The absolute length of such devices could be strongly reduced by operating in a strong coupling regime. The lowest limit for the device length is then dictated by the highest amount of gain and loss available in the system.

Many other avenues yet to be explored (quantum aspects, noise) should now have a more precise frame to carry out investigation of these novel systems.