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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 18 — Sep. 9, 2013
  • pp: 21651–21668
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Switching using PT symmetry in plasmonic systems: positive role of the losses

Anatole Lupu, Henri Benisty, and Aloyse Degiron  »View Author Affiliations


Optics Express, Vol. 21, Issue 18, pp. 21651-21668 (2013)
http://dx.doi.org/10.1364/OE.21.021651


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Abstract

We analyze the operation of 2 × 2 switches composed of two coupled waveguides operating on the basis of parity-time (PT) symmetry: the two waveguides differ through their gain or loss factors and not through the real part of their propagation constant. Plasmonics constitutes a preferred application for such systems, since combination of plasmonics with gain is increasingly mastered. The exact PT-symmetric case (gain and loss of identical absolute value) is considered as well as various unbalanced cases, thanks to their respective switching diagrams. Although perfect signal-conserving cross and bar states are not always possible in the latter cases, they can nevertheless form the basis of very good switches if precise design rules are followed. We draw from the analysis what the optimal configurations are in terms of, e.g., guide gain or gain-length product to operate the switch. Many analytical or semi-analytical results are pointed out. A practical example based on the coupling of a long-range surface-plasmon-polariton and a polymeric waveguide having gain is provided.

© 2013 OSA

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

1. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998). [CrossRef]

35

35. M. Kulishov, B. Kress, and R. Slavík, “Resonant cavities based on Parity-Time-symmetric diffractive gratings,” Opt. Express 21(8), 9473–9483 (2013). [CrossRef] [PubMed]

], 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.

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.

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)
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 βeff1 and βeff2 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 g12 = 1.5 using the same gain axis; (e) “Uncalibrated case with g12=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.
. 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)].

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

11. A. Guo, G. J. Salamo, 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(9), 093902 (2009). [CrossRef] [PubMed]

,13

13. T. Kottos, “Broken symmetry makes light work,” Nat. Phys. 6(3), 166–167 (2010). [CrossRef]

,14

14. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010). [CrossRef]

] 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

48. H. Benisty and M. Besbes, “Plasmonic inverse rib waveguiding for tight confinement and smooth interface definition,” J. Appl. Phys. 108(6), 063108 (2010). [CrossRef]

] 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 Tij, in the (gain/loss, length) axis (instead of {index detuning/length} for classical couplers [52

52. H. Kogelnik and R. V. Schmidt, “Switched directional couplers with alternating Δβ,” IEEE J. Quantum Electron. 12(7), 396–401 (1976). [CrossRef]

]), where lines or points for the various switching conditions are made apparent. In the spirit of [52

52. H. Kogelnik and R. V. Schmidt, “Switched directional couplers with alternating Δβ,” IEEE J. Quantum Electron. 12(7), 396–401 (1976). [CrossRef]

] 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 βj in the presence of the combined gain/loss is:
δ=β1+ig1(β2iχ2)2=β1β22+iΔim
(1)
where g1 is the gain of the first waveguide, χ2 (≡−g2) is the loss of the second waveguide and the quantity Δim defined by:
Δim=g1+χ22
(2)
is the combined imaginary detuning, i.e. |gain| + |loss|. Our choice is to use the concept of “amplitude gain” to get more compact expressions without having confusing factors of 2. The transfer matrix M(z), with coefficients mij, of such a directional coupler is [52

52. H. Kogelnik and R. V. Schmidt, “Switched directional couplers with alternating Δβ,” IEEE J. Quantum Electron. 12(7), 396–401 (1976). [CrossRef]

]:
M(z)=(cos(Ωz)iδΩsin(Ωz)iκΩsin(Ωz)iκΩsin(Ωz)cos(Ωz)+iδΩsin(Ωz))exp(g1χ22z)
(3)
Ω=δ2+κ2
(4)
where z is the propagation distance, κ is the coupling coefficient and Lc = π/2κ is the coupling length corresponding to a complete crossover from one guide to the other for δ = 0 (Ω = κ).

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

Tij=|Mij|2
(5)

Figure 2
Fig. 2 Color map of a transmission Tij(L,g1) in a dB scale. The thin black solid lines on the maps correspond to the iso-level curves Tij = 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, Tij = 0 to Tii = 0, Tij = 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).
shows the values of Tij 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 Lc from one waveguide to the other. The coupler is in the perfect cross state ⊗ whenever a complete energy crossover between waveguides (Tii = 0, Tij = 1) takes place. As is well known, this requires that:

κL=(2m+1)π2,m=1,2,3,...
(6)

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

κL=mπ,m=0,1,2,3,...
(7)

The perfect switching operation in a conservative directional coupler, achieved for example by refractive index tuning [52

52. H. Kogelnik and R. V. Schmidt, “Switched directional couplers with alternating Δβ,” IEEE J. Quantum Electron. 12(7), 396–401 (1976). [CrossRef]

], 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. Tij = 1 where i≠j,). These branches, corresponding to the iso-level curves on the Tij 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.

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≥2Lc, 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 (g12) = 0, from the transmission matrix of Eq. (1), satisfying the conditions for a perfect bar state (T11 = 1, Ө, T12 = 0, ⊗) requires that:
{|cos(ΩL)+ΔimΩ1sin(ΩL)|=1κΩ1sin(ΩL)=0
(8)
where using Eqs. (1) and (2), Eq. (4) allows expressing the sine and cosine arguments as:

ΩL=κ2Δim2L
(9)

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≥Lc 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:
{|κΩ1sin(ΩL)|=1cos(ΩL)+ΔimΩ1sin(ΩL)=0
(10)
The system of Eqs. (10) can be cast in the following form:

{|sin(ΩL)|=Ωκcos(ΩL)=Δimκ
(11)

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≥2Lc. 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<2Lc. S1(L) = 1 when 2Lc≤L<3Lc. In the interval 3Lc≤L<4Lc 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 Lc as suggested by the examples of concatenated segments of Fig. 2.

3.3 Injection into the lossy waveguide

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

52. H. Kogelnik and R. V. Schmidt, “Switched directional couplers with alternating Δβ,” IEEE J. Quantum Electron. 12(7), 396–401 (1976). [CrossRef]

].

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 ΔimL 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 (Δim = 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 2Lc, L = 2mLc. From Eq. (8), the detuning Δim needed for perfect switching, starting from Δim = 0, is then given by the m-dependent solution of the transcendental equation:

tan(m2π2Δim2L2)=m2π2Δim2L21
(12)

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 ΔimL and sine is ΩL. The plot naturally enforces Eq. (9), stating Ω2 = κ2im2. The condition T11 = 0 reads tan(ΩL) = −Ω/Δim. Remarking that L = 2Lc = π/κ, we have tan(Ωπ/κ) = -Ω/Δim. Figure 3(a)
Fig. 3 (a) Trigonometric picture illustrating Eq. (9). (b) Cross and bar states intensity variation with gain g12 = g1).
tells us that Ω/κ = sinψ, while Ω/Δim = 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,

πsinψ=πψ
(13)

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 = Δim/κ = 0.6765, thus ΔimL = 0.6765π = 2.125.

For higher m, and with more generality, as detailed in Appendix A, Eq. (13) becomes γπsinψ=mπψ[see Eq. (20)] where:

γ=L2Lc=mκκ2χ22
(14)

Practically, the ΔimL product grows typically like m2/3 for m<~10 values.

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

6. M. Kulishov, J. M. Laniel, N. Bélanger, and D. V. Plant, “Trapping light in a ring resonator using a grating-assisted coupler with asymmetric transmission,” Opt. Express 13(9), 3567–3578 (2005). [CrossRef] [PubMed]

], 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

7. C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, “Faster than Hermitian Quantum Mechanics,” Phys. Rev. Lett. 98(4), 040403 (2007). [CrossRef] [PubMed]

,26

26. H. Ramezani, J. Schindler, F. M. Ellis, U. Günther, and T. Kottos, “Bypassing the bandwidth theorem with PT symmetry,” Phys. Rev. A 85(6), 062122 (2012). [CrossRef]

]. 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)
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) T12,cross gain penalty in dB as a function of loss/gain ratio ξ (log scale).
-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

11. A. Guo, G. J. Salamo, 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(9), 093902 (2009). [CrossRef] [PubMed]

] 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[-(g12)L/2] to all Tij. 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 Tij = 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.

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 2Lc = 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, Δim, and the same κ. T12,cross (cross state) is just given by the same quantity (κ/Ω)2 sin2L) exp[(g12)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 T12,cross then compensate each other. With the rule that χ2 = ξ g1, we write ΔimL = (1 + ξ)g1L/2 = κL cosψ = π cosψ and get the value of g1L. Injecting it in the last factor exp[(g12)L] = exp[((1−ξ) g1L] and converting to dB, we thus find:

T12,cross(dB)=10[log(10)1]2πcosψ1ξ1+ξ=18.41ξ1+ξ
(15)

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 Δim, 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 = 2Lc can for instance be written as g1 = cosψ(2πL−1)(1 + ξ)−1 by just applying ΔimL = πcosψ with the explicit expression Δim = 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

14. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010). [CrossRef]

]. 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

48. H. Benisty and M. Besbes, “Plasmonic inverse rib waveguiding for tight confinement and smooth interface definition,” J. Appl. Phys. 108(6), 063108 (2010). [CrossRef]

,50

50. H. Benisty and M. Besbes, “Confinement and optical properties of the plasmonic inverse-rib waveguide,” J. Opt. Soc. Am. B 29(4), 818–826 (2012). [CrossRef]

] or LRSPP where metal losses are fixed [36

36. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B 61(15), 10484–10503 (2000). [CrossRef]

,37

37. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of asymmetric structures,” Phys. Rev. B 63(12), 125417 (2001). [CrossRef]

,44

44. A. Degiron, S.-Y. Cho, T. Tyler, N. M. Jokerst, and D. R. Smith, “Directional coupling between dielectric and long-range plasmon waveguides,” New J. Phys. 11(1), 015002 (2009). [CrossRef]

,45

45. I. De Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nat. Photonics 4(6), 382–387 (2010). [CrossRef]

]. 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
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 Tij = 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 T1j diagrams are denoted Lp. The vertical dotted line corresponds to the “exact PT symmetry” operation point.
. 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, Tij = 0) or cross (Tii = 0, Tij = 1) cases. Hence, the design procedure is now reduced to choose among the discrete loss-dependent lengths Lp2) illustrated in Fig. 5. For larger gain g12, 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 Tij = 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.

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(εSU8). 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.
.

6. Numerical example with LRSPP and dielectric waveguides

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

44. A. Degiron, S.-Y. Cho, T. Tyler, N. M. Jokerst, and D. R. Smith, “Directional coupling between dielectric and long-range plasmon waveguides,” New J. Phys. 11(1), 015002 (2009). [CrossRef]

] following the prescriptions of [53

53. A. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer, 2000).

].

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
Fig. 8 Cross and bar states intensity variation with optimized fixed gain (g1 = 0.71κ) and variable loss χ2.
. 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.

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

49. H. Benisty, A. Degiron, A. Lupu, A. De Lustrac, S. Chénais, S. Forget, M. Besbes, G. Barbillon, A. Bruyant, S. Blaize, and G. Lérondel, “Implementation of PT symmetric devices using plasmonics: principle and applications,” Opt. Express 19(19), 18004–18019 (2011). [CrossRef] [PubMed]

,51

51. H. Benisty, C. Yan, A. Degiron, and A. T. Lupu, “Healing near-PT-symmetric structures to restore their characteristic singularities: Analysis and examples,” J. Lightwave Technol. 30(16), 2675–2683 (2012). [CrossRef]

].

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.

Appendix A: fixed losses PTSC bar→cross switching

As outlined in section 5, in the case of the fixed losses PTSC, the principle of lossless operation consists in equilibrating losses by gain at the initial perfect bar state Ө (T11=1, T12=0), then by a further increase of the gain to perform a switching to the cross state ⊗ with an amplified output light intensity (T11=0, T12>1). On the basis of Eqs. (8) and (10), the necessary conditions to be fulfilled are:

{|cos(ΩΘL)+ΔimΘΩΘsin(ΩΘL)|=1|cos(ΩL)+ΔimΩsin(ΩL)|=0
(16)

The upper equation of the system (16) is satisfied whenever
κ2(g1Θ+χ22)2L=mπ
(17)
where m = 1,2,3,…

By taking into account that the gain is equal to losses (g1Ө2) at the initial bar state, the PTSC matching length is:

L=mπκ2χ2
(18)

The detuning required for switching to the cross state Δim can then be obtained as a solution of the lower equation of the system (16):

tan((κL)2(ΔimL)2)=(κL)2(ΔimL)21
(19)

By means of the trigonometric representation illustrated by Fig. 3(a), Eq. (19) can be cast in a more convenient form for roots-finding:
sinψ=mπψγπ
(20)
g1=2κcosψχ2
(21)
where the integer m ≤ γ = L/2Lc.

It can be seen that Eq. (12) corresponding to the lowest switching length L=2Lc in a perfectly balanced PTSC is just a particular case of Eq. (19) for m=1 and χ2=g1.

Appendix B: fixed losses PTSC cross→bar switching

On the basis of Eqs. (8) and (11) the necessary conditions to perform a switching from a perfect cross state ⊗ (T11=0, T12=1) to a bar state Ө (T11>1, T12=0) in the case of a fixed loss level are:

{cos(ΩL)=ΔimκκΩΘsin(ΩΘL)=0
(22)

By taking into account that gain is equal to losses (Δim2) at the initial cross state, the PTSC matching length is given by the upper equation of the system (B1):

L=(m+1)πacos(χ2/κ)κ2χ22
(23)

which is more complex than Eq. (18). The required gain and detuning for switching to the bar state are then obtained using the lower equation of the system (22):

ΔimΘ=κ2(mπL)2
(24)
g1Θ=2ΔimΘχ2
(25)

Appendix C: fixed gain PTSC bar→cross switching

At variance with appendices A and B, the gain is now fixed and losses are variable. On the basis of Eqs. (8) and (11), the necessary conditions to perform a switching from an initial amplified bar state Ө (T11>1, T12=0) to a final perfect cross state ⊗ (T11=0, T12=1) are:

{κΩΘsin(ΩΘL)=0cos(ΩL)=Δimκ
(26)

Since there are no losses in the initial bar state, the bar detuning is ΔimӨ=g1. From the upper equation of the system (C1) it follows that:

(κL)2=(g1L)2+(mπ)2
(27)

As at the cross state the losses are equal to the gain, the detuning Δim=2ΔimӨ. Using the lower equation of the system (26):
cos((κL)24(g1L)2)=2g1LκL
(28)
and from Eq. (27) we obtain the amplification level required for switching g1L as the solution of the transcendental equation:

cos((mπ)23(g1L)2)=2g1L(mπ)2+(g1L)2
(29)

Finally, the PTSC length is then directly obtained from Eq. (27).

References and links

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C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998). [CrossRef]

2.

C. M. Bender, “Making Sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70(6), 947–1018 (2007). [CrossRef]

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A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT -symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38(9), L171–L176 (2005). [CrossRef]

4.

M. Greenberg and M. Orenstein, “Irreversible coupling by use of dissipative optics,” Opt. Lett. 29(5), 451–453 (2004). [CrossRef] [PubMed]

5.

M. Kulishov, J. M. Laniel, N. Bélanger, J. Azaña, and D. V. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Express 13(8), 3068–3078 (2005). [CrossRef] [PubMed]

6.

M. Kulishov, J. M. Laniel, N. Bélanger, and D. V. Plant, “Trapping light in a ring resonator using a grating-assisted coupler with asymmetric transmission,” Opt. Express 13(9), 3567–3578 (2005). [CrossRef] [PubMed]

7.

C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, “Faster than Hermitian Quantum Mechanics,” Phys. Rev. Lett. 98(4), 040403 (2007). [CrossRef] [PubMed]

8.

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007). [CrossRef] [PubMed]

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S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101(8), 080402 (2008). [CrossRef] [PubMed]

10.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008). [CrossRef] [PubMed]

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A. Guo, G. J. Salamo, 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(9), 093902 (2009). [CrossRef] [PubMed]

12.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103(3), 030402 (2009). [CrossRef] [PubMed]

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T. Kottos, “Broken symmetry makes light work,” Nat. Phys. 6(3), 166–167 (2010). [CrossRef]

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C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010). [CrossRef]

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M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, “PT optical lattices and universality in beam dynamics,” Phys. Rev. A 82(1), 010103 (2010). [CrossRef]

16.

J. Čtyroký, V. Kuzmiak, and S. Eyderman, “Waveguide structures with antisymmetric gain/loss profile,” Opt. Express 18(21), 21585–21593 (2010). [CrossRef] [PubMed]

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S. Longhi, “Invisibility in PT -symmetric complex crystals,” J. Phys. A: Math. Theor. 44(48), 485302 (2011). [CrossRef]

18.

A. E. Miroshnichenko, B. A. Malomed, and Yu. S. Kivshar, “Nonlinearly-PT -symmetric systems: spontaneous symmetry breaking and transmission resonances,” Phys. Rev. A 84(1), 012123 (2011). [CrossRef]

19.

Y. D. Chong, L. Ge, and A. D. Stone, “PT-symmetry breaking and laser-absorber modes in optical scattering systems,” Phys. Rev. Lett. 106(9), 093902 (2011). [CrossRef] [PubMed]

20.

M.-A. Miri, A. Regensburger, U. Peschel, and D. N. Christodoulides, “Optical mesh lattices with PT –symmetry,” Phys. Rev. A 86(2), 023807 (2012). [CrossRef]

21.

H. Ramezani, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Exceptional-point dynamics in photonic honeycomb lattices with PT symmetry,” Phys. Rev. A 85(1), 013818 (2012). [CrossRef]

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S. Longhi and G. Della Valle, “Photonic realization of PT-symmetric quantum field theories,” Phys. Rev. A 85(1), 012112 (2012). [CrossRef]

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A. A. Sukhorukov, S. V. Dmitriev, S. V. Suchkov, and Y. S. Kivshar, “Nonlocality in PT-symmetric waveguide arrays with gain and loss,” Opt. Lett. 37(11), 2148–2150 (2012). [CrossRef] [PubMed]

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M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009). [CrossRef] [PubMed]

44.

A. Degiron, S.-Y. Cho, T. Tyler, N. M. Jokerst, and D. R. Smith, “Directional coupling between dielectric and long-range plasmon waveguides,” New J. Phys. 11(1), 015002 (2009). [CrossRef]

45.

I. De Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nat. Photonics 4(6), 382–387 (2010). [CrossRef]

46.

R.-M. Ma, R. F. Oulton, V. J. Sorger, G. Bartal, and X. Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. 10(2), 110–113 (2011). [CrossRef] [PubMed]

47.

A. Castanié and D. Felbacq, “Confined plasmonic modes in a nanocavity,” Opt. Commun. 285(16), 3353–3357 (2012). [CrossRef]

48.

H. Benisty and M. Besbes, “Plasmonic inverse rib waveguiding for tight confinement and smooth interface definition,” J. Appl. Phys. 108(6), 063108 (2010). [CrossRef]

49.

H. Benisty, A. Degiron, A. Lupu, A. De Lustrac, S. Chénais, S. Forget, M. Besbes, G. Barbillon, A. Bruyant, S. Blaize, and G. Lérondel, “Implementation of PT symmetric devices using plasmonics: principle and applications,” Opt. Express 19(19), 18004–18019 (2011). [CrossRef] [PubMed]

50.

H. Benisty and M. Besbes, “Confinement and optical properties of the plasmonic inverse-rib waveguide,” J. Opt. Soc. Am. B 29(4), 818–826 (2012). [CrossRef]

51.

H. Benisty, C. Yan, A. Degiron, and A. T. Lupu, “Healing near-PT-symmetric structures to restore their characteristic singularities: Analysis and examples,” J. Lightwave Technol. 30(16), 2675–2683 (2012). [CrossRef]

52.

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53.

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OCIS Codes
(250.5300) Optoelectronics : Photonic integrated circuits
(130.4815) Integrated optics : Optical switching devices
(250.5403) Optoelectronics : Plasmonics
(080.6755) Geometric optics : Systems with special symmetry

ToC Category:
Integrated Optics

History
Original Manuscript: July 11, 2013
Revised Manuscript: August 22, 2013
Manuscript Accepted: August 26, 2013
Published: September 6, 2013

Citation
Anatole Lupu, Henri Benisty, and Aloyse Degiron, "Switching using PT symmetry in plasmonic systems: positive role of the losses," Opt. Express 21, 21651-21668 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-18-21651


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References

  1. C. M.  Bender, S.  Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998). [CrossRef]
  2. C. M.  Bender, “Making Sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70(6), 947–1018 (2007). [CrossRef]
  3. A.  Ruschhaupt, F.  Delgado, J. G.  Muga, “Physical realization of PT -symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38(9), L171–L176 (2005). [CrossRef]
  4. M.  Greenberg, M.  Orenstein, “Irreversible coupling by use of dissipative optics,” Opt. Lett. 29(5), 451–453 (2004). [CrossRef] [PubMed]
  5. M.  Kulishov, J. M.  Laniel, N.  Bélanger, J.  Azaña, D. V.  Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Express 13(8), 3068–3078 (2005). [CrossRef] [PubMed]
  6. M.  Kulishov, J. M.  Laniel, N.  Bélanger, D. V.  Plant, “Trapping light in a ring resonator using a grating-assisted coupler with asymmetric transmission,” Opt. Express 13(9), 3567–3578 (2005). [CrossRef] [PubMed]
  7. C. M.  Bender, D. C.  Brody, H. F.  Jones, B. K.  Meister, “Faster than Hermitian Quantum Mechanics,” Phys. Rev. Lett. 98(4), 040403 (2007). [CrossRef] [PubMed]
  8. R.  El-Ganainy, K. G.  Makris, D. N.  Christodoulides, Z. H.  Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007). [CrossRef] [PubMed]
  9. S.  Klaiman, U.  Günther, N.  Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101(8), 080402 (2008). [CrossRef] [PubMed]
  10. K. G.  Makris, R.  El-Ganainy, D. N.  Christodoulides, Z. H.  Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008). [CrossRef] [PubMed]
  11. A.  Guo, G. J.  Salamo, M.  Volatier-Ravat, V.  Aimez, G. A.  Siviloglou, D. N.  Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009). [CrossRef] [PubMed]
  12. O.  Bendix, R.  Fleischmann, T.  Kottos, B.  Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103(3), 030402 (2009). [CrossRef] [PubMed]
  13. T.  Kottos, “Broken symmetry makes light work,” Nat. Phys. 6(3), 166–167 (2010). [CrossRef]
  14. C. E.  Rüter, K. G.  Makris, R.  El-Ganainy, D. N.  Christodoulides, M.  Segev, D.  Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010). [CrossRef]
  15. M. C.  Zheng, D. N.  Christodoulides, R.  Fleischmann, T.  Kottos, “PT optical lattices and universality in beam dynamics,” Phys. Rev. A 82(1), 010103 (2010). [CrossRef]
  16. J.  Čtyroký, V.  Kuzmiak, S.  Eyderman, “Waveguide structures with antisymmetric gain/loss profile,” Opt. Express 18(21), 21585–21593 (2010). [CrossRef] [PubMed]
  17. S.  Longhi, “Invisibility in PT -symmetric complex crystals,” J. Phys. A: Math. Theor. 44(48), 485302 (2011). [CrossRef]
  18. A. E.  Miroshnichenko, B. A.  Malomed, Yu. S.  Kivshar, “Nonlinearly-PT -symmetric systems: spontaneous symmetry breaking and transmission resonances,” Phys. Rev. A 84(1), 012123 (2011). [CrossRef]
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