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Journal of the Optical Society of America B

Journal of the Optical Society of America B

| OPTICAL PHYSICS

  • Editor: Henry van Driel
  • Vol. 29, Iss. 4 — Apr. 1, 2012
  • pp: 818–826
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Confinement and optical properties of the plasmonic inverse-rib waveguide

H. Benisty and M. Besbes  »View Author Affiliations


JOSA B, Vol. 29, Issue 4, pp. 818-826 (2012)
http://dx.doi.org/10.1364/JOSAB.29.000818


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Abstract

Plasmonic inverse-rib optical waveguides, consisting of a high-index inverse rib embedded in low-index medium above a flat metallic surface, are investigated under four aspects: (i) the optimal angle θ of the rib sidewall for tight modal confinement is assessed, (ii) the effect of the geometric parameters and the wavelength on propagation losses is given, (iii) we use a 3D simulation to assess how well light from an emitting dipole is captured by such a tightly guiding structure, and (iv) we show that for two such parallel hybrid waveguiding systems, when one of them has added gain, we have a plasmonic version of the PT-symmetric waveguide arrangement, and we additionally show that complex gain is needed to restore a truly exceptional point in its propagation constant evolution.

© 2012 Optical Society of America

1. INTRODUCTION

Plasmonic waveguides capitalize on the superior confinement properties of metal confinement to attain very small effective areas. Such miniaturization is interesting either for implementing photonic functions at the nanoscale (e.g., in nanobiophotonics) or for improving fundamental light–matter interaction processes such as spontaneous emission (SpE). Plasmonic lasers, so-called “spasers,” emerged in 2009 [1

1. R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009). [CrossRef]

,2

2. M. A. Noginov, G. Zhul, 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, 1110–1112 (2009). [CrossRef]

]. In one preferred form, they exploit the special guidance that arises between a metallic plane and a high-index channel, separated by a thin lower index gap [3

3. R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photon. 2, 496–500 (2008). [CrossRef]

]. This is a class of hybrid plasmonic waveguiding, with assistance from high-index dielectric confinement and thus mitigation of the loss/confinement compromise [3

3. R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photon. 2, 496–500 (2008). [CrossRef]

5

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

].

We have recently proposed, inspired by this spaser achievement, a hybrid geometry shown in Fig. 1(a) that we call the plasmonic inverse-rib optical waveguide (PIROW) [5

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

]. An important aspect of our PIROW proposal is that the metal can remain unpatterned, while the fabrication technology can be made very simple, direct, and versatile through exposure and spin-coating of known available resists, say polymethyl methacrylate. In a slightly simplified view, a low-index (L) resist is coated, exposed, and developed so as to form a trench, but not traversing down to the metal. Then a high-index resist (H) can nearly planarize the structure and fill the trench. A tightly confined plasmonic mode then appears at the tip of the trench cross section, typically with TM-like polarization. It is rapidly evanescent in both xy directions of the plane. The metal/H/L sequence helps confining the large field region to the L gap below the inverse ridge. This configuration increases crucial figures of merits such as the Purcell factor, and is certainly advantageous to elaborate devices with subwavelength resolution. Our proposed implementation also makes it easy to modulate the guide parameters by structuring only the low-index medium. It can easily be exploited to take advantage of the rich background of distributed Bragg reflector (DBR) structures or of 2D photonic crystals.

Currently, hybrid plasmonic waveguide proposals are emerging, using a low-index dielectric between a metal and a waveguiding higher index dielectric [6

6. D. X. Dai and S. L. He, “A silicon-based hybrid plasmonic waveguide with a metal cap for a nano-scale light confinement,” Opt. Express 17, 16646–16653 (2009).

9

9. A. V. Krasavin and A. V. Zayats, “Numerical analysis of long-range surface plasmon polariton modes in nanoscale plasmonic waveguides,” Opt. Lett. 35, 2118–2120 (2010). [CrossRef]

]. They often assume “square” geometries that are derived from simplified processing considerations, such as a preference for vertical etching and silicon-oriented applications. Our purpose, exploiting simpler technology and targeting near-infrared wavelengths for biophotonics, is complementary. So, clearly, a few clever aspects outlined in these proposals could be combined with ours. We believe that the smooth interfaces of the PIROW remain a crucial asset.

Channeling or concentrating a point emitter’s light in a single mode, the so-called β SpE factor, is relevant for applications to sensing, e.g., capturing light with subwavelength resolution from fluorescent probes attached to analytes. It is well documented that this β factor is controlled by the environment-dependent radiative lifetime, the well-known Purcell factor. We therefore calculate the β factor and the Purcell factor at key locations, and we test how much metal proximity quenches radiative emission.

2. ANGLE-DEPENDENT CONFINEMENT

Here, we vary the angle θ of the PIROW rib sidewall. The angle and all notations are defined in Fig. 1(b). The other parameters are the same as in our first study [5

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

], namely nL=1.5 and nH=2.0. A gold surface at 633 nm is chosen, with the indices of Johnson and Christy [24

24. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]

]. Other indices are fixed because they have much weaker spectral variations (polymer, solgel). We use the same 2D vectorial solver developed in-house, as in [5

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

,25

25. J.-M. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley–IEEE, 2002).

], deployed under a MATLAB environment. The thickness of the L index is 90 nm, and the top high-index layer is 40 nm thick outside the rib. The rib width Ltip is taken as 20 nm (as it was in [5

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

], but with θ=15°) wide. The rib height is 80 nm so as to leave a d=10nm low-index gap. In this regime, we have local fields in the tip area mainly determined by electrostatic response to the metal charges with only minimal retardation effects.
Fig. 1. (a) Structure of the PIROW, the red spot being indicative for the confined mode profile and (b) optogeometric parameters of the rib and layer sequence.

The profile along y thus retains the predominance of the low-index peak field region, even for the limit case θ=0° (vertical walls), which provides only weak high-index confinement. The profiles in the lateral x direction, Fig. 2(b), make it clear what is the key compromise for strong confinement. The field is maximally pinched around its central peak for the lowest θ values (say 0°–10°), but the tails are still large. For larger angles, the tails decrease, while the central peak at high field values remains narrow and only broadens for a sidewall angle θ>70°.

An important measure of the degree of confinement is the effective area of the mode. At the present stage, an exact calculation is not critical, and for more clarity [26

26. R. F. Oulton, G. Bartal, D. F. P. Pile, and X. Zhang, “Confinement and propagation characteristics of subwavelength plasmonic modes,” New J. Phys. 10, 105018 (2008). [CrossRef]

], we simply determine the extent Δxα and Δyα of the regions of each chosen axis [Ey(0,y) or Ey(x,d/2)] for which the inequality |Ey|>αEymax is satisfied with a parametric α ratio. We note that the field Ey(0,y) is nearly constant and does not decay much below Eymax across the low-index gap. We simply define the mode area as Sxy(θ,d,α)=ΔxαΔyα. If the ratio α is not too small, Δyα remains essentially the gap width itself, Δyα=d. Hence the x confinement is the main characteristic. Only for relatively low α does Δyα extend above d, in which case our definition is still of great relevance, but a more accurate study could be welcome for details.

These effective area data are gathered in Fig. 2(d). The five curves correspond to the increasing ratio α, namely, α=0.45, 0.5, 0.6, 0.7, and 0.8. The three bottom curves (solid lines) correspond to the regime where Δyα=d. The two top curves (dashed lines) correspond to the regime where Δyα>d. The two corresponding products (Δxα×d) are plotted as a dashed–dotted line to allow for comparison. The main striking point here is that for a not-too-high-ratio, say α=0.60, the effective area is optimally confined typically for θ around 40°. The minimal effective area for the dashed curves (lower α ratio) occurs at a larger angle: first, for this α regime, one looks at the tails of the profile, for which a high effective index is favorable, which comes together with a larger angle θ. And second, for the dashed curves, the decay in the high-index medium plays a role and adds a similar trend [more decay for a larger effective index and a higher θ, Fig. 2(a)].

The typical effective areas for Sxy(θ,d,α) for ratio α=0.6 or 0.7 are 5 to 10 104μm2, which is the equivalent of 30nm×30nm or less.

We checked the influence of the low-index medium, performing a similar study for nL=1.50 instead of 1.40. The effective index curve is indicated by a dashed line in Fig. 2(c). The simple trend, with an offset of Δn0.1 on the low-index/low-angle side suggests that, as far as design is concerned, extrapolating an effective index from such limited data can be effective to orient modeling and material choice in a substantial range.

In Fig. 3, we compare the confinement curves for these two cases by plotting side by side the plot of Fig. 2(d) and an equivalent one for nL=1.40. There are several trends. Focusing first on the Δxα=0.7(θ) for both indices (associated to halved |E|2), we see a small advantage for the lower index, with an area 450nm2 instead of 550nm2. The advantage is more obvious for smaller α, i.e., if we look further into the mode tail. The advantage for Δxα=0.45(θ) in terms of area ΔxαΔyα (dashed lines in the graphs) can be almost a factor of 2, 2500nm2, instead of about 4600nm2. Note also that the optimal confinement angle generally shifts to larger angles for the smaller α values. This is because the mode index is higher. It implies a faster lateral decay in the low-index material, which is, however, appearing clearly only if we look far enough in the tails and not too close to the tip, as the large angles tend to limit the field profile downward curvature around the tip (for a given gap thickness).

Fig. 3. Comparison of confinement for two low-index values differing by Δn=0.1, as a function of the inverse-rib sidewall angle: (a) nL=1.40 and (b) nL=1.50.

3. LOSSES OF A PLASMONIC INVERSE-RIB WAVEGUIDE

The 633 nm He–Ne laser wavelength is convenient in several experiments. We hence start by looking in Fig. 4 at the imaginary effective index as a function of PIROW angle for this case and for the index nL equal to 1.40. The trend is quite smooth, increasing gently at larger angles. The large angle limits the tip effect and, thus, forces the field to interact with the metal on a broader x range. Note that the trend is similar to the trend of Δxα=0.9(θ), suggesting that the key factor to minimize losses is to get the tightest lateral (x) confinement, whatever length the field tail has along y—Fig. 2(b). At the typical 45° angle where confinement is optimal under broad criteria, the losses are 10%–15% above their minimum value (obtained at vanishing θ).
Fig. 4. Losses as a function of the inverse-rib angle at 633 nm, given through the imaginary effective index. Note however the substantial real effective index variation on the same range in Fig. 2.

In both Figs. 5 and 6, the inset translates the imaginary index into an absorption length and compares the result to a penetration length of a Bragg mirror, calculated with the naive formula for the indicated index contrast (as done in [5

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

]) Δneff. A shallower angle brings some more margin for Au (Fig. 5) to get Bragg reflection at 633 nm for instance. But Ag gives the largest margin, allowing modest contrasts to provide Bragg reflections down to roughly λ=550nm.
Fig. 5. Losses as a function of wavelength for a 45° PIROW and a 15° PIROW as indicated, using gold. Inset, the associated absorption lengths that become larger than a typical penetration length in a periodic system of given modulation Δneff at wavelengths between 600 and 700 nm.
Fig. 6. (a) Effective index versus wavelength for a 45° PIROW with Au (top curve) or Ag (lower curve) and (b) same as Fig. 5, but comparing Ag (dashed line) and Au (solid line) losses Im(neff). The inset transforms the same comparison into an absorption length one, as in Fig. 5. Note that Ag may behave well at wavelengths as short as 550–600 nm.

We wish in the near future to calculate what degree of reflection is provided by various shape modulations and how radiation losses due to scattering at transitions can be minimized, a familiar issue of the resonant waveguide grating studies (see, for instance, the collective COST 268 study [27

27. J. Ctyroky, S. Helfert, R. Pregla, P. Bienstman, R. Baets, R. De Ridder, R. Stoffer, G. Klaasse, J. Petracek, P. Lalanne, J. P. Hugonin, and R. M. De La Rue, “Bragg waveguide grating as a 1D photonic band gap structure: COST 268 modelling task,” Opt. Quantum Electron. 34, 455–470 (2002).

]). Alternatives are to vary the gap width, tip width, or tip angle, or a combination of these. Fabrication constraints do not allow a 3D management of the shape, so the dose variations in e-beam lithography should result in a combination of the variation of the three parameters.

4. LIGHT CONCENTRATION CAPABILITIES

The enhanced confinement of hybrid plasmonic waveguides translates into attracting possibilities to channel SpE of emitters situated in them or nearby. The overall enhancement of electromagnetic channels for SpE can be seen through the Purcell factor, Fp=τ1/τo1, the ratio of SpE rate in the structure τ1 given as an inverse lifetime, to that in the bulk material where it actually lies, τo1. In our case, we expect the guided PIROW mode to be the main actor to enhance SpE rate. A high Purcell factor represents a favorable factor for good light collection only if it does not correspond to extra dissipation in metal, but to actual collection in a guided mode. The associated measure of the degree of collection, more precisely of its ratio to all radiation channels in a single mode, is the “SpE factor,” commonly denoted as the β factor.

Here the distinction among channels may look somewhat ambiguous because in the long-range (several micrometers), all the hybrid mode energy ends up being dissipated in the metal. However, for short lengths, about 1 μm, light channeled in a guided mode starts to be reasonably distinct in space from that emitted elsewhere, while absorption of the mode has still been very modest. We used a 3D finite-element method, which was developed in-house using a LU solver [25

25. J.-M. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley–IEEE, 2002).

]. It includes an electric dipole source to measure these effects. As for geometry, we investigate the emission for a dipole situated at two sets of positions: the plane at midheight of the low-index dielectric gap and the vertical symmetry plane. In both cases, only a quarter of the real space can be modeled, thanks to the two vertical structure symmetry planes. We only use a vertical dipole, even at x0, but we force solutions to be electrically odd or even at the xz symmetry plane and combine the resulting fields adequately to get the full solution (neither odd nor even in general). We model a system with the following parameters:

The gap height is d=10nm, tip width is Ltip=20nm, rib height is 80 nm, and the top high-index layer is 40 nm thick. The high index itself is nH=2.0, and the low index is nL=1.4 to favor a tighter mode profile fitting in a small simulation box. We study the case of modest loss, which means larger wavelength, but we go no further than λ=700nm so that far-field effects are reached as closely as possible for the same reasons of limited simulation box. The domain of study, i.e., four times the computational domain given the symmetries, is 1.0μm×1.4μm (along the guide axis)×0.42μm (height), comprising 0.2 μm of gold in height. We also make use of an embedded subdomain of collection with reduced size to check that the emitted power is no more in the near-field regime but essentially in the propagation regime at the edge of the large domain. The expected difference between this subdomain and the nominal one is then essentially the weak absorption in the waveguide, only a few percent, whereas nonguided near-field components would still feed the metal’s Joule losses. This subdomain is here 0.8μm×1.1μm (along the guide axis)×0.42μm (height).

Fig. 7. Purcell factor FP for a vertical dipole at wavelength λ=700nm in a PIROW: (a) dipole at variable height, either in the symmetry plane x=0 or along the rib edge and (b) dipole lying along x at height z=d, the height of the tip bottom. Insets, scanned positions.

To further study the situation of emitters at the tip, we show in Fig. 7(b) the Purcell factor along the line at height z=d, still for a vertical dipole. In agreement with the above trends, it increases slightly when going to the tip’s very corner. Then it increases when going sideways to the low index, but it decays further away from the tip. The evolution can be explained by considering the component of the dipole normal to the 45° tilted interface, thus a large component for which the situation amounts to a dipole normal to a surface between two dielectrics. In such a case, the Purcell factor counterintuitively peaks on the low-index side and drops on the high-index side, essentially because the dielectric image dipoles are in phase and out of phase respectively. This is documented in detail since, e.g., Lukosz and Kunz’s analytical work [28

28. W. Lukosz and R. E. Kunz, “Fluorescence lifetime of magnetic and electric dipoles near a dielectric interface,” Opt. Commun. 20, 195–199 (1977). [CrossRef]

]. Here we have a corner, so the dielectric image effect is not as perfect, but given the 135° inner angle of the high dielectric, the picture holds. Such effects might also be invoked to explain the low Purcell factor Fp<1 at the rib’s basis, the extreme right of Fig. 7(a).

Fig. 8. Light capture efficiency by a PIROW on both sides for a dipole lying along the vertical direction at x=0. Insets show (left) the scanned position and the nominal and (right) the two “nested” collection boxes used to check that extraction is in the far-field regime.

For a dipole lying in the low-index gap, close to the metal, this capture is weak because of the metal proximity, typically 30% on the average. The nearest emitter locations in the low-index gap tend to feed the metal almost exclusively, thus mitigating the interest of the larger Purcell factor. After a noticeable change of slope upon crossing upward the L/H interface, the collected fraction still grows, and a good compromise typically arises 10 nm above the tip interface, 20 nm above the metal, where the Purcell factor is still reasonably large (around 6–7). Thus, we confirm in our novel structure the trend, now well-established [29

29. R. Esteban, T. V. Teperik, and J. J. Greffet, “Optical patch antennas for single photon emission using surface plasmon resonances,” Phys. Rev. Lett. 104, 026802 (2010). [CrossRef]

,30

30. G. W. Ford and W. H. Weber, “Electromagnetic interaction of molecules with metal surfaces,” Phys. Rep. 113, 195–287 (1984). [CrossRef]

], according to which the optimal distance to capture an emitter’s light enhanced near a metallic lies in the range 1020nm for the range of wavelength investigated.

5. COUPLED GUIDES WITH GAIN AND LOSS: ATTAINMENT OF EXCEPTIONAL POINT

Fascinating avenues have been recently opened regarding optical systems with PT symmetry and fields which are symmetry-breaking in these structures [11

11. 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, 3567–3578 (2005). [CrossRef]

21

21. C. T. West, T. Kottos, and T. Prosen, “PT-symmetric wave chaos,” Phys. Rev. Lett. 104, 054102 (2010).

]. Systems with balanced gain and losses with a symmetry plane that exchanges them provide a realization of PT symmetry. The eigenvalues of these structures (wave vectors for a frequency in real space) can remain real in the presence of gain, up to an exceptional point where complex eigenvalues appear, with a singular behavior, quadratic instead of linear versus gain around this point. The possibilities explored until recently did not comprise plasmonic structures, or more precisely only “passive PT-symmetric” structures with metal [17

17. A. Guo, G. J. Salamo, R. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).

], but with no or little gain, thus missing the exceptional point. Plasmonic structures have, in practice, fixed losses. We recently pointed out that by proper adjustment of coupling constant to losses in a pair of coupled guide, one of which has such fixed losses, a singular point with the transition from zero to nonzero imaginary eigenvectors could occur, with a potential for enhanced modulation [22

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

]. We used the so-called LRSPP (long-range surface plasmon polariton) supported by thin gold films and coupled to dielectric waveguides [23

23. 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, 015002 (2009). [CrossRef]

] to demonstrate the plasmonics+PT-symmetry-breaking combination. We also introduced coupled PIROW waveguides, but we did not discuss actual simulations with gain in one of them.

We discuss a PIROW-based version of such systems here. We first discuss a basic configuration that might naturally arise if localized photopumping is performed, which is to pump only the inverse rib of the gain-carrying waveguide. More realistically, we also include pumping in the shaded region of Fig. 9(a) above the inverse rib, within circular boundaries centered on the rib apex, as would qualitatively occur with reasonably localized photopumping of the active high-index material.

Our current understanding of this discrepancy with an ideal singularity is that instead of having a real coupling constant as initially assumed in coupled mode theory (CMT) of the coupled guides, we have a complex coupling constant, which becomes more complex as gain grows on the gain-carrying waveguide. More precisely, what counts in solving the CMT model is the product of the two coupling constants K=κabκba. And it is this product that becomes complex when the system is not a perfect PT-symmetric system, notably due to geometric reasons. Here metallic losses are present on both sides and dielectric gain on one side only, making the situation quite nonideal with respect to PT symmetry.

Prompted by this general idea, we attempted to “heal” the coupling constant to make it real by introducing a compensating region. A change of the real part of the index was added so as to tune the ingredients of K=κabκba. Furthermore, because K becomes complex due to the variable added gain, the index modification has to be proportional to this gain; in other words, a complex index variation has to be introduced instead of gain, with a given complex angle. We chose here to modify only the nH region above the inverse rib, shaded in Fig. 9(a), and not to modify the inverse-rib region, i.e., to leave it with the same index Re(n)=(nH) and Im(nH) varying as previously, for simplicity. Based on the above discussion, we represent the effect of pumping as an adjustable combination of the imaginary and real part to the complex index, of the form (n˜H)=(nH)+ig(1+iα), α being here a coefficient <1. Such a modulation could arise in practice from electro-optical effects if a different dye or a different pumping wavelength is used in this area.

The difficulty of imposing a particular change of both the real and imaginary index in a prescribed and abruptly bounded region can, in practice, be replaced by two feasible tasks: (i) obtainment of a proper knowledge of all the electro-optical properties as a function of the local pumping intensity on a model system and (ii) choice of a smooth distribution of excitation intensity (optical pumping) across the waveguide section, as can be provided by few optogeometric parameters (central beam position, peak wavelength, and use of local absorption and field concentration/cancellation effects).

6. CONCLUSIONS

In this study, we have addressed issues that arose from our PIROW proposal [5

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

], a hybrid waveguide with limited losses that can, in principle, be elaborated with simple technology, including a number of basic functions. The first part of the study has clarified that the optimum PIROW angle is around 35°–50°, which can be understood as an essentially electrostatic effect: a sharper angle does not bring near the metal enough high-index material to shrink the mode, thus a rather deconfined mode is formed; conversely, a larger angle delocalizes the mode laterally.

Finally, we have capitalized on our recent approach [22

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

] to fully consider the benefit of mimicking perfect PT-symmetric systems with two coupled waveguides, one with the fixed plasmonic losses and the other having added optical gain in and around the inverse-rib region. Without precautions, the singularity appears to be dangerously smoothed, which means that the capability to transform a small gain variation into a large difference in the coupled waveguide transfer function could seem jeopardized. However, a singularity “healing” mechanism was proposed, whereby pumping is assumed to induce a complex index change, typically ig(10.174i) instead of the pure imaginary contribution ig in our case. It could restore the singularity nearly entirely. Work is in progress to provide a more general basis to this mechanism.

ACKNOWLEDGMENTS

We thank Yan Chen for useful discussions on PT-symmetric systems during his master’s first year internship.

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S. Klaiman and L. S. Cederbaum, “Non-Hermitian Hamiltonians with space–time symmetry,” Phys. Rev. A 78, 062113 (2008).

13.

S. Klainman, U. Günther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).

14.

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

15.

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

16.

J. J. Chen, Z. Li, S. Yue, and Q. H. Gong, “Hybrid long-range surface plasmon-polariton modes with tight field confinement guided by asymmetrical waveguides,” Opt. Express 17, 23603–23609 (2009).

17.

A. Guo, G. J. Salamo, R. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).

18.

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

19.

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

20.

A. A. Sukhorukov, Z. Xu, and Y. Kivshar, “Nonlinear breaking of PT symmetry in coupled waveguides with balanced gain and loss,” in Nonlinear Photonics, OSA Technical Digest (CD) (Optical Society of America, 2010), paper NTuC19.

21.

C. T. West, T. Kottos, and T. Prosen, “PT-symmetric wave chaos,” Phys. Rev. Lett. 104, 054102 (2010).

22.

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

23.

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, 015002 (2009). [CrossRef]

24.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]

25.

J.-M. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley–IEEE, 2002).

26.

R. F. Oulton, G. Bartal, D. F. P. Pile, and X. Zhang, “Confinement and propagation characteristics of subwavelength plasmonic modes,” New J. Phys. 10, 105018 (2008). [CrossRef]

27.

J. Ctyroky, S. Helfert, R. Pregla, P. Bienstman, R. Baets, R. De Ridder, R. Stoffer, G. Klaasse, J. Petracek, P. Lalanne, J. P. Hugonin, and R. M. De La Rue, “Bragg waveguide grating as a 1D photonic band gap structure: COST 268 modelling task,” Opt. Quantum Electron. 34, 455–470 (2002).

28.

W. Lukosz and R. E. Kunz, “Fluorescence lifetime of magnetic and electric dipoles near a dielectric interface,” Opt. Commun. 20, 195–199 (1977). [CrossRef]

29.

R. Esteban, T. V. Teperik, and J. J. Greffet, “Optical patch antennas for single photon emission using surface plasmon resonances,” Phys. Rev. Lett. 104, 026802 (2010). [CrossRef]

30.

G. W. Ford and W. H. Weber, “Electromagnetic interaction of molecules with metal surfaces,” Phys. Rep. 113, 195–287 (1984). [CrossRef]

31.

Z. Han, A. Y. Elezzabi, and V. Van, “Experimental realization of subwavelength plasmonic slot waveguides on a silicon platform,” Opt. Lett. 35, 502–504 (2010). [CrossRef]

OCIS Codes
(130.3120) Integrated optics : Integrated optics devices
(230.7380) Optical devices : Waveguides, channeled
(240.6680) Optics at surfaces : Surface plasmons
(250.3680) Optoelectronics : Light-emitting polymers
(350.4238) Other areas of optics : Nanophotonics and photonic crystals
(130.4110) Integrated optics : Modulators

ToC Category:
Plasmonics

History
Original Manuscript: November 30, 2011
Manuscript Accepted: January 2, 2012
Published: March 30, 2012

Citation
H. Benisty and M. Besbes, "Confinement and optical properties of the plasmonic inverse-rib waveguide," J. Opt. Soc. Am. B 29, 818-826 (2012)
http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-29-4-818


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References

  1. R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009). [CrossRef]
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  3. R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photon. 2, 496–500 (2008). [CrossRef]
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  5. H. Benisty and M. Besbes, “Plasmonic inverse rib waveguiding for tight confinement and smooth interface definition,” J. Appl. Phys. 108, 063108 (2010). [CrossRef]
  6. D. X. Dai and S. L. He, “A silicon-based hybrid plasmonic waveguide with a metal cap for a nano-scale light confinement,” Opt. Express 17, 16646–16653 (2009).
  7. M. Fujii, J. Leuthold, and W. Freude, “Dispersion relation and loss of subwavelength confined mode of metal-dielectric-gap optical waveguides,” IEEE Photon. Technol. Lett. 21, 362–364 (2009). [CrossRef]
  8. X. Y. Zhang, A. Hu, J. Z. Wen, T. Zhang, X. J. Xue, Y. Zhou, and W. W. Duley, “Numerical analysis of deep sub-wavelength integrated plasmonic devices based on semiconductor-insulator-metal strip waveguides,” Opt. Express 18, 18945–18959 (2010).
  9. A. V. Krasavin and A. V. Zayats, “Numerical analysis of long-range surface plasmon polariton modes in nanoscale plasmonic waveguides,” Opt. Lett. 35, 2118–2120 (2010). [CrossRef]
  10. J. Ctyroky, V. Kuzmiak, and S. Eyderman, “Waveguide structures with antisymmetric gain/loss profile,” Opt. Express 18, 21585–21593 (2010). [CrossRef]
  11. 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, 3567–3578 (2005). [CrossRef]
  12. S. Klaiman and L. S. Cederbaum, “Non-Hermitian Hamiltonians with space–time symmetry,” Phys. Rev. A 78, 062113 (2008).
  13. S. Klainman, U. Günther, and N. Moiseyev, “Visualization of branch points in PT-symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
  14. K. G. Makris, R. El-Gaininy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008). [CrossRef]
  15. O. Bendix, R. Fleischmann, T. Kottos, and B. Shapior, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009). [CrossRef]
  16. J. J. Chen, Z. Li, S. Yue, and Q. H. Gong, “Hybrid long-range surface plasmon-polariton modes with tight field confinement guided by asymmetrical waveguides,” Opt. Express 17, 23603–23609 (2009).
  17. A. Guo, G. J. Salamo, R. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
  18. T. Kottos, “Broken symmetry makes light work,” Nat. Phys. 6, 166–167 (2010). [CrossRef]
  19. C. E. Rüter, K. G. Makris, R. El-Gaininy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys. 6, 192–195 (2010). [CrossRef]
  20. A. A. Sukhorukov, Z. Xu, and Y. Kivshar, “Nonlinear breaking of PT symmetry in coupled waveguides with balanced gain and loss,” in Nonlinear Photonics, OSA Technical Digest (CD) (Optical Society of America, 2010), paper NTuC19.
  21. C. T. West, T. Kottos, and T. Prosen, “PT-symmetric wave chaos,” Phys. Rev. Lett. 104, 054102 (2010).
  22. H. Benisty, A. Degiron, A. Lupu, A. De Lustrac, S. Chenais, S. Forget, M. Besbes, G. Barbillon, A. Bruyant, S. Blaize, and G. Lerondel, “Implementation of PT symmetric devices using plasmonics: principle and applications,” Opt. Express 19, 18004–18019 (2011). [CrossRef]
  23. 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, 015002 (2009). [CrossRef]
  24. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]
  25. J.-M. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley–IEEE, 2002).
  26. R. F. Oulton, G. Bartal, D. F. P. Pile, and X. Zhang, “Confinement and propagation characteristics of subwavelength plasmonic modes,” New J. Phys. 10, 105018 (2008). [CrossRef]
  27. J. Ctyroky, S. Helfert, R. Pregla, P. Bienstman, R. Baets, R. De Ridder, R. Stoffer, G. Klaasse, J. Petracek, P. Lalanne, J. P. Hugonin, and R. M. De La Rue, “Bragg waveguide grating as a 1D photonic band gap structure: COST 268 modelling task,” Opt. Quantum Electron. 34, 455–470 (2002).
  28. W. Lukosz and R. E. Kunz, “Fluorescence lifetime of magnetic and electric dipoles near a dielectric interface,” Opt. Commun. 20, 195–199 (1977). [CrossRef]
  29. R. Esteban, T. V. Teperik, and J. J. Greffet, “Optical patch antennas for single photon emission using surface plasmon resonances,” Phys. Rev. Lett. 104, 026802 (2010). [CrossRef]
  30. G. W. Ford and W. H. Weber, “Electromagnetic interaction of molecules with metal surfaces,” Phys. Rep. 113, 195–287 (1984). [CrossRef]
  31. Z. Han, A. Y. Elezzabi, and V. Van, “Experimental realization of subwavelength plasmonic slot waveguides on a silicon platform,” Opt. Lett. 35, 502–504 (2010). [CrossRef]

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