Structural slow waves: parallels between photonic crystal and

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Structural slow waves: parallels between photonic crystal and plasmonic waveguides Philippe Lalanne, Stéphane Coudert, guillaume duchateau, Stefan Dilhaire, and Kevin Vynck ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.8b01337 • Publication Date (Web): 26 Nov 2018 Downloaded from http://pubs.acs.org on November 27, 2018

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ACS Photonics

Structural slow waves: parallels between photonic crystal and plasmonic waveguides Philippe Lalanne*1, Stéphane Coudert2, Guillaume Duchateau2, Stefan Dilhaire3, Kevin Vynck1 1LP2N, Institut d’Optique Graduate School, CNRS, Univ. Bordeaux, 33400 Talence, France 2Université de Bordeaux-CNRS-CEA, Centre Lasers Intenses et Applications, UMR 5107, 351 Cours de la Libération, 33405 Talence, France 3Laboratoire Onde et Matière d’Aquitaine (LOMA), UMR 5798, CNRS-Université de Bordeaux, 33400 Talence, France. * E-mail: [email protected] Abstract: The slowdown of propagating electromagnetic waves by engineering waveguide structures is receiving considerable attention in nanophotonics. Two types of structures are independently investigated, namely diffraction-limited photonic-crystal waveguides and highly-confined plasmonic waveguides. Here we propose a unified viewpoint on slow waveguide structures by analyzing both approaches within a single theoretical framework. This allows us to clarify the physical mechanisms underlying slowness – thereby highlighting their inherent differences – and to remove some inaccurate ideas on the topic. We further discuss the strengths and weaknesses of plasmonic and photonic slow waveguides, assess the technical and scientific challenges recently solved and critically identify those that still stand in the way. Keywords: slow waveguide, slow surface plasmon, slow photonic-crystal waveguide, LDOS enhancement, roughness-induced backscattering

Introduction

In general, the velocity at which a pulse of light propagates through a medium (group velocity) ―1 is given by 𝑣𝑔 = 𝑐𝑛𝑔 ―1 = 𝑐(𝑛 + 𝜔𝑑𝑛 𝑑𝜔) , where 𝑐 is the velocity of light in vacuum, 𝜔 is the light angular frequency, 𝑛 is the refractive index of the medium, and 𝑛𝑔 denotes the group index. Theoretically, there is no fundamental limit on the value to which light may be slowed down. Very low propagation velocities (17 m/s)1 and even stopped light2 have been reported in ultra-cold atomic gases. These ultra-slow regimes are reached when the velocity of light pulses can be described fully in terms of an extremely frequency-dependent refractive index of the material, offering a sharp dip in the absorption or gain spectrum in a narrow spectral region. Simple saturation effects can lead to such behavior, as well as more advanced effects such as electromagnetically induced transparency.3,4 Slowness induced by material dispersion (𝑑𝑛 𝑑𝜔) will not be discussed further in this Perspectives article. It is believed that if slow light is delivered at a micro or nanoscale on-chip, for instance in a fully integrated configuration compatible with standard technological platforms, science and technology will be strongly impacted. This route is mainly followed by relying on structural dispersion. It refers to situations in which the velocity of light pulses is modified by structuring and/or assembling matter at the micro or nanoscale. The group velocity is no longer related to

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the refractive index 𝑛 of a medium, but to an effective index 𝑛eff that depends on 𝜔. In marked contrast with material slowness, structural slowness is accompanied by a strong electromagnetic field enhancement. There are essentially two schemes for structural slowness: (i) photonic-crystal waveguides, e.g. single-row-missing photonic-crystal waveguides or coupled optical resonators5, and (ii) plasmonic waveguides, e.g. metal/insulator/metal (MIM) or insulator/metal/insulator (IMI) heterostructures, metal nanowires, or hyperbolic materials.6 We will indifferently use the terms ‘slow photon’ or ‘slow photonic waveguide’ for dielectric structures and consistently ‘slow plasmon’ or ‘slow plasmonic waveguide’ for metallic ones, and ‘slow wave’ or ‘slow waveguide’ in general terms. Photonic-crystal and plasmonic slow waves share many common features, starting from the field-enhancement property, and are studied, quite naturally, for the same applications in science and technology, e.g. light-harvesting,7 biosensing,8 nonclassical light sources,9,10 lasing,5,11-14 quantum information,15,16 nonlinearities and switching,17-20 integrated photonic circuits21,22 … Photonic crystals are definitely more advantageous for applications such as delay lines, buffering or bit memory,23 whereas plasmonic structures are considered to be more promising for nanoscale applications.24,10,25-29

Fig. 1 – Structural slow electromagnetic waves. (a) Slowness at the microscale with photonic periodic waveguides. It is achieved for lattice periods that are comparable to the wavelength and the transversal confinement is limited by the diffraction. The slow-mode intensity pattern is periodic along the direction of propagation. (b) Slowness at the nanoscale with plasmonic translation-invariant waveguides. Slow plasmons are achieved with heterostructures having at least one transverse dimension much smaller than the wavelength. The slow-mode intensity pattern is uniform along the direction of propagation. (courtezy S.I. Bozhevolnyi).

In photonics, the energy is purely electromagnetic. Slowness originates from waves that bounce back and forth as they propagate through a periodic structure (see Fig. 1a for an illustration). It immediately appears that a phase matching condition should be satisfied and that the characteristic dimension of the structure, i.e. the period a, should be comparable to the wavelength λ, i.e. 𝑎 ≈ λ (2𝑛). Quite differently, slowness in plasmonics does not necessarily require periodic back-reflections and can be implemented at deep subwavelength scales with translational-invariant waveguides.30-34 Slow surface plasmons result from a coupling of electromagnetic fields with the free electrons of metals. As a result, slow plasmons appear when the characteristic dimensions are much smaller than the wavelength, for instance with a MIM dielectric gap (see Fig. 1b for an illustration). Consequently, contrary to photonic slow light that occurs in a quite narrow frequency range with effective mode areas of the order of λ2, slow plasmons are inherently broadband with deep-subwavelength mode areas.


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The literature on structural slow waves appears to be disjointed between slow plasmons and slow photons. Review articles, wherein applications of slow waves are analyzed and discussed, 16,19,5,18,22,35,15,36,6 are available for each individual scheme, but to our knowledge, there is no comprehensive overview of structural slow waves in general. Our primary motivation is to address this shortcoming. By treating absorption loss for plasmons as a perturbation, the same theoretical approach, notably the Lorentz reciprocity theorem in its conjugate form, applies indifferently to each scheme. The emergence of a “unified” vision of both schemes is additionally favored by the systematic reference to a single and common physical parameter to quantify the slowness, the group index 𝑛𝑔. Section 2 is concerned with the study of the emblematic example of gap modes of MIM waveguides with classical electrodynamic models. The derivation is found in many textbooks on electrodynamics. Our prime interest is to look more specifically to what is happening for small gaps when the group velocity vanishes and to introduce, with a concrete example, many ingredients that will be discussed afterwards more abstractly. Although the results are well known, it is recommended that potential readers do not jump over this Section. Section 3 is devoted to the basics of the electrodynamics of slow modes. We review the two physical mechanisms underlying slowness and analyze their inherent differences. First, we clarify how ultraslow transport is achieved by associating two counter-propagative power flows. It is generally admitted that the two power flows exactly balance at vanishing speeds. For instance, for slow plasmons, owing to the negative contribution, the power flow in the metal is expected to balance the positive-permittivity dielectric cladding layers.6 We revisit this widely admitted picture for photons and plasmons, showing that the power flows are not perfectly balanced and that the slight difference is important, as for instance it allows perfect adiabatic tapering. Then, we derive the scaling laws with 𝑛𝑔 of the electric and magnetic fields of slow modes, showing that they are considerably different for plasmons and photons. Their knowledge is very important since many physical effects in the linear and nonlinear regimes depend on how the electric field increases with 𝑛𝑔. Section 4 highlights the advantages and drawbacks of plasmonic and photonic slowness. It provides a comprehensive picture of their inherent forces and limitations, including the bandwidth, the attenuation due to field enhancement and fabrication imperfections (roughness, absorption …), which are known to considerably reduce the propagation distance of slow modes in realistic structures.37,31 We also try to understand why slow plasmons seem to promise much for reaching ultraslow regimes and even possibly for stopping light,6 whereas imperfections are known to ruin any hope of observing ultraslow transport with photonic waveguides.37,38 Section 5 addresses the issue of engineering the local density of states (LDOS) with slow waveguides on the illustrative and important example of light emission. Slowness is indeed attached to field enhancement and may thus be used to boost light emission. After a rapid analysis of light emission in slow photonic waveguides, which is nowadays well understood, we focus on emission in slow plasmonic waveguides. The analysis starts with the textbook case of a single metal/insulator (MI) interface, for which we highlight the role of slow plasmons for quenching the emission. The case of MIMs is considered afterwards. We evidence that the large LDOS of the slow gap plasmon mode for tiny insulator gaps counterbalance quenching and further discuss how this effect may be exploited for optimizing the efficiency of photon/plasmon sources. Section 6 finally points out some perspectives for the use of slow waveguides in devices and summarizes the work.


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In the following, it might appear that the Ohmic loss is somewhat neglected. For instance, absorption is disregarded in Sections 2 and 3 and is treated as a perturbation in Section 4. There is a good reason for not rooting complex permittivities directly from the beginning into Maxwell’s equations. This simplifies the mathematical treatment of the next Sections and allows us to use exactly the same Hermitian theoretical formalism to analyze photons and plasmons, making the intercomparison more quantitative. We would like to emphasize that (i) the role of Ohmic losses will be discussed whenever they impact the reasoning, (ii) the perturbation treatment of Section 4 is exact for translation invariant waveguides, and exact numerical values will be provided for the main formula. The importance of Ohmic losses, which are critical and sometimes underestimated in the literature, is therefore not overlooked here.

Slow gap plasmons in metal/insulator/metal waveguides

In this Section, we study the properties of slow gap modes guided in MIM hetero-structures with tiny gaps employing a classical electrodynamic model. Our prime intention is to introduce, on a concrete example, the concepts and electromagnetic formalisms, e.g. complex group indices, antiparallel power flows, Lorentz reciprocity theorem, which will be used further at a more abstract level in the following Sections. A second motivation is to have a complete analysis of an important element which plays a key role in nanophotonics, e.g. for energy transport though metal slits or grooves in metallic thin films, artificial magnetism at optical frequency, anomalous dispersion in hyperbolic media. MIM heterostructures support both transverse-electric (TE) and transverse-magnetic (TM) gap modes.30,32 TE-modes may propagate over distances that may even exceed decay lengths observed for TM-modes. Here, we are concerned by the fundamental gap-plasmon TM mode (see Fig. 2), which does not exhibit cutoff and offers slow propagation as the gap width becomes much smaller than the wavelength.

Figure 2. Symmetric gap plasmon mode of MIM waveguides translationally invariant along the 𝑦 axis. (a) 𝑧 is the direction of propagation and 𝑥 denotes the transverse direction. 𝜀𝑚 < 0 and 𝜀𝑑 > 0 are the dielectric permittivities of the metal and insulator. As the gap width 𝑔 becomes much smaller than the wavelength, the mode has no cutoff and the propagation constant 𝛽 diverges as 𝑔 ―1. (b) Group index for 𝜀𝑑 ∂𝜀𝑚

= 2.25, 𝜀𝑚 = ― 28.0 + 1.5𝑖 and ∂𝜆 = ―84.23 + 2.63𝑖 (silver at 𝜆 = 800 nm39) in the limit of small gaps (i.e. large 𝛽’s). Black circles are numerical data obtained by computing the complex-valued effective index 𝛽/𝑐 with the aperiodic rigorous


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(

coupled-wave analysis,40 and then the group index 𝑛𝑔 = 𝛽/𝑐 +

).

𝜔 𝑑𝛽 𝑐 𝑑𝜔

For ultrasmall gap widths, gap plasmons are sensitive to the electron density profile at the metal surfaces and classical electrodynamic models cease to be valid. With a fully quantum mechanical approach,41 the effects of electron spill-out on gap plasmons have been analyzed and it has been shown that the mode index of ultranarrow gap plasmons converges to the bulk refractive index in the limit of vanishing gap, thereby rectifying the unphysical divergence found with classical models. These effects that take place at the MI interfaces, 1-2 Å from the walls, and introduce additional Ohmic losses are however negligible for the gap widths considered in the following, as we checked with simplified nonlocal models.42 Solving the time-harmonic source-free Maxwell’s equations for the MIM geometry is classical, as it looks very similar to the textbook case of dielectric slab waveguides for TM polarization. A small additional complexity, related to the definition of the square root function in the complex plane, should be addressed if the metal loss is not neglected Im 𝜀𝑚 ≠ 0. We will simply consider the case of 𝜀𝑚 negative and real, and introduce absorption with perturbation theory, thereby making all the following quantities, e.g. the propagation constant 𝛽, the group index 𝑛𝑔, also real. The magnetic field of the symmetric gap-mode for TM polarization, propagating along the positive 𝑧 direction in a MIM waveguide with gap width 𝑔, can be written as

{

( ( (

)) + exp ( ― 𝛾𝑑(𝑥 + 𝑔2)) ) exp(𝑖𝛽𝑧), 𝑔 𝐻𝑦(𝑥) = ℎ0(1 + exp ( ― 𝛾𝑑𝑔))exp ( ― 𝛾𝑚(|𝑥| ― 2)) exp(𝑖𝛽𝑧), 𝑔

𝐻𝑦(𝑥) = ℎ0 exp 𝛾𝑑 𝑥 ― 2

(

𝑔

|𝑥| ≤ 2

𝑔

|𝑥| > 2

,

(1)

𝜔2 1/2

with 𝛾𝛼 = 𝛽2 ― 𝜀𝛼 𝑐2

)

, with 𝛼 ≡ 𝑚, 𝑑. Note that 𝛾𝑚 and 𝛾𝑑 are both positive. The magnetic

field amplitude ℎ0 is a constant that will be normalized afterwards. By using the tangential-field 𝑔

continuity equation for 𝐸𝑧 at 𝑥 =± 2, we obtain the transcendental dispersion equation 𝛾𝑚

𝛾𝑑

― 𝜀𝑚(1 + exp ( ― 𝛾𝑑𝑔)) = 𝜀𝑑(1 ― exp ( ― 𝛾𝑑𝑔)).

(2)

Effective index. Equation (2) can be solved accurately by numerically computing 𝛽(𝜔). It is 𝜔 easily solved analytically in the limit of small gaps, 𝑐 𝑔 ≪ 1. By assuming that 𝛾𝑑𝑔 ≪ 1 and 𝛾𝑑 ≈ 𝛾𝑚 ≈ 𝛽 ,we get 𝛽𝑔 = ―

2𝜀𝑑

𝜀𝑚 .

(3)

Equation (3) is important as it directly shows that, as the gap-plasmon spatial extension is shrunk by the gap size, the gap-plasmon wavelength 2𝜋 𝛽 can be as small as a few nanometers, which is much smaller than that in vacuum or any known dielectric medium. This is the physical origin of the subwavelength localization beyond the diffraction limit (that is, the diffraction limit in a dielectric medium). Equation (3) also implies that both transverse wave-vectors, 𝛾𝑑 and 𝛾𝑚, scale as 𝑔 ―1. Field profile and mode normalization. Always in the limit of small gaps, the gap-plasmon field distribution takes the following asymptotic form


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𝐻𝑦(𝑥) = 2ℎ0 𝑒𝑥𝑝(𝑖𝛽𝑧) 𝐸 (𝑥) = 2𝛽ℎ0 (𝜔𝜀 𝜀 ) 𝑒𝑥𝑝(𝑖𝛽𝑧) 𝑥

0 𝑑

𝐸𝑧(𝑥) = 𝑖𝛽ℎ0 (𝜔𝜀0𝜀𝑑)[𝑒𝑥𝑝 (𝛾𝑑(𝑥 ― 𝑔 2)) ― 𝑒𝑥𝑝 ( ― 𝛾𝑑(𝑥 + 𝑔 2)) ] 𝑒𝑥𝑝(𝑖𝛽𝑧)

,

(4a)

𝑔

for |𝑥| ≤ 2, and 𝐻𝑦(𝑥) = 2ℎ0𝑒𝑥𝑝 ( ―𝛾𝑚(|𝑥| ― 𝑔 2)) 𝑒𝑥𝑝(𝑖𝛽𝑧) 𝐸 (𝑥) = 2𝛽ℎ0 (𝜔𝜀 𝜀 )𝑒𝑥𝑝 ( ―𝛾 (|𝑥| ― 𝑔 2)) 𝑒𝑥𝑝(𝑖𝛽𝑧) 𝑥

0 𝑚

𝑥 2𝑖𝛾 ℎ 𝑚 0 |𝑥|

𝐸𝑧(𝑥) =

𝑚

,

(4b)

(𝜔𝜀0𝜀𝑚)𝑒𝑥𝑝 ( ―𝛾𝑚(|𝑥| ― 𝑔 2)) 𝑒𝑥𝑝(𝑖𝛽𝑧)

𝑔

for |𝑥| > 2. We straightforwardly infer that the Poynting-vector 𝑧-component in the gap and in 2 the metals have opposite directions, and that the associated power flows are ―4ℎ0 (𝜔𝜀0𝜀 ) 𝑚

2ℎ20

―2ℎ20

(𝜔𝜀0𝜀𝑚), respectively, so that the net power flow is (𝜔𝜀0𝜀𝑚). Normalizing this and net positive flow to 1, we find that ℎ0 is independent of 𝑔 and must be chosen so that 1/2 ℎ0 = ( ― 𝜔𝜀0𝜀𝑚 2) .

(5)

Absorption. Because we neglect absorption, Im 𝜀𝑚 = 0 and Eq. (3) predicts a real propagation constant. The effect of absorption can be taken into account by applying the Poynting theorem and perturbation theory to a lossy waveguide (Δ𝜀 = Im(𝜀𝑚) ≠ 0) with the mode profile of Eq. (4b). The integration of the exponentially-decaying field in the metal leads to a complex propagation constant, whose imaginary part is given by 𝜀𝑑 Im(𝜀𝑚)

Im 𝛽 = 2

| 𝜀𝑚 | 2

𝑔 ―1,

(6)

showing that the imaginary part of 𝛽 also scales as 𝑔 ―1. It is interesting to note that this result can be obtained directly from Eq. (3) by taking a complex-valued 𝜀𝑚. In fact, one can show that Eq. (3) rigorously holds in presence of absorption.30 Group index. Using the expression of the group velocity 𝑛𝑔 = 𝑐∂𝛽 ∂𝜔, Eq. (3) eventually leads to the following formula for the group index 𝑛𝑔 = ―

𝜆2 𝜀𝑑 ∂𝜀𝑚 𝑔 𝜋𝜀2𝑚 ∂𝜆 .

(7)

Like for 𝛽, Eq. (7) is valid in the presence of absorption. We just need to plug in the complexvalues permittivity of the metal to obtain a complex group velocity 𝑐 𝑛𝑔, whose real and imaginary parts represent the velocity and the damping of the light pulse, at least when the pulse attenuation is weak. The predictive force of Eq. (7), in its complex-valued version, is tested in Fig. 2b for a silver MIM waveguide at a wavelength of 800 nm.

Electromagnetism of slow guided modes

This Section is devoted to the derivation of some fundamentals of the electrodynamics of slow modes. For this derivation, absorption loss due to metal in plasmonic waveguides does not play an important role and can be disregarded. The net benefits are that the group velocity coincides with the energy velocity, providing a simpler interpretation, and that the same intuitive theoretical formalism can be used for both plasmonic and photonic slow waveguides, enabling a direct comparison of each scheme.


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General electromagnetic theorem for slow modes We start by deriving a relation between the power flow and the group velocity of slow guided modes. Throughout the manuscript, we place no restriction on the geometry, and we denote by 𝜀0𝜀(𝑥,𝑦) the arbitrary cross-section permittivity distribution of the plasmonic waveguide. The waveguide permeability is equal to the vacuum permeability 𝜇0 everywhere for the sake of simplicity. Our analysis relies on basic electromagnetic laws of 𝑧-invariant waveguides.43 We consider a guided mode at frequency 𝜔,

[𝐄𝑛(𝐫),𝐇𝑛(𝐫)] = [𝒆(𝑥,𝑦),𝒉(𝑥,𝑦)] exp 𝑖(𝛽𝑧 ― 𝜔𝑡),

(8)

with a field distribution that is exponentially decaying away in the claddings and a propagation constant denoted by 𝛽. Note that we drop the index “𝑛” in 𝒆(𝑥,𝑦) and 𝒉(𝑥,𝑦) for simplicity. For any infinitely-large transverse cross-section Σ of the waveguide and for Im (𝜀) = 0, the Lorentz reciprocity theorem applied to waveguides43 leads to the simple relationship 1 ∬ 2 Σ Re(𝒆

d𝜔

[

∂(𝜔𝜀) ∂𝜔

× 𝒉 ∗ ) ⋅ 𝐳 d𝑥d𝑦 = d𝛽 ∬Σ 𝜀0

]

⋅ |𝒆|2 + 𝜇0|𝒉|2 d𝑥d𝑦.

(9)

For photonic waveguides, the modes are Bloch modes. The latter are pseudo-periodic functions of 𝑧, with a field distribution still given by Eq. (8), except that 𝒆(𝑥,𝑦) and 𝒉(𝑥,𝑦) are now periodic functions of 𝑧, e.g. 𝒆(𝑥,𝑦,𝑧 + 𝑎) = 𝒆(𝑥,𝑦,𝑧). Accounting for this extra-periodicity, Eq. (9) becomes44 1 ∬ 2 Σ Re(𝒆

1 d𝜔

(

∂(𝜔𝜀) 2 ∂𝜔 |𝒆|

× 𝒉 ∗ ) ⋅ 𝐳 d𝑥d𝑦 = 4𝑎 d𝛽 ∭ 𝜀0

)

+ 𝜇0|𝒉|2 𝑑𝑥𝑑𝑦𝑑𝑧,

(10)

where the triple integral runs over any unit cell of the periodic waveguide delimited by two infinitely-large cross-sections separated by the period 𝑎. The energetic interpretation of the classical equations (9) and (10) evidences that the group velocity d𝜔 d𝛽 of the wave packet is equal to the velocity at which energy is conveyed by the mode along the waveguide. Physics of slow waves There are different ways to understand the wave-deceleration in slow waveguides. For photonic waveguides, deceleration is due to a distributed back reflection, and for plasmonic waveguides an equivalent could be a negative Goos-Hänchen phase shifts at the metal/insulator interfaces. Perhaps, a most unified way, that we will use hereafter, consists in considering that deceleration originates from antiparallel power flows. As beautifully evidenced by coupled-wave theory,45 slow light in photonic waveguides results from two counterpropagating waves, each carrying some energy, which together constitute a quasi-stationary pattern.45,46 Similarly, for plasmonic waveguides, because the permittivities of metallic and dielectric layers are opposite, the 𝑧-component of the Poynting vector is positive in dielectric layers, whereas it is negative in the metallic layers. In this picture, ultimate slowdowns as the group (or the energy) velocity approaches zero are often seen as resulting from a perfect balance, for photons when the quasi-stationary pattern of photonic waveguides becomes perfectly stationary, or for plasmons when the energy flows in the metal and in the dielectrics are exactly opposite. Albeit intuitive and widespread, see for instance the introduction of the recent review article,6 this picture however carries some flaw. The power flows indeed never exactly cancel out, even as the energy velocity asymptotically tends towards zero. A convincing counterexample is found by considering the MIM gap mode, for which, as shown above, the power flows in the dielectric gap and in the metal clads are


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2 2 respectively 𝑃for = ―4ℎ0 (𝜔𝜀0𝜀𝑚) > 0 and 𝑃back = 2ℎ0 (𝜔𝜀0𝜀𝑚) < 0 in the limit of small gap widths (i.e. 𝑃for = 2 and 𝑃back = ―1 for a normalized gap mode). The energy flows do not balance: no matter the smallness of 𝑣𝑔 and no matter the convention for the normalization, the power flow in the gap is twice larger than the power flow in the metal on that example. For a general and intuitive argument clearly evidencing that power flows cannot exactly cancel, imagine an adiabatic taper that progressively transforms a fast mode with a power flow 𝛷𝑓 incident from the left into an outgoing slow mode with a power flow 𝛷𝑠, see Fig. 3. Further wrongly assume that the opposite flows of the slow mode on the right side of the taper tend to exactly balance as 𝑣𝑔→0 and 𝛷𝑠→0. Applying now the Poynting theorem to the blue boxes, 𝛷1 + 𝛷𝑓 + 𝛷𝑠 = 0, one gets 𝛷1 = ― 𝛷𝑓 since 𝛷𝑠→0: we arrive at the contradictory conclusion that any slowdown is automatically accompanied by radiation on the lateral sides of the taper. Perfectly (or adiabatically) stopping light would then be impossible. This is indeed wrong, as we know from slowdown practice for lossless 1D systems for which there is no loss by absorption nor by scattering in the clads, by construction. It is important to find out where we get the misconception of the exactly balanced flow and to understand in detail the principle of adiabatic tapering. For that purpose, refer to Eqs. (9) or (10) and consider that the power flow (left-hand-side term) is constant (𝛷𝑠 = ― 𝛷𝑓) for any cross-section of the taper, even for instance for cross-section planes close to the tip apex where the group velocity vanishes. The right-hand-side term of the equation should then remain constant as 𝑣𝑔→0, implying that the surface integral of the electromagnetic energy density ∂(𝜔𝜀) ∬Σ 𝜀0 ∂𝜔 |𝒆|2 + 𝜇0|𝒉|2 d𝑥d𝑦 should diverge with 𝑛𝑔, both for photonic and plasmonic

[

]

waveguides. There is no problem with energy conservation as the product of the vanishing velocity and the diverging electromagnetic energy density remains constant. The fact that the divergence of the electromagnetic density is not observed in practice, because of absorption loss or imperfections, will be discussed in Section 4.

Figure 3 – Impossibility of adiabatically slowing down up to vanishing 𝑣𝑔 (stopping light) if modes with 𝑣𝑔 = 0 have a null power flow. Indeed, perfect adiabaticity is theoretically possible as shown with studies on 1D lossless thin-film slow light channels, for which 𝛷1 is null by construction.

Scaling laws at small group velocities Now that the physics of slow waves is clarified, let us see how the electromagnetic fields scale as 𝑣𝑔 vanishes. The power flow being nonzero even for vanishing group velocities, slow modes


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ACS Photonics

can be normalized such that their power flow is one. This is the classical convention adopted thereafter. We start by slow photons. Since the left-hand side of Eq. (10) equals one, the equality of the electric and magnetic energy densities directly implies that both the electric and magnetic fields scale as

|𝐄𝑛|,|𝐇𝑛| ∝ 𝑛𝑔.

(11)

To derive Eq. (11), we have additionally neglected the spatial mode profile dispersion. This assumption, which is only approximately valid for intermediately small values of the group velocity, becomes perfectly valid for lower speeds as 𝑣𝑔 only significantly decreases in a narrow spectral range. This well-known scaling law,38 illustrated in Fig. 4a, represents the building block for slow light applications based on a strong boost of light-matter interactions.

Figure 4 – Field enhancement of normalized slow waves with unitary power flows. (a) Photonic case: both the electric and magnetic field intensities scale as the first power of 𝑛𝑔. Since the taper is almost perfect (0.997 efficiency for this 2D simulation47), a strong 20 intensity enhancement is observed between the slow mode (𝑛𝑔 = 100) and the fast one (𝑛𝑔 ≈ 3). Note that the expected 30 enhancement is not really implemented because the transverse profiles of the slow and fast modes are slightly different. (b) Plasmonic MIM case: because of the additional transversal geometry change (gap width reduction) as the group velocity is lowered, the electric field intensity (not the magnetic one) increases at a much faster rate, proportionally to (𝑛𝑔)2. The normalized intensity profiles are computed with the aperiodic Fourier Modal Method.40

For translation-invariant plasmonic waveguides, Eq. (9) holds and by assuming unitary power flows, we obtain 𝑐 𝑛𝑔

1

(

∂𝜔𝜀

)

× 4∬ 𝜀0 ∂𝜔 |𝒆|2 + 𝜇0|𝒉|2 𝑑𝑥𝑑𝑦 = 1.

(12)

In sharp contrast with the photonic case, it is essential to realize that one cannot obtain directly the scaling laws for 𝒆 or 𝒉 from Eq. (12), owing to the fact that one should change at least one transverse characteristic dimension of the 𝑧-invariant plasmonic waveguide to tune down the group velocity, thereby radically changing the transverse mode profile. At large 𝑛𝑔 for deep subwavelength confinements, self-sustained energy oscillations between magnetic and electric energies no longer hold, and the oscillations are restored by considering the kinetic energy of


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the free carriers involved in the subwavelength plasmon mode.3 A precise analysis of the scaling of 𝒆 and 𝒉 with 𝑛𝑔 requires knowing the exact geometry but, as shown in Section 2 on the textbook example of MIM waveguides with asymptotically small dielectric gaps, the magnetic energy becomes negligible and the general trend is imposed by the balance between the electron-kinetic and electric-field energies48

|𝐄𝑛| ∝ 𝑛𝑔,|𝐇𝑛| ∝ 𝐶𝑡𝑒,

(13)

for normalized plasmonic modes. Noticeably, the electric-field increase rate for small group velocities is much larger for plasmons than for photons. The reason comes from the additional transversal geometry change (gap width reduction) that is accompanying plasmon slowdowns, as illustrated in Fig. 4b. Based on the previously presented difference in scaling between plasmonic and photonic slow waves, it is interesting to reconsider how the antiparallel power flows scale with 𝑛𝑔. There are indeed differences between the photonic and plasmonic cases. Let us denote again by 𝑃for and 𝑃back, the positive powers flowing forward or backward. The ratio 𝑢 = 𝑃back/𝑃for (0 ≤ 𝑢 < 1) is an important parameter that quantifies the slowness. For photonic waveguide modes operating outside a photonic gap, 𝑢 is strictly smaller than 1, and for a normalized Bloch mode with a unitary power flow (𝑃for ― 𝑃back = 1), we have 𝑃for ≈ 𝑃back ∝ 𝑛𝑔 ≫ 1 with (𝑃back ― 𝑃for)/𝑃for ∝ 𝑣𝑔 𝑐, since the electromagnetic fields scale proportionally to 𝑛𝑔. This implies that, in the slow light regime, the quasi standing-wave patterns of normalized photonic waveguide modes are composed of two counter-propagating waves, each carrying a huge power flow that diverges linearly with 𝑛𝑔. The plasmonic case is markedly different. As the group velocity vanishes, the plasmonic electric field scales with 𝑛𝑔, but since the confinement also scales with 1/𝑛𝑔, 𝑃back and 𝑃for both become constant. For instance, for the MIM waveguide studied in Section 2, 𝑃for = 2 and 𝑃back = 1 for a normalized gap mode. In sharp contrast with photonic waveguides, each flow becomes independent of 𝑛𝑔 in the limit of small gaps. By way of summary, Table 1 provides a recapitulation of main properties of structural slow photons and slow plasmons. photonic waveguide

plasmonic waveguide < 30 (𝑛𝑔 is always a complex-valued number)

|𝑬|

20-100 for PhCWs5 (coupled resonators may offer larger values) ∝ 𝑛𝑔

|𝑯|

∝ 𝑛𝑔

∝ 𝐶𝑡𝑒

𝑃for,𝑃back

∝ 𝑛𝑔

∝ 𝐶𝑡𝑒

∝ 𝑛𝑔―2 (𝔅 ≈ 0.004 for PhC band-edge slow light at 𝑛𝑔 = 50)

∝ 𝐶𝑡𝑒 (𝔅 = ―1/3 for MIM)

Realistic 𝑛𝑔

∆𝜔 ∆𝑛𝑔 𝔅= 𝜔 𝑛𝑔 Main attenuation factor

∝ 𝑔 ―1 ∝ 𝑛𝑔

clad radiation ( ∝ 𝑛𝑔) for small 𝑛𝑔, backscattering ( ∝ 𝑛2𝑔) at large 𝑛𝑔.

absorption ( ∝ 𝑛2𝑔) for 𝑛𝑔 < 30, backscattering ( ∝ 𝑛4𝑔) for 𝑛𝑔 > 30.49

50 µ𝑚 < 𝐿𝑝 < 500 µm for 𝑛𝑔 = 100 @

𝐿𝑝 ≈ 300 nm for 𝑛𝑔 = 10, barely 100 nm


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propagationlength

λ = 1500 nm according to the state-ofthe-art fabrication techniques.

for 𝑛𝑔 = 30 (@ λ = 800 nm).

Table 1. Scaling of the main properties of structural slow light and slow plasmons. Representative measured values are given in square brackets.

Limitations of real waveguides

Despite their interesting physical properties, slow waves suffer in practice from important limitations, in particular regarding the operation bandwidth and the propagation distance. The available bandwidth, that is, the information content that can be slowed down or stored, is limited by dispersion, whereas the propagation distance, which defines the storage time and the interaction length, is limited by bulk-material absorption and scattering loss due to fabrication imperfections. The latter are well documented in the context of photonic-crystal waveguides38,50-52. For plasmonic waveguides, it is often assumed that the attenuation is dominantly due to absorption and there are few studies on roughness-induced attenuation. It is important to have the intrinsic differences and limitations in mind to optimally exploit slowness with the appropriate scheme in applications. The following subsections briefly review key aspects. Bandwidth A major source of limitation of slow-light structure is dispersion. For any system, the operating frequency is fixed, but in general, the transported signal possesses a bandwidth and one should consider how the slowness changes over the bandwidth. To quantify the limiting impact of group-velocity dispersion, it is convenient to define a dimensionless coefficient 𝔅 as ∆𝜔 𝜔

=𝔅

∆𝑛𝑔

𝑛𝑔 ,

(14)

which quantifies the variation ∆𝜔 of 𝜔 due to a small variation ∆𝑛𝑔 at a fixed operating value 𝑛𝑔. Large bandwidths are achieved for large 𝔅’s, and reversely small bandwidths for small values. 𝔅 can be evaluated by considering the dispersion relation 𝜔(𝛽). By derivation, it is possible to infer another relation linking the frequency 𝜔 and the group velocity, or equivalently the group index 𝑛𝑔. Consider first a photonic waveguide with a quadratic dispersion curve 𝜔(𝛽) = (𝛽 ― 𝜋/𝑎)2 2𝑚 + 𝜔0 near a band edge 𝜔0, where 𝑎 is the period and 𝑚 =

∂2𝜔 ―1

( ) ∂𝑘2

is the

effective photon mass, which describes the flatness of the dispersion curve. For 𝑛𝑔 = (𝑚𝑐2 2)1/2(𝜔 ― 𝜔0)1/2, we find 𝑚𝑐2

𝔅 = 𝜔 𝑛2 ,

(15)

0 𝑔

which evidences that 𝔅 scales with 𝑛𝑔―2. The bandwidth near a Brillouin zone boundary therefore rapidly decreases as the operating group velocity decreases. To estimate 𝔅 for slow plasmons, we conveniently use the dispersion relation of MIM waveguides, 𝛽𝑔 = ―

(

2𝜀𝑑

2𝑐𝜀𝑑 𝑔) 𝜀𝑚 , see Eq. (3). We obtain 𝑛𝑔 = (

d𝜀𝑚 d𝜔

)

𝜀2𝑚 , and using a Drude

model 𝜀𝑚 = 𝜀∞(1 ― 𝜔2𝑝 𝜔2), 𝔅 then reads as 𝔅 = ―1/3 ,

(16)


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and is independent of 𝑛𝑔. The bandwidth of photonic waveguides with quadratic dispersion curves can be enhanced by engineering to locally flatten the dispersion curve,53 but even so, slow photonic waveguides operate in a quite narrow frequency range that remains much smaller than that of slow plasmonic waveguides Propagation length Since light cannot propagate indefinitely at low speed due to fabrication imperfections (surface roughness) or intrinsic material limitations (Ohmic losses), the delay is finite. In general, the longer the delay for a given device, the narrower the bandwidth, which is why the so-called ‘delay–bandwidth product’ is such a useful figure of merit. Let us then consider the propagation length 𝐿𝑝 (defined for a 1 𝑒 field attenuation) of slow waveguides. Qualitatively first, the impact of imperfections can be estimated by considering a local permittivity change Δε, a “Dirac” defect, in a waveguide. In the Born approximation, this defect acts as a local current source, 𝐉 = ―𝑗𝜔𝛥𝜀𝐄𝑛𝛿(𝑟 ― 𝑟0), which is proportional to Δε and to the driving field (the slow incident mode 𝐄𝑛). As illustrated in Fig. 5, this local source may radiate into the cladding, induce absorption if the cladding or the defect absorb, or backscatter light into the waveguide by exciting the counter-propagating slow mode. The power 𝑃𝑟𝑎𝑑 radiated into the cladding is proportional to |𝐉|2 and since 𝐉 is proportional to 𝐄𝑛, 𝑃rad ∝ |𝐄𝑛|2. Similarly, since the power 𝑃abs absorbed by the defect is proportional to 𝐉 ⋅ 𝐄𝑛, 𝑃abs ∝ |𝐄𝑛|2. The absorption by metal cladding for plasmonic waveguides deserves a special care.54 The backscattered power 𝑃back is proportional to |𝐄𝑛 ⋅ 𝑱|2 (the excitation modal coefficient of the 1

normalized back-propagating mode is 4𝐄𝑛 ⋅ 𝑱),44 so that 𝑃back ∝ |𝐄𝑛|4. These are only general trends, but they all tell us that the influence of material loss or fabrication imperfection on mode structure is strong at small group velocities, where even a tiny amount of loss produces large changes in the dispersion, such that the group velocity is never zero.55-57 We will investigate this aspect into more details now.

Figure 5 – Attenuation channels for a tiny imperfection characterized by a localized permittivity change Δ𝜀𝛿(𝒓 ― 𝒓𝟎) in a slow waveguide. Energy can be either radiated into the cladding, backscattered or absorbed if Im(Δ𝜀) ≠ 0. Radiation into the cladding is equivalent to absorption in metal clads for MIM waveguides.

Roughness-induced attenuation of photonic waveguides. The propagation of light in real photonic waveguides is well documented in the literature; absorption can be neglected with photonic materials, so that roughness of the etched pattern is the main limiting technological factor. Since |𝐄𝑛| ∝ 𝑛𝑔, we have that 𝑃rad ∝ 𝑛𝑔 and 𝑃back ∝ 𝑛2𝑔, so that the corresponding attenuation lengths 𝐿𝑝 scale with 𝑛𝑔―1 and 𝑛𝑔―2, respectively.38 These simple scaling laws have been confirmed by ab-initio electromagnetic computations performed with Bloch-modeexpansion methods for realistic waveguide geometries incorporating roughness on the


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ACS Photonics

sidewalls of the etched holes.50 Theory predicts typical values for the propagation-length that vary between 𝐿𝑝 = 50 and 500 µm for 𝑛𝑔 = 100, and between 𝐿𝑝 = 103 and 104 µm for 𝑛𝑔 = 30, depending on the accuracy of the fabrication processes generally of the order of one nanometer or less. These theoretical predictions are confirmed by experimental transmission and reflection measurements performed with 100 ― 1000𝜆 long waveguides,37,58,18,51,23 showing a weak reflected signal for low group indices and a weak transmitted one at large group indices. Absorption loss in plasmonic waveguides. The extremely tight confinement in plasmonic components is due to the large wavevectors of plasmons. It is inevitably accompanied by substantial absorption loss, essentially given by the inverse of the damping rate of the metal. Thus, it is fundamentally difficult to achieve a long propagation distance at slow speeds. To analyze the attenuation due to metal absorption, one naturally may consider the expression of the absorbed power 𝑃abs = ω 2∬𝜀0Im(𝜀𝑚)|𝐄𝑛|2𝑑𝑆, where the integral runs over the transverse cross-section of the plasmonic waveguide. Since 𝐄𝑛 ∝ 𝑛𝑔 for MIM waveguides, see Eq. (13), we may infer that 𝑃abs ∝ 𝑛2𝑔 at first glance. This however neglects the dispersion of the transverse mode profile, and therefore largely overestimates the absorption. Instead, we directly rely on Eq. (9), which is valid for all reciprocal geometries, and we find that the integral of |𝒆|2 or |𝐄𝑛|2 over a cross-section scales as 𝑛𝑔. This evidences that 𝑃abs actually scales as 𝑛𝑔. More insight into the role of mode-profile dispersion can be gained by considering the textbook case of MIM waveguides. We showed in Section 2 that, as the gap width decreases, the gap-plasmon effective index 𝑛eff = 𝑐𝛽 𝜔 ∝ 𝜆 𝑔 increases, see Eq. (3). Thus, the penetration 𝑐

―1

depth of the gap plasmon mode into the metal, 𝜔(𝑛2eff ― 𝜀𝑚) , decreases and becomes shorter than the skin depth. The effective area contributing to the cross-section integral in 𝑃abs = ω 2 ∬𝜀0Im(𝜀𝑚)|𝐄𝑛|2𝑑𝑆 diminishes. It is thus the field confinement in the immediate vicinity of the MI interfaces that is responsible for the 𝑛𝑔-scaling of the absorbed power, and not the 𝑛2𝑔scaling of our naive approach. Despite this lowering, the attenuation due to absorption is drastic in plasmonic waveguide. For instance, for an Ag/SiO2/Ag waveguide with 𝜀Ag = ―27 + 𝑖1.5 at λ = 800, one finds 𝐿𝑝 ≈ 300 nm for 𝑛𝑔 = 10 and barely 100 nm for 𝑛𝑔 = 30. Roughness-induced attenuation in plasmonic waveguides. Since 𝑃back ∝ |𝐄𝑛|4 and |𝐄𝑛| ∝ 𝑛𝑔 for plasmonic waveguides (compared to |𝐄𝑛| ∝ 𝑛𝑔 for photonic waveguides), roughnessinduced backscattering is plasmonic waveguides is scaling as ∝ 𝑛𝑔4 and is expected to significantly alter the propagation of slow plasmons. Moreover, since 𝑃abs only scales as 𝑛𝑔 comparatively, one may wonder if backscattering may become the dominant attenuation mechanism at small group velocities. This possibility has not been explored experimentally. Theoretically since imperfections at metal-dielectric interfaces result in a large Δ𝜀, the local current source 𝑱 is not simply proportional to the plasmon mode field and local-field corrections should be considered.59 Additionally, the plasmon-mode profile disperses as the group velocity is lowered. This may explain why theoretical analyses on roughness-induced attenuation in plasmonic waveguides are rare. Imperfections in MIM waveguides with large gap widths (operating away from the slow regime) have first been documented in Ref. 60; a strong impact has been predicted but for relatively large roughness ( ≈ 4 nm) obtained for films deposited by use of conventional ebeam evaporation. Slow propagation and small roughness have been analyzed49 for either


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chemically-synthesized metal films or flakes, for which the metal surfaces are essentially atomically flat, except for some potential monoatomic adlayer defects, or high-quality polycrystalline metal films composed of large grains with a small residual roughness and separated by deep valleys that are remnants of grain boundaries.61 The predictive conclusions are that backreflection due to imperfection is completely negligible for chemically-synthesized metal films. For polycrystalline films, backreflection is predicted to be much stronger than for photonic crystal waveguides, in accordance with the scaling laws, and become more impactful than absorption for very small speed regimes (𝑛𝑔 > 30). We might think that slow photons are much more sensitive to roughness than slow plasmons. This is actually not true, as evidenced by the different scaling laws in 𝑛𝑔4 for plasmons and 𝑛𝑔2 for photons. The strong impact of roughness is in fact hidden by absorption for usual slowness (𝑛𝑔 ≈ 5 ― 20). Despite having an obvious weaker scaling ∝ 𝑛2𝑔 than backscattering, absorption is the dominant attenuation channel.49 The reason is simply that absorption ( ∝ 𝑛𝑔) occurs over the entire metal volume, whereas backscattering ( ∝ 𝑛𝑔4) is due only to surface-localized defects. Impossibility to reach ultraslow regimes Some of the first experimental investigations of the speed of light transport in monomode photonic-crystal waveguides reported group velocities of 𝑐/50 and 𝑐/150,37 whereas in other types of experiments group velocities lower than 𝑐/1000 were seemingly observed.62 For slow plasmons, experiments seemed to evidence that novel metallic waveguide structures drastically reduce the propagation speed and may even stop broadband light, producing the so-called ‘‘trapped rainbows’’.63,64 As we will now show the very notion of group velocity loses all meaning at such small group velocities for both photons and plasmons. There are two regimes for slow photons.65,56,52 Far from the band edges, where the concept of group velocity applies, tiny fabrication errors introduce random spatial dephasing of the propagating field and, as the interference between multiply-scattered propagating waves may be destructive, propagating light is exponentially damped with an attenuation coefficient proportional to 𝑛2𝑔 due to backscattering.38 This well-known scaling is valid only in a double limit, when the disorder level tends towards zero at a fast-enough rate compared to 𝑣𝑔 to guarantee that the impact of random imperfections on transport remains perturbative.52 In practice, the disorder level, albeit very weak, is fixed and given by the employed nanofabrication technique. A sudden breakdown of the perturbative regime, which inevitably ceases to be valid at small speeds, thus eventually occurs. In this “ultraslow” regime, light propagating in the randomly-perturbed periodic waveguide not only experiences random phase-shifts, but also strong backreflection due to 𝑣𝑔-enhanced impedance mismatches or stop-band reflection for slow light operated near band edges.52 The backreflection results in strong interference between multiply-scattered waves, leading to the formation of wavelengthscale localized modes, similar in nature to photonic-crystal microcavity modes. In this regime, the wave number k cannot be considered as a good quantum number anymore and the group velocity evidently looses physical significance.56,57 The same physics hold for slow plasmons, except that the situation is even worse since absorption loss additionally comes into play and the localization regime is even not reached in practice; the plasmon is damped in space or in time before forming disordered resonant cavities. The literature is quite misleading on that matter.


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ACS Photonics

Light emission and LDOS engineering in slow waveguides

Controlling light emission at the single photon level is one of the major challenges in modern photonics, with great promises for quantum communications, quantum computing and optoelectronics thanks to devices such as highly-efficient non-classical light sources, lowthreshold lasers and single-photon switches or diodes. Probably the most important problem to address to reach this aim is the realization of efficient coupling between an emitter and a single guided mode. This coupling is described by the 𝛽-factor that is the ratio between the power emitted into the mode of interest and the total emitted power. It may thus be made close to unity by boosting the emission into the mode and/or reducing the emission into the other decay channels. Quantitatively, the amplitude coupling coefficient between a dipole-current 𝐉0𝛿 (𝐫 ― 𝐫0) and translation invariant43 or periodic waveguides waveguide44 is proportional to the electric field of the normalized mode at the source position 𝐫0. Slow modes with large field enhancements ( ∝ 𝑛𝑔 for photons and ∝ 𝑛𝑔 for plasmons) are thus expected to offer nearunity couplings. In this Section, we review the basic principles underlying light emission in slow waveguides and identify the most promising strategies for near-unity 𝛽-factors.

Figure 6 – Efficient funelling of the spontaneaous decay of individual optical emitters into slow modes. (a) Slow photons: near-unity funelling into a single slow mode can be achieved provided that the LDOS of the slow mode is much higher than the LDOS of the clad modes. (b) Slow plasmons: even if tiny gaps lead to gap plasmons with extremely large LDOS, near-unity funelling is prohibited because of near-field nonradiative decay in the metal (quenching).

Emission in slow photonic waveguides The possibility of achieving near-unity 𝛽-factors is well comprehended for slow photonic waveguides, see Fig. 6a. On the one hand, the enhancement (Purcell effect) of the local density of states (LDOS) scales with 𝑛𝑔 in 1D systems, thereby boosting the emission into the slow mode when approaching the band-edge. On the other hand, photonic-crystal waveguides offer great flexibility in loss engineering, such that it is also possible to drastically reduce emission into the other decay channels.44 𝛽-factors above 90% can hence be achieved even without strong LDOS enhancement, i.e. by operating at moderate 𝑛𝑔 values. Unsurprisingly, the main limitation in terms of performance comes from fabrication imperfections, which yield the formation of localized modes with unpredictable spatial positions, resonant frequencies and quality factors.66 This limitation may be mitigated by using waveguides shorter than the localization length. In addition, when using emitters with a sharp spectral response, it is necessary to match the emission frequency with a desired 𝑛𝑔. One can


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show that the highest 𝑛𝑔 that may be aimed at in presence of disorder scales as 𝑚,67 with 𝑚 the effective photon mass. One should therefore preferably opt for photonic waveguides with a large effective photon mass, i.e. flat dispersion curves, for optimal performance. To date, near-unity 𝛽-factors (𝛽 ≈ 0.98) have been reported with quantum dots embedded in semi-closed photonic-crystal waveguides operating at 𝑛𝑔 > 50.9 Such a high efficiency is achieved only on a narrow spectral range, yet it may be broadened up to a few tens of nanometers at the cost of a slightly lower 𝛽-factor by operating at lower 𝑛𝑔 values. As an alternative approach to embedded quantum dots, important efforts are currently being made to trap cold atoms near photonic structures.68 A recent study reports 𝛽 ≈ 0.5 for cold atoms trapped in the so-called corrugated “alligator” waveguide designed to achieve long-range interactions between distant atoms for 𝑛𝑔 ≈ 11.69 The structure suffers from a much lower effective photon mass and fewer degrees of freedom for design compared to a photoniccrystal-type waveguide,67 suggesting that higher performances may be achieved with new geometries. Emission in slow plasmonic waveguides Compared to light emission in photonic structures, the decay rate of emitters into slow plasmons70 is larger and the bandwidth is definitely much broader. The only grey area is the efficiency issue because of the prominence of near-field non-radiative decay in the metal for tiny gaps, see Fig. 6b. The whole literature is not always clear on the role played by plasmons, especially “slow” plasmons with large parallel momenta, into the decay process. In the remainder of this Section, we analyze this issue, step by step, first revisiting the classical light emission by a dipole-current located just above a single MI interface, and then considering the case of light emission by a dipole-current located in an MIM waveguide, the objective being to completely clarify the role of slow plasmons. Dipolar emission near a metal surface. This problem has been reviewed by many authors, including.71 It contains a rich variety of physical phenomena. In particular, we know that the lifetime of an atom placed close to the surface can be typically reduced by several orders of magnitude,72 implying that the LDOS close to an interface is drastic. The reduction is due to several decay channels. A part of the oscillating dipole decay is coupled into radiation modes detected in the far field, another part launches surface plasmons on the surface, and another significant part is directly transformed by near-field coupling into Ohmic loss in the metal just beneath the source, resulting in a quenching of the far-field emission especially when the dipole is very close to the surface.72 The plasmon dispersion relation of flat MI interfaces is well known73 𝑘=

𝜔 𝑐

𝜀𝑑𝜀𝑚 (𝜀 + 𝜀 ), 𝑚 𝑑

(17)

with 𝑘 the parallel in-plane wavevector, 𝜔 the frequency, 𝜀𝑚 and 𝜀𝑑 the relative permittivities of 2 the metal and dielectric materials. We will use a Drude metal 𝜀𝑚 = 1 ― 𝜔𝑝 (𝜔2 ― 𝑖𝛾𝜔) for illustration hereafter. Since 𝜀𝑚 is complex, either 𝑘 or 𝜔 is complex and two interpretations, in relation with slow plasmons, are usually considered. The first interpretation relies on the dispersion relation obtained by fixing the frequency to a real value (imposed by the driving dipole) and by directly solving Eq. (17) for complex 𝑘’s, denoted with a tilde to emphasize their complex nature  the imaginary part of 𝑘 is related to the inverse of the attenuation length of the plasmon mode. The dispersion relation Re 𝑘 ―𝜔 is


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shown in Fig. 7b. This plasmon mode is the analogue of the photonic-crystal mode of the previous Section or the gap plasmon mode analyzed in Section 2; it is a guided mode with a spatial damping due to absorption. Because realistic values for the loss are taken into account in Fig. 7, the relation exhibits a 𝜀𝑑 𝜔 back bending for a relatively low value of Re 𝑘, Re 𝑘 ≈ ( ) 𝑝 𝑐. In this real frequency 2 1 + 𝜀𝑑

representation, no slow plasmons with large 𝑘 are launched,73 even when the emission frequency is closed to 𝜔𝑝 (1 + 𝜀𝑑)1/2 and the dipole is very close to the surface. This result contrasts with the emission in MIM waveguides with small gaps, presented below, for which we will show that guided gap plasmons with large 𝑘’s are excited at any frequency. Another representation of the same physics relies on the quasinormal modes (QNM)74 that are the natural plasmon resonances with a complex frequency 𝜔 of the MI interface. The QNMs possess a sinusoidal spatial dependence parallel to the surface (no spatial damping) and a finite lifetime proportional to the inverse of Im 𝜔. The relation dispersion, Re 𝜔 ―𝑘, shown in Fig. 7c, is calculated by solving the transcendental Eq. (17) at complex frequencies for fixed 𝑘. Now, plasmon quasinormal modes with arbitrary large 𝑘’s (since quantum effects are not considered71) are obtained for a certain complex frequency 𝜔SP, such that 𝜀𝑚(𝜔SP) + 𝜀𝑑 = 0 i.e. ωSP = ωp (1 + εd) + 𝑖γ/2 for γ ≪ ωp. The quasinormal modes form a complete basis, so that the field radiated by the dipole at the (real) excitation frequency 𝜔 can be reconstructed from a sum over all QNMs.74 For illumination by a local oscillating dipole-current placed in the near field, QNMs with arbitrarily large 𝑘’s are excited. Intuitively, we anticipate that all modes are excited in phase at the dipole position, so that their respective contributions add up constructively especially just beneath the dipole, where strong absorption takes place, as evidenced by the hot spot in Fig. 7a. Away from the source on the surface, all the sinusoidal QNMs are not phased and the sum of their field is no longer constructive, so that the contribution of high ―𝑘 modes to the reconstructed field rapidly decreases away from the dipole in the metal, as shown in Fig. 7a. Moreover, very high ―𝑘 modes have a very fast transversal decay ( ∝ 𝑘 ―1) and are thus excited only when the dipole is very close to the surface. This implies that, as the distance 𝑑 between the dipole and the surface decreases, the hot spot becomes more intense and localized. Thus, with the QNM interpretation, in contrast with the guided mode interpretation, high ―𝑘 slow plasmons play an essential (albeit detrimental) role in the light emission process; quenching drastically increases when 𝑑 decreases (the decay rate scales as (𝜔𝑑/𝑐) ―3 in the static limit71) and when the dipole frequency 𝜔 matches Re 𝜔SP, since the excitation coefficients of QNMs with a frequency 𝜔 are proportional to (𝜔 ― 𝜔 ) ―1.74 The intuitive description of the role of “slow” plasmons has been carefully confirmed with numerical simulations obtained for a silver nanorod in air. By computing several hundreds of QNMs, it has been shown that the hot spot can be accurately reconstructed in the QNM basis up to 𝑑 as small as 1.5 nm, see Fig. 3 in Ref. 75. The spectrum of the plasmon eigenfrequencies does not form a continuum for the nanorod like for the infinite flat interface. Rather they take discrete values. However, for very small separation distance between the dipole and the antenna, the dipole “sees” an almost flat surface. The nanorod spectrum presents an accumulation point at the plasmon resonance frequency of the flat interface for which 𝜀𝑚(𝜔SP) + 𝜀𝑑 = 0,75 showing that these nanorod QNMs are indeed very similar to those of the flat surface.


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Figure 7 – Slow plasmons on single metal-dielectric interfaces. (a) Radiation of a vertical dipole current above an Ag/polymer interface. The color scale represents log10(|E|2) of the total electric field for a dipole-metal separation distance 𝑑 = 4 nm. The oscillating dipole decay is coupled into radiation modes detected in the far field, into surface plasmons launched on the surface, or directly transformed by near field coupling as Joule heating in the metal just beneath the source. (b) Usual dispersion relation, Re 𝑘 ―𝜔, of surface plasmons at a flat interface between a Drude-metal 2 with a relative permittivity 𝜀𝑚 = 1 ― 𝜔𝑝 (𝜔2 ― 𝑖𝛾𝜔) and a dielectric material with a relative permittivity 𝜀𝑑 = 1. (c) Quasinormal mode dispersion relation, Re 𝜔 ―𝑘. Slow plasmons with large 𝑘 show up, but their lifetime is vanishingly small. The horizontal asymptote (dashed curve) occurs for a complex frequency 𝜔SP verifying 𝜀𝑚(𝜔SP) + 𝜀𝑑 = 0. (b) and (c): Both curves are obtained by solving Eq. (17). 𝜔𝑝 is the plasma frequency and 𝛾 = 0.02𝜔𝑝. The black line is the light line of the air clad.

Dipolar emission in MIM waveguides. Let us now consider the emission of an oscillating dipole in the dielectric gap of a MIM waveguide, in the limit of tiny gaps (see Fig. 6b). The conclusions on the role of slow plasmons will be completely different. Indeed, in full analogy with the single-interface or antenna cases, the non-radiative decay in the metal is again present. However, as discussed above, the slow gap plasmons (𝐑𝐞 𝒌 ―𝝎) that are launched in the gap have an extremely large LDOS. Actually, it can be shown analytically that the non-radiative decay rate (quenching) and the slow-plasmon decay rate both scale as (𝒌𝟎𝒈) ―𝟑. None of the two decay channels prevail as the gap width 𝒈 becomes very small. The branching ratio 𝜼 (the fraction of the total decay that effectively decay into the gap mode) becomes an intrinsic quantity as it is independent of 𝒈, depending solely on 𝜺𝒅 and on the metal loss 𝐈𝐦 𝜺𝒎, 𝜼 ≈

(𝟏 +

𝐈𝐦 𝜺𝒎 ―𝟏 𝟐𝜺𝒅

)

.54,76

The good news is that the emission is no longer quenched for tiny gaps so that, provided that the gap plasmon mode is efficiently tapered into a photonic mode,54 large efficiencies are achievable. The downside of the existence of a branching ratio is that the efficiency cannot reach unity. This imposes a limitation on the 𝛽-factors that usually not exceed 80% for MIM77,78,79,54 or IMI80,81 waveguides, and remain significantly smaller than their photonics counterparts.


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Perspectives and conclusions

The applications of slow photonic waveguides in both linear and nonlinear optics are so diverse that it is impossible to cover them all even briefly. Slow photons are useful to realize compact all-optical buffer memories, which is one of the most important missing components for the construction of various all-optical processing devices such as photonic routers. They are also useful to funnel light emission into a single channel and to enhance various light–matter interactions in nonlinear optics. We have no choice but to note that slow-photon applications rarely feature a speed reduction greater than 10–20. For instance, moderate 𝑛𝑔 values ( ≈ 30) are used in photonic-crystal and coupled-resonator waveguides to implement delay lines over millimeter distances.82,83,23 For short devices such as low-power optical switches,20 typical 𝑛𝑔 values of ≈ 50 are met in practice. Interesting perspectives may arise with regimes of smaller group velocities. Remembering that the attenuation length 𝐿𝑝 of photonic waveguides at λ = 1500 nm and 𝑛𝑔 = 100 varies from 50 to 500 µm according to the fabrication technology and that 𝐿𝑝 ∝ 𝑛𝑔―2, we anticipate that slow light at 𝑛𝑔 = 300–500 may propagate over 10-period long distances before backscattering takes place. Since remarkably short and efficient tapers have been designed47,84 (see Fig. 4a) and successfully implemented in many slow-light experiments,9,85-87 devices that feature a sudden reduction of the group velocity, followed by a few-periods-long ultra-slow section and a reciprocal acceleration to return to the initial speed appear within reach. Applications may include all-optical processing on chip, biosensors or new resonator-like devices with anomalous step-like spectral transmission. However, as ultra-slow photonic waveguides and microcavities have a lot in common, especially for large 𝑛𝑔 when it becomes difficulty to engineer waveguide dispersion,84 attention should be paid not to repeat earlier works with high-Q microcavities. Comparatively, slow-plasmon studies are more prospective. Slow plasmons can only propagate over very small distances, due to absorption. Their applications must therefore inevitably rely on components with very short lengths, not longer than a few hundreds of nanometers. Two main slow-plasmon components have emerged in the past decade, namely nanofocusing tapers and slow-plasmon nanocavities or nanoantennas. Nanofocusing devices that taper and concentrate optical energy into nanoscale objects and structures probably constitute the most important device using slow plasmons. Various geometries based on MIM and IMI configurations have been successfully implemented, e.g. conical metal rods with a rounded tip for near-field imaging, metal wedges on substrates such as in Fig. 3, or exotic structures such as metal–insulator–insulator (MII) wedges88 or tapered chains of coupled nanoparticles,89 see Ref. 36 for a review. The whole literature suggests that adiabatic tapering into deep-subwavelength slow plasmons is relatively easy to implement, since for sufficiently short tapers, the local field enhancement along the taper could efficiently compensate for the dissipative reduction of the plasmon amplitude. Nanofocusing devices have been used in many applications, yet many new ones appear every year, for high-harmonic generation, hot-electron plasmon-assisted generation, see Ref. 63 for a review. The second important slow-plasmon device is nanocavity or nanoantenna. At deep subwavelength dimensions, these components are essentially “Fabry-Perot cavities” involving slow plasmons that propagate in a slow waveguide and are reflected at the waveguide terminations. For instance, the dipolar resonance of a simple nanorod immersed in a dielectric background can be considered as resulting from the bouncing back and forth of two IMI


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plasmon modes between the wire facets. Similarly, two MIM plasmon modes are involved into the dipole and quadrupole modes of cut-wire-pair resonators and nanopatch antennas. As the transverse dimensions (the rod diameter or gap width) are scaled down, the physics is the same. The fundamental IMI or MIM plasmon modes experience a monotonic increase in transverse localization and effective index, see Eq. (3), implying that by reducing the nanoresonator length by the same scaling factor, the resonance wavelength is maintained.90,91 This simple model that promotes slow plasmons as an integral part of the near-field response of deep-subwavelength nanoresonators predicts well the resonance frequency. Pushed further, it also explains how important quantities such as the Q factor, the absorption or the facet reflectivity scale.90 In particular, it predicts that the Purcell factor scales as 𝑔 ―3,54 making slowplasmon nanoresonators or nanoantennas a serious candidate for an ultimate control of light emission 91,92,10, especially if an accurate control of the conversion of the slow gap-plasmons into free-space photons is implemented by engineering the tapered facets to boost the photon generation yield. In summary, we have provided a comparative overview of the physical mechanisms by which wave-deceleration is achieved in plasmonic and photonic waveguides. The scaling laws of various important physical quantities (field enhancement, bandwidth, attenuation) as a function of 𝑛𝑔 have been presented, see Table 1. Slow plasmonic and photonic waveguides share many common features, but they have also distinct properties. We think that the keyfundamental property responsible for the difference, notably for the mode intensities ( ∝ 𝑛𝑔 for slow photons and ∝ 𝑛2𝑔 for slow plasmons), is that the slow mode of photonic waveguides retains the same profile as light is slowed down, whereas the plasmon mode profile drastically changes for plasmonic waveguides. Slow-plasmon devices are generally operating at much lower 𝑛𝑔 than slow-photon devices ( 𝑛𝑔 = 15 corresponds to insulator gap width of 𝑔 = 2 nm at visible wavelengths). One should nevertheless remember that slow plasmons and photons have different field-enhancement scalings (|𝐄𝑛| ∝ 𝑛𝑔 for plasmons and |𝐄𝑛| ∝ 𝑛𝑔 for photons), so that it is rather illogical to compare plasmonic and photonic devices for the same 𝑛𝑔. In this respect, a plasmonic taper operating at 𝑛𝑔 = 15 would be equivalent to a photonic taper operating at 𝑛𝑔 ≈ 225 in our analogy. Towards strong field enhancement and ultimate miniaturization, the recent literature has promoted slow plasmons as the pinnacle of slowness in nanophotonics. Indeed, fieldenhancement and bandwidth are decisive figures of merits that render plasmon wavedeceleration incomparably more attractive than photon wave-deceleration. However, from other perspectives, the benefit of the deep-subwavelength confinement (or miniaturization) brought by plasmons is not as clear. Consider all-optical switches, perhaps the most important building blocks for on-chip all-optical processing. Switching is generally achieved by modulating the cavity transmission intensity by illuminating the cavity with a control light beam, which shifts the resonance wavelength via the intensity dependent refractive index of the cavity material. The required switching power scales as 𝑉 𝑄2,18 because the light intensity in the cavity is proportional to 𝑄 𝑉, and the required wavelength shift is proportional to 𝑄 ―1. In contrast to spontaneous rate (Purcell) enhancements,10 it thus becomes unclear if plasmons, with their drastic reduction of 𝑉 often counter-balanced by an inevitable 𝑄 reduction due to absorption, can really be competitive in devices exploiting, e.g. optical nonlinearities. We expect that this short comparison leads to a balanced point of view between slow plasmonic and photonic waveguides.


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Acknowledgements

The work was partly supported by the French State, managed by the French National Agency for Research (ANR) in the frame of the “Investments for the Future” Programme IdEx Bordeaux – LAPHIA (Grant No. ANR-10-IDEX-03-02) and by Bordeaux University.

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fig 2 275x112mm (150 x 150 DPI)


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fig 4 254x103mm (150 x 150 DPI)


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fig 6 215x79mm (150 x 150 DPI)


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TOC 118x59mm (150 x 150 DPI)


Structural slow waves: parallels between photonic crystal and

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