Plasmons Compressing the Light – a Jewel in the Treasure Chest of Mark

Nanophotonics 2021; ▪▪▪(▪▪▪): 1–12

Perspective

Jacob B. Khurgin* Plasmons compressing the light – a jewel in the treasure chest of Mark Stockman’s legacy https://doi.org/10.1515/nanoph-2021-0251 structures has been discussed for many years [3], but it took Received May 19, 2021; accepted June 17, 2021; Mark Stockman to bring it closer to reality when he first published online July 1, 2021 introduced concepts of adiabatic focusing [4] and focusing using selfsimilar chains of nanoparticles [5]. These ideas Abstract: Among all the contributions made by Mark have been further developed by the community and have Stockman, his work on concentrating the light energy to established the means of both effective focusing of light unprecedented densities is one of the most remarkable into miniscule volumes [6] and, most remarkably, accel- achievements. Here it is briefly reviewed and a relatively erating by orders of magnitudes the spontaneous emission novel, intuitive, and physically transparent interpretation rates via the Purcell effect [7, 8]. These achievements are of nanofocusing using the effective volume of hybrid already finding applications in nanoscale sensing and in coupled modes formalism is presented and the role of the development of efficient single photon sources. Landau damping as the main limiting factor is highlighted. In 2003 Mark Stockman, working with colleagues Keywords: adiabatic; impedance matching; nanofocusing; Kuiru Li and David Bergman, was the first one to come up plasmonics. with this groundbreaking idea that using a chain of selfsimilar (i.e. having the same shape) nanoparticles with progressively decreasing sizes can focus the light into the 1 Introduction gap between the smallest nanospheres where the local fields are enhanced by orders of magnitude [5]. In this work Mark and his colleagues skillfully utilized the fact that the Professor Mark I. Stockman of Georgia State University, a resonant frequencies of subwavelength localized surface towering figure in photonics and a force behind numerous plasmons (LSPs) depend only on the shape and not on the trailblazing advances in nanophotonics, and more specif- size of the modes, and that allows efficient resonant ically plasmonics, passed away in November 2020. It will transfer of power along the chain in both directions; i.e. it take time for the world of science to recognize what a sig- can enhance both absorption and emission of light. nificant and irreparable loss it has been and to properly Shortly thereafter, in 2004 Mark expanded the concept assess the breadth and depths of Mark’s contributions of adiabatic light concentration to propagating surface made over the course of five decades. Still, it is not too early plasmon polaritons (SPPs) in the tapered plasmonic to recognize Mark’s key contributions that have the deepest waveguide, and it is this type of light concentration that impact on the course of progress in photonics. has become known as “nanofocusing” in a narrow sense. Mark’s work on coherent sources of surface plasmons As the width of the waveguide and the effective wavelength (SPASER) [1, 2] is probably the most widely recognized one, of SPP are reduced, the impedance gradually increases and but it would be a great disservice to his legacy to omit other, reaches values almost comparable to those of atomic and in my view, no less important contributions to nano- molecular dipoles. The key to efficient transfer of energy is photonics. Among them, his innovative ideas in the field of the fact that the width (and hence impedance) are chang- nanoscopy and nanofocusing do stand out and thus ing adiabatically; hence the reflection is almost nonexis- deserve a special attention. The whole idea of nano- tent and the only limitation to efficiency is metal focusing (i.e. concentrating the light into subwavelength absorption. In his work Mark has taken previous work on volumes) using photonic and plasmonic nanoscale polariton nanofocusing [9] farther along to practical implementations achieved since his pioneering contribu-

*Corresponding author: Jacob B. Khurgin, Johns Hopkins University, tion [6]. Baltimore, MD, 212118, USA, E-mail: [email protected]. https://orcid. Concurrently with the aforementioned work on light org/0000-0003-0725-8736 concentration, Mark Stockman (in collaboration with Peter

Nordlander’s group at Rice University) has pointed that coupled oscillator model will be described to explain the significant enhancement can be attained in plasmonic di- energy concentration in plasmonic dimers and trimers. A mers. As discussed in [10] the dimer acts as an optical an- novel concept of coupled mode having two different tenna with low impedance while its gap mode acts as a “effective volumes”–one as a cavity and one as an an- high impedance cavity [11] and was exploited in achieving tenna – will be developed and related to the local density of record high emission rates for quantum emitters [8]. states. A coupled mode model will be then further extended And yet, despite the large number of experimental to the “hybrid modes” incorporating quantum oscillators and results, computer models, and theoretical analyses, the metallic nanoparticles. It will be shown that ultimately the question of what kind of ultimate enhancement can be degree of enhancement is limited by the minimum size of the attained using the principle of adiabatic nanofocusing re- smallest nanoparticle (∼10 nm) at which the onset of Landau mains open. Furthermore, various groups approach it from damping broadens the linewidth of resonance and thus limits different angles. Some, approaches that can be traced to the enhancement. The last section contains the conclusions electromagnetic (EM) theory emphasize the impedance that further emphasize foresightofMarkStockmanandhis matching between antenna and atom or molecule (to contribution to nanophotonics. which we shall refer as a quantum oscillator) [12–14], while others concentrate on local density of states [15], and yet other approaches treat the complex plasmonic structures 2 Quantum oscillator and its as coupled oscillators [11, 16]. The nanophotonics community could benefit from a unified theory of adiabatic impedance nanofocusing (and the reverse process of emission enhancement) which would show the equivalence of The quantum oscillator, which can be an atom, a molecule, different approaches using a very simple and transparent and exciton, or a quantum dot, and can be both an model in a way Mark Stockman always presented his work, absorber and emitter, was first treated as a two-level sys- i.e. with clarity, depth, and succinctness. These days tem by Greffet et al. in [14]. The impedance was introduced computing power at our disposal allows us to resolve many using Green’s function and related to local density of physical problems with speed and efficiency which would states. Here the somewhat different approach leading to have been seen beyond the realm of possible only a few the same circuit model is considered. In the presence of decades ago. But with the growing preponderance of nu- external field Eext with frequency ω the two level system merical method something is in danger of being lost. That with resonant frequency ω21 and transition matrix element something is the ability to present a simple a clear picture r12 = 〈1|r|2〉, shown in Figure 1a, develops a dipole moment of the physics behind complicated phenomena, and it is 2e2r2 ω p = 12 21 E , (1) this ability that made Mark Stockman stand out among his a ℏ(ω2 − ω2 − iωγ ) ext 21 a peers. Alas, Mark is no longer available to undertake this task, so, this paper is an attempt to explain light con- where γa is the dephasing rate. The physical size (effective certation on nanoscale to a wide audience to the best of my radius) of the oscillator can be approximated as ra = 1/2 rather modest abilities, with a hope that this work will 〈1|r2|1〉 and then the current “flowing through the quan- serve as a tribute to Mark’s rich legacy. tum oscillator” is Ia∼− iωpa/2ra, while the voltage on it is This work has been conceived as neither a review nor a roughly Va∼2raEa. Hence the impedance can be found as comprehensive tutorial. It combines previously known V G−1r2 a 2 0 a 2 2 Za = = [γ ω + i(ω − ω )] (2) concepts with a number of new insights which may help I ω ωπr2 a 21 a 21 12 the reader to get an intuitive feel for nanofocusing and 2 −5 appreciate Mark Stockman’s foresight. Plasmonics is a where G0 = 2e /h = 7.75 × 10 S is quantum conductance. f = m ω r2 / ℏ truly multidisciplinary science and the same problem of Introducing the oscillator strength 12 2 0 12 12 3 and light concentrating can be approached from the point of using the typical relation between transition frequency and 2 2 view of electrical engineer (impedance matching), quan- the effective radius ℏω21∼ℏ /2m0ra we obtain tum optics (density of states), or physical optics (absorp- G−1 γ ω ω 0 21 tion and emission cross-sections). Here I attempt to show Za = [ + i − i ] (3) 3πf12 ω21 ω21 ω how all these approaches lead to the same results, which, in my view has certain pedagogical value. First I briefly with three terms inside the brackets corresponding to the review the idea of nanofocusing as impedance matching resistance, inductance, and capacitance, all connected in between free space and the quantum oscillator. Then a series (Figure 1a), as first noted in [14]. J.B. Khurgin: Plasmons compressing the light 3

E forbidden transition. (Note that this mismatch is essentially a ratio between the diameter of diffraction limited focused p beam and resonant absorption cross-section of the atom or C L 21 - + molecule indicating the connection between the electrical engineering and physical optics approaches). Clearly, to Za match impedances one has to resort to using an antenna that serves as impedance transformer. Such a transformer E can be a nanoparticle supporting the LSP. Intuitively the dimensions of that mode should be on the order of the geometric mean between wavelength and atomic dipole, i.e. few tens of nanometers.

E LK Z Cmetmem t Cd p 3 Nanofocusing as impedance matching

Figure 1: (a) Two-quantum oscillator and its circuit model, Let us now determine the impedance of an LSP of the (b) localized surface plasmon (LSP) mode on a metal nanoparticle plasmonic nanoparticle shown in Figure 1b. A spherical and its circuit model. nanoparticle with a diameter d is shown in Figure 1b, but On resonance one obtains the theory developed here can be easily applied to the other G−1 η shapes. Following the theory of Engheta and Alu [12], we 2 0 0 Za = = (4) consider a nanoparticle made from a metal with dielectric 3π Qaf12 6πα0Qaf12 2 2 constant ϵm(ω)=ϵrb − ωp/(ω + jωγ), where εrb is the where Qa = ω /γ, α = 1/137 is the ﬁne structure constant 12 0 “background” permittivity associated with interband and η = 377Ω is the vacuum impedance. With a reason- 0 transitions, ωp is the plasma frequency, and γ is the scat- ably high Q the impedance of the quantum oscillator is on tering rate in metal, surrounded by a dielectric with relative the order of η0, however when one considers transfer of permittivity εd. The AC conductivity of metal is energy in space, it is the impedance per unit area that 2 σ(ω) =−iωϵ (ω)ϵ matters [14, 17], i.e. dividing (2) by ra obtains on resonance m 0 2 2 −1 ωpω ωpγ Va G η 2 0 0 =−iωϵrbϵ + i ϵ + ϵ (8) Z′ = = = (5) 0 2 2 0 2 2 0 a I πQr2 πα Q r2 ω + γ ω + γ a 12 2 0 a 12

The first term corresponds to the capacitance, Cmat the second to the kinetic inductance, LK and the last one to the r12∼1Å for the allowed transitions in the visible/near infrared (IR) region of spectrum and assuming Q in the 101– resistance ρ, all connected in parallel as displayed in 3 2 Figure 1b. In addition, there is also a capacitance associ- 10 range one obtains Za′ ∼1 − 100kΩ/nm . This result is fi C similar to the one obtained in [14] using Green’s function, ated with the eld in the dielectric d. The capacitance d albeit obtained using classical dipole analogy. At the same scales with diameter while inductance and resistance are ω ≫ γ time the impedance per unit area of the free space is [14] inversely proportional to it. Overall, assuming we can write for the impedance ω2 π 2 2 Z′ = = η ∼1mΩ/nm (6) 2 2 0 3 2 0 ω ω γ 6πϵ0c 3λ −1 p p Z (ω) =−iω(ϵrb + βϵd)ϵ BCd + i ϵ BLd + ϵ BLd (9) 0 ω 0 ω2 0 The impedance mismatch between the quantum B B oscillator and free space is therefore where coefficients c and L are shape-dependent coefficients, both on the order of unity and coefficient β cor- Z′ 2 λ2 fi a ∼ , (7) responds to the fraction of electric eld in the dielectric. Z′ π2α Q r2 0 4 0 12 Clearly the resonance occurs when ω or anywhere between 6 and 8 orders of magnitude, even for √P̅̅̅̅̅̅̅̅ ω = ω0 = (10) the allowed transition, and more than that for weaker, (BC/BL) ϵrb + βϵd 4 J. B. Khurgin: Plasmons compressing the light

For the spherical particle one can show that BC =BL and circuit theory, one can expect to achieve impedance match β =2 so that a familiar expression for the LSP resonance when√̅̅̅̅ the impedance of LSP mode in nanoparticle ′ emerges Z = Za′Zo′, and characteristic size of the nanoparticle is

ωp 1/6 ω = √̅̅̅̅̅̅̅ d∼V1/3∼[λ4r2 Q2Q α / π2] ∼ − 0 (11) p 12 a 0 4 20 40 nm (15) ϵrb + 2ϵd for a wide range of Q’s. These numbers are not far from the On resonance ones found optimal in [19, 21], but it has also been shown this λ2 ω λ2 Z(ω ) = p = p Qη does not lead to the absorption (emission) enhancement 0 0 (12) 2 4 5 2πλ0ϵ0γcBLd 2πλ0dBL exceeding 4Q i.e.afactorofabout10–10 . Therefore, if one uses the circuit analogy further, the impedance matching will As one can see, the impedance decreases dramatically improve if one uses chain impedance transformers as shown with the size of the nanoparticle. It should also be noted in Figure 2c – a suggestion first made by Mark Stockman and that the resonant frequency does not depend on the actual coworkers in [5]. The crude estimate of the field enhancement size of the nanoparticle because in (9) we neglect the is then that each successive step of nanofocusing enhances “normal” inductance due to magnetic field induced by the absorption by another factor of 4A2Q2 and in principle one current in the metal, LM∼μ d “connected in series” with 0 can expect a really enormous enhancement of absorption and L ∼ /ω2 ϵ d L L kinetic inductance K 1 p 0 . One can see that K= M Purcell factor. But that assessment neglects the fact that the d∼λ / π when p 2 , i.e. is equal to the skin depth which range of the particle dimensionsislimitedonbothsides.As amounts to less than 100 nm. Therefore the above results size of nanoparticle exceeds about 100 nm, radiative loss are valid only for d < 100 nm. For larger nanoparticles the becomesthechiefsourceofdamping.Atthesametimewhen resonance shifts to the red part of the spectrum and even- thenanoparticledimensionsbecomesmallerthan5–10 nm, tually the impedance becomes Landau damping [22–25], which also can be thought of as a i ω2 γ surface-assisted absorption [26, 27] sets in causing increase of −1 p Z (ω)∼ − iω(ϵrb + βϵd)ϵ d + + ϵ d , (13) 0 ωμ d ω2 0 loss and reduction of Q. 0 The rest of this paper is devoted to exploring focusing ω ∼c/ϵ1/2d leading to the resonance condition 0 d , i.e. reso- with plasmonic dimers [28] and trimers [29]. While using nance being determined by the size (length) rather than by impedances conveniently relates the issue to the classical the material. In other words, one deals with an antenna impedance matching in electrical engineering, it is also rather than with a “plasmon.” But as long as we operate in worthwhile to explore the same issue from a different point the plasmonic regime the impedance (12) decreases with of view – by developing an analytical coupled mode model −1 the size of nanoparticle as Z∼d . for the chain of nanoparticles [30–32]. Previously, this Now, when one considers impedance mismatch one model has only been used for plasmonic dimers and the has to realize that the energy transfer takes place only role of damping was not explored, but in this work it is across a fraction of the surface; hence it makes sense to extended to three and more nanoparticles. Doing so allows define the impedance per unit area as shown in [14] one to see clearly how the adiabaticity allows one to in- λ2 crease degree of field enhancement, and to concurrently ′ p Q Z = 4Z/πd2∼c η (14) explore the limits of this enhancement λ2 Vω 0 i.e. proportional to the effective local density of states ρ∼Q/Vω. Therefore, matching the impedance is equivalent – to matching the densities of states, a fact first noticed by 4 Coupled modes theory Novotny [15, 18]. If one places the quantum oscillator near plasmonic dimer the surface of the nanoparticle as shown in Figure 2a, the local field enhancement will facilitate energy transfer to First, one shall note that the dipole moment of the LSP on a the quantum oscillator thus enhancing absorption [19]. The nanosphere of Figure 2a is same enhancement will of course be observed in the ϵ (ω) − ϵ 3 m d emission as spontaneous emission gets enhanced by the p(t) = 4πϵ0ϵdr Eext(t) (16) ϵm(ω) + 2ϵd Purcell effect [20, 21]. From the point of view of circuit modeling, the nanoparticle serves as an impedance where Eext(t) is the external field. By assuming that one transformer as outlined in Figure 2b. Intuitively following operates close to the resonance, ϵm(ω0) ≈ −2ϵd, we obtain J.B. Khurgin: Plasmons compressing the light 5

Eextext Z 0 ImpedanceImpedance Quantum Free Z E (b) transformertransformer oscillator maxma Space - + (a) - + Z0 Za Za Z Veﬀ

Figure 2: (a) LSP mode enhancing field near Free - + the quantum oscillator, (b) circuit model of (c) Space absorption/emission enhancement as impedance matching and (c) impedance Z0 Impedance Transformer Za matching with a chain of nanoparticles.

ω2 It is also possible to relate a to the magnitude of the p(t) ≈ α 0 E (t) (17) 0ω2 − ω2 − jωγ ext 0 electric ﬁeld on the surface of the nanoparticle, where the LSP resonant frequency ω is given in (11) and 1/2 0 2 2 −1/2 Emax = ( ) (ϵ0V) a , (24) the DC polarizability is ϵd 2 + ϵrb/ϵd 9ϵ ϵ2 α = 0 d V (18) and introduce the effective volume ϵrb + 2ϵd Vϵd(2 + ϵrb/ϵd) ϵ = ϵ = α = ϵ V Veff = (25) (for the case when rb d 1, 3 0 ). It is clear that (17) 4 is a solution of a differential equation for a driven harmonic oscillator so that

2 2 1 2 d p dp a = ϵ ϵdE V (26) =−ω2 p − γ + αω2 E (19) 2 0 max eff dt2 0 dt 0 ext Finally, one obtains Now, it is possible to relate the dipole and the total 1/2 energy of the LSP mode as ϵ0ϵdVeff α1 = A( ) (27) p2 p2 2 ULSP = (2 + ϵrb/ϵd) = (2 + ϵrb/ϵd) πϵ ϵ r3 ϵ V 24 0 d 18 0 Where the “material” factor is = |a|2 (20) ϵ A = 6 d + ϵ ϵ (28) where a new variable a is a square root of energy in LSP 2 rb/ d mode (classical analogue of plasmon annihilation oper- For ϵrb = ϵd = 1, A=2, and for other, non-spherical ator). Now shapes A, renamed as “material and shape factor”,will still 1/2 a iα Q 18ϵ V be on the order of unity. Note that at resonance = 1 Eext, p = a( 0 ) (21) 2 + ϵrb/ϵd and therefore E = iF E (29) and (19) becomes max 1 ext 2a a where the maximum enhancement by a single nanoparticle d 2 d 2 =−ω a − γω + α1ω Eext, (22) F =AQ. Thus with a single nanoparticle maximum ﬁeld dt2 0 dt 0 1 enhancement is just on the order of Q factor, i.e. no more where than 10–20 times. Note that the ratio between the effective 1/2 volume and nanoparticle volume (25) can be written as ϵd α = ϵ ( ) (ϵ V)1/2 2 1 3 d 0 (23) Veff = 3Vϵd/2A, i.e. the two volumes roughly have the same 2(2 + ϵrb/ϵd) order of magnitude. 6 J. B. Khurgin: Plasmons compressing the light

2 Let us now consider coupling between two nano- d as das α + α =−ω2 a − γω + κω2 a + 1√̅ 2ω2 E 2 0 s 0 s 0 ext particles separated by distance r12, as shown in Figure 3a. dt dt 2 The ﬁeld of the larger nanoparticle at the center of the (34) 2a a α − α d a 2 d a 2 1 2 2 smaller nanoparticle is =−ω aa − γω − κω aa + √̅ ω E dt2 0 dt 0 2 0 ext 1/2 3V1 A V1, eff E = E , = ( ) a (30) 12 max 1 πr3 πr3 ϵ2 ϵ ϵ 1 The “effective polarizabilities” for two super-modes 4 12 12 d 2 0 d are and the coupling coefficient is obtained by substituting α + α ϵ ϵ V 1/2V1/2 + V1/2 ϵ ϵ V 1/2 (30) into (22) 1 2 0 d eff 1,eff 2,eff 0 d s, ant αs = √̅ = A( ) √̅ = A( ) 2 2 2 2 ϵ ϵ V 1/2 A V 1/2 κ = α E /a = A( 0 d 2, eff ) ( 1, eff ) 12 2 12 1 πr3 ϵ2 ϵ ϵ 1/2 2 12 d 2 0 d α1 − α2 ϵ0ϵdVa, ant αa = √̅ = A( ) A2 (V V )1/2 A2 ϵ2 (V V )1/2 2 2 = 1, eff 2, eff = 3 d 4 1 2, 3 2 1/3 1/3 3 2 πr ϵd 2ϵd 2A 3 (V + V ) (35) 12 1 2 K1/2 = A , where the “effective volume as an antenna” has been 3 (31) (1 + K1/3) introduced as

1 1/2 1/2 2 1 − 2 where K = V1/V2. It is easy to see that κ21=κ12=κ, hence one V = (V ± V ) = V ( ± K 1/2) s(a), ant 1, eff 2, eff 1, eff 1 (36) can write coupled mode equations 2 2 V ≈ d2a da Clearly, for vastly dissimilar nanoparticles s( a), ant 1 =−ω2 a − γω 1 + κω2 a + α ω2 E V / ’ dt2 0 1 dt 0 2 1 0 ext 1, eff 2 hence the dimer s mode performance as an antenna (32) is determined by the larger nanoparticle in the dimer as d2a da 2 =−ω2 a − γω 2 + κω2 a + α ω2 E seen in Figure 3a. The steady state solutions of (34) are dt2 0 2 dt 0 1 2 0 ext ω2 0 √̅̅̅̅ 1 Now if one introduces symmetric and antisymmetric as =αs Eext ≈αs Eext ω2 (1−κ)−ω2 −jωγ 2( 1−κ −ω/ω )−jQ−1 super-modes 0 0 ω2 a + a a − a 0 1 a = 1√̅ 2, a = 1√̅ 2 , aa =αa E ≈αa √̅̅̅̅ E s a (33) ω2 ( +κ)−ω2 −jωγ ext +κ −ω ω −jQ−1 ext 2 2 0 1 2( 1 / 0) (37) adding and subtracting two lines in (32) yields Now, effective volume as a cavity for each mode can be E Eext ext found from the fact that the energy is equally divided be- κ tween two nanoparticles, and since the maximum field is at Vant Emax the surface of a smaller nanoparticle, (a) Emax V1 V2 V eﬀ −1 cavity Vs(a), eff = 2V2, eff = 2V1, eff K (38) κ antenna The dimer mode behaves as a large antenna, and at r12 thesametimeitbehavesasasmallcavity,whichis E Eext ext exactly what is needed for strong field enhancement κ κ Vant as shown schematically on the right hand side of Emax (b) Emax Figure 3a. V1 V2 V cavity When the light with a resonance frequency ω = ωs = 3 Veﬀ √̅̅̅̅ ω0 1 − κ is incident onto the dimer, the amplitudes of LSP κ κ antenna in two modes are

as = jαsQE Figure 3: Energy transfer in (a) plasmonic dimer and (b) plasmonic ext trimer. On the right hand side the couple modes are represented as jαaQEext (39) aa = √̅̅̅̅ √̅̅̅̅ , oscillators with large volume as an antenna Vant (important for 2jQ( 1 + κ − 1 − κ ) + 1 coupling to the free space) and small volume as a cavity Veff (important for coupling to quantum oscillators). and the maximum field in the symmetric mode is J.B. Khurgin: Plasmons compressing the light 7

1/2 1/2 K ϵ0ϵdVs, ant 2 ). Obviously this cannot be accomplished in a dimer Emax, s = iQA( ) ( ) Eext 2 ϵ0ϵdVs, eff and one should consider moving on to more complex 1/2 schemes. Vs, ant = jF1( ) Eext (40) Vs, eff The enhancement occurs because for the supermode 5 Extension of coupled mode the antenna volume is determined primarily by the larger nanoparticls, while the effective cavity volume by the small theory to three and more nanoparticle. The additional enhancement factor is nanoparticles V 1/2 s, ant 1 −1/2 1/2 Fs = ( ) = (1 + K )K (41) The task of reducing the volume of smallest nanoparticle Vs, 2 eff while keeping coupling strong can be accomplished by

From this expression it follows that Fs can become inserting an intermediary nanoparticle between the “an- K arbitrary large as increases. However, there is also the tenna” V1 and “cavity” V3 = V1/K with its volume V2 = asymmetric mode, and in that mode the dipole of small 1/2 −1/2 (V1V3) = K as shown in Figure 3c. The coupled nanoparticle is directed against the one excited in the equations are symmetric mode, i.e. 2a a d 1 2 d 1 2 2 √̅̅̅̅ √̅̅̅̅ − =−ω a − γω + κω a + α ω E 1 −1/2 1/2 1 2 0 1 t 0 2 1 0 ext Fa =− (1 − K )K (2iQ( 1 + κ − 1 + κ ) + 1) (42) dt d 2 2a a d 2 2 d 2 2 2 2 Substituting (31) we obtain for the added enhancement =−ω a2 − γω + κω a1 + κω a3 + α2ω Eext (45) dt2 0 dt 0 0 0 in dimer, 2a a ⎡ d 3 2 d 3 2 2 =−ω a3 − γω + κω a2 + α3ω Eext K1/2 ⎢ dt2 0 dt 0 0 F = ⎢( +K−1/2) 2 ⎣ 1 2 where according to (31) ⎤ K1/4 (1−K−1/2) ⎥ κ = A ⎥ 3 (46) − √̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ √̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ⎦ (1 + K1/6) −3 −3 2iQ( 1+AK1/2(1+K1/3) − 1−AK1/2(1+K1/3) )+1 For large K coupling κ ≈ K−1/4 i.e. it is stronger than in (43) the dimer. The three super-modes are

In Figure 4a the total dimer enhancement Fd = |F F | is 1 2 a − a plotted for two different Q-factor for gold Q=10 and silver a = 1√̅ 3 ; 0 2 Q=20 (also shown as dashed liens is the enhancement by a √̅ a + a + a |F | K 1 2 2 3 single nanoparticle 1 )As increases, i.e. the volume of a+ = (47) 2 smaller nanoparticle decreases, the coupling decreases as √̅ −1/2 a − a + a κ∼K Fa Fs 1 2 2 3 .Asaresult, approaches ,henceonesimulta- a− = , neously excites both modes, and the enhancement saturates. 2 √̅ √̅ 2 ThesaturationsetsinwhenQκ∼1orK∼(QA) .ForK >> 1 with resonance frequencies ω0, ω0 − 2κ, and ω0 + 2κ respectively and polarizabilities K1/2 2iQAK1/2 F ≈ [ + K−1/2] ≈ iQA = F 2 2 1 (44) 1/2 2 ( + K1/3)3 α − α ϵ ϵ V 1 α = 1√̅ 2 = A( 0 d 0, ant) 0 2 2 The total enhancement in dimer is therefore √̅ 1/2 2 2 2 α + α + α ϵ ϵ V Fd = |F | = Q A fi 1 2 2 3 0 d +, ant 1 . Thus in the dimer the eld enhancement α+ = = A( ) (48) can be thought to occur in two steps – first large nano- 2 2 √̅ particle gets excited and then its field excites the smaller 1/2 α1 − 2α2 + α3 ϵ0ϵdV−, ant QA α− = = A( ) one, each step has the same enhancement, .Onecan 2 2 see now that in order to provide larger enhancement one must increase coupling (i.e. reduce ratio K)whilekeep- The effective volumes as antennas for three modes, ing smaller nanoparticle volume small (i.e. having a high shown in Figure 3b are 8 J. B. Khurgin: Plasmons compressing the light

(a) (b) Ag dimer Ag trimer

Au dimer Au trimer Single nanosphere

Ag dimer Au trimer

Au dimer Single nanosphere

Figure 4: Field enhancement in (a) plasmonic dimers, (b) plasmonic trimers, and (c) zoomed in practical region where enhancement is not impeded by Landau damping.

1 2 1 2 V = (V1/2 − V1/2 ) = V ( − K−1/2) the angstrom-size quantum oscillators. The maximum 0, ant 1, eff 3, eff 1, eff 1 2 2 fields on resonance are √̅ 1 −1/4 −1/2 2 1/2 V+, = V , (1 + 2 K + K ) (49) V ant 4 1 eff E = iQA( 0, ant) E max, 0 V ext √̅ 0, eff 1 2 V = V ( − K−1/4 + K−1/2) (52) −, ant 1, eff 1 2 iQA V 1/2 4 √̅̅̅̅̅̅̅ ±, ant Emax, ± = √̅ ( ) Eext 2jQ(1 − 1 ∓ 2κ ) + 1 V±, eff Once again, the effective volume as an antenna (i.e. effective dipole) is determined by the largest nanoparticle Thus additional field enhancement with trimer an- hence the trimer can effectively couple to the free space. tenna is therefore Operating at resonance frequency we immediately obtain √̅ K1/2 + K−1/4 + K−1/2 −1/2 1 2√̅̅̅̅̅̅̅ a0 = jαsQEext F3 = [2(1 − K ) − √̅ 4 2jQ(1 − 1 − 2κ ) + 1 jαaQEext √̅ (53) a = √̅̅̅̅̅̅̅ −1/4 −1 2 + √̅ 1 − 2K + K / 2jQ(1 − 1 − 2κ ) + 1 (50) − √̅̅̅̅̅̅√̅̅ ] 2jQ(1 − 1 + 2κ ) + 1 jαaQEext a− = √̅̅̅̅̅̅√̅̅ 2jQ(1 − 1 + 2κ ) + 1 In Figure 4b the total dimer enhancement Ft = |F1F3| are plotted for two different Q-factor for gold and silver For The effective volumes as cavities are K >>1 the added enhancement saturates at −1 V0, eff = 2V3, eff = 2V1, eff K ; V±, eff = 4V3, eff 2 F3 = (QA + 1/4)QA ≈ (QA) (54) −1 = V , K 4 1 eff ; (51) 3 and the total trimer field enhancement is Ft =(QA) .Itis These volumes, determined by the smallest nano- not difﬁcult to make a conjecture that by going to larger particle are small which is advantageous for coupling to number of nanoparticle in a chain, an even larger J.B. Khurgin: Plasmons compressing the light 9

√̅̅̅̅̅ N enhancement (QA) can be attained. But the question is ℏω/2 a = U1/2 = p how larger can K be, i.e. how large can be V1 and how small a a a (56) er12 can be VN? Maximum size of V1 is on the scale of λ/10 − λ/5, i.e. he point where radiative damping reduces Q. One as- Substituting it into Eq. (1) one obtains 3 3 √̅̅̅̅ sumes it is anywhere between (50 nm) and (100 nm) , 2 1 er12ω V ( )3 ( )3 aa = E (57) minimum size of N is between 5nm and 10 nm , i.e. 2 2 ext 2ℏω [ω − ω − iωγa] where Landau damping becomes the dominant damping 12 mechanism reducing Q. Therefore, maximum K is only and the differential equation for aa is × 3 about 2 10 .Then,ascanbeseeninFigure4cfor 2a a d a 2 d 2 spherical nanoparticle there is very little difference in =−ω aa − γaω + αaω Eext (58) dt2 12 dt the additional enhancement between dimer and trimer and also for Au and Ag. The additional enhancement is where √̅̅̅̅ anywhere between the factor of 12–15 for dimer and 18– αa = er12/ 2ℏω (59) 20 for trimer. This perhaps explains why trimers and more complex structures have not found many applica- If we now compare (59) with (27) we can even introduce tions as slightly improved enhancement does not justify the effective volume for the quantum oscillator increased complexity. At any rate, the total field e2r2 12 −2 −1 2 enhancement in dimer or trimer reaches about 300 for Au V , a = ≈ 2A ϵ α λr (60) eff A2ϵ ϵ ℏω d 0 12 and 800 for Ag which enhances the absorption by as 0 d much as 105 for Au and 5 × 105 for Ag. It is important to Since in the visible/near IR region the dipole for an note here that Mark Stockman was one of the first to note allowed transition r12∼1Å, the effective volume is on the that Landau damping can limit degree of enhancement scale of about 10 nm3 and less so for the weaker transitions. attainable in plasmonics [33]. Thus, two level system embedded in the dielectric can be treated in the same way as any other oscillator, described by its effective volume, polarizability, resonant frequency, 6 Effective volume for quantum and damping rate. Now, the coupling of energy into a quantum absorber/emitter and its use for absorber using a nanoantenna consisting of N nano- quantifying absorption and spheres can be approached as coupling into a hybrid supermode with N+1entities– N nanospheres and a emission enhancement quantum oscillator. Such a hybrid mode has effective dipole that exceeds r by many orders of magnitude. In As has been mentioned above, it had been shown that 12 Figure 5a a dimer antenna with nanospheres of volumes quantum entity, emitter, or absorber can be represented V and V coupled into the quantum absorber with byacircuitmodelwithL,C,andRisseriesandcharac- 1 2 effective volume V a is represented as a hybrid trimer. terized by impedance Z(ω). Here and alternative is pre- eff, If the absorber is resonant with the antenna, ω =ω ,one sented. This alternative consists of representing 12 0 can see that the optimum coupling will be achieved quantum emitter/absorber as a classical dipole oscil- 1/2 when V , =(V , V , a) . The coupling coefficient, lator, characterized (in addition to resonance frequency 2 eff 1 eff eff according to (46) is κ = AK1/4(1 + K1/6)−3 where K = ω and damping rate γa) by the effective volume V a. 12 eff, V /V This model can then be applied to describe absorption 1, eff a, eff and by performing all the steps outlined in a and emission by quantum oscillator as well as its previous section one obtains for the amplitude of a = F α E coupling to the antennas. The energy of a quantum quantum oscillator a t a ext. Introducing the in- I = E2 ϵ1/2/η oscillator and its dipole are related as tensity of the incoming light in ext d 0, the rate of energy transfer to the atomic oscillator is mω2p2 ℏω p2 a a 2 Ua = = = |aa| (55) 2 2 2 U e2r2 2e f 2 e r d a 2 2 2 12 12 12 = γ|aa| = γQ |F (K)| I = σ I (61) dt 3 1/2 in abs in ℏωϵ0cϵd Therefore, we can introduce the “amplitude” of the atomic motion as a square root of its energy, or, essentially, In the absence of nanoantenna the absorption cross a classical analogue of annihilation operator. section is 10 J. B. Khurgin: Plasmons compressing the light

I 10 5 in (b) 20 (a) κ κ 2 15

|a3| cle (nm) V V - + 4 1 2 10 10 onEnhancement nanopar κ κ 5 Size of a smaller Absorp 10 3 0 10 -4 10 -3 10 -2 10 -1 10 0 10 1 Oscillator strength

Figure 5: (a) Nanofocusing in a hybrid trimer consisting of a plasmonic dimer and quantum oscillator. (b) Enhancement of absorption. Shaded region represents a practical range of enhancement not impeded by excessive Landau damping.

e2r2 12 the hard limit is set by Landau damping that requires the σa = Q (62) 1/2 ℏϵ0cϵd smallest nanoparticle (or tip of waveguide) to be at least 5–10 nm to avoid excessive loss. Therefore, the absorption cross-section with dipole antenna gets enhanced σ = |F (K)|2σ abs 3 a (63) 7 Conclusions Note that here the decay rates of atomic oscillator and In this work Mark Stockman’s contributions to just one area of LSP has been assumed to be equal. If they are not equal, one photonics has been highlighted. Thanks to him, the idea of has to introduce the weighted average γ which will change nanofocusing took root in the community, and plenty of the results but not all that significantly unless γa > γ,whichis theoretical and experimental works have followed. Perhaps highly unlikely given broad linewidths of LSP modes. The inevitably, the original idea often became compromised by question is how large can the ratio K be? Assuming that 3 3 excessive hype and unsustainable claims, but the core of V , ∼(100 nm) and expressing V , a ≈ 10 nm × f using 1 eff eff 12 Mark predictions has been indeed confirmed. Plasmonics (60) we obtain K ∼105/f . The enhancement is plotted max 12 undeniably offers unique chance to confine light into versus oscillator strength in Figure 5. Note that oscillator minuscule volumes and achieve very strong fields which can strength may exceed unity due to small effective mass as is be used for probing the matter on a nanoscale [36], if not for the case in semiconductor quantum dots [34]. As it is true for more exotic applications in photonic integrated circuits and the entire field of plasmonics, the enhancement is the stron- light sources. Excessive loss due to Landau damping does gest for the weak absorbers with smaller oscillator strength prevent us from fully realizing the nanofocusing potential, [35]. Unfortunately, once again, the largest enhancements are butitIsnotentirelyimpossiblethattheissueoflosswillbe not attainable since the required size of the smaller nano- addressed in the future. Before I conclude, I should note that particle V1/3, also shown in Figure 5b becomes so small (less 2 Mark Stockman himself never oversold his ideas and habit- than 10 nm) that Landau damping reduces Q-factor dramat- ually offered a sober and critical assessment of trendy topics, ically. As a result, the largest enhancement for Au is on the be that negative refraction [37, 38], epsilon near zero mate- scale of 2.5 × 104 and for Ag is should be 4 times as large. The rials [39], spoof plasmons [40] and other. His critical eye and enhancement (63) represent the ratio of the squares of sober mind will be just as missed as his unabashed enthu- effective dipole moments of a quantum absorber/emitter siasm for science, never ending quest for discovery, and his and the hybrid mode. Obviously, the enhancement of the zest for life which he preserved all the way to the end. rate of spontaneous emission will be similar, as explained in [21]. The enhancement of Raman scattering the can reach 1010. In the end, whether it is efficiency of Acknowledgement: As always, in this work I was aided, absorption or emission using adiabatic concentration, supported and comforted by my steadfast colleagues Prof. J.B. Khurgin: Plasmons compressing the light 11

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