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 self- similar (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 Open Access. © 2021 Jacob B. Khurgin, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 International License. 2 J. B. Khurgin: Plasmons compressing the light 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 com- munity 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).