An intriguing branching of the maximum position of the absorption cross section in Mie theory explained
11 An intriguing branching of the maximum position ofthe absorption cross section in Mie theory explained I LIA
L. R
ASSKAZOV , P. S
COTT C ARNEY , AND A LEXANDER M OROZ The Institute of Optics, University of Rochester, Rochester, NY 14627, USA Wave-scattering.com (e-mail: [email protected]) * Corresponding author: [email protected] May 18, 2020
A potential control over the position of maxima of scat-tering and absorption cross-sections can be exploited tobetter tailor nanoparticles for specific light-matter inter-action applications. Here we explain in detail the mech-anism of an appreciable blue shift of the absorptioncross-section peak relative to a metal spherical parti-cle localized surface plasmon resonance (LSPR) and re-maining scattering and extinction cross sections. Sucha branching of cross sections maxima requires a cer-tain threshold value of size parameter x ≈ and isa prerequisite for obtaining high fluorescence enhance-ments, because the spectral region of high radiative rateenhancement becomes separated from the spectral re-gion of high non-radiative rate enhancement. A con-sequence is that the maximum of the absorption crosssection cannot be used as the definition of the LSPR po-sition for x (cid:38) . The electromagnetic theory of light scattering from small par-ticles with sizes comparable with the wavelength of the incidentillumination has the long history [1]. In a number of applica-tions, the control of the absorption is preferable, which led to theextensive theoretical and numerical works in this direction [2–8],as well as various experimental applications [9–12]. Our recentstudy [13] has revealed that optimal metallo-dielectric core-shellparticles designed to obtain the highest possible fluorescenceenhancements have the spectral region of high radiative rate en-hancement red -shifted and well separated from the blue -shiftedspectral region of high non-radiative rate enhancement [Fig. 413]. Herein and below, any red or blue shift, unless specifiedotherwise, refers relative to the particle localized surface plas-mon resonance (LSPR). For the purpose of this letter, a LSPRwill be identified as a maximum of the total (extinction) crosssection. The maximum of radiative rate enhancement coincideswith the maximum of near-field (NF) [13] and the NF peak redshift has been known for a long time [14]. Contrary to that,there seem to be no reports of a blue shift of the maximum ofthe absorption cross section, σ abs , responsible for the maximumof non-radiative rate enhancement [13]. Below we explain indetail the mechanism underlying the puzzling and appreciable,yet so far overlooked, blue shift of the absorption cross-sectionpeak. Because Fig. 1 shows that such a pronounced blue shift Fig. 1. (a) Fundamental cross sections of a homogeneous Auparticle of radius r s =
170 nm calculated via Mie theory.Whereas the maxima and minima of the extinction ( σ ext ) andscattering ( σ sca ) cross-sections occur in unison, the absorp-tion ( σ abs ) cross section does not exhibit any maximum at thedipole ( (cid:96) =
1) and quadrupole ( (cid:96) =
2) peaks of σ ext . The firstmaximum of σ abs is largely blue shifted and occurs nearly atthe octupole ( (cid:96) =
3) peak of σ ext . The panels (b)-(d) show indetail the cross sections in the respective angular-momentumchannels. The quasi-static LSPR positions ω (cid:96) , defined implic-itly by ε s = − ( (cid:96) + ) / (cid:96) , are shown by pink vertical lines. Notethe reshuffling of the natural order of the maxima of σ abs of therespective multipoles.of the absorption cross section is intrinsic already for a homoge-neous spherical metal particle of sufficiently large radius r s , weshall here, for the sake of simplicity, focus only on homogeneous particles.The resulting cross sections in the Mie theory are given asan infinite sum over all momentum channels (cid:96) ≥ (cid:96) of one of polariza-tions ( A = E for electric (or TM) polarization, and A = M formagnetic (or TE) polarization) contributes the following partialamount to the resulting cross sections shown in Fig. 1(a), a r X i v : . [ phy s i c s . op ti c s ] M a y σ sca ; (cid:96) = π k | T A (cid:96) | , (1) σ abs ; (cid:96) = π k (cid:16) − | + T A (cid:96) | (cid:17) , (2) σ ext ; (cid:96) = − π k (cid:60) ( T A (cid:96) ) , (3) where T A (cid:96) are the T-matrix elements, (cid:60) takes the real part, k = π / λ is the wavenumber, and λ is the incident wavelengthin the host medium. For (cid:96) =
1, the above expressions canbe easily rephrased in terms of a particle polarizability, α , onsubstituting 2 ik α /3 for T E . In the case of a homogeneoussphere, the respective T-matrix elements in a given (cid:96) -th angularmomentum channel are [Eqs. (2.127) 15] T A (cid:96) = − m [ xj (cid:96) ( x )] (cid:48) j (cid:96) ( x s ) − j (cid:96) ( x )[ x s j (cid:96) ( x s )] (cid:48) m [ xh (cid:96) ( x )] (cid:48) j (cid:96) ( x s ) − h (cid:96) ( x )[ x s j (cid:96) ( x s )] (cid:48) , (4) where x = kr s is the dimensionless size parameter, r s is thesphere radius, x s = x √ ˜ ε s , where ˜ ε s = ε s / ε h is the relative di-electric contrast, with ε s ( ε h ) being the sphere (host) permittivity.In what follows, the host will be assumed to be air ( ε h = m = m = ˜ ε s for electric polarization, j (cid:96) and h (cid:96) = h ( ) (cid:96) are theconventional spherical functions (see Sec. 10 of Ref. [17]), andprime denotes the derivative with respect to the argument. TheDrude model is used throughout this letter for Au permittivity, ε = ε ∞ − ω p ω ( ω + i γ ) , (5) with high-frequency permittivity limit ε ∞ = ω p = γ = ε h = r s (cid:39)
70 nm, the maxima of all cross sections of the dipole termof the Mie series experience initially monotonically increasing red shift from the initial quasi-static LSPR position at ω . There-after the maximum of the absorption cross section (2) graduallyreverses its red shift relative to ω into an increasing blue shift,which for r s ≈
100 nm can be as large as 100 nm. The positionof the maximum of the absorption cross section becomes even-tually nearly constant with increasing size parameter x in a blue shifted region (centered around the frequency implicitly givenby Eq. (11) below) relative to ω .As suggested by Figs. 2(c) and 3, the modified long-wavelength approximation (MLWA) agrees very well with theexact Mie theory over entire size range up to x ≈ ∼ x ) and radiative reac-tion ( ∼ x ) terms. This is why the MLWA can account for a size-dependent red shift of the dipole LSPR, whereas the Rayleighapproximation cannot. Here we use the following MLWA formof the T -matrix which is valid in any given channel (cid:96) (cf. Eq. Fig. 2.
Evolution of the spectral positions of the maxima ofabsorption, extinction and scattering cross sections with in-creasing sphere radius r s of Au sphere in air for dipole (cid:96) = ω with increasing r s ,the absorption cross-section peak reverses its initial red shiftto an appreciable blue shift which remains essentially stablefor r s (cid:38)
100 nm. Note in passing that a spectral gap betweenthe LSPR (determined as the extinction cross section peak posi-tion) and the absorption cross-section peak position can be aslarge as 2 eV for r s ≈
200 nm.
Fig. 3.
Evolution of the spectral positions of the maxima ofquadrupole ( (cid:96) =
2) and octupole ( (cid:96) =
3) absorption, extinctionand scattering cross sections with increasing sphere radius r s .(A3) of [21] for (cid:96) = T E (cid:96) ∼ iR ( x ) F + D ( x ) − iR ( x ) , F = ˜ ε s + (cid:96) + (cid:96) , D ( x ) = (cid:18) (cid:96) − (cid:96) + ε s + (cid:19) ( (cid:96) + )( (cid:96) + ) (cid:96) ( (cid:96) − )( (cid:96) + ) x , R ( x ) = (cid:96) + (cid:96) ( (cid:96) − ) !! ( (cid:96) + ) !! ( ˜ ε s − ) x (cid:96) + · (6) Eq. (6) is derived by using asymptotic expressions for sphericalBessel and Hankel functions used in (4). It makes transparentthat T E (cid:96) in any given channel is determined solely by a sizeindependent quasi-static Fröhlich term F , a dynamic depolariza-tion term D ( ∼ x ), and a radiative reaction term R ( ∼ x (cid:96) + ).The vanishing of the size independent F in the denominatoryields the usual quasi-static Fröhlich LSPR condition, which determines the quasi-static LSPR frequencies ω (cid:96) . In the caseof Drude fit (5) of ε s one finds ω (cid:96) = ω p / (cid:112) ε ∞ + [( (cid:96) + ) ε h / (cid:96) ] .One refers, somewhat misleadingly, to a unitarity , if the substitu-tion of a given approximation to T E (cid:96) into the above equationsyields σ ext ; (cid:96) = σ sca ; (cid:96) + σ abs ; (cid:96) . The MLWA can be shown to satisfyunitarity. In contrast, the usual Rayleigh limit, which amountsto setting D ( x ) = R ( x ) ≡ T E in (6), yields a purely imaginary T E (cid:96) for real ε s (i.e., a purely real polarizability) and violates the unitarity [21].In what follows we shall focus on the dipole MLWA. Thehigher order multipole MLWA can be treated similarly. Onsubstituting Eq. (6) into Eq. (1) – Eq. (3), one finds the followingcross sections of the dipole MLWA contribution: σ abs ;1 = π k x (cid:0) x + (cid:1) (cid:61) ( ˜ ε s ) (cid:12)(cid:12)(cid:12) ˜ ε s + − ( ˜ ε s − ) x − i ( ˜ ε s − ) x (cid:12)(cid:12)(cid:12) , (7) σ sca ;1 = π k x | ˜ ε s − | (cid:12)(cid:12)(cid:12) ˜ ε s + − ( ˜ ε s − ) x − i ( ˜ ε s − ) x (cid:12)(cid:12)(cid:12) , (8) σ ext ;1 = π k x (cid:0) x + (cid:1) (cid:61) ( ˜ ε s ) + x | ˜ ε s − | (cid:12)(cid:12)(cid:12) ˜ ε s + − ( ˜ ε s − ) x − i ( ˜ ε s − ) x (cid:12)(cid:12)(cid:12) , (9) where (cid:61) ( ˜ ε s ) denotes the imaginary part of ˜ ε s . For (cid:61) ( ˜ ε s ) =
0, thecommon denominator | ∆ | of the dipole MLWA cross-sections(7)-(9) vanishes at ˜ ε s ≈ − − x (10) up to the order x , in which case ∆ ≈ O ( x ) . For (cid:61) ( ˜ ε s ) (cid:54) =
0, onecan approximate | ∆ | as | ∆ | ≈ (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − x (cid:19) (cid:61) ( ˜ ε s ) + x (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) x (cid:61) ( ˜ ε s ) (cid:12)(cid:12)(cid:12)(cid:12) .Eq. (10) imposes restriction on the real part of ˜ ε s [Eq. (B1) 21]explaining the observed initial size-dependent red shift of allcross sections (Fig. 2(a)).At the MLWA branching point at r s ≈
90 nm in Fig. 2(b),where the peak position of σ abs ;1 begins to deviate from theremaining cross sections, the size parameter x ≈ x ≈ x s = √ ˜ ε s x s ≈ r s ≈
70 in the Mie theory in Fig. 2(a)). Theappearance of the branching point and ensuing blue shift of theabsorption cross section (2) can be explained within the dipoleMLWA as follows:(i) For x (cid:29) x the maxima positions are governed by the firsttwo terms of the complex root of ∆ on the rhs of Eq. (10), aswas already observed by Bohren and Huffman [Sec. 12.1.116], yielding the red-shifted behaviour of the maximumof 1/ ∆ relative to ω with increasing x . The latter is thechief cause of the synchronized red-shifted behaviour of themaxima of all three fundamental scattering cross-sectionrelative to ω with increasing x up to x ≈
1. This is why,unlike the Rayleigh approximation, the MLWA can accountfor the size-dependent red shift of the dipole LSPR.(ii) As soon as x ∼ x , the x -term of ∆ in Eqs. (7) – (9) beginsto dominate. One can achieve another maximum of 1/ ∆ ifthe x -term is rendered as small as possible, i.e. ˜ ε s has to beideally +
1. The latter can, in the case of dissipative media
Fig. 4.
Evolution of the blue-shifted maximum of the dipole σ abs ;1 with increasing x . (a) The minimum of denominator and(b) a corresponding value of the numerator for absorptionand scattering cross sections of Eq. (7) and Eq. (8). A doublepeak of 1/ | ∆ | for r s =
170 nm, with the maximum at thefrequency implicitly given by Eq. (11), can be clearly identified.Note that the numerator of σ sca ;1 is smaller than that of σ abs ;1 at (11). Panels (a) and (b) also illustrate the well-known factwhy smaller particles have much larger absorption than largerparticles [16].(Au), be in principle achieved only with a suitably tailoredgain. In the present case we are left with the condition (cid:60) ( ˜ ε max ; abs ) ≈ (11) which means that the blue-shifted maximum of σ abs ;1 coin-cides with the condition of minimizing the radiative reac-tion (Fig. 4(a)). This causes a maximum of σ abs ;1 but not of σ sca ;1 ((Fig. 1). The reason why the other maximum of 1/ ∆ does not cause a maximum of σ sca ;1 is that the latter has | ˜ ε s − | in its numerator, which becomes very small, in con-trast to σ abs ;1 having (cid:61) ( ˜ ε s ) in its numerator (Fig. 4(b)). Note, (cid:60) ( ε max ; abs ) ≈ − r s ≈ (cid:60) ( ε max ; abs ) ≈ r s (cid:38)
107 nm.The condition (11) explains why the position of the blue-shiftedmaximum of σ abs ;1 remains substantially constant with increas-ing r s , soon after it branched off from the position of maximaof other two cross-sections. With the parameters of the Drudefit (5), the zero of (cid:60) ( ˜ ε s ) occurs at ω z = ω p / √ ε ∞ = ω p / √ ≈ (cid:60) ( ˜ ε s ) = ω rc = ω p / √ ≈ σ abs ;1 stabilizes around ω rc with increas-ing x . Obviously, the blue shift of absorption is possible onlyif (cid:61) ( ˜ ε s ) at the frequency implicitly given by Eq. (11) is suffi-ciently small, as in the present case of Au. Provided that (cid:61) ( ˜ ε s ) at (11) is sufficiently large, this may prevent ∆ from acquiringa local minimum and the blue shift of absorption may be ab-sent. The situation for (cid:96) > (cid:96) = R in Eq. (6) maintains its factor˜ ε s −
1. A slight modification to (cid:96) = (cid:96) -dependent factor ( (cid:96) + ) / [ (cid:96) ( (cid:96) − ) !! ( (cid:96) + ) !! ] of R ( x ) in Eq. (6) decreases much faster with increasing (cid:96) than the (cid:96) -dependent fac-tor ( (cid:96) + )( (cid:96) + ) / [ (cid:96) ( (cid:96) − )( (cid:96) + )] of D ( x ) . Therefore, thethreshold size parameter x value required for the MLWA σ abs ; (cid:96) to exhibit a blue-shifted maximum necessarily increases with in-creasing (cid:96) . Again, this is clearly seen in Fig. 3. Consequently, thebuild-up of the blue-shifted maxima of σ abs ; (cid:96) is not uniform in (cid:96) .For a given fixed x , only the lowest multipoles have blue-shiftedmaximum, whereas the maxima of the remaining multipoles continue to be red-shifted. The latter causes a rearrangement ofthe natural order of the (cid:96) -pole absorption maxima: the lowestabsorption peak of σ abs may be due to (cid:96) = (cid:96) = (cid:96) =
1, 2 appear (Fig. 1).Overall, the above condition (11) defines a rare location at whichthe total σ abs ( (cid:29) σ sca ) is the dominant contribution to σ ext andthe single-scattering albedo (the ratio of scattering efficiency tototal extinction efficiency) acquires its minimum (Fig. 1(a)).To conclude, the properties of small metal particles continueto surprise. Whenever one thinks a full understanding has beenreached, some unexpected connection, or an unnoticed prop-erty appears. We have demonstrated that at the size parametervalue x ≈ σ abs ;1 can be appreciably blue shifted relative to the quasi-static po-sition ω of the dipole LSPR. An obvious consequence is thatthe maximum of σ abs ;1 can no longer be used as the definitionof the LSPR position for x (cid:38) σ abs ; (cid:96) foran (cid:96) -pole increases with increasing (cid:96) . The latter causes a rear-rangement of the natural order of (cid:96) -pole absorption maxima(e.g. the lowest absorption peak of σ abs may be due to octupole( (cid:96) =
3) as shown in Fig. 1(a)). The blue shift of absorption is pos-sible only if (cid:61) ( ˜ ε s ) at the frequency implicitly given by Eq. (11)is sufficiently small, as in the present case of Au nanosphere.Our results bring us one step closer to an ideal world scenario,where one would be able to achieve a control over the position ofmaxima of basic cross-sections in order to better tailor nanoparti-cles to specific light-matter interaction applications. They couldbe of immediate interest not only for enhanced fluorescence[13], but also for nonlinear optics [6], surface enhanced Ramanspectroscopy [22, 23], heat management, thermophotovoltaic,photothermal imaging [9], thermoplasmonics, such as thermallyenhanced surface-chemistry [10], plasmonic heating [11], opto-plasmonic evaporation, or solar vapor generation enabled bynanoparticles [12].It is worthwhile to mention that recent work [24] deals witha related issue within the electrostatic approximation (i.e. theRayleigh limit), without any use of MLWA, and without anycomparison with the exact Mie results. Under these limitations,it is difficult to argue for a blue shift if already the well-knownsize-dependent red shift (10) relative to ω (cid:96) cannot be properlyaccounted for. REFERENCES
1. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallö-sungen,” Annalen der Physik , 377–445 (1908).2. R. Fleury, J. Soric, and A. Alù, “Physical bounds on absorption andscattering for cloaked sensors,” Phys. Rev. B , 045122 (2014).3. V. Grigoriev, N. Bonod, J. Wenger, and B. Stout, “Optimizing nanoparti-cle designs for ideal absorption of light,” ACS Photonics , 263–270(2015).4. K. Ladutenko, P. Belov, O. Peña-Rodríguez, A. Mirzaei, A. E. Mirosh-nichenko, and I. V. Shadrivov, “Superabsorption of light by nanoparti-cles,” Nanoscale , 18897–18901 (2015). 5. A. Mirzaei, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar,“Superabsorption of light by multilayer nanowires,” Nanoscale , 17658–17663 (2015).6. J. Butet, K.-Y. Yang, S. Dutta-Gupta, and O. J. F. Martin, “Maximizingnonlinear optical conversion in plasmonic nanoparticles through idealabsorption of light,” ACS Photonics , 1453–1460 (2016).7. A. E. Miroshnichenko and M. I. Tribelsky, “Ultimate Absorption in LightScattering by a Finite Obstacle,” Phys. Rev. Lett. , 033902 (2018).8. M. I. Tribelsky and A. E. Miroshnichenko, “Universal tractable modelof dynamic resonances and its application to light scattering by smallparticles,” arXiv preprint: 2004.10569 (2020).9. D. Boyer, P. Tamarat, A. Maali, B. Lounis, and M. Orrit, “Photothermalimaging of nanometer-sized metal particles among scatterers,” Science , 1160–1163 (2002).10. W. Ni, H. Ba, A. A. Lutich, F. Jäckel, and J. Feldmann, “Enhancingsingle-nanoparticle surface-chemistry by plasmonic overheating in anoptical trap,” Nano Lett. , 4647–4650 (2012).11. N. Harris, M. J. Ford, and M. B. Cortie, “Optimization of plasmonicheating by gold nanospheres and nanoshells,” J. Phys. Chem. B ,10701–10707 (2006).12. O. Neumann, A. S. Urban, J. Day, S. Lal, P. Nordlander, and N. J. Halas,“Solar vapor generation enabled by nanoparticles,” ACS Nano , 42–49(2013).13. S. Sun, I. L. Rasskazov, P. S. Carney, T. Zhang, and A. Moroz, “Thecritical role of shell in enhanced fluorescence of metal-dielectric core-shell nanoparticles,” arXiv preprint:2003.11850 (2020).14. B. J. Messinger, K. U. von Raben, R. K. Chang, and P. W. Barber,“Local fields at the surface of noble-metal microspheres,” Phys. Rev. B , 649–657 (1981).15. R. G. Newton, Scattering Theory of Waves and Particles (Springer,Berlin, Heidelberg, 1982).16. C. F. Bohren and D. R. Huffman,
Absorption and scattering of light bysmall particles (Wiley-VCH Verlag GmbH, Weinheim, Germany, 1998).17. M. Abramowitz and I. A. Stegun,
Handbook of Mathematical Functions (Dover Publications, New York, 1973).18. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,”Phys. Rev. B , 4370–4379 (1972).19. M. Meier and A. Wokaun, “Enhanced fields on large metal particles:dynamic depolarization,” Opt. Lett. , 581–583 (1983).20. K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The opticalproperties of metal nanoparticles: The influence of size, shape, anddielectric environment,” J. Phys. Chem. B , 668–677 (2003).21. A. Moroz, “Depolarization field of spheroidal particles,” J. Opt. Soc. Am.B , 517–527 (2009).22. T. van Dijk, S. T. Sivapalan, B. M. DeVetter, T. K. Yang, M. V. Schul-merich, C. J. Murphy, R. Bhargava, and P. S. Carney, “Competitionbetween extinction and enhancement in surface-enhanced Ramanspectroscopy,” J. Phys. Chem. Lett. , 1193–1196 (2013).23. S. T. Sivapalan, B. M. DeVetter, T. K. Yang, T. van Dijk, M. V. Schul-merich, P. S. Carney, R. Bhargava, and C. J. Murphy, “Off-resonancesurface-enhanced Raman spectroscopy from gold nanorod suspen-sions as a function of aspect ratio: Not what we thought,” ACS Nano ,2099–2105 (2013).24. T. Yezekyan, K. V. Nerkararyan, and S. I. Bozhevolnyi, “Maximizingabsorption and scattering by spherical nanoparticles,” Opt. Lett.45