Plasmonic focus * Localized surface plasmon resonances * A brief compendium on small metal particles * Fine metallic nanoparticles * Decay rates * Surface scattering correction * Dynamic depolarization

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The performance of individual metal nanoparticles (MNPs) is crucially influenced by the localized surface plasmon resonance (LSPR) homogeneous line width or, alternatively, the LSPR dephasing time. So far, two main paths have been pursued in order to improve over the performance of individual spherical metal nanoparticles MNPs: (1) various core-shell MNP morphologies and (2) non-spherical MNP external shapes such as rods, cubes, and prisms. About a third alternative path read here.

Elementary properties of localized surface plasmon resonances (LSPR) of small metal (nano)particles

The field of small metal particles has been exploding in recent years, with the ongoing development resulting in many exciting applications in biology, energy conversion, medicine, sensing, etc. At a first glance, basic theoretical principles underlying the properties of localized surface plasmon resonances (LSPR) should have been known and well understood by now. Remember an excellent monograph by C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, New York, 1998). However, even for a spherical particle, the small particle limit is not as simple as it seems. The reason being that the respective coefficients of the long wavelength expansion in terms of power series in the size parameter x=ka have eps+2 in denominator, and thus become singular at the proximity of a surface plasmon resonance. [See e.g. such an expansion for the Mie's coefficient a_1 on p. 295 of R. Gans and H. Happel, Zur Optik kolloidaler Metallösungen, Annalen der Physik IV. Folge 29, pp. 277-300 (1909), which is reproduced as Eq. (11) in W. T. Doyle and A. Agarwal, Optical extiction of metal spheres, J. Opt. Soc. Am. 55, 305-308 (1965).] The sections below are intended to highlight and summarize some of the basic theoretical principles underlying the optical properties of small metal particles in their applications involving a modification of decay rates and scattering, because, at times, some of the properties repeatedly continue of having been rediscovered till nowdays.

Radiative reaction correction

An oscillating dipole loses its energy due to the radiation it emits. As a result, the dipole oscillations are damped. This radiation damping is described by the Abraham-Lorentz equation, which can be found in any standard textbook on classical electrodynamics, such as that by J. D. Jackson [see Sec. 16.2 and Eqs. (16.8-9) therein]. Such a radiation damping applies to any oscillating dipole, be it an elementary molecular dipole or the dipole induced on a small (nano)particle. Since the induced dipole moment of a small (nano)particle is directly related to its polarizability, such a radiation damping translates to that of a particle polarizability. This point has been first emphasized as early as in 1982 by Wokaun, Gordon, and Liao [1] [see Eqs. (1) and (2) therein], although, as referred to by Ohtaka and Inoue [3] [see Eq. (2.7) therein], it was anticipated much earlier in the textbook by Landau and Lifshitz [2]. Not aware of the above work, the necessity of the radiative reaction term in the polarizability has been rediscovered 6 years later by B. T. Draine [4] within the coupled dipole approximation (CDA).

Theoretically, the radiation damping term is necessary for the extinction cross section to be the sum of the scattering and absorption cross sections, which is sometime misleadingly referred to as unitarity in the dipole approximation (see also Appendix C of [5]).


  1. A. Wokaun, J. P. Gordon, and P. F. Liao, Radiation Damping in Surface-Enhanced Raman Scattering, Phys. Rev. Lett. 48, 957-960 (1982) (published 5 April 1982).
  2. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Oxford, Pergamon, 1959) p 186.
  3. K. Ohtaka and M. Inoue, Electromagnetic aspects of the enhanced Raman scattering by a molecule adsorbed on a polarisable sphere, J. Phys. C: Solid State Phys. 15 6463-6480 (1982) (received 21 May 1982).
  4. B. T. Draine, The discrete-dipole approximation and its application to interstellar graphite grains, Astrophys. J. 333, 848-872 (1988).
  5. A. Moroz, Depolarization field of spheroidal particles, J. Opt. Soc. Am. B 26(3), 517-527 (2009).
    (Accompanying F77 code sphrd.f to determine extinction efficiency in various long-wavelength approximations discussed in the article. See here for on-line supplementary material. )

Dynamic depolarization

As observed by Meier and Wokaun [1], starting from the electrostatic solution at the value of size parameter x=ka=0, the maximum electric field intensity enhancement at the surface of a sphere is seen first to increase with sphere radius a. At a=12.5 nm, the intensity enhancement is 7.8% larger than the electrostatic limit. A closer examination reveals that, for silver, this maximum occurs at lambda = 357 nm, which is slightly longer than the electrostatic value, lambda =355 nm. For larger particles the enhancement strongly decreases; it shifts to longer wavelengths and is broadened. A second maximum at shorter wavelength is seen at a>30 nm.

From the monograph by Bohren&Huffman [see Eq. (12.13) in Sec. 12.1.1], the surface plasmon position of a spherical particle shows the following size dependence:

epsilon_r = - [2 + (12/5) x**2] epsilon_h          (R1)

where epsilon_r is the dielectric constant of a sphere at the resonance position and epsilon_h is the dielectric constant of the surrounding medium.

The above size dependences cannot be accounted for by a polarizability comprising only the k**3-dependent radiative reaction term. One needs in addition a k**2-dependent dynamic depolarization term [1,3].

All the terms in the electron dipole polarizability are included if the electron dipole polarizability is determined directly from the dipole T-matrix element on using the relation

alpha = -i (6*pi/k**3) T = -i (3/2*k**3) T,          (R2)

where the first (second) expression holds in the SI (gauss) units. (A free T-matrix code for metallic particles is available in Ref. [4] herein below.)

As shown by Dungey and Bohren [5], the use of such a polarizability brings a significant improvement within the coupled/discrete dipole approximation (CDA/DDA) over the schemes incorporating only the radiative reaction term.

It is worth of reminding here that the static sphere polarizability

alpha = a**3 * (epsilon-epsilon_h)/(epsilon+2*epsilon_h),

which is the Rayleigh (x ---> 0) limit of the expression (R2), is not unitary (see also Appendix C of [6]) and is insufficient in describing essential features of small metal naparticles, such as the size-dependence of a red-shift of the surface plasmon resonance (SPR) (see Eq. (R1) and Sec. 12.1.1. of [2])

The Meier and Wokaun recipe [1] is based upon two hypothesis. The first hypothesis is that the polarization is homogeneous over the volume of the particle. The second hypothesis, which can be viewed as a continuation of the first one, is that the same dipole moment is assigned to each volume element of the particle. Given the above two hypothesis, Ref. [6] provides an exact solution for the case of a spheroid and, in doing so, it also amends the original Meier and Wokaun result for a sphere [1].

In Ref. [6], the dynamic k**2-dependent depolarization component of E_d has been shown to depend on dynamic geometrical factors, which can be expressed in terms of the standard geometrical factors of electrostatics. The Meier and Wokaun recipe itself was shown to be equivalent to a long-wavelength limit of the Green's function technique. The resulting Meier and Wokaun long-wavelength approximation (MWLWA) exhibits a red-shift compared against exact T-matrix results. The red-shift is a hallmark of the Meier and Wokaun long-wavelength approximation: it will persist unless the assumption of a uniform field E_i inside a particle is relaxed [6]. For a sphere it is possible to get rid of the red-shift by assuming rather simple form of a weak nonuniformity of the field E_i inside a particle that can be fully accounted for by a renormalization of the dynamic geometrical factors [6].

Interestingly enough, for electric field oriented along the spheroid rotational axis, the work by Kuwata et al [7] has anticipated dynamic depolarization factors by purely empirical formula obtained by best fit to numerical results, in which case dynamic depolarization factors were fitted by a polynom of the 3rd order in the standard geometrical factor of electrostatics L_z. In contrast, analytic results of Ref. [6] suggest that a linear L_z dependence, but with eccentricity-dependent coefficients, may be enough.

The long-wavelength approximations of Refs. [6,7] accurately match the SPR position, height, and linewidth of the exact results for noble particles with an equivalent-volume-sphere radius of up to 50 nm in the visible. Note in passing that the linewidth Gamma directly determines the plasmon dephasing time T_2=2 hbar/Gamma, where hbar is the Planck constant, the quality factor Q of the resonance at the SPR frequency omega_r via the formula Q = omega_r/Gamma, and the local field enhancement factor |f| (in a harmonic model |f| = Q).

Recently, the results of Ref. [6] have been extended to provide an approximation for the polarizability for large metallic spheroidal nanoshells [8].


  1. M. Meier and A. Wokaun, Enhanced fields on large metal particles: dynamic depolarization, Opt. Lett. 8, 581-583 (1983).
  2. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, New York, 1998).
  3. M. Kerker, D.-S. Wang, and H. Chew, Surface enhanced Raman scattering (SERS) by molecules adsorbed at spherical particles: errata, Appl. Opt. 19, 4159-4174 (1980).
  4. A. Moroz, Improvement of Mishchenko's T-matrix code for absorbing particles, Appl. Opt. 44(17), 3604-3609 (2005).
    (Accompanying F77 code is available here together with routines zger.f and zsur.f; see also routines zge.f and zsu.f for multiplication from the other side.)
  5. C. E. Dungey and C. F. Bohren, Light scattering by nonspherical particles: a refinement to the coupled-dipole method, JOSA A 8, 81-87 (1991).
  6. A. Moroz, Depolarization field of spheroidal particles, J. Opt. Soc. Am. B 26(3), 517-527 (2009).
    (Accompanying F77 code sphrd.f to determine extinction efficiency in various long-wavelength approximations discussed in the article. See here for on-line supplementary material. )
  7. H. Kuwata, H. Tamaru, K. Esumi, and K. Miyano, Resonant light scattering from metal nanoparticles: Practical analysis beyond Rayleigh approximation, Appl. Phys. Lett. 83(22), 4625-4627 (2003).
  8. H. Y. Chung, P. T. Leung, and D. P. Tsai, Dynamic modifications of polarizability for large metallic spheroidal nanoshells, J. Chem. Phys. 131, 124122 (2009).

Surface scattering corrections to the dielectric function of nanoparticles

The dielectric constant of a small metallic particle is dependent on the size and shape of the particle via lowering of the effective mean free path L of conduction electrons due to collisons of the conduction electrons with the particle surface (the free-path effect), resulting in the so-called size-corrected dielectric function. Indeed, the damping constant in the Drude theory is the inverse of the collision time of the conduction electrons, and the latter decreases because of additional collisions with the boundary of the particle. Obviously, this dependence is important, when the particle size is smaller than the conduction electron mean free path (52 nm for silver [1,3], 42 nm for gold [2], and 45 nm for Cu [3]).

The free-path effect can be incorporated in the dielectric constant of free-electron metals according to the recipe by Kreibig and Fragstein [1]. Although the effect of the mean free path limitation on the real part of the dielectric function is slight, the effect on the imaginary part is often substantial. For instance, for small silver particles near a SPR, the imaginary part is enhanced by more than 10% for particles of radius 100 nm, whereas the imaginary part is twice the bulk value for the radius of about 11.5 nm. The effect of the decreased mean free path is to increase the width and lower the peak height of the surface plasmon absorption (see Sec. 12.1.6 of Bohren&Huffman [4]).

For a homogeneous sphere of a radius R, essentially two different classical surface scattering models were proposed: (i) diffusive and (ii) isotropic scattering models.

Independently of the above scattering models, researches working in the field of ergodic billiards have also figured out a mean free path in a classical ergodic billiard. For a homogeneous sphere of radius R one obtains L = R in the diffusive case and L = 4R/3 in the isotropic and billiard cases. The latter results has also been recovered by Coronado and Schatz within a geometrical probability approach [9].

In Ref. [10], the mean free path for a spherical shell geometry was calculated (i) on assuming Euler diffusive surface scattering and (ii) under the assumption of a homogeneous and isotropic electron distribution within a shell. Further, the general billiard formula of Refs. [7,8] was confirmed in the nanoshell case by an explicit averaging procedure. The extra cos(theta)-factor in the billiard measure has been interpreted as that the (electron) enclosure surface is a Lambertian surface. The latter is distinguished by a fact that the radiant intensity observed from it is directly proportional to the cosine of the angle theta between the observer's line of sight and the surface normal, and hence when an area element on the surface is viewed from any angle, it has the same radiance. (An example a perfect Lambertian radiator is a black body, and, to a large extent, the Sun in the visible spectrum.) Consequently, the billiard measure can be interpreted as that the electron scattering events on the domain surface conspire to produce a diffusive Lambertian surface. The condition of specular reflections can then be abandoned.

Let D be the shell thickness and R the outer shell radius. Preliminary experimental results, which were performed exclusively within the range D/R < 0.25 and which shown a linear dependence of L on D, are compatible with the billiard, or Lambertian, scattering model and rule out both the Euler diffusive scattering and the isotropic scattering. The billiard model characterized by the Lambertian scattering on the electron enclosure surface is distinguished by the following properties in the nanoshell case [10]:

The success of the Euler diffusive scattering model in the homogeneous sphere case seems to be purely accidental, since all the classical scattering models yield a linear dependence of the mean free path on the sphere radius with minor differences in the slope. Contrary to homogeneous sphere case, the mean free path in the shell geometry turns out to be much more sensitive to underlying model assumptions of electron surface scattering. Consequently, future experiments on single and well-controlled dielectric core-metal shell nanoparticles can much easier differentiate in between competing models of surface scattering. This would enable one to assess more precisely the contribution of other mechanisms, such as chemical interface damping [12], to overall plasmon resonance damping.

Quite amazingly, although Lermé et al [13] did not emphasize it, their experimental data regarding silver nanospheres fit nicely to the billiard model [10]. The latter yields for the mean-free path the value of 4R/3 (e.g. eq. 141 of Ref. [10]) which, when compared with eq 1 of [13], would have yielded g=3/4=0.75. The latter value is almost in the middle of an overall experimental value of Access1 determined by Lermé et al [13]. Compared to the billiard model, the diffusive model of Euler and Kreibig would yield for the mean-free path the value of R, or g=1 [10].

As suggested to me by Professor C. F. Bohren, the history of the above Eqs. (C1) and (C2) goes much further and derives from early results by Cauchy, who studied the average projected area Access1 of convex bodies [14,15]. The relevant Cauchy theorem states that in three spatial dimensions [14,15]

Access1                   (C3)

where S is the corresponding surface area of a convex body. The average is taken over all possible orientations in space. Other way round, the average projection of a convex body onto a plane is equal to one quarter of its surface area. In reactor and in other branches of physics the mean-free path of a single particle bouncing off the walls of a convex body has been then heuristically determined as [16-19]

Access1                   (C4)

One can verify that on substituting for Access1 from (C3) into (C4) one recovers (C1). This relation has also been used in the derivation of the Sabine equation in acoustics, using a geometrical approximation of sound propagation.

In two dimensions, the average projected area Access1 of a convex body reduces to an average projected length [20]

Access1                   (C5)

The average chord length is thus, with V the area of the figure and S the circumference of the figure, given by

Access1                   (C6)

which is identical to (C2).

However, a core-shell problem becomes the problem of calculating the mean-free path in a circular shell region that is (i) neither convex nor (ii) simply connected. This means that the early results by Cauchy [14,15], Czuber [16], Dirac [17], Weinberg and Wigner [18], and others [9,18-20], which yield the mean free path in terms of the geometrical parameters of a convex domain, can no be longer applied. Nevertheless, the validity of the geometric formula (C6) can still be established in the case of the billiard scattering [21]. In analogy with the 3D case [10], a different result is then obtained in the diffusive case [21].


  1. U. Kreibig, Electronic properties of small silver particles: the optical constants and their temperature dependence, J. Phys. F: Met. Phys. 4, 999-1014 (1974).
  2. U. Kreibig and C. V. Fragstein, The limitation of electron mean free path in small silver particles, Z. Physik A 224(4), 307-323 (1969).
  3. F. W. Reynolds and G. R. Stilwell, Mean Free Paths of Electrons in Evaporated Metal Films, Phys. Rev. 88, 418-419 (1952).
  4. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, New York, 1998).
  5. J. Euler, Infrared properties of metals and the mean free paths of conduction electrons, Z. Phys. 137, 318-332 (1954).
  6. U. Kreibig, Small silver particles in photosensitive glass: Their nucleation and growth, Appl. Phys. 10(3), 255-264 (1976).
  7. N. Chernov, Entropy, Lyapunov Exponents and Mean Free Path for Billiards, J. Stat. Phys. 88(1-2), 1-29 (1997). [ps]
  8. N. Chernov and R. Markarian, on-line book Introduction to the Ergodic Theory of Chaotic Billiards.
  9. E. A. Coronado and G. C. Schatz, Surface plasmon broadening for arbitrary shape nanoparticles: A geometrical probability approach, J. Chem. Phys. 119(7), 3926-3934 (2003).
  10. A. Moroz, Electron mean-free path in a spherical shell geometry, J. Phys. Chem. C 112(29), 10641-10652 (2008).
    (The article has been supplemented with a detailed Electronic Supporting Information. See also an accompanying F77 code fsc.f to calculate the mean-free path for various model cases discussed in the article.)
  11. G. Raschke, S. Brogl, A. S. Susha, A. L. Rogach, T. A. Klar, J. Feldmann, B. Fieres, N. Petkov, T. Bein, A. Nichtl, and K. Kürzinger, Gold Nanoshells Improve Single Nanoparticle Molecular Sensors, Nano Lett. 4(10), 1853-1857 (2004).
  12. H. Hövel, S. Fritz, A. Hilger, U. Kreibig, and M. Vollmer, Width of cluster plasmon resonances: Bulk dielectric functions and chemical interface damping, Phys. Rev. B 48, 18178-18188 (1993) [pdf].
  13. J. Lermé, H. Baida, C. Bonnet, M. Broyer, E. Cottancin, A. Crut, P. Maioli, N. Del Fatti, F. Valle, and M. Pellarin, Size Dependence of the Surface Plasmon Resonance Damping in Metal Nanospheres, J. Phys. Chem. Lett. 1(19), 2922-2928 (2010).
  14. A. Cauchy, Memoire sur la rectification des courbes et la quadrature des surfaces courbes (Paris, 1832). Reproduced in Oevres Complčtes, 1st series, Vol. 2, p. 167 (Gauthiers Villars, Paris 1908).
  15. A. Cauchy, C. R. Acad. Sci., Paris, 13, 1060 (1841). A short account of his results [14] in the form of theorems was later published in the Comptes rendus, but no formal proof was given.
  16. E. Czuber, Zur Theorie der geometrischen Wahrscheinlichkeiten, Sitzungsberichte der mathematisch-naturwissenschaftliche Klasse der Kaiserlichen Akademie der Wissenschaften Wien 90(2), 719-742 (1884).
  17. P. A. M. Dirac, Approximate rate of neutron multiplication for a solid of arbitrary shape and uniform density, British Report MS-D-5, Part I, 1943.
  18. V. Vouk, Projected area of convex bodies, Nature 162, 330-331 (1948).
  19. A. M. Weinberg and E. P. Wigner, The Physical Theory of Neutron Chain Reactors, (The University of Chicago Press, Chicago, 1958).
  20. W. J. M. de Kruijf and J. L. Kloosterman, On the average chord length in reactor physics, Ann. Nucl. Energy 30(5), 549-553 (2003).
  21. A. Moroz, Electron mean-free path in metal coated nanowires, J. Opt. Soc. Am. B 28(5), 1130-1138 (2011).
    (The article has been supplemented with a detailed Supporting Information. See also an accompanying F77 code fsc2d.f to calculate the mean-free path for various model cases discussed in the article.)

Decay rates of atoms and molecules in a proximity of small (nano)particles

Although Pol and Bremmer [1,2] and Fock [3] treated the problem of diffraction of electromagnetic waves radiating from a point source outside an absorbing sphere as early as in 1937 and 1946, respectively, eventual application to the problem of atomic dipoles interacting with small micro- and nano-particles followed much later with the work by Ruppin [4] and Chew [5,6]. Kim et al [7] then worked out a comprehensive treatment of the classical decay rates for a molecule in the vicinity of a spherical surface through the application of the work of Pol and Bremmer [1,2] and that of Fock [3]. This theory took full advantage of the Hertz vector formalism, which was claimed to be mathematically simpler than the Lorenz-Mie approach, which uses the vector harmonic expansions. Results were obtained for both radiative and nonradiative transfer when the molecule is located outside or inside the surface. All the above results were tailored for homogeneous particles.

I'm proud to provide in my articles [8,9] an extension for the classical electromagnetics problem of a dipole radiating inside and outside a stratified sphere consisting of an arbitrary number of concentric spherical shells. There is no limitation on the dipole position, the number of the concentric shells, the shell medium, nor on the sphere radius. Electromagnetic fields were determined anywhere in the space, the time-averaged angular distribution of the radiated power, the total radiated power, Ohmic losses due to an absorbing shell, and Green's function were calculated. The results were applied to inelastic light scattering (fluorescence and Raman), the radiative and nonradiative normalized decay rates, and frequency shift. Using correspondence principle, the radiative decay rate was calculated from the Poynting vector, whereas the nonradiative decay rate was calculated from the Ohmic losses inside a sphere absorptive shell. Fast and numerically stable transfer-matrix solution has been incorporated into the respective F77 codes CHEW and CHEWFS, which are freely available here and here, respectively.

In Ref. [9] frequency shifts, radiative decay rates, the Ohmic loss contribution to the nonradiative decay rates, fluorescence yield, and photobleaching of a two-level atom radiating anywhere inside or outside a complex spherical nanoshell with a dielectric core and three alternating gold and silica concentric spherical shells were determined. An application of the above codes for a homogeneous metal nanoparticle can be found in Ref. [10].

Coincidentally, one of the core-shell particle geometries studied in my article [8] correspond to that of a recently reported spaser-based nanolaser [11].


  1. B. van der Pol and H. Bremmer, Diffraction of electromagnetic waves from an electrical point source round a finitely conducting sphere, with applications to radio telegraphy and the theory of the rainbow I, Phil. Mag. 24, 141-176 (1937).
    (The very first calculations of radiative decay rates for vertical (radial) dipole).
  2. B. van der Pol and H. Bremmer, Diffraction of electromagnetic waves from an electrical point source round a finitely conducting sphere, with application to radio telegraphy and theory of the rainbow. II, Phil. Mag. 24, 825-864 (1937).
    (The very first calculations of radiative decay rates for vertical (radial) dipole).
  3. V. A. Fock, The fields of a vertical and horizontal dipole above the earth surface, Soviet. Phys. JETP 19, 916-929 (1946).
    (The very first calculations of radiative decay rates for horizontal (tangential) dipole).
  4. R. Ruppin, Decay of an excited molecule near a small metal sphere, J. Chem. Phys. 76, 1681-1684 (1982).
  5. H. Chew, Transition rates of atoms near spherical surfaces, J. Chem. Phys. 87, 1355-1360 (1987).
  6. We study the transition rates of atoms inside and outside dielectric spheres

  7. H. Chew, Radiation and lifetimes of atoms inside dielectric particles, Phys. Rev. A 38, 3410-3416 (1988).
  8. Y. S. Kim, P. T. Leung, and T. F. George, Classical decay rates for molecules in the presence of a spherical surface: A complete treatment, Surf. Sci. 195, 1-14 (1988).
  9. A. Moroz, A recursive transfer-matrix solution for a dipole radiating inside and outside a stratified sphere, Ann. Phys. (NY) 315(2), 352-418 (2005). (published online on 7 October 2004) [story behind this article]
    (accompanying F77 code CHEW and limited MS Windows executable)
  10. A. Moroz, Spectroscopic properties of a two-level atom interacting with a complex spherical nanoshell, Chem. Phys. 317(1), 1-15 (2005). (published online on 9 August 2005) [quant-ph/0412094] (see its on-line Appendix A for a MS Windows executable; source F77 code CHEWFS)
  11. A. Moroz, Non-radiative decay of a dipole emitter close to a metallic nanoparticle: Importance of higher-order multipole contributions,
    Opt. Commun. 283(10), 2277-2287 (2010). [arXiv:0909.4878]
    (accompanying F77 code CRMNT and limited MS Windows executable)
  12. M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, Demonstration of a spaser-based nanolaser, Nature 460, 1110-1112 (2009).

Multilayered nanoshells

Recently, Alu and Engheta [1] have brought multilayered plasmonic shells into the spotlight as a model system for optical invisibility cloak. However, when dealing with dispersive and absorbing shells, one has to take into account not only surface scattering corrections to the dielectric function (btw., there is no discussion of this point in Ref. [1]) but also another subtle feature: In the case of an absorbing shell it may happen that the linearly independent spherical Bessel functions j_l and y_l (see Sec. 10 of Ref. [2] for notations) are in fact related in finite mathematics by

y_l   approx   i*j_l          (R3)

up to almost all significant digits in double precision!!!

This pathological behaviour can easily be demonstrated on the example of j_0 and y_0. Recall that j_0=(e^{iz}-e^{-iz})/(2iz) and y_0=-(e^{iz}+e^{-iz})/(2z). Thus, for sufficiently large Im z >0, each of j_0 and y_0 would be dominated by the e^{-iz}/(2z) term, resulting in j_0 ~ i e^{-iz}/(2z) and y_0 ~ - e^{-iz}/(2z) that implies the above relation (R3). Consequently, if the spherical Hankel functions h_l of the first kind are formed as the sum [2]

h_l  =   j_l + i*y_l,

their precision may be drastically compromised. Indeed, the above relation would then for Im z >> 1 determine a much smaller number (recall h_0=-ie^{iz}/z << 1) from the sum of two much bigger numbers (recall j_0 ~ i e^{-iz}/(2z) >> 1 and y_0 ~ - e^{-iz}/(2z) >> 1). Therefore, it is always recommended to determine h_l by a direct independent recurrence, such as that proposed by Mackowski et al [3] [see recurrences (63),(64) therein].

Otherwise (see Sec. 4.6 and model calculations for a nanoshell C in Ref. [4]) the radiative decay for the interior of a nanoshell may differ by up to four-orders of magnitude from the correct one. If one can perform calculations in an extended precision, this pathological behaviour can be largely overcome, yet the independent recurrence by Mackowski et al [3] is still highly recommended.


  1. A. Alu and N. Engheta, Multifrequency optical iInvisibility cloak with layered plasmonic shells, Phys. Rev. Lett. 100, 113901 (2008).
  2. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1973)
  3. D. W. Mackowski, R. A. Altenkirch, and M. P. Menguc, Internal absorption cross sections in a stratified sphere, Appl. Opt. 29(10), 1551-1559 (1990).
  4. A. Moroz, Spectroscopic properties of a two-level atom interacting with a complex spherical nanoshell, Chem. Phys. 317(1), 1-15 (2005). (published online on 9 August 2005) [quant-ph/0412094] (see its on-line Appendix A for a MS Windows executable; source F77 code CHEWFS)

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