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  • Editor: Bernard Kippelen
  • Vol. 19, Iss. S4 — Jul. 4, 2011
  • pp: A772–A785
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Convergence of vector spherical wave expansion method applied to near-field radiative transfer

Karthik Sasihithlu and Arvind Narayanaswamy  »View Author Affiliations


Optics Express, Vol. 19, Issue S4, pp. A772-A785 (2011)
http://dx.doi.org/10.1364/OE.19.00A772


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Abstract

Near-field radiative transfer between two objects can be computed using Rytov’s theory of fluctuational electrodynamics in which the strength of electromagnetic sources is related to temperature through the fluctuation-dissipation theorem, and the resultant energy transfer is described using the dyadic Green’s function of the vector Helmholtz equation. When the two objects are spheres, the dyadic Green’s function can be expanded in a series of vector spherical waves. Based on comparison with the convergence criterion for the case of radiative transfer between two parallel surfaces, we derive a relation for the number of vector spherical waves required for convergence in the case of radiative transfer between two spheres. We show that when electromagnetic surface waves are active at a frequency the number of vector spherical waves required for convergence is proportional to Rmax /d when d/Rmax → 0, where Rmax is the radius of the larger sphere, and d is the smallest gap between the two spheres. This criterion for convergence applies equally well to other near-field electromagnetic scattering problems.

© 2011 OSA

1. Introduction

Electromagnetic scattering from a sphere is a well studied topic since the seminal work of Mie [1

1. G. Mie, “Contributions on the optics of turbid media, particularly colloidal metal solutions” Tech. Rep. SAND78-6018, Translated by P. Newman, Sandia Labs (1978), (an english translation of G. Mie’s 1908 paper.).

]. The sphere is one of the few bodies for which the scattering behavior has been thoroughly investigated and hence can be used as an approximation to model interaction of electromagnetic radiation with particulate matter. Study of electromagnetic scattering from two spheres and in general from multiple spheres is equally important as it is necessary to understand the effect of multiple scattering from closely spaced particles. The first computationally viable solution to the problem of electromagnetic scattering by two spheres of arbitrary size was put forward by Bruning and Lo [2

2. J. H. Bruning and Y. Lo, “Multiple scattering by spheres” Tech. Rep. Antenna Laboratory Report No. 69-5, University of Illinois, Urbana, Illinois (1969).

,3

3. J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres Part I–multipole expansion and ray-optical solutions” IEEE Trans. Antenn. Propag. AP-19, 378–390 (1971). [CrossRef]

]. Prior works [4

4. C. Liang and Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481 (1967).

,5

5. R. Crane, “Cooperative scattering by dielectric spheres,” Tech. Rep., Lincoln Laboratory, M.I.T, Lexington, MA (1967).

] involved the time consuming computation of Wigner’s 3-j symbols [6

6. E. Merzbacher, Quantum Mechanics (Wiley, 1997).

] while applying the translational addition theorem introduced by Stein [7

7. S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).

] and Cruzan [8

8. O. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).

]. Bruning and Lo overcame this problem by introducing an efficient recurrence relation to compute the translation coefficients.

In the analysis of scattering by spheres, depending on the location of the point at which the scattered electric and magnetic fields are calculated (or measured), far-field or near-field effects can dominate. For a sphere of radius R and excitation source of wavelength λ, near-field effects are dominant in regions that satisfy the condition Rrλ. The far-field approximation is sufficient when (1) rR, and (2) rλ. Here r is the distance between the center of the sphere and the point at which the scattered fields are calculated. Some of the practical applications of mulitple sphere scattering, ranging from the study of scattering from aerosols [9

9. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic,1969).

] and soot particles [10

10. D. W. Mackowski, “Electrostatics analysis of radiative absorption by sphere clusters in the Rayleigh limit: application to soot particles.” Appl. Opt. 34, 3535–3545 (1995). [CrossRef] [PubMed]

] to applications like study of light scattering from comet dust [11

11. H. Kimura, L. Kolokolova, and I. Mann, “Light scattering by cometary dust numerically simulated with aggregate particles consisting of identical spheres,” Astron. Astrophys. 449, 1243–1254 (2006). [CrossRef]

], involve far-field phenomena – i.e., near-field effects are unimportant. The near-field response of a particle or surface to an electromagnetic excitation source is crucial in many applications, e.g. in surface enhanced Raman spectroscopy [12

12. P. L. Stiles, J. A. Dieringer, N. C. Shah, and R. P. Van Duyne, “Surface-enhanced Raman spectroscopy.” Annu. Rev. Anal. Chem. 1, 601–626 (2008). [CrossRef]

, 13

13. A. Campion and P. Kambhampati, “Surface-enhanced Raman scattering,” Chem. Soc. Rev. 27, 241 (1998). [CrossRef]

], near-field scanning optical microscopy [14

14. D. W. Pohl, W. Denk, and M. Lanz, “Optical stethoscopy: Image recording with resolution λ/20,” Appl. Phys. Lett. 44, 651–653 (1984). [CrossRef]

, 15

15. U. Durig, D. W. Pohl, and F. Rohner, “Near-field optical-scanning microscopy,” J. Appl. Phys. 59, 3318–3327 (1986). [CrossRef]

], plasmonics and sub-wavelength optics [16

16. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics.” Nature 424, 824–830 (2003). [CrossRef] [PubMed]

18

18. S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Nat. Photonics 3, 388–394 (2009). [CrossRef]

], van der Waals and Casimir forces [19

19. K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and Casimir forces revisited in the near field,” Surf. Sci. Rep. 57, 59–112 (2005). [CrossRef]

], near-field thermal radiative transfer [19

19. K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and Casimir forces revisited in the near field,” Surf. Sci. Rep. 57, 59–112 (2005). [CrossRef]

, 20

20. M. Francoeur and M. P. Menguc, “Role of fluctuational electrodynamics in near-field radiative heat transfer,” J. Quant. Spectrosc. Radiat. Transfer 109, 280–293 (2008). [CrossRef]

], and near-field thermal imaging [21

21. Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P. Lemoine, K. Joulain, J. Mulet, Y. Chen, and J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature 444, 740–743 (2006). [CrossRef] [PubMed]

, 22

22. A. Kittel, U. Wischnath, J. Welker, O. Huth, F. Ruting, and S. Biehs, “Near-field thermal imaging of nanostructured surfaces,” Appl. Phys. Lett. 93, 193109 (2008).

]. In particular, the near-field contribution to the thermal radiative transfer between polar dielectric surfaces (like SiO2, SiC, etc) separated by a vacuum gap is dominated by tunneling of electromagnetic surface modes [23

23. D. Polder and M. Van Hove, “Theory of radiative heat transfer between closely spaced bodies,” Phys. Rev. B 4, 3303–3314 (1971). [CrossRef]

25

25. J. B. Pendry, “Radiative exchange of heat between nanostructures,” J. Phys.: Condens. Matter 11, 6621–6633 (1999). [CrossRef]

]. These modes are characterized by the presence of large energy density at the interface between the dielectric medium and vacuum and decay rapidly with distance from the surface [19

19. K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and Casimir forces revisited in the near field,” Surf. Sci. Rep. 57, 59–112 (2005). [CrossRef]

].

Near-field effects lead to enhancement of thermal radiative transfer beyond the limits imposed by Planck’s theory of thermal radiation [26

26. M. Francoeur, M. Menguc, and R. Vaillon, “Near-field radiative heat transfer enhancement via surface phonon polaritons coupling in thin films,” Appl. Phys. Lett. 93, 043109 (2008). [CrossRef]

32

32. P. Ben-Abdallah and K. Joulain, “Fundamental limits for noncontact transfers between two bodies,” Phys. Rev. B 82, 121419 (2010). [CrossRef]

] and can be exploited to increase the efficiency and power density of thermophotovoltaic [33

33. A. Narayanaswamy and G. Chen, “Surface modes for near field thermophotovoltaics,” Appl. Phys. Lett. 82, 3544–3546 (2003). [CrossRef]

35

35. M. Laroche, R. Carminati, and J.-J. Greffet, “Near-field thermophotovoltaic energy conversion,” J. of Appl. Phys. 100, 063704 (2006). [CrossRef]

] as well as thermoelectric energy conversion devices [36

36. R. Yang, A. Narayanaswamy, and G. Chen, “Surface-plasmon coupled nonequilibrium thermoelectric refrigerators and power generators,” J. Comput. Theor. Nanos. 2, 75–87 (2005).

]. Experimental confirmation of near-field enhancement of thermal radiation beyond Planck’s limit has been possible by measuring radiative transfer between a microsphere and a flat surface [27

27. A. Narayanaswamy, S. Shen, and G. Chen, “Near–field radiative heat transfer between a sphere and a substrate,” Phys. Rev. B 78, 115303 (2008). [CrossRef]

, 28

28. S. Shen, A. Narayanaswamy, and G. Chen, “Surface phonon polaritons mediated energy transfer between nanoscale gaps,” Nano Lett. 9, 2909–2913 (2009). [CrossRef] [PubMed]

, 30

30. E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J.-J. Greffet, “Radiative heat transfer at the nanoscale,” Nat. Photonics 3, 514–517 (2009). [CrossRef]

, 37

37. C. Otey and S. Fan, “Exact microscopic theory of electromagnetic heat transfer between a dielectric sphere and plate,” Arxiv preprint arXiv:1103.2668(2011).

], and between two parallel surfaces [38

38. C. Hargreaves, “Radiative transfer between closely spaced bodies,” Philips Res. Rep. Suppl. 5, 1–80 (1973).

40

40. R. Ottens, V. Quetschke, S. Wise, A. Alemi, R. Lundock, G. Mueller, D. Reitze, D. Tanner, and B. Whiting, “Near-field radiative heat transfer between macroscopic planar surfaces,” Arxiv preprint arXiv:1103.2389 (2011).

]. However, with parallel surfaces, it has not been possible to explore near-field effects at sub-micron gaps. For this reason, numerical models of near-field radiative transfer between spherical surfaces are important. A method similar to the Mie theory, based on the vector spherical wave expansion, has been used to model such phenomena when spherical particles are involved [41

41. A. Narayanaswamy and G. Chen, “Thermal near–field radiative transfer between two spheres,” Phys. Rev. B 77, 075125 (2008). [CrossRef]

]. We shall focus here on the numerical convergence of the vector spherical wave expansion method when near-field effects due to electromagnetic surface modes on the surface of a sphere are dominant. In particular, we will demonstrate the importance of a new convergence criteria by calculating the enhanced thermal radiative transfer between two silica spheres due to the tunneling of electromagnetic surface modes.

1.1. Vector spherical wave expansion

Let us consider the problem of scattering of an electromagnetic plane wave from a spherical particle in a homogeneous medium (vacuum) to give a brief introduction to the vector spherical wave expansion method that is used in this work. The origin of a spherical coordinate system, with respect to which the position vector r = (r, θ, ϕ) is defined, is located at the center of the sphere. The vector spherical waves, which are the fundamental solutions of the vector Helmholtz equation:
××A(r)k2A(r)=0,
(1)
are given by:
Mlm(p)(kr)=zl(p)(kr)Vlm(2)(θ,ϕ)
(2)
Nlm(p)(kr)=ζl(p)(kr)Vlm(3)(θ,ϕ)+zl(p)(kr)krl(l+1)Vlm(1)(θ,ϕ)
(3)
where Mlm(p)(kr) and Nlm(p)(kr) are vector spherical waves of order (l, m). l can take integer values from 0 to ∞. For each l, |m| ≤ l. The superscript p refers to the radial behavior of the waves. For p = 1, the M and N waves are regular waves and remain finite at the origin and zl(1)(kr) is the spherical bessel function of order l. For p = 3, the M and N waves are outgoing spherical waves that are singular at the origin and zl(3)(kr) is the spherical Hankel function of the first kind of order l. The radial function ζl(p)(x)=1xddx(xzl(p)(x)). Vlm(1)(θ,ϕ), Vlm(2)(θ,ϕ), and Vlm(3)(θ,ϕ) are vector spherical harmonics of order (l, m) and can be expressed in terms of the spherical harmonics Ylm(θ, ϕ) and their derivatives [41

41. A. Narayanaswamy and G. Chen, “Thermal near–field radiative transfer between two spheres,” Phys. Rev. B 77, 075125 (2008). [CrossRef]

, 42

42. J. D. Jackson, Classical Electrodynamics (John Wiley, 1998).

].

1.2. Convergence criteria

Irrespective of single sphere or multiple sphere scattering, as a practical matter, the infinite series in Eq. (4) and Eq. (5) need to be truncated (in indices l and m), retaining only enough terms necessary to ensure a sufficiently accurate approximation. The number of terms to retain depends on different length scales pertinent to the problem. For far-field scattering by a single sphere, the relevant length scales are the radius of the sphere, R, and the wavelength of incident radiation, λ. The number of terms for convergence Nconv is given by [44

44. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1998). [CrossRef]

, 48

48. W. Wiscombe, “Mie scattering calculations: advances in technique and fast, vector-speed computer codes,” Tech. rep., NCAR/TN-140+ STR, National Center for Atmospheric Research, Boulder, Colorado (1996).

, 49

49. W. Chew, E. Michielssen, J. Song, and J. Jin, Fast and efficient algorithms in computational electromagnetics, (Artech House, Inc., 2001).

]
Nconv=a+x+bx1/3
(6)
where x = 2π R/λ, and a is 1 or 2 and b is 4 or 4.05, depending on x. These values of a and b are obtained empirically and the resulting expansion gives an error less than 0.01 %. For far-field scattering by two spheres shown in Fig. 1(a), the relevant length scales are the radii of spheres R 1 and R 2, the wavelength λ, and the center-to-center distance between the spheres, D. In this case, a different criterion has been proposed [50

50. N. A. Gumerov and R. Duraiswami, “Computation of scattering from n spheres using multipole reexpansion,” J. Acoust. Soc. Am. 112, 2688–2701 (2002). [CrossRef]

]
Nconv=eπDλ,
(7)
where e is the base of natural logarithm. While the convergence criteria in Eq. (6) and (7) are relevant for far-field scattering, near-field effects lead to several peculiarities and hence we can expect different criteria for the number of terms of convergence. This is especially true when surface plasmon and/or phonon polaritons lead to enormous enhancement of the field amplitude near the interfaces. Hence for problems involving near-field effects the role of the gap d between the bodies can be expected to be prominent in deciding the number of terms for convergence.

Fig. 1 (a) The configuration of two spheres for which the study is performed. (b) The variation of real and imaginary part of the dielectric function (ε) of silica in the frequency range under consideration.

The fact that scattering and absorption of evanescent waves lead to increased contributions from higher order terms has been recognized previously. While analyzing scattering and extinction by small particles, Quinten et al [51

51. M. Quinten, A. Pack, and R. Wannemacher, “Scattering and extinction of evanescent waves by small particles,” Appl. Phys. B 68, 87–92 (1999). [CrossRef]

] noted the increase in the contributions from higher order modes for scattering and absorption by evanescent waves as compared to propagating waves. Yannopapas and Vitanov [52

52. V. Yannopapas and N. Vitanov, “Spontaneous emission of a two-level atom placed within clusters of metallic nanoparticles,” J. Phys.: Condens. Matter 19, 096210 (2007). [CrossRef]

] noticed difficulty in attaining convergence while calculating local density of states (LDOS) near the surface of a metallic sphere. However an explicit form for the number of terms for convergence along the lines of Eq. (6) or (7) has not been proposed for near-field scattering. A brief mention of a convergence criteria was made by Narayanaswamy and Chen in their analysis of surface phonon polariton mediated near-field radiative transfer between two closely spaced spheres [41

41. A. Narayanaswamy and G. Chen, “Thermal near–field radiative transfer between two spheres,” Phys. Rev. B 77, 075125 (2008). [CrossRef]

]. While a scaling form for the number of terms of convergence was proposed, a more detailed error analysis was not pursued. In this paper a comprehensive error analysis has been made for computation of near-field radiative transfer between two silica spheres of equal radii. Based on the error analysis we propose a criterion for the number of terms required to attain an error of less than 1%. We also extend this formulation for the case of two spheres of unequal radii.

2. Two sphere problem

2.1. Convergence of summation over l

To find the number of l terms required in Eq. (8) and Eq. (9) for convergence of near-field quantities for the two sphere problem, comparison is drawn with the well understood case of near-field transfer between two half-spaces [23

23. D. Polder and M. Van Hove, “Theory of radiative heat transfer between closely spaced bodies,” Phys. Rev. B 4, 3303–3314 (1971). [CrossRef]

, 24

24. J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Enhanced radiative transfer at nanometric distances,” Microscale Thermo. Eng. 6, 209–222 (2002). [CrossRef]

, 54

54. A. Narayanaswamy and G. Chen, “Direct computation of thermal emission from nanostructures,” Annual Reviews of Heat Transfer (Begell House, 2005), vol. 14, pp. 169–195.

]. In the latter case, the summations in Eq. (8) and Eq. (9) are replaced by integrals of the form 0dkinf(kin)exp(kinz), where kin is the in-plane wavevector, z is the spacing between the two half-spaces, and f (kin) is an appropriately defined function. It is seen that only wavevectors satisfying the condition kin ≲ 1/z contribute to the integral. The equivalent of the in-plane wavevector for a sphere is the wavelength of spatial variation of the field on the surface of the sphere. For l = Nconv, the periodicity of the spatial variation is determined by the behavior of YNconv,m, |m| ≤ Nconv. The smallest wavelength on the surface is given by 2πR/Nconv, resulting in a maximum in-plane wavevector-equivalent of Nconv/R. By analogy with the convergence requirement for two half-spaces, we obtain Nconv/R ∼ 1/d or
Nconv=CRd,
(11)
where C is a constant (or a weak function of R) that is dependent on the desired accuracy of the conductance. This criterion is sufficient for the analysis of radiative transfer between spheres of submicron radii where the radiative transfer is dominated by near-field effects. However, for spheres of larger radii where the contribution from propagating waves is not negligible, we have observed that the above criterion is not sufficient to attain convergence and the equation needs to be modified to :
Nconv=CRd+eπDλ,
(12)
with the additional term taken from Eq. (7).

In order to quantify the error due to retaining contributions only from wave functions with lN in Eq. (8) and Eq. (9), the conductance G and spectral conductance Gω are plotted as a function of N in Fig. 2 and Fig. 3 respectively. It is seen that an exponentially decaying function of the form G(N) = G + ae bN matches the variation of G (and Gω) adequately, where G denotes the value of G as N → ∞, and a and b are constants. The relative error E(N) (in %) for total conductance, defined as E(N) = (G(N) – G )/G , is also plotted in Fig. 2. The conductance G(N) and error E (N) for R = 10 μm and R = 25 μm, with d/R = 0.01, are shown in Fig. 2(a) and 2(b) respectively. The difference between the convergence of the numerical method at nonresonant (0.1005 eV) and resonant (0.061 eV) frequencies is illustrated by plotting the spectral conductance for R = 10 μm and d/R = 0.01 as a function of N in Fig. 3(a) and Fig. 3(b). Also plotted are the relative errors in spectral conductance at these frequencies.

Fig. 2 Convergence of conductance (on the left axis) and error (on the right axis) shown for (a) R = 10 μm spheres at d = 100 nm and (b) R = 25 μm spheres at d = 250 nm. The solid line through the relative error data points is included to illustrate the exponentially decaying trend.
Fig. 3 Convergence of spectral conductance (on the left axis) and error (on the right axis) shown for R = 10 μm spheres for d = 100 nm at (a) a nonresonant frequency (0.1005 eV) and (b) a resonant frequency (0.061 eV).

The exponential decay in the error values is observed for both these frequencies. As expected, convergence to a given relative error requires a larger value of N at a resonant frequency than at a non-resonant frequency. For instance, at N = 2R/d the relative error for the resonant frequency is ≈ 3 % while it is ≈ 0.2 % at the non-resonant frequency.

While Eq. (12) proposes a scaling form for Nconv, the constant C needs to be quantified for attaining a given relative error. This can be obtained by analyzing the variation of Nconv with R/d for spheres of different radii. Figure 4 shows the dependence of Nconv to attain 1 % error on R/d for spheres of both submicron radii and larger radii at the resonant frequency (0.061 eV). It is apparent that C assumes a constant value (≈ 2.72) at the resonant frequency. For total conductance, C = 2 is sufficient to attain 1% error as seen in Fig. 2. It should be noted that these values of C are particular to the case of radiative transfer between silica spheres. This constant is expected to vary weakly with the dielectric properties of the material(s) of the spheres. However the scaling form shown in Eq. (11) was proposed without taking into consideration the material of the spheres and hence can be used for any material that supports surface phonon or plasmon polaritons.

Fig. 4 Variation of Nconv with R/d for two equal-sized spheres with R = 500 nm, 1 μm, 15 μm and 25 μm.

For the sake of completion we have also analyzed the convergence of conductance between spheres of unequal radii. For two spheres of radii R 1 and R 2 with R 1 < R 2, Nconv can be expected to scale with R 2 as:
Nconv=CR2d+eπDλ,
(13)
The variation of spectral conductance Gω as a function of N for two spheres with R 1 = 2 μm, R 2 = 40 μm and d = 200 nm at a nonresonant frequency (0.1005 eV) and a resonant frequency (0.061 eV) are shown in Fig. 5(a) and Fig. 5(b) respectively. Due to computational constraints the maximum number of terms considered for this study has been limited to N = 3R 2/d. From Fig. 5(b) we note that at the resonant frequency the value of constant C in Eq. (13), for 1% error, turns out approximately to be the same as the value obtained for equal radii [see Fig. 3(b)]. Hence we conclude that Eq. (13) is expected to hold true even in the limiting case of R 1 → 0, which implies that to determine the scattered field due to excitation by a dipole current source at a distance z(≪ λ) from the surface of a sphere of radius R, Nconv scales as R/z. This can be used as a criterion for the convergence of the number of terms to be retained in the summation over l while calculating quantities such as LDOS near the surface of a sphere, like in [52

52. V. Yannopapas and N. Vitanov, “Spontaneous emission of a two-level atom placed within clusters of metallic nanoparticles,” J. Phys.: Condens. Matter 19, 096210 (2007). [CrossRef]

]. Since the convergence criterion in Eq. (13) depends on R 2, we anticipate that the method outlined in Ref. [41

41. A. Narayanaswamy and G. Chen, “Thermal near–field radiative transfer between two spheres,” Phys. Rev. B 77, 075125 (2008). [CrossRef]

] cannot be extended directly to the case of near-field radiative transfer between a sphere and a parallel surface [37

37. C. Otey and S. Fan, “Exact microscopic theory of electromagnetic heat transfer between a dielectric sphere and plate,” Arxiv preprint arXiv:1103.2668(2011).

].

Fig. 5 Convergence of spectral conductance shown for R 1 = 2 μm and R 2 = 40 μm spheres for d = 200 nm at (a) a nonresonant frequency (0.1005 eV) (b) a resonant frequency (0.061 eV).

2.2. Convergence of summation over m

In addition to truncating the infinite series in Eq. (8) and (9) with respect to l, it is also necessary to truncate the series in m. Since each value of m contributes independent of other values of m to the spectral conductance (and conductance), we can write Gω=m=0max(l)Gω(m), where max(l) is the maximum value of l used in the computation. The variation of Gω(m) with m between two spheres of radii R = 25 μm and gap d = 250 nm at a resonant frequency (ω = 0.061 eV) and a nonresonant frequency (ω = 0.1005 eV) are shown in Fig. 6(a) and Fig. 6(b) respectively. We notice that there is an approximately exponential decay [ Gω(m)Aexp(Bm)] in the contributions from higher values of m. The values of B for the resonant and nonresonant frequency are shown in Fig. 6(a) and Fig. 6(b) respectively. This observation enables us to propose an empirical criterion for the number of terms Mconv to be retained in summation over m as:
Gω(Mconv)=0.005Gω(0)
(14)
i.e only wavefunctions with contribution to spectral conductance higher than 0.5% of the contribution from m = 0 are used for the series summation in m in Eq. (8) and Eq. (9). For the case considered in Fig. 6, the error due to retaining contributions only from the wavefunctions satisfying Eq. (14) is ≈ 0.028% at the resonant frequency and ≈ 0.024% at the nonresonant frequency.

Fig. 6 Contribution to spectral conductance from each value of m for R = 25 μm and d = 250 nm at (a) a resonant frequency (0.061 eV) (b) nonresonant frequency (0.1005 eV). The rate of exponential decay (B) for higher values of m at the resonant frequency is also shown.

3. Concluding remarks

To summarize, we have investigated the numerical convergence of vector spherical wave expansion technique applied to near-field electromagnetic scattering. The conclusions of this study are as follows:
  1. The number of vector spherical waves required for numerical convergence of near-field radiative thermal conductance between two closely spaced spheres of equal size that support surface polaritons is given by Nconv=CRd+eπDλ, where C is a dimensionless number that depends on the desired accuracy. For spheres of unequal radii, R is replaced by the radius of the larger sphere.
  2. Contributions from larger values of m decay exponentially with m and the summation over m can be truncated at a value of m = MconvNconv.
  3. The convergence criteria developed here are also applicable to other near-field scattering problems where a new length scale is introduced in lieu of d. For example, to determine the LDOS at a point at a distance z(≪ λ) from the surface of a sphere, the convergence criterion would be Nconv=CRz+eπDλ.

Acknowledgments

This work is funded partially by National Science Foundation Grant CBET-0853723.

References and links

1.

G. Mie, “Contributions on the optics of turbid media, particularly colloidal metal solutions” Tech. Rep. SAND78-6018, Translated by P. Newman, Sandia Labs (1978), (an english translation of G. Mie’s 1908 paper.).

2.

J. H. Bruning and Y. Lo, “Multiple scattering by spheres” Tech. Rep. Antenna Laboratory Report No. 69-5, University of Illinois, Urbana, Illinois (1969).

3.

J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres Part I–multipole expansion and ray-optical solutions” IEEE Trans. Antenn. Propag. AP-19, 378–390 (1971). [CrossRef]

4.

C. Liang and Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1481 (1967).

5.

R. Crane, “Cooperative scattering by dielectric spheres,” Tech. Rep., Lincoln Laboratory, M.I.T, Lexington, MA (1967).

6.

E. Merzbacher, Quantum Mechanics (Wiley, 1997).

7.

S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).

8.

O. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).

9.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic,1969).

10.

D. W. Mackowski, “Electrostatics analysis of radiative absorption by sphere clusters in the Rayleigh limit: application to soot particles.” Appl. Opt. 34, 3535–3545 (1995). [CrossRef] [PubMed]

11.

H. Kimura, L. Kolokolova, and I. Mann, “Light scattering by cometary dust numerically simulated with aggregate particles consisting of identical spheres,” Astron. Astrophys. 449, 1243–1254 (2006). [CrossRef]

12.

P. L. Stiles, J. A. Dieringer, N. C. Shah, and R. P. Van Duyne, “Surface-enhanced Raman spectroscopy.” Annu. Rev. Anal. Chem. 1, 601–626 (2008). [CrossRef]

13.

A. Campion and P. Kambhampati, “Surface-enhanced Raman scattering,” Chem. Soc. Rev. 27, 241 (1998). [CrossRef]

14.

D. W. Pohl, W. Denk, and M. Lanz, “Optical stethoscopy: Image recording with resolution λ/20,” Appl. Phys. Lett. 44, 651–653 (1984). [CrossRef]

15.

U. Durig, D. W. Pohl, and F. Rohner, “Near-field optical-scanning microscopy,” J. Appl. Phys. 59, 3318–3327 (1986). [CrossRef]

16.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics.” Nature 424, 824–830 (2003). [CrossRef] [PubMed]

17.

A. Zayats, I. Smolyaninov, and A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005). [CrossRef]

18.

S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Nat. Photonics 3, 388–394 (2009). [CrossRef]

19.

K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and Casimir forces revisited in the near field,” Surf. Sci. Rep. 57, 59–112 (2005). [CrossRef]

20.

M. Francoeur and M. P. Menguc, “Role of fluctuational electrodynamics in near-field radiative heat transfer,” J. Quant. Spectrosc. Radiat. Transfer 109, 280–293 (2008). [CrossRef]

21.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P. Lemoine, K. Joulain, J. Mulet, Y. Chen, and J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature 444, 740–743 (2006). [CrossRef] [PubMed]

22.

A. Kittel, U. Wischnath, J. Welker, O. Huth, F. Ruting, and S. Biehs, “Near-field thermal imaging of nanostructured surfaces,” Appl. Phys. Lett. 93, 193109 (2008).

23.

D. Polder and M. Van Hove, “Theory of radiative heat transfer between closely spaced bodies,” Phys. Rev. B 4, 3303–3314 (1971). [CrossRef]

24.

J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Enhanced radiative transfer at nanometric distances,” Microscale Thermo. Eng. 6, 209–222 (2002). [CrossRef]

25.

J. B. Pendry, “Radiative exchange of heat between nanostructures,” J. Phys.: Condens. Matter 11, 6621–6633 (1999). [CrossRef]

26.

M. Francoeur, M. Menguc, and R. Vaillon, “Near-field radiative heat transfer enhancement via surface phonon polaritons coupling in thin films,” Appl. Phys. Lett. 93, 043109 (2008). [CrossRef]

27.

A. Narayanaswamy, S. Shen, and G. Chen, “Near–field radiative heat transfer between a sphere and a substrate,” Phys. Rev. B 78, 115303 (2008). [CrossRef]

28.

S. Shen, A. Narayanaswamy, and G. Chen, “Surface phonon polaritons mediated energy transfer between nanoscale gaps,” Nano Lett. 9, 2909–2913 (2009). [CrossRef] [PubMed]

29.

A. Narayanaswamy, S. Shen, L. Hu, X. Chen, and G. Chen, “Breakdown of the planck blackbody radiation law at nanoscale breakdown of the planck blackbody radiation law at nanoscale gaps,” Appl. Phys. A 96, 357–362 (2009). [CrossRef]

30.

E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J.-J. Greffet, “Radiative heat transfer at the nanoscale,” Nat. Photonics 3, 514–517 (2009). [CrossRef]

31.

S. Biehs, E. Rousseau, and J. Greffet, “Mesoscopic description of radiative heat transfer at the nanoscale,” Phys. Rev. Lett. 105, 234301 (2010). [CrossRef]

32.

P. Ben-Abdallah and K. Joulain, “Fundamental limits for noncontact transfers between two bodies,” Phys. Rev. B 82, 121419 (2010). [CrossRef]

33.

A. Narayanaswamy and G. Chen, “Surface modes for near field thermophotovoltaics,” Appl. Phys. Lett. 82, 3544–3546 (2003). [CrossRef]

34.

S. Basu, Z. Zhang, and C. Fu, “Review of near-field thermal radiation and its application to energy conversion,” Int. J. Energy Res. 33, 1203–1232 (2009). [CrossRef]

35.

M. Laroche, R. Carminati, and J.-J. Greffet, “Near-field thermophotovoltaic energy conversion,” J. of Appl. Phys. 100, 063704 (2006). [CrossRef]

36.

R. Yang, A. Narayanaswamy, and G. Chen, “Surface-plasmon coupled nonequilibrium thermoelectric refrigerators and power generators,” J. Comput. Theor. Nanos. 2, 75–87 (2005).

37.

C. Otey and S. Fan, “Exact microscopic theory of electromagnetic heat transfer between a dielectric sphere and plate,” Arxiv preprint arXiv:1103.2668(2011).

38.

C. Hargreaves, “Radiative transfer between closely spaced bodies,” Philips Res. Rep. Suppl. 5, 1–80 (1973).

39.

L. Hu, A. Narayanaswamy, X. Chen, and G. Chen, “Near-field thermal radiation between two closely spaced glass plates exceeding plancks blackbody radiation law,” Appl. Phys. Lett. 92, 133106 (2008). [CrossRef]

40.

R. Ottens, V. Quetschke, S. Wise, A. Alemi, R. Lundock, G. Mueller, D. Reitze, D. Tanner, and B. Whiting, “Near-field radiative heat transfer between macroscopic planar surfaces,” Arxiv preprint arXiv:1103.2389 (2011).

41.

A. Narayanaswamy and G. Chen, “Thermal near–field radiative transfer between two spheres,” Phys. Rev. B 77, 075125 (2008). [CrossRef]

42.

J. D. Jackson, Classical Electrodynamics (John Wiley, 1998).

43.

J. Stratton, Electromagnetic Theory (Wiley-IEEE Press, 2007).

44.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1998). [CrossRef]

45.

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, 1995).

46.

W. C. Chew, “Efficient ways to compute the vector addition theorem,” J. Electromagn. Wave 7, 651–665 (1993). [CrossRef]

47.

W. C. Chew, “Derivation of the vector addition theorem,” Microw. Opt. Technol. Let. 3, 256–260 (1990). [CrossRef]

48.

W. Wiscombe, “Mie scattering calculations: advances in technique and fast, vector-speed computer codes,” Tech. rep., NCAR/TN-140+ STR, National Center for Atmospheric Research, Boulder, Colorado (1996).

49.

W. Chew, E. Michielssen, J. Song, and J. Jin, Fast and efficient algorithms in computational electromagnetics, (Artech House, Inc., 2001).

50.

N. A. Gumerov and R. Duraiswami, “Computation of scattering from n spheres using multipole reexpansion,” J. Acoust. Soc. Am. 112, 2688–2701 (2002). [CrossRef]

51.

M. Quinten, A. Pack, and R. Wannemacher, “Scattering and extinction of evanescent waves by small particles,” Appl. Phys. B 68, 87–92 (1999). [CrossRef]

52.

V. Yannopapas and N. Vitanov, “Spontaneous emission of a two-level atom placed within clusters of metallic nanoparticles,” J. Phys.: Condens. Matter 19, 096210 (2007). [CrossRef]

53.

W. Chew, “A derivation of the vector addition theorem,” Microw. Opt. Technol. Let. 3, 256–260 (1990). [CrossRef]

54.

A. Narayanaswamy and G. Chen, “Direct computation of thermal emission from nanostructures,” Annual Reviews of Heat Transfer (Begell House, 2005), vol. 14, pp. 169–195.

OCIS Codes
(030.5620) Coherence and statistical optics : Radiative transfer
(240.6690) Optics at surfaces : Surface waves
(260.2160) Physical optics : Energy transfer
(290.4020) Scattering : Mie theory
(290.4210) Scattering : Multiple scattering
(290.6815) Scattering : Thermal emission

ToC Category:
Radiative Transfer

History
Original Manuscript: May 9, 2011
Revised Manuscript: May 26, 2011
Manuscript Accepted: May 27, 2011
Published: June 6, 2011

Virtual Issues
Vol. 6, Iss. 8 Virtual Journal for Biomedical Optics

Citation
Karthik Sasihithlu and Arvind Narayanaswamy, "Convergence of vector spherical wave expansion method applied to near-field radiative transfer," Opt. Express 19, A772-A785 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-S4-A772


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References

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  14. D. W. Pohl, W. Denk, and M. Lanz, “Optical stethoscopy: Image recording with resolution λ/20,” Appl. Phys. Lett. 44, 651–653 (1984). [CrossRef]
  15. U. Durig, D. W. Pohl, and F. Rohner, “Near-field optical-scanning microscopy,” J. Appl. Phys. 59, 3318–3327 (1986). [CrossRef]
  16. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics.” Nature 424, 824–830 (2003). [CrossRef] [PubMed]
  17. A. Zayats, I. Smolyaninov, and A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005). [CrossRef]
  18. S. Kawata, Y. Inouye, and P. Verma, “Plasmonics for near-field nano-imaging and superlensing,” Nat. Photonics 3, 388–394 (2009). [CrossRef]
  19. K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and Casimir forces revisited in the near field,” Surf. Sci. Rep. 57, 59–112 (2005). [CrossRef]
  20. M. Francoeur and M. P. Menguc, “Role of fluctuational electrodynamics in near-field radiative heat transfer,” J. Quant. Spectrosc. Radiat. Transfer 109, 280–293 (2008). [CrossRef]
  21. Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P. Lemoine, K. Joulain, J. Mulet, Y. Chen, and J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature 444, 740–743 (2006). [CrossRef] [PubMed]
  22. A. Kittel, U. Wischnath, J. Welker, O. Huth, F. Ruting, and S. Biehs, “Near-field thermal imaging of nanostructured surfaces,” Appl. Phys. Lett. 93, 193109 (2008).
  23. D. Polder and M. Van Hove, “Theory of radiative heat transfer between closely spaced bodies,” Phys. Rev. B 4, 3303–3314 (1971). [CrossRef]
  24. J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Enhanced radiative transfer at nanometric distances,” Microscale Thermo. Eng. 6, 209–222 (2002). [CrossRef]
  25. J. B. Pendry, “Radiative exchange of heat between nanostructures,” J. Phys.: Condens. Matter 11, 6621–6633 (1999). [CrossRef]
  26. M. Francoeur, M. Menguc, and R. Vaillon, “Near-field radiative heat transfer enhancement via surface phonon polaritons coupling in thin films,” Appl. Phys. Lett. 93, 043109 (2008). [CrossRef]
  27. A. Narayanaswamy, S. Shen, and G. Chen, “Near–field radiative heat transfer between a sphere and a substrate,” Phys. Rev. B 78, 115303 (2008). [CrossRef]
  28. S. Shen, A. Narayanaswamy, and G. Chen, “Surface phonon polaritons mediated energy transfer between nanoscale gaps,” Nano Lett. 9, 2909–2913 (2009). [CrossRef] [PubMed]
  29. A. Narayanaswamy, S. Shen, L. Hu, X. Chen, and G. Chen, “Breakdown of the planck blackbody radiation law at nanoscale breakdown of the planck blackbody radiation law at nanoscale gaps,” Appl. Phys. A 96, 357–362 (2009). [CrossRef]
  30. E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J.-J. Greffet, “Radiative heat transfer at the nanoscale,” Nat. Photonics 3, 514–517 (2009). [CrossRef]
  31. S. Biehs, E. Rousseau, and J. Greffet, “Mesoscopic description of radiative heat transfer at the nanoscale,” Phys. Rev. Lett. 105, 234301 (2010). [CrossRef]
  32. P. Ben-Abdallah and K. Joulain, “Fundamental limits for noncontact transfers between two bodies,” Phys. Rev. B 82, 121419 (2010). [CrossRef]
  33. A. Narayanaswamy and G. Chen, “Surface modes for near field thermophotovoltaics,” Appl. Phys. Lett. 82, 3544–3546 (2003). [CrossRef]
  34. S. Basu, Z. Zhang, and C. Fu, “Review of near-field thermal radiation and its application to energy conversion,” Int. J. Energy Res. 33, 1203–1232 (2009). [CrossRef]
  35. M. Laroche, R. Carminati, and J.-J. Greffet, “Near-field thermophotovoltaic energy conversion,” J. of Appl. Phys. 100, 063704 (2006). [CrossRef]
  36. R. Yang, A. Narayanaswamy, and G. Chen, “Surface-plasmon coupled nonequilibrium thermoelectric refrigerators and power generators,” J. Comput. Theor. Nanos. 2, 75–87 (2005).
  37. C. Otey and S. Fan, “Exact microscopic theory of electromagnetic heat transfer between a dielectric sphere and plate,” Arxiv preprint arXiv:1103.2668(2011).
  38. C. Hargreaves, “Radiative transfer between closely spaced bodies,” Philips Res. Rep. Suppl. 5, 1–80 (1973).
  39. L. Hu, A. Narayanaswamy, X. Chen, and G. Chen, “Near-field thermal radiation between two closely spaced glass plates exceeding plancks blackbody radiation law,” Appl. Phys. Lett. 92, 133106 (2008). [CrossRef]
  40. R. Ottens, V. Quetschke, S. Wise, A. Alemi, R. Lundock, G. Mueller, D. Reitze, D. Tanner, and B. Whiting, “Near-field radiative heat transfer between macroscopic planar surfaces,” Arxiv preprint arXiv:1103.2389 (2011).
  41. A. Narayanaswamy and G. Chen, “Thermal near–field radiative transfer between two spheres,” Phys. Rev. B 77, 075125 (2008). [CrossRef]
  42. J. D. Jackson, Classical Electrodynamics (John Wiley, 1998).
  43. J. Stratton, Electromagnetic Theory (Wiley-IEEE Press, 2007).
  44. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1998). [CrossRef]
  45. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, 1995).
  46. W. C. Chew, “Efficient ways to compute the vector addition theorem,” J. Electromagn. Wave 7, 651–665 (1993). [CrossRef]
  47. W. C. Chew, “Derivation of the vector addition theorem,” Microw. Opt. Technol. Let. 3, 256–260 (1990). [CrossRef]
  48. W. Wiscombe, “Mie scattering calculations: advances in technique and fast, vector-speed computer codes,” Tech. rep. , NCAR/TN-140+ STR, National Center for Atmospheric Research, Boulder, Colorado (1996).
  49. W. Chew, E. Michielssen, J. Song, and J. Jin, Fast and efficient algorithms in computational electromagnetics , (Artech House, Inc., 2001).
  50. N. A. Gumerov and R. Duraiswami, “Computation of scattering from n spheres using multipole reexpansion,” J. Acoust. Soc. Am. 112, 2688–2701 (2002). [CrossRef]
  51. M. Quinten, A. Pack, and R. Wannemacher, “Scattering and extinction of evanescent waves by small particles,” Appl. Phys. B 68, 87–92 (1999). [CrossRef]
  52. V. Yannopapas and N. Vitanov, “Spontaneous emission of a two-level atom placed within clusters of metallic nanoparticles,” J. Phys.: Condens. Matter 19, 096210 (2007). [CrossRef]
  53. W. Chew, “A derivation of the vector addition theorem,” Microw. Opt. Technol. Let. 3, 256–260 (1990). [CrossRef]
  54. A. Narayanaswamy and G. Chen, “Direct computation of thermal emission from nanostructures,” Annual Reviews of Heat Transfer (Begell House, 2005), vol. 14, pp. 169–195.

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