OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 3 — Feb. 4, 2008
  • pp: 1820–1835
« Show journal navigation

Multiple-multipole simulation of optical near-fields in discrete metal nanosphere assemblies

Wei-Yin Chien and Thomas Szkopek  »View Author Affiliations


Optics Express, Vol. 16, Issue 3, pp. 1820-1835 (2008)
http://dx.doi.org/10.1364/OE.16.001820


View Full Text Article

Acrobat PDF (1076 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We applied a multiple-multipole method to calculate the field enhancement of discrete metal nanosphere assemblies due to plasma resonance, thus performing the first full electromagnetic simulation of a variety of nanoparticle assemblies for efficient field focusing, including the self-similar geometric series of spheres first proposed by Li, Stockman and Bergman. Our study captures electromagnetic resonance effects important for optimizing nanoparticle assemblies to achieve maximum electric field focusing. We predict optical frequency electric fields can be enhanced in gold nanoparticle assemblies in aqueous solution by the order of ~450, within a factor of 2 of that achievable in silver nanostructures. We find that both absorption and far-field scattering resonances of nanoparticle assemblies must be carefully interpreted when inferring near-field focusing properties.

© 2008 Optical Society of America

1. Introduction

An important problem in optics is to “see” beyond the diffraction limit of light using optical near-fields [1

1. L. Novotny and B. Hecht, Principles of Nano-Optics (University Press, Cambridge, 2006).

]. Plasma resonance in metal nanostructures provides one means to efficiently couple far-field radiation to nanometer scale volumes, at a so-called “hot-spot”, to produce such effects as surface-enhanced Raman scattering (SERS). SERS is an enhancement of Raman scattering due in large part to enhancement of local electric fields. The Raman shift induced by molecule specific vibrations renders SERS useful in molecular sensing [2

2. K. Kneipp, M. Moskovits, and H. Kneipp, Surface-Enhanced Raman Scattering: Physics and Applications (Springer, Berlin, 2006). [CrossRef]

]. The Raman scattered optical power due to SERS scales with the fourth power of electric field enhancement,

PS(vS)Efocal4E04
(1)

where νs is the scattered photon frequency.

For this reason, the question of how metal particles should be arranged with respect to each other to produce the strongest field enhancement at the “hot-spots” remains an important one. It is known that linear chains of silver nanoparticles provide a way to concentrate the electromagnetic field in nanoscale regions between the particles [3

3. L. A. Sweatlock, S. A. Maier, and H. A. Atwater, “Highly confined electromagnetic fields in arrays of strongly coupled Ag nanoparticles,” Phys. Rev. B 71, 235408 (2005). [CrossRef]

]. A more efficient field enhancing structure consists of a self-similar chain of particles with decreasing diameters and gaps [4

4. K. Li., M. I. Stockman, and D. J. Bergman, “Self-similar chain of metal nanospheres as an efficient nanolens,” Phys. Rev. Lett. 91, 227402 (2003). [CrossRef] [PubMed]

]. Tapered, continuous plasmonic waveguides are a third instance of efficient nanofocusing structures [5

5. M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93, 137404 (2004). [CrossRef] [PubMed]

]. Moreover, the local optical field can also be controlled using a dielectric core-metal shell nanoparticle [6

6. J. B. Jackson, S. L. Westcott, L. R. Hirsch, J. L. West, and N. J. Halas, “Controlling the surface enhanced Raman effect via the nanoshell geometry,” Appl. Phys. Lett. 82, 257–259 (2003). [CrossRef]

].

Our study is motivated by recent advances in the fabrication of metal nanostructures. Plasmon resonance in noble metal bispheres was measured by Reinhard, et al., [7

7. B. M. Reinhard, M. Siu, H. Argarwal, A. P. Alivisatos, and J. Liphardt, “Calibration of dynamic molecular rulers based on plasmon coupling between gold nanoparticles,” Nano Lett. 5, 2246–2252 (2005). [CrossRef] [PubMed]

] to demonstrate nanometer scale distance calibration. Further, the ability to organize gold nanoparticles into discrete and well-controlled structures using programmable DNA templates [8

8. F. Aldaye and H. F. Sleiman, “Dynamic DNA templates for discrete gold nanoparticles assemblies: Control of geometry, modularity, write/erase and structural switching,” J. Am. Chem. Soc. 129, 4130–4131 (2007). [CrossRef] [PubMed]

] provides the possibility for rational design and synthesis of efficient near-field focusing structures for applications such as SERS.

In our work, we have simulated the optical response of discrete assemblies of nanospheres in linear, self-similar and other arrangements using a multiple-multipole technique (also known as the T-matrix method) [9–11

9. Y. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995). [CrossRef] [PubMed]

]. The mathematical construction of the multiple-multipole method has already been established and simulations on some simple geometries have been performed [12–16

12. Y. Xu, “Calculation of the addition coefficients in electromagnetic multisphere-scattering theory,” J. Comput. Phys. 127, 285–298 (1996). [CrossRef]

]. The most relevant is the recent work by Pellegrini, et al., in which metal bispheres and clusters of small spheres dressing a large sphere have been analyzed. Thus far, there has been no full electromagnetic analysis of the focusing properties of sphere assemblies explicitly designed for efficient field focusing, such as the self-similar structure. The quasi-static approximation or dipole approximation are often made in studies of light scattering by assemblies of spherical nanoparticles; Khlebtsov et al. have conclusively shown that a multipole approach is needed for accurate prediction of far-field light scattering even for simple sub-wavelength bispheres [17

17. B. Khlebtsov, A. Melnikov, V. Zharov, and N. Khlebtsov, “Absorption and scattering of light by a dimer of metal nanospheres: comparison of dipole and multipole approaches,” Nanotechnology 17, 1437–1445 (2006). [CrossRef]

]. Our work reveals, for the first time, the complex resonances of various nanoparticle assemblies, particularly the efficient self-similar structure. We provide estimates of electric near-field focusing in these structures without recourse to quasi-static or dipole approximations.

We describe in section 2 the multiple-multipole method, including the recursion relations used for efficient numerical calculation. Simulation results for scattering, absorption and local field enhancement are presented and discussed in section 3 for a variety of structures: nanosphere assemblies arranged in linear chains, self-similar series and in discrete approximations of the truss, cone and tetrahedron. Conclusions drawn from our numerical work are summarized in section 4.

2. Numerical technique

2.1 Multiple-multipoles

We solved the electromagnetic scattering problem of aggregates of Ns small spheres, each characterized by a normalized radius xj=kaj=2πmmediumaj/λvacuum, a normalized position R j=(Xj,Yj,Zj) and a relative refractive index mj=mabsj/mmedium . We use harmonic time dependence exp(-iωt) for all field quantities. Mie theory accurately describes the scattering of uncoupled spheres, but generalization of Mie theory to multiple-multipoles is necessary to account for successive light scattering amongst nanospheres at sub-wavelength separations.

Fig. 1. Depiction of the general problem. Each sphere is characterized by a radius aj and refractive index mj (left). The rotation scheme of a bisphere jl to the overall coordinate system (right).

The multiple-multipole theory, also known as the T-matrix method [9–11

9. Y. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995). [CrossRef] [PubMed]

] begins with an expansion of the incident, scattered, and internal (to the sphere) electromagnetic field into the well known vector spherical harmonic basis N (1), M (1), N (3), M (3) [9

9. Y. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995). [CrossRef] [PubMed]

,18–20

18. H.C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

] about each sphere:

Escaj=n=1m=nniEmn[amnjNmn(3)(j)+bmnjMmn(3)(j)]
Eincj=n=1m=nniEmn[pmnjNmn(1)(j)+qmnjMmn(1)(j)]
Eintj=n=1m=nniEmn[dmnjNmn(1)(j)+cmnjMmn(1)(j)]
(2)
Hscaj=1ηn=1m=nnEmn[bmnjNmn(3)(j)+amnjMmn(3)(j)]
Hincj=1ηn=1m=nnEmn[qmnjNmn(1)(j)+pmnjMmn(1)(j)]
Hintj=1ηjn=1m=nnEmn[cmnjNmn(1)(j)+dmnjMmn(1)(j)]
(3)

where amnj,bmnj,pmnj,qmnj,cmnj and dmnj are the expansion coefficients, η=η 0/mmedium is the wave impedance of the ambient medium, ηj=η/mj is the wave impedance in metal sphere j, and N (1), M (1) indicate vector spherical harmonics centered on the origin of sphere j. We assume here that the vector harmonics are defined at each sphere with respect to a common set of direction axes x, y and z. The complete scattered electric field is given by a sum over all Ns spheres,

Esca=iNsEscaj.
(4)

We assume a plane wave polarized in the +x-direction and propagating in the +z-direction, so that the non-trivial incident field coefficients are,

p1n0=q1n0=exp(iZj)2,p1n0=q1n0=exp(iZj)2n(n+1)
(5)

where we take as the spatial coordinates of the origin of sphere j with respect to the system origin. The prefactor Emn=|E 0|in(2n+1)(n-m)!/(n+m)!, where |E 0| is the magnitude of the incident wave, is introduced as a normalization constant to maintain numerical stability [9

9. Y. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995). [CrossRef] [PubMed]

] and to maintain consistency with the single-sphere Mie theory, where m=±1. In the above, the order m and the degree n are integers such that -nmn, and describes the multipole (i.e. dipole, quadrupole, etc.) nature of the electromagnetic field dressing each sphere. In numerical work, the harmonic degree is truncated to a finite number 1≤nN.

The central numerical problem is the efficient calculation of the scattered field coefficients amnj,bmnj in a self-consistent manner that accounts for successive scattering between all nanospheres. Imposing the usual boundary conditions for electromagnetic fields at the surfaces of the spherical particles gives a set of inhomogeneous linear equations for the coefficients amnj,bmnj [9

9. Y. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995). [CrossRef] [PubMed]

,10

10. D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A 433, 599–614 (1991). [CrossRef]

],

amnj=αnj(pmnjj=1jlNsv=1μ=vvEμvEmn[aμvlAmnμv(Rjl)+bμvlBmnμv(Rjl)])
bmnj=βnj(qmnjj=1jlNsv=1μ=vvEμvEmn[aμvlBmnμv(Rjl)+bμvlAmnμv(Rjl)])
(6)

where the quantities αjn and βjn are the single sphere Lorenz-Mie coefficients and Amnμν and Bmnμν are the coefficients of the vector harmonic addition theorem [9

9. Y. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995). [CrossRef] [PubMed]

,10

10. D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A 433, 599–614 (1991). [CrossRef]

,12

12. Y. Xu, “Calculation of the addition coefficients in electromagnetic multisphere-scattering theory,” J. Comput. Phys. 127, 285–298 (1996). [CrossRef]

] that are functions of the inter-origin vector R jl=R l-R j.

The system of Eq. (6) is often re-written in block form in which the matrix T represents an interaction operator for the vector harmonic coefficients associated with each sphere,

aj+i=1ijNsTjiai=pjor(1+T)a=p
(7)

with the scattering vector aj=[amnjbmnj] , the incident field vector pj[anjpmnjβnjqmnj] and T ji being composed of elements of the form (Eμν/Emn)αnjAmnμν . In the limit of weak interaction between spheres, Mie theory is asymptotically recovered, a=(1-T+T 2-T 3+…) p. The strong interaction between metal nanospheres in close proximity, illuminated close to plasmon-polariton resonance, prohibits the use of such an expansion. In our work, the linear system of Eq. (7) was solved using a biconjugate gradient method with commercially available software.

2.2 Numerical evaluation of coefficients

The Lorenz-Mie coefficients for sphere j depend on the normalized radius xj and relative refractive index mj as,

αnj=mjψn(xj)ψn(mjxj)ψn(xj)ψn(mjxj)mjξn(xj)ψn(mjxj)ξn(xj)ψn(mjxj)
βnj=ψn(xj)ψn(mjxj)mjψn(xj)ψn(mjxj)ξn(xj)ψn(mjxj)mjξn(xj)ψn(mjxj)
(8)

where ψn(z)=zjn(z) and ξn(z)=zh (1) n(z), with jn and h(1)n the spherical Bessel and spherical Hankel functions of first kind.

The vector harmonic addition coefficients are defined implicitly through the vector harmonic addition theorem [9

9. Y. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995). [CrossRef] [PubMed]

,10

10. D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A 433, 599–614 (1991). [CrossRef]

,12

12. Y. Xu, “Calculation of the addition coefficients in electromagnetic multisphere-scattering theory,” J. Comput. Phys. 127, 285–298 (1996). [CrossRef]

,21

21. D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A 11, 2851–2861 (1994). [CrossRef]

] for decomposing the harmonics N (3) (j), M (3) (j) about an origin at R j into harmonics N (1) (l), M (1) (l) about the origin R l,

Mmn(3)(j)=v=1μ=vv[Aμvmn(Rjl)Mμv(1)(l)+Bμvmn(Rjl)Nμv(1)(l)]
Nmn(3)(j)=v=1μ=vv[Bμvmn(Rjl)Mμv(1)(l)+Aμvmn(Rjl)Nμv(1)(l)].
(9)

A variety of methods can be used to calculate the tensor coefficients. Efficient numerical evaluation of these addition coefficients is critical. In our work we have combined several techniques briefly described in the following. We have used Mackowski’s strategy of decomposing general translation coefficients into rotations and axial translations,

Aμvmn(Rjl)(Rμm(n))1Aμvmn(0,0,Rjl)Rμm(n)
Bμvmn(Rjl)(Rμm(n))1Bμvmn(0,0,Rjl)Rμm(n)
(10)

as illustrated in Fig. 1. The rotation matrix Rμm(n)(α,β,γ) is defined by the Euler angles α=acos(Xjl/|R jl|)/sin β, β=acos(Zjl/|R jl|) and γ=0 for rotating the angular momentum z-axis into alignment with the inter-origin vector R jl=R l-R j. The rotation matrix elements are,

Rμm(n)(α,β,γ)=EmnEμnFμnFmnDμm(n)(α,β,γ)
(11)

with normalization constants

Emn=E0in(2n+1)(nm)!(n+m)!
(12)
Fmn=(1)m(2n+1)(nm)!(4π(n+m)!)
(13)

and the rotation operator of quantum mechanics [22

22. A. R. Edmond, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, 1957).

]

Dμm(n)(α,β,γ)=eikγdμm(n)(β)eimα
(14)
dμm(n)(β)=[(n+μ)!(nμ)!(n+m)!(nm)!]12(cosβ2)μ+m(sinβ2)μmPnμ(μm,μ+m)(cosβ)
(15)

composed of the Jacobi Polynomial evaluated directly as,

Pn(α,β)(x)=2nv=0n(n+αv)(n+βnv)(x1)nv(x+1)v.
(16)

The symmetry relations dμm(n)(β)=(1)μmdμm(n)(β) and dμm(n)(β)=(1)μmdmμ(n)(β) were used to further reduce computational time.

The coefficients for positive axial translation of vector harmonics by a distance |R jl| were calculated using the following expressions [12

12. Y. Xu, “Calculation of the addition coefficients in electromagnetic multisphere-scattering theory,” J. Comput. Phys. 127, 285–298 (1996). [CrossRef]

],

Amnmv=(1)m+n2n+12n(n+1)p=nvn+v(1)q[n(n+1)+v(v+1)p(p+1)]a(m,n,m,v,p)hp(1)(Rjl)
(17)
Bmnmv=(1)m+n2n+12n(n+1)2imRjlp=nvn+v(1)qa(m,n,m,v,p)hp(1)(Rjl)
(18)
q=(n+vp)2
(19)

where the Gaunt coefficients ap=a(m′,n,-m′,ν,p), p=n+ν, n+ν-2,… |n-ν|, were calculated using Bruning and Lo’s recursive technique [11

11. J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres Part I – Multipole Expansion and Ray-Optical Solutions,” IEEE Tran. Antennas Propag.AP-19, 378–390 (1971). [CrossRef]

],

αp3ap4(αp2+αp14m2)ap2+αpap=0
(20)
αp=[(n+v+1)2p2][p2(nv)2]4p21
(21)
an+v=(2n1)!!(2v1)!!(2n+2v1)!!(n+v)!(nm)!(v+m)!
(22)
an+v2=(2n+2v3)(2n1)(2v1)(n+v)[nvm2(2n+2v1)]an+v
(23)

where (2q-1)!!=(2q-1)(2q-3)…3·1;(-1)!!≡1. The simplicity of our chosen strategy arises from the fact that axial translations are diagonal in harmonic order m=µ, while the rotations are diagonal in harmonic degree η=ν.

Once the scattered wave coefficients amnj,bmnj are calculated, quantities of direct physical interest are evaluated easily. The electromagnetic fields are given by Eqs. (3) and (4), so that the electric field enhancement, defined as the ratio of |E sca+E inc|/|E inc|, can be evaluated. The time average Poynting vector is also directly evaluated <S>=1/2Re{E.H*}. Finally, the normalized far-field scattering, extinction and absorption cross-sections (also known as the efficiencies) were calculated [9

9. Y. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995). [CrossRef] [PubMed]

,10

10. D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A 433, 599–614 (1991). [CrossRef]

],

Qsca=4πGk2n=1m=nnn(n+1)(2n+1)(nm)!(n+m)!(amnT2+bmnT2)
(24)
Qext=4πGk2n=1m=nnn(n+1)(2n+1)(nm)!(n+m)!Re{pmn0*amnT+qmn0*bmnT}
(25)
Qabs=QextQsca
(26)

where G=πjNS(aj)2 is the geometric area of each assembly under broadside illumination. Eqs. (24) and (25) make use of a single set of total scattered field coefficients amnT,bmnT for an expansion about a sphere (centered at R l) that is chosen as an origin,

amnT=j=1Nsv=1μ=vv[Amnμv(Rjl)aμvj+Bmnμv(Rjl)bμvj]
bmnT=j=1Nsv=1μ=vv[Amnμv(Rjl)bμvj+Bmnμv(Rjl)aμvj].
(27)

where the Amn'μν,Bmn'μν coefficients are calculated as Amn'μν,Bmn'μν , but with spherical Hankel functions hn(1) replaced by spherical Bessel functions jn in Eqs. (17) and (18), as appropriate to translation of outward propagating vector harmonics N (3), M (3) [12

12. Y. Xu, “Calculation of the addition coefficients in electromagnetic multisphere-scattering theory,” J. Comput. Phys. 127, 285–298 (1996). [CrossRef]

].

3. Results and analysis

3.1 Geometries setup and convergence

The simulations were performed on systems of gold and silver nanoparticles assembled into various geometries and immersed in an aqueous medium (mmedium=1.3342). The dielectric properties of the silver and gold nanoparticles were assumed to be that of bulk material; measured bulk dielectric values reported by Johnson and Christy [23

23. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]

] were used in all calculations and interpolated as needed. The problem setup is depicted in Fig. 1, where the incident field is propagating in the z-direction and polarized along the x-direction. The structures are always oriented in such a way that the maximum charge polarization, along x, is obtained. The seed structure in the center of our systems, where the “hot-spot” is located, consists of a bisphere whose spheres have 5nm radii and a separation of 0.5nm – close to the limit of what DNA based self-assembly might achieve. Field enhancement is taken as the ratio of the local electric field at the “hot-spot”, the center of the seed bisphere, to the electric field strength in the illuminating field.

While the multiple-multipole theory has been described in the exact case where all harmonic degrees and orders are included, practical computation truncates harmonic degree to 1≤nN. In our work, to achieve convergence in near field enhancement at the level of <5%, we have used N=10. This corresponds to a total of 2N(N+1)=220 vector harmonic expansion components per sphere. An example of the convergence in field enhancement is given in Fig. 2 below. The contribution of high degree multipoles is obviously important for the accurate prediction of resonant wavelengths at which field enhancement is maximum, in agreement with Khlebtsov, et al., findings that the dipole approximation is inadequate. The physical explanation is simple, electric fields confined to a characteristic size d are decomposed into harmonics extending to up to degree Nαa/d, where a is particle radius.

Fig. 2. An example showing the convergence of near-field enhancement at the focal point for varying maximum degree N of spherical harmonics included in the simulation (Eqs. (4), (5) specifically). The structure employed is a self-similar series of 12 nanospheres with a geometric growth factor κ=0.831 (see section 3.3 for definition). Both the cases of gold and silver particles are illustrated.

3.2 Field enhancement in linear chains

The linear chain geometry simulated here consists of identical gold or silver nanospheres with 5nm radius and interparticle gap of 0.5nm, shown in Fig. 3. Field enhancement is maximum at the middle gap. As the number of spheres Ns is increased from 2 to 20, the resonant wavelength is red-shifted and the quality factor of the field enhancement resonance increases, as shown in Figs. 4 and 5.

Fig. 3. Linear chain geometries of identical gold or silver nanospheres.
Fig. 4. Enhancement of near-field at focal point among linear chains with varying number of gold nanospheres.
Fig. 5. Enhancement of near-field at focal point among linear chains with varying number of silver nanospheres.

The field enhancement is calculated as the ratio of the electric field magnitude at the focal spot of the structure to the incident plan wave electric field magnitude. The linear particle chain functions as an optical antenna, similar to the continuous metal wire exhibiting a plasma resonance [24

24. L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. 98, 266802 (2007). [CrossRef] [PubMed]

], but with an effective resonance determined by the phase velocity of a plasma excitation propagating along a chain of particles. The field enhancement response of gold linear chains is suppressed at free space optical wavelengths shorter than 600nm, due to a reduction in gold dielectric quality factor, Q=d(ωε’)//2ε” where ε’, ε” are real and imaginary parts of the dielectric constant [25

25. F. Wang and Y. Ron Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. 97, 206806 (2006). [CrossRef] [PubMed]

], resulting from the onset of interband absorption at 2eV [23

23. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]

]. The greater material quality factor of silver versus gold is further evidenced by the sharper resonances in field enhancement observed for the silver particle array.

3.3 Field enhancement in self-similar structures with constant number of spheres

The self-similar configuration of metal nanospheres first proposed by Li, Stockman and Bergman [4

4. K. Li., M. I. Stockman, and D. J. Bergman, “Self-similar chain of metal nanospheres as an efficient nanolens,” Phys. Rev. Lett. 91, 227402 (2003). [CrossRef] [PubMed]

] was shown to be efficient at focusing optical near fields using quasi-static simulations. The self-similar structure is defined with a central bisphere with sphere radii a0 (taken to be 5nm) and interparticle gap d0 (taken to be 0.5nm). Successive spheres are added to the structure in geometric series with radii aj and nearest neighbour gap dj enlarged by a factor 1/κ, such that a j+1=aj/κ and d j+1=dj/κ. Note that in the limit where κ=1, the linear chain is recovered. We compared the response of arrays with a fixed number of spheres Ns=4 and 6 and varying geometric factor κ and length L, illustrated to scale in Fig. 6 and 7. Since the complexity of self-assembly processes increases with number of particles per discrete unit, optimizing field enhancement for fixed particle number Ns is important.

Fig. 6. A 4-sphere with varying geometric progression growth factors κ=1, 0.5, 0.3, 0.2, 0.143 and 0.111.
Fig. 7. A 6-sphere with varying geometric progression growth factors κ=1, 0.768, 0.558, 0.456, and 0.393.
Fig. 8. Enhancement of near-field at focal point among various geometric progressions of 4 gold nanospheres as the geometric parameter κ is varied.
Fig. 9. Enhancement of near-field at focal point among various geometric progressions of 4 silver nanospheres. as the geometric parameter κ is varied.
Fig. 10. Enhancement of nearfield at focal point among various geometric progressions of 6 gold nanospheres as the geometric growth parameter κ is varied.
Fig. 11. Enhancement of nearfield at focal point among various geometric progressions of 6 silver nanospheres as the geometric growth parameter κ is varied.

The simulated field enhancements for 4-sphere and 6-sphere structures composed of gold and silver particles are plotted in Figs. 8–10. The higher material Q of silver compared to gold is apparent in the widths of field enhancement resonances, and also in the clear presence of higher order resonances for silver structures but not gold. Interband absorption in gold above 2eV suppresses field enhancement at λ<600nm. The improved efficiency of the geometric series of spheres over the linear chain is due to the increased metallic volume that expels a greater amount of optical energy, and the reduced number of gaps into which optical energy is focused. Importantly, these simulated field enhancement plots reveal that the geometric scale factor κ (which implicitly defines structure length for a fixed number of spheres Ns) that maximizes field enhancement is a non-trivial function of both sphere material and number of spheres. Full electromagnetic simulation without resorting to the quasi-static approximation is required to accurately model the resonances of the plasma wave excitation along a self-similar structure.

3.4 Field enhancement in self-similar structures of fixed length

Simulation of structures of fixed length and varying growth factor κ and sphere number Ns, illustrated in Figs. 12, were also performed. The optimization of structures of fixed linear extent is important for in-vivo applications [26

26. I. H. El-Sayed, X. Huang, and M. A. El-Sayed, “Selective laser photo-thermal therapy of epithelial carcinoma using anti-EGFR antibody conjugated gold nanoparticles,” Cancer Lett. 239, 129–135 (2006). [CrossRef]

], where the retention or expulsion of particles is dependent on size. The electric field enhancement is plotted in Figs. 13–16 for self-similar structures with length 167.5nm (209.5nm), the length of linear chains of 16 (20) spheres of 5nm radius and 0.5nm interparticle gap.

Fig. 12. Geometric progression of nanospheres in self-similar structures whose total lengths are fixed to that of a 16-sphere linear chain (167.5nm).
Fig. 13. Focal point electric fieldenhancement for gold self-similar structures equal in length to a 16-sphere linear chain (167.5nm).
Fig. 14. Focal point electric fieldenhancement for silver selfsimilar structures equal in length to a 16-sphere linear chain (167.5nm).

The field enhancement resonances are red-shifted as the structures approach linear chains (geometric factor κ→1). The red-shift can be interpreted as arising from the increase in hybridization of the plasma resonances associated with individual spheres, and hence a reduction in frequency of the lowest order plasma oscillation distributed over the structure [27

27. E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419–422 (2003). [CrossRef] [PubMed]

]. The linear chain does not provide the same field enhancement as self-similar structures with κ<1. The precise value of κ (which implicitly defines number of spheres Ns for a fixed length structure) that maximizes field enhancement depends upon both sphere material and physical length of the structure, but the general trend is improved efficiency with smaller κ (fewer spheres Ns).

Fig. 15. Focal point electric fieldenhancement for gold self-similar structures equal in length to a 20-sphere linear chain (209.5nm).
Fig. 16. Focal point electric fieldenhancement for silver selfsimilar structures equal in length to a 20-sphere linear chain (209.5nm).

3.5 Near-field distribution in self-similar structures

The spatial distribution of electric field enhancement |E sca+E inc|/|E inc| in the vicinity of the self-similar structures is useful in explaining the role of structure in near-field enhancement and the importance of vector multipole calculation. Field enhancements in the x-z and x-y planes (the x-z plane is the plane of incidence) for self-similar gold structures of 167.5nm length are presented on a logarithmic scale in Figs. 17, 18. The internal field of each sphere is not shown. Each field enhancement plot is taken with incident light at the wavelength that maximizes field enhancement at the focal spot of the structure, summarized in Table 1. These plots reveal the “hot-spots” associated with each structure, with those structures that approach the linear chain having a preponderance of “hot-spots”. Structures with small κ are dominated by the large volume spheres, which serve to enhance field in a reduced number of “hot-spots”. The higher dielectric quality factor Q of silver versus gold is evident in the increased field enhancement about the silver structures.

Fig. 17. The electric field enhancement in the xz-plane for self-similar gold assemblies of length 167.5nm.

The near-field plots also reveal the importance of multiple-multipole calculations. The incident side of the nanospheres shows field enhancement associated with the charge polarization induced by the incident electric field. Although this induced charge distribution is dipolar in nature, the field distribution about spheres is clearly not dipolar due to the strong coupling amongst them. Furthermore, there is a clear distinction in Fig. 17 between the incident side and the shadow side of the nanospheres, a direct manifestation of the retardation of fields. The near-fields of Fig. 17 indicate clearly that as the condition for the quasi-static approximation, x≪2πmmediuma/λ, is better satisfied, by smaller spheres (in the linear chains), field retardation is reduced and the distinction between incident side near-field and shadow side near-field is diminished.

Fig. 18. The electric field enhancement in the xy-plane for self-similar gold assemblies of length 167.5nm.

Table 1:. Summary of resonance peaks for self-similar structures of length 167.5nm taken from data plotted in Figs. 13,14,21,22. Near-field distributions are plotted at near-field enhancement peaks.

table-icon
View This Table

The retardation of fields is also seen in the time average Poynting vector <S>=1/2Re{E×H*}. The Poynting vector magnitude is plotted in Fig. 19 for self similar gold structures of 167.5 nm length. Energy is clearly drawn into the structures on the incident side, casting shadow regions for spheres as small as 5nm in radius. It should be noted that although isolated spheres may be well modeled under the quasi-static approximation, the complete structure need not be well modeled using a quasi-static approximation. Shadows in the Poynting vector are indeed visible for each structure in Fig. 19.

Fig. 19. The time-average Poynting vector magnitude in the xz-plane for self-similar gold assemblies of length 167.5nm. The reference incident power is situated at -2.58=log (1/η).

The higher order resonances of self-similar structures are complex in nature. The spatial distribution of near-field enhancement in the incident plane for a silver Ns=8, κ=0.672 selfsimilar structure is plotted in Fig. 20 for the three lowest order resonances in near-field enhancement (taken from Fig. 16). The three near-field resonances appear to be distinguished by different hybridizations of dipolar plasma excitations on the individual spheres. The geometric series structure does not lend itself to a simple quantitative description of resonances expected for a linear chain.

Fig. 20. Electric field enhancement in the xz-plane for the silver Ns=8-self-similar structure at three different wavelengths. a) 542nm, b) 474nm c) 440nm.

3.6 Far-field properties of self-similar structures

The absorption efficiency and near-field enhancement are resonant at approximately equal wavelengths (refer to Table 1). The co-location of resonances is in agreement with studies of small individual spheres [29

29. B. J. Messinger, K. U. von Raben, R. K. Chang, and P. W. Barber, “Local fields at the surface of noblemetal microspheres,” Phys. Rev. B 24, 649–657 (1981). [CrossRef]

]. We also note that linear structures have Ns-1 gaps where field enhancement is comparable to that at the central focus (Figs. 17, 18), resulting in significantly increased absorption at the near-field enhancement resonance in comparison to structures with κ<1. Strong interband absorption at wavelengths λ < 600nm is evident for gold particle assemblies with small κ, where absorption is expected to be dominated by the largest spheres.

The resonances in scattering efficiency of self-similar structures in the linear chain limit κ→1 closely follow the resonances in field enhancement and absorption efficiency. However, in the limit of small geometric factor κ, the scattering efficiency peaks acquire a significant red-shift with respect to the near-field enhancement. The physical origin of this red-shift is unknown, but has been observed in numerical studies of single sphere scattering [29

29. B. J. Messinger, K. U. von Raben, R. K. Chang, and P. W. Barber, “Local fields at the surface of noblemetal microspheres,” Phys. Rev. B 24, 649–657 (1981). [CrossRef]

]. The red-shift was found in [29

29. B. J. Messinger, K. U. von Raben, R. K. Chang, and P. W. Barber, “Local fields at the surface of noblemetal microspheres,” Phys. Rev. B 24, 649–657 (1981). [CrossRef]

] to increase with sphere size, in agreement with our finding of increased red-shift with decreasing κ (and hence larger spheres in the structure). Elastic light scattering data, such as that collected with dark-field microscopy, will therefore need to be carefully interpreted when inferring near-field enhancement properties of self-similar structures with geometric factor κ≪1.

Fig. 21. Absorption efficiency among various geometric progressions of gold nanospheres whose lengths are fixed to that of a 16-sphere linear chain.
Fig. 22. Absorption efficiency among various geometric progressions of silver nanospheres whose lengths are fixed to that of a 16-sphere linear chain.
Fig. 23. Scattering efficiency among various geometric progressions of gold nanospheres whose lengths are fixed to that of a 16-sphere linear chain.
Fig. 24. Scattering efficiency among various geometric progressions of silver nanospheres whose lengths are fixed to that of a 16-sphere linear chain.

3.7 Field enhancement in other geometries

Three other geometries were investigated, as illustrated in Fig. 29: the discrete nanosphere approximations of a cone, pyramid and triangular truss. The near-field enhancement versus wavelength for structures composed of 5nm radius gold nanoparticles with 0.5nm minimum interparticle gaps are illustrated in Fig. 30. As in the case of linear chains, the preponderance of interparticle gaps results in field enhancement inferior to the self-similar structure.

Fig. 29. Discrete approximation of a) cone, b) pyramid and c) triangular truss.
Fig. 30. Enhancement of nearfield at focal point among the cone, pyramid and truss geometries.

4. Conclusions

We have used a multiple-multipole method to characterize the near-field enhancement properties of discrete assemblies of metal nanospheres. We have found that the self-similar geometric series of spheres first proposed by Li, Stockman and Bergman were significantly more efficient than linear chains and other structures (cones, pyramids, triangular trusses). Examination of the spatial distribution of the near-field about self-similar structures demonstrates clearly the need to go beyond dipole approximations and quasi-static approximations to accurately capture the optical properties of these structures. We have found the near-field enhancement of self-similar structures to exhibit complex resonant behaviour. The rational design of such structures – to target a particular operating wavelength for instance – will require full electromagnetic modeling such as that of the multiple-multipole method. We anticipate that recent advances in DNA based self-assembly techniques [7

7. B. M. Reinhard, M. Siu, H. Argarwal, A. P. Alivisatos, and J. Liphardt, “Calibration of dynamic molecular rulers based on plasmon coupling between gold nanoparticles,” Nano Lett. 5, 2246–2252 (2005). [CrossRef] [PubMed]

,8

8. F. Aldaye and H. F. Sleiman, “Dynamic DNA templates for discrete gold nanoparticles assemblies: Control of geometry, modularity, write/erase and structural switching,” J. Am. Chem. Soc. 129, 4130–4131 (2007). [CrossRef] [PubMed]

] for fabrication of discrete metal nanoparticle structures will lead to the physical realization of the efficient near-field focusing structures analyzed herein. Furthermore, we have found that farfield scattering properties, which have been traditionally used to infer near-field enhancement properties of individual metal nanospheres, will require more careful analysis in the case of self-similar structures with a small geometric factor κ.

Acknowledgment

This work was supported by Le Fonds québécois de la recherche sur la nature et les technologies (FQRNT) and the Canadian Institute for Advanced Research (CIFAR). We are also grateful for the discussions with Prof. Andrew G. Kirk and Prof. Hanadi Sleiman, both from McGill University. We also thank the McGill University Photonic Systems Group for providing computer resources.

References and links

1.

L. Novotny and B. Hecht, Principles of Nano-Optics (University Press, Cambridge, 2006).

2.

K. Kneipp, M. Moskovits, and H. Kneipp, Surface-Enhanced Raman Scattering: Physics and Applications (Springer, Berlin, 2006). [CrossRef]

3.

L. A. Sweatlock, S. A. Maier, and H. A. Atwater, “Highly confined electromagnetic fields in arrays of strongly coupled Ag nanoparticles,” Phys. Rev. B 71, 235408 (2005). [CrossRef]

4.

K. Li., M. I. Stockman, and D. J. Bergman, “Self-similar chain of metal nanospheres as an efficient nanolens,” Phys. Rev. Lett. 91, 227402 (2003). [CrossRef] [PubMed]

5.

M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93, 137404 (2004). [CrossRef] [PubMed]

6.

J. B. Jackson, S. L. Westcott, L. R. Hirsch, J. L. West, and N. J. Halas, “Controlling the surface enhanced Raman effect via the nanoshell geometry,” Appl. Phys. Lett. 82, 257–259 (2003). [CrossRef]

7.

B. M. Reinhard, M. Siu, H. Argarwal, A. P. Alivisatos, and J. Liphardt, “Calibration of dynamic molecular rulers based on plasmon coupling between gold nanoparticles,” Nano Lett. 5, 2246–2252 (2005). [CrossRef] [PubMed]

8.

F. Aldaye and H. F. Sleiman, “Dynamic DNA templates for discrete gold nanoparticles assemblies: Control of geometry, modularity, write/erase and structural switching,” J. Am. Chem. Soc. 129, 4130–4131 (2007). [CrossRef] [PubMed]

9.

Y. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995). [CrossRef] [PubMed]

10.

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. London Ser. A 433, 599–614 (1991). [CrossRef]

11.

J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres Part I – Multipole Expansion and Ray-Optical Solutions,” IEEE Tran. Antennas Propag.AP-19, 378–390 (1971). [CrossRef]

12.

Y. Xu, “Calculation of the addition coefficients in electromagnetic multisphere-scattering theory,” J. Comput. Phys. 127, 285–298 (1996). [CrossRef]

13.

F. J. Garcia de Abajo, “Multiple scattering of radiation in clusters of dielectrics,” Phys. Rev. B 60, 6086–6102 (1999). [CrossRef]

14.

G. Pellegrini, G. Mattei, V. Bello, and P. Mazzoldi, “Interacting metal nanoparticles: Optical properties from nanoparticle dimers to core-satellite systems,” Mat. Sci. Eng. C 27, 1347–1350 (2007). [CrossRef]

15.

H. Xu, “Calculation of the near field of aggregates of arbitrary spheres,” J. Opt. Soc. Am. A 21, 804–809 (2004). [CrossRef]

16.

R.-L. Chern, X.-X. Liu, and C.-C. Chang, “Particle plasmons of metal nanospheres: Application of multiple scattering approach,” Phys. Rev. E 76, 016609 (2007). [CrossRef]

17.

B. Khlebtsov, A. Melnikov, V. Zharov, and N. Khlebtsov, “Absorption and scattering of light by a dimer of metal nanospheres: comparison of dipole and multipole approaches,” Nanotechnology 17, 1437–1445 (2006). [CrossRef]

18.

H.C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

19.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

20.

G. Arfken and H. J. Weber, Mathematical Methods for Physicists, 6th. ed., (Academic, Orlando, 2005).

21.

D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A 11, 2851–2861 (1994). [CrossRef]

22.

A. R. Edmond, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, 1957).

23.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]

24.

L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. 98, 266802 (2007). [CrossRef] [PubMed]

25.

F. Wang and Y. Ron Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. 97, 206806 (2006). [CrossRef] [PubMed]

26.

I. H. El-Sayed, X. Huang, and M. A. El-Sayed, “Selective laser photo-thermal therapy of epithelial carcinoma using anti-EGFR antibody conjugated gold nanoparticles,” Cancer Lett. 239, 129–135 (2006). [CrossRef]

27.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419–422 (2003). [CrossRef] [PubMed]

28.

C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, and J. Feldmann, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88, 077402 (2002). [CrossRef] [PubMed]

29.

B. J. Messinger, K. U. von Raben, R. K. Chang, and P. W. Barber, “Local fields at the surface of noblemetal microspheres,” Phys. Rev. B 24, 649–657 (1981). [CrossRef]

OCIS Codes
(240.5420) Optics at surfaces : Polaritons
(240.6680) Optics at surfaces : Surface plasmons
(290.4020) Scattering : Mie theory
(160.4236) Materials : Nanomaterials
(250.5403) Optoelectronics : Plasmonics
(240.6695) Optics at surfaces : Surface-enhanced Raman scattering

ToC Category:
Optics at Surfaces

History
Original Manuscript: December 12, 2007
Revised Manuscript: January 23, 2008
Manuscript Accepted: January 24, 2008
Published: January 25, 2008

Virtual Issues
Vol. 3, Iss. 3 Virtual Journal for Biomedical Optics

Citation
Wei-Yin Chien and Thomas Szkopek, "Multiple-multipole simulation of optical nearfields in discrete metal nanosphere assemblies," Opt. Express 16, 1820-1835 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-3-1820


Sort:  Year  |  Journal  |  Reset  

References

  1. L. Novotny and B. Hecht, Principles of Nano-Optics (University Press, Cambridge, 2006).
  2. K. Kneipp, M. Moskovits, and H. Kneipp, Surface-Enhanced Raman Scattering: Physics and Applications (Springer, Berlin, 2006). [CrossRef]
  3. L. A. Sweatlock, S. A. Maier, and H. A. Atwater, "Highly confined electromagnetic fields in arrays of strongly coupled Ag nanoparticles," Phys. Rev. B 71, 235408 (2005). [CrossRef]
  4. K. Li. and M. I. Stockman, and D. J. Bergman, "Self-similar chain of metal nanospheres as an efficient nanolens," Phys. Rev. Lett. 91, 227402 (2003). [CrossRef] [PubMed]
  5. M. I. Stockman, "Nanofocusing of optical energy in tapered plasmonic waveguides," Phys. Rev. Lett. 93, 137404 (2004). [CrossRef] [PubMed]
  6. J. B. Jackson, S. L. Westcott, L. R. Hirsch, J. L. West, and N. J. Halas, "Controlling the surface enhanced Raman effect via the nanoshell geometry," Appl. Phys. Lett. 82, 257-259 (2003). [CrossRef]
  7. B. M. Reinhard, M. Siu, H. Argarwal, A. P. Alivisatos, and J. Liphardt, "Calibration of dynamic molecular rulers based on plasmon coupling between gold nanoparticles," Nano Lett. 5, 2246-2252 (2005). [CrossRef] [PubMed]
  8. F. Aldaye and H. F. Sleiman, "Dynamic DNA templates for discrete gold nanoparticles assemblies: Control of geometry, modularity, write/erase and structural switching," J. Am. Chem. Soc. 129, 4130-4131 (2007). [CrossRef] [PubMed]
  9. Y. Xu, "Electromagnetic scattering by an aggregate of spheres," Appl. Opt. 34, 4573-4588 (1995). [CrossRef] [PubMed]
  10. D. W. Mackowski, "Analysis of radiative scattering for multiple sphere configurations," Proc. R. Soc. London Ser. A 433, 599-614 (1991). [CrossRef]
  11. J. H. Bruning and Y. T. Lo, "Multiple scattering of EM waves by spheres Part I - Multipole Expansion and Ray-Optical Solutions," IEEE Tran.Antennas Propag. AP-19, 378-390 (1971). [CrossRef]
  12. Y. Xu, "Calculation of the addition coefficients in electromagnetic multisphere-scattering theory," J. Comput. Phys. 127, 285-298 (1996). [CrossRef]
  13. F. J. Garcia de Abajo, "Multiple scattering of radiation in clusters of dielectrics," Phys. Rev. B 60, 6086-6102 (1999). [CrossRef]
  14. G. Pellegrini, G. Mattei, V. Bello, and P. Mazzoldi, "Interacting metal nanoparticles: Optical properties from nanoparticle dimers to core-satellite systems," Mat. Sci. Eng. C 27, 1347-1350 (2007). [CrossRef]
  15. H. Xu, "Calculation of the near field of aggregates of arbitrary spheres," J. Opt. Soc. Am. A 21, 804-809 (2004). [CrossRef]
  16. R.-L. Chern, X.-X. Liu and C.-C. Chang, "Particle plasmons of metal nanospheres: Application of multiple scattering approach," Phys. Rev. E 76, 016609 (2007). [CrossRef]
  17. B. Khlebtsov, A. Melnikov, V. Zharov, and N. Khlebtsov, "Absorption and scattering of light by a dimer of metal nanospheres: comparison of dipole and multipole approaches," Nanotechnology 17, 1437-1445 (2006). [CrossRef]
  18. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  19. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  20. G. Arfken and H. J. Weber, Mathematical Methods for Physicists, 6th. ed., (Academic, Orlando, 2005).
  21. D. W. Mackowski, "Calculation of total cross sections of multiple-sphere clusters," J. Opt. Soc. Am. A 11, 2851-2861 (1994). [CrossRef]
  22. A. R. Edmond, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, 1957).
  23. P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972). [CrossRef]
  24. L. Novotny, "Effective wavelength scaling for optical antennas," Phys. Rev. Lett. 98, 266802 (2007). [CrossRef] [PubMed]
  25. F. Wang and Y. Ron Shen, "General properties of local plasmons in metal nanostructures," Phys. Rev. Lett. 97, 206806 (2006). [CrossRef] [PubMed]
  26. I. H. El-Sayed, X. Huang and M. A. El-Sayed, "Selective laser photo-thermal therapy of epithelial carcinoma using anti-EGFR antibody conjugated gold nanoparticles," Cancer Lett. 239, 129-135 (2006). [CrossRef]
  27. E. Prodan, C. Radloff, N. J. Halas and P. Nordlander, "A hybridization model for the plasmon response of complex nanostructures," Science 302, 419 - 422 (2003). [CrossRef] [PubMed]
  28. C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, and J. Feldmann, "Drastic reduction of plasmon damping in gold nanorods," Phys. Rev. Lett. 88, 077402 (2002). [CrossRef] [PubMed]
  29. 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 24, 649-657 (1981). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited