## Parametric plasmonics and second harmonic generation in particle chains |

Optics Express, Vol. 19, Issue 27, pp. 25843-25853 (2011)

http://dx.doi.org/10.1364/OE.19.025843

Acrobat PDF (961 KB)

### Abstract

Parametric optics and second harmonic generation in pure plasmonic particle chains are studied. By a proper design of the plasmonic particle geometry, the modes supported by the chain can achieve phase-matching conditions. Then the magnetic-field dependence of the plasmon electric susceptibility can provide the nonlinearity and the coupling mechanism leading to parametric processes, sum frequency and second harmonic generation. Hence, chains of plasmonic particles can support parametric optics and higher harmonic generation by using its own modes only. Since the second order nonlinearity involves both electric and magnetic fields, the SHG reported here is supported also by centrosymmetric particle chains.

© 2011 OSA

## 1. Introduction

1. M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” Opt. Lett. **23**(17), 1331–1333 (1998). [CrossRef]

4. Andrea Alu and Nader Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B **74**, 205436 (2006). [CrossRef]

*λ*, and then the total width of the modes can be much smaller than

*λ*. Hence the name “Sub-Diffraction Chains” (SDC). SDCs are potential candidates for dense integration of optical systems, and were proposed as guiding structures, junctions, and couplers [1

1. M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” Opt. Lett. **23**(17), 1331–1333 (1998). [CrossRef]

5. D. V. Orden, Y. Fainman, and V. Lomakin, “Optical waves on nanoparticle chains coupled with surfaces,” Opt. Lett. **34**(4), 422–424 (2009). [CrossRef] [PubMed]

6. D. V. Orden, Y. Fainman, and V. Lomakin, “Twisted chains of resonant particles: optical polarization control, waveguidance, and radiation,” Opt. Lett. **35**(15), 2579–2581 (2010). [CrossRef] [PubMed]

7. Y. Hadad and Ben Z. Steinberg, “Magnetized spiral chains of plasmonic ellipsoids for one-way optical waveguides,” Phys. Rev. Lett. **105**, 233904 (2010). [CrossRef]

_{3}) [8

8. Zi-jian Wu, Xi-kui Hu, Zi-yan Yu, Wei Hu, Fei Xu, and Yan-qing Lu, “Nonlinear plasmonic frequency conversion through quasiphase matching,” Phys. Rev. B **82**, 155107 (2010). [CrossRef]

9. A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, “Quadratic phase matching in nonlinear plasmonic nanoscale waveguides,” Opt. Express **17**(22), 20063–20068 (2009). [CrossRef] [PubMed]

10. G. Bachelier, I. R. Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Multipolar second-harmonic generation in noble metal nanoparticles,” J. Opt. Soc. Am. B **25**(6) 955–960 (2008). [CrossRef]

16. S. Kujala, B. K. Canfield, and M. Kauranen, “Multipole interference of second harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. **98**, 167403 (2007). [CrossRef] [PubMed]

17. T. Utikel, T. Zentgraf, T. Paul, C. Rockstuhl, F. Lederer, M. Lippitz, and H. Giessen, “Towards the origin of the nonlinear response in hybrid plasmonic systems,” Phys. Rev. Lett. **106**, 133901 (2011). [CrossRef]

**r**↦ −

**r**) one has

**p**↦ −

**p**and

**E**↦ −

**E**, so we must have

13. J. Nappa, G. Revillod, I. R. Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Electric dipole origin of the second harmonic generation of small metallic particles,” Phys. Rev. B **71**, 165407 (2005). [CrossRef]

14. J. Shan, J. I. Dadap, I. Stiopkin, G. A. Reider, and T. F. Heinz, “Experimental study of optical second-harmonic scattering from spherical nanoparticles,” Phys. Rev. A **73**, 023819 (2006). [CrossRef]

*only if*the nonlinear dipole response depends solely on the electric field. Under space inversion

**H**↦

**H**, so SHG

*can be supported by centrosymmetric particles if the nonlinear dipole response depends on*

**E**

*and*

**H**(e.g. it involves terms as

**H**×

**E**due to Lorentz force). As we shall see, the underlying physics associated with SHG in our chains is in line with these general observations.

*polarization*of another mode. The physical process underlying the non-linearity is schematized in Fig. 1(b). Two important points should be emphasized. First, it is essentially a Lorentz force term, so SHG is supported even with centrosymmetric particles such as the ellipsoids shown in the figure. Second, the nonlinearity at any given particle in the chain, is excited due to magnetic fields created by its neighbors that can be viewed as retarded-field non-local contributions (in analogy with bulk-theories, it comes from other particles in the bulk). Again consistently with previous publications non-local dipole contributions can support SHG in centrosymmetric particles. Hence, our ellipsoids-based SDC can amplify or generate optical signals constructively

*using its own modes only*.

3. M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B **62**(24), R16356 (2000). [CrossRef]

6. D. V. Orden, Y. Fainman, and V. Lomakin, “Twisted chains of resonant particles: optical polarization control, waveguidance, and radiation,” Opt. Lett. **35**(15), 2579–2581 (2010). [CrossRef] [PubMed]

*D*is much smaller than the wavelength so it can be considered as an infinitesimal dipole, and the inter-particle distance

*d*is large compare to

*D*. Studies show excellent agreement with exact solutions even when

*d*and

*D*are of the same order [19

19. S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B **67**, 205402 (2003). [CrossRef]

*n*-th particle is mainly due to its two nearest neighbors, namely due to particles

*n*± 1. Since the electric near field of an infinitesimal dipole behaves essentially as (

*kr*)

^{−3}, this approximation holds very well for SDCs with inter-particle distance

*d*≪

*λ*. Indeed it was shown that chain dispersions resulting from the NNA are in excellent agreement with those of the full theory if

*d*/

*λ*≤ 0.1 [4

4. Andrea Alu and Nader Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B **74**, 205436 (2006). [CrossRef]

*d*/

*λ*≈ 0.25 [3

3. M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B **62**(24), R16356 (2000). [CrossRef]

3. M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B **62**(24), R16356 (2000). [CrossRef]

4. Andrea Alu and Nader Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B **74**, 205436 (2006). [CrossRef]

## 2. Formulation

*α*is subject to an exciting electric field whose local value in the

*absence of the particle*is

**E**

*, its response is described by the electric dipole*

^{L}**p**=

*α*

**E**

*. The matrix (or tensor) polarizability of a general ellipsoid made of an anisotropic dielectric material with electric matrix-susceptibility*

^{L}*χ*can be found in [20

20. A. H. Sihvola, *Electromagnetic Mixing Formulas and Applications*, Electromagnetic Waves Series (IEE1999). [CrossRef]

**62**(24), R16356 (2000). [CrossRef]

**74**, 205436 (2006). [CrossRef]

*x,y,z*axes, its polarizability

*α*is Here

**I**

_{3}is the 3 by 3 identity matrix,

*V*= 4

*πa*/3 is the ellipsoid volume and

_{x}a_{y}a_{z}*a*,

_{x}*a*,

_{y}*a*are its semiaxes.

_{z}**L**= diag(

*N*,

_{x}*N*,

_{y}*N*) is the depolarization matrix whose entries are obtained by elliptic integrals and satisfy Σ

_{z}*= 1 [20*

_{u}N_{u}20. A. H. Sihvola, *Electromagnetic Mixing Formulas and Applications*, Electromagnetic Waves Series (IEE1999). [CrossRef]

*N*= 1/3∀

_{u}*u*. A

*prolate*ellipsoid (

*a*>

_{x}*a*=

_{y}*a*=

_{z}*a*), has

*N*=

_{y}*N*= (1 –

_{z}*N*)/2,

_{x}*N*= (1 –

_{x}*e*

^{2}){ln[(1 +

*e*)/(1 –

*e*)] – 2

*e*}/(2

*e*

^{3}), where

*a*= 2

_{x}*a*has

*N*≈ 0.1736,

_{x}*N*≈ 0.4132. An

_{z}*oblate*ellipsoid (

*a*=

_{x}*a*=

_{y}*a*>

*a*) has

_{z}*N*=

_{x}*N*= (1 –

_{y}*N*)/2,

_{z}*N*= (1 +

_{z}*e*

^{2})(

*e*– arctan

*e*)/

*e*

^{3}, where

*a*=

_{z}*a*/3 has

*N*≈ 0.6354,

_{z}*N*≈ 0.1823.

_{x}*χ*of magnetized plasmons is obtained by applying Lorentz force. Up to first order in the magnetization field

**H**, it is given by (see appendix) where the scalar

*χ*is the non-magnetized plasma susceptibility [18]. Up to a constant factor, the matrix operator

_{ee}**B**

*is equivalent to a vector multiplication by the magnetic field*

_{H}**H**from left, and × is the vector product. The time constant

*τ*represents material loss. In practical situations this loss is orders of magnitude larger than the particle radiation loss [3

**62**(24), R16356 (2000). [CrossRef]

*χ*is valid for non-magnetized metals. No assumption is made about the rate of change of

_{ee}**B**; Lorentz force holds for any time-scale. The expression above can be obtained also as a first order approximation (in

**B**) of the magnetized plasma susceptibility given in [18].

**p**

*be the*

_{n}*n*-th particle dipole moment. Under the DDA and the NNA, it is excited only by its two nearest neighbors. Hence it is governed by the difference equation [3

**62**(24), R16356 (2000). [CrossRef]

*kd*≪ 1 and use the expression for the near field of an infinitesimal dipole [18]) where

**S**

*is the nearest-neighbors sum, Equations (1)–(5) constitute a starting point for various parametric processes in particle chains under the discrete dipole and nearest neighbor approximations.*

_{n}### 2.1. Dispersion and phase matching

*τ*→ 0), and when the non-linearity due to

**B**

*can be neglected. By substituting the solution into Eq. (4) and using the linear part of Eq. (2), we obtain the*

_{H}*three independent*dispersion relations, governing the transverse (

*x,y*) and longitudinal (

*z*) independent polarizations here (

*σ*,

_{x}*σ*,

_{y}*σ*) = (

_{z}*σ*,

*σ*,−2

*σ*), with

*σ*=

*V*/(2

*πd*

^{3}), where

*σ*<

*N*∀

_{u}*u*. We look for solutions that satisfy the SHG condition or the sum frequency generation (SFG) condition, For spherical particles

*N*= 1/3∀

_{u}*u*, so Eq. (7) reduces to the known dispersions [3

**62**(24), R16356 (2000). [CrossRef]

**74**, 205436 (2006). [CrossRef]

*x̂*polarized mode at its central frequency, and for wave#3 a

*ẑ*polarized mode [see Fig. 1(b)]. By imposing Eq. (8) on the corresponding dispersions in Eq. (7), we obtain

*It is all about chain-particle geometry*. Using the expressions for the prolates and oblates

*N*’s [see discussion after Eq. (1)] and

_{u}*σ*, the ellipsoid parameter

*a*/

_{x}*a*and particles separation parameter

_{z}*a*/

_{x}*d*satisfying the above equation were computed for geometrically feasible configurations (

*d*> 2

*a*). The results are shown in Fig. 2(a). For general ellipsoids, more solutions may be available.

_{z}*x̂*-polarized wave as mode#1, but now with

*ŷ*,

*ẑ*-polarized modes#2,3, respectively. With the dispersions in Eq. (7) and with

*β*

_{1}=

*π*/2, Eq. (9) becomes where we used Σ

*N*= 1. Solutions for oblates are shown in Fig. 2(b). Again, other solutions are available for general ellipsoids and/or other polarizations and/or other values of

_{u}*β*

_{1}.

### 2.2. Nonlinear chain dynamics

**H**of the mode under the NNA has not been studied before. The NNA validity for

**H**is not as transparent as for

**E**, since for electric dipole the near H-field goes up as

*r*

^{−2}while the near E-field as

*r*

^{−3}. Hence, below we first make an exact evaluation of the mode magnetic field (i.e. considering contributions from all the chain particles), and then compare it to the magnetic field obtained under the NNA (i.e. contributions from the two closest neighbors only). We show the range of parameters for which the two results do not differ much (the specific examples shown later, however, use the exact field).

*nd*) due to a single electric dipole

**p**

*at (0, 0,*

_{m}*md*), is given by [18] where

*d*=

_{mn}*d*|

*m*–

*n*|,

*g*(

*x*) =

*e*/(4

^{ikx}*πx*), and sgn(

*u*) =

*u*/|

*u*|. The magnetic field at the location of the

*n*-th dipole is given by summing

**H**

*in Eq. (12) over*

_{nm}*m, m*≠

*n*, and using Eq. (6) for

**p**

*. The self magnetic field of the*

_{m}*n*-th particle,

**H**

*, is excluded as it averages to zero over the particle volume. The result is where and where*

_{nn}*Li*is the

_{s}*s*-th order Polylogarithm function [21] defined as

*Li*(

_{s}*e*) are expressable in terms of the Clausen’s integral and series, for which efficient summation formulas are available (see Sec. 27.8 in [22]). The exact

^{iθ}**H**

*in absolute value and in units of*

_{n}*ẑ*×

**p**

_{n}ck^{3}/(4

*π*) is shown in Fig. 3(a). For

*kd*≪ 1 the magnetization strength is maximal for

*β*≈ 0.4

*π*. For larger values of

*kd*another maximum emerges along the light line

*β*=

*kd*. Note however that the chain supports propagating modes in the form of Eq. (6) only for

*β*>

*kd*[4

**74**, 205436 (2006). [CrossRef]

**p**

*=*

_{m}**p**

_{0}

*e*,

^{imβ}*β*≈

*π*/2. The magnetic field is still given by the sum in Eq. (13), with Eq. (12). Since

*β*≈

*π*/2 and due to the sgn term in Eq. (12), contributions from odd neighbor-pairs

*m*=

*n*± (2

*ℓ*+ 1),

*ℓ*= 0,1,... add up

*constructively*at

*n*; most important, this includes the nearest neighbor-pair

*n*±1 whose contribution is the strongest possible. Likewise, contributions from even neighbor-pairs

*m*=

*n*± 2

*ℓ*,

*ℓ*= 1,2,... add up

*destructively*and cancel out. The physical picture is schematized in Fig. 1(b). Consistent with the NNA (valid for

*kd*≪ 1), we keep only the contributions from the pair

*m*=

*n*±1 and neglect the rest; the next non-zero term is an order of magnitude weaker. We end up with an expression similar to that of Eq. (13), where

*F*(

*kd*,

*β*) is approximated by The exact

*F*and its NNA are compared in Fig. 3(b). For

*β*/

*π*≈ 0.4 : 0.5 the error of the NNA is less then 10% for

*kd*up to 0.4.

**I**

_{3}+

*χ*

**L**and rearrange), where

*τ*

^{−1}is the loss term,

*σ*is defined after Eq. (7), the vector Ψ represents the non-linear dynamics [use vector identities e.g. (

**a**×

**b**) ×

**c**=

**b**(

**a**·

**c**) –

**a**(

**b**·

**c**) etc], and Under the DDA and NNA, Eqs. (17)–(18) constitute a self-consistent formulation for general second order parametric optics in plasmonic SDC’s.

## 3. Second harmonic generation

*x̂*polarized SDC mode with

*ω*

_{1},

*β*

_{1}satisfying the corresponding linear ideal (lossless) dispersion relation Eq. (7). Likewise, let wave#3 be a

*ẑ*polarized SDC mode, with

*ω*

_{3},

*β*

_{3}satisfying the corresponding dispersion relation in Eq. (7)

*and*the SHG phase matching condition in Eq. (8). This is possible by virtue of the results shown in Fig. 2. We express these modes in a slight generalization of Eq. (6), where

*i*= 1,3 depend on

*n*to allow for possible loss, and for gain or depletion due to mutual interactions. We turn to obtain the dynamic equation for the

*ω*

_{3}oscillations from the formulation in Eqs. (17)–(18). For compactness, we also define To get both sides of Eq. (17) to oscillate at

*ω*

_{3}= 2

*ω*

_{1}the linear terms should incorporate only wave#3, and Ψ should incorporate only wave#1. The former is

*ẑ*-polarized. The surviving

*ẑ*directed terms in Ψ that contribute to

*ω*

_{3}oscillations are the second one (

*A*

_{1}=

*A*(

*k*

_{1}

*d*,

*β*

_{1}) where

*A*(

*kd*,

*β*) is defined in equations Eq. (16) (this is because the magnetization field, presented by

*A*, is due to wave#1). Likewise, to get both sides of Eq. (17) to oscillate at

*ω*

_{1}the linear terms should incorporate only wave#1, and Ψ should incorporate only multiplications between waves#1 and #3. The only surviving terms are

### 3.1. Non depleted pump with β_{1} = π/2 in lossless chain

*exact*solution to Eq. (25), that satisfies the initial condition in Eq. (24), is given by [use (

*ω*

_{3}/

*ω*)

_{p}^{2}=

*N*+ 2

_{z}*σ*,

*β*

_{3}= 2

*β*

_{1}=

*π*, as implied by Eqs. (7)–(8)) and their solution in Fig. 2] It is seen that the SDC’s second-harmonic wave grows as

*n*

^{2}- a quadratic growth typical to SHG.

### 3.2. Chain with loss

*but still with the exact phase matching conditions in Eq. (8)*. Note that the mode magnetization (which carries the gain) is maximized at

*β*

_{1}≈ 0.4

*π*- see Fig. 3. We “generalize” the non-depleted pump assumption to hold for a lossy medium: we assume that the pump attenuation is mainly due to loss mechanism, and the attenuation due to energy transfer to wave# 3 can be neglected. Hence, we have for the pump: The attenuation factor

*γ*can be computed by generalizing the dispersion relation to hold for complex

*β*[4

**74**, 205436 (2006). [CrossRef]

*γ*≈ 0.1 : 0.15 [1

1. M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” Opt. Lett. **23**(17), 1331–1333 (1998). [CrossRef]

**62**(24), R16356 (2000). [CrossRef]

*e*

^{−2γn}. A straightforward substitution of

*ae*

^{−2γn}into the equation shows that the particular solution is given by The second term is the homogeneous solution that satisfies the homogeneous counterpart (no forcing) of Eq. (29). It has the form

*r*is obtained by substituting this solution into the homogeneous equation and looking for the roots of the characteristic polynomial Note that

*r*

_{1}

*r*

_{2}=

*e*

^{−2iβ3}. A careful examination shows that one of the roots, say

*r*

_{1}, satisfies |

*r*

_{1}| < 1. Thus, the solution that satisfies Eq. (29)

*and*the boundary conditions in Eq. (28) is It is interesting to point out that the solution for lossless chain with

*β*

_{1}=

*π*/2 studied in Sec.3.1 cannot be obtained by a mere substitution of

*β*

_{1}=

*π*/2, 1/

*τ*= 0 in Eq. (33). For

*β*

_{1}=

*π*/2, 1/

*τ*= 0 the characteristic polynomial has a higher order root multiplicity, in which case the solution must be written in terms of powers of

*n*.

*local*ratio between the pump wave and the second harmonic. This ratio is given by Hence, there is a critical value for

*r*

_{1}: if |

*r*

_{1}| >

*e*

^{−γ}this ratio increases unboundedly as the waves propagate along the chain, signifying an efficient energy transfer from the pump to the second harmonic. Clearly, for very large

*n*the non-depleted assumption in lossy chain, defined at this subsection onset, ceases to be valid.

*local*ratio between the signal and the pump as stated before; (2) the value of the signal when the typical loss decay is “cleaned out” (due to the multiplication by

*e*). As will be shown in the example below, the result is exponentially increasing, with typical numerical values shown in Fig. 4.

^{γn}*x̂*polarized pump wave propagates with

*β*

_{1}= 0.4

*π*. To achieve SHG phase-matching, we choose

*a*/

_{x}*d*= 0.5, and we use Fig. 2(a) and get

*a*/

_{x}*a*≈ 2.93, ⇒

_{z}*N*≈ 0.112. Solving for the corresponding complex dispersion relation with Ag loss factor 1/(

_{x}*τω*) = 2 · 10

_{p}^{−3}we get

*ω*

_{1}= 0.3323

*ω*and

_{p}*γ*= 0.143. For Ag,

*ω*≈ 8.6 · 10

_{p}^{15}rad/sec, hence

*ω*

_{1}= 2.86 · 10

^{15}so

*λ*

_{1}= 660nm. The prolates are spaced by

*d*= 80nm, hence

*a*≈ 40nm,

_{x}*a*≈ 14nm, and

_{z}*σ*=

*V*/(2

*πd*

^{3}) = 0.0097. We used these parameters to compute the SHG in Eqs. (34)–(35), with the exact magnetization field in Eqs. (13)–(15). The results are shown in Fig.4 for

*r*

_{1}| = 0.8915 >

*e*

^{−γ}= 0.8668, thus the local ratio between the second harmonic and the pump increases along the array.

## 4. Summary

## A. Derivation of Eq. (2)

**H**-field assumption. Since the force exerted by

**H**on the free charge is

*velocity dependent*, and since

**H**is assumed to be weak, we present the charge displacement as

**r**=

**r**

_{1}+

**r**

_{2}where the former is due to the electric field

**E**and the latter is a correction due to

**H**and due to the movement exerted by

**E**. Hence, up to first order in

**H**, the Lorentz force induced electron displacement is governed by We assume that

**E**and

**H**oscillate at frequencies

*ω*

_{1}and

*ω*

_{2}, respectively (these are not Phasors; time dependence is included). Hence

**r**

_{1}and

**r**

_{2}possess

*e*

^{−iω1t}and

*e*

^{−i(ω1+ω2)t}time dependencies, respectively. Equation (36) reads now where

*ω*

_{3}=

*ω*

_{1}+

*ω*

_{2}. Since charge displacement is proportional to dipole volume density, we have where

**P**

_{1,2}are the dipole volume densities associated with the displacements

**r**

_{1,2}respectively. The scalar

*ω*

_{1}and

*ω*

_{2}imply that

## Acknowledgments

## References and links

1. | M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” Opt. Lett. |

2. | S. A. Tretyakov and A. J. Viitanen, “Line of periodically arranged passive dipole scatterers,” Electrical Engineering |

3. | M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B |

4. | Andrea Alu and Nader Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” Phys. Rev. B |

5. | D. V. Orden, Y. Fainman, and V. Lomakin, “Optical waves on nanoparticle chains coupled with surfaces,” Opt. Lett. |

6. | D. V. Orden, Y. Fainman, and V. Lomakin, “Twisted chains of resonant particles: optical polarization control, waveguidance, and radiation,” Opt. Lett. |

7. | Y. Hadad and Ben Z. Steinberg, “Magnetized spiral chains of plasmonic ellipsoids for one-way optical waveguides,” Phys. Rev. Lett. |

8. | Zi-jian Wu, Xi-kui Hu, Zi-yan Yu, Wei Hu, Fei Xu, and Yan-qing Lu, “Nonlinear plasmonic frequency conversion through quasiphase matching,” Phys. Rev. B |

9. | A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, “Quadratic phase matching in nonlinear plasmonic nanoscale waveguides,” Opt. Express |

10. | G. Bachelier, I. R. Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Multipolar second-harmonic generation in noble metal nanoparticles,” J. Opt. Soc. Am. B |

11. | Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B |

12. | H. Husu, J. Makitalo, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Particle plasmon resonance in L-shaped gold nano-particles,” Opt. Express |

13. | J. Nappa, G. Revillod, I. R. Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Electric dipole origin of the second harmonic generation of small metallic particles,” Phys. Rev. B |

14. | J. Shan, J. I. Dadap, I. Stiopkin, G. A. Reider, and T. F. Heinz, “Experimental study of optical second-harmonic scattering from spherical nanoparticles,” Phys. Rev. A |

15. | B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in noncentrosymmetric nanodimers,” Nano Letters |

16. | S. Kujala, B. K. Canfield, and M. Kauranen, “Multipole interference of second harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. |

17. | T. Utikel, T. Zentgraf, T. Paul, C. Rockstuhl, F. Lederer, M. Lippitz, and H. Giessen, “Towards the origin of the nonlinear response in hybrid plasmonic systems,” Phys. Rev. Lett. |

18. | J. D. Jackson, |

19. | S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B |

20. | A. H. Sihvola, |

21. | Leonard Lewin, |

22. | M. Abramowitz and I. A. Stegun, |

**OCIS Codes**

(190.2620) Nonlinear optics : Harmonic generation and mixing

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(250.5403) Optoelectronics : Plasmonics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: June 13, 2011

Revised Manuscript: September 15, 2011

Manuscript Accepted: September 19, 2011

Published: December 5, 2011

**Citation**

Ben Z. Steinberg, "Parametric plasmonics and second harmonic generation in particle chains," Opt. Express **19**, 25843-25853 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-27-25843

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### References

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- Leonard Lewin, Polylogarithms and Associated Functions (Elsevier1981).
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