Modified long wavelength approximation for the optical response of a graded index. plasmonic nanoparticle. Republic of China OR , U. S. A.

Transcription

1 Modified long wavelength approximation for the optical repone of a graded index plamonic nanoparticle H. Y. Chung 1*, P. T. Leung 2,3# and D. P. Tai 1 1 Department of Phyic, National Taiwan Univerity, Taipei 10617, Taiwan, Republic of China 2 Department of Phyic, Portland State Univerity, P. O. Box, Portland, OR , U. S. A. 3 Intitute of Optoelectronic Science, National Taiwan Ocean Univerity, Keelung, Taiwan 202, Republic of China Abtract The optical repone of graded index pherical metallic nanoparticle i tudied in the modified long wavelength approximation (MLWA) with electrodynamic effect accounted for to the lowet order of the invere of the wavelength. An effective medium approach i adopted which lead to the concluion that the firt order dynamical effect will enter mainly via the polarizability and not the effective dielectric function of the ytem. Numerical tudie uing variou graded Drude function how that thee effect are not only ignificant for particle of large ize, but can alo be appreciable for maller particle with varying index profile. Correponding author: * b @ntu.edu.tw # hopl@pdx.edu 1

2 Introduction Application of the ditinct propertie of inhomogeneou optical material ha been of prime ignificance ince the early day of fiber optic 1, panning a large cope of interet from the well-etablihed reearch in gradient optic 2 to the recent exploion in the development of metamaterial 3. With the emerging field of tranformation optic etablihed 4 it i believed that it will ultimately be poible to achieve almot unlimited control of the propagation of light via pecific deign of dielectric medium, exploiting both the electric and magnetic propertie of both metallic (plamonic) and nonmetallic material 5. Of a relatively imple kind of inhomogeneou ytem i the functionally graded material 6, and in particular, the graded index particle 7. With the firt fabrication of thee material reported more than two decade ago 6, intriguing optical application of thee inhomogeneou particle are till being dicovered in recent time 8. However, to thi date, theoretical undertanding of the optical propertie of thee particle beyond the quai-tatic approximation i baed mainly on numerical imulation 8 ; and analytic model available o far in the literature are largely limited to tatic dielectric repone of thee particle 9. It i well-known that the full optical (electrodynamic) olution for even a homogeneou pherical particle the famou Mie theory 10 i rather nontrivial, and it took everal decade for theorit to complete a program of extenion of thi olution to a general tratified pherical multilayered ytem 11. Although the theory ha alo been generalized in the literature to the cae of a graded phere with a radial continuouly-varying index profile 12, one till ha to reort to a highly numerical approach in olving the Maxwell wave equation within thi inhomogeneou ytem. Neverthele, in the cae when the particle ize i much maller than the 2

3 incident wavelength of the light, impler quai-tatic olution for the dipole repone of the particle have been obtained in different approache 9. Thee include both the olution of the electrotatic field equation and the approach uing the differential effective dipole approximation (DEDA), repectively, and comparable accuracy between the two method ha alo been demontrated 9. It wa then hown later that thee two approache are indeed equivalent to one another 13. Thi i not urpriing ince in the DEDA, the graded-index particle i modeled by a multi-hell procedure, and the effective dipole polarizability o derived i guaranteed to produce identical olution a obtained from olving the quai-tatic field equation, due to the uniquene in the olution of the boundary value problem 13. Moreover, thee quai-tatic olution will become inaccurate for large particle ize, or a we hall ee below, even for maller particle with a varying index profile. It i the purpoe of our preent work to extend the previou quai-tatic theory 9, 13 for thee graded index particle o that they can be applied to particle of larger ize and/or material with rapidly varying index profile. We hall focu on inhomogeneou plamonic particle, and our approach i baed on the o-called Modified Long Wavelength Approximation (MLWA) 14 which ha been hitherto etablihed in the literature only for homogeneou phere and pheroidal particle 15, 16, and array of them 17, 18 ; a well a tratified pherical (and pheroidal) ytem 19, 20. We hall firt etablih the model and then apply it to the calculation of the extinction cro ection for variou graded index phere, through which the failure of the quai-tatic reult will be demontrated. 3

4 Theory 14, 15 The main idea of the original MLWA i to obtain the lowet order correction to the quai-tatic theory from the finitene of the wavelength, by modifying the electrotatic depolarization field inide a polarizable phere to include (i) the dynamic depolarization due to retardation and (ii) the radiation damping field acting on the induced dipole in the phere. Such modification are hence limited only to the dipole repone of the phere, ince the above two modification are obtained from the full dynamical (retarded) field of the induced electric dipole within the phere 15. Depite uch limitation, however, it ha been checked for conitency with the lowet order reult from the exact Mie theory for a pherical particle 15, 16, and ha been proven to be ufficiently accurate when compared to exact calculation for ingle pherical metal particle up to ize of ~ 100 nm for optical wavelength 21. In a recent work on the optical propertie of a nanomatryohka 20, we have generalized the original MLWA to a pherical multi-hell tructure with the application of an effective medium treatment previouly introduced in the literature 22. Thi treatment aume a heterogeneou ytem of multilayered hell to be effectively replaced by a homogeneou one with a uniform dielectric function, which i determined by equating the polarizabilitie of the two ytem under the hypothetical condition when the outide (hot) medium i replaced by the ame material a that of the outermot hell in the ytem 20,22. The reult o obtained are hence limited to particle of mall dimenion and pherical ymmetry (o that they can be characterized by imple polarizabitie) although we have alo extended the method to treat pheroidal hell in our previou work 19. Here we hall extend the idea further to treat a graded index phere with a radially-varying dielectric function () r which can a well be diipative (i.e. complex) and diperive [i.e. ( r, ) ]. Our goal i 4

5 to find an effective dielectric function, which can be ued to calculate the polarizability of the original phere uing the reult for a homogeneou phere. For a given radiu r, let u conider a homogeneou phere of the ame radiu ( r ) with a dielectric function denoted by ( r). Note that the argument r of indicate the radiu of the effective homogeneou phere, which hould not be confued with the patial coordinate appeared in the original dielectric function () r. According to the MLWA, the dipole polarizability of uch a phere in a homogeneou hot medium h can be expreed a follow 15 : () r h 3 () r r, (1) 3 ( r)[ ( r) ] h h where 2 3 ( r) 1 q i2 q / 3, and q 2 r /. Note that Eq. (1) lead back to the imple electrotatic reult for. Note alo that the ( r) to be determined according to the approach decribed above will be independent of the hot function h ince it i obtained under the condition when the hot i replaced by the material of the outermot layer of the graded ytem a explained above [20, 23, 24, 26] (ee Eq. (6) below). Hence, for the phere of radiu r dr [Fig 1(a)], the polarizability can be expreed a: ( r dr) h 3 ( r dr) ( r dr). (2) 3 ( r dr)[ ( r dr) ] h h On the other hand, the ytem in Fig. 1(a) can alo be treated equivalently a a phere of radiu r coated with an infiniteimal thin hell of thickne dr and dielectric function ( r dr) a illutrated in Fig. 1(b). Hence if we now imagine a hypothetical ituation when the hot medium i completely replaced by a medium of dielectric function ( r dr), both ytem in Fig. h 5

6 1(a) and 1(b) will become equivalent a illutrated in Fig. 1(c) and 1(d). Applying thi equivalence with the reult given in Eq. (1) and (2), we obtain: ( r) ( r dr) 3 ( r dr) ( r dr) 3 r ( r dr). 3 ( r dr) ( r)[ ( r) ( r dr)] 3 ( r dr) ( r dr)[ ( r dr) ( r dr)] Now with a firt order Taylor expanion a follow: (3) ( r dr) dr ( r dr) dr, (4) ( r dr) dr where the argument r i omitted on the right hand ide for implicity and ( ' ) d / dr, Eq. (3) finally lead to the following differential equation for : d ( ) [(3 r )( ) 9 ], (5) dr 3r or equivalently: d ( ) [(3 2 q )( ) 9 ], (6) dr 3r where,,, q are all function of r. The boundary condition i lim ( r) lim ( r). Note r 0 r 0 that Eq. (6) together with Eq. (1) form our main reult which contain the dynamic correction 2 to the effective dielectric function in the MLWA imply in a ingle term ( q ). Furthermore, a we hall limit our application of the MLWA to particle with ize of about 10% of the wavelength, we thu expect q 2 1 for mot application of Eq. (6) and thu the effect of the MLWA on will be een to be negligible in general a illutrated in the following numerical 6

7 analyi. In fact, one can how that in the limit q 0, Eq. (6) indeed reduce to the previou quai-tatic reult obtained in Ref. [9] a follow. Thu we have from Eq. (6) with q 0 : d 1 ( )( 2 ). (7) dr r Next we expre the dipole factor b introduced in Ref. [9] in term of a follow: h b 2 h, (8) which yield 1 2b h. (9) 1 b Subtitute Eq. (9) into Eq. (7) lead to d 1 2b 1 1 2b 1 2b h h h 2 dr 1 b r 1 b 1 b, (10) which implifie to db 1 [(1 2 b ) h (1 b ) ][(1 2 b ) h 2(1 b ) ], (11) dr 3r h a reult identical to the DEDA reult obtained in Ref. [9]. But ince the DEDA i an exact reult in agreement with the olution from the field equation 13, our MLWA reult in Eq. (5) and (6) will erve to modify the exact quai-tatic reult a well. 7

8 Model and numerical calculation To illutrate the dynamical effect, we here perform ome numerical tudie with imple profile function for the dielectric function () r uing our reult in Eq. (6) and how that the effect of the 2 m 2 q term i minimal. For a power law dependence () r cr, the zeroth order ( q = 0) olution of (6) can be obtained in the following form 9 : m () r zr, (12) with c 2 z [ (1 m) m 2 m 9]. (13) 2 Uing Eq. (12) and (13) into the RHS of Eq. (6), we can obtain the firt order olution for via one imple integration. In addition, Eq. (6) can alo be olved accurately for thi imple profile function of () r uing a numerical olver via direct integration of the nonlinear differential equation. Figure 2 how the reult for obtained by the three approache a decribed above. We have fixed the radiu of the inhomogenou phere to be a 0.1 and have normalized the reult with repect to ( a). It i obviou that the MLWA effect on the effective dielectric function are inignificant for uch phere ize. We hall thu ignore thee effect and imply apply Eq. (12) and (13) into the MLWA polarizability (i.e. Eq. (1)) to tudy the effect on the extinction efficiency of a relatively large graded phere in the following, and hall focu on graded plamonic particle with a dielectric function of the following form 23 : n ( r, ) Nr ( ), (14) Ag 8

9 where N i a normalization parameter, and the metallic (ilver) dielectric function ha the following Drude form 19 2 : 1 / ( ) with Ag p i 16 1 p and B F / v a, where 13 1 B and 15 vf nm/. Figure 3 how the pectral extinction efficiency 24 for two particle ize with a linear radial profile function ( m 1), where reult are compared between the preent MLWA and the quai-tatic model. In addition, the accuracy of the MLWA reult i confirmed via a comparion with thoe obtained from a multilayer approximation 8 baed on the Mie theory 11. Thee Mie reult are obtained by approximating the phere a a tratified ytem of a number of dicrete layer with dielectric function determined a the mean value of the function for each of the layer. It i oberved that except for the higher order multipole peak, the MLWA reult are much cloer to the Mie reult, with both howing a ignificant red-hift and diminution in the dipole urface plamon peak compared to that obtained from the quai-tatic theory. The inaccuracy of thi latter theory i further manifeted when the phere radiu increae from 15 nm to 30 nm a expected. Figure 4 how the dynamic effect due to the variation in the profile function for a fixed phere ize of mall radiu (10 nm), where the power m of the profile function i varied. While it i often taken for granted that the quai-tatic reult hould be good for uch a mall ize of the phere, we how here that uch an aumption may not be valid for a mall graded-index particle with a varying index profile. Indeed, although we ee from Fig. 4(a) that all the three calculation (tatic, MLWA, and Mie) are very cloe for a homogeneou particle ( m 0); the quai-tatic reult become rather inaccurate for the ame mall-ized particle with a varying index profile ( m 1,2 ) a hown in Fig. 4(b) and 4(c). Thi i conitent with what wa 9

10 oberved previouly in the nanomatryohka multi-hell ytem [20]. We alo ee that while the MLWA reult again compare reaonably well with the Mie reult, the relative error between each of the quai-tatic and MLWA reult and the Mie reult become more ignificant when we go from the linear to the quadratic profile for thi rather mall 10 nm radiu phere. Hence one mut be very critical with the accuracy of the quai-tatic theory when applied to gradedindex particle. Concluion We have in thi work preented a relatively imple but more accurate model for the calculation of the optical repone of a graded index phere with a radially-varying dielectric profile of not-too-mall ize. Baed on the MLWA, we have introduced firt order correction from retardation effect leading to ignificant improvement on the quai-tatic reult. We have found that thee correction are not only important for particle of larger ize, but alo even for mall particle with a varying index profile. Thu, except for very mall and uniform particle, the MLWA will lead to ignificant red-hift, broadening, and diminution in the extinction pectrum for graded index particle; yielding value cloer to the exact reult obtained from fully dynamical calculation. To our knowledge, ince the olution to the Maxwell equation in the inhomogeneou medium i rather nontrivial 12, one ha to often reort to either an allnumeric approach or omething like a dicrete layer approximation to theoretically tudy light cattering from thee particle when their ize are not too mall 8. Thu our preent work will be ueful to provide an analytic model for the tudy of optical interaction with thee particle beyond the quai-tatic theory 9, with accuracy poibly extended to particle ize of around 10% of the wavelength of the incident light 21. For more complicated radial index profile for which 10

11 Eq. (6) may not be able to yield analytical olution (even in the q 0 limit), the optical repone can till be accounted for via numerical computation baed on Eq. (1) and (6) in a way more efficient than numerically implementing the full Mie theory 8, 12, 20. Of coure, the MLWA alo ha it obviou limitation to be applicable only to particle of not-too-large ize a well a it inability to account for higher mulipole excitation of the particle. In addition, it alo cannot be applied to tranparent particle although thi i not our concern here with plamonic nanoparticle. The reaon i that an unrealitic aborption cro ection can be deduced from the complex MLWA polarizability (Eq. (1)), due to the preence of the radiation damping term (i.e. the q 3 term in ( r) ), wherea thi term vanihe for a tranparent particle. It will be our future endeavor to extend our preent effective medium model o that ome of thee limitation can be overcome. Acknowledgment We would like to thank Profeor K. W. Yu of the Chinee Univerity of Hong Kong for ueful communication. We alo acknowledge the financial upport from National Science Council, Taiwan, ROC, under Grant number NSC M , I , 98-EC-17- A-09-S1-019 and NSC M MY3. 11

12 Reference 1. A. W. Snyder and J. D. Love (1983) Optical Waveguide Theory, Chapman & Hall, New York. 2. C. Gomez-Reino, M. V. Perez, and C. Bao (2002) Gradient-Index Optic, Fundamental and Application, Springer-Verlag, Berlin. 3. W. Cai and V. M. Shalaev (2009) Optical Metamaterial: Fundamental and Application, Springer, New York. 4. U. Leonhardt and T. G. Philbin, (2009), Chapter 2 Tranformation Optic and the Geometry of Light, Prog. Opt. 53, M. Wegener and S. Linden, (2010), Shaping optical pace with metamaterial, Phy. Today 63(10), M. Yamanouchi, M. Koizumi, T. Hirai, and I. Shiota, (1990), in Proceeding of the Firt International Sympoium on Functionally Graded Material, Sendai, Japan. 7. See, e.g., Z. F. Sang and Z. Y. Li, (2005), Effective negative refractive index of graded granular compoite with metallic magnetic particle, Phy. Lett. A 334(5-6), S. C. Kong, A. Taflove, and V. Backman, (2009), Quai one-dimenional light beam generated by a graded-index microphere, Opt. Exp. 17(5), L. Dong, G. Q. Gu, and K. W. Yu, (2003), Firt-principle approach to dielectric repone of graded phere particle, Phy. Rev. B 67(22), G. Mie, (1908), Contribution on the optic of turbid media, particularly colloidal metal olution, Ann. Phy. 2, R. Bhandari, (1965), Scattering coefficient for a multilayered phere: analytic expreion and algorithm, Appl. Opt. 24(13), ; Z. S. Wu and Y. P. Wang, (1991), 12

13 Eletromagnetic cattering for multilayered phere: Recurive algorithm, Radio Sci. 26(6), ; B. R. Johnon, Light cattering by a multilayer phere, (1996), Appl. Opt. 35(18), ; B. R. Johnon, (1999), Exact theory of electromagnetic cattering by a heterogeneou multilayer phere in the infinite-layer limit: effective-media approach, J. Opt. Soc. Am. A 16(4), ; W. Yang, (2003), Improved recurive algorithm for light cattering by a multilayered phere, App. Opt. 42(9), P. J. Wyatt, (1962), Scattering of electromagnetic plane wave from inhomogeneou pherically ymmetric object, Phy. Rev. 127(5), K. W. Yu and G. Q. Gu, (2005), Effective conductivity of compoite of graded pherical particle, Phy. Lett. A 345(4-6), K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, (2003), The optical propertie of metal nanoparticle: the influence of ize, hape, and dielectric environment, J. Phy. Chem. B 107(3), M. Meier and A. Wokaun, (1983), Enhanced field on large metal particle: dynamic depolarization, Opt. Lett. 8(11), A. Moroz, (2009), Depolarization field of pheroidal particle, J. Opt. Soc. Am. B 26(3), M. Meier and A. Wokaun, and P. F. Liao, (1985), Enhanced field on rough urface: dipolar interaction among particle of ize exceeding the Rayleigh limit, J. Opt. Soc. Am. B 2(6), R. Kullock, S. Graftrom, P. R. Evan, R. J. Pollard, and L. M. Eng, (2010), Metallic nanorod array: negative refraction and optical propertie explained by retarded dipolar interaction, J. Opt. Soc. Am. B 27(9),

14 19. H. Y. Chung, P. T. Leung, and D. P. Tai, (2009), Dynamic modification of polarizability for large metallic pheroidal nanohell, J. Chem. Phy. 131(12), H. Y. Chung, G. Y. Guo, H. P. Chiang, D. P. Tai, and P. T. Leung, (2010), Accurate decription of the optical repone of a multilayered pherical ytem in the long wavelength approximation, Phy. Rev. B 82(16), H. Merten, A. F. Koenderink, and A. Polman, (2007), Plamon-enhanced luminecence near noble- metal nanophere: Comparion of exact theory and an improved Gerten and Nitzan model, Phy. Rev. B 76(11), N. V. Vohchinnikov and J. S. Mathi, (1999), Calculating cro ection of compoite intertellar grain, Atrophy Journal 526(1), ; J. Li, G. Sun, and C. T. Chan, (2006), Optical propertie of photonic crytal compoed of metal-coated phere, Phy. Rev. B 73(7), Note that the ubject of graded plamonic tructure i of great recent interet (ee the comprehenive review by J. J. Xiao, K. Yahubo and K. W. Yu (2010) in K. N. Heley (ed), Plamonic: Theory and Application, Nova Sci, Hauppauge, Chapter 13. Here we hall limit ourelve to a toy model tudy adopting a graded Drude model in our numerical illutration. Different form of graded Drude model have been tudied Huang and Yu, (2004), Optical nonlinearity enhancement of graded metallic film, Appl. Phy. Lett. 85(1), The extinction efficiency i defined a the um of the aborption and cattering efficiencie: Q 8 Im a and ab Qc where Q ab 2 Q c a 14

15 Figure caption 1. Illutration of the effective-medium approach (ee text for explanation). 2. The effective dielectric function for a graded phere with a power law profile function m () r cr and radiu of a 0.1 according to the MLWA precription. Reult are normalized to ( a), and are hown for the full numerical, the zeroth order, and the firt order olution, from which the effect on due to a finite wavelength i een to be minimal within the MLWA (color online). 3. Comparion of the extinction efficiencie computed uing the quai-tatic (dotted), the MLWA (dahed) and the tratified Mie (olid) model, repectively, for a graded phere with radiu of (a) 15 nm and (b) 30 nm. The dielectric function of the phere i taken to have the form ( r) c( ) r, where c( ) ( ) / 30 in unit of nm -1. Ag 4. The extinction efficiencie of a graded phere of a dielectric function ( r) c( ) r m with (a) m 0, (b) m 1 and (c) m 2. The radiu of the phere i fixed to be 10 nm. 15

16 Fig. 1 16

17 Fig. 2 17

18 Fig. 3 18

19 Fig. 4 19