Numerial study of surfae plasmon enaned nonlinear absorption and refration Dana C. Kolgraf-Owens 1* and Pieter G. Kik 1, 1 CREOL and FPCE: Te College of Optis and Potonis, University of Central Florida, 4 Central Florida Boulevard, Orlando, FL 3816 Also at Pysis Department, University of Central Florida, 4 Central Florida Boulevard, Orlando, FL 3816 * Corresponding autor: kolgraf@reol.uf.edu Abstrat: Maxwell Garnett effetive medium teory is used to study te influene of silver nanopartile indued field enanement on te nonlinear response of a Kerr-type nonlinear ost. We sow tat te omposite nonlinear absorption oeffiient, β, an be enaned relative to te ost nonlinear absorption oeffiient near te surfae plasmon resonane of silver nanopartiles. Tis enanement is not due to a resonant enanement of te ost nonlinear absorption, but rater due to a pasesifted enanement of te ost nonlinear refrative response. Te enanement ours at te expense of introduing linear absorption, α, wi leads to an overall redued figure of merit β /α for nonlinear absorption. For tin (< 1 µm) omposites, te use of surfae plasmons is found to result in an inreased nonlinear absorption response ompared to tat of te ost material. 8 Optial Soiety of Ameria OCIS odes: (6.65) Effetive medium teory; (16.145) Artifiially engineered materials; (16.433) Nonlinear optial materials; (16.436) Nanomaterials Referenes and links 1. L. Franois, M. Mostafavi, J. Belloni, and J. A. Delaire, "Optial limitation indued by gold lusters: Meanism and effiieny," Pys. Cem. Cem. Pys. 3, 4965-4971 (1).. L. Franois, M. Mostafavi, J. Belloni, J. F. Delouis, J. Delaire, and P. Feneyrou, "Optial limitation indued by gold lusters. 1. Size effet," J. Pys. Cem. B 14, 6133-6137 (). 3. F. E. Hernandez, W. Sensky, I. Coanosi, D. J. Hagan, and E. W. Van Stryland, "India ink/arbon disulfide reates laser safety devie," Laser Fous World 37, 15 (1). 4. F. E. Hernandez, W. Sensky, I. Coanosi, D. J. Hagan, and E. W. Van Stryland, "Visosity dependene of optial limiting in arbon blak suspensions," Appl. Opt. 41, 113-117 (). 5. F. E. Hernandez, S. S. Yang, V. Dubikovskiy, I. W. Sensky, E. W. Van Stryland, and D. J. Hagan, "Dual Foal Plane Visible Optial Limiter," J. Nonlinear Opt. Pys. Mater. 9, 43 (). 6. X. Sun, R. Q. Yu, G. Q. Xu, T. S. A. Hor, and W. Ji, "Broadband optial limiting wit multiwalled arbon nanotubes," Appl. Pys. Lett. 73, 363-3634 (1998). 7. Y. P. Sun and J. E. Riggs, "Organi and inorgani optial limiting materials. From fullerenes to nanopartiles," Int. Rev. Pys. Cem. 18, 43-9 (1999). 8. G. Wang and W. F. Sun, "Optial limiting of gold nanopartile aggregates indued by eletrolytes," J. Pys. Cem. B 11, 91-95 (6). 9. J. E. Sipe and R. W. Boyd, "Nonlinear Suseptibility of Composite Optial-Materials in te Maxwell Garnett Model," Pys. Rev. A 46, 1614-169 (199). 1. M. I. Stokman, K. B. Kurlayev, and T. F. George, "Linear and nonlinear optial suseptibilities of Maxwell Garnett omposites: Dipolar spetral teory," Pys. Rev. B 6, 1771-1783 (1999). 11. D. D. Smit, G. Fiser, R. W. Boyd, and D. A. Gregory, "Canellation of potoindued absorption in metal nanopartile omposites troug a ounterintuitive onsequene of loal field effets," J. Opt. So. Am. B 14, 165-1631 (1997). 1. A. E. Neeves and M. H. Birnboim, "Composite strutures for te enanement of nonlinear-optial suseptibility," J. Opt. So. Am. B 6, 787-796 (1989). 13. A. A. Salisi, G. Compagnini, L. D'Urso, and O. Puglisi, "Nonlinear optial ativity in Ag-SiO nanoomposite tin films wit different silver onentration," Appl. Surf. Si. 6, 37-41 (4). #965 - $15. USD Reeived 1 May 8; revised 4 Jun 8; aepted 8 Jun 8; publised 3 Jul 8 (C) 8 OSA 7 July 8 / Vol. 16, No. 14 / OPTICS EXPRESS 1683
14. S. Qu, C. Du, Y. Song, Y. Wang, Y. Gao, S. Liu, Y. Li, and D. Zu, "Optial nonlinearities and optial limiting properties in gold nanopartiles proteted by ligands," Cem. Pys. Lett. 356, 43-48 (). 15. S. Debrus, J. Lafait, M. May, N. Pinon, D. Prot, C. Sella, and J. Venturini, "Z-san determination of te tird-order optial nonlinearity of gold:silia nanoomposites," J. Appl. Pys. 88, 4469-4475 (). 16. Y. Hosoya, T. Suga, T. Yanagawa, and Y. Kurokawa, "Linear and nonlinear optial properties of sol-gelderived Au nanometer-partile-doped alumina," J. Appl. Pys. 81, 1475-148 (1997). 17. H. B. Liao, R. F. Xiao, J. S. Fu, H. Wang, K. S. Wong, and G. K. L. Wong, "Origin of tird-order optial nonlinearity in Au:SiO omposite films on femtoseond and pioseond time sales," Opt. Lett. 3, 388-39 (1998). 18. E. Cattaruzza, G. Battaglin, F. Gonella, G. Mattei, P. Mazzoldi, R. Polloni, and B. F. Sremin, "Fast tirdorder optial nonlinearities in metal alloy nanoluster omposite glass: negative sign of te nonlinear refrative index," Appl. Surf. Si. 47, 39-395 (5). 19. K. Uida, S. Kaneko, S. Omi, C. Hata, H. Tanji, Y. Asaara, A. J. Ikusima, T. Tokizaki, and A. Nakamura, "Optial nonlinearities of a ig onentration of small metal partiles dispersed in glass: opper and silver partiles," J. Opt. So. Am. B 11, 136-143 (1994).. O. Maruyama, Y. Senda, and S. Omi, "Non-linear optial properties of titanium dioxide films ontaining dispersed gold partiles," J. Non-Cryst. Solids 59, 1-16 (1999). 1. C. Flytzanis, F. Hae, M. C. Klein, D. Riard, and R. Roussignol, "Nonlinear Optis In Composite Materials," in Progress In Optis XXIX, E. Wolf, ed. (Elsevier Siene Publisers B.V., 1991), pp. 31-411.. F. Hae, D. Riard, and C. Flytzanis, "Optial Nonlinearities of Small Metal Partiles - Surfae-Mediated Resonane and Quantum Size Effets," J. Opt. So. Am. B 3, 1647-1655 (1986). 3. F. Hae, D. Riard, C. Flytzanis, and U. Kreibig, "Te Optial Kerr Effet in Small Metal Partiles and Metal Colloids - te Case of Gold," Appl. Pys. A-Materials Siene & Proessing 47, 347-357 (1988). 4. D. Riard, P. Roussignol, and C. Flytzanis, "Surfae-mediated enanement of optial pase onjugation in metal olloids," Opt. Lett. 1, 511-513 (1985). 5. R. A. Ganeev, A. I. Ryasnyansky, S. R. Kamalov, M. K. Kodirov, and T. Usmanov, "Nonlinear suseptibilities, absorption oeffiients and refrative indies of olloidal metals," J. Pys. D 34, 16-1611 (1). 6. A. Samo, "Dispersion of refrative properties of solvents: Cloroform, toluene, benzene, and arbon disulfide in ultraviolet, visible, and near-infrared," J. Appl. Pys. 94, 6167-6174 (3). 7. Dr. Sott Webster, CREOL and FPCE: Te College of Optis and Potonis, University of Central Florida, 4 Central Florida Blvd., Orlando, FL 3816 (personal ommuniation, 7). 8. P. B. Jonson and R. W. Cristy, "Optial-Constants Of Noble-Metals," Pys. Rev. B 6, 437-4379 (197). 9. R. del Coso and J. Solis, "Relation between nonlinear refrative index and tird-order suseptibility in absorbing media," J. Opt. So. Am. B 1, 64-644 (4). 1. Introdution Te fabriation of materials tat exibit strong nonlinear absorptive properties at visible and near-infrared frequenies as been of interest for several years [1-8]. Su materials would enable ontrol over te maximum transmitted irradiane or fluene troug a material, wi ould be used for example to minimize optially indued damage wile still allowing nondamaging levels of radiation to pass troug [3, 5]. One approa to aieve tis involves te use of materials tat exibit relatively large two-poton absorption (TPA). Due to te small values of te nonlinear absorption oeffiient in most materials, te nonlinear absorption tat an be aieved in tis manner is generally weak. A ommon approa to inrease te nonlinear absorption is to use lenses tat fous te inident ligt onto te nonlinearly absorbing sample [3, 5], resulting in a larger irradiane and terefore induing a larger nonlinear absorption. A signifiant drawbak of tis approa is te relatively ompliated and bulky nature of te resulting optial system. To simplify su systems it would be benefiial to design materials wit a strongly enaned nonlinear optial (NLO) response, potentially removing te need for external lenses. An alternative approa to aieving an enaned nonlinear response involves te use surfae plasmon resonanes. A surfae plasmon (SP) is a olletive osillation of free arges in a ondutor tat, wen resonantly exited, an indue strongly enaned loal fields. For example, isolated silver nanopartiles in water an produe loal field enanements at optial frequenies in te range of 1-4. As we will sow, tis field enanement an signifiantly inrease te magnitude of te nonlinear refrative index. In order to evaluate te effet of SPindued field enanement on te nonlinear absorption of a ost material, we investigate te #965 - $15. USD Reeived 1 May 8; revised 4 Jun 8; aepted 8 Jun 8; publised 3 Jul 8 (C) 8 OSA 7 July 8 / Vol. 16, No. 14 / OPTICS EXPRESS 1684
arateristis of a omposite onsisting of a small volume filling fration of metal nanopartiles (NPs) in a nonlinear ost material. Te NLO properties of su dilute metallodieletri omposites ave been studied extensively bot teoretially [9-1] and experimentally [13-5]. Many of te experiments reported in literature onsider te nonlinear response at a single wavelengt or disuss te influene of surfae plasmons on te nonlinear response of te omposite material wile ignoring te effet of te assoiated anges to te linear response on te figure of merit β /α were β is te effetive omposite nonlinear absorption oeffiient and α is te omposite linear absorption oeffiient. In te present teoretial study we investigate te nonlinear optial properties of a metal dieletri omposite over a broad range of frequenies and expliitly evaluate te effet of surfae plasmon-indued linear absorption on te nonlinear absorptive performane. In partiular we onsider te effet of surfae plasmons on te nonlinear response of te ost material, ignoring any nonlinearity of te metal itself. We analytially alulate te linear optial response of te omposite material and evaluate its effet on te maximum nonlinear absorption tat an be aieved in a omposite material of finite tikness. Using a realisti model system onsisting of metal nanopartiles embedded in a realisti igly nonlinear ost material, we sow tat te SP mediated field enanement in a nonlinear refrative ost results in a large positive nonlinear absorption. Furtermore we sow tat for very tin samples in wi te SP indued linear absorption is relatively low, te total nonlinear absorption of te omposite an be improved signifiantly by te addition of metal nanopartiles. Finally, we sow tat te presene of a strong SP related linear absorption in tese omposites puts a pratial upper bound on te tikness of te surfae plasmonenaned nonlinear absorbing omposite.. Computation of effetive medium parameters.1. Model system In order to study te effet of surfae plasmons on te nonlinear absorption of a omposite material, we investigate a model system tat is expeted to provide a large and fast nonlinear response. Te ost material is osen to be arbon disulfide (CS ) wi is a material wit a large intrinsi nonlinear absorption ross setion at te surfae plasmon resonane for Ag. Te linear properties for CS are taken from [6] and te omplex nonlinear refrative index of CS is taken to be n =6 1-15 m /W wit β =.6 m/gw, as obtained by z-san measurements using low repetition rate fs pulses at a enter wavelengt of 44 nm [7]. In te present study tese values are assumed to be onstant at wavelengts lose te nanopartile resonane (38-5nm). To enane te nonlinear response of CS, we onsider 1 nm diameter sperial silver nanopartiles, wi are well known to support surfae plasmon resonanes in te ultraviolet to blue spetral region wit large loal field orretion fators. For te desription of te silver nanopartiles in te frequeny range of interest we use a surfae sattering orreted Drude model fit to literature values obtained by Jonson and Cristy [8]. Te omplex dieletri funtion of te silver inlusions ε i as a funtion of angular frequeny ω [rad/s] is given by εi = ε ω p [ ω + i( Γ + Γs ) ω] (1) using fitting values of ε =5.451, ω p =1.474 1 16 [rad/s], and Γ = 8.354 1 13 [s -1 ], were Γ is te fitted bulk eletron sattering rate. Te estimated surfae sattering rate is given by Γ s = Aυ f /r =.8 1 14, were A is a onstant set to 1, υ f = 1.39 1 6 m/s is te Fermi veloity in silver, and r is te radius of te partile. As will be sown below, te enanement of te nonlinear absorption oeffiient depends strongly on te exat values used for te omplex dieletri funtion of te metal. Tis makes te onsideration of surfae sattering essential wen alulating te nonlinear response of silver nanopartile-based omposites. For example, a nanopartile diameter as large as 35 nm is suffiient to ange te total damping rate by 1% relative to te measured bulk values. #965 - $15. USD Reeived 1 May 8; revised 4 Jun 8; aepted 8 Jun 8; publised 3 Jul 8 (C) 8 OSA 7 July 8 / Vol. 16, No. 14 / OPTICS EXPRESS 1685
. Linear effetive medium properties Te linear optial properties of a system onsisting of a low volume fration of noninterating small sperial partiles embedded in a ost material are well desribed by Maxwell Garnett (MG) teory [9, 1], under ertain limiting onditions. Te first ondition is tat te optial response of te system an be well approximated by te eletrostati limit. Tis requires te nanopartiles to be mu smaller tan te inident wavelengt in te ost material as well as smaller tan te skin dept. Full field simulations of Ag partiles in CS under plane wave illumination (not sown) indiate tat bot requirements are satisfied for 1 nm diameter silver partiles. Te seond assumption is tat neigboring partiles do not interat. Tis requirement an be satisfied by working wit low volume filling frations. In tis study we ave assumed a volume filling fration, f, of 1-3 unless oterwise stated. For 1 nm diameter partiles, tis orresponds to a typial enter-to-enter partile spaing of te order of 1 nm. Under tese onditions, te dieletri funtion of a MG omposite material, ε, is given by ε 1+ fγ ε ε i = ε γ = () 1 fγ ε i + ε were ε i is te dieletri funtion of te inlusion (ere Ag), and ε is te dieletri funtion of te ost material (ere CS ). Te parameter γ is a field enanement fator representing te ratio of te internal partile response field, resulting from indued surfae arge, relative to te inident field. 8 4 (a) γ' γ" γ -4 α (x1 3 ) [m -1 ] 6 4 (b) 38 4 4 44 46 48 5 λ [nm] Fig. 1. (a) Real and imaginary parts of te loal field orretion fator and (b) omposite linear absorption. Figure 1(a) sows a plot of te real and imaginary parts of γ obtained from Eq. (). A resonane feature is observed at a wavelengt of 45 nm. Tis resonane oinides wit te frequeny at wi te denominator of te expression for γ reaes a minimum, wi ours wen ε i ' = -ε ' wit ε' indiating te real part of te respetive dieletri funtions. Tis feature is due to te exitation of te Fröli mode or te nanopartile plasmon resonane, in wi te eletrons osillate in a dipolar manner, as sown in te inset in Fig. 1(b). In tis and furter plots, te dased vertial line indiates te nanopartile plasmon resonane frequeny, and te dotted vertial lines represent te alf widt at alf maximum points of te plasmon resonane, or more preisely, of te imaginary part of te field enanement fator, #965 - $15. USD Reeived 1 May 8; revised 4 Jun 8; aepted 8 Jun 8; publised 3 Jul 8 (C) 8 OSA 7 July 8 / Vol. 16, No. 14 / OPTICS EXPRESS 1686
γ. Note tat on resonane, γ is almost entirely imaginary, indiating tat te response field inside and outside te partile exibit lose to a 9 pase differene relative to te inident ligt (a 9 pase lead inside te partile, and a 9 pase delay outside te partile along te diretion of polarization). Tis omplex nature of te partile response as an important onsequene: even if ε as a negligible imaginary omponent, te dieletri funtion of te omposite, ε, will ontain a nonzero imaginary part, resulting in a finite absorption oeffiient. In Fig. 1(b) we sow te resulting omposite linear absorption oeffiient, α. Near resonane, α reaes a peak value of ~7 m -1. Tis sows one major allenge in te use of surfae plasmons to enane nonlinear absorption: te presene of resonantly enaned linear absorption limits te useful sample tikness. Tis will be disussed in more detail below..3 Nonlinear effetive medium properties Te enaned fields tat our near metal nanopartiles as a result of te surfae plasmon resonane give rise to an enaned nonlinear polarization of te ost. Te MG teory as been extended to ompute te resulting nonlinear suseptibility enanement fator g of te tird order suseptibility of a metal-dieletri omposite assuming tat te metal response is entirely linear, and tat te ost exibits a nonlinear refrative index [9, 1]. Te resulting tird order suseptibility enanement fator is given by g χ χ 1 ε + ε ε + ε = (1 f )[8 f (1 + f + 5 3ε 3ε 3 + 6 f (1 + f ) γ γ + f (1 + f ) γ + 18 f ( γ + γ ) + 5] f ) γ γ were γ is defined in Eq. (). At low filling frations (i.e. f«1 and ε ε ) tis equation simplifies to g 3 [ 8γ γ + 6γ γ + γ + 18( γ + )] χ f = 1+ γ. (4) χ 5 Te total enanement of te tird order suseptibility is seen to depend on te field enanement fator aording to terms of magnitude γ 4, γ 3, and γ. Tis ompliated form is a onsequene of te inomogeneous distribution of te eletri field outside te polarized nanopartile. Note tat te maximum value of γ ours near resonane and sales wit 1/ε i. As a result, te magnitude of te nonlinear enanement fator depends strongly on te imaginary part of te dieletri funtion. Altoug te trends observed in tis study are general, te results will depend on te exat values of imaginary part of te dieletri funtion of te metal and of te ost. In order to evaluate te NLO refrative properties of te omposite material, we alulate te omplex nonlinear refrative index η = n + iκ based on te omposite nonlinear suseptibility χ = g χ as obtained from Eq. and using te known dieletri funtions. Sine our omposite medium as a omplex linear refrative index, we use te general equation relating te tird order suseptibility to te nonlinear omplex refrative index, wi is given in SI units by [9] m η 3 = 1 W 4ε η κ i χ n were n and κ are te real and imaginary parts of te linear refrative index respetively and χ is te nonlinear suseptibility in units of m /V. We an now define a nonlinear refrative index enanement fator, g, wi is given by (5) #965 - $15. USD Reeived 1 May 8; revised 4 Jun 8; aepted 8 Jun 8; publised 3 Jul 8 (C) 8 OSA 7 July 8 / Vol. 16, No. 14 / OPTICS EXPRESS 1687
g η, η 1 i( κ / n ) = = g (6) η, η 1 i( κ / n ) Note tat wit te exeption of extremely lossy omposites, te frequeny dependent nonlinear index enanement fator, g, will be virtually idential to te tird order suseptibility enanement fator, g. NLO index enanement fator a 8 4-4 b -8-1 g ' g " 9 1 6 a 15 3 18 6 1 b 1 33 4 3 7 38 4 4 44 46 48 5 λ [nm] Fig.. Real and imaginary parts of te nonlinear refrative index enanement fator g of 1 nm diameter Ag partiles embedded in CS. Figure sows te omplex nonlinear index enanement fator, g, for our omposite as obtained using Eq. ( (6). Near resonane, te nonlinear refrative index enanement fator is a omplex quantity, wi is a onsequene of te previously disussed frequeny dependent pase sift between te inident field and te loal field. Te inset sows te same data displayed in terms of omplex pase and amplitude, wit inreasing wavelengt orresponding to lokwise rotation along te urve. Tese results an be used to predit te influene of te surfae plasmon resonane on te response of a nonlinear ost material. Sine g is negative over most of te plasmon resonane, te use of a ost material wit a large positive κ (and tus a large effetive nonlinear absorption oeffiient) will in fat result in a omposite tat as a large negative κ orresponding to a ig effetive saturable absorption at te surfae plasmon resonane frequeny. Tese results demonstrate tat surfae plasmons do not diretly enane te nonlinear absorption oeffiient of a ost material, but rater provide a pase-sifted enanement of te omplex nonlinear refrative properties of te ost. As a result, enaned nonlinear absorption an be aieved in a narrow frequeny range (a) below te plasmon resonane by using a positive nonlinear refrative ost, (b) above te plasmon resonane by using a negative nonlinear refrative ost, or, in priniple, () at te plasmon resonane by using a saturable absorbing ost. To our knowledge, owever, an instantaneous, omogeneous saturable absorbing ost does not exist. Figure 3 sows te alulated omposite nonlinear refrative index, n, (solid line, panel (a)) and te nonlinear absorption β (solid line, panel (b)) as a funtion of wavelengt for Ag nanopartiles in CS. As we stated in setion.1, tese and subsequent alulations assume #965 - $15. USD Reeived 1 May 8; revised 4 Jun 8; aepted 8 Jun 8; publised 3 Jul 8 (C) 8 OSA 7 July 8 / Vol. 16, No. 14 / OPTICS EXPRESS 1688
tat te values of n and β do not ange over te wavelengt range of interest. At te SP resonane frequeny te omposite exibits strongly negative nonlinear refration, despite te positive nonlinear refrative properties of te ost. At frequenies sligtly below te SP resonane te omposite an be seen to exibit strong nonlinear absorption, wit a β tat is ~ times larger tan β. Tis is an impressive result, given te small metal fill fration of f = 1-3. As disussed above, it is important to realize tat tis large β value is not due to a resonant enanement of β, but rater due to a pase-sifted enanement of te nonlinear refrative response of te ost. To igligt tis, te dotted lines in Fig. 3 sow te nonlinear refrative properties of te omposite tat result exlusively from te enanement of β, wile te dased line sows te nonlinear refrative properties of te omposite tat result exlusively from te enanement of n,. From Fig. 3 it is lear tat te omposite nonlinear response is dominated by te enaned nonlinear refrative response of te ost material for most frequenies around te SP resonane. Note tat te frequeny bandwidt over wi enaned β values are obtained is less tan alf te SP resonane widt, due to te more tan quadrati dependene of te nonlinear index enanement on te field enanement fator. n (x1-14 ) [m /W] 3-3 -6 (a) total n ontribution β ontribution β [m/gw] 1-1 (b) 38 4 4 44 46 48 5 λ [nm] Fig. 3. (a) Calulated nonlinear refrative index and (b) nonlinear absorption oeffiient of a Ag-CS nanoomposite. Te separate ontributions from te ost nonlinear refrative index and ost nonlinear absorption are indiated by te dased and dotted urves respetively. Te appearane of nonlinear absorption in a omposite material onsisting of linear nanopartiles and a nonlinear refrative ost is ounterintuitive, but an be understood by onsidering te effet of te ost refrative index on te plasmon resonane frequeny. From Eq. () it an easily be seen tat te surfae plasmon resonane sifts to lower frequenies as te refrative index of te ost is inreased, taking into aount tat te metal dieletri funtion beomes more negative at lower frequenies. For a nonlinear ost wit a positive value of n, a ig irradiane is aompanied by an inrease in te refrative index of te ost, resulting in a redsift of te plasmon resonane. Tis simple observation an qualitatively explain te appearane of nonlinear absorption. To illustrate tis point, Fig. 4(a) sows te alulated linear absorption spetrum (solid blak line) of an Ag-CS omposite at a fill fration of 1-3, as well as te orresponding absorption spetrum wen te refrative index of te ost is artifiially inreased by.1 (gray solid line). To improve te visibility of te small resulting anges in te absorption spetrum, te values ε Ag are redued by a fator ten and a large index ange of.1 is used. Te absorption spetrum learly exibits a redsift as a result of te refrative index inrease. Tis sift an be seen to lead to a derease in absorption at ig frequenies (downward arrow), orresponding to apparent saturable #965 - $15. USD Reeived 1 May 8; revised 4 Jun 8; aepted 8 Jun 8; publised 3 Jul 8 (C) 8 OSA 7 July 8 / Vol. 16, No. 14 / OPTICS EXPRESS 1689
absorption. Similarly, at low frequenies tis same sift leads to an inrease in absorption (upward arrow), orresponding to nonlinear absorption. Tese observations are in agreement wit te trends observed in te alulated omposite nonlinear absorption resulting from n, as sown in Fig. 4(b). 1 (a) n =n CS n =n CS +.1 () n =n CS n =n CS +.1i 1 α [μm -1 ] 5 5 α [μm -1 ].1 (b) β (β =) (d) β (n, =).1 β [m/mw]. -.1. -.1 β [m/mw] 4 45 43 λ [nm] 4 45 43 435 λ [nm] Fig. 4. (a) Calulated linear absorption of a Ag-CS omposite wit and witout a small inrease in te ost refrative index. (b) Calulated omposite nonlinear absorption oeffiient onsidering only te nonlinear refrative index ontribution of te ost. () Calulated linear absorption of a Ag-CS omposite wit and witout a small inrease in te absorption oeffiient of te ost. (d) Calulated omposite nonlinear absorption oeffiient onsidering only te nonlinear absorption ontribution of te ost. Similarly, te effet of a nonlinear absorptive ost on te omposite nonlinear response an be understood by onsidering its effet on te surfae plasmon linewidt. For a ost wit a positive value of β, a ig irradiane is aompanied by an inrease in te ost absorption. From Eq. (), it an be seen tat te introdution of an imaginary ontribution to te ost dieletri funtion results in a broader and weaker surfae plasmon related absorption response in te omposite. To illustrate tis, Fig. 4() again sows te linear absorption alulated for a Ag-CS omposite (solid blak line), as well as for a omposite were te imaginary part of te ost refrative index as been artifiially inreased by.1 (solid grey line). From Fig. 4() we see tat te resulting broadening of te plasmon resonane leads to a derease in absorption near te enter of te resonane, orresponding to an effetive saturable absorption. At frequenies away from te surfae plasmon resonane, a sligt inrease in absorption is observed, orresponding to nonlinear absorption. Tese observations are in agreement wit te trends observed in te alulated omposite nonlinear absorption resulting from β, sown for omparison in Fig. 4(d). Similar arguments may be used to understand te effet of a nonlinear inlusion on te omposite nonlinear response. Note tat tis analysis ignores te inomogeneous nature of te eletri field trougout te omposite, and as su only provides a qualitative understanding of te proesses involved. Fig. 4(a) does point out an important effet: at te ig index anges sown, te nonlinear absorption is learly no longer linearly dependent on te inident irradiane. Tis suggests tat onsidering only tird order nonlinearities is only aurate wen te indued plasmon resonane frequeny sift is mu smaller tan te plasmon resonane linewidt. #965 - $15. USD Reeived 1 May 8; revised 4 Jun 8; aepted 8 Jun 8; publised 3 Jul 8 (C) 8 OSA 7 July 8 / Vol. 16, No. 14 / OPTICS EXPRESS 1683
3. Evaluation of plasmon enaned nonlinear absorption 3.1 Figure of merit (FOM) In order to evaluate te nonlinear optial response of different MG omposites, in Fig. 5(a) we sow te omputed β for fill frations, f, ranging from 1-6 to 1-3. At low filling frations te omposite nonlinear absorption losely resembles tat of te ost. As te fill fration is inreased, te nonlinear absorption oeffiient of te omposite is seen to ange signifiantly at frequenies near te plasmon resonane. For tis omposite, tese anges beome signifiant only at fill frations larger tan ~1-4. Te peak enanement of β ontinues to inrease as te fill fration is raised. Tis suggests tat te strongest nonlinear optial absorption will be aieved at te largest possible fill fration. However as disussed in te introdution, te presene of an SP resonane also results in enaned linear absorption, putting an upper bound on te maximum pratial tikness of te omposite. For tis reason a figure of merit (FOM) is ommonly employed defined as FOM = β /α in order to reflet te fat tat te maximum useful tikness z max is approximately 1/α. β /α [m/gw] β [m/gw] 1-1 - 1 1-1 1-1 -3 (a) (b) f=1-3 f=1-4 f=1-5 f=1-6 f=1-4, ε " i =ε" Ag / 1-4 38 4 4 44 46 48 5 λ [nm] Fig. 5. (a) Nonlinear absorption oeffiient β of a silver-cs omposite for several silver filling frations and (b) orresponding figure of merit. Figure 5(b) sows te alulated frequeny dependent FOM of a CS -Ag nanopartile omposite material for several filling frations based on te alulated β and te linear absorption oeffiient α alulated using Eq. (). Peraps surprisingly, despite te observed strong enanements in β and n, te igest figure of merit is observed for te lowest metal fill fration and away from te SP resonane. Tis sows tat if te use of tik samples is aeptable, it is preferable to avoid te use of surfae plasmon mediated field enanement, sine in tis ase te benefits of enaning te nonlinear optial response are outweiged by te introdution of strong linear loss. We will disuss tis in more detail in te next setion. Note tat a negative FOM is obtained at frequenies just above te SP resonane for te 1-4 and 1-3 filling frations, orresponding to an indued effetive saturable absorption, rater tan te desired enaned nonlinear absorption. It sould be pointed out tat for a given ost, an independent metod to inrease te peak β is to redue te loss in te metal (Eqs. and (6)). Tis inrease in β would be aompanied by a orresponding redution in te resonane bandwidt and an inrease in te peak α (Eq. ()). To illustrate tis effet, te dotted lines in #965 - $15. USD Reeived 1 May 8; revised 4 Jun 8; aepted 8 Jun 8; publised 3 Jul 8 (C) 8 OSA 7 July 8 / Vol. 16, No. 14 / OPTICS EXPRESS 16831
Fig. 5 sow te frequeny dependent β and FOM at a fill fration f=1-4 wen ε i is redued by a fator of. Figure 5(a) demonstrates tat tis redution in loss leads to an inrease in te peak value of β of lose to an order of magnitude. In addition, from Fig. 5(b), it an be seen tat te redution in loss leads to an overall improvement of te figure of merit, indiating tat te inrease in linear absorption is outweiged by te inrease in nonlinear absorption. 3. Nonlinear transmission In order to exemplify te interplay between te omposite linear and nonlinear absorption, we alulate te transmitted irradiane (I out ) as a funtion of inident irradiane (I in ) at a wavelengt of 435 nm. Tis orresponds to te wavelengt at wi β is approximately maximum for all fill frations (see Fig. 5(a)), and as su te urves represent a best ase senario. In te presene of bot linear and nonlinear absorption, te transmitted irradiane is given by I α I in out = (7) αl βiin ( α βiin) e were L is te sample tikness. Note tat tis equation assumes normal inidene plane wave illumination. Figure 6(a) sows te nonlinear transmission for a 1 nm tik omposite Ag-CS layer. Tis tikness is suffiiently small to prevent any signifiant linear absorption for te fill frations used, as evidened by te fat tat at low inident irradiane te transmitted irradiane is independent of fill fration. For referene te linear transmission orresponding to zero absorption is indiated by te dased line. At iger irradianes we see tat I out begins to deviate from linear transmission as a result of te finite β. Te point at wi te transmitted irradiane deviates from a linear response by 5% is indiated by te solid irles. As te irradiane inreases furter, te total transmitted irradiane is seen to saturate. A similar trend is observed for all fill frations, owever as te fill fration is inreased, te transmitted irradiane is seen to saturate at lower input irradiane, and te maximum transmitted irradiane is redued. Bot tese effets are due to te fat tat β inreases as f inreases, as was observed in Fig. 5(a). Tese results sow tat for tin layers, te nonlinear absorption an be enaned by te addition of metal nanopartiles, witout signifiantly affeting te linear transmission. Figure 6(b) sows te alulated nonlinear transmission for a 1 µm tik omposite Ag- CS layer. Two important differenes are observed ompared to te tin layer results sown in Fig. 6(a). First, for small fill frations (top urves) we see tat a lower saturation irradiane is aieved tan in Fig. 6(a) due to te longer interation lengt. For low metal fill frations, te nonlinear transmission is largely due to te response of te ost material, wit a negligible ontribution from te small amount of metal in te omposite. As te fill fration is inreased to 1-3, te onset of nonlinear beavior is seen to sift to lower irradiane, as was observed in Fig. 6(a). In tis ase owever, te linear transmission of tese layers is also found to derease dramatially as a result of te surfae plasmon indued linear absorption. Tese results igligt te main allenge in te use of surfae plasmons for te enanement of nonlinear refrative and absorptive appliations: altoug te nonlinear response an be signifiantly enaned by te plasmon resonane, te indued linear loss plaes an upper bound on te useful sample tikness for a fixed metal fill fration. It sould be pointed out tat te irradianes assumed ere in order to observe nonlinear transmission in very tin (<1 µm) layers exeed 1 GW/m. At tese extremely ig irradiane values nonlinear proesses oter tan Kerr nonlinearities are expeted to dominate, inluding dramati termal effets and possibly ionization and break down. It sould terefore be empasized tat tese urves indiate te limits of wat an be aieved in su dilute systems purely based on a Kerr response in te ost material. To experimentally observe tese effets, tiker layers or iger fill frations are likely needed, and speial are #965 - $15. USD Reeived 1 May 8; revised 4 Jun 8; aepted 8 Jun 8; publised 3 Jul 8 (C) 8 OSA 7 July 8 / Vol. 16, No. 14 / OPTICS EXPRESS 1683
will ave to be taken to minimize termal effets, inluding te use of femtoseond radiation at low repetition rates. 1 8 (a) t = 1 nm (b) t = 1 μm 1 6 I out [GW/m ] 1 4 1 1 linear ost f = 1e-6 f = 1e-5 f = 1e-4 f = 1e-3 1-1 1 1 4 1 6 1 1 1 4 1 6 1 8 I in [GW/m ] Fig. 6. Transmitted irradiane as a funtion of inident irradiane at a wavelengt of 435 nm for different filling frations for an Ag-CS omposite wit a tikness of (a) 1nm (tin layer limit) and (b) 1 µm. Te dased line represents a linear response wit zero absorption. Te results presented sow tat surfae plasmons on metal partiles an be used to strongly modify te nonlinear refrative properties of dilute metallodieletri omposites. Based on our findings it is lear tat te development of su systems for real appliations sould fous on te minimization of plasmon enaned linear absorption wile maintaining a signifiant enanement of te nonlinear refrative response. Future work will fous on evaluating te response of omposites tat are outside te appliability of te Maxwell Garnett teory, su as dense strutures ontaining non-sperial elements. Su dense plasmon enaned nonlinear absorbing media ould provide an improved balane between linear and nonlinear absorption. 4. Conlusion Te influene of plasmon resonant silver nanopartiles on te linear and nonlinear effetive medium properties of CS in te Kerr regime is explored for a Maxwell Garnett type omposite. We demonstrate tat te nonlinear absorption oeffiient of a Ag-CS omposite an be enaned relative to te nonlinear absorption oeffiient of CS over a narrow spetral range near te surfae plasmon resonane. It is sown tat tis enanement is not due to te enanement of te ost nonlinear absorption oeffiient but rater due to a pase sifted enanement of te ost nonlinear refrative index. Furtermore it is sown tat tis effet ours at te expense of te introdution of a surfae plasmon related linear absorption to te omposite resulting in an overall redued FOM for omposites ontaining metal nanopartiles. Te indued linear absorption as te effet of plaing an upper limit on te useful sample tikness. For a volume filling fration of 1-3 tis results in an upper useful sample tikness of ~1 µm at te wavelengt of maximum linear absorption. For tin samples (linear transmission > 1/e) te total nonlinear absorption of te omposite an be enaned relative to tat of te ost alone at te same tikness. However, due to te dependene of te total nonlinear absorption on te sample tikness, a stronger nonlinear absorption an be obtained by avoiding te use of silver nanopartile plasmon resonanes and making use of a tiker layer. For a given ost, reduing te imaginary part of te metal dieletri funtion #965 - $15. USD Reeived 1 May 8; revised 4 Jun 8; aepted 8 Jun 8; publised 3 Jul 8 (C) 8 OSA 7 July 8 / Vol. 16, No. 14 / OPTICS EXPRESS 16833
results in a stronger but narrower resonane, and an improved ratio between te nonlinear and linear absorption oeffiient. Aknowledgments We would like to tank Dr. Sott Webster, Dr. Eri Van Stryland and Dr. David Hagan for elpful disussions. Tis material is based upon work supported in part by te U. S. Army Resear Offie under ontrat/grant number 537-CH-MUR. #965 - $15. USD Reeived 1 May 8; revised 4 Jun 8; aepted 8 Jun 8; publised 3 Jul 8 (C) 8 OSA 7 July 8 / Vol. 16, No. 14 / OPTICS EXPRESS 16834