Optical properties of one-dimensional soft photonic crystals with ferrofluids

Transcription

1 Front. Phys., 2013, 81: 1 19 DOI /s REVIEW ARTICLE Optial properties of one-dimensional soft photoni rystals with ferrofluids Chun-Zhen Fan 1,2,, Er-Jun Liang 1,, Ji-Ping Huang 2, 1 Shool of Physial Siene and Engineering, and Key Laboratory of Materials Physis of Ministry of Eduation of China, Zhengzhou University, Zhengzhou , China 2 State Key Laboratory of Surfae Physis and Department of Physis, Fudan University, Shanghai , China hunzhen@zzu.edu.n, ejliang@zzu.edu.n, jphuang@fudan.edu.n Reeived Deember 17, 2012; aepted Deember 26, 2012 We review the reent theoretial study on the optial properties of one-dimensional soft photoni rystals 1D SPCs with ferrofluids. The proposed struture is omposed of alternating ferrofluid layers and dieletri layers. For the ferrofluid, single domain ferromagneti nanopartiles an align to a hain under the stimuli of an external magneti field, thus hanging the mirostruture of the system. Meanwhile, nonlinear optial responses in ferrofluids are also briefly reviewed. Keywords soft photoni rystals, ferrofluids, band gap PACS numbers Qs, Cb Contents 1 Introdution 1 2 Fundamentals Transfer matrix method Magneti struture anisotropi fator 3 3 Magneto-ontrollable SPCs based on ferrofluids Formulism Numerial results and disussions 5 4 Magneto-ontrollable SPCs with the inorporation of the absorption effet Formulism Numerial results and disussions 8 5 1D graded SPCs with multilayer struture Formulism Numerial results and disussions 11 6 Seond-harmoni generation with magnetoontrolling abilities in ferrofluids Formulism Numerial results and disussions 15 7 Conlusions 16 Aknowledgements 16 Referenes 16 1 Introdution Photoni rystals PCs are periodi arrays of material with different refrative index [1, 2]. It was first put forth independently by Yablonovith and John in 1987 [3, 4]. The major funtionality of the PCs is the strong distortion or modifiation of dispersion, whih an influene the behavior of photons. PCs have attrated a wide researh attentions owing to their unique light manipulation abilities, demonstrating wide appliations in optial sensor, optial filter and various optial integrated iruit devies [5 7]. They are usually omposed of the solid material like silion [8, 9], whih an be realized through the ombination with semiondutor material to form the Si-SiO 2,SiO 2 -GaAs or Si-AlGaAs struture by the dry ethhing method or the wet eletrohemial proess [10]. However, it is an alternative to fabriate PCs with olloidal suspension like the soft photoni rystals SPCs at a low ost and fast response, whih an be realized through the self-assembled olloidal mirospheres or tuned with different external stimuli [11 13]. Until now, muh work has been arried out to study the SPCs, like the dually tunable SPCs omposed of thermosensitive gel partiles [14], soft glass photoni rystal fibers [15], and the hydrogel olloidal rystals ahieved Higher Eduation Press and Springer-Verlag Berlin Heidelberg 2013

2 2 Chun-Zhen Fan, Er-Jun Liang, and Ji-Ping Huang, Front. Phys., 2013, 81 by simply varying the water ontent of the mirogel pellet before annealing and rystallization [16]. In this work, we review our reent theoretial study in SPCs, whih is omposed of alternating ferrofluid layers and dieletri layers. In general, ferrofluids is a type of olloidal suspension, in whih single domain ferromagneti nanopartiles about 10 nm in diameter randomly disperse in polar or nonpolar liquid [17 26]. Typially, ferrofluids onsists of ferromagneti partiles like Co or Fe 3 O 4 or ferrimagneti nanopartiles like CoFe 2 O 4 [27], oated with surfatant to prevent their aggregation due to the van der Waals and magneti fores [27 30]. This stable suspension would be disturbed with the appearane of the external magneti field, resulting in a redistribution of the these ferromagneti nanopartiles in the suspension. Finally, they form linear hains or olumns. Due to the ontatless ontrol of magneti fields and fast response, magneti field ontrollable assembly has been regarded as a onvenient and effiient method to reate ordered olloidal arrays and modulate their photoni properties. Muh work has been arried out to study the magnetially ontrollable PCs through the selfassembly of olloidal spheres [19], where the harged polystyrene partiles ontaining nanosale iron oxide nanopartiles or the spinel ferrites material were synthesized through emulsion polymerization. And the lattie spaing an be altered by magneti fields [22]. Due to the unique properties of SPCs, their wide appliations inlude the magneto-optial devies [28, 29, 31 33], biomedial treatment [34, 35], aner diagnosis [36] and hypothermal treatment [37]. In this review, we theoretially investigate the optial properties in 1D SPCs [38 42]. The proposed SPCs is omposed of alternating ferrofluid layers and dieletri layers. Due to the magneti field indued anisotropi property of ferrofluids, the photoni band gap an effetively be tuned with the initiation of an external magneti field and the volume fration of the nanopartiles. The absorption effet in the ferrofluids layer is onsidered with ferromagneti nanopartiles oated with metalli layer. The omplex wave vetor results in the appearane of an additional band gap in the lower frequeny region. Moreover, these band gaps blueshift when the external magneti field is enhaned, and redshift when the thikness of metalli layer is inreased. Graded ferrofluids layer in the SPCs is investigated through a varying number of deposited ferrofluids layers. Regarding eah layer in the multilayer struture as a series of apaitane, the effetive dieletri onstant is derived. Numerial results have shown that the position and width of the photoni band gap an be effetively modulated by varying the number of graded omposite layer, the volume fration of nanopartiles and the external stimuli. Finally, the nonlinear optial response in the ferrofluids is studied, where ferromagneti nanopartiles are oated by a nonmagneti nanoshell with an intrinsi seond-harmoni generation SHG suseptibility in a nonmagneti host fluid. The SHG of suh materials possess magneti-field ontrollabilities, redshift, and enhanement. This review is organized as follows. In Setion 2, we introdue the transfer matrix method and the magneti filed indued struture anisotropi fator α, whih are frequently employed in our theoretial study. In Setion 3, we study the magneti field ontrollable photoni band gap in the SPCs with ferrofluids, see the original researh work in Ref. [38]. In Setion 4, the omplex wave vetor in the ferrofluids layer is investigated, where the ferromagneti nanopartile is oated with metalli layer, see the original researh work in Ref. [39]. In Setion 5, the graded multilayer struture of the ferrofluids is studied, see the original researh work in Ref. [40]. In Setion 6, the nonlinear optial materials with an enhaned SHG suseptibility is presented, see the original researh work in Ref. [41]. This review ends with a summary in Setion 7. 2 Fundamentals 2.1 Transfer matrix method Under normal inidene, the shemati light pathway between two surfaes surfae I and surfae II with different refration index is illustrated in Fig. 1, where n i is the refration index of the inident spae, n 1 is the refration index of material 1, n o is the refration index of the outgoing spae. The inident beam undergoes an external refletion at the first interfae I, and the transmitted beam undergoes an internal refletion and transmission at the seond interfae II. Dynami properties of the n i n 1 n o H Ⅰ t Ⅱ E E E E i E t1 E H t2 E i2 EE H E Fig. 1 Shemati view of the light pathway between two surfaes surfae I and surfae II with different refration index is shown. n i is the refration index of the inident spae, n 1 is the refration index of material 1, n o is the refration index of the outgoing spae. Under normal inidene, the inident beam undergoes an external refletion at the first interfae I and the transmitted beam undergoes an internal refletion and transmission at the seond interfae II. The omponents of the E and H field are also illustrated. H H E H E r1 E i1 E r2 H

3 Chun-Zhen Fan, Er-Jun Liang, and Ji-Ping Huang, Front. Phys., 2013, 81 3 eletri field and magneti field are ontained with boundary onditions. Namely, normal omponents of DB and tangential omponents of EH areontinuous, whih an be expliitly expressed as [43, 44] E I = E i + E r1 = E t1 + E i1 E II = E i2 + E r2 = E t2 1 H I = H i os θ i H r1 os θ i = H t1 os θ 1 H i1 os θ 1 H II = H i2 os θ 1 H r2 os θ 1 = H t2 os θ o 2 where E i is the inident eletri field, E r1 E t1 istherefleted transmitted eletri filed at interfae I, E r2 E t2 is the refleted transmitted eletri filed at interfae II. The definition of the H filed is the same as that of the E field. Using the relation H = ne ɛ 0 /μ 0 and the boundary onditions Eq. 2, the magneti field an be written as a funtion of the eletri field: H I = ɛ 0 /μ 0 n i os θ i E i E r1 = ɛ 0 /μ 0 n 1 os θ 1 E t1 E i1 H II = ɛ 0 /μ 0 n 1 os θ 1 E i2 E r2 3 = ɛ 0 /μ 0 n o os θ o E t2 4 Assuming η = n ɛ 0 /μ 0, thus η i n i os θ i ɛ0 /μ 0, η 1 n 1 os θ 1 ɛ0 /μ 0 and η o n o os θ o ɛ0 /μ 0. Eq. 3 and Eq. 4 an be simplified as H I = η i E i E r1 =η 1 E t1 E i1 H II = η 1 E i2 E r2 =η o E t2 5 Additionally, there is a phase differene δ between E i2 and E t1, where δ = k i Δ = 2π/λn i t os θ t1. Thus, E i2 = E t1 e iδ and E i1 = E r2 e iδ. Substitute E i2 and E i1 into Eq. 1 and Eq. 5, we an get the following relations, E I = E II os δ + H II isinδ/η 1 H I = E II iη 1 sin δ + H II os δ 6 Rewrittentheminmatrixform, EI os δ isinδ/η1 = H I iη 1 sin δ os δ = M E II H II where os δ isinδ/η1 M = iη 1 sin δ os δ EII H II 7 8 Now the harateristi equation between two layers is obtained [Eq. 7]. Eah layer of the multilayers has its own transfer matrix and the overall transfer matrix of the N periods PCs is the produt of individual transfer matries. For N unit layer, it leads to the matrix equation, M = N M i 9 i Using the boundary onditions, the transfer matrix ould be rewritten in the form: m11 m 12 M = 10 m 21 m 22 The oeffiients of refletion and transmission are defined as 2η i t = 11 η i m 11 + η i η o m 12 + m 21 + η o m 22 r = η im 11 + η i η o m 12 m 21 η o m η i m 11 + η i η o m 12 + m 21 + η o m 22 These expressions an be used for any number of layers in PCs to determine the oeffiients of transmission T = t 2 and refletion R = r 2 [2, 45]. For the 1D PCs with double layers, the dispersion relation admits the following form oskd =osk 1 d 1 osk 2 d 2 1 k1 + k 2 sink 1 d 1 sink 2 d k 2 k 1 whih is got by solving the non-zero solution of det M e ikd = 0 [46]. where d 1 d 2 is the thikness of orresponding layers. d is the lattie onstant and d = d 1 + d 2. k represents the wave vetor in the PCs, k 1 = ω ɛ 1 / denotes the loal wave vetor of layer 1, and k 2 = ω ɛ 2 / is the loal wave vetor of layer 2. ω is the frequeny of light or eletromagneti waves, and is the speed of light in vauum. 2.2 Magneti struture anisotropi fator For the ferrofluids, ferromagneti nanopartiles around 10 nm in diameter randomly distributed in the suspension. When the external magneti field H was applied, ferromagneti nanoaprtiles will align along the field diretion, thus hanging the mirostruture of the system orrespondingly [47 53]. The shemati struture of the ferromagneti nanopartile under the influene of the magneti field is shown in Fig. 2. Let α represent the field indued struture anisotropi fator. In detail, α denotes the loal field fators α and α for longitudinal and transverse field ases, respetively [54]. Here the longitudinal or transverse field ase orresponds to the fat that the E-field of the light is parallel or perpendiular to the partile hain. There is a

4 4 Chun-Zhen Fan, Er-Jun Liang, and Ji-Ping Huang, Front. Phys., 2013, 81 3 Magneto-ontrollable SPCs based on ferrofluids Fig. 2 Ferromagneti nanopartiles under the influene of the external magneti field H is illustrated. And these ferromagneti nanopartile form hains along the H diretion. α is the magneti field indued struture anisotropi fator. It has the longitudinal α and α perpendiular omponents respetively. sum rule for α and α, α +2α = 3. The parameter α measures the degree of anisotropy indued by the applied magneti field H [55]. More preisely, the degree of the field-indued anisotropy is measured by how muh α deviates from unity, 0 <α < 1 for transverse field ases and 1 <α < 1.5 for longitudinal field ases. As H inreases α and α should tend to 0 and 1.5, respetively, whih is indiative of the formation of more and more partile hains as evident in experiments. A rude estimate of α an be obtained from the ontribution of hains, namely, α = 1 n kv 0 V k HN k 14 φ k=1 where φ denotes the volume fration of the strutured partiles in the suspension, N k the depolarization fator for a hain with k strutured partiles, V 0 the volume of one partile in suspension onsidered to be spherial and all idential, and V k H the density of the hain whih is a funtion of H [56, 57]. It is noteworthy that for given φ, V k H also depends on the dipolar oupling onstant whih relates the dipole dipole interation energy of two ontating partiles to the thermal energy. Speifially, if there is no external magneti field H =0, the magneti moments within the ferrofluids partiles are randomly distributed, whih indiates that no net measurable magnetization an be obtained. It leads to α = α = 1. But when an external magneti field is applied, the magneti moments of the partiles orient along the field diretion of the externally applied field. It means that the alignment of magneti moments is very sensitive to the external magneti field [58]. Therefore, α should be a funtion of external magneti fields H. Here we present a lass of magneto-ontrollable SPCs based on ferrofluids [38]. The proposed struture is omposed of alternating ferrofluid layers and dieletri layers. Periodi array of dieletri satters and the observation of the strong loalization of photons are the essential properties of PCs in analogy to the allowed forbidden energy bands gaps of semiondutors [59]. This funny field has inspired great interest in reent years beause of their potential ability to ontrol the propagation of eletromagneti waves or modify the density of eletromagneti states inside the rystal [14, 60, 61]. PCs with omplete band gaps have many appliations [62], inluding the fabriation of lossless dieletri mirrors and resonant avities for light [63]. In addition, the visibility of graphene layers was found to be enhaned greatly when one-dimensional 1D PCs were added [64], and the linear and nonlinear optial properties of 1D PCs ontaining ZnO defets were also studied [65]. Reently the eletri tunability of PCs with nonlinear omposites was theoretially explored [66]. Experimentally, magneti-field tunable photoni stop bands in a stak of ontainers with magnetizable ferromagneti spheres were studied [67]. Colloidal photoni rystals, whih were magnetially fabriated by using superparamagneti magnetitenanorystal lusters, an also realize stop bands overing the entire visible spetrum [68]. Usually a tunable band gap in PCs is ahieved by varying the lattie onstant or symmetry of the perfetly ordered rystal by using different methods suh as elasti stress struture transformation [69], temperature tuning [70], femtoseond diret writing tehnique [71] or magneti eletri fields [67, 72]. Meanwhile, the following developments in PCs are also very fasinating, suh as tunable photoni band in two-dimensional photoni rystal slab waveguides by atomi layer deposition [73], tunability of PCs due to the eletrially indued birefringene of the twisted dieletri grains [74] and eletrial tuning of three-dimensional photoni rystals using polymer-dispersed liquid rystals [75]. In this setion we exploit a lass of magnetoontrollable 1D SPCs, whih are made of olloidal ferrofluids. 3.1 Formulism For1DSPCswithdoublelayersseeFig.3,theloal field fator indued by the external magneti field in the suspension an be obtained by the Ewald Kornfeld formula [54]. If there is no speial instrutions, we shall use α to denote both α and α in the following. Here we

5 Chun-Zhen Fan, Er-Jun Liang, and Ji-Ping Huang, Front. Phys., 2013, 81 5 onsider the obalt nanopartiles suspended in the water. as ɛ 2 = ɛ 0. E x^ H y^ z H e x^ Fig. 3 Shemati graph showing one-dimensional soft photoni rystals 1D SPCs ontaining periodi double layers, one of whih is a ferrofluid layer layer 1 and the other of whih is a dieletri layer layer 2. In the ferrofluid layer, ferromagneti spherial nanopartiles in irle are embedded in a arrier fluid in the presene of x-direted external magneti field H e. An eletromagneti wave is inident on the 1D SPCs along the z axis, whose eletrifield E and magneti-field H omponents are direted along the x and y axes, respetively. For model alulations, we take layer 2 to be air, the arrier fluid to be water, and the suspended nanopartiles to be obalt. Reprodued from Ref. [38]. 3.2 Numerial results and disussions Now we are in a position to do numerial alulations in our proposed 1D SPCs. Figure 4 shows the alulated band struture of the model system. When the magnitude of the applied magneti field H e inreases from zero to a large value namely, the longitudinal loal H e fator α hanges from 1.0 to 0.2, we find that H e has a signifiant effet on the band gaps and auses them to move towards a higher frequeny region blue shift. Next, we display the first band gap width Δ in Fig. 5. The Δ is shown to get larger when α approahes zero step by step. Alternatively, the band gap width is aused to inrease when the magnitude of the external magneti field H e inreases. Due to the different polarization of the inident light, the effetive dieletri onstant of the omposite ɛ e will be ɛ or ɛ.hereweuseɛ to represent ɛ and ɛ,whih an be given through the wellknown Maxwell Garnett theory [55, 76 78]: ɛ e ɛ 2 = p ɛ 1 ɛ 2 15 αɛ e +3 αɛ 2 ɛ 1 +2ɛ 2 where p is the volume fration of the nanopartiles in the omposite suspension. ɛ 1 is the dieletri onstant of the nanopartiles. ɛ 2 is the dieletri onstant of host fluid water. Here we take ɛ 1 of the obalt nanopartiles as ɛ 0, whih is haraterized by using the inident wavelength λ = 661 nm [79], and the dieletri onstant of water is 1.77ɛ 0. From Eq. 15 we know the anisotropy fator α indued in this omposite has a diret effet on the effetive dieletri onstant of the omposite. Thus, the refrative index of this layer is tuned orrespondingly. Then we use the transfer matrix method [80] to study the band strutures of 1D periodi photoni strutures. For 1D SPCs with double layers, the dispersion relation is in the form as oskd =osk 1 d 1 osk 2 d 2 1 k1 + k 2 sink 1 d 1 sink 2 d k 2 k 1 where d 1 d 2 is the thikness of orresponding layers. Here we take layer 1 as the ferrofluid omposites and layer 2 as the air. d is the lattie onstant and d = d 1 +d 2. k represents the wave vetor in the PCs, k 1 = ω ɛ 1 / denotes the loal wave vetor of layer 1, and k 2 = ω ɛ 2 / is the loal wave vetor of layer 2. ω is the frequeny of light or eletromagneti waves, and is the speed of light in vauum. The dieletri onstant of air is taken Fig. 4 Band struture of the model 1D SPCs Fig. 3 for different α. Parameters: p =0.25 and d 1 = d 2 =0.5a. Reprodued from Ref. [38]. Fig. 5 The first band gap width of the model 1D SPCs Fig. 3 vs. α. Parameters: p =0.25 and d 1 = d 2 =0.5a. Reprodued from Ref. [38]. Figure 6 shows the obalt onentration effet on the band struture of the model 1D SPCs. In this ase, we set H e =0.Whenp gets larger i.e., the ferrofluid beomes denser, the first and seond band gaps get wider and move towards the lower frequeny region red shift.

6 6 Chun-Zhen Fan, Er-Jun Liang, and Ji-Ping Huang, Front. Phys., 2013, 81 Here, p = 0 represents the ase that there are no obalt nanopartiles. In this ase, the band gaps are aused to appear by the ombination of pure-water layers and air layers. Our results show that the omposite effet in ferrofluid layers plays an important role as p is non-zero. In suh ases, if an external magneti field H e is applied, it an lead to a tunable effetive dieletri onstant or refrative index of the ferrofluid layer, thus yielding magneto-ontrollable band gaps. Fig. 6 Band struture of the model 1D SPCs for various volume fration p. Parameters: α =1andd 1 = d 2 =0.5a. Here, α =1 represents that there is no external magneti field H e. Reprodued from Ref. [38]. In this setion, we have only alulated the ase of the E omponent being parallel with the nanopartile hain, i.e., longitudinal E ase. For the transverse E ase, the orresponding band gaps an also be tuned. However, for the same H e, the tunability for the transverse E ase is less and opposite, due to the existene of the sum rule α +2α =3. In addition, the formation of nanopartile hains an be weakly affeted by the appliation of inident eletromagneti waves e.g., of light [81]. Apparently, this does not affet the present results at all. 4 Magneto-ontrollable SPCs with the inorporation of the absorption effet Until now one-dimensional, two-dimensional, and threedimensional PCs have been widely studied [82 84]. Theoretially, the plane wave expansion [85], the transfer matrix [86], and the finite differene time methods [87] have been well developed to study the propagational properties of photons. Experimentally, the lithography [88, 89], the self-assembly [11], and the olloidal rystal template methods are developed for fabriation of PCs with well ontrolled struture [90]. Among them, the ability to get tunable band gap is a topi of muh interest, whih is originated from the variation of refrative index or the symmetry of materials. With the utilization of omposite mirostrutures [91] tunable photoni band gaps an be realized by different external fators suh as an eletri [92], magneti field [38], temperature [93] or mehanial fore [94]. Mostly, eah omponent of PCs is a nonabsorbent material, so the wave vetor in the system is frequeny independent. If the metalli part is inserted in the periodi struture, the pathway of the eletromagneti field will be different [95 97]. Here we investigate the 1D SPCs based on ferrofluids with onsideration of the absorption effet, where the ferromagneti nanopartiles oated by the metalli layer see Fig. 7. It is omposed of a ferrofluids layer and an air layer. a is the thikness of the ferrofluids layer, and b is the thikness of the air. The filling fator representing the ratio of the ferrofluids in the lattie is defined as ν = a/d. Forthe air layer, the refration index is taken as 1.0. In our model system, ferromagneti linear nanopartiles of linear dieletri onstant oated with a nonmagneti metalli nonlinear shell are investigated in ferrofluids. New nanopartiles magneti-metalli omposite ombining an optial signature with magneti response are partiularly useful [98 101]. For the magnetinon metalli shell, they are oated by stabilizing surfatant layers to prevent the aggregation of the nanosized magneti partile under van der Waals fores. To investigate the EM wave propagation in this magneti field indued anisotropi struture, we hoose the magnetimetalli ore-shell struture. Synthesis of the metalli Au, Ag oated obalt nanopartiles an be realized by partial replaement reation [35, 102, 103]. Through this homogeneous non-aqueous approah, magneti oremetalli shell omposite forms the stable ferrofluids. 4.1 Formulism We theoretially investigate the properties of optial propagation in one-dimensional SPCs based on ferrofluids by using the transfer matrix method. Speifially, ferrofluids are omposed of the suspended ferromagneti nanopartiles oated with silver, whih has a frequeny dependent dieletri funtion. Inorporation of the absorption effet in the 1D SPCs, the omplex wave vetor an be expressed as [104] oskd=f 1 ω+if 2 ω 17 where K represents the omplex wave vetor. d is the lattie onstant. f 1 ω andf 2 ω denote the real and imaginary part, respetively. To get the expression of f 1 ω and f 2 ω, we transform Eq. 16 as the following: oskd=os n os n1 n 2 + n 2 n 1 ωb n 2 sin n 1

7 Chun-Zhen Fan, Er-Jun Liang, and Ji-Ping Huang, Front. Phys., 2013, 81 7 Fig. 7 Shemati view of 1D SPCs, omposed of alternating ferrofluid layers and dieletri layers. For the ferrofluids, the ferromagneti nanopartiles are suspending in the host liquid. Here we take it as the obalt nanopartiles ore oated by the silver shell. When an external magneti filed H e is applied, the ore-shell nanopartiles will align along the magneti field diretion and form hains. An eletromagneti wave is inident on the 1D SPCs along the z axis, whose eletri-field E and magnetifield H omponents are direted along the x and y axes for the longitudinal L field ases, and along the y and x diretion for the transverse T field ases. The thikness of the ferrofluids layer is a. The thikness of the air layer is b. Hene, the lattie onstant is d = a + b. Reprodued from Ref. [39]. ωb sin n 2 18 For simpliity, we take the dieletri layer 2 as air in our alulation; atually we an also take it as SiO 2, ZnSe, GaAs and so on. Substitute n 1 = n r +in i, n 2 = 1, ξ r =Re n n 1 and ξ i =Im n 1 + 1n1 into Eq. 18. It beomes oskd=os n r +in i ωb os 1 2 ξ r +iξ i sin n r +in i ωb sin 19 Simplify Eq. 19, and we finally get [ oskd= os n r osh n i ] ωb i sin n r sinh n i os 1 ωb 2 ξ r +iξ i sin [ sin n r osh n i ] +i os n r sinh n i 20 Thus, ωb f 1 ω =os n r osh n i os 1 ωb 2 ξ r sin sin n r osh n i + 1 ωb 2 ξ i sin os n r sinh n i f 2 ω = sin n r 1 ωb 2 ξ r sin 1 ωb 2 ξ i sin osh n i sinh os sin n i n r n r os sinh ωb n i As we know, the real part of the omplex wave vetor K, namely k r, orresponds to the wave number in the 1D SPCs, while the imaginary part k i determines the absorption oeffiient β. It an be expressed as β =2k i 23 Simplify os K by using K = k r + ik i and we obtain os K =osk r +ik i =osk r osh k i isink r sinh k i 24 The real part of the photoni band struture for given wave number k r an be found based on Eqs. 17 and 24 by seeking the frequenies ω whih follow the equation: osk r d=f 1 ω [ ] f 2 1/2 2 ω sin 2 k r d In our numerial alulations, the dieletri onstant of the silver ɛ 1ω is a funtion of ω. It follows a Drude dieletri funtion, whih is valid for noble metals within the frequeny range of interest. ɛ 1ω an be expressed as [41] ɛ 1 2 p ω ω =ɛ [ɛ0 ɛ ] ωω + iγ 26 where ω p is the bulk plasmon frequeny, ɛ is the highfrequeny limit dieletri onstant, ɛ0 is the stati dieletri onstant, and γ is the ollision frequeny. Speifially, for silver, ɛ =5.45, ɛ0 = 6.18, and ω p = rad/s [105]. In addition, we take γ =0.01ω p a typial value for metals. For the obalt nanopartile, we take ɛ 1 = i. It should be reasonable to see the ɛ 1 as frequeny-independent, sine it only varies very slightly with the frequeny of external fields. Thus, the equivalent linear dieletri onstant ɛ 1 ω for the ore-shell omposite nanopartile an be obtained through the Maxwell-Garnett formula [55, 106], whih desribes a two-omponent omposite where many partiles of the dieletri onstant ɛ p and the volume fration v p are randomly embedded in a host medium. Thus, ɛ 1 ω ɛ 1ω ɛ ɛ 1 ω+2ɛ =1 f 1 ɛ 1ω 1 ω ɛ 1 +2ɛ 1 ω 27

8 8 Chun-Zhen Fan, Er-Jun Liang, and Ji-Ping Huang, Front. Phys., 2013, 81 where f is the volume ratio of the nanoshell to the whole oated nanopartile. It equals to f =1 r 3 /R 3 =1 t 3. Shell thikness parameter t = r/r, wherer relates to the radius of the magneti ore. R is the radius of the ore-shell omposites. The thinner the shell layer, the larger t is. The Maxwell-Garnett formula follows a wellknown asymmetrial effetive medium theory, and may thus be valid for a low onentration of nanopartiles in the omposites [107]. The effetive linear dieletri onstant of the whole suspension under the present onsideration ɛ e ω an be given by the developed Maxwell-Garnett approximation whih works for suspensions with field-indued anisotropi strutures [106]: ɛ e ω ɛ 2 = p ɛ 1ω ɛ 2 28 αɛ e ω+3 αɛ 2 ɛ 1 ω+2ɛ 2 ɛ 2 is a frequeny-independent dieletri onstant of the host liquid, and ɛ 2 =1.77 for water. The volume fration of the nanopartiles in the suspension is p, and here we take p =0.18. The parameter α measures the degree of strutural anisotropy due to the formation of nanopartile hains, whih are indued by the external magneti field H e. The aim of the intended introdution of α into Eq. 28 is to inlude the field-indued anisotropy in the system, at least qualitatively. In detail, α denotes the loal field fators, α and α for longitudinal and transverse field ases, respetively. Shemati struture of the loal magneti field anisotropy fator α and α is shown in Fig. 8. As for the refraitve index n 1 of the ferrofluids layer, it an be easily obtained from expression n 1 = ɛ e, where ɛ e is the effetive dieletri onstant of the ferrofluids layer. Thus, n r =Re ɛ e andn i =Im ɛ e. the struture anisotropy indued by the external magneti field makes the dispersion urve blueshift moving to the high frequeny region. Meanwhile the absorption oeffiient dereases and the absorption urves are not ontinuous. To fully understand the origin of suh phenomena we an take a look at the inset Figs. 9 a and b. Inset a shows the real part of the omplex wavevetor k r d/π as a funtion of the frequeny. An additional band gap appears due to the dissipation oeffiient γ in Eq. 26 in the low frequeny region. Now let us take a look at the distribution of f 1 ω as a funtion of frequeny, whih is shown as inset b. It represents the real part of the omplex wave vetor oskd, and its osillation defines the frequeny regions in whih the argument Kd takesrealvaluesthe regions where f 1 ω 1. In other words, the band gaps happen in the frequeny region where f 1 ω > 1. When α gradually dereases, the distribution of f 1 ω blueshift moving to the higher frequeny region. For the transverse T field ases the behaviors of the dispersion urve is similar to those for L field ases not shown here. Thus, the enhaned magnitude of the external magneti filed α dereases or α inreases auses the band gaps of 1D SPCs blueshift. Fig. 8 Shemati graph shows the loal magneti fator along the major α or minor axis α respetively. The whole nanopartiles hain an be treated as an equivalent spheroid. Reprodued from Ref. [39]. 4.2 Numerial results and disussions Figure 9 shows the absorption oeffiient log βof 1D SPCs as a funtion of the frequeny ωd/2π. When the external magneti field H e is applied, the oated nanopartiles will form hains along the magneti field diretion hanging the mirostruture of the system aordingly. For the sake of numerial alulations, we take t =0.5 andν =0.5. First, we onsider the longitudinal L field ases as shown in Fig. 9. When the loal magneti fator α gradually dereases from 1.0 to 0.02, Fig. 9 For the L field ases, the absorption oeffiient log β as a funtion of the frequeny ωd/2π for various α.inseta the dispersion relation of the wave vetor k rd/π, insetb real part of oskd, namely, f 1 ω. The square, irle and triangular dots are orresponding to α =0.02, 0.4, 1.0 respetively. Parameters: t =0.5 andν =0.5. Reprodued from Ref. [39]. Figure 10 demonstrates the absorption oeffiient log βof1dspcsasafuntionofthefrequeny ωd/2π depending on the thikness parameter t. For the L field ases, α =0.4 andν =0.5. When the thikness of the shell inreases, t will derease. From Fig. 10, one an learly see that band gaps shift to the low frequeny region when t inreases from 0.1, 0.3, to 0.8. That means, the thinner shell layer of silver gives rise to the redshift ofband gaps. Also the largert leads to higher absorption oeffiient log β. Inset a illustrates the distribution of k r d/π as a funtion of frequeny. These band gaps in Fig. 10 are aused by the frequeny range

9 Chun-Zhen Fan, Er-Jun Liang, and Ji-Ping Huang, Front. Phys., 2013, 81 9 where f 1 ω > 1 as shown in inset b. Fig. 10 For the L field ases, the absorption oeffiient log β as a funtion of the frequeny ωd/2π for various t. Inseta the dispersion relation of the wave vetor k rd/π, inset b real part of oskd, namely, f 1 ω. The square, irle and triangular dots are orresponding to t = 0.1, 0.3, 0.8 respetively. Parameters: α =0.4 andν =0.5. Reprodued from Ref. [39]. The influene of filling fator ν on the dispersion urve is illustrated in Fig. 11. Here we keep the parameters α =0.4 andt =0.5. By tuning the filling fator ν from 0.1, 0.3 to 0.5, we an find a redshift moving to the low frequeny region of the band gap of the 1D SPCs as shown in Fig. 11. It is evident that the absorption oeffiient beomes larger when ν inreases Fig. 11. The distribution of k r d/π and f 1 ω areshownasinseta and b respetively. in Fig. 12a for the L field ase where α =0.4, t =0.5. ν is taken from 0.0 to 1.0. It an be observed that the variation of the seond band gap width is losely related to ν. It got inreased when ν is smaller than 0.35, but when ν is larger than 0.35 the width is redued. The situation is the same for the T field ase as shown in Fig. 12b. Based on the transfer matrix method, we theoretially studied the omplex wave vetor in 1D SPCs, whih are omposed of the periodi ferrofluids layer and air layer. The behavior in the 1D SPCs is quite different from the real wave vetor in the PCs. Numerial results show that struture anisotropy indued by the external magneti field auses the dispersion urve to blueshift when α approahes 0.0, or α is lose to 1.5. Also by inreasing the thikness parameter t and the filling fator ν the band gaps will redshift. These band gaps in this proposed 1D SPCs struture are aused by the osillating f 1 ω within the frequeny region where f 1 ω > 1. An additional band gap ours in the low frequeny region due to the dissipation fator γ in the dieletri funtion of silver. Also the band width variations are alulated. Our results presented in this setion provide a guideline for designing the potential photoni devies based on the ferrofluids and it is helpful for improving the quality of these devies. Fig. 11 For the L field ases, the absorption oeffiient log β as a funtion of the frequeny ωd/2π for various ν. Inseta the dispersion relation of the wave vetor k rd/π, inset b real part of oskd, namely, f 1 ω. The square, irle and triangular dots are orresponding to ν = 0.1, 0.3, 0.5 respetively. Parameters: α =0.4 andt =0.5. Reprodued from Ref. [39]. We have also determined the variation of the band gap width as a funtion of the loal magneti fator α, the thikness ratio t and the filling fator ν of the ferrofluids layer. The band gap width tends to inrease when the magnitude of the loal magneti fator α dereases or t inreases not shown here. Speifially, the band gap width variation as a funtion of filling fator ν are shown Fig. 12 3D illustration of the dispersion relation k rd/π as a funtion of the ferrofluids filling fration ν and frequeny ωd/2π. For the filling fator, ν is taken from 0.01 to 1.0. a for the L field ases, α =0.4. b for the T field ases, α =1.2. Parameter: t = 0.5. Reprodued from Ref. [39].

10 10 Chun-Zhen Fan, Er-Jun Liang, and Ji-Ping Huang, Front. Phys., 2013, D graded SPCs with multilayer struture PCs have attrated wide researh attention owing to their unique light manipulation abilities, demonstrating wide appliations in optial sensor, optial filter and various optial integrated iruit devies [5]. To fabriate them, both top down lithographi tehnique and bottom up method nanosale self-assembled method [108] have been put forward to realize different strutures like the slab [109], multilayer [110], ylinders array [111], or even the fratal strutures [112]. Among them, one most studied struture is in the form of planar films [113], where the periodi alternating layers is obtained by bottom-up assembly. This proedure is based on sequential spin-oating suspensions of nanopartiles on the substrates. With suh a method, simple, stable, inexpensive PCs are obtained. Meanwhile, modulation of the refrative index is ahieved by alternating the size, type, onentrations of nanopartiles in omposite layer or by ontrolling the number of the deposited layers [114, 115]. Moreover, PCs are able to be realized through a ombination with hannels, walls, or mirofluidi hips [116], where olloids are fored through the area with pressure. In most ases, the omposite layer is a suspension of monodispersed olloid, onstituted by dieletri SiO 2 or TiO 2 partiles [114, 117]. If small metalli nanopartiles are introdued into a suspension, the third order Kerrtype nonlinearity of omposites have to be inluded, resulting in an enhanement in the loal eletri field [55, 92]. Simulation results have shown that tunable plasmon resonant effets arise with gold nanopartiles inserted into a periodi PCs [118]. Here we study the optial properties in 1D SPCs with graded multilayer. Graded materials are the ones whose material properties are spatial dependent. It has long been reognized that gradients in layer omposition an improve the performane of a material [ ], and open up the additional possibility of shaping the refrative ontrast [ ]. Let us onsider a graded multilayer struture with N layers, the dieletri onstant vary ontinuously along the z diretion and in the ith layer the dieletri onstant is expressed as ɛ i,z.the shemati struture of our proposed 1D GSPCs is illustrated in Fig. 13. It is omposed of one omposite layer and one air layer. Three layers with different nanopartile volume fration to represent the graded omposite layer in Fig. 13. Speifily, Ag nanopartile embedded in the TiO 2 is studied in the omposite layer. 5.1 Formulism For 1D PCs with an infinite period, the whole struture an be regarded as an series of multilayer struture, ating as a LC iruit. The reiproal of the equivalent apaitane C eqv of multiple apaitors onneted in series is the sum of the reiproals of the individual apaitanes C n. It an be expressed as 1/C eqv = 1/C 1 +1/C /C n = n i=1 1/C n, where C = ɛs/4πk d s. ɛ is the dieletri onstant, s is the apaitor plates of area, d s is the distane between apaitor plates, k is the eletrostati onstant. Thus, by using the equivalent apaitane of series ombination, the optial response of the graded omposite layer ɛ 1 an be expressed as 1 ɛ 1 = 1 L L 0 dz ɛ i i, z 29 where L is the thikness of the graded film, and it is divided into N layers. Due to the periodiity of the two refrative index materials, the multilayer struture reates a periodi potential for photons in one dimension. Eletromagneti waves of a given frequeny propagating through a stak of layers at normal inidene is onsidered [125, 126]. By resorting to the solution of 1D Maxwell s equations, the eletri and magneti fields omponent along the inident light diretion an be related via the transfer matrix. Fig. 13 Shemati struture of 1D GSPCs is shown. It onsists of the omposite layer and the air layer. The width of the air layer and the omposite is a and b respetively. In the omposite layer, three layers with different volume fration are illustrated. Reprodued from Ref. [40]. As metalli nanopartiles are inluded in the omposite film, the loal eletri field enhanement indued by the third order nonlinearity has to be onsidered. With Ref. [92] the loal eletri field inside the nanopartial an be written as ɛ 1 = ɛ Ag +ɛ k χ 3,whereχ 3 is the third order nonlinearity, ɛ k is equal to 9 ɛ TiO2 / 2 + pɛ TiO2 + 1 pɛ Ag E 0 and E 0 is the external eletri field. The effetive dieletri onstant of the omposite layer ɛ e an be given through the Maxwell Garnett theory [55]: ɛ e ɛ TiO2 ɛ e +2ɛ TiO2 = p ɛ 1 ɛ TiO2 ɛ 1 +2ɛ TiO 2 30 where p is the volume fration of the nanopartiles in

11 Chun-Zhen Fan, Er-Jun Liang, and Ji-Ping Huang, Front. Phys., 2013, the omposite suspension. ɛ 1 is the dieletri onstant of the nanopartiles. ɛ TiO2 is the dieletri onstant of TiO 2 anditisfixedas2.4ɛ 0. ɛ 1 is taken as 9.564ɛ 0 for Ag nanopartiles, whih is haraterized by using the inident wavelength λ = 497 nm [79]. 5.2 Numerial results and disussions In this setion, we present our numerial results Figs of 1D GSPCs onsisting of alternating air layers and Ag nanopartile based omposite layers. With the inorporation of metalli nanopartiles, loal eletri field indued by the external stimuli has to be inluded. Let x represent χ 3 E 0. The refration index of the omposite layers as a funtion of the external eletri field stimuli x is illustrated in Fig. 14. It shows that tunable refration ontrast an be obtained when x is hanging from 0.0 to 1.0, indiating an enhaned magnitude of external field. With a different magnitude of the external field, the refration index an be effetively tuned, suh spatial modulation opens up a forbidden gap in the eletromagneti dispersion relation. results are shown in Fig. 16. For the graded profile with 2layers,p =0.1, 0.6; 5 layers, p =0.1, 0.3, 0.5, 0.7, 0.9; 10 layers, p =0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0. With Eq. 30, our results indiate that spetral position of the band gap an effetively be tuned by hanging the volume fration in eah omposite layer. Fig. 15 The dispersion relation in graded multilayer struture with different N is illustrated, where the omposite layer has 2, 5, 10 layers respetively. Parameters: p = layers, x =0.1, 0.6; 5 layers, x =0.1, 0.3, 0.5, 0.7, 0.9; 10 layers, x =0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0. Reprodued from Ref. [40]. Fig. 14 Refration index of the omposite layer n as a funtion of the magnitude of the external eletri field x is shown. Parameters: p = Reprodued from Ref. [40]. Figure 15 shows the dispersion relation of the 1D GPCs with different number of omposite layer N 2, 5, 10. For the graded profile with 2 layers, x =0.1, 0.6; 5layers,x =0.1, 0.3, 0.5, 0.7, 0.9; 10 layers, x =0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0. In eah omposite layer, the volume fration of the nanopartiles is fixed as p =0.15. Eq. 29 indiates that the omposite layer with different number of layers will result in a hange of the refration index in eah layer. From Fig. 15, we an find that the photoni band gap is moving towards the low frequeny region when the number of layer N is losing to 10. And the photoni band width is getting wider with smaller N = 2. Thus, the band gap position and band gap width an be effetively tuned by hanging N. The same as Fig. 15, but in eah omposite layer, the volume fration of the nanopartiles are different and the Fig. 16 The dispersion relation in graded multilayer struture with different p is illustrated, where the omposite layer has 2, 5, 10 layers respetively. Parameters: x = layers, p =0.1, 0.6; 5layers,p =0.1, 0.3, 0.5, 0.7, 0.9; 10 layers, p =0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0. Reprodued from Ref. [40]. In Fig. 15 and Fig. 16, the graded profile ontinuous variation in the magnitude of x or p is investigated. Now we fous on the linear profile, namely, the multilayer struture of PCs is purely onstituted by the air layer and omposite layer with a fixed x or p. Fig. 17a shows that the band gap position is moving towards the high frequeny region with an enhaned magnitude of x. But the band gap width is a onstant even with different x as shown in Fig. 17b, whih is quite different from that of graded profile Fig. 15. With a variation of volume fration p in the omposite layer, the results

12 12 Chun-Zhen Fan, Er-Jun Liang, and Ji-Ping Huang, Front. Phys., 2013, 81 Fig. 17 The dispersion relation in multilayer struture with different external stimuli x =0.0, x =0.2, x =0.5 is shown in a. The position and width of the first band gap in the multilayer struture as a funtion of the applied field x is shown in b. Parameters: p =0.15, a =0.5, b =0.5. Reprodued from Ref. [40]. Fig. 18 The dispersion relation in multilayer struture with different volume fration of the nanopartile p = 0.05, p = 0.15, p =0.3 is shown in a. The position and width of the first band gap in the multilayer struture as a funtion of the volume fration p is shown in b. Parameters: x =0.1, a =0.5, b =0.5. Reprodued from Ref. [40]. are shown in Fig. 18. With a low volume fration, the photoni band gap position [Fig. 18a] moves towards the low frequeny region. The first band widths are also alulated in Fig. 18b in green urve. It is found that the band gap width is still invariant in the linear profile even with different p. In Fig. 19a, the dispersion relation with linear profile through the variation of the orresponding thikness of the two alternating layers is explored. Here we keep the lattie onstant d as a onstant. When the thikness of the omposite layer d 2 is taken from 0.5, 0.7 to 0.9, the band gap position will move towards the high frequeny region. Meanwhile, the first band gap width is getting larger with an inreased d 2, as shown in Fig. 19b. As we know, PCs are designed to affet the propagation properties of photons. The periodi spatial modulation opens up a forbidden gap in the eletromagneti dispersion relation. In our proposed 1D GSPCs, the refration index an be effetively tuned by the external eletri field, the volume fration of the nanopartiles and the number of graded omposite layer. 6 Seond-harmoni generation with magnetoontrolling abilities in ferrofluids In this part, the nonlinear photoni responses of soft materials based on ferrofluids are presented [41]. We first introdue seond harmoni rystals with magnetoontrolling abilities about the proposed soft material. Theoretial [127] and experimental [ ] reports suggested that spherial partiles exhibit a rather unexpeted and nontrivial behavior, seond harmoni generation SHG, due to the broken inversion symmetry at partile surfaes, despite their entral symmetry whih seemingly prohibits seond-order nonlinear effets. In olloidal suspensions, the SHG response for entrosymmetri partiles was experimentally reported [128]. Most reently, the SHG from entrosymmetrial struture has reeived extensive attention e.g., see Refs. [ ]. In view of reent advanements in the fabriation of nanoshells [134, 135] and single domain ferromagneti nanopartiles [136], we shall theoretially suggest a lass of nonlinear optial materials in whih single domain ferromagneti nanopartiles oated by a nonmagneti nanoshell with

13 Chun-Zhen Fan, Er-Jun Liang, and Ji-Ping Huang, Front. Phys., 2013, an intrinsi SHG suseptibility are suspended in a nonmagneti host fluid Fig. 20. For suh a material, there is not only an inident light, but also an external magneti field H. The latter yields the formation of hains of oated and thus yields the effetive SHG with magnetifield ontrollabilities [ ]. This kind of SHG is expeted to reeive a broad interest in the physis, optis, and engineering ommunities, beause it is diffiult or impossible to ahieve onventional, naturally ourring materials or random omposites [55, 106, ]. 6.1 Formulism When a olletion of objets e.g., oated nanopartiles or nanopartile hains whose size and spaing are muh smaller than the wavelength λ of an inident light, the light passing through the struture annot tell the differene, and hene the inhomogeneous struture an be seen as a homogeneous one [144]. In this regard, to investigate the SHG responses of the proposed material, we are allowed to average over inhomogeneous oated nanopartiles or nanopartile hains, oneptually replaing the inhomogeneous objets by a homogeneous material Figs. 21 and 22. Fig. 19 The dispersion relation in multilayer struture with different thikness in the omposite layer is shown in a. Parameters: a =0.5, b =0.5; a =0.3, b =0.7; a =0.1, b =0.9. The position and width of the first band gap as a funtion of the thikness in the multilayer struture is shown in b. Parameters: x = 0.1, p = Reprodued from Ref. [40]. Fig. 20 Design for a nonlinear optial material, whih is subjeted to an external magneti field H. or : Longitudinal or perpendiular field ases orresponding to the fat that the E field of an inident light is parallel or perpendiular to the nanopartile hain. Reprodued from Ref. [41]. R r Fig. 21 Shemati graph showing the equivalene between a oated inhomogeneous nanopartile left and a homogeneous nanopartile right. Reprodued from Ref. [41]. Let us onsider a linear ferromagneti spherial nanopartile with dieletri onstant ɛ 1 and radius r, whih is oated by a nonlinear optial nonmagneti nanoshell e.g., noble metals like silver or gold with frequeny-dependent dieletri onstant ɛ 1ω and intrinsi SHG suseptibility d ijk s 2ω; ω, ω witheahofthe supersripts running over the three Cartesian indies. Here ω denotes the angular frequeny of a monohromati external eletri field, and the radius of the whole oated nanopartile is represented as R in the following. All the oated nanopartiles are suspended in a linear nonmagneti host fluid of ɛ 2. In the nanoshell, the loal onstitutive relation between the displaement field D s and the eletri field E s in the stati ase is given by Ds i = ɛ 1ω ij Es j + d ijk s 2ω; ω, ωese j s k j jk i = x, y, z 31 where D i s and Ei s are the ith omponent of D s and E s, respetively. Here ɛ 1 ωij = ɛ 1 ωδ ij denotes the linear dieletri onstant, whih is assumed for simpliity to be isotropi. Upon ertain symmetry, one an have R

14 14 Chun-Zhen Fan, Er-Jun Liang, and Ji-Ping Huang, Front. Phys., 2013, 81 d iii d xxx s s 2ω; ω, ω 0i = x, y, z 32 2ω; ω, ω =d yyy s 2ω; ω, ω=d zzz s 2ω; ω, ω 33 If a monohromati external field is applied, the nonlinearity in the system will generally generate loal potentials and fields at all harmoni frequenies. For a finite-frequeny external eletri field of the form E 0 = E 0 ωe iωt +.., the equivalent and effetive SHG suseptibilities for the oated nanopartile and the whole suspension, d zzz 1 2ω; ω, ω [Eq. 35] and d zzz e 2ω; ω, ω [Eq. 40], an be Reprodued by onsidering the volume average of the displaement field at the frequeny 2ω in the inhomogeneous medium. The eletri field E s in the nanoshell an be alulated [140] using standard eletrostatis, by solving the orresponding Maxwell equation E s =0, whih implies that E s = φ, whereφ is an eletri potential. Next, the equivalent linear dieletri onstant ɛ 1 ω for the oated nanopartile an be given by the Maxwell Garnett formula [106]: ɛ 1 ω ɛ 1 ω ɛ ɛ 1 ω+2ɛ =1 f 1 ɛ 1 ω 1 ω ɛ 1 +2ɛ 1 ω 34 where f =1 r 3 /R 3 is the volume ratio of the nanoshell to the whole oated nanopartile. The Maxwell-Garnett formula is a well-known asymmetrial effetive medium theory, and may thus be valid for a low onentration of nanopartiles in omposites [107]. While treating a single oated nanopartile with full range 0 f 1, the Maxwell Garnett formula [Eq. 34] holds for the alulation of ɛ 1 ω indeed due to the natural existene of asymmetry in the oated nanopartile. The solution of E s in Ref. [140] an be used to derive the equivalent SHG suseptibility for the oated nanopartile, d iii 1 d iii 1 2ω; ω, ω, whihanbeexpressedas 2ω; ω, ω=fdiii s 2ω; ω, ω z E 2 s,j2ω Es,j ω s 35 E 0,i 2ω E 0,i ω j=x where s denotes a volume average over the nanoshell. To show the feature of the proposed material, we assume the optial responses [namely, ɛ 1 ω andd iii 1 2ω; ω, ω] of an equivalent spheroid or a hain see Fig. 21 to be the same as those of eah oated nanopartile inside the spheroid or hain. For onveniene, the suspension is further assumed to be the one that ontains idential equivalent spheroids with geometrial depolarization fator α or α along major or minor axis Fig. 22. In the following, α is also alled loal magneti field fators, beause, from the physial point of view, the spheroids or hains are just formed due to the appliation of external magneti fields. In this onnetion, the summation term in Eq. 35 admits {Π2ωΠ 2 ω + 4/5rR 3 [Π2ωp 2 s ω + 2Πωp s2ωp s ω] + 8/35r 3 + R 3 /r 6 R 6 p s 2ωp 2 sω}/[e 0,i 2ωE0,i 2 ω] where Πω =T s ωe 0,i ω p s ω =b s ωr 3 T s ωe 0,i ω 36 with b s ω =ɛ 1 ɛ 1ω/ɛ 1 +2ɛ 1ω 37 and T s ω =[Θω+2b s ωθω 11 f] 1.Here Θω =[ɛ 1 ω+2ɛ 2]/3ɛ 2. Fig. 22 Shemati graph showing the equivalene between nanopartile hains and spheroids with geometrial major-axis or minoraxis depolarization fator α or α. The major axis is parallel to external magneti fields. α is also alled loal magneti field fators. Reprodued from Ref. [41]. In the following analysis, we assume the spheroidal partile possess the same linear response as the oated partile. But, for the nonlinear response, to inlude the shape effet, let us introdue a loal magneti field fator α that denotes α and α for longitudinal and transverse field ases, respetively Figs. 21 and 22. There is asumruleforα and α, α +2α = 1 [145]. The parameter α measures the degree of strutural anisotropy due to the formation of nanopartile hains, whih is indued to appear by the external magneti field H. More preisely, the degree of the magneti-field-indued anisotropy is measured by how muh α deviates from 1/3, 1/3 < α < 1 and 0 < α < 1/3. As H inreases, α and α should tend to 1 and 0, respetively, whih is indiative of the formation of longer nanopartile hains or equivalent spheroids. Therefore, α should be a funtion of external magneti fields H. Speifially, for H =0thereisα = α =1/3, whih orresponds to an isotropi system in whih all the oated nanopartiles are randomly distributed in the suspension. After the introdution of α, while we assume the spheroid possess the same nonlinear response as the oated partile, for more aurate estimation of the nonlinear response of the spheroid, let us replae Θω =[ɛ 1ω+2ɛ 2 ]/3ɛ 2 with Θω =[ɛ 2 + αɛ 1 ω ɛ 2]/ɛ 2. That is, the shape effet of spheroids has been inluded. Apparently, the substitution of α = 1/3 into Eq. 35 yields the same expression as Eq. 15 in Ref. [140] in whih a random omposite of partiles with nonlinear non-metalli shells was investigated. Alternatively, aording to the alulation of major-axis depolarization fator L of prolate