Analytical model for optical bistability in nonlinear metal nano-antennae involving Kerr materials

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1 Analytial model for optial bistability in nonlinear metal nano-antennae involving Kerr materials Fei Zhou, 1 Ye Liu, 1 Zhi-Yuan Li, 1, * and Younan Xia 1 Laboratory of Optial Physis, Institute of Physis, Chinese Aademy of Sienes, P.O. Box 63, Beijing 119, China Department of Biomedial Engineering, Washington University, St. Louis 6313, Missouri, USA *lizy@aphy.iphy.a.n Abstrat: Optial bistability at nanosale is a promising way to realize optial swithing, a key omponent of integrated nanophotoni devies. In this work we present an analytial model for optial bistability in a metal nano-antenna involving Kerr nonlinear medium based on detailed analysis of the orrelation between the inident and intion light intensity under surfae plasmon resonane (SPR). The model allows one to onstrut a lear piture on how the threshold, ontrast, and other harateristis of optial bistability are influened by the nonlinear oeffiient, inident light intensity, loal field enhanement fator, SPR peak width, and other physial parameters of the nano-antenna. It shows that the key towards low threshold power and high ontrast optial bistability in the nanosystem is to redue the SPR peak width. This an be ahieved by reduing the absorption of metal materials or introduing gain media into nanosystems. 1 Optial Soiety of Ameria OCIS odes: (4.668) Surfae plasmons; (19.145) Bistability; (19.37) Kerr effet. Referenes and links 1. H. M. Gibbs, Optial Bistability: Controlling Light with Light, Quantum eletronis priniples and appliations (Aademi Press, 1985). H. Nihei, and A. Okamoto, Photoni rystal systems for high-speed optial memory devie on an atomi sale, Pro. SPIE 4416, (1). 3. G. Assanto, Z. Wang, D. J. Hagan, and E. W. Vanstryland, All-optial modulation via nonlinear asading in type II seond-harmoni generation, Appl. Phys. Lett. 67(15), 1 1 (1995). 4. D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel kin, Ultrafast optial swithing in three-dimensional photoni rystals, Phys. Rev. Lett. 91(1), 1393 (3). 5. G. Priem, P. Dumon, W. Bogaerts, D. Van Thourhout, G. Morthier, and R. Baets, Optial bistability and pulsating behaviour in silion-on-insulator ring resonator strutures, Opt. Express 13(3), (5). 6. F. Y. Wang, G. X. Li, H. L. Tam, K. W. Cheah, and S. N. Zhu, Optial bistability and multistability in onedimensional periodi metal-dieletri photoni rystal, Appl. Phys. Lett. 9(1), 1119 (8). 7. M. F. Yanik, S. H. Fan, and M. Soljai, High-ontrast all-optial bistable swithing in photoni rystal miroavities, Appl. Phys. Lett. 83(14), (3). 8. M. F. Yanik, S. H. Fan, M. Soljaić, and J. D. Joannopoulos, All-optial transistor ation with bistable swithing in a photoni rystal ross-waveguide geometry, Opt. Lett. 8(4), (3). 9. G. A. Wurtz, R. Pollard, and A. V. Zayats, Optial bistability in nonlinear surfae-plasmon polaritoni rystals, Phys. Rev. Lett. 97(5), 574 (6). 1. C. J. Min, P. Wang, C. C. Chen, Y. Deng, Y. H. Lu, H. Ming, T. Y. Ning, Y. L. Zhou, and G. Z. Yang, Alloptial swithing in subwavelength metalli grating struture ontaining nonlinear optial materials, Opt. Lett. 33(8), (8). 11. Y. Shen, and G. P. Wang, Optial bistability in metal gap waveguide nanoavities, Opt. Express 16(1), (8). 1. N. Large, M. Abb, J. Aizpurua, and O. L. Muskens, Photoondutively loaded plasmoni nanoantenna as building blok for ultraompat optial swithes, Nano Lett. 1(5), (1). 13. P. Mühlshlegel, H. J. Eisler, O. J. F. Martin, B. Heht, and D. W. Pohl, Resonant optial antennas, Siene 38(578), (5). 14. O. L. Muskens, V. Giannini, J. A. Sánhez-Gil, and J. Gómez Rivas, Strong enhanement of the radiative deay rate of emitters by single plasmoni nanoantennas, Nano Lett. 7(9), (7). (C) 1 OSA 1 June 1 / Vol. 18, No. 13 / OPTICS EXPRESS 13337

2 15. J. W. Liaw, Analysis of a bowtie nanoantenna for the enhanement of spontaneous emission, IEEE J. Sel. Top. Quantum Eletron. 14(6), (8). 16. Y. Liu, F. Qin, F. Zhou, and Z. Y. Li, Ultrafast and low-power photoni rystal all-optial swithing with resonant avities, J. Appl. Phys. 16(8), 831 (9). 17. B. T. Draine, and P. J. Flatau, Disrete-dipole approximation for sattering alulations, J. Opt. So. Am. A 11(4), (1994). 18. F. Zhou, Z. Y. Li, Y. Liu, and Y. N. Xia, Quantitative analysis of dipole and quadrupole exitation in the surfae plasmon resonane of metal nanopartiles, J. Phys. Chem. C 11(51), 33 4 (8). 19. A. Alù, and N. Engheta, Tuning the sattering response of optial nanoantennas with nanoiruit loads, Nat. Photonis (5), (8).. J. Berthelot, A. Bouhelier, C. Huang, J. Margueritat, G. Colas-des-Frans, E. Finot, J.-C. Weeber, A. Dereux, S. Kostheev, H. I. E. Ahrah, A.-L. Baudrion, J. Plain, R. Bahelot, P. Royer, and G. P. Wiederreht, Tuning of an optial dimer nanoantenna by eletrially ontrolling its load impedane, Nano Lett. 9(11), (9). 1. Z. Y. Li, and Y. N. Xia, Metal nanopartiles with gain toward single-moleule detetion by surfae-enhaned Raman sattering, Nano Lett. 1(1), (1). 1. Introdution It has been ommonly aknowledged that all-optial devies at the mirometer and nanometer sales is a promising way towards realization of n-generation ultrafast ommuniation and signal proessing systems beyond today s miroeletronis devies, whih have gradually enountered limitation in bandwidth and speed. Optial swithing is an essential omponent in the all-optial network. A feasible approah to all-optial swithing is based on optial bistability, an important subjet in nonlinear optis [1]. Optial bistability offers many intriguing appliations, suh as optial memory [], optial transistor [3], all-optial swithing [4], and among others. In reent years there has been a great interest in exploring and realizing optial bistability in nonlinear nanophotoni systems. Optial bistability has been predited by theory or demonstrated by experimental studies to exist in waveguide-ring resonators [5], photoni rystal avities [6 8], plasmoni rystals [9], subwavelength metalli gratings [1], metal gap waveguide nanoavities [11], and nanoantenna with amorphous silion filled in the gap [1]. It is important to ahieve a deeper understanding of the basi physis of optial bistability at the nanosale in order to design and realize high-performane nanophotoni swithing devies. The physis of optial bistability in a lassial optial resonant system suh as Fabry-Perot etalon has been well established and an be desribed by simple analytial models [1]. However, so far a similar analytial model has not yet been available for nonlinear nanophotoni systems. Reently we suessfully worked out an analytial model for desribing the optial bistability of a metal nano-antenna involving a Kerr nonlinear material, whih will be addressed in this letter. We propose to investigate the optial bistability of an optial nano-antenna [13 15] beause this struture an provide very strong field enhanement in the enter gap due to surfae plasmon resonane (SPR) and the SPR peak is very sensitive to the refrative index hange of the Kerr material used to fill the gap. The analytial model an help us obtain deep insight into the fundamental physis of optial bistability at nanosale. This paper is arranged as follows. In Se. we present a detailed proess of how we reah suh an analytial solution of optial bistability in the nonlinear nano-antenna struture. In Se. 3 we briefly disuss what the analytial model an do to reveal the key fators for a deep insight of the underlying physis of optial bistability in nonlinear nanosystems. In Se. 4 we make a brief summary of this paper.. Derivation of the analytial model The nano-antenna struture we studied is depited in Fig. 1. Silver is hosen as the material for the arms beause of its low absorption. In the gap between the two arms, a Kerr nonlinear material is introdued. We use polystyrene with the following optial parameters: a linear 1 refrative index n = 1.59, and a Kerr nonlinear oeffiient n = m W [16]. In our study, we are interested in the bistability between the inident optial intensity I and the (C) 1 OSA 1 June 1 / Vol. 18, No. 13 / OPTICS EXPRESS 13338

3 intion power W. In our simulation, the oordinates are established as shown in Fig. 1. The inident light propagates along the -z-axis, and its eletri field E is along the x-axis, whih is a longitudinal polarization. The total length of the nano-antenna is 13 nm, and the width is nm. The thikness of polystyrene embedded in the enter gap is 1 nm. In the following, we will derive the analytial expressions to reveal the physial proess of optial bistability. Fig. 1. Shemati struture of the nonlinear nano-antenna system. As the first step, we temporarily disregard the nonlinear property of polystyrene. The alulated intion spetrum is displayed in Fig.. The disrete dipole approximation (DDA) method [17,18] is adopted in our simulations. In Fig., the blak irles are the alulated intion ross setion values, showing strong SPR. The red line is the urve fitted with a Lorentz model, whih an perfetly desribe the intion spetrum. The Lorentz funtion is C A w = π 4 λ λ + w where C is the intion ross setion, w is the full width at half maximum (FWHM) of the SPR peak, λ is the inident light wavelength, λ is the SPR peak wavelength, and A is a fitting parameter related to the magnitude of the peak. In our ase, the fit results are 3 A=.65 µm, w=.493 µm, and λ =.6638 µm. Extintion Cross Setion (µm ) Wavelength (µm), C Fit urve Fig.. Extintion spetrum of nano-antenna. Blak irles are the results simulated with DDA method, and the red line is the fit urve with Lorentz funtion. Exellent agreement is obtained. In order to deal with the nonlinear proess, we need to study the eletri field distribution in the gap. We simulate the eletri field distribution at the wavelength of resonane (663.8 nm). The results are depited in Fig. 3. The field distributions in yz-planes (Fig. 3a) and xyplanes (Fig. 3b) are both shown. Meanwhile, we give the field enhanement fator along the x-axis and y-axis as depited in Fig. 3() and (d). The field enhanement fators along x-axis and y-axis with an arbitrarily seleted wavelength (7 nm) are also plotted in Fig. 3() and (d). At both wavelengths, the eletri field is quite uniform for most part in the gap exept for the position very lose to the edge of the silver bars. Here, we only selet two wavelengths to show the property of uniform eletri field in the gap, but the same results are found at other (1) (C) 1 OSA 1 June 1 / Vol. 18, No. 13 / OPTICS EXPRESS 13339

4 wavelengths. So, without losing generality, the field in the gap an be onsidered as a uniform eletri field. Consequently the refrative index hange in the gap region is also onsidered to be uniform. y-axis (nm) (b) z-axis (nm) - - (a) y-axis (nm) x-axis (nm) I/I nm 7. nm gap region () x-axis (nm) (d) I/I gap region nm 7. nm y-axis (nm) Fig. 3. Eletri field distributions (in unit of I ) with different wavelengths of λ = nm: (a) yz-plane and (b) xy-plane. () and (d) are the field enhanement fator I I along x and y axes (the blak lines). The fields for the wavelength of λ = 7 nm are also plotted (the red lines). The eletri field distribution in the gap region shows high uniformity. As is well known, the loal field enhanement in the nano-antenna is aused by SPR. The ent of SPR an be haraterized by the intion effiieny of the nanopartile, whih is proportional to C. So the field enhanement fator I I (with I being the loal field intensity) is a funtion of C. To get the analytial expression between I I and C, we alulate the loal field enhanement fator in the gap and the intion ross setion for wavelength from 5 to 9 nm. The relationship between I I and C is shown in Fig. 4, where the blak dots are for the wavelength at the left side of the resonant peak, while the red dots for the right side. At a resonant wavelength of around 664 nm, the intion ross setion is maximal, and the field enhanement fator reahes its maximum of around 4. With the deviation away from this resonant wavelength, the eletri field enhanement fator beomes lower. The overall urve in Fig. 4 an be approximately desribed by a linear expression: I I = βc, () 4 - where β is a linear fitting parameter about µm for this ase. The linear approximation does not influene the physial essene of the nano-antenna system and it makes possible simple analytial solution of the nonlinear bistability problem. 4 3 DDA Result: Left side DDA Result: Right side Fit Curve I/I C (µm ) Fig. 4. Relationship between the field enhanement fator I I and the intion ross setion C. The blak dots are for the wavelength at the left side of the resonant peak, while the red dots for the right side. The fit urve by a linear funtion is denoted as blue line. Previous studies showed that the resonant wavelength of nano-antennae an be tuned by hanging the load [19,]. In our ase, the resonant wavelength is simply tuned by the refrative index hange of the Kerr material in the gap. Now we turn to study the shift of (C) 1 OSA 1 June 1 / Vol. 18, No. 13 / OPTICS EXPRESS 1334

5 resonant peak with different refrative indies for the entral nonlinear material in details. As is shown before, the refrative index distribution in the gap an be onsidered uniform. So we hange the refrative index in the gap region uniformly, and alulate their intion spetra, whih are shown in Fig. 5(a). As the refrative index in the gap inreases, the resonant peak shifts to longer wavelengths while the shape of the intion spetrum hanges little. We rat the wavelength of resonant peak versus the refrative index in the gap region, and the result is shown in the Fig. 5(b). Very good linear relationship is found. So the peak wavelength an be expressed as λi = λ + αni, where λ I is the resonant peak wavelength with n I, whih is the nonlinear refrative index in the gap. λ and α are the linear fit parameters. On the other hand, the nonlinear refrative index is diretly proportional to the loal field intensity, whih is ni = n + ni. So the final equation is: λ = λ + α + = λ + α (3), I n ni n I In our ase shown in Fig. 5, α =.6379 µm. Up to this point, we have analytial expressions for the intion spetrum [Eq. (1)], the field enhanement fator versus the intion ross setion [Eq. ()], and the shift of resonant wavelength under different inident intensities [Eq. (3)]. Considering that the optial bistability we study here is the relationship between the input intensity and intion power, we transform the intion ross setion C into the intion power W : W = C I (4) By replaing λ in Eq. (1) with λ I and substituting Eqs. (), (3) and (4) into Eq. (1), we an finally obtain the relation between W and I :. λ λ αn βw w π 4 + I = W. (5) A w Equation (5) is the analytial expression of optial bistability between the inident intensity I and the intion power W. Extintion Cross Setion (µm ) (a) n=1.59 n=1.7 n=1.8 n=1.9 n=. n=.1 n=. n=.3 n= Wavelength (µm) SPR peak position (µm) (b) DDA Result Fit Curve Refrative Index n I Fig. 5. (a) Shift of intion spetra with the hange of refrative index in the gap region. (b) The relationship between the SPR peak wavelength and the orresponding refrative index in the gap region. The blak dots are the simulated results with DDA method, and the red line is the fitted urve. Very good linear relationship is found. 3. Key fators of optial bistability Now that we have derived the analytial formalism of optial bistability in the nonlinear nanoantenna, we proeed to disuss several key fators from the analytial model. The first thing that we onern is the riterion for optial bistability. When optial bistability exists, there must be three values of W orresponding to an inident intensity I. Two of them orrespond to the real and stable physial states, while the other one with value in the middle (C) 1 OSA 1 June 1 / Vol. 18, No. 13 / OPTICS EXPRESS 13341

6 orresponds to a virtual and unstable physial state. Taking the derivative of I with respet to W in Eq. (5), we an get the following expression: di π = 4( λ λ) + w 16αβ n( λ λ) W + 1 α β nw. dw Aw The ondition for the existene of optial bistability is that has two distint roots, whih an be satisfied when di (6) dw = (7) ( λ λ ) = 16α β n 4 3w >. (8) From this equation we find that λ λ > 3w must be satisfied to assure the existene of optial bistability in the nano-antenna system. For positive nonlinearity with n > it means that λ > λ + 3w, while for negative nonlinearity with n < we must have λ < λ 3w. In our nano-antenna struture, the resonant peak is loated at around 664 nm with FWHW of about 49 nm. So the ritial ondition for optial bistability beomes λ > 75 nm. I W (Watt).3..1 λ=755 nm λ=745 nm λ=735 nm λ=75 nm λ=715 nm λ=75 nm λ=695 nm λ=685 nm λ=675 nm λ=665 nm I (MW/m ) Fig. 6. Relationship between the intion power and the inident intensity at different wavelengths. The W is peak intion power. We plot the intion powers versus the inident intensities at different wavelengths in Fig. 6. Eah olor represents an individual inident wavelength. From the figure, we an find that the ritial wavelength for bistability is somewhere between 75 nm (the yan line) and 715 nm (the magenta line). For shorter inident wavelengths than the ritial wavelength, there are no bistable states; while for longer ones, the bistable states an be observed. The total refrative index hange of the Kerr material required to maintain optial bistablity is on the order of.1. The above analytial model an provide muh more information than the riterion of optial bistability. We an get the two roots of Eq. (7) diretly as: W W,1, ( λ λ ) ( p) = 1 1, 3αβ n ( λ λ ) ( p) = 1+ 1, 3αβ n (9) (C) 1 OSA 1 June 1 / Vol. 18, No. 13 / OPTICS EXPRESS 1334

7 3w where p= 4 λ λ Eq. (9) into Eq. (5):. The orresponding inident intensity an be obtained by substituting 3 8π λ λ 1 I1 = p p p 7Aαβ nw , 3 8π λ λ 1 I = + p p p 7Aαβ nw ( I 1, W,1 ) and ( I, W, ) are the two ritial points of the bistability urves, and I 1 represents the threshold power of optial bistability at a given inident wavelength. The ontrast of bistability signal is haraterized by and W W,,1 ( W, + W,1) 1 = 1 p, ( p) 3/ I 1 I1 = I + I 1+ p It is seen that the ontrasts of both I and W are only dependent on the parameter p. To inrease the ontrast of the two states, the only hoie is either to make the resonant peak narrower or to selet an inident wavelength farther away from the resonant wavelength. On the other hand, as shown in Eq. (1), the threshold intensity is inversely proportional to A, α, β, and n, whih means that the pump intensity an be dereased with larger A, α, β, and n. In other words, the threshold pump power an be redued by inreasing the magnitude and sensitivity of the SPR peak, the loal field enhanement fator, and the Kerr nonlinear oeffiient. The maximum enhanement fator at the resonant wavelength is nearly 4 for the nanoantenna, whih an redue the threshold power effiiently. However, the FWHM of intion spetrum is relatively large as nearly 5 nm. This is the main problem for low-power and high-ontrast optial bistability with nano-antenna aording to the above analytial model. The threshold power is around 8 MW m at wavelength 755 nm. If a lower threshold power needs to be implemented, the intion spetrum with SPR should be narrower. Our previous study shows that the line width of intion spetrum is losely related the absorption of metal nanopartiles [1]. If the absorption of a metal material is redued, the line width of the SPR peak an be ompressed. One way to ahieve this is to shift the operation wavelength of optial funtionality to longer wavelengths. Another way is to take a metal material with smaller intrinsi absorption. Here we propose an alternative way. Moreover, it has been shown that a proper gain, whih an be supplied by fluoresene moleules or quantum dots, an effetively ompensate for the absorption of metal, and lead to a very narrow spetrum width [1]. We alulate the intion spetra of the nano-antenna with different gain oeffiient k in the gap region, where the medium has a omplex refrative index of ni ik. The results are shown in Fig. 7(a). With the gain medium, the line width of intion spetra redues signifiantly. At k =.5, the line width dereases to about 1.5 nm, a quarter of the value without gain. In addition, the introdution of a gain medium does not influene the position of intion peak and the speed at whih the peak shifts when the refrative index is hanged. It just ompresses the line width of intion spetrum. In Fig. 7(b), we show the relationship between the line width and the gain. (1) (11) (1) (C) 1 OSA 1 June 1 / Vol. 18, No. 13 / OPTICS EXPRESS 13343

8 oeffiient. When k grows from zero to.5, the line width narrows nearly linearly. The alulation results of the optial bistability urve with k =.5 are displayed in Fig. 7. We find that the threshold power is redued to only 7 MW m at a wavelength of 683 nm, and the neessary total refrative index hange of the Kerr material requested to maintain optial bistability is redued to a level on the order of.1. W (Watt) λ=683 nm λ=681 nm λ=679 nm λ=677 nm λ=675 nm λ=673 nm λ=671 nm λ=669 nm λ=667 nm λ=665 nm Width (nm) 5 Peak Position Fit Curve (a) I (MW/m ) (b) gain oeffiient 4. Conlusions Fig. 7. (a) Optial bistability urves at different wavelengths with a gain oeffiient of k =.5 for the material in the gap. (b) The dependene of SPR peak width on the gain oeffiient. In summary, we have presented an analytial model for the optial bistability in a nonlinear metal nano-antenna struture. The model learly reveals how the performane of bistability is orrelated with the physial properties of the nanosystem. Aording to the model, the key towards optial bistability of low pump power and high ontrast is to redue the SPR peak width of the nanosystems by designing appropriate nanosystems with low absorption or with gain. Our study an help to explore nonlinear nanophotoni systems for appliations in ompat low-power bistable all-optial devies and storage. Aknowledgment This work was supported by the National Natural Siene Foundation of China (NNSFC) under grants and , and the National Key Basi Researh Speial Foundation of China under grant 6CB391. (C) 1 OSA 1 June 1 / Vol. 18, No. 13 / OPTICS EXPRESS 13344

Optical transistor action by nonlinear coupling of stimulated emission and coherent scattering

Optical transistor action by nonlinear coupling of stimulated emission and coherent scattering Optial transistor ation by nonlinear oupling of stimulated emission and oherent sattering David L. Andrews * and David S. Bradshaw Nanostrutures and Photomoleular Systems, Shool of Chemistry, University