Harmonic generation in metallic, GaAs-filled nanocavities in the enhanced transmission regime at visible and UV wavelengths

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1 Harmonic generation in metallic, GaAs-illed nanocavities in the enhanced transmission regime at visible and UV wavelengths M. A. Vincenti, 1, D. de Ceglia, 1 V. Roppo, and M. Scalora 3 1 AEgis Technologies Group, 41 Jan Davis Dr., Huntsville, AL 3586, USA Universitat Politècnica de Catalunya, Departament de Física i Enginyeria Nuclear, Rambla Sant Nebridi, 8 Terrassa, Spain 3 Charles M. Bowden Research Center AMSRD-AMR-WS-ST, RDECOM, Redstone Arsenal, Alabama 35898, USA Abstract: We have conducted a theoretical study o harmonic generation rom a silver grating having slits illed with GaAs. By working in the enhanced transmission regime, and by exploiting phase-locking between the pump and its harmonics, we guarantee strong ield localization and enhanced harmonic generation under conditions o high absorption at visible and UV wavelengths. Silver is treated using the hydrodynamic model, which includes Coulomb and Lorentz orces, convection, electron gas pressure, plus bulk χ (3) contributions. For GaAs we use nonlinear Lorentz oscillators, with characteristic χ () and χ (3) and nonlinear sources that arise rom symmetry breaking and Lorentz orces. We ind that: (i) electron pressure in the metal contributes to linear and nonlinear processes by shiting/reshaping the band structure; (ii) TE- and TM-polarized harmonics can be generated eiciently; (iii) the χ () tensor o GaAs couples TE- and TM-polarized harmonics that create phase-locked pump photons having polarization orthogonal compared to incident pump photons; (iv) Fabry- Perot resonances yield more eicient harmonic generation compared to plasmonic transmission peaks, where most o the light propagates along external metal suraces with little penetration inside its volume. We predict conversion eiciencies that range rom 1 6 or second harmonic generation to 1 3 or the third harmonic signal, when pump power is GW/cm. 11 Optical Society o America OCIS codes: (19.6) Harmonic Generation; (19.435) Nonlinear Optics at Suraces; (4.3695) Linear and Nonlinear light scattering at suraces (5.5); Diraction and Grating; (4.668) Surace Plasmon; (5.664) Subwavelength Structures. Reerences and links 1. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wol, Extraordinary optical transmission through subwavelength hole arrays, Nature 391(6668), (1998).. L. Salomon, F. Grillot, A. V. Zayats, and F. de Fornel, Near-ield distribution o optical transmission o periodic subwavelength holes in a metal ilm, Phys. Rev. Lett. 86(6), (1). 3. Y. Liu, and S. Blair, Fluorescence enhancement rom an array o subwavelength metal apertures, Opt. Lett. 8(7), (3). 4. D. J. Park, S. B. Choi, Y. H. Ahn, F. Rotermund, I. B. Sohn, C. Kang, M. S. Jeong, and D. S. Kim, Terahertz near-ield enhancement in narrow rectangular apertures on metal ilm, Opt. Express 17(15), (9). 5. A. Nahata, R. A. Linke, T. Ishi, and K. Ohashi, Enhanced nonlinear optical conversion rom a periodically nanostructured metal ilm, Opt. Lett. 8(6), (3). 6. M. Airola, Y. Liu, and S. Blair, Second-harmonic generation rom an array o sub-wavelength metal apertures, J. Opt. A, Pure Appl. Opt. 7(), S118 S13 (5). 7. A. Lesuler, L. K. S. Kumar, and R. Gordon, Enhanced second harmonic generation rom Nanoscale doublehole arrays in gold ilm, Appl. Phys. Lett. 88(6), 6114 (6). # $15. USD Received Dec 1; accepted 1 Jan 11; published 19 Jan 11 (C) 11 OSA 31 January 11 / Vol. 19, No. 3 / OPTICS EXPRESS 64

2 8. J. A. H. van Nieuwstadt, M. Sandtke, R. H. Harmsen, F. B. Segerink, J. C. Prangsma, S. Enoch, and L. Kuipers, Strong modiication o the nonlinear optical response o metallic subwavelength hole arrays, Phys. Rev. Lett. 97(14), 1461 (6). 9. N. Rakov, F. E. Ramos, and M. Xiao, Strong second-harmonic radiation rom a thin silver ilm with randomly distributed small holes, J. Phys. Condens. Matter 15(3), L349 L35 (3). 1. T. Xu, X. Jiao, and S. Blair, Third-harmonic generation rom arrays o sub-wavelength metal apertures, Opt. Express 17(6), (9). 11. N. N. Lepeshkin, A. Schweinsberg, G. Piredda, R. S. Bennink, and R. W. Boyd, Enhanced nonlinear optical response o one-dimensional metal-dielectric photonic crystals, Phys. Rev. Lett. 93(1), 139 (4). 1. D. T. Owens, C. Fuentes-Hernandez, J. M. Hales, J. W. Perry, and B. Kippelen, A comprehensive analysis o the contributions to the nonlinear optical properties o thin Ag ilms, J. Appl. Phys. 17(1), (1). 13. D. Krause, C. W. Teplin, and C. T. Rogers, Optical surace second harmonic measurements o isotropic thinilm metals: Gold, silver, copper, aluminum, and tantalum, J. Appl. Phys. 96(7), 366 (4). 14. F. Xiang Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, Surace and bulk contributions to the second-order nonlinear optical response o a gold ilm, Phys. Rev. B 8(3), 334 (9). 15. Y. R. Shen, The Principles o Nonlinear Optics, (Wiley Classics Library, Wiley, New York, ). 16. D. Maystre, M. Neviere, and R. Reinisch, Nonlinear polarization inside metals: a mathematical study o the ree electron model, Appl. Phys., A Mater. Sci. Process. 39(), (1986). 17. M. A. Vincenti, V. Petruzzelli, A. D'Orazio, F. Prudenzano, M. J. Bloemer, N. Aközbek, and M. Scalora, Second harmonic generation rom nanoslits in metal substrates: applications to palladium-based H sensor, J. Nanophotonics (1), 1851 (8). 18. F. I. Baida, and D. Van Labeke, Light transmission by subwavelength annular aperture arrays in metallic ilms, Opt. Commun. 9(1-3), 17 (). 19. F. I. Baida, D. Van Labeke, G. Granet, A. Moreau, and A. Belkhir, Origin o the super-enhanced light transmission through a -D metallic annular aperture array: a study o photonic bands, Appl. Phys. B 79(1), 1 8 (4).. W. Fan, S. Zhang, N. C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, Second Harmonic generation rom a nanopatterned isotropic nonlinear material, Nano Lett. 6(5), (6). 1. E. H. Barakat, M. P. Bernal, and F. I. Baida, Second harmonic generation enhancement by use o annular aperture arrays embedded into silver and illed by lithium niobate, Opt. Express 18(7), (1).. W. Fan, S. Zhang, K. J. S. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, Second harmonic generation rom patterned GaAs inside a subwavelength metallic hole array, Opt. Express 14(1), (6). 3. N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, Optical harmonic generation in relection rom media with inversion symmetry, Phys. Rev. 174(3), (1968). 4. J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, Analysis o second-harmonic generation at metal suraces, Phys. Rev. B 1(1), (198). 5. J. E. Sipe, and G. I. Stegeman, Surace Polaritons: Electromagnetic Waves at Suraces and Interaces, V. M. Agranovich and D. Mills, eds., (North-Holland, Amsterdam, 198). 6. M. Corvi, and W. L. Schaich, Hydrodynamics model calculation o second harmonic generation at a metal surace, Phys. Rev. B 33(6), (1986). 7. A. Eguiluz, and J. J. Quinn, Hydrodynamic model or surace plasmon in metals and degenerate semiconductors, Phys. Rev. B 14(4), (1976). 8. M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, Second and Third Harmonic Generation in Metal-Based Structures, Phys. Rev. A 8(4), 4388 (1). 9. N. Bloembergen, and P. S. Pershan, Light Waves at the Boundary o Nonlinear Media, Phys. Rev. 18(), 66 6 (196). 3. W. Glenn, Second-harmonic generation by picosecond optical pulses, IEEE J. Quantum Electron. 5(6), 84 9 (1969). 31. J. T. Manassah, and O. R. Cockings, Induced phase modulation o a generated second-harmonic signal, Opt. Lett. 1(1), (1987). 3. S. L. Shapiro, Second harmonic generation in LiNbO3 by picosecond pulses, Appl. Phys. Lett. 13(1), 19 (1968). 33. L. D. Noordam, H. J. Bakker, M. P. de Boer, and H. B. van Linden van den Heuvell, Second-harmonic generation o emtosecond pulses: observation o phase-mismatch eects, Opt. Lett. 15(4), 1464 (199). 34. N. C. Kothari, and X. Carlotti, Transient second-harmonic generation: inluence o eective group-velocity dispersion, J. Opt. Soc. Am. B 5(4), 756 (1988). 35. V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J. W. Haus, O. G. Kosareva, and V. P. Kandidov, Role o phase matching in pulsed second-harmonic generation: Walk-o and phase-locked twin pulses in negative-index media, Phys. Rev. A 76(3), 3389 (7). 36. V. Roppo, M. Centini, D. de Ceglia, M. A. Vincenti, J. W. Haus, N. Akozbek, M. J. Bloemer, and M. Scalora, Anomalous momentum states, non-specular relections, and negative reraction o phase-locked, secondharmonic pulses, Metamaterials (Amst.) (-3), (8). # $15. USD Received Dec 1; accepted 1 Jan 11; published 19 Jan 11 (C) 11 OSA 31 January 11 / Vol. 19, No. 3 / OPTICS EXPRESS 65

3 37. M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, and M. Scalora, Inhibition o linear absorption in opaque materials using phase-locked harmonic generation, Phys. Rev. Lett. 11(11), (8). 38. V. Roppo, C. Cojocaru, F. Raineri, G. D Aguanno, J. Trull, Y. Halioua, R. Raj, I. Sagnes, R. Vilaseca, and M. Scalora, Field localization and enhancement o phase-locked second- and third-order harmonic generation in absorbing semiconductor cavities, Phys. Rev. A 8(4), (9). 39. V. Roppo, J. Foreman, N. Akozbek, M. A. Vincenti, and M. Scalora, Third Harmonic Generation at 3nm in the Metallic Regime o GaP, (1). 4. E. D. Palik, Handbook o Optical Constants o Solids (Academic Press, London-New York (1985). 41. M. A. Vincenti, M. De Sario, V. Petruzzelli, A. D Orazio, F. Prudenzano, D. de Ceglia, N. Akozbek, M. J. Bloemer, P. Ashley, and M. Scalora, Enhanced transmission and second harmonic generation rom subwavelength slits on metal substrates, Proc. SPIE 6987, 6987O, 6987O-9 (8). 4. J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, Transmission resonances on metallic gratings with very narrow slits, Phys. Rev. Lett. 83(14), (1999). 43. Q. Cao, and Ph. Lalanne, Negative role o surace plasmons in the transmission o metallic gratings with very narrow slits, Phys. Rev. Lett. 88(5), 5743 (). 44. P. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, Perturbative approach or surace plasmon eects on lat interaces periodically corrugated by subwavelength apertures, Phys. Rev. B 68(1), 1544 (3). 45. Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, Transmission o light through a periodic array o slits in a thick metallic ilm, Opt. Express 13(1), (5). 46. D. Paciici, H. J. Lezec, H. A. Atwater, and J. Weiner, Quantitative Determination o Optical Transmission through Subwavelength Slit Arrays in Ag ilms: The Essential role o Surace Wave Intererence and Local Coupling between Adjacent Slits, Phys. Rev. B 77(11), (8). 47. R. W. Boyd, Nonlinear Optics, (Academic Press, New York, 3) 48. V. Roppo, F. Raineri, C. Cojocaru, R. Raj, J. Trull, I. Sagnes, R. Vilaseca, and M. Scalora, Generation eiciency o the Second Harmonic Inhomogeneous Component, arxiv: v1, (1). 49. R. S. Bennink, Y. K. Yoon, R. W. Boyd, and J. E. Sipe, Accessing the optical nonlinearity o metals with metaldielectric photonic bandgap structures, Opt. Lett. 4(), (1999). 5. J. Olivares, J. Requejo-Isidro, R. del Coso, R. de Nalda, J. Solis, C. N. Aonso, A. L. Stepanov, D. Hole, P. D. Townsend, and A. Naudon, Large enhancement o the third-order optical susceptibility in Cu-silica composites produced by low-energy high-current ion implantation, J. Appl. Phys. 9(), 164 (1). 51. J. M. Ballesteros, and R. Serna, J. Soli's, C. N. Aonso, A. K. Petord-Long, D. H. Osborne, and R. F. Haglund, Pulsed laser deposition o Cu:AlO3 nanocrystal thin ilms with high third-order optical susceptibility, Appl. Phys. Lett. 71, 445 (1997). 1. Introduction Since the irst observation o enhanced optical transmission (EOT) [1] eorts have multiplied to prove that the conditions or EOT coincide with strong ield localization on the metal surace and in proximity o the apertures [ 4]. Second harmonic generation (SHG) has been observed or a single aperture surrounded by grooves [5], and or arrays o sub-wavelength holes o dierent shapes [6] arranged in either periodic or irregular patterns [5 9]. Third harmonic generation (THG) has been demonstrated or a gold ilm patterned with nano-holes [1]. Metals like silver are centrosymmetric and lack a second order nonlinear term. However, they possess a relatively ast third order nonlinear response that may be among the largest o any known material (χ (3) ~ esu) that arises rom smearing o the Fermi surace, and slower thermal contributions [11,1], and is certain to dominate third order processes. SHG arises rom a combination o symmetry breaking at the surace and rom volume contributions, in part due to the magnetic Lorentz orce, and calculations are perormed by introducing eective surace and volume sources each having suitable weight [13 16]. Another relevant eature in harmonic generation rom sub-wavelength patterned metal is the nature and topology o the apertures. Field localization and harmonic generation are signiicantly dierent whether (slits, annular structures) or not (holes) resonant modes are excited inside the aperture. The ability o slits and annular structures to support TEM-like resonant modes [17 19] indeed allows more opportunities to eiciently generate harmonic ields when the apertures are illed with nonlinear materials like LiNbO 3 or GaAs [ ]. This notwithstanding, a number o important aspects are ignored in theoretical descriptions: (1) detailed dynamical contributions o the metal to SHG and THG rom electron gas pressure, convection, inner core electrons and a χ (3) response; () harmonic generation due to # $15. USD Received Dec 1; accepted 1 Jan 11; published 19 Jan 11 (C) 11 OSA 31 January 11 / Vol. 19, No. 3 / OPTICS EXPRESS 66

4 symmetry breaking and magnetic orces at work in GaAs; (3) phase-locking between the pump and the harmonics that allows generation in wavelength ranges below the absorption edge; (4) spectral shits due to electron gas pressure and third order processes in metal and semiconductor sections o the grating. These elements complicate the theoretical picture, and have not been investigated in this context. At the same time, these aspects have the potential to make these structures unctional in regimes where absorption is substantial and always decisive or many potential applications. Our study o harmonic generation unolds without imposing any separation between surace and volume sources by adopting the hydrodynamic model [3 7] to describe ree (conduction) electrons in the metal, by making no a priori assumptions about charge or current distributions, and by including Coulomb (electric), Lorentz (magnetic), convective, electron gas pressure and linear and nonlinear contributions to the dielectric constant o the metal arising rom inner core electrons, as outlined in reerence [8]. When harmonic generation is tackled in plasmonic contexts there is an understandable but critical tendency to simpliy the approach by ocusing on the nonlinear proprieties o the material that ills the cavity, and by ignoring any role the metal may play other than being a mere vessel [ ]. Indeed surace sources and magnetic orces that drive bound electrons in the active material (e.g. GaAs) are routinely overlooked, along with electron gas pressure and harmonic generation that arises rom the surrounding metal walls. Yet, these same elements can play a catalytic role by activating new interaction channels and should be investigated. Another ingredient that emerges as pivotal at wavelengths below the absorption edge is a phase-locking mechanism that dominates harmonic generation in the phase-mismatched regime, and renders materials transparent at the harmonic wavelengths. In bulk materials this is exempliied by the existence o a double-peaked generated SH signal. The evidence or this phenomenon, which is produced by the homogeneous and inhomogeneous solutions o the wave equation [9 3], can be ound in experimental works carried out in bulk media, where large phase and group velocity mismatches between the undamental ield (FF) and the SH waves allows the observation o two distinct SH pulses traveling at dierent phase and group velocities [33 35]. The homogeneous solution propagates with the phase and group velocity dictated by material dispersion, and walks o rom the pump. The inhomogeneous solution is trapped by the pump and is impressed with the pump's phase and group velocity. This peculiar behavior corresponds to a phase-locking mechanism that occurs in negative index [35,36] and absorbing materials [37,38] as the FF co-propagates with harmonics tuned below the absorption edge [34 38]. The eect persists in a GaAs cavity, with improvements predicted and observed or SH (61nm) and TH (48nm) eiciencies [38]. There is evidence that phaselocking and induced transparency also occur in ranges where the dielectric constant o materials like GaP displays metallic behavior (3nm) [39]. In what ollows we tune the pump at 164nm, where GaAs is transparent, so that SH and TH all in ranges where absorption is dominant: the components that survive experience dramatic enhancement o conversion eiciencies.. Linear response o a silver metal grating illed with GaAs We irst examined the properties o a single slit o size a illed with GaAs carved on an otherwise smooth silver [4] layer having thickness w (Fig. 1(a)). At the FF ε GaAs (164nm) ~1.1. In the absorbing region ε GaAs (53nm) ~ i.86 and ε GaAs (354nm)~ i We optimized the linear transmission using incident TM-polarized light (H-ield points into the page x-axis in Fig. 1(a)). The slit supports TEM-like modes that exhibit ield intensities more than 1 times larger than input intensities [41,4]. We varied silver ilm thickness and aperture size and obtained a transmission map that reveals the resonant nature o the structure (Fig. 1(b)). Simulations were then carried out on an array o slits 6nm wide on a 1nmthick ilm. The periodicity p o the array was varied rom nm to 3nm. In Fig. (a) we show the transmission or TM- (red line square markers) and TE-polarizations (blue line # $15. USD Received Dec 1; accepted 1 Jan 11; published 19 Jan 11 (C) 11 OSA 31 January 11 / Vol. 19, No. 3 / OPTICS EXPRESS 67

5 Transmission TE [%] Transmission TM [%] Total Transmission [%] Total Transmission [%] circle markers), normalized with respect to the energy that impinges on the geometrical area o the slits. In Fig. (b) we report the total transmittance as a unction o wavelength or p = 54nm, near the transmission maximum. An EOT o ~8% turns into a total transmission o 3% with a ~1nm-wide Fabry-Perot (FP) resonance centered at 164nm. The second peak near 35nm is due mostly to the intrinsic transparency o silver. Plasmonic eatures (gap and resonance near 55nm) are scarcely visible and have little impact on harmonic generation compared to readily available and much more prominent cavity modes. H TM GaAs Ag E TM E TE S TM H TE a y S TE y y z zx (a) w (a) (b) (b) Fig. 1. (a) Sketch o a single slit o size a illed with GaAs and milled in a silver ilm o thickness w; (b) Transmission map at λ = 164nm or a single slit on a silver substrate. We assume the ields are incident normal to the grating TM TE Wavelength [nm] Pitch Size [nm] (a) Wavelength [nm] Fig.. (a) Transmission versus pitch size p at 164nm or TM (red line square markers, right axis) and TE (blue line circle markers, let axis) polarizations. Transmission minima occur when p matches a multiple o the plasmon wavelength; (b) Transmission vs. wavelength when p = 54nm. Inset: plasmonic gap and resonances. I the periodicity is chosen to be a multiple o the surace plasmon wavelength o the unperturbed dielectric/metal interace then a plasmonic band gap appears [43 46]. Slits have no cut-o or TM-polarized light, and a TEM-like resonant mode is always available, e.g. Figure 1(b). The intererence o cavity and surace modes modulates the linear transmission proile Fig. (a) and opens a gap when the bare surace plasmon wavelength matches array periodicity. Transmission values or an incident, TE-polarized pump ield Fig. (a) are well below 1% or large periodicities, and approach 1% when slit-to-slit distance is small. The (b) # $15. USD Received Dec 1; accepted 1 Jan 11; published 19 Jan 11 (C) 11 OSA 31 January 11 / Vol. 19, No. 3 / OPTICS EXPRESS 68

6 reason or the large dierence between the two polarizations is due to the act that resonant FP modes are not accessible to TE-polarized light, which at 164nm are well below cut-o. 3. Linear and nonlinear models or Silver and GaAs We now illustrate linear and nonlinear dynamics o a sub-wavelength patterned silver ilm illed with GaAs in the pulsed regime, and identiy the origins o the generated signals. The model is outlined in details elsewhere [8], and here we summarize the most salient points. The hydrodynamic model describes the metal, and includes electric and magnetic orces, convection and electron gas pressure. We study a wavelength range where interband transitions are important (visible, UV) and use a Drude-Lorentz model to account or core electron contributions to the linear dielectric constant and to harmonic generation. GaAs is modeled as nonlinear Lorentz oscillators with absorption resonances at ~4nm and ~68nm [4], and all electrons are under the inluence o electric and magnetic orces: we account or surace and volume contributions in all sections o the grating. Both silver and GaAs are assigned χ () (zero or the centrosymmetric metal) and χ (3) tensors typical o their crystallographic groups. Conduction electrons in the metal are described as ollows [8]: n e, e e 1 m c m c m c n e, P P E E P P H P P P P 5E 1 E / m c n e F P P P F 3m c 9 (1) In (Eq. (1) e, m, and n, are the electron's charge, eective mass, and density in the conduction band, E F is the Fermi energy, and c is the speed o light in vacuum. For good conductors the ratio E / m c is between 1 5 and 1 3, depending on eective masses, densities, and Fermi velocities. We choose E F F / m c 1. The Lorentz orce, P H, is 4 accompanied by a quadrupole-like Coulomb term proportional to ~ P and E P, convective terms P P P P, and linear and nonlinear pressure terms proportional to P P, respectively. As we will see later, in nanocavity environments the linear pressure term can shit the band structures by tens o nanometers. I we assume that the ields and their respective polarizations and magnetizations may be decomposed as a superposition o harmonics, (Eq. (1) then represents a set o three coupled equations [8]. We distinguish between TE- and TM-polarized ields and deine: it it it it 3 3it 3 3it ie e E e E e E TEx TEx TEx TEx e E e TEx ETEx e E x it it it it 3 3it 3 3it E Ey je e TMy ETMy e E e TMy ETMy e E e TMy ETMy e (a) E i t z it i t i t 3 3i t 3 3i t k E e TMz ETMz e E e TMz ETMz e E e TMz ETMz e j it it it it 3 3it 3 3it i H e H e H e H e H e H e TMx TMx TMx TMx TMx TMx H x it it it it 3 3it 3 3it H H H e y TEy HTEy e H e TEy H TEy e H e TEy H TEy e (b) H i t z it i t i t 3 3i t 3 3i t k H e TEz HTEz e H e TEz H TEz e H e TEz H TEz e The polarization directions are noted in Fig. 1(a). For TE-polarization the H-ields point along the y- and z- directions; or TM-polarization the E-ield has components along y and z. The oscillator model is exempliied by the ollowing scaled polarization equations that # $15. USD Received Dec 1; accepted 1 Jan 11; published 19 Jan 11 (C) 11 OSA 31 January 11 / Vol. 19, No. 3 / OPTICS EXPRESS 69

7 describe generic, rapidly varying ield envelope unctions valid in the pump depletion regime [8]: 1 E P b, P ip,, H b b b, n e E P,, e e P ip H b b, b P P P E b, b, b,, b, b, (3a) m c m c m c b b E P,, b,3 b i P P H b b P 3iP b,3 b,3 H E P 3 b, E P b, P ip,, H b b n e e 1, b e P P P E E P P ip H m c m c 3 m c b b b b b b b b b,,,,,,,3,, 3 b b b 3i 3 P P H E P b,3,3 3 b, 1 n e e E P i, b b, e P P H P P P E b,3 b,3 b,3, b,3 b,3 3 m c m c m c i P P H E P The scaled coeicients are b, b, b b b b, b, b, b, N b Ni (3b) (3c), ( ( N) i N) ; N is an, N, b b integer that denotes the given harmonic order; the subscript b stands or bound. For urther details about the model and the integration scheme employed we direct the reader to reerence [8]. Bulk second and third order nonlinearities may be introduced directly into each o (Eqs. () (3) 3 (3), or by deining a nonlinear polarization in the usual way, i.e. PNL E E..., as we do. The second order polarization vector o GaAs may be written as ollows [47]: () P x EE y z () PNL, y d 14 Ex Ez (4) () P EE z x y Substituting the E-ield vector deined in (Eq. () into (Eq. (4) leads to the ollowing equations: 3 3 it ETMy ETMz ETMy ETMz ETMy ETMz ETMy ETMz e () 3 3 it PNL, x d14 ETMy ETMz ETMy ETMz ETMy ETMz e (5a) 3it ETMyETMz ETMy ETMz e E E E E E E E E e () 3 3 it PNL, y =d14 ETEx ETMz ETEx ETMz ETEx ETMz e 3it ETExETMz ETEx ETMz e 3 3 it TEx TMz TEx TMz TEx TMz TEx TMz (5b) # $15. USD Received Dec 1; accepted 1 Jan 11; published 19 Jan 11 (C) 11 OSA 31 January 11 / Vol. 19, No. 3 / OPTICS EXPRESS 7

8 E E E E E E E E e () 3 3 it PNL, z d14 ETEx ETMy ETEx ETMy ETEx ETMy e 3it ETExETMy ETEx ETMy e 3 3 it TEx TMy TEx TMy TEx TMy TEx TMy (5c) A cursory inspection o (Eqs. (5) might lead one to conclude that i a TM-polarized ield were incident on GaAs the only surviving source term would be a TE-polarized SH signal, namely i t E E i e in (Eq. (5a). Fortunately, the picture is ar more interesting than it appears at TMy TMz irst sight. Nonlinear source terms in both (Eqs. (1) and (3)), i.e. ree and bound charges in the metal and bound charges in GaAs (via derivatives o the type PbN, and magnetic terms) give rise to TM-polarized harmonics. A more careul analysis o (Eqs. (5) then reveals that the TM-polarized pump and its harmonics serve as nonlinear sources or all TE-polarized ields, including the pump. The production o a TE-polarized pump ield does not require the presence o the metal, and initiates with the mere introduction o the magnetic Lorentz orce in (Eqs. (3). Once TE-polarized ields are generated, all interaction channels become active and the generation o all harmonic ields is assured. However, the generation o downconverted, orthogonally polarized pump photons depends on the structure o the χ () tensor. For example, a χ () tensor whose only non-zero components are d 11, d, and d 33 cannot couple TM- to TE-polarized pump photons. We will return to this issue below. The description o χ (3) contributions may begin with the general expansion o the third order polarization as ollows [47]: P E E E (6) (3) i i, j, k, l j k l j1,3 k1,3 l1,3 For a material like GaAs having cubic symmetry o the type 43m, (Eq. (6) reduces to: P E 3 E E 3 E E (3) (3) 3 (3) (3) x x x x x x x x y y y x x x z z z x P E 3 E E 3 E E (3) (3) 3 (3) (3) y y y y y y x x y y x y y y z z z y P E 3 E E 3 E E (3) (3) 3 (3) (3) z z z z z z z z x x x z z z y y y z For the metal the situation is similar, except that or isotropic crystal symmetry the relations between the tensor components allow one to write: P E E E E E (3) (3) 3 x Ag x y x z x P E E E E E (3) (3) 3 y Ag y x y z y P E E E E E (3) (3) 3 z Ag z x z y z It is evident that substitution o (Eqs. () or the electric ield into (Eqs. (7-8) leads to χ (3) contributions to all harmonic components, with sel- and cross-phase modulation o the ields along with terms that couple orthogonal polarization states. The introduction o phase modulation eects on the pump ields is important, especially in metals [11,1], because given the right combination o intense ields and large χ (3) band shits may be substantial. 4. Nonlinear results or Silver gratings illed with GaAs Whether it is due to vertical or horizontal resonances or a combination o both, EOT at near- IR, visible and UV wavelengths is characterized by ield localization, absorption, and penetration inside the metal because in these ranges metals display dielectric constants o (7) (8) # $15. USD Received Dec 1; accepted 1 Jan 11; published 19 Jan 11 (C) 11 OSA 31 January 11 / Vol. 19, No. 3 / OPTICS EXPRESS 71

9 Conversion Eiciency Conversion Eiciency Conversion Eiciency Conversion Eiciency order unity. The interaction o light with ree and bound electrons in metals becomes more eicient especially i the light is concentrated in small volumes. We consider the same system described in Figs. 1 and. Incident pulses are 1s with peak intensities ~GW/cm. Pulse duration is such that the FP resonance is nearly resolved. To avoid conusion we temporarily introduce bulk quadratic and cubic nonlinearities only in GaAs. Later we will relax this condition. For illustration purposes we choose χ () = d 14 = d 5 = d 36 = 1pm/V, (3) (3) (3) (3) (3) (3) (3) (3) (3) 19 and (m /V ). An x x x x y y y y z z z z x x y y x x z z y y z z z z y y z z x x x x y y incident TM-polarized ield generates at least our harmonic ields: TM-polarized SH and TH (Figs. 3(a)-3(b)), and TE-polarized SH and TH (Figs. 4(a)-4(b)). I these results are examined together with Fig. (a) they reveal that the nonlinear response is inluenced by the linear properties: all the generated harmonics experience the same orbidden states o the incident pump ield. 1.6x1-9 1.x1-9 TM-polarized SHG T R (a) 4x1-7 3x1-7 TM-polarized THG T R (b).8x1-9 x1-7.4x1-9 1x1-7.x x Pitch Size (nm) Pitch Size (nm) Fig. 3. TM-polarized (a) SH and (b) TH transmitted (blue line circle markers), relected (red line square markers) conversion eiciencies. SH energies are comparable or p>5nm and p<5nm. However, or p>5nm SH light immediately leaves the grating; or p<5nm the ields linger near the grating and are re-absorbed by the metal. TE-polarized SHG T R (a) 8x1-15 6x1-15 TE-polarized THG T R (b) 1.x1-9 4x1-15.5x1-9 x Pitch Size (nm) Pitch Size (nm) Fig. 4. TE-polarized (a) second and (b) third harmonic transmitted (blue line circle markers), relected (red line square markers) total conversion eiciencies. The account o the dynamics that we provided in Figs. 3 or TM-polarized SH ields also applies to the TE-polarized SH signal. This situation occurs because the harmonics operate at twice and three times the requency o the pump, but experience the same index o reraction as the pump. Once the ields are generated and leave the slit's proximity, where coupling is largest, the harmonics are # $15. USD Received Dec 1; accepted 1 Jan 11; published 19 Jan 11 (C) 11 OSA 31 January 11 / Vol. 19, No. 3 / OPTICS EXPRESS 7

10 inluenced and constrained by the size o the wavelength relative to the grating periodicity. For example, the TM-polarized SH is inhibited beginning when p matches the unperturbed air/silver plasmon wavelength o the pump and SH. The same phenomenon is evident also or THG with appropriate pitch values. Although the choice χ () = 1pm/V in GaAs leads to predicted TE- and TM-polarized SH eiciencies having similar values, eiciencies or TM-polarized SHG [] have not been reported. The eiciency that we predict or the SH TM-polarized signal (which is independent o GaAs and arises mostly rom the metal) is nearly 1 times larger than SHG rom smooth metal layers, and 1 times larger than SHG rom metal-dielectric stacks [8]. The lack o plasmonic resonances at the SH and TH wavelengths (see Fig. (b)) suggests that this behavior depends almost exclusively on (FP) cavity-q [41], ield overlap, and phase-locking. 5. Spectral eatures and ield proiles We now examine the individual eatures o the harmonic signals emitted by the grating, including spectra and ield proiles. We choose pump pulses having peak intensity GW/cm and 5s duration, incident on a grating with p = 59nm. This pitch optimizes SHG o both polarizations and places plasmonic eatures away rom all wavelengths. For GaAs we choose χ () = 1pm/V and χ (3) = 1 18 m /V, consistent with more recent experimental observations o harmonic generation in similar wavelength ranges [37,38]. In Figs. 5-6 we show relected and transmitted spectra or all generated ields. We note that the spectra display a small shit between transmission and relection maxima but have similar amplitudes. In Fig. 6 we show the spectra or TE-polarized pump photons. The conversion eiciencies that we predict are already comparable to TE-polarized, TH conversion eiciencies. Albeit relatively small, the eiciency o this novel down-conversion process may be enhanced in bulk GaAs or by pumping the grating with TM-polarized SH and TH seed ields, as (Eq. (5a) suggests, via the 3 3 i t TMy TMz TMy TMz term: i E E E E e. In Fig. 7 we show snapshots o the corresponding ield proiles inside the nanocavity when the peak o the pulse reaches the grating (see ields dynamics in Media 1). All ields are well-localized inside the cavity, including the TH tuned at 354nm. Similar resonant behavior was obtained or planar cavities with harmonic ields tuned below the absorption edge [38,48]. One o our objectives is to also show that phase-locking [9 35] is playing a non trivial role. A smooth, 1nm-thick GaAs layer is only % transparent at 53nm, and completely opaque at 354nm. In a multi-pass geometry or a resonant cavity environment [48] the homogenous portion o the SH signal is removed more eiciently compared to bulk, so that all generated components that survive in the nonlinear medium propagate under phase-locking conditions. In our case Fig. (b) linear transmission is less than.5% at 53nm. At 354nm the grating is a bit transmissive thanks to the natural transparency o silver, but GaAs remains completely opaque. # $15. USD Received Dec 1; accepted 1 Jan 11; published 19 Jan 11 (C) 11 OSA 31 January 11 / Vol. 19, No. 3 / OPTICS EXPRESS 73

11 Conversion Eiciency Transmitted TM-polarized Pump Conversion Eiciency Conversion Eiciency Conversion Eiciency Conversion Eiciency 6x1-1 (a) TM-SH Transm itted R elected 1.x1-5 (b) TM-TH Transm itted R elected 4x1-1.8x1-5 x1-1.4x Wavelength (nm ) Wavelength (nm ).5x1-9.x1-9 (c) TE-SH Transm itted R elected 5x1-15 4x1-15 (d) TE-TH Transm itted R elected 1.5x1-9 3x x1-9 x1-15.5x1-9 1x Wavelength (nm ) Wavelength (nm ) Fig. 5. TM-polarized SH (a) and TH (b) transmitted and relected conversion eiciency spectra, normalized with respect to the spectrum o the transmitted pump ield. (c) and (d): same as (a) and (b) but or TE-polarized ields. Predicted conversion eiciency o the TMpolarized TH signal is remarkably high. 1.x1-15.8x1-15 TE-Transmitted TE-Relected TM-pump x x x Wavelength (nm) Fig. 6. Spectra o the transmitted, TM-polarized pump ield (black line triangle markers), and TE-polarized, transmitted (blue line circle markers) and relected (red line square markers) down-converted pump photons. Conversion eiciencies are relatively small, but are nevertheless similar to those o TE-polarized THG. # $15. USD Received Dec 1; accepted 1 Jan 11; published 19 Jan 11 (C) 11 OSA 31 January 11 / Vol. 19, No. 3 / OPTICS EXPRESS 74

12 Fig. 7. Pump and harmonic ield intensities inside and near the nano-cavities. The magnetic ield intensities are depicted or TM-polarization; the transverse electric ield is shown or TEpolarization. The pump magnetic ield intensity (a) is ampliied 5 times; the transverse pump electric ield (not shown) is ampliied by approximately two orders o magnitude. This combination gives way to TH conversion eiciencies that are unusually large (~1 5 ). Fields dynamics is shown in Media 1. More convincing numerical evidence o phase-locking may be achieved by increasing substrate thickness up to 17nm, and by reducing the width o the slits down to nm, so that we are still operating under resonant conditions (see Fig. 1(b)). As a result conversion eiciencies and localization properties vary little compared to Figs. 6-7, except or an increased number o longitudinal peaks, as the ields resonate inside the cavity even though all the TE-generated ields are ar below cut-o. This is a sure sign that phase-locking is the mechanism that drives the harmonic ields to resonate, despite the act that the cavity should resonate only at the pump requency [38,48]. 6. Nonlinear results or Silver grating illed with a diagonal nonlinear material We now show that it is possible to improve SH and TH conversion eiciencies i we assume the medium has a χ () tensor whose only non-zero components are d 11 = d = d 33, and i a χ (3) is triggered inside the metal. The second order polarization o Eq. (4) takes the ollowing orm: () P x Ex () PNL, y d 11 Ey. (9) () P z Ez Unlike Eqs. (5), Eq. (9) allows ull exploitation o transverse and longitudinal ield localization, since in a cavity environment orthogonally polarized ields may not have similar amplitudes and localization properties. The third order nonlinear coeicient in metal sections (3) (3) is chosen as in Eq. (8). For our calculations we use ~ 1, (3) 18 where GaAs 1 m / V, as beore. Our choice corresponds to (3) ~ esu, which (3) 6 is much smaller than what was reported or Cu ( Cu ~ 1 esu) [48 51] in a multilayer environment [11], but is in line with nonlinearities reported or silver particles. For illustration purposes we ill the cavity with a material that has the same index o reraction as GaAs, the χ () o Eq. (9), and we compare eiciencies with all else being equal. In Figs. 8 we depict the conversion eiciencies or TM-polarized harmonics (TE-polarized ields remain null) as a unction o pulse duration. The igures show that eiciencies increase as a unction o pulse Ag Ag GaAs # $15. USD Received Dec 1; accepted 1 Jan 11; published 19 Jan 11 (C) 11 OSA 31 January 11 / Vol. 19, No. 3 / OPTICS EXPRESS 75

13 SHG Converison Eiciency THG Conversion Eiciency width and saturate or pulses longer than ~6s. This behavior is typical o cavity phenomena [48]. The most important eatures in Figs. 8 are perhaps the total eiciencies, i.e. ~1 6 or SHG and ~1 3 or THG (see ields dynamics in Media ). By choosing a suitable material and by allowing a non-zero χ (3) in the metal results in exceptional conversion eiciencies in wavelength ranges that are usually deemed inaccessible, independent o phase matching conditions and absorption at the harmonic wavelengths. Finally in Fig. 9 we show typical ield proiles that correspond to Fig. 8. The igure should be compared directly with the corresponding intensities o Fig. 7. While in Fig. 7 most o the TM-polarized SH-signal came mostly rom the metal, now the SH ield originates mostly rom the bulk nonlinearities o GaAs. Nevertheless, in Fig. 9(b) one can still see remnants o the localization displayed in Fig. 7(b), since the metal remains an active participant. In contrast, THG originates at the walls o the slit: the ield spills into the cavity, becomes phaselocked and resonates with the pump, leading to large conversion eiciencies in the UV range..x x1-6 Total Transmitted Relected.4.3 Total Transmitted Relected 1.x1-6..5x Pulse Duration (s) Pulse Duration (s) Fig. 8. SH (a) and TH (b) conversion eiciencies vs. pulse duration. A diagonal χ () tensor boosts SHG by at least three orders o magnitude compared to GaAs thanks to the ull exploitation o ield localization properties. Allowing a non-zero χ (3) in the metal improves THG well in excess o one order o magnitude compared to GaAs alone. Fig. 9. Typical pump (a), SH (b) and TH (c) magnetic ield intensities when the peak o a 5s pulse reaches the grating. One should compare the ield with the corresponding harmonics in Fig. 7. While the entire metal surace has a non-zero χ (3), only nonlinear sources on the metal walls inside the cavity matter to the process. Fields dynamics is shown in Media. # $15. USD Received Dec 1; accepted 1 Jan 11; published 19 Jan 11 (C) 11 OSA 31 January 11 / Vol. 19, No. 3 / OPTICS EXPRESS 76

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