Near-Þeld localization of ultrashort optical pulses in transverse 1-D periodic nanostructures

We present a transverse 1-D periodic nanostructure which exhibits lateral internal Þeld localization for normally incident ultrashort pulses, and which may be applied to the enhancement of nonlinear optical phenomena. The peak intensity of an optical pulse propagating in the nanostructure is approximately 12 times that of an identical incident pulse propagating in a bulk material of the same refractive index. For second harmonic generation, an overall enhancement factor of approximately 10.8 is predicted. Modeling of pulse propagation is performed using Fourier spectrum decomposition and Rigorous Coupled-Wave Analysis (RCWA). © 2000 Optical Society of America OCIS codes: (050.1970) diffractive optics; (190.4420) nonlinear optics, transverse effects in; (230.3990) microstructure devices; (999.9999) ultrashort pulses; (999.9999) optical nonlinearity enhancement References and links 1. P. Lalanne and J.-P. Hugonin, ÒHigh-order effective-medium theory of subwavelength gratings in classical mounting: application to volume holograms,Ó J. Opt. Soc. Am. A 15, 1843Ð1851 (1998). 2. J. N. Mait, D. W. Prather, and M. S. Mirotznik, ÒDesign of binary subwavelength diffractive lenses by use of zeroth-order effective-medium theory,Ó J. Opt. Soc. Am. A 16, 1157Ð1167 (1999). 3. F. Xu, R.-C. Tyan, P.-C. Sun, Y. Fainman, C.-C. Cheng, and A. Scherer, ÒForm-birefringent computer-generated holograms,Ó Opt. Lett. 21, 1513Ð1515 (1996). 4. R.-C. Tyan, A. A. Salvekar, H.-P. Chou, C.-C. Cheng, A. Scherer, P.-C. Sun, F. Xu, and Y. Fainman, ÒDesign, fabrication and characterization of form-birefringent multilayer polarizing beam splitter,Ó J. Opt. Soc. Am. A 14, 1627Ð1636 (1997). 5. J. E. Sipe and R. W. Boyd, ÒNonlinear susceptibility of composite optical materials in the Maxwell Garnett model,Ó Phys. Rev. A 46, 1614Ð1629 (1992). 6. R. W. Boyd and J. E. Sipe, ÒNonlinear optical susceptibilities of layered composite materials,Ó J. Opt. Soc. Am. B 11, 297Ð303 (1994). 7. G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, and L. A Weller-Brophy, ÒEnhanced nonlinear optical response of composite materials,Ó Phys. Rev. Lett. 74, 1871Ð1874 (1995). 8. R. S. Bennink, Y.-K. Yoon, R. W. Boyd, and J. E. Sipe, ÒAccessing the optical nonlinearity of metals with metaldielectric photonic bandgap structures,Ó Opt. Lett. 24, 1416Ð1418 (1999). 9. K. P. Yuen, M. F. Law, K. W. Yu, and P. Sheng, ÒEnhancement of optical nonlinearity through anisotropic microstructures,Ó Opt. Comm. 148, 197Ð207 (1998). 10. H. Ma, R. Xiao, and P. Sheng, ÒThird-order optical nonlinearity enhancement through composite microstructures,Ó J. Opt. Soc. Am. B 15, 1022Ð1029 (1998). 11. M. G. Moharam and T. K. Gaylord, ÒDiffraction analysis of dielectric surface-relief gratings,Ó J. Opt. Soc. Am. 72, 1385Ð1392 (1982) 12. N. Chateau and J.-P. Hugonin, ÒAlgorithm for the rigorous coupled-wave analysis of grating diffraction,Ó J. Opt. Soc. Am. A 11, 1321Ð1331 (1994). 13. M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, ÒStable implementation of the rigorous coupledwave analysis for surface-relief gratings: enhanced transmittance matrix approach,Ó J. Opt. Soc. Am. A 12, 1077Ð 1086 (1995). 14. L. Li, ÒFormulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,Ó J. Opt. Soc. Am. A 13, 1024Ð1035 (1996). 15. M. Schmitz, R. Bruer, O. Bryngdahl, ÒComment on numerical stability of rigorous differential methods of diffraction,Ó Opt. Comm. 124, 1Ð8 (1996). #22950 $15.00 US Received June 02, 2000; Revised July 26, 2000 (C) 2000 OSA 31 July 2000 / Vol. 7, No. 3 / OPTICS EXPRESS 123 16. R. Tyan, ÒDesign, modeling and characterization of multifunctional diffractive optical elements,Ó Ph.D. Thesis, University of California, San Diego (1998). 17. W. Nakagawa, R.-C. Tyan, P.-C. Sun, F. Xu, and Y. Fainman, ÒUltrashort pulse propagation in near-Þeld periodic diffractive structures using Rigorous Coupled-Wave Analysis,Ó submitted to J. Opt. Soc. Am. A (2000).


Introduction
Near-Þeld effects in subwavelength diffractive optical structures (optical nanostructures) have played an important role in a variety of optics and photonics technologies, including artiÞcial materials [1,2], polarization-selective devices based on form birefringence [3,4] and nonlinearity enhancement in composite materials [5Ð10].In each of these cases, the unique optical properties of the device derive from the strong inßuence of the nanoscale structure on the optical near Þelds.For a variety of applications, it is desirable to employ these near-Þeld nanostructures in conjunction with ultrashort optical pulsesÑfor example, in integrated optical communications systems to increase data rates or to enable all-optical switching through nonlinear optical effects.However, to the best of our knowledge, the interaction of ultrashort pulses and near-Þeld optical nanostructures has not yet been thoroughly investigated.
In this manuscript, we apply our recently-developed tool for the analysis of ultrashort pulse propagation in periodic diffractive nanostructures to study an important near-Þeld effect: lateral localization of ultrashort optical pulses propagating in a transverse 1-D periodic nanostructure.Ultrashort optical pulses are widely used in the study of nonlinear optical phenomena due to their extremely high peak powerÑsince the output of a nonlinear optical process scales as the square or higher power of the input, temporal localization of the pulse energy results in an overall enhancement of the nonlinear effect.Similarly, transverse localization of the pulse energy in a nanostructure will lead to a further increase in the peak power of the pulse, and consequently further enhancement of the nonlinear effect.We present a design for a subwavelength 1-D periodic nanostructure which exhibits strong transverse Þeld localization in the high index region for a TE-polarized incident optical pulse.
In order to investigate ultrashort laser pulse propagation in optical nanostructures, we have developed a modeling tool to analyze the interaction of ultrashort optical pulses with periodic subwavelength diffractive structures based on the integration of two proven analysis methods: Fourier spectrum decomposition and the Rigorous Coupled-Wave Analysis method (RCWA) [11].In Section 2, we brießy describe this extension of the RCWA method to model ultrashort pulse propagation.In Section 3, we describe the transverse Þeld localization structure, and in Section 4, we discuss its application to the enhancement of nonlinear optical phenomena.A summary of the results and conclusions are presented in Section 5.

Modeling method
The Rigorous Coupled-Wave Analysis [11] method is a well-established tool for characterization of monochromatic wave propagation through periodic diffractive nanostructures.However, since an ultrashort pulse contains a broad optical frequency spectrum, it is necessary to extend the existing RCWA method to analyze polychromatic Þelds.We consider the propagation of an inÞnite periodic sequence of Gaussian ultrashort pulses and use Fourier methods to obtain the corresponding spectrum, consisting of a number of discrete frequency components.These discrete components are independently analyzed using a modiÞed RCWA technique (various modiÞcations have been made to the original RCWA method [11] to facilitate stable numerical solution [12Ð15] and explicit computation of the internal Þelds of the nanostructure [16,17]).The resulting monochromatic diffracted Þelds are then coherently superimposed to produce the time-domain solution, revealing the interaction between the ultrashort pulses and the periodic diffractive nanostructure.
We assume an incident inÞnite periodic train of Gaussian pulses as shown in Eq. ( 1), corresponding to the output of typical ultrashort pulse laser systems: (1) where and are space and time coordinates, respectively, indicates the polarization of the incident pulse, is the center frequency of the pulse, is the wave vector corresponding to the center frequency of the pulse, is a unit vector in the direction of , is the group velocity of the pulse, is the width parameter of the Gaussian pulse envelope, is the time at which the pulse peak arrives at , is the temporal separation between pulses in the incident pulse train, indicates the convolution operation, and is an integer.Taking the Fourier transform of Eq. ( 1) and imposing a Þnite truncated bandwidth centered at , we obtain , where and .Eq. ( 2) is a Þnite, discrete frequency-domain representation of the incident Þelds having discrete frequency components over index n (where ) at frequencies .Thus, for each component, the RCWA method can be applied to solve for the diffracted Þelds.
For each frequency component, the RCWA method yields the resulting reßected, transmitted, or internal Þelds of the grating.To obtain the time domain solution for the Þelds, we must coherently superpose the Þelds corresponding to each frequency component.This superposition corresponds to the inverse of the Fourier transform performed in obtaining Eq. (2) from Eq. (1).Since the frequency-domain representation of the Þelds in Eq. ( 2) is discrete, we apply a discrete inverse Fourier transform, yielding the Þeld at a particular point in space and time.In practice, the superposition is carried out for an array of points, and the results introduced into a variety of visualization tools for analysis and interpretation.
In the following example, we assume a typical femtosecond laser pulse of width 167 fs FWHM, corresponding to a Gaussian width parameter of sec.The temporal separation between pulses in the incident pulse train is assumed to be ps, corresponding to a frequency sampling interval of rad.We also choose a truncated bandwidth of rad, corresponding to discrete spectral components.Although a frequency-dependent material refractive index can easily be incorporated due to the spectral decomposition, in this manuscript the effects of material dispersion are omitted for simplicity.

Transverse Þeld localization
We apply this tool to analyze pulse propagation in a subwavelength 1-D periodic nanostructure which exhibits strong transverse Þeld localization within each period of the structure.For a center wavelength of µm, we have chosen the subwavelength 1-D grating to have a period of µm and a Þll factor of , as shown in Fig. 1.For clarity, the depth of the structure is chosen to be µm to avoid the introduction of interference effects in the propagation direction.The refractive indices of the grating materials are assumed to be in the high index region (corresponding to the properties of GaAs) and in the air gap.The incident pulse train is assumed to be normally incident.Figs.2a and 2b respectively show the propagation of TE-and TM-polarized 167 fs FWHM ultrashort optical pulses through the nanostructure described above.Note that although the incident pulses have uniform transverse proÞles, inside the grating structure the pulses exhibit strong transverse localization.In the TE case, the majority of the pulse energy is localized in the high refractive index region, while in the TM case the pulse energy is found in the air gap.
The peak value of the squared magnitude of the electric Þeld ( ) inside the nanostructure occurs for the TE polarization, and is approximately 2.4 times that of the incident pulse.Despite the localization inside the structure, the transmitted and reßected pulses have uniform transverse proÞles due to the subwavelength scale of the nanostructure.The TE-and TM-polarized pulses exhibit radically different transverse proÞles due to the differing boundary conditions imposed on the near Þelds by the nanostructure.For the TM polarization (i.e. with electric Þeld in the -direction as shown in Fig. 1), the boundary condition at the grating groove interfaces requiring continuity of the normal component of the electric displacement results in a stronger electric Þeld in the low index region of the grating.Since the grating has a subwavelength period, this effect results in transverse localization of most of the pulse energy in the low refractive index material, with the highest Þeld magnitude being observed nearest the interfaces.Alternatively, for the TE polarization (i.e. with electric Þeld in the -direction as shown in Fig. 1), the boundary conditions require continuity of the tangential electric Þelds, imposing no particular transverse proÞle on the Þeld.However, the mode structure of the coupled waveguide array results in transverse localization of the Þeld energy in the high refractive index material, in a similar fashion to the mode proÞle of a singlemode slab waveguide.Since most commonly used bulk nonlinear optical materials tend to have relatively high indices of refraction, it is the TE polarization caseÑwhere the pulse energy is localized in the high refractive index region of the gratingÑthat is of interest.

Enhancement of nonlinear optical phenomena
In order to apply the transverse localization effect to the enhancement of nonlinear optical phenomena, we must Þrst Þnd the optimum transverse Þeld proÞle.Due to the near-Þeld nature of the transverse localization effect, we expect the degree of Þeld localization to be extremely sensitive to the dimensions of the structure.Fig. 3 shows the transverse proÞle of at the pulse peak for Þve nanostructures having the same period ( µm) but differing Þll factors: 1%, 3%, 6%, 9%, and 12%, as well as the bulk case (100%).For very small Þll factors (e.g. the 1% case), the Þelds cannot vary signiÞcantly on a substantially subwavelength scale, resulting in a nearly uniform transverse Þeld proÞle (the width of the high refractive index region is for 1% Þll factor).As the Þll factor increases, the localization effect strengthens, reaching its maximum value of approximately 2.5 times that of the incident pulse at a Þll factor of 6%.As the Þll factor continues to increase, however, the increasing volume fraction of the high refractive index material results in a diminishing peak .The peak values for Þll factors varying from 1% to 12% are shown in Fig. 4. For Þll factors larger than 12%, multiple transverse modes exist, signiÞcantly reducing the peak .For the 9% Þll factor case (corresponding to the results of Fig. 2), the peak value of is approximately 2.4 times that of the incident pulse, and over 12 times that of the bulk material case.Fig. 4 shows the pulse group velocity inside the nanostructure as a function of the Þll factor.As the Þll factor increases from 1% to 12%, the group velocity decreases from almost to roughly , where is the speed of light.Thus, as the fraction of the pulse energy contained within the high refractive index material increases, the group velocity of the pulse in the nanostructure decreases.This behavior is similar to the dependence of the mode propagation speed on the guide thickness in a single-mode slab waveguide.In applying the transverse Þeld localization effect to the enhancement of nonlinear optical phenomena, we must also consider the volume fraction of nonlinear material.The intensity output of the second-harmonic generation (SHG) process is proportional to .By integrating across the fraction of the grating period corresponding to the high refractive index material and comparing with the bulk case, we can obtain an effective SHG enhancement factor.The effective SHG enhancement factor for Þll factors varying from 1% to 12% is shown in Fig. 5.A Þll factor , corresponding to the results of Figs. 1 and 2, yields the maximum value of the SHG enhancement factor: approximately 10.8.Fig. 5. SHG enhancement factor for the nanostructure of Fig. 1, with Þll factor varying from 1% to 12%.The enhancement factor is computed by integrating over the part of the grating period corresponding to the high refractive index material, and normalizing to the bulk case.

Conclusions
We have presented an example optical nanostructure which imposes transverse near-Þeld localization on incident ultrashort optical pulses.The elevation of the peak intensity of the pulse due to the transverse localization effect is used to complement the high peak power of an ultrashort optical pulse, with an application for enhancement of nonlinear optical phenomena.In addition, we brießy described the modeling method used to produce these results, consisting of an extension of the well-established RCWA method for the analysis of ultrashort optical pulses using Fourier spectrum decomposition.As this example suggests, interesting new phenomena and practical photonic devices may emerge from the combination of ultrashort optical pulses and near-Þeld diffractive nanostructures.

Fig. 2 .
Fig. 2. Normalized squared magnitude of the electric Þeld of an ultrashort optical pulse (center wavelength 1.0 µm, 167 fs FWHM) propagating inside the structure shown in Fig. 1.Each movie shows one period of an inÞnitely periodic structure for: (a) (2.5 MB) TE polarized incident pulse and (b) (2.0 MB) TM polarized incident pulse.

Fig. 3 .
Fig. 3. Transverse proÞles of the squared Þeld magnitude at the pulse peak in one period of an inÞnitely periodic nanostructure for several Þll factors.The grating period is 0.65 µm, and the normally incident pulse has a FWHM of 167 fs.The colored vertical lines indicate the respective boundaries of the high index region of the structure for the Þve fractional Þll factors.

Fig. 4 .
Fig. 4. Group velocity and peak Þeld magnitude squared as a function of Þll factor for a 167 fs FWHM pulse propagating in the structure of Fig. 1.