T Grosjean

M Mivelle

F I Baida

G W Burr

U C Fischer

Diabolo Nanoantenna for Enhancing and Confining the Magnetic Optical Field

 An incident wave generates on such structures an electromagnetic near-field which is highly localized in the vicinity of the metal structure. We address here the choice of shapes and arrangement that can create a highly confined and enhanced magnetic field in the optical frequency range on the background of a much lower electric field.

 The ability of some circular nanostructures to develop magnetic moments offers the possibility to locally enhance the magnetic optical near-fields. This was shown for example in the case of apertures of a diameter small compared to the vacuum wavelength in a infinitely conducting film 4-6 or in a real metal screen, 7 as well as in the case of metal structures such as rings, 8 tori, 9 cylinders, 10 and split ring resonators. 11

On the other hand, it is well-known that a large electric field enhancement is created in the air gap between two closely spaced metal nanoparticles, an effect which has long been exploited in the field of surface-enhanced Raman spectroscopy. 12 Recently the bowtie nanoantenna 13,14 and bowtie nanoaperture 15 were shown to be very efficient nanostructures for creating a highly confined and enhanced electric field at optical frequencies in the small gap between their two triangular metal elements. These results lead to a new structure for enhanced magnetic fields through Babinet's principle which relates the scattered electromagnetic field of an aperture in an infinitely thin ideally conducting film and the one of the complementary structure. In its rigorous form, the principle states that the scattered fields are the same if we exchange the electric field of one case with the magnetic field of opposite phase of the other case, for both the exciting and the scattered fields. [START_REF] Jackson | Classical Electrodynamics[END_REF] It is expected that Babinet's principle holds in an approximate way for real metal films of a finite thickness. In this paper, we examine by numerical calculations the near-field of the complementary bowtie nanoaperture, which we call the "diabolo nanoantenna". Our results show that this structure is capable of significantly enhancing and confining the optical magnetic field.

Principle. The geometry of the proposed 2D diabolo nanoantenna is shown in Figure 1a. Following Babinet's principle ABSTRACT: In this Letter, we introduce a new nanoantenna concept aimed at generating a single magnetic hot spot in the optical frequency range, thus confining and enhancing the magnetic optical field on the background of a much lower electric field. This nanoantenna, designed by applying Babinet's principle to the bowtie nanoaperture, takes the shape of a diabolo. It differs from the well-known bowtie nanoantenna in that the opposing pair of metal triangles are electrically connected through their facing tips. Thus instead of a large charge density accumulating at the air gap of the bowtie nanoantenna, leading to a large electric field, a high optical current density develops within the central "metal gap" of the diabolo nanoantenna, leading to a large magnetic field. Numerical simulation results on the first nanodiabolo geometries show a 2900-fold enhancement of the magnetic field at a wavelength of 2540 nm, confined to a 40-by-40 nm region near the center of the nanoantenna. KEYWORDS: Optical magnetic field, nanoantenna, plasmon resonance, polarization, electromagnetism applied to the bowtie nanoaperture, this structure consists of a pair of planar metal triangles joined together through their facing tips. The junction is considered to be a cubic metal particle (called "metal gap" in the following). A magnetic hot spot is generated at the metal gap when the incident electric field is linearly polarized along the axis of symmetry of the nanoantenna that crosses the pair of triangles (called y axis in the following).

The basic principle of the diabolo nanoantenna arises from the electrical connection of two facing metal triangles, which in the bowtie nanoantenna would be electrically isolated. The charge accumulation which occurs at the apexes of the two closely spaced metal triangles of the bowtie nanoantenna (see Figure 1b) is converted into a highly localized enhancement of the optical current density along the metal gap (Figure 1c). The strong capacitive effect which drove the bowtie nanoantenna and led to giant electric-field enhancement is changed to an inductive effect associated with a strong local enhancement of the magnetic field all around the metal gap. The metal triangles act as an electric funnel which reinforces the optical current density into the metal gap. Following Amp ere's law, the magnetic field should be azimuthally polarized around this central nanowire. Note that metal conductivity plays a crucial role in the performances of the diabolo nanoantenna since the enhanced magnetic field, which is continuous at the metal interface, can easily penetrate into the metal and lead to noticeable dissipation effects.

Simulation. As a first demonstration of the properties of the diabolo nanoantenna, the electromagnetic response of the structure is simulated using the FDTD method. [START_REF] Taflove | Computational Electrodynamics: The Finite-Difference Time-Domain Method[END_REF] The nanoantenna is considered to be deposited onto a dielectric substrate and illuminated with a plane wave propagating inside the substrate at normal incidence (Figure 2).The input electric field is linearly polarized along the y axis.

The model used for the simulations consists of a volume spanning (2 μminX and Y about the gap of the nanoantenna and -1t o2μmi nZ. The refraction index of the substrate is chosen to be equal to 1.5. A diabolo nanoantenna with 45°flare angles and a length D of 310 nm lies on the glass substrate. The electromagnetic response of the structure is studied for various metals (gold, aluminum, and silver), various sizes of the metal gap (G), and thicknesses of the nanoantenna (T). The permittivity of the metals is defined with a Drude model. Note that the height of the metal gap (along the z axis) is given by T whereas G defines both the width and the length of the square nanowire section. All six boundaries of the computation volume are terminated with Convolutional-PMLs 17 in order to avoid parasitic unphysical reflections around the structure. The nonuniform grid resolution varies from 25 nm for portions at the periphery of the simulation to 5 nm in the region of the nanoantenna. The results presented here show the enhancements of the electric and magnetic intensities due to the presence of the diabolo nanoantenna. These enhancements are the field intensities diffracted by the diabolo nanoantenna at a given plane (or position) normalized by the intensities calculated at the same plane (or position) without the presence of the metal structure.

Results and Discussions. As predicted, an enhanced magnetic field is highly confined all around the metal gap (Figure 3a,b) and is azimuthally polarized in the (XZ) plane (inset of Figure 3a). The enhancement of the magnetic intensity reaches a value of 230 at a distance of 10 nm far from the gap (Figure 3b) but becomes higher than 510 very close to the structure (Figure 3a). The enhancement of the electric intensity arises mainly at the end faces of the nanoantenna and is about 35 times smaller than the enhancement of the magnetic intensity at a distance of 10 nm beyond the metal gap. Figure 3e shows the normalized intensity of the magnetic field for the x polarization of the incident electric field as compared to the y polarization in Figure 3b for otherwise identical conditions. The hot spot at the center of the structure is canceled and the intensity maximum becomes 88 times smaller. These polarization properties allow the magnetic hot spot to be easily and rapidly tuned by rotating the input polarization direction.

Figure 4a reports the magnetic spectral responses of various diabolo nanoantennas of length D = 310 nm and with parameters G and T equal to 50 and 50 nm (solid curve), 50 and 20 nm (dashed curve), 20 and 50 nm (dotted curve), and 20 and 20 nm (dashed-dotted curve). Figure 4b shows the magnetic response of the structure made of gold (solid curve), aluminum (dashed curve), and silver (dotted curve) for G = T = 50 nm and D = 310 nm. In all cases, the nanoantennas are excited with a single temporal pulse described by a linearly polarized electric field along the y axis. The time-varying magnetic field is calculated at a point placed 10 nm far from the center of the metal gap, in air. The spectrum is calculated by a simple Fourier transform of this result and is normalized by the spectrum of the magnetic field at the same position without the metal layer.

These results clearly show that the magnetic hot spot is generated resonantly by the diabolo nanoantenna through the excitation of a plasmon mode. On the one hand, Figure 4a shows that the surface area of the cross section of the metal gap that is perpendicular to the current flow is the key geometrical parameter in the definition of the amplitude of the enhanced magnetic field. Any decrease in the surface area leads to an increase of the magnetic field and two geometries of the metal gap that lead to the same surface generate magnetic effects of the same level but not at the same wavelength (see dashed and dotted curves of Figure 4a). Thus, the maximum enhancement of the magnetic intensity, which is about 510 for a gold diabolo nanoantenna with G = T = 50 nm Figure (3a), becomes higher than 2900 when G = T = 20 nm (see Figure 3f). Note that the enhancement of the magnetic field in circular apertures small compared to the wavelength is instead a decreasing function of the aperture size. [START_REF] Bethe | [END_REF] On the other hand, an increase of metal conductivity induces stronger current density in the structure and therefore higher inductive effect in the gap zone. Figure 4b shows that the enhancement of the magnetic intensity is better for silver than for gold and aluminum, which have lower conductivities in the near-infrared region. When the cross section of the metal gap becomes smaller than twice the skin depth, resistivity increases and produces a higher electric field in and around (by continuity) the optical nanowire. We see in panels a and f of Figure 3 that the enhanced magnetic field does not show a constant amplitude all around the metal gap. The different dielectric constants at both sides of the diabolo nanoantenna (the structure lies on a glass substrate) and the asymmetric illumination can contribute to this phenomenon. However, there is no simple interpretation of the fact that the higher magnetic field is located at the glass/metal interface in Figure 3a and at the metal/air interface in Figure 3f, and this phenomenon will have to be investigated in more detail.

The underlying physics of the diabolo nanoantenna is that of a short-circuit between two oppositely charged metal bodies. Under illumination with an electric field polarized along the antenna y axis, charge accumulations of opposite signs arise at the end faces of the structure. These charge effects lead to high enhancements of the electric field on both sides of the diabolo nanoantenna, as shown in Figure 3d. The resulting potential difference between these two electrically connected metal structures generates currents which are guided across the metal gap. The double-funnel geometry of the diabolo nanoantenna concentrates and amplifies with high efficiency the current density in the metal gap. Following Amp ere's law, the resulting local increase of the current intensity leads to a strong enhancement of the surrounding magnetic field (as shown in Figure 3a,b). Note that the charge accumulations on the side edges of the structure, the currents into the metal gap, and the metal conductivity may find a direct relationship with the capacitance, inductance and resistance of an equivalent RLC circuit, respectively. Figure 5 shows a comparative study of the magnetic spectral responses of a gold diabolo nanoantenna with G = T = 20 nm and D = 310 nm and a 455 nm long gold nanowire whose square cross section is 20 nm large. Figure insets represent the transverse distributions of the magnetic intensity, simulated 10 nm beyond the structures, at their common resonance wavelength of 2570 nm (scale bar = 200 nm). These results show an enhancement 4-fold smaller for the nanowire than for the diabolo nanoantenna with a loss of confinement along the nanowire main axis, indicating the importance of the triangular portions of the diabolo antenna.

The highly confined distribution of the magnetic field around the metal gap (Figure 3a) shows that the plasmon mode sustained by the diabolo nanoantenna is associated with an effective wavelength much smaller than the vacuum wavelength. This may explain why the plasmon mode is excited at a length of the diabolo nanoantenna considerably smaller than the vacuum wavelength. This property could be due to the triangular geometries of the diabolo nanoantenna since triangular and conical structures are known to sustain plasmon modes of very small wavelengths, leading for example to the superfocusing effect. 18 While the 2900-fold enhancement of the optical magnetic field offered by the diabolo nanoantenna is large, it is significantly smaller than the typical electric field enhancements offered by the bowtie nanoantenna. 19 In order to explain the more modest resonance enhancement offered by the diabolo nanoantenna, we have to consider the different boundary conditions. For the diabolo nanoantenna, the largest magnetic fields are tangential to the metal surface, while at the tips of the bowtie structure, the largest electric fields are perpendicular to the metal surface. Since the perpendicular electric field has a discontinuity across the interface, the electric field in the metal is smaller than the field just outside and thus the losses in the metal are small. On the other hand, the tangential magnetic field is continuous across the interface and high magnetic fields exist in the metal leading to stronger losses. For this reason only a moderate resonance enhancement can be obtained for diabolo nanoantenna as compared to the bowtie nanoantenna. The losses in the metal are associated with imaginary part of the dielectric constant, which tends to increase with decreasing wavelength. Therefore the resonance enhancement gets smaller and smaller with decreasing wavelength.

Conclusion. In conclusion we have shown that the diabolo nanoantenna is an interesting structure for creating a highly localized magnetic field with a high field enhancement in the near-infrared region. Structures of this type can be readily fabricated, either by a focused ion beam or electron beam lithography, similar to the way the dipole and bowtie nanoantennas were produced. 20,21 Alternately, modified nanosphere lithography could be employed. 22 The existence of an enhanced magnetic field can be tested by using a SNOM configuration which allows us to measure the vectorial magnetic and electric nearfields. 23,24 We intend to use such structures as a detector for the optical magnetic field and for the magneto-optic modulation of plasmon polaritons at near-infrared or even visible frequencies, which is not possible by using ferromagnets. 25 The diabolo nanoantenna may be also relevant for inducing magnetic transitions in matter. 26

  Figure 3 shows the near-field distributions of (a, b) the magnetic-field and (c, d) the electricfield enhancements generated by a gold diabolo nanoantenna at a wavelength of 1940 nm. The structure under study is characterized by G = T = 50 nm and D = 310 nm. Intensity plots are given along (a, c) the longitudinal (XZ) and (b, d) the transverse (XY) planes. The (XZ) plane crosses the diabolo nanoantenna at the middle of the metal gap whereas the (XY) plane is taken 10 nm beneath the nanoantenna output plane, in air.

Figure 1 .

 1 Figure 1. (a) 3D view of the diabolo nanoantenna and its complementary bowtie nanoaperture. (b, c) Physical principles of the bowtie and diabolo nanoantennas, respectively.

Figure 2 .

 2 Figure 2. Scheme of the simulation model under study: (a) top view in the (XY) plane and (b) side view along the (XZ) plane.

Figure 4 .

 4 Figure 4. (a) Comparison of the magnetic spectral responses of gold diabolo nanoantennas with D = 310 nm and G and T equal to 50 and 50 nm (solid curve), 50 and 20 nm (dashed curve), 20 and 50 nm (dotted curve), and 20 and 20 nm (dashed-dotted curve). (b) Magnetic spectral responses of diabolo nanoantennas made of gold (solid line), aluminum (dashed line), and silver (dotted line) with G = 50 nm, T = 50 nm, and D = 310 nm.

Figure 3 .

 3 Figure 3. (a-d) Normalized intensity distributions of (a, b) the magnetic field and (c, d) the electric field of a gold diabolo nanoantenna (G = T = 50 nm, D = 310 nm) excited with a plane wave whose electric field is linearly polarized along the y axis (λ = 1940 nm). Intensity plots along (a, c) the longitudinal (XZ) plane and (b, d) along the transverse (XY) plane. Inset of (a): vectorial map of the magnetic field around the metal gap, in the (XZ) plane. (e) Enhancement of the magnetic intensity for an incident electric field polarized perpendicularly to the y axis. Image size: 700 nm Â700 nm. (f) Same configuration as (a) with G = T =20 nm and λ = 2570 nm. Image size: 300 nm  300 nm.

Figure 5 .

 5 Figure 5. Comparison of the magnetic spectral responses of a gold diabolo nanoantenna with D = 310 nm and G = T = 20 nm (solid curve) and a 455 nm long gold nanowire of 20 nm large square cross section (dashed curve).Figure insets: transverse distributions of the magnetic intensity simulated at the common resonance wavelength of the two structures (scale bar 200 nm).

  Figure 5. Comparison of the magnetic spectral responses of a gold diabolo nanoantenna with D = 310 nm and G = T = 20 nm (solid curve) and a 455 nm long gold nanowire of 20 nm large square cross section (dashed curve).Figure insets: transverse distributions of the magnetic intensity simulated at the common resonance wavelength of the two structures (scale bar 200 nm).

ACKNOWLEDGMENT

This work is funded by the "Agence Nationale de la Recherche" under Contract No. ANR07-NANO-036. It is supported by the "Pôle de comp etitivit e Microtechnique".

AUTHOR INFORMATION

Corresponding Author *E-mail: thierry.grosjean@univ-fcomte.fr.


Member, IEEE John M Dudley

Stéphane Coen

Numerical Simulations and Coherence Properties of Supercontinuum Generation in Photonic Crystal and Tapered Optical Fibers

Keywords: Nonlinear pulse propagation, optical fibers, photonic crystal fibers, supercontinuum generation, Raman solitons

Numerical simulations have been used to study broad-band supercontinuum generation in optical fibers with dispersion and nonlinearity characteristics typical of photonic crystal or tapered fiber structures. The simulations include optical shock and Raman nonlinearity terms, with quantum noise taken into account phenomenologically by including in the input field a noise seed of one photon per mode with random phase. For input pulses of 150-fs duration injected in the anomalous dispersion regime, the effect of modulational instability is shown to lead to severe temporal jitter in the output, and associated fluctuations in the spectral amplitude and phase across the generated supercontinuum. The spectral phase fluctuations are quantified by performing multiple simulations and calculating both the standard deviation of the phase and, more rigorously, the degree of first-order coherence as a function of wavelength across the spectrum. By performing simulations over a range of input pulse durations and wavelengths, we can identify the conditions under which coherent supercontinua with a well-defined spectral phase are generated.

I. INTRODUCTION

P HOTONIC CRYSTAL FIBERS (PCF) commonly consist of a central-fused silica core surrounded by a regular array of air holes running along the fiber length. The number, size, and orientation of the air-holes around the core provide additional degrees of freedom not present in conventional fibers, and can be manipulated to yield novel confinement and dispersion characteristics. PCFs have thus been shown to exhibit several remarkable properties, including "endlessly single-mode" behavior [START_REF] Knight | All-silica single-mode optical fiber with photonic crystal cladding[END_REF], [START_REF] Birks | Endlessly single-mode photonic crystal fiber[END_REF], novel group velocity dispersion (GVD) characteristics such as zero-dispersion wavelengths (ZDW) in the near-infrared [START_REF] Mogilevtsev | Group velocity dispersion in photonic crystal fibers[END_REF]- [START_REF] Knight | Anomalous dispersion in photonic crystal fiber[END_REF], and enhanced effective nonlinearity due to the reduced mode size in the core [START_REF] Broderick | Nonlinearity in holey optical fibers: Measurement and future opportunities[END_REF]. With fiber structures designed so that the ZDW is shifted within the operating bandwidths of high-power near-infrared femtosecond lasers, these properties have been exploited to demonstrate soliton propagation effects around 850 nm [START_REF] Wadsworth | Soliton effects in photonic crystal fibers at 850 nm[END_REF], and the generation of broad-band supercontinuum (SC) spectra spanning over an octave [START_REF] Ranka | Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm[END_REF], [START_REF] Ranka | Nonlinear interactions in air-silica microstructure optical fibers[END_REF]. Such broad-band SC spectra have already found application in optical coherence tomography [START_REF] Hartl | Ultrahigh resolution optical coherence tomography using continuum generation in an air-silica microstructure optical fiber[END_REF] and, particularly, in optical frequency metrology where they have allowed successful bridging of the optical-microwave frequency gap [START_REF] Jones | Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis[END_REF]- [START_REF] Diddams | Direct link between microwave and optical frequencies with a 300 THz femtosecond laser comb[END_REF]. Similar broad-band SC have also been demonstrated in standard fiber tapered to a strand diameter of 2 m where the tapering introduces dispersion and nonlinearity characteristics similar to those of PCF [START_REF] Birks | Supercontinuum generation in tapered fibers[END_REF], [START_REF] Man | Fabrication of indefinitely long tapered fibers for supercontinuum generation[END_REF].

Numerical simulations of SC generation in PCF have also recently been reported, agreeing well with experimental observations and providing an improved understanding of the underlying spectral broadening mechanisms involved [START_REF] Gaeta | Supercontinuum generation in microstructured fibers[END_REF]- [START_REF] Husakou | Supercontinuum generation of higherorder solitons by fission in photonic crystal fibers[END_REF]. However, an aspect of SC generation in PCFs which has not yet been the subject of detailed study is the sensitivity of the spectral broadening processes to input pulse noise. This is nonetheless of particular importance, because key applications of octave-spanning SC sources such as precision optical frequency metrology require highly coherent SC light with limited pulse to pulse variations and reduced timing jitter [START_REF] Holzwarth | White-light frequency comb generation with a diode-pumped CrLiSAF laser[END_REF]. In this regard, we note that SC generation in standard optical fiber has previously been the subject of much research in a telecommunications context, where spectrally sliced broad-band SC have been proposed as sources for wavelength-division-multiplexing applications [START_REF] Islam | Femtosecond distributed soliton spectrum in fibers[END_REF]- [START_REF] Tamura | Fundamentals of stable continuum generation at high repetition rates[END_REF]. Significantly, in this previous work, numerical simulations have shown that the generation of broad-band spectra using picosecond and subpicosecond pulse pumping in the presence of noise can, in fact, be associated with significant temporal jitter and a poorly defined spectral phase across the generated broad-band spectrum [START_REF] Nakazawa | Coherence degradation in the process of supercontinuum generation in an optical fiber[END_REF], [START_REF] Nowak | Stable supercontinuum generation in short lengths of conventional dispersion-shifted fiber[END_REF], [START_REF] Kubota | Analyses of coherence-maintained ultrashort optical pulse trains and supercontinuum generation in the presence of soliton-amplified spontaneous emission interaction[END_REF].

In this paper, we present results of numerical simulations of SC generation in PCF and tapered fibers which study, in particular, the fluctuations in both the temporal and spectral properties introduced by the presence of noise on the input pulse. The paper is organized as follows. In Section II, we describe our numerical model, and give the fiber parameters used in the simulations. In order to discuss the main physical mechanisms involved in SC generation, Section III.A presents results for input pulses injected in the fiber anomalous dispersion regime in the absence of noise. In Section III.B, the effect of input pulse noise is then discussed, and shown to be associated with fluctuations in the spectral phase across the SC. By quantifying the phase fluctuations using the mutual degree of coherence calculated over an ensemble of simulations, the influence of the input pulse wavelength and duration on the SC coherence is studied, allowing optimal experimental conditions for coherent SC generation to be identified. Section IV discusses several implications of these results and concludes the paper.

II. NUMERICAL MODEL

Our numerical model uses a generalized scalar propagation equation suitable for studying broad-band pulse evolution in optical fibers [START_REF] Coen | White light supercontinuum generation with 60-ps pump pulses in a photonic crystal fiber[END_REF], [START_REF] Blow | Theoretical description of transient stimulated Raman scattering in optical fibers[END_REF] (1)

Here, is the electric field envelope, the 's are the usual dispersion coefficients at center frequency , and is the nonlinear coefficient, with m /W the nonlinear refractive index and the fiber effective area. The right hand side of (1) models self-phase modulation (SPM), self-steepening, optical shock formation, and intrapulse Raman scattering. The response function includes both instantaneous electronic and delayed Raman contributions, with representing the contribution of the Raman response to the instantaneous nonlinear polarization. For , we used the experimentally determined Raman response of silica fibers. Fiber loss was neglected. The presence of quantum noise was taken into account phenomenologically by including in the input field a noise seed of one photon per mode with random phase [START_REF]Nonlinear Fiber Optics[END_REF].

The fiber parameters were based on a 10-cm pure fused silica strand of diameter 2.5 m surrounded by air. This is similar to the tapered fiber described in [START_REF] Birks | Supercontinuum generation in tapered fibers[END_REF], in which SC spectra with more than an octave of bandwidth were generated. The fiber dispersion for the fundamental HE mode was calculated using standard techniques [START_REF] Snyder | Optical Waveguide Theory: Chapman and Hall[END_REF], and for comparison with previously published dispersion data in PCF and tapered fiber [START_REF] Birks | Supercontinuum generation in tapered fibers[END_REF], [START_REF] Ranka | Optical properties of highdelta air-silica microstructure fibers[END_REF], Fig. 1 plots the calculated GVD in terms of both the dispersion parameter (in ps/nm/km) and (in ps /m). Since both the dispersion and nonlinearity characteristics of this strand (in particular a ZDW at 780 nm) are nearly identical to those of a large air-fill fraction PCF [START_REF] Birks | Supercontinuum generation in tapered fibers[END_REF], the simulation results presented below can be considered generally applicable both to tapered fibers and PCF.

The effect of dispersion in the simulations was found to be accurately modelled over the range 300-1600 nm using terms up to in the Taylor series expansion, with higher order terms having negligible influence. The particular values at 850 nm were ps m , ps m , ps m , ps m , ps m , ps m . The integration of the propagation equation was carried out using the enhanced split-step-Fourier scheme described in [START_REF] Blow | Theoretical description of transient stimulated Raman scattering in optical fibers[END_REF], and typically required at least time and frequency discretization points and a longitudinal step size 10 m in order to avoid numerical instabilities [START_REF]Nonlinear Fiber Optics[END_REF], [START_REF] Matera | Sideband instability induced by periodic power variation in long-distance fiber links[END_REF]. With these fiber parameters, the numerical simulations based on (1) will be seen to adequately model the experimentally re- ported SC structure and bandwidth for 50-150-fs input pulses. The use of shorter pulses or longer fiber lengths, however, may require effects such as fiber loss, polarization coupling [START_REF] Golovchenko | Unified analysis of four photon mixing, modulational instability and stimulated Raman scattering under various polarization conditions in fibers[END_REF], and the wavelength dependence of the effective area [START_REF] Chernikov | Femtosecond soliton propagation in fibers with slowly decreasing dispersion[END_REF] to be included. For some input pulse and fiber parameter ranges, it may even be necessary to use alternative numerical methods based on the direct solution of Maxwell's equations [START_REF] Husakou | Supercontinuum generation of higherorder solitons by fission in photonic crystal fibers[END_REF].

III. SIMULATION RESULTS

A. Results in the Absence of Input Pulse Noise

Simulations were first carried out in the absence of noise on the input pulse, using hyperbolic secant input pulses with a peak power of 10 kW and a wavelength of 850 nm in the fiber anomalous dispersion regime. These input pulse parameters are typical of those used in experimental studies [START_REF] Birks | Supercontinuum generation in tapered fibers[END_REF]. Fig. 2 shows simulation results for input pulse durations (FWHM) of (a) 50 fs and (b) 150 fs, plotting the spectral and temporal evolution at different points along the fiber.

Although the higher nonlinearity associated with PCF and tapered fibers leads to enhanced signatures of nonlinear and dispersive interactions during propagation, we stress that ultrashort pulse propagation in optical fibers has been the subject of extensive research for over 20 years [START_REF] Nakatsuka | Nonlinear picosecond pulse propagation through optical fibers with positive group velocity dispersion[END_REF], and many of the observed spectral and temporal features seen in the simulation results in Fig. 2 (and in experiments) can be readily interpreted in terms of well-understood physics [START_REF]Nonlinear Fiber Optics[END_REF], [START_REF] Alfano | The Supercontinuum Laser Source[END_REF]. In particular, the dramatic spectral broadening observed during the SC generation process arises from the interaction between the fiber dispersion and various nonlinear processes, including SPM, four-wave mixing (FWM), self-steepening, and stimulated Raman scattering [START_REF]Nonlinear Fiber Optics[END_REF]. For pulses injected in the fiber anomalous dispersion regime (such as in our simulations), the initial stage of propagation is associated with strong temporal compression and approx- imately symmetrical spectral broadening, followed by more severe pulse breakup due to the development of ultrafast temporal oscillations on the injected pulse envelope from modulational instability (MI)-like effects. These processes are largely determined by the interaction of SPM and GVD, although higher order dispersion plays an increasingly important role as new spectral components are generated in the vicinity of the fiber ZDW and introduces some spectral asymmetry.

Referring back to Fig. 2, the onset of the pulse breakup can be seen to occur after 2 cm for the 50-fs input pulse and after 4 cm for the 150-fs pulse. The fact that the 50-fs input pulse experiences significantly more spectral broadening and compression in the first few centimeters of propagation is consistent with the increased effect of SPM on shorter pulses. The spectral broadening continues with further propagation and, quantifying the spectral width at the 20-dB point as in [START_REF] Birks | Supercontinuum generation in tapered fibers[END_REF], we find that both output spectra in Fig. 2 extend over an octave from 550-1100 nm. However, although the output spectra are essentially continuous over this range, the spectral intensity does exhibit a somewhat complex structure, with particularly distinct peaks on the long wavelength side of the spectrum in the anomalous dispersion regime. Such structure has been previously reported in experiments, for example Fig. 5(b) of [START_REF] Ranka | Nonlinear interactions in air-silica microstructure optical fibers[END_REF], Fig. 4 of [START_REF] Diddams | Toward the ultimate control of light: Optical frequency metrology and the phase control of femtosecond pulses[END_REF] and Fig. 6 of [START_REF] Birks | Supercontinuum generation in tapered fibers[END_REF]. In the time domain, the development of this spectral structure is seen to be associated with the formation of a series of distinct high intensity pulses on the trailing edge of the residual input pulse.

As discussed by Islam et al. in [START_REF] Islam | Femtosecond distributed soliton spectrum in fibers[END_REF], these distinct pulses are Raman solitons whose generation is seeded from the MI-induced oscillations developed on the propagating pulse envelope. As they propagate in the fiber, they undergo a continuous shift to longer wavelengths because of the soliton self-frequency shift, and walk off temporally from the pump. Fig. 3 shows in detail the output spectral and temporal profiles to illustrate more clearly the relationship between the spectral and temporal features of these Raman solitons. In particular, numerical filtering allows the anomalous dispersion regime spectral peaks to be associated with the corresponding time-domain Raman solitons, shown as the labeled features A and B on the figure. In fact, we note that for the 150-fs case, the time-domain manifestation of the spectral feature A (between 920-1050 nm) actually corresponds to two Raman solitons separated by 240 fs; the corresponding spectral peaks partially overlap and thus exhibit a 13-nm ( 4.2 THz) spectral modulation.

The figure shows some additional interesting features. For the 150-fs case, the strong temporal oscillations due to MI are identified on the envelope of the residual input pulse, and for the 50-fs case, we have also identified a normal dispersion regime peak around 600 nm (labeled DW), which arises from the transfer of energy from the anomalous dispersion regime via the generation of a low amplitude dispersive-wave [START_REF] Wai | Nonlinear pulse propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fibers[END_REF]. The associated time-domain manifestation of this in terms of a low amplitude broad pulse is clearly identified in the figure .  Although such dispersive wave generation also occurs for the 150-fs input pulse, the spectral and temporal signatures in this case are less distinct and therefore not specifically identified on the figure. Of course, with significant spectral content present on both sides of the fiber ZDW at 780 nm, FWM and cross-phase modulation are also likely to play an important role in determining the overall form of the SC spectrum, but a quantitative discussion of these effects in terms of the results in Fig. 3 is not considered here.

B. Results in the Presence of Input Pulse Noise

The simulations described above for input 50-and 150-fs pulses were then repeated with the inclusion of a one photon per mode noise level on the input field. Multiple simulations were performed under identical conditions except for the initial random noise present on the input field. For the injected 50-fs pulses, no significant differences in either the spectral or temporal characteristics from run to run were observed. However, for the injected 150-fs pulses, the presence of input pulse noise led to significant run to run differences in both the output spectra and corresponding temporal profiles.

Fig. 4 shows results for five such simulations using 150-fs pulses, clearly showing run to run variations in the structure of the observed spectra and temporal intensity. The significant jitter in the temporal position of the generated Raman solitons is particularly apparent, associated with shifts in the wavelengths of the corresponding peaks in the spectrum. These observations are consistent with previous studies of SC generation in conventional fibers, which have shown that broad-band spectral generation seeded from MI is very sensitive to noise on the input pulse [START_REF] Islam | Femtosecond distributed soliton spectrum in fibers[END_REF]- [START_REF] Kubota | Analyses of coherence-maintained ultrashort optical pulse trains and supercontinuum generation in the presence of soliton-amplified spontaneous emission interaction[END_REF].

As well as leading to the averaging of the temporal or spectral characteristics which would be experimentally measured with integrated or multishot detectors, this jitter is also associated with coherence degradation due to severe fluctuations in the spectral phase at each wavelength. This coherence degradation can be examined directly by calculating the standard deviation of the spectral phase across the output spectrum from a large number of independent simulations. An additional measure of the coherence degradation can be obtained by using a technique recently developed in an experimental study of SC generation in bulk media [START_REF] Bellini | Phase-locked white-light continuum pulses: Toward a universal optical frequency-comb synthesizer[END_REF]. This method uses a modified Young's two source experiment where the two spatially separated sources which interfere are independently generated SC which yield a polychromatic interference pattern when combined together in the far-field. The spectral resolution of this pattern reveals distinct fringes at each wavelength in the SC spectrum, with the wavelength dependence of the fringe visibility providing a rigorous measure of the local coherence properties, since it is directly related to the modulus of the complex degree of (mutual) coherence between the independent SC sources. This measure of coherence degradation is calculated numerically using the definition [START_REF] Birks | Endlessly single-mode photonic crystal fiber[END_REF] Here, the angle brackets denote an ensemble average over independently generated pairs of SC spectra , and is the time measured at the scale of the temporal resolution of the spectrometer used to resolve these spectra. In practice, we have applied the ensemble average on the results of many simulations which yielded 200 pairs of SC obtained from input pulses with different random quantum noise. Also, since we are mainly interested in the wavelength dependence of the coherence, we have calculated the modulus at , which would simply correspond to the fringe visibility at zero path difference in the Young's two source experiment described above. We also note that is calculated over a 250-GHz numerical bandwidth, consistent with the spectral resolution used in typical SC experiments [START_REF] Ranka | Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm[END_REF]. Additional simulations have also verified that the wavelength dependence of (which is discussed in detail below) is independent of the particular simulation resolution used.

Fig. 5 shows the results of this analysis applied to the 150-fs results discussed above. Fig. 5(a) and (d) show, respectively, the mean spectrum and temporal intensity profile calculated over the ensemble of simulations. It is clear that there is significant averaging of the fine structure which was previously apparent in Fig. 3(b). (Interestingly, it was noted as early as 1989 that, even though the individual spectra may possess considerable structure, the averaging over Raman soliton pulses frequency shifted by different amounts because of noise can lead to an integrated flat smooth spectrum [START_REF] Islam | Femtosecond distributed soliton spectrum in fibers[END_REF].)

In terms of the coherence degradation across the spectrum, Fig. 5(b) shows the calculated spectral phase deviation at each wavelength in the spectrum. Although there is a slight decrease in around 850 nm, the wavelength dependence of the phase stability is not clearly manifested in this figure. As expected, however, variations in the phase stability are more strongly manifested in the degree of mutual coherence plotted in Fig. 5(c), and therefore it is this which we use in the remainder of the paper to discuss the coherence degradation in more detail.

In particular, to study how the decoherence is correlated with the evolution of the injected input pulse, we have plotted in Fig. 6 typical temporal and spectral characteristics at different propagation distances. The top curves show the temporal intensity distribution obtained from one particular simulation (single shot), while the bottom curves show the spectral characteristics calculated from the ensemble average. Here, we plot both the mean spectrum (left axis) and (right axis). During the initial stage of evolution at cm when the input 150-fs pulse undergoes significant compression and associated spectral broadening, over the major part of the SC. However, with an increased propagation distance of cm where the effects of MI and pulse breakup are clearly manifested on the temporal intensity profile, there is significant coherence degradation, with a reduced . The increased spectral broadening with further propagation is associated with continued coherence degradation so that over most of the SC spectrum at the fiber output for cm.

Significantly, setting the Raman gain to zero and repeating these simulations yielded similar coherence properties, implying that it is the sensitivity of MI to input pulse noise and not any effect due to stimulated Raman scattering which is the primary cause of the coherence degradation under our conditions. This suggests that superior coherence properties would be expected for SC generated using shorter input pulses where SPM plays a more significant role in the spectral broadening, and the effects of MI are reduced. Indeed, this is consistent with our observation that there was negligible run to run jitter with 50-fs input pulses, and confirmed more quantitatively in Fig. 7, which compares the simulation results for 150-fs input pulses with those obtained with shorter input pulse durations of 100 and 50 fs (the wavelength and peak power were maintained at 850 nm and 10 kW, respectively). It is clear that, although the spectral broadening in all cases is comparable, the coherence properties improve significantly as the input pulse duration is decreased and, indeed over more than an octave for the shortest input pulses of 50 fs.

Additional confirmation of the influence of MI on the coherence degradation is presented in Fig. 8, which shows the results of simulations using 10-kW peak power 150-fs input pulses injected over a range of different wavelengths. With an input wavelength of 740 nm in the normal dispersion regime where MI is completely inhibited, there is negligible coherence degradation. The generated spectral width of 350 nm at the 20-dB level is, however, significantly less than that obtained with an anomalous dispersion regime pump. The significant coherence degradation observed at 780 nm (the ZDW) and 820 nm (anomalous dispersion) is consistent with the increased influence of MI at these longer wavelengths.

IV. DISCUSSION AND CONCLUSION

The results presented here have several important implications. Firstly, it is clear that the bandwidth and apparent uniformity of a generated SC spectrum are in themselves insufficient measures of its suitability for applications in frequency metrology. Because of the sensitivity of the spectral broadening mechanisms to noise, spectral uniformity can arise as a result of averaging effects, and broad bandwidth SC can, in fact, be associated with large spectral phase fluctuations. With the power and fiber parameters used in our simulations, the results suggest that suitable sources for metrology spanning over an octave in bandwidth and possessing excellent coherence properties require input pulses of duration 50 fs or less. Indeed, we note that this is the pulse duration range which has been used in successful PCF-based metrology experiments to date [START_REF] Holzwarth | White-light frequency comb generation with a diode-pumped CrLiSAF laser[END_REF]. We also note that the results presented here illustrate the importance of using stabilized pump lasers in order to obtain the best coherence properties.

In addition, the use of the complex degree of coherence provides a convenient means of quantifying the coherence degradation across SC spectra generated in PCF and tapered fibers, and we anticipate that its experimental characterization will be readily implemented. We also note that the loss of coherence may prove a limiting factor in experiments using multishot meaurement techniques to perform intensity and phase characterization of SC generation, and that in the presence of significant phase instability, single-shot characterization would be required.

In the particular context of SC generation in PCF, an important extension of the work described in this paper is to consider in detail how the spectral and temporal SC chararteristics vary with changes in the fiber structure. Although a complete analysis of the propagation problem is, of course, possible via the direct solution of Maxwell's equations, for many PCF structures based in a regular geometrical air-hole distribution, the fiber characteristics can be readily included in the extended NLSE model above via modified fiber dispersion and effective area parameters [START_REF] Mortensen | Effective area of photonic crystal fibers[END_REF]. In the case of fiber structures associated with very complex dispersion characteristics, however, it is important to ensure that the number of terms in the Taylor series expansion is sufficiently high, so that the dispersion is modeled accurately over the wavelength range of interest.

Finally, although we have discussed some strong qualitative similarities between our simulation results and experiments, it is appropriate here to consider whether a more quantitative analysis is possible. In this regard, it is important to note that the SC generation process is highly sensitive to the input pulse characteristics, and experimental uncertainties in parameters such as the injected pulse power preclude a more quantitative comparison of our results with published SC generation results. We note, nonethless, that in a recent experiment, studying Raman soliton propagation in PCF where the injection conditions and even the presence of input pulse chirp were carefully characterized, very good quantitative agreement between the extended NLSE model and experiment was reported [START_REF] Washburn | Tunable near-infared femtosecond soliton generation in photonic crystal fibers[END_REF].

Fig. 1 .

 1 Fig. 1. Calculated GVD characteristics for the fused silica fiber strand used in numerical simulations.

Fig. 2 .

 2 Fig. 2. Evolution of (a) 50-and (b) 150-fs input pulses along 10 cm of fiber, showing both the spectral (left) and temporal (right) properties.

Fig. 3 .

 3 Fig. 3. Detail of output characteristics for (a) 50-and (b) 150-fs input pulses. (i) and (ii) show the spectra on logarithmic and linear scales respectively, while (iii) shows the temporal intensity. The various labeled features are discussed in the text.

Fig. 4 .

 4 Fig. 4. Spectra (left) and temporal intensities (right) for five simulations using a 10-kW peak power 150-fs input pulse at a wavelength of 850 nm.

Fig. 5 .

 5 Fig. 5. Averaged output characteristics for multiple simulations of an injected 150-fs pulse in the presence of a random one photon per mode noise level. The spectral characteristics show (a) the calculated mean spectrum, (b) the standard deviation of the spectral phase at each wavelength in the spectrum (in units of 2), and (c) the corresponding degree of mutual coherence (defined in the text). The temporal characteristics in (d) show the calculated mean intensity profile.

Fig. 6 .

 6 Fig.6. For propagation distances of (a) 2 cm, (b) 5 cm, and (c) 10 cm, the top curves show the temporal intensity distribution obtained from one simulation, while the bottom curves show the mean spectrum (left axis) and degree of coherence (right axis) calculated from an ensemble average.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Mean spectrum (left axis) and calculated degree of coherence (right axis) obtained with 10-kW pulses at 850 nm for the different input pulse durations shown.

J. M. Dudley is with the Laboratoire d'Optique P. M. Duffieux, Université de Franche-Comté, 25030 Besançon, France (e-mail: john.dudley@univfcomte.fr).S. Coen is with the Service d'Optique et Acoustique, Université Libre de Bruxelles, B-1050 Brussels, Belgium.

ACKNOWLEDGMENT

The authors would like to thank S. Ralph, B. Washburn, P. Lacourt, L. Provino, and R. Trebino for stimulating discussions.


F Chinesta
email: chinesta@ec-nantes.fr

A Poitou

S Chatel

J A Garcia

L L Gascon

ON THE EQUIVALENT IN-PLANE PERMEABILITY

Keywords: RTM, Permeability, Separated representations, Model reduction

Simulation of RTM processes is usually performed in 2D. Thus, only a mesh of the middle plane is needed with the associated degrees of freedom savings with respect to fully 3D modeling. However such modeling needs the definition of an equivalent in-plane permeability representing the ignored dimension (the thickness). The definition of such permeability is not a trivial task because each ply in the thickness direction can be anisotropic, being the principal anisotropy direction different from one ply to the neighbor plies. In this work we propose a novel fully 3D modeling whose computational cost is equivalent to a 2D solution. It allows addressing properly the equivalent in-plane permeability issue.

INTRODUCTION

In general the simulation of RTM processes assumes a 2D flow model. The most usual model results of combining the Darcy's law and the flow incompressibility:

0 p          v K v
that results in the second order BVP:

  0 p     K
The main issue in defining this model concerns the definition of the permeability tensor K . Different techniques exist, but in that follows we are assuming that an averaged permeability has been determined for each type of reinforcement architecture. First we are considering a laminate composed of many layers with the same principal directions, whose permeability tensor writes 0 0 0 0

               

K

Now, we consider the flow in a plate mould, being the permeability principal directions parallel to the mould walls. In this case the permeability tensor becomes diagonal. The simplest scenario concerns    and 0   . In this case the flow is unidirectional. When the ply is placed after performing a certain in-plane rotation to the principal permeability directions, the  component of the permeability tensor is no more zero. In that case when we impose a constant injection pressure, being the pressure in the opposite mould boundary zero, the resulting pressure field is depicted in Fig. 1. We can conclude that if the principal directions of the permeability of each layer are aligned along the coordinate axes, even if the permeability evolves in the thickness direction, the flow remains unidirectional. The situation is radically different if the permeability also evolves in the x-direction. When we assumes an evolution of   , x z  as depicted in figure 5, the pressure field that results, illustrated in figure 6, implies the velocity field depicted in figure 7, which exhibit noticeable deviations with respect to an unidirectional flow. These simple examples have proven that in general laminates, involving many layers with different reinforcement orientations the situation could be quite complex and the assumption of a simple average of the permeability of the different layers could be a crude approximation that deserves a deeper analysis. For this purpose a fully 3D simulation could be the best alternative, but an accurate representation of the thickness direction could become too expensive from the computational view point. An appealing alternative consists in applying a separated representation of the pressure field involved in the porous media flow modelling. This kind of approximation allows computing 3D solutions with a computational cost characteristic of 2D simulations, that is, of standard RTM simulations. In the next section we summarized the main elements of such technique, deeply described in many of our former works [START_REF] Ammar | A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids[END_REF][START_REF] Ammar | A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. Part II: transient simulation using space-time separated representations[END_REF], and that is here particularized to the treatment of the Darcy's model defined in plate moulds. Its extension to shell geometries is quite direct.

PGD IN PLATE DOMAINS

In what follows we are illustrating the construction of the Proper Generalized Decomposition of a model defined in a plate domain

I    with 2    and   0, I H  :   0 p     K (1)
We consider that the laminate is composed of P different anisotropic plies each one characterized by a well defined permeability tensor   , i x y K -it is assumed constant in the ply thickness-. Moreover, without a loss of generality, we assume the same thickness for the different layers of the laminate, that we denotes by h. Thus, we can define a characteristic function representing the position of each layer:

1 1 ( ) 0 i i i z z z z otherwise         , 1, , i P  L (2)
where  

1 i z i h    . Now, the laminated
permeability can be given in the following separated form:

      1 , , i P i i i x y z z       K K x (3)
      1 , i N i i i p z X Z z      x x (5)
In what follows we are illustrating the construction of one such decomposition. For this purpose we assume that at iteration n N  the solution is already known:

      1 , i n n i i i p z X Z z      x x (6)
and that at the present iteration we look for the solution enrichment:

    1 , , ( ) ( )
n n p z p z R S z     x x x (7)
The test function involved in the weak form is searched under the form:

  * * * , ( ) ( ) ( ) ( ) p z R S z R S z     x x x (8)
By introducing Eqs. ( 7) and ( 8) into (4) it results:

°°°°°* * * * * * * * n R S R S R S d dS dS dS R R R dz dz dz R S R S d dS dS R R dz dz                                                                                                           K Q ( 9 )
where ° denotes the plane component of the gradient

operator °  , T x y       and n Q denotes the flux at iteration n: °        1 i i i n n i i i X Z z dZ z X dz                   x Q K x ( 1
    1 n R X   x x and     1 n S z Z z   .
We are illustrating each one of the just referred steps:

1. Computing   R x from   S z : When   S z is known the test function reduces to:   * * , ( ) ( ) p z R S z   x x (11)
and the weak form (9) reduces to:

°°°* * * * n R S R S d dS dS R R dz dz R S d dS R dz                                                             K Q (12)
Now, as all the functions involving the coordinate z are known, they could be integrated in

  0, I H  . Thus, if
we consider:

T         k K k k ( 1 3 ) with xx x y yx yy k k k k        k , xz yz k k        k and zz k   , then we can define: 2 0 0 2 0 0 z H z H z z x z H z H T z z dS S dz Sdz dz dS dS Sdz dz dz dz                                   k K k k (14) and °    0 0 1 0 0 z H z H i i i N z z x n i z H z H i i T i i z z dZ SZ dz S dz dz X X dZ dS dS Z dz dz dz dz dz                                                       k x Q x k k (15) that allows writing equation (12) into the form °°°* * * * x x n R R d R R R d R                                         K Q (16)
that defines an elliptic 2D problem defined in the middle plane of the plate.

Computing

 

S z from   R x : When   R x is known the test function writes:   * * , ( ) ( ) p z R S z   x x (17)
and the weak form (9) reduces to:

°°°* * * * n R S R S d dS dS R R dz dz R S d dS R dz                                                              K Q (18)
Now, as all the functions involving the in-plane coordinates   , x y  x are known, they could be integrated in  . Thus, using the previous notation, we can define:

°  °    °  °    2 z R R d R Rd R Rd R d                                     k K k k (19) and °°  °°1 i i i i N z n i i i i R X d R Xd Z dZ dz R X d RX d                                                              k Q k k ( 20
                                                K Q (21)
that defines a one-dimensional BVP.

CONCLUSIONS

This paper comes back to a recurrent issue in the numerical modelling of RTM flows, the one related to the pertinence of using 2D flow models making use of an averaged permeability of the different layers involved in the laminate. In complex situations significant deviations could be found, these deviations are being studied at present, and could justify the use of a fully 3D modelling. However, 3D simulations are reputed expensive from the point of view of the computational resources required for addressing complex scenarios. The use of separated representations as the ones involved in the proper generalized decompositions -PGD-could be an appealing alternative for addressing 3D models with a cost characteristic of 2D simulations. The application of the PGD on the fully 3D simulation of flows encountered in RTM processes constitutes a work in progress.

Figure 1 :

 1 Figure 1: Pressure distribution

Figure 2 :

 2 Figure 2: Evolution of the permeability in the thickness

Figure 3 :Figure 4 :

 34 Figure 3: Pressure distribution

Figure 5 :

 5 Figure 5: Evolution of the permeability

  z is searched under the separated form:

  0 ) Now, as the enrichment process is non-linear we propose to search the couple of functions   R x and   S z by applying an alternating direction fixed point algorithm. Thus, assuming   R x known, we compute   S z , and then we update   R x . The process continues until reaching convergence. The converged solutions allow defining the next term in the finite sums decomposition: