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Rigorous Optical Modeling of Elliptical Photonic Nanowires Amine Ben Salem, Rim Cherif, and Mourad Zghal
Abstract—We analyze the optical properties including chromatic dispersion, birefringence, and nonlinear coefficient dependence on the ellipticity of photonic nanowires. We propose a linear approximation to determine the equivalent-circular photonic nanowire exhibiting similar optical characteristics with the elliptical nanowire. in elliptical We find strong birefringence up to the order of photonic nanowires that could be very attractive for optical fiber sensors and stable combs. We also investigate the effect of the ellipticity on the supercontinuum generation which is found to be detrimental to the spectral broadening. Index Terms—Birefringence, ellipticity, photonic nanowire, supercontinuum (SC) generation.
I. INTRODUCTION
P
HOTONIC nanowires have attracted significant interest with their subwavelength diameter dimensions due to their unique properties and wide range of applications [1], [2]. They enable the exploration of new operation regimes where strong field confinement, enhanced light-matter interactions, and group velocity dispersion (GVD) engineering are combined to allow high nonlinear interactions [2]. Tapering is a commonly used method for reducing the optical fiber dimension to obtain a core diameter less than 1 m. These structures present ideal devices for nonlinear processes such as soliton self-compression and supercontinuum (SC) generation with very low input pulse energy and in short fiber lengths [3]–[6]. Accurate analysis of the nanowires’ optical properties is required to successfully model and determine the optical propagation based on the real waveguide shape which could present geometrical imperfections [7]–[11]. In fact, it has been shown that by breaking the circular symmetry observed in the core region, birefringent and dispersive properties can be highly modified. Most achieved works have demonstrated that by introducing elliptical air holes in the photonic crystal fiber cladding, strong [8]–[10] birefringence can be achieved up to the order of if a small-sized elliptical air hole is introduced in or even the core as a deflected core [11]. Therefore, photonic nanowires Manuscript received February 08, 2012; revised April 01, 2012; accepted April 02, 2012. Date of publication April 06, 2012; date of current version May 07, 2012. This work was supported in part by “Institut Télécom” through the “Futur et Ruptures” program. The authors are with GreS’Com Laboratory, Engineering School of Communication of Tunis, University of Carthage, 2083 Ariana, Tunisia (e-mail:
[email protected];
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JLT.2012.2193873
with high birefringence up to have been recently demonstrated theoretically and experimentally by simply increasing the ellipticity and constructing an air-cladding elliptical core from tapering standard single-mode fibers [12]. In this paper, we propose a rigorous characterization of the optical properties including chromatic dispersion, birefringence, and nonlinear coefficient dependence on the ellipticity of photonic nanowires. This is the first time to our knowledge that a linear approximation to determine the equivalent-circular photonic nanowire exhibiting similar optical characteristics with the elliptical nanowire is proposed. Then, nonlinear characterization in the new designed elliptical nanowires is examined by studying the SC generation. The remainder of this paper is organized as follows. Section II presents the analysis of the optical properties of the fundamental mode in elliptical photonic nanowires including chromatic dispersion, birefringence and mode confinement. In Section III, we evaluate the nonlinear coefficient of elliptical photonic nanowires, investigate the nonlinear propagation in the anomalous dispersion regime, and analyze the effect of the ellipticity on the generated spectra. II. LINEAR OPTICAL CHARACTERIZATION PHOTONIC NANOWIRES
OF
ELLIPTICAL
With their high-contrast refractive index between the core and the air-cladding and the small core diameter less than 1 m, photonic nanowires are modeled as circular rods in air [3]. Thus, Maxwell’s equations can be reduced to the following Helmholtz equation satisfied by the vectorial electric field [3] (1) is the free space wavenumber, is the where refractive index profile of the photonic nanowire, and is the propagation constant of the optical mode. By applying a fullvectorial finite element method with dense meshes of – elements according to the dimensions of the photonic nanowire, we calculate the effective index of the fundamental mode HE . In fact, by dividing the fiber cross section into curvilinear hybrid edge/nodal elements and applying the finite element procedure, we obtain the following eigenvalue equation [13]: (2) and are the finite element matrices, and is where the discretized electric field vector consisting of the edge and nodal variables. Resolving (2) in the silica wavelength range
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BEN SALEM et al.: RIGOROUS OPTICAL MODELING OF ELLIPTICAL PHOTONIC NANOWIRES
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Fig. 1. Photonic elliptical nanowires.
gives the effective index of the fundamental mode as a function of the optical wavelength varying from 300 to 1800 nm. For our simulations, we define “ ” and “ ” to be the major and the minor diameters of the photonic nanowire (see Fig. 1), respectively. Then, we investigate the effect of increasing the ellipticity , by fixing “ ” and varying “ ”, on the optical properties of the fundamental mode HE including the birefringence, the chromatic dispersion, and the mode confinement. During the simulation, the diameter of the air-cladding is set to be much bigger than the major core diameter “ ” ensuring high accuracy for the optical properties determination. The study is performed for different photonic nanowires pumped at nm; the pump wavelength which covers larger anomalous GVD and gives efficient high nonlinear interactions [5].
Fig. 2. Chromatic dispersion of the x-polarized HE nanowires with different ellipticity
mode of photonic nm.
A. Chromatic Dispersion The GVD referred to as chromatic dispersion is determined from the second derivative of the mode effective index as a function of the wavelength. It is given by (3) where c is the velocity of light in vacuum. The calculated chromatic dispersions of the x-polarized HE of elliptical photonic nanowires with a fixed major diameter nm are presented in Fig. 2. They show the existence of two zero dispersion wavelengths (ZDW) which is one of the major properties of photonic nanowires. This can be justified by the dominance of the waveguide dispersion over the material dispersion when reducing the core nanowire diameter. We find that the slight increase of the ellipticity ( varies from 1 to 0.8) shifts the ZDW toward the blue light region. Therefore, at nm, the GVD value of the photonic nanowires is reduced from 267 to 107 ps/(nm km) when reaching high ellipticity . By performing an intensive study and calculating the GVD for different photonic nanowires, we find that for an introduced elliptical stress, an equivalent circular nanowire diameter with similar GVD profile, exhibiting relative errors not exceeding 5% for the GVD values and 1% for the ZDW positions, can be determined from a homothetic reduction of the original circular nanowire. Through a series of numerical calculations, we found that, for , the scaling homothetic coefficient “ ”
Fig. 3. (a) Equivalent circular nanowire diameter determined by a homothetic reduction “h” as a function of the ellipticity. (b) Equivalent nanowires exand hibiting similar GVD profiles found for .
of the photonic nanowire can be linearly related to the ellipticity and fitted to the following equation: (4) From (4), one can accurately determine the equivalent nanowire diameter defined as , exhibiting similar GVD profile as the elliptical nanowire, as seen in Fig. 3(a). For instance, elliptical core photonic nanowires with and nm [see Fig. 3(b)], exhibit similar GVD profiles as equivalent nanowires with diameters nm
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Fig. 4. Birefringence of elliptical photonic nanowires at
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nm.
and nm, respectively. Thus, the GVD curve of any photonic nanowire with an elliptical core can be deduced by calculating the corresponding GVD-circular photonic nanowire. Then, determining an equivalent circular nanowire diameter will prove important to avoid incorrect optical modeling of an elliptical nanowire if it is considered as a circular one with an “ ”-diameter.
Fig. 5. Fractional power of the HE mode inside the core of various phonm. tonic nanowire diameters with different ellipticities at
To obtain straightforward information about the power distribution in the radial direction, we evaluate the fractional power inside the core defined by
(5)
B. Birefringence The birefringence can be evaluated as the difference between the two effective indices of the x- and y-polarized modes. Fig. 4 shows the calculated birefringence of elliptical photonic nanowires with a maximum 20% reduction of the minor diameter “ ” to the major one “ ” We found that the birefringence increases dramatically with an increase of the ellipticity ( decreases from 1 to 0.8), and could therefore, a very high birefringence of the order of be achieved. We notice that, for each ellipticity value, an optimal nanowire diameter exhibiting the highest birefringence value is found. Therefore, increasing the ellipticity enlarges the optimal , strong birefrinbirefringent nanowire diameter. For is evaluated for the 550 nm diamgence of eter. For such high birefringence achieved in different elliptical nanowires, potential applications can be driven such as polarization maintaining fibers for high-bit-rate communication systems by guiding only through one polarization state of the fundamental mode and then eliminating the polarization crosstalk, as well as the polarization mode dispersion. Therefore, by maintaining the state of polarization during the light transmission, birefringent photonic nanowires show to be very attractive for optical fiber sensors and stable combs.
is the longitudinal component of the Poynting vector where and is the cross-sectional position vector. Fig. 5 depicts the calculated fractional power of the x-polarized mode inside the core for different photonic nanowire ellipticities. We mention that very small difference is found with the fractional powers calculated for the y-polarized mode. As we can see, the higher the ellipticity, the lower the core power then, the stronger the evanescent field. Therefore, 50% of the total light propagating in the evanescent field is obtained for the 400 nm circular photonic nanowire; however, increasing the ellipticity from 1 to 0.8 enhances the evanescent field with 13% more from the core fractional power. III. NONLINEAR PROPAGATION IN ELLIPTICAL PHOTONIC NANOWIRES A. Effective Nonlinearities Since a significant fraction of the optical mode propagates in the air as the evanescent field, an accurate estimation of the nonlinear coefficient is needed. The nonlinear coefficient is defined as [19]
C. Evanescent Field One of the fascinating properties of photonic nanowires is the existence of the evanescent field surrounding the core which is considered as a significant fraction of the optical mode propagating in the air outside the core. This makes photonic nanowires very attractive and well suited for sensing applications [14], [15]. A number of nanofiber devices based on evanescent coupling have been demonstrated, including coiled resonators [16], interferometers [17], and lasers [18].
(6) where
is the nonlinear index inside the core and outside. The calculation of the nonlinear coefficient of the HE mode (see Fig. 6) shows that photonic nanowires, with their small core diameters, exhibit tighter light confinement and very high nonlinear coefficient than other fibers. We find that the
BEN SALEM et al.: RIGOROUS OPTICAL MODELING OF ELLIPTICAL PHOTONIC NANOWIRES
TABLE I DISPERSION PARAMETERS AND INPUT SOLITON ORDERS ELLIPTICAL PHOTONIC NANOWIRES
Fig. 6. Nonlinear coefficient of the HE nanowire diameters with different ellipticities at
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FOR
DIFFERENT
mode of various photonic nm.
mode field diameter (MFD) follows the core diameter when it is larger than the wavelength of the propagating light . However, as the core becomes smaller than , the MFD reaches a minimum and then the nonlinear coefficient reaches a maximum for a specific waveguide diameter. We find that increasing the ellipticity enlarges the optimal nanowire diameter exhibiting the highest nonlinearity and enhances the nonlinearities for photonic nanowires with diameters nm. Nonlinear coefficients of 507 (W km) , 537.3 (W km) , 553.7 (W km) , and 569.7 (W km) are evaluated for the 800 nm photonic nanowires with ellipticities of 1, 0.9, 0.85, and 0.8, respectively. B. SC Generation Considering the determined optical waveguide properties of the photonic nanowires, the investigation of the nonlinear propagation is based on the resolution of the generalized nonlinear Schrödinger equation (GNLSE) given by [20]
(7) where is the slowly varying envelope, is the loss coefficient, and is the th-order dispersion coefficients. is the pump central frequency. The nonlinear response function includes the electronic and nuclear (Raman) contributions. The fractional contribution is taken and for , we used the experimentally determined Raman response of silica fibers. The resolution of the GNLSE is performed by the symmetrized split step Fourier method [20]. We based our study on the generation of SC in the anomalous GVD for different elliptical photonic nanowires aiming to analyze the impact of such ellipticity on the generated spectra. The 800 nm pump wavelength is selected with an input pulse full-width at half-maximum (FWHM) duration fs, which are found to be suitable parameters for
Fig. 7. Generated spectra on the HE polarization in 800 nm photonic nanowires with different ellipticity values after 1.15 mm propagation distance.
generating efficient SC [5]. We consider the injection of an input soliton order having an envelope field expression given by [20] where is the peak power, is the input soliton duration defined as , and is the chirp parameter controlling the initial chirp. The soliton order is defined as where is the dispersion length and is the nonlinear length. The initial chirp is set in order to optimize the SC generation as demonstrated by the authors in [5]. The loss is mitigated since millimeter nanowire length is considered. The GVD parameter and the third-order dispersion as well as the input soliton order for the considered elliptical photonic nanowires are listed in Table I. According to the optimization study presented in [5], we recall that efficient soliton self-compression and broadband SC was generated in a circular 800 nm diameter photonic nanowire with the aforementioned input parameters in only 1.15 mm nanowire length. The input energy was taken as 2.5 nJ and the nonlinear interaction corresponds to the injection of an input soliton order of . We notice that the scalar approach for the resolution of the nonlinear Schrödinger equation can be applied for the elliptical photonic nanowires ( from 1 to 0.8) with nm, since we find that two separately launched pulses on each axis (fast and slow) are totally decoupled in time after 1.15 mm propagation distance [21]. Fig. 7 depicts the generated spectra on the HE polarization in the photonic nanowires with different ellipticities. For the nanowire with , efficient soliton self-compression and broadband SC generation are shown due to the large GVD region and low third-order dispersion [2], [4], [5]. Therefore, we demonstrate
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that the increase of ellipticity degrades the generated bandwidth of the SC. This spectral narrowing can be justified by the reduction of the GVD region as well as the increase of the third-order dispersion, so that the temporal pulse does not undergo optimal compression and the efficient soliton compression cannot be achieved. Thus, for nonlinear applications, particular careful control of the manufacturing process of the nanowires is required. IV. CONCLUSION We have reported on the impact of the elliptical core imperfection on the optical properties of photonic nanowires. A linear approximation has been proposed to determine the equivalent circular nanowire diameter for an elliptical one exhibiting similar GVD profile. We demonstrate that high ellipticity makes the nanowire highly birefringent with a blue-shifted ZDW. We find strong birefringence up to the order of in elliptical photonic nanowires that could be very attractive for optical fiber sensors and stable combs. The analysis of the SC generation has been conducted for different elliptical photonic nanowires highlighting on the detrimental impact of the ellipticity on the spectral broadening. Thus, for such application, a careful control of the manufacturing process of the nanowires is required. An understanding of the sensitivity of the optical properties to the nanowire ellipticity will prove helpful for future nanowire designs. REFERENCES [1] L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature, vol. 426, pp. 816–819, 2003. [2] M. A. Foster, A. C. Turner, M. Lipson, and A. L. Gaeta, “Nonlinear optics in photonic nanowires,” Opt. Exp., vol. 16, pp. 1300–1320, 2008. [3] L. Tong, J. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Exp., vol. 12, pp. 1025–1035, 2004. [4] M. Foster, A. L. Gaeta, Q. Cao, and R. Trebino, “Soliton-effect compression of supercontinuum to few-cycle durations in photonic nanowires,” Opt. Exp., vol. 13, pp. 6848–6855, 2005. [5] A. Ben Salem, R. Cherif, and M. Zghal, “Low-energy single-optical-cycle soliton self-compression in photonic nanowires,” J. Nanophoton., vol. 5, pp. 059506-1–059506-6, 2011.
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