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904 J. Opt. Soc. Am. B/Vol. 27, No. 5 /May 2010 Lu et al.

Optimization of supercontinuum generation inair–silica nanowires

Hua Lu, Xueming Liu,* Yongkang Gong, Xiaohong Hu, and Xiaohui Li

State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and Precision Mechanics,Chinese Academy of Sciences, Xi’an 710119, China

*Corresponding author: liuxm@opt.ac.cn

Received January 6, 2010; revised February 26, 2010; accepted February 26, 2010;posted March 1, 2010 (Doc. ID 122101); published April 14, 2010

An effective hybrid genetic algorithm (GA) for optimizing air–silica nanowires, the incident pulse, and super-continuum (SC) generation, is proposed in this paper. Based on the proposed algorithm, the dispersion andnonlinearity of air–silica nanowires, as well as the duration and chirp of incident pulses, are optimized toachieve SC generation with a broader, smoother, and more intense spectrum. It is found that the optimizedspectrum becomes smoother from 740 to 1500 nm and is broadened by 300 nm. Meanwhile, the spectral inten-sity in the range of 450–945 nm is significantly increased by a factor of 10. © 2010 Optical Society of America

OCIS codes: 060.4370, 060.4005, 150.1135.

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. INTRODUCTIONhe nonlinear broadening of the injected spectrum is the

nherent property of nonlinear optics and has been stud-ed for years. The first report of supercontinuum (SC) gen-ration in bulk glass was by Alfano and Shapiro in 19701,2]. Since then, SC has been generated in other opticalonlinear media, including gases, solids, liquids, andarious kinds of waveguides [3]. Until the late 1990s, aew kind of optical waveguide, i.e., photonic crystal fiberPCF), was found and used to generate ultrabroadbandigh-intensity spectra through SC generation [3–6]. Soar, a number of research studies on SC generation inCF have been done numerically and experimentally. Thenderlying physical mechanisms of spectral broadeningenerally include self- and cross-phase modulation [7,8],ispersion wave generation [9], four-wave mixing [10],odulation instability [11], soliton formation and fission

10,12–14], and stimulated Raman scattering [15,16]. Theactors influencing SC generation contain the incidentulse peak power, pulse duration, pulse initial chirp,ulse wavelength, fiber type, pulse propagation distance,nd pulse noise, etc. [3,17–20].Recently, photonic nanowires with core diameter below�m have attracted more and more attention and been

tudied intensively [21–23]. Because of the inherent largeispersion, high nonlinearity, and maximal light confine-ent [22], the nanowires have been utilized to investigate

onlinear optical phenomena, especially SC generation23,24]. SC generation has applications in many fieldsuch as spectroscopy, pulse compression, frequency me-rology, and optical coherence tomography [25,26].

In this paper, we utilize the generalized nonlinearchrödinger equation (GNLSE) as a feasible method toodel SC generation [27,28]. Many algorithms, such as

imulated annealing method [29], and genetic algorithmsGAs) [30], can be applied to optimize the SC spectrum.As have a number of advantages including the weak

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ondition (discontinuity of functions), but the traditionalAs may trap in a local optimum of the search space be-

ause of the well-known drawbacks, i.e., weak exploita-ion capability and premature convergence [30]. Althoughhe simulated annealing method can avoid local conver-ence, it cannot simultaneously obtain the multiple op-ima that are useful in some optimization problems.

In this paper, aiming to optimize the SC spectrum, weropose an effective hybrid GA including such techniquess scaling, sharing, clustering, and elitist replacement.hese techniques extend the traditional GAs to avoid lo-al trap by forcing GAs to maintain a diverse populationf members throughout the research. This method canompletely solve the complicated problems of multimodalunction optimization and can obtain both the global andocal optima [31]. Two test functions are presented toerify the proposed hybrid GA. We calculate the nonlinearoefficient and group-velocity dispersion (GVD) throughoynting vector and Maxwell’s equations, respectively, asell as numerically simulate SC generation in air–silicaanowires by the GNLSE. The optimized results indicatehat there exist the optimal core diameter, nanowireength, pulse duration, and initial chirp. Our work is ben-ficial to the design of SC generation and the in-depth re-earch of photonic nanowires.

. SUPERCONTINUUM GENERATION INIR–SILICA NANOWIRES. Group-Velocity Dispersionne can reduce Maxwell’s equations to the followingelmholtz equation satisfied by the electric fields [7]:

�� 2 + n2k 2 − �2�e� = 0. �1�

t 0

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Lu et al. Vol. 27, No. 5 /May 2010/J. Opt. Soc. Am. B 905

ere, n is the refractive index of the dielectric material,0=2� /�0 is the free-space wave vector, and � is theropagation constant. The eigenvalue equation of Eq. (1)or fundamental mode (HE11 modes) is described as fol-ows [32]:

� J1��U�

UJ1�U�+

K1��W�

WK1�W��� J1��U�

UJ1�U�+

n22K1��W�

n12WK1�W��

= � �

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. �2�

ere, U=a�k02n1

2−�2�1/2, W=a��2−k02n2

2�1/2, and=k0a�n1

2−n22�1/2. J1�U� and K1�W� are the first-kind

essel and modified Bessel functions, respectively. n1 ishe refractive index of the silica core (1.4533 at the wave-ength of 800 nm), n2 is the refractive index of the airladding (assumed to be 1.0) [32]. We can numericallyolve the eigenvalue equation of fundamental mode (HE11ode) expressed by Eq. (2) and obtain the propagation

onstant �. The GVD of photonic nanowires can then bechieved by using the following dispersion formula [33]:

vg =dw

d�= −

2�c

�2

d�

d�, �3�

D =d�vg

−1�

d�. �4�

ere, � is the wavelength, c is the light speed in vacuum,nd vg is the group velocity.The GVD curves of the air–silica nanowires with differ-

nt core diameters are plotted in Fig. 1 [22]. The air–silicaanowires show very large normal dispersion �D�0� inhe ultraviolet (UV) and infrared region. To generate SCith femtosecond pulse in the anomalous dispersion re-ion, we choose the incident pulse center wavelength at00 nm and the core diameter at 800 nm. The GVDalue of the air–silica nanowire is estimated to be31.7 ps·nm−1·km−1.

ig. 1. (Color online) GVD in air–silica nanowires with00–900 nm core diameter. When the pulse wavelength is cho-en at 800 nm, the nanowires locate in the anomalous dispersionegion.

. Effective Nonlinearitywing to the large core-cladding refractive-index differ-nce and ultrasmall core diameter, the air–silica nano-ires have tighter light confinement and higher nonlin-ar coefficient than conventional fibers [32]. Therefore,uch kind of fiber is very suitable for research ofonlinear-optical phenomena . The effective nonlinearityescribed by nonlinear parameter � is defined as [34]

� =2�

�·� n2Sz

2d2r

�� Szd2r�2

�2�

�·

�0

a

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0

2�

d�

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a

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2�

d� +�a

�

Sz2rdr�0

2�

d��2 .

�5�

ere, n2=2.610−20 m2·W−1 is the nonlinear refractivendex [7]. We assume that the nonlinear refractive indexf air is negligible compared to that of silica, so the inte-ral in the numerator is evaluated only over the core re-ion [34].

The nonlinear parameter of air–silica nanowires for an00 nm wavelength is plotted in Fig. 2. In our simulation,he core diameter is selected to be 800 nm and the corre-ponding nonlinear parameter is estimated to be07.3 W−1·km−1. The effective modal area is Aeff2�n2 /�0�=0.4 �m2 for an 800 nm core-diameter nano-ire with injected 800 nm wavelength.

. Simulation of Supercontinuum Generationonsidering the waveguide properties of air–silica nano-ires, the theoretical model must include all higher-orderffects when ultrashort pulse is input. Our simulationsre based on the GNLSE [7]:

�A

�z+

2A − i

k�2

ik�k

k!

�kA

�Tk

= i��A2A +i

w0

�

�T�A2A� − TRA

�A2

�T � , �6�

ig. 2. (Color online) Diameter-dependent nonlinearity of air–ilica nanowires for 800 nm wavelength.

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906 J. Opt. Soc. Am. B/Vol. 27, No. 5 /May 2010 Lu et al.

here all higher-order effects are included such asigher-order dispersion, self-steepening (SS), and intra-ulse Raman scattering (IRS). Meanwhile, the fiber loss s also considered. A=A�z , t� is the electric field envelope,nd � is the nonlinear coefficient. �k are the coefficients inhe Taylor series expansion of propagation constant � athe input pulse’s central frequency w0. T= t−�1z is therame of reference moving with the pulse. TR is treated asfitting parameter.The GNLSE is solved by predictor–corrector split-step

ourier method [27]. The input pulse is assumed to be un-hirped hyperbolic secant pulse with central wavelength0=800 nm, (i.e., the central wavelength of a Ti:sapphireaser). P0=3 kW, I0=P0 /Aeff=0.75 TW·cm−2, and T060 fs ��FWHM=106 fs�. The two- to six-order dispersionarameters are considered to exactly simulate SC genera-ion in air–silica nanowires. The particular values at00 nm are �2=−78.65 ps2·km−1, �3=5.34610−3 ps3·km−1, �4=4.18810−4 ps4·km−1, �5=−1.87810−6 ps5·km−1, �6=7.6610−9 ps6·km−1, respectively.he characteristic self-steepening parameter=1/ �w0 ·T0� and TR that governs the effects of IRS are.0071 and 3 fs, respectively. Based on the flame-brushapering technique, photonic nanowires with uniform di-meter and low loss have been fabricated successfully.herefore, the loss coefficient is set to be.002 dB·mm−1 [23]. When the nanowire length z=0.15LD=6.87 mm (LD=T0

2 / �2=45.8 mm is the dispersionength), the output spectrum is obtained and depicted inig. 3. We can see that a broad spectrum is generated andovers the wavelength range from 700 to 1500 nm.

. HYBRID GENETIC ALGORITHM FORPTIMIZING SUPERCONTINUUMENERATIONenerally, a good SC spectrum should possess a broad,

mooth, and intense profile. In this paper, aiming to opti-ize the nanowire parameters with a good SC spectrum,e propose an effective hybrid GA. Considering the com-rehensive feature, the objective function can be chosens

F =�w1

w2

�I�w�dw, �7�

ig. 3. (Color online) The spectral profiles of SC in an 800 nmore-diameter nanowire after propagating 0 and 6.87 mm,espectively.

here 1 and 2 are the minimum and maximum fre-uencies in the spectrum we have interest in. I� � is thentensity of SC obtained by solving Eq. (6). In our simu-ation, 1 and 2 are assumed to be frequencies at wave-engths of 1800 and 450 nm, respectively. The proceduref an implementation case of our hybrid GA is given asollows:

(1) Initialization and coding. Each feasible solutions coded as an individual, namely, by chromosome. Theopulation is initialized; i.e., we randomly generate N100 individuals with the least interval of 0.0001 be-

ween the given lower and upper bounds of variables.hen, the individuals are encoded from the binary to realariables and the corresponding fitness values are ob-ained. We choose the shortest normalized Niche radius=0.01. The number of elite Me is set to be 3, and theumber of peak centers (i.e., niches) Mp=3.(2) Selection, crossover, mutation. According to the

btained fitness values, N individuals are selected byournament selection method. Then offspring is createdy using enhanced edge recombination crossover betweenach stochastic pair with the crossover rate Pc=0.8. Eachit in every chromosome mutates with the rate Pm=0.01.hese procedures of selection, crossover, and mutationre consistent with those of the traditional GAs [30].(3) Calculating fitness and elite replacement. All

ndividuals in the new population are encoded from bi-ary to real variables. The raw fitness values of all indi-iduals, i.e., the object values, are calculated. Throughinear scaling, final fitness values fj �j=1,2, . . . ,N� are ob-ained. Then, an individual with the lowest fitness valuen each niche is replaced by the best individual in the cor-esponding niche of father generation.

(4) Clustering and sharing. The obtained populations sorted by ascending order of fitness value. Adaptive Mpeak centers are found out; then the rest of the individu-ls are placed into the nearest (Euclidean distance) peakenter. The niche number �mj� of each individual is calcu-ated (mj is equal to the individual count in each peakenter) followed by the shared fitness value of each indi-idual, i.e., fj�= fj /mj �j=1,2, . . . ,N�. Each peak can be re-arded as a niche in the multimodal domain. Throughharing, as one of the niching techniques, we can retainopulation diversity effectively.(5) Terminating conditions. When the generation

umber �g� is equal to 60, the hybrid GA is terminatednd the optimal results of all niches are obtained. Other-ise, the procedure is reiterated from step (2).

o justify the proposed GA, two examples are given. Theest functions are as follows:

y = cos4�5��x0.75 − 0.05�� 0 � x � 0.45, �8�

y = x + 10 sin�5x� + 7 cos�4x� 1.0 � x � 4.5. �9�

he proposed GA method is used to optimize the two ob-ective functions above. The optimized results are de-icted in Fig. 4, where each symbol ‘ *’ represents the op-imal solution. From Fig. 4(a) we can see that the threeaxima are obtained simultaneously, with x values of

.0184, 0.1575, and 0.3444, respectively. The multiple

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Lu et al. Vol. 27, No. 5 /May 2010/J. Opt. Soc. Am. B 907

axima of Eq. (9) are also achieved as shown in Fig. 4(b).he simulation results verify that the hybrid GA is robustnd has excellent capability to search the global and localptimal solutions simultaneously.

. RESULTShe dispersion and nonlinear coefficient in air–silicaanowires closely correlate with their core diameter. Inur optimization, we consider the structure of nanowiresthe core diameter �Dcore� and length �z�), and the proper-ies of the incident pulse (the initial chirp �C� and dura-ion �T0�) simultaneously. Here, Dcore is set from00 to 900 nm, z from 3 to 10 mm, C from −5 to 5, T0rom 40 to 80 fs, Me=3, and Mp=3, respectively. The cor-esponding particular values and nonlinear coefficientan be seen in Table 1. The other parameters are theame as those shown in Section 2. Figure 5 depicts the op-imal SC spectra with the object values of 1.84781018,.86871018, and 1.98951018, respectively. The corre-ponding parameters are Dcore=717.71 nm, z=6.68 mm,=−0.30, T0=79.72 fs; Dcore=847.09 nm, z=5.80 mm,=−1.30, T0=76.95 fs; and Dcore=838.00 nm,=5.47 mm, C=−0.200, T0=78.98 fs. Compared with theriginal spectrum with the object value of 1.35431018 inig. 3 �z=6.87 mm�, the optimized spectrum is broadenedy approximately 300 nm and becomes smoother from40 to 1500 nm. Meanwhile, the spectral intensity in theange of 450–945 nm is improved by a factor of 10 andan be seen in Fig. 6.

Table 1. Values and Nonlinear Coefficientswith Wave

Dcore (nm) 700 900

�2 �ps2/km� −31.42 −88.68�3 �ps3/km� −0.1772 0.0842�4 �ps4/km� 9.61010−4 1.33910−4

�5 �ps5/km� −3.46510−6 −8.72510−7

�6 �ps6/km� 1.21810−8 3.96710−9

� �W−1/m� 0.5883 0.4339

ig. 4. (Color online) Searching the maxima of Eq. (8) and Eq.9) through the proposed hybrid GA. (a) x values: 0.0184, 0.1575,nd 0.3444. (b) x values: 1.5736, 2.9135, and 4.2045.

. CONCLUSIONSe have numerically simulated the GVD and nonlinear

oefficient in air–silica nanowires by using Maxwell’squations and the Poynting vector, and investigated SCeneration by solving GNLSE with higher-order disper-ion and intrapulse Raman scattering. An effective hybridenetic algorithm (GA) for optimizing SC generation haseen proposed. Numerical simulations show that our op-imized algorithm is able to avoid trapping in local opti-al solutions and overcomes the inherent drawbacks ofeak exploitation capability of traditional GAs. Espe-

ially, our algorithm has the advantages of obtaining thelobal and local optimal solutions simultaneously. Whenhe pump is with 3 kW peak power at 800 nm wavelength,he optimized core diameter, nanowire length, pulse du-ation, and initial chirp are 838.00 nm, 5.47 mm, 78.98 fs,nd −0.200, respectively. The optimized spectrum be-omes smoother from 740 to 1500 nm and is broadened by00 nm. Moreover, the spectral intensity in the range of50–945 nm is improved by a factor of 10.

CKNOWLEDGMENTShis work was supported by the “Hundreds of Talentsrograms” of the Chinese Academy of Sciences and by theational Natural Science Foundation of China (NSFC)nder grants 10874239 and 10604066.

e Boundary and Optimal Core Diametersh 800 nm

717.71 847.09 838.00

−44.41 −86.10 −85.11−0.1329 0.0508 0.0434

8.33410−4 2.54610−4 2.80310−4

−3.10410−6 −1.28910−6 −1.37610−6

1.12010−8 5.396510−9 5.69810−9

0.5741 0.4711 0.4785

ig. 5. (Color online) The spectra after optimization. The corre-ponding parameters used are (a) Dcore=717.71 nm, z=6.68 mm,=−0.30, T0=79.72 fs, (b) Dcore=847.09 nm, z=5.80 mm,=−1.30, T0=76.95 fs, and (c) Dcore=838.00 nm, z=5.47 mm,=−0.200, T0=78.98 fs.

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908 J. Opt. Soc. Am. B/Vol. 27, No. 5 /May 2010 Lu et al.

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