Soliton-self compression in highly nonlinear chalcogenide photonic nanowires with ultralow pulse energy

doi:10.1364/OE.19.019955

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Soliton-self compression in highly nonlinear chalcogenide photonic nanowires with ultralow pulse energy

Amine Ben Salem, Rim Cherif, and Mourad Zghal »View Author Affiliations

Optics Express, Vol. 19, Issue 21, pp. 19955-19966 (2011)
http://dx.doi.org/10.1364/OE.19.019955

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▸ Article Outline
– Abstract
1. Introduction
2. Theory and numerical analysis
3. Numerical results
4. Conclusion
– References and links

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Abstract

We design As₂Se₃ and As₂S₃ chalcogenide photonic nanowires to optimize the soliton self-compression with short distances and ultralow input pulse energy. We numerically demonstrate the generation of single optical cycle in an As₂S₃ photonic nanowire: a 5.07 fs compressed pulse is obtained starting from 250 fs input pulse with 50 pJ in 0.84 mm-long As₂S₃ nanowire. Taking into account the high two photon absorption (TPA) coefficient in the As₂Se₃ glass, accurate modeling shows the compression of 250 fs down to 25.4 fs in 2.1 mm-long nanowire and with 10 pJ input pulse energy.

1. Introduction

Highly nonlinear optical waveguides have great interest for compact, low power and all optical nonlinear applications. In fact, photonic nanowires with diameters smaller than the wavelength of the guided light attract considerable interest due to their unique optical and nonlinear properties for wide range of applications [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 426(6968), 816–819 (2003). [CrossRef] [PubMed]

]. They are obtained by tapering optical fibers which is the commonly used method for reducing optical fiber dimensions and engineering the waveguide dispersion [2

M. A. Foster, A. C. Turner, M. Lipson, and A. L. Gaeta, “Nonlinear optics in photonic nanowires,” Opt. Express 16(2), 1300–1320 (2008). [CrossRef] [PubMed]

–4

Y. K. Lizé, E. C. Mägi, V. G. Ta’eed, J. A. Bolger, P. Steinvurzel, and B. J. Eggleton, “Microstructured optical fiber photonic wires with subwavelength core diameter,” Opt. Express 12(14), 3209–3217 (2004). [CrossRef] [PubMed]

]. These structures are not only suitable for nanophotonic devices but also enable nonlinear process such as soliton-effect compression and broadband supercontinuum (SC) generation at low input pulse energies [5

R. R. Gattass, G. T. Svacha, L. M. Tong, and E. Mazur, “Supercontinuum generation in submicrometer diameter silica fibers,” Opt. Express 14(20), 9408–9414 (2006). [CrossRef] [PubMed]

S. G. Leon-Saval, T. A. Birks, W. J. Wadsworth, P. St. J. Russell, and M. W. Mason, “Supercontinuum generation in submicron fibre waveguides,” Opt. Express 12(13), 2864–2869 (2004). [CrossRef] [PubMed]

]. Generation of few-optical cycles directly from oscillators requires special designed cavities with high reflectivity dispersion compensating mirrors, and initial energy in the order of hundreds of µJ. Therefore, the used compression technique consists of using external passive or active dispersion compensation schemes in order to compress the laser pulse initially broadened by self-phase modulation (SPM) in a waveguide. In both techniques, the required dispersion compensation requires complex electronically controlled feedback systems. An impressive method toward the monocycle regime relying on the soliton-effect compression has been used to successfully compress low-energy ps-pulses without additional dispersion compensation schemes [7

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45(13), 1095–1098 (1980). [CrossRef]

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, and W. J. Tomlinson, “Extreme picosecond pulse narrowing by means of soliton effect in single-mode optical fibers,” Opt. Lett. 8(5), 289–291 (1983). [CrossRef] [PubMed]

]. This method exploits the property of propagating high-order soliton in the anomalous dispersion regime in optical fibers where efficient compression results from the interplay between the dispersion and the self phase modulation effects. Therefore scaling laws have been developed for the soliton-self compression parameters, for hyperbolic-secant pulses, with numerical and analytical methods [8

C. M. Chen and P. L. Kelley, “Nonlinear pulse compression in optical fibers: scaling laws and numerical analysis,” J. Opt. Soc. Am. B 19(9), 1961–1967 (2002). [CrossRef]

The introduction of highly nonlinear photonic nanowires with anomalous group velocity dispersion at visible and near-infrared wavelengths has enormously contributed in the study of soliton-self compression techniques and the generation of few to single optical cycles. Specifically, the broad region of anomalous dispersion regime can be shifted for photonic nanowires and extends into the visible for silica nanowires and into the midinfrared for chalcogenide nanowires. Large anomalous group velocity dispersion region with low third-order dispersion are required to achieve efficient soliton-self compression [2

M. A. Foster, A. C. Turner, M. Lipson, and A. L. Gaeta, “Nonlinear optics in photonic nanowires,” Opt. Express 16(2), 1300–1320 (2008). [CrossRef] [PubMed]

,10

M. Foster, A. L. Gaeta, Q. Cao, and R. Trebino, “Soliton-effect compression of supercontinuum to few-cycle durations in photonic nanowires,” Opt. Express 13(18), 6848–6855 (2005). [CrossRef] [PubMed]

]. In the 800 nm region, Foster et al. [10

], experimentally demonstrated soliton-self compression by performing a cross correlation frequency resolved optical grating (XFROG) measurements. They showed the generation of 6.8 fs compressed pulse at low input energy from initial 70 fs in 980 nm core diameter air-silica photonic nanowire. In addition, they theoretically predicted soliton-self compression of 30 fs input pulse to 1.8 fs in 650 μm-long 800 nm-diameter air-silica nanowire.

Most achievements in ultrafast nonlinear optics have been made in photonic nanowires based on silica glass but studies on highly nonlinear glasses have been recently started too. Chalcogenide glasses based on As₂S₃ and As₂Se₃ have been the subject of intense investigations due to their high nonlinear coefficient n₂ (100-1000 times larger than of silica glass), low linear loss and low to moderate two photon absorption (TPA) coefficient [11

J. M. Harbold, F. O. Ilday, F. W. Wise, J. S. Sanghera, V. Q. Nguyen, L. B. Shaw, and I. D. Aggarwal, “Highly nonlinear As-S-Se glasses for all-optical switching,” Opt. Lett. 27(2), 119–121 (2002). [CrossRef] [PubMed]

]. Recent works have showed the generation of SC in such photonic chalcogenide As₂S₃ and As₂Se₃ nanowires with ultra low input energy at a pump wavelength of 1550 nm [12

D. I. Yeom, E. C. Mägi, M. R. E. Lamont, M. A. F. Roelens, L. Fu, and B. J. Eggleton, “Low-threshold supercontinuum generation in highly nonlinear chalcogenide nanowires,” Opt. Lett. 33(7), 660–662 (2008). [CrossRef] [PubMed]

,13

D. D. Hudson, S. A. Dekker, E. C. Mägi, A. C. Judge, S. D. Jackson, E. Li, J. S. Sanghera, L. B. Shaw, I. D. Aggarwal, and B. J. Eggleton, “Octave spanning supercontinuum in an As₂S₃ taper using ultralow pump pulse energy,” Opt. Lett. 36(7), 1122–1124 (2011). [CrossRef] [PubMed]

]. However, to our knowledge, soliton self-compression in such highly nonlinear nanowires has not been studied so far.

In this paper, we investigate the linear and the nonlinear properties of As₂Se₃ and As₂S₃ chalcogenide photonic nanowires for soliton self-compression. In Section 2, we study the theory and the numerical modeling behind accurate calculation of the optical properties of photonic nanowires including chromatic dispersion, effective mode area and nonlinear coefficient. In addition, the cut-off condition of single mode operation in photonic nanowires is determined. Section 3 shows the results of the optical properties modeling and analyzes the soliton-self compression in chalcogenide photonic nanowire glasses. Aiming to optimize the soliton-effect compression in photonic nanowire, we select the 800 nm core diameter nanowire size for which large region of the anomalous group velocity dispersion and low third-order dispersion are found at pump wavelength of 1550 nm for the air-chalcogenide (As₂Se₃ and As₂S₃) photonic nanowires. We show that we can generate less than one optical cycle in 800 nm diameter As₂S₃ nanowires at very low pulse energy of pJ levels. In fact, pumping the 800 nm air-As₂S₃ nanowire at λ_p = 1550 nm with 50 pJ energy allows the compression of a pre-chirped 250 fs down to 5.07 fs. Therefore, soliton-effect compression in the anomalous dispersion regime in the 800 nm diameter As₂Se₃ nanowire has been studied and shows the possibility of generating 25.4 fs starting from 250 fs with 10 pJ energy at λ_p = 1550 nm. The effect of TPA has been also studied since As₂Se₃ presents a high TPA coefficient comparing to As₂S₃.

2. Theory and numerical analysis

2.1 Mode propagation

Since they exhibit large refractive index difference between the core and the air-cladding, photonic nanowires with core diameters less than one micron are modeled as circular rods in air [14

L. Tong, J. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Express 12(6), 1025–1035 (2004). [CrossRef] [PubMed]

]. Thus, Maxwell’s equations can be reduced to the following Helmholtz equation satisfied by the vectorial electric field E; it is given by [14

L. Tong, J. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Express 12(6), 1025–1035 (2004). [CrossRef] [PubMed]

\nabla^{2} E + (n^{2} k_{0}^{2} - β^{2}) E = 0

(1)

where

\nabla

is the Laplacian operator,

k_{0} = 2 π / λ

is the free space wavenumber, n = n(r) is the refractive index profile of the photonic nanowire and β is the propagation constant of the optical mode. The eigenvalue equations of Eq. (1) for the hybrid modes HE_νm and EH_νm are given by [14

L. Tong, J. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Express 12(6), 1025–1035 (2004). [CrossRef] [PubMed]

[\frac{J_{υ}^{'} (U)}{U J_{υ}^{} (U)} + \frac{K_{υ}^{'} (W)}{W K_{υ}^{} (W)}] [\frac{J_{υ}^{'} (U)}{U J_{υ}^{} (U)} + {(\frac{n_{c l}}{n_{c}})}^{2} \frac{K_{υ}^{'} (W)}{W K_{υ}^{} (W)}] = {(\frac{ν β}{k_{0} n_{c}})}^{2} {(\frac{V}{U W})}^{4}

(2)

where J_v is the Bessel function of the first kind, K_v is the modified Bessel function of the second kind,

U = d \sqrt{k_{0}^{2} n_{c}^{2} - β^{2}} / 2

W = d \sqrt{β^{2} - k_{0}^{2} n_{c l}^{2}} / 2

V = k_{0} d \sqrt{n_{c}^{2} - n_{c l}^{2}} / 2

, and d is the nanowire core diameter. We assume that the index of the air-cladding n_cl is 1.0, and use the Sellmeier-type dispersion formula to obtain the refractive index n_c of the chalcogenide core material. The V-number is the normalized frequency and related to the numerical aperture

N A = \sqrt{n_{c}^{2} - n_{c l}^{2}}

. It is expressed by

V = π d / (λ . N A)

Like conventional optical fibers, photonic nanowires experience single-mode (HE₁₁) guiding for V < 2.405.

2.2 Chromatic dispersion

In order to determine the chromatic dispersion of the chalcogenide nanowires, we need to calculate the effective index

n_{e f f} = β λ / (2 π)

of the fundamental mode HE₁₁. We apply a full vectorial finite element method (FEM) with dense meshes made up of 10⁴~10⁵ elements according to the dimensions of the photonic nanowire. 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 [15

M. Zghal and R. Cherif, “Impact of small geometrical imperfections on chromatic dispersion and birefringence in photonic crystal fibers,” Opt. Eng. 46(12), 128002 (2007). [CrossRef]

[K] {E} - k_{0}^{2} n_{e f f}^{2} [M] {E} = {0}

(3)

where [K] and [M] are the finite element matrices, {E} is the discretized electric field vector consisting of the edge and nodal variables. Resolving Eq. (1) in the As-S and As-Se wavelength ranges gives the effective index n_eff of the fundamental mode as a function of the optical wavelength. Thus, the group velocity dispersion (GVD) referred to as chromatic dispersion (D_c) can be determined from the second derivative of the mode effective index as a function of the wavelength. It is given by:

D_{c} = - \frac{λ}{c} \frac{d_{}^{2} n_{e f f}}{d λ_{}^{2}}

(4)

where c is the velocity of light in vacuum.

2.3 Evanescent field and effective nonlinearity

With their small core dimensions and the large core-cladding refractive-index difference, chalcogenide nanowires exhibit tighter light confinement and higher nonlinear coefficient than standard fibers. However, in nanowires, the optical mode can extend evanescently outside the core boundaries, and then an accurate estimation of the nonlinear coefficient γ is needed. The nonlinear coefficient γ is then defined as [16

M. A. Foster, K. D. Moll, and A. L. Gaeta, “Optimal waveguide dimensions for nonlinear interactions,” Opt. Express 12(13), 2880–2887 (2004). [CrossRef] [PubMed]

γ = \frac{2 π}{λ} \frac{\int n_{2} . S_{z}^{2} d^{2} r}{{(\int S_{z}^{} d^{2} r)}^{2}}

(5)

where S_z is the longitudinal component of the Poynting vector, n₂ is the nonlinear index and r is the cross sectional position vector. Because of a significant fraction of the optical mode propagates in the air outside the core and n₂ of the cladding (air) is negligible compared to that of the core glass, the integrals in the numerator are evaluated only over the glass core region. The integrals in the denominator are evaluated over the total transverse section of the fiber. In addition, to obtain complete information about the power residing in the evanescent field, we evaluate the fractional power η_EF outside the core. It is given by

η_{EF} = 1 - η

where η is the fractional power inside the core and it is expressed by:

η = \frac{\int_{0}^{d / 2} \int_{0}^{2 π} S_{z} . r d r . d φ}{\int_{0}^{\infty} \int_{0}^{2 π} S_{z} . r d r . d φ}

(6)

2.4 Nonlinear propagation

Considering the optical waveguide properties of the chalcogenide photonic nanowire, the investigation of the nonlinear propagation and the soliton-self compression are based on the resolution of the generalized nonlinear Schrödinger equation (GNLSE). The authors in ref [17

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78(17), 3282–3285 (1997). [CrossRef]

]. attempt to overcome the limitations of the slowly-varying envelope approximation (SVEA) and have demonstrated that the concept of envelope and first-order equation can be used even in the extreme case of single-optical-cycle-regime. The GNLSE is given by [18

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, 2007).

\begin{array}{l} \frac{\partial A}{\partial z} = - \frac{α}{2} A + \sum_{m \geq 0}^{} \frac{j^{m + 1} β_{m}}{m!} \frac{\partial^{m} A}{\partial t^{m}} + j (γ + j \frac{α_{2}}{2 A_{e f f}}) (1 + \frac{j}{ω_{0}} \frac{\partial}{\partial t}) \\ \times (A (z, t) \int_{0}^{+ \propto} R (t') {| A (z, t - t') |}^{2} d t') \end{array}

(7)

where A(z,t) is the slowly varying envelope, α is the loss coefficient and β_m the m^th-order dispersion coefficients. α₂ is the TPA coefficient of the core glass and ω₀ is the pump central frequency. The nonlinear response function R(t) includes the instantaneous and the delayed Raman contributions. It is given by [18

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, 2007).

R (t) = (1 - f_{R}) δ (t) + f_{R} h_{R} (t)

(8)

f_R is the fractional contribution of the material. The delayed Raman response h_R(t) is expressed through the Green’s function of the damped harmonic oscillator [19

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25(12), 2665–2673 (1989). [CrossRef]

h_{R} (t) = \frac{τ_{1} ² + τ_{2} ²}{τ_{1} τ_{2} ²} \exp (- \frac{t}{τ_{2}}) \sin (\frac{t}{τ_{1}})

(9)

where the parameters τ₁ and τ₂ correspond respectively to the inverse of the phonon oscillation frequency and the bandwidth of the Raman gain spectrum.

The resolution of the GNLSE is performed by the symmetrized split step Fourier method. The 4th-order Runge-Kutta algorithm was used in our calculations. To have accurate numerical results, 2¹³ time and frequency discretization points and a longitudinal step size <1 µm were used in our simulations. To start the resolution, we need to set the input pulse shape and duration. We consider the injection of an input soliton order N having an envelope field expression given by [18

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, 2007).

A (0, t) = \sqrt{P_{0}} \sec h (\frac{t}{T_{0}}) \exp (- i C \frac{t^{2}}{2 T_{0}^{2}})

(10)

where P₀ is the peak power and T₀ is the input soliton duration defined as T_FWHM /1.763. T_FWHM is the input pulse full width at half maximum (FWHM) duration. C is the chirp parameter controlling the initial chirp. The soliton order N is defined as [18

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, 2007).

]

N^{2} = L_{D} / L_{N L} = T_{0}^{2} γ P_{0} / | β_{2} |

where

L_{D} = T_{0}^{2} / | β_{2} |

is the dispersion length and

L_{N L} = {(γ P_{0})}^{- 1}

is the nonlinear length.

3. Numerical results

In this section, we determine the optical properties of air-chalcogenide nanowires which are produced from tapering conventional single mode fibers. We investigate the optical field distribution in air-As₂Se₃ and air-As₂S₃ nanowires and determine the cut-off condition for single mode operation in these photonic nanowires. Then, we analyze the soliton-self compression in both As₂Se₃ and As₂S₃ air-chalcogenide photonic nanowires. The study is based on the optimization of initial chirp in order to generate efficient soliton-self compression with the lowest input pulse energy.

3.1 Air-As₂Se₃ nanowires

We have carried out a rigorous calculation of the chromatic dispersion of the different air-As₂Se₃ photonic nanowires by using the Sellmeier equation of the As₂Se₃ glass defined as

n (λ) = \sqrt{A + B λ^{2} / (λ^{2} - D) + C λ^{2}}

where A = −4.5102, B = 12.0582, C = 0.0018 µm⁻² and D = 0.0878 µm² [20

R. Cherif, A. Ben-Salem, M. Zghal, P. Besnard, T. Chartier, L. Brilland, and J. Troles, “Highly nonlinear As₂Se₃-based chalcogenide photonic crystal fiber for midinfrared supercontinuum generation,” Opt. Eng. 49(9), 095002 (2010). [CrossRef]

]. The unit of λ is µm. Figure 1 depicts the chromatic dispersion as a function of the wavelength of air-As₂Se₃ nanowires with core diameters ranging from 600 nm to 1000 nm.

Fig. 1 Calculated chromatic dispersion of air-As₂Se₃ nanowires with core diameters ranging from 600 nm to 1000 nm.

As we can see from Fig. 1, increasing the air-As₂Se₃ nanowire diameter shifts the first zero dispersion wavelength of the nanowire toward midinfrared wavelengths which enlarges the region of the anomalous dispersion regime. As the core diameter is reduced to sub-micron dimensions, the waveguide dispersion becomes dominant over the material dispersion (bulk As₂Se₃) allowing the overall-GVD to be highly engineered. We find that the 900 nm air-As₂Se₃ has a zero dispersion wavelength around 1550 nm. The 800 nm air-As₂Se₃ presents a positive chromatic dispersion value at 1550 nm and low third-order dispersion (β₂ = −0.48 ps².m⁻¹ and β₃ = 7.79 × 10⁻³ ps³.m⁻¹). Consequently, pumping at this wavelength in the anomalous dispersion regime is attractive to simulate soliton-effect compression. Before investigating the nonlinear propagation in such air-As₂Se₃ nanowires, one should characterize the optical properties of the air-As₂Se₃ nanowires under investigation.

Figure 2(a) depicts the calculated fractional powers inside and outside the core as a function of the air-As₂Se₃ nanowire diameter at 1550 nm. The dominance of the evanescent field begins to appear and exceeds 50% of the total light with nanowire diameters less than 370 nm. The single mode operation is found in nanowires with diameters less than d_SM = 448 nm. At this critical diameter, more than 85% of the light is inside the core. For the 800 nm air-As₂Se₃ nanowire, 98% of the total light propagates inside the core. Figure 2(b) shows the calculated nonlinear coefficient γ of the different air-As₂Se₃ nanowires and their effective mode area

A_{e f f} = {| \int S_{z} d A |}^{2} / \int {| S_{z} |}^{2} d A

at λ_p = 1550 nm. We notice that the mode confinement determined by the effective mode area becomes stronger when reducing the core diameter of the air-As₂Se₃ nanowire. This behavior continues linearly to evolve and shows an optimal nanowire size exhibiting the minimum effective mode area / the highest effective nonlinearity with a core diameter of about 450 nm. After this optimum value, the mode confinement diverges rapidly. This is justified by the fact that the air-As₂Se₃ nanowire no longer tightly confines the light so that the evanescent field starts to dominate. For the 800 nm air-As₂Se₃ nanowire, the effective mode area is evaluated to be A_eff = 0.3 µm². Thus, a very high nonlinear coefficient is exhibited and found to be around 145.3 W⁻¹.m⁻¹ by taking n₂ = 1.1 × 10⁻¹⁷ m².W⁻¹ [21

R. E. Slusher, G. Lenz, J. Hodelin, J. Sanghera, L. B. Shaw, and I. D. Aggarwal, “Large Raman gain and nonlinear phase shifts in high-purity As₂Se₃ chalcogenide fibers,” J. Opt. Soc. Am. B 21(6), 1146–1155 (2004). [CrossRef]

Fig. 2 (a) Fractional powers inside and outside the core and (b) effective mode area (A_eff) and nonlinear coefficient (γ) of various air-As₂Se₃ nanowire diameters at λ_p = 1550 nm.

After determining the optical properties of air-As₂Se₃ photonic nanowires, we introduce them in the GNLSE as inputs to investigate the nonlinear propagation in the 800 nm air-As₂Se₃ nanowire. As we found that the 800 nm air-As₂Se₃ performs in the anomalous dispersion regime around a pump wavelength of 1550 nm, we select, based on the availability, a femtosecond laser delivering 250 fs FWHM at 1550 nm [10

]. The Raman response parameters of the As₂Se₃ glass are set to be τ₁ = 23fs, τ₂ = 164.5fs and f_R = 0.148 [22

A. Ben-Salem, R. Cherif, and M. Zghal, “Raman response of a highly nonlinear As₂Se₃-based chalcogenide photonic crystal fiber,” Proc. PIERS, 1256–1260, Marrakesh, Morocco (2011).

]. The coefficient loss α = 1 dB.m⁻¹ [23

C. Xiong, E. Magi, F. Luan, A. Tuniz, S. Dekker, J. S. Sanghera, L. B. Shaw, I. D. Aggarwal, and B. J. Eggleton, “Characterization of picosecond pulse nonlinear propagation in chalcogenide As₂S₃ fiber,” Appl. Opt. 48(29), 5467–5474 (2009). [CrossRef] [PubMed]

Taking into account the high two photon absorption coefficient (α₂ = 2.5 × 10⁻¹² m/W [21

]) and the high nonlinearity exhibited in As₂Se₃ glass, we select a very low input pulse energy to characterize the soliton-self compression in the 800 nm air-As₂Se₃ nanowire. We set the input pulse energy to 10 pJ at a pump wavelength of 1550 nm and inject an input hyperbolic-secant pulse initially chirped with C = −0.3 which is found to optimize the temporal compression. This optimal value of the chirp is determined by carrying out several calculations, at a fixed input pulse energy, in which we vary the initial chirp and register the shortest FWHM duration of the generated compressed pulse T_comp, as seen in Fig. 3 . We find that positive chirp degrades the temporal compression while negative chirp enhances the soliton-effect compression (C = −0.3 provides maximum temporal compression).

Fig. 3 Impact of the initial chirp on the compression efficiency.

Figure 4(a) shows a maximum generated compressed pulse with a duration of 25.4 fs in 2.1 mm nanowire length. The optimal soliton-self compression length is numerically determined from the simulations ensuring that a maximum-compressed and preserved soliton-shape is reached [24

A. Ben Salem, R. Cherif, and M. Zghal, “Generation of few optical cycles in air-silica nanowires,” Proc. SPIE 8001, 80011J (2011). [CrossRef]

] without the presence of high pulse intensity fluctuations or peak perturbations. Good agreement is found with the analytical model giving optimal distance proposed in [9

C. M. Chen and P. L. Kelley, “Nonlinear pulse compression in optical fibers: scaling laws and numerical analysis,” J. Opt. Soc. Am. B 19(9), 1961–1967 (2002). [CrossRef]

]. This nonlinear interaction corresponds to the injection of a soliton order N = 14.5. We characterize the compression efficiency by the evaluation of two parameters namely the compression factor F_c and the quality factor Q_c. The compression factor F_c, defined as the ratio of the FWHM pulse duration at the beginning and the end of the nanowire, is found to be 9.84. The correspondent quality factor _{$Q_{c} = P_{c o m p} / (P_{0} F_{c})$} [10

] which is equal to the peak power of the compressed pulse (P_comp) normalized to the input peak power (P₀) and the compression factor (F_c) is evaluated to be 0.21. Spectral evolution of the initial pre-chirped hyperbolic-secant pulse is depicted in Fig. 4(b).

Fig. 4 (a) Temporal and (b) spectral evolution of 250 fs input pulse with 10 pJ input energy in the 800 nm air-As₂Se₃ nanowire at λ_p = 1550 nm, with and without TPA.

The supercontinuum is generated with taking into account the effect of TPA (for which we extracted the maximum compressed pulse) and then without TPA and we notice its big effect on the spectral generated bandwidth. In fact, an extra 500 nm bandwidth is generated when one neglects the TPA. Thus, a perfect modeling has to consider the effect of TPA in order to obtain correct results simulating the nonlinear propagation in highly nonlinear air-As₂Se₃ nanowires. We notice the generation of over 700 nm bandwidth of near infrared SC showing a symmetrical broadening which confirms that self phase modulation is the dominant effect giving such broadening and leading to temporal soliton-self compression.

3.2 Air-As₂S₃ nanowires

To evaluate the chromatic dispersion in air-As₂S₃ photonic nanowires, we used the Sellmeier equation of the As₂S₃ glass defined as

n (λ) = \sqrt{1 + \sum_{i} A_{i} λ^{2} / (λ^{2} - λ_{i}^{2})}

where A₁ = 1.898367, A₂ = 1.922297, A₃ = 0.87651, A₄ = 0.11887, A₅ = 0.95699, λ₁ = 0.15 µm, λ₂ = 0.25µm, λ₃ = 0.35µm, λ₄ = 0.45µm and λ₅ = 27.3861µm [25

M. Bass, Handbook of Optics (McGraw Hill, 1997).

]. The unit of λ is µm.

Figure 5 depicts the calculated chromatic dispersion of air-As₂S₃ nanowires. We notice the similar chromatic dispersion behavior as air-As₂Se₃ nanowires: when the diameter increases the region of the anomalous dispersion regime shifts toward infrared wavelengths. Air-As₂S₃ nanowires with diameters ranging from 600 nm to 1000 nm exhibit positive chromatic dispersion around 1550 nm. We found very low GVD and third-order dispersion coefficients (β₂ = −1.54 ps².m⁻¹ and β₃ = 5.65 × 10⁻³ ps³.m⁻¹) for the 800 nm air-As₂S₃ nanowire. Pumping at this wavelength will excite soliton-self compression in the anomalous dispersion regime in the 800 nm air-As₂S₃ nanowire. A detailed study of the fundamental mode has been achieved for air-As₂S₃ nanowires.

Fig. 5 Calculated chromatic dispersion of air-As₂S₃ nanowires with core diameters ranging from 600 nm to 1000 nm.

Figure 6(a) shows the calculated fractional powers inside and outside the core at a wavelength of 1550 nm as a function of the air-As₂S₃ nanowire diameter. The evanescent field starts to dominate and exceeds 50% of the total light with nanowire diameters less than 420 nm. With a critical diameter d_SM = 533 nm, air-As₂S₃ nanowires experience fundamental mode propagation. At this diameter, more than 82% of the light propagates inside the core. For the 800 nm air-As₂Se₃ nanowire, 4% of the light resides in the evanescent field while 96% of the remaining light propagates inside the core. Thus, high nonlinear coefficient is exhibited and evaluated to be around 48.7 W⁻¹.m⁻¹ by taking n₂ = 4 × 10⁻¹⁸ m².W⁻¹ [26

D.-P. Wei, T. Galstian, I. Smolnikov, V. Plotnichenko, and A. Zohrabyan, “Spectral broadening of femtosecond pulses in a single-mode As-S glass fiber,” Opt. Express 13(7), 2439–2443 (2005). [CrossRef] [PubMed]

Fig. 6 (a) Fractional powers inside and outside the core and (b) effective mode area (A_eff) and nonlinear coefficient (γ) of different air-As₂S₃ nanowires at λ_p = 1550 nm.

In fact, Fig. 6(b) shows the calculated nonlinear coefficient of different air-As₂Se₃ nanowires and their effective mode area at 1550 nm. Similar behavior of the nonlinear coefficient is depicted for air-As₂S₃ nanowires as the air-As₂Se₃ nanowires. The highest effective nonlinearity is exhibited for an optimal nanowire size of 540 nm. We find that the highest effective nonlinearity and the mode confinement depend on the contrast of the core-cladding refractive-index difference. In fact, when the refractive index difference Δn between the core and the air-cladding increases for a certain optical wavelength, the optimal photonic nanowire diameter presenting the highest effective nonlinearity shifts and becomes smaller. This is the case at 1550 nm for As₂S₃ (Δn = 1.43) and As₂Se₃ (Δn = 1.83) glasses showing optimal nanowire diameters of 540 nm and 450 nm, respectively. More light confinement with 2% for the 800 nm air-As₂Se₃ nanowire is found compared to that in the 800 nm air-As₂S₃ nanowire.

We are now in the process of investigating the nonlinear propagation in the 800 nm air-As₂S₃ nanowire. As the 800 nm air-As₂Se₃, we select the femtosecond laser delivering 250 fs FWHM pulse at 1550 nm. We take τ₁ = 15.5fs, τ₂ = 230.5fs and f_R = 0.1 to model the Raman response function of the As₂S₃ glass and set α = 1 dB.m⁻¹ [23

]. Because of the high nonlinearity exhibited in the 800 nm air-As₂S₃ nanowire, a very low input pulse energy of 50 pJ is taken. This corresponds to the excitation of an input soliton order of N = 10.56. In order to optimize the soliton-self compression, the input hyperbolic-secant pulse is chosen initially chirped with C = −0.2 (Similar approach has been performed to determine the optimal chirp value). Figure 7 shows a maximum compressed pulse with duration of 5.07 fs in 0.84 mm nanowire length. Thus, less than single optical cycle is generated at a pump wavelength of 1550 nm. Good agreement was found with the analytical expression determining the optimal soliton-self compression distance [9

C. M. Chen and P. L. Kelley, “Nonlinear pulse compression in optical fibers: scaling laws and numerical analysis,” J. Opt. Soc. Am. B 19(9), 1961–1967 (2002). [CrossRef]

]. A compression factor F_c = 49.3 and a quality factor Q_c = 0.28 are achieved.

Fig. 7 Temporal evolution of 250 fs input pulse with 50 pJ input energy in the 800 nm air-As₂S₃ nanowire at λ_p = 1550 nm.

Figure 8 depicts the spectral evolution and approximately two octave spanning supercontinuum is generated in the 800 nm air-As₂S₃ with only 50 pJ input energy. We notice that the impact of the TPA is not discussed here because As₂S₃ glass exhibits a very low TPA coefficient comparing to that of the As₂Se₃ glass and simulations show no effect on the generated SC. We recall that previous experimental work has been performed at the same pump wavelength of 1550 nm using short segment of 10 cm of a highly nonlinear silica fiber with sub-nJ laser source [27

B. Kibler, R. Fischer, P.-A. Lacourt, F. Courvoisier, R. Ferriere, L. Larger, D. N. Neshev, and J. M. Dudley, “Optimised one-step compression of femtosecond fibre laser soliton pulses around 1550 nm to below 30 fs in highly nonlinear fibre,” Electron. Lett. 43(17), 915–916 (2007). [CrossRef]

]. Although 30 fs compressed pulse has been generated, air-chalcogenide nanowires (as demonstrated in both air-As₂Se₃ and air-As₂S₃ structures) show shorter compressed pulse in only few millimeter-long fibers at pJ energy levels. Thus, chalcogenide photonic nanowires are very promising for efficient broadband soliton-self compression with short lengths and very low input pulse energy.

Fig. 8 Spectral evolution of 250 fs input pulse with 50 pJ input energy in the 800 nm air-As₂S₃ nanowire at λ_p = 1550 nm.

4. Conclusion

We have performed a numerical study of soliton self-compression in photonic nanowires. Detailed studies of the optical properties have been achieved in chalcogenide photonic nanowires based on As₂S₃ and As₂Se₃ glasses. Most attractive optical properties of photonic nanowire are the existence of the evanescent field surrounding the nanowire and the strong radial confinement of the light which make photonic nanowires well suited for an efficient and controlled interaction of guided light with matter and perfect devices for sensing applications [28

G. Brambilla, “Optical fibre nanotaper sensors,” Opt. Fiber Technol. 16(6), 331–342 (2010). [CrossRef]

]. Soliton-effect compression has been investigated in chalcogenide photonic nanowires and pulses in the monocycle regime have been generated in As₂S₃ nanowire while 25.4fs compressed pulse is generated in the As₂Se₃ with consideration of TPA. Thus, very high compression factors of 49.3 and 9.84 have been achieved starting from 250 fs for chalcogenide (As₂S₃ and As₂Se₃) nanowires, respectively. All the compressed pulses are obtained at very low input pulse energy with pJ levels in just few millimeters-long nanowires. The study was based on the adjustment of the initial chirp in order to maximize the compression. In addition, the generation of midinfrared two octaves spanning SC around 1550 nm in only 0.84 mm air-As₂S₃ nanowire is shown. Chalcogenide photonic nanowires are very promising devices for designing midinfrared light coherent sources with short lengths and low powers. They are suitable waveguides for high soliton-effect compression which opens new horizon toward extreme nonlinear optics and attosecond physics.

Acknowledgments

The work is partially supported by the “Institut Télécom” through the “Futur et Ruptures” PhD thesis of A. Ben-Salem. The authors thank Professor John Dudley from Institut Femto-ST, Université de Franche-Comté, Besançon, France for helpful discussions and collaboration.

References and links

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 426(6968), 816–819 (2003). [CrossRef] [PubMed]
2.	M. A. Foster, A. C. Turner, M. Lipson, and A. L. Gaeta, “Nonlinear optics in photonic nanowires,” Opt. Express 16(2), 1300–1320 (2008). [CrossRef] [PubMed]
3.	E. C. Mägi, P. Steinvurzel, and B. J. Eggleton, “Tapered photonic crystal fibers,” Opt. Express 12(5), 776–784 (2004). [CrossRef] [PubMed]
4.	Y. K. Lizé, E. C. Mägi, V. G. Ta’eed, J. A. Bolger, P. Steinvurzel, and B. J. Eggleton, “Microstructured optical fiber photonic wires with subwavelength core diameter,” Opt. Express 12(14), 3209–3217 (2004). [CrossRef] [PubMed]
5.	R. R. Gattass, G. T. Svacha, L. M. Tong, and E. Mazur, “Supercontinuum generation in submicrometer diameter silica fibers,” Opt. Express 14(20), 9408–9414 (2006). [CrossRef] [PubMed]
6.	S. G. Leon-Saval, T. A. Birks, W. J. Wadsworth, P. St. J. Russell, and M. W. Mason, “Supercontinuum generation in submicron fibre waveguides,” Opt. Express 12(13), 2864–2869 (2004). [CrossRef] [PubMed]
7.	L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45(13), 1095–1098 (1980). [CrossRef]
8.	L. F. Mollenauer, R. H. Stolen, J. P. Gordon, and W. J. Tomlinson, “Extreme picosecond pulse narrowing by means of soliton effect in single-mode optical fibers,” Opt. Lett. 8(5), 289–291 (1983). [CrossRef] [PubMed]
9.	C. M. Chen and P. L. Kelley, “Nonlinear pulse compression in optical fibers: scaling laws and numerical analysis,” J. Opt. Soc. Am. B 19(9), 1961–1967 (2002). [CrossRef]
10.	M. Foster, A. L. Gaeta, Q. Cao, and R. Trebino, “Soliton-effect compression of supercontinuum to few-cycle durations in photonic nanowires,” Opt. Express 13(18), 6848–6855 (2005). [CrossRef] [PubMed]
11.	J. M. Harbold, F. O. Ilday, F. W. Wise, J. S. Sanghera, V. Q. Nguyen, L. B. Shaw, and I. D. Aggarwal, “Highly nonlinear As-S-Se glasses for all-optical switching,” Opt. Lett. 27(2), 119–121 (2002). [CrossRef] [PubMed]
12.	D. I. Yeom, E. C. Mägi, M. R. E. Lamont, M. A. F. Roelens, L. Fu, and B. J. Eggleton, “Low-threshold supercontinuum generation in highly nonlinear chalcogenide nanowires,” Opt. Lett. 33(7), 660–662 (2008). [CrossRef] [PubMed]
13.	D. D. Hudson, S. A. Dekker, E. C. Mägi, A. C. Judge, S. D. Jackson, E. Li, J. S. Sanghera, L. B. Shaw, I. D. Aggarwal, and B. J. Eggleton, “Octave spanning supercontinuum in an As₂S₃ taper using ultralow pump pulse energy,” Opt. Lett. 36(7), 1122–1124 (2011). [CrossRef] [PubMed]
14.	L. Tong, J. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Express 12(6), 1025–1035 (2004). [CrossRef] [PubMed]
15.	M. Zghal and R. Cherif, “Impact of small geometrical imperfections on chromatic dispersion and birefringence in photonic crystal fibers,” Opt. Eng. 46(12), 128002 (2007). [CrossRef]
16.	M. A. Foster, K. D. Moll, and A. L. Gaeta, “Optimal waveguide dimensions for nonlinear interactions,” Opt. Express 12(13), 2880–2887 (2004). [CrossRef] [PubMed]
17.	T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78(17), 3282–3285 (1997). [CrossRef]
18.	G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, 2007).
19.	K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25(12), 2665–2673 (1989). [CrossRef]
20.	R. Cherif, A. Ben-Salem, M. Zghal, P. Besnard, T. Chartier, L. Brilland, and J. Troles, “Highly nonlinear As₂Se₃-based chalcogenide photonic crystal fiber for midinfrared supercontinuum generation,” Opt. Eng. 49(9), 095002 (2010). [CrossRef]
21.	R. E. Slusher, G. Lenz, J. Hodelin, J. Sanghera, L. B. Shaw, and I. D. Aggarwal, “Large Raman gain and nonlinear phase shifts in high-purity As₂Se₃ chalcogenide fibers,” J. Opt. Soc. Am. B 21(6), 1146–1155 (2004). [CrossRef]
22.	A. Ben-Salem, R. Cherif, and M. Zghal, “Raman response of a highly nonlinear As₂Se₃-based chalcogenide photonic crystal fiber,” Proc. PIERS, 1256–1260, Marrakesh, Morocco (2011).
23.	C. Xiong, E. Magi, F. Luan, A. Tuniz, S. Dekker, J. S. Sanghera, L. B. Shaw, I. D. Aggarwal, and B. J. Eggleton, “Characterization of picosecond pulse nonlinear propagation in chalcogenide As₂S₃ fiber,” Appl. Opt. 48(29), 5467–5474 (2009). [CrossRef] [PubMed]
24.	A. Ben Salem, R. Cherif, and M. Zghal, “Generation of few optical cycles in air-silica nanowires,” Proc. SPIE 8001, 80011J (2011). [CrossRef]
25.	M. Bass, Handbook of Optics (McGraw Hill, 1997).
26.	D.-P. Wei, T. Galstian, I. Smolnikov, V. Plotnichenko, and A. Zohrabyan, “Spectral broadening of femtosecond pulses in a single-mode As-S glass fiber,” Opt. Express 13(7), 2439–2443 (2005). [CrossRef] [PubMed]
27.	B. Kibler, R. Fischer, P.-A. Lacourt, F. Courvoisier, R. Ferriere, L. Larger, D. N. Neshev, and J. M. Dudley, “Optimised one-step compression of femtosecond fibre laser soliton pulses around 1550 nm to below 30 fs in highly nonlinear fibre,” Electron. Lett. 43(17), 915–916 (2007). [CrossRef]
28.	G. Brambilla, “Optical fibre nanotaper sensors,” Opt. Fiber Technol. 16(6), 331–342 (2010). [CrossRef]

OCIS Codes
(160.4330) Materials : Nonlinear optical materials
(190.4370) Nonlinear optics : Nonlinear optics, fibers
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons
(320.5520) Ultrafast optics : Pulse compression
(320.7140) Ultrafast optics : Ultrafast processes in fibers
(320.6629) Ultrafast optics : Supercontinuum generation

ToC Category:
Nonlinear Optics

History
Original Manuscript: August 10, 2011
Revised Manuscript: September 16, 2011
Manuscript Accepted: September 20, 2011
Published: September 27, 2011

Citation
Amine Ben Salem, Rim Cherif, and Mourad Zghal, "Soliton-self compression in highly nonlinear chalcogenide photonic nanowires with ultralow pulse energy," Opt. Express 19, 19955-19966 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-21-19955

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References

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,” Nature426(6968), 816–819 (2003). [CrossRef] [PubMed]
M. A. Foster, A. C. Turner, M. Lipson, and A. L. Gaeta, “Nonlinear optics in photonic nanowires,” Opt. Express16(2), 1300–1320 (2008). [CrossRef] [PubMed]
E. C. Mägi, P. Steinvurzel, and B. J. Eggleton, “Tapered photonic crystal fibers,” Opt. Express12(5), 776–784 (2004). [CrossRef] [PubMed]
Y. K. Lizé, E. C. Mägi, V. G. Ta’eed, J. A. Bolger, P. Steinvurzel, and B. J. Eggleton, “Microstructured optical fiber photonic wires with subwavelength core diameter,” Opt. Express12(14), 3209–3217 (2004). [CrossRef] [PubMed]
R. R. Gattass, G. T. Svacha, L. M. Tong, and E. Mazur, “Supercontinuum generation in submicrometer diameter silica fibers,” Opt. Express14(20), 9408–9414 (2006). [CrossRef] [PubMed]
S. G. Leon-Saval, T. A. Birks, W. J. Wadsworth, P. St. J. Russell, and M. W. Mason, “Supercontinuum generation in submicron fibre waveguides,” Opt. Express12(13), 2864–2869 (2004). [CrossRef] [PubMed]
L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett.45(13), 1095–1098 (1980). [CrossRef]
L. F. Mollenauer, R. H. Stolen, J. P. Gordon, and W. J. Tomlinson, “Extreme picosecond pulse narrowing by means of soliton effect in single-mode optical fibers,” Opt. Lett.8(5), 289–291 (1983). [CrossRef] [PubMed]
C. M. Chen and P. L. Kelley, “Nonlinear pulse compression in optical fibers: scaling laws and numerical analysis,” J. Opt. Soc. Am. B19(9), 1961–1967 (2002). [CrossRef]
M. Foster, A. L. Gaeta, Q. Cao, and R. Trebino, “Soliton-effect compression of supercontinuum to few-cycle durations in photonic nanowires,” Opt. Express13(18), 6848–6855 (2005). [CrossRef] [PubMed]
J. M. Harbold, F. O. Ilday, F. W. Wise, J. S. Sanghera, V. Q. Nguyen, L. B. Shaw, and I. D. Aggarwal, “Highly nonlinear As-S-Se glasses for all-optical switching,” Opt. Lett.27(2), 119–121 (2002). [CrossRef] [PubMed]
D. I. Yeom, E. C. Mägi, M. R. E. Lamont, M. A. F. Roelens, L. Fu, and B. J. Eggleton, “Low-threshold supercontinuum generation in highly nonlinear chalcogenide nanowires,” Opt. Lett.33(7), 660–662 (2008). [CrossRef] [PubMed]
D. D. Hudson, S. A. Dekker, E. C. Mägi, A. C. Judge, S. D. Jackson, E. Li, J. S. Sanghera, L. B. Shaw, I. D. Aggarwal, and B. J. Eggleton, “Octave spanning supercontinuum in an As2S3 taper using ultralow pump pulse energy,” Opt. Lett.36(7), 1122–1124 (2011). [CrossRef] [PubMed]
L. Tong, J. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Express12(6), 1025–1035 (2004). [CrossRef] [PubMed]
M. Zghal and R. Cherif, “Impact of small geometrical imperfections on chromatic dispersion and birefringence in photonic crystal fibers,” Opt. Eng.46(12), 128002 (2007). [CrossRef]
M. A. Foster, K. D. Moll, and A. L. Gaeta, “Optimal waveguide dimensions for nonlinear interactions,” Opt. Express12(13), 2880–2887 (2004). [CrossRef] [PubMed]
T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett.78(17), 3282–3285 (1997). [CrossRef]
G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, 2007).
K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron.25(12), 2665–2673 (1989). [CrossRef]
R. Cherif, A. Ben-Salem, M. Zghal, P. Besnard, T. Chartier, L. Brilland, and J. Troles, “Highly nonlinear As2Se3-based chalcogenide photonic crystal fiber for midinfrared supercontinuum generation,” Opt. Eng.49(9), 095002 (2010). [CrossRef]
R. E. Slusher, G. Lenz, J. Hodelin, J. Sanghera, L. B. Shaw, and I. D. Aggarwal, “Large Raman gain and nonlinear phase shifts in high-purity As2Se3 chalcogenide fibers,” J. Opt. Soc. Am. B21(6), 1146–1155 (2004). [CrossRef]
A. Ben-Salem, R. Cherif, and M. Zghal, “Raman response of a highly nonlinear As2Se3-based chalcogenide photonic crystal fiber,” Proc. PIERS, 1256–1260, Marrakesh, Morocco (2011).
C. Xiong, E. Magi, F. Luan, A. Tuniz, S. Dekker, J. S. Sanghera, L. B. Shaw, I. D. Aggarwal, and B. J. Eggleton, “Characterization of picosecond pulse nonlinear propagation in chalcogenide As2S3 fiber,” Appl. Opt.48(29), 5467–5474 (2009). [CrossRef] [PubMed]
A. Ben Salem, R. Cherif, and M. Zghal, “Generation of few optical cycles in air-silica nanowires,” Proc. SPIE8001, 80011J (2011). [CrossRef]
M. Bass, Handbook of Optics (McGraw Hill, 1997).
D.-P. Wei, T. Galstian, I. Smolnikov, V. Plotnichenko, and A. Zohrabyan, “Spectral broadening of femtosecond pulses in a single-mode As-S glass fiber,” Opt. Express13(7), 2439–2443 (2005). [CrossRef] [PubMed]
B. Kibler, R. Fischer, P.-A. Lacourt, F. Courvoisier, R. Ferriere, L. Larger, D. N. Neshev, and J. M. Dudley, “Optimised one-step compression of femtosecond fibre laser soliton pulses around 1550 nm to below 30 fs in highly nonlinear fibre,” Electron. Lett.43(17), 915–916 (2007). [CrossRef]
G. Brambilla, “Optical fibre nanotaper sensors,” Opt. Fiber Technol.16(6), 331–342 (2010). [CrossRef]

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Soliton-self compression in highly nonlinear chalcogenide photonic nanowires with ultralow pulse energy

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Abstract

1. Introduction

2. Theory and numerical analysis

2.1 Mode propagation

2.2 Chromatic dispersion

2.3 Evanescent field and effective nonlinearity

2.4 Nonlinear propagation

3. Numerical results

3.1 Air-As₂Se₃ nanowires

3.2 Air-As₂S₃ nanowires

4. Conclusion

Acknowledgments

References and links

References

Appl. Opt.

Electron. Lett.

IEEE J. Quantum Electron.

J. Opt. Soc. Am. B

Nature

Opt. Eng.

Opt. Express

Opt. Fiber Technol.

Opt. Lett.

Phys. Rev. Lett.

Proc. SPIE

Other

Cited By

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Optics Express

Soliton-self compression in highly nonlinear chalcogenide photonic nanowires with ultralow pulse energy

Article Tools

Article Navigation

Abstract

1. Introduction

2. Theory and numerical analysis

2.1 Mode propagation

2.2 Chromatic dispersion

2.3 Evanescent field and effective nonlinearity

2.4 Nonlinear propagation

3. Numerical results

3.1 Air-As2Se3 nanowires

3.2 Air-As2S3 nanowires

4. Conclusion

Acknowledgments

References and links

References

Appl. Opt.

Electron. Lett.

IEEE J. Quantum Electron.

J. Opt. Soc. Am. B

Nature

Opt. Eng.

Opt. Express

Opt. Fiber Technol.

Opt. Lett.

Phys. Rev. Lett.

Proc. SPIE

Other

Cited By

Figures

3.1 Air-As₂Se₃ nanowires

3.2 Air-As₂S₃ nanowires