1. Introduction

Metal nanoparticles (NP) play a central role in the emerging field of nanooptics and plasmonics. The interaction between light and metal NPs is dominated by charge-density oscillations on the closed surface of the particles, or localized plasmon resonances and leads to large field enhancements in the near field of the NPs. These elementary electronic excitations have been the subject of extensive research, both fundamental and with a view to applications (for a review see e.g [1].). To remark a few examples of the diverse fields of applications, localized plasmons allow greatly increased signal strengths in Raman spectroscopy and surface spectroscopy enabling the detection of single molecules [2], significant enhancement of the emission rate of fluorescent molecules or quantum dots [3] and high-harmonic generation by nJ pulses directly from a laser oscillator without amplifier [4]. A second important property of metal NPs is its intrinsically high nonlinear coefficient, which is about seven orders of magnitude higher than in fused silica. Recently we predicted for aqueous colloids containing silver nanoparticles that plasmon enhancements can also be used for the generation of octave spanning supercontinuum generation [5] by low-intense fs pulses.

The generation of coherent white-light (or supercontinuum generation) in microstructure fibers by fs and ps pulses with lowest pulse energy in the range of 1 nJ [6] has attracted much attention and found many applications (a review is presented in Ref [7].). The physical mechanism of the dramatic spectral broadening process in this coherent white-light source is related with the soliton dynamics in the anomalous dispersion region of the fibers [8]. Recently much effort has been paid to lower the power or energy threshold necessary to generate octave-spanning spectra by reducing the diameter to submicrometers [9] or using nanowires made of materials with high nonlinearities [10]. The reduction of the power threshold is particularly important for the realization of cost-effective and compact SC sources. Since lasers with GHz repetition rates exhibit only low power, a different motivation is connected with the aim to increase the resolution in laser frequency metrology by using laser frequency combs at GHz repetition without the use of a Fabry-Perot cavity.

In this paper we follow an alternative concept for lowering the threshold for supercontinuum generation and present a comprehensive investigation of low-threshold supercontinuum generation in glasses doped with silver-NPs. The linear loss, group velocity dispersion and the complex nonlinear coefficient are calculated in dependence on the filling factor and the frequency. The high intrinsic value of the nonlinear coefficient of the silver nanoparticles and the plasmon resonance enhancement lead to a very sensitive frequency dependence and a dramatic increase of the effective nonlinear index of the composite with maximum values up to –0.1 cm²/W for moderate filling factors of about 10⁻³. Despite the relatively high loss the propagation of fs pulses at 800 nm with low intensities in the range of tens of GW/cm² through a NP-doped glass layer leads to octave-spanning blue-shifted supercontinuum generation extending from 512 to 925 nm.

2. Linear and nonlinear optical properties of the composite

First, we study the linear and nonlinear optical properties of the composite. For the case when the average distance between the inclusions and their size is much less than the light wavelength, the optical properties of a composite can be described by the effective medium theory in terms of the characteristic parameters of the nanoparticles and the host material. For spherical nanoparticles with low filling factor and uncorrelated positions the well-established formalism of Maxwell-Garnett [11] can be applied up to relatively large filling factors f of about 0.2. In this model the effective dielectric constant of the composite is given by ${\epsilon }_{eff}={\epsilon }_h(1+2\sigma f)/(1-\sigma f)$ with $\sigma =({\epsilon }_i-{\epsilon }_h)/({\epsilon }_i+2{\epsilon }_h)$. Here${\epsilon }_i$ and${\epsilon }_h$are the dielectric functions of the NPs (inclusions) and host material, correspondingly. We use a Sellmeyer-type expression for the dielectric function of glass (for details and coefficients see [12]) and for the dielectric function of silver the Drude model for the dielectric function with ${\epsilon }_i=1-{\omega }_p^2/({\omega }^2+i\nu \omega )$is used, with parameters for the collision rate ν = 0.076 fs⁻¹ and the plasma frequency ω _p = 10.5 fs⁻¹ [13]. The loss due to scattering on inclusions is negligible for inclusion diameter below 20 nm. In Fig. 1(a),(b) the group velocity dispersion coefficient (GVD) β′′(ω) = ∂² k _eff /∂ω ², k _eff = n _eff (ω)ω/c, n _eff(ω) = (ε _eff)^1/2 of the composite is presented as a function of the wavelength for different filling factors f. Figure 1(a) demonstrates that for low filling factors, below f = 10⁻³ the GVD coefficient approaches that of the host material andchanges the sign around 1270 nm. As can be seen in Fig. 1(b), for large filling factor the dispersion is normal over the whole considered spectral range and strongly increases near theplasmonic resonance. The strong wavelength dependence of the linear loss can be seen from Fig. 2 for different filling factors f. The losses reach their maximum near the resonance wavelength, at 425 nm. For large filling factors the loss has very high values in the range of 0.1-1 μm⁻¹, implying an effective propagation length L_eff of only few microns. We can also evaluate the effective third-order susceptibility of a composite using the generalized Maxwell Garnet model. For spherical nanoparticles the effective third-order susceptibility χ_3,eff is given by [14]

 

Fig. 1 (Color online) Dispersion coefficient β′′(ω) = ∂² k_eff /∂ω² of silica glass doped with silver NPs for different filling factors f.

Download Full Size | PDF

 

Fig. 2 (Color online) Linear loss coefficient α = Im(k_eff (ω)) of silica glass doped with silver NPs for different filling factors f.

Download Full Size | PDF

with $P=(1-\sigma f)({\epsilon }_i+2{\epsilon }_h)/3{\epsilon }_h\text{and }\sigma =({\epsilon }_i-{\epsilon }_h)/({\epsilon }_i+2{\epsilon }_h)$ as above.

The susceptibility of the host silica glass is taken to be ${\chi }_{\text{h}}=2.233\cdot {\text{10}}^{\text{-18}}{\text{ m}}^{\text{2}}{\text{/V}}^{\text{2}}.$While there exists a certain discrepancy in the literature, the value was${\chi }_{\text{i}}=(-6.3+1.9i)\cdot {\text{10}}^{\text{-16}}{\text{ m}}^{\text{2}}{\text{/V}}^{\text{2}}$used for the susceptibility of silver, which is consistent with observations of several groups with reliable experimental techniques [15–17]. A remarkable fact from these observations is that silver NPs exhibit a self-defocusing nonlinearity with a negative sign of Re(χ_i). Figure 3 displays the nonlinear coefficient ${\text{n}}_{\text{2,eff}}\text{= 3}{\chi }_{\text{3,eff}}\text{/(4}{\epsilon }_{\text{0}}\text{c}{\epsilon }_{\text{eff}}\text{)}$ of the composite as a function of wavelength for three different filling factors. We can clearly see the sign change of the nonlinearity coefficient at wavelengths around 820 nm for the filling factor f = 10⁻⁵ and at 1350 nm for the filling factor f = 10⁻³ and its dramatic increase of its magnitude with maximum values up to −0.1 cm²/W for moderate filling factors of about 10⁻³. For a higher filling factor f = 0.1 the nonlinearity of the composite is mostly dominated by nanoparticles contribution and is thus self-defocusing for all the presented wavelengths. Such behavior is explained by the large contribution of the NP nonlinearity as the wavelength approaches the resonance and the field in the NPs is strongly enhanced, while for larger wavelength and for lower filling factors the nonlinearity of the host material plays the dominant role. In Fig. 3(b) we depict the imaginary part of the refractive index which increases as wavelength approaches the resonance but is negative for low filling factors and large wavelengths. This does not imply a nonphysical nonlinear gain but rather a slight reduction of the total loss as the intensity increases.

 

Fig. 3 (Color online) Real (a) and imaginary (b) part of the nonlinear coefficient n_2,eff of the composite as a function of wavelength for different filling factors f.

Download Full Size | PDF

Mathematically it is expressed by the fact that Im[3χ_3,eff/(4ε₀cε_eff)] can be negative owing to positive Im[ε_eff] even though Im[χ_3,eff] is positive.

3. Pulse propagation through the composite and supercontinum generation

We now study the pulse evolution and spectral broadening of fs pulses propagating through the above-described bulk composite medium containing silver NPs. The effective propagation length is below 1 mm for the cases considered in our paper, therefore the diffraction of the beam can be neglected for a beam radius above 30 μm. The pulse evolution in the plane-wave geometry is simulated by the first-order propagation equation without slowly-varying envelope approximation-accounting for dispersion to all orders, neglecting the back-propagating waves and scattering on inclusions [8]

Here ${\chi }_{3,eff}\left(\omega \right)$ is given by Eq. (1) and $\tilde{F}$ denotes the Fourier transform. Equation (2) describes the evolution of a pulse directly in terms of the electric field instead of the envelope and allows to include the frequency dependence of the nonlinear coefficient without any approximation. We use the split-step Fourier method for the numerical solution of the Eq. (2).

As the pulse propagates through the composite it experiences significant spectral broadening despite the relatively high loss. The evolution of the spectrum is illustrated in Fig. 4 for an input wavelength of 830 nm. One can see that the spectrum significantly extends to the blue side, leading to an asymmetric shape, since the broadening on the red side is small. After an optimum propagation length is reached (in this case it is around 300 μm), the spectral width gradually decreases. The mechanism of spectral broadening can be explained by self-phase modulation (SPM) under the influence of plasmon enhancement by the NPs leading to a sensitive frequency dependence of the nonlinear coefficient and an asymmetric shape not typical for SPM.

 

Fig. 4 (Color online) Pulse evolution in fused silica with silver NPs with filling factors f = 10⁻³ for 20 fs pulse with central wavelengths of 830 nm.

Download Full Size | PDF

While the components at the red side of the input frequency cannot broaden owing to the low values of the nonlinear coefficient, those on the blue side are significantly broadened and extend into spectral regions with even higher values of the nonlinear coefficient. This is stopped only by the high losses near the plasmonic resonance.

If we use a composite with a higher filling factor f = 0.1, almost octave broadening can be reached even after few microns of propagation, as Fig. 5 displays. The input pulses where localized at three wavelengths of 830 nm, 1300 nm and 1550 nm. For the wavelengths of 830 nm and 1300 nm we can see a significant broadening of almost one octave extending from 515 nm to 925 nm and from 830 nm to 1545 nm, correspondingly, achieved after a propagation of correspondingly 4 and 6 μm, but for the wavelength of 1550 nm the broadening is small because the nonlinearity at this wavelength is much smaller than at shorter wavelengths. Working with lower filling factor we still achieve broadening after longer propagation distance. In case of very low filling factor of f = 10⁻⁵ the required input power was increased to 300 GW/cm² in order to achieve a spectrum extending from 580 nm to 930 nm after a propagation of 200 μm. For longer input pulses the spectra becomes narrower as Fig. 6 shows, however, the reduction of the spectral width is much weaker than in typical SPM broadening without plasmon enhancement.

 

Fig. 5 (Color online) Supercontinuum generation in fused silica with silver NPs at different wavelengths. For 20 fs pulses with an input central wavelengths of 830 nm, 1300 nm and 1550 nm and intensity of 30 GW/cm² the spectrum are shown after the propagation of a distance of 4 μm for an input wavelengths of 830 nm and after the propagation of a distance of 6 μm for the wavelengths of 1300 nm and 1550 nm. The filling factor is f = 0.1.

Download Full Size | PDF

 

Fig. 6 (Color online) Spectrum for input pulses of 40 fs (solid curve) and 100 fs (dashed curve). The propagation length is 17 μm and the filling factor is f = 0.1.

Download Full Size | PDF

The spectrum extends from 600 to 910 nm for a pulse with an input FWHM of 40 fs (solid curve), which is only slightly smaller than for the case of a 20-fs input pulse. The reason is that due to the strong plasmon enhancement even weak spectral components in the range around 700 nm will extend to smaller wavelengths up to the range of about 550 nm. For a 100-fs pulse the spectrum remains relatively narrow.

The broad spectrum and the smooth phase imply the possibility of pulse compression by an external compensation of the chirp. To demonstrate the possibility of pulse compression by using a layer of glass doped with silver NPs we consider a 40-channel spatial light modulator with the working range from 460 to 1800 nm. We assume that the phase can be controlled independently in each channel to compensate the chirp achieved during broadening. The resulting optimum temporal shape for the optimum compensating phases in the channels is shown in Fig. 7 with a compression to a FWHM of 5.5 fs.

 

Fig. 7 (Color online) Spectrum (a), phase (b), and temporal shape (c) of the compressed pulse after chirp compensation by a 40-channel modulator.

Download Full Size | PDF

4. Conclusion

In conclusion, we have studied linear loss, GVD and the nonlinear refraction index of silica glasses doped with silver nanoparticles and the supercontinuum generation by low-intense fs pulses in this material. The high intrinsic value of the nonlinear coefficient of the silver nanoparticles and the plasmon resonance enhancement lead to a very sensitive frequency dependence and a large increase of the effective nonlinear index with maximum values up to 0.1 cm2/W. Octave-spanning supercontinuum generation is predicted by pump pulses with 20 - 40 fs duration and a low intensity in the range of 30 GW/cm² after a propagation length from 4 to 300 μm.