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Spatial phase modulation due to quintic and septic nonlinearities in metal colloids

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Abstract

We report spatial self- and cross-phase modulation effects in metal-dielectric nanocomposites (MDNCs) whose nonlinear (NL) response is dominated by the quintic or septic refractive nonlinearity. The MDNCs consist of silver nanoparticles (NPs) suspended either in acetone or carbon disulfide and their effective NL susceptibilities are controlled by adjusting the volume fraction occupied by the NPs and the incident laser intensity. A theoretical treatment based on the Maxwell-Garnett model was developed to include contributions up to the seventh-order susceptibility showing a very good agreement with the experimental results.

© 2014 Optical Society of America

1. Introduction

Metal-dielectric nanocomposites (MDNCs) is attracting large interest due to their high nonlinear (NL) optical susceptibility, fast response and the possibility of controlling the optical behavior changing the volume fraction, f, of the metallic nanoparticles (NPs). Glasses, polymers and liquids containing metallic NPs are illustrative examples of MDNCs of interest for photonics [110]. The NL response of a MDNC is described by effective susceptibilities containing information on the host and the NPs. Therefore, the induced polarization of a MDNC is described by a power series in the optical field having the effective susceptibilities, χeff(2N+1), N=1,2,3,, as coefficients of the expansion. Up to now the majority of studies were focused on χeff(3) in centro-symmetric MDNCs [714].

Nowadays, due to the development of short pulse lasers it is possible to use large optical fields without destruction of samples and the contributions of high-order nonlinearities (HON) become detectable. Due to the mismatch between the dielectric function of the NPs and the host, there is an enhancement of the electromagnetic field that depends on the laser frequency and its detuning from the NPs localized surface plasmon resonance (LSPR). Therefore, high values of the effective susceptibilities are observed that may favor detection of effects related to HON either associated to direct NL processes or due to cascade processes [1520].

Control of the MDNCs’ effective susceptibilities changing the volume fraction, f, was reported for colloids with metallic NPs [1416] and it was demonstrated a way to cancel the third-order refractive index, n2Reχeff(3), to obtain a system with non-nulled n4Reχeff(5) [21].

In this paper we apply the procedure of [21] and report the investigation of spatial self-phase modulation (SPM) and spatial cross-phase modulation (XPM) in MDNCs presenting quintic and septic refractive nonlinearities, in the absence of contributions due to the lower order NL refractive indices. The experimental results were corroborated with calculations based on a generalization of the NL Maxwell-Garnett (MG) model [22].

Section 2 describes the samples preparation and the experimental setups used. Section 3 presents a generalization of the MG model to include high order contributions and experiments of SPM and XPM based on HON are reported. By changing the value of f and the laser intensity it was possible to demonstrate nonlinearity management of the MDNCs studied. Section 4 presents a summary of the results.

2. Experimental details

Colloids containing silver NPs suspended in acetone and carbon disulfide (CS2) were prepared by chemical reduction methods, following the procedures of [23] and [14], respectively. Sample A (Ag NPs in acetone) was obtained by diluting 90 mg of AgNO3 in 500 ml of water at 100°C; 10 ml of 1% sodium citrate solution was added for reduction the Ag+ ions, and later was boiled and strongly stirred for 1 h. The preparation of sample B (Ag NPs in CS2) consisted of adding dropwise 3.75 ml of a 0.03 mol/l AgNO3 aqueous solution to a 0.05 mol/l N(C8H17)4Br solution in toluene at 100 drops/min. After 10 min, 50 μl of dodecanethiol (dodecanethiol-to-silver molar ratio of 2:1) were introduced in the mixture. Then 3.1 ml of a freshly prepared NaBH4 aqueous solution (0.4 mol/l) was rapidly added to the mixture. The reacting medium was stirred for 3 h, and the organic layer was extracted. The resulting dodecanethiol-stabilized Ag NPs were precipitated by adding ethanol and cooling to −18 °C for 4 h. Finally, the NPs were centrifuged, washed several times with ethanol and redispersed in toluene. Homogeneous distributions of spherical NPs with average diameter of (9.0 ± 2.2) nm and (6.0 ± 3.0) nm for sample A and sample B, respectively, were obtained after laser ablation of the pristine colloids for 1 h using a Nd: YAG laser (532 nm, 8 ns, 85 mJ/pulse, 10 Hz) following the method of [24].

Colloids with f varying from 0.5×105 to 2.5×104 for sample A, and from 0.2×105 to 4.5×105 for sample B, were obtained by adding 20 µl to 300 µl of the Ag-water suspension in 1 ml of acetone, and adding 2 µl to 40 µl of the Ag-toluene suspension in 1 ml of CS2, respectively.

The linear absorption spectra of the samples were measured from 200 nm to 800 nm using a commercial spectrophotometer. The light source for the NL measurements was the second harmonic of a Q-switched and mode-locked Nd: YAG laser (80 ps, 532 nm). Single pulses at 10 Hz were selected using a pulse-picker.

Effective NL refractive indices and NL absorption coefficients were measured using the Z-scan technique [25]. The laser beam was focused by a 10 cm focal distance lens (confocal length: 4.7 mm) on a sample with thickness of 1 mm, contained in a quartz cell. Slow photodetectors placed in the far-field region with adjustable apertures in front of them were used to measure the beam intensity transmitted by the sample. The aperture radius, ra, is related to its transmittance by S=1exp(2ra2/wa), with wa being the beam radius at the aperture positions. Closed-aperture (CA), S<1, and open-aperture (OA), S=1, schemes were used to determine the NL refractive indices and the NL absorptive coefficients. A reference channel was used to improve the signal-to-noise ratio as in [26].

SPM experiments were performed by focusing the laser beam with a 2 cm focal length lens, producing a beam waist of 7 µm at the focus. The far-field diffraction patterns were captured by a CCD camera located 20cm from the exit plane of the sample (cell thickness: 1 mm). For the XPM experiments, counter-propagating pump and probe beams were used with a 5 cm long cell. A CCD camera, located 5cm from the exit plane of the sample, monitored the transverse beam profile.

3. Results and discussion

3.1 Generalized Maxwell-Garnett model

To understand the experimental results we developed an extension of the Maxwell-Garnett model [22] including the contributions of χeff(3), χeff(5) and χeff(7). The colloid is supposed to be homogeneous, f1, the NPs are spherical and their diameter, a, is smaller than their relative distance, b. The light wavelength, λ, satisfies λ>b>a. The optical field, E0, is considered uniform on each particle. Under these conditions the optical polarization, in the quasi-static approximation, can be written as P=Ph+1Vi=1Nppi, where Ph is associated to the host, Np is the number of NPs inside the volume V and pi=εhσiE0 is the induced dipole moment of each NP; σi=3viβ is the NP polarizability, with β=(εnpεhεnp+2εh), where εnp (εh) is the dielectric function of the NPs (host) and vi is the NP volume; β depends on |E0| through εnp and εh. Hence, the optical polarization is given by:

P=εhL{εhεhL[χh+3βf1βf]}E0=εhLχ(|E0|2)E0,
where f=1Vi=1Npvi is the volume fraction, χh is the host susceptibility, εnp and εh can be expressed as a sum of the linear and NL contributions as εh,np=εh,np(L)+εh,np(NL), where the NL terms are given by:
εnp(NL)=εhL[34χnp(3)|Enp|2+58χnp(5)(|Enp|2)2+3564χnp(7)(|Enp|2)3],
εh(NL)=εhL[34χh(3)|E0|2],
with χnp(i), i=3,5,7, being the i-th susceptibility of the NPs and χh(3) belongs to the host, |E0|2 correspond to the mean square modulus of the applied electric field. The contributions of χh(2N+1), N2, were neglected considering the experimental results herein reported. The numerical coefficients of Eqs. (2) and (3) correspond to the degeneracy factors for the i-th order process in the convention of [27]. Due to the small value of f the intensity dependent susceptibility is written as χ(|E0|2)=χh+3βf where the NL terms in f were neglected because: (βf)n1+βf, n2. The mean squared modulus of the electric field inside the NP is represented by |Enp|2=|η|2|E0|2 and η=3εh/(εnp+2εh) is the local field factor.

The effective NL susceptibilities of different orders are determined by performing two series expansions in terms of |E0|2 and |Enp|2. The first expansion consists in expressing χ(y) as a Taylor series up to third order in y=|E0|2 that assumes the form

χ(y)=χ(0)+([χ(y)]y)y=0y+12(2[χ(y)]y2)y=0y2+16(3[χ(y)]y3)y=0y3,
with the coefficients of the expansion obtained by introducing Eqs. (2) and (3) in Eq. (1) and calculating the derivatives of χ(y) with respect to y. Since f1, the coefficients in Eq. (4) can be written as
([χ(y)]y)y=0=34χh(3)+34fL2|L|2χnp(3),
(2[χ(y)]y2)y=0=54fL2|L|4χnp(5)38fL3|L|4(χnp(3))2,
(3[χ(y)]y3)y=0=932fL4|L|6(χnp(3))3158fL3|L|6(χnp(3)χnp(5))+10532fL2|L|6χnp(7),
where L=3εh(L)/(εnp(L)+2εh(L)). Equations (5)(7) were derived neglecting χh(5) and χh(7), and terms proportional to (χh(3))2 and (χh(3)χnp(3)) due to the magnitude of χh(3)=+1.67×1021(m2/V2) [28] for sample A, and χh(3)=+2.92×1020+i3.50×1022(m2/V2) [25, 29] for sample B, that are approximately five orders of magnitude smaller than χnp(3).

In order to obtain an expression of the electric field inside the NPs in terms of E0, a second expansion was performed for |η|2 as function of z=|Enp|2 written as

|Enp|2(|η|2)z=0{1+12[2(|η|2)z2]z=0|E0|2+14[2(|η|2)z2]2z=0(|E0|2)2}|E0|2,
considering (|η|2/z)z=0=0 and |Enp|2=|η|2|E0|2. Introducing Eqs. (2) and (3) in the expression for |η|2 and calculating the derivatives with respect to z, we obtain

|Enp|2|L|2{112|L|2Re[LχNP(3)]|E0|2+14|L|4(Re[LχNP(3)])2|E0|4}|E0|2.

The expansion of χ as a Taylor series up to third order in z2 is given by

χ(z2)=χ(0)+([χ(z2)][z2])z2=0z2+12(2[χ(z2)][z2]2)z2=0(z2)2+16(3[χ(z2)][z2]3)z2=0(z2)3,
and substituting Eq. (9) in Eq. (10) we have

χ(|E0|2)=χ(0)+|L|2([χ(z2)][z2])z2=0|E0|2+12|L|4(2[χ(z2)][z2]2[χ(z2)][z2]Re[LχNP(3)])z2=0|E0|4+16|L|6(3[χ(z2)][z2]332[χ(z2)][z2]2Re[LχNP(3)]+32[χ(z2)][z2](Re[LχNP(3)])2)z2=0|E0|6.

Expressing the effective dielectric function of the MDNC as εeff=εh+noddn!21n[(n1)/2]![(n+1)/2]!χeff(n)|E0|n1, we obtain the effective susceptibility as a function of |E0|2 that can be written, according to [27], as

χeff(|E0|2)=χeff(1)+34χeff(3)|E0|2+58χeff(5)|E0|4+3564χeff(7)|E0|6.

Finally, by comparison of Eqs. (11) and (12), using the coefficients given by Eqs. (5)-(7) , we obtain expressions for the effective third-, fifth- and seventh-order susceptibilities as

χeff(3)=fL2|L|2χnp(3)+χh(3),
χeff(5)=fL2|L|4χnp(5)610fL3|L|4(χnp(3))2310fL|L|6|χnp(3)|2,
χeff(7)=fL2|L|6χnp(7)+1235fL4|L|6(χnp(3))3+335f|L|8[4L2χnp(3)+|L|2(χnp(3))*]|χnp(3)|247fL|L|6[2L2χnp(3)+|L|2(χnp(3))*]χnp(5),

It can be seen from Eq. (13) that χeff(3) may be cancelled adjusting f and the result is independent of the laser intensity; however, Reχeff(3) and Imχeff(3) are canceled for different f values. Equation (14) and (15) indicate that the NPs contribution to χeff(5) and χeff(7) are due to χnp(3), χnp(5), and χnp(7), with different powers. Observe that the contributions of the NPs susceptibilities are enhanced due to the high powers of L and |L|. It is important to notice that in the case of fifth- and seventh-order effective susceptibilities, we have to vary f and the incident intensity, to determine the conditions for destructive interference between the susceptibilities of different orders. We remark that χ(|E0|2) is linearly proportional to f because terms proportional to fn, n2, were neglected. This result is corroborated by the experimental measurements reported below corresponding to 106<f<104.

The NL response of the NPs is mainly due to the s-electrons because the d-band is 4eV above the Fermi level and the energy of the incident photons is 2.34eV.

3.2 Z-scan experiments and theoretical analysis

Figure 1(a) and 1(b) show the linear absorbance spectra of samples A and B for f=4.0×105. The features at 400nm are due to the LSPR in the silver NPs. Also shown in Fig. 1 are the spectra of pure acetone and CS2 that present large transparency window from the near ultraviolet/visible to the near infrared. The LSPR is broader in the CS2 based colloid due to the large chemical interaction of sulfur and the Ag surface; this is a typical result for Ag-CS2 colloid with homogeneous NPs size distribution as illustrated by the transmission electron microscope image presented in [14].

 figure: Fig. 1

Fig. 1 Linear absorption spectra of the colloids with f=4.0×105 and solvents: (a) sample A (cell thickness: 1 mm) and (b) sample B (cell thickness: 5 mm).

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Figure 2 shows the Z-scan traces for samples A and B. Figure 2(a) illustrates the results obtained for sample A using the CA scheme corresponding to f=0.8×105, 1.3×105, 2.0×105 and 3.0×105. OA Z-scan profiles did not show NL absorption for these volume fractions. However, for 3.3×105<f5.3×105 the colloid presented saturated absorption as shown in Fig. 2(b) due to the small detuning between the laser frequency and the LSPR. Z-scan measurements were also performed for sample B with f varying from 0.4×105 to 4.5×105 as displayed in Figs. 2(c) and 2(d), respectively.

 figure: Fig. 2

Fig. 2 Normalized Z-scan profiles obtained for different NPs volume fractions f. For sample A (laser peak intensity: 5GW/cm2): (a) closed-aperture scheme; (b) open-aperture scheme. For sample B (laser peak intensity: 0.2GW/cm2): (c) closed-aperture and (d) open-aperture scheme.

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Sign-reversal of the NL refractive index as a function of f can be seen for both samples in the interval 1.3×105<f<3.3×105.

Experiments were also performed using different laser intensities. The samples were investigated in order to determine n2, n4, and n6 as well as to the illustrate their behavior for the f value that corresponds to n2=0. Figure 3(a) shows the NL refractive response of sample A for f=1.6×105 (n2=0 and n4=+3.2×1025cm4/W2). For laser intensities up to 4GW/cm2 no feature is observed in the CA Z-scan trace while for 9GW/cm2 a positive effective NL refractive index is observed due to the HON. On the other hand, when f1.6×105 and for laser intensity of 2GW/cm2 larger CA Z-scan signal, dominated by the negative third-order nonlinearity, is observed [21]. This behavior is illustrated in Fig. 3(b) which shows the Z-scan trace obtained for 4GW/cm2. For larger intensities clear indication of HON contribution is observed in the Z-scan traces as illustrated in Fig. 3(b) corresponding to n2=5.5×1015, n4=+2.1×1024cm4/W2 and n6=2.6×1034cm6/W3 for f=5.0×105. The values of n2, n4, and n6 mentioned above were obtained as described below. The results for sample B is analogous to the ones shown in Fig. 3.

 figure: Fig. 3

Fig. 3 Closed-aperture Z-scan profiles, for sample A, obtained for different laser peak intensities with two NPs volume fractions: (a) f=1.6×105 and (b) f=5.0×105. The solid lines were obtained using Eq. (16).

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The solid lines in Fig. 3 correspond to the normalized laser transmittance, measured in the far field as function of the sample's position, described by [15]

T(z,ΔΦ0)1+N=13(4N)ΔΦ0(2N+1)z/z0[(z/z0)2+(2N+1)2][(z/z0)2+1]N,
where ΔΦ0(2N+1)=kn2NIN[1exp(Nα0L)]/(Nα0), k=2πn0/λ, n2N is the (2N+1)th refractive index, λ is the laser wavelength, L is the sample length, α0 is the linear absorption coefficient and z0 is the Rayleigh length of the focused beam. For both samples it was observed that for the various NPs volume fractions the normalized peak-to-valley transmittance change, |ΔTPV|, obtained from the CA Z-scan traces, present a polynomial behavior versus I. From these measurements the NL parameters of third and higher order were obtained, as in [15, 16]. A combination of the Levenberg-Marquardt and the least-squares minimum method were used to minimize errors.

For sample B a NL refractive medium with n2=0, n4=+1.3×1022cm4/W2 and n6=5.7×1031cm6/W3 was obtained for f=1.8×105. The OA profiles, shown in Figs. 2(b) and 2(d), correspond to α2=4.9×1010cm/W, α4=+1.4×1019cm3/W2 and α6=1.7×1029cm5/W3 for sample A with f=5.0×105; and α2=1.0×109cm/W, α4=+3.9×1018cm3/W2 and α6=1.3×1026cm5/W3 for sample B with f=2.0×105. The results for sample A are the same reported in [21].

Further results for sample B are summarized in Fig. 4. Figure 4(a) illustrates the destructive interference between the n2Reχeff(3) and n4Reχeff(5) contributions versus f. For f=3.3×105 we obtain a NL refractive MDNC with n6=1.1×1030cm6/W3 (septic refractive nonlinearity). Analogously, a NL refractive MDNC with n4=+1.1×1022cm4/W2 (quintic refractive nonlinearity), is shown in Fig. 4(b) for f=1.5×105 due to destructive interference between the third- and seventh-order contributions. Figure 4(c) exhibits a NL absorptive behavior with α6=7.5×1027cm5/W3, for f=1.2×105 and I=2.5×108W/cm2. Using Eqs. (13)-(15) we obtained the third-order susceptibility of the host, χh(3)=2.9×1020+i3.41×1022(m2/V2)in agreement with [25, 29], and the NL susceptibilities for the NPs given by χnp(3)=6.4×1016+i2.0×1016(m2/V2), χnp(5)=1.37×1033i2.2×1033(m4/V4) and χnp(7)=4.1×1051+i5.1×1051(m6/V6) for the sample B. Notice that the value of χnp(3) is in agreement with [14] that was determined in the absence of HON. We recall that the silver NPs susceptibility depends on the stabilizing agents attached to the NPs surface as well as the host solvent [30, 31]. It is important to observe that n2, n4 and n6 depend linearly with f as obtained in Eqs. (13)-(15). Terms proportional to f2 due to cascade contributions [18] were not observed in the presents experiments.

 figure: Fig. 4

Fig. 4 Dependence of the effective third-, fifth- and seventh-order coefficients of sample B as a function of the volume fraction, f. Notice that the sample presents: septic refractive nonlinearity at f=3.3×105 (a) and quintic refractive nonlinearity at f=1.5×105 (b). Figures (a) and (b) were obtained with laser peak intensity of 108W/cm2. When f=1.2×105 and the laser peak intensity is 2.5×108W/cm2, the NL absorption is due to Imχeff(7) (c).

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3.3 Spatial self-phase modulation (SPM) in a colloid with quintic refractive nonlinearity

SPM is a phenomenon due to interaction of a laser beam with the NL medium inducing an intensity dependent refractive index that is spatially modulated. Focusing/defocusing and conical diffraction due to SPM were observed for various materials (see for instance [3240]). For high intensities SPM is influenced by HON and the simultaneous presence of third- and fifth-order nonlinearities allow the propagation of spatial solitons in liquids [41].

Here, to investigate the SPM effects in a quintic composite (n2=n6=0;n40), we examined the transverse beam profile in the far-field region after propagation through the NL samples. Figures 5(a)-5(c) show diffraction patterns observed in the beam transmitted by sample A (with f=1.6×105) due to the SPM effect. The f value used corresponds to a medium having NL response dominated by Reχeff(5) with Reχeff(3)=0. The diffraction patterns observed in the far-field consist of concentric rings that vary in number, thickness and positions depending on the fifth-order NL phase-shift, ΔΦ0(5), and laser intensity.

 figure: Fig. 5

Fig. 5 Diffraction patterns of a Gaussian beam in a quintic refractive composite placed in the (a)–(f) focal plane and (g)–(j) around the focal plane position. Laser peak intensities: (a) I1=70GW/cm2; (b) I2=90GW/cm2; (c) I3=100GW/cm2. Experimental (black line) and theoretical (red line) intensity distribution versus the radial coordinates for (d) I1; (e) I2; (f) I3. Sample placed 0.3 mm (g) before and (h) after of the focal plane with laser peak intensity: I=70GW/cm2 at the focus. Intensity distribution as function of the radial coordinates considering (i) R=0.4mm; (j) R=+0.4mm. The red curves were obtained using the value of n4 determined in the Z-scan experiments.

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To describe the conical diffraction we used the Fraunhofer approximation of the Fresnel-Kirchhoff diffraction integral, given by

I=I0|0J0(kθr)exp[r2wp2iϕ(r)]rdr|2,
where, I0=4π2|E(0,z0)exp[α0L/2]/iλD|, E(0,z0) is the electric field at the beam axis, z0 is the focal plane position, θ is the far-field diffraction angle, and J0(kθr) is the first kind zero-order Bessel function [42]. ϕ(r)=kn0r2/2R+ΔΦ0exp[2r2/wp2] represents to the total phase-shift, including the Gaussian phase due to the linear propagation plus the transverse NL phase-shift, r is the radial distance from the laser axis, R is the wavefront radius of curvature, wp is the beam radius at the entrance plane of the NL medium, and D is the distance from the exit plane of the cell and the far-field detection plane.

Figures 5(d)-5(f) show the intensity distribution versus the radial coordinate (black line) obtained from CCD images processed as intensity matrices. The red lines correspond to the numerical results obtained from Eq. (17), considering the NL phase-shift due to n4 that was obtained from the Z-scan experiments. A low-intensity background is observed in the experimental profile due to linear light scattering. Notice that the number of rings, their thickness and the spacing are in good agreement with the numerical results.

Figures 5(g) and 5(h) illustrate the SPM effect in sample A for f=1.6×105. Notice that different diffraction patterns are observed when the sample is located of 0.3 mm before (R<0) and 0.3 mm after (R>0) the focal plane, respectively. The curves in Figs. 5(i)-5(j) show that the analysis of the experimental images (black line) are consistent with the theoretical result (red line), for R±0.4mm. The present experimental results are in agreement with the numerical results of [32] that predicted a far-field diffraction pattern formation that depends on the sign of the product between the wavefront curvature radius and the NL phase-shift induced in the sample. It is important to notice that diffraction patterns, due to pure χeff(5), are reported here for the first time, presenting good agreement with theoretical predictions.

3.4 Spatial cross-phase modulation (XPM) experiment in a colloid with septic nonlinearity

To investigate the XPM due to the septic refractive index (n2=n4=0;n60) the laser beam was split into pump and probe beams with beam's waist of w0=100μm (Rayleigh length 6cm) in the entrance face of the sample. The two beams propagate in opposite directions along sample B (cell length: 5 cm). The pump beam intensity Ipump=108W/cm2 and f=3.3×105 were chosen; the probe-to-pump intensities ratio was 1:10 to ensure that the NL phase shift is due to the pump beam only.

Figure 6 shows the pump beam profile after propagation through sample B with f=3.3×105. Figure 6(a) exhibits the beam profile at the far-field region for Ipump=106W/cm2, while Fig. 6(b) shows the spatial broadening by a factor of 2, for Ipump=108W/cm2, due to n6=1.1×1030cm6/W3. Figure 6(c) shows the intensity distribution of the pump beam transverse profile (black lines), obtained from Figs. 6(a)-6(b). The red and blue lines correspond to the calculated profile for pump intensity of 106 and 108W/cm2, respectively, using the Eq. (18) considering E2=0.

 figure: Fig. 6

Fig. 6 Pump beam profile after propagation through sample B, for f=3.3×105: (a) I1=106W/cm2; (b) I2=108W/cm2. (c) Self-defocusing traces due to χeff(7) obtained from (a) and (b). The red and blue lines are theoretical results for the respective intensities.

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The XPM effects due to χeff(7) were also studied by changing the relative distance (x/w0) between the centers of the incident pump and probe beams. Figures 7(a)-7(c) shows the experimental probe beam profiles after propagation through sample B, with f=3.3×105, in presence of pump beam, Ipump=108W/cm2, for different values of (x/w0). The pink and white lines represent the positions of incident probe and pump beams, respectively. Figures 7(d)-7(f) show the intensity profile of the probe beam without the pump beam (red dashed lines) and in the presence of the pump beam (black lines), obtained from the images shown in Figs. 7(a)-7(c). Notice in particular a partial focusing of the probe beam induced by the pump beam although the sample presents negative value of n6. This effect is analogous to the third-order induced focusing effect reported in [39] where the sample presented a self-defocusing behavior at the laser frequency used.

 figure: Fig. 7

Fig. 7 Experimental probe beam profile due to χeff(7) for a distance of (a) x=0; (b) x=w0 and (c) x=2.2w0 between the centers of the incident pump (white line) and probe (pink line) beams; (d)-(f) Normalized intensity distribution of the probe beam transverse profile without the pump beam (red dashed line) and with the pump beam (black line); (g)-(i) Theoretical probe beam profile calculated using Eqs. (18) and (19) with the NL coefficients determined in the Z-scan experiments for the relative distances of (a)-(c). Pump beam intensity: 108W/cm2; probe beam intensity: 107W/cm2.

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The mathematical description of the XPM experiment is provided by the coupled NL Schrödinger equations, Eqs. (18) and (19), considering two counter-propagating beams. Using the slowly varying approximation, the equations are

2ikE1z+ΔE1=ω2c2[3χeff(3)(|E1|2+2|E2|2)E1+10χeff(5)(|E1|4+6|E1|2|E2|2+3|E2|4)E1+35χeff(7)(|E1|6+18|E1|2|E2|4+12|E1|4|E2|2+4|E2|6)E1],
2ikE2z+ΔE2=ω2c2[3χeff(3)(2|E1|2+|E2|2)E2+10χeff(5)(3|E1|4+6|E1|2|E2|2+|E2|4)E2+35χeff(7)(4|E1|6+12|E1|2|E2|4+18|E1|4|E2|2+|E2|6)E2],
where E1 and E2 are the optical field amplitudes of the pump and probe beams, respectively; Δ is the transverse Laplacian operator, ω is the laser frequency and c is the speed of light in vacuum. The values of χeff(3), χeff(5) and χeff(7), for each volume fraction f, is obtained from Eqs. (13)-(15).

Figures 7(g)-7(i) show the theoretical probe beam transverse profile, obtained from Eqs. (18) and (19), using the NL parameters measured in the Z-scan experiments (n2=n4=0;n6=1.1×1030cm6/W3) when f=3.3×105, Iprobe=107W/cm2 and Ipump=108W/cm2. A good agreement is observed between experimental and theoretical results that support the given interpretation.

4. Conclusion

In summary, we reported conditions for obtaining a MDNC presenting either fifth- or seventh-order refractive index, adjusting the volume fraction occupied by silver NPs with respect to the host volume and the incident laser intensity. Spatial self- and cross-phase modulation were demonstrated due to the effective fifth- and seventh-order nonlinearities and their characterization were reported. The experimental procedure to control the dominant nonlinearity was supported by the Maxwell-Garnett model that was generalized to include effects of fifth- and seventh-order.

Acknowledgments

We acknowledge financial support from the Brazilian agencies Conselho Nacional de Desenvolvimento Cientifico e Tecnológico (CNPq) and the Fundação de Amparo à Ciência e Tecnologia do Estado de Pernambuco (FACEPE). The work was performed in the framework of the National Institute of Photonics (INCT de Fotônica) project and PRONEX/CNPq/FACEPE.

References and links

1. M. Eichelbaum, K. Rademann, A. Hoell, D. M. Tatchev, W. Weigel, R. Stößer, and G. Pacchioni, “Photoluminescence of atomic gold and silver particles in soda-lime silicate glasses,” Nanotechnology 19(13), 135701 (2008). [CrossRef]   [PubMed]  

2. S. Nakashima, K. Sugioka, K. Tanaka, M. Shimizu, Y. Shimotsuma, K. Miura, K. Midorikawa, and K. Mukai, “Plasmonically enhanced Faraday effect in metal and ferrite nanoparticles composite precipitated inside glass,” Opt. Express 20(27), 28191–28199 (2012). [CrossRef]   [PubMed]  

3. J. P. Huang and K. W. Yu, “Enhanced nonlinear optical responses of materials: composite effects,” Phys. Rep. 431(3), 87–172 (2006). [CrossRef]  

4. S. Mohan, J. Lange, H. Graener, and G. Seifert, “Surface plasmon assisted optical nonlinearities of uniformly oriented metal nano-ellipsoids in glass,” Opt. Express 20(27), 28655–28663 (2012). [CrossRef]   [PubMed]  

5. A. P. Kulkarni, K. M. Noone, K. Munechika, S. R. Guyer, and D. S. Ginger, “Plasmon-enhanced charge carrier generation in organic photovoltaic films using silver nanoprisms,” Nano Lett. 10(4), 1501–1505 (2010). [CrossRef]   [PubMed]  

6. L. Lu, Z. Luo, T. Xu, and L. Yu, “Cooperative plasmonic effect of Ag and Au nanoparticles on enhancing performance of polymer solar cells,” Nano Lett. 13(1), 59–64 (2013). [CrossRef]   [PubMed]  

7. R. A. Ganeev, A. I. Ryasnyansky, A. L. Stepanov, and T. Usmanov, “Saturated absorption and nonlinear refraction of silicate glasses doped with silver nanoparticles at 532 nm,” Opt. Quantum Electron. 36(10), 949–960 (2004). [CrossRef]  

8. R. A. Ganeev and A. I. Ryasnyansky, “Nonlinear optical characteristics of nanoparticles in suspensions and solid matrices,” Appl. Phys. B 84(1–2), 295–302 (2006). [CrossRef]  

9. K.-H. Kim, A. Husakou, and J. Herrmann, “Saturable absorption in composites doped with metal nanoparticles,” Opt. Express 18(21), 21918–21925 (2010). [CrossRef]   [PubMed]  

10. K.-H. Kim, A. Husakou, and J. Herrmann, “Slow light in dielectric composite materials of metal nanoparticles,” Opt. Express 20(23), 25790–25797 (2012). [CrossRef]   [PubMed]  

11. J. Jayabalan, A. Singh, S. Khan, and R. Chari, “Third-order nonlinearity of metal nanoparticles: isolation of instantaneous and delayed contributions,” J. Appl. Phys. 112(10), 103524 (2012). [CrossRef]  

12. L. De Boni, E. C. Barbano, T. A. de Assumpção, L. Misoguti, L. R. P. Kassab, and S. C. Zilio, “Femtosecond third-order nonlinear spectra of lead-germanium oxide glasses containing silver nanoparticles,” Opt. Express 20(6), 6844–6850 (2012). [CrossRef]   [PubMed]  

13. C. B. de Araújo, T. R. Oliveira, E. L. Falcão-Filho, D. M. Silva, and L. R. P. Kassab, “Nonlinear optical properties of PbO–GeO2 films containing gold nanoparticles,” J. Lumin. 133, 180–183 (2013). [CrossRef]  

14. E. L. Falcão-Filho, C. B. de Araújo, A. Galembeck, M. M. Oliveira, and A. J. G. Zarbin, “Nonlinear susceptibility of colloids consisting of silver nanoparticles in carbon disulfide,” J. Opt. Soc. Am. B 22(11), 2444–2449 (2005). [CrossRef]  

15. E. L. Falcão-Filho, C. B. de Araújo, and J. J. Rodrigues Jr., “High-order nonlinearities of aqueous colloids containing silver nanoparticles,” J. Opt. Soc. Am. B 24(12), 2948–2956 (2007). [CrossRef]  

16. E. L. Falcão-Filho, R. Barbosa-Silva, R. G. Sobral-Filho, A. M. Brito-Silva, A. Galembeck, and C. B. de Araújo, “High-order nonlinearity of silica-gold nanoshells in chloroform at 1560 nm,” Opt. Express 18(21), 21636–21644 (2010). [CrossRef]   [PubMed]  

17. J. Jayabalan, “Origin and time dependence of higher-order nonlinearities in metal nanocomposites,” J. Opt. Soc. Am. B 28(10), 2448–2455 (2011). [CrossRef]  

18. K. Dolgaleva, H. Shin, and R. W. Boyd, “Observation of a microscopic cascaded contribution to the fifth-order nonlinear susceptibility,” Phys. Rev. Lett. 103(11), 113902 (2009). [CrossRef]   [PubMed]  

19. G. Stegeman, D. G. Papazoglou, R. Boyd, and S. Tzortzakis, “Nonlinear birefringence due to non-resonant, higher-order Kerr effect in isotropic media,” Opt. Express 19(7), 6387–6399 (2011). [CrossRef]   [PubMed]  

20. K. Dolgaleva, R. W. Boyd, and J. E. Sipe, “Cascaded nonlinearity caused by local-field effects in the two-level atom,” Phys. Rev. A 76(6), 063806 (2007). [CrossRef]  

21. A. S. Reyna and C. B. de Araújo, “Nonlinearity management of photonic composites and observation of spatial-modulation instability due to quintic nonlinearity,” Phys. Rev. A 89(6), 063803 (2014). [CrossRef]  

22. J. E. Sipe and R. W. Boyd, “Nonlinear susceptibility of composite optical materials in the Maxwell Garnett model,” Phys. Rev. A 46(3), 1614–1629 (1992). [CrossRef]   [PubMed]  

23. P. C. Lee and D. Meisel, “Adsorption and surface-enhanced Raman of dyes on silver and gold sols,” J. Phys. Chem. 86(17), 3391–3395 (1982). [CrossRef]  

24. A. M. Brito-Silva, L. A. Gómez, C. B. de Araújo, and A. Galembeck, “Laser ablated silver nanoparticles with nearly the same size in different carrier media,” J. Nanomater. 2010, 142897 (2010). [CrossRef]  

25. M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990). [CrossRef]  

26. H. Ma, A. S. L. Gomes, and C. B. de Araújo, “Measurements of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. 59(21), 2666–2668 (1991). [CrossRef]  

27. P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University Press, 1990).

28. R. L. Sutherland, Handbook of Nonlinear Optics (Dekker, 1996).

29. X. Liu, S. Guo, H. Wang, N. Ming, and L. Hou, “Investigation of the influence of finite aperture size on the Z-scan transmittance curve,” J. Nonlinear Opt. Phys. Mater. 10(4), 431–439 (2001). [CrossRef]  

30. L. A. Gómez, C. B. de Araújo, A. M. Brito-Silva, and A. Galembeck, “Influence of stabilizing agents on the nonlinear susceptibility of silver nanoparticles,” J. Opt. Soc. Am. B 24(9), 2136–2140 (2007). [CrossRef]  

31. L. A. Gómez, C. B. de Araújo, A. M. Brito-Silva, and A. Galembeck, “Solvent effects on the linear and nonlinear optical response of silver nanoparticles,” Appl. Phys. B 92(1), 61–66 (2008). [CrossRef]  

32. L. Deng, K. He, T. Zhou, and C. Li, “Formation and evolution of far-field diffraction patterns of divergent and convergent Gaussian beams passing through self-focusing and self-defocusing media,” J. Opt. A, Pure Appl. Opt. 7(8), 409–415 (2005). [CrossRef]  

33. M. H. M. Ara, Z. Dehghani, R. Sahraei, A. Daneshfar, Z. Javadi, and F. Divsar, “Diffraction patterns and nonlinear optical properties of gold nanoparticles,” J. Quant. Spectrosc. Radiat. Transf. 113(5), 366–372 (2012). [CrossRef]  

34. M. A. R. C. Alencar and C. B. de Araújo, “Conical diffraction instability due to cross-phase modulation in Kerr media,” J. Opt. Soc. Am. B 23(2), 302–307 (2006). [CrossRef]  

35. E. V. G. Ramirez, M. L. A. Carrasco, M. M. M. Otero, S. C. Cerda, and M. D. I. Castillo, “Far field intensity distributions due to spatial self phase modulation of a Gaussian beam by a thin nonlocal nonlinear media,” Opt. Express 18(21), 22067–22079 (2010). [CrossRef]   [PubMed]  

36. S. D. Durbin, S. M. Arakelian, and Y. R. Shen, “Laser-induced diffraction rings from a nematic-liquid-crystal film,” Opt. Lett. 6(9), 411–413 (1981). [PubMed]  

37. G. Wang, S. Zhang, F. A. Umran, X. Cheng, N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014). [CrossRef]  

38. T. Myint and R. R. Alfano, “Spatial phase modulation from permanent memory in doped glass,” Opt. Lett. 35(8), 1275–1277 (2010). [CrossRef]   [PubMed]  

39. J. M. Hickmann, A. S. L. Gomes, and C. B. de Araújo, “Observation of spatial cross-phase modulation effects in a self-defocusing nonlinear medium,” Phys. Rev. Lett. 68(24), 3547–3550 (1992). [CrossRef]   [PubMed]  

40. R. Zamiri, A. Zakaria, M. B. Ahmad, A. R. Sadrolhosseini, K. Shameli, M. Darroudi, and M. A. Mahdi, “Investigation of spatial self-phase modulation of silver nanoparticles in clay suspension,” Optik (Stuttg.) 122(9), 836–838 (2011). [CrossRef]  

41. E. L. Falcão-Filho, C. B. de Araújo, G. Boudebs, H. Leblond, and V. Skarka, “Robust two-dimensional spatial solitons in liquid carbon disulfide,” Phys. Rev. Lett. 110(1), 013901 (2013). [CrossRef]   [PubMed]  

42. J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

References

  • View by:

  1. M. Eichelbaum, K. Rademann, A. Hoell, D. M. Tatchev, W. Weigel, R. Stößer, and G. Pacchioni, “Photoluminescence of atomic gold and silver particles in soda-lime silicate glasses,” Nanotechnology 19(13), 135701 (2008).
    [Crossref] [PubMed]
  2. S. Nakashima, K. Sugioka, K. Tanaka, M. Shimizu, Y. Shimotsuma, K. Miura, K. Midorikawa, and K. Mukai, “Plasmonically enhanced Faraday effect in metal and ferrite nanoparticles composite precipitated inside glass,” Opt. Express 20(27), 28191–28199 (2012).
    [Crossref] [PubMed]
  3. J. P. Huang and K. W. Yu, “Enhanced nonlinear optical responses of materials: composite effects,” Phys. Rep. 431(3), 87–172 (2006).
    [Crossref]
  4. S. Mohan, J. Lange, H. Graener, and G. Seifert, “Surface plasmon assisted optical nonlinearities of uniformly oriented metal nano-ellipsoids in glass,” Opt. Express 20(27), 28655–28663 (2012).
    [Crossref] [PubMed]
  5. A. P. Kulkarni, K. M. Noone, K. Munechika, S. R. Guyer, and D. S. Ginger, “Plasmon-enhanced charge carrier generation in organic photovoltaic films using silver nanoprisms,” Nano Lett. 10(4), 1501–1505 (2010).
    [Crossref] [PubMed]
  6. L. Lu, Z. Luo, T. Xu, and L. Yu, “Cooperative plasmonic effect of Ag and Au nanoparticles on enhancing performance of polymer solar cells,” Nano Lett. 13(1), 59–64 (2013).
    [Crossref] [PubMed]
  7. R. A. Ganeev, A. I. Ryasnyansky, A. L. Stepanov, and T. Usmanov, “Saturated absorption and nonlinear refraction of silicate glasses doped with silver nanoparticles at 532 nm,” Opt. Quantum Electron. 36(10), 949–960 (2004).
    [Crossref]
  8. R. A. Ganeev and A. I. Ryasnyansky, “Nonlinear optical characteristics of nanoparticles in suspensions and solid matrices,” Appl. Phys. B 84(1–2), 295–302 (2006).
    [Crossref]
  9. K.-H. Kim, A. Husakou, and J. Herrmann, “Saturable absorption in composites doped with metal nanoparticles,” Opt. Express 18(21), 21918–21925 (2010).
    [Crossref] [PubMed]
  10. K.-H. Kim, A. Husakou, and J. Herrmann, “Slow light in dielectric composite materials of metal nanoparticles,” Opt. Express 20(23), 25790–25797 (2012).
    [Crossref] [PubMed]
  11. J. Jayabalan, A. Singh, S. Khan, and R. Chari, “Third-order nonlinearity of metal nanoparticles: isolation of instantaneous and delayed contributions,” J. Appl. Phys. 112(10), 103524 (2012).
    [Crossref]
  12. L. De Boni, E. C. Barbano, T. A. de Assumpção, L. Misoguti, L. R. P. Kassab, and S. C. Zilio, “Femtosecond third-order nonlinear spectra of lead-germanium oxide glasses containing silver nanoparticles,” Opt. Express 20(6), 6844–6850 (2012).
    [Crossref] [PubMed]
  13. C. B. de Araújo, T. R. Oliveira, E. L. Falcão-Filho, D. M. Silva, and L. R. P. Kassab, “Nonlinear optical properties of PbO–GeO2 films containing gold nanoparticles,” J. Lumin. 133, 180–183 (2013).
    [Crossref]
  14. E. L. Falcão-Filho, C. B. de Araújo, A. Galembeck, M. M. Oliveira, and A. J. G. Zarbin, “Nonlinear susceptibility of colloids consisting of silver nanoparticles in carbon disulfide,” J. Opt. Soc. Am. B 22(11), 2444–2449 (2005).
    [Crossref]
  15. E. L. Falcão-Filho, C. B. de Araújo, and J. J. Rodrigues., “High-order nonlinearities of aqueous colloids containing silver nanoparticles,” J. Opt. Soc. Am. B 24(12), 2948–2956 (2007).
    [Crossref]
  16. E. L. Falcão-Filho, R. Barbosa-Silva, R. G. Sobral-Filho, A. M. Brito-Silva, A. Galembeck, and C. B. de Araújo, “High-order nonlinearity of silica-gold nanoshells in chloroform at 1560 nm,” Opt. Express 18(21), 21636–21644 (2010).
    [Crossref] [PubMed]
  17. J. Jayabalan, “Origin and time dependence of higher-order nonlinearities in metal nanocomposites,” J. Opt. Soc. Am. B 28(10), 2448–2455 (2011).
    [Crossref]
  18. K. Dolgaleva, H. Shin, and R. W. Boyd, “Observation of a microscopic cascaded contribution to the fifth-order nonlinear susceptibility,” Phys. Rev. Lett. 103(11), 113902 (2009).
    [Crossref] [PubMed]
  19. G. Stegeman, D. G. Papazoglou, R. Boyd, and S. Tzortzakis, “Nonlinear birefringence due to non-resonant, higher-order Kerr effect in isotropic media,” Opt. Express 19(7), 6387–6399 (2011).
    [Crossref] [PubMed]
  20. K. Dolgaleva, R. W. Boyd, and J. E. Sipe, “Cascaded nonlinearity caused by local-field effects in the two-level atom,” Phys. Rev. A 76(6), 063806 (2007).
    [Crossref]
  21. A. S. Reyna and C. B. de Araújo, “Nonlinearity management of photonic composites and observation of spatial-modulation instability due to quintic nonlinearity,” Phys. Rev. A 89(6), 063803 (2014).
    [Crossref]
  22. J. E. Sipe and R. W. Boyd, “Nonlinear susceptibility of composite optical materials in the Maxwell Garnett model,” Phys. Rev. A 46(3), 1614–1629 (1992).
    [Crossref] [PubMed]
  23. P. C. Lee and D. Meisel, “Adsorption and surface-enhanced Raman of dyes on silver and gold sols,” J. Phys. Chem. 86(17), 3391–3395 (1982).
    [Crossref]
  24. A. M. Brito-Silva, L. A. Gómez, C. B. de Araújo, and A. Galembeck, “Laser ablated silver nanoparticles with nearly the same size in different carrier media,” J. Nanomater. 2010, 142897 (2010).
    [Crossref]
  25. M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
    [Crossref]
  26. H. Ma, A. S. L. Gomes, and C. B. de Araújo, “Measurements of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. 59(21), 2666–2668 (1991).
    [Crossref]
  27. P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University Press, 1990).
  28. R. L. Sutherland, Handbook of Nonlinear Optics (Dekker, 1996).
  29. X. Liu, S. Guo, H. Wang, N. Ming, and L. Hou, “Investigation of the influence of finite aperture size on the Z-scan transmittance curve,” J. Nonlinear Opt. Phys. Mater. 10(4), 431–439 (2001).
    [Crossref]
  30. L. A. Gómez, C. B. de Araújo, A. M. Brito-Silva, and A. Galembeck, “Influence of stabilizing agents on the nonlinear susceptibility of silver nanoparticles,” J. Opt. Soc. Am. B 24(9), 2136–2140 (2007).
    [Crossref]
  31. L. A. Gómez, C. B. de Araújo, A. M. Brito-Silva, and A. Galembeck, “Solvent effects on the linear and nonlinear optical response of silver nanoparticles,” Appl. Phys. B 92(1), 61–66 (2008).
    [Crossref]
  32. L. Deng, K. He, T. Zhou, and C. Li, “Formation and evolution of far-field diffraction patterns of divergent and convergent Gaussian beams passing through self-focusing and self-defocusing media,” J. Opt. A, Pure Appl. Opt. 7(8), 409–415 (2005).
    [Crossref]
  33. M. H. M. Ara, Z. Dehghani, R. Sahraei, A. Daneshfar, Z. Javadi, and F. Divsar, “Diffraction patterns and nonlinear optical properties of gold nanoparticles,” J. Quant. Spectrosc. Radiat. Transf. 113(5), 366–372 (2012).
    [Crossref]
  34. M. A. R. C. Alencar and C. B. de Araújo, “Conical diffraction instability due to cross-phase modulation in Kerr media,” J. Opt. Soc. Am. B 23(2), 302–307 (2006).
    [Crossref]
  35. E. V. G. Ramirez, M. L. A. Carrasco, M. M. M. Otero, S. C. Cerda, and M. D. I. Castillo, “Far field intensity distributions due to spatial self phase modulation of a Gaussian beam by a thin nonlocal nonlinear media,” Opt. Express 18(21), 22067–22079 (2010).
    [Crossref] [PubMed]
  36. S. D. Durbin, S. M. Arakelian, and Y. R. Shen, “Laser-induced diffraction rings from a nematic-liquid-crystal film,” Opt. Lett. 6(9), 411–413 (1981).
    [PubMed]
  37. G. Wang, S. Zhang, F. A. Umran, X. Cheng, N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
    [Crossref]
  38. T. Myint and R. R. Alfano, “Spatial phase modulation from permanent memory in doped glass,” Opt. Lett. 35(8), 1275–1277 (2010).
    [Crossref] [PubMed]
  39. J. M. Hickmann, A. S. L. Gomes, and C. B. de Araújo, “Observation of spatial cross-phase modulation effects in a self-defocusing nonlinear medium,” Phys. Rev. Lett. 68(24), 3547–3550 (1992).
    [Crossref] [PubMed]
  40. R. Zamiri, A. Zakaria, M. B. Ahmad, A. R. Sadrolhosseini, K. Shameli, M. Darroudi, and M. A. Mahdi, “Investigation of spatial self-phase modulation of silver nanoparticles in clay suspension,” Optik (Stuttg.) 122(9), 836–838 (2011).
    [Crossref]
  41. E. L. Falcão-Filho, C. B. de Araújo, G. Boudebs, H. Leblond, and V. Skarka, “Robust two-dimensional spatial solitons in liquid carbon disulfide,” Phys. Rev. Lett. 110(1), 013901 (2013).
    [Crossref] [PubMed]
  42. J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

2014 (2)

A. S. Reyna and C. B. de Araújo, “Nonlinearity management of photonic composites and observation of spatial-modulation instability due to quintic nonlinearity,” Phys. Rev. A 89(6), 063803 (2014).
[Crossref]

G. Wang, S. Zhang, F. A. Umran, X. Cheng, N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

2013 (3)

E. L. Falcão-Filho, C. B. de Araújo, G. Boudebs, H. Leblond, and V. Skarka, “Robust two-dimensional spatial solitons in liquid carbon disulfide,” Phys. Rev. Lett. 110(1), 013901 (2013).
[Crossref] [PubMed]

L. Lu, Z. Luo, T. Xu, and L. Yu, “Cooperative plasmonic effect of Ag and Au nanoparticles on enhancing performance of polymer solar cells,” Nano Lett. 13(1), 59–64 (2013).
[Crossref] [PubMed]

C. B. de Araújo, T. R. Oliveira, E. L. Falcão-Filho, D. M. Silva, and L. R. P. Kassab, “Nonlinear optical properties of PbO–GeO2 films containing gold nanoparticles,” J. Lumin. 133, 180–183 (2013).
[Crossref]

2012 (6)

2011 (3)

2010 (6)

2009 (1)

K. Dolgaleva, H. Shin, and R. W. Boyd, “Observation of a microscopic cascaded contribution to the fifth-order nonlinear susceptibility,” Phys. Rev. Lett. 103(11), 113902 (2009).
[Crossref] [PubMed]

2008 (2)

M. Eichelbaum, K. Rademann, A. Hoell, D. M. Tatchev, W. Weigel, R. Stößer, and G. Pacchioni, “Photoluminescence of atomic gold and silver particles in soda-lime silicate glasses,” Nanotechnology 19(13), 135701 (2008).
[Crossref] [PubMed]

L. A. Gómez, C. B. de Araújo, A. M. Brito-Silva, and A. Galembeck, “Solvent effects on the linear and nonlinear optical response of silver nanoparticles,” Appl. Phys. B 92(1), 61–66 (2008).
[Crossref]

2007 (3)

2006 (3)

R. A. Ganeev and A. I. Ryasnyansky, “Nonlinear optical characteristics of nanoparticles in suspensions and solid matrices,” Appl. Phys. B 84(1–2), 295–302 (2006).
[Crossref]

J. P. Huang and K. W. Yu, “Enhanced nonlinear optical responses of materials: composite effects,” Phys. Rep. 431(3), 87–172 (2006).
[Crossref]

M. A. R. C. Alencar and C. B. de Araújo, “Conical diffraction instability due to cross-phase modulation in Kerr media,” J. Opt. Soc. Am. B 23(2), 302–307 (2006).
[Crossref]

2005 (2)

L. Deng, K. He, T. Zhou, and C. Li, “Formation and evolution of far-field diffraction patterns of divergent and convergent Gaussian beams passing through self-focusing and self-defocusing media,” J. Opt. A, Pure Appl. Opt. 7(8), 409–415 (2005).
[Crossref]

E. L. Falcão-Filho, C. B. de Araújo, A. Galembeck, M. M. Oliveira, and A. J. G. Zarbin, “Nonlinear susceptibility of colloids consisting of silver nanoparticles in carbon disulfide,” J. Opt. Soc. Am. B 22(11), 2444–2449 (2005).
[Crossref]

2004 (1)

R. A. Ganeev, A. I. Ryasnyansky, A. L. Stepanov, and T. Usmanov, “Saturated absorption and nonlinear refraction of silicate glasses doped with silver nanoparticles at 532 nm,” Opt. Quantum Electron. 36(10), 949–960 (2004).
[Crossref]

2001 (1)

X. Liu, S. Guo, H. Wang, N. Ming, and L. Hou, “Investigation of the influence of finite aperture size on the Z-scan transmittance curve,” J. Nonlinear Opt. Phys. Mater. 10(4), 431–439 (2001).
[Crossref]

1992 (2)

J. E. Sipe and R. W. Boyd, “Nonlinear susceptibility of composite optical materials in the Maxwell Garnett model,” Phys. Rev. A 46(3), 1614–1629 (1992).
[Crossref] [PubMed]

J. M. Hickmann, A. S. L. Gomes, and C. B. de Araújo, “Observation of spatial cross-phase modulation effects in a self-defocusing nonlinear medium,” Phys. Rev. Lett. 68(24), 3547–3550 (1992).
[Crossref] [PubMed]

1991 (1)

H. Ma, A. S. L. Gomes, and C. B. de Araújo, “Measurements of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. 59(21), 2666–2668 (1991).
[Crossref]

1990 (1)

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

1982 (1)

P. C. Lee and D. Meisel, “Adsorption and surface-enhanced Raman of dyes on silver and gold sols,” J. Phys. Chem. 86(17), 3391–3395 (1982).
[Crossref]

1981 (1)

Ahmad, M. B.

R. Zamiri, A. Zakaria, M. B. Ahmad, A. R. Sadrolhosseini, K. Shameli, M. Darroudi, and M. A. Mahdi, “Investigation of spatial self-phase modulation of silver nanoparticles in clay suspension,” Optik (Stuttg.) 122(9), 836–838 (2011).
[Crossref]

Alencar, M. A. R. C.

Alfano, R. R.

Ara, M. H. M.

M. H. M. Ara, Z. Dehghani, R. Sahraei, A. Daneshfar, Z. Javadi, and F. Divsar, “Diffraction patterns and nonlinear optical properties of gold nanoparticles,” J. Quant. Spectrosc. Radiat. Transf. 113(5), 366–372 (2012).
[Crossref]

Arakelian, S. M.

Barbano, E. C.

Barbosa-Silva, R.

Blau, W. J.

G. Wang, S. Zhang, F. A. Umran, X. Cheng, N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

Boudebs, G.

E. L. Falcão-Filho, C. B. de Araújo, G. Boudebs, H. Leblond, and V. Skarka, “Robust two-dimensional spatial solitons in liquid carbon disulfide,” Phys. Rev. Lett. 110(1), 013901 (2013).
[Crossref] [PubMed]

Boyd, R.

Boyd, R. W.

K. Dolgaleva, H. Shin, and R. W. Boyd, “Observation of a microscopic cascaded contribution to the fifth-order nonlinear susceptibility,” Phys. Rev. Lett. 103(11), 113902 (2009).
[Crossref] [PubMed]

K. Dolgaleva, R. W. Boyd, and J. E. Sipe, “Cascaded nonlinearity caused by local-field effects in the two-level atom,” Phys. Rev. A 76(6), 063806 (2007).
[Crossref]

J. E. Sipe and R. W. Boyd, “Nonlinear susceptibility of composite optical materials in the Maxwell Garnett model,” Phys. Rev. A 46(3), 1614–1629 (1992).
[Crossref] [PubMed]

Brito-Silva, A. M.

A. M. Brito-Silva, L. A. Gómez, C. B. de Araújo, and A. Galembeck, “Laser ablated silver nanoparticles with nearly the same size in different carrier media,” J. Nanomater. 2010, 142897 (2010).
[Crossref]

E. L. Falcão-Filho, R. Barbosa-Silva, R. G. Sobral-Filho, A. M. Brito-Silva, A. Galembeck, and C. B. de Araújo, “High-order nonlinearity of silica-gold nanoshells in chloroform at 1560 nm,” Opt. Express 18(21), 21636–21644 (2010).
[Crossref] [PubMed]

L. A. Gómez, C. B. de Araújo, A. M. Brito-Silva, and A. Galembeck, “Solvent effects on the linear and nonlinear optical response of silver nanoparticles,” Appl. Phys. B 92(1), 61–66 (2008).
[Crossref]

L. A. Gómez, C. B. de Araújo, A. M. Brito-Silva, and A. Galembeck, “Influence of stabilizing agents on the nonlinear susceptibility of silver nanoparticles,” J. Opt. Soc. Am. B 24(9), 2136–2140 (2007).
[Crossref]

Carrasco, M. L. A.

Castillo, M. D. I.

Cerda, S. C.

Chari, R.

J. Jayabalan, A. Singh, S. Khan, and R. Chari, “Third-order nonlinearity of metal nanoparticles: isolation of instantaneous and delayed contributions,” J. Appl. Phys. 112(10), 103524 (2012).
[Crossref]

Cheng, X.

G. Wang, S. Zhang, F. A. Umran, X. Cheng, N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

Cheng, Y.

G. Wang, S. Zhang, F. A. Umran, X. Cheng, N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

Coghlan, D.

G. Wang, S. Zhang, F. A. Umran, X. Cheng, N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

Daneshfar, A.

M. H. M. Ara, Z. Dehghani, R. Sahraei, A. Daneshfar, Z. Javadi, and F. Divsar, “Diffraction patterns and nonlinear optical properties of gold nanoparticles,” J. Quant. Spectrosc. Radiat. Transf. 113(5), 366–372 (2012).
[Crossref]

Darroudi, M.

R. Zamiri, A. Zakaria, M. B. Ahmad, A. R. Sadrolhosseini, K. Shameli, M. Darroudi, and M. A. Mahdi, “Investigation of spatial self-phase modulation of silver nanoparticles in clay suspension,” Optik (Stuttg.) 122(9), 836–838 (2011).
[Crossref]

de Araújo, C. B.

A. S. Reyna and C. B. de Araújo, “Nonlinearity management of photonic composites and observation of spatial-modulation instability due to quintic nonlinearity,” Phys. Rev. A 89(6), 063803 (2014).
[Crossref]

C. B. de Araújo, T. R. Oliveira, E. L. Falcão-Filho, D. M. Silva, and L. R. P. Kassab, “Nonlinear optical properties of PbO–GeO2 films containing gold nanoparticles,” J. Lumin. 133, 180–183 (2013).
[Crossref]

E. L. Falcão-Filho, C. B. de Araújo, G. Boudebs, H. Leblond, and V. Skarka, “Robust two-dimensional spatial solitons in liquid carbon disulfide,” Phys. Rev. Lett. 110(1), 013901 (2013).
[Crossref] [PubMed]

E. L. Falcão-Filho, R. Barbosa-Silva, R. G. Sobral-Filho, A. M. Brito-Silva, A. Galembeck, and C. B. de Araújo, “High-order nonlinearity of silica-gold nanoshells in chloroform at 1560 nm,” Opt. Express 18(21), 21636–21644 (2010).
[Crossref] [PubMed]

A. M. Brito-Silva, L. A. Gómez, C. B. de Araújo, and A. Galembeck, “Laser ablated silver nanoparticles with nearly the same size in different carrier media,” J. Nanomater. 2010, 142897 (2010).
[Crossref]

L. A. Gómez, C. B. de Araújo, A. M. Brito-Silva, and A. Galembeck, “Solvent effects on the linear and nonlinear optical response of silver nanoparticles,” Appl. Phys. B 92(1), 61–66 (2008).
[Crossref]

L. A. Gómez, C. B. de Araújo, A. M. Brito-Silva, and A. Galembeck, “Influence of stabilizing agents on the nonlinear susceptibility of silver nanoparticles,” J. Opt. Soc. Am. B 24(9), 2136–2140 (2007).
[Crossref]

E. L. Falcão-Filho, C. B. de Araújo, and J. J. Rodrigues., “High-order nonlinearities of aqueous colloids containing silver nanoparticles,” J. Opt. Soc. Am. B 24(12), 2948–2956 (2007).
[Crossref]

M. A. R. C. Alencar and C. B. de Araújo, “Conical diffraction instability due to cross-phase modulation in Kerr media,” J. Opt. Soc. Am. B 23(2), 302–307 (2006).
[Crossref]

E. L. Falcão-Filho, C. B. de Araújo, A. Galembeck, M. M. Oliveira, and A. J. G. Zarbin, “Nonlinear susceptibility of colloids consisting of silver nanoparticles in carbon disulfide,” J. Opt. Soc. Am. B 22(11), 2444–2449 (2005).
[Crossref]

J. M. Hickmann, A. S. L. Gomes, and C. B. de Araújo, “Observation of spatial cross-phase modulation effects in a self-defocusing nonlinear medium,” Phys. Rev. Lett. 68(24), 3547–3550 (1992).
[Crossref] [PubMed]

H. Ma, A. S. L. Gomes, and C. B. de Araújo, “Measurements of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. 59(21), 2666–2668 (1991).
[Crossref]

de Assumpção, T. A.

De Boni, L.

Dehghani, Z.

M. H. M. Ara, Z. Dehghani, R. Sahraei, A. Daneshfar, Z. Javadi, and F. Divsar, “Diffraction patterns and nonlinear optical properties of gold nanoparticles,” J. Quant. Spectrosc. Radiat. Transf. 113(5), 366–372 (2012).
[Crossref]

Deng, L.

L. Deng, K. He, T. Zhou, and C. Li, “Formation and evolution of far-field diffraction patterns of divergent and convergent Gaussian beams passing through self-focusing and self-defocusing media,” J. Opt. A, Pure Appl. Opt. 7(8), 409–415 (2005).
[Crossref]

Divsar, F.

M. H. M. Ara, Z. Dehghani, R. Sahraei, A. Daneshfar, Z. Javadi, and F. Divsar, “Diffraction patterns and nonlinear optical properties of gold nanoparticles,” J. Quant. Spectrosc. Radiat. Transf. 113(5), 366–372 (2012).
[Crossref]

Dolgaleva, K.

K. Dolgaleva, H. Shin, and R. W. Boyd, “Observation of a microscopic cascaded contribution to the fifth-order nonlinear susceptibility,” Phys. Rev. Lett. 103(11), 113902 (2009).
[Crossref] [PubMed]

K. Dolgaleva, R. W. Boyd, and J. E. Sipe, “Cascaded nonlinearity caused by local-field effects in the two-level atom,” Phys. Rev. A 76(6), 063806 (2007).
[Crossref]

Dong, N.

G. Wang, S. Zhang, F. A. Umran, X. Cheng, N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

Durbin, S. D.

Eichelbaum, M.

M. Eichelbaum, K. Rademann, A. Hoell, D. M. Tatchev, W. Weigel, R. Stößer, and G. Pacchioni, “Photoluminescence of atomic gold and silver particles in soda-lime silicate glasses,” Nanotechnology 19(13), 135701 (2008).
[Crossref] [PubMed]

Falcão-Filho, E. L.

Galembeck, A.

Ganeev, R. A.

R. A. Ganeev and A. I. Ryasnyansky, “Nonlinear optical characteristics of nanoparticles in suspensions and solid matrices,” Appl. Phys. B 84(1–2), 295–302 (2006).
[Crossref]

R. A. Ganeev, A. I. Ryasnyansky, A. L. Stepanov, and T. Usmanov, “Saturated absorption and nonlinear refraction of silicate glasses doped with silver nanoparticles at 532 nm,” Opt. Quantum Electron. 36(10), 949–960 (2004).
[Crossref]

Ginger, D. S.

A. P. Kulkarni, K. M. Noone, K. Munechika, S. R. Guyer, and D. S. Ginger, “Plasmon-enhanced charge carrier generation in organic photovoltaic films using silver nanoprisms,” Nano Lett. 10(4), 1501–1505 (2010).
[Crossref] [PubMed]

Gomes, A. S. L.

J. M. Hickmann, A. S. L. Gomes, and C. B. de Araújo, “Observation of spatial cross-phase modulation effects in a self-defocusing nonlinear medium,” Phys. Rev. Lett. 68(24), 3547–3550 (1992).
[Crossref] [PubMed]

H. Ma, A. S. L. Gomes, and C. B. de Araújo, “Measurements of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. 59(21), 2666–2668 (1991).
[Crossref]

Gómez, L. A.

A. M. Brito-Silva, L. A. Gómez, C. B. de Araújo, and A. Galembeck, “Laser ablated silver nanoparticles with nearly the same size in different carrier media,” J. Nanomater. 2010, 142897 (2010).
[Crossref]

L. A. Gómez, C. B. de Araújo, A. M. Brito-Silva, and A. Galembeck, “Solvent effects on the linear and nonlinear optical response of silver nanoparticles,” Appl. Phys. B 92(1), 61–66 (2008).
[Crossref]

L. A. Gómez, C. B. de Araújo, A. M. Brito-Silva, and A. Galembeck, “Influence of stabilizing agents on the nonlinear susceptibility of silver nanoparticles,” J. Opt. Soc. Am. B 24(9), 2136–2140 (2007).
[Crossref]

Graener, H.

Guo, S.

X. Liu, S. Guo, H. Wang, N. Ming, and L. Hou, “Investigation of the influence of finite aperture size on the Z-scan transmittance curve,” J. Nonlinear Opt. Phys. Mater. 10(4), 431–439 (2001).
[Crossref]

Guyer, S. R.

A. P. Kulkarni, K. M. Noone, K. Munechika, S. R. Guyer, and D. S. Ginger, “Plasmon-enhanced charge carrier generation in organic photovoltaic films using silver nanoprisms,” Nano Lett. 10(4), 1501–1505 (2010).
[Crossref] [PubMed]

Hagan, D. J.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

He, K.

L. Deng, K. He, T. Zhou, and C. Li, “Formation and evolution of far-field diffraction patterns of divergent and convergent Gaussian beams passing through self-focusing and self-defocusing media,” J. Opt. A, Pure Appl. Opt. 7(8), 409–415 (2005).
[Crossref]

Herrmann, J.

Hickmann, J. M.

J. M. Hickmann, A. S. L. Gomes, and C. B. de Araújo, “Observation of spatial cross-phase modulation effects in a self-defocusing nonlinear medium,” Phys. Rev. Lett. 68(24), 3547–3550 (1992).
[Crossref] [PubMed]

Hoell, A.

M. Eichelbaum, K. Rademann, A. Hoell, D. M. Tatchev, W. Weigel, R. Stößer, and G. Pacchioni, “Photoluminescence of atomic gold and silver particles in soda-lime silicate glasses,” Nanotechnology 19(13), 135701 (2008).
[Crossref] [PubMed]

Hou, L.

X. Liu, S. Guo, H. Wang, N. Ming, and L. Hou, “Investigation of the influence of finite aperture size on the Z-scan transmittance curve,” J. Nonlinear Opt. Phys. Mater. 10(4), 431–439 (2001).
[Crossref]

Huang, J. P.

J. P. Huang and K. W. Yu, “Enhanced nonlinear optical responses of materials: composite effects,” Phys. Rep. 431(3), 87–172 (2006).
[Crossref]

Husakou, A.

Javadi, Z.

M. H. M. Ara, Z. Dehghani, R. Sahraei, A. Daneshfar, Z. Javadi, and F. Divsar, “Diffraction patterns and nonlinear optical properties of gold nanoparticles,” J. Quant. Spectrosc. Radiat. Transf. 113(5), 366–372 (2012).
[Crossref]

Jayabalan, J.

J. Jayabalan, A. Singh, S. Khan, and R. Chari, “Third-order nonlinearity of metal nanoparticles: isolation of instantaneous and delayed contributions,” J. Appl. Phys. 112(10), 103524 (2012).
[Crossref]

J. Jayabalan, “Origin and time dependence of higher-order nonlinearities in metal nanocomposites,” J. Opt. Soc. Am. B 28(10), 2448–2455 (2011).
[Crossref]

Kassab, L. R. P.

C. B. de Araújo, T. R. Oliveira, E. L. Falcão-Filho, D. M. Silva, and L. R. P. Kassab, “Nonlinear optical properties of PbO–GeO2 films containing gold nanoparticles,” J. Lumin. 133, 180–183 (2013).
[Crossref]

L. De Boni, E. C. Barbano, T. A. de Assumpção, L. Misoguti, L. R. P. Kassab, and S. C. Zilio, “Femtosecond third-order nonlinear spectra of lead-germanium oxide glasses containing silver nanoparticles,” Opt. Express 20(6), 6844–6850 (2012).
[Crossref] [PubMed]

Khan, S.

J. Jayabalan, A. Singh, S. Khan, and R. Chari, “Third-order nonlinearity of metal nanoparticles: isolation of instantaneous and delayed contributions,” J. Appl. Phys. 112(10), 103524 (2012).
[Crossref]

Kim, K.-H.

Kulkarni, A. P.

A. P. Kulkarni, K. M. Noone, K. Munechika, S. R. Guyer, and D. S. Ginger, “Plasmon-enhanced charge carrier generation in organic photovoltaic films using silver nanoprisms,” Nano Lett. 10(4), 1501–1505 (2010).
[Crossref] [PubMed]

Lange, J.

Leblond, H.

E. L. Falcão-Filho, C. B. de Araújo, G. Boudebs, H. Leblond, and V. Skarka, “Robust two-dimensional spatial solitons in liquid carbon disulfide,” Phys. Rev. Lett. 110(1), 013901 (2013).
[Crossref] [PubMed]

Lee, P. C.

P. C. Lee and D. Meisel, “Adsorption and surface-enhanced Raman of dyes on silver and gold sols,” J. Phys. Chem. 86(17), 3391–3395 (1982).
[Crossref]

Li, C.

L. Deng, K. He, T. Zhou, and C. Li, “Formation and evolution of far-field diffraction patterns of divergent and convergent Gaussian beams passing through self-focusing and self-defocusing media,” J. Opt. A, Pure Appl. Opt. 7(8), 409–415 (2005).
[Crossref]

Liu, X.

X. Liu, S. Guo, H. Wang, N. Ming, and L. Hou, “Investigation of the influence of finite aperture size on the Z-scan transmittance curve,” J. Nonlinear Opt. Phys. Mater. 10(4), 431–439 (2001).
[Crossref]

Lu, L.

L. Lu, Z. Luo, T. Xu, and L. Yu, “Cooperative plasmonic effect of Ag and Au nanoparticles on enhancing performance of polymer solar cells,” Nano Lett. 13(1), 59–64 (2013).
[Crossref] [PubMed]

Luo, Z.

L. Lu, Z. Luo, T. Xu, and L. Yu, “Cooperative plasmonic effect of Ag and Au nanoparticles on enhancing performance of polymer solar cells,” Nano Lett. 13(1), 59–64 (2013).
[Crossref] [PubMed]

Ma, H.

H. Ma, A. S. L. Gomes, and C. B. de Araújo, “Measurements of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. 59(21), 2666–2668 (1991).
[Crossref]

Mahdi, M. A.

R. Zamiri, A. Zakaria, M. B. Ahmad, A. R. Sadrolhosseini, K. Shameli, M. Darroudi, and M. A. Mahdi, “Investigation of spatial self-phase modulation of silver nanoparticles in clay suspension,” Optik (Stuttg.) 122(9), 836–838 (2011).
[Crossref]

Meisel, D.

P. C. Lee and D. Meisel, “Adsorption and surface-enhanced Raman of dyes on silver and gold sols,” J. Phys. Chem. 86(17), 3391–3395 (1982).
[Crossref]

Midorikawa, K.

Ming, N.

X. Liu, S. Guo, H. Wang, N. Ming, and L. Hou, “Investigation of the influence of finite aperture size on the Z-scan transmittance curve,” J. Nonlinear Opt. Phys. Mater. 10(4), 431–439 (2001).
[Crossref]

Misoguti, L.

Miura, K.

Mohan, S.

Mukai, K.

Munechika, K.

A. P. Kulkarni, K. M. Noone, K. Munechika, S. R. Guyer, and D. S. Ginger, “Plasmon-enhanced charge carrier generation in organic photovoltaic films using silver nanoprisms,” Nano Lett. 10(4), 1501–1505 (2010).
[Crossref] [PubMed]

Myint, T.

Nakashima, S.

Noone, K. M.

A. P. Kulkarni, K. M. Noone, K. Munechika, S. R. Guyer, and D. S. Ginger, “Plasmon-enhanced charge carrier generation in organic photovoltaic films using silver nanoprisms,” Nano Lett. 10(4), 1501–1505 (2010).
[Crossref] [PubMed]

Oliveira, M. M.

Oliveira, T. R.

C. B. de Araújo, T. R. Oliveira, E. L. Falcão-Filho, D. M. Silva, and L. R. P. Kassab, “Nonlinear optical properties of PbO–GeO2 films containing gold nanoparticles,” J. Lumin. 133, 180–183 (2013).
[Crossref]

Otero, M. M. M.

Pacchioni, G.

M. Eichelbaum, K. Rademann, A. Hoell, D. M. Tatchev, W. Weigel, R. Stößer, and G. Pacchioni, “Photoluminescence of atomic gold and silver particles in soda-lime silicate glasses,” Nanotechnology 19(13), 135701 (2008).
[Crossref] [PubMed]

Papazoglou, D. G.

Rademann, K.

M. Eichelbaum, K. Rademann, A. Hoell, D. M. Tatchev, W. Weigel, R. Stößer, and G. Pacchioni, “Photoluminescence of atomic gold and silver particles in soda-lime silicate glasses,” Nanotechnology 19(13), 135701 (2008).
[Crossref] [PubMed]

Ramirez, E. V. G.

Reyna, A. S.

A. S. Reyna and C. B. de Araújo, “Nonlinearity management of photonic composites and observation of spatial-modulation instability due to quintic nonlinearity,” Phys. Rev. A 89(6), 063803 (2014).
[Crossref]

Rodrigues, J. J.

Ryasnyansky, A. I.

R. A. Ganeev and A. I. Ryasnyansky, “Nonlinear optical characteristics of nanoparticles in suspensions and solid matrices,” Appl. Phys. B 84(1–2), 295–302 (2006).
[Crossref]

R. A. Ganeev, A. I. Ryasnyansky, A. L. Stepanov, and T. Usmanov, “Saturated absorption and nonlinear refraction of silicate glasses doped with silver nanoparticles at 532 nm,” Opt. Quantum Electron. 36(10), 949–960 (2004).
[Crossref]

Sadrolhosseini, A. R.

R. Zamiri, A. Zakaria, M. B. Ahmad, A. R. Sadrolhosseini, K. Shameli, M. Darroudi, and M. A. Mahdi, “Investigation of spatial self-phase modulation of silver nanoparticles in clay suspension,” Optik (Stuttg.) 122(9), 836–838 (2011).
[Crossref]

Sahraei, R.

M. H. M. Ara, Z. Dehghani, R. Sahraei, A. Daneshfar, Z. Javadi, and F. Divsar, “Diffraction patterns and nonlinear optical properties of gold nanoparticles,” J. Quant. Spectrosc. Radiat. Transf. 113(5), 366–372 (2012).
[Crossref]

Said, A. A.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

Seifert, G.

Shameli, K.

R. Zamiri, A. Zakaria, M. B. Ahmad, A. R. Sadrolhosseini, K. Shameli, M. Darroudi, and M. A. Mahdi, “Investigation of spatial self-phase modulation of silver nanoparticles in clay suspension,” Optik (Stuttg.) 122(9), 836–838 (2011).
[Crossref]

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

Shen, Y. R.

Shimizu, M.

Shimotsuma, Y.

Shin, H.

K. Dolgaleva, H. Shin, and R. W. Boyd, “Observation of a microscopic cascaded contribution to the fifth-order nonlinear susceptibility,” Phys. Rev. Lett. 103(11), 113902 (2009).
[Crossref] [PubMed]

Silva, D. M.

C. B. de Araújo, T. R. Oliveira, E. L. Falcão-Filho, D. M. Silva, and L. R. P. Kassab, “Nonlinear optical properties of PbO–GeO2 films containing gold nanoparticles,” J. Lumin. 133, 180–183 (2013).
[Crossref]

Singh, A.

J. Jayabalan, A. Singh, S. Khan, and R. Chari, “Third-order nonlinearity of metal nanoparticles: isolation of instantaneous and delayed contributions,” J. Appl. Phys. 112(10), 103524 (2012).
[Crossref]

Sipe, J. E.

K. Dolgaleva, R. W. Boyd, and J. E. Sipe, “Cascaded nonlinearity caused by local-field effects in the two-level atom,” Phys. Rev. A 76(6), 063806 (2007).
[Crossref]

J. E. Sipe and R. W. Boyd, “Nonlinear susceptibility of composite optical materials in the Maxwell Garnett model,” Phys. Rev. A 46(3), 1614–1629 (1992).
[Crossref] [PubMed]

Skarka, V.

E. L. Falcão-Filho, C. B. de Araújo, G. Boudebs, H. Leblond, and V. Skarka, “Robust two-dimensional spatial solitons in liquid carbon disulfide,” Phys. Rev. Lett. 110(1), 013901 (2013).
[Crossref] [PubMed]

Sobral-Filho, R. G.

Stegeman, G.

Stepanov, A. L.

R. A. Ganeev, A. I. Ryasnyansky, A. L. Stepanov, and T. Usmanov, “Saturated absorption and nonlinear refraction of silicate glasses doped with silver nanoparticles at 532 nm,” Opt. Quantum Electron. 36(10), 949–960 (2004).
[Crossref]

Stößer, R.

M. Eichelbaum, K. Rademann, A. Hoell, D. M. Tatchev, W. Weigel, R. Stößer, and G. Pacchioni, “Photoluminescence of atomic gold and silver particles in soda-lime silicate glasses,” Nanotechnology 19(13), 135701 (2008).
[Crossref] [PubMed]

Sugioka, K.

Tanaka, K.

Tatchev, D. M.

M. Eichelbaum, K. Rademann, A. Hoell, D. M. Tatchev, W. Weigel, R. Stößer, and G. Pacchioni, “Photoluminescence of atomic gold and silver particles in soda-lime silicate glasses,” Nanotechnology 19(13), 135701 (2008).
[Crossref] [PubMed]

Tzortzakis, S.

Umran, F. A.

G. Wang, S. Zhang, F. A. Umran, X. Cheng, N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

Usmanov, T.

R. A. Ganeev, A. I. Ryasnyansky, A. L. Stepanov, and T. Usmanov, “Saturated absorption and nonlinear refraction of silicate glasses doped with silver nanoparticles at 532 nm,” Opt. Quantum Electron. 36(10), 949–960 (2004).
[Crossref]

Van Stryland, E. W.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

Wang, G.

G. Wang, S. Zhang, F. A. Umran, X. Cheng, N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

Wang, H.

X. Liu, S. Guo, H. Wang, N. Ming, and L. Hou, “Investigation of the influence of finite aperture size on the Z-scan transmittance curve,” J. Nonlinear Opt. Phys. Mater. 10(4), 431–439 (2001).
[Crossref]

Wang, J.

G. Wang, S. Zhang, F. A. Umran, X. Cheng, N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

Wei, T.-H.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

Weigel, W.

M. Eichelbaum, K. Rademann, A. Hoell, D. M. Tatchev, W. Weigel, R. Stößer, and G. Pacchioni, “Photoluminescence of atomic gold and silver particles in soda-lime silicate glasses,” Nanotechnology 19(13), 135701 (2008).
[Crossref] [PubMed]

Xu, T.

L. Lu, Z. Luo, T. Xu, and L. Yu, “Cooperative plasmonic effect of Ag and Au nanoparticles on enhancing performance of polymer solar cells,” Nano Lett. 13(1), 59–64 (2013).
[Crossref] [PubMed]

Yu, K. W.

J. P. Huang and K. W. Yu, “Enhanced nonlinear optical responses of materials: composite effects,” Phys. Rep. 431(3), 87–172 (2006).
[Crossref]

Yu, L.

L. Lu, Z. Luo, T. Xu, and L. Yu, “Cooperative plasmonic effect of Ag and Au nanoparticles on enhancing performance of polymer solar cells,” Nano Lett. 13(1), 59–64 (2013).
[Crossref] [PubMed]

Zakaria, A.

R. Zamiri, A. Zakaria, M. B. Ahmad, A. R. Sadrolhosseini, K. Shameli, M. Darroudi, and M. A. Mahdi, “Investigation of spatial self-phase modulation of silver nanoparticles in clay suspension,” Optik (Stuttg.) 122(9), 836–838 (2011).
[Crossref]

Zamiri, R.

R. Zamiri, A. Zakaria, M. B. Ahmad, A. R. Sadrolhosseini, K. Shameli, M. Darroudi, and M. A. Mahdi, “Investigation of spatial self-phase modulation of silver nanoparticles in clay suspension,” Optik (Stuttg.) 122(9), 836–838 (2011).
[Crossref]

Zarbin, A. J. G.

Zhang, L.

G. Wang, S. Zhang, F. A. Umran, X. Cheng, N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

Zhang, S.

G. Wang, S. Zhang, F. A. Umran, X. Cheng, N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

Zhou, T.

L. Deng, K. He, T. Zhou, and C. Li, “Formation and evolution of far-field diffraction patterns of divergent and convergent Gaussian beams passing through self-focusing and self-defocusing media,” J. Opt. A, Pure Appl. Opt. 7(8), 409–415 (2005).
[Crossref]

Zilio, S. C.

Appl. Phys. B (2)

R. A. Ganeev and A. I. Ryasnyansky, “Nonlinear optical characteristics of nanoparticles in suspensions and solid matrices,” Appl. Phys. B 84(1–2), 295–302 (2006).
[Crossref]

L. A. Gómez, C. B. de Araújo, A. M. Brito-Silva, and A. Galembeck, “Solvent effects on the linear and nonlinear optical response of silver nanoparticles,” Appl. Phys. B 92(1), 61–66 (2008).
[Crossref]

Appl. Phys. Lett. (2)

G. Wang, S. Zhang, F. A. Umran, X. Cheng, N. Dong, D. Coghlan, Y. Cheng, L. Zhang, W. J. Blau, and J. Wang, “Tunable effective nonlinear refractive index of graphene dispersions during the distortion of spatial self-phase modulation,” Appl. Phys. Lett. 104(14), 141909 (2014).
[Crossref]

H. Ma, A. S. L. Gomes, and C. B. de Araújo, “Measurements of nondegenerate optical nonlinearity using a two-color single beam method,” Appl. Phys. Lett. 59(21), 2666–2668 (1991).
[Crossref]

IEEE J. Quantum Electron. (1)

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[Crossref]

J. Appl. Phys. (1)

J. Jayabalan, A. Singh, S. Khan, and R. Chari, “Third-order nonlinearity of metal nanoparticles: isolation of instantaneous and delayed contributions,” J. Appl. Phys. 112(10), 103524 (2012).
[Crossref]

J. Lumin. (1)

C. B. de Araújo, T. R. Oliveira, E. L. Falcão-Filho, D. M. Silva, and L. R. P. Kassab, “Nonlinear optical properties of PbO–GeO2 films containing gold nanoparticles,” J. Lumin. 133, 180–183 (2013).
[Crossref]

J. Nanomater. (1)

A. M. Brito-Silva, L. A. Gómez, C. B. de Araújo, and A. Galembeck, “Laser ablated silver nanoparticles with nearly the same size in different carrier media,” J. Nanomater. 2010, 142897 (2010).
[Crossref]

J. Nonlinear Opt. Phys. Mater. (1)

X. Liu, S. Guo, H. Wang, N. Ming, and L. Hou, “Investigation of the influence of finite aperture size on the Z-scan transmittance curve,” J. Nonlinear Opt. Phys. Mater. 10(4), 431–439 (2001).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

L. Deng, K. He, T. Zhou, and C. Li, “Formation and evolution of far-field diffraction patterns of divergent and convergent Gaussian beams passing through self-focusing and self-defocusing media,” J. Opt. A, Pure Appl. Opt. 7(8), 409–415 (2005).
[Crossref]

J. Opt. Soc. Am. B (5)

J. Phys. Chem. (1)

P. C. Lee and D. Meisel, “Adsorption and surface-enhanced Raman of dyes on silver and gold sols,” J. Phys. Chem. 86(17), 3391–3395 (1982).
[Crossref]

J. Quant. Spectrosc. Radiat. Transf. (1)

M. H. M. Ara, Z. Dehghani, R. Sahraei, A. Daneshfar, Z. Javadi, and F. Divsar, “Diffraction patterns and nonlinear optical properties of gold nanoparticles,” J. Quant. Spectrosc. Radiat. Transf. 113(5), 366–372 (2012).
[Crossref]

Nano Lett. (2)

A. P. Kulkarni, K. M. Noone, K. Munechika, S. R. Guyer, and D. S. Ginger, “Plasmon-enhanced charge carrier generation in organic photovoltaic films using silver nanoprisms,” Nano Lett. 10(4), 1501–1505 (2010).
[Crossref] [PubMed]

L. Lu, Z. Luo, T. Xu, and L. Yu, “Cooperative plasmonic effect of Ag and Au nanoparticles on enhancing performance of polymer solar cells,” Nano Lett. 13(1), 59–64 (2013).
[Crossref] [PubMed]

Nanotechnology (1)

M. Eichelbaum, K. Rademann, A. Hoell, D. M. Tatchev, W. Weigel, R. Stößer, and G. Pacchioni, “Photoluminescence of atomic gold and silver particles in soda-lime silicate glasses,” Nanotechnology 19(13), 135701 (2008).
[Crossref] [PubMed]

Opt. Express (8)

S. Nakashima, K. Sugioka, K. Tanaka, M. Shimizu, Y. Shimotsuma, K. Miura, K. Midorikawa, and K. Mukai, “Plasmonically enhanced Faraday effect in metal and ferrite nanoparticles composite precipitated inside glass,” Opt. Express 20(27), 28191–28199 (2012).
[Crossref] [PubMed]

L. De Boni, E. C. Barbano, T. A. de Assumpção, L. Misoguti, L. R. P. Kassab, and S. C. Zilio, “Femtosecond third-order nonlinear spectra of lead-germanium oxide glasses containing silver nanoparticles,” Opt. Express 20(6), 6844–6850 (2012).
[Crossref] [PubMed]

K.-H. Kim, A. Husakou, and J. Herrmann, “Saturable absorption in composites doped with metal nanoparticles,” Opt. Express 18(21), 21918–21925 (2010).
[Crossref] [PubMed]

K.-H. Kim, A. Husakou, and J. Herrmann, “Slow light in dielectric composite materials of metal nanoparticles,” Opt. Express 20(23), 25790–25797 (2012).
[Crossref] [PubMed]

E. L. Falcão-Filho, R. Barbosa-Silva, R. G. Sobral-Filho, A. M. Brito-Silva, A. Galembeck, and C. B. de Araújo, “High-order nonlinearity of silica-gold nanoshells in chloroform at 1560 nm,” Opt. Express 18(21), 21636–21644 (2010).
[Crossref] [PubMed]

G. Stegeman, D. G. Papazoglou, R. Boyd, and S. Tzortzakis, “Nonlinear birefringence due to non-resonant, higher-order Kerr effect in isotropic media,” Opt. Express 19(7), 6387–6399 (2011).
[Crossref] [PubMed]

E. V. G. Ramirez, M. L. A. Carrasco, M. M. M. Otero, S. C. Cerda, and M. D. I. Castillo, “Far field intensity distributions due to spatial self phase modulation of a Gaussian beam by a thin nonlocal nonlinear media,” Opt. Express 18(21), 22067–22079 (2010).
[Crossref] [PubMed]

S. Mohan, J. Lange, H. Graener, and G. Seifert, “Surface plasmon assisted optical nonlinearities of uniformly oriented metal nano-ellipsoids in glass,” Opt. Express 20(27), 28655–28663 (2012).
[Crossref] [PubMed]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

R. A. Ganeev, A. I. Ryasnyansky, A. L. Stepanov, and T. Usmanov, “Saturated absorption and nonlinear refraction of silicate glasses doped with silver nanoparticles at 532 nm,” Opt. Quantum Electron. 36(10), 949–960 (2004).
[Crossref]

Optik (Stuttg.) (1)

R. Zamiri, A. Zakaria, M. B. Ahmad, A. R. Sadrolhosseini, K. Shameli, M. Darroudi, and M. A. Mahdi, “Investigation of spatial self-phase modulation of silver nanoparticles in clay suspension,” Optik (Stuttg.) 122(9), 836–838 (2011).
[Crossref]

Phys. Rep. (1)

J. P. Huang and K. W. Yu, “Enhanced nonlinear optical responses of materials: composite effects,” Phys. Rep. 431(3), 87–172 (2006).
[Crossref]

Phys. Rev. A (3)

K. Dolgaleva, R. W. Boyd, and J. E. Sipe, “Cascaded nonlinearity caused by local-field effects in the two-level atom,” Phys. Rev. A 76(6), 063806 (2007).
[Crossref]

A. S. Reyna and C. B. de Araújo, “Nonlinearity management of photonic composites and observation of spatial-modulation instability due to quintic nonlinearity,” Phys. Rev. A 89(6), 063803 (2014).
[Crossref]

J. E. Sipe and R. W. Boyd, “Nonlinear susceptibility of composite optical materials in the Maxwell Garnett model,” Phys. Rev. A 46(3), 1614–1629 (1992).
[Crossref] [PubMed]

Phys. Rev. Lett. (3)

J. M. Hickmann, A. S. L. Gomes, and C. B. de Araújo, “Observation of spatial cross-phase modulation effects in a self-defocusing nonlinear medium,” Phys. Rev. Lett. 68(24), 3547–3550 (1992).
[Crossref] [PubMed]

K. Dolgaleva, H. Shin, and R. W. Boyd, “Observation of a microscopic cascaded contribution to the fifth-order nonlinear susceptibility,” Phys. Rev. Lett. 103(11), 113902 (2009).
[Crossref] [PubMed]

E. L. Falcão-Filho, C. B. de Araújo, G. Boudebs, H. Leblond, and V. Skarka, “Robust two-dimensional spatial solitons in liquid carbon disulfide,” Phys. Rev. Lett. 110(1), 013901 (2013).
[Crossref] [PubMed]

Other (3)

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University Press, 1990).

R. L. Sutherland, Handbook of Nonlinear Optics (Dekker, 1996).

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Figures (7)

Fig. 1
Fig. 1 Linear absorption spectra of the colloids with f = 4.0 × 10 5 and solvents: (a) sample A (cell thickness: 1 mm) and (b) sample B (cell thickness: 5 mm).
Fig. 2
Fig. 2 Normalized Z-scan profiles obtained for different NPs volume fractions f. For sample A (laser peak intensity: 5 G W / c m 2 ): (a) closed-aperture scheme; (b) open-aperture scheme. For sample B (laser peak intensity: 0.2 G W / c m 2 ): (c) closed-aperture and (d) open-aperture scheme.
Fig. 3
Fig. 3 Closed-aperture Z-scan profiles, for sample A, obtained for different laser peak intensities with two NPs volume fractions: (a) f = 1.6 × 10 5 and (b) f = 5.0 × 10 5 . The solid lines were obtained using Eq. (16).
Fig. 4
Fig. 4 Dependence of the effective third-, fifth- and seventh-order coefficients of sample B as a function of the volume fraction, f. Notice that the sample presents: septic refractive nonlinearity at f = 3.3 × 10 5 (a) and quintic refractive nonlinearity at f = 1.5 × 10 5 (b). Figures (a) and (b) were obtained with laser peak intensity of 10 8 W / c m 2 . When f = 1.2 × 10 5 and the laser peak intensity is 2.5 × 10 8 W / c m 2 , the NL absorption is due to Im χ e f f ( 7 ) (c).
Fig. 5
Fig. 5 Diffraction patterns of a Gaussian beam in a quintic refractive composite placed in the (a)–(f) focal plane and (g)–(j) around the focal plane position. Laser peak intensities: (a) I 1 = 70 G W / c m 2 ; (b) I 2 = 90 G W / c m 2 ; (c) I 3 = 100 G W / c m 2 . Experimental (black line) and theoretical (red line) intensity distribution versus the radial coordinates for (d) I 1 ; (e) I 2 ; (f) I 3 . Sample placed 0.3 mm (g) before and (h) after of the focal plane with laser peak intensity: I = 70 G W / c m 2 at the focus. Intensity distribution as function of the radial coordinates considering (i) R = 0.4 m m ; (j) R = + 0.4 m m . The red curves were obtained using the value of n 4 determined in the Z-scan experiments.
Fig. 6
Fig. 6 Pump beam profile after propagation through sample B, for f = 3.3 × 10 5 : (a) I 1 = 10 6 W / c m 2 ; (b) I 2 = 10 8 W / c m 2 . (c) Self-defocusing traces due to χ e f f ( 7 ) obtained from (a) and (b). The red and blue lines are theoretical results for the respective intensities.
Fig. 7
Fig. 7 Experimental probe beam profile due to χ e f f ( 7 ) for a distance of (a) x = 0 ; (b) x = w 0 and (c) x = 2.2 w 0 between the centers of the incident pump (white line) and probe (pink line) beams; (d)-(f) Normalized intensity distribution of the probe beam transverse profile without the pump beam (red dashed line) and with the pump beam (black line); (g)-(i) Theoretical probe beam profile calculated using Eqs. (18) and (19) with the NL coefficients determined in the Z-scan experiments for the relative distances of (a)-(c). Pump beam intensity: 10 8 W / c m 2 ; probe beam intensity: 10 7 W / c m 2 .

Equations (19)

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P = ε h L { ε h ε h L [ χ h + 3 β f 1 β f ] } E 0 = ε h L χ ( | E 0 | 2 ) E 0 ,
ε n p ( N L ) = ε h L [ 3 4 χ n p ( 3 ) | E n p | 2 + 5 8 χ n p ( 5 ) ( | E n p | 2 ) 2 + 35 64 χ n p ( 7 ) ( | E n p | 2 ) 3 ] ,
ε h ( N L ) = ε h L [ 3 4 χ h ( 3 ) | E 0 | 2 ] ,
χ ( y ) = χ ( 0 ) + ( [ χ ( y ) ] y ) y = 0 y + 1 2 ( 2 [ χ ( y ) ] y 2 ) y = 0 y 2 + 1 6 ( 3 [ χ ( y ) ] y 3 ) y = 0 y 3 ,
( [ χ ( y ) ] y ) y = 0 = 3 4 χ h ( 3 ) + 3 4 f L 2 | L | 2 χ n p ( 3 ) ,
( 2 [ χ ( y ) ] y 2 ) y = 0 = 5 4 f L 2 | L | 4 χ n p ( 5 ) 3 8 f L 3 | L | 4 ( χ n p ( 3 ) ) 2 ,
( 3 [ χ ( y ) ] y 3 ) y = 0 = 9 32 f L 4 | L | 6 ( χ n p ( 3 ) ) 3 15 8 f L 3 | L | 6 ( χ n p ( 3 ) χ n p ( 5 ) ) + 105 32 f L 2 | L | 6 χ n p ( 7 ) ,
| E n p | 2 ( | η | 2 ) z = 0 { 1 + 1 2 [ 2 ( | η | 2 ) z 2 ] z = 0 | E 0 | 2 + 1 4 [ 2 ( | η | 2 ) z 2 ] 2 z = 0 ( | E 0 | 2 ) 2 } | E 0 | 2 ,
| E n p | 2 | L | 2 { 1 1 2 | L | 2 Re [ L χ N P ( 3 ) ] | E 0 | 2 + 1 4 | L | 4 ( Re [ L χ N P ( 3 ) ] ) 2 | E 0 | 4 } | E 0 | 2 .
χ ( z 2 ) = χ ( 0 ) + ( [ χ ( z 2 ) ] [ z 2 ] ) z 2 = 0 z 2 + 1 2 ( 2 [ χ ( z 2 ) ] [ z 2 ] 2 ) z 2 = 0 ( z 2 ) 2 + 1 6 ( 3 [ χ ( z 2 ) ] [ z 2 ] 3 ) z 2 = 0 ( z 2 ) 3 ,
χ ( | E 0 | 2 ) = χ ( 0 ) + | L | 2 ( [ χ ( z 2 ) ] [ z 2 ] ) z 2 = 0 | E 0 | 2 + 1 2 | L | 4 ( 2 [ χ ( z 2 ) ] [ z 2 ] 2 [ χ ( z 2 ) ] [ z 2 ] Re [ L χ N P ( 3 ) ] ) z 2 = 0 | E 0 | 4 + 1 6 | L | 6 ( 3 [ χ ( z 2 ) ] [ z 2 ] 3 3 2 [ χ ( z 2 ) ] [ z 2 ] 2 Re [ L χ N P ( 3 ) ] + 3 2 [ χ ( z 2 ) ] [ z 2 ] ( Re [ L χ N P ( 3 ) ] ) 2 ) z 2 = 0 | E 0 | 6 .
χ e f f ( | E 0 | 2 ) = χ e f f ( 1 ) + 3 4 χ e f f ( 3 ) | E 0 | 2 + 5 8 χ e f f ( 5 ) | E 0 | 4 + 35 64 χ e f f ( 7 ) | E 0 | 6 .
χ e f f ( 3 ) = f L 2 | L | 2 χ n p ( 3 ) + χ h ( 3 ) ,
χ e f f ( 5 ) = f L 2 | L | 4 χ n p ( 5 ) 6 10 f L 3 | L | 4 ( χ n p ( 3 ) ) 2 3 10 f L | L | 6 | χ n p ( 3 ) | 2 ,
χ e f f ( 7 ) = f L 2 | L | 6 χ n p ( 7 ) + 12 35 f L 4 | L | 6 ( χ n p ( 3 ) ) 3 + 3 35 f | L | 8 [ 4 L 2 χ n p ( 3 ) + | L | 2 ( χ n p ( 3 ) ) * ] | χ n p ( 3 ) | 2 4 7 f L | L | 6 [ 2 L 2 χ n p ( 3 ) + | L | 2 ( χ n p ( 3 ) ) * ] χ n p ( 5 ) ,
T ( z , Δ Φ 0 ) 1 + N = 1 3 ( 4 N ) Δ Φ 0 ( 2 N + 1 ) z / z 0 [ ( z / z 0 ) 2 + ( 2 N + 1 ) 2 ] [ ( z / z 0 ) 2 + 1 ] N ,
I = I 0 | 0 J 0 ( k θ r ) exp [ r 2 w p 2 i ϕ ( r ) ] r d r | 2 ,
2 i k E 1 z + Δ E 1 = ω 2 c 2 [ 3 χ e f f ( 3 ) ( | E 1 | 2 + 2 | E 2 | 2 ) E 1 + 10 χ e f f ( 5 ) ( | E 1 | 4 + 6 | E 1 | 2 | E 2 | 2 + 3 | E 2 | 4 ) E 1 + 35 χ e f f ( 7 ) ( | E 1 | 6 + 18 | E 1 | 2 | E 2 | 4 + 12 | E 1 | 4 | E 2 | 2 + 4 | E 2 | 6 ) E 1 ] ,
2 i k E 2 z + Δ E 2 = ω 2 c 2 [ 3 χ e f f ( 3 ) ( 2 | E 1 | 2 + | E 2 | 2 ) E 2 + 10 χ e f f ( 5 ) ( 3 | E 1 | 4 + 6 | E 1 | 2 | E 2 | 2 + | E 2 | 4 ) E 2 + 35 χ e f f ( 7 ) ( 4 | E 1 | 6 + 12 | E 1 | 2 | E 2 | 4 + 18 | E 1 | 4 | E 2 | 2 + | E 2 | 6 ) E 2 ] ,

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