Nonlinear polarization instability in cubic-quintic plasmonic nanocomposites

Albert S. Reyna; Emeric Bergmann; Pierre-François Brevet; Cid B. de Araújo

doi:10.1364/OE.25.021049

1. Introduction

The nonlinear optical polarization induced by a laser beam propagating through a medium with inversion symmetry can be described by an expansion in powers series of the electric field where the even-order terms are null, unless the symmetry is disturbed. The nonzero expansion coefficients, $χ^{(2 N + 1)}$ , with $N = 1, 2, \dots$ , called as $(2 N + 1) - t h$ order susceptibility, provide a measure of how strong is the light-matter interaction [1]. The majority of articles in the literature are related to the cubic nonlinearity, due to $χ^{(3)}$ , which is the parameter that describes phenomena such as third-order self-focusing, coherent anti-Stokes Raman scattering and two-photon absorption, among other phenomena [2]. However, in several cases, the odd higher-order nonlinearities (HON) are essential for characterization of the nonlinear response [3–11]. For instance, novel effects such as liquid light condensates [12], high-harmonic generation [13], filamentation [14,15], stable propagation of two dimensional spatial solitons in isotropic media [16–18] and optical rogue waves generation [19] are associated to HON. In addition, the HON contributions are essential for modelling three-body interactions in Bose-Einstein condensates [20,21], Bose gas with hard-core contact interactions and spin-polarized Fermi gas in the collisional regime [22], higher-order modulation instability [23], stability conditions for optical solitons [24,25], management of the solitary-wave solutions of the nonlinear Schrödinger equation [26] and several other physical situations.

One important consequence of HON is the nonlinear birefringence effect which is manifested by the polarization rotation of an elliptically polarized light beam along its propagation pathway [27]. This effect can be very much influenced by HON when the highly nonlinear materials currently available are excited with moderate laser intensities or even when gases are excited with intense laser pulses [3–11,13,28]. For instance, in [28], the authors were able to identify HON contributions for the nonlinear birefringence in air and determined the nonlinear refractive indices associated to third-, fifth-, seventh-, ninth- and eleventh-order susceptibilities of the main air components (nitrogen, oxygen and argon). Subsequently, a simple formalism was presented for precise characterization of the non-resonant nonlinear birefringence [29]. Moreover, HON play an important role on the modulation instability effects, at moderate or high light intensities, since their contributions are quite sensitive to any perturbation of the optical field. Indeed, recent studies on metamaterials predict that the modulation instability gain is enhanced due to the simultaneous contributions of HON (such as quintic nonlinearity) or saturable nonlinearity and high-order dispersions [30–32]. Those studies show relevant results related to solitons generation due to the modulation instability effects induced by HON.

Nowadays, it is well known that further advances in the telecommunications area require in-depth studies of light propagation in highly nonlinear optical fibers [33]. Indeed, the third-order nonlinear birefringence has been widely studied with important applications in all-optical devices [34–36]. However, an important limitation of optical fibers in all-optical circuits is related to the long propagation distances required to obtain induced polarization rotation. To mitigate this problem, highly nonlinear-core waveguides were developed allowing an increase of the polarization rotation of the coupled light field with small propagation distances [37]. A particular example was reported using an optical capillary where the core was filled with nitrobenzene that is highly nonlinear [38]. Therefore large nonlinear phase shifts of $Δ ϕ^{(N L)} \approx 12 π$ in the infrared were obtained for propagation distances of ~10 cm.

On the other hand, metal-dielectric nanocomposites (MDNCs) consisting of a dielectric host containing metal nanoparticles emerged as nonlinear systems with high optical susceptibility and ultrafast response [8–11,39–42] that can be used as excellent platforms to study many associated HON phenomena. The nonlinear response of the MDNCs is described in terms of effective nonlinear susceptibilities that include contributions of the metal nanoparticles and their host. In particular, silver-nanocolloids have been studied under different conditions, by varying intrinsic parameters (such as: size, shape, environment and volume fraction of metal NPs) and extrinsic parameters (such as: laser wavelength, intensity, pulses duration and repetition rate), exhibiting important HON contributions. The exploitation of HON allowed enhancement of the nonlinear response [42,43] and even suppression of nonlinear absorption effects [44] by using a nonlinearity management procedure, which consists in varying the nanoparticles volume fraction, f, and the incident laser intensity [42–44]. In this sense, metal-nanocolloids are strong candidates to investigate the contributions of HON to nonlinear birefringence.

In the present paper, we demonstrate large nonlinear phase-shifts contributed by the third- and fifth-order nonlinearities when picosecond laser pulses propagate in a 9 cm long capillary filled with silver nanoparticles suspended in liquid carbon disulfide (CS₂). Control of the nonlinear response was obtained by varying the nanoparticles volume fraction from $f = 1.0 \times 10^{- 5}$ to $f = 4.5 \times 10^{- 5}$ , obtaining $Δ ϕ^{(N L)} \approx 20 π$ for peak intensities of tens of MW/cm². In this range of nanoparticles concentration and laser intensity, silver-nanocolloids behave like cubic-quintic media where, depending on the values of f, the nonlinear response is dominated by the third- or fifth-order nonlinearity. The experimental results were modeled by two coupled differential nonlinear equations that describe the evolution of the right- and left-circular optical field polarization in the samples. The gain spectra of the modulation instability were obtained, showing that the instabilities increase significantly due to the presence of the quintic nonlinearity.

The paper is organized as follows: Section 2 describes the samples preparation and identifies their linear and nonlinear optical parameters, as well as describes the experimental setups used. Section 3 presents details of the theory used for modeling the experimental results. Section 4 presents the experiments and numerical simulations made. A summary of the results is presented in Section 5.

2. Experimental details

Silver-nanocolloids were prepared following the chemical synthesis route described in [42–44]. As a result, a pristine colloid was obtained with a non-homogeneous size distribution of silver nanoparticles. Subsequently, the pristine colloid was subjected to laser ablation using the second harmonic beam from a Q-switched Nd:YAG laser (10 Hz, 8 ns, 85 mJ/pulse) for 1 hour in order to obtain a homogeneous nanoparticles size distribution by photofragmentation. During the laser ablation process the colloid was slowly stirred according to the procedure described in [45].

Optical absorbance spectra of the samples, before and after photofragmentation of the original nanoparticles synthesized, were measured using a commercial spectrophotometer. The absorption feature related to the localized surface plasmons resonance (LSPR) centered at ~400 nm was observed. As in our previous experiments [42–44] we observed a smaller LSPR linewidth after photofragmentation that indicates a homogeneous distribution of the silver nanoparticles sizes. This was corroborated by transmission electron microscopy that showed a homogeneous distribution of spherical nanoparticles with average diameter of $(6.0 \pm 3.0) n m$ .

The setup used to study the intensity-dependent birefringence is illustrated by Fig. 1(a). The second harmonic of a Q-switched and mode-locked Nd: YAG laser (532 nm, 80 ps, 10 Hz) was used to excite the samples with maximum pulse energy of 10 µJ. A system composed of a Glan prism (P) located between two $λ / 2$ plates was used to control the incident power and to rotate the polarization axis of a linearly polarized laser pulse by the azimuth angle $θ$ . A 40x microscope objective (L1) was used to couple the laser light into a fused-silica hollow capillary $(n_{c l a d d i n g} = 1.46)$ with inner (outer) diameter of 2 µm (285 µm). The capillary with length of 9 cm had at the entrance and exit faces two reservoirs with the input and output sides of the capillary touching the reservoirs windows. The silver-nanocolloid was filled in one of the reservoirs, and using a strong air pressure the colloid was conveyed through the capillary to the other reservoir, in order to guarantee that the capillary is completely filled with the metal colloid (no air bubbles were observed in the end of the filling process). A 20x microscope objective (L2) was used to collimate the laser light after the propagation along the capillary. The laser beam was split by a polarizing beam splitter cube (PBS) to separate the vertical (V) and horizontal (H) polarization components that were monitored by fast detectors D1 and D2, respectively. A reference detector (RD) was used to compensate for the laser intensity fluctuations. Figure 1(b) displays the inner diameter of the capillary, in a fragment of 5 mm, showing small asymmetries that induce linear birefringence in the sample. The inset of Fig. 1(b) shows an image of the capillary core (length: 1 mm), obtained using an optical microscope.

Fig. 1 (a) The experimental setup: polarizer (P), beam splitter (BS), spherical lenses with f = 5 cm (L), 40x microscope objective (L1), 20x microscope objective (L2), polarizing beam splitter cube (PBS) and reference detector (RD). The transmitted light with vertical and horizontal polarization was captured in the fast detectors D1 and D2, respectively. (b) Inner diameter of capillary, in a portion of 5 mm, showing small asymmetries. The inset is an optical microscope image of a small section of the hollow capillary core (length: 1 mm).

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Two experiments were performed to study the nonlinear birefringence induced in the samples. In one experiment made to analyze the optical transmission as a function of the optical field polarization, the laser beam intensity was chosen (by adjusting the first $λ / 2$ plate and the polarizer) and the incident polarization direction was rotated by the second $λ / 2$ plate using a motorized rotation stage, with steps of ~3.6 degrees. The second experiment, aiming the investigation of the transmittance intensity dependence was performed by rotating the first $λ / 2$ plate using another motorized rotation stage with intensity steps of ~350 kW/cm², maintaining fixed the incident electric field polarization.

In order to evaluate the contributions of the third- and fifth-order susceptibilities on the polarization instability effect, we used pure CS₂ (host) and silver-nanocolloids with four different volume fractions, managed to display specific nonlinear responses. The corresponding values of $α_{0}$ , $β^{(2)}$ , $χ_{x x x x}^{(3)}$ and $χ_{x x x x x x}^{(5)}$ are displayed in Table 1, as obtained from previous experiments [42–44]. Sample A (S-A), corresponding to pure CS₂ $(f = 0)$ , exhibits only focusing third-order nonlinearity, for the intensities used in this work. The other four samples (S-B, S-C, S-D and S-E), containing silver nanoparticles with $1.0 \times 10^{- 5} \leq f \leq 4.5 \times 10^{- 5}$ , display contributions of third- and fifth-order susceptibilities which depend on the f value [42–44]. Notice that S-B represents a cubic-quintic (self-focusing) medium where the nonlinear refraction is dominated by the positive signal of $Re [χ_{x x x x}^{(3)}]$ . On the other hand, S-D and S-E correspond to cubic-quintic (defocusing-focusing) media because their nonlinear responses exhibit a negative value of $Re [χ_{x x x x}^{(3)}]$ and positive value of $Re [χ_{x x x x x x}^{(5)}]$ . However, S-C exemplifies a pure refractive quintic medium since $Re [χ_{x x x x}^{(3)}] = 0$ , due to the destructive interference of the third-order susceptibilities of the host and the nanoparticles [42,43]. The change from the self-focusing to self-defocusing behavior in each case is due to the balance between the contributions of positive third-order refractive index of the host (CS₂) and negative third-order refractive index of the silver nanoparticles, as detailed in [43].

Table 1. Linear absorption coefficient^a, second-order dispersion coefficient^a, third and fifth-order susceptibilities^b for pure CS₂ (S-A) and silver-colloids with different nanoparticles volume fraction.

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3. Nonlinear birefringence in a capillary filled with an isotropic cubic-quintic medium

3.1 General description

The complex electric field associated with an optical wave with arbitrary polarization can be written as

E (r, t) = \frac{1}{2} (\hat{x} E_{x} + \hat{y} E_{y}) \exp (- i ω_{0} t) + c . c,

where

E_{x}

and

E_{y}

are the complex amplitudes of the field components in the x- and y-axis with the carrier frequency

ω_{0}

.

The nonlinear light propagation inside the colloid hosted by the capillary can be modeled by using the Helmholtz equation given by

\nabla^{2} E (r, ω) + k_{0}^{2} ε (ω) \cdot E (r, ω) = 0,

where

E

and

ε

represent the frequency-dependent field and the dielectric tensor, respectively.

\nabla^{2}

is the Laplacian operator,

k_{0} = ω_{0} / c

,

ω_{0}

is the laser frequency and

c

is the speed of light in vacuum.

As follows from Eqs. (1) and (2) the field components obeys the Helmholtz equation:

\nabla^{2} E_{μ} (r, ω) + k_{0}^{2} ε_{μ σ} (ω) E_{σ} (r, ω) = 0,

with

μ

and

σ

equal to x or y. In the principal-axis of the system, the dielectric tensor is represented as a diagonal matrix

(ε_{μ σ} \to ε_{μ σ} δ_{μ, σ})

, with

ε_{x x} = ε_{x}

and

ε_{y y} = ε_{y}

being the only nonzero components. This complex dielectric function

(ε_{μ})

is associated to the refractive index,

n_{μ}

, and the absorption coefficient,

α_{μ}

, of the material through the expression

ε_{μ} = {(n_{μ} + i \frac{α_{μ}}{2 k_{0}})}^{2}

. For media exhibiting nonlinear response, the dielectric function can be expressed in terms of the linear and nonlinear contributions as

ε_{μ} (ω) = ε_{μ}^{L} (ω) + ε_{μ}^{N L} (ω)

, where the nonlinear term is related to the nonlinear optical polarization by:

P_{μ}^{N L} (ω) = ε_{0} ε_{μ}^{N L} (ω) E_{μ} (ω),

where

ε_{μ}^{N L} (ω)

contains the contributions of third- and higher-order susceptibilities, as can be observed below. In particular, silver nanoparticles suspended in liquid CS₂ behave as a cubic-quintic medium in the experimental conditions used here, according to [42,43], and therefore, the nonlinear polarization is described by the sum of the third- and fifth-order contributions,

P_{μ}^{N L} = P_{μ}^{(3)} + P_{μ}^{(5)}

,with:

P_{μ}^{(3)} (ω) = 3 ε_{0} {2 χ_{x x y y}^{(3)} (ω) E_{μ} (ω) [E (ω) \cdot E^{*} (ω)] + χ_{x y y x}^{(3)} (ω) E_{μ}^{*} (ω) [E (ω) \cdot E (ω)]},

\begin{array}{l} P_{μ}^{(5)} (ω) = 10 ε_{0} {\frac{10}{3} χ_{x x y y x x}^{(5)} (ω) {| E_{μ} (ω) |}^{2} [E (ω) \cdot E (ω)] E_{μ}^{*} (ω) \\ + \frac{5}{3} χ_{x x y y y y}^{(5)} (ω) \sum_{σ = x, y} {| E_{σ} (ω) |}^{4} E_{μ} (ω)}, \end{array}

where

χ^{(3)}

and

χ^{(5)}

are the third- and fifth-order susceptibilities. A detailed explanation how to obtain Eqs. (5) and (6), as well as higher-order nonlinear polarizations are given in [29]. By considering Eqs. (4)-(6) we obtain an expression for the dielectric function in the form:

\begin{array}{l} ε_{μ}^{N L} (ω) = 3 [(2 χ_{x x y y}^{(3)} (ω) + χ_{x y y x}^{(3)} (ω)) {| E_{μ} (ω) |}^{2} + 2 χ_{x x y y}^{(3)} (ω) \sum_{σ = x, y} {| E_{σ} (ω) |}^{2} (1 - δ_{μ, σ})] \\ + 10 [\frac{10}{3} χ_{x x y y x x}^{(5)} (ω) {| E_{μ} (ω) |}^{4} + \frac{5}{3} χ_{x x y y y y}^{(5)} (ω) ({| E_{μ} (ω) |}^{4} + \sum_{σ = x, y} {| E_{σ} (ω) |}^{4} (1 - δ_{μ, σ}))] \\ + [3 χ_{x y y x}^{(3)} (ω) + 10 \frac{10}{3} χ_{x x y y x x}^{(5)} (ω) {| E_{μ} (ω) |}^{2}] \sum_{σ = x, y} {[E_{σ} (ω)]}^{2} (1 - δ_{μ, σ}) \frac{E_{μ}^{*} (ω)}{E_{μ} (ω)} . \end{array}

The Helmholtz equation [Eq. (3)] can be solved by the separation of variables method for each component of the electric field:

E_{μ} (r, ω) = F (x, y) A_{μ} (z, ω - ω_{0}) \exp (i β_{0, μ} z),

where

F (x, y)

is the transverse mode pattern supported by the capillary,

A_{μ} (z, ω - ω_{0})

is the slowly varying amplitude and

β_{0, μ} = β_{μ} (ω_{0})

is the propagation constant for

μ = x, y

. Substituting Eq. (8) in Eq. (3) and separating the terms dependent upon the propagation distance

(z)

and the transverse coordinates

(x, y)

, we obtain

2 i β_{0, μ} \frac{\partial A_{μ}}{\partial z} + (β_{μ}^{2} - β_{0, μ}^{2}) A_{μ} = 0,

\frac{\partial^{2} F}{\partial x^{2}} + \frac{\partial^{2} F}{\partial y^{2}} + [ε_{μ} (ω) k_{0}^{2} - β_{μ}^{2}] F = 0,

where the slowly varying envelope approximation

[(\partial^{2} A_{μ} / \partial z^{2}) < < β_{0, μ} (\partial A_{μ} / \partial z)]

was made to obtain Eq. (9).

β_{μ} = β_{μ} (ω)

represents the frequency-dependent wavenumber, where the zero-order term of the series corresponds to

β_{0, μ}

. The dielectric function in Eq. (10) can be approximated by

ε_{μ} = {(n_{0, μ} + Δ N_{μ})}^{2} \approx {(n_{0, μ})}^{2} + 2 (n_{0, μ}) (Δ N_{μ}),

where

Δ N_{μ} = i α_{0, μ} / (2 k_{0}) + [Δ n_{μ} + i Δ α_{μ} / (2 k_{0})]

contains a dissipative term and a small perturbation induced by the nonlinearity,

n_{0, μ} (α_{0, μ})

is the linear refractive index (linear absorption coefficient) and

Δ n_{μ} (Δ α_{μ})

represents the contributions of the nonlinear refractive index (nonlinear absorption coefficient). It is important to note that HON are included in the first order perturbation,

Δ N_{μ}

, as shown in Eq. (12). However

{(Δ N_{μ})}^{2}

and high-order perturbation were neglected in Eq. (11) because, for the intensities used in this work, the contributions to the linear dielectric function are negligible since

[{(Δ N_{μ})}^{2}

is ~5 orders of magnitude smaller than

(Δ N_{μ})]

. The linear and nonlinear dielectric functions are given by

ε_{μ}^{L} = [n_{0, μ} + i α_{0, μ} / (2 k_{0})]

and

ε_{μ}^{N L} = 2 (n_{0, μ}) [Δ n_{μ} + i Δ α_{μ} / (2 k_{0})]

, and by substituting in Eqs. (7) and (8), we have

\begin{array}{l} (Δ n_{μ} + i \frac{Δ α_{μ}}{2 k_{0}}) = \frac{3}{2 n_{0, μ}} {| F |}^{2} [(2 χ_{x x y y}^{(3)} + χ_{x y y x}^{(3)}) {| A_{μ} |}^{2} + 2 χ_{x x y y}^{(3)} \sum_{σ = x, y} {| A_{σ} |}^{2} (1 - δ_{μ, σ})] \\ + \frac{10}{2 n_{0, μ}} {| F |}^{4} [\frac{10}{3} χ_{x x y y x x}^{(5)} {| A_{μ} |}^{4} + \frac{5}{3} χ_{x x y y y y}^{(5)} ({| A_{μ} |}^{4} + \sum_{σ = x, y} {| A_{σ} |}^{4} (1 - δ_{μ, σ}))] \\ + \frac{1}{2 n_{0, μ}} {| F |}^{2} [3 χ_{x y y x}^{(3)} + 10 \frac{10}{3} χ_{x x y y x x}^{(5)} {| F |}^{2} {| A_{μ} |}^{2}] \sum_{σ = x, y} {[A_{σ} (ω)]}^{2} (1 - δ_{μ, σ}) \frac{A_{μ}^{*}}{A_{μ}} e x p [2 i (β_{0, σ} - β_{0, μ}) z] . \end{array}

β_{μ}

can be determined by solving Eq. (10) using the first-order approximation

(β_{μ} \to β_{μ} + Δ β_{μ})

, where

F

and the eigenvalue

β_{μ}^{2} \to β_{μ}^{2} + 2 β_{μ} (Δ β_{μ})

are represented by

F = F^{(0)} + ξ F^{(1)} + \dots,

β_{μ}^{2} = {(β_{μ})}^{2} + ξ 2 β_{μ} (Δ β_{μ}) + \dots,

where

ξ

is a perturbation parameter which can vary over a continuous range of values from 0 (no perturbation) to 1 (full perturbation). The term

{(Δ β_{μ})}^{2}

was disregarded by the same reason as

{(Δ N_{μ})}^{2}

, as mentioned earlier. Inserting Eqs. (13) and (14) into the eigenvalue Eq. (10), and using Eq. (11), we obtain

ξ^{0} : [(\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + n_{0, μ}^{2} k_{0}^{2}) - β_{μ}^{2}] F^{(0)} = 0,

ξ^{1} : [(\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + n_{0, μ}^{2} k_{0}^{2}) - β_{μ}^{2}] F^{(1)} + [2 (n_{0, μ}) (Δ N_{μ}) k_{0}^{2} - 2 β_{μ} Δ β_{μ}] F^{(0)} = 0.

In order to find an expression for $F^{(0)}$ , Eq. (15) was written in cylindrical coordinates $(ρ, ϕ, z)$ and by using the separation of variables $F^{(0)} (ρ, ϕ) = F^{(0)} (ρ) \exp (i m ϕ)$ , it was possible to recognize Eq. (15) as the modified Bessel differential equation: $\frac{\partial^{2} F^{(0)}}{\partial ρ^{2}} + \frac{1}{ρ} \frac{\partial F^{(0)}}{\partial ρ} + (n_{0, μ}^{2} k_{0}^{2} - β_{μ}^{2} - \frac{m^{2}}{ρ^{2}}) F^{(0)} = 0$ . Then, a solution valid for the region filled with the nonlinear medium, corresponding to the capillary core $(n_{c o r e} = n_{0, μ})$ is

F_{c o r e}^{(0)} (ρ) = C_{1} J_{m} (ρ \sqrt{n_{0, μ}^{2} k_{0}^{2} - β_{μ}^{2}}),

where

J_{m} (x)

is the first kind m-order Bessel function. For the region formed by the capillary wall

(n_{0, μ} = n_{c l a d d i n g})

,

F^{(0)}

should decay exponentially with

ρ

. A function that describes such behavior is the modified Bessel function of second kind, given by

F_{c l a d d i n g}^{(0)} (ρ) = C_{2} K_{m} (ρ \sqrt{β_{μ}^{2} - n_{c l a d d i n g}^{2} k_{0}^{2}}) .

The values of $C_{1}$ and $C_{2}$ are determined by considering the boundary conditions.

From Eq. (16), corresponding to the first-order of perturbation, we get

Δ β_{μ} = k_{0} \frac{\iint (Δ N_{μ}) {| F^{(0)} |}^{2} d x d y}{\iint {| F^{(0)} |}^{2} d x d y},

where

F^{(0)}

is given by Eqs. (17) and (18), and

Δ N_{μ}

was defined right after Eq. (11). Therefore, to the first order in the perturbation, the nonlinearity

(Δ N_{μ})

does not affect the modal distribution [Eqs. (17) and (18)], but modifies the wavenumber through

Δ β_{μ}

and consequently the evolution of the field amplitude,

A_{μ}

. In this way, by considering the first-order approximation of the wavenumber

(β_{μ} \to β_{μ} + Δ β_{μ})

into Eq. (9) we obtain the expression

\frac{\partial A_{μ}}{\partial z} = i (β_{μ} + Δ β_{μ} - β_{0, μ}) A_{μ},

where the approximation

β_{μ}^{2} - β_{0, μ}^{2} \approx 2 β_{0, μ} (β_{μ} - β_{0, μ})

was used. Nevertheless, as an exact functional form of

β_{μ}

is generally unknown, an expansion in Taylor series around the frequency

ω_{0}

is made to obtain the more specialized expression

β_{μ} (ω) = β_{0, μ} + β_{μ}^{(1)} (ω - ω_{0}) + \frac{1}{2} β_{μ}^{(2)} {(ω - ω_{0})}^{2} + O [{(ω - ω_{0})}^{3}],

where

β_{μ}^{(m)} = {(d^{m} β_{μ} / d ω^{m})}_{ω = ω_{0}}

,

m = 1, 2

, represent the dispersion terms and the last term represents higher-order contributions that in the present case can be neglected. Then, substituting Eq. (21) in Eq. (20) and using Eqs. (12) and (19), we obtain two coupled differential equations that describe the evolution of the two polarization components along a capillary filled with a cubic-quintic isotropic media. Both equations can be summarized as:

\begin{array}{l} \frac{\partial A_{μ}}{\partial z} + β_{μ}^{(1)} \frac{\partial A_{μ}}{\partial t} + i \frac{β^{(2)}}{2} \frac{\partial^{2} A_{μ}}{\partial t^{2}} + \frac{α_{0}}{2} A_{μ} \\ = i \frac{k_{0}}{2 n_{0, μ}} 3 F^{(1)} {[(2 χ_{x x y y}^{(3)} + χ_{x y y x}^{(3)}) {| A_{μ} |}^{2} + 2 χ_{x x y y}^{(3)} \sum_{σ = x, y} (1 - δ_{μ, σ}) {| A_{σ} |}^{2}] A_{μ} \\ + χ_{x y y x}^{(3)} [\sum_{σ = x, y} (1 - δ_{μ, σ}) {(A_{σ})}^{2}] A_{μ}^{*} \exp [2 i (β_{0, σ} - β_{0, μ}) z]} \\ + i \frac{k_{0}}{2 n_{0, μ}} 10 F^{(2)} {[\frac{10}{3} χ_{x x y y x x}^{(5)} {| A_{μ} |}^{4} + \frac{5}{3} χ_{x x y y y y}^{(5)} [{| A_{μ} |}^{4} + \sum_{σ = x, y} (1 - δ_{μ, σ}) {| A_{σ} |}^{4}]] A_{μ} \\ + \frac{10}{3} χ_{x x y y x x}^{(5)} [\sum_{σ = x, y} (1 - δ_{μ, σ}) {| A_{μ} |}^{2} {(A_{σ})}^{2}] A_{μ}^{*} \exp [2 i (β_{0, σ} - β_{0, μ}) z]}, \end{array}

with

μ = x, y, F^{(1)} = (\iint {| F^{(0)} |}^{4} d x d y) / {(\iint {| F^{(0)} |}^{2} d x d y)}^{2}

and

F^{(2)} = (\iint {| F^{(0)} |}^{6} d x d y) / {(\iint {| F^{(0)} |}^{2} d x d y)}^{3}

. The modal birefringence of the capillary is expressed by

Δ β_{0} = β_{0, x} - β_{0, y} = \frac{2 π}{λ} | n_{x} - n_{y} |

, which leads to different group velocities for the two polarization components

(β_{x}^{(1)} \neq β_{y}^{(1)})

. However,

β^{(2)}

and

α_{0}

are assumed to be the same for both polarization components.

For convenience, we rewrite Eq. (22) using the circularly polarized components $A_{\pm} = 2^{- \frac{1}{2}} ({\bar{A}}_{x} \pm i {\bar{A}}_{y})$ , where ${\bar{A}}_{x} = A_{x} \exp (i Δ β z / 2)$ and ${\bar{A}}_{y} = A_{y} \exp (- i Δ β z / 2)$ , and after performing some manipulations we obtain:

\begin{array}{l} \frac{\partial A_{\pm}}{\partial z} + \frac{1}{2} [β_{+}^{(1)} \frac{\partial A_{\pm}}{\partial t} + β_{-}^{(1)} \frac{\partial A_{\mp}}{\partial t}] + \frac{i}{2} β^{(2)} \frac{\partial^{2} A_{\pm}}{\partial t^{2}} + \frac{α_{0}}{2} A_{\pm} \\ = \frac{i}{2} (Δ β_{0}) A_{\mp} + i \frac{3 ω_{0}}{n_{0} c} F^{(1)} [χ_{x x y y}^{(3)} ({| A_{+} |}^{2} + {| A_{-} |}^{2}) + χ_{x y y x}^{(3)} {| A_{\mp} |}^{2}] A_{\pm} \\ + i \frac{5 ω_{0}}{2 n_{0} c} F^{(2)} {\frac{10}{3} χ_{x x y y x x}^{(5)} [{| A_{+} + A_{-} |}^{2} {(A_{+} + A_{-})}^{*} \mp {| A_{+} - A_{-} |}^{2} {(A_{+} - A_{-})}^{*}] A_{\mp} \\ + \frac{5}{6} χ_{x x y y y y}^{(5)} [{| A_{+} + A_{-} |}^{4} + {| A_{+} - A_{-} |}^{4}]} A_{\pm}, \end{array}

with

β_{\pm}^{(1)} = β_{x}^{(1)} \pm β_{y}^{(1)}

.

3.2 Nonlinear birefringence in capillaries filled with silver-nanocolloids

Silver nanocolloids exhibit fast electronic nonlinear response but their nonlinear optical properties depend on the shape, size and volume fraction of the silver nanoparticles, the laser frequency (which may excite resonant or non resonant processes), and the mismatch between the dielectric functions of the nanoparticles and their host [47–50]. In particular, the complex nonlinear susceptibilities, which correspond to non resonant processes of electronic origin, obey the relationship: $χ_{x x y y}^{(3)} = χ_{x y y x}^{(3)} = 1 / 3 χ_{x x x x}^{(3)}$ and $χ_{x x y y x x}^{(5)} = χ_{x x y y y y}^{(5)} = 1 / 5 χ_{x x x x x x}^{(5)}$ [29].

For the case of a capillary filled with silver nanocolloids, the induced birefringence is due to the refractive index shift $(Δ n)$ induced by the nanoparticles nonlinearity. For light peak intensity of tens of MW/cm² (used in this work), we have $Δ n < 10^{- 5}$ , corresponding to a weak birefringence. Therefore, we may assume $β_{x}^{(1)} \approx β_{y}^{(1)} = β^{(1)}$ , and since the difference between the group velocities, $v_{g} = {[β^{(1)}]}^{- 1}$ , in both directions is negligible, we can introduce $v_{g} = {[β^{(1)}]}^{- 1}$ as an intrinsic variable by performing the transformation of Eq. (23) to the pulse reference frame (retarded frame) with $(z, t) \to (z, τ = t - z / v_{g})$ . Thus, we obtain

\begin{array}{l} \frac{\partial A_{\pm}}{\partial z} + \frac{i}{2} β^{(2)} \frac{\partial^{2} A_{\pm}}{\partial τ^{2}} + \frac{α_{0}}{2} A_{\pm} \\ = \frac{i}{2} (Δ β_{0}) A_{\mp} + i \frac{ω_{0}}{n_{0} c} F^{(1)} χ_{x x x x}^{(3)} [({| A_{+} |}^{2} + {| A_{-} |}^{2}) + {| A_{\mp} |}^{2}] A_{\pm} \\ + i \frac{5 ω_{0}}{12 n_{0} c} F^{(2)} χ_{x x x x x x}^{(5)} {4 [{| A_{+} + A_{-} |}^{2} {(A_{+} + A_{-})}^{*} \mp {| A_{+} - A_{-} |}^{2} {(A_{+} - A_{-})}^{*}] A_{\mp} \\ + [{| A_{+} + A_{-} |}^{4} + {| A_{+} - A_{-} |}^{4}]} A_{\pm}, \end{array}

where the dispersion coefficients for silver nanocolloids can be obtained by using the Maxwell-Garnett model [51]. Then, the effective dielectric function for a macroscopically isotropic medium is given by

ε_{e f f} (λ, f) = ε_{h} (λ) [1 + \frac{3 Θ (λ) f}{1 - Θ (λ) f}],

with

Θ (λ) = [ε_{N P} (λ) - ε_{h} (λ)] {[ε_{N P} (λ) + 2 ε_{h} (λ)]}^{- 1}

, where

ε_{N P}

and

ε_{h}

are the linear dielectric functions of the silver nanoparticles and the host, respectively, and f is the volume fraction occupied by the silver nanoparticles. For the experiments reported in this article, liquid CS₂ was the solvent (host) and its dielectric function according to [52] has the form

\begin{matrix} ε_{h} (λ) = {[n_{C S_{2}} (λ)]}^{2} \\ = {[1.580826 + \frac{1.52389 \times 10^{- 2}}{λ^{2}} + \frac{4.8578 \times 10^{- 4}}{λ^{4}} + \frac{8.2863 \times 10^{- 5}}{λ^{6}} + \frac{1.4619 \times 10^{- 5}}{λ^{8}}]}^{2} . \end{matrix}

On the other hand, the wavelength-dependent dielectric function of the silver nanoparticles, in the visible light region, can be adequately described by using the Drude's free-electron model [53] given by

ε_{N P} (λ) = (1 - \frac{λ^{2}}{λ_{p}^{2}}) + i (\frac{1}{2 π c τ_{r}} \frac{λ^{3}}{λ_{p}^{2}}),

where

λ_{p}^{- 2} = ρ e^{2} / π m_{0} c^{2}

,

ρ

and

e

are the density and charge of the conduction electrons,

m_{0}

is their effective mass, and

τ_{r}

is the relaxation time of the electrons in the metal. Hence, the dielectric function can be written as

ε_{N P} (λ) = (1 - 52.51 λ^{2}) + i (0.899 λ^{3})

where we consider the values of

λ_{p} = 0.138 μ m

and

τ_{r} = 31 f s

obtained in [54] by fitting the reflection and transmission measurements of a silver thin film using the Drude free-electron theory. Therefore, by substituting Eqs. (26) and (27) in the effective dielectric function expression [Eq. (25)], and defining the effective refractive index by

n_{e f f} (λ, f) = 2^{- \frac{1}{2}} \sqrt{Re [ε_{e f f} (λ, f)] + | ε_{e f f} (λ, f) |}

, we obtain the dispersion coefficients

β^{(1)} (λ, f) = \frac{1}{c} [n_{e f f} (λ, f) - λ \frac{d [n_{e f f} (λ, f)]}{d λ}],

β^{(2)} (λ, f) = \frac{λ^{3}}{2 π c^{2}} \frac{d^{2} [n_{e f f} (λ, f)]}{d λ^{2}} .

For instance, when $λ = 532 n m$ and $f = 10^{- 5}$ we determined $β^{(1)} = 6.06 n s / m$ and $β^{(2)} = 0.717 p s^{2} / m$ . Since high-order dispersion of silver nanocolloids is not reported in the literature its contribution was neglected in the deduction of Eq. (24).

An important point to highlight here is that the Z-scan experiments, previously performed for the nonlinear characterization of silver nanocolloids, were limited to a single wavelength. Therefore, in our analysis it is not possible to identify the type of non resonant processes contributing to the effective third- and fifth-order susceptibilities, shown in Table 1. Thus, for our particular case, the second and third term of the right hand in the Eq. (24) represent the contributions of all possible non resonant cubic effects such as stimulated Raman, Rayleigh-wing scattering as well as non resonant fifth-order effects.

3.3 Linear Stability Analysis

In order to analyze the effects of temporal-modulation instability (TMI) of a short laser pulse propagating in a cubic-quintic medium we performed a typical linear perturbation analysis of the pulse amplitude. For simplicity, we assume that the field polarization is oriented along the fast axis and thus $A_{x} = 0$ . Therefore, considering a Gaussian laser pulse profile, the field amplitude is represented by

A_{\pm} (z, τ) = \pm i B (z, τ) \exp (- \frac{α_{0} z}{2} - i \frac{Δ β_{0}}{2} z) \exp (- \frac{τ^{2}}{τ_{0}^{2}}),

where

B (z, τ) = B_{0} \exp [Λ (τ) z + i Φ (τ) z]

is the unperturbed pulse amplitude.

B_{0}

is the incident power at

z = 0

,

τ_{0}

is the initial pulse duration,

Λ (τ)

and

Φ (τ)

are given by

Λ (τ) = - \frac{B_{0}^{2} ω_{0}}{c} [3 F^{(1)} Im (χ^{(3)}) + 20 B_{0}^{2} F^{(2)} Im (χ^{(5)}) \exp (- 2 \frac{τ^{2}}{τ_{0}^{2}})] \exp (- 2 \frac{τ^{2}}{τ_{0}^{2}}),

\begin{array}{l} Φ (τ) = - \frac{1}{2} β^{(2)} [- \frac{2}{τ_{0}^{2}} (1 - \frac{2 τ^{2}}{τ_{0}^{2}})] \\ + \frac{B_{0}^{2} ω_{0}}{c} [3 F^{(1)} Re (χ^{(3)}) + 20 B_{0}^{2} F^{(2)} Re (χ^{(5)}) \exp (- 2 \frac{τ^{2}}{τ_{0}^{2}})] \exp (- 2 \frac{τ^{2}}{τ_{0}^{2}}) . \end{array}

The modulated amplitude of the laser pulse may be written as the superposition of perturbed and unperturbed amplitudes, as

a_{\pm} (z, τ) = \pm i {B_{0} \exp [Λ (τ) z] + B_{1, \pm} (z, τ)} \exp (- \frac{α_{0} z}{2} - i \frac{Δ β_{0}}{2} z) \exp [i Φ (τ) z] \exp (- \frac{τ^{2}}{τ_{0}^{2}}),

where

B_{1} (z, τ)

is the complex perturbed beam amplitude. A linear stability analysis was performed in order to examine the evolution of the perturbation

B_{1} (z, τ)

and the stability of the solution (33). Thus, by introducing Eq. (33) in Eq. (24) and linearizing as a function of

B_{1, \pm} (z, τ)

, we obtain

\begin{array}{l} i {\frac{\partial B_{1, \pm}}{\partial z} + i Φ B_{1, \pm} \mp \frac{i}{2} (Δ β_{0}) (B_{1, +} - B_{1, -})} \\ - \frac{1}{2} β^{(2)} {\frac{\partial^{2} B_{1, \pm}}{\partial τ^{2}} + 2 [i z \frac{\partial Φ}{\partial τ} - \frac{2 τ}{τ_{0}^{2}}] \frac{\partial B_{1, \pm}}{\partial τ} + i z B_{1, \pm} [\frac{\partial^{2} Φ}{\partial τ^{2}} - \frac{4 τ}{τ_{0}^{2}} \frac{\partial Φ}{\partial τ} + i z {(\frac{\partial Φ}{\partial τ})}^{2}] + \frac{2}{τ_{0}^{2}} [\frac{2 τ^{2}}{τ_{0}^{2}} - 1] B_{1, \pm}} \\ = - \frac{ω_{0}}{c} {| B_{0} |}^{2} F^{(1)} χ_{x x x x}^{(3)} {5 B_{1, \pm} + 2 [B_{1, +} + B_{1, -}]} \exp (2 Λ z - α_{0} z - 2 \frac{τ^{2}}{τ_{0}^{2}}) \\ - \frac{5 ω_{0}}{12 c} F^{(2)} χ_{x x x x x x}^{(5)} {16 {| B_{0} |}^{4} [3 B_{1, \pm} + 2 (B_{1, +} + B_{1, -} + B_{1, \mp})] \\ + 48 {| B_{0} |}^{2} B_{0}^{2} {[B_{1, +} + B_{1, -}]}^{*}} \exp (4 Λ z - 2 α_{0} z - 4 \frac{τ^{2}}{τ_{0}^{2}}) . \end{array}

Solution of Eq. (34) were found by neglecting further variations in the pulse shape and considering the perturbed wave amplitude to be a sinusoidal function of $z$ and $τ$ , that is, $B_{1, \pm} = u_{\pm} \exp [i (K z - Ω τ)] + i v_{\pm} \exp [- i (K z - Ω τ)]$ , with $K$ and $Ω$ being the wave number and frequency of the perturbed wave amplitude, respectively. Nontrivial solutions are obtained only when the perturbation satisfies the following dispersion relation

K^{2} + [2 β^{(2)} Ω (z \frac{\partial Φ}{\partial τ})] K - M = 0,

where

\begin{array}{l} M = Re (N^{2}) - 2 β^{(2)} Ω [z \frac{\partial Φ}{\partial τ} Re (N) - \frac{2 τ}{τ_{0}^{2}} Im (N)] \\ + {[40 \frac{ω_{0}}{c} B_{0}^{2} {| B_{0} |}^{2} F^{(2)} \exp (4 Λ z - 2 α_{0} z - 4 \frac{τ^{2}}{τ_{0}^{2}})]}^{2} Re [{(χ_{x x x x x x}^{5})}^{2}], \end{array}

\begin{array}{l} N = Φ + \frac{1}{2} β^{(2)} {- Ω [Ω + 2 i (i z \frac{\partial Φ}{\partial τ} - \frac{2 τ}{τ_{0}^{2}})] + i z [\frac{\partial^{2} Φ}{\partial τ^{2}} - \frac{4 τ}{τ_{0}^{2}} \frac{\partial Φ}{\partial τ} + i z {(\frac{\partial Φ}{\partial τ})}^{2}] + \frac{2}{τ_{0}^{2}} [\frac{2 τ^{2}}{τ_{0}^{2}} - 1]} \\ - 3 \frac{ω_{0}}{c} {| B_{0} |}^{2} \exp (2 Λ z - α_{0} z - 2 \frac{τ^{2}}{τ_{0}^{2}}) [3 F^{(1)} χ_{x x x x}^{(3)} + 20 {| B_{0} |}^{2} F^{(2)} χ_{x x x x x x}^{(5)} \exp (2 Λ z - α_{0} z - 2 \frac{τ^{2}}{τ_{0}^{2}})] . \end{array}

The local gain spectra of TMI as a function of the frequency shift along the fast axis was obtained from the imaginary part of $K$ $[g (Ω, z) = 2 Im (K)]$ , as illustrated in Fig. 2. We remark that the local gain depends on the z position because the field amplitude varies along the propagation distance due to dissipative terms. The TMI gain curves were calculated for a peak intensity of $42 M W / c m^{2}$ that corresponds to the pulse intensity at $τ = 0.1 τ_{0}$ , after a propagation distance of 1.5 mm. In the normal-dispersion regime $(β^{(2)} > 0)$ , the S-A sample (self-focusing cubic medium) presents a very small modulation instability gain compared to the other samples, and for this reason its gain spectrum is not shown in Fig. 2. In fact, numerical simulations were performed for pulse propagation with distance of 9 cm (capillary length) showing that the modulation instability effect inside S-A is negligible, as shown below (see Fig. 3). However, under the same conditions, side-bands in the local gain spectrum of S-B, which is also a focusing medium, are observed displaying a growth of the modulation instability attributed to the quintic nonlinearity which is positive and larger than in S-A. The local gain curve obtained for S-C (focusing refractive quintic media without cubic contribution) clearly corroborates that the growth rate of TMI strongly depends on the quintic nonlinearity contribution, since the refractive cubic nonlinearity is null, as indicated in Table 1. Although in the S-D and S-E, the refractive cubic nonlinearity is negative (defocusing), the refractive quintic nonlinearity dominates the effects of modulation instability since its contribution is enhanced due to the larger dependence with the electric field. As in the case of metamaterials discussed in [30] distortions in the sidebands are due to the linear losses, but enhanced by nonlinear absorption contributions. To obtain an analytical expression for the total gain accumulated after a propagation distance L, it is necessary to integrate the local gain with respect to z in the interval from 0 to L [55]. Notice however that by performing the numerical simulation based on the nonlinear propagation equations this was automatically considered.

Fig. 2 Local gain spectra of modulation instability versus the frequency shift along the fast axis, for the five samples.

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Fig. 3 Numerical pulse shape evolution for sample A (S-A), sample B (S-B), sample C (S-C), sample D (S-D) and sample E (S-E), with input intensity of 60 MW/cm². Pulse duration: 80 ps. Propagation length: 9 cm.

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4. Results and discussions

The evolution of the x- and y-components of the electric field, influenced by the nonlinear birefringence of the 9 cm long capillary filled with silver nanoparticles suspended in CS₂, was analyzed by solving numerically the two coupled equations [Eq. (24)], using the compact finite-difference method based on the Crank–Nicolson scheme [56].

Figure 3 shows numerical results that represent the pulse shape evolution, for the five samples, with input intensity of 60 MW/cm². For all cases, the input beam has a Gaussian profile with pulse duration of 80 ps. A temporal window of 400 ps, with steps of 0.1 ps, was used for the numerical simulations. For the S-A sample (black line) the pulse propagates keeping its Gaussian shape, but suffering a small broadening from 80 ps to ~89 ps. Due to the small nanoparticles concentration the contribution of the fifth-order nonlinearity is negligible in this case. On the other hand, the pulse propagation inside sample S-B (red line) is influenced by TMI, in accordance with the gain spectrum obtained in Fig. 2. In this case the nonlinear refractive index $n_{2} \propto Re [χ_{x x x x}^{(3)}]$ of S-B is smaller than for sample S-A [see Table 1] and the main contribution for TMI is due to $n_{4} \propto Re [χ_{x x x x x x}^{(5)}]$ , as previously deduced from the linear stability analysis. To corroborate our interpretation, we studied the pulse propagation in a pure quintic refractive medium (blue line), corresponding to sample S-C. The TMI effect increased due to the larger $n_{4}$ than in the samples S-A and S-B. Moreover, larger TMI effect is observed in the samples S-D (green line) and S-E (pink line), corresponding to more concentrated silver nanocolloids. Intensity losses along the propagation, in all samples, are due to linear and nonlinear absorptions. It is important to emphasize that both nonlinear refraction and absorption terms contribute to the stable or unstable propagation of optical fields, under suitable conditions. For instance, the nonlinear absorption, i.e. the imaginary part of the susceptibilities, contributes to the stability of a beam propagating in a self-focusing medium (S-A and S-B) by arresting the catastrophic collapse, as shown in [16,17]. Moreover, in defocusing cubic media (S-C, S-D and S-E), the diffraction of a beam is compensated by the focusing quintic nonlinearity [18].

Figure 4 shows the normalized transmittance as a function of the incident polarization azimuth angle, θ, for all samples. From top to bottom, the incident peak intensities correspond to 6, 24, 42 and 60 MW/cm², in each row. For I ≤ 6 MW/cm² [first row of Fig. 4], all samples behave as linear isotropic media. As a consequence, the normalized experimental transmittance exhibits a $\cos^{2} θ$ dependence (black circles) for the vertical (V)-polarization (captured in the detector D1) and a $\sin^{2} θ$ dependence (red squares) for the horizontal (H)-polarization (captured in the detector D2). Black and red lines represent the normalized transmittance for the V- and H-polarization, obtained by numerical solution of Eq. (24) and using the coefficients of Table 1. At I = 24 MW/cm² [second row of Fig. 4], the transmittance response as a function of θ shows significant nonlinear contributions, dominated by third-order nonlinearity. For the cubic and cubic-quintic self-focusing samples, S-A and S-B, respectively, the transmittance response displays a lower variation with θ which increases with χ⁽³⁾, in comparison to the first row of Fig. 4. The quintic self-focusing sample (S-C) transmittance shows small variation with θ at 24 MW/cm² because the contributions of χ⁽³⁾ is null and χ⁽⁵⁾ is very small. In contrast, the cubic-quintic (defocusing-focusing) samples (S-D and S-E) show a larger dependence as a function of the incident polarization azimuth angle. With the increase of the incident intensity, the refractive index variation between the slow and fast axis of the capillary increases or decreases depending of the sign of total nonlinear susceptibility and the direction of the incident field polarization in the transverse plane. In this way, rotation of the incident polarization direction, at high intensities, generate multiple regions where the variation of the nonlinear birefringence increases or decreases the polarization instability effects. As a consequence, regions of small and large response are simultaneously observed for high intensities by varying the incident polarization azimuth angle, as shown in the third and fourth rows of Fig. 4 that corresponds to intensities of 42 MW/cm² and 60 MW/cm², respectively. Note that in the last two rows of Fig. 4, the fifth-order contribution is very important to increase the modulation instability effect, in agreement with the Fig. 2. Numerical simulations of Eq. (24), represented by the solid lines, were made using as initial condition a 80 ps Gaussian pulse. However, the relative orientation of the capillary fast- and slow-axis in relation to the laboratory frame (x, y) was treated as a free parameter. Values of $Δ β_{0}$ between 0.07 and 0.12, which correspond to $| n_{x} - n_{y} | \approx 10^{- 6}$ , were used to obtain a better fit of the experimental data. These values are reasonable since the refractive index variation produced by the nonlinearities, $Δ n = n_{2} I + n_{4} I^{2}$ , are of the same order of magnitude.

Fig. 4 Normalized transmittance as a function of the incident polarization azimuth angle, θ, for (a) sample A, (b) sample B, (c) sample C, (d) sample D and (e) sample E. From top to bottom, the incident peak intensities are 6, 24, 42 and 60 MW/cm², in each row. Black circles and red squares correspond to vertical and horizontal polarization transmittance, respectively. Solid lines were obtained from numerical solutions of Eq. (24) and the parameters of Table 1.

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Figure 5 shows the transmittance behavior, in the V-polarization (black circles) and H-polarization (red squares), for incident peak intensities between 0.1 MW/cm² and 70 MW/cm². An incident azimuth angle of 43° in relation to V-polarization was used for all samples. For non-birefringent materials or materials exhibiting only linear birefringence, the transmittance remains constant for different intensities. However, Fig. 5(a), corresponding to S-A (pure CS₂), shows modulation of the transmittance response with increasing of the intensity, induced by the cubic nonlinearity. Each oscillation observed in the transmittance signal corresponds to $~ 2 π$ phase-shift. It is possible to observe that with the addition of silver nanoparticles [Figs. 5(b)–5(e)], the modulation increases due to the large effective nonlinear susceptibility of the samples. It is worth noting that the fifth-order nonlinearities are essential for the increasing of the modulation, as shown in Fig. 5(c), which corresponds to a refractive quintic medium with $Re [χ^{(3)}] = 0$ . A maximum nonlinear phase shift of ~20π was observed for intensities of 70 MW/cm², using the S-E sample, as shown in Fig. 5(e). The black and red solid lines represent the numerical simulations of Eq. (24), showing a good agreement with the experimental results. The blue dashed lines in Figs. 5(b)–5(e) display the transmittance behavior neglecting the $Re [χ^{(5)}]$ contribution, showing that the modulation due to the nonlinear birefringence is highly modified due to the fifth-order nonlinearity.

Fig. 5 Vertical (black circles) and horizontal (red squares) polarization transmittance as a function of the incident peak intensities variation, for (a) sample A, (b) sample B, (c) sample C, (d) sample D and (e) sample E. Black and red solid lines were obtained from numerical solutions of Eq. (24) and the parameters of Table 1. Blue dashed lines represent the numerical solutions of Eq. (24) neglecting the contribution of $Re (χ^{(5)})$ .

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Small discrepancies between the experimental and theoretical results are due to the capillary used for all experiments that supports few propagation modes (V-number is ~9), while in the theoretical description it was assumed, for simplicity, that the propagation occurs in a single-mode capillary. Therefore, coupling effects between the different spectral modes were neglected in the numerical simulations. The free parameters used in the simulation, such as the focusing angle of the beam entering in the capillary and the linear refractive index dependence with f are not expect to contribute for relevant discrepancies between the experimental and numerical results.

5. Summary

In summary, the nonlinear birefringence effect due to cubic and quintic nonlinearities was investigated in a 9 cm long capillary filled with silver nanoparticles suspended in CS₂. For a given optical field amplitude the intensity-dependent birefringence was varied by changing the volume fraction occupied by the silver nanoparticles from 0 (pure CS₂) to $4.5 \times 10^{- 5}$ . Two experimental schemes were used to analyze the transmittance response as a function of the incident polarization azimuth angle and the light intensity, in order to identify the contributions of the third- and fifth-order susceptibilities on the polarization instability effect. To compare with the experimental results, a model describing the evolution of the optical field polarization was developed considering the refractive and dissipative contributions due to the third- and the fifth-order susceptibility. The model describes how the gain spectrum of the modulation instability increases significantly with the presence of the quintic nonlinearity. In addition, numerical simulations were performed considering the dispersion coefficients as well as the linear and nonlinear susceptibilities for the colloids studied, showing good agreement with the experimental results.

Finally we want to recall that the nonlinear birefringence effect is attractive for exploitation in all-optical switches. For the samples used in this work, a large nonlinear phase-shift (~20π) was observed in diluted silver-nanocolloids with $f = 4.5 \times 10^{- 5}$ . However, more concentrated samples will present larger nonlinear phase-shift due to the increased third- and fifth-order nonlinearities and may present further interesting behavior not observed in the present experimental conditions. Also we recall that by using the nonlinearity management procedure of [42–44], it is possible to improve the figures-of-merit of all-optical switches based on metal-dielectric nanocomposites [44].

Funding

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq); Fundação de Amparo à Ciência e Tecnologia do Estado de Pernambuco (FACEPE); National Institute of Photonics (INCT de Fotônica– INFO); PRONEX/CNPq/FACEPE; CAPES-COFECUB Program.

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Sample	Volume fraction (f)		$β^{(2)}$ (ps²m⁻¹)	$χ_{x x x x}^{(3)}$ (x10⁻²⁰ m²V⁻²)	$χ_{x x x x x x}^{(5)}$ (x10⁻³⁵ m⁴V⁻⁴)
S-A	0	0	0.717	2.92 + i 0.04	(2.9 + i 0.9) x 10^{-2 (c)}
S-B	1.0 x 10⁻⁵	9.02	0.717	1.79 - i 0.20	2.05 + i 0.20
S-C	1.8 x 10⁻⁵	16.23	0.718	- i 0.41	2.98 + i 0.31
S-D	3.0 x 10⁻⁵	27.05	0.719	−2.07 - i 0.65	6.36 + i 0.64
S-E	4.5 x 10⁻⁵	40.57	0.720	−4.89 - i 1.02	9.04 + i 0.87

Nonlinear polarization instability in cubic-quintic plasmonic nanocomposites

Abstract

1. Introduction

2. Experimental details

3. Nonlinear birefringence in a capillary filled with an isotropic cubic-quintic medium

3.1 General description

3.2 Nonlinear birefringence in capillaries filled with silver-nanocolloids

3.3 Linear Stability Analysis

4. Results and discussions

5. Summary

Funding

References and links

Cited By

Figures (5)

Tables (1)

Equations (37)

Optics Express