Nonlinear polarization evolution of hybridly polarized vector beams through isotropic Kerr nonlinearities

Bing Gu; Bo Wen; Guanghao Rui; Yuxiong Xue; Jun He; Qiwen Zhan; Yiping Cui

doi:10.1364/OE.24.025867

1. Introduction

The polarization-dependent optical nonlinearities that are associated with the tensor nature of the third-order susceptibility χ⁽³⁾ of optical materials are of direct relevance to technological applications in nonlinear photonics devices. One of the interesting effects is the nonlinear ellipse rotation that occurs when an intense elliptically polarized beam interacts with an isotropic medium owing to the existence of $χ_{x y y x}^{(3)}$ [1]. The nonlinear ellipse rotation effect has been exploited to discriminate the origin of optical nonlinearity [2, 3], to characterize the third-order nonlinear susceptibility tensor [4, 5], and in the nonlinear polarization switching [6, 7].

Since Callen et al. [8] observed the far-field annular intensity patterns of a Gaussian beam passing through carbon disulfide (CS₂) in 1967, the spatial self-phase modulation (SSPM) effect has been observed in many materials, such as nematic liquid crystal film [9], carbon nanotube solutions [10], and hybrid materials [11]. Moreover, this nonlinear optical effect has been recently reported in two-dimensional nanomaterials, including graphene dispersions [12], layered transition metal dichalcogenides [13], and MoS₂ nanoflake solutions [14]. On the other hand, researchers exploited several physical mechanisms of the novel SSPM effects, including photothermal effect [8], thermal nonlinearity with gravitational effect [10], and nonlocal optical nonlinearity [15].

Up to now, most of investigations on both the nonlinear ellipse rotation and SSPM effects were excited by the scalar light field with homogeneously polarization, such as linearly, circularly, or elliptically polarized beams. It is noteworthy that the cylindrical vector beam with localized linear polarization has excited many novel nonlinear optical effects, such as second-harmonic generation [16], modulation instability [17], vectorial self-diffraction [18], and optical limiting [19]. Very recently, we have reported the polarization characteristics of radially polarized beams induced by an anisotropic Kerr nonlinearity [20]. As a novel kind of vector field, hybridly polarized vector beams with spatially variant hybrid state of polarization (SoP) have recently received significant attentions because of their interesting applications in particle orientation analysis [21], optical trapping [22], and controllable field collapse [23]. Apparently, the interaction of hybridly polarized vector beams with matters differs from that of radially polarized beams, although the understanding of the related nonlinear optical effect is seldom in the literature [23]. More importantly, one may expect the appearance of novel nonlinear optical behavior than contains both nonlinear ellipse rotation and vectorial SSPM effects.

In this work, for the first time to our knowledge, we report the vectorial SSPM effect and nonlinear ellipse rotation of hybridly polarized vector beams through an isotropic nonlinear Kerr medium. As the experimental evidence, we investigate the SSPM effect of the femtosecond-pulsed hybridly polarized vector beams in carbon disulfide at 800 nm. The results demonstrate that the vectorial SSPM intensity patterns, the SoP distribution, as well as the spin angular momentum (SAM) flux of a hybridly polarized vector beam could be manipulated by tuning the magnitude of the isotropic optical nonlinearity with the existence of nonlinear susceptibility tensor $χ_{x y y x}^{(3)}$ .

2. Theory

The electric field distribution of a hybridly polarized vector beam can be expressed as [24,25]

\vec{E} (ρ, φ) = A (ρ) (e^{i φ} {\vec{e}}_{x} + e^{- i φ} {\vec{e}}_{y}) .

Here the polar coordinate system (ρ, φ) describes the vector beam illuminating the lens.

{\vec{e}}_{x}

and

{\vec{e}}_{y}

are the unit vectors in the Cartesian coordinate system. A(ρ) represents the radial-dependent amplitude distribution in the cross section of the vector field. For the lowest-order Laguerre-Gaussian (LG) focused by a thin lens with the focal length of f, we have

A (ρ) \propto ρ / ω \exp [- ρ^{2} / ω^{2} - i k ρ^{2} / (2 f)]

, where ω is the radius of the input elegant Gaussian beam, k = 2π/λ is the wave vector, and λ is the laser wavelength.

According to the vectorial Rayleigh-Sommerfeld formulas [26] under the paraxial approximation, similar to the previous works reported in [20, 27], we obtain the electric field along the + z direction with the coordinate origin at the focal plane of a thin lens as

\vec{E} (r, θ, z) = \sqrt{\frac{η}{π}} \frac{2 E_{0} f g_{1}}{μ^{3 / 2} ω^{3} (f + z)} e^{i k (f + z) - 2 η} [e^{i θ} {\vec{e}}_{x} + e^{- i θ} {\vec{e}}_{y}],

where μ = 1/ω² + ikz/[2f(f + z)] and η = k²r²/[8μ(f + z)²]. Here the polar coordinate system (r, θ) represents the focused vector beam at the z plane. E₀ denotes the peak electric field amplitude of the vector beam at the focus. g₁ = 2.06637 is a normalized constant determined by the condition of (|E_f|²/|E₀|²)_max = 1, where E_f = E(r, θ, 0). It is noted that Eq. (2) just describes the steady-state case. Since the temporal profile of the laser pulses is not taken into consideration in the analysis. As described by Eq. (2), the focused hybridly polarized vector beam preserves the initial distribution of SoP at any propagation position in free space.

The electric field of such a focused vector beam can be decomposed into a linear combination of left-hand (LH) and right-hand (RH) circular components as

\vec{E} (r, θ, z) = E_{+} {\vec{σ}}_{+} + E_{-} {\vec{σ}}_{-},

where

E_{\pm} = \sqrt{2 η} E_{0} f g_{1} e^{i k (f + z) - 2 η} [e^{i θ} \mp i e^{- i θ}] / [\sqrt{π} μ^{3 / 2} ω^{3} (f + z)]

.

{\vec{σ}}_{+} = ({\vec{e}}_{x} + i {\vec{e}}_{y}) / \sqrt{2}

and

{\vec{σ}}_{-} = ({\vec{e}}_{x} - i {\vec{e}}_{y}) / \sqrt{2}

are the LH and RH circular polarization unit vectors.

Now we consider that LH and RH circular polarized beams propagate along the + z axis within an isotropic nonlinear medium (for example, CS₂). Under the thin-sample approximation and the slowly varying envelope approximation, the complex field at the exit plane of the sample in terms of $E_{\pm} (r, θ, z)$ is given at the entrance plane of the sample as [28]

{\vec{E}}_{e} (r, θ, z) = E_{+} e^{i Δ ϕ_{+}} {\vec{σ}}_{+} + E_{-} e^{i Δ ϕ_{-}} {\vec{σ}}_{-},

where Δϕ _± = kΔn _± L and L is the sample thickness. The changes in the refractive indexes Δn _±of two circular components are given by [1]

Δ n_{\pm} = \frac{2 π}{n_{0}} [A (| E_{\pm} |^{2} + | E_{\mp} |^{2}) + B | E_{\mp} |^{2}],

where n₀ is the linear refractive index of the sample.

A = 6 Re [χ_{x x y y}^{(3)}]

and

B = 6 Re [χ_{x y y x}^{(3)}]

are two independent tensor components of the third-order nonlinear susceptibility in an isotropic medium. Depending on the nature of the physical origin of the optical nonlinearity, one takes the ratio between these two components as B/A = 0 for electrostriction or thermal nonlinearity, B/A = 1 for nonresonant electron nonlinearity, and B/A = 6 for molecular orientation [28].

As described by Eq. (5), the changes in refractive index Δn _±for two circular components are different. Defining the ellipticity as e = (|E₊|-|E_-|)/(|E₊| + |E_-|), we obtain

Δ n_{\pm} = \frac{2 π}{n_{0}} [A + \frac{B}{2} \frac{{(1 \mp e)}^{2}}{(1 + e^{2})}] I,

where I = |E₊|² + |E_-|² is the total intensity of the focused hybridly polarized vector beam. From Eq. (2), the intensity can be written as

I (r, z) = \frac{8 I_{0} g_{1}^{2} f^{2} \sqrt{η η *}}{π {| μ |}^{3} ω^{6} {(f + z)}^{2}} e^{- 2 η - 2 η *} .

Here the asterisk denotes the complex conjugate of the complex number. I₀ = ε/(π^1/2g₁²ω₀²τ) is the peak intensity at the focus, where ε is the incident energy, ω₀ = λf /(πω) is the waist radius of the focused vector beam, and τ is the half-width at e⁻¹ of the maximum for the laser pulse duration.

From Eq. (6), we determine the third-order nonlinear refractive indexes related to LH and RH components by

n_{2}^{\pm} = \frac{2 π}{n_{0}} [A + \frac{B}{2} \frac{{(1 \mp e)}^{2}}{(1 + e^{2})}] .

We can see that the nonlinear refractive index strongly depends on the ellipticity e of the light field. Some interesting cases of Eq. (8) are as follows. For a circularly polarized beam (e =+ 1 and −1 for LH and RH circular polarized beams, respectively), the nonlinear refractive index is given by n₂^cir = (2π/n₀)A. Clearly, the n₂^cir value of a circularly polarized beam depends on A but not B. For a linearly polarized beam with e = 0, one gets n₂^lin = (2π/n₀)(A + B/2). In this case, the absolute value of n₂^lin reaches the maximum. For the beam with arbitrary ellipticity, the nonlinear refractive index is described by Eq. (8).

Based on the vectorial Rayleigh-Sommerfeld formulas under the paraxial approximation [26], we obtain the electric field distribution in the far-field observational plane by

{\vec{E}}_{a} (r_{a}, ϕ, d) = - \frac{i k e^{i k D}}{2 π D} \int_{0}^{\infty} \int_{0}^{2 π} {\vec{E}}_{e} (r, θ, z) \exp (\frac{i k r^{2}}{2 D}) \exp (- \frac{i k r r_{a}}{D} \cos (θ - ϕ)) r d r d θ,

where D = d-z, d is the distance from the focal plane to the far-field observational plane. Equation (9) gives a general electric field distribution of a hybridly polarized vector beam passing through an isotropic nonlinear Kerr medium.

3. Numerical results

To investigate the effects of an isotropic nonlinear Kerr medium on hybridly polarized vector beams, the typical parameters are taken for numerical simulations as λ = 800 nm, f = 200 mm, ω = 2.5 mm, ω₀ = 20.4 μm, z = 0, and d = 200 mm in the entire analysis [18]. Depending on the optical nonlinear mechanisms, we can take B/A = 0, B/A = 1, and B/A = 6 [28]. Besides, we define the peak nonlinear refractive phase shift for a linearly polarized beam at the focus, as ΔΦ₀ = kn₂^linI₀L.

Figure 1 illustrates the intensity patterns of hybridly polarized vector beams at the far-field observational plane when an isotropic Kerr medium is located at the focal plane of the lens with different values of B/A and ΔΦ₀. Owing to the refractive-index changes induced by the optical nonlinearity, the far-field intensity patterns of hybridly polarized vector beams after passing through isotropic Kerr media exhibit a central dark spot surrounded by the multiple concentric ring structures. For the case of B/A = 0, as described by Eq. (8), the nonlinear effect does not influence on the SoP distribution of the incident light field. As a result, the transmitted light field preserves the initial polarization distribution during the propagation. Moreover, the far-field intensity pattern exhibits a multiple concentric ring structure with circular symmetry as shown in the first column of Fig. 1. Interestingly, for the case of B/A≠0, the far-field intensity patterns exhibit a square-like distribution with four-fold rotational symmetry, as displayed in the second and third rows of Fig. 1. When B/A = 1, more light energy is diffracted into the outer rings as the value of ΔΦ₀ increases. However, for the case of B/A = 6, the intensity patterns are nearly independent of the value of ΔΦ₀ except for the periphery distribution of the diffracted light. With increasing the value of B/A from 0 to 6 for the fixed phase shift ΔΦ₀, more light energy is diffracted into the inner ring because the relative contribution of B increases in the whole nonlinearity for the fixed value of ΔΦ₀. The far-field SSPM effect can be understood as follows. The light field with a structured distribution of SoP via the isotropic nonlinear effect produces a structured phase described by Eq. (6) in the hybridly polarized vector beam. The structured phase shift modulates the propagation behavior of the beam itself, resulting in the structured far-field intensity pattern.

Fig. 1 Normalized far-field intensity patterns and schematics of SoPs (Deep red: LH polarization; Deep blue: RH polarization; White lines: linear polarization) of hybridly polarized vector beams for isotropic Kerr media with different values of B/A and ΔΦ₀.

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Next, we study the polarization evolutions of hybridly polarized vector beams after passing through isotropic Kerr media. By calculating the Stokes parameters of the electric field, we can derive the orientation angle Ψ and the ellipticity angle χ of the localized polarization ellipse by

\tan 2 Ψ = S_{2} / S_{1},

\sin 2 χ = S_{3} / S_{0},

where S_i represents four Stokes parameters. Accordingly, we map the distribution of SoPs in the far-field observational plane, as superimposed in Fig. 1. Clearly, the intensity patterns of hybridly polarized vector beams induced by isotropic nonlinearities have abundant SoPs (including linear polarization, RH and LH elliptical polarizations) in the cross section of the field. As shown in Fig. 1, the SoP distributions in the far-field observational plane have two-fold rotational symmetry. Besides, the orientation of polarization ellipse at both sides of the localized circular polarization is orthogonal. Interestingly, there exist three nonlinear eigenpolarizations, namely, localized linear polarization, LH or RH circular polarizations located at the azimuthal angle with respect to the x direction as ϕ = mπ/4 (where m = 0,1,…,7). These localized eigenpolarizations of the vector beam do not change while propagating through the isotropic Kerr medium. The result is agreement with that of the scalar light beam [29, 30]. More importantly, for the localized elliptical polarization, as predicted by Eq. (8), there is an anisotropy that produces a nonlinear phase change, which is only depends on the

B

coefficient instead of

A

, resulting in the nonlinear ellipse rotation.

To illustrate the influence of the nonlinear effect on the localized polarization ellipse, as an example, we present the ΔΦ₀-dependence of the orientation angle Ψ and the ellipticity angle χ of the localized polarization ellipse located at the point of x = 4 mm and y = 2 mm on the far-field observational plane of a hybridly polarized vector beam for B/A = 1, as shown in Fig. 2. For the sake of comparison, the results for the case of B/A = 0 are also shown by the circles in Fig. 2. Different from that the polarization ellipse is independent of the nonlinear effect for B/A = 0, both the orientation and ellipticity of the localized polarization ellipse strongly depend on the value of ΔΦ₀ for B/A≠0. That is, the desirable polarization (both the orientation and ellipticity of the polarization ellipse) is predicted by changing the magnitude of optical nonlinearity. However, for elliptically polarized light, only the orientation of the polarization ellipse changes if the imaginary part of $χ_{x y y x}^{(3)}$ is zero [30].

Fig. 2 ΔΦ₀-dependence of (a) the orientation angle Ψ and (b) the ellipticity angle χ of the localized polarization ellipse at the point (x = 4 mm, y = 2 mm) on the far-field observational plane of a hybridly polarized vector beam for an isotropic Kerr medium with the value of B/A = 1.

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It is well known that the optical SAM is associated with the polarization distribution of light field. For the transverse electric field described by Eq. (9), the optical SAM flux density can be expressed as [31]

S_{z} = \frac{{\vec{E}}^{*} \times \vec{E}}{i \iint {\vec{E}}^{*} \cdot \vec{E} d x d y} = \frac{S_{3}}{\iint {| \vec{E} |}^{2} d x d y} {\vec{e}}_{z},

where

S_{3} = i (E_{x} E_{y}^{*} - E_{x}^{*} E_{y})

is the third Stokes parameter. As described by Eq. (12), the local SAM flux density along the + z direction is equal to S₃ normalized to the power of the light field. Figure 3 presents the SAM flux distributions at the far-field observation plane of hybridly polarized vector beams for isotropic nonlinear media with different values of B/A and ΔΦ₀. Similar to the SoP distributions shown in Fig. 1, the SAM flux distributions exhibit the radially-modulated fan-shaped structures and have two-fold rotational symmetry. With increasing the value of B/A from 0 to 6 for the fixed ΔΦ₀, more SAM flux concentrates to the central region. The interaction of structured light field with isotropic medium changes the SoP distribution of the hybridly polarized vector beam in the far-field plane, resulting in a redistribution of the SAM flux. As the value of |ΔΦ₀| increases, the magnitude of SAM flux decreases because more light energy is diffracted into the periphery of the intensity pattern. It should be noted that the localized polarization ellipse induced by isotropic refractive nonlinearity only redistributes the SAM flux, whereas the total SAM remains conservation and is always equal to zero.

Fig. 3 SAM flux redistributions at the far-field observational plane of hybridly polarized vector beams for isotropic Kerr media with different values of B/A and ΔΦ₀.

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4. Experiments

To verify the above-mentioned effects, we carry out the experiment with the femtosecond-pulsed hybridly polarized vector beams in CS₂. We choose CS₂ due to its well-characterized exhibition of isotropic Kerr nonlinearity. The nonlinear sample CS₂ is contained in 2 mm thick quartz cell at room temperature and standard atmosphere. The laser source used in our experiments is a Ti:sapphire regenerative amplifier (Coherent Inc.), operating at central wavelength of 800 nm with pulse duration of 180 fs, repetition rate of 1 kHz, and near-Gaussian spatial and temporal profiles. After passing through a quarter wave plate, the circularly polarized laser is obtained. To determine the value of B/A in CS₂, we perform the closed-aperture Z-scan experiments at the intensity of 23 GW/cm² using both linearly and circularly polarized laser pulses as shown the squares and circles in Fig. 4, respectively. Then we extract the nonlinear refractive indexes of n₂^lin = 2.92 × 10⁻⁶ cm²/GW and n₂^cir = 1.36 × 10⁻⁶ cm²/GW. Accordingly, we determine B/A = 2.3 ± 0.2 for CS₂ at 800 nm, which is consistent with the ones reported previously [32].

Fig. 4 The closed-aperture Z-scan traces of CS₂ at the intensity of 23 GW/cm² under the excitation of linearly and circularly polarized laser pulses.

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Based on the principle of the wavefront reconstruction, we generate the femtosecond-pulsed hybridly polarized vector beams by a common path interferometer implemented with a computer-controlled spatial light modulator [24, 25]. The generated hybridly polarized vector beams have near lowest-order LG spatial distribution and Gaussian temporal profile, with pulse energy of 6.0 μJ and pulse duration of ~340 fs. The hybrid polarized beam is focused by an achromatic lens with focal length of f = 150 mm, producing the waist radius at the focus ω₀≈21 μm. The nonlinear sample is located at the focal plane. A detector (Beamview, Coherent Inc.) is placed at the observational plane with a distance of d = 60 mm from the distance from the focal plane to the detector plane.

The experiments are performed at three intensities of I₀ = 70, 105, and 126 GW/cm². Correspondingly, the measured intensity patterns of femtosecond-pulsed hybridly polarized vector beams passing through CS₂ are shown in Fig. 5. With the known parameters (ΔΦ₀ and B/A) under our experimental condition, we simulate the far-field intensity patterns with the nonlinear sample at the focal plane, as shown in the second row of Fig. 5. Interestingly, we experimentally observe the far-field intensity patterns exhibiting a square-like distribution with four-fold rotational symmetry, which arises from the SSPM effect of the interaction of hybridly polarized vector beam with isotropic Kerr nonlinearity. Clearly, the theoretical simulations are in agreement with the experimental observations, implying that our theoretical analysis is reasonable. It should be noted that the discrepancy between the experiment and theory in the SSPM intensity patterns is apparent. This difference is anticipated for the following reasons. A monochromatic beam with a single wavelength passing through the nonlinear medium is considered in the theoretical framework; whereas a femtosecond pulse train at a central wavelength with a spectral bandwidth of tens-of-nanometers is used in the experimental measurements. The hybridly polarized vector beam with lowest-order LG spatial profile is assumed in the analysis; while the femtosecond-pulsed vector beam with near lowest-order LG configuration is generated. Besides, the intensity fluctuation of the output beam is inevitable and the instability of the system for generating vector beams is existent [33].

Fig. 5 Experimentally measured and theoretically predicted intensity patterns of femtosecond-pulsed hybridly polarized vector beams passing through CS₂ with different values of ΔΦ₀.

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5. Conclusions

In summary, we investigated the far-field SSPM effect and nonlinear ellipse rotation of hybridly polarized vector beams induced through isotropic Kerr nonlinearities. Based on the vectorial Rayleigh-Sommerfeld formulas under the paraxial condition, we obtained the analytical expression of the paraxial focal field of the hybridly polarized vector beam. After introducing the third-order nonlinear refractive index of an isotropic medium excited by the vector field with arbitrary ellipticity, we numerically simulated the vectorial SSPM intensity patterns and SAM flux redistribution of the hybridly polarized vector beam. Experimentally, we observed the SSPM effect of the femtosecond-pulsed hybridly polarized vector beams in CS₂ at 800 nm, which is in agreement with the theoretical predictions. Our results demonstrate that the SSPM intensity pattern, the SoP distribution, and the SAM flux of a hybridly polarized vector beam could be manipulated by tuning the isotropic optical nonlinearity, which may find interesting applications in nonlinear mechanism analysis, nonlinear characterization technique, and SAM manipulation.

Funding

National Natural Science Foundation of China (NSFC) (Grant Nos: 11474052, 11504049, 11174160); National Key Basic Research Program of China (Grant No: 2015CB352002); National Key Laboratory of Science and Technology on Vacuum Technology and Physics (ZWK1608).

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Nonlinear polarization evolution of hybridly polarized vector beams through isotropic Kerr nonlinearities

Abstract

1. Introduction

2. Theory

3. Numerical results

4. Experiments

5. Conclusions

Funding

References and links

Cited By

Figures (5)

Equations (12)

Optics Express