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Inhomogeneously polarized optical waves form a class of nonlinear vector wave propagation that has not been widely studied in the literature. We find a modulation instability only when the wave has nonzero ellipticity in a medium where the Kerr nonlinearity possesses opposite handness. Under the modulation instability the wave develops an azimuthally periodic shape with two or four peaks.
©2006 Optical Society of America
1. Introduction
The vector nature of light wave is characterized by its state of polarization. Light wavefronts with spatially homogeneous polarization are most commonly studied in the past. This includes linear, circular and elliptical polarizations. Traditionally, a spatially inhomogeneous polarization distribution was considered as a nuisance or aberration in the optical design and has not drawn much attention. However, there is an increasing interest in this type of polarization recently, mostly enabled by the advances made in micro-fabrication and theoretical modeling techniques that were not previously available.
One example that has attracted much of the interest is the so-called cylindrical vector beams. Cylindrical vector beams are solutions of the Maxwell equations that obey cylindrical symmetry both in amplitude and polarization. These beams can be divided into radial polarization, azimuthal polarization and generalized cylindrical polarization, according to the actual polarization pattern. Owing to their peculiar linear properties, especially when focused under high numerical aperture (NA) conditions, light waves with these polarization patterns have been the subject of many papers and found uses in sub-wavelength imaging, laser micromachining, and particle trapping and manipulation [1–9]. Nonlinear optics of vector waves has been carefully studied [10], but cylindrical polarization has only recently been reported for defocusing Kerr media in the non-paraxial propagation [11]. Generally, optical vortex solitons have been treated as scalar waves that have a cross-sectional shape similar to a dark soliton, but possess a spiral phase component [12–13].
An azimuthal instability has been widely studied for homogeneously polarized or scalar waves in nonlinear media that support solitons [14–16]. Soto-Crespo et al. have identified and analyzed an azimuthal instability for scalar Gaussian beams in a saturable nonlinear medium and identified a recurring azimuthal symmetry breaking. They have also been analyzed and observed in quadratic nonlinear media [17]. Azimuthons were recently analyzed and defined in Ref. (16) as localized waves that appear by deformations of optical vortices.
In this paper we explore the effect that the self-focusing Kerr nonlinearity has on a vector wave that is inhomogeneously polarized. The azimuthal modulation instability is distinguished from the vortex soliton nonlinearities by several features, including initial polarization state and the nonlinear circular birefringence. The equations are described in Section 2 and results are presented in Section 3 starting with the homogeneously polarized wave in the nonlinear medium to define the analogous problem to the subsequent sub-section describing the cylindrical vector waves.
2. Simulations
We begin with the components of the vector wave equation expressed in linear polarization basis.
where α denotes x or y. The first term on the right hand side has a transverse Laplacian operator that describes diffraction. The transverse field amplitude, longitudinal and transverse coordinates are scaled. The transverse coordinates are scaled to the width, w, of the initial Gaussian envelope function; the longitudinal coordinate is scaled by the diffraction length zd=2w2k0 , where k0 is the wavenumber in the medium. The field is scaled to the Kerr nonlinearity denoted in Boyd [19] by A, so that , where Ephys denotes the “physical” field in Gaussian units and ω is the angular frequency of the wave. The equations use the paraxial approximation and assume pulse dispersion is negligible. Temporal propagation effects are included when the longitudinal coordinate is assumed to be in the co-propagating reference frame.
The nonlinear medium has a Kerr nonlinearity with the circular birefringence polarization coupling term expressed by a B coefficient, which is scaled to the A coefficient. B introduces circular birefringence to the nonlinear propagation effects [19] and vector propagation effects for homogeneous polarizations were experimentally studied [20, 21]. The B coefficient has values B=0 for an electrostrictive nonlinearity, B=1 for a nonresonant electronic nonlinearity, and B=6 for a molecular orientational nonlinearity [18, 19]. In this paper we examine only the last case in detail, but contrast it with results for the scalar wave and results for the case B=0. The most succinct form of the equations is expressed in the circular polarization basis, which is
Without diffraction both circular polarizations are locally conserved. We numerically integrate these equations using a second-order split step operator technique similar to the technique described in [22]. We integrated both polarization representations described by Eqs. (1) and (2) for comparison and found identical results. For the fast Fourier transforms we generally use a transverse 256x256 grid, but also we have run simulations with a 512x512 grid.
The initial conditions are a modified Gaussian shape, since the field vanishes at x=y=0 for the radial polarization case. We start with a cylindrical vector beam that passes through a wave plate with retardance φ. For the Cartesian components of the field we have
and E0 is the field amplitude variable. The angle θ is the rotation of the local polarization orientation vector from radial polarization (θ=0) to tangential polarization state (θ=π/2) [4]. The angle φ is the phase retardance due to the phase-plate element, for a quarterwave plate φ=π/2 and for a half-wave plate φ=π. When φ=0 the beam is cylindrically polarized.
3. Results
The appearance of modulation instabilities for the nonlinear Schrödinger equation and many generalizations is well studied in the literature [10]. The differences for this case are in the initial cylindrical polarization state and the addition of the B coefficient. We found that the B coefficient is responsible for the unusual modulation instability whose existence we examine in this paper.
3.1. Homogeneous linear polarization case
The homogeneous linear polarized field result is well known in the literature and corresponds to a scalar form of the equations, so we briefly examine it in this subsection to draw a contrast to the cylindrically polarized vector case. The collapse of the beam due to the strong self-focusing nonlinearity places limitations on the input intensity. For the Gaussian beam the Townes profile plays a role in the shape of the collapsing beam [10, 23]. If the initial field is a donut shape
A peak develops on-axis due to the Poisson spot phenomenon and then evolves during propagation analogous to a Gaussian beam. For comparison of the instability threshold we use the scalar form of Eq. (1).
This would be the same result for a vector wave linearly polarized, say along the x-axis. Figure 1 shows the initial donut shaped intensity profile on the left with E0=1.5 and B=6. The initial intensity is close to the critical power threshold for self-focusing collapse for this case. After propagating a distance z=2 the profile takes on a symmetric shape with a peak centered at r=0. The appearance of the central maximum is a consequence of linear diffraction effects and it is well-known as the Poisson spot phenomenon. The Poisson spot phenomenon can be avoided by adding a phase rotation around the origin, i.e. topological charge, which is identical to the optical vortex soliton case, except for a nonlinear coefficient having the opposite sign.
Fig. 1. Scalar case showing donut initial profile (right) and final profile (left). The parameters are B=6 and E0=1.5.
3.2. Cylindrically polarized vector wave
For the cylindrically polarized vector case we use an initial intensity profile that leads to the same intensity profile as the scalar case. The initial conditions are expressed by Eqs. (3) and (4). The results of propagation are quite distinct though. The rotation of the phase around the origin leads to a topological charge effect analogous to the optical vortex [12–13] that eliminates the Poisson spot phenomenon found for the scalar case.
We first consider the case B=0. The vector wave propagation depends on neither the phase retardance nor the angular orientation of the polarization. A representative result of this case is shown in Fig. 2. The initial field amplitude was chosen to be identical to the homogeneous polarization result. The intensity remains symmetric and the field diffracts during propagation. The maximum amplitude of the wave show on the right in Fig. 2 is monotonically falling.
Fig. 2. ector wave propagation for the case B=0 and E0=1.5. The intensity remains rotationally symmetric and the intensity monotonically falls. Other parameters are θ=0, and φ=π/4.
This case is now contrast to the case with B=6. Here the propagation phenomena are quite striking. First we consider propagation as the amplitude is changed for a fixed phase retardance of 45° and radial polarization (θ=0) in Fig. 3.The radial symmetry of the intensity is broken by the retardance phase. Two lobes appear along the x-axis and grow in strength as energy is redirected into new transverse spatial wave vector components. This is a modulation instability that is absent without diffraction. The instability is also absent when the phase retardance vanishes.
Fig. 3. Intensity versus amplitude after propagation a distance z=2 through the medium. Other parameters are: B=6 θ=0 and φ=π/4 (Movie).
The beam in Fig. 3 collapses into two separate beams at high amplitudes and eventually undergoes the same collapse due to self-focusing as the scalar case. Figure 4 shows the beam propagation as a function of z. The amplitude of E0=2 is chosen.
Fig. 4. Evolution of the initial beam on the axis for the following parameter values: B=6, E0=2, θ=0, φ=π/4 (Movie).
For this case the amplitude that collapse is observed around E0=2.2, as shown in Fig. 5 where the maximum intensity normalized to the initial intensity is plotted for five initial amplitudes spanning the range (2.1, 2.5). The fields undergo an initial rise due to the modulation instability, but below a critical power the intensity increase is dominated by a subsequent diffraction dominated regime. The data indicates that the critical power lies in the range of the field amplitudes between 2.3 (dotted) and 2.4 (dashed), and it is a function of the retardance angle.
Fig. 5. The maximum intensity relative to its initial value versus z. The initial amplitudes are near the critical power for the given retardance values. The critical power lies between the amplitude values 2.3, shown by a dotted line, and 2.4, shown by a dashed line. The parameters are B=6, θ=0, and φ=π/4.
The phase retardance angle determines whether the modulation instability is observable. The modulation gain is small for values of φ close to zero; for most values the modulation instability is dominated by two initial peaks form on the circle where the intensity is maximum; they are angularly diametrically opposed. The angle they appear at depends on the value of θ. At φ=π/2 four equal peaks angularly spaced by π/2 are formed. The animation in Fig. 6 demonstrates the modulation instability profile’s dependence on the retardance angle for an initial amplitude of 2. The propagation distance is z=2.
Fig. 6. Intensity profile as a function of the phase retardance angle φ for B=6, E0=2 and θ=0. The propagation distance is z=2 (Movie).
The polarization state is characterized through the Stokes vector, which varies point by point for these beams. The third component of the Stokes vector is defined by
It determines the ellipticity of the beam and is the one component we examine here. Figure 7 shows the positions of maximum ellipticity on axes rotated 45° from the Cartesian coordinates, i.e. they lie between the intensity peaks; in other words the modulation instability redistributes the power flow to the regions where the field is linearly polarized.
The orientation of the peaks depends on the orientation on the initial cylindrical polarization, as Fig. 8 illustrates. The peaks rotate around the axis and are proportional to angle θ. For θ=0 the peaks lie on the x axis and for θ=π/2 the peaks lie on the y axis.
Fig. 7. The component, S3 , of the Stokes vector defined in Eq. (7) after propagating to z=2; the change of S3 with retardation angle is shown. The parameter values are: B=6, E0=2 and θ=0 (Movie).
Fig. 8. Intensity profile as a function of the cylindrical orientation angle θ for B=6, E0=2 φ=π/4. The propagation distance is z=2 (Movie).
Observing the modulation instability described here requires a simple experimental setup. Forming the initial cylindrical vector polarization is described in the literature, such as Refs.[1–2] and the cylindrical vector beam passes through a wave plate before it propagates into the nonlinear medium. For example, the material carbon disulfide (CS2) has a molecular rotational nonlinearity (B=6A) with a Kerr coefficient n2=3.2x10-14 cm2/W. The Kerr coefficient is related to the A and B coefficients by n2=4π2(A+B/2)/ c, where n0 is the linear refractive index of CS2. The critical power for self-focusing in liquid CS2 is about 27 KW. In our scaled units with the 50 micron beam waist and 1 micron wavelength, the diffraction length is zd=3.14 cm. The corresponding intensity at the critical power is 344 MW/cm2; this is a relatively low intensity and corresponds to a nonlinear index change of 1.1x10-5. We observe the modulation instability at powers that are smaller than the critical value. The scaled peak intensity 0.02 in Fig. 2 corresponds to a physical intensity of 8.4 MW/cm2.
4. Conclusion
The cylindrical vector generalization of a vortex optical soliton exhibits interesting pattern formation through an instability responsible for the axial breakup of the initial waveform. When the B coefficient vanishes, the modulation instability is not observed. The collapse of the wave proceeds uniformly. For nonzero B the modulation instability is observed only when the initial field has a phase retardance, i.e. S 3≠0; a wave plate is imposed on the initial field.
Changing the orientation of the cylindrical polarization from radial (θ=0) to tangential (θ=π/2) rotates the angle of the intensity peaks by the same angle. The peaks appear at initially zero phase retardance points. As the phase-plate retardance is increased, the initial instability creates two peaks of equal height, which are positioned opposite to one another on a circle, at the radius of the intensity maximum.
As the initial fluctuation saturates a second pair of peaks were observed and grew in a position that was rotated by π/2 from the initial two peaks. For a phase-plate retardance of π/2 four equal height peaks grow from the initial intensity profile. For higher intensities the profiles undergo self-focusing collapse at an intensity that is higher than for the scalar case.
While the beam propagation described only through a Kerr nonlinearity still collapses, the threshold for collapse occurs at a higher total power than for the homogeneous polarized field. The collapse also depends on the input polarization state. The modulation instability should be observable in materials with a molecular orientational Kerr nonlinearity, such as CS2 and the effects reported here would be observable in the MW/cm2 intensity range.
The exploration of further nonlinear optical properties of inhomogeneously polarized vector waves wills undoubted yield further surprises. These results could impact applications in areas already fruitful for cylindrical waves, such as optical traps, super-resolution spectroscopy, and controlled laser damage in optical materials. The cylindrically polarized beam with φ=0 collapses uniformly above the critical power and this could be useful for optical traps that would confine the specimens more tightly and the beam would taper its cross-sectional area. Optical damage with cylindrical beams in materials could controllably write two or four damage spots to design coupled optical waveguiding and the longitudinal intensity component of a collapsing wave could be defined within a smaller area yielding greater resolution.
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