## Abstract

We investigate the far-field vectorial self-diffraction behavior of a cylindrical vector field passing though an optically thin Kerr medium. Theoretically, we obtain the analytical expression of the focal field of the cylindrical vector field with arbitrary integer topological charge based on the Fourier transform under the weak-focusing condition. Considering the additional nonlinear phase shift photoinduced by a self-focusing medium, we simulate the far-field vectorial self-diffraction patterns of the cylindrical vector field using the Huygens-Fresnel diffraction integral method. Experimentally, we observe the vectorial self-diffraction rings of the femtosecond-pulsed radially polarized field and high-order cylindrical vector field in carbon disulfide, which is in good agreement with the theoretical simulations. Our results benefit the understanding of the related spatial self-phase modulation effects of the vector light fields, such as spatial solitons, self-trapping, and self-guided propagation.

© 2011 OSA

## 1. Introduction

Interaction of intense laser with matter changes the refractive index of a material and could result in a lot of spatial self-phase modulation effects, such as spatial solitons, self-trapping, and self-guided propagation. Among these effects, the self-diffraction effect has attracted extensively interest since Callen *et al*. [1] observed the far-field annular intensity distribution of a Gaussian laser beam passing through carbon disulfide in 1967. Afterwards, the far-field annular self-diffraction patterns were found in many materials, such as nematic liquid crystals [2], azo-doped polymer film [3], absorbing self-defocusing media [4], carbon nanotube solutions [5], hybrid materials [6], and photopolymer [7]. At the same time, the formation and evolution of the self-diffraction patterns were extensively investigated [8–11]. Besides, the underlying mechanisms of the novel self-diffraction phenomena, including photothermal effect [1], nonlocal optical nonlinearities [12], and thermal nonlinearity with gravitational effect [5], have been exploited. Up to now, most of investigations were devoted to exploring self-diffraction behaviors of linearly polarized Gaussian beams. Correspondingly, the scalar self-diffraction phenomena were widely studied.

Recently, cylindrical vector fields, which have the inhomogeneous distribution of states of polarization in the cross-section of field, have become a subject of rapidly growing interest, due to the unique features compared with homogeneously polarized fields and novel applications in various realms [13]. Under the vector field excitation, many novel nonlinear optical effects, such as second-harmonic generation [14], third-harmonic generation [15], and self-focusing dynamics [16], have been studied. Consequently, one may expect the appearance of novel effects induced by the vector light fields.

In the present article, for the first time to our knowledge, we investigate the far-field vectorial self-diffraction patterns of a cylindrical vector field passing though an optically thin Kerr medium. We theoretically obtain the focal field of the cylindrical vector field with arbitrary integer topological charge and simulate the far-field vectorial self-diffraction patterns induced by a self-focusing medium. We experimentally observe the vectorial self-diffraction rings of the femtosecond-pulsed cylindrical vector fields in carbon disulfide. The experimental observations are in good agreement with the theoretical simulations.

## 2. Theory

In general, a cylindrical vector field can be written as [13]

*ρ*and

*φ*are the polar radius and azimuthal angle in the polar coordinate system, respectively. Here

**P**(

*φ*) is the unit vector describing the distribution of the states of polarization of the vector field,

*m*is the topological charge, and

*φ*

_{0}is the initial phase [17].

**ê**

_{x}and

**ê**

_{y}are the unit vectors in the Cartesian coordinate system, respectively. Interestingly, two extreme cases of vector fields describing by Eq. (1) are the radially and azimuthally polarized vector fields when

*m*= 1 with

*φ*

_{0}= 0 and

*π*/2, respectively. In the case of

*m*= 0, Eq. (1) describes the horizontal and vertical linearly-polarized fields, for

*φ*

_{0}= 0 and

*φ*

_{0}=

*π*/2, respectively.

*A*(

*ρ*) stands for the amplitude distribution in the cross-section of the cylindrical vector field. Under the uniform-intensity illumination, we have

*A*(

*ρ*) =

*A*

_{0}within the region of 0 ≤

*ρ*≤

*a*, where

*a*is the radius of the cylindrical vector field.

The focused field distribution of the cylindrical vector field by a lens with a focal length of *f* under the weak-focusing condition can be written, according to the Fourier transform, as follows

*k*= 2

*π*/

*λ*is the wave vector and

*λ*is the wavelength of the used laser in free space. A Cartesian system (x′,y′) and a corresponding polar coordinate system (

*r,*

*ψ*) are attached in the rear focal plane of the lens. After integrating Eq. (2) over

*φ*for an integer

*m*≥ 0, we yield the focused field distribution

*ω*

_{0}=

*λf*/2

*a*is the beam waist,

*E*

_{0}denotes the electric field amplitude at the focus,

_{1}

*F*

_{2}[·] is the generalized hypergeometric function, and

*A*is a normalized constant obtained by the condition of (|

_{m}*E*|

_{f}^{2}/|

*E*

_{0}|

^{2})

_{max}= 1. The first five coefficients of

*A*are listed in Table 1. Setting

_{m}*m*= 0 in Eq. (3), the field distribution has the well-known Airy spot profile that describes the focused field of the top-hat beam [18].

Interestingly, the focused field profile of the cylindrical vector field is the so-called doughnut light field with the central dark spot and a single outer bright ring, as shown in Fig. 1. This doughnut field has found some exciting applications such as optical trapping [19] and manipulating nanoobjects [20]. In addition, the radius of the doughnut field increases as the topological charge *m* of the cylindrical vector field increases.

For the vector field with a temporal Gaussian pulse profile, the peak intensity at the focus can be written as
${I}_{0m}=2\sqrt{\pi \text{ln}2\varepsilon}/{\omega}_{0}^{2}{A}_{m}^{2}{\tau}_{F}$, where *ɛ* is the incident energy and *τ _{F}* is the full width at half maximum of the pulse duration.

As described by Eq. (3), the focused vector field belongs to a kind of local linearly polarized vector field. For the sake of simplicity, we consider only that an optically thin isotropic Kerr sample (its thickness is much less than the Rayleigh length of the light field) is located at the focal plane. Correspondingly, the field **E*** _{e}*(

*r,*

*ψ*) at the exit plane of the sample can be given as

*n*

_{2}is the third-order nonlinear refraction index, and

*L*is the sample thickness. We now define the peak nonlinear refractive phase shift at the focus, as ΔΦ

*=*

_{m}*kn*

_{2}

*I*

_{0m}

*L*.

Based on the Huygens-Fresnel diffraction integral method, we obtain the field distribution in the far-field observation plane attached to a Cartesian system (x″,y″) and a corresponding polar coordinate system (*r _{a},ϕ*), which has a distance

*d*from the exit plane of the sample

*J*(·) is the Bessel function of the first kind of

_{m}*m*th order. In the above theoretical analysis, we only consider the optical field under the steady-state condition. The temporal profile of the laser pluses have been omitted. In fact, we can easily extend the steady-state results to transient effects induced by a pulse train by using the time-averaging approximation. For the cylindrical vector field with a temporally Gaussian pulses, we yield the average nonlinear refractive phase shift at the focus as $\u3008\mathrm{\Delta}{\mathrm{\Phi}}_{m}\u3009=\mathrm{\Delta}{\mathrm{\Phi}}_{m}/\sqrt{2}$.

To investigate the characteristics of the far-field intensity of the vector field when the nonlinear sample is located at the rear focal plane of the lens, we take the parameters as *φ*_{0} = 0, *λ* = 804 nm, *ω*_{0} = 65 *μ*m, and *d* = 180 mm. The top row of Fig. 2 presents the numerical simulations of the far-field patterns of the vector fields for different topological charges *m* at a fixed value ΔΦ* _{m}* =

*π*. For the sake of comparison, the far-field pattern of the scalar linearly-polarized top-hat beam (i.e., the case of

*m*= 0) is also shown in the first column of Fig. 2. Compared with the single ring structures in the far-field patterns of the vector fields in the absence of the nonlinearity, as shown in Fig. 1, the far-field patterns induced the nonlinearity exhibit the multiple concentric ring structures, originating from the refractive-index changes self-induced by the nonlinearity. To identify the polarization features of the far-field multiple ring patterns induced by the nonlinearity, a vertical polarizer is used, and the corresponding patterns are displayed in the middle row of Fig. 2. To more clearly show the the properties of the far-field multiple ring patterns, the bottom row also shows the corresponding intensity profiles along the horizontal center lines of the intensity patterns in the top (or middle) row. For the case of the scalar top-hat beam, as shown in the first column, we find the complete extinction. For the case of vector fields, the far-field intensity patterns induced by the nonlinearity behind the vertical polarizer appear the radial extinction and exhibit the the radially-modulated fan-shaped structures. Moreover, the number of the extinction directions is the same as the topological charge

*m*. Evidently, such a extinction nature is the same input vector fields as reported in Ref. 21. The results imply that the nonlinearity has no influence on the distributions of states of polarization for the vector fields whereas influences on the far-field intensity distributions. The phenomena can be understood as follows. The focused vector field induces a change in the refractive index of the sample by Δ

*n*(

*r*) =

*γ*

*I*(

*r*). As a result, the different locations of the field cross-section in the radial direction experience the different nonlinear phase shifts of ΔΦ(

*r*) =

*kn*

_{2}

*I*(

*r*)

*L*, resulting in the self-phase modulation. If the phase difference between the two locations in the far-field plane is ΔΦ(

*r*) =

*h*

*π*,

*h*being an even or odd integer, the constructive or destructive interference takes place, respectively, giving rise to the appearance of self-diffraction patterns with the multiple concentric ring structures.

It is interesting to investigate the dependence of the far-field intensity distributions of the vector fields passing through a thin Kerr medium on the nonlinear phase shift. As an example, Fig. 3 illustrates the far-field patterns of the vector field with *m* = 2 and *φ*_{0} = 0 at different values of ΔΦ_{2}, under the conditions of *λ* = 804 nm, *ω*_{0} = 65 *μ*m, and *d* = 180 mm. In principle, the change of the nonlinear phase shift can originate from the different nonlinearity (i.e. different nonlinear medium) for a given laser intensity or the different laser intensity for a given nonlinear medium and both. One can see that there occurs always a dark spot in the central zones of the far-field patterns and its size is almost independent of the nonlinear phase shift ΔΦ_{2}. As ΔΦ_{2} increases, the number of bright diffraction rings around the dark spot increases and the more light energy is diffracted into the outer rings.

## 3. Experiment

It is completely different from the illumination with homogeneously polarized field that a cylindrical vector field results in a vectorial self-diffraction effect, as described in Eq. (6) and illustrated in Figs. 2 and 3. In what follows, we experimentally verify this effect by performing the femtosecond-pulsed cylindrical vector fields in carbon disulfide. The experimental arrangement is illustrated in Fig. 4. The light source used in our experiments is a Ti:sapphire laser (Coherent Inc.). Based on the principle of the wavefront reconstruction and by using the configuration in Refs. [17, 21], we generated the femtosecond-pulsed cylindrical vector fields for different topological charges *m* with the fixed pulse energy of *ɛ* = 1.3 *μ*J, the pulse duration of *τ _{F}* ≃ 70 fs, and the repetition rate of 1 kHz at the central wavelength of

*λ*= 804 nm. In addition, it should be emphasized that the generated femtosecond-pulsed cylindrical vector field has a top-hat spatial distribution with a diameter of 2

*a*= 3.7 mm and a near-Gaussian temporal profile. The cylindrical vector field was focused by a lens with a focal length of

*f*= 300 mm, producing the beam waist of

*ω*

_{0}= 65

*μ*m at the focus (the Rayleigh length

*z*

_{0}= 16.5 mm). As the nonlinear medium, the carbon disulfide was contained in 5 mm thick quartz cell at room temperature and standard atmosphere. The wall thickness of the quartz cell was 1 mm. The sample was located at the focal plane. A detector (Beamview, Coherent Inc.) was placed at the observation plane with a distance of

*d*= 180 mm from the sample to probe the transmitted intensity distribution. Considering the interface losses of the light energy, we determine the optical intensities for vector field with

*m*= 1 and 2 within the solution as

*I*

_{01}= 74 and

*I*

_{02}= 42 GW/cm

^{2}, respectively.

The far-field intensity patterns for the radially polarized field (namely, *m* = 1 and *φ*_{0} = 0) without and with the nonlinear sample at the focal plane are shown in the middle row of Fig. 5. To identify the vectorial properties of the far-field field, a horizontal polarizer was used. Correspondingly, the results are also displayed in Fig. 5. Most importantly, as shown in Fig. 5, we experimentally detected the vectorial self-diffraction ring, for the first time to our knowledge. For a high-order cylindrical vector field with *m* = 2 and *φ*_{0} = 0, the corresponding experimental results are displayed in the middle row of Fig. 6. It should be pointed out that the saturated signals exist in the observed experimental results due to the energy saturation of the detector.

As is well known, carbon disulfide and the quartz wall exhibit the isotropic self-focusing effect with a nonlinear refractive index of *n*_{2} = 3×10^{−6} cm^{2}/GW and *n _{q}* = 3.3×10

^{−7}cm

^{2}/GW at around 800 nm under the femtosecond pulse excitation [22, 23], respectively. Under our experimental condition, the average nonlinear refractive phase shifts for the cylindrical vector fields with

*m*= 1 and 2 arising from the carbon disulfide were estimated to be 〈ΔΦ〉 = 1.93

*π*and 1.10

*π*, respectively. The phase shifts originating from the quartz walls were so small, ∼ 4%, that it should not induce a significant change of the far-field intensity. Hence, the contribution of the quartz walls was not taken into consideration in the analysis. Using Eq. (6) and the known experimental parameters, we simulate the far-field self-diffraction patterns without and with the nonlinear sample at the focal plane, as shown in the lower rows of Figs. 5 and 6. Clearly, the theoretical simulations are consistent with the experimental observations, implying that our theoretical analysis is reasonable and could gain an insight on the underlying mechanisms for the observed vectorial self-diffraction effect. It should be noted that the discrepancy between the theory and experiment in the periphery of the self-diffraction patterns is apparent. This difference is anticipated for the following reason. A monochromatic field is considered for a single defined wavelength and a unique peak intensity in the theoretical simulations; whereas a femtosecond pulse train at a central wavelength with a spectral bandwidth of tens-of-nanometers is used in the measurements. Furthermore, the peak intensity considerably changes over the pulse temporal envelope.

Both the theoretical and experimental results demonstrate that the far-field of the focused cylindrical vector field remains its polarized feature without the disturbance of optical nonlinearity. Besides, the far-field self-diffraction field induced by the Kerr nonlinearity also holds the cylindrical vector polarized feature and exhibits the multiple diffraction ring structures. Our results should benefit the understanding of the related spatial self-phase modulation effects of the vector light fields.

## 4. Conclusion

In summary, we have theoretically and experimentally investigated the vectorial self-diffraction effect of a cylindrical vector field passing though a self-focusing medium. We have presented the analytical focal field of the cylindrical vector field with arbitrary integer topological charge and obtained the far-field vectorial self-diffraction patterns using the Huygens-Fresnel diffraction integral method. Moreover, we have observed the femtosecond-pulsed cylindrical vector field induced the vectorial self-diffraction ring from carbon disulfide, which is in good agreement with the theoretical simulations.

## Acknowledgments

This work was supported by the National Basic Research Program of China (Grant No. 2012CB921900), the National Science Foundation of China (Grants No. 11174160 and No. 11174157), and the Program for New Century Excellent Talents in University (NCET-10-0503).

## References and links

**1. **W. R. Callen, B. G. Huth, and R. H. Pantell, “Optical patterns of thermally self-defocused light,” Appl. Phys. Lett. **11**, 103–105 (1967). [CrossRef]

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

**3. **P. F. Wu, B. Zou, X. Wu, J. Xu, X. Gong, G. Zhang, G. Tang, and W. Chen, “Biphotonic self-diffraction in azo-doped polymer film,” Appl. Phys. Lett. **70**, 1224–1226 (1997). [CrossRef]

**4. **R. G. Harrison, L. Dambly, D. J. Yu, and W. P. Lu, “A new self-diffraction pattern formation in defocusing liquid media,” Opt. Commun. **139**, 69–72 (1997). [CrossRef]

**5. **W. Ji, W. Z. Chen, S. H. Lim, J. Y. Lin, and Z. X. Guo, “Gravitation-dependent, thermally-induced self-diffraction in carbon nanotube solutions,” Opt. Express **14**, 8958–8966 (2006). [CrossRef]

**6. **M. Trejo-Durán, J. A. Andrade-Lucio, A. Martinez-Richa, R. Vera-Graziano, and V. M. Castaño, “Self-diffracting effects in hybrid materials,” Appl. Phys. Lett. **90**, 091112 (2007). [CrossRef]

**7. **A. B. Villafranca and K. Saravanamuttu, “Spontaneous and sequential transitions of a Gaussian beam into diffraction rings, single ring and circular array of filaments in a photopolymer,” Opt. Express **19**, 15560–15573 (2011). [CrossRef]

**8. **E. Santamato and Y. R. Shen, “Field-curvature effect on the diffraction ring pattern of a laser beam dressed by spatial self-phase modulation in a nematic film,” Opt. Lett. **9**, 564–566 (1984). [CrossRef]

**9. **D. J. Yu, W. P. Lu, and R. G. Harrison, “Analysis of dark spot formation in absorbing liquid media,” J. Mod. Opt. **45**, 2597–2606 (1998). [CrossRef]

**10. **L. G. Deng, K. N. He, T. Z. Zhou, and C. D. 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**, 409–415 (2005). [CrossRef]

**11. **C. M. Nascimento, M. A. R. C. Alencar, S. Chávez-Cerda, M. G. A. da Silva, M. R. Meneghetti, and J. M. Hickmann, “Experimental demonstration of novel effects on the far-field diffraction patterns of a Gaussian beam in a Kerr medium,” J. Opt. A: Pure Appl. Opt. **8**, 947–951 (2006). [CrossRef]

**12. **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**, 22067–22079 (2010). [CrossRef]

**13. **Q. W. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. **1**, 1–57 (2009). [CrossRef]

**14. **A. Ohtsu, Y. Kozawa, and S. Sato, “Calculation of second-harmonic wave pattern generated by focused cylindrical vector beams,” Appl. Phys. B **98**, 851–855 (2010). [CrossRef]

**15. **S. Y. Yang and Q. W. Zhan, “Third-harmonic generation microscopy with tightly focused radial polarization,” J. Opt. A: Pure Appl. Opt. **10**, 152103(2008). [CrossRef]

**16. **A. A. Ishaaya, L. T. Vuong, T. D. Grow, and A. L. Gaeta, “Self-focusing dynamics of polarization vortices in Kerr media,” Opt. Lett. **33**, 13–15 (2008). [CrossRef]

**17. **X. L. Wang, Y. N. Li, J. Chen, C. S. Guo, J. P. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express **18**, 10786–10795 (2010). [CrossRef]

**18. **W. Zhao and P. Palffy-Muhoray, “Z-scan technique using top-hat beams,” Appl. Phys. Lett. **63**, 1613–1615 (1993). [CrossRef]

**19. **Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express **18**, 10828–10833 (2010). [CrossRef]

**20. **T. Züchner, A. V. Failla, and A. J. Meixner, “Light microscopy with doughnut modes: a concept to detect, characterize, and manipulate individual nanoobjects,” Angew. Chem. Int. Ed. **50**, 5274–5293 (2011). [CrossRef]

**21. **X. L. Wang, J. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. **32**, 3549–3551 (2007). [CrossRef]

**22. **X. Q. Yan, X. L. Zhang, S. Shi, Z. B. Liu, and J. G. Tian, “Third-order nonlinear susceptibility tensor elements of CS_{2} at femtosecond time scale,” Opt. Express **19**, 5559–5564 (2011). [CrossRef]

**23. **B. Gu, Y. Wang, J. Wang, and W. Ji, “Femtosecond third-order optical nonlinearity of polycrystalline BiFeO_{3},” Opt. Express **17**, 10970–10975 (2009). [CrossRef]