Polarization of the third harmonic emission from the filamentation of femtosecond cylindrical vector beams

Dongwei Li; Dongwei Li; Dongwei Li; Zuoqiang Hao; Zuoqiang Hao; Zuoqiang Hao; Zuoqiang Hao; Jianghao Li; Jianghao Li; Jianghao Li; Yangjian Cai; Yangjian Cai; Yangjian Cai; Yangjian Cai; Lanzhi Zhang; Lanzhi Zhang; Lanzhi Zhang; Lanzhi Zhang

doi:10.1364/OE.507793

1. Introduction

The intense third harmonic (TH) with high conversion efficiency has attracted significant interest due to its potential applications in various fields, including fluorescence microscopy [1], atmospheric remote sensing of pollutants [2], high-resolution biological imaging [3], and microwave channeling [4]. Previous studies have focused on enhancing the intensity of the converted TH by improving the conversion efficiency between the infrared (IR) femtosecond laser pulse and TH using techniques such as two-colored filaments [5,6], thin fiber [6], pressure gradient [7], and interacted plasma [8]. Besides, the spatial and angular distribution, as well as the generation mechanism of the TH, has been investigated [9]. However, most of these studies only focus on Gaussian-shaped TH generation using femtosecond filamentation, while the generation of structured TH has seldom been investigated. With the promising advancements in optical manipulation [10], super-resolution imaging [11], sharper focusing [12], quantum optics [13], and spin-orbit photonics [14], there is a growing interest in vector optical fields with spatially varying polarization distributions. A theoretical model was developed and discussed in [15] to predict the generation of vortex TH through the co-propagation of two-color vortex filamentation. Subsequently, an experimental demonstration of TH vortex beam generation was conducted [16]. The experiment confirmed the preservation of the orbital angular momentum in the conical TH, however, the polarization distribution of the generated TH has not yet been studied and analyzed.

The generation of optical vector fields in deep ultraviolet range has been a challenging issue due to the phase-matching condition that generally restricts nonlinear interaction to a specific polarization state. Consequently, the conversion from IR into the UV wavelengths becomes more difficult. Recently, vector TH with a visible light wavelength of 520 nm has been generated using an IR laser operating at 1560 nm and a quasi-periodically poled crystal [17]. However, nonlinear crystals require complex designs to accommodate different initial laser conditions, such as wavelengths and intensity distributions, in order to match the complex phase conditions of the crystals. Moreover, the pulse inevitably undergoes broadening within the crystal due to the group velocity dispersion, which makes the generation of femtosecond-level vector TH more challenging.

In our recent study, we successfully generated an intense vector supercontinuum radiation through femtosecond filamentation. The results show that the vector supercontinuum maintains almost the same polarization distribution as that of the pump pulses [18]. Based on this observation, it can be inferred that the vector TH could also be generated using the same method. In this work, we present a proof-of-concept demonstration of femtosecond vector TH radiation at a center wavelength of 262 nm, generated from a fundamental wave (FW) with a wavelength of 790 nm. We explore the spatial polarization characteristics of the vector TH for different input laser energies by analyzing the Stokes parameters. The result demonstrates a notable preservation of the vector polarization distribution, exhibiting a strong similarity to that of the FW. However, we observe a decrease in the matching ratio between the measured azimuthal angles and corresponding theoretical values as the input laser energy increases.

2. Experimental setup

Figure 1(a) illustrates the experimental setup used in this study. An amplified Ti:sapphire femtosecond laser system with a central wavelength of 790 nm, pulse duration of 50 fs, and repetition rate of 1 kHz is used as the laser source. The output Gaussian beam has a horizontal polarization and passes through a q-plate to generate different vector beams, such as azimuthally, radially, cylindrical, and 2-order polarized beams. The initial horizontal polarization of the FW beam is quantified by calculating its degree of polarization, which is found to be 0.99. The high value indicates that the initial polarization of FW is perfectly linear. After passing through the q-plate and the lens, the degree of polarization of the vector beam has a slight decrease to 0.91 [18]. The decrease may be induced by optical elements, such as the lens, polarizer, and q-plate [19]. The q-plate, made of nematic liquid crystals, is a typical optical element for the generation of vector beams. Through precise modulation of liquid crystal molecular alignment, a localized half-wave is achieved at each point within the q-plate, which can produce a vector beam when linearly polarized light is incident [20]. Subsequently, the vector beam is focused by a lens with a focal length of 300 mm to induce air filamentation. A charge-coupled device (CCD) is used to capture the far-field image of the TH and the side luminescence image of the filamentation, respectively. Four high-reflection mirrors (M1-M4) are used to filter out the FW. Several suitable filters are placed before the CCD camera to record the intensity profile accurately, where the QB21 and bandwidth filters (265 nm ± 20 nm) are used to minimize the influence of the FW. A Stokes measurement system consisting of a quarter wavelength plate (QWP) and an ultra-broadband wire-grid polarizer is used to obtain the Stokes parameters. The TH is then directed into an integrating sphere connected to a spectrometer by using a flexible mirror of M5. Additionally, the TH output power is measured using a power meter, which is placed in the same position as the integrating sphere.

Fig. 1. (a) Schematic experimental setup (EA: energy attenuator; q-plate: QP; L1, L2, and L3: lens; M1-M5: high reflection mirrors of 266 nm; IS: integrating sphere; QWP: quarter wavelength plate; WG: ultra-broadband wire-grid polarizer; SM: spectrometer; F: filters). (b) The polarization and (c) azimuthal angle distribution of the TH generated with the radial, azimuthal, cylindrical, and 2-order vector optical fields, respectively. Input laser energy is 300 µJ.

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3. Results and discussion

To evaluate the polarization distribution of the generated TH, we measure the Stokes parameters [21] which can be obtained by measuring the intensity of the optical field (I_0°, I_45° I_90°, and I_135°) using a wire-grid polarizer with angles set at 0°, 45°, 90°, and 135°, respectively. The Stokes components of the optical field for radially polarized FW and TH vector field can be expressed as

(1)$$\left\{ {\begin{array}{l} {{S_0} = {I_{{0^\circ }}} + {I_{{{90}^\circ }}}}\\ {{S_1} = {I_{{0^\circ }}} - {I_{{{90}^\circ }}}}\\ {{S_2} = {I_{{{135}^\circ }}} - {I_{{{45}^\circ }}}}\\ {{S_3} = {I_{{0^\circ },{{135}^\circ }}} - {I_{{0^\circ },{{45}^\circ }}}} \end{array}} \right.$$

where, I_{0°, 135°} and I_{0°, 45°} represent the intensity of the optical field when employing an additional quarter wave plate set to the horizontal direction. In this case, the azimuthal angles and ellipticity of the generated TH optical field can be calculated by

(2)$$\psi = \frac{1}{2}{tan ^{ - 1}}\left( {\frac{{{S_2}}}{{{S_1}}}} \right)$$

(3)$$\varepsilon = \textrm{}\frac{1}{2}{sin ^{ - 1}}\textrm{}\left( {\frac{{{S_3}}}{{{S_0}}}} \right)$$

By using the Stokes method, the polarization and azimuthal angle distributions of the TH can be determined. Figure 1(b) shows the calculated polarization distributions of the FW and the corresponding TH fields under different polarization conditions. The azimuthal angle distribution of the TH generated with the radial, azimuthal, cylindrical, and 2-order vector optical fields is also shown in Fig. 1(c). The azimuthal angle distributions of the TH optical fields induced by different vector polarizations are consistent with those reported previously [22], Besides, the polarization distributions exhibit some distortions due to the influence of filamentation. As observed in Fig. 1(b), the TH optical field generated by filamentation presents a multi-ring intensity distribution, and the elliptical polarization distribution appears on the localized region of TH vector optical field. Moreover, we can also observe that the polarization direction has a slight change as compared to the theoretical vector beam. To clearly show the variation of polarization direction in the TH optical field, the polarization azimuthal angle distributions are plotted in Fig. 1(c), where some oscillating and periodic patterns can be observed in the TH optical field.

The TH spectra generated from laser beams with different polarization distributions and input laser energies are plotted in Fig. 2(a). The TH spectra from the horizontally polarized Gaussian beam is also included for comparison. The central wavelength of the TH spectral is 262 nm, falling within the deep UV region. For the azimuthally and radially polarized beams, the TH spectra are almost identical with pump energy of 300 µJ or 500 µJ. However, the spectral intensity of these beams is considerably lower than that of Gaussian beam with horizontal polarization. It can also be seen clearly that the TH spectrum becomes broader as the pump energy increases from 300 µJ to 500 µJ, although it is still narrower than that of horizontally polarized Gaussian beam. Here, we focus on the case of radially polarized beams, due to their identical spectra compared to the case of azimuthally polarized beams. The TH spectra from radially polarized beams as a function of input laser energy are plotted in Fig. 2(b). The TH spectral signal becomes observable when the pump laser energy reaches 140 µJ. As the laser energy is further increased, the TH spectrum experiences broadening, accompanied by a gradual increase in intensity, which should be mainly due to the cross-phase modulation between the FW and TH pulses [2]. In addition, the center wavelength of the TH remains constant at 262 nm. We further capture the side luminescence images of filamentation of laser beams with typical laser energies, which are shown in the inset of Fig. 2(c). As can be seen, the fluorescence intensity increases significantly with the pump laser energy increases. The detailed evolution of the on-axis intensity of filamentation along the propagation distance is plotted in Fig. 2(c). Both the length and intensity of the filamentation are increased with the increase of incident laser energy. Thus, it can be deduced that the change in the TH spectrum is mainly due to the enhancement of filamentation. Furthermore, the TH conversion efficiencies for Gaussian and radially polarized beams are calculated based on measured powers of the TH beams using the power meter (Fig. 2(d)). For the case of the Gaussian beam, it can be seen that the conversion efficiency increases gradually to 0.023% as the laser energy is increased to 300 µJ. The conversion efficiency remains almost unchanged as the laser energy is increased from 300 µJ to 450 µJ. However, beyond 450 µJ, the conversion efficiency starts to decrease. It has been reported that the conversion efficiency of TH emission is proportional to the product of the laser intensity and the effective interaction length [23]. In the initial stage, when the laser energies are relatively low, both the laser intensity and interaction length increase gradually as the input laser energy is increased (see Fig. 2(c)), resulting in a continuous increase in the conversion efficiency. However, when the input laser energy is high enough, despite the further extension of the effective interaction length, the laser intensity inside filamentation becomes clamped. The clamped intensity plays a crucial role in the generation of TH. Zhu et al. demonstrated that the TH generation is primarily governed by the intensity of the FW beam rather than its energy [24]. Consequently, in the filamentation where the laser intensity is clamped, the intensity of the TH radiation becomes restricted. Therefore, as the FW energy is further increased, the conversion efficiency of TH emission decreases correspondingly. As a consequence, further enhancement of the conversion efficiency is hindered. For the case of radially polarized beam, the maximum conversion efficiency is observed to be 0.015% when the input laser energy reaches 500 µJ, which is noticeably lower than that of the Gaussian beam. This difference can be attributed to the significantly shorter filamentation length generated by radially polarized beam compared to that by the Gaussian beam [18].

Fig. 2. (a) Spectra of the TH versus input laser energy for different laser beams. (b) The spectra of TH induced by radially polarized beams with different input laser energies. (c) On-axis maximum intensity of the filamentation of horizontally polarized Gaussian laser beams with different laser energies; (d) Measured conversion efficiency of the TH generated by radially and horizontally polarized laser beams.

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To investigate the polarization characteristics of the TH beams, Stokes parameters are measured and displayed in Fig. 3. For the FW beam (the first column of Fig. 3), the initial optical field S₀ shows a typical intensity distribution of radially polarized beam with a polarization singularity in the center, as shown in Fig. 3(a1). The Stokes parameters S₁ and S₂ of the vector beam exhibit typical four-petal shapes. We can also see from Fig. 3(a4) that there is almost no circular polarization component (S₃≈0). For the TH optical field, a notable feature is the emergence of a multi-ring structure, as can be seen from the images of parameter S₀. When the input laser energy is increased to 500 µJ (Fig. 3(e1)), these multiple rings become superimposed. The generated multi-ring structure on the TH optical field is similar to the observation in the vortex beam [16]. The normalized Stokes parameters S₁ and S₂ of the TH optical field maintain the similar four-petal pattern. However, as shown in Figs. 3(c4)–3(e4), the Stokes parameter S₃ does not equal zero, which indicate the existence of circular polarization component. Hence, the TH beams exhibit a hybridized polarization distribution characterized by a multiple rings pattern, particularly noticeable when the laser energy reaches a higher value.

Fig. 3. The images of normalized Stokes polarization parameters (a1)-(e1) S₀, (a2)-(e2) S₁/S₀, (a3)-(e3) S₂/S₀, S₃/S₀ (a4)-(e4) of radially polarized beam with false color for the FW and TH lights with different input laser energies from 200 µJ to 500 µJ.

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We further investigate the distribution, ellipticity and angle of the polarization for the FW and TH beams with different input laser energies. The first column of Fig. 4 presents two-dimensional polarization distributions of FW and TH fields with different input laser energies. These distributions are obtained by calculating the CCD data with a grid of 10 × 10 pixels. It can be seen that the FW has a perfect radial polarization distribution, and the generated TH also maintains the radially polarized distribution, when the input laser energy is relatively low. As the input laser energy is increased to 300 µJ, elliptical polarizations begin to emerge at the outer ring of the TH field, while the overall optical field predominantly preserves the radially polarized distribution. The second column of Fig. 4 shows two-dimensional polarization angles for these beams. We can observe that the polarization distribution of the TH shows good agreement with that of the radially polarized beam [22]. When compared to the FW case, the polarization direction of the TH undergoes some changes as the input laser energy is increased from 200 µJ to 500 µJ. Furthermore, As the laser pulse energy increases, the filamentation becomes denser, which will lead to a stronger birefringence effect [25], resulting in an increase in the ellipticity of the TH optical field, as can be seen from the third column of Fig. 4.

Fig. 4. The first column ((a1)-(d1)): The polarization distributions of (a1) FW and TH fields with different input laser energies of (b1)-(b3) 200 µJ, (c1)-(c3) 300 µJ, and (d1)-(d3) 500 µJ, respectively. The second and third columns are corresponding directions and ellipticity of polarization for the FW and TH beams, respectively.

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To quantify the polarization characteristics of the TH field, we evaluate the variation of ellipticity and direction of the polarization in the ring regions of these FW and TH beams. The ring regions are marked by black dashed circles on the TH optical fields shown in Figs. 4(a2)–4(d2). The average and standard deviation of the ellipticity are defined as follows:

(4)$$\bar{\varepsilon } = \frac{1}{{360}}\mathop \sum \nolimits_{i = 0}^{359} {\varepsilon _i}$$

(5)$${\varepsilon _\sigma } = \sqrt {\frac{1}{{360}}\mathop \sum \nolimits_{i = 0}^{359} {{({{\varepsilon_i} - \bar{\varepsilon }} )}^2}} , $$

where, i and ${\varepsilon _i}$ are angles and ellipticity of the polarization, respectively. The results are plotted in Fig. 5(a). The polarization ellipticity of the FW is calculated to be 0.049 ± 0.03, with a maximum deviation of 0.11 (the ellipticity of an ideal radially polarized beam is 0). Furthermore, we evaluate the ellipticity of the TH beams at different input laser energies, and obtain 0.034 ± 0.027, 0.079 ± 0.012, and 0.090 ± 0.11, for the three input energies. The corresponded maximum deviation relative to the theoretical radially beam is 0.10, 0.30, 0.44, respectively. The results demonstrate that the ellipticity increases with the increase of the input laser energy. One main reason is that there exits perturbation in the initial FW laser polarization, i.e., incident laser is not perfectly linearly polarized light. The linear polarization with this perturbation appears to be unstable during filamentation processes, and the deviation will increase especially when the input laser power is relatively high [26].

Fig. 5. (a) The ellipticity, (b) azimuthal angles and (c) the matching ratios of azimuthal angle for FW and TH with different input laser energies, respectively.

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Furthermore, the polarization angles of FW and TH optical fields are also analyzed in the same ring regions, as shown in Fig. 5(b). we can also see that the polarization direction of FW beam has a typical change along the circle, while the polarizations of TH optical fields have some deviations. To evaluate the change of polarizations directions, we define the matching ratio between the experimental and theoretical polarizations angles, which can be written as

(6)$$R({\%} )= 100 - \frac{1}{{360}}\mathop \sum \nolimits_{i = 0}^{359} \frac{{|{\psi_i^{ex} - \psi_i^{th}} |}}{{90}} \times 100, $$

where, $\psi _i^{ex}$ and $\psi _i^{th}$ are the experimental and theoretical azimuthal angles, and $\psi _i^{th}(\varphi )= P\varphi + {\varphi _0}$, where $\varphi $ and ${\varphi _0}$ are the tangential and initial angle, respectively. From the Eq. (6), the matching ratio R of FW is calculated ∼91.1%. For the cases of three typical input laser energies, the R of TH is 90.2%, 86.3%, 81.9% respectively. Moreover, the matching ratio R has also been analyzed for the whole beam cross section by calculating at different radii. The ratios as a function of radius for the FW and TH with different laser energies is plotted in Fig. 5(c). We can see that for the FW beam, the matching ratio R is better than 90% within the whole cross section of the beams except the center region. The mismatching of azimuthal angles is caused by the uncertainty of polarization near polarization singularities. For the TH beams, the ratios are better than ∼80% except the beam center region, when the laser energy is 200 µJ and 300 µJ. As the laser energy is increased to 500 µJ, the mismatching region increases obviously. In general, the vector polarization distribution can be preserved by the TH. However, the change of the polarization is attributed to the perturbation of intense filamentation where the initial polarization imperfection is amplified by the phase modulation of two perpendicular components during the nonlinear propagation. As a result, the azimuthal polarization angles become uniformly [27,28]. Notably, as observed in Figs. 4(a2)–4(d2), as well as Fig. 5(a) and 5(b), the polarization of TH has significant oscillating or periodic structures. The distortion of the polarization in TH field mainly arises from the birefringence effect occurring in the filamentation. Consequently, this birefringence effect induces variations in both the polarization angle and the ellipticity of the TH field. [25,29–31]. The Further studies are still needed to reveal the underlying mechanism.

4. Conclusion

In conclusion, we experimentally generate a TH vector optical field at the center wavelength of 262 nm through femtosecond filamentation. By analyzing the measured Stokes parameters of the TH, we demonstrate that the TH preserves predominantly the vector polarization distribution of the FW. In addition, the ellipticity of the vector TH increases, while the matching ratio between the experimental and theoretical results decreases, with increasing input laser energy. Our work provides an effective and straightforward method to generate femtosecond TH vector beams in the UV range, which facilitates potential applications such as micro/nanofabrication and super-resolution microscopy.

Funding

National Natural Science Foundation of China (12204282, 12074228, 12304325, 12004221, 12192254, 12104267, 11974218); National Key Research and Development Program of China (2019YFA0705000, 2022YFA1404800); Local Science and Technology Development Project of the Central Government (YDZX20203700001766); Natural Science Foundation of Shandong Province (ZR2021MA023); Taishan Scholar Project of Shandong Province (tsqn201812043, tsqnz20221132); Innovation Group of Jinan (2020GXRC039).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Polarization of the third harmonic emission from the filamentation of femtosecond cylindrical vector beams

Abstract

1. Introduction

2. Experimental setup

3. Results and discussion

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Equations (6)

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