Tight focusing properties and focal field tailoring of cylindrical vector beams generated from a linearly polarized coherent beam array

Yuqiu Zhang; Tianyue Hou; Hongxiang Chang; Tao Yu; Tao Yu; Qi Chang; Man Jiang; Pengfei Ma; Rongtao Su; Pu Zhou

doi:10.1364/OE.417038

1. Introduction

Polarization is an important vectorial feature of light. As a type of vector beams (VBs) with nonuniform polarization distributions in space domain, the so-called cylindrical vector beams (CVBs) with polarization singularities have both cylindrical symmetry polarization distributions and hollow-shaped intensity distributions [1]. In recent years, the CVBs have attracted substantial attention for their unique polarization properties and potential applications including a long-range tractor beam for airborne light-absorbing particles [2], optical communication [3], and material processing [4,5]. Particularly, the tight focusing properties of CVBs attract many researchers owing to their unique focal patterns near the focal plane [6]. For example, a radial polarized beam can achieve a stronger longitudinal component, and a smaller focus size can be obtained compared with a linearly polarized beam [7,8]. In addition, an azimuthally polarized beam can generate an annular-shaped focus [9]. On account of the unique tight focusing properties, CVBs are widely applied in numerous applications, such as high-resolution optical imaging [10,11], optical trapping [12,13], data storage [14], and microparticle optical manipulation [15–17].

Driven by the increasingly demanding applications, the generation of CVBs has been reported widely. The earliest experiment of CVBs generation was based on the calcite crystal placed in a telescope of the cavity [18]. With the development of the micromachining technology, the diffractive phase plate or polarization selective end mirror devices have been used for CVBs generation [19,20]. Despite the rapid development of various active generation methods of CVBs, the variety of the output mode is limited by the resonator. Thus, some other more flexible methods of CVBs generation, including spatial light modulators (SLMs) [21–23], q-plates [24], meta-surfaces [25,26], long-period fiber grating [27] and all-fibered [28–31], have been proposed in recent years. The SLM-based approach offers a great deal of flexibility and allows the generation of any desired spatial mode. However, this approach is limited by the efficiency and output power. On the other hand, as for the fiber-based approaches, the output power of a single fiber laser is limited by nonlinear effects and mode instability, which somewhat limits the use of CVBs in material processing and optical manipulation.

As for the enhancement of output power of CVBs, coherent beam combining (CBC) may make it possible to achieve high output power while maintaining good beam quality [32,33]. In terms of CBC of high-power amplifiers and large array, high average output power of 16 kW and combination of over 100 beams of the fiber laser have been demonstrated recently [34,35]. Up to now, the generation of structured light beams based on the CBC technology, such as vortex beam [36–40], Bessel-Gaussian beam [41], airy beam [42], and orbital angular momentum (OAM) beam array [43] has been proposed theoretically or experimentally. Particularly in 2009, Kurti et al. have demonstrated the generation of CVBs by combination of fiber laser array experimentally for the first time [44]. According to their approach, each beamlet of radial Gaussian beam array is linear polarization, and the technique has the advantages of simplicity, direct control over mode generation, and high output power. In 2012, Ma et al. put forward an new architecture based on coherent polarization beam combination (CPBC) technique to generate high purely CVBs successfully [45]. In conclusion, it can be expected to achieve high-power focal field tailoring based on the CBC technology by using SLMs to actively control the polarization and phase of the signal light.

In this paper, we have investigated the tight focusing properties of combined CVBs based on fiber laser array and realized the flexible control of focal field. Based on the Richards-Wolf vectorial diffraction method, the analytical expression of focusing field distribution of combined CVBs has been derived. And then, we compared the differences between ideal CVBs and combined CVBs. Some valuable conclusions have been drawn to obtain the desired focusing field distribution according to the analyses and simulations. Furthermore, the specific examples of focal field tailoring are presented by adjusting the initial polarization rotation of separate sub-apertures. These results could provide a valuable guidance for tailoring structured light beams in future applications.

2. Tight focusing field distributions of the combined CVBs

The schematic diagram of CVBs focused by a high numerical aperture (NA) objective lens is shown in Fig. 1. The linear polarized Gaussian array beam with radial distribution is focused by a high-NA lens, where the radius of effective exit sub-aperture is R, and the distance of the center of beamlets is 2R. The incident field distribution of N-th linear polarized Gaussian circular array can be written as

(1)$${\overrightarrow E _0}(r,\varphi ,0) = \left[ {\begin{array}{c} {{E_r}}\\ {{E_\varphi }} \end{array}} \right] = {t_m}(r,\varphi )\sum\limits_{n = 1}^N {\exp \left[ { - \frac{{{r^2} + r_0^2 + 2r{r_0}\cos (\varphi - {\varphi_n})}}{{w_0^2}}} \right]} \left[ {\begin{array}{c} {\cos ({\varphi_n} + {\varphi_0} - \varphi )}\\ {\sin ({\varphi_n} + {\varphi_0} - \varphi )} \end{array}} \right],$$

where w₀ is the waist width of the laser beam, r₀ is the radius of the array, φ_n = 2π(n-1)/N and n is integer. Here, each sub-aperture has a polarization rotation by φ₀ from its radial direction, so that φ₀=0 and π/2 represent radial polarization and azimuthal polarization, respectively.

Fig. 1. Schematic diagram for tight focusing of linearly polarized coherent beams array.

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The hard aperture truncated function can be expressed by a finite sum of complex Gaussian functions

(2)$${t_m}(r,\varphi ) = \sum\limits_{j = 1}^M {{B_j}\exp \left\{ { - \frac{{{C_j}}}{{{R^2}}}[{{r^2} + r_0^2 - 2r{r_0}\cos(\varphi - {\varphi_n})} ]} \right\}} ,$$

where B_j and C_j are the coefficients obtained by Ref. [46], and M = 16 nearly describes a hard aperture. $\delta = {w_0}/R$ is the truncation parameter.

In general, the vector diffraction theory under tight focusing of a high-NA objective lens was proposed by Richard and Wolf [47,48]. According to the theory, the electric field distribution of linear polarized Gaussian array beam with radial distribution near the focus can be expressed as

(3)$$\begin{array}{l} {\overrightarrow E ^{(S)}} = \left[ {\begin{array}{c} {E_r^{(S)}}\\ {E_\varphi^{(S)}}\\ {E_z^{(S)}} \end{array}} \right] = \frac{{ - \textrm{i}f}}{\lambda }\int_0^{{\theta _{\max }}} {d\theta \int_0^{2\pi } {{l_\textrm{0}}(\theta ,\varphi )\sqrt {\cos \theta } \sin \theta } } \\ \times {\rm{exp}} [{\textrm{i}k({z_S}\cos \theta - {r_S}\sin \theta \cos(\varphi - {\varphi_s}))} ]\left[ {\begin{array}{c} {{E_r}\cos \theta \cos ({\varphi - {\varphi_s}} )- {E_\varphi }\sin (\varphi - {\varphi_s})}\\ {{E_r}\cos \theta \sin ({\varphi - {\varphi_s}} )+ {E_\varphi }\cos (\varphi - {\varphi_s})}\\ {{E_r}\sin \theta } \end{array}} \right]\textrm{d}\varphi \end{array} , $$

where, $E_r^{(S)}$, $E_\varphi ^{(S)}$ and $E_z^{(S)}$ represent the radial, azimuthal and longitudinal components under the cylindrical coordinate system. θ denotes the focusing angle (the angle between the optical axis and the propagation direction), and ${\theta _{\max }} = {\arcsin ^{ - 1}}(NA)$ is the maximum convergence angle determined by the NA of objective lens. For an aplanatic lens, $r = f \cdot \sin \theta $ is the sine condition of the objective, where f is the focal length. In this paper, the Bessel-Gaussian beam is considered as the pupil apodization function which can be written as [48]

(4)$${l_0}(\theta ) = \exp \left[ { - {\beta^2}{{\left( {\frac{{\sin \theta }}{{\sin {\theta_{\max }}}}} \right)}^2}} \right]{J_1}\left( {2\beta \frac{{\sin \theta }}{{\sin {\theta_{\max }}}}} \right),$$

where β is the lens filling factor which is the ratio between pupil radius and beam waist, J₁ is the Bessel function of the first kind and first order. The electric field can be further simplified according to the following mathematical transformation formulas [1]

(5)$$\int_0^{2\pi } {\cos } (m\xi ){\textrm{e}^{\textrm{i}t\cos (\xi - \gamma )}}\textrm{d}\xi = 2\pi {\textrm{i}^m}{J_m}(t)\cos (m\gamma ),$$

(6)$$\int_0^{2\pi } {\sin } (m\xi ){\textrm{e}^{\textrm{i}t\cos (\xi - \gamma )}}\textrm{d}\xi = 2\pi {\textrm{i}^m}{J_m}(t)\sin (m\gamma ),$$

(7)$${J_m}( - t) = {( - 1)^m}{J_m}(t).$$

Where, J_n denotes a Bessel function of the first kind of order n. In order to simplify Eq. (3) to the Bessel function relation expression of Eq. (5), the Eq. (3) can be converted according to the trigonometric formula [49]

(8)$$a\cos x + b\cos y = t\sqrt {1 + {{\left( {\frac{s}{t}} \right)}^2}} \cos \left[ {\frac{1}{2}(x \mp y) + \arctan \left( {\frac{s}{t}} \right)} \right],$$

where $s = (a - b)\sin [{(x \pm y)\textrm{/2}} ]$ and $t = (a + b)\cos [{(x \pm y)\textrm{/2}} ]$. Thus, the $\cos(\varphi - {\varphi _n})$ and $\cos(\varphi - {\varphi _s})$ items in Eq. (3) can be rewritten as

(9)$$\begin{array}{l} \exp \left[ {\frac{{2r{r_0}\cos (\varphi - {\varphi_n})}}{{w_0^2}}\textrm{ + }{C_j}\frac{{2r{r_0}\cos(\varphi - {\varphi_n})}}{{{R^2}}}} \right] \times \exp [{ - \textrm{i}k{r_S}\sin \theta \cos(\varphi - {\varphi_s})} ]\\ = \exp \left\{ { - \textrm{i}{T_j}\cos \left\{ {\varphi - \left[ {\frac{1}{2}({{\varphi_n} - {\varphi_s}} )- \arctan \left( {\frac{{{s_1}}}{{{t_1}}}} \right)} \right]} \right\}} \right\}, \end{array}$$

where ${T_j} = {t_1}\sqrt {1 + {{\left( {\frac{{{s_1}}}{{{t_1}}}} \right)}^2}}$,${s_1} = ({a_1} - {b_1})\sin [{({\varphi_s} - {\varphi_n})\textrm{/2}} ]$,${t_1} = ({a_1} + {b_1})\cos [{({\varphi_s} - {\varphi_n})\textrm{/2}} ]$,${b_1} = k{r_s}\sin \theta $, and ${a_1} = 2\textrm{i}r{r_0}({C_j}w_0^2 + {R^2})/(w_0^2{R^2})$. Substituting Eqs. (4)–(9) into Eq. (3), the electric field distribution of the radial distribution of linearly polarized Gaussian array beam near the focus can be expressed as

(10)$$\begin{array}{l} {\overrightarrow E ^{(S)}} = \left[ {\begin{array}{c} {E_r^{(S)}}\\ {E_\varphi^{(S)}}\\ {E_z^{(S)}} \end{array}} \right] ={-} \textrm{i}\mathrm{\pi }f\int_\textrm{0}^{{\theta _{\max }}} {\sum\limits_{n = 1}^N {\sum\limits_{j = 1}^M {{l_\textrm{0}}(\theta ,\varphi )\sqrt {\cos \theta } \sin \theta {t_m}(r,\varphi )} } } \\ \times \exp \left[ { - \frac{{{C_j}w_0^2 + {R^2}}}{{{R^2}w_0^2}}({{f^2}{{\sin }^2}\theta + r_0^2} )} \right]\exp (\textrm{i}kz\cos \theta )\\ \times \left[ {\begin{array}{c} {\cos {\varphi_p}({\cos \theta - 1} ){J_0}({{T_j}} )- \cos ({{\varphi_p} - 2{\varphi_S} - 2{\varphi_{nS}}} )({\cos \theta + 1} ){J_2}({{T_j}} )}\\ {\sin {\varphi_p}({\cos \theta + 1} ){J_0}({{T_j}} )+ \sin ({{\varphi_p} - 2{\varphi_S} - 2{\varphi_{nS}}} )({\cos \theta - 1} ){J_2}({{T_j}} )}\\ { - 2\textrm{i}\cos {\varphi_p}\sin \theta {J_1}({{T_j}} )} \end{array}} \right]\textrm{d}\theta . \end{array}$$

It is noted that the intensity distribution of the focal optical field can be obtained by $I = {|{E_r^{(S)}} |^2} + {|{E_\varphi^{(S)}} |^2} + {|{E_z^{(S)}} |^2}$.

3. Tight focusing properties of the combined CVBs near the focus

In this section, we investigate the properties of tight focusing intensity distribution of the combined CVBs with different polarization distributions. In the following calculations, the parameters are taken as β = 1, f = 1 mm, $\lambda = 1.064\textrm{nm}$,$\delta = 0.9$, N = 8. The incident fields of combined radially and azimuthally polarized incident vector beams and their corresponding tight focused fields for different values of R and NA are shown in Figs. 2 and 3, respectively. It should be noted that the transverse coordinates of focal plane are normalized by wavelength λ and the intensity distribution are normalized by its maximum value of total intensity in the following illustrations. It can be seen that the focal shapes of combined radially polarized beams and combined azimuthally polarized beams are circular and annulus respectively, which are consistent with the results of tight focusing properties of ideal CVBs. However, for the unmatched values of R and NA, the focal shape appears side lobes around the main lobe (see R=1.6 mm, NA=0.7 or R=2 mm, NA=0.8). To be specific, the desired focal shapes with efficient sidelobes suppression are shown within the red dotted line. As can be seen that the focal shape size decreases with increasing NA. Furthermore, to get a desired focus shape, the value of NA should increase with the increase of R. The physical reason is that when the effective exit sub-aperture increases, a larger NA is needed to focus incident fields better (when the focal length keeps unchanged). Thus, it is necessary to choose the matching parameters of NA and R to obtain a desired focal spot.

Fig. 2. Intensity distributions of combined radially polarized beams at the initial plane and focal plane for different values of NA and R.

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Fig. 3. Intensity distributions of combined azimuthally polarized beams at the initial plane and focal plane for different values of NA and R.

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To investigate the influence of truncation parameter on the focal shape, the intensity distributions of combined azimuthally polarized beams at the focal plane for different values of beam width w₀ are shown in Fig. 4. In the following calculation, the parameters of the simulation are as follows: R=3.2 mm and NA = 0.9. The initial values of w₀ and their corresponding values of δ are shown in Fig. 4. The results demonstrate that the truncation parameter has little influence on the main lobe, and the outermost ring gradually becomes weak as δ increases. The main reason is that the radius of beam array r₀ keeps unchanged, and the parameters of NA and R can still match as w₀ changes. As we all know, the truncation parameter has a marked impact on the side lobe for coherent combined beam array. Hence, the truncation parameter δ is taken as 0.9 to reduce the loss of laser power and to still keep a greater fill factor.

Fig. 4. Intensity distributions of combined azimuthally polarized beams at the incident plane and focal plane, respectively.

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Intensity distributions and illustrations of the polarization direction at the focal plane for different values of φ₀ are shown in Fig. 5. φ₀ = 0 and π/2 correspond to radially and azimuthally polarized beams, respectively. We can see that the focal shape gradually shifts from Gaussian-like to a flat-topped distribution and then to an annular distribution with increasing initial polarization rotation φ₀. Therefore, we can obtain a focal shape we wanted by adjusting the initial polarization orientation of each sub-aperture. It can be seen from Figs. 5(d)–5(f), the polarization orientation of main lobe at the focal plane is consistent with the initial polarization of combined CVBs, that is to say, the focal plane is radially polarized and azimuthally polarized when φ₀ = 0 and π/2 respectively. In contrast, the polarization orientation of the first side lobe is opposite to the initial polarization orientation of the combined CVBs.

Fig. 5. Intensity distributions and illustrations of the polarization direction at the focal plane for different values of φ₀.

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The evolution of focal shape can be embodied by the variation of the field of azimuthal, radial and longitudinal components. 2D intensity distributions of azimuthal, radial and longitudinal components for different values of φ₀ are shown in Fig. 6. When φ₀ = 0, the azimuthal component approaches to zero and there are only radial and longitudinal components. As φ₀ increases, the azimuthal component becomes stronger, and the radial and longitudinal components become weaker. As a consequence, only the azimuthal component exists when φ₀ increases to π/2. The 1D intensity distributions for azimuthal, radial and longitudinal components are plotted in Fig. 7. To better illustrate the difference between the combined CVBs and the ideal CVBs, the Intensity distributions and 1D intensity distributions of ideal CVBs are shown in Fig. 8. The calculating parameters of the ideal CVBs and the combined CVBs keep the same, including the pupil apodization. As shown in Figs. 5 and 8, it can be seen that the focal shapes of combined CVBs are consistent with that of ideal CVBs except that the annular side lobes are obvious especially for combined azimuthally polarized beam. The flat-topped conditions of the combined CVBs and the ideal CVBs are slightly different, and the flat-topped condition of the ideal CVBs can be found to be at φ₀ =π/5. It is worth noting that the intensity of radial component of the combined CVBs is approximately one fifth of the longitudinal component (see Fig. 7(a)), thus the width of main lobe is slightly larger than that of longitudinal component. In contrast, the intensity of radial component of the ideal CVBs is almost one half of the longitudinal component (see Fig. 8(d)), that means, compared with ideal radially polarized beam, a smaller beam focal spot width generated by combined radially polarized beam can be obtained.

Fig. 6. Intensity distributions of azimuthal, radial and longitudinal components for different values of φ₀ at the focal plane.

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Fig. 7. 1D intensity distributions at the focal plane for different values of φ₀.

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Fig. 8. (a)-(c) Intensity distributions, and (d)-(f) 1D intensity distributions of ideal CVBs at the focal plane for different values of φ₀.

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The intensity distributions for different φ₀ in r-z plane are presented in Fig. 9. One can see that the beam width of annular shape (Fig. 9(c)) is approximately twice as large as that of Gaussian-like (Fig. 9(a)), which is different from the result of ideal CVBs. The main reason is that the radial component intensity of the combined radially polarized beam is much smaller than that of ideal radially polarized beam, and thus the beam size of combined radially polarized beam is closer to the size of azimuthal component. In our calculation, the flat-topped condition is found to be at φ₀ = π /4.

Fig. 9. Intensity distributions for different φ₀ in r-z plane.

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Due to the technique of producing CVBs is by using separate sub-apertures in this paper, the obvious advantage is that the polarized orientation of each aperture can be set to any direction. Thus, some peculiar focal shapes can be obtained by adjusting the initial polarized orientation. Some of the results are shown in Fig. 10. The focal shapes are composed by eight radially distributed spots (see Figs. 10(a2) and 10(b2)) when the adjacent polarized orientations are opposite to each other (see Figs. 10(a1) and 10(b1)). Two separated spots (see Figs. 10(c2) and 10(d2)) can be obtained by reversing the polarized orientation of two apertures on the opposite side of the circle array (see Figs. 10(c1) and 10(d1)). The polarization of focal spot is not radial or azimuthal distributions (see Figs. 10(a3)–10(d3), and it means that the radial and azimuthal components at the focal plane are not pure.

Fig. 10. Incident intensity distributions, tight focused fields and illustration of the polarization patterns for arbitrary polarization distribution.

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4. Conclusions

The tight focusing properties of the combined CVBs has been investigated in detail based on the coherent beam combining technology. The CBC technology offers the generation of high-power focal spots and flexible control of polarization orientation simultaneously in this paper. The analytical expression of the tight focusing field of the combined CVBs has been derived according to the Richard-Wolf vector diffraction integral. It is demonstrated that the desired focusing shapes can be obtained by adjusting the parameters (e.g., effective exit sub-aperture R, numerical aperture NA and truncation parameter δ) of the combined CVBs. It has been found the truncation parameter has little influence on the side lobes. Besides, the polarization orientation of main lobe at the focal plane is coincide to the initial polarization orientation of combined CVBs. With the increase of the initial polarization rotation φ₀, the width of the focal spot increases, and the focal shape shifts from Gaussian-like to a flat-topped distribution and then to an annular distribution. The width of annular shape is approximately twice as large as that of Gaussian-like shape. In our calculation, the flat-topped condition is found to be at φ₀ = π /4. Focal field tailoring can be realized by adjusting the initial polarization rotation of each aperture. It is obvious that the controlling of structured fields can be more flexible with the increasing number of sub-apertures. In the future work, combined with the active phase control, a more complex vector beam could be expected to achieve. The results in this paper can be crucial for the applications including industrial manufacturing, microlithography and multi-particle manipulation.

Funding

National Natural Science Foundation of China (61705264, 61705265).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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Tight focusing properties and focal field tailoring of cylindrical vector beams generated from a linearly polarized coherent beam array

Abstract

1. Introduction

2. Tight focusing field distributions of the combined CVBs

3. Tight focusing properties of the combined CVBs near the focus

4. Conclusions

Funding

Disclosures

References

Cited By

Figures (10)

Equations (10)

Optics Express