Abstract
A scalar polymorphic beam is designed with independent control of its intensity and phase along a strongly focused laser curve of arbitrary shape. This kind of beam has been found crucial in the creation of freestyle laser traps able to confine and drive the motion of micro/nanoparticles along reconfigurable 3D trajectories in real time. Here, we present and experimentally prove the concept of vector polymorphic beam adding the benefit of independent design of the light polarization along arbitrary curves. In particular, we consider polarization shaped tangential and orthogonal to the curve that are of high interest in optical manipulation and laser micromachining. The vector polymorphic beam is described by a surprisingly simple closedform expression and can be easily generated by using a computer generated hologram.
Introduction
Shaping a coherent laser beam with independent control of its intensity, phase and polarization is a long posted problem of high interest in science and technology. This is particularly important in areas such as optical manipulation of micro/nanoparticles and light material processing, where laser beams strongly focused in the form of diffractionlimited patterns such as points and curves are required. For example, it is known that the threedimensional (3D) high intensity gradients of a focused laser beam yield optical forces responsible for stable 3D trapping of particles^{1}. The phase gradients of the beam can be also exploited to exert optical forces able to drive the motion of the particles along different trajectories^{2,3,4,5}. The combined use of high intensity and phase gradients allows for improving laser micromachining tools^{6,7}. It has been also reported that vector Gaussian beams with radial and/or tangential (azimuthal) polarization have advantages for laser material processing (micromachining such as drilling and cutting), e.g.: tangential polarization provides increased laser microdrilling velocities and generation of thin capillaries with high aspect ratios in thick sheets^{8,9}. Moreover, it has been demonstrated that femtosecond vortex pulses tightly focused onto the surface of dielectric media allows creating subwavelength ripples whose orientation depends on the polarization direction^{10,11}. The ability to change the polarization state in the focal plane by tuning the vortex topological charge^{11} adds new income in the development of such micro and nanoscale surface structuring.
In general, a vector beam E(x, y) = A(x, y)e with arbitrary complex field amplitude A(x, y) and polarization distribution e = (a_{1}(x, y), a_{2}(x, y)), at a given transverse xyplane, can be generated by using computer generated holograms (CGHs) addressed into programmable spatial light modulators (SLMs) as reported elsewhere^{12,13,14,15,16}. Here e is a Jones vector where a_{1,2}(x, y) are complex functions such that a_{1}(x, y)^{2} + a_{2}(x, y)^{2} = 1. Indeed, a combination of two collinear beams E_{1}(x, y) = A(x, y)a_{1}(x, y)e_{1} and E_{2}(x, y) = A(x, y)a_{2}(x, y)e_{2} with orthogonal linear polarization states e_{1} = (1, 0) and e_{2} = (0, 1) forms a vector beam. Circular polarized beams \({{\bf{e}}}_{\mathrm{1,2}}^{c}=(1,\pm \,i)\) can be also used for this task. The superposition of the beams E_{1} and E_{2} created by using two SLMs is often achieved applying a MachZehnder interferometer^{16} or a commonpath interferometer with a Ronchi grating^{17} as the one sketched in Fig. 1(a). Numerous vector beams have been created by using this approach^{12,13,14,15,16}, however, the challenging problem of the design and generation vector beams strongly focused in form of a diffractionlimited light curve of arbitrary shape with independent control of its intensity, phase and polarization distributions along it has not been completely solved. While in^{18}, an inverse design based on a numerical calculation procedure has been proposed for complete 2D shaping of the optical focal field with the prescribed distribution of intensity, phase and polarization, however, an analytical expression for the required beam simplifying the light curve generating process has not been found. On the other hand in^{19}, using the expression for scalar curved beams proposed by us in^{20}, a particular case of polarization shaping of curved vector beams in three dimensions has been demonstrated. Note that the approach reported in^{19} does not allow shaping an arbitrary phase along the curve.
In this work we present a kind of vector beam, referred to as vector polymorphic beam, that solves this challenging problem of beam shaping. It is based on its scalar analogous beam^{21}, which can be strongly focused in form of diffractionlimited light curve of arbitrary shape with the following key properties: high 3D intensity gradients, independent design of the intensity and phase distributions (both of them can be arbitrary) along the curve according to the considered application. The scalar polymorphic beam has been used to create freestyle laser traps providing both optical confinement and transport of micro/nanoparticles along reconfigurable trajectories^{5,20,22}. Apart from optical manipulation, scalar polymorphic beams have been also applied for laser printing of plasmonic nanoparticles^{23} illustrating their possible application for singleshot laser lithography, micromachining (drilling and marking)^{7,8,9}, etc. Its design has an inherent versatility that can be exploited to create other types of coherent beams such as electron beams shaped along curves, as it has been shown in^{24}. Another advantage is that this kind of beam shaping technique does not require the use of iterative algorithms, thus enabling direct and fast generation of such laser curves. Moreover, it can be easily implemented by using a CGH addressed into a conventional programmable SLM.
In next section we propose a simple technique for the generation of vector polymorphic beams with tailored polarization distribution along the light curve, thus enlarging the practical applications of their scalar counterparts. In section 3 the technique is experimentally verified. The article ends with conclusion remarks.
Principle of the Technique
The vector polymorphic beam is defined by:
with e_{1,2} corresponding to orthogonal polarization components, for example, linear e_{1} = (1, 0) and e_{2} = (0, 1) or circular \({{{\bf{e}}}^{{\rm{c}}}}_{\mathrm{1,2}}\) = (1, ±i) ones. Specifically, g_{1,2}(t) = g(t)a_{1,2}(t) are weight functions defining the intensity and phase distributions along the curve, while a_{1,2}(t) controls the polarization along it assuming a_{1}(t)^{2} + a_{2}(t)^{2} = 1. Here, R(t) is the radius of an arbitrary 2D curve given in polar coordinates that can be either closed (T = 2π) or open, f is a normalization constant, T stands for the maximum value of the azimuthal angle t, while k = 2π/λ with λ being the light wavelength.
Let us first briefly recall the main characteristics of the scalar polymorphic beam^{21}
used in the design of the vector polymorphic beam Eq. (1). To create the light curve, the polymorphic beam is Fourier transformed^{21}:
by using a convergent lens of focal length f. Thus, the shape of the beam \(\tilde{E}(u,v)\) in the focal plane is described by the 2D curve written in parametric form as c(t) = (u(t), v(t)), with u(t) = −R(t) cos t and v(t) = −R(t) sin t. While, the complex weight function
controls the amplitude and phase distributions along the curve. Specifically, the field amplitude distribution along the curve is given by \(\tilde{E}(u(t),\,v(t))=g(t)/\kappa {\bf{c}}^{\prime} (t)\), where: \({\bf{c}}^{\prime} (t)=\sqrt{R^{\prime} {(t)}^{2}+R{(t)}^{2}}\) with c′(t) = dc(t)/dt, and κ = L/λf with \(L={\int }_{0}^{T}{\bf{c}}^{\prime} (\tau ){\rm{d}}\tau \) being the curve length^{21}. While, the phase of g(t) is controlled by the real function S(t) describing the phase variation along the curve. Note that the parameter l defines the phase accumulation along the entire curve^{21}. For closed curves the phase accumulation is 2πl and l corresponds to the vortex topological charge^{25}. For instance, a light curve with uniform intensity is obtained by using g(t) = E_{0}κc′(t) (with dimension of electric field) while
sets a uniform phase distribution along the curve c(t). A non uniform phase shaped along the curve can be easily obtained by using, for example, the following constraint
with α being a real number^{21}. For instance, in Fig. 1(b) it is shown a scalar polymorphic beam focused in form of triangularlike curve with a topological charge l = 8, for the case of uniform [Eq. (5)] and nonuniform [Eq. (6) with α = 2] phase distribution. Note that the change of either the phase distribution and/or value of l does not alter the shape and size of the beam curve, independently whether the curve is closed or open as the spiral one demonstrated in Fig. 1(c).
Such a versatile control of amplitude and phase along the curve results crucial for creating vector polymorphic beams. For example, to adapt the polarization e(t) = (a_{1}(t), a_{2}(t)) to the curve shape, the phase of a_{1,2}(t) can be expressed similar to Eq. (4) as it follows: exp[i2πp_{1,2}σ_{1,2}(t)] with σ_{1,2}(t) = S_{1,2}(t)/S_{1,2}(T) and p_{1,2} being real numbers. Note that σ_{1,2}(t) ∈ [0, 1] and the variation of the Jones vector along the curve can be also uniform if S_{1,2}(t) is described by Eq. (5), or nonuniform when it is described for example by Eq. (6). Thus, when \({a}_{1}(t)={a}_{2}(t)=\mathrm{1/}\sqrt{2}\), σ_{1,2}(t) = σ(t) and p_{2} = −p_{1} = p the polarization defined by the Jones vector \({\bf{e}}(t)=({e}^{{\rm{i}}2\pi p\sigma (t)},\,{e}^{{\rm{i}}2\pi p\sigma (t)})/\sqrt{2}\) performs a 2p rotation along two meridians of the polarization Poincaré sphere with an azimuthal angle distant 180° between each other. In this case the global phase of the vector beam is given by the phase of g(t), Eq. (4). Other combinations of a_{1,2}(t), allows for more complex movements along the polarization sphere.
A polarization tangential to the curve results more relevant in practical applications such as laser material processing and micromachining. Indeed, as pointed out in^{8,9} tangential polarization yields improved laser drilling on materials. Phase gradients of scalar vortex beams have been also proved successful for clearer and smoother processed surfaces^{6}. Therefore, a vector polymorphic beam with both polarization and phase gradient tangential to the curve opens up promising perspectives. In this case the vector polymorphic beam has to be created with
when using orthogonal linear polarization components. Indeed, the Jones vector e(t) = (a_{1}(t), a_{2}(t)) is tangential to the curve c(t). We recall that the weight functions are g_{1,2}(t) = g(t)a_{1,2}(t) and therefore it is possible obtain the tangential polarization independently of the intensity and phase prescribed by g(t) along the curve. To set the polarization orthogonal to the curve, a_{1}(t) and a_{2}(t) given by Eq. (7) have to be exchanged.
As we have previously mentioned, orthogonal left and righthand circular polarization components \({{\bf{e}}}_{\mathrm{1,2}}^{c}\) = (1, ±i) can be also used to generate a vector polymorphic beam. In this case, in order to create a polarization tangential to the curve, the required functions for two orthogonal components of the beam are defined by
Experimental Results
The experimental setup sketched in Fig. 1(a) has been used to generate the vector beams Eq. (1) considered here. It consists of a programmable SLM (Holoeye PLUTO, pixel size of 8 μm) in which a phaseonly CGH encoding the beam components as E_{1}(x, y) exp (i2πx/Λ) + E_{2}(x, y) exp (−i2πx/Λ) has been addressed by using the approach reported in^{26}. We recall that the weight functions are g_{1,2}(t) = g(t)a_{1,2}(t) and therefore the polarization information of a_{1,2}(t) has been also included into the CGH encoding E_{1,2}(x, y). This CGH allows for generating the beams \({\tilde{E}}_{\mathrm{1,2}}(u,v)\) spatially separated at the focal plane of the convergent lens L1, where they are respectively modulated by two halfwave plates (HWP1 and HWP2) in order to obtain the required orthogonal linear polarization components, see Fig. 1(a). If orthogonal left and righthand circular polarization components are used to generate a vector polymorphic beam then the HWPs of the setup have to be replaced by quarterwave plates. The beam components \({\tilde{E}}_{\mathrm{1,2}}(u,v)\) are combined by using another convergent lens L2 (in our case L1 and L2 are identical, working together as a 4f system) and a diffraction grating of period Λ (in our case a Ronchi Ruling grating of 20 lp/mm, Edmund Optics). Thus, the focused vector polymorphic beam \({\tilde{E}}_{1}(u,v){{\bf{e}}}_{1}+{\tilde{E}}_{2}(u,v){{\bf{e}}}_{2}\) (the laser curve) is obtained at the focal plane of the lens L3, where its intensity distribution has been recorded by a digital camera (color CMOS, Thorlabs, pixel size of 4.7 μm). In our case the analyzer has been set into a programmable rotation stage (Newport URS100BCC).
In the considered examples, the curve is described by
known as Superformula^{27}, that allows for straightforward generation of a large variety of shapes where the real numbers in q = (a, b, n_{1}, n_{2}, n_{3}, m) are the design parameters of the curve and ρ(t) is a nonperiodic function of t required for the construction of asymmetric and spirallike curves (e.g.: ρ(t) ∝ e^{βt} or ρ(t) ∝ t^{β}). For example, with q = (1, 1, 1, 1, 1, 0) and constant ρ(t) = ρ_{0} a circle of radius R(t) = ρ_{0} is obtained, while for other values of q a variety of closed polygons of different symmetry are easily generated^{21}.
Let us first consider the experimental examples displayed in Fig. 1(d,e) corresponding to a triangularlike and spiral curves. In this case the vector polymorphic beam focuses into the curve with uniform intensity distribution and constant phase, while its polarization \({\bf{e}}(t)=({e}^{{\rm{i}}2\pi p\sigma (t)},\,{e}^{{\rm{i}}2\pi p\sigma (t)})/\sqrt{2}\) varies along the curve following the meridian path on the polarization Poincaré sphere as shown in Fig. 1(f). Specifically, Fig. 1(d) shows the experimental results obtained for the triangularlike curve with l = 0, p_{2} = −p_{1} = p = 8 and σ(t) yielding uniform [Eq. (5)] and nonuniform [Eq. (6) with α = 2] variation of the polarization along it, see second and third rows respectively. Note that in this case the analyzer has been set at 45° and therefore the measured intensity distribution shows 2p = 16 fringes distributed along the curve. The same polarization configurations have been prescribed in the case of the spiral curve, see Fig. 1(e). We underline that in the case of uniform variation of σ(t) the polarization distribution periodically changes along the curve (t ∈ [0, T]): it is linearly polarized at 45° and −45° in the points where σ(t) = n/16 and σ(t) = 1/32 + n/16, correspondingly (with n = 0, 1, 2, ..., 15). While right(left)hand circular polarization is obtained in the points where σ(t) = 1/64 + n/16 (and σ(t) = 3/64 + n/16), see also Fig. 1(f). In the case of nonuniform variation of σ(t) [described by Eq. (6)] this transformation in the polarization is accelerated along the curve.
The examples considered in Fig. 1 have mostly fundamental character. Now we turn to practically important cases: Polymorphic beams with both uniform intensity and phase distribution and linear polarization tangential to the curve. In Fig. 2 and Supplementary Video 1, the rotation of the analyzer indicates that the polarization has been set tangential to the curve as it follows from the further analysis of the Stokes components. Indeed, by calculating the first three Stokes parameters S_{0} = I(0°) + I(90°), S_{1} = I(0°) − I(90°), and S_{2} = I(45°) − I(135°), −where I(θ) stands for the measured intensity distribution when the analyzer is set at an angle θ with respect the horizontal axis − one derives that \({S}_{0}^{2}={S}_{1}^{2}+{S}_{2}^{2}\) and therefore the fourth parameter is S_{3} = 0. It means that the polarization is linear in all the points of the curve and forms an angle θ = (arctan(S_{2}/S_{1}))/2 with horizontal axis as it is shown in the last row of Fig. 2. Note that an uniform phase distribution, given by the function S(t) in Eq. (5) with l = 8, has been used in all the beams displayed in Fig. 2. For example, the uniform phase distribution prescribed in Fig. 2(a,b) is the same as the one displayed in Fig. 1(b,c), respectively.
Discussions
The vector polymorphic beam can be easily created by using a hologram and provides a direct (noniterative) way for confining light in the form of a diffractionlimited 2D curve of arbitrary shape and size, with independent control of the intensity, phase and polarization distributions. These degrees of freedom are demanded by relevant applications as for example laser material processing^{7,8,9,23} and optical manipulation of micro/nanoparticles^{5,20,22}. Here, the versatility in the design and generation of the vector polymorphic beams have been illustrated in several examples using a straightforward experimental setup. More sophisticated setups, as the ones reported in^{9}, can be applied instead for practical implementations of the polymorphic beams. Industrial applications of the polymorphic beams could require a diffraction optical element instead of a liquid crystal SLM for hologram encoding.
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Acknowledgements
The Spanish Ministerio de Economía y Competitividad is acknowledged for the project TEC201457394P.
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J.A.R. and T.A. developed the idea discussed the results and wrote the manuscript. J.A.R designed and performed the experiments as well as programs.
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Correspondence to José A. Rodrigo.
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vector polymorphic beams whose polarization has been set tangential to curves
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Rodrigo, J.A., Alieva, T. Vector polymorphic beam. Sci Rep 8, 7698 (2018). https://doi.org/10.1038/s41598018261269
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