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Abstract
In this paper we compare the intensity distributions in the paraxial and tightly focused regimes corresponding to a double ring perfect optical vortex (DR-POV). Using the scalar diffraction theory and the Richards-Wolf formalism, the fields in the back focal plane of a low and high (tight focusing) NA lens are calculated. In the paraxial case we experimentally observed a DR-POV whose rings enclose a dark zone thanks to the destructive interference introduced by a π phase shift. In the tightly focused regime, however, the numerical simulations showed that the intensity near the focus is influenced by the input field polarization and it is not intuitive. In both cases we found that the dark region subtended between the rings has a minimal width that is inversely proportional to the pupil radius of the system, reaching 0.42λ for the radially polarized DR-POV. For the tightly focused case, we calculated the optical forces in the transversal and longitudinal coordinates exerted on a metallic particle. As a result, it is theoretically demonstrated that the circularly polarized DR-POV can trap Au metallic particles in 3D using a light wavelength close to its resonance.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
An optical vortex is a beam with spiral wavefront that revolves along the propagation beam axis a number of $2\pi$ phase jumps (topological charge) exhibiting a central phase singularity with null intensity [1]. These beams are able to transfer orbital angular momentum (OAM) to small particles and posses interesting properties when focused, finding application in optical manipulation [2], cold atom capture [3], stimulated emission depletion (STED) microscopy and high resolution imaging [4–6], among others.
Recently, manipulation of metallic nanoparticles using wavelenghts close to its resonance has received much attention because their properties are attractive in a variety of applications such as Surface Enhanced Raman Spectroscopy (SERS) [7] or fluorescent biological labelling [8]. Unfortunately, under this condition the particle’s absorption and scattering increase and a careful selection of wavelength and beam structure have to be done to overcome thermal instabilities. In this field, optical vortices have been put to use as well [9–13], but only [9,13] tackle wavelengths close to the resonance. In these works, the metallic nanoparticles rotate around a single intensity trajectory thanks to OAM transfer.
Commonly, transversal vortex intensity and its topological charge are strongly coupled making it difficult to control them separately. In 2013 it was introduced a vortex named Perfect Optical Vortex (POV) [14] that represents the Fourier transformation of a Bessel beam, that has been subject of intense study in both coherent and incoherent fields [15–18]. Ideally, the POV is a radial Dirac delta whose radius is independent of the topological charge. Based on the POV, a beam called double ring perfect optical vortex (DR-POV) was proposed [19] to trap low refractive index particles within its two rings. This beam was made of two equal charged POVs having orthogonal circular polarization and reached a ring separation of 25$\mu$m at $\lambda$=488nm. Earlier, a concentric vortex trap possessing equal intensity rings is reported in [20]. In this work two vortex beams with unequal topological charges are focused to the radii 2.9$\mu$m and 6.4$\mu$m using linearly polarized visible light at 532nm to study the hydrodynamic motion of dielectric spherical beads of 0.99$\mu$m. A two ringed trap for metallic nanoparticles has not been reported to date. When several metallic particles are in close proximity, the total resonant response of the system changes with respect to one particle [21]. Hence, it would be very interesting to count with a rotating concentric optical trap wherein the metallic particles could interact. Despite [19,20] treat vortex formation destined to optical manipulation, only [20] includes some tight focusing considerations. Tight focused beams are produced in high numerical aperture (NA) systems like the objectives routinely used in optical manipulation experiments. In the tightly focused regime the Fresnel and Fraunhofer diffraction integrals are not valid and the Debye integrals have to be used to describe the spectrum formation accurately [22,23]. In this regime the initial polarization state of the beam has to be taken into account.
In its original form, POV [14] is derived via the scalar diffraction formula that doesn’t cover the high numerical aperture systems used in optical tweezers setup. Tight focusing of vector vortex beams including POV have been studied in several works [24–28]. In these works emphasis in tight spot or flattop beam formation is given. Other structures formed by focusing vector vortex beams such as multiple longitudinal spots [29], polygon-like curves [30] and petal-like beams [31–33] have been devised however, to the best of our knowledge tight focusing formation of a DR-POV has not been investigated.
In this paper we generate a DR-POV superposing two equally polarized POV beams with the same topological charge that are dephased by $\pi$ radians. As a result of destructive interference, the dark hollow region between the two rings is preserved with a high contrast, achieving closer ring separations as compared to previous DR-POV [19]. We study the DR-POV behaviour in the paraxial and tightly focused regimes, showing that in this last case the DR-POV beam can trap metallic nanoparticles close to its resonant wavelength stably in 3D.
2. Paraxial case
2.1 Theoretical background
Scalar diffraction theory is a valuable tool to explain light transmission through linear and invariant optical systems. Such situation arises when: a) the medium is homogeneous and isotropic, b) the aperture’s area is large compared with light wavelength $\lambda$, i.e., the diffraction angles caused by the aperture are small [34]. This last case is what is known as paraxial regime.
As it is well known, when an object with the transmittance function $U(r,\varphi )$ is illuminated by a completely coherent uniform plane wave, the signal in the back focal plane of the converging lens is given by the 2-D Fourier transform whose polar form is written as [34]
2.2 Numerical results
The Fourier transforms of Eq. (4) are shown in Fig. 1 for the optical parameters $\lambda$=633nm, $f$=1m, $R$=2.048mm, $\rho _{1}$=6.6mm, $\rho _{2}$=7mm and $m$=5. One observes that both spectra produce DR-POVs that possess topological charge $m$=5. Figures 1(b) and 1(d) reveal that in the vortex difference the phase in the rings is out phase while in the vortex sum it is in phase.
Fig. 1. Normalized intensity and phase profile for a)-b) DR-POV (difference), c)-d) DR-POV (sum). Radii and topological charge $\rho _{1}$=6.6mm, $\rho _{2}$=7mm, $m$=5, respectively.
The 1D intensity profiles taken at the center of Figs. 1(a) and 1(c) are shown for different radial separations in Fig. 2. In general, the vortex difference performs better at producing two concentric rings with a darker inter radial zone than the vortex sum DR-POV. This behaviour is specially notorious at closer distances where the $\pi$ phase difference in $U_{-}(r,\varphi )$ leads to destructive interference of the principal lobes of each POV when they overlap, in contrast to $U_{+}(r,\varphi )$. It is worth noting that in a non diffractive system ($R=\infty$), both the vortex sum and the vortex difference would produce two delta rings that never touch each other except when $\rho _{1}=\rho _{2}$. The presence of the pupil $R$ produces the widening and appearance of lobes in the DR-POVs rings due to convolution with the circ(.) Fourier spectrum.
Fig. 2. a-d) Intensity profiles for DR-POVs with $m$=5 and different $\rho _{1}$, $\rho _{2}$ values. Vortex sum (blue line), vortex difference (red line).
In view of its capability to reach closer peak distances, from now on we will focus on the DR-POV obtained by the Fourier transform of $U_{-}(r,\varphi )$. The behaviour of the DR-POV for different topological charges is plotted in Fig. 3(a) for the previously mentioned optical parameters ($\lambda$, $f$, $R$), $\rho _{1}$=6.8mm and $\rho _{2}$=7mm, respectively. The Fig. 3(a) shows that an increased topological charge produces a minimal widening of the rings and also that the maxima positions are not exactly $\rho _{1}, \rho _{2}$. As said before, the field overlapping between the POVs introduced by the convolution with the Fourier transform of the pupil function moves the peak positions from the intended values $\rho _{1}, \rho _{2}$ in a non linear manner. Figure 3(b) plots the actual peak separation in the DR-POV rings versus $\Delta \rho =|\rho _{1}-\rho _{2}|$ value in $U_{-}(r,\varphi )$ using the pupil radii $R$=[1.024, 2.048]mm. The plot demonstrates that for a fixed pupil radius, the peak distance is insensitive on the topological charge value except when the obstruction in the entrance signal is significant, as occurred in $m=30$ and $R=$1.024mm. The plot also shows that the ring distance reduces with $\Delta \rho$ until it reaches a stationary value. This minimum value originates when the principal lobes of each POV interfere. As $R$ grows the slopes in the principal lobes become stepper and allow a smaller minimal distance. The expression for this limit was calculated from the simulations to be
where in the case of Fig. 1 and Fig. 2 the value $\Delta \rho _{\mathrm {min}}\approx$ 0.21mm.Fig. 3. a) DR-POV intensity for different topological charges. b) DR-POV actual peak distance versus $\Delta \rho$ value for two values of $m$ and $R$.
2.3 Experiments and results
The proposed DR-POV transmittance $U_{-}(r,\varphi )$ was coded in a phase only spatial light modulator (SLM) by means of the pulse width modulation technique [35]. In our experimental setup we used the PLUTO-VIS SLM Holoeye with 1920$\times$1080 pixel resolution and $8\mu$m pixel pitch. The input source was a HeNe laser $\lambda$=633nm polarized in the direction of the SLM active axis and the focal distance of the Fourier transforming lens was $f$=1m. The CCD detector was the Sony SSC-C374 with 720x480 pixel resolution and sensor size 6.3 x 4.7mm.
Figure 4 contains the results related to the experimental DR-POV generation. Figure 4(a) shows an example of the transmittance function $U_{-}(r,\varphi )$ coded into the SLM with support radius $R$=512pix, $\rho _{1}$=1.85mm, $\rho _{2}$=2mm and topological charge $m$=10. The experimental intensity patterns produced by our DR-POV with $\rho _{1}$=1.85mm, $\rho _{2}$=2mm and topological charges $m=[10, 30]$ are shown in Figs. 4(b) and 4(c). It can be seen that both beams possess a dark inter radial zone and this feature does not change significantly with $m$ value. We verify the topological charge interfering the DR-POV with a spherical phase front [19] and the result is shown in Figs. 4(d) and 4(e). The number of fringes coincides with the value $m$=10 and 30, respectively. Finally, the intensity profile comparison between the DR-POVs is shown in Fig. 4(f). We can see that the peak distance in both cases is 0.11mm in agreement with the $\Delta \rho _{\mathrm {min}}\approx$0.1mm obtained from the Eq. (5) and that the contrast is high independently of the topological charge value employed.
Fig. 4. a) $U_{-}(r,\varphi )$ signal coded into the SLM with $\rho _{1}$=1.85mm, $\rho _{2}$=2mm and topological charge $m$=10 (carrier signal not shown). b), c) DR-POV experimental intensity. d), e) DR-POV interference pattern with a spherical wave. b), d) $m$=10 and c), e) $m$=30. f) Intensity comparison 1D.
In summary it should be noted that $U_{-}(r,\varphi )$ is an idealized model that employs pure Bessel beams instead of Bessel-Gauss profiles describing more realistic laser illumination. As a consequence, each POV profile turns into a modified Bessel function $I_{m}(.)$ that is never truly independent of the topological charge as POV’s definition Eq. (2) states. This fact is described extensively in [36]. However, since the ratio between each POV’s radius to its ring width is bigger than unity the topological charge influence in the DR-POV radii is negligible. Additionally, the POV radius and ring width vary depending on the generating device employed (spiral phase plates, optimal optical elements and axicons) and exact expressions can be found in [15,37].
3. Tightly focusing case
3.1 Theoretical background
When the diffracting aperture is comparable to the light’s wavelength, the vectorial nature of light (polarization) can not be neglected. Due to the strong boundary effects the full Maxwell equations have to be employed to calculate the intensity distribution at the Fourier plane accurately. This vectorial treatment is adequate to study the diffraction in the non paraxial i.e., tight focusing regime such as in high numerical aperture NA objectives or radio antennas. Recently, based on the formalism developed by Richards and Wolf [22], a generalization of the vectorial diffraction integrals was reported to include cylindrical vector beams with both phase and polarization singularities [38]. Thanks to this work the spectrum calculation of tightly focused vortex beams with different polarization states is carried out easily through three integrals defined later on. Letting $m, n$ be two positive integer numbers denoting the topological charge and polarization order, the expressions for the components of the $\mathbf {E}, \mathbf {H}$ fields near the focus for azimuthally polarized light are
In Eqs. (6)-(9), the diffraction integrals along $z$ from which the output field components are calculated are
3.2 Numerical results
Substituting Eq. (11) into integrals Eq. (10) we numerically obtain the transverse and longitudinal intensity profiles corresponding to a focused DR-POV with different polarization states that are shown in Fig. 5. For the homogeneous polarization states we choose the linear $y$-polarized beam and the right circular polarization, respectively. For the non homogeneous case we simulated the azimuthal and radial polarization states. The simulations employed the optical parameters $\lambda$=500nm, $n_{1}$=1.518 (oil microscope objective), NA=$n_{1}$, $f$=300$\mu$m and $\theta _{\mathrm {max}}$=$\pi /2$. The images correspond to DR-POVs having $\rho _1$=6.6$\mu$m, $\rho _2$=7$\mu$m and topological charge $m$=5.
Fig. 5. Intensity $I=|E_{x}|^{2}+|E_{y}|^{2}+|E_{z}|^{2}$ of the DR-POV for different polarization states, topological charge $m$=5 and radii $\rho _1$=6.6$\mu$m, $\rho _2$=7$\mu$. a-d) Transversal plane $z=0$. e-h) Longitudinal plane $y=0$.
From Figs. 5(a) and 5(b) we see that the linear and circular polarizations appear to be better suited to preserve two rings in contrast to the radial and azimuthal polarizations Figs. 5(c) and 5(d). In Figs. 5(e) and 5(f) the longitudinal extent where the rings are present is around 1$\mu$m$\approx$1.9$\lambda$ that is within the range of other focused vector beams [27,45]. Although not shown, the transversal phases posses phase profiles akin to the paraxial case depicted in Fig. 1(b). For the $y$-polarized light it were the $E_{y}$ and $H_{x}$ components, for the right circular polarization it were $E_{x}$, $E_{y}$, $H_{x}$ and $H_{y}$ components, for the azimuthal polarization it was $H_{z}$ and for the radial polarization it was $E_{z}$, respectively. We evaluate each of the four polarized DR-POVs and present the plots of its shape at different values of $\rho _{1}$, $\rho _{2}$, topological charge $m$ and its actual ring distance as a function of $\Delta \rho$. We begin by presenting the results corresponding to the homogeneous polarized DR-POVs in Figs. 6 and 7. As can be seen, the linear $y$-polarized DR-POV profile is not axially symmetric due to the fact that the linear polarization is not axially symmetric with the polar angle $\varphi$ and it has $|2(n-1)|=2$ discontinuities [38]. In contrast, the circular light DR-POV intensity profile is axially symmetric because the circular polarization state possesses this quality. However, as the rings approach the contrast is lower compared to what can be achieved in the linear $y$-polarization $y$=0 cut. In both cases, the shape endurance with respect of $m$ is fairly good, being the circular polarization a little more sensitive to $m$. Finally, the peak distance for the two cases follows a similar behaviour with respect to the paraxial case in Fig. 3(b). However, according to the simulations the minimum ring distance has a different scaling factor and is given by the expression
Fig. 6. a-d) Transversal intensity profiles $z$=0 for the linear $y$-polarized DR-POV with $m$=50 and different $\rho _{1}$, $\rho _{2}$ values. e) Intensity profile as a function of $m$. f) DR-POV actual peak distance versus $\Delta \rho$ for two values of $m$ and NA.
Fig. 7. a-d) Transversal intensity profiles $z$=0 for the right circular polarized DR-POV with $m$=50 and different $\rho _{1}$, $\rho _{2}$ values. e) Intensity profile as a function of $m$. f) DR-POV actual peak distance versus $\Delta \rho$ for two values of $m$ and NA.
Now, we focus on the non homogeneous polarized DR-POVs shown in Figs. 8 and 9. Alike the circular polarization case, the azimuthal and radial polarized DR-POVs are axially symmetric however, in contrast to its homogeneous counterpart, they show rather complex intensity profiles and its shape is very sensitive to the topological charge $m$. It can be seen that the azimuthal polarized DR-POV tends to fuse in a single ring while the radial polarized DR-POV does not so. Noteworthy, the increment in $m$ improves the two-ringed structure appearance in the radial polarization case. This effect is due to the transmittance function Eq. (11), that acquires a bigger dark core as $m$ grows. In turn, less light from low aperture angles arrives to the Fourier plane, improving the beam quality as it is done mechanically in other works [46]. The plots of the actual peak distance confirms that the azimuthal polarized DR-POV withstands only a limited range of $\Delta \rho$ values. On the other hand, the behaviour of the radial case is similar to the homogeneous polarized DR-POVs, however, the minimal ring distance was slightly less reaching 0.21$\mu$m or $0.42\lambda$ when NA=1$\times n_{1}$ is maximum. At low values of topological charge $m$=5, the intensity profiles for the azimuthal and radial polarized DR-POVs presented multiple peaks and the actual peak separation could not be determined. In Fig. 8(f) the peak distance was taken as the distance between the center position of the main lobes.
Fig. 8. a-d) Transversal intensity profiles $z$=0 for the azimuthally polarized DR-POV with $m$=50 and different $\rho _{1}$, $\rho _{2}$ values. e) Intensity profile as a function of $m$. f) DR-POV actual peak distance versus $\Delta \rho$ for two values of NA.
Fig. 9. a-d) Transversal intensity profiles $z$=0 for the radially polarized DR-POV with $m$=50 and different $\rho _{1}$, $\rho _{2}$ values. e) Intensity profile as a function of $m$. f) DR-POV actual peak distance versus $\Delta \rho$ for two values of NA.
We turn our attention to the force exerted by the DR-POV into a metallic Rayleigh particle along the transversal and longitudinal coordinates in Fig. 10. Due to we are interested in axially symmetric movable rings we analyse the circularly polarized DR-POV with $\rho _{1}$=6.8$\mu$m, $\rho _{2}$=7$\mu$m and topological charge $m$=[5, 50]. We consider that the medium is water $n_{\mathrm {med}}=1.33$, $\epsilon _{\mathrm {med}}=1.77$, the laser power is $P$=100mW, $\lambda$=500nm, NA=1.518 and $f$=300$\mu$m. As the particle we choose a gold sphere of radius $a$=50nm with $\epsilon _{\mathrm {par}}$=-2.68-3.19$i$ at $\lambda$=500nm, that is close to its resonance wavelength. Although the particle radius is over the Rayleigh range $\lambda /20$, the Eqs. (14)-(15) give a rough approximate of the force (see curve G, Fig. 2 in [41]). Figure 10 shows that in the transversal coordinate the circular polarized DR-POVs can trap Au particles into their bright rings because in those positions the total force vanishes with a negative derivative. However, in order that the trap is effectively stable in 3D dimension the longitudinal coordinate must also present a stable equilibrium point. It can be seen in Figs. 10(b) and 10(c) that the circularly polarized DR-POV is effectively 3D stable. The stability in all cases is due to a large gradient force component. It is noteworthy the fact that the increment in the topological charge gives a bigger force magnitude around the equilibrium point, contrary to other works [13]. The force magnitude around the equilibrium point is calculated as $\mathrm {min}(|F_{\mathrm {min}}|,|F_{\mathrm {max}}|)$ defined for asymmetric force distributions [47]. For the topological charge $m$=5, the transversal and longitudinal forces reached [0.18, 0.12]pN and for $m$=50 they were [0.3, 1.1]pN, respectively. However, for a more precise analysis are required other methods that take into account the finite particle size and thermodynamic effects.
Fig. 10. Total force $F_{\mathrm {grad}}+F_{\mathrm {scat}}$ experienced by an Au particle $a$=50nm trapped in a circularly polarized DR-POV with radii $\rho _{1}$=6.8$\mu$m, $\rho _{2}$=7$\mu$m at $\lambda$=500nm. a) Transversal plane. b, c) Longitudinal plane.
4. Conclusion
We have proposed a method to generate a DR-POV beam with controllable ring positions using the difference of two coherent Bessel vortex beams with equal polarization and the same topological charges. We investigated the Fourier plane intensity of this beam under the paraxial and tightly focused regimes and found that the two rings are separated thanks to the presence of destructive interference introduced by a $\pi$ phase shift. In the case of homogeneous polarized light, the DR-POV has an excellent shape endurance against the topological charge alike the POV beam. For the non homogeneous polarized light we found that there is a complex interplay between the topological charge and numerical aperture to achieve the best result. In the case of radially polarized light we encountered that the two rings are preserved when $|\rho _{1}-\rho _{2}|\rightarrow 0$, reaching 0.21$\mu$m at 500nm. We demonstrated theoretically that the circularly polarized DR-POV is able to stably 3D trap metallic particles in its two rings when the wavelength is close to the particle’s resonance, opening the door to novel trapping schemes and applications. We think that the DR-POV can be useful in eliciting new fluorescence patterns in STED microscopy, as a hot spot in electric field enhancement for Surface Enhanced Raman Scattering (SERS), the study of the interaction of metallic resonant nanoparticles and even laser drilling.
Acknowledgments
C. Rickenstorff gratefully acknowledges Oscar Martínez Matos for his valuable critics and comments and to Mr. V. V. Kotlyar and Mr. S. S. Stafeev for their theoretical support and the correction of $H_{z}$ expression in circular polarization. GSO aknowledges to SNI and VIEP-BUAP.
Disclosures
The authors declare no conflicts of interest.
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