1. Introduction

Optical vortices, also known as phase singularities, are ubiquitous in many areas of optical physics. They denote regions in space where three or more light fields interfere and cause a region of intensity minimum. At such points the phase is singular in nature hence the term phase singularity [1, 2]. Such singularities are not restricted to the optical domain but may be found in fluid dynamics [1], Bose-Einstein Condensates (BEC) [3] and crystallography [4] amongst other areas. Optical singularities are typically associated with light fields that possess inclined wavefronts: it is this inclination that results in an azimuthal component of the Poynting vector and leads to orbital angular momentum in the light field.

Fundamental studies with optical vortices in monochromatic light fields have led to numerous physical insights into the orbital angular momentum of light (OAM) [5, 6], wavefront studies [7] and spatial and temporal coherence studies [8, 9]. Optical vortices have found novel applications such as quantum entanglement [10], phase contrast imaging [11], optical micromanipulation [12] and optical speckle fields [13]. In particular, optical vortices have had a strong impact in the field of optical micromanipulation [6, 12, 14–19]. Optical micromanipulation is established as a powerful technique by which we may influence the motion of microscopic particles through the use of the gradient and scattering forces. In the most popular manifestation, optical tweezers [20], particles are confined to the regions of highest light intensity. The use of light fields with embedded vortices in optical micromanipulation has led to the study of optical angular momentum of light, in particular, for Laguerre-Gaussian (LG) [6, 14] or high order Bessel light fields [18, 19]. Colloidal microparticles may be trapped in such vortex beams due to the gradient force and may rotate around the annular intensity profile of the light field due to the angular momentum transfer by from scattering or absorption [14, 18, 19]. Typically we may controllably load the annular intensity profile of single ringed LG beams with a known number of colloidal particles. Notably studies into multiple forward scattering from optically trapped colloids held in such an optical vortex beam have so far not been performed. This is a potentially an important area as the far field interference pattern from multiply trapped colloids contains key information that correlates to the phase structure and helicity of the field (vortex beam), and indeed yields key signatures relating to the coherence properties of the trapping light.

In this paper, we investigate the interference effects originating from the forward scattering of multiple colloids (two and more) optically trapped in a Laguerre-Gaussian beam. We study how the forward scattering light field from each colloid interferes and yields rich intensity patterns in the far-field, through both numerical simulation and experiments. The far-field interference patterns are found to have direct correlation with the azimuthal phase and helicity of the trapping beam in the absence of such prior knowledge. A spiral wave pattern is formed when the annular intensity ring of the LG beam is completely filled with colloids. When two colloidal spheres are optically trapped at arbitrary points around the circumference of such a LG mode, we realize a variation of Young’s slits type experiment at the microscopic scale that allows us to record the azimuthal phase variation around the beam circumference in situ. This therefore also yields a signature of the spatial coherence of the vortex light field. Furthermore, we demonstrate that when three or more colloids are trapped in the LG beam, a two-dimensional landscape of unity charge vortices is formed in the far-field.

2. Numerical simulation

In this work, each microsphere being trapped in the LG beam generates its own diffracted wave that propagates to the far field. We numerically model the far-field diffraction of the microspheres located in a Gaussian beam adapting the approach of Chevaillier et al. [21]. We treat the microsphere as a diffracting disk in the Mie Regime and calculate the intensity pattern for the case of the sphere diffracting a Laguerre-Gaussian beam. In contrast to previous studies that treat solely a single microsphere in Gaussian beam, our model extends to numerous colloidal particles positioned on the annular intensity ring of the LG beam, thus taking into account multiple interference between the microspheres confined in the annular intensity rings of the LG beams.

Laguerre-Gaussian beams are denoted by two integer indices p and l: p+1 denotes the number of radial nodes whereas l denotes the number of cycles of azimuthal 2π phase shift around the mode circumference. Single ringed LG beams (p=0) are considered in this work. A key characteristic of an LG beam or optical vortex field is its unique helical phase structure, denoted in the mode description by e^ilϕ where ϕ is the azimuthal phase variation around the circumference of the optical field and l denotes the number of cycles of azimuthal 2π phase shift around the beam [5, 6]. To simulate the propagation dynamics of a single ringed LG beam filled with microspheres, we make use of numerical calculations based on Fourier optics and the analytical equation of a LG beam diffracted by a disk. The numerical method works on the proposed Fourier decomposition of a given amplitude distribution into plane waves in x, y and z. The plane wave of each vector space is propagated a certain distance and then a superposition formed to realize the far-field intensity pattern. This technique has been acknowledged as a useful and important tool for detail analysis of wave propagation [22]. The final mathematical expression of a propagating light field E(x ₀,y ₀,0) propagated towards the far-field at incremental z values, ∆z is given as

where k_x, k_y are the transverse wave vectors, the longitudinal wave vector $k_z=\sqrt{{\left(\frac{2\pi }{\lambda }\right)}^2-k_x^2-k_y^2}$ where λ is the wavelength of the monochromatic optical field.

Hence, by replacing the E(x ₀,y ₀,0) with an analytical expression for an LG beam containing diffracting disks, we are able to numerically simulate the propagation of the LG beams diffracted by microspheres towards the far-field in free space. By using a split-step Fourier method based on Eq. (1), we numerically propagate the LG beam containing microspheres in its annular intensity ring towards the far-field (Fraunhofer plane) with increments of z values, given as ∆z. Since the microspheres are well described in the Mie Regime and the scattering angle is small, we consider mainly diffracted rays from the microspheres in the simulation. The scattering geometry is illustrated in Fig. 1. The result of the numerical simulation of nine microspheres scattering the trapping light is presented in Fig. 1(B) to Fig. 1(D).

 

Fig. 1. (A) shows the scattering geometry of nine disks placed off the central axis of the LG₀₃ beam (red annular ring). b_o denote the beam waist of the LG beam and d denotes the distances from the beam to the microsphere (treated as a diffracting disk) and Z_R denotes far-field. The scattering geometry are calculated in Cartesian coordinates (x,y,z). (B) to (E) shows the numerical simulation of interference between the nine Mie spheres placed off the beam axis of LG₀₃ beam as it propagates to the far-field, Z_R denotes far-field.

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3. Experimental setup

Figure 2 shows the experimental setup. We make use of an expanded single ring LG beam profile within which we trap the colloidal particles. The experimental setup, shown in Fig. 2, is carried out without the use of high numerical aperture objectives. Instead we use a planoconvex lens (L1) to focus a low divergence LG beam onto the sample plane. An approximation to a single ringed LG beam is generated using a highly efficient blazed computer generated hologram (CGH) [18] on glass, illuminated by a 1W, TEM₀₀ beam at wavelength of 1070 nm laser beam (10W, CW, Yb-fiber laser (IPG Corporation)). The LG beams generated for the experiments described are of radial index p=0, azimuthal index l=1 and l=3. The LG beam is expanded to fill up the back aperture of the plano-convex lens, L1, and hence achieve a focused spot with diameter of around 15μm. The role of the lens (L2) is to relay the scattered light that is collected by a 60X microscope objective (OB, NA = 0.85), onto the observation plane (CCD camera). By shifting the position of the relay len, we can image the optically trapped particles in the LG beam. We can also observe the different propagation plane towards the far-field (Fraunhofer diffraction patterns) using the relay lens.

 

Fig. 2. Experimental setup. A TEM₀₀ beam is directed onto a high efficiency blazed Computer Generated Hologram (CGH) that creates a LG beam in its first order of azimuthal index l=3 and p=0 that diffracts at an angle to the incident beam. The LG beam expanded to fill approximately the back aperture of the plano-convex lens, L1 of focal length 25.4mm. The diffraction pattern is collected from OB, 60X microscope objective (NA = 0.85), and relay far-field diffraction pattern of the colloids onto the imaging CCD through a second plano-convex lens, L2 of focal length 50mm, at its focal plane (Fraunhofer diffraction patterns). Inset: LG beam (l=3 and p=0) trapping 9 spheres, each of diameter 6.84μm.

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The optical power required to trap a circular arrangement of colloidal microparticles is typically around 100mW. The colloidal samples used are silica microspheres of 6.84μm diameter dispersed in a D₂0 solution to reduce heating effects and placed in a cylindrical sample chamber of diameter 1cm and depth 100μm. The size of the microsphere is chosen to approximately match the width of the annular intensity profile of the LG beam. The trapping light field is almost total covered by the microspheres. The inset of Fig. 2 shows a LG beam (l=3, p=0) trapping nine silica colloids of 6.84μm around the beam circumference.

By translating the beam or sample stage, we are able to controllably load the annular beam profile of the LG mode with colloidal particles ranging in number from two to nine. Transfer of the orbital angular momentum of light through scattering is observed (only when the annular rings are totally filled) but at a very slow rate due to our enlarged trapping beam and relatively low power density for these given sphere sizes [19]. Thus we can experimentally adjust the colloids position (static) at will around the LG beam profile as shown in Fig 3(A) Fig 3(E).

 

Fig. 3. (A-E) , LG beam (l=3, p=0) trapping 2 to six silica colloids of 6.84μm around its annular intensity profile in a controlled manner using optical gradient forces.

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4. Results

In this section, we present interference patterns observed beyond the plane of our trapped colloids. We examine the patterns observed at approximately half the Rayleigh range of the light field and then we further present interference patterns observed in the back focal plane (Fraunhofer plane) when forward scattering of multiple colloids interferes in the far-field. Firstly, for a completely filled annular ring with colloids, a spiral wave pattern is observed at half of the Rayleigh range. Secondly, when three or more colloids are trapped, we observed an optical vortex landscape in the Fraunhofer plane. Finally, by comparing the far-field interference pattern from two trapped colloids in a Gaussian and an LG beam, we are able to observe a manifestation of the famous Young’s slits experiment with optically trapped particles and observe features related to the azimuthal phase difference around the LG beam circumference.

4.1 Spiral wave pattern

We investigate the interference pattern of the colloidal geometry where nine colloidal microsphere (each of diameter 6.84μm) fills up the annular intensity rings. The central diffraction pattern of the LG beam (l=3, p=0) beam diffracting from a fully filled microsphere are traced at z_R/2, where z_R is the Rayleigh range.

 

Fig. 4. shows the simulation and experimental results of the spiral wave pattern. (A), (C) (simulation) and (B), (D) (experiment). The spiral intensity arms indicate the helicity of the trapping light. (A) and (B) are for as trapping beam of l=+3, (C) and (D) l=-3. The nine spiral arms correspond to the nine colloids trapped along the annular intensity profile, denoted by the red lines.

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From Fig. 4, we notice that (through simulation and experiment) the number of spiral arms is directly correlated to the number of colloids being trapped within the LG beam. In this case, there are nine colloids being trapped around the LG beam (l=3, p=0) thus resulting in nine spiral intensity arms. Upon closer examination, we notice a distinct similarity to the spiral interference pattern typically observed between a LG beam and a plane wave [12]. We further investigate the behavior of the pattern by reversing the helicity of the LG beam. By reversing the helicity of the LG beam (l= -3, p=0) using a dove prism, the spiral intensity profile is itself reversed, as shown in Figs. 4. (A), 4(B) (for positive helicity) and Figs. 4(C), 4(D) (for negative helicity). The number of spiral arms is directly indicative of the number of single colloids being trapped around the intensity ring profile. In this case, we have nine colloids being trapped, corresponding to the nine spiral intensity arms. The distinct spiral winding direction (indicated by the red lines) that is formed directly indicates the helicity of the LG beam and this is further verified by numerical simulations.

4.2 Two dimensional optical vortices light field

Optical vortices are formed by the superposition of three or more plane waves [2]. Recently, spatial light modulators [23, 24] have been used to generate controlled multiple beam interference which in turn can generate two or three-dimensional landscape of propagating vortex lattices. In this section, we show that the interference pattern of the far-field interference from multiple colloids can generate a two dimensional landscape of optical vortices. Here we explore the patterns observed in the far field at the back focal plane of the system. Again we load between two to nine particles into the LG beam (l=3, p=0) beam. The interference of the forward scattering of the trapped colloidal crystals generates interesting interferometric patterns. Figure 5(A) to Fig. 5(F) show that the far-field intensity patterns consist of a large array of patterns with regions of high and low intensity, ranging from simple interference fringes to complex intensity landscapes. The shows data from numerical simulation (intensity (i), phase (ii)) and experimental (intensity, (iii)) data side by side.

 

Fig. 5. (A, B,C, D, E) shows the (i) simulation of the intensity , (ii) simulation of the phase map and (iii) experiment of the far-field diffraction pattern of two, three, four, five, six microspheres filling up the annular intensity ring respectively. (F) shows the full circular colloidal crystals formed in the annular intensity ring, fully filled with nine spheres (see inset of Fig. 2). The red spot & dotted lines denote the high intensity regions in the far-field pattern on the simulation and experimental results. On the other hand, the low intensity regions (around the high intensity regions) contain several unity charged vortices.

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Figures 5 (A)–(F) (i) & (iii), show comparison of the intensity pattern of the experimental data with the simulation result and good agreement is observed. Notably in Fig. 5(A), we see that when two colloids are trapped on opposing sides of the LG beam (see Fig. 3(A)), the far-field diffraction exhibit linear intensity fringes rather akin to a Young’s slits experiment at the microscopic scale. When three colloids are trapped at about 60° to one another around the circumference, they form a triangular crystal structure (see Fig. 3(B)). The far-field diffraction forms a two-dimensional optical vortex lattice, when matched to the phase plot as shown in Fig. 5(B). For Fig. 5(C), we trapped two sets of colloids (each containing two microspheres) 180° from each other (on each side of the beam), (see Fig. 3(C)) and a tilted two dimensional optical vortex lattice forms in the far-field. In Fig. 5(D), we show the formation of fork-like intensity formed in the far-field when two sets of colloids (one containing two microspheres and the other contain three microspheres) are being trapped on opposing sides of the intensity profile (see Fig. 3(D)). Likewise, when two set of colloids (each comprising of three microspheres) are being trapped on opposing sides of the intensity profile (see Fig. 3(E)), intensity rings coupled with snake-like intensity patterns at its side are formed in the far-field as shown in Fig. 5(E). In Fi.g 5(F), we show that when a circular colloidal structure is being formed within the annular intensity profile of the LG beam (see inset Fig. 2), the diffraction patterns shows circularly arranged intensity spots around an annular intensity profile.

Through the phase plot from the numerical simulation, we further noticed that when 3 or more spheres are loaded symmetrically around the ring of the optical vortex beam, the intensity pattern in the far-field displays periodic areas of minimum intensity points. From the simulation of the phase plot, we can trace the intensity minimum point to the location of the “fork” within the interference pattern (the tilted phase profile of the far-field interference field), which is the location of an optical vortex. Hence in short, the far-field intensity of an LG beam (l=3, p=0) beam diffracted by 3 or more colloidal particles trapped within annular intensity pattern contains a large amount of unity and opposite charge vortex pairs. This is seen in Fig. 5 (B) (ii) to Fig. 5(F) (ii). Therefore, from the simulation and experimental results, we show that the diffracting light fields from the microspheres are very much like numerous individual spherical beams that interfere as plane waves in the far-field. Thus by simply controlling the trapped position of the colloids in a LG beam, a series of complex two dimensional optical vortex landscapes can be generated, without the need to employ any complex mathematical algorithm or any complex multiple beam interferometer setups.

4.3 Young slits experiment with optically trapped particles

In Fig. 5(A), we have observed a microscopic version of Young’s double slit experiment here performed by two optically trapped colloids. Thus, we further investigated these interference intensity fringes, where the two colloids may serve as the phase sampling points within the beam profile. The far-field interference pattern from the two colloids (which acts like a microscopic self aligned aperture) may provide direct correlation to azimuthal phase shift in the LG beam. To verify this, we first trap two colloids within a standard Gaussian beam (with approximate zero azimuthal phase shift) and then two colloids within an LG beam (l=1, p=0) (with a single helical phase ramp from 0 to 2π around the beam circumference) and observe the far field interference pattern in each case. The optical trapping system permits us to place the two objects at will at any azimuthal position within the LG beam (l=1, p=0) beam profile, as shown before in Fig. 3.

Young’s slits is a map of the spatial coherence of a light field that is well known to result in co-sinusoidal fringes modulated by a sinc function [25]. Thus with the added azimuthal phase variation from the LG beam itself, dependent upon the relative position of each sphere around the ring, as illustrated in Fig 6, we predict the observation of linear fringes shifted by π due to the azimuthal variation around the beam circumference. By observing the far field interference pattern of a Gaussian beam (Fig. 7(C)) and a LG beam (l=1, p=0) (Fig. 7(B)), where each of the beam has been aperture by two 6.84μm silica spheres, we are able to detect the effect of the displacement and separation of the intensity fringes solely due to the azimuthal phase variation. Since the intensity of the superposition of two fields in a Young’s double slit is directly proportional to ${\cos }^2\left(\frac{\alpha }{2}\right),$, the intensity variation of the double slit interference fringes with an extra phase shift may be given by:

where α is the optical phase difference of the two apertures (which are the 6.84μm silica spheres) and Δϕ is the additional azimuthal phase difference between the two colloids.

 

Fig. 6. is a schematic illustration of two spheres trapped on opposite sides of a LG beam with a linear helical phase ramp varying from 0 to 2π (l=1) around the mode circumference

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In both simulation and experiment, the interference fringes are obtained in the same way as described in previous section. However in this instance, we made use of a LG beam (l=1, p=0). This yields a 2π linear azimuthal phase ramp around the beam circumference. The small separation between the colloids, due to the smaller annular intensity profile of the LG beam (l=1, p=0), means we are able to obtain a relatively similar pitch of interference fringes compared to interference fringes from the Gaussian beam. The experimental and numerically evaluated interference fringes are shown in Fig. 7. We see that the interference fringes are shifted by π, compared to a Gaussian beam, when an LG beam propagates through the two colloids. Though there is a strong central intensity due to the propagating Gaussian and LG beam in each instance, it is clear that from the diffracting interference fringes; an obvious π phase increment is observed as expected (emphasized using the dashed red line). We also notice a slight tilt in the interference fringes, which we attribute to the finite sampling area of the beam due the finite size of each colloid. The shift in the fringe correlates to the relative phase difference sampled by the two colloids is π as shown in Fig. 6. Experimentally, we have found that the average fringe visibility to be 0.67. Thus illustrates a high fringe visibility/contrast and stability of the interference fringes.

By simply trapping two micron spheres colloids (each of which serves as a micron-sized aperture) and analyzing the far-field interference pattern, one may potentially map out the azimuthal phase of an arbitrary complex beam. The phase mapping resolution may benefit from improvement with use of different sphere sizes. We propose that such a microscopic optical trapping interferometry technique may be used to map out the localized optical phase structure of a more complex beam such as a modulated optical vortex [26], higher-order Bessel beam [18], Mathieu beams [27] and Ince-Gaussian beams [28]. Additionally such controlled phase sampling techniques using optically trapped colloids as micron-sized apertures may be implemented to probe the spatial coherence of polychromatic light fields [29] in an optical trapping system.

 

Fig. 7. Microscopic Young’s slits with optically trapped particles. (A) and (B) shows far-field diffraction (simulation) of two colloids trapped in a Gaussian beam and a LG beam (l=1, p=0), (C) and (D) shows the experimental results respectively. Red line indicates the shift in the fringes due to the π phase between the two opposing points in the LG beam.

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5. Conclusion

In conclusion, we have shown that colloidal particles trapped within a LG beam create interesting light patterns towards the far-field, some of which have direct correlation with the azimuthal phase of the trapping beam. As the annular intensity ring is filled up with the colloids, a unique spiral intensity pattern is formed that is characteristic of the helicity of the trapping beam. More importantly, by simply analyzing the far-field interference of two individual colloids trapped within a LG beam, we have realized for the first time, a microscopic version of the famous Young’s slits experiment which is able to detect localized azimuthal phase difference between two opposing points on the beam circumference. By simply controlling the trapped position of three or more colloids in a LG beam, a series of complex two dimensional optical vortex landscapes can be generated. Furthermore, in contrast to previous work [30], our study represents the first study of the far-field diffraction patterns from controlled loading of colloidal particles in a LG beam through the optical trapping technique. Future studies could focus on the experimental mapping of the phase structure and spatial coherence of more complex light fields using the optical trapped colloids as microscopic Young’s slits.