1. Introduction

Angular momentum possessed by a light beam can be classified as spin angular momentum [1] and orbital angular momentum [2] each due to the polarization states or the helical phase structure of the light, respectively. Researchers used the optical tweezers systems to study the light-mater interaction, and to transfer the angular momentum of light to the trapped particles [3, 4]. The spin angular momentum instigates the rotation of the trapped particle on its own axis and the orbital angular momentum causes the particle to orbit around the beam axis [5, 6]. The orbital angular momentum of light can be transferred through scattering [7] and absorption [3, 8] and the transfer of spin angular momentum is through absorption [8] and birefringence [4]. It is cumbersome to measure accurately the transfer of orbital angular momentum to the trapped particle through scattering. Moreover the transfer of spin angular momentum by birefringence depends on the polarization change of the incident light after passing through a birefringent particle, which again depends on the thickness of the birefringent particle and wavelength of light. Absorption is a common and uniform mode of transfer for both orbital and spin angular momentum of light.

The orbital angular momentum transferred to a trapped particle in optical tweezers can be intrinsic and extrinsic in nature contrary to the spin angular momentum, which is always intrinsic [5]. In optical tweezers the intrinsic and the extrinsic nature of the orbital angular momentum is defined based on the position of the trapped particle. If a particle is trapped at the center of the beam (on-axis) the orbital angular momentum transferred is referred as intrinsic and if the particle is trapped at off-axis, the transfer of orbital angular momentum is extrinsic [5, 6]. Hence it becomes significant to determine the position of a trapped particle as it affects the transfer of angular momentum of light. When the particles are absorptive in nature and on-axis trapping is desired then both the spin and orbital angular momentum are intrinsic in nature and act in an interchangeable fashion and hence can be added or cancelled from the total angular momentum [8, 9].

Trapping of metallic particles having complex refractive index was achieved by scanning laser beam (TEM₀₀ mode) in circular fashion [10]. Recently researchers proposed optical spanner [8] or optical tweezers systems using singular beam to trap absorptive particles using LG beams. The absorptive particles are trapped at central dark or minimum intensity region of the singular beams. Using optical spanner different combinations of angular momentum was transferred [8] like orbital plus spin angular momentum, orbital minus spin angular momentum and pure orbital angular momentum. However the transfer of ‘pure’ on-axis spin angular momentum to the trapped absorptive particle is not possible using conventional optical spanner, as the default orbital angular momentum possessed by the singular beam would always be transferred to the trapped particle. Recently the transfer of pure orbital angular momentum (decoupling of orbital and spin angular momentum) to the metallic particle was reported, where the mode of transfer of angular momentum was scattering [11]. However, using highly converging Gaussian beam the trapping of an absorptive particle against the sample stage was reported [12–13].

We propose a novel optical tweezers system using a self-imaged bottle beam to trap an absorptive particle and to transfer the ‘pure’ on-axis spin angular momentum. Our results distinguish from the work reported in Ref. [12–13] as the absorptive particle was trapped at the central dark spot of the self-imaged bottle beam and we exploit the inverted optical tweezers system. The trapping efficiency of an absorptive particle at null-intensity point is higher than the Gaussian beam as explained in Ref. [12]. Furthermore, the self-imaged bottle beam belong to the family of propagation-invariant beams [14] and hence inherent the interesting property of self-reconstruction. The self-imaged bottle beam [15–17] has embedded optical bottles or null intensity points along the propagation, which are used to trap and to transfer the ‘pure’ spin angular momentum to the absorptive particles. This bottling effect of the self-imaged bottle beam is due to the destructive interference of the superimposed Bessel beam and not due to the singularity, hence there is no default orbital angular momentum carried by the self-imaged bottle beam.

2. Self-imaged bottle beam

Gratings illuminated with a highly spatially coherent plane wave produce self-images and self-image-like field distributions in certain planes behind the grating. This self-imaging phenomenon is also known as the Talbot effect [18]. The self-imaging effect is also explained as the coherent superposition of non-diffracting beams [19]. The self-imaging effect was further used to generate self-imaged bottle beam [17]. This bottling effect appears when two or more superimposed beams interfere destructively to generate bottles or a null intensity points at the center of the beam. As described in Ref. 17 the intensity of the self-imaged bottle beam can be given as: -

The Bessel beams superimposed in the above equation are of the zeroth order and the propagation constants of Bessel beams are k_r1, k_r2 and k_z1, k_z2. A’ is an amplitude factor, generally taken as 1 and Δ is an arbitrary additional phase change in the optical system.

We fabricated efficient holograms to generate the self-imaged bottle beams. The phase distribution required for a self-imaged bottle beam is taken from Eq. (1). After the quantization of the phase values into binary, e-beam lithography was employed to generate a mask followed by fabrication using UV exposure. We used Q-2001CT mask aligner (Quintel Corporation) with wavelength 365nm and irradiance of 15mw/cm² to expose the photoresist film (AZ5214). Since the speed and duration of the spin coating, the exposure time and the development time determine the profile heights of the photoresist film; we calibrated these parameters to get a precise phase distribution on the film. The adopted parameters in our fabrication process were exposure duration of 15s, spin speed of 3500 rpm, development time of 15s. The hologram made of photoresist is a phase-only type, which would have higher diffraction efficiency than its transmission counterpart.

Free-space propagation of the self-imaged bottle beam generated from the hologram is shown in Fig. 1. The self-imaged bottle beam varies its transverse intensity profile periodically with alternative bright and dark central spot. One full period of the self-imaged bottle beam is shown in Fig. 1 where the bottle is obtained in Figs. 1(c–e) and the bright spot is shown in Figs. 1(a, b, f). As reported in Ref. [17] we obtained the self-imaged beam with many bottles embedded in it. Fig. 2 shows a movie (sequence of the still images) of the free-space propagation of the self-imaged bottle beam embedding two bottles.

3. Experiment

We used a self-imaged bottle beam in optical tweezers system for the first time to the best of our knowledge to trap an absorptive particle. As shown in Fig. 1 the transverse intensity profile of the self-imaged bottle beam varies periodically along the propagation axis contrary to the beams used in the conventional optical tweezers system (Gaussian, LG and Bessel beams). Hence it becomes significant to image the correct point of the self-imaged bottle beam onto the sample stage to obtain desired trapping results. Figure 3(a, b) shows two distinct transverse intensity profile of the self-imaged bottle beam imaged at the sample stage, which could be used for different trapping applications. Figure 3(a) corresponds to the bright spot of the self-imaged bottle beam as shown in Fig. 1(a), which is similar to the intensity profile of the Bessel beam and hence can be used to trap and levitate high-index microparticles [20]. Figure 3(b) represents the dark spot or the destructive interference point of the self-imaged bottle beam imaged onto the sample stage corresponding to Fig. 1(d), which can be used to trap the absorptive particles.

 

Fig. 1. Free space propagation of the self-imaged bottle beam. One full period of the self-imaged bottle beam is shown above with the bottle obtained in Fig. (c-e).

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Fig. 2 (1.32 Mb) Movie of free-space propagation of the self-imaged bottle beam. The movie shows the self-imaged bottle beam embedding two bottles.

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Fig. 3. Transverse intensity profile of the self-imaged bottle beam imaged at the sample stage, (a) Bright spot, (b) Dark spot, respectively.

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Figure 4 shows the experimental set-up of the optical tweezers system using a self-imaged bottle beam. The light from the Nd:YVO₄ infrared laser (1 W, 1064nm) was expanded ten times by the lenses f ₁ and f ₂ before incident to the quarter waveplate and the fabricated hologram. The circular polarized self-imaged bottle beam was telescoped down by ten times using telescopic arrangement (lenses f ₃ and f ₄) to the sample stage to reduce the central spot of the beam to about 8~10 μm. An x-y translation was provided to the last lens (f ₄) to manipulate the beam spot over the sample stage and the x-y-z translation was provided to the sample stage. An IR-filter was used before CCD to observe clearly the rotation of the trapped particle. The schematic diagram shown in Fig. 4 illustrates the oscillating intensity profile of the self-imaged bottle, where black regions of the beam shows the destructive interference and hence the dark spots. By varying the distance between the hologram and the lens f ₃ it was possible to image different points of the self-imaged bottle beam onto the sample stage as shown in Fig. 3. For the trapping of an absorptive particle the point shown in Fig. 3 (b) was used in the experiment, which is the destructive interference point and highlighted as point A in Fig. 4. The parameters used for the hologram (k_r1=1/pixel and k_r2=0.2/pixel and size of each pixel was 15 μm) gave bottling effect for a length of 2cm in the free-space propagation and using ten times telescopic arrangement this bottling effect was reduced to a distance of 0.2mm at the samples stage. By varying the radial wavevectors (k_r1 and k_r2) of the self-imaged bottle beam we can pre-determine the bottle dimensions and location [17].

 

Fig. 4 Experimental set-up of a self-imaged bottle beam based optical tweezers system. Absorptive particles were stably trapped in the bottle (Point A) and the transfer of spin angular momentum was accomplished.

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Copper oxide (CuO) was used as the probing absorptive particle of diameter around 4~6 μm and little detergent was added to the sample to increase the mobility of the particles. The self-imaged bottle beam is propagation invariant beam where the diffraction of the central core is negligible as compared to the whole beam. The bottling effect appears in the central core of the self-imaged bottle beam (as shown in Fig. 1(d) and Fig. 3(b)) and it generates 3-dimensional dark spot contrary to the singular beams like LG or higher-order Bessel beam [5–9]. Hence using the inverted optical tweezers system we obtained the transverse trapping of an absorptive particle at the central dark spot of the self-imaged bottle beam by imaging the embedded bottle on the sample stage. By introducing the quarter waveplate the spin angular momentum of circular polarized light was transferred to the CuO particle due to the absorption of light energy by the particle causing its rotation around its own axis as shown in Figs. 5(a–f). The rotation of the CuO particle was due to the transfer of the ‘pure’ spin angular momentum of the light as a self-imaged bottle beam does not possess any singularity and hence no transfer of the default orbital angular momentum of the light. The torque on the absorptive particle is given as: -

where P_abs is the power absorbed by the particle and σ_z is +1and -1 for right and left circularly polarized light, respectively and 0 for plane-polarized light and ω is the frequency of the light. The torque experienced by the particle allows it to overcome the drag force and a particle rotates with constant angular velocity.

 

Fig. 5 (a–f) Transfer of ‘pure’ on-axis spin angular momentum from a circular polarized self-imaged bottle beam to an absorptive particle.

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4. Discussion

Allen and Padgett recently put across an important question where they reported the possibility where the matter (preferably an atom as a probing particle) may respond to the local value of spin angular momentum, which could be varying in magnitude and sign at different positions of the wavefront when spin angular momentum is transferred using singular beams [21]. They validate this based on the fact that the spin angular momentum depends on the intensity gradient, which is influenced by the singularity of the host beam and depends on mod (l), where l is the topological charge of singularity. Using the self-imaged bottle beam optical tweezers we propose a novel way for on-axis trapping of an absorptive particle and eliminates the default dependence of spin angular momentum on the singularity of the host beam. However, it could still be possible that an atom trapped in a self-imaged bottle beam could respond to the local spin angular momentum at various positions of the beam, as the bottling effect is obtained due to the destructive interference of two superimposed Bessel beams which might causes different intensity gradient at various locations of the wavefronts. Although as suggested in Ref. [21], for the particles of size in micrometers the observation of such local spin angular momentum is not substantial and hence, the self-imaged bottle beam based optical tweezers system can be used for the transfer of ‘pure’ on-axis spin angular momentum to the microscopic absorptive particles.

5. Summary

In conclusion, we propose and validate the novel optical trapping system using self-imaged bottle beam where an absorptive particle was trapped and the transfer of spin angular momentum was accomplished without the default transfer of the orbital angular momentum possessed by the singular beams used conventionally to manipulate the absorptive particles.