1. Introduction

The dipole or gradient force of light is at the heart of the field of optical micromanipulation. The most important manifestation of this force is in the single beam optical trap, known commonly as “optical tweezers” [1–3], which is now an established tool across a wide range of sciences [4]. It has permitted innovative studies in the biosciences, particularly in molecular motors [5]. Optical traps may be multiplexed using interferometric, acousto-optic or holographic methods to create many trap sites simultaneously [6–10]. In such extended arrays of traps one typically ignores the light redistribution by the individual trapped microparticles (e.g. by scattering and/or refocusing) and the consequences for other microparticles in close proximity. However, this very interaction can lead to the phenomena of ‘optical binding’ whereby this light redistribution actually dictates the equilibrium position of each sphere in the array. Two versions of this phenomena have been observed, namely backscattering of an incident laser leading to very close (1 micron or less) arrangements of microparticles [11], and forward scattering of light in a counter-propagating (CP) beam geometry has resulted in another form of binding where particles are held up to ten times their diameter away from one another [12]. To date this has been backed up with qualitative modeling [13–15] and an observation of bistability in the trap positions [14]. In particular, this form of binding has implications for the creation of small optically trapped crystals and for co-operative studies in colloidal matter. It also presents an interesting and key challenge both from a theoretical and experimental standpoint in optical micromanipulation.

In this paper, we present data on one-dimensional arrays of microparticles in a dual-beam fiber optical trap. Our primary goal in this work is to perform an in situ visualization and analysis of the optical binding phenomena. We achieve this with the development of the first ever dual-beam fiber trap operating with a femtosecond pulse laser. The trap operates in an environment with a fluorescent dye added to the host medium. Two-photon excitation of the dye by the ultrashort pulsed trapping laser [16] permits us to map the light redistribution of around each trapped microparticle. We can thus observe the binding process in real time and a comparison with a numerical model is presented. To proceed, we first discuss our physical model that underpins our subsequent experimental observations.

2. Theory

Our model comprises two monochromatic laser fields of frequency ω counter-propagating along the z-axis which interact with a system of N transparent dielectric spheres of refractive-index n_s , radius R , and which are immersed in a host medium of refractive-index n_h . The monochromatic electric field is expressed as a sum of positive and negative frequency components as

where x̂ is the unit polarization vector of the field, ε_±(r⃗) are the slowly varying electric field amplitudes of the right or forward propagating (+) and left or backward propagating (-) fields, and k = n_hω/c is the wave-vector of the field in the host medium. The incident fields are assumed to be collimated Gaussians at longitudinal coordinates z = 0 for the forward field and z = D_f for the backward field

where r ² = x ² + y ², w ₀ is the initial Gaussian spot size, P ₀ is the input power in each beam, and D_f is the distance between the fiber ends. It is assumed that all the spheres are contained between the beam waists within the length D_f ≫ R. For the case of a single field we choose the forward propagating field envelope ε₊(r⃗).

In the present work we assume that the dielectric spheres, representing the trapped microparticles, are held at fixed positions {r⃗_j }, j = 1,2,...N. The dielectric spheres then provide a spatially inhomogeneous refractive-index distribution, which is of the form

where θ(R-|r⃗ - r⃗_j |) is the Heaviside step function which is unity within the sphere of radius R centered on r⃗ = r⃗_j , and zero outside. Then following standard procedures [13] the CP fields are found to obey the paraxial wave equations

where ${\nabla }_{\perp }^2$ = ∂²/∂x ² + ∂²/∂y ² is the transverse Laplacian describing beam diffraction. The paraxial wave equations (4) are to be solved subject to the boundary conditions (2), and we have done this using the split-step beam propagation method. We remark that our equations are valid in the Mie size regime where the sphere diameter is larger than the wavelength 2R > λ [1]. Furthermore, in keeping with the experimental conditions considered here the refractive-index difference between the sphere and host medium is small, Δn = (n_s-n_h) <0.1, meaning that there is negligible backscatter from the spheres, and the scattering is dominantly in the forward direction, the so-called Mie effect [17]. More specifically, the focal length for a sphere in the small-angle approximation neglecting higher-order aberrations is f = R/ (2Δn) [18], and since the sphere only focuses rays that pass through it within the sphere radius R away from the axis we can introduce an effective numerical aperture for the sphere NA = sin (θ_max) = R/f = 2Δn, with θ_max the maximum ray deflection angle due to the sphere. Since Δn < 0.1 for the experiments and simulations presented here, the spheres act as low NA < 0.2 focusing elements that can be well treated using scalar paraxial theory. This is true since the maximum deflection angle θ_max ≪ 1 due to the sphere is small, so that initial paraxial rays will remain paraxial and the incident polarization state of the field will be mainly unchanged.

In this paper, our main interest is in the field distributions involved in optical binding as opposed to the self-consistent determination of the micro particle spacing, so the positions of the sphere centers z_j , j = 1,2,...N are taken as input parameters from the experiment for the cases of both optical binding or by a ‘helper’ optical tweezers with its axis aligned orthogonal to the z-axis. Since the CP fields are assumed incoherent, the total field intensity is I_field (x,y,z) = |ε ₊(x,y,z)|² + |ε _-(x,y,z)|². In the experiment the detected quantity is the two-photon fluorescence signal from the fluorescein as imaged and collected along the x-axis. The experimentally detected two-photon fluorescence signal, adapted from [19], is then proportional to

and we use this to compare the predicted fluorescence profiles from the numerical simulations with the experimental measurements. Although our model cannot predict the absolute magnitude of the fluorescence signal, comparing the observed and numerical spatial profiles of the fluorescence provides information of the beam profiles. In what follows the calculated fluorescence intensity distributions in the (y-z) plane are shown as false color images with the maximum field strength being normalized to the color red.

3. Experiment

A dual-beam fiber optical trap was used for all experimental studies [20]. A Titanium-Sapphire (Ti:Sa) femtosecond laser at a central wavelength of λ = 800 nm (p-polarized) with 95 fs output pulses at a repetition rate of 80 Mhz, average power ∼1 W (pulse energy of 12.5 nJ, 131 kW peak power), was used to operate the fiber trap. The light was coupled into two single mode fibers (Thorlabs 780HP for 780 to 970 nm; mode field diameter 5.0±0.5μm at 850 nm; Numerical Aperture 0.13; Attenuation <3.5 dB/km at 850nm) via a λ/2 plate and a polarizing beam splitter. The optical power emerging from each fiber could be adjusted with the neutral density filters to ensure equal field distribution of 40 mW (see Fig. 1).

 

Fig. 1. Fiber optical trap setup: Light at 800 nm from a Ti:Sa femtosecond laser is coupled via ND filters into fiber F1 and F2 to ensure equal power distribution. Inset shows fiber trap side view: The array is formed in the gap between the two fibers (F1 and F2) with D being the separation of the spheres and D_f the fiber separation. A second helper tweezers is coupled into the observation microscope via dichroic beam splitter to hold a sphere in the beam or to initiate the array. Images were taken through the microscope via the CCD camera in front of which a lens could be flipped to achieve varying image magnification.

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By choosing the optical path difference (5 cm) of the CP beams larger than the laser coherence length (calculated pulsed laser coherence length < 50 μm), standing wave effects [21] were avoided. One fiber (F1) was mounted on a cover slip which was in a fixed positioned above the imaging setup, the second fiber (F2) was mounted on a XYZ stage and could be aligned with F1 much like in a fiber to fiber pig tailing setup of a distance between the two fiber ends (D_f) of about 50 to 100 μm. A separate ytterbium fiber laser (IPG Photonics) at λ = 1070 nm was introduced into the sample chamber orthogonal to the beams creating the fiber trap. This beam was tightly focused via a microscope objective and additionally created a separate “helper” optical tweezers that permitted loading of the optically bound array [22].

The imaging system consisted out of a 100× long working distance microscope objective (Mitutoyo) or alternatively a 60× microscope (Newport) and a CCD camera (Watec WAT 902DM2S), which was connected to a computer with frame grabber card to capture the images. In the data presented the microscopic field of view for the 100× objective was, however, not always sufficient to get an image showing the field distribution in the array and both fiber ends at the same time. To ensure experimental data was acquired when the array center was at D_f/2, and thereby assuring a centro-symmetric intensity distribution, a lens was flipped in front of the CCD camera and the power distribution readjusted via the λ/2 plate when necessary. A short pass filter was used to block out both trapping and tweezing wavelengths and solely pass the two-photon excitation light. The experiment was illuminated from above the cover slip and could be switched off to capture the two-photon images. Data analysis of these images was performed by utilizing a Lab VIEW script to determine the positions of the beads with an error of better than ±0.5 μm within a frame [14]. A similar script was used obtain the line profile of the fluorescence intensity distribution as well as the experimental false color images from the grey scale two-photon image which are shown in the following graphs but are not to scale.

4. Two-photon excitation

We next consider the pertinent parameters for the first demonstration of the femtosecond fiber trap. To date virtually all optical trapping experiments have incorporated continuous wave laser sources. Standard single beam traps have used femtosecond sources recently for simultaneous trapping and nonlinear excitation [23]. When considering the femtosecond fiber trap, group velocity dispersion (GVD) and nonlinear optical effects such as self phase modulation (SPM) become an issue. In our experiment, the fiber length was between 30 cm and 40 cm to keep these effects at a minimum. To compare the change in pulse duration, we measured the pulse duration via intensity autocorrelation prior to and subsequent to propagation in a 375 mm length of fiber. An input pulse of 95 fs duration increased to 800 fs after the fiber thus showing the dramatic effects of GVD. Before the fiber, the spectrum is of Gaussian shape, and after the fiber, the spectrum broadened significantly by a factor of approximately 2.6 due to SPM. Although the pulse duration and spectrum were increased, a two-photon fluorescence signal was still readily observed from the dye within the sample medium, indicating that the average intensity of 40 mW after the fiber, which corresponds to a pulse energy of 0.5 nJ and pulse peak power of 0.6 kW, are still sufficient to obtain two-photon fluorescence. When the Ti:Sa laser was operated in the continuous-wave (cw) regime no signal was detected by the CCD camera and this is interpreted as evidence for two-photon excitation in our experiment. A comparative measurement with a 30 mm shorter fiber showed no significant change in pulse duration and spectrum. It should be noted that nonlinear processes occur approximately within the first few mm of fiber [24, 25]. However, with a fiber length exceeding 60 cm a fluorescence signal was not obtained.

The host medium for the microparticles was prepared following previous studies [14, 15]; A de-ionized water and sucrose mixture is used to produce a variable host refractive-index, and fluoresce in (broad excitation band centered around 480 nm and emission band centered around 530 nm) was added to the sample [16] with a relatively high concentration of approximately 150±30 mg/l, since our peak pulse power is relatively low compared to that of Ref. [26]. Adding fluorescein to the host medium did not change its refractive index. It is known that fluorescein marker concentration fluctuations can lead to linear deviations [26] in the observed two-photon signal between experimental realizations. In contrast, excitation power variations, due to laser fluctuations between measurements and readjustments to the field distribution to center the array via the λ/2 plate, have a quadratic influence [27, 28] on the observed two-photon signal. Since the power fluctuations were small, we therefore approximated the deviations as linear. It is important to note that the model does not account for the experimental changes in the two-photon signal due to the variations in marker concentration and excitation power. However, since the theory cannot fix the absolute fluorescence signal strengths, the numerical results were linearly scaled to allow for comparison of the spatial profiles of the fluorescence.

5. Visualization of the light redistribution in optical binding

We now move on to discuss the visualization of the optical binding and its accompanying light intensity redistribution. As a first step we permitted light to solely enter one of the fibers and used the helper tweezers to hold a single microsphere (N = 1) in the path of the single laser field. We scanned this 5.17 μm silica sphere through the emerging field and compared the observed light pattern with a simulation. The movie (left side of Fig. 2) shows good agreement with the experiment and similar simulations conducted in Ref. [29]. When the bead is not fully centered light is diffracted away from the beam path creating a cone of low intensity light emerging from the rim of the bead, which is shown in a transverse line profile plot from a to b along the y-axis in Fig. 2 (right).

 

Fig. 2. [1.38MB] Diffraction pattern of a 5.17 μm silica sphere being scanned along the y-axis by an optical tweezers. The sphere position (overlaid white circle) is at approximately 50±1 μm from the beam waist. The movie shows the experimental images and the theoretical simulation. The insert on the right shows the intensity distribution along the y-axis from a to b at 8±1 μm after the sphere which has an offset from the beam axis of 3±0.5 μm (experiment: blue dots; theory: red line). Light is diffracted away from the beam path creating a valley of low intensity light.

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To elucidate the influence of the refractive-index difference between the sphere and the host medium on our results we present in Fig. 3 a comparison of a single 5.17 μm diameter sphere diffracting the fiber output beam path at a distance of 54±1 μm from the beam waist for (A) Δ_n = n_s - n_h = 0.07, and (B) Δn = 0.05. (Note that for 3 μm and 5.17 μm diameter silica spheres in water and a free space wavelength of 0.8 μm, the sphere diameter is almost five and nine times the wavelength in the host medium, so we are well in the Mie size regime per our usage.)

 

Fig. 3. Diffraction pattern of a 5.17 μm silica sphere which is held by optical tweezers at 54±1 μm from the beam waist in a single beam, originating from the left side of images. Comparison between different refractive index mismatches Δn:

1. Δn = 0.07. 1) On-axis intensity distribution (blue – experimental data; red – theoretical prediction). 2) Theoretical simulation of diffraction pattern, and 3) False color images of two-photon fluorescence.
2. Δn = 0.05. 1) On-axis intensity, 2) Theoretical simulation, and 3) False color images of two-photon fluorescence.

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For each example plot 1 shows a comparison of the experimental and numerical on-axis fluorescence signals (y = 0), whereas plots 2 and 3 show the numerical and experimental fluorescence profiles over the (y-z) plane, respectively (This numerical labeling of the plots is used in all subsequent plots). These results show that the higher the refractive-index difference the more the light is refocused after the sphere, as expected intuitively, and this observation is at the heart of the interpretation of how optical binding works, at least in the Mie size regime considered here. More specifically, the focusing length of the sphere in the small-angle approximation (measured from the sphere center) is f = R/(2Δn) which yields f ≈ 18 microns, in reasonable agreement with the results in Fig. 3 where a focus appears at around 70 microns (note that the slight inaccuracy of the small-angle approximation to the focal length does not imply failure of the scalar paraxial wave theory used here which accurately captures the higher-order aberrations for low NA optical elements.)

5.1 Optically bound arrays

Next, we consider the case of optically bound arrays with CP laser fields. The number of spheres in an array and the associated separation is dependent on various parameters such as wavelength, waist size and separation, refractive-index difference and sphere diameter. This has previously been investigated [12–15] and will not be further dealt with in this publication. Here our primary focus is the field redistribution that accompanies optical binding and its measurement using two-photon fluorescence. In previous studies, we have observed that the optically bound array spacing can vary due to variations in sphere size and/or refractive index, even for spheres from the same batch that should nominally have the same properties. To avoid this we shall consider single realizations of each array, so that the array spacing is fixed, and this allows for comparison between theory and experiment.

Optical binding with CP fields arises from the fact that the net force acting on each sphere has two components deriving from the force exerted from each laser field. Considering the case of two spheres for illustration, a give sphere will experience a direct force from the field emanating from the closest laser fiber end, and a second oppositely directed force from the refocused laser field emanating from the other fiber. Balancing of these two forces is the usual explanation of how optical binding can arise, and the extension to more particles follows. Fig. 4(a) is for the example of an optically bound array of two spheres (N=2), and shows the intensity distribution profile for two 3 μm spheres with a separation of 8 μm, and very good agreement is obtained between the numerical and experimental profiles.

 

Fig. 4. Optically bound arrays:

1. Array of two 3 μm spheres with a separation of 8 μm with Δn = 0.06. 1) On-axis intensity distribution showing the full waist separation of 72 μm (blue – experimental data; red – theoretical prediction). 2) Theoretical simulation of diffraction pattern in a 2 sphere array, 3) False color image of two-photon fluorescence.
2. Same array as in A) with left propagating field blocked, and for a time such that the sphere separation is 9 μm. 2) Theoretical simulation of the diffraction pattern, 3) false color image of two-photon fluorescence beam coming from left side of picture.
3. Array of three 3 μm spheres with a separation of 5 μm with Δn = 0.05 and a waist separation of 100 μm. 1) On-axis intensity comparison between theory and experiment which is being cut off at 60 μm. 2) and 3) theoretical and experimental images of diffraction pattern.
4. Array of four 3 μm sphere array with a separation of 12 μm with Δn = 0.01 with a waist separation of 85 μm. 1) On-axis intensity plot, 2) Theoretical image matching, 3) experimental false color image of two-photon fluorescence with right and left hand side of beam being partly cut off.

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In particular, the profiles clearly show that the intensity is refocused after the spheres, in keeping with the physical picture of optical binding. 4(b) is the same as Fig. 4(a) except the left propagating laser field has been turned off causing the particles to be propelled to the right due to imbalance of the optical forces now acting on the spheres, and for an elapsed time such that the particle spacing had increased to 9 μm. Once again, very good agreement is obtained between theory and experiment, and this example further shows that two-photon fluorescence can be used as a tool to obtain real-time monitoring of the dynamics of optically bound arrays.

To obtain binding for larger arrays the refractive-index difference needed to be lowered in order to inhibit the collapse of the array into a closed chain [12, 15]. Qualitatively we may explain this as follows: a lower refractive-index difference subsequently causes less light being refocused by a sphere (as was shown in Fig. 3) onto its nearest neighbor in the array. For the case of 3 μm sized spheres this means that by decreasing the refocusing effect of each individual sphere, the balance of the forces from both CP fields is still maintained for a higher number of spheres N. The results in Figs. 4 (c) and 4(d) are for optically bound arrays with N=3 and N=4 particles, with excellent overall agreement seen in all cases. Fig. 4(c) shows three spheres bound in an array, and when Δn is changed to 0.05. The separation of the spheres decreased to 5 μm and did not permit the optical field to emerge further between the spheres. With a refractive-index difference of 0.01, N=4 spheres can be bound in an array whilst having a very large spacing of 12 μm, as shown in Fig. 4(d). Here the reduced refocusing effect can be clearly seen at each end of the array, where the on-axis intensity peak has significantly decreased in comparison to Figs. 4(a) or 4(c). These examples amply demonstrate that imaging of two-photon fluorescence is a reliable tool for visualizing the redistribution of intensity in optical binding of arrays, and that our model for the field propagation is valid in the Mie size regime considered here.

5.2 Optical bistability in bound matter

We have recently shown that optically bound arrays can also exhibit the phenomena of bistability due to feedback. As the sphere position influences the diffractive refocusing of the field and vice versa, this may result in more than one stable sphere separation [14]. In Fig. 5(a) and (b) two such stable positions for 2.3 μm spheres are shown. Comparing both graphs shows that with a small separation the intensity peak after the light exits the array is very pronounced which indicates a high intensity and a strong optical gradient. For the large separation, those peaks have significantly decreased permitting the field intensity between the spheres to evolve.

 

Fig. 5. [1.31MB] On-axis intensity distribution of two 2.3 μm sphere array exhibiting bistability with a separation of a) 5 μm, and B) 16 μm with Δn = 0.065 and a waist separation of 90 μm. The respective on-axis intensity plots show a slight disagreement for a separation of 16 μm in B1) which is caused by sphere size variation within the sample batch. Respective intensity planes are shown in A2) and B2) for the simulations and A3) and B3) from the experiment. The movie shows an array of two 2.3 μm spheres having a separation of 16 μm, which are being guided to the right fiber facet. After the left propagating beam is reintroduced, they exhibit a separation of 5 μm. To get a clearer image of the spheres the microscope illumination was used, but the two-photon signal of the beam in the medium can still be observed.

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6. Conclusion

In this paper, we have reported the development of a femtosecond fiber optical trap that permits the first direct visualization of the process of optical binding for micro particles, including the light distribution within an array and the light redistribution in optical guiding. A numerical model for diffraction and beam propagation has allowed a direct comparison between experiment and theory for the resultant light redistribution in these cases. It also permitted validation of the physical model and our numerical model in the Mie size regime. Visualization of the light distribution within an optically bound array confirms that diffractive refocusing of the incident light fields is the key issue for array formation. We anticipate that the two-photon imaging methods reported here will have broader use in the field of optical binding and diagnosing the physics of optical micromanipulation.