1. Introduction

Since the invention of optical tweezers in the 1970s [1, 2], trapping and manipulating dielectric microscopic particles with lasers have found numerous applications in biomedical research and materials science. Nowadays optical micromanipulation is carried out in many experimental configurations, which create light traps using a variety of approaches, for instance using optical fibers [3], direct beam interference [4, 5], or spatial light modulators (SLM) [6, 8, 7, 9].

In the context of optical micromanipulation the forces exerted by optical fields are often conceptually divided into two parts: the gradient force and the scattering force. Microscopically this distinction between the (conservative) dipole forces arising from intensity gradients and the mechanical effects originating from the scattering of light by the microscopic particle is a natural “partition” of the forces. In the case of a single dipole, e.g., the scattering or “radiation pressure” force always points into the direction of the incident photon flux and the gradient force into the direction of the intensity gradient [10]. For larger dielectric particles of arbitrary shape these definitions are less natural, and different communities may mean different things by “scattering forces”, for example depending on whether the particle can be considered to be a small perturbation for the light fields or not. For the trapping of large spherical particles which can approximately be described by the ray optics model, the scattering and gradient forces of each single ray basically act along or orthogonally to the ray propagation direction, i.e. can be interpreted as the axial and transverse component, respectively [11].

Apart from specific configurations, where scattering forces can be exploited to trap and even stretch particles [2, 12], most techniques rely on the gradient force to perform stable trapping. Especially within single-beam traps, scattering forces act destabilizing and thus are often undesired. Nonetheless, for specific applications it has been shown that scattering forces can be quite useful. For instance, they can form “force vortices” which act as an “optical spanner” [13]. They can also cause spinning of birefringent particles [14] or drive specifically manufactured micro-machines [15]. And it seems there will be an even broader spectrum of possibilities, if one could only control the scattering force more precisely: Since the direction of the scattering force is – besides particle properties – determined by the local wavefront shape, controlling the light field’s amplitude and phase is a precondition to maintain full control over the scattering force. Unfortunately, the use of phase-only spatial light modulators (SLM) in holographic as well as direct projection methods does generally not allow one to control amplitude and phase of the trapping field simultaneously. Although there are methods to shape the amplitude profile of the first diffraction order with a pure phase modulating element [16, 17, 18], and also methods which employ two SLMs to generate complex-valued fields [19, 20], they usually imply a significant loss of light.

Here we demonstrate the creation of optical “trapping patterns” which exert user-defined scattering forces with a phase-only SLM. In particular we demonstrate the creation of line traps, which provide transportation of micro particles on freely designable pathways. Previous experiments have also demonstrated particle transport driven by a phase gradient along the ring-shaped cross-section of Laguerre-Gaussian modes [21]. Another approach presented by Curtis and co-workers [22] allows the design of more general pathways, however still with the restriction to closed paths. Moreover, the transverse scattering force component is not independent of the spatial structure of the path. Recently, a method to create arbitrary light structures with freely programmable phase profiles has been demonstrated [23]. Though powerful in many respects, the method might not be optimal for the design of high-contrast amplitudes, since amplitude-shaping is based on the omission of pixels.

We use a specific method for shaping complex light fields [24, 25] the applicability of which to liquid crystal SLMs has been investigated recently by our group [26]. The technique is based upon two cascaded phase diffractive elements, arranged in an optical 2-f setup, where the first element is designed to create the desired amplitude in the plane of the second element, which finally “imprints” the desired phase. The method is in principle lossless and can accommodate high amplitude-contrast requirements.

2. Methodic principle and practical implementation

Figure 1(a) outlines the principle of the technique. Amplitude and phase of the light field are shaped in two subsequent steps by two phase diffractive elements (P1 and P2). Both elements are arranged in a 2-f setup, i.e., they are placed in the front and back focal plane of a convex lens. Consequently, the Fourier Transform of the field in the plane of P1 emerges in the plane of P2. The principle of the method is explained in the following practical example.

Let us consider that we want to create a light field a(x,y)=|a(x,y)|exp[iϕ(x,y)] in the object plane, which collects particles like a funnel, i.e. which “grasps” and guides them towards a specific point in the object plane. Such a light field represents the “reticle” shown in Fig. 1(b). The four bright lines confine particles by gradient forces. To maintain the “collecting” property, we design the phase of the light field to have a gradient pointing towards the reticle center, which corresponds to the transverse component of the linear photon momentum p_t(x,y)=h̄∇_t ϕ(x,y), where the index t denotes the transverse component, i.e. the component within the microscope object plane. It should be mentioned that the resulting scattering force on trapped microscopic particles strongly depends on specific particle properties, such as shape and refractive index. In some specific cases, particles can also travel against the direction of p_t [27]. However, such a behaviour is only expected from particles of highly irregular shape.

 

Fig. 1. (a) Principle of the method. The reticle in the object plane are created by the two phase masks P1 and P2, which are subsequently arranged in a 2-f setup. P1 generates the modulus |A(u,ν)| in the plane of P2, where P2 shapes the desired phase Φ(u,ν). (b) Schematic of a trapping light field, the amplitude (upper image) and phase of which are designed to collect and guide microparticles to the center of the “crosshairs”.

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The creation of the complex field a(x,y)=|a(x,y)|exp[iϕ(x,y)] in the object plane requires a definite complex field A(u,ν)=|A(u,ν)|exp[iΦ(u,ν)] in the plane of P2, which can be calculated by mathematically back-propagating a(x,y). In the most common case, this plane is conjugate to the object plane, thus A(u,ν) corresponds to the optical Fourier transform [28] of a(x,y). Hence the task is to create A(u,ν) in the plane of P2 in order to obtain the desired complex field a(x,y) in the object plane. Since P2 is a phase diffractive element, it can be programmed to shape the phase part Φ(u,ν) directly, but unfortunately not the modulus |A(u,ν)|. The key idea now is to design P1 such that a readout reproduces |A(u,ν)| in the plane of P2 as exactly as possible. There exist a couple of so-called phase retrieval algorithms which can be utilized for this design process [29]. The “random” phase Θ(u,ν), which usually results from such optimization procedures, cannot be controlled, but it is known. Thus, in the plane of P2 the desired phase Φ(u,ν) can be restored by calculating P2 as follows:

where Θ describes the non-controllable phase that is produced by P1 in the plane of P2, and mod2π{…} symbolizes the “modulo-” operation that “cuts” the phase of P2 into slices of 2π.

Figure 2 illustrates how we realized a practical implementation of the method. By introducing a concave mirror into the beam path, it is possible to utilize one single phase modulator to display P1 and P2, which are adjacently arranged at the modulator panel. Our SLM is a HEO 1080 P phase-only modulator from Holoeye Photonics AG, which can display phase patterns showing up to 41% diffraction efficiency (defined as the intensity ratio of first diffraction order to the incident light) at 1064 nm. The setup is almost identical to that introduced in Ref. [26]. A minor but important difference is that we designed P1 as an “off-axis” structure this time, such that its zeroth diffraction order does not focus on P2. This is advantageous, because the relatively high laser power required for optical trapping could otherwise lead to irreversible damage of the liquid crystal in the SLM. According to our measurements, slight (reversible) power-induced changes of the liquid crystal birefringence begin to appear after a few minutes of exposure to a Gaussian beam of 2 W CW power with a 2.5 mm waist. In our experiments, the phase mask P1 was optimized in 15 iterative steps of the Gerchberg-Saxton algorithm [30].

 

Fig. 2. Experimental setup. The complex trapping field is created stepwise: Illuminating pattern 1 with a collimated CW Ytterbium fiber laser creates the amplitude profile A(u,ν) in the plane of pattern 2, which reshapes the “random” phase to Φ(k_x,k_y). The modulated laser beam is subsequently coupled into a microscope objective, finally taking the desired form a(x,y) exp[iϕ(x,y)] in the object plane.

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Strictly speaking, the fact that the spherical mirror is inclined with respect to the beam axis introduces wavefront aberrations such as coma. Implementing a beamspitter would allow an orthogonal geometry and thus avoid these aberrations. However, this would also imply a reduced light efficiency. Because the mirror tilt angle is small (below 1 degree), we decided to accept the aberrations for the sake of a maximal light throughput. Furthermore, P1 and P2 are not exactly located in conjugate planes, which is due to the SLM tilt with respect to the beam axis. In our case, the tilt angle is in the range of 2-3 degrees and the assumption of conjugate planes leads to an only minor quality reduction. A corresponding modification of the hologram computation could compensate the effect, however at the cost of higher numerical expenses.

The sensitivity of the setup to alignment errors is discussed in Ref. [26].

3. Experimental results

Figure 3 documents the practical realization of the particle-collecting reticle discussed in the former section. The experimental situation is described by the sketch in the upper left corner. The upper right images show the reflection of the light field on a mirror, revealing good intensity reconstruction. The accuracy of the programmed phase was tested by direct interaction of the reticle with microparticles. The objective used is a Zeiss Neofluar 100× with a numerical aperture (NA) of 1.3. The movie strip of Fig. 3 shows four silica microbeads of 2 µm diameter, each trapped in a different “arm” of the reticle. First, the transverse phase gradient of the structure was programmed to point outwards with a modulus of 2π/µm. The according transverse radiation pressure pushed the silica beads towards the outer edges of the trapping field (first frame). Inversion of the phase gradient caused the particles to move towards the center (second and third frame). For silica particles in water, which show a relative refractive index of 1.43/1.33, the axial gradient force is sufficient to overcome the opposing axial scattering force – thus the reticle is able to confine these particles three-dimensionally. This was demonstrated by lifting the trapped microspheres about 8 µm deep into the water volume. The axial shift can be judged by some untrapped beads in the lower left edge, which appear out-of-focus in the last three frames. Finally, the phase gradient was once more inverted and the particles moved back to the outer edges of the reticle. A real-time video documentation of the experiment is attached (reticle.avi). The laser power output used for this experiment was 1.6 W, which corresponds to a net power of approximately 100 mW in the object plane.

 

Fig. 3. Experimental realization of the “reticle” trap. Upper left image: Experimental situation. Trapping is performed from below through a glass cover slip with a Zeiss Neofluar 100× objective, NA=1.3. Upper right image: Trap, projected at a mirror. Movie strip (frames taken out of the file reticle.avi, size 3 MB): Four trapped silica microbeads are alternately pushed in and out by changing the sign of the transverse photon momentum. The particles are three-dimensionally trapped, which is demonstrated by lifting them off the ground of the object chamber (note the encircled untrapped particles in the lower left corner getting out of focus). [Media 1]

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A second experiment was performed with polystyrene beads located at a microscopic air-liquid interface [31], using a Zeiss Achroplan 63× air objective with a NA of 0.95. The interface is located within a tiny hole (300 µm diameter) – small enough to prevent leakage – in the bottom of a stainless steel fluid container and can be accessed from below with an inverted microscope (see Fig. 4). Such a setup allows detailed examinations of interface properties [32]. Moreover, for hydrophobic polystyrene particles, the exerted surface tension acts stabilizing against axial scattering forces. To demonstrate the ability of the method, we designed a “light square” and an asymmetric “curvy” structure resembling Austria in shape which trap particles (about 3 µm in diameter) and push them along their boundaries by scattering forces. The corresponding transverse phase gradient has a modulus of 3.6×2π/µm. Again, the intensity reconstructions are almost speckle-free. The movie strips in Fig. 4 indicate that – in the case of the square trap – the obtained particle speed (mean value 20 µm/s) is much higher than that of the silica spheres in Fig. 3, although the programmed phase gradient is smaller. This is – besides the lower infrared absorption of the used air objective – mainly explained by the higher number of beads (more beads gain more light momentum) and the higher refractive index of the particles (1.59 for polystyrene beads), which implies a higher scattering force. Because this also holds in axial direction, the gradient force by itself is now no longer sufficient to confine the particles stably in the object plane. Hence for such particles another stabilizing force is required, which in our case is the surface tension. The laser power output used for the experiments with polystyrene beads was 1.4 W, which corresponds to a net power of approximately 150 mW in the object plane.

 

Fig. 4. Upper left image: Trapping is performed at an air-liquid interface, which axially stabilizes the hydrophobic microparticles (polystyrene beads, 3 µm diameter). Upper right image:“Square trap”. The microscopic light square traps particles and moves them along its boundaries. The corresponding movie strip shows it “in action” (movie file square.avi, size 2.8 MB). The guiding performance of a asymmetric structure is demonstrated by the lower movie strip (movie file austria.avi, size 3.1 MB). [Media 2][Media 3]

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4. Summary and discussion

Recently we have examined the applicability of a method for generating complex wavefronts to liquid crystal SLMs. Here we demonstrate the usefulness of this technique for the optical manipulation of microscopic dielectric particles that can be trapped by pre-defined optical “trapping patterns”. Similar to the method introduced in Refs. [17, 23], the method permits independent control over amplitude and phase of a light field, but with potentially higher efficiency. The technique makes use of two independent diffractive elements located in two distinct optical planes: an “amplitude shaper” phase hologram in an image plane and a “phase shaper” hologram in a Fourier plane, thus the commonly used “Fourier manipulation techniques” (e.g. gratings and lenses for three-dimensional trap shifting) are extended by possibilities to alter the trapping field via the element in the image plane (P1). For instance, lateral shifts of the trapping field can be directly achieved by a corresponding motion of P1, which enables real-time steering similar to that obtainable with a “Fresnel” setup [9]. We implemented the method using a single user-programmable phase-only liquid crystal SLM.

Several alternative approaches for creating complex light fields with single phase masks have been suggested in the past [16, 17, 18]. Furthermore, phase retrieval techniques can be also employed to reconstruct a complex amplitude field directly [33], however only in a restricted area and with unavoidable light losses.

Theoretically, the method described here would enable the generation of arbitrary light fields with almost 100% intensity efficiency. In our specific case, the maximal obtainable efficiency is about 17%, which is mainly caused by the relatively high absorption by our SLM. With no absorption, the obtained efficiency would be…

However, this technical restriction will be less severe in future generations of light modulators. In fact, even current systems claim to have light efficiencies of up to 80%, which would enable the presented “complex” holographic optical tweezers to reach up to 64% efficiency.

The method is well suited to create high contrast amplitudes in the plane of P2, which is especially advantageous in the case of P2 being located in the Fourier plane, where field amplitudes often have a large dynamic range. However, high intensities may destroy the liquid crystal – hence, in some cases, P2 should rather be placed in a Fresnel than in the Fourier plane, where the amplitude contrast is typically smaller.

The presented “complex” holographic tweezers might be useful in different tasks concerning optical trapping and manipulating microscopic particles or atoms, since its shows higher flexibility than commonly used techniques, which typically only allow controlling the amplitude of light fields. For instance, one could use specifically tailored phase landscapes for transporting, sorting or controlled aggregation of microparticles.