1. Introduction

Recent years have seen the development of robust and diverse optical micromanipulation techniques, developing in parallel to the growth of increasingly complex laser modes that carry angular momentum. Optical trapping is now used to position and manipulate a wide variety of objects, including dielectrics, metals, absorbing objects, biological systems, and materials exhibiting birefringence. The laser modes include optical vortices, such as Laguerre-Gauss modes, which carry orbital angular momentum (OAM) [1], and beams with varying degrees of elliptical polarization, which carry spin angular momentum (SAM) [2, 3]. The mechanism for transfer of angular momentum to a micro object depends on the type of momentum. OAM, in which the angular momentum comes from the helical structure of the wave fronts of the beam, transfers through absorption of photons [4, 5], while SAM is carried in circularly and elliptically polarized light, and transferred to birefringent materials [6, 7].

With this great variety of laser modes and manipulations, an efficient technique for measuring the mode of a laser beam without altering the mode would be useful in mode generation as well as in applications of these modes. Mode measurement using traditional polarimetry requires optical analyzers placed in the beam before or after the trapping region, which cannot be done simultaneously with optical manipulation. The focus of this work is to characterize the distinct motion of birefringent calcite crystals in elliptically polarized light through the transfer of SAM. Birefringent calcite crystals are an ideal system to work with, because they respond in predictable ways to polarization. We describe the theoretical foundations, including the torques acting on a birefringent crystal in polarized light, and model the instantaneous rotational frequency. We then demonstrate the motion of calcite crystals under particular ellipticities and polarization orientations of the laser mode. Through this work, we aim to make progress toward the development of a technique for identifying the polarization of the laser mode (including ellipticity and orientation) in the trapping focus of a beam when the optic axis of the crystal is known.

2. Theoretical considerations

Calcite is one of the birefringent polymorphs of calcium carbonate, with a single optic axis, and distinct ordinary and extraordinary indices of refraction, n_o and n_e. The extraordinary direction is along the c-axis (optic axis) of the crystal, and the ordinary direction is perpendicular to the plane of the c-axis. An ellipse, called the principal section, can be defined with the extraordinary direction along one axis and the ordinary direction along the other axis. Because the ordinary and extraordinary directions have different indices of refraction, light with polarization components lying along these two directions will experience a phase difference between the two components, which will affect the polarization, or ellipticity, of the light.

A change in the polarization of the transmitted light means the electric field has changed, and it therefore has a different angular momentum than the incident light. This change in angular momentum is transferred to the birefringent crystal and is equivalent to a torque on the crystal [8]. The angular momentum of the light is proportional to its ellipticity; the beam has the greatest angular momentum when the light is completely circularly polarized, and zero angular momentum when linearly polarized. Therefore, there is the potential for greater spin angular momentum transfer when the light is more elliptically polarized.

The polarization orientation of the laser mode also directly affects the instantaneous crystal rotation. There are two particular angles relevant to the crystal motion, as shown in Fig. 1(a). The angle ϕ is the angle, relative to the incident light’s linear polarization direction, of the quarter wave plate (QWP) generating the elliptical or circular polarization. The QWP is positioned directly below the focusing microscope objective, as shown in Fig. 1(b) (the experimental arrangement will be explained in detail in Sec. 3). When ϕ = 0 the light remains linearly polarized, at ϕ = π/4 the light is circularly polarized, and between these angles it is elliptically polarized. Ellipticity ϵ of the light is defined as ϵ = b/a, where b is the minor axis and a is the major axis of the ellipse. The angle θ lies between the crystal optic axis and the QWP axis. The QWP axis will define the major axis of the generated polarization ellipse for ϕ = 0 to ϕ = π/4, and the minor axis for ϕ = π/4 to π/2. Therefore, this is the same as defining the angle of the crystal optic axis relative to the major axis of the ellipse for ϕ < π/4 and relative to the minor axis for ϕ > π/4.

 

Fig. 1 Experimental arrangement. (a) Definition of angles θ and ϕ. (b) Laser light is focused from below through a sample and onto a QPD (quadrant photodiode). Illuminating white light shines from above through the sample and is imaged onto a camera.

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When elliptically polarized light passes through a calcite crystal, two things will happen. The crystal will experience a transfer of angular momentum (dependent on ellipticity) that causes it to rotate in a direction determined by the handedness of the light. It will also align its optic axis along the polarization direction of the light (the major axis of the ellipse defining the polarization) [7]. Our work explores the effects of these two components of torque. We call the torque due to the spin angular momentum carried by the beam the spin torque, τ_s, and the torque due to the tendency of the optic axis of the crystal to align with the polarization direction of the light the alignment torque, τ_a. The net torque is then given by τ_net = −τ_a + τ_s, where ${\tau }_a=\frac{\mathit{ϵ}{E}_0^2}{2{\omega }_l}\sin (\mathrm{\Gamma })\cos (2\varphi )\sin (2\theta )$ and ${\tau }_s=\frac{\mathit{ϵ}{E}_0^2}{2{\omega }_l}(1-\cos (\mathrm{\Gamma }))\sin (2\varphi )$ [7, 9]. The constant Γ, a function of the birefringence of the material and the thickness d of the crystal, is given by Γ = kd(n_o − n_e). The optical frequency of the light is indicated here as ω_l in order to distinguish it from rotational frequencies under discussion.

The crystal behavior can be understood by examining the net torque on the crystal and the interplay between the two torques. Depending on the momentary (instantaneous) orientation of the crystal, these two torques act either in the same direction or opposite directions, and so can act to reduce or increase the net torque and rotational frequency. A crystal rotating with nonuniform circular motion accelerates when both torques act in the same direction, and continues to accelerate until the two torques act in opposition. Therefore, we expect it to slow down for values of θ between 0 and π/2, and then increase, reaching a maximum at the orientation where the alignment torque changes direction (θ = π). In general, the rotational motion is non-uniform, but there are special cases. A crystal with relatively high rotational frequencies does not exhibit significant non-uniform motion. Also, if the light is completely circular (if ϕ = π/4), then there is no alignment torque.

The torque also varies in a sinusoidal way with the thickness d of the crystal, as the phase change of the light is proportional to d. The spin torque is a maximum for thicknesses equivalent to odd multiples of a half-wave plate, where light would experience a complete phase-flip (for example, from left-circularly polarized to right-circularly polarized), and a minimum for thicknesses equal to that of a full-wave rotation, where light experiences no net phase change. The alignment torque, on the other hand, is zero for both a half- and full-wave phase shift, and alternating positive and negative for quarter-wave thicknesses. This strong dependence on thickness makes precise experimental verification difficult, as the crystals we are currently working with are highly non-uniform.

The previous discussion illustrates the importance of the alignment torque, and that the rotation frequency is constant with neither time nor orientation. There are two rotational frequencies of interest: the average and the instantaneous rotational frequencies. The average rotational frequency has been discussed in a previous work [10]. Here, we are concerned with the instantaneous rotational frequency of the crystal. In particular, we consider the instantaneous frequency of the crystal at slow speeds (under about 3Hz) for which there is a visually observable change in speed. In this regime, it is possible to determine the handedness of the light, the degree of ellipticity, and the orientation of the ellipse.

If the crystal rotating in an elliptically polarized beam is modeled as a rigid body experiencing an angle-dependent torque, the complete dynamics of the crystal rotation are given by a first-order differential equation in the rotation frequency ω [11]

 

Fig. 2 Numerical solution to Eq. (1) for (a) angular velocity (ω) vs. time, and (b) angular velocity vs. θ. The QWP angle is ϕ = π/3.

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3. Experimental methods

We use an inverted microscope arrangement to manipulate the calcite crystals, send an adjustable polarization mode through the microscope objective as the manipulating beam, and simultaneously record the transmitted light on a photodiode and the imaged sample on a camera. The experimental setup is shown in Fig. 1(b). A 660 nm diode laser is initially linearly polarized; the polarization is rotated with a quarter-wave plate, which is set at an angle ϕ relative to the linear polarization direction. Power going to the sample can range from 2 mW to 50 mW, depending on the crystal size and desired rotation rate. The polarization mode is focused into the sample region with a 100X, 1.3 NA microscope objective. The manipulation beam is collected above the sample with a condenser and sent to a quadrant photodiode. The sample is also illuminated with a white-light source from above, and then imaged through a dichroic mirror onto a camera. The video data recorded from the camera gives us information about the orientation of the crystal at any time.

Our sample is composed of a glass cover slip below a standard glass microscope slide, with a spacer between the glass surfaces made of double-sided tape. The enclosed sample region contains distilled deionized water, a small amount of a wetting agent, and calcite crystals. The calcite is created by crushing larger calcite crystals into 2 µm to 15 µm pieces. This gives a large range of crystal shapes and sizes.

4. Results

What follows is a characterization of the crystal motion using video and quadrant photodiode measurements, and a comparison between modeled and measured behavior. We demonstrate the correspondence between change in angular velocity and particular crystal orientation. Several crystals were trapped and rotated in our elliptically polarized beam. We analyze one crystal, trapped under different polarizations, in this discussion. With each crystal, we increment the polarization of the light by changing the angle ϕ between the incident polarization and quarter-wave plate axis, and record video and photodiode data simultaneously for each polarization. We determine the crystal’s optic axis by noting its orientation when ϕ = 0. The angle θ is then defined relative to that orientation.

We see a good correspondence between the anticipated dynamics and the measured nonuniform rotation. Shown in Fig. 3 is the transmitted intensity of light passing through a calcite crystal under circular polarization, ϕ = π/4 [Fig. 3(a)] and elliptical polarization ϕ = π/3 [Fig. 3(b)]. The laser illumination in the sample was 8 mW, and the the crystal is approximately 5 µm thick, and 5 µm by 7 µm across. The average rotational frequency is 0.42 Hz for the circularly polarized light, and 0.35 Hz for the elliptically polarized light with ϕ = π/3. Under a circularly polarized driving beam, the oscillations in the photodiode signal are due to the rotation of the crystal: the irregular shape of the crystal causes more and less light transmitted as it rotates, although because of the irregularities, not in a purely sinusoidal way. In circularly polarized light, we see one oscillation in the amount of transmitted light for each full crystal rotation, as is expected for a uniformly rotating object. With an elliptically polarized driving beam, the crystal moves more quickly twice per rotation, as discussed in Sec. 2, which is recorded in the photodiode signal as a rapid change in transmitted light. The overall change in light transmission due to the irregularly-shaped rotating crystal affects the relative amplitudes of the two periodicities.

 

Fig. 3 Transmitted photodiode signal vs. time for calcite rotating under (a) circularly polarized light (ϕ = π/4), and (b) elliptically polarized light (ϕ = π/3). The blue asterisks show the orientations of the crystal where rotation frequency is greatest. The inset shows the experimentally determined orientation of the major axis of the polarization ellipse. (c) Calcite crystal in elliptically polarized light, ϕ = π/3. The QWP axis is defined as ϕ, and the long axis of the polarization ellipse is π/2 relative to ϕ. The green arrow indicates the crystal optic axis in each frame. The crystal moves fastest in frames one (13.5 s), three (17.5 s), and five (21.1 s), and slowest in frames two (16.1 s) and four (20.1 s). For all figures, the laser power is 8 mW, and the crystal is approximately 5 µm by 7 µm by 5 µm thick.

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The analysis of the measured photodiode data and video images can be used to either find the optic axis of a crystal, or find the light polarization direction. When the polarization direction is known, but the crystal does not have clearly defined structure, the optic axis can be found by observing the crystal orientation at the times of fastest rotation. When the crystal optic axis direction is known, the photodiode and video data can be used to determine the polarization direction. We illustrate the latter process by comparing the transmitted light of Fig. 3(b) to the crystal motion in Fig. 3(c). Times at which the calcite is moving the most quickly or slowly are found from the extremum of the rate of change of the photodiode signal in Fig. 3(b), indicated with the blue asterisks. The rotational orientation of the calcite axis at these times is used to determine the polarization direction of the light. In Fig. 3(c), the first, third and fifth frames are at times at which the crystal is rotating most quickly (the blue asterisks at 13.5 s, 17.5 s, 21.1 s), whereas the second and third frames are at times at which the crystal is moving slowly (16.1 s, 20.1 s). The green arrows in these frames show the direction of the previously-determined optic axis of the crystal. The direction of the green arrows for the fast motion enables us to determine the polarization axis of the light to be 58° relative to vertical. The direction of the polarization ellipse, determined in this manner, is shown in the inset in Fig. 3(b). For comparison, the known direction of the QWP axis, ϕ, and its perpendicular are depicted by the black arrows in Fig. 3(c). For ϕ = π/3, the polarization axis is perpendicular to the QWP axis. The green arrows align quite closely with ϕ + π/2 for fast motion, and with ϕ for slow motion, as expected. In contrast with traditional polarimetry, in which waveplates, a polarizer, and a detector are placed in the beam, this technique gives a measurement of the beam in the manipulation location without altering or blocking the mode. This initial study was done with randomly crushed calcite crystals, which are in no way uniform throughout the sample. Thus, making absolute predictions of the light polarization is difficult, because of the random nature of the crystal shape and size.

The nonuniform rotation of the calcite crystal is demonstrated in more detail in Fig. 4. The manipulating laser is elliptically polarized with the quarter-wave plate angle ϕ = π/3. In Fig. 4(a), the measured and predicted orientation angles θ of the crystal’s optic axis are denoted by the red x’s and blue line, respectively. Note that the orientation angle θ changes at different rates as the crystal rotates, increasing and decreasing twice in each period. This behavior is again seen in Fig. 4(b) where the change in θ is plotted directly vs. time. The measured rotational frequency (in red) in Fig. 4(b) are found by first smoothing the θ values using a triangular window of width five and then calculating Δθ/Δt. The predicted rotational frequencies ω (in blue) are a numerical solution to Eq. (1) with ϕ = π/3 and parameters given in Sec. S:theory, as shown in Fig. 2(a). The predicted crystal axis orientations θ in Fig. 4(a) are calculated from this ω. Comparisons of the experimental and theoretical data show that while there is considerable fluctuation in the experimental data, the dynamics over time can be matched with predictions.

 

Fig. 4 Crystal optic axis angle θ vs. time (a), and angular frequency vs. time (b).

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5. Conclusion

Calcite, with its birefringence, asymmetry, and defined optic axis, exhibits interesting and informative rotational motion in an elliptically polarized laser mode. We have compared rotational motion of a nonregular crystal, including orientation angle and instantaneous rotational frequency, as measured by transmitted intensity and video, with predictions. The measurements confirm that the calcite moves faster when its optic axis is aligned with the polarization axis of the mode.

The theoretical and experimental approaches described here give us the tools to predict the polarization state of light, given a known crystal axis, based on crystal rotation. This is a step towards future work on mapping the ellipticity of complex polarization modes. With an assembly of crystals of uniform size and thickness, each crystal would respond consistently to the same illumination. Such a collection of crystals may be placed in a polarization singularity, where the polarization varies across the mode (see [10]). The crystal motions and orientations at different transverse positions would give information about the different ellipticities in the beam.