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Abstract
We demonstrate the polarization-dependent torque on rhombohedral calcite in an optical trap. Our precipitate technique produces regular crystals approximately 10μm on all edges. The regularity of the crystal shape makes it possible to visually identify the optical axis as well as the orientation of the polarization axes. When a rhombohedral crystal is trapped in an elliptically polarized beam, it orients itself with its optic axis approximately parallel to the beam axis. While in this orientation, the total torque increases and decreases relative to three extraordinary and ordinary axes of the crystal. We measure this axis-dependent calcite rotation through video analysis and model the dependence of the torque on the crystal orientation. The ability to predict the motion of calcite gives an analytical tool for applications such as fluid stirring or “lab on a chip” systems.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical tweezers, the trapping and manipulating of tiny objects, uses the momentum carried in shaped laser beams to explore the effects of light on matter. This way of controlling objects without physically touching them is used in a great variety of contexts such as stretching cells to learn about cell membranes, pushing nanospheres through constricted flow areas such as blood vessels, rotating asymmetric objects, and creating an entire “lab on a chip” where light makes many functions occur in a tiny area [1,2]. Modes carrying defined polarization (vector modes) offer a valuable parameter with which to manipulate materials, particularly those which are optically birefringent. The learning that can occur through activating birefringent materials or using those materials to shed light on polarization is unlimited. Recent examples exploring this interaction include investigations into the structure of mollusk nacre, spinning liquid crystals, and measuring polarization with calcite rotation [3–5].
We examine the rotation of calcite crystals in polarized light with the goal of developing a way of moving more than one crystal (having multiple activities on the same “chip”) in the same focused laser beam [6]. The unique behavior of calcite we show here offers multiple avenues for manipulation. We have seen that perfect rhombohedrons of calcite position themselves in an optical trap such that the optical axis is along the beam propagation direction (beam axis). In this position, when illuminated with shaped laser light, they rotate in an unusually non-uniform way. The torque on calcite crystals in an optical trap changes dramatically depending on the crystal position. The two ways that light affects calcite crystals are due to its internal crystal structure and to its unusual asymmetric shape. Both of these contributing factors have been explored in detail. Rotation rates of birefringent objects in polarized light depend on the ellipticity and orientation of the polarization [5, 7], and shape-birefringent objects (cylinders, rods) have an aspect-ratio-dependent orientation [8]. When calcite is positioned with its optic axis along the beam axis, more than one ordinary and extraordinary refractive index directions affect the motion, and the crystal moves with a surprising sequence of rotational speeds. We illustrate this behavior through video image analysis and model the orientation-dependent torque.
2. Theory
Calcite is a calcium carbonate (CaCO3) polymorph with negative uniaxial birefringence. It grows with a crystal structure that gives it the characteristic rhombohedral shape we see in Fig. 1 with corners at 78° and 102° angles. Uniaxial birefringent materials have two distinct polarization axes, referred to as the ordinary and extraordinary axes, and corresponding indices of refraction no and ne. For negative uniaxial crystals, no > ne. The optic axis, defined such that a transverse electric field propagating along it will experience only one index of refraction, is indicated in the figure as a cylinder running from the far “blunt” corner to slightly off-corner on the near side [9].
The ordinary and extraordinary axes are designated such that no obeys Snell’s law and ne is a function of θp, the angle between the optic axis and the propagation direction of light. The relationship between no, ne, and θp is given by
which is significant as we discuss the incident angles on multiple faces in Section 5.The motion of a birefringent crystal is driven by the change in electric field as it passes through the crystal [7]. The polarization axes encountered by the electric field as well as the angle of incidence on the crystal and the thickness of the crystal determine the phase and direction change of the field. This change in polarization creates a net torque that is the sum of an alignment torque τa and a spin torque τs.
whereThe constant Γ = kd(no − ne) is a function of the birefringence of the material and the thickness d of the crystal. The optical frequency of the light is indicated here as ωl in order to distinguish it from rotational frequencies under discussion. In this equation θ is the angle between the polarization angle of the light and the extraordinary axis of the crystal, and ϕ is related to the ellipticity of the light. Spin torque is greatest when the polarization is circular (ϕ = π/4), and alignment torque is greatest when the optic axis is oriented π/4 from the major axis of the polarization ellipse. There is always a lowest-energy position of the crystal such that light passing through it experiences the least change in momentum, and this is clearly different for the two cases in Fig. 2.
Fig. 2 Calcite positioned in two ways in an optical trap. Ordinary (green) and extraordinary (blue) axes are drawn for each face presented to the laser’s electric field. The beam propagation direction is out of the page and sample polarization ellipse is drawn in red. Definition of the angle θ is shown relative to the face three extraordinary axis.
When calcite is positioned with two faces parallel to the plane of the electric field, there are a maximum of two polarization axes (ordinary and extraordinary) and indices of refraction that the field may encounter, as shown in Fig. 2(a). The beam axis is directed perpendicular to the page in this figure (irradiance shaded in red) and an arbitrarily chosen polarization ellipse with a horizontal major axis is in the plane of the figure (drawn in red). The angle θ for τa is shown relative to the polarization ellipse. The extraordinary axis (blue) lies in a plane containing the optic axis and has a component running from one “blunt” corner to another, and the angle θ is defined relative to this axis. The ordinary axis (green) has a component that is perpendicular to the extraordinary axis. When the calcite is positioned with its optic axis along the propagation direction (beam axis) there are three faces presented to the electric field (Fig. 2(b)). In this position the ordinary and extraordinary directions become much more complex. There are three distinct planes of principal sections, defined such that they contain the optic axis and are perpendicular to two faces of the calcite crystal [9]. An extraordinary axis lies in the plane of the principal section and the ordinary axis lies perpendicular to both the extraordinary direction and the principal section. Each of these ordinary and extraordinary axes for the faces labeled 1, 2, and 3 are shown in Fig. 2(b). The optic axis passes between faces one and two, and does not touch face three.
The difference between the polarization axes presented to the electric field in the two positions in Fig. 2 is the fundamental reason behind the difference in the rotational motion. For the case in Fig. 2(a), with the electric field interacting with only one set of ordinary and extraordinary axes, the alignment torque oscillates with θ, the angle between the extraordinary axis and the polarization axis (Fig. 3). This results in a simple variation of the torque and angular acceleration. For linearly polarized light, the alignment torque makes the crystal align with the extraordinary axis along the direction of polarization, and a crystal will generally settle in this particular orientation. With elliptically polarized light, both the spin torque and the alignment torque contribute to the total torque acting on the crystal. When the spin torque is large enough to overcome the alignment torque the crystal rotates, moving fastest when its extraordinary axis is aligned with the polarization direction [5]. The rotation therefore depends on the degree of ellipticity, which is given by the angle phi. When the crystal is positioned instead as in Fig. 2(b), the crystal aligns in linearly polarized light with the polarization direction along the extraordinary axis of the third face and directly between the axes of the first and second faces. When the polarization is elliptical the net torque is much more complex. In Section 5 we describe in detail this torque and the contributions from three different sets of birefringent axes.
Fig. 3 Alignment torque on one set of birefringent axes. The torque is a maximum and minimum at θ = π/4 and θ = 3π/4 and changes direction when the extraordinary axis is parallel or perpendicular to the polarization axis.
3. Experimental methods
3.1. Crystal production
In previous work our samples were composed of irregular pieces of calcite made by crushing a larger crystal; these pieces did not have a specific shape or consistent size. To perform precise experiments, we investigated ways of making calcite pieces of similar sizes and with recognizable form and edges. We executed numerous trials of precipitating calcium carbonate by creating equimolar solutions of calcium chloride and sodium carbonate to form a precipitate. We then filtered out the precipitate, dried the sample, and observed which polymorph it contained, as calcium carbonate forms into any of three polymorphs of aragonite, calcite, or vaterite depending on variables such as temperature, concentration, and pressure [10]. By testing different molarities we found that the molarity that resulted in samples of purely calcite was 0.038M at room temperature (approximately 27°C). A lower temperature and higher molarity resulted in a sample of mainly vaterite [11]. This crystal precipitate technique produces very uniform calcite crystals. In each precipitate process, the crystal size is approximately the same, but this size will vary from one precipitate to the next depending on temperature and time left in solution. A scanning electron microscope image of a large number of crystals from one precipitate process is in Fig. 4.
Fig. 4 An SEM image of calcite crystals made through precipitate technique. The average size of crystals in this sample is 10μm on an edge.
Examples of each of the polymorphs of calcium carbonate are shown in Fig. 5. From left to right they are: calcite positioned with two faces parallel to the image plane, vaterite, and aragonite. The calcite positioned parallel to the image plane clearly shows the 78° and 102° angles characteristic of the calcite rhomb.
Fig. 5 Three polymorphs of calcium carbonate grown through precipitate technique in our lab. From left to right, they are: calcite with two faces parallel to the page, vaterite, and aragonite.
3.2. Optical apparatus
Our optical tweezers arrangement consists of an inverted microscope system with a 100×, 1.3 NA microscope objective focusing light into the sample region. Our light source is a 660nm diode laser with power levels in the sample from 10mW to 80mW. The laser is initially linearly polarized; we control the polarization with a quarter-wave plate with its fast axis at an angle ϕ from the incident polarization direction. Our sample of calcite suspended in water mixed with a wetting agent is approximately 100μm thick, bounded by a glass microscope slide and a glass cover slip on the top and bottom and enclosed on four sides with double sided tape. We collect sample image information with a Basler Ace camera recording data with LabVIEW at 30 fps.
3.3. Video image processing
The raw image data we acquire is short videos in which we can clearly observe our calcite rotating at varying speeds. To retrieve instantaneous rotational speed information, we process the data using Python and MATLAB. The program written to analyze the raw data processes each individual frame by utilizing a variety of methods from the OpenCV library. First, bilateral filtering is applied to the frames, which are then converted to grayscale. Next, the Otsu method of thresholding is performed to convert our frames to binary images consisting of black and white pixels. Using this method, any pixel below a certain calculated intensity is set to 0 (black), and those above the calculated intensity are set to 255 (white). To look for the crystal’s edges the Canny edge detection method is applied, followed by dilation to reduce breaks in the contours. The program then looks for the largest contour, which in our case would be the crystal, by calculating each contour’s enclosed area. The smallest area rectangle necessary to enclose the crystal is then calculated and applied to each frame. The corners of the rectangle are stored in an array and subsequently written to a file. This array is analyzed in MATLAB to find the rotational speed.
The rotation rate obtained through analyzing frame-by-frame information is a novel technique and one that could be applied to real-time speed analysis (if the frame rate is faster than the object speed). This technique also has a few limitations. Since the edge-detection method is based on thresholding and our videos have varying levels of illumination, it occasionally finds a rectangle that is not related to the crystal shape. These data points have to be identified and sorted out of the data files. In addition, it is important to note that rotations in the video plane will be accurately measured, but if the object is rotating out of the plane of the video the measured change in position would not accurately reflect the rotation speed. However, in our particular situation, while our calcite crystals do have a small out-of-plane motion component, the estimated change to our rotation rate due to this component is within our measurement uncertainty.
4. Results
Calcite rhombohedrons exhibit a characteristic rotational pattern in our optical trap regardless of laser power and ellipticity. The pattern of the rotation is also the same at different locations within the sample region. An easily identifiable rotational speed occurs when the optic axis lies at least partially transverse to the beam axis, as discussed in Section 2 and shown in Fig. 2(a). The average speed of rotation in this position has a linear dependence on laser power and sinusoidal dependence on ellipticity. With a lowest energy state occurring when the extraordinary axis is positioned parallel to the polarization axis, there is a torque on the crystal towards this orientation. This results in a nonuniform rotational behavior that makes the crystal speed up and slow down as the extraordinary axis approaches and moves away from the polarization major axis. This behavior has been described previously [5]. Calcite with its optic axis parallel to the beam axis, as in Fig. 2(b), always exhibits a complex repeating pattern. Instead of a cycle of fast and slow motions with π symmetry, we see an unexpected cycle of rotational speeds with only a 2π symmetry, moving quickly at one orientation, very slowly at exactly π away, and with other rotational rates between.
We can clearly see the difference between these two types of motion by looking at one crystal which we captured rotating first in one and then in the other configuration. It is typically difficult to compare these positions, as the more stable position is that as in Fig. 2(b). In this instance, because the trapping was against the surface of the sample region, the crystal was initially pushed with one face against the surface as in Fig. 2(a). At a later time it experienced some laser fluctuation and then settled into its stable position. The gradient force is similar in both positions, as the crystal has approximately the same thickness and width in either orientation. The rotation rates for this crystal, approximately 5μm on an edge, are shown in Fig. 6. The slow average rotation rates (the period is approximately 3.0 seconds for this motion) makes the random signal fluctuations relatively high, but the periodic pattern is still apparent. When the crystal is positioned with two faces parallel to the plane of the electric field (Fig. 2(a)) so its optic axis lies partially in the transverse plane (red line), we see a faster rotation rate every π alternating with a slower rotation rate at π/2. The same crystal then positioned itself with its optic axis primarily along the beam axis, and the motion changed significantly. In this position the crystal shows a very slow rotation rate every 2π and a fast rotation rate at exactly π away. Between the primary alternating speeds there are two other rotation speeds, although only one slow position at approximately 0.25π before the slow rate is distinct in this example. Other variations are more apparent in the following figures.
Fig. 6 Rotation speeds for a crystal positioned in elliptically polarized light at 40° with its optic axis transverse to the beam axis (red) and parallel to the beam axis (black).
The complexity of the on-axis rotational behavior is shown further in Fig. 7 and Fig. 8. The instantaneous speed of a crystal rotating in elliptically polarized light set with a quarter-wave plate at −40° from the incident linear polarization is shown in Fig. 7. This large crystal was trapped three-dimensionally in the center of our sample; therefore, we can be sure that there is no surface friction affecting the motion. The same crystal in the same sample position rotating in elliptically polarized light set with the quarter-wave plate at −30° is shown in Fig. 8. In each case we show the plots of rotational speed with θ and with time. Plots of the rotational speed with θ are critical when considering the exact position of the crystal and plots with time offer a sense of the non-uniform but periodic motion as well as the overall rotation rates. For example, as the ellipticity increases, the period goes from approximately 4 seconds (Fig. 7) to approximately 6 seconds (Fig. 8). The distinctive features of both plots are that the instantaneous rotational speed is very slow once every full cycle, beginning at θ = 0, and is faster at π, 3π, etc. After each significantly slow position there is a fast-slow alternation before the next fast position, and then a slow-fast alternation to complete the full cycle.
Fig. 7 Calcite 10 μm on an edge in elliptically polarized light at −40°. Polarization ellipse drawn in red. Rotational speeds are found from video using image analysis.
Fig. 8 Calcite 10 μm on an edge in elliptically polarized light at −30°. Polarization ellipse drawn in red. Rotational speeds are found from video using image analysis.
The crystal is in specific positions at each significant slow and fast rotational speed. To illustrate the consistency of this behavior Fig. 9 shows seven frames corresponding to the distinct rotation rates for the motion measured in Fig. 8. The crystal is rotating in the positive θ direction (counter-clockwise) from frames 9(a) – 9(g). Frame 9(a) corresponds with the slowest rotational position at θ = 0, where the extraordinary axis of face three is aligned with the laser polarization axis. Frames 9(b) and 9(c) are for approximately 0.40π and 0.60π, and frame 9(d) corresponds to the fastest rotational position at π, which is π from the first frame. Frames 9(e) and 9(f) are at approximately 1.4π and 1.6π. Notice that in frame 9(a) the crystal is tilted up along the laser polarization direction such that face three is more exposed to the laser illumination, and in frame 9(d) it is rotated by π and tilted the opposite way so faces one and two are more exposed to the light. We model this behavior in more detail in the analysis of Section 5.
Fig. 9 Video frames from significant positions corresponding to the rotational motion in Fig 8. White bar in frame (a) gives a 10μm scale. Red arrows in frames and red polarization ellipse indicate the polarization direction at −30° from the vertical.
5. Analysis
The complex behavior shown in Section 4 is driven by the alignment torque and a changing laser flux on the crystal faces. When the calcite aligns itself as on the right in Fig. 2 with its optic axis approximately along the beam axis, it rotates about a slightly tilted axis, as shown in Fig. 10. Rotating about an axis tilted with respect to the beam axis, the flux changes as more and less of each crystal face is visible to the beam. For example, where the crystal moves the most slowly, face three presents with more flux going through it (see frame 9(a) in Fig. 9). At the opposite rotation, faces one and two present more surface and the crystal has a faster speed (frame 9(d)). These two positions are indicated in Fig. 10 for the electric field direction drawn in red, where on the left face three has more surface presented and the rotation rate is slower, and on the right faces one and two have more surface and the rotation is faster. In these figures the optic axis is drawn running through the crystal and emerging out of the page below and to the right of the center of the crystal. Each face also has a unique incident angle and a unique set of ordinary and extraordinary axes. Therefore, the alignment torque τa in Eq. (2) is different for each face as the crystal rotates. The spin torque τs depends on ϕ and Γ. The value of Γ is dependent on the difference in refractive indices for each face. While this is also different for each face, it does not change with the rotation angle θ, and hence τs adds a constant rotational acceleration. This will affect the overall rotational speed of the crystal, but not the dynamics within each rotational period. Another consideration is that we are working with polarized beams, and at the focus of a beam polarization becomes undefined. This would clearly affect the polarization-dependent motion of the crystal. However, the crystal sizes are quite large in our examples (5μm – 15μm), significantly larger than the Rayleigh range of the focused beam, and the bulk of the material in the beam does experience a defined polarization.
Fig. 10 Positions for a slow (left) and fast (right) rotation speed given the polarization direction drawn in red. The line representing the tilted optic axis, about which the crystal rotates, is drawn running through the crystal and emerging below and to the right of center.
To model the total torque and changing flux we assume that each face of the crystal is illuminated fully for exactly half of the rotation, θ = −π/2 to π/2, and is illuminated by a particular factor C of this light from θ = π/2 to 3π/2, with θ defined as shown in Fig. 2(b). The angle θp for each face is measured such that the plane of incidence contains the extraordinary axis. Faces one and two have an angle relative to the optic axis of θp = 35.5°, and for face three θp = 48.2°. The ordinary index of refraction is a constant n0 = 1.658 and the extraordinary index is calculated from Eq. (1) to be ne = 1.649 for faces one and two and ne = 1.639 for face three.
We plot the alignment torque on each face (blue, red, and green lines) and the total alignment torque (black) at different orientations in Fig. 11(a). In these plots we use a factor of C = 0.50, modeling such that half of the full illumination reaches the crystal on half of the rotation. There are several angles at which the net torque changes sign, which should correspond with extremes in rotation rates. To better visualize the rotation rates at these transition points, the calculated torque summed over θ is shown in Fig. 11(b). The lowest rotation rates are expected at 0, 2π, and 4π, and the fastest at π and 3π. Between each of these positions there are cycles of less extreme faster and slower motion.
Fig. 11 Total alignment torque (left) and angular velocity (right) as a function of rotation angle θ.
This model predicts the changes in measured rotation rates that we see. To show the accurate match with our data, we plot the calculated angular velocity overlaid with our experimental results. Both the model and the experimental data identify θ = 0 as corresponding to the position where the extraordinary axis for face three is aligned with the major axis of the polarization ellipse. For our crystal rotating in elliptically polarized light at −40° and at −30° we see a match with the most significant features (Fig. 12). In the case of greater ellipticity (Fig. 12(b)), both the angular positions and the relative angular velocities are a good fit. For lower ellipticity and faster overall rotation (Fig. 12(a)), the angular positions at which the maxima and minima are reached match, although the experimental velocities are more regularly periodic than the model.
Fig. 12 Calcite 10μm on an edge rotating in elliptically polarized light; measurements (black) normalized and overlaid with model of angular speed of crystal (blue).
Our model is effective for understanding the rotational positions at which crystals larger than the beam waist (and hence much larger than λ) move faster and slower, although it does not precisely predict the relative amplitudes of the rotation rates. In particular, as the polarization becomes more circular, the spin torque contribution is more significant and the rotation rate becomes closer to periodic (with three periods for each full rotation). Additional improvements can be made to this model, including taking into account the change in the thickness of the crystal for each face as it rotates. Another effect is that since the motion is very slow, the inertia that these large crystals have and the drag forces acting on the crystals may be significant, particularly when the crystal briefly becomes virtually stationary. However, given these considerations, our theoretical match describes the non-uniform rotational behavior across elllipticities and crystal positions within our sample region.
6. Conclusion
Trapped and rotating rhombohedral calcite crystals show a complex periodic behavior previously unseen with birefringent systems. Our calcite is grown through a precipitate technique that results in regular crystals, and rotation rates are processed with video image analysis. We model the rotational behavior of calcite positioned with its optic axis along the beam axis, showing that the torque is governed by alignment between three extraordinary axes and the polarization direction as well as the laser flux reaching each face. Our model fits significant features of the rotation, and further investigation will produce a more precise match to the variation in rotation rate amplitudes. We also anticipate positioning multiple crystals in the same polarization mode and observing their behavior. Understanding the dependence of this motion on the direction and ellipticity of the laser polarization offers enticing approaches to polarization measurements and to modulated manipulation of matter in vector modes.
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