Abstract:
A method for optical lift includes receiving illumination in a first direction on at least one of two different surface profiles of one or more cambered refractive objects. The one or more cambered refractive objects are rotated to a position of stable rotational equilibrium in response to the received illumination. The one or more cambered refractive objects are moved in a second direction non-parallel direction from the first direction.

Description:
[0001]    This application claims the benefit of U.S. Provisional Patent Application Ser. No. 61/419,721, filed Dec. 3, 2010, which is hereby incorporated by reference in its entirety. 
     
    
     FIELD 
       [0002]    This invention relates to optical lift apparatuses that position and transport objects using non-contact optical forces and methods thereof. 
       BACKGROUND 
       [0003]    Optical tweezers use a highly focused beam of light to provide an attractive or repulsive force to physically hold and move microscopic dielectric objects. Typically, this beam of light is focused by sending it through a microscope objective or other short focal length lens. The narrowest point of the focused beam of light, known as the beam waist, contains a very strong electric field gradient. Dielectric particles are attracted along the gradient to the region of strongest electric field which is the center of the beam of light. The beam of light also tends to apply a force on particles in the beam along the direction of beam propagation. This is known as the scattering force and results in the particle being displaced slightly downstream from the exact position of the beam waist. 
         [0004]    Unfortunately, optical tweezers are limited to the small volume of light in the focal region and therefore are unsuitable for large areas of volumetrically dispersed objects. Additionally, optical tweezers do not work well across large depths. Further, the components needed to make optical tweezers are expensive. 
       SUMMARY 
       [0005]    An optical lift apparatus includes one or more cambered refractive objects. Each of the one or more cambered refractive objects has at least two different surface profiles to configure each of the one or more cambered refractive objects to rotate into a position of stable rotational equilibrium and have a lift force in a non-parallel direction with respect to an incoming direction of illumination applied to one or more of the different surface profiles. 
         [0006]    A method for making an optical lift apparatus includes providing one or more cambered refractive objects. At least two different surface profiles are formed on each of the one or more cambered refractive objects to configure each of the one or more cambered refractive objects to rotate into a position of stable rotational equilibrium and have a lift force in a non-parallel direction with respect to an incoming direction of illumination applied to one or more of the different surface profiles. 
         [0007]    A method for optical lift includes receiving illumination in a first direction on at least one of two different surface profiles of one or more cambered refractive objects. The one or more cambered refractive objects are rotated to a position of stable rotational equilibrium in response to the received illumination. The one or more cambered refractive objects are moved in a second direction non-parallel direction from the first direction. 
         [0008]    This technology provides a number of advantages including providing optical lift apparatuses which can position and transport an optical structure using non-contact optical forces without the need for focused beams. Additionally, unlike optical tweezers, with this technology an intensity gradient is not required and can be negligible while still achieving a transverse force. This technology can be used in a variety of different applications, including providing optical lift to power micro-machines, transport microscopic particles in a liquid, and to augment the design of solar sails for interstellar space travel. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0009]    The patent or application file contains at least one drawing executed in color. Copies of this patent or patent application publication with color drawing(s) will be provided by the Office upon request and payment of the necessary fee. 
           [0010]      FIG. 1  is a block diagram of an exemplary optical lift apparatus; 
           [0011]      FIG. 2  is a perspective view of an exemplary optical sail; 
           [0012]      FIG. 3  is a functional diagram illustrating exemplary transport of the optical structure illustrated in  FIG. 1 ; 
           [0013]      FIGS. 4   a - 4   c  are partial cross-sectional views and partial ray tracing illustrations of different angles of attack for the optical structure shown in  FIG. 1  in water; 
           [0014]      FIG. 5  is a graph of angle of attack, a, and corresponding lift angle, Θ, as a function of relative refractive index, m, for two stable orientations; and 
           [0015]      FIG. 6  is a time lapsed, composite image illustrating exemplary transport of the optical structure shown in  FIG. 1 . 
       
    
    
     DETAILED DESCRIPTION 
       [0016]    An exemplary optical lift apparatus  10  is illustrated in  FIG. 1 . This exemplary optical lift apparatus  10  includes an optical structure  12 ( 1 ), an illumination device  14 , and an illumination control computing device  16 , although the optical lift apparatus can comprise other types and numbers of device, components, and other elements in other configurations. This exemplary technology provides a number of advantages including providing optical lift apparatuses which can position and transport an optical structure using non-contact optical forces without the need for focused beams and with a negligible field gradient. 
         [0017]    Referring more specifically to  FIG. 1 , the optical structure  12 ( 1 ) is a semicircular rod having a flat surface  20  and a curved surface  22 , although other types and numbers of cambered refractive objects having at least two different outer surface profiles can be used. One or more outer surfaces of the optical structure  12 ( 1 ) may be chemically functionalized to allow other bodies, such as proteins by way of example, to stick to the optical structure  12 ( 1 ) and therefore also be transported. By way of example only, DNA strands or long chain molecules can serve as a tether to connect at least one of the outer surfaces of the optical structure  12 ( 1 ) to other structures, such as biological tissue. 
         [0018]    Referring to  FIG. 2 , an optical sail  24  has an array of optical structures  12 ( 1 )- 12 ( 4 ) connected together to from a monolithic structure to allow the simultaneous propulsion and steering of the solar sail  24 , although other types and number of structures and other elements can be used. Optical structures  12 ( 2 )- 12 ( 4 ) are each the same in structure and operation as optical structure  12 ( 1 ), although other types and combinations of optical structures which are configured in other manners can be used. Solar sails can by way of example be used for transporting a small payload in free space. In this example, the flat outer surface profiles  20  of the optical structures  12 ( 1 )- 12 ( 4 ) are aligned to face in substantially the same direction, although the optical sail  24  could have other configurations. The array of the optical structures  12 ( 1 )- 12 ( 4 ) may optionally be mounted on a frame (not shown) or fabricated in a thin plastic membrane (not shown) to achieve a large force to mass ratio. Depending on the particular application, the optical structures  12 ( 1 )- 12 ( 4 ) separately or together, also may have one or more reflective or absorbing materials or layers, such as high reflection anti-reflection coatings by way of example, positioned to enhance either the lifting force or the scattering force. Additionally, one or more electro-optic elements (not shown), such as a shutter apparatus comprising one or more liquid crystal shutters or one or more mechanical louvers by way of example, may optionally be connected to the optical sail  24  and positioned between the light source and the optical sail  24  to control the exposure to incident light on the optical sail  24 . For example, sections of the optical sail  24  can be oriented so that when exposed to light by the optional shutter system can be steered to the right and other sections of the optical sail can be oriented so that when exposed to light by the optional shutter system can be steered to the left by way of example. Similarly, sections of the optical sail  24  can be oriented so that when exposed to light by the optional shutter system can be steered up and other sections of the optical sail can be oriented so that when exposed to light by the optional shutter system can be steered down by way of example. Further, the optical sail  24  as well as any of the other exemplary optical structures  12 ( 1 )- 12 ( 4 ), separately or together, could be used without a controlled source of light and instead could be with uncontrolled sources of light, such as sunlight by way of example only. 
         [0019]    Referring back to  FIG. 1 , the illumination device  14  is positioned to direct uniform illumination towards the optical structure  12 ( 1 ), although other types and numbers of illumination devices could be used. In this example, the illumination device  14  is a coherent light source, such as a laser by way of example, although an incoherent light source, such as an arc lamp by way of example, or other types of light sources can be used. 
         [0020]    The illumination control computing device  16  may be used to control the illumination device  14 , although other manners for controlling the illumination device  14  can be used. In this example, the illumination control computing device  16  includes a central processing unit (CPU) or processor  26 , a memory  28 , a user input device  30 , a display  32 , and an interface device  34  which are coupled together by a bus or other link, although other numbers and types of systems, devices, components, and elements in other configurations and locations can be used. The processor  26  executes a program of stored instructions for one or more aspects of the present technology as described and illustrated by way of the examples herein including application of illumination to position and transport a cambered optical structure, although other types and numbers of processing devices and logic could be used and the processor could execute other numbers and types of programmed instructions. 
         [0021]    The memory  28  stores these programmed instructions for one or more aspects of the present technology as described and illustrated by way of the examples herein, although some or all of the programmed instructions could be stored and executed elsewhere. A variety of different types of memory storage devices, such as a random access memory (RAM) or a read only memory (ROM) in the system or a floppy disk, hard disk, CD ROM, DVD ROM, or other computer readable medium which is read from and written to by a magnetic, optical, or other reading and writing system that is coupled to the processor  26 , can be used for the memory  28 . 
         [0022]    The user input device  30  is used to input data and/or selections, such as when to apply illumination, although the user input device could be used to input other types of requests and data and interact with other elements. The user input device  30  can include keypads, touch screens, and/or vocal input processing systems although other types and numbers of user input devices can be used. The display  32  is a computer monitor, although other types and numbers of displays could be used. 
         [0023]    The interface device  34  is used to operatively couple and communicate between the illumination control computing device  16  and the illumination device  14  via a communications network, although other types and numbers of communication networks or systems with other types and numbers of connections and configurations can be used. 
         [0024]    Although an example of the illumination control computing device  16  is illustrated and described herein, this device can be implemented on any suitable computer system or computing device. It is to be understood that the computing device in the example described herein is for exemplary purposes, as many variations of the specific hardware and software used to implement the examples are possible, as will be appreciated by those skilled in the relevant art(s). 
         [0025]    Furthermore, the computing device of the example may be conveniently implemented using one or more general purpose computer systems, microprocessors, digital signal processors, and micro-controllers, programmed according to the teachings of the examples, as described and illustrated herein, and as will be appreciated by those ordinary skill in the art. 
         [0026]    In addition, two or more computing systems or devices can be substituted for the computing device in the example. Accordingly, principles and advantages of distributed processing, such as redundancy and replication also can be implemented, as desired, to increase the robustness and performance of the devices and systems of the examples. The examples may also be implemented on computer device or devices that extend across any suitable network using any suitable interface mechanisms and communications technologies, including by way of example only telecommunications in any suitable form (e.g., voice and modem), wireless communications media, wireless communications networks, cellular communications networks, G3 communications networks, Public Switched Telephone Network (PSTNs), Packet Data Networks (PDNs), the Internet, intranets, and combinations thereof. 
         [0027]    The example may also be embodied as a non-transitory computer readable medium having instructions stored thereon for one or more aspects of the present technology as described and illustrated by way of the examples herein, as described herein, which when executed by a processor, cause the processor to carry out the steps necessary to implement the methods of the examples, as described and illustrated herein. 
         [0028]    Referring to  FIG. 3 , a functional diagram of how the optical structure  12 ( 1 ) may be lifted from a surface and transported with a beam of light from illumination device  14  controlled by illumination control computing device  16  is illustrated. The optical structure  12 ( 1 ) is illuminated with light from the illumination device  12 . For example, uniform or weakly focused illumination may be used, either from a coherent light source such as a laser, or an incoherent light source, such as an arc lamp, can be used. The optical structure  12 ( 1 ) causes the reflection, transmission, and refraction of light, thereby changing the momentum of the incident rays. The difference in momentum is imparted to the optical structure  12 ( 1 ), causing the optical structure  12 ( 1 ) to change its spatial position and angular orientation. When light illuminates the optical structure  12 ( 1 ) at particular angles, the optical structure  12 ( 1 ) experiences a constant force in a direction that has components that are parallel and perpendicular to the direction of the incident beam of light. The particular lift height and translation distance of the optical structure  12 ( 1 ) is controlled by the duration of the optical pulse of light, shown in the exemplary intensity v. time graph, applied by the illumination device  14  from control signals from the illumination control computing device  16  and also from the shape of the optical structure  12 ( 1 ). This optical lift may be used to transport either microscopic objects, such as biological tissue, or other particles in fluid, micromachines, attached to chemically functionalized surfaces or a macroscopic object, such as solar sail illustrated back in  FIG. 2 . 
         [0029]    Referring to  FIGS. 4   a - 4   c,  partial cross-sectional views and partial ray tracing illustrations of different angles of attack for the exemplary optical structure  12 ( 1 ) in water are illustrated. As illustrated, the optical structure  12 ( 1 ) comprising a semi-cylindrical rod will rotate into a position of stable equilibrium. In  FIG. 4   a the angle of attack, α, is − 45 degrees, in  FIG. 4   b  the angle of attack, α, is 0 degrees, and in  FIG. 4   c  the angle of attack, α, is 45 degrees by way of example only The net force, F, and the rotational direction owing to torque, T, are indicated in  FIGS. 4   a - 4   c.  The force vectors are drawn from the center of mass, which is located a distance 4R/3π along the bisector below the flat surface, where R is the radius of the semi-circle. The lift (levitation) component of force is in the x (z) direction. The lift component of force is in the positive x-direction when α=±45°, whereas it is in the negative x-direction when α=0°. The latter case is consistent with the direction of lift for a conventional airfoil. The optical structure  12 ( 1 ) experiences a non-zero torque in all three examples. The horizontal arrows (purple) in  FIGS. 4   a - 4   c  suggest counter-clockwise rotation when α=−45°, and clockwise rotation when α=0°, and α=45°. One may expect to find attack angles where the torque vanishes, providing stable lift, or where the lift force vanishes, producing stall. The optical structure  12 ( 1 ) comprising the semi-cylindrical rod also experiences a forward scattering force for any attack angle, which corresponds to drag in aerodynamics 
         [0030]    Collimated rays from the illumination device  14  are illustrated in red are incident from the left side in each of  FIGS. 4   a - 4   c.  Green lines represent the Minkowski radiation pressure force for each ray, reflected, refracted, and transmitted rays are shown in red, the net force vector is shown in blue, and the direction of rotation owing to torque is shown in purple. The lift force is positive in  FIG. 4   a  and  FIG. 4   c  and is negative in  FIG. 4   b.  Torque causes the particle to rotate counter-clockwise in  FIG. 4   a  and clockwise in  FIG. 4   b  and  FIG. 4   c.  In the example illustrated in  FIGS. 4   a - 4   c,  arbitrary spatial units are given. 
         [0031]    Owing to the light-induced torque, the relative angle of attack, α, (i.e., the angle subtended by the incident rays and the flat surface  20  of the optical structure  12 ( 1 )) changes until the optical structure  12 ( 1 ) reaches a position of stable rotational equilibrium. The particular angle of attack depends on the refractive index of the optical structure  12 ( 1 ) and the surrounding material, as well as the shape of the optical structure  12 ( 1 ). For example, the orientation displayed in  FIG. 4   b  with an angle of attack (α=0) is stable when the refractive index of the optical structure  12 ( 1 ) is 1.65 times that of the surrounding medium. For the optical structure  12 ( 1 ) having a semicircular cross-section, the lift component of force (e.g., the vector component that is perpendicular to the direction of the incident light rays) is predicted to be optimal when the refractive index ratio is about 1.20. 
         [0032]    Accordingly, in these examples the optical structure  12 ( 1 ) comprising a semi-cylindrical rod will simultaneously torque about the y-axis toward a stable angle of attack and torque about the x-axis to align with the rays. The former will be energetically favorable if the length, L, of the optical structure  12 ( 1 ) is much greater than its diameter, 2R, e.g., if I x =ML 2 /12&gt;&gt;I y =MR 2 /2+M(4R/3π) 2 , where M is the mass of the optical structure  12 ( 1 ). A ray tracing model confirmed that optical lift strongly dominates the tendency of the optical structure  12 ( 1 ) to initially rotate in the direction of the beam when L&gt;20R. Even with L≈3R, as in the experimental demonstration, the alignment with the beam axis (as evidenced by an apparent shortening of the optical structure  12 ( 1 )) is a weak effect. When gravity, van der Waals force, or surface tension are included, the levitation force and beam-aligning torque may be negligible at a physical surface, and the optical structure  12 ( 1 ) may simply slide along the surface. 
         [0033]    The power required to achieve an optical lift effect may be determined by estimating the work done by raising the center of mass a distance h. Assume that the optical structure  12 ( 1 ) comprises the semi-cylindrical rod of radius R, length L, and density, ρ, has settled to the bottom of chamber filled with liquid of density ρ 0 , with the curved side of the optical structure  12 ( 1 ) in contact with the chamber. Optical forces will cause the optical structure  12 ( 1 ) to rotate by an angle θ, raising the center of mass a distance, h, where h/R=√{square root over (1+ε 2 −2ε cos θ)}+ε−1 and ε=4/3π. In the small angle approximation we write h/R≈(θ/θ 0 ) 2  where θ 0 =2(1−ε)/ε. 
         [0034]    Both the levitation force (F z ) and the torque do work to raise the effective mass, μ=μρ 0 πR 2 L/2, against gravity, where η=(ρ−ρ 0 )/ρ 0  is a buoyance parameter. Thus, μgh=Tθ+F z h=(n 1 P/c)(RQ T θ+Q z h). Hence the power required to achieve an optical lift effect is approximately P=(μgc/Q T n 1 θ 0   2 )θ, assuming the torque-related energy is greater than the levitation energy. For an optical structure  12 ( 1 ) having an effective mass of 100 picograms and an average torque efficiency of 10%, this amounts to a power across the rod of 0.4 mW if θ=30°. The total power of the illuminating beam from illumination device  14  must be proportionately larger—by a factor of the beam and rod cross-section ratio. Therefore, several tens of milliwatts of collimated light may be sufficient to observe the torque and transverse motion associated with optical lift. 
         [0035]    Referring to  FIG. 5 , a graph illustrating angle of attack, α, and corresponding lift angle, Θ, plotted as a function of relative refractive index, m, for two stable orientations are illustrated. A stall condition (zero lift angle) occurs for orientation- 1  when m≈1.2, whereas a large lift angle Θ≈60° is predicted for orientation- 2 . Stall occurs for orientation- 2  when m&gt;1.4. 
         [0036]    Stable rotational equilibrium is particularly important for applications that benefit from uniform motion. This requires both T y =0 and ∂T y /∂α&lt;0. In this example, four attack angles, α 1 , α 2 , and 180°−α 1,2 , where the semi-cylinder experiences stable non-zero lift forces were discovered. These occur when the relative refractive index, m=n particle /n host  falls an within the range of values from unity to 1.4, as shown in  FIG. 5 . When m&gt;1.4, only two angles, α 1  and 180°−α 1  provide stable lift. The corresponding lift angles plotted in  FIG. 5 , Θ=arctan(F x /Fz), exceed 60° in some cases. Remarkably, the stable lift force can therefore exceed the scattering force by more than 70%. The values of α 1  may be positive or negative, whereas α 2  is always negative. Similarly, Θ 1  may be positive or negative, whereas Θ 2 ≧0. The horizontal orientation, α 1 =0, is stable at m=1.65, resulting in a lift angle of Θ 1 =−32°. Also illustrated in  FIG. 5  is that when the relative index has a value of m=1.2, the lift force stalls at α 1 =−11.4°. For isotropic materials the equilibrium and lift angles were computed to be weakly sensitive to the polarization state of the incident beam. Further calculations in the range 0&lt;m&lt;1 found the rod to be stable while erect (α 1,2 =±90°), although there was zero lift at these orientations. Accordingly, optical lift may be optimized for a given application by controlling not only the particle shape, but also optical properties such as refraction, reflection, and absorption Although in this example, the relative refractive index was about 1.2, the relative refractive index of the optical structure can range between about 1.0 and 2.0 and in some examples as illustrated in  FIG. 5  between about 1.05 and 1.4. 
         [0037]    For practical application, it is desirable to characterize and optimize the efficiency of the momentum transfer process. In this example, the ideal system is treated as having two dimensions, ignoring possible rotations of the optical structure into the direction of the beam of light. For incident rays directed along the z-axis, the net force and torque may be expressed in terms of efficiency parameters, Q: 
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         [0000]    where P is the total beam power, Q x =Q sin Θ corresponds to the efficiency of lift, Q z =Q cos Θ corresponds to the forward scattering or levitation efficiency, Q=√{square root over (Q x   2 +Q z   2 )}, and Q T  is the efficiency of torque. Computed efficiency values for the rotational equilibrium states α 1  and α 2 , shows that lift is the dominant component of force for m&lt;1.35. In particular, the magnitude of the lift efficiency is greatest at m=1.2 for the α 2 ≈−40° state, and vanishes at this value of m for the α 1 ≈−11° state. Accordingly, 10-20% of the incident beam momentum may be converted into a stable lift force for a dielectric rod having a semi-circular cross-section. 
         [0038]    Referring to  FIG. 6 , a time lapsed composite image of exemplary transport of the optical structure  12 ( 1 ) is illustrated. In this example, a laser beam having a wavelength of 975 nm from the illumination device  14  was weakly focused to a diameter d=50 μm on a 150 μm thick optical structure  12 ( 1 ) using a lens (not shown) of focal length f=60 mm. White light images were recorded through a 40× microscope objective onto a camera (not shown), with the laser wavelength (λ=975 μm) filtered out. The samples were mounted on a three-axis translation stage. 
         [0039]    The time-lapsed composite image (1.67 seconds per shot) in  FIG. 6  of the optical structure  12 ( 1 ) comprises the semi-cylindrical rod lifting sideways from left to right near the bottom of a glass chamber, owing to a transverse optical lift force. The value of the relative refractive index of the optical structure  12 ( 1 ) immersed in the chamber was m=1.2. The optical structure  12 ( 1 ) initially experiences a torque, then exhibits a distinct translation, with a component of velocity, v x , directed toward the right of the illustrated image. At roughly 130 mW the optical structure  12 ( 1 ) comprising the semi-cylindrical rod rapidly rotated to a stable attack angle, and then simultaneously underwent lift and levitation as expected. Under similar experimental conditions, micro-spheres did not exhibit this effect. The optical structure  12 ( 1 ) comprising the semi-cylindrical rod was not attracted to the center of the laser beam, as would have been the case if a strong transverse gradient force were present. The defocusing of the optical structure  12 ( 1 ) comprising the semi-cylindrical rod in  FIG. 6  is attributed to the levitation component of force. The maximum transverse speed was about 3.5 μm/s, and the average levitation speed was about 2.5 μm/s, resulting in a lift angle in the range of about Θ=55°. As expected, the particle speed was greater in the central region of high power density. 
         [0040]    Accordingly, as illustrated by this example a transparent, refractive optical structure  12 ( 1 ) in the shape of the semi-cylindrical rod or cambered light foil experiences an optically-induced lift force, accompanied by a rotation to a stable orientation, when exposed to a uniform incident light field. With this technology, a lift force, i.e., a force perpendicular to the direction of the incident light beam from the illumination device  14  and an orientation of stable equilibrium results when the optical structure  12 ( 1 ) is exposed to a uniform light field. This effect is based on the theory of radiation pressure. 
         [0041]    An examination of the Kutta-Joukowski theorem of aerodynamic lift provides support for this optical lift. It states that the lift force in the x-direction is related to the pressure p at every point on a wing: 
         [0000]        F   x =           p{circumflex over (n)}·{circumflex over (x)}da    (1)
 
         [0000]    where {circumflex over (n)} is the normal vector of the wing surface, and da is an area element on the wing surface. In optics, the Minkowski expression for the local force at a dielectric interface is always normal to the surface, and it is therefore equivalent to the factor p{circumflex over (n)} da in equation (1). Numerical computations of the radiation pressure using ray tracing methods, followed by numerical integration, affords a simple means of computing the optical lift on an arbitrarily shaped particle, assuming the particle size is much greater than the wavelength of the illuminating beam. Smaller particles require a Lorenz-Mie approach to account for resonant effects. Owing directly to the Poynting theorem, the net force and torque do not depend on whether the Minkowski or Abraham momentum formulation is used. By computing both values, which disagree when too few rays are considered, an arbitrary degree of agreement and validity with this technology was achieved by increasing the number of rays. 
         [0042]    The momentum change of a ray may be determined by accounting for the direction and strength of each transmission and reflection event. The force on a pencil of rays of incident power P j , owing to both reflection and refraction at a single dielectric interface, may be expressed according to the Minkowski interpretation: 
         [0000]                =−( P   j   /c )( n   2,j  cos θ 2,j (1− R   j )− n   1,j  cos θ 1.j (1+ R   j )) {circumflex over (n)}   j    (2)
 
         [0000]    where n 1,j  and n 2,j  are the respective indexes of refraction of the incident and refracted ray, and likewise, θ 1,j  and θ 2,j  are angles of incidence and refraction as governed by Snell&#39;s law (n 1,j  sin θ 1,j =n 2,j  sin θ 2,j ), {circumflex over (n)} j  is the outward normal unit vector, pointing from the interface, toward the material of index n 1,j , R j  is the angle-dependent and polarization-dependent reflection coefficient, and c is the speed of light in vacuum. According to Newton&#39;s third Law of Motion, the force on the body from an individual ray is given by          =         . Refraction and the ensuing displacement of the rays, along with reflection, may produce a torque about the particle center of mass. The net force and torque on the body may be computed by summing over a large number of rays: 
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         [0000]    The moment arm           is a vector pointing from the particle center of mass to the point of intersection of the j th  ray. 
         [0043]    Accordingly, as illustrated in the examples herein refractive rods having an asymmetric camber may exhibit a transverse lift force, along with a forward scattering force, when exposed to a uniform collimated beam of light. Although this phenomenon is an optical analog to aerodynamic lift, it is refraction and reflection, rather than the Bernoulli principle, which accounts for the lift force. As many as four angles of attack that provide uniform lift without tumbling. 
         [0044]    Having thus described the basic concept of the invention, it will be rather apparent to those skilled in the art that the foregoing detailed disclosure is intended to be presented by way of example only, and is not limiting. Various alterations, improvements, and modifications will occur and are intended to those skilled in the art, though not expressly stated herein. These alterations, improvements, and modifications are intended to be suggested hereby, and are within the spirit and scope of the invention. Additionally, the recited order of processing elements or sequences, or the use of numbers, letters, or other designations therefore, is not intended to limit the claimed processes to any order except as may be specified in the claims. Accordingly, the invention is limited only by the following claims and equivalents thereto.