Abstract:
A method and system for manipulating object using a three dimensional optical trap configuration. By use of selected hologram on optical strap can be configured as a preselected three dimensional configuration for a variety of complex uses. The system can include various optical train components, such as partially transmissive mirrors and Keplerian telescope components to provide advantageously three dimensional optical traps.

Description:
CROSS-REFERENCE TO RELATED PATENT APPLICATIONS 
     This application is a continuation of U.S. application Ser. No. 11/974,716, filed Oct. 16, 2007, which claims priority from U.S. Provisional Application 60/852,252, filed Oct. 17, 2006, both of which are incorporated herein by reference in their entirety. 
     This invention is directed toward volumetric imaging of holographic optical traps. More particularly, the invention is directed to a method and system for creating arbitrary pro-selected three-dimensional (3D) configurations of optical traps having individually specified optical characteristics. Holographic techniques are used to modify individual trap wavefronts to establish pre-selected 3D structures having predetermined properties and are positionable independently in three dimensional space to carry out a variety of commercially useful tasks. 
    
    
     The United States Government has certain rights in this invention pursuant to a grant from the National Science Foundation through grant number DMR-0451589. 
    
    
     BACKGROUND OF THE INVENTION 
     There is, a well developed technology of using single light beams to form an optical trap which applies optical forces from the focused beam flight to confine an object to a particular location in space. These optical traps, or optical tweezers, have enabled fine scale manipulation of objects for a variety of commercial purposes. In addition, line traps, or extended optical tweezers, have been created which act as a one dimensional potential energy landscape for manipulating mesoscopic objects. Such line traps can be used to rapidly screen interactions between colloidal aid biological particles which find uses in biological research, medical diagnostics and drug discovery. However, these applications require methods of manipulation diagnostics and drug discovery. However, these applications require methods of manipulation for projecting line traps with precisely defined characteristics which prevent their use in situations with high performance demands. Further, the low degrees of freedom and facility of use for such line traps reduces the ease of use and limits the types of uses available. 
     SUMMARY OF THE INVENTION 
     The facility and range of applications of optical traps is greatly expanded by the method and system of the invention in which 3D intensity distributions are created by holography. These 3D representations are created by holographically translating optical traps through an optical train&#39;s focal plane and acquiring a stack of two dimensional images in the process. Shape phase holography is used to create a pre-selected 3D intensity distribution which has substantial degrees of freedom to manipulate any variety of object or mass for any task. 
     Various aspects of the invention are described hereinafter; and these and other improvements are described in greater detail below, including the drawings described in the following section. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  illustrates an optical train for performing a method of the invention; 
         FIG. 2A  illustrates a particular optical condition with z&lt;0 for an objective lens in the system of  FIG. 1 ;  FIG. 2B  illustrates the optical condition for z=0 for the objective lens of  FIG. 1  and  FIG. 2C  illustrates the optical condition for z&gt;0 for the objective lens of  FIG. 1 ; 
         FIG. 3A  illustrates a 3D reconstruction of an optical tweezer propagating along the z axis;  FIG. 3B  illustrates a cross-section of  FIG. 3A  along an xy plane;  FIG. 3C  illustrates a cross-section of  FIG. 3A  along a yz plane;  FIG. 3D  illustrates a cross-section of  FIG. 3A  along an xz plane;  FIG. 3E  illustrates a volumetric reconstruction of 35 optical tweezers arranged in a body-centered cubic lattice of the type shown in  FIG. 3F ; 
         FIG. 4A  illustrates a 3D reconstruction of a cylindrical lens line optical tweezer;  FIG. 4B  illustrates a cross-section of  FIG. 4A  along an xy plane;  FIG. 4C  illustrates a cross-section of  FIG. 4A  along a yz plane; and  FIG. 4D  illustrates a cross-section of  FIG. 4A  along an xz plane; and 
         FIG. 5A  illustrates a 3D reconstruction of a holographic optical trap featuring diffraction-limited convergence to a single focal plane;  FIG. 5B  illustrates a cross-section of  FIG. 5A  along a xy plane;  FIG. 5C  illustrates a cross-section of  FIG. 5A  along a yz plane; and  FIG. 5D  illustrates a cross-section of  FIG. 5A  along an xz plane. 
     
    
    
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
     An optical system for performing, methods of the invention is illustrated generally at  10  in  FIG. 1 . A beam of light  20  is output from a frequency-doubled solid-state laser  30 , preferably a Coherent Verdi system operating at a wavelength of λ=532 nm. The beam of light  20  is directed to an input pupil  40  of a high-numerical-aperture objective lens  50 , preferably a Nikon 100×Plan Apo, NA 1.4, oil immersion system that focuses the beam of light  20  into an optical trap (not shown). The beam of light  20  is imprinted with a phase-only hologram by a computer-addressed liquid-crystal spatial light modulator  60  (“SLM  60 ”), preferably a Hamamatsu X8267 PPM disposed in a plane conjugate to the objective lens&#39;  50  input plane. Computer  95  executes conventional computer software to generate the appropriate hologram using the SLM  60 . As a result, the light field, ψ(r), in the objective lens&#39;  50  focal plane is related to the field ψ(ρ) in the plane of the SLM  60  by the Fraunhofer transform, 
                       ψ   ⁡     (   r   )       =       -     ⅈ     λ   ⁢           ⁢   f         ⁢       ∫   Ω             ⁢       ψ   ⁡     (   ρ   )       ⁢           ⁢     exp   ⁡     (       -   ⅈ     ⁢       2   ⁢   π       λ   ⁢           ⁢   f       ⁢     r   ·   ρ       )       ⁢       ⅆ   2     ⁢   ρ             ,           (   1   )               
where f is the objective&#39;s focal length, where Ω is the optical train&#39;s aperture, and where we have dropped irrelevant phase factors. Assuming that the beam of light  20  illuminates the SLM  60  with a radially symmetric amplitude profile, u(ρ), and uniform phase, the field in the SLM&#39;s plane may be written as,
 
ψ(ρ)= u (ρ)exp( i φ(ρ)),  (2)
 
where φ(ρ) is the real-valued phase profile imprinted on the beam of light  20  by the SUM  60 . The SLM  60  in our preferred form of the system  10  imposes phase shifts between 0 and 2 π radians at each pixel of a 768×768 array. This two-dimensional phase array can be used to project a computer-generated phase-only hologram, φ(ρ), designed to transform the single optical tweezer into any desired three-dimensional configuration of optical traps, each with individually specified intensities and wavefront properties.
 
     Ordinarily, the pattern of holographic optical traps would be put to use by projecting it into a fluid-borne sample mounted in the objective lens&#39;  50  focal plane. To characterize the light field, we instead mount a front-surface mirror  70  in the sample plane. This mirror  70  reflects the trapping light back into the objective lens  50 , which transmits images of the traps through the partially reflecting mirror  70  to a charge-coupled device (CCD) camera  80 , preferably a NEC TI-324AII. In our implementation, the objective lens  50 , the camera  80  and camera eyepiece (not shown), are mounted in a conventional optical Microscope (not shown) and which is preferably a Nikon TE-2000U. 
     Three-dimensional reconstructions of the optical traps&#39; intensity distribution can be obtained by translating the mirror  70  relative to the objective lens  50 . Equivalently, the traps can be translated relative to the mirror  70  by superimposing the parabolic phase function, 
                         φ   z     ⁡     (   ρ   )       =     -         πρ   2     ⁢   z       λ   ⁢           ⁢     f   2             ,           (   3   )               
onto the hologram φ 0 (ρ) encoding a particular pattern of traps. The combined hologram, φ 0 (ρ)=φ 0 (ρ)+φ z (ρ) mod 2 π, projects the same pattern of traps as φ 0 (ρ) but with each trap translated by−z along optical axis  90  of the system  10 . The resulting image obtained from the reflected light represents a cross-section of the original trapping intensity at distance z from the focal plane of the objective lens  50 . Translating the traps under software control by computer  95  is particularly convenient because it minimizes changes in the optical train&#39;s properties due to mechanical motion and facilitates more accurate displacements along the optical axis  90 . Images obtained at each value of z are stacked up to yield a complete volumetric representation of the intensity distribution.
 
     As shown schematically in  FIGS. 2A-2C , the objective lens  50  captures essentially all of the reflected light for z&lt;0. For z&gt;0, however, the outermost rays of the converging trap are cut off by the objective lens&#39;  50  output pupil  105 , and the contrast is reduced accordingly. This could be corrected by multiplying the measured intensity field by a factor proportional to z for z&gt;0. The appropriate factor, however, is difficult to determine accurately, so we present only unaltered results. 
       FIG. 3A  shows a conventional optical tweezer  100  reconstructed in the manner described hereinbefore and displayed as an isointensity surface at 5 percent peak intensity and in three cross-sections ( FIGS. 3B-3D ). The representation in  FIG. 3A  is useful for showing the overall structure of the converging light, and the cross-sections of  FIGS. 3B-3D  provide an impression of the three dimensional light field that will confine an optically trapped object. The angle of convergence of 63° in immersion oil obtained from these data is consistent with an overall numerical aperture of 1.4. The radius of sharpest focus, r min ≈0.2 μm, is consistent with diffraction-limited focusing of the beam of light  20 . 
     These results highlight two additional aspects of this reconstruction technique. The objective lens  50  is designed to correct for spherical aberration when the beam of light  20  passing through water is refracted by a glass coverslip. Without this additional refraction, the projected optical trap  100  actually is degraded by roughly 20λ of spherical aberration, introduced by the objective lens  50 . This reduces the apparent numerical aperture and also extends the trap&#39;s focus along the z axis. The trap&#39;s effective numerical aperture in water would be roughly 1.2. The effect of spherical aberration can be approximately corrected by pre-distorting the beam of light  20  with the additional phase profile, 
                         φ   a     ⁡     (   ρ   )       =       a     2       ⁢     (       6   ⁢     x   4       -     6   ⁢     x   2       +   1     )         ,           (   4   )               
the Zernike polynomial describing spherical aberration. The radius, x, is measured as a fraction of the optical train aperture, and the coefficient α is measured in wavelengths of light. This procedure is used to correct for small amount of aberration present in practical optical trapping systems to optimize their performance.
 
     This correction was applied to an array  110  of 35 optical tweezers shown as a three-dimensional reconstruction in  FIG. 3E . These optical traps  100  are arranged in a three-dimensional body-centered cubic (BCC) lattice  115  shown in  FIG. 3F  with a 10.8 μm lattice constant. Without correcting for spherical aberration, these traps  100  would blend into each other along the optical axis  90 . With correction, their axial intensity gradients are clearly resolved. This accounts for holographic traps&#39; ability to organize objects along the optical axis. 
     Correcting for aberrations reduces the range of displacements, z, that can be imaged. Combining φ a  (ρ) with φ z (ρ) and φ 0 (ρ) increases gradients in φ(ρ), particularly for larger values of ρ near the edges of the diffraction optical element. Diffraction efficiency falls off rapidly when |∇φ(ρ)| exceeds 2π/Δρ, the maximum phase gradient that can be encoded on the SLM  60  with pixel size Δρ. This problem is exacerbated when φ 0 (ρ) itself has large gradients. In a preferred embodiment more complex trapping patterns without aberration are prepared. In particular, we use uncorrected volumetric imaging to illustrate the comparative advantages of the extended optical traps  100 . 
     The extended optical traps  100  have been projected in a time-shared sense by rapidly scanning a conventional optical tweezer along the trap&#39;s intended contour. A scanned trap has optical characteristics as good as a point-like optical tweezer, and an effective potential energy well that can be tailored by adjusting the instantaneous scanning rate Kinematic effects due to the trap&#39;s motion can be minimized by scanning rapidly enough. For some applications, however, continuous illumination or the simplicity of an optical train with no scanning capabilities can be desirable. 
     Continuously illuminated line traps have been created by expanding an optical tweezer  125  along one direction (see  FIG. 4A ). This can be achieved, for example, by introducing a cylindrical lens component such as by element  130  (see  FIG. 1 ) into the objective&#39;s input plane. Equivalently, a cylindrical-lens line tweezer can be implemented by encoding the function φ c (ρ)=πz 0 ρ x   2 /(λƒ 2 ) on the SLM  60 . The result, shown in  FIGS. 4A-4D  appears best useful in the plane of best focus, z=z 0 , with the point-like tweezer having been extended to a line with nearly parabolic intensity and a nearly Gaussian phase profile. The three-dimensional reconstruction, however, reveals that the cylindrical lens component merely introduces a large amount of astigmatism into the beam of light  20 , creating a second focal line perpendicular to the first. This is problematic for some applications because the astigmatic beam&#39;s axial intensity gradients are far weaker than a conventional optical tweezer&#39;s. Consequently, cylindrical-lens line traps typically cannot localize objects against radiation pressure along the optical axis  90 . 
     Replacing the single cylindrical lens with a cylindrical Keplerian telescope for the element  130  eliminates the astigmatism and thus creates a stable three-dimensional optical trap. Similarly, using the objective lens  50  to focus two interfering beams creates an interferometric optical trap capable of three-dimensional trapping. These approaches, however, offer little control over the extended traps intensity profiles, and neither affords control over the phase profile. 
     Shape-phase holography provides absolute control over both the amplitude and phase profiles of an extended form of the optical trap  100  at the expense of diffraction efficiency. It also yields traps with optimized axial intensity gradients, suitable for three-dimensional trapping. If the line trap is characterized by an amplitude profile ũ(ρ x ) and a phase profile {tilde over (p)}(ρ x ) along the {circumflex over (ρ)} x  direction in the objective&#39;s focal plane, then the field in the SLM plane is given from Eq. (1) as,
 
ψ(ρ)= u (ρ x )exp( ip (ρ x )),  (5)
 
where the phase p(ρ x ) is adjusted so that u(ρ x )≧0. Shape-phase holography implements this one-dimensional complex wavefront profile as a two-dimensional phase-only hologram,
 
                     φ   ⁡     (   ρ   )       =     {             p   ⁡     (     ρ   x     )       ,             S   ⁡     (   ρ   )       =   1                 q   ⁡     (   ρ   )       ,               S   ⁡     (   ρ   )       =   0     ,                     (   6   )               
where the shape function S(ρ) allocates a number of pixels along the row ρ y  proportional to u(ρ x ). One particularly effective choice is for S(ρ) to select pixels randomly along each row in the appropriate relative numbers. The unassigned pixels then are given values q(ρ) that redirect the excess light away from the intended line. Typical results are presented in  FIG. 5A .
 
     Unlike the cylindrical-lens trap, the holographic line trap  130  in  FIGS. 5A-5D  focuses as a conical wedge to a single diffraction-limited line in the objective&#39;s focal plane. Consequently, its transverse angle of convergence is comparable to that of an optimized point trap. This means that the holographic line trap  120  has comparably strong axial intensity gradients, which explains its ability to trap objects stably against radiation pressure in the z direction. 
     The line trap&#39;s transverse convergence does not depend strongly on the choice of intensity profile along the line. Its three-dimensional intensity distribution, however, is very sensitive to the phase profile along the line. Abrupt phase changes cause intensity fluctuations through Gibbs phenomenon. Smoother variations do not affect the intensity profile along the line, but can substantially restructure the beam. The line trap  120  created by the cylindrical lens element  130  for example, has a parabolic phase profile. Inserting this choice into Eq. (2) and calculating the associated shape-phase hologram with Eqs. (1) and (6) yields the same cylindrical lens phase profile. This observation opens the door to applications in which the phase profile along a line can be tuned to create a desired three-dimensional intensity distribution, or in which the measured three-dimensional intensity distribution can be used to assess the phase profile along the line. These applications will be discussed elsewhere. 
     The foregoing description of embodiments of the present invention have been presented for purposes of illustration and description. It is not intended to be exhaustive or to limit the present invention to the precise form disclosed, and modifications and variations are possible in light of the above teachings or may be acquired from practice of the present invention. The embodiments were chosen and described in order to explain the principles of the present invention and its practical application to enable one skilled in the art to utilize the present invention in various embodiments, and with various modifications, as are suited to the particular use contemplated.