Robustness of Holographic Optical Traps Against Phase Scaling Errors
Abstract.
Complex threedimensional patterns of multifunctional optical traps can be encoded in phaseonly computergenerated holograms and projected with the holographic optical trapping technique. The trapforming holograms, in turn, are implemented as diffractive optical elements whose phase transfer functions generally do not faithfully reproduce the design. We demonstrate that phase encoding errors reduce the overall intensities of the projected traps but, remarkably, do not affect their positions, relative intensities or mode structure. We exploit this robust performance to implement dualcolor holographic optical tweezers with a single hologram.
Holographic optical traps use phaseonly holograms to form large arrays of optical traps from a single input laser beam. By combining the beamsplitting and wavefrontshaping capabilities of computer generated holograms, holographic traps can be arranged in arbitrary threedimensional configurations, with each trap having independently specified characteristics, including relative intensity and mode structure. The unsurpassed control over the microscopic world afforded by this technique has been widely adopted for fundamental research in softmatter systems and for biomedical and industrial applications (1).
In principle, holographic trapping patterns can be projected with absolute fidelity to design and nearideal efficiency. Practical diffractive optical elements (DOEs), however, seldom offer the requisite continuously varying phase profiles, and almost never provide precisely the phase pattern required for in a given design. This has been recognized as a central problem for holographic projection systems since the introduction of the kinoform (2). Imperfectly imprinting the designed phase pattern onto the input beams wavefronts degrades the projected intensity patterns. To quantify this, we adapt the classic expansion of the projected field into generalized conjugates of the designed field (3); (4). This analysis demonstrates that the performance of optimized (5) holographic trapping systems is remarkably robust against phase defects, and further suggests useful extensions of the technique.
A standard holographic optical trapping system (6); (7); (8); (9); (10) is powered by a collimated laser beam, which is relayed to the input pupil of a highnumericalaperture lens such as a microscope objective lens. This lens focuses the beam to a diffractionlimited spot at a location determined by the beam's angle of incidence and degree of collimation at the lens' input pupil. Such a focused spot acts as a singlebeam optical gradient force trap known as an optical tweezer (11) and is capable of capturing and holding mesoscopic objects in three dimensions. Placing a wavefrontshaping hologram in a plane conjugate to the input pupil transforms the single optical tweezer into a pattern of holographic optical traps whose number, threedimensional configuration, relative and absolute intensities, and mode structure all are encoded in the hologram. Deficiencies in the hologram's implementation might reasonably be expected to degrade all of these characteristics. An analysis based on scalar diffraction theory shows otherwise.
The complex field , in a plane at distance from the focal plane of a lens of focal length is related to the field in the lens' input plane by the Fresnel diffraction integral (12),
(1) 
where we have suppressed overall phase factors and assumed . Here, and are the realvalued amplitude and phase profiles, respectively, of the input beam at position in the input pupil , as shown in Fig. 1, and is the wavenumber of light of wavelength . A diffractive optical element (DOE) imposes the additional phase profile , which ideally would correspond to the computergenerated hologram encoding a desired pattern of traps. In practice, differs from the , so that differs from the planned trapping pattern in the plane of best focus.
Particularly when considering DOEs that encode threedimensional trapping configurations, identifying the plane of best focus is the first step in assessing performance. If a simple beamsplitting DOE is illuminated with a collimated beam, we may take , and the resulting pattern of traps comes to sharpest focus when the remaining dependent phase term in the integrand of Eq. (1) vanishes, which occurs in the plane . The optimized holographic trapping technique (5) instead uses a slightly converging beam with
(2) 
which shifts the plane of best focus to , for . Artifacts due to imperfect phase modulation need not focus in the same plane, as we will see, and the resulting axial displacement can minimize their influence. Offaxis displacements have long been used to improve kinoforms' reconstruction performance, and have proven useful for improving the mode quality of holographically projected optical traps (13). Axial displacements (14); (15); (5) offer additional benefits for holographic trapping.
Our implementation of optimized holographic optical trapping is described in detail in Ref. (5), and is built around a NA 1.4 Plan Apo oil immersion objective lens mounted in a Nikon TE2000U inverted optical microscope. Laser light at a wavelength of 532 provided by a Coherent Verdi laser, is imprinted with computergenerated holograms by a Hamamatsu X826716 spatial light modulator (SLM) which acts as a DOE with a array of phase pixels. The focused optical traps are imaged by placing a mirror in the objective lens' focal plane and capturing the reflected light with an NEC TI324AII chargecoupled device (CCD) camera.
Most DOEs, including our SLM, can impose only a limited range of phase delays, which ideally corresponds to one wavelength of light, so that . Introducing the DOE's phase transfer function, , such that , and noting that is a periodic function of with period , we may expand the DOE's contribution to the field's phase factor in a Fourier series (3); (4)
(3) 
with coefficients
(4) 
We accordingly define the generalized th order conjugate fields,
(5) 
The projected field is then
(6) 
For example, if a DOE created for one wavelength of light, , is illuminated with another, , then with and
(7) 
Comparable results may be obtained for more general phase transfer functions, including those featuring discrete phase levels. The most stringent test, imperfect binary phase holograms with
(8) 
are described in this way with coefficients
(9) 
These terms also fall off with order as . Best performance, in this case, is obtained with .
Equation (6) reveals that the projected image
(10) 
includes a proportion of the intended intensity profile, , even if the DOE imperfectly implements the requisite hologram. This firstorder image is a faithful, undistorted realization of the designed pattern whose accuracy is limited only by errors in calculating and physical imperfections in the optical train. The other terms in Eq. (10) represent artifacts introduced by the DOE's phase transfer function .
The term in Eq. (10) describes the undiffracted portion of the input beam, which typically comes to a focus in the center of the plane . Because it receives a fixed proportion of the light, the resulting “central spot” can be brighter than any of the intended traps in .
Generalized conjugate fields are related by in the plane so that the associated images , are related by point reflection through the origin, as shown in Fig. 2. Furthermore, because multiplying by proportionately increases the hologram's spatial frequency. The remaining terms in Eq. (10) thus describe a hierarchy of “ghost” images at locations dictated by integer scale dilations, point inversions, and their superpositions. Ghosts generally act as unintended traps. If they coincide with intended traps, however, the resulting interference can cause large deviations in the traps' relative intensities.
In conventional holographic optical traps whose DOE is illuminated with collimated light, the entire hierarchy of conjugate fields is focused into the same plane. The central spot, the ghosts, and the undesirable superpositions thus maximally affect the trapping pattern.
The optimized holographic trapping system eliminates most of these defects. Here, the input beam's curvature is offset by a compensating Fresnel lens function,
(11) 
added to the trapforming hologram and implemented by the DOE (5). This shifts the trapping pattern a distance back along the optical axis toward the focal plane. Because also is affected by the DOE's phase transfer function, it contributes to the hierarchy of conjugate fields in Eq. (5). Noting also that shows that the th order generalized conjugate field comes to best focus in the plane . Both the central spot and the ghost images therefore are projected away from the intended trapping pattern, and spurious superpositions are strongly suppressed. Artifacts due to practical limitations of the DOE's phase transfer function therefore should have a minimal influence on the number, configuration, or relative intensity of traps in an optimized holographic trapping system. This is consistent with the observed performance of such systems (5). The principal ramification of a nonideal is a reduction in the overall intensity of the projected trapping pattern.
Adding increases the complexity of the projected hologram, which can challenge the capabilities of DOE technologies with limited spatial bandwidths. Systematic metrics for assessing hologram complexity relative to DOE capabilities have yet to be developed. Consequently, the practical limitations of the optimized holographic trapping technique cannot yet be assessed a priori. Nevertheless, complex threedimensional optimized trapping patterns consisting of hundreds of independent traps have been created with a array of phase pixels (16).
The final characteristic of holographically projected traps that we will consider is their mode structure. Conventional optical tweezers typically are formed from collimated TEM modes with planar wavefronts. More exotic traps such as optical vortices (13); (17); (18) and Bessel beams (19); (20) derive their interesting and useful properties from the detailed structure of their wavefronts. The structure necessary to create such traps can be imposed on a TEM beam by a modeforming hologram. Indeed, the beamsplitting and modeforming operations can be combined in a single computergenerated hologram to create arrays of multifunctional optical traps (8); (10); (5). Here again, the phase transfer function, , of the DOE can affect the fidelity with which a particular mode is projected, and thus can influence the associated trap's functionality.
As a practical example, we consider optical vortices, torqueexerting traps created by focusing helical light beams. Helical modes are characterized by an overall phase factor , where is the azimuthal angle about the optical axis. The integer winding number sets the pitch of the helix, and is often referred to as the topological charge (21). The helical topology suppresses the intensity along the axis of such a beam, not because the amplitude vanishes but rather because of destructive interference due to the coincidence of all phases there. An optical vortex, therefore, focuses to a dark spot surrounded by a bright ring of light. In optical vortices created by imposing a helical phase profile on a Gaussian beam, the ring's radius scales linearly with topological charge (22); (23); (24).
Now we consider what happens to an optical vortex designed to have winding number when projected by a nonideal DOE. Because the Fourier coefficients in Eq. (6) fall off with index, we approximate the field in the focal plane by the principal terms
(12) 
Taking for an optical vortex centered within a standard holographic optical trapping system, the conjugate field is
(13) 
The resulting intensity distribution,
(14) 
is characterized by azimuthal intensity modulations with both fold and fold symmetry. Taking , and defining and these terms' relative amplitudes are , , and . Their relative phases are given by
(15) 
and .
In principle, the central spot's amplitude profile, , is sharply peaked around the optical axis and so should not overlap substantially with the optical vortex's ringlike profile, . The amplitude of the fold intensity corrugation resulting from their interference therefore should be negligible for . Even so, holographically projected optical vortices such as the example in Fig. 3(a) often are surrounded by radial spokes extending to very large radii. These outer spokes are projected from the hologram's central region (25), whose features typically are too fine to be reproduced faithfully by a pixellated DOE (22). The undersampled phase pattern near the optical axis acts as a diffuser and scatters light to larger radii where it contributes to the visible spokes. Both these and the optical vortex's higherorder diffraction rings can be eliminated by excising the central region of the modeforming phase mask (25); (26).
Whereas the fold features are due in large part to the DOE's pixellated structure, the fold corrugation results from interference between the principal and conjugate fields. This corrugation, which also can be seen in Fig. 3(a), has been described before and significantly affects the dynamics of objects trapped on the circumference of an optical vortex (22). It can be minimized by displacing the principal vortex away from the center of the field of view (13); (25). Ideally, this eliminates modulation of the optical vortex's circumferential intensity profile altogether, as shown in Fig. 3(b). Because optical vortices cover a larger area than conventional optical tweezers, however, some interference with neighboring and ghost traps can occur in more complex configurations (27). Optimizing the phase transfer function to minimize these interactions thus is more important in creating multifunctional optical traps than in projecting arrays of conventional optical tweezers.
The expansion in generalized conjugate fields introduced in Eqs. (4), (5) and (6) clearly demonstrates that imperfections in a DOE's phase transfer function only minimally influence the number, distribution, relative intensities and mode structure of optimized holographic traps encoded in a computergenerated hologram. This robustness suggests a strategy for projecting holographic traps in multiple wavelengths simultaneously. Because of the wavelength dependence of in Eq. (2), beams of different wavelengths would focus to different planes in an optimized holographic optical trapping system, even with achromatic optics. Separate holograms can be calculated for each wavelength, each with the appropriate displacement along the optical axis, and the results added to create a multiwavelength hologram that projects distinct patterns of traps in each color. As in previous approaches to multiwavelength holography (28); (29), all patterns are projected in each wavelength. This is less of a problem for holographic trapping than for data multiplexing (28) or image formation (29) because the unintended patterns in each color are displaced out of the plane of best focus, and typically out of the sample altogether. The result is that only the designed patterns in each color will be projected into the focal volume, as shown in Fig. 4. Suppressing the unintended patterns in each wavelength has been demonstrated for color separation gratings (30). The same methods could be applied to more demanding multicolor holographic trapping applications. Polychromatic holographic trapping by superposition complements recent demonstrations of whitelight optical vortices (31) and Bessel beams (32), which also have applications to optical trapping.
The ability to project multifunctional optical traps in multiple wavelengths should facilitate simultaneous manipulation and photochemical transformation of lightsensitive systems. This would be useful for noninvasive intracellular surgery and for assembling and photochemically bonding threedimensional heterostructures (16). Multicolor arrays also will be useful for sorting objects by their absorptivity or index of refraction, for example in holographically implemented optical fractionation (33). Combining these functions on a single DOE would simplify the implementation by projecting all wavelengths along a single path.
In summary, we have shown that an optimized holographic optical trapping system's performance is remarkably insensitive to details of the DOE's phase transfer function. This accounts for the success of early implementations (6); (9) whose DOEs were not accurately tuned to the wavelength of input light. It also suggests new applications, such as multicolor trapping, and opportunities for simplified implementation of dynamic holographic optical trapping systems.
This work was supported by NSF Grants Number DMR0451589 and DBI0233971. S.L. acknowledges support from the Kessler Family Foundation.
References

(1)
D. G. Grier. “A revolution in optical manipulation.” Nature 424, 810–816 (2003).

(2)
L. B. Lesem, P. M. Hirsch and J. A. Jordan. “The kinoform: A new wavefront reconstruction device.” IBM J. Res. Develop. 13, 150–155 (1969).

(3)
J. W. Goodman and A. M. Silverstri. “Some effects of Fourierdomain phase quantization.” IBM J. Res. Dev. 14, 478 (1970).

(4)
J. P. Riley and F. N. Birkett. “A reflection kinoform for use with a CO laser.” Optica Acta 24, 999–1009 (1977).

(5)
M. Polin, K. Ladavac, S.H. Lee, Y. Roichman and D. G. Grier. “Optimized holographic optical traps.” Opt. Express 13, 5831–5845 (2005).

(6)
E. R. Dufresne and D. G. Grier. “Optical tweezer arrays and optical substrates created with diffractive optical elements.” Rev. Sci. Instrum. 69, 1974–1977 (1998).

(7)
M. Reicherter, T. Haist, E. U. Wagemann and H. J. Tiziani. “Optical particle trapping with computergenerated holograms written on a liquidcrystal display.” Opt. Lett. 24, 608–610 (1999).

(8)
J. Liesener, M. Reicherter, T. Haist and H. J. Tiziani. “Multifunctional optical tweezers using computergenerated holograms.” Opt. Commun. 185, 77–82 (2000).

(9)
E. R. Dufresne, D. Altman and D. G. Grier. “Brownian dynamics of a sphere in a slit pore.” Europhys. Lett. 53, 264–270 (2001).

(10)
J. E. Curtis, B. A. Koss and D. G. Grier. “Dynamic holographic optical tweezers.” Opt. Commun. 207, 169–175 (2002).

(11)
A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm and S. Chu. “Observation of a singlebeam gradient force optical trap for dielectric particles.” Opt. Lett. 11, 288–290 (1986).

(12)
J. W. Goodman. Introduction to Fourier Optics (McGrawHill, New York, 1996), 2nd ed.

(13)
H. He, N. R. Heckenberg and H. RubinszteinDunlop. “Optical particle trapping with higherorder doughnut beams produced using high efficiency computer generated holograms.” J. Mod. Opt. 42, 217–223 (1995).

(14)
N. R. Heckenberg, R. McDuff, C. P. Smith, H. RubinszteinDunlop and M. J. Wegener. “Laser beams with phase singularities.” Opt. Quantum Elect. 24, S951–S962 (1992).

(15)
A. Jesacher, S. Furhpater, S. Bernet and M. RitschMarte. “Size selective trapping with optical ”cogwheel” tweezers.” Opt. Express 12, 4129–4135 (2004).

(16)
Y. Roichman and D. G. Grier. “Holographic assembly of quasicrystalline photonic heterostructures.” Opt. Express 13, 5434–5439 (2005).

(17)
N. B. Simpson, L. Allen and M. J. Padgett. “Optical tweezers and optical spanners with LaguerreGaussian modes.” J. Mod. Opt. 43, 2485–2491 (1996).

(18)
K. T. Gahagan and G. A. Swartzlander. “Optical vortex trapping of particles.” Opt. Lett. 21, 827–829 (1996).

(19)
A. Vasara, J. Turunen and A. T. Friberg. “Realization of general nondiffracting beams with computergenerated holograms.” J. Opt. Soc. Am. A 6, 1748–1754 (1989).

(20)
J. Arlt, V. GarcesChavez, W. Sibbett and K. Dholakia. “Optical micromanipulation using a Bessel light beam.” Opt. Commun. 197, 239–245 (2001).

(21)
L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw and J. P. Woerdman. “Orbital angularmomentum of light and the transformation of LaguerreGaussian laser modes.” Phys. Rev. A 45, 8185–8189 (1992).

(22)
J. E. Curtis and D. G. Grier. “Structure of optical vortices.” Phys. Rev. Lett. 90, 133901 (2003).

(23)
J. E. Curtis and D. G. Grier. “Modulated optical vortices.” Opt. Lett. 28, 872–874 (2003).

(24)
S. Sundbeck, I. Gruzberg and D. G. Grier. “Structure and scaling of helical modes of light.” Opt. Lett. 30, 477–479 (2005).

(25)
C.S. Guo, X. Liu, J.L. He and H.T. Wang. “Optimal annulus structures of optical vortices.” Opt. Express 12, 4625–4634 (2004).

(26)
K. Ladavac and D. G. Grier. “Colloidal hydrodynamic coupling in concentric optical vortices.” Europhys. Lett. 70, 548–554 (2005).

(27)
K. Ladavac and D. G. Grier. “Microoptomechanical pump assembled and driven by holographic optical vortex arrays.” Opt. Express 12, 1144–1149 (2004).

(28)
J. Bengtsson. “Kinoforms designed to produce different fanout patterns for two wavelengths.” Appl. Opt. 37, 2011–2020 (1998).

(29)
U. Levy, E. Marom and D. Mendlovic. “Simultaneous multicolor image formation with a single diffractive optical element.” Opt. Lett. 26, 1149–1151 (2001).

(30)
L. L. Doskolovich, N. L. Kazanskiy, V. A. Soifer, P. Perlo and P. Repetto. “Design of DOEs for wavelength division and focusing.” J. Mod. Opt. 52, 917–926 (2005).

(31)
J. Leach and M. J. Padgett. “Observation of chromatic effects near a whitelight vortex.” New J. Phys. 5, 154 (2003).

(32)
P. Fischer, C. T. A. Brown, J. E. Morris, C. LópezMariscal, E. M. Wright, W. Sibbett and K. Dholakia. “White light propagation invariant beams.” Opt. Express 13, 6657–6666 (2005).

(33)
K. Ladavac, K. Kasza and D. G. Grier. “Sorting by periodic potential energy landscapes: Optical fractionation.” Phys. Rev. E 70, 010901(R) (2004).