Dexterous holographic trapping of dark-seeking particles with Zernike holograms
Abstract
The intensity distribution of a holographically-projected optical trap can be tailored to the physical properties of the particles it is intended to trap. Dynamic optimization is especially desirable for manipulating dark-seeking particles that are repelled by conventional optical tweezers, and even more so when dark-seeking particles coexist in the same system as light-seeking particles. We address the need for dexterous manipulation of dark-seeking particles by introducing a class of “dark” traps created from the superposition of two out-of-phase Gaussian modes with different waist diameters. Interference in the difference-of-Gaussians (DoG) trap creates a dark central core that is completely surrounded by light that can trap dark-seeking particles rigidly in three dimensions. DoG traps can be combined with conventional optical tweezers and other types of traps for use in heterogeneous samples. The ideal hologram for a DoG trap being purely real-valued, we introduce a method based on the Zernike phase-contrast principle to project real-valued holograms with the phase-only diffractive optical elements used in standard holographic optical trapping systems. Zernike phase holography should be useful for improving the diffraction efficiency of holograms projected with phase-only diffractive optical elements.
I Introduction
Holographic optical trapping uses the forces and torques exerted by computer-generated holograms to manipulate microscopic objects. Most of the literature of holographic trapping focuses on micromanipulation of dielectric particles with refractive indexes higher than the refractive index of the medium, ${n}_{p}>{n}_{m}$. Such high-index particles tend to be drawn toward regions of high light intensity, such as the focal point of strongly focused optical tweezers. Low-index particles, reflecting particles and particles that absorb light all tend to be repelled by bright light, and therefore are difficult to manipulate with standard optical traps. Successful two-dimensional manipulation of dark-seeking particles has been achieved in the dark regions of interference patterns [1] and the dark core of optical vortices [2, 3]. Full three-dimensional control has been demonstrated with cages of light created with rapidly scanned optical tweezers [4], and with optical bottles created from superpositions of Bessel beams [5, 6, 7]. The bright cages that define these traps all have dark gaps through which trapped particles can escape. Proposals to close the gaps have focused on the properties of vector beams of light with non-trivial polarization structure [8, 9]. These vector traps, however, cannot be projected with standard holographic trapping systems.
Here, we report a class of optical traps for dark-seeking particles that is based on scalar diffraction theory and so is compatible with standard holographic trapping techniques. The ideal hologram encoding these traps is entirely real-valued, which poses a challenge for standard implementations that rely on phase-only spatial light modulators. We therefore introduce a method to transform amplitude-only holograms into phase-only holograms for convenient projection at high diffraction efficiency. The resulting dark tweezers can be combined with conventional bright tweezers in a standard holographic trapping system to enable manipulation of heterogeneous colloidal dispersions. We demonstrate the dark traps’ capabilities through experimental studies on a model system composed of micrometer-scale silica spheres and polystyrene spheres co-dispersed in an aqueous solution of dimethylsulfoxide (DMSO). The silica spheres have a lower refractive index than this medium and thus are dark-seeking while the polystyrene spheres have a higher index and are light-seeking.
II Dark optical tweezers
Conventional optical tweezers are created by bringing a Gaussian laser beam to a diffraction-limited focus with a high-numerical-aperture (NA) lens. Analogous dark optical tweezers can be created by superposing two confocal Gaussian beams with equal amplitudes, different waist diameters and a relative phase of $\pi \mathrm{rad}$. The scalar amplitude profile of such a superposition in the focal plane of the lens is
$$u(\mathbf{r})={u}_{0}\left[\mathrm{exp}\left(-\frac{{r}^{2}}{4{\alpha}^{2}}\right)-\mathrm{exp}\left(-\frac{{r}^{2}}{4{\beta}^{2}}\right)\right],$$ | (1) |
where $\beta $ is the radius of the dark core and $\alpha $ is the radius of the enclosing region of light. As in the case of conventional optical tweezers, $\beta $ is constrained by the Abbe diffraction limit to $\beta \ge \lambda /2$ for light of wavelength $\lambda $ in the medium. The dark trap furthermore requires $\alpha >\beta $, with the difference ideally exceeding $\lambda /2$. Similar modes have been described previously [10] but do not appear to have been used to create optical traps.
Although the superposition described by Eq. (1) could be implemented with conventional optical elements, it is more conveniently projected with a holographic optical trapping system [11, 12] such as the example shown schematically in Fig. 1(a). The standard implementation imprints a hologram on the wavefronts of a conventional laser using a phase-only spatial light modulator (SLM) and then relays the modified beam to an objective lens that focuses it into a sample. The ideal hologram for the difference-of-Gaussians trap described by Eq. (1) therefore may be computed as the Fourier transform of the field in the focal plane [13],
$h(\mathbf{r})$ | $={\displaystyle \frac{1}{{u}_{0}}}{\displaystyle \int u(\mathbf{x})\mathrm{exp}\left(-i\frac{k}{f}\mathbf{r}\cdot \mathbf{x}\right){d}^{2}x}$ | (2a) | ||
$={\alpha}^{2}\mathrm{exp}\left(-{\displaystyle \frac{{k}^{2}}{{f}^{2}}}{\alpha}^{2}{r}^{2}\right)-{\beta}^{2}\mathrm{exp}\left(-{\displaystyle \frac{{k}^{2}}{{f}^{2}}}{\beta}^{2}{r}^{2}\right),$ | (2b) |
where $f$ is the focal length of the lens and $k=2\pi {n}_{m}/\lambda $ is the wave number of light in a medium of refractive index ${n}_{m}$. Unfortunately, the hologram in Eq. (2b) is purely real-valued and so cannot be projected with a standard phase-only SLM.
III Zernike holograms
A complex-valued hologram may be factored into real-valued amplitude and phase profiles,
$$h(\mathbf{r})=u(\mathbf{r})\mathrm{exp}(i\varphi (\mathbf{r})),$$ | (3) |
both of which can be imprinted onto the wavefronts of a laser beam using advanced projection techniques [14, 15]. Most holographic trapping systems, however, rely on phase-only diffractive optical elements that only modify the phase profile. A common expedient is to ignore the amplitude profile by setting $u(\mathbf{r})=1$, and to imprint only the phase profile, $\varphi (\mathbf{r})$, onto the laser beam’s wavefronts [16, 17, 18]. The resulting phase-only hologram,
$$H(\mathbf{r})=\mathrm{exp}\left(i\varphi (\mathbf{r})\right),$$ | (4) |
is a superposition of the ideal hologram, $h(\mathbf{r})$, with an error field,
$$\mathrm{\Delta}h(\mathbf{r})=\left[1-u(\mathbf{r})\right]\mathrm{exp}(i\varphi (\mathbf{r})).$$ | (5) |
The effect of $\mathrm{\Delta}h(\mathbf{r})$ on the projected optical trapping pattern depends on the nature and symmetries of the ideal pattern [18].
The hologram that projects one optical trap can be combined with holograms for other traps by superposing their fields in the hologram plane,
$$h(\mathbf{r})=\sum _{n=1}^{N}{\alpha}_{n}{h}_{n}(\mathbf{r}),$$ | (6) |
where ${h}_{n}(\mathbf{r})$ is the ideal hologram for the $n$-th trap and ${\alpha}_{n}$ is a complex coefficient setting the relative amplitude and phase of that trap. Traps can be translated by $\mathrm{\Delta}\mathbf{r}$ in three dimensions by adding a suitable parabolic profile [17],
$${\varphi}_{\mathrm{\Delta}\mathbf{r}}(\mathbf{r})=\left(-\frac{k}{f}\mathbf{r}+\frac{k{r}^{2}}{2{f}^{2}}\widehat{z}\right)\cdot \mathrm{\Delta}\mathbf{r},$$ | (7) |
As the complexity of a trapping pattern increases, the amplitude profile, $u(\mathbf{r})$, develops increasingly rapid spatial variations. The error term, $\mathrm{\Delta}h(\mathbf{r})$, therefore tends to redirect light away from the intended trapping pattern and outward toward the edges of the instrument’s field of view. This means that the phase-only hologram, $H(\mathbf{r})$, can project a near-ideal rendition of the intended trapping pattern within a limited volume. In such cases, the error term principally reduces the hologram’s diffraction efficiency into the desired mode by redirecting light elsewhere.
While the simple phase-only conversion described by Eq. (4) is fast and effective, more sophisticated algorithms [19] can refine a phase-only hologram to improve diffraction efficiency and to mitigate artifacts due to overlap between the intended and error patterns. These refinements typically are too slow for real-time operation, however, and are reserved for the most exacting applications.
Purely real-valued holograms pose a particular challenge to phase-only projection. Sign changes in $h(\mathbf{r})$ can be absorbed into the phase profile with Euler’s theorem, thereby ensuring that the amplitude, $u(\mathbf{r})$, is non-negative [20]. Amplitude variations can be approximated by multiplexing the desired hologram with another grating to deflect light. These methods either discard most of the light in the field of view or discard most information about the amplitude profile. They typically create holograms with undesirably low diffraction efficiency.
We improve both the fidelity and the diffraction efficiency of both real- and complex-valued holograms by transferring more information about the amplitude profile into the phase profile using a variant of Zernike’s phase-contrast approximation,
$$u(\mathbf{r})\approx i\left[1-\mathrm{exp}(iu(\mathbf{r}))\right],$$ | (8) |
under the assumption that $$. Using the phase profile to encode amplitude information complements Zernike’s original purpose, which was to explain how phase variations contribute to observable contrast in microscope images. The phase profile implied by Eq. (8),
$${\varphi}_{Z}(\mathbf{r})=\frac{1}{2}u(\mathbf{r}),$$ | (9a) | ||
serves as a phase-only approximation to the real-valued amplitude profile, yielding the phase-only Zernike approximation to the ideal complex-valued hologram, | |||
$${H}_{Z}(\mathbf{r})=\mathrm{exp}\left(i\left[{\varphi}_{Z}(\mathbf{r})+\varphi (\mathbf{r})\right]\right).$$ | (9b) | ||
The associated error field, | |||
$$\mathrm{\Delta}{h}_{Z}(\mathbf{r})\approx \left[1-\left(1-\frac{i}{2}\right)u(\mathbf{r})\right]{e}^{i\varphi (\mathbf{r})},$$ | (9c) | ||
improves upon the standard result because its complex amplitude tends to cancel ghost traps and other projection artifacts. |
Figure 1(b) shows the phase hologram encoding a dark trap that is obtained by applying Eq. (III) to the purely real-valued hologram for a DoG trap, Eq. (2b). Although it superficially resembles a standard Fresnel lens, the pattern of concentric phase rings has very different behavior.
Figure 1(c) and 1(d) show volumetric reconstructions [21] of the dark trap projected by the hologram in Fig. 1(b). The trap is created by imprinting the hologram on the wavefronts of a TEM${}_{00}$ laser beam at a vacuum wavelength of $1064\text{}\mathrm{nm}$ (fiber laser, IPG Photonics, YLR-LP-SF) using a liquid crystal spatial light modulator (Holoeye PLUTO). The modified beam is relayed to the input pupil of an objective lens (Nikon Plan Apo, $100\times $, numerical aperture 1.4, oil immersion) that focuses the light into the intended optical trap with a focal length of $f=180\text{}\mathrm{\mu}\mathrm{m}$. The trapping beam is diverted into the objective lens with a dichroic beamsplitter (Semrock) that has a reflectivity of $99.5\text{}\mathrm{\%}$ at the trapping wavelength.
Images of the projected intensity pattern are obtained by mounting a front-surface mirror in the focal plane of the objective lens [21]. The reflected light is collected by the objective lens, and a small proportion passes through the dichroic mirror. This transmitted light is collected with a $200\text{}\mathrm{mm}$ tube lens and is recorded with a video camera (FliR Flea3, monochrome) with an effective system magnification of $0.048\text{}\mathrm{\mu}\mathrm{m}/\mathrm{pixel}$. Transverse intensity slices, such as the example in Fig. 1(c), are obtained by translating the trap in steps of $\mathrm{\Delta}z=48\text{}\mathrm{nm}$ along the axial direction using Eq. (7) [21]. A stack of slices then is combined to obtain the axial section that is presented in Fig. 1(d). These images confirm that the Zernike hologram for a DoG trap successfully projects a beam of light that focuses to a dark volume surrounded by light on all sides.
IV Simultaneous manipulation of dark- and light-seeking particles
Figure 2 presents holographically measured trajectories of high- and low-index particles being translated simultaneously in a conventional optical tweezer and a DoG trap, respectively. The figure also includes one frame from the holographic video (Supplementary Video 1) that was used to measure the particles’ trajectories. The particles are dispersed in an aqueous solution of dimethylsufoxide (DMSO, Acros Organics, CAS Number 67-68-5, Lot Number A0277547) with a measured refractive index of ${n}_{m}=1.53$. The low-index particle is composed of silica (Bangs Laboratories, SS05N, Lot Number 4186) with a radius of ${a}_{p}=1.15\text{}\mathrm{\mu}\mathrm{m}$, and a nominal refractive index of ${n}_{{\text{SiO}}_{2}}=1.43$. The high-index particle is composed of polystyrene (Thermo Fisher Scientific, 5100B Lot Number 42393) with a radius of ${a}_{p}=0.5\text{}\mathrm{\mu}\mathrm{m}$ and a nominal refractive index of ${n}_{\text{PS}}=1.60$. A mixed dispersion of these spheres is contained in the volume created by sealing the edges of a glass coverslip to the face of a glass microscope slide. This chamber has an optical path length of $H=25\text{}\mathrm{\mu}\mathrm{m}$.
In-line holograms of the particles are recorded by illuminating the sample with the collimated beam from a diode laser (Coherent Cube) operating at a vacuum wavelength of $\lambda =447\text{}\mathrm{nm}$. Light scattered by the particles interferes with the rest of the beam in the focal plane of the microscope. The intensity of the magnified interference pattern is recorded by the video camera at $30\text{}\mathrm{frames}/\mathrm{s}$. The camera’s $10\text{}\mathrm{\mu}\mathrm{s}$ exposure time is short enough to avoid motion blurring [22, 23]. Each holographic snapshot can be analyzed with predictions of Lorenz-Mie theory to measure each particle’s diameter, refractive index and three-dimensional position relative to the center of the microscope’s focal plane [24]. We use these capabilities to differentiate the two types of particles on the basis of their refractive index and to track their three-dimensional motions as they are manipulated in holographic optical traps.
The discrete plot symbols in Fig. 2 represent the measured three-dimensional trajectories of a (low-index) silica sphere and a (high-index) polystyrene sphere as they are transported in opposite directions around nominally circular trajectories in the $yz$ plane. The particles follow the programmed trap trajectories at $5.0\text{}\mathrm{\mu}\mathrm{m}\text{}{\mathrm{s}}^{-1}$ given an estimated laser power of $100\text{}\mathrm{mW}$ for the conventional optical trap and $50\text{}\mathrm{mW}$ for the DoG trap.
Although the two traps were programmed to move in circles of equal radius, the recorded paths are elliptical. The two particles move over equal horizontal ranges, but the polystyrene particle moves further than expected in the axial direction and the silica particle moves less far. Similar axial deviations have been reported for particles in conventional optical tweezers [25] and can be ascribed to differences in the particles’ buoyant densities and to changes in the traps’ Rayleigh ranges as they are displaced along the optical axis.
V Assessing performance
Thermally-driven fluctuations in the particles’ positions can be used to measure the stiffness of the traps, and therefore their trapping efficiency. Treating the sphere’s thermal fluctuations within the trap as an Ornstein-Uhlenbeck process, an $N$-step discretely sampled trajectory, $\{{x}_{n}\}$, along $\widehat{x}$ yields an estimate for the associated stiffness [19],
$${\kappa}_{x}\pm \mathrm{\Delta}{\kappa}_{x}=\frac{{k}_{B}T}{{c}_{0}}\pm \frac{{k}_{B}T}{{c}_{0}}\sqrt{\frac{2}{N}\left(1+\frac{2{c}_{1}^{2}}{{c}_{0}^{2}-{c}_{1}^{2}}\right)},$$ | (10a) | ||
where ${k}_{B}T$ is the thermal energy scale at absolute temperature $T$ and | |||
$${c}_{j}=\frac{1}{N}\sum _{n=1}^{N}{x}_{n}{x}_{(n+j)modN}$$ | (10b) |
is the autocorrelation of $\{{x}_{n}\}$ at lag $j$. Similar estimates can be computed for the trap stiffness along the other Cartesian coordinates. Equation (V) is suitable for interpreting holographically measured single-particle trajectories because the measurement error, ${\sigma}_{x}=5\text{}\mathrm{nm}$, is smaller than the typical scale of thermal fluctuations, $\mathrm{\Delta}x=\sqrt{2({c}_{0}-{c}_{1})}\ge 70\text{}\mathrm{nm}$, over the range of laser powers considered [19]. A $2000$-frame holographic video recorded over one minute provides enough information to measure ${\kappa}_{x}$ to within $1\text{}\mathrm{\%}$.
Figure 3(a) shows how the measured stiffness of a DoG trap depends on laser power and particle diameter for low-index silica spheres. The trap used for these measurements has $\alpha =1.92\text{}\mathrm{\mu}\mathrm{m}$ and $\beta =\alpha /2$. The power, $P$, projected into the trap is measured with a slide-mounted thermal power sensor (Thorlabs S175C). As might reasonably be expected, the trap stiffness increases linearly with $P$ in all three coordinates for each of the three particle sizes. Values for the transverse stiffness plotted in Fig. 3(a) reveal no anisotropy, and thus no dependence on the trap’s polarization, which is directed along $\widehat{x}$. The axial stiffness, plotted in Fig. 3(b), also increases linearly with $P$ over the range studied and, not surprisingly, differs significantly from the transverse stiffness. Fig. 3(c) and Fig. 3(d) show respectively how the transverse and axial trapping efficiencies, ${Q}_{\u27c2}={\partial}_{P}({\kappa}_{x}+{\kappa}_{y})/2$ and ${Q}_{z}={\partial}_{P}{\kappa}_{z}$, depend on particle radius, ${a}_{p}$.
Interestingly, the measured transverse trapping efficiency, ${Q}_{\u27c2}$, depends nonmonotonically on particle size. We account for this observation with a simplified model of the particles’ interaction with the DoG trap’s intensity profile. The particle’s potential energy at distance $r$ from the center of the dark well depends on its overlap with the light, which we model as | ||||
$U(r)$ | $=A{\displaystyle {\int}_{0}^{2\pi}}\mathit{d}\theta {\displaystyle {\int}_{0}^{{a}_{p}}}\mathit{d}xx\zeta (x){u}^{2}(\mathbf{x}-\mathbf{r})$ | (11a) | ||
$=4\pi {a}_{p}^{3}{u}_{0}^{2}A\left[g(r,{\displaystyle \frac{1}{{\alpha}^{2}}})+g(r,{\displaystyle \frac{1}{{\beta}^{2}}})-2g(r,{\displaystyle \frac{1}{2{\alpha}^{2}}}+{\displaystyle \frac{1}{2{\beta}^{2}}})\right],$ | (11b) | |||
where $\zeta (x)=2\sqrt{{a}_{p}^{2}-{x}^{2}}$ is the chord length a distance $x$ from the center of a sphere of radius ${a}_{p}$, and where we have defined | ||||
$$g(r,\frac{1}{{\alpha}^{2}})={e}^{\frac{{r}^{2}}{4{\alpha}^{2}}}{\int}_{0}^{1}y\sqrt{1-{y}^{2}}{I}_{0}\left(\frac{r{a}_{p}}{2{\alpha}^{2}}y\right){e}^{-\frac{{a}_{p}^{2}}{4{\alpha}^{2}}{y}^{2}}\mathit{d}y.$$ | (11c) | |||
The integrand in Eq. (11c) includes the modified Bessel function, ${I}_{0}(\cdot )$, and is most readily computed numerically. We treat the overall scale of the trapping potential, $A$, as an adjustable parameter. Typical potential energy curves are plotted in Fig. 3(e). |
The trapping efficiency is obtained from Eq. (V) by fitting the potential minimum to a parabola using the planned intensity distribution, ${u}^{2}(\mathbf{r})$, and particle radius, ${a}_{p}$, as inputs. Planned and measured intensity distributions are compared in Fig. 3(f) for the trap used for Fig. 3. The dependence of the in-plane trapping efficiency on particle radius is plotted as a blue dashed curve in Fig. 3(c), using only $A$ as an adjustable parameter. The model predicts that the DoG trap is optimally stiff when trapping low-index dielectric particles that are slight larger than trap’s dark core, in agreement with experimental observations.
VI Manipulating absorbing particles
In addition to low-index particles and high-index particles, the DoG trap can manipulate particles that absorb light strongly. Such particles are repelled by conventional optical tweezers through transfer of momentum from absorbed light. Absorption also mediates heating, which can propel the particles through self-thermophoresis [26, 27], and in extreme cases can destroy the particles or boil the fluid medium [28, 29].
We have demonstrated the ability of DoG traps to stably trap and transport absorbing particles through experiments on composite particles made of hematite cubes embedded in dielectric spheres and superparamagetic particles composed of hematite nanoparticles dispersed within dielectric spheres. The dielectric spheres used for these demonstrations are created by emulsion polymerization of 3-methacryloxypropyl trimethoxysilane (TPM) [30], an organosilicate with a refractive index of ${n}_{p}=1.495$ [31]. Hematite absorbs infrared light strongly, and both types of hematite-loaded particles tend to be ejected from conventional optical tweezers. Both types of particles are readily trapped and transported in three dimensions with DoG traps. Supplementary Video 2 shows the simultaneous three-dimensional control over a $1\text{}\mathrm{\mu}\mathrm{m}$-diameter polystyrene particle and a $1.8\text{}\mathrm{\mu}\mathrm{m}$ TPM particle enclosing a $0.8\text{}\mathrm{\mu}\mathrm{m}$ hematite cube.
VII Discussion
This study introduces a class of “dark” optical traps that are created by superposing two Gaussian modes of different waist diameters and opposite phases. The hologram encoding these Difference-of-Gaussians (DoG) traps is purely real-valued. We therefore introduce a technique based on the Zernike phase-contrast approximation to project real-valued holograms with phase-only diffractive optical elements, such as the spatial light modulators commonly used in holographic optical trapping systems. We have demonstrated DoG traps’ ability to trap dark-seeking particles and to move them in three dimensions. We further have characterized the efficiency of DoG traps for localizing low-index particles and have explained the observed nonmonotonic dependence of the transverse trapping efficiency on particle size. When a DoG trap is optimally matched to the trapped particle, its trapping efficiency is comparable to that of conventional optical tweezers for light-seeking particles.
Funding
This work was supported by the National Science Foundation through Award No. DMR-2104836. The integrated instrument for holographic trapping and holographic microscopy used in this study was constructed with support of the MRI program of the NSF under Award Number DMR-0922680 and is maintained as shared instrumentation by the Center for Soft Matter Research at NYU. The authors gratefully acknowledge the support of the nVidia Corporation through the donation of the Titan Xp and Titan RTX GPUs used for this research.
Acknowledgments
We are grateful to Shahrzad Zare, who contributed to the initial stages of this work.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
The open-source software used to project holographic optical traps, record in-line holographic microscopy data and analyze those data is available online at https://github.com/davidgrier/.
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