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Optical forces arising from phase gradients

Yohai Roichman [1], Bo Sun [2], Yael Roichman [1], Jesse Amato-Grill [1] and David G. Grier [1]
Department of Physics [1], Department of Chemistry [2] and Center for Soft Matter Research, New York University, New York, NY 10003


Abstract:

We demonstrate both theoretically and experimentally that phase gradients in a light field can be used to create a new category of optical traps complementary to the more familiar intensity-gradient traps known as optical tweezers. We further show that phase-gradient forces generally violate conservation of energy, and briefly discuss some ramifications of their non-conservativity.

Light's ability to exert forces has been recognized since Kepler's De Cometis of 1619 described the deflection of comet tails by the sun's rays. Maxwell showed that the momentum flux in a beam of light is proportional to the intensity and can be transferred to illuminated objects, resulting in radiation pressure that pushes objects along the direction of propagation. This Letter demonstrates that phase gradients can redirect radiation pressure to create optical force fields transverse to the optical axis. Photon orbital angular momentum (OAM) (1) is one experimentally realized (2,3) example of this phenomenon. We then put phase-gradient forces to work by combining them with intensity gradients in holographically projected light fields to create a new category of extended optical traps with tailored force profiles.

Figure 1: (a) Schematic representation of the optimized holographic optical trapping system using shape phase holography to project extended optical traps. (b) Experimental realization of a holographic line trap carrying a phase gradient in the $ \hat{\boldsymbol{x}}$ direction, imaged in the plane of best focus. (c) Focal pattern of a holographic ring trap with $ \ell = 30$ .
\begin{figure}\centering
\includegraphics[width=\columnwidth]{schematic5}
\end{figure}

The vector potential describing a beam of light of frequency $ \omega$ and polarization $ \hat{\boldsymbol{\varepsilon}}(\vec{r})$ may be written as

$\displaystyle \vec{A}(\vec{r},t) = u(\vec{r}) \, e^{i \Phi(\vec{r})} e^{-i \, \omega t} \, \hat{\boldsymbol{\varepsilon}}(\vec{r}),$ (1)

where $ u(\vec{r})$ is the real-valued amplitude and $ \Phi(\vec{r})$ is the real-valued phase. We assume for simplicity that the light is linearly polarized so that $ \hat{\boldsymbol{\varepsilon}}(\vec{r})$ is real. For a plane wave propagating in the $ \hat{\boldsymbol{z}}$ direction, $ \Phi(\vec{r}) = kz$ , where $ k = n_m \, \omega/c$ is the light's wavenumber, $ c$ is the speed of light in vacuum, and $ n_m$ is the refractive index of the medium. Imposing a transverse phase profile $ \varphi(\vec{r})$ on the wavefronts of such a beam yields

$\displaystyle \Phi(\vec{r}) = k_z(\vec{r}) z + \varphi(\vec{r}),$ (2)

where $ \hat{\boldsymbol{z}} \cdot \nabla \varphi = 0$ . The direction of the wavevector, $ \vec{k}(\vec{r}) = k_z(\vec{r}) \, \hat{\boldsymbol{z}} + \nabla
\varphi$ , now varies with position, subject to the constraint $ k^2 =
\left\vert \vec{k} \right\vert ^2 = k_z^2 + \left\vert \nabla \varphi \right\vert ^2$ , which applies in the paraxial limit, $ k \gg \left\vert \nabla \varphi \right\vert $ . The associated electric and magnetic fields are given in the Lorenz gauge by

$\displaystyle \vec{E}(\vec{r},t)$ $\displaystyle = -\frac{\partial}{\partial t} \vec{A}(\vec{r},t)$   and (3)
$\displaystyle \vec{H}(\vec{r},t)$ $\displaystyle = \frac{1}{\mu} \, \nabla \times \vec{A}(\vec{r},t),$    

where $ \mu$ is the magnetic permeability of the medium, which we assume to be homogeneous and isostropic. Following Abraham's formulation (4), the momentum flux carried by the beam is

$\displaystyle \vec{g}(\vec{r}) = \frac{1}{c^2} \, \Re\left\{ \vec{E}^\ast \times \vec{H} \right\} = \frac{k}{n_m \mu c} \, I(\vec{r}) \, \nabla \Phi,$ (4)

where $ I(\vec{r}) = \left\vert u(\vec{r}) \right\vert ^2$ is the light's intensity, and where we have employed the gauge condition $ \nabla \cdot \vec{A} = 0$ .

The momentum flux separates into an axial component $ \vec{g}_z(\vec{r}) = k k_z I(\vec{r}) \, (n_m \mu c)^{-1} \,
\hat{\boldsymbol{z}}$ and an additional component (5)

$\displaystyle \vec{g}_\perp(\vec{r}) = \frac{k}{n_m \mu c} \, I(\vec{r}) \, \nabla \varphi$ (5)

transverse to the optical axis. Small illuminated objects thus experience a transverse force proportional to $ \vec{g}_\perp(\vec{r})$ .

More than a decade ago, Allen and coworkers (1) pointed out that the helical phase profile, $ \varphi(\vec{r}) = \ell \theta,$ imbues a beam of light with an OAM flux, $ \vec{r} \times
\vec{g}_\perp$ , amounting to $ \ell \hbar$ per photon. Here, $ \theta$ is the azimuthal angle around the optical axis, and $ \ell $ is an integer describing the wavefronts' helical pitch. This OAM is distinct from the photons' intrinsic spin angular momentum (7,6) Through it, even linearly polarized light can exert a torque around the optical axis (8,2). Equation (5) reveals this to be a manifestation of the more general class of transverse forces arising from phase gradients.

Figure 2: Dependence of mean velocity $ v$ on phase gradient $ q$ . Insets: Axial slices in the $ xz$ plane through volumetric images of the line trap's three-dimensional intensity distribution at two values of $ q$ .
\begin{figure}\centering\includegraphics[width=\columnwidth]{vdv3}
\end{figure}

Intensity gradients also exert forces on illuminated objects (9). In this case, the dipole moment induced in the object responds to gradients in the field, yielding a force proportional to the gradient of the intensity, which therefore is manifestly conservative (10). For a small sphere, the intensity-gradient force has the form (11,9),

$\displaystyle \vec{F}_\nabla(\vec{r}) = n_m \, \frac{k^2 a^3}{2} \, \left(\frac{m^2 - 1}{m^2 + 2}\right) \, \nabla I,$ (6)

where $ m = n_p/n_m$ is the ratio of the particle's refractive index, $ n_p$ , to the medium's, $ n_m$ . Unlike $ \vec{g}$ , $ \vec{F}_\nabla$ can be directed up the optical axis. The resulting axial restoring force is the basis of single-beam optical traps (9).

Because beams of light have gradients in both the intensity and the phase, the total optical force is not conservative. This is evident because

$\displaystyle \nabla \times \vec{g} = \frac{k}{\mu n_m} \, (\nabla I) \times (\nabla \Phi)$ (7)

does not vanish in general. Although Ashkin pointed out that optical traps exert non-conservative forces (10), virtually all subsequent reports have treated optical tweezers as (conservative) potential energy wells.

We realize phase-gradient forces using a new class of extended optical traps created through shape-phase holography (14,12,13) in an optimized holographic optical trapping system (15) Our apparatus, shown schematically in Fig. 1(a), uses a phase-only spatial light modulator (SLM) (Hamamatsu X7690-16) to imprint computer-generated holograms on the wavefronts of a slightly converging laser beam (Coherent Verdi 5W) at a vacuum wavelength of 532 nm. The modified beam is relayed to an objective lens (Nikon Plan Apo, $ 100\times$ oil immersion, NA 1.4) that focuses it into the intended three-dimensional optical trapping pattern. A wavelength-selective polarizing beam splitter reflects the laser light into the objective's input pupil and allows images at other wavelengths to pass through to a CCD camera (NEC TI-324AII).

The holograms designed for this study bring laser light to a focus along one-dimensional curves, $ C$ , embedded in the three-dimensional focal volume of the objective lens. Each hologram also encodes a designated intensity profile $ I(s)$ and phase profile $ \varphi(s)$ along the arclength $ s$ of $ C$ . This is accomplished by numerically back-projecting (16) the intended field along $ C$ onto the plane of the SLM to obtain the ideal complex-valued hologram, $ \psi(\vec{r}) = b(\vec{r}) \, \exp(i p(\vec{r}))$ . The real-valued amplitude, $ b(\vec{r})$ , is defined to be non-negative by absorbing sign changes into $ p(\vec{r})$ . The shape-phase algorithm (14,12,13) then assigns the phase shifts $ p(\vec{r})$ to the SLM's pixels with a probability proportional to $ b(\vec{r})$ . An alternate phase pattern imprinted on unassigned pixels is used to divert excess light away from the intended trapping pattern. (12).

The images in Figs. 1(b) and (c) show a focused line trap (12) and ring trap (14), respectively, each designed to have uniform intensity and uniform phase gradients. These images were obtained by placing a mirror in the microscope's focal plane to capture an image of the reflected light. Because these extended traps come to a diffraction-limited focus, their axial intensity gradients are steep enough to trap particles in three-dimensions against radiation pressure (9). To study the phase gradient force predicted by Eq. (5), we used digital video microscopy (17) to track micrometer-scale colloidal spheres moving within these traps.

Figure 3: (a) Phase-gradient barrier and (b) phase-gradient well created along a uniformly bright line trap. (1) Two 1.5  $ \unit{\mu m}$ diameter silica spheres trapped on the line. (2) The uniform in-plane intensity of the focused line. (3) Axial section through the measured intensity, showing the divergence (a3) and convergence (b3) resulting from the imposed phase profile. Scale bar indicates 5  $ \unit{\mu m}$ .
\begin{figure}\centering
\includegraphics[width=\columnwidth]{parabolic4}
\end{figure}

In the case of the line trap, we first subjected the trapped particle to linear phase gradients, $ \nabla \varphi = q \hat{\boldsymbol{x}}$ , over the range $ q = \pm 12\,\unit{radians/\ensuremath{\unit{\mu m}}\xspace }$ . The insets to Fig. 2 show axial sections through volumetric reconstructions of the trap's three-dimensional intensity distribution (13) for two different values of $ q$ . The diffraction-limited focal line remains in the $ xy$ plane despite the imposed phase gradient. The beam's direction of propagation, however, deviates from $ \hat{\boldsymbol{z}}$ by the angle $ \sin^{-1}(q/k)$ . This tilt directs a component of the beam's wave vector along $ \hat{\boldsymbol{x}}$ , thereby creating a longitudinal component of the radiation pressure. The uniform inclination evident from the images in Fig. 2 confirms both the phase gradients' magnitude and also their uniformity.

The line trap was projected into an aqueous dispersion of colloidal silica spheres $ 2a = 1.53~\ensuremath{\unit{\mu m}}\xspace $ in diameter sealed into the 40  $ \unit{\mu m}$ thick gap between a glass microscope slide and a #1 glass coverslip. Focusing the trap near the sample's midplane avoids reflections from the glass-water interface and minimizes hydrodynamic coupling to the walls. Equation (5) and the Stokes mobility law for a colloidal sphere then suggest that a trapped particle's speed, $ v$ , should be proportional to $ q$ .

To test this prediction, we measured the time required for a single sphere to travel the length, $ L = 5~\ensuremath{\unit{\mu m}}\xspace $ , of a 100 mW trap as the sign of $ q$ was flipped 20 times for each value of $ \left\vert q \right\vert $ . The observed root-mean-square off-line excursions of roughly 200 nm suggest axial and lateral trap stiffnesses comparable to those of a point-like optical tweezer powered by 1 mW. Under these conditions, the trapped sphere traveled along the line at speeds up to $ 2~\unit{\ensuremath{\unit{\mu m}}\xspace /s}$ when subjected to the largest phase gradients. Results obtained by systematically varying $ q$ are plotted in Fig. 2. They show the anticipated linear dependence, except very near $ q = 0$ , where phase-gradient forces are too weak to overcome localized pinning centers created by small uncorrected intensity variations.

More complicated phase gradients give rise to more interesting physical effects. The particles shown in Fig. 3 also are trapped along a uniformly bright line trap of length $ L = 10~\ensuremath{\unit{\mu m}}\xspace $ . This line, however, has a parabolic phase profile along its length, $ \varphi(x) = \pm (qx)^2$ that is predicted to force objects either out to the ends of the line or toward its center depending on the sign. The images in Fig. 3(a1) and (b1) demonstrate this effect for a pair of trapped colloidal spheres. Axial sections through the three-dimensional intensity distribution in Figs. 3(a3) and (b3) show that the phase-gradient barrier in Fig. 3(a) results from light diverging along the line's length, while the well in Fig. 3(b) results from the projection of converging rays. So long as the particles are rigidly confined to the uniformly bright focal line, Eq. (7) suggests that the phase-gradient force approximates a conservative one-dimensional potential energy landscape.

Figure 4: Colloidal transport driven by azimuthal phase gradients in holographic ring traps. (a) Computed axial section through a holographic ring trap of radius $ R = 20~\ensuremath{\unit{\mu m}}\xspace $ and helicity $ \ell = 30$ . (b) Volumetric representation of the measured three-dimensional intensity field in a holographic ring trap of radius $ R = 20~\ensuremath{\unit{\mu m}}\xspace $ and $ \ell = 10$ . Data points show the peak speed $ v$ of a single colloidal silica sphere circulating around the ring in (b) as a function of topological charge $ \ell $ .
\begin{figure}\centering\includegraphics[width=\columnwidth]{ring2}
\end{figure}

Like holographic line traps, holographic ring traps, such as the example in Fig. 1(c), can be endowed with arbitrary phase profiles, including the uniform azimuthal phase gradient, $ \varphi(\vec{r}) = \ell \theta$ , that defines a helical mode. A helical profile, by itself, causes a beam to focus into a ring of light, forming a torque-exerting optical trap known as an optical vortex (2). Whereas the radius of an optical vortex, $ R_\ell$ , is proportional to its helicity (7,18), holographic ring traps can be projected with any desired radius, $ R$ , independent of $ \ell $ (14). This facilitates systematic studies of colloidal transport under varying phase gradients. Also unlike optical vortices, holographic ring traps have strong enough axial intensity gradients to trap objects in three dimensions. This can be seen in the computed axial section in Fig. 4(a) in which the trap appears as two bright focal spots on the mid-line. Imposing a helical phase profile on a ring trap suppresses the beam's axial intensity through destructive interference, diverting it instead to a radius, $ R_\ell$ from the axis (7,18). If the ring's radius $ R$ exceeds the vortices', $ R_\ell$ , the converging helical beam focuses not only to the intended ring trap, but also to two conventional optical vortices above and below the focal plane, which appear as bright features in Fig. 4(a). This structure also is evident in the ring trap's measured three-dimensional intensity field (13) in Fig. 4(b). The optical vortices' comparatively weak axial intensity gradients can be clearly seen in Fig. 4(a).

A ring trap with a uniform azimuthal phase gradient exerts a torque about its axis. We demonstrated this by tracking a single colloidal silica sphere of radius $ a = 0.76~\ensuremath{\unit{\mu m}}\xspace $ circulating around a holographic ring trap of radius $ R = 2.6~\ensuremath{\unit{\mu m}}\xspace $ projected into the midplane of a 40  $ \unit{\mu m}$ thick sample. The trapped particle was subjected to azimuthal phase gradients in the range $ \ell = \pm 50$ , and its peak speed was measured (17) to within 10% for each value of the helicity. The results are plotted in Fig. 4.

Like optical vortices, holographic ring traps carrying orbital angular momentum are subject to $ \ell $ -fold and $ 2\ell$ -fold azimuthal intensity variations due to non-ideal phase scaling (19) that trap the particle for $ \left\vert \ell \right\vert < \ell_c$ (7,20). For $ \left\vert \ell \right\vert > \ell_c$ , however, the particle's peak speed increases linearly with $ \left\vert \ell \right\vert $ , consistent with the predictions of Eq. (5). Intermittent circulation near $ \left\vert \ell \right\vert = \ell_c$ gives rise to large velocity fluctuations characterized by giant enhancement of the particle's effective diffusion coefficient (20). Disorder in the effective force landscape also gives rise to interesting collective dynamics for multiple particles trapped on the ring, including transitions among periodic, chaotic and weakly chaotic steady states (21). Phase-gradient forces in holographic ring traps therefore provide useful model systems for studying fundamental problems in nonequilibrium statistical mechanics. They also promise practical applications as the basis for microscopic pumps (22), mixers (23), and optomechanical micromachines (24). Azimuthal phase gradients also can be used to endow a holographic ring trap with more complicated force profiles, even if the ring's intensity is uniform.

We have demonstrated that phase gradients in a beam of light give rise to forces transverse to the optical axis, and that these forces can be harnessed for novel optical traps. A well-established example, photon OAM in helical modes, is shown to be a particular manifestation of this general effect. Tuning optical traps' force profiles with phase gradients will be useful for manipulating microscopic objects, and will greatly facilitate rapid measurements of colloidal interactions, for example. Although phase-gradient forces generally are non-conservative, they can act as conservative force fields on appropriately restricted manifolds. More generally, optical forces' non-conservativity may engender interesting effects in illuminated particles' dynamics, including departures from Boltzmann statistics for systems nominally in equilibrium. Finally, we note in passing that phase gradients give rise to spatial variations in the polarization, which we have not considered in this Letter. Although optically isotropic materials are not influenced by polarization gradients, anisotropic materials can be. Phase-directed polarization gradients therefore should provide additional independent avenues for controlling microscopic systems.

This work was supported by the National Science Foundation through Grant Number DMR-0606415. We are grateful to Marco Polin for enlightening conversations.




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David G. Grier 2007-10-29