## Abstract

We demonstrate simultaneous holographic optical trapping and optical image processing using a single-phase diffraction pattern displayed on a liquid crystal spatial light modulator (SLM). The ability of modern SLMs to provide multiorder phase shifts represents a degree of freedom that allows the calculation of diffraction patterns that act in precisely defined but different ways on light beams of different wavelengths. We exploit this property to calculate a single-phase hologram that shapes multiple optical traps at 785 nm while performing double-helix point spread function engineering at 532 nm. Both channels are independent to a large degree and have efficiencies of about 75% compared to the ideal diffractive patterns.

© 2014 Optical Society of America

Multiorder diffractive optical elements (DOEs), i.e., elements exhibiting phase shifts of up to multiples of $2\pi $, combine diffractive and refractive properties. In the past, this fact has been used to design diffractive structures that are achromatic for certain design wavelengths [1] or to encode different information for different wavelengths within a single diffractive pattern (DP) [2]. Recently, we have shown that the ability of modern liquid crystal spatial light modulators (SLMs) to provide large phase shifts is sufficient to implement these designs with good quality [3]. We achieved diffraction efficiencies of more than 80% for two-color projection and about 60% for three-color projection using a single DP displayed on a SLM.

In this Letter, we would like to address a specific application, namely to use a
*single* SLM-based DP to perform holographic optical trapping of
microparticles *and* optical image processing at the same time. In
particular, we generate a DP that shapes four separated optical traps for a readout
wavelength of 785 nm while transforming the imaging point spread function (PSF)
from the Airy disc into a double helix at a readout wavelength of 532 nm. A PSF
engineered this way is known to provide direct and accurate information about the axial
position of a (in this case optically trapped) microparticle. We further demonstrate the
combination of optical trapping with standard widefield and spiral phase contrast
imaging [4]. In principle, almost any
configuration of traps can be combined with any kind of imaging modality that can be
realized with a SLM.

Figure 1 illustrates the principle of encoding different information for two wavelengths in a single DP. Here, only a single pixel is considered. For a two-color readout, each pixel should ideally provide two independent phase values, which are represented by a target vector ${\mathrm{\Phi \u20d7}}_{\mathrm{target}}$ in a 2D phase space. We further have the degree of freedom to add multiples of $2\pi $ to each component of the phase vector, which in the example of Fig. 1 results in a total number of six target vectors. Each of these six vectors is a perfect solution. The SLM phase-modulation capability (denoted as “SLM phase space” in the figure) represents a line in this 2D phase space, with a slope that is determined by the thickness and wavelength-dependent refractive index of the liquid crystal. The end of this line marks the maximal achievable phase shifts at the considered wavelengths, which are for our SLM (Hamamatsu X10468-01) $5.99\pi $ at 532 nm and $3.65\pi $ at 785 nm. The voltage on the pixel is now chosen such that the target phase vector is approximated as good as possible, i.e., such that the length $|{\mathrm{\Phi \u20d7}}_{\mathrm{SLM}}-{\mathrm{\Phi \u20d7}}_{\mathrm{target}}|$ is minimized, where ${\mathrm{\Phi \u20d7}}_{\mathrm{SLM}}$ corresponds to the actual phase pair that can be realized with the SLM. It shall be noted that the error definition can also be altered in favor of either of the wavelengths, i.e., one can minimize the error for one wavelength at the cost of the quality in the other.

We combined a trapping and an imaging pattern using the scheme explained above. The former was designed to shape four separated foci in the focal space of the objective lens ($60\times $, NA 1.35) using a weighted Gerchberg–Saxton algorithm [5]. The foci build the edges of a square in $x\u2013y$ but lie at slightly different axial positions. They act as optical traps for microparticles if a sufficiently strong laser is used to read out the hologram. The imaging DP turns the PSF into a double helix that revolves around its axis under defocus [6]. The design we used was recently introduced by Prasad [7]. A double helix is special because the geometric angle of a line connecting both lobes gives direct and precise information about the axial position of the imaged particle. In an optical tweezers setup, it thus allows for precise axial force measurements [8].

Before combining the holograms, we had to decide on the wavelengths for which each of
them should be effective. Choosing the wavelengths means choosing the slope of the SLM
phase space in Fig. 1. We chose 785 nm
for the trapping and 532 nm for the imaging hologram. The phase patterns for 532
and 785 nm as well as the combined hologram are shown in Fig. 2. The pattern-combination process inevitably
produces small phase errors, i.e., the phase patterns that are actually
“seen” by the readout wavelengths differ from the original designs. These
errors reduce the diffraction efficiency and cause some stray light. Which form the
stray light takes, i.e., whether it appears pseudorandomly diffused or has a rather
defined form, depends on the structure of the holograms that are combined. The bottom
two images in Fig. 2 show the effective DPs
as seen by readout light at 532 and 785 nm. To calculate the expected diffraction
efficiencies of the combined hologram at the readout wavelengths, it is helpful to treat
the phase errors for each wavelength as separate patterns ${\mathrm{\Phi}}_{\mathrm{err},\lambda}$ that lie on top of the ideal phase holograms. Then, the
achievable efficiencies correspond to the fractions of light that are
*not* scattered by the error patterns, i.e., their relative-zero
diffraction-order strengths:

The combined hologram of Fig. 2 was displayed on a phase-only SLM (Hamamatsu X10468-01) that was integrated in a combined imaging and trapping setup. A sketch of the setup is shown in Fig. 3. For imaging we used unpolarized light from a tungsten lamp that was band filtered to a central wavelength of 532 and 3 nm width. A polarizer in front of the camera lens blocked the “wrong” polarization component, which is not affected by the SLM. For trapping we used a diode laser at 785 nm (XTRA from Toptica Photonics) that provided up to 200 mW of optical power in a single spatial mode. A dichroic mirror was used to separate imaging and trapping light. For the sample we used polystyrene microbeads (3 μm diameter) immersed in water.

Figure 4(a) shows four beads trapped in the foci shaped by the combined hologram at 785 nm. Each bead appears as two bright lobes in the image, which is a consequence of the double-helix PSF engineering performed by the same hologram at 532 nm. The rotational angle of a line connecting a lobe pair contains direct and precise information about the axial position of the corresponding bead and thus allows for precise axial-force measurements [8]. When the trapping hologram is combined with an imaging hologram containing only a constant phase, the beads appear in widefield mode. This is shown in Fig. 4(b). By switching between the two combined holograms on the SLM it is possible to toggle between the two imaging modalities ([Media 1]). During this process the beads remain stably trapped.

We compared the imaging and trapping performances of our combined hologram with the corresponding ideal DPs, which are shown in the top row of Fig. 2. The trapping performance was evaluated by reading out the combined and ideal trapping holograms with the 785 nm laser and imaging the shaped intensity patterns in an intermediate image plane. Both holograms shape four distinct foci [(Figs. 5(a) and 5(b))], albeit those of the combined hologram are dimmer, with a relative strength of 78% (ratio of integrated intensities in the foci). This value is somewhat less than the simulation result of 91%, which might be explained by the higher complexity of the combined hologram, which contains more wrapping lines (i.e., lines along which the phase pattern shows discontinuities). Furthermore, the phase steps along these lines can have magnitudes of up to $3,65\pi $ rather than only $2\pi $, which implies more extended “flyback zones” that reduce the diffraction efficiency.

The imaging performance was evaluated by comparing the maximal lobe intensities in double-helix-filtered bead images using the combined and ideal imaging holograms [(Figs. 5(c) and 5(d))]. The lobes produced by the combined hologram have a relative intensity of about 75% compared to those shaped by the ideal hologram, which is again less than the expected 95%.

Nevertheless, the combined hologram provides highly efficient and independent phase modulation at both wavelengths, and it should be possible to improve further on the diffraction efficiency by incorporating the real SLM properties into the hologram design [9] in order to compensate for pixel-fringing effects.

The possibility to combine almost arbitrary trap configurations and imaging methods is further emphasized in Fig. 6. In this experiment, four polystyrene beads (1.75 μm diameter) were optically trapped in a common axial plane and simultaneously imaged under spiral phase contrast [4], which isotropically highlights phase and amplitude edges in microscopic samples. A movie that shows how the trapped particles are moved within the sample chamber is also provided ([Media 2]). We further simulated combinations of spiral phase imaging with different configurations of optical traps and calculated the expected efficiencies for the imaging and trapping paths (see right side of [Fig. 6]). This should, for instance, allow an estimate of how the trapping power will vary when one trap is moved across the focal region by continuously updating trapping patterns and combining them with the spiral phase imaging mask. The results indicate that the efficiencies can vary by about 5 to 10%. The efficiency variations stayed in that range even when the trapping configurations were more dramatically changed, e.g., when some traps were added or removed.

We have presented combined optical trapping and image processing using a single-phase diffractive pattern displayed on a SLM. By exploiting wavelength dispersion and the ability of modern SLMs to produce phase shifts much larger than $2\pi $, we designed a hologram that interacts in two defined but completely different ways on the trapping and imaging light fields. In particular, we demonstrated double-helix imaging of multiple trapped microbeads, where both tasks, i.e., trap shaping and PSF engineering, were performed by a single hologram. The diffraction efficiencies of this combined hologram were measured to be about 75% compared to the ideal double-helix imaging and trapping patterns. We further demonstrated the combination of spiral phase imaging with optical trapping.

We would like to emphasize that the presented method for generating combined trapping and imaging holograms represents a general pathway for diffractive color multiplexing. We already demonstrated its use for multicolor projection [3], but indeed there are many other potential applications. For instance, it is also possible to implement highly efficient multi-wavelength optical tweezers, where one wavelength is used to trap particles while the other does targeted Raman or fluorescence excitation or photoporation. It also allows for independent beam shaping and aberration correction on excitation and emission light in various microscopy techniques, such as fluorescence imaging with sufficiently large Stokes shifts, harmonic imaging, or coherent anti-Stokes Raman imaging. There is also potential for mutispot stimulated emission depletion (STED) microscopy, where arrays of exciting foci and depleting optical vortices can be shaped by a single DP displayed on a SLM.

Although in principle any set of diffractive patterns for different readout wavelengths can be combined with the presented method, the quality of the result will depend on the patterns themselves as well as the readout wavelengths. For instance, it is obvious that the result will probably be unsatisfactory if the readout wavelengths are too similar (e.g., 530 and 535 nm). The slope of the one-dimensional SLM phase space (the red line in [Fig. 1]) would then be close to one. However, cases for which this slope is equal or close to a small integer number or its inverse (e.g., 2 or 1/2) are also unfavorable, because then the total phase space is effectively reduced. For our SLM such an unfavorable wavelength pair is 465 and 785 nm, with respective maximal phase modulation depths of $3.65\pi $ and $7.24\pi $.

This work was supported by the ERC Advanced Grant 247024
*catchIT*.

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