Tunable two-element diffractive lenses with optimal bandwidth utilization.

Previous work has shown that a set of two diffractive optical elements arranged in series can form a diffractive lens with variable optical power that can be tuned by relative rotation of its two sub-elements about their common central axis. However, previous designs of these diffractive optical elements did not take advantage of the full spatial resolution required for the fabrication process because the corresponding sub-elements consisted of both, regions with very high phase gradients (requiring full resolution) and other extended regions with nearly vanishing phase gradients, where the available resolution is "wasted". Here, an advanced design is proposed that exploits the full spatial bandwidth of the production system. This is done by increasing the polar (angular) phase gradient of each sub-element such that it approaches the usually much larger radial phase gradient. A pair of these new sub-elements then composes a diffractive lens that has the same tuning range of its optical power than a standard tunable diffractive lens, but advantageously achieved within a much smaller relative rotation range. This has advantages in systems where high speed tuning of optical power is required, and in systems where the rotation angle is limited.

Zoom system by rotation of toroidal lenses.

In a recent publication [Appl. Opt.57, 8087 (2018).] a zoom system based on rotating toroidal lenses had been theoretically suggested. Here we demonstrate two different experimental realizations of such a system. The first consists of a set of four individually rotatable cylindrical lenses, and the second of four rotatable diffractive optical elements with phase structures corresponding to "saddle-lenses". It turns out that image aberrations produced by the refractive zoom system are considerably reduced by the diffractive system.

Diffractive tunable lens for remote focusing in high-NA optical systems.

Remote focusing means to translate the focus position of an imaging system along the optical axis without moving the objective lens. The concept gains increasing importance as it allows for quick 3D focus steering in scanning microscopes, leaves the sample region unperturbed and is compatible with conjugated adaptive optics. Here we present a novel remote focusing approach that can be used in conjunction with high numerical aperture optics. Our method is based on a pair of diffractive elements, which jointly act as a tunable auxiliary lens. By changing the mutual rotation angle between the two elements, we demonstrate an axial translation of the focal spot produced by a NA = 0.95 air objective (corresponding to NA = 1.44 for an oil immersion lens) over more than 140 µm with largely maintained focus quality. We experimentally show that for the task of focus shifting, the wavefront produced by the high-NA design is superior to those produced by a parabolic lens design or a regular achromatic lens doublet.

Zoomable telescope by rotation of toroidal lenses.

A novel type of a continuously zoomable telescope is based on two pairs of adjacent toroidal lenses ("saddle lenses") in combination with standard optical components. Its variable magnification is adjusted by a mere rotation of the four saddle lenses around the optical axis. This avoids the necessity of classical zoom systems to shift multiple lenses along the longitudinal axis of the setup. A rotationally tunable pair of saddle lenses consists of two individual saddle lenses (also known as quadrupole lenses, or biconic lenses), which are arranged directly behind each other, acting as a "combi-saddle lens." The transmission function of such a combi-saddle lens corresponds to that of a single saddle lens, but with an adjustable optical power which depends on the mutual rotation angle between its two components. The optical system contains two of these combi-saddle lenses, and acts as a cylindrical Kepler telescope in one plane, and as a cylindrical Galilei telescope in the orthogonal plane. The two orthogonal Kepler/Galilei telescopes stay aligned and change their magnification factors in the same way when the telescope is zoomed by adjusting the optical powers of the two combi-saddle lenses. Altogether this produces a sharp image, which is mirrored with respect to the axis of the Kepler telescope. Thus, in addition to the zooming capabilities of the telescope, it is also possible to rotate the resulting image by a rotation of the whole telescope, or of all included saddle lenses. The operation principle of the telescope will be explained in both a ray-optical and a wave-optical description.

Multi-color operation of tunable diffractive lenses.

Rotationally tunable diffractive optical elements (DOEs) consist of two stacked diffractive optical elements which are rotated with respect to each other around their central optical axis. The combined diffractive element acts as a highly efficient diffractive lens, which changes its optical power as a function of the mutual rotation angle. Here we show that the principle can be extended to produce polychromatic tunable lenses, i.e. lenses which have the same optical power, and the same diffraction efficiency within the full tuning range at three or more selectable wavelengths. The basic principle is to use higher order DOEs, which will be polychromatic at harmonics of a fundamental wavelength. The method can be applied to other types of optical elements which are tunable by rotation, like axicons, or generalized lenses with arbitrary radial phase profiles, or to elements tunable by a mutual translation, like diffractive Alvarez lenses.

Programmable freeform optical elements.

Modern liquid crystal spatial light modulators (SLMs) are capable of shifting the optical path length by some microns, which corresponds to phase shifts of several multiples of 2π. We use this capability to display freeform optical elements (FOEs) on a SLM, as largely smooth phase variations with only a small number of wrapping lines. These FOEs can be programmed to generate so-called caustic intensity distributions, which may be real images reconstructed at a selected position in front of the SLM surface. In contrast to standard diffractive structures, reconstruction of the freeform images is non-dispersive (i.e. white light images can be programmed), free of speckle, and its efficiency does not depend on the wavelength. These features promise novel applications in image projection, and various application fields of SLMs in microscopy.

Modified Alvarez lens for high-speed focusing.

We present a modified configuration of a tunable Alvarez lens with a refocusing frequency of 1 kHz or more. In contrast to the classic Alvarez lens, the approach does not utilize a translational motion of two sub-lenses with respect to each other, but uses a 4f-setup to image two diffractive sub-lenses onto each other. Hereby focus tuning is achieved by rotating a galvo-mirror which affects the overlap of the two sub-lenses which together form an effective lens of refractive power which depends on the rotation angle of the galvo-mirror. We have demonstrated tuning of the optical power in a system where the diffractive Alvarez lens is realized by an LCOS-SLM. We consider our Alvarez setup especially suitable for applications where high refocusing rates are important, as for example in 3D life cell monitoring or tracking.

Colored point spread function engineering for parallel confocal microscopy.

Using the color selectivity of a spatial light modulator (SLM) for both, tailoring the excitation beam at one wavelength, and multiplexing the image at the red-shifted fluorescence wavelength, it is possible to parallelize confocal microscopy, i.e. to simultaneously detect an axial stack (z-stack) of a sample. For this purpose, two diffractive patterns, one steering the excitation light, and the other manipulating the emission light, are combined within the same area of the SLM, which acts as a pure phase modulator. A recently demonstrated technique allows one to combine the patterns with high diffraction efficiency and low crosstalk, using the extended phase shifting capability of the SLM, which covers multiples of 2π at the respective wavelengths. For a first demonstration we compare standard confocal imaging with simultaneous image acquisition in two separate sample planes, which shows comparable results.

Tilt-effect of holograms and images displayed on a spatial light modulator.

We show that a liquid crystal spatial light modulator (LCOS-SLM) can be used to display amplitude images, or phase holograms, which change in a pre-determined way when the display is tilted, i.e. observed under different angles. This is similar to the tilt-effect (also called "latent image effect") known from various security elements ("kinegrams") on credit cards or bank notes. The effect is achieved without any specialized optical components, simply by using the large phase shifting capability of a "thick" SLM, which extends over several multiples of 2π, in combination with the angular dependence of the phase shift. For hologram projection one can use the fact that the phase of a monochromatic wave is only defined modulo 2π. Thus one can design a phase pattern extending over several multiples of 2π, which transforms at different readout angles into different 2π-wrapped phase structures, due to the angular dependence of the modulo 2π operation. These different beams then project different holograms at the respective readout angles. In amplitude modulation mode (with inserted polarizer) the intensity of each SLM pixel oscillates over several periods when tuning its control voltage. Since the oscillation period depends on the readout angle, it is possible to find a certain control voltage which produces two (or more) selectable gray levels at a corresponding number of pre-determined readout angles. This is done with all SLM pixels individually, thus constructing different images for the selected angles. We experimentally demonstrate the reconstruction of multiple (Fourier- and Fresnel-) holograms, and of different amplitude images, by readout of static diffractive patterns in a variable angular range between 0° and 60°.

Adjustable diffractive spiral phase plates.

We report on the fabrication and the experimental demonstration of Moiré diffractive spiral phase plates with adjustable helical charge. The proposed optical unit consists of two axially stacked diffractive elements of conjugate structure. The joint transmission function of the compound system corresponds to that of a spiral phase plate where the angle of mutual rotation about the central axis enables continuous adjustment of the helical charge. The diffractive elements are fabricated by gray-scale photolithography with a pixel size of 200 nm and 128 phase step levels in fused silica. We experimentally demonstrate the conversion of a TEM(00) beam into approximated Laguerre-Gauss (LG) beams of variable helical charge, with a correspondingly variable radius of their ring-shaped intensity distribution.

How to use a phase-only spatial light modulator as a color display.

We demonstrate that a parallel aligned liquid crystal on silicon (PA-LCOS) spatial light modulator (SLM) without any attached color mask can be used as a full color display with white light illumination. The method is based on the wavelength dependence of the (voltage controlled) birefringence of the liquid crystal pixels. Modern SLMs offer a wide range over which the birefringence can be modulated, leading (in combination with a linear polarizer) to several intensity modulation periods of a reflected light wave as a function of the applied voltage. Because of dispersion, the oscillation period strongly depends on the wavelength. Thus each voltage applied to an SLM pixel corresponds to another reflected color spectrum. For SLMs with a sufficiently broad tuning range, one obtains a color palette (i.e., a "color lookup-table"), which allows one to display color images. An advantage over standard liquid crystal displays (LCDs), which use color masks in front of the individual pixels, is that the light efficiency and the display resolution are increased by a factor of three.

Colour hologram projection with an SLM by exploiting its full phase modulation range.

We demonstrate independent and simultaneous manipulation of light beams of different wavelengths by a single hologram, which is displayed on a phase-only liquid crystal spatial light modulator (SLM). The method uses the high dynamic phase modulation range of modern SLMs, which can shift the phase of each pixel in a range between 0 up to 10π, depending on the readout wavelength. The extended phase range offers additional degrees of freedom for hologram encoding. Knowing the phase modulation properties of the SLM (i.e. the so-called lookup table) in the entire exploited wavelength range, an exhaustive search algorithm allows to combine different independently calculated 2π-holograms into a multi-level hologram with a phase range extending over several multiples of 2π. The combined multi-level hologram then reconstructs the original diffractive patterns with only small phase errors at preselected wavelengths, thus projecting the desired image fields almost without any crosstalk. We demonstrate this feature by displaying a static hologram at an SLM which is read out with an incoherent red-green-blue (RGB) beam, projecting a color image at a camera chip. This is done for both, a Fourier setup which needs a lens for image focusing, and in a "lensless" Fresnel setup, which also avoids the appearance of a focused zero-order spot in the image center. The experimentally obtained efficiency of a two-colour combination is on the order of 83% for each wavelength, with a crosstalk level between the two colour channels below 2%, whereas a three-colour combination still reaches an efficiency of about 60% and a crosstalk level below 5%.

Axial super-localisation using rotating point spread functions shaped by polarisation-dependent phase modulation.

We present an approach for point spread function (PSF) engineering that allows one to shape the optical wavefront independently in both polarisation directions, with two adjacent phase masks displayed on a single liquid-crystal spatial light modulator (LC-SLM). The set-up employs a polarising beam splitter and a geometric image rotator to rectify and process both polarisation directions detected by the camera. We shape a single-lobe ("corkscrew") PSF that rotates upon defocus for each polarisation channel and combine the two polarisation channels with a relative 180° phase-shift on the computer, merging them into a single PSF that exhibits two lobes whose orientation contains information about the axial position. A major advantage lies in the possibility to measure and eliminate the aberrations in the two polarisation channels independently. We demonstrate axial super-localisation of isotropically emitting fluorescent nanoparticles. Our implementation of the single-lobe PSFs follows the method proposed by Prasad [Opt. Lett.38, 585 (2013)], and thus is to the best of our knowledge the first experimental realisation of this suggestion. For comparison we also study an approach with a rotating double-helix PSFs (in only one polarisation channel) and ascertain the trade-off between localisation precision and axial working range.

Broadband suppression of the zero diffraction order of an SLM using its extended phase modulation range.

The diffraction efficiency of a hologram displayed on a phase-only spatial light modulator (SLM) is maximal, if the SLM modulates the phase of the diffracted beam in a range between 0 and 2π. However, if the readout wavelength changes, or a broadband beam is used, due to dispersion this ideal modulation range cannot be maintained, which leads to lower diffraction efficiency and to the appearance of an undesired intense zero diffraction order. Here we show how an SLM with an extended phase modulation range of 4π can be used to display on-axis holograms with a strong suppression of the zero diffraction order in a wide spectral range, extending over 200 nm. The basic idea is to transform the original on-axis hologram into an off-axis hologram by adding a blazed grating and performing a modulo 2π operation, and then transforming it back by adding the conjugate grating, but without performing a subsequent modulo operation. The final hologram then spans over a phase range of 4π. The total diffracted field corresponds to that of the original on-axis hologram, but now the zero-order Fourier component is diffracted away from the optical axis. The same principle can be used to entangle the on-axis hologram with other phase structures, e.g. a random phase mask or a second hologram structure, followed by a subsequent addition of the conjugate mask, which may also suppress higher diffraction orders. The reconstructed holograms show a strong contrast enhancement in a broad wavelength range.

Lensless imaging through thin diffusive media.

Objects imaged through thin scattering media can be reconstructed with the knowledge of the complex transmission function of the diffuser. We demonstrate image reconstruction of static and dynamic objects with numerical phase conjugation in a lensless setup. Data is acquired by single shot intensity capture of an object coherently illuminated and obscured by an inhomogeneous medium, i.e. light diffracted at a specimen is scattered by a polycarbonate diffuser and the resulting speckle field is recorded. As a preparational step, which has to be performed only one time before imaging, the complex speckle field diffracted by the diffuser to the camera chip is measured interferometrically, which allows to reconstruct the transmission function of the diffuser. After insertion of the specimen, the speckle field in the camera plane changes, and the complex field of the sample can be reconstructed from the new intensity distribution. After initial interferometric measurement of the diffuser field, the method is robust with respect to a subsequent misalignment of the diffuser. The method can be extended to image objects placed between a pair of thin scattering plates. Since the object information is contained in a single speckle intensity pattern, it is possible to image dynamic processes at video rate.

Dispersion tuning with a varifocal diffractive-refractive hybrid lens.

We present a hybrid diffractive-refractive optical lens doublet consisting of a varifocal Moiré Fresnel lens and a polymer lens of tunable refractive power. The wide range of focal tunability of each lens and the opposite dispersive characteristics of the diffractive and the refractive element are exploited to obtain an optical system where both the Abbe number and the refractive power can be changed separately. We investigate the performance of the proposed hybrid lens at zero overall refractive power by tuning the Abbe number of a complementary standard lens while maintaining a constant overall focal length for the central wavelength. As an application example, the hybrid lens is used to tune to an optimal operating regime for quantitative phase microscopy based on a two-color transport of intensity (TIE) approach which utilizes chromatic aberrations rather than intensity recordings at several planes to reconstruct the optical path length of a phase object.

Combined holographic optical trapping and optical image processing using a single diffractive pattern displayed on a spatial light modulator.

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.

Demonstration of focus-tunable diffractive Moiré-lenses.

In an earlier publication [Appl. Opt. 47, 3722 (2008)] we suggested an adaptive optical lens, which consists of two cascaded diffractive optical elements (DOEs). Due to the Moiré-effect the combined optical element acts as a Fresnel zone lens with a refractive power that can be continuously adjusted by a mutual rotation of the two stacked DOEs. Here we present an experimental realization of this concept. Four designs of these Moiré-DOEs (MDOEs) were fabricated in thin (0.7 mm) glass slides by lithography and subsequent etching. Each element was realized as a 16 phase level DOE designed for 633 nm illumination. Our experimental investigation shows that the Moiré-lenses have a broad adjustable refractive power range with a high efficiency, which allows one to use them for flexible beam steering and for imaging applications.

Mapping of phase singularities with spiral phase contrast microscopy.

In spiral phase contrast (SPC) microscopy the edge-enhancement is typically independent of the helicity of the phase vortex filter. Here we show that for layered specimens containing screw-dislocations, as are e.g. present in mica or some crystallized organic substances, the intensity distribution in the filtered image acquires a dependence on the rotational direction of the filter. This allows one to map the distribution of phase singularities in the topography of the sample, by taking the intensity difference between two images recorded with opposite handedness. For the demonstration of this feature in a microscopy set-up, we encode the vortex filter as a binary off-axis hologram displayed on a spatial light modulator (SLM) placed in a Fourier plane. Using a binary grating, the diffraction efficiencies for the plus and minus first diffraction orders are equal, giving rise to two image waves which travel in different directions and are Fourier filtered with opposite helicity. The corresponding two images can be recorded simultaneously in two separate regions of the camera chip. This enables mapping of dislocations in the sample in a single camera exposure, as was demonstrated for various transparent samples.

Quantitative single-shot imaging of complex objects using phase retrieval with a designed periphery.

Measuring transmission and optical thickness of an object with a single intensity recording is desired in many fields of imaging research. One possibility to achieve this is to employ phase retrieval algorithms. We propose a method to significantly improve the performance of such algorithms in optical imaging. The method relies on introducing a specially designed phase object into the specimen plane during the image recording, which serves as a constraint in the subsequent phase retrieval algorithm. This leads to faster algorithm convergence and improved final accuracy. Quantitative imaging can be performed by a single recording of the resulting diffraction pattern in the camera plane, without using lenses or other optical elements. The method allows effective suppression of the "twin-image", an artefact that appears when holograms are read out. Results from numerical simulations and experiments confirm a high accuracy which can be comparable to that of phase-stepping interferometry.