Control of wave-front propagation with diffractive elements

Combined diffractive optical elements with adjustable optical properties controlled by a relative rotation: tutorial

A pair of adjacent transmissive diffractive optical elements (DOEs) forms a combined DOE with tunable optical properties, as, for example, a diffractive lens with an adjustable focal length. The optical properties are controlled by a relative movement of the two DOEs, such as a translation or a rotation around the optical axis. Here we discuss various implementations of this principle, such as tunable diffractive lenses, axicons, vortex plates, and aberration correction devices. We discuss the limits of the tuning range and of diffraction efficiency. Furthermore, it is demonstrated how chromatic aberrations can be suppressed by using multi-order DOEs.

Azimuthal multiplexing 3D diffractive optics

Diffractive optics have increasingly caught the attention of the scientific community. Classical diffractive optics are 2D diffractive optical elements (DOEs) and computer-generated holograms (CGHs), which modulate optical waves on a solitary transverse plane. However, potential capabilities are missed by the inherent two-dimensional nature of these devices. Previous work has demonstrated that extending the modulation from planar (2D) to volumetric (3D) enables new functionalities, such as generating space-variant functions, multiplexing in the spatial or spectral domain, or enhancing information capacity. Unfortunately, despite significant progress fueled by recent interest in metasurface diffraction, 3D diffractive optics still remains relatively unexplored. Here, we introduce the concept of azimuthal multiplexing. We propose, design, and demonstrate 3D diffractive optics showing this multiplexing effect. According to this new phenomenon, multiple pages of information are encoded and can be read out across independent channels by rotating one or more diffractive layers with respect to the others. We implement the concept with multilayer diffractive optical elements. An iterative projection optimization algorithm helps solve the inverse design problem. The experimental realization using photolithographically fabricated multilevel phase layers demonstrates the predicted performance. We discuss the limitations and potential of azimuthal multiplexing 3D diffractive optics.

Design of task-specific optical systems using broadband diffractive neural networks

Deep learning has been transformative in many fields, motivating the emergence of various optical computing architectures. Diffractive optical network is a recently introduced optical computing framework that merges wave optics with deep-learning methods to design optical neural networks. Diffraction-based all-optical object recognition systems, designed through this framework and fabricated by 3D printing, have been reported to recognize hand-written digits and fashion products, demonstrating all-optical inference and generalization to sub-classes of data. These previous diffractive approaches employed monochromatic coherent light as the illumination source. Here, we report a broadband diffractive optical neural network design that simultaneously processes a continuum of wavelengths generated by a temporally incoherent broadband source to all-optically perform a specific task learned using deep learning. We experimentally validated the success of this broadband diffractive neural network architecture by designing, fabricating and testing seven different multi-layer, diffractive optical systems that transform the optical wavefront generated by a broadband THz pulse to realize (1) a series of tuneable, single-passband and dual-passband spectral filters and (2) spatially controlled wavelength de-multiplexing. Merging the native or engineered dispersion of various material systems with a deep-learning-based design strategy, broadband diffractive neural networks help us engineer the light-matter interaction in 3D, diverging from intuitive and analytical design methods to create task-specific optical components that can all-optically perform deterministic tasks or statistical inference for optical machine learning.

Deconvolution approach for 3D scanning microscopy with helical phase engineering

RESCH (refocusing after scanning using helical phase engineering) microscopy is a scanning technique using engineered point spread functions which provides volumetric information. We present a strategy for processing the collected raw data with a multi-view maximum likelihood deconvolution algorithm, which inherently comprises the resolution gain of pixel-reassignment microscopy. The method, which we term MD-RESCH (for multi-view deconvolved RESCH), achieves in our current implementation a 20% resolution advantage along all three axes compared to RESCH and confocal microscopy. Along the axial direction, the resolution is comparable to that of image scanning microscopy. However, because the method inherently reconstructs a volume from a single 2D scan, a significantly higher optical sectioning becomes directly visible to the user, which would otherwise require collecting multiple 2D scans taken at a series of axial positions. Further, we introduce the use of a single-helical detection PSF to obtain an increased post-acquisition refocusing range. We present data from numerical simulations as well as experiments to confirm the validity of our approach.

Fast and accurate propagation of coherent light

We describe a fast algorithm to propagate, for any user-specified accuracy, a time-harmonic electromagnetic field between two parallel planes separated by a linear, isotropic and homogeneous medium. The analytical formulation of this problem (ca 1897) requires the evaluation of the so-called Rayleigh-Sommerfeld integral. If the distance between the planes is small, this integral can be accurately evaluated in the Fourier domain; if the distance is very large, it can be accurately approximated by asymptotic methods. In the large intermediate region of practical interest, where the oscillatory Rayleigh-Sommerfeld kernel must be applied directly, current numerical methods can be highly inaccurate without indicating this fact to the user. In our approach, for any user-specified accuracy ϵ>0, we approximate the kernel by a short sum of Gaussians with complex-valued exponents, and then efficiently apply the result to the input data using the unequally spaced fast Fourier transform. The resulting algorithm has computational complexity [Formula: see text], where we evaluate the solution on an N×N grid of output points given an M×M grid of input samples. Our algorithm maintains its accuracy throughout the computational domain.

Cascaded diffractive optical elements for improved multiplane image reconstruction

Computer-generated phase-only diffractive optical elements in a cascaded setup are designed by one deterministic and one stochastic algorithm for multiplane image formation. It is hypothesized that increasing the number of elements as wavefront modulators in the longitudinal dimension would enlarge the available solution space, thus enabling enhanced image reconstruction. Numerical results show that increasing the number of holograms improves quality at the output. Design principles, computational methods, and specific conditions are discussed.

Tailored optical force fields using evolutionary algorithms

We introduce a method whereby the electromagnetic field that governs the force on a Rayleigh particle can be tailored such that the resultant force field conforms to a desired geometry. The electromagnetic field is expanded as a set of vector spherical wavefunctions (VSWFs) that describe the field over all space. Given the incident field, the resultant force on a given Rayleigh particle can be calculated throughout a volume of interest. We use an evolutionary algorithm (EA) to search the space of coefficients governing the VSWFs for those that produce the desired force field. We demonstrate how Maxwell's equations will support an "optical tunnel" that guides particles to a trap location while at the same time preventing particles outside the tunnel from approaching the trap. This result is of interest because the field is impressed throughout the domain; that is to say, once the field is generated, no additional control is required to guide the particles.

Phase and amplitude imaging from noisy images by Kalman filtering

We propose and demonstrate a computational method for complex-field imaging from many noisy intensity images with varying defocus, using an extended complex Kalman filter. The technique offers dynamic smoothing of noisy measurements and is recursive rather than iterative, so is suitable for adaptive measurements. The Kalman filter provides near-optimal results in very low-light situations and may be adapted to propagation through turbulent, scattering, or nonlinear media.

Improving the system stability of a digital Shack-Hartmann wavefront sensor with a special lenslet array

There has been very limited study on the stability of a Shack-Hartmann wavefront sensor (SHWS) since its emergence in the early 1970s. In this paper, through experimental study of the system stability of a digital SHWS, a special lenslet array with long focal range is designed and implemented with a spatial light modulator to improve the system performance. Diffractive lenses with long focal length range can provide pseudo-nondiffracting beams and a long range of focusing plane. The performance and effect of the modified SHWS with this lenslet array are investigated, and the experimental results show that the system stability and measurement repeatability are not sensitive to the sensing distance and stay at an acceptable level.

Beam propagation modeling of modified volume Fresnel zone plates fabricated by femtosecond laser direct writing

Light diffraction by volume Fresnel zone plates (VFZPs) is simulated by the Hankel transform beam propagation method (Hankel BPM). The method utilizes circularly symmetric geometry and small step propagation to calculate the diffracted wave fields by VFZP layers. It is shown that fast and accurate diffraction results can be obtained with the Hankel BPM. The results show an excellent agreement with the scalar diffraction theory and the experimental results. The numerical method allows more comprehensive studies of the VFZP parameters to achieve higher diffraction efficiency.

High-efficiency rotating point spread functions

Rotating point spread functions (PSFs) present invariant features that continuously rotate with defocus and are important in diverse applications such as computational imaging and atom/particle trapping. However, their transfer function efficiency is typically very low. We generate highly efficient rotating PSFs by tailoring the range of invariant rotation to the specific application. The PSF design involves an optimization procedure that applies constraints in the Gauss-Laguerre modal plane, the spatial domain, and the Fourier domain. We observed over thirty times improvement in transfer function efficiency. Experiments with a phase-only spatial light modulator demonstrate the potential of high-efficiency rotating PSFs.

Axial field shaping under high-numerical-aperture focusing

Kant reported [J. Mod. Optics 47, 905 (2000)] a formulation for solving the inverse problem of vector diffraction, which accurately models high-NA focusing. Here, Kant's formulation is adapted to the method of generalized projections to obtain an algorithm for designing diffractive optical elements (DOEs) that reshape the axial point-spread function (PSF). The algorithm is applied to design a binary phase-only DOE that superresolves the axial PSF with controlled increase in axial sidelobes. An 11-zone DOE is identified that axially narrows the PSF central lobe by 29% while maintaining the sidelobe intensity at or below 52% of the peak intensity. This DOE could improve the resolution achievable in several applications without significantly complicating the optical system.

Computer-generated volume holograms fabricated by femtosecond laser micromachining

We define computer-generated volume holograms (CGVHs) as arbitrary 3D refractive index modulations designed to perform optical functions based on diffraction, scattering, and interference phenomena. CGVHs can differ dramatically from classical volume holograms in terms of coding possibilities, and from thin computer-generated holograms in terms of efficiency and selectivity. We propose an encoding technique for designing such holograms and demonstrate the concept by scanning focused femtosecond laser pulses to produce localized refractive index modifications in glass. These CGVHs show a significant increase in efficiency with thickness. Consequently, they are attractive for photonic integration with free-space and guided-wave devices, as well as for encoding spatial and temporal information.

Polarization selective computer-generated holograms realized in glass by femtosecond laser induced nanogratings

We demonstrate polarization selective computer-generated holograms (PSCGH) for visible light operation fabricated in glass by a femtosecond laser. For this purpose we create arrays of tailored microwaveplates by controlling the laser formation of nanogratings embedded in fused silica. A birefringent cell-oriented encoding method adapted to the characteristics of the physical writing process is proposed and implemented. According to this method, each cell contains a micro-waveplate with controlled phase retardation and orientation. A detour of each microwaveplate, combined with the orientation of its principal optical axis, simultaneously realizes a different phase function for each polarization. PSCGH's are attractive for integration with other free-space and guided-wave devices embedded in glass.

Iterative algorithms for holographic shaping of non-diffracting and self-imaging light beams

We have developed iterative algorithms for the calculation of holograms for non-diffracting or self-imaging light beams. Our methods make use of the special Fourier-space properties of the target beams. We demonstrate experimentally the holographic generation of perhaps the most challenging type of beam: a self-imaging beam shaped in more than one plane. Potential applications include the generation of light "crystals" for optical trapping and atomic diffraction studies.

White light propagation invariant beams

Propagation invariant light fields such as Bessel light beams are of interest in a variety of current areas such as micromanipulation of atoms and mesoscopic particles, laser plasmas, and the study of optical angular momentum. Considering the optical fields as a superposition of conical waves, we discuss how the coherence properties of light play a key role in their formation. As an example, we show that Bessel beams can be created from temporally incoherent broadband light sources including a halogen bulb. By using a supercontinuum source we elucidate how the beam behaves as a function of bandwidth of the incident light field.

Spatiotemporal control of ultrashort optical pulses by refractive-diffractive-dispersive structured optical elements

Structured optical elements that control the spatial and temporal characteristics of femtosecond light pulses are analyzed and synthesized. We show that unique spatiotemporal effects can be attained based on the diffraction, refraction, and dispersive effects that appear in the femtosecond regime. We argue that the design requirements for ultrafast optics are beyond the achromatization considerations that are usually applied to incoherent illumination because of the need to consider coherent effects. Despite fundamental limitations in the space-time control of ultrashort pulses, we show the potential of this technique to improve simultaneously the spatial and the temporal resolution of a lens and to generate ultrafast pulse sequences.

Electromagnetic degrees of freedom of an optical system

We present a rigorous electromagnetic formalism for defining, evaluating, and optimizing the degrees of freedom of an optical system. The analysis is valid for the delivery of information with electromagnetic waves under arbitrary boundary conditions communicating between domains in three-dimensional space. We show that, although in principle there is an infinity of degrees of freedom, the effective number is finite owing to the presence of noise. This is in agreement with the restricted classical theories that showed this property for specific optical systems and within the scalar and paraxial approximations. We further show that the best transmitting and receiving functions are the solutions of well-defined eigenvalue equations. The present approach is useful for understanding and designing modern optical systems for which the previous approaches are not applicable, as well as for application in inverse and synthesis problems.

Axially symmetric on-axis flat-top beam

A synthesis method for arbitrary on-axis intensity distributions from axially symmetric fields is developed in the paraxial approximation. As an important consequence, a new pseudo-nondiffracting beam, the axially symmetric on-axis flat-top beam (AFTB), is given by an integral transform form. This AFTB is completely determined by three simple parameters: the central spatial frequency S(c), the on-axis flat-top length L, and the on-axis central position z(c). When LS(c) >> 1, this AFTB can give a nearly flat-top intensity distribution on the propagation axis. In particular, this AFTB approaches the nondiffracting zero-order Bessel J0 beam when L--> infinity. It is revealed that the superposition of multiple AFTB fields can give multiple on-axis flat-top intensity regions when some appropriate conditions are satisfied.

Conversion of bright nondiffracting beams into dark nondiffracting beams by use of the topological properties of polarized light

We present a method for the generation of an axial phase dislocation on a wave front, which is induced by topological properties of polarized light. This effect is shown to be useful for conversion of bright nondiffracting beams into dark nondiffracting beams. Experiments showing the generation of dark nondiffracting beams have been performed.

Polarized pseudonondiffracting beams generated by polarization-selective diffractive phase elements

The concept and the generation of polarized pseudonondiffracting beams (PNDB's) from polarization-selective diffractive phase elements (DPE's) are presented for what we believe is the first time in a monochromatic illuminating system. The polarized PNDB's behave as segmented almost constant axial-intensity distributions with individual different polarization states in different segments. The pure polarization state in each segment can be arbitrarily preset. The design of polarization-selective DPE's is achieved with the use of the conjugate-gradient method. The simulation results show that the DPE's we designed can successfully implement the desired polarization modulation. Furthermore the PNDB's characteristics of axial-intensity uniformity and beamlike shape are well achieved with the DPE's we designed.

Implementation of pseudo-nondiffracting beams by use of diffractive phase elements

We report the experimental implementation of pseudo-nondiffracting beams by use of diffractive phase elements (DPE's). Based on the conjugate-gradient method presented in J. Opt. Soc. Am. A 15, 144-151 (1998), these DPE's are designed and fabricated on a flat quartz substrate. The experimental results show that the performance of the fabricated DPE's is in good agreement with the theoretical prediction.

Pattern generation with an extended focal depth

The depth of focus of light patterns can be extended, within given tolerances, beyond the classical limits. For a quantitative evaluation we introduce a degree of depth-of-focus extension and a three-dimensional energy-distribution efficiency. The basic limitations involved in depth-of-focus extension are discussed. A coherent system in which the input is optimized for a desired output pattern is presented. An example of a pattern containing diffraction-limited line segments and a 4 times improvement in depth of focus is demonstrated. This task is much more difficult than generating patterns of isolated light spots in which the depth of focus is extended beyond an order of magnitude.

Diffractive phase elements that synthesize color pseudo-nondiffracting beams

The design of diffractive phase elements (DPE's) for generating color pseudo-nondiffracting beams (PNDB's) in a multiple-wave illuminating system by the conjugate-gradient method is described. The axial-intensity distributions for dual-color PNDB's generated by the DPE's are shown. The color PNDB's behave as segmented axial-intensity distributions; in each of segments only one color persists. The sequence of wavelength components in the color PNDB's can be arbitrarily preset. Three-dimensional plots of a dual-color PNDB indicate the characteristics of a beam with high transverse resolution.

On-axis computer-generated holograms for three-dimensional display

The feasibility of on-axis synthetic near-field amplitude holograms for three-dimensional display applications is demonstrated. An iterative optimization algorithm is used that generates an object-dependent diffuser that utilizes the phase and, to some extent, amplitude freedoms in the reconstruction plane. The discrimination between twin images and undiffracted terms is thus improved. The on-axis approach presents important advantages: a low coherence requirement for the illuminating source, a lower spacebandwidth and higher viewing angle than with the off-axis alternatives. Defocusing and parallax are experimentally attained with an extended white-light source and a lensless setup.

Dark beams with a constant notch

Dark beams are wave fields carrying information in dark regions. We experimentally demonstrate dark beams that maintain a constant notch shape and size along a predetermined domain in free space. The energy is concentrated around the dark region with negligible sidelobes. These beams are generated with low-information-content phase-only diffractive elements in an on-axis configuration.

Nondiffracting interference patterns generated with programmable spatial light modulators

Nondiffracting beams are of interest for optical metrology applications because the size and shape of the beams do not change as the beams propagate. We have created a generating pattern consisting of a linear combination of two nondiffracting patterns. This pattern forms a nondiffracting interference pattern that appears as a circular array of nondiffracting spots. More complicated multiplexed arrays are also constructed that simultaneously yield two different nondiffracting patterns. We generate these Bessel function arrays with a programmable spatial light modulator. Such arrays would be useful for angular alignment and for optical interconnection applications.

Design of diffractive phase elements that produce focal annuli: a new method

A new optimization method based on the general theory of amplitude-phase retrieval is proposed for designing the diffractive phase elements (DPE's) that produce focal annular patterns. A set of equations for determining the phase distribution of the DPE is given. The profile of a surface-relief DPE can be designed with an iterative algorithm. Numerical calculations are carried out for several examples. A comparison of the performance of the DPE's designed with the Gerchberg-Saxton algorithm and the new algorithm is presented. The effect of quantization of the phase distribution of the DPE's on the results is also investigated. The results show that the new algorithm can successfully achieve the design of the DPE's that convert the uniform incident beam into the focal annular patterns.

Pseudonondiffracting slitlike beam and its analogy to the pseudonondispersing pulse

A new nonspreading beam is proposed for the case in which diffraction occurs only in one transverse coordinate. The beam has the shape of a pulse in one dimension and is constant in the other (slitlike shape). The intensity of the pulse's peak remains almost constant along a finite interval on the propagation axis. The proposed beam is analyzed and demonstrated experimentally. The analogy between this beam and the temporal pulse in a dispersive medium is discussed.