Synthesis of an arbitrary axial field profile by computer-generated holograms

Coordinate-transformation iteration algorithm for phase modulation of arbitrary axial light distribution

Axial light distribution modulation is widely applied in optical tweezers, hard-brittle material cutting, multilayer laser direct writing, etc. To generate arbitrary axial light distribution, the coordinate-transformation iteration (CTI) algorithm is presented. The CTI algorithm unifies equations in low and high numerical aperture (NA) scenarios, using the same iterative algorithm to produce phase computer-generated holograms. In a low NA scenario, twin-foci, flattop, and sin² distributions have been achieved. In high NA scenarios, multirings, multifoci, and needle-like distributions have been realized in simulation with specific polarized incident beams. Since the CTI algorithm is inherently an efficient one-dimensional phase retrieval algorithm that is not limited by NA, this method has the potential to become a well-received solution for axial light distribution modulation.

Wavefront shaping through emulated curved space in waveguide settings

The past decade has witnessed remarkable progress in wavefront shaping, including shaping of beams in free space, of plasmonic wavepackets and of electronic wavefunctions. In all of these, the wavefront shaping was achieved by external means such as masks, gratings and reflection from metasurfaces. Here, we propose wavefront shaping by exploiting general relativity (GR) effects in waveguide settings. We demonstrate beam shaping within dielectric slab samples with predesigned refractive index varying so as to create curved space environment for light. We use this technique to construct very narrow non-diffracting beams and shape-invariant beams accelerating on arbitrary trajectories. Importantly, the beam transformations occur within a mere distance of 40 wavelengths, suggesting that GR can inspire any wavefront shaping in highly tight waveguide settings. In such settings, we demonstrate Einstein's Rings: a phenomenon dating back to 1936.

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.

Theory of "frozen waves": modeling the shape of stationary wave fields

In this work, starting by suitable superpositions of equal-frequency Bessel beams, we develop a theoretical and experimental methodology to obtain localized stationary wave fields (with high transverse localization) whose longitudinal intensity pattern can approximately assume any desired shape within a chosen interval 0 < or = z < or = L of the propagation axis z. Their intensity envelope remains static, i.e., with velocity v = 0, so we have named "frozen waves" (FWs) these new solutions to the wave equations (and, in particular, to the Maxwell equation). Inside the envelope of a FW, only the carrier wave propagates. The longitudinal shape, within the interval 0 < or = z < or = L, can be chosen in such a way that no nonnegligible field exists outside the predetermined region (consisting, e.g., in one or more high-intensity peaks). Our solutions are notable also for the different and interesting applications they can have--especially in electromagnetism and acoustics--such as optical tweezers, atom guides, optical or acoustic bistouries, and various important medical apparatuses.

Longitudinal spatial coherence applied for surface profilometry

A method of optical coherence profilometry, believed to be new, is demonstrated. This method is based on the spatial, rather than the temporal, coherence phenomenon. Therefore the proposed interferometric system is illuminated by a quasi-monochromatic spatial incoherent source instead of a broadband light source. The surface profile is measured by means of shifting the spatial degree of coherence gradually along its longitudinal axis while keeping the optical path difference between the measured surface and a reference plane constant. Experimental proof of the new principle is presented.

Marginal phase correction of truncated Bessel beams

Approximate analytic expressions are obtained for evaluating the axial intensity and the central-lobe diameter of J0 Bessel beams transmitted through a finite-aperture phase filter. A reasonable quality factor governing the axial-intensity behavior of a phase-undistorted truncated Bessel beam is found to be the inverse square root of the Fresnel number defined, for a given aperture, from the axial point of geometrical shadow. Additional drastic reduction of axial-intensity oscillations is accomplished by using marginal phase correction of the beam instead of the well-known amplitude apodization. A procedure for analytically calculating an optimal monotonic slowly varying correction phase function is described.

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.

Iterative algorithm for determining optimal beam profiles in a three-dimensional space

A new, to our knowledge, iterative algorithm for achieving optimization of beam profiles in a three-dimensional volume is presented. The algorithm is based on examining the region of interest at discrete plane locations perpendicular to the propagation direction. At each such plane an intensity constraint is imposed within a well-defined transverse spatial region of interest, whereas the phase inside that region as well as the complex amplitude outside the region is left unchanged from the previous iteration. Once the optimal solution is found, the mask that generates the desired distribution can be readily implemented with a planar diffractive optical element such as a computer-generated hologram. Several computer simulations verified the utility of the proposed approach.

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.

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.

Design of a phase-filtering mask and an aspherical lens for uniform irradiance and phase on target

We propose two designs with which one can obtain a circular light disk with both uniform irradiance and phase, namely, a homodisk on a plane target from a TEM(0, 0) Gaussian beam. The first design is pure-phase filtering masks based on wave theory and computer simulation. The second design, an aspherical lens, satisfies the two requirements successively. We used geometrical optics to obtain uniform irradiance with the aspherical lens. The theoretical results of both designs are reasonably good.

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.

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.

Snake beam: a paraxial arbitrary focal line

The creation of paraxial arbitrary focal lines by a Fourier computer-generated hologram is demonstrated. The desired focal line is represented by a series of connected straight line segments, each of which is implemented by a radial harmonic function located on a different radial portion of the entire hologram. Each subhologram is multiplied by appropriate linear and quadratic phase functions and is shifted by some distance from the center. The two phase factors determine the location of each line segment, while the in-plane shift determines the tilt angle of the segment.

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.

Control of wave-front propagation with diffractive elements

In principle, diffractive elements can be designed to control a propagating wave in three dimensions, but no general procedure has been proposed. We present an iterative method that is suitable for the design of such diffractive elements. The resulting complex amplitude distribution will approach any arbitrary requirement, as much as is permitted by basic physical principles. The problem of "nondiffracting-beam" propagation is analyzed as a special case, and intensity peaks are generated that propagate as much as 4 m with a bounded width. The generality of the method, as compared with other techniques, is beams.