M. Soskind

Shaping propagation invariant laser beams

Abstract. Propagation-invariant structured laser beams possess several unique properties and play an important role in various photonics applications. The majority of propagation invariant beams are produced in the form of laser modes emanating from stable laser cavities. Therefore, their spatial structure is limited by the intracavity mode formation. We show that several types of anamorphic optical systems (AOSs) can be effectively employed to shape laser beams into a variety of propagation invariant structured fields with different shapes and phase distributions. We present a propagation matrix approach for designing AOSs and defining mode-matching conditions required for preserving propagation invariance of the output shaped fields. The propagation matrix approach was selected, as it provides a more straightforward approach in designing AOSs for shaping propagation-invariant laser beams than the alternative technique based on the Gouy phase evolution, especially in the case of multielement AOSs. Several practical configurations of optical systems that are suitable for shaping input laser beams into a diverse variety of structured propagation invariant laser beams are also presented. The laser beam shaping approach was applied by modeling propagation characteristics of several input laser beam types, including Hermite–Gaussian, Laguerre–Gaussian, and Ince–Gaussian structured field distributions. The influence of the Ince–Gaussian beam semifocal separation parameter and the azimuthal orientation between the input laser beams and the AOSs onto the resulting shape of the propagation invariant laser beams is presented as well.

Formation and propagation of Bessel beams: practical considerations

Bessel beams belong to a class of propagation invariant, structured laser beams, and are used in various applications, including particle micro-manipulation, optical coherence tomography, optical metrology, and high resolution microscopy. In practice, Bessel beams are produced by optical fields with finite lateral dimensions propagating through finite aperture optical components, such as axicons. However, field distortions and component misalignments influence the shape of the resulting beams. In this paper, we present the influence of the beam shape and ellipticity, beam forming optics imperfections, and component misalignments on the shape of the resulting Bessel beams along the direction of propagation. Our results demonstrate that even modest fabrication errors in the input field distribution or component formation can significantly increase undesirable axial intensity oscillations in the resulting Bessel beams. Misalignments between the axicon and incoming laser beam can additionally lead to a decrease in the maximum axial intensity of the propagating beam.

Super-resolution microscopy employing propagation-invariant laser beams

We propose a novel super-resolution scanning microscopy technique employing higher-order propagationinvariant laser beams. The technique is capable of resolving objects with lateral dimensions smaller than that of the focal spot size defined by a propagating TEM00 (single mode) Gaussian beam. The field distributions at the object plane are produced by employing a spatial phase modulator. The acquired signals from the localized laser beam nodes are employed in image reconstruction, resulting in post-processed super-resolved images. The desired increase in spatial resolution is associated with an increase in the time required to spatially probe the region of interest covered by the propagating optical field. Our technique is based on a single propagating laser field, and is therefore significantly simpler to implement than techniques employing composite laser fields, such as STED (stimulated emission depletion) microscopy.

Coherent superposition of propagation-invariant laser beams

The coherent superposition of propagation-invariant laser beams represents an important beam-shaping technique, and results in new beam shapes which retain the unique property of propagation invariance. Propagation-invariant laser beam shapes depend on the order of the propagating beam, and include Hermite-Gaussian and Laguerre-Gaussian beams, as well as the recently introduced Ince-Gaussian beams which additionally depend on the beam ellipticity parameter. While the superposition of Hermite-Gaussian and Laguerre-Gaussian beams has been discussed in the past, the coherent superposition of Ince-Gaussian laser beams has not received significant attention in literature. In this paper, we present the formation of propagation-invariant laser beams based on the coherent superposition of Hermite-Gaussian, Laguerre-Gaussian, and Ince-Gaussian beams of different orders. We also show the resulting field distributions of the superimposed Ince-Gaussian laser beams as a function of the ellipticity parameter. By changing the beam ellipticity parameter, we compare the various shapes of the superimposed propagation-invariant laser beams transitioning from Laguerre-Gaussian beams at one ellipticity extreme to Hermite-Gaussian beams at the other extreme.

Understanding Laser Beam Quality Beyond M2

The laser beam M2 quality parameter is based on the second moments’ theory, as defined by ISO standards, and provides a common approach for defining the propagation characteristics of laser beams as a whole. At the same time, the M2 parameter fails to quantitatively distinguish the quality of laser beams with different spatial characteristics. For example, several laser beams with very different spatial profiles may have the same M2 value. To overcome this ambiguity, a different beam quality criterion is introduced, allowing for a quantitative definition of both the structured laser beam shape and its propagation characteristics. This criterion, called the encircled power M2 (EPM2), bridges the gap between the M2 quality parameter and the structured laser beam shape. Based on several examples we demonstrate the utility of EPM2 as applied to characterization of several structured laser beam types.

Practical considerations of producing propagation invariant laser beams

Due to their several unique properties, propagation invariant laser beams (PILBs) are playing an increasingly important role in several photonics applications. This paper describes some practical aspects of producing propagation invariant laser beams with different symmetries and structured light field distributions. Both intra-cavity and extra-cavity schemes can be employed. In the case of extra-cavity implementations, we show several practical layouts of anamorphic optical systems (AOSs) for shaping the propagation-invariant laser beams based on the size and waist locations of the input laser beams. We also establish matrix equations required to quantify the influence of input PILBs lateral and angular misalignments with respect to the AOSs onto the spatial characteristics of the produced output PILBs, and present examples of the resulting PILBs affected by the input PILBs misalignments.

Coherent radiation enhancement for laser beam shaping applications

We describe a novel coherent radiation enhancement technique and its application to laser beam shaping. The technique is based on coherent transformations of the propagating radiation employing amplitude and/or phase structures, and produces localized radiation enhancements in the output plane. The described technique offers significant flexibility in generating a variety of output laser beam shapes. Employment of electronically controllable spatial light modulators in place of the phase or amplitude structures allows dynamic adjustments of the output laser beam patterns. We demonstrate the influence of various parameters on the resulting output radiation enhancements, including the effects of the shape of the propagating radiation as well as the shape and size of the phase or amplitude structures. Our results indicate that by appropriately selecting the phase and amplitude characteristics of the structures employed during the beam shaping, a significant increase in the resulting peak intensities of the shaped beams is achieved.

Producing Bessel beams based on incoherent superposition of laser fields

Bessel beams belong to a class of propagation invariant, structured beams, and are used in a variety of applications, including particle micro-manipulation, optical coherence tomography, optical metrology, and high resolution microscopy. In practical applications, Bessel beams are formed by the interaction of optical fields with finite lateral dimensions. In this paper, we discuss the formation and propagation characteristics of Bessel beams based on input field distributions defined by Laguerre-Gaussian beams of different orders. We present the influence of the beam order on the shape and the axial intensity distribution of the resulting Bessel beams. One of the limiting factors in the applications of Bessel beams is related to the variations in the axial intensity distribution of the produced beams. We show that the incoherent superposition of input Laguerre-Gaussian beams of different orders can resolve the above limitation and produce Bessel beams with uniform peak intensity distributions over an extended axial distance.

Producing superresolved point-spread functions using a phase modulation technique

In this paper we will be discussing different techniques for producing superresolved point-spread functions (PSFs) that are based on amplitude and phase pupil masks. We propose a novel method of producing elongated superresolved pointspread functions (PSFs) using a combination of a vortex phase modulation technique and elliptical amplitude masking at the pupil of an optical system. When compared to diffraction-limited PSFs produced by an optical system with the same pupil ellipticity, the proposed method produces a significant reduction in PSF width. The proposed technique can be applied to a variety of applications, including scanning microscopy and optical micromanipulation, as well as highdensity optical data storage.