Gauss Light

[Pasty Luckenbaugh]

Inoptics, a Gaussian beam is an idealized beam of electromagnetic radiation whose amplitude envelope in the transverse plane is given by a Gaussian function; this also implies a Gaussian intensity (irradiance) profile. This fundamental (or TEM00) transverse Gaussian mode describes the intended output of many lasers, as such a beam diverges less and can be focused better than any other. When a Gaussian beam is refocused by an ideal lens, a new Gaussian beam is produced. The electric and magnetic field amplitude profiles along a circular Gaussian beam of a given wavelength and polarization are determined by two parameters: the waist w0, which is a measure of the width of the beam at its narrowest point, and the position z relative to the waist.[1]

Since the Gaussian function is infinite in extent, perfect Gaussian beams do not exist in nature, and the edges of any such beam would be cut off by any finite lens or mirror. However, the Gaussian is a useful approximation to a real-world beam for cases where lenses or mirrors in the beam are significantly larger than the spot size w(z) of the beam

Fundamentally, the Gaussian is a solution of the axial Helmholtz equation, the wave equation for an electromagnetic field. Although there exist other solutions, the Gaussian families of solutions are useful for problems involving compact beams.

The equations below assume a beam with a circular cross-section at all values of z; this can be seen by noting that a single transverse dimension, r, appears. Beams with elliptical cross-sections, or with waists at different positions in z for the two transverse dimensions (astigmatic beams) can also be described as Gaussian beams, but with distinct values of w0 and of the z = 0 location for the two transverse dimensions x and y.

The Gaussian beam is a transverse electromagnetic (TEM) mode.[2] The mathematical expression for the electric field amplitude is a solution to the paraxial Helmholtz equation.[1] Assuming polarization in the x direction and propagation in the +z direction, the electric field in phasor (complex) notation is given by:

Although there are other modal decompositions, Gaussians are useful for problems involving compact beams, that is, where the optical power is rather closely confined along an axis. Even when a laser is not operating in the fundamental Gaussian mode, its power will generally be found among the lowest-order modes using these decompositions, as the spatial extent of higher order modes will tend to exceed the bounds of a laser's resonator (cavity). "Gaussian beam" normally implies radiation confined to the fundamental (TEM00) Gaussian mode.

The geometric dependence of the fields of a Gaussian beam are governed by the light's wavelength λ (in the dielectric medium, if not free space) and the following beam parameters, all of which are connected as detailed in the following sections.

Because the divergence is inversely proportional to the spot size, for a given wavelength λ, a Gaussian beam that is focused to a small spot diverges rapidly as it propagates away from the focus. Conversely, to minimize the divergence of a laser beam in the far field (and increase its peak intensity at large distances) it must have a large cross-section (w0) at the waist (and thus a large diameter where it is launched, since w(z) is never less than w0). This relationship between beam width and divergence is a fundamental characteristic of diffraction, and of the Fourier transform which describes Fraunhofer diffraction. A beam with any specified amplitude profile also obeys this inverse relationship, but the fundamental Gaussian mode is a special case where the product of beam size at focus and far-field divergence is smaller than for any other case.

Laser beam quality is quantified by the beam parameter product (BPP). For a Gaussian beam, the BPP is the product of the beam's divergence and waist size w0. The BPP of a real beam is obtained by measuring the beam's minimum diameter and far-field divergence, and taking their product. The ratio of the BPP of the real beam to that of an ideal Gaussian beam at the same wavelength is known as M2 ("M squared"). The M2 for a Gaussian beam is one. All real laser beams have M2 values greater than one, although very high quality beams can have values very close to one.

As derived by Saleh and Teich, the relationship between the ingoing and outgoing beams can be found by considering the phase that is added to each point ( x , y ) \displaystyle (x,y) of the gaussian beam as it travels through the lens.[15] An alternative approach due to Self is to consider the effect of a thin lens on the gaussian beam wavefronts.[16]

Hermite Gaussian modes, with their rectangular symmetry, are especially suited for the modal analysis of radiation from lasers whose cavity design is asymmetric in a rectangular fashion. On the other hand, lasers and systems with circular symmetry can better be handled using the set of Laguerre-Gaussian modes introduced in the next section.

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Significance: Typical light sheet microscopes suffer from artifacts related to the geometry of the light sheet. One main inconvenience is the non-uniform thickness of the light sheet obtained with a Gaussian laser beam.

Aim: We developed a two-photon light sheet microscope that takes advantage of a thin and long Bessel-Gauss beam illumination to increase the sheet extent without compromising the resolution.

Approach: We use an axicon lens placed directly at the output of an amplified femtosecond laser to produce a long Bessel-Gauss beam on the sample. We studied the dopaminergic system and its projections in a whole cleared mouse brain.

Results: Our light sheet microscope allows an isotropic resolution of 2.4 μm in all three axes of the scanned volume while keeping a millimetric-sized field of view, and a fast acquisition rate of up to 34 mm2/s. With slight modifications to the optical setup, the sheet extent can be increased to 6 mm.

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We report, in theory and experiment, on a novel class of controlled light capsules with nearly perfect darkness, directly employing intrinsic properties of modified Bessel-Gauss beams. These beams are able to naturally create three-dimensional bottle-shaped region during propagation as long as the parameters are properly chosen. Remarkably, the optical bottle can be controlled to demonstrate various geometries through tuning the beam parameters, thereby leading to an adjustable light capsule. We provide a detailed insight into the theoretical origin and characteristics of the light capsule derived from modified Bessel-Gauss beams. Moreover, a binary digital micromirror device (DMD) based scheme is first employed to shape the bottle beams by precise amplitude and phase manipulation. Further, we demonstrate their ability for optical trapping of core-shell magnetic microparticles, which play a particular role in biomedical research, with holographic optical tweezers. Therefore, our observations provide a new route for generating and controlling bottle beams and will widen the potentials for micromanipulation of absorbing particles, aerosols or even individual atoms.

The rapidly growing optical tweezers techniques offer a precise and controllable access to manipulation of objects on micro- and nano-scale, resulting in many significant insights into events and processes on microscopic level occurring in biological, physical and chemical worlds1. Conventional optical tweezers confine objects in high-intensity region by counterbalancing scattering and gradient forces2,3, so the captured objects are usually restricted to transparent particles with refractive index greater than that of the surrounding medium. However, the optical manipulation of absorbing particles or low-index particles still remains a challenge though many attempts have been made4,5,6, holding back the raising interest of optical tweezers in relevant applications. Fortunately, a novel trapping configuration based on optical bottles or light capsules, composed of a huge darkness volume surrounded by higher intensity light barriers, can readily enable a stable confinement and guiding of light-absorbing particles7,8,9,10,11,12. Inside the light capsules, absorbing particles are repelled by high-intensity barrier and pushed towards regions of lower light intensity due to the thermal photophoretic force7,13. This will minimize the optical damage induced by light heating compared with the particles confined at high intensity region, which will be of particular interest to the in-vivo cell trapping experiments14,15. In addition, the optical bottle beams have also been exploited to capture cold atoms with decreased photon scattering and decoherence rates16,17,18,19,20, low-index particles21 and even chiral microparticles22. Furthermore, the concept of bottle beams is extended to areas of acoustic waves23 and plasmonic surface beams24,25,26. All these expansions will open new avenues to applications of biomedical imaging, object manipulation and cloaking.

Over the past years, several optical schemes have been proposed to realize light capsules. Some make use of the coherent superposition of two vortex beams11,20, Gaussian beams18, Laguerre-Gaussian beams27,28, or Bessel beams29 and some take the advantage of moir-techniques10, speckle fields30 or abruptly autofocusing beams31. They either require fine alignment of the optical system or lack the versatility. All these methods mentioned above are concentrated on scalar light fields, while the polarization capacity of light recently has also been adopted to create vectorial bottle beams with controllable light distributions32,33. This approach can adjust the bottle shape, but the main problem is that the generation of spatially variant vector optical fields is not a trivial task34,35,36.

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