laguerre-gaussianbeam

With an accout for my.chemeurope.com you can always see everything at a glance – and you can configure your own website and individual newsletter.

Gaussianbeampropagation simulation

The peak intensity at an axial distance z from the beam waist is calculated using L'Hôpital's rule as the limit of the enclosed power within a circle of radius r, divided by the area of the circle πr2:

In optics, a Gaussian beam is a beam of electromagnetic radiation whose transverse electric field and intensity (irradiance) distributions are described by Gaussian functions. Many lasers emit beams with a Gaussian profile, in which case the laser is said to be operating on the fundamental transverse mode, or "TEM00 mode" of the laser's optical resonator. When refracted by a lens, a Gaussian beam is transformed into another Gaussian beam (characterized by a different set of parameters), which explains why it is a convenient, widespread model in laser optics.

The objective (lens closest to the specimen) focuses on the specimen outside the focal point creating a real image.  This image from the objective actually increases the detail or resolving power of that specimen.  Resolution of the microscope is what allows the human eye to see detail they cannot see with the naked eye.  It allows the viewer to enter the microworld.  The higher the objective lens the better the resolution.  The eyepiece does not contribute anything new to the image; it simply spreads out the details.  This is referred to as empty magnification. This is why eyepieces are always less than 20 times magnification.

高斯光束

Different types of lenses in the microscope can cause rays to travel in different directions depending on the angle of the incident or source rays.  Light rays going through the lens can cause the light to converge or diverge, depending on whether the lens is concave or convex.  Biconvex (converging) lenses are thickest at the center and biconcave (diverging) are thinnest at the center.  There are many varieties of lenses that can be utilized with an optic system.

Light going through a double convex (biconvex) lens will converge at a focal point.  If a biconvex lens is near an object inside its focal point, a virtual upright image can be seen.   The lenses of the microscope’s eyepiece (closest to your eye) create a virtual image because your eye is within the focal point.  The eyepiece will only enlarge the image of the specimen.

The mathematical function that describes the Gaussian beam is a solution to the paraxial form of the Helmholtz equation. The solution, in the form of a Gaussian function, represents the complex amplitude of the electric field, which propagates along with the corresponding magnetic field as an electromagnetic wave in the beam.

Gaussianbeamq parameter

Gaussianbeamcalculator

The geometry and behavior of a Gaussian beam are governed by a set of beam parameters, which are defined in the following sections.

The focal length or focal distance is the distance between the center of a converging thin lens and the point at which parallel rays of incident light converge; or the distance between the center of a diverging lens and the point from which parallel rays of light appear to diverge. The point at which it intersects the focal plane is called the "focal point." The distance from the lens to the image is called the "optical element-image distance."

COMSOL Gaussianbeam

laguerre-gaussian modes

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.

Since the gaussian beam model uses the paraxial approximation, it fails when wavefronts are tilted by more than about 30° from the direction of propagation[1]. From the above expression for divergence, this means the Gaussian beam model is valid only for beams with waists larger than about 2λ/π.

The parameter w(z) approaches a straight line for . The angle between this straight line and the central axis of the beam is called the divergence of the beam. It is given by

For a Gaussian beam, the complex electric field amplitude, measured in volts per meter, at a distance r from its centre, and a distance z from its waist, is given by

Besselbeam

For a Gaussian beam propagating in free space, the spot size w(z) will be at a minimum value w0 at one place along the beam axis, known as the beam waist. For a beam of wavelength λ at a distance z along the beam from the beam waist, the variation of the spot size is given by

The complex beam parameter plays a key role in the analysis of gaussian beam propagation, and especially in the analysis of optical resonator cavities using ray transfer matrices.

Because of this property, a Gaussian laser beam that is focused to a small spot spreads out rapidly as it propagates away from that spot. To keep a laser beam very well collimated, it must have a large diameter.

where w(z) is the radius at which the field amplitude and intensity drop to 1/e and 1/e2, respectively. This parameter is called the beam radius or spot size of the beam. E0 and I0 are, respectively, the electric field amplitude and intensity at the center of the beam at its waist, that is E0 = | E(0,0) | and I0 = I(0,0). The constant is the characteristic impedance of the medium in which the beam is propagating. For free space, .

The peak intensity is thus exactly twice the average intensity, obtained by dividing the total power by the area within the radius w(z).