Gaussian beam

The optical concept of Gaussian beams (also called Gaussian beam ) combines methods of radiation and wave optics for the description of light propagation. It is an approximate solution of the paraxial Helmholtz equation. A Gaussian beam is characterized by a transverse profile according to a Gaussian curve ( the amplitude of the electromagnetic field decreases with the distance to the axis of propagation exponentially ) and a longitudinal Lorentz profile ( he is at a point of the waist, focused and " bleeds " with increasing distance from her ).

Gaussian beams describe particularly well the light emission of many laser (see beam quality ), but they can also be used in many other situations of electromagnetic radiation. They are particularly interesting because they allow phase considerations such as wave optics, but obey simple calculation methods of ray optics.

Mathematical Description

For the mathematical description of a Gaussian beam preferably cylindrical coordinates are used. The coordinate system is chosen so that the propagation direction is the Z axis and the beam waist at the origin is located at. The complex amplitude of the electric field, taking account of the phase as a function of the distance from the z-axis and the distance from the waist is given by the function:

The phase space approaches at a large distance from the waist of a spherical wave. With the approximations of the functions given below and for large, the phase factor is:

This result is in fact also obtained upon development of the source distance in the phase factor of a spherical wave. - However, the characteristic of the Gaussian beam phase reduction of after complete passage shows through the waist of the rotationally symmetric fundamental mode the significant difference between the point-symmetric beam which spherical wave and the directed axially symmetric beams see below Gouy phase.

Which belongs to the field strength intensity is:

Here, the imaginary unit, the circuit and the wave number and the values ​​at the point. The parameters of functions and describe the geometry of the Gaussian beam and are discussed below.

Transverse profile

As already mentioned, the beam has a Gaussian transverse profile, according to a Gaussian curve. Thus as the beam radius w is defined at a particular value of the distance z to the z axis, at which the amplitude to 1 / e (about 36 %), the intensity of 1 / e ², has been made. The minimum beam radius (ie at z = 0 ) is present at the waist of the beam is denoted by. As a function of distance z along the axis of the beam radius in the near field will behave according to

With the Rayleigh length

Axial profile

The spacing of the Rayleigh length from the beam waist of the beam is to

Widened. The Rayleigh length is thus the distance at which the reflecting surface has doubled relative to the smallest waist.

The distance between the left and right point is called bi-or confocal parameter:

Thus, the amplitude is therefore dropped at a certain z -coordinate of the times. This corresponds to a Lorentz profile.

Curvature

The exponential functions with imaginary exponents determined in the phase of the wave. The parameter determines vividly how much the phase is delayed at off-axis points, ie, how much the wave fronts are curved, and therefore is called radius of curvature. It is calculated as

Right in the beam waist for the radius of curvature is infinite and there are plane wave fronts. In comparison to the planar homogeneous shaft, however, the intensity profile is perpendicular to the propagation direction is not constant, and therefore, the beam waist outside the diverge and the wavefronts bend.

Divergence

If the course of for, he approaches a straight line - this shows the connection to the ray optics on. How much of the Gaussian beam is, therefore expands transversely, then can be defined by the angle (more precisely, ' slope ', because due to beam parameter product for small beam waists possible) specify between this line and the z- axis, this is called the divergence:

This relationship leads to the effect that the divergence increases with a strong focus: Where the narrow beam waist, the beam extends at great distances apart greatly. So you have to find a compromise between focus and reach.

Gouy phase

A term of the wave phase of the Gaussian beam is called the Gouy phase:

The phase difference of the fundamental mode corresponds to the transition from flipping the focus according to the classical ray optics.

When complete passage of the Gaussian beam by his waist paraxial beam experiences compared to the plane wave corresponding to a half-wavelength phase shift lower in the case of rotationally symmetric fundamental mode.

First, Louis Georges Gouy observed experimentally in 1890, the first surprising effect. Gaussian beams are according to the Fourier theorem a superposition of plane waves tilt modes. The inclined to the beam axis spectral components propagate - measured in the z- direction - obviously with a smaller phase shift compared to a paraxial wave. The steady incline range gives superimposed on the observed finite phase reduction.

Matrizenoptik

When a Gaussian beam on lenses or mirrors falls, the resulting beam is a Gaussian beam again. Thus, the rules of Matrizenoptik from classical optics can be completely transferred. If we define the parameter, the ABCD matrix of an optical element acts on him in accordance with

Complicated combinations of optical elements can be summarized in a matrix. For the calculation of laser resonators and beam paths which is a big advantage.

Derivation

As a starting point is Maxwell's equations from which a wave equation can be derived for electromagnetic waves:

A general approach for solving this equation is

With the polarization. Insertion of the approach into the wave equation yields the Helmholtz equation for the scalar amplitude of the wave

With the circular wave number. A solution of this equation would be, for example, the plane waves, but they have the problem that they are defined on the whole space, whereas laser beams are spatially very limited. It therefore makes sense for the field strength of the approach

To choose. This is a harmonious spatial oscillation in the direction of propagation before and two ( yet ) any shapes in the transverse plane (perpendicular to the propagation direction ). This approach continues to apply to the entire space, it is therefore yet another assumption is made, the so-called Paraxialnäherung (English slowly varying envelope approximation) of the Helmholtz equation applies when

With the meaning that the boundary of the beam along the direction of flight changes only slowly. Onset of the approach to the Helmholtz equation, the derivation run as far as possible, applying the approximation ( terms with more than one z- derivative equal to zero set ) leads to the differential equation

Which can be separated into two independent equations:

Are the solution of these equations

Where and are the Hermite polynomials. These solutions provide the general description of the transverse modes of a laser beam dar. However, the Gaussian beam is just the solution for for the Hermite polynomials one. Using cylindrical coordinates and inserting the solutions in the approach provides the above- mentioned field distribution: the TEM00 mode or Gaussian beam.