Gaussian beam

The Gaussian beam (also Gaussian bundle ) is a concept of paraxial optics for describing the propagation of light, in which methods of ray and wave optics are combined . In cross section, the Gaussian beam shows a profile according to a Gaussian curve with a width that varies along the axis of propagation. The beam tapers approximately linearly until it reaches the narrowest point, which is known as the focus or waist , and then grows again as well. Along the axis of propagation, the spatial intensity of the beam shows a Lorentz profile , the maximum is at the point of the waist. The electromagnetic field of the Gaussian beam results from the Maxwell equations for constant frequency ω, i.e. from the Helmholtz equation , according to a paraxial approximation. With a given direction of propagation and wavelength, the Gaussian beam is completely determined by specifying the location and the beam diameter of the waist.

Gaussian rays describe the light emission of many lasers particularly well (see diffraction index ), but they can also be used in many other electromagnetic radiation situations. They are particularly interesting because on the one hand they obey the simple calculation methods of ray optics, but on the other hand they also allow phase considerations as in wave optics.

Mathematical description

Gaussian beam (schematic) with dimensions, beam limitation, wavefronts and intensity profile

Cylinder coordinates are preferably used for the mathematical description of a Gaussian beam . The coordinate system is chosen so that the direction of propagation is the z-axis and the beam waist is in the coordinate origin . The complex amplitude of the electric field taking into account the phase as a function of the distance to the z-axis and the distance to the waist is described by the function: ${\displaystyle z=0}$ ${\displaystyle r}$ ${\displaystyle z}$

${\displaystyle E(r,z)=E_{0}\;{\frac {w_{0}}{w(z)}}\cdot \mathrm {e} ^{-\left({\frac {r}{w(z)}}\right)^{2}}\cdot \mathrm {e} ^{-ik{\frac {r^{2}}{2R(z)}}}\cdot \mathrm {e} ^{-i(kz-\zeta (z))}}$

The phase surface approaches that of a spherical wave at a large distance from the waist. With the approximations of the functions given below and for large , the phase factor becomes: ${\displaystyle R(z)}$ ${\displaystyle \zeta (z)}$ ${\displaystyle |z|\gg z_{R}}$

${\displaystyle \mathrm {e} ^{-i[k(z+{\frac {r^{2}}{2z}})\mp {\frac {\pi }{2}}]}}$

This result is in fact also on the development of the source distance in the phase factor receive a spherical wave: . - However, the phase reduction characteristic of the Gaussian beam after complete passage through the waist of the rotationally symmetrical basic mode shows the significant difference between the point-symmetrically radiating spherical wave and the directed, axially symmetrical beam, see Gouy phase below . ${\displaystyle \textstyle \rho }$ ${\displaystyle \mathrm {e} ^{-ik\rho }}$ ${\displaystyle \textstyle \rho ={\sqrt {r^{2}+z^{2}}}\approx z+r^{2}/(2z)}$ ${\displaystyle \pi }$

The intensity associated with the field strength is:

${\displaystyle I(r,z)={\frac {1}{2}}c_{0}\varepsilon _{0}\left|E(r,z)\right|^{2}=I_{0}\left({\frac {w_{0}}{w(z)}}\right)^{2}\mathrm {e} ^{-{\frac {2r^{2}}{w(z)^{2}}}}}$

Here, the imaginary unit , the circular wave number and or the values at the point . The parameter functions , and describe the geometry of the Gaussian beam and are explained below. ${\displaystyle i}$ ${\displaystyle k={\tfrac {2\pi }{\lambda }}}$ ${\displaystyle E_{0}}$ ${\displaystyle I_{0}}$ ${\displaystyle (r=0,z=0)}$ ${\displaystyle w(z)}$ ${\displaystyle R(z)}$ ${\displaystyle \zeta (z)}$

Transversal profile

As already mentioned, the Gaussian beam has a transverse profile according to a Gaussian curve . At a certain value, the beam radius is defined as the distance to the axis at which the amplitude has dropped to 1 / e (approx. 37%), the intensity to 1 / e² (approx. 13.5%). The minimum beam radius that is at the waist of the beam (i.e. at ) is denoted by. Depending on the distance along the axis, the beam radius then behaves accordingly in the near field ${\displaystyle w}$ ${\displaystyle z}$ ${\displaystyle z}$ ${\displaystyle z=0}$ ${\displaystyle w_{0}}$ ${\displaystyle z}$

${\displaystyle w(z)=w_{0}\,{\sqrt {1+{\left({\frac {z}{z_{R}}}\right)}^{2}}}}$

with the Rayleigh length

${\displaystyle z_{R}={\frac {\pi \cdot w_{0}^{2}}{\lambda }}}$.

Axial profile

The beam is open at a distance of the Rayleigh length from the beam waist

${\displaystyle w(\pm z_{R})=w_{0}{\sqrt {2}}}$

widened. The Rayleigh length is therefore the distance at which the beam area has doubled in relation to the smallest waist.

The distance between the left and right point with is called the bi- or confocal parameter: ${\displaystyle |z|=z_{R}}$

${\displaystyle b=2z_{R}={\frac {2\pi w_{0}^{2}}{\lambda }}}$

So the amplitude has dropped to a factor of -fold at a certain z-coordinate . This corresponds to a Lorentz profile. ${\displaystyle |E(0,z_{R})|=E_{0}{\tfrac {w_{0}}{w(z_{R})}}={\tfrac {1}{\sqrt {2}}}\,E_{0}}$ ${\displaystyle {\tfrac {1}{\sqrt {2}}}}$

curvature

Radius of the wave fronts over the direction of propagation. For the radius of curvature becomes infinite, for large z there is a proportional dependence. The smallest radius of curvature is included . ${\displaystyle z\rightarrow 0}$ ${\displaystyle z_{R}}$

The exponential functions with imaginary exponents determine the phase position of the wave . The parameter clearly determines how much the phase is delayed at off-axis points, i.e. how much the wave fronts are curved, and is therefore called the radius of curvature . He calculates too ${\displaystyle (r,z)}$ ${\displaystyle R(z)}$

${\displaystyle R(z)=z\,\left(1+{\left({\frac {z_{R}}{z}}\right)}^{2}\right)}$.

Directly in the beam waist for the radius of curvature is infinite and there are plane wave fronts. Compared to the plane homogeneous wave , however, the intensity profile perpendicular to the direction of propagation is not constant, which is why the beam diverges outside the waist and the wave fronts curve. ${\displaystyle z=0}$

divergence

If you look at the course of for , it approaches a straight line - this shows the connection to the ray optics. How strongly the Gaussian beam runs, i.e. it expands transversely, can then be indicated by the angle (more precisely: 'slope', since due to the beam parameter product also possible for small beam waistlines ) between this straight line and the z-axis, this is called the divergence : ${\displaystyle w(z)}$ ${\displaystyle z\gg z_{R}}$ ${\displaystyle w_{0}\theta _{\mathrm {div} }={\tfrac {\lambda }{\pi }}M^{2}}$ ${\displaystyle \mathrm {Winkel} >\pi /2}$ ${\displaystyle w_{0}}$ ${\displaystyle \rightarrow \tan(\mathrm {Winkel} )}$

${\displaystyle \theta _{\mathrm {div} }={\frac {\Theta }{2}}=\arctan \left({\frac {w(z)}{z}}\right)\ \ {\overset {z\gg z_{R}}{\approx }}\ \ \arctan \left({\frac {\lambda }{\pi w_{0}}}\right)}$

This relationship leads to the effect that the divergence increases with strong focusing: If the beam waist is narrow, the beam spreads greatly apart at great distances. So you have to find a compromise between focus and range.

Gouy phase

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

${\displaystyle \zeta (z)=\arctan \left({\frac {z}{z_{R}}}\right)}$

The phase difference from the basic mode at the transition from to corresponds to the turning over in the focus according to the classic ray optics. ${\displaystyle \pi }$ ${\displaystyle z\ll 0}$ ${\displaystyle z\gg 0}$

When the Gaussian bundle completely passes through its waist, the paraxial ray experiences a phase shift that is half a wavelength smaller than that of the plane wave in the case of the rotationally symmetrical fundamental mode. ${\displaystyle e^{-ikz}}$

Louis Georges Gouy first observed the initially surprising effect experimentally in 1890. According to the Fourier theorem, Gaussian beams are a superposition of inclination modes of plane waves. The spectral components inclined to the beam axis propagate - measured in the z-direction - apparently with a smaller phase shift compared to an axially parallel wave. The constant inclination spectrum shows the observed finite phase reduction superimposed.

Die optics

If a Gaussian beam falls on lenses or mirrors, the resulting beam is again a Gaussian beam. This means that the rules of matrix optics can be completely transferred from classic optics. If the parameter is defined , the ABCD matrix of an optical element acts on it accordingly ${\displaystyle q(z)=z+iz_{R}}$

${\displaystyle q_{1}(z)={\frac {Aq_{0}+B}{Cq_{0}+D}}}$

Complicated combinations of optical elements can be combined into a matrix. This is a great advantage for the calculation of laser resonators and beam paths.

Derivation

The starting point is the Maxwell equations , from which a wave equation for electromagnetic waves can be derived:

${\displaystyle c^{2}\Delta {\vec {E}}({\vec {x}},t)={\ddot {\vec {E}}}({\vec {x}},t)}$

A general approach to solving this equation is

${\displaystyle {\vec {E}}({\vec {x}},t)={\vec {e}}E({\vec {x}})e^{i\omega t}}$

with the polarization . Inserting the approach into the wave equation yields the Helmholtz equation for the scalar amplitude of the wave ${\displaystyle {\vec {e}}}$

${\displaystyle \Delta E({\vec {x}})=-k^{2}E({\vec {x}})}$

with the circular wavenumber . A solution to this equation would be, for example, the plane waves, but these have the problem that they have the same amplitude in the entire space, while laser beams are spatially very limited. It is therefore useful for the field strength approach ${\displaystyle k=\omega /c}$

${\displaystyle E({\vec {x}})=E_{0}X(x,z)Y(y,z)e^{-ikz}}$

to choose. This specifies a harmonic, spatial oscillation in the direction of propagation as well as two (so far) arbitrary shapes in the transverse plane (perpendicular to the direction of propagation). This approach continues to apply to the entire space, which is why another assumption is made, the so-called paraxial approximation ( slowly varying envelope approximation ) of the Helmholtz equation , for which applies

${\displaystyle k{\frac {\partial }{\partial z}}\gg {\frac {\partial ^{2}}{\partial z^{2}}},{\frac {\partial X}{\partial z}}{\frac {\partial Y}{\partial z}}}$

meaning that the profile of the beam changes only slowly along the direction of propagation. Inserting the approach into the Helmholtz equation, executing the derivation as far as possible, applying the approximation (setting terms with more than one z-derivative equal to zero) leads to the differential equation

${\displaystyle Y{\frac {\partial ^{2}X}{\partial x^{2}}}+X{\frac {\partial ^{2}Y}{\partial y^{2}}}-2ikY{\frac {\partial X}{\partial z}}-2ikX{\frac {\partial Y}{\partial z}}=0}$

which can be separated into two independent equations:

${\displaystyle {\begin{aligned}\left({\frac {\partial ^{2}}{\partial x^{2}}}-2ik{\frac {\partial }{\partial z}}\right)X(x,z)&=0\\\left({\frac {\partial ^{2}}{\partial y^{2}}}-2ik{\frac {\partial }{\partial z}}\right)Y(y,z)&=0\end{aligned}}}$

Solutions to these equations are

${\displaystyle {\begin{aligned}X_{m}(x,z)&={\sqrt {\frac {w_{0}}{w(z)}}}H_{m}\left({\frac {{\sqrt {2}}x}{w(z)}}\right)\exp \left(-{\frac {x^{2}}{w^{2}(z)}}-i{\frac {kx^{2}}{2R(z)}}+i{\frac {2m+1}{2}}\zeta (z)\right)\\Y_{n}(y,z)&={\sqrt {\frac {w_{0}}{w(z)}}}H_{n}\left({\frac {{\sqrt {2}}y}{w(z)}}\right)\exp \left(-{\frac {y^{2}}{w^{2}(z)}}-i{\frac {ky^{2}}{2R(z)}}+i{\frac {2n+1}{2}}\zeta (z)\right)\end{aligned}}}$

where and are the Hermite polynomials . These solutions represent the various transverse modes of a laser beam. The Gaussian beam is the solution for for which the Hermite polynomials are one. Using cylindrical coordinates and inserting the solutions in the approach supplies the field distribution mentioned at the beginning: the TEM ₀₀ mode or Gaussian beam. ${\displaystyle H_{m}}$ ${\displaystyle H_{n}}$ ${\displaystyle m=n=0}$

literature

• Dieter Meschede : Optics, light and laser . 2nd Edition. BG Teubner, Munich 2005, ISBN 3-519-13248-6 .
• Anthony E. Siegman : Lasers . 1st edition. University Science Books, 1986, ISBN 0-935702-11-3 , Wave Optics and Gaussian Beams chapter (American English).
• Eugene Hecht: optics . 4th edition. Oldenbourg Wissenschaftsverlag, Munich 2005, ISBN 3-486-27359-0 .
• Herwig Kogelnik , Tingye Li : Laser beams and Resonators . In: Applied Optics . Vol. 5, No. 10 , 1966, pp. 1550-1567 , doi : 10.1364 / AO.5.001550 .

Web links

• Focusing and expansion of laser radiation, Linos catalog excerpt (PDF; 84 kB)
• Gaussian Beam Propagation, Melles-Griot catalog excerpt (PDF; 276 kB)
• Gaussian rays: visualizations for amplitude, phase and electromagnetic field strength