Gaussian beam

In Optical, a Gaussian beam is a particular solution of the equation of propagation of Helmholtz (as well as a Onde planes) within the framework of the paraxial Approximation. This model produces a better description of coherent radiations as the laser beams although it is incomplete in the treatment of the Diffraction.

More specifically, a Gaussian beam is a beam whose evolution of the transverse profile of amplitude according to the space propagation is proportional to a function Gaussienne, for example a function of Gauss-Hermit.

Definitions of a Gaussian beam

There exist several ways of defining a Gaussien beam. Historically, the Gaussiens beams were used in optics as a solution of the equation of propagation within the framework of the paraxial Approximation. The paraxial approximation supposes a low divergence of the beam compared to its axis of propagation. The generally allowed observation angle maximum is about 20 degrees.

Other approaches coming from electromagnetism make it possible to obtain a formulation of Gaussiens beams. Thus, one can define the Gaussian beams monomode and multimode as being a particular case in the paraxial approximation of one or more not source complexes.

Another solution can consist in extending the formalism of the rays of the geometrical Optique to the complex rays, i.e. with rays whose position, direction and matrix of curve can be complex. Lastly, one can also define a Gaussien beam starting from his spectral representation. By defining a field whose amplitude is Gaussienne on a plan, one can express by using a Specter of plane waves of this distribution of amplitude the field propagated in an unspecified point.

Propagation of the Gaussian beams

The complex Electric field of a Gaussian Beam measured (in Volt S by Meter) with $\backslash ,\; r$ of the center of the beam and with $\backslash ,\; z$ of its origin is:

$E\; (R,\; Z)\; =\; E\_0\; \backslash \; frac\; \{w\_0\}\; \{W\; (Z)\}\; \backslash \; exp\; \backslash \; left\; (\backslash \; frac\; \{-\; r^2\}\; \{w^2\; (Z)\}\backslash \; right)\; \backslash \; exp\; \backslash \; left\; (-\; ikz\; -\; ik\; \backslash \; frac\; \{r^2\}\; \{2R\; (Z)\}\; +i\; \backslash \; zeta\; (Z)\; \backslash \; right)\; \backslash ,$

And the distribution of the temporal average intensity (or Radiance), measured in Watt S per square meter is:

$I\; (R,\; Z)\; =\; \{\; |E\; (R,\; Z)|^2\; \backslash \; over\; 2\; \backslash \; eta\}\; =\; I\_0\; \backslash \; left\; (\backslash \; frac\; \{w\_0\}\; \{W\; (Z)\}\; \backslash \; right)\; ^2\; \backslash \; exp\; \backslash \; left\; (\backslash \; frac\; \{-\; 2r\; ^2\}\; \{w^2\; (Z)\}\; \backslash \; right)\; \backslash ,$

Where:

• $\backslash ,\; I\; =\; \backslash \; sqrt\; \{-\; 1\}\; \backslash ,$ is the imaginary part
• $\backslash ,\; K\; =\; \{2\; \backslash \; pi\; \backslash \; over\; \backslash \; lambda\}$ is the Nombre of wave (in Radian S per meter).
• $\backslash ,\; W\; (Z)$ is the distance to the center of the beam axis where the amplitude to electric field decreases by 1 E , which corresponds to a reduction in the intensity by (1 E ) ². This parameter is called the width of the beam
• E ₀ and I ₀ is respectively the amplitude and the intensity of the electric field in the center of the beam in the beginning. I.e. $E\_0\; \backslash \; \backslash \; stackrel\; \{\backslash \; mathrm\; \{def\}\}\; \{=\}\; \backslash \; |E\; (0,0)|$ and $I\_0\; \backslash \; \backslash \; stackrel\; \{\backslash \; mathrm\; \{def\}\}\; \{=\}\; \backslash \; I\; (0,0)$
• $\backslash \; eta\; \backslash ,$ is the constant characteristic of the impedance of the media crossed by the wave. The Impedance characteristic of the vacuum, $\backslash \; eta\; =\; \backslash \; eta\_0\; \backslash \; approx\; 377\; \backslash \; \backslash \; mathrm\; \{ohms\}$

Parameters of the beam

The geometry and of behavior of a Gaussian beam depends on various parameters which one will define below.

Beamwidth

For a Gaussian beam being propagated in the vacuum, the width of the beam $\backslash ,\; W\; (Z)$ will be with a minimal value of W ₀ at its origin. For a Wavelength λ and a distance Z along the beam axis, the variation of the width of the beam will be:

$w\; (Z)\; =\; w\_0\; \backslash ,\; \backslash \; sqrt\; \{1+\; \{\backslash \; left\; (\backslash \; frac\; \{Z\}\; \{z\_0\}\; \backslash \; right)\}^2\}\; \backslash .$

Where the origin of axis Z is defined, without loss of general information, like the point of origin and:

$z\_0\; =\; \backslash \; frac\; \{\backslash \; pi\; w\_0^2\}\; \{\backslash \; lambda\}$

It is what is called the carried Rayleigh or the Depth of field .

Range of Rayleigh and parameter confocal

At a distance from origin equal to Z ₀, the width of the beam W is:

$W\; (\backslash \; pm\; z\_0)\; =\; w\_0\; \backslash \; sqrt\; \{2\}\; \backslash ,$

The distance in + and - Z ₀ is called the parameter confocal:

$b\; =\; 2\; z\_0\; =\; \backslash \; frac\; \{2\; \backslash \; pi\; w\_0^2\}\; \{\backslash \; lambda\}\; \backslash .$

Radius of curvature

R ( Z ) is the Radius of curvature of the wave front of the beam. Its value is a function of the position like:

$R\; (Z)\; =\; Z\; \backslash \; left\; 1+\; \{\backslash \; left\; (\backslash \; frac\; \{z\_0\}\; \{Z\}\; \backslash \; right)\}^2\; \}\; \backslash right\; \backslash \; .$

Divergence of the beam

The parameter $w\; (Z)$ approaches a straight line for $z\; \backslash \; gg\; z\_0$. The angle between this straight line and the central axis of the beam are called the Divergence beam and are given by:

$\backslash \; theta\; \backslash \; simeq\; \backslash \; frac\; \{\backslash \; lambda\}\; \{\backslash \; pi\; w\_0\}\; \backslash \; qquad\; (\backslash \; theta\; \backslash \; mathrm\; \{\backslash \; in\; \backslash \; radians.\})$

The aperture of the beam since its origin is thus:

$\backslash \; Theta\; =\; 2\; \backslash \; theta\; \backslash$

• This property of a Gaussian beam returns a laser beam very widened so at its origin it has a very small diameter. So that there remains constant at a long distance, it is necessary thus that this beam is of large diameter at its origin.

• Like models it Gaussian uses a paraxial approximation, it does not apply more when one looks in a point where the wave front makes more butt 30° with the direction of propagation. Starting from the definition of the divergence, this wants to say that the Gaussian model is valid only for one beam with a width at the origin moreover than 2 λ / π .
• the quality of a beam is calculated by the product of its divergence and its width in the beginning. This number obtained on a real beam is compared with that of a Gaussian ideal beam of the same wavelength. The report/ratio of these two numbers is called the M ² and must tend towards 1 ideally.

Phase of Gouy

The times longitudinal of the phase of the wave or Phase of Gouy of the beam is:

$\backslash \; zeta\; (Z)\; =\; \backslash \; arctan\; \backslash \; left\; (\backslash \; frac\; \{Z\}\; \{z\_0\}\; \backslash \; right)\; \backslash .$

Parameter complexes beam

As the electric field comprises an imaginary part:

$Q\; (Z)\; =\; Z\; +\; q\_0\; =\; Z\; +\; iz\_0\; \backslash .$

That one often calculates like:

$\{1\; \backslash \; over\; Q\; (Z)\}\; =\; \{1\; \backslash \; over\; Z\; +\; iz\_0\}\; =\; \{Z\; \backslash \; over\; z^2\; +\; z\_0^2\}\; -\; I\; \{z\_0\; \backslash \; over\; z^2\; +\; z\_0^2\}\; =\; \{1\; \backslash \; over\; R\; (Z)\}\; -\; I\; \{\backslash \; lambda\; \backslash \; over\; \backslash \; pi\; w^2\; (Z)\}$

Z complex obtained plays a crucial role in the analysis of the properties of the Gaussian beam, especially in that of the resonant cavities and the matrices of transfer of radiation.

Power and intensity

Power by an opening

The power P (in Watt S) passing by a hole of ray R in a transverse plan with the propagation and a distance Z is:

$P\; (R,\; Z)\; =\; P\_0\; \backslash \; left\; 1\; -\; e^\; \{-\; 2r\; ^2/w^2\; (Z)\}\; \backslash \; right\; \backslash \; qquad\; \backslash \; begin\; \{boxes\}\; \backslash \; end\; \{boxes\}$ Where $P\_0\; =\; \{1\; \backslash \; over\; 2\}\; \backslash \; pi\; I\_0\; w\_0^2$ is the total power transmitted by the beam.

It is found that:

• For a hole of ray $r\; =\; W\; (Z)\; \backslash ,$, the fraction of the transmitted power is:

$\{P\; (Z)\; \backslash \; over\; P\_0\}\; =\; 1\; -\; e^\; \{-\; 2\}\; \backslash \; approx\; 0.865\; \backslash .$

• Environ 95% of the power of the beam will pass by a hole having $r\; =\; 1.224\; \backslash \; cdot\; W\; (Z)\; \backslash ,$.

Average and maximum intensity

The maximum intensity on the beam axis to $\backslash ,\; z$ of the origin is calculated by using the Règle of the Hospital for the integration of the power included/understood in the circle of radius $\backslash ,\; r$ divided by surface underlain by $\backslash ,\; \backslash \; pi\; r^2$:

$I\; (0,\; Z)\; =\; \backslash \; lim\_\; \{R\; \backslash \; to\; 0\}\; \backslash \; frac\; \{P\_0\; \backslash \; left\; 1\; -\; e^\; \{-\; 2r\; ^2/w^2\; (Z)\}\; \backslash \; right\}\; \{\backslash \; pi\; r^2\}$

= \ frac {P_0} {\ pi} \ lim_ {R \ to 0} \ frac {\ left - (- 2) (2r) e^ {- 2r ^2/w^2 (Z)} \ right} {w^2 (Z) (2r)} = {2P_0 \ over \ pi w^2 (Z)}.

FOOT-NOTE: The maximum power is thus the double of the average power obtained by the division of the total power by $w^2\; (Z)$.

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