The below equations are applicable for the beam on both sides of the transforming lens as shown in the figure
above.
Equation 1 – Beam Propagation
Given the three values (W
0
, Z
0
, and Θ), the beam width (W) can be predicted at any location (z) along the axis
of propagation via the beam propagation equation:
𝑊(𝑧)
2
= 𝑊
0
2
+ 𝛩
2
(𝑧 − 𝑍
0
)
2
Equation 2 – Beam Propagation Ratio
The beam propagation ratio (M
2
) is defined by the equation:
𝑀
2
=
𝑊
0
𝛩𝜋𝑛
4𝜆
Where:
𝜆
The laser wavelength in a vacuum
𝑛
The index of refraction of the medium (assumed to be ~1)
Equation 3 – The Rayleigh Length
The Rayleigh length is defined by the equation:
𝑍
𝑅
=
𝑊
0
𝛩
An M
2
Tutorial
5.4
In some cases the preceding equations are used to characterize a propagating laser. The propagation
parameters W
0
, Z
0
, and Θ are found using numerical analysis, and from these values M
2
, K, BPP, and Z
R
are
calculated. Measuring and finding the beam propagation parameters for a given laser system is difficult, if not
impossible. Factors that limit the ability to characterize a laser include:
The laser waist is in a non-accessible location, such as inside the laser cavity.
The waist does not exist in real space, but rather behind the laser cavity in virtual space.
The beam has a large W
0
making it difficult to locate with great precision. This causes a decrease in Θ
and an increase in Z
R
.
The beam is too large to realistically measure using readily available detection methods.
These difficulties are solved by creating a measurable artificial waist via a simple focusing optic. This artificial
waist retains all the fundamental properties of the original beam. In this new form, the beam is measured and
mathematically transformed to reveal the input parameters.
To obtain a correct transform of the laser requires the focusing lens to be aberration free. Aberrations degrade
beam quality and produce inaccurate results. To minimize this degradation, it is recommended to use a thin
plano-convex lens with an f# ≥ 20.
Summary of Contents for BeamSquared
Page 10: ......
Page 15: ...BeamSquared Optical Train Dimensions 1 4 ...
Page 75: ...400mm Focal Length Lens M2 1 500mm Focal Length Lens M2 1 ...
Page 87: ......