OPHIR BeamSquared, User Manual

Page: 79 / 112

Chapter 6 M
2
Computations
This section describes the basic calculations used to compute the M
2
results. No attempt is made to disclose
every possible feature of the algorithms employed, but rather to convey the techniques used that allow the
reader to verify conformity to the ISO procedure. All optical and physical calibration values are also discussed
in this section.
The computational methods described here apply equally to all of the operating modes in BeamSquared.
Curve Fitting 6.1
The collected data points are fit to the hyperbolic beam propagation equation ( Equation 4 , repeated below)
using a non-linear least squares technique.
𝑊
02
(𝑧)
2
= 𝑊
02
2
+ 𝛩
2
2
(𝑧 − 𝑍
02
)
2
The results of the fit yield values for W
02
, Z
02
, and Θ
2
in both the X and Y axes.
M
2
, K Factor, and BPP 6.1.1
Equation 5 – M
2
and K Factors
The M
2
or K factor is computed from the values obtained from the curve fit as:
𝑀
2
=
1
𝐾
=
𝑊
02
𝛩
2
𝜋𝑛
4𝜆
Where:
𝜆 The laser wavelength in a vacuum
𝑛 The index of refraction of the medium (assumed to be ~1)
Equation 6 – BPP Factors
BPP is computed from the M
2
results.
𝐵𝑃𝑃 =
𝑀
2
𝜆
𝜋
Translation Equations 6.1.2
The real laser beam’s parameters are computed from the following three equations:
Full Divergence Angle 6.1.2.1
Equation 7 – Full Divergence Angle
𝛩
1
=
𝑊
𝑓2
𝑓
𝜆
Where:
𝑊
𝑓2
The beam width at the focal length
𝑓
𝜆
The focal length of the lens at the laser wavelength λ