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Beamage-M² User Manual Revision 6 47
APPENDIX A. ISO11146 AND ISO11670 DEFINITIONS
The beam centroid coordinates are given by:
𝑥̅(𝑧) =
∫ ∫
𝐸(𝑥, 𝑦, 𝑧)𝑥𝑑𝑥𝑑𝑦
∞
−∞
∞
−∞
∫ ∫ 𝐸(𝑥, 𝑦, 𝑧)𝑑𝑥𝑑𝑦
∞
−∞
∞
−∞
𝑦̅(𝑧) =
∫ ∫
𝐸(𝑥, 𝑦, 𝑧)𝑦𝑑𝑥𝑑𝑦
∞
−∞
∞
−∞
∫ ∫
𝐸(𝑥, 𝑦, 𝑧)𝑑𝑥𝑑𝑦
∞
−∞
∞
−∞
The beam widths are defined as an
“extent of a power density distribution in a cross section of beam based
on the centered second-
order moments of the power density distribution.”
The second-order moments of the power density distribution are given by:
𝜎
𝑥
2
(𝑧) =
∫ ∫
𝐸(𝑥, 𝑦, 𝑧)(𝑥 − 𝑥̅)
2
𝑑𝑥𝑑𝑦
∞
−∞
∞
−∞
∫ ∫ 𝐸(𝑥, 𝑦, 𝑧)𝑑𝑥𝑑𝑦
∞
−∞
∞
−∞
𝜎
𝑦
2
(𝑧) =
∫ ∫
𝐸(𝑥, 𝑦, 𝑧)(𝑦 − 𝑦̅)
2
𝑑𝑥𝑑𝑦
∞
−∞
∞
−∞
∫ ∫
𝐸(𝑥, 𝑦, 𝑧)𝑑𝑥𝑑𝑦
∞
−∞
∞
−∞
𝜎
𝑥𝑦
2
(𝑧) =
∫ ∫
𝐸(𝑥, 𝑦, 𝑧)(𝑥 − 𝑥̅)(𝑦 − 𝑦̅)𝑑𝑥𝑑𝑦
∞
−∞
∞
−∞
∫ ∫
𝐸(𝑥, 𝑦, 𝑧)𝑑𝑥𝑑𝑦
∞
−∞
∞
−∞
The beam widths are given by:
𝑑
𝜎
𝑥
= 2√2 {(𝜎
𝑥
2
+ 𝜎
𝑦
2
) + 𝛾 [(𝜎
𝑥
2
− 𝜎
𝑦
2
)
2
+ 4(𝜎
𝑥𝑦
2
)
2
]
1
2
}
1
2
𝑑
𝜎
𝑦
= 2√2 {(𝜎
𝑥
2
+ 𝜎
𝑦
2
) − 𝛾 [(𝜎
𝑥
2
− 𝜎
𝑦
2
)
2
+ 4(𝜎
𝑥𝑦
2
)
2
]
1
2
}
1
2
where:
𝛾 =
𝜎
𝑥
2
− 𝜎
𝑦
2
|𝜎
𝑥
2
− 𝜎
𝑦
2
|
The major axis is the width’s maximum whereas the minor axis is the width’s minimum.
The effective diameter of the beam is an
“extent of a circular power density having an ellipticity greater than
0.87. […] If the ellipticity is larger than 0.87, the beam profile may be considered to be of circular symmetry
at that measuring location and the beam diameter can be obtained from:”
𝑑
𝜎
= 2√2(𝜎
𝑥
2
+ 𝜎
𝑦
2
)
1/2
The beam ellipticity is the
“ratio between the minimum and maximum widths”.
The beam orientation is the
“
angle between the x-axis
[…] and that of the principal axis of the power density
distribution which is closer to the x-axis
.”
From this definition, the angle is comprised between 45° and -45°:
𝜑(𝑧) =
1
2
arctan (
2𝜎
𝑥𝑦
2
𝜎
𝑥
2
− 𝜎
𝑦
2
)
The beam’s divergences transformed by an aberration-free focusing element of focal length
f are provided
by the following equations:
𝜃
𝑥
=
𝑑
𝜎
𝑥
𝑓