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{{Short description|
The '''beam diameter''' or '''beam width''' of an [[Light beam|electromagnetic beam]] is the diameter along any specified line that is perpendicular to the beam axis and intersects it. Since beams typically do not have sharp edges, the diameter can be defined in many different ways. Five definitions of the beam width are in common use: [[#D4σ or second-moment width|D4σ]], 10/90 or 20/80 [[#Knife-edge width|knife-edge]], [[#1/e2 width|1/e<sup>2</sup>]], [[#Full width at half maximum|FWHM]], and [[#D86 width|D86]]. The beam width can be measured in units of length at a particular plane perpendicular to the beam axis, but it can also refer to the angular width, which is the angle subtended by the beam at the source. The angular width is also called the [[beam divergence]].
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===Full width at half maximum===
{{details|Full width at half maximum|Half-power point#Antennas}}
The simplest way to define the width of a beam is to choose two diametrically opposite points at which the [[irradiance]] is a specified fraction of the beam's peak irradiance, and take the distance between them as a measure of the beam's width. An obvious choice for this fraction is
=== 1/e<sup>2</sup> width ===
The 1/e<sup>2</sup> width is equal to the distance between the two points on the marginal distribution that are 1/e<sup>2</sup> = 0.135 times the maximum value. In many cases, it makes more sense to take the distance between points where the intensity falls to 1/e<sup>2</sup> = 0.135 times the maximum value. If there are more than two points that are 1/e<sup>2</sup> times the maximum value, then the two points closest to the maximum are chosen. The 1/e<sup>2</sup> width is important in the mathematics of [[Gaussian beam]]s, in which the intensity profile is described by <math>I(r) = I_{0} \left( \frac{w_0}{w} \right)^2 \exp \! \left( \! -2 \frac{r^2}{w^2}\right ) </math>.
The American National Standard Z136.1-2007 for Safe Use of Lasers (p. 6) defines the beam diameter as the distance between diametrically opposed points in that cross-section of a beam where the power per unit area is 1/e (0.368) times that of the peak power per unit area. This is the beam diameter definition that is used for computing the maximum permissible exposure to a laser beam. In addition, the Federal Aviation Administration also uses the 1/e definition for laser safety calculations in FAA Order JO 7400.2, Para. 29-1-5d.<ref>[https://www.faa.gov/documentLibrary/media/Order/7400.2L_Bsc_w_Chg1_dtd_10-12-17.pdf FAA Order JO 7400.2L, Procedures for Handling Airspace Matters], effective 2017-10-12 (with changes), accessed 2017-12-04</ref>
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Measurements of the 1/e<sup>2</sup> width only depend on three points on the marginal distribution, unlike D4σ and knife-edge widths that depend on the integral of the marginal distribution. 1/e<sup>2</sup> width measurements are noisier than D4σ width measurements. For [[transverse mode|multimodal]] marginal distributions (a beam profile with multiple peaks), the 1/e<sup>2</sup> width usually does not yield a meaningful value and can grossly underestimate the inherent width of the beam. For multimodal distributions, the D4σ width is a better choice. For an ideal single-mode Gaussian beam, the D4σ, D86 and 1/e<sup>2</sup> width measurements would give the same value.
For a Gaussian beam, the relationship between the 1/e<sup>2</sup> width and the full width at half maximum is <math>2w = \frac{\sqrt 2\ \mathrm{FWHM}}{\sqrt{\ln 2}} = 1.699 \times \mathrm{FWHM}</math>, where <math>2w</math> is the full width of the beam at 1/e<sup>2</sup>.<ref name=zemax>{{cite web |url=
=== D4σ or second-moment width ===
The D4σ width of a beam in the horizontal or vertical direction is 4 times σ, where σ is the [[standard deviation]] of the horizontal or vertical marginal distribution respectively. Mathematically, the D4σ beam width in the ''x'' dimension for the beam profile <math> I(x,y) </math> is expressed as<ref name=Siegman>{{cite web |first=A. E. |last=Siegman |url=http://www.stanford.edu/~siegman/beams_and_resonators/beam_quality_tutorial_osa.
:<math> D4\sigma = 4 \sigma = 4 \sqrt{\frac{\int_{-\infty}^\infty \int_{-\infty}^\infty I(x,y) (x - \bar{x})^2 \,dx \,dy} {\int_{-\infty}^\infty \int_{-\infty}^\infty I(x,y)\, dx \,dy}}, </math>
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Before the advent of the [[charge-coupled device|CCD]] beam profiler, the beam width was estimated using the knife-edge technique: slice a laser beam with a razor and measure the power of the clipped beam as a function of the razor position. The measured curve is the integral of the marginal distribution, and starts at the total beam power and decreases monotonically to zero power. The width of the beam is defined as the distance between the points of the measured curve that are 10% and 90% (or 20% and 80%) of the maximum value. If the baseline value is small or subtracted out, the knife-edge beam width always corresponds to 60%, in the case of 20/80, or 80%, in the case of 10/90, of the total beam power no matter what the beam profile. On the other hand, the D4σ, 1/e<sup>2</sup>, and FWHM widths encompass fractions of power that are beam-shape dependent. Therefore, the 10/90 or 20/80 knife-edge width is a useful metric when the user wishes to be sure that the width encompasses a fixed fraction of total beam power. Most CCD beam profiler's software can compute the knife-edge width numerically.
=== Fusing knife-edge
The main drawback of the knife-edge technique is that the measured value is displayed only on the scanning direction, minimizing the amount of relevant beam information. To overcome this drawback, an innovative technology offered commercially allows multiple directions beam scanning to create an image like beam representation.<ref>Aharon. "[http://www.novuslight.com/laser-beam-profiling-and-measurement_N678.html Laser Beam Profiling and Measurement]"</ref>
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The D86 width is defined as the diameter of the circle that is centered at the centroid of the beam profile and contains 86% of the beam power. The solution for D86 is found by computing the area of increasingly larger circles around the centroid until the area contains 0.86 of the total power. Unlike the previous beam width definitions, the D86 width is not derived from marginal distributions. The percentage of 86, rather than 50, 80, or 90, is chosen because a circular Gaussian beam profile integrated down to 1/e<sup>2</sup> of its peak value contains 86% of its total power. The D86 width is often used in applications that are concerned with knowing exactly how much power is in a given area. For example, applications of high-energy [[laser weapon]]s and [[lidar]]s require precise knowledge of how much transmitted power actually illuminates the target.
==== ISO11146 beam width for elliptic beams ====
The definition given before holds for stigmatic (circular symmetric) beams only. For astigmatic beams, however, a more rigorous definition of the beam width has to be used:<ref name="ISO11146-3">ISO 11146-3:2004(E), "Lasers and laser-related equipment — Test methods for laser beam widths, divergence angles and beam propagation ratios — Part 3: Intrinsic and geometrical laser beam classification, propagation and details of test methods".</ref>
:<math> d_{\sigma x} = 2 \sqrt{2} \left( \langle x^2 \rangle + \langle y^2 \rangle + \gamma \left( \left( \langle x^2 \rangle - \langle y^2 \rangle \right)^2 + 4 \langle xy \rangle^2 \right)^{1/2} \right)^{1/2} </math>
and
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:<math> \gamma = \sgn \left( \langle x^2 \rangle - \langle y^2 \rangle \right) = \frac{\langle x^2 \rangle - \langle y^2 \rangle}{|\langle x^2 \rangle - \langle y^2 \rangle|}. </math>
Using this general definition, also the beam
:<math> \phi = \frac{1}{2} \arctan \frac{2 \langle xy \rangle}{\langle x^2 \rangle - \langle y^2 \rangle }.</math>
==Measurement==
International standard ISO 11146-1:2005 specifies methods for measuring beam widths (diameters), [[beam divergence|divergence angles]] and beam propagation ratios of laser beams (if the beam is stigmatic) and for general astigmatic beams ISO 11146-2 is applicable.<ref name="11146-1">ISO 11146-1:2005(E), "Lasers and laser-related equipment — Test methods for laser beam widths, divergence angles and beam propagation ratios — Part 1: Stigmatic and simple astigmatic beams."</ref><ref name="11146-2">ISO 11146-2:2005(E), "Lasers and laser-related equipment — Test methods for laser beam widths, divergence angles and beam propagation ratios — Part 2: General astigmatic beams."</ref> The D4σ beam width is the ISO standard definition and the measurement of the [[Beam parameter product|M
The other definitions provide complementary information to the D4σ. The D4σ and knife-edge widths are sensitive to the baseline value, whereas the 1/e<sup>2</sup> and FWHM widths are not. The fraction of total beam power encompassed by the beam width depends on which definition is used.
The width of laser beams can be measured by capturing an image on a [[camera]], or by using a [[laser beam profiler]].
==See also==
*[[Beam waist]]
*[[Fresnel zone]]
==References==
{{Reflist}}
[[Category:Antennas (radio)]]
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