Beam diameter

The beam diameter or beam width of an 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 obtained in many different ways. Five definitions of the beam width are in common use: D4σ, 10/90 or 20/80 knife-edge, 1/e², FWHM, and D86.

Beam diameter is usually used to characterize electromagnetic beams in the optical regime, and occasionally in the microwave regime, that is, cases in which the aperture from which the beam emerges is very large with respect to the wavelength.

Beam diameter usually refers to a beam of circular cross section, but not necessarily so. A beam may, for example, have an elliptical cross section, in which case the orientation of the beam diameter must be specified, for example with respect to the major or minor axis of the elliptical cross section. The term "beam width" may be preferred in applications where the beam does not have circular symmetry.

Width definitions

FWHM

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 ½, in which case the diameter obtained is the full width of the beam at half its maximum intensity (FWHM).

1/e² width

In many cases, it makes more sense to take the distance between points where the intensity falls to 1/e² = 0.135 times the maximum value. If there are more than two points that are 1/e² times the maximum value, then the two points closest to the maximum are chosen. The 1/e² width is important in the mathematics of Gaussian beams.

Measurements of the 1/e² 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/e² width measurements are noisier than D4σ width measurements. For multimodal marginal distributions (a beam profile with multiple peaks), the 1/e² 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. Conversely, for a single mode pure Gaussian waveform, D4σ and the 1/e² width measurements are equivalent.

D4σ or second moment width

D4σ is shorthand for the diameter that is 4 times σ, where σ is the standard deviation of the horizontal or vertical marginal distribution. Mathematically, the D4σ beam width in the x-dimension for the beam profile $I(x,y)$ is expressed as

$D4\sigma =4\sigma =4{\sqrt {\frac {\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }I(x,y)(x-{\bar {x}})^{2}dxdy}{\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }I(x,y)dxdy}}}$,

where

${\bar {x}}={\frac {\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }I(x,y)xdxdy}{\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }I(x,y)dxdy}}$

is the centroid of the beam profile in the x-direction.

When a beam is measured with a laser beam profiler, the wings of the beam profile influence the D4σ value more than the center of the profile since the wings are weighted by the square of its distance, x², from the center of the beam. If the beam does not fill more than a third of the beam profiler’s sensor area, then there will be a significant number of pixels at the edges of the sensor that register a small baseline value (the background value). If the baseline value is large or if it is not subtracted out of the image, then the computed D4σ value will be larger than the actual value because the baseline value near the edges of the sensor are weighted in the D4σ integral by x². Therefore, baseline subtraction is necessary for accurate D4σ measurements. The baseline is easily measured by recording the average value for each pixel when the sensor is not illuminated. The D4σ width, unlike the FWHM and 1/e² widths, is meaningful for multimodal marginal distributions — that is, beam profiles with multiple peaks — but requires careful subtraction of the baseline for accurate results. The D4σ is the ISO international standard definition for beam width.

Knife-edge width

Before the advent of the CCD beam profiler, the beam width was estimated using the knife-edge technique. The technique is as follows: 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/e², 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.

D86 width

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 strange percentage of 86, rather than 50, 80, or 90, is chosen because a circular Gaussian beam profile integrated down to 1/e² 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 weapons and lidars require precise knowledge of how much transmitted power actually illuminates the target.

Measurement

International standard ISO 11146-1:2005 specifies methods for measuring beam widths (diameters), divergence angles and beam propagation ratios of laser beams (if the beam is stigmatic) and for general astigmatic beams ISO 11146-2 is applicable. The D4σ beam width is the ISO standard definition and the measurement of the M² beam quality parameter requires the measurement of the D4σ widths.

The other definitions provide complementary information to the D4σ. The D4σ and knife-edge widths are sensitive to the baseline value, whereas the 1/e² and FWHM widths are not. The fraction of total beam power encompassed by the beam width depends on which definition is used.

See also

• Beam divergence
• Laser beam profiler