Beam diameter
Encyclopedia
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 defined 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. 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.
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.
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 (−3 dB
), in which case the diameter obtained is the full width of the beam at half its maximum intensity (FWHM). This is also called the half-power beam width (HPBW).
s.
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. For an ideal single-mode Gaussian beam, the D4σ and the 1/e2 width measurements would give the same value.
of the horizontal or vertical marginal distribution. Mathematically, the D4σ beam width in the x-dimension for the beam profile
is expressed as
,
where
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.
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/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.
s require precise knowledge of how much transmitted power actually illuminates the target.
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.
Light beam
A light beam or beam of light is a narrow projection of light energy radiating from a source into a beam. Sunlight is a natural example of a light beam when filtered through various mediums...
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, 10/90 or 20/80 knife-edge, 1/e2, FWHM, and 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
Angular diameter
The angular diameter or apparent size of an object as seen from a given position is the “visual diameter” of the object measured as an angle. In the vision sciences it is called the visual angle. The visual diameter is the diameter of the perspective projection of the object on a plane through its...
, which is the angle subtended by the beam at the source.
Beam diameter is usually used to characterize electromagnetic beams in the optical regime, and occasionally in the microwave
Microwave
Microwaves, a subset of radio waves, have wavelengths ranging from as long as one meter to as short as one millimeter, or equivalently, with frequencies between 300 MHz and 300 GHz. This broad definition includes both UHF and EHF , and various sources use different boundaries...
regime, that is, cases in which the aperture from which the beam emerges is very large with respect to the wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
.
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.
Full width at half maximum
The simplest way to define the width of a beam is to choose two diametrically opposite points at which the irradianceIrradiance
Irradiance is the power of electromagnetic radiation per unit area incident on a surface. Radiant emittance or radiant exitance is the power per unit area radiated by a surface. The SI units for all of these quantities are watts per square meter , while the cgs units are ergs per square centimeter...
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 (−3 dB
Decibel
The decibel is a logarithmic unit that indicates the ratio of a physical quantity relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities...
), in which case the diameter obtained is the full width of the beam at half its maximum intensity (FWHM). This is also called the half-power beam width (HPBW).
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 beamGaussian beam
In optics, a Gaussian beam is a beam of electromagnetic radiation whose transverse electric field and intensity distributions are well approximated by Gaussian functions. Many lasers emit beams that approximate a Gaussian profile, in which case the laser is said to be operating on the fundamental...
s.
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
Transverse mode
A transverse mode of a beam of electromagnetic radiation is a particular electromagnetic field pattern of radiation measured in a plane perpendicular to the propagation direction of the beam...
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. For an ideal single-mode Gaussian beam, the D4σ and the 1/e2 width measurements would give the same value.
D4σ or second moment width
D4σ is shorthand for the diameter that is 4 times σ, where σ is the standard deviationStandard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...
of the horizontal or vertical marginal distribution. Mathematically, the D4σ beam width in the x-dimension for the beam profile
is expressed as,where
is the centroid
Centroid
In geometry, the centroid, geometric center, or barycenter of a plane figure or two-dimensional shape X is the intersection of all straight lines that divide X into two parts of equal moment about the line. Informally, it is the "average" of all points of X...
of the beam profile in the x-direction.
When a beam is measured with a laser beam profiler
Laser beam profiler
A laser beam profiler captures, displays, and records the spatial intensity profile of a laser beam at a particular plane transverse to the beam propagation path. Since there are many types of lasers — ultraviolet, visible, infrared, continuous wave, pulsed, high-power, low-power — there is an...
, 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 CCDCharge-coupled device
A charge-coupled device is a device for the movement of electrical charge, usually from within the device to an area where the charge can be manipulated, for example conversion into a digital value. This is achieved by "shifting" the signals between stages within the device one at a time...
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/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 lidarLIDAR
LIDAR is an optical remote sensing technology that can measure the distance to, or other properties of a target by illuminating the target with light, often using pulses from a laser...
s 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 anglesBeam divergence
The beam divergence of an electromagnetic beam is an angular measure of the increase in beam diameter or radius with distance from the optical aperture or antenna aperture from which the electromagnetic beam emerges. The term is relevant only in the "far field", away from any focus of the beam...
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
Beam parameter product
In laser science, the beam parameter product is the product of a laser beam's divergence angle and the radius of the beam at its narrowest point . The BPP quantifies the quality of a laser beam, and how well it can be focused to a small spot.A Gaussian beam has the lowest possible BPP,...
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.