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M squared
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In laser science, the parameter M2, also known as the beam propagation ratio or beam quality factor is a measure of laser beam quality. It represents the degree of variation of a beam from an ideal Gaussian beam.[1] It is calculated from the ratio of the beam parameter product (BPP) of the beam to that of a Gaussian beam with the same wavelength. It relates the beam divergence of a laser beam to the minimum focussed spot size that can be achieved. For a single mode TEM00 (Gaussian) laser beam, M2 is exactly one. Unlike the beam parameter product, M2 is unitless and does not vary with wavelength.
The M2 value for a laser beam is widely used in the laser industry as a specification, and its method of measurement is regulated as an ISO standard.[2]
Measurement
[edit]
There are several ways to define the width of a beam. When measuring the beam parameter product and M2, one uses the D4σ or "second moment" width of the beam to determine both the radius of the beam's waist and the divergence in the far field.[1]
M2 can be measured by placing an array detector or scanning-slit profiler at multiple positions within the beam after focusing it with a lens of high optical quality and known focal length. To properly obtain M2, the following steps must be followed:[3]
- Measure the D4σ widths at 5 axial positions near the beam waist (the location where the beam is narrowest).
- Measure the D4σ widths at 5 axial positions at least one Rayleigh length away from the waist.
- Fit the 10 measured data points to where is half of the beam width, and is distance in the direction of beam propagation, with the location of the beam waist with width .[4] Fitting the 10 data points yields M2, , and . Siegman showed that all beam profiles — Gaussian, flat-top, TEMxy, or any shape — must follow the equation above provided that the beam radius uses the D4σ definition of the beam width.[1]
Using other definitions of beam width gives results that may be inaccurate for some beam profiles. In practice, one could use a single measurement at the waist to obtain the waist diameter, a single measurement in the far field to obtain the divergence, and then use these to calculate the M2. The procedure above gives a more accurate result, however.
Utility
[edit]M2 is useful because it reflects how well a collimated laser beam can be focused to a small spot, or how well a divergent laser source can be collimated. It is a better guide to beam quality than Gaussian appearance because there are cases in which a beam can look Gaussian, yet have an M2 value far from unity.[1] Likewise, a beam intensity profile can appear very "un-Gaussian", yet have an M2 value close to unity.
The quality of a beam is important for many applications. In fiber-optic communications beams with an M2 close to 1 are required for coupling to single-mode optical fiber.
M2 determines how tightly a collimated beam of a given diameter can be focused: the diameter of the focal spot varies as M2,[5] and the irradiance scales as 1/M4. For a given laser cavity, the output beam diameter (collimated or focused) scales as M, and the irradiance as 1/M2. This is very important in laser machining and laser welding, which depend on high fluence at the weld location.
Generally, M2 increases as a laser's output power increases. It is difficult to obtain excellent beam quality and high average power at the same time due to thermal lensing in the laser gain medium.
Embedded Gaussian
[edit]Real laser beams are often non-Gaussian, being multi-mode or mixed-mode. Multi-mode beam propagation can be modeled by considering an imaginary "embedded" Gaussian, whose beam waist is M times smaller than that of the multimode beam. The diameter of the multimode beam is then M times that of the embedded Gaussian beam everywhere, and the divergence is M times greater,[1] but the wavefront curvature is the same. The multimode beam has M2 times the beam area but 1/M2 less beam intensity than the embedded beam. This holds true for any given optical system, and thus the minimum (focussed) spot size or beam waist of a multi-mode laser beam is M times that of the embedded Gaussian beam waist.
See also
[edit]References
[edit]- ^ a b c d e Siegman, A. E. (October 1997). "How to (Maybe) Measure Laser Beam Quality" (PDF). Archived from the original (PDF) on June 4, 2011. Retrieved Feb 8, 2009. Tutorial presentation at the Optical Society of America Annual Meeting, Long Beach, California. See also A. E. Siegman, “Defining, measuring, and optimizing laser beam quality”, Proc. SPIE 1868, 2 (1993); https://doi.org/10.1117/12.150601
- ^ "Lasers and laser-related equipment – Test methods for laser beam widths, divergence angles and beam propagation ratios". ISO Standard. 11146. 2005.
{{cite journal}}: Cite journal requires|journal=(help) - ^ 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".
- ^ See Siegman (1997), p. 9. There is a typo in the equation on page 3. Correct form comes from equations on page 9.
- ^ Thériault, Gabrielle (September 08, 2023). "The beginner’s guide to laser beam quality and M2 measurement", Gentec-EO. Retrieved 9 May 2025.
M squared
View on Grokipediafrom Grokipedia
M², also known as the beam quality factor or beam propagation factor, is a dimensionless parameter in laser optics that quantifies the quality of a laser beam by comparing its divergence and focusability to those of an ideal diffraction-limited Gaussian beam of the same wavelength.[1][2] For a perfect single-mode TEM00 Gaussian beam, M² equals 1, indicating the theoretical minimum divergence and spot size; values greater than 1 signify deviations due to multimode content, aberrations, or other imperfections that increase beam spread.[3][1]
The parameter is formally defined by the International Organization for Standardization (ISO) in standard 11146 as the ratio of the actual beam parameter product—defined as the product of the beam waist radius w0 and the far-field divergence half-angle θ—to that of an ideal Gaussian beam.[1][2] Mathematically, this is expressed as M² = (π w0 θ) / λ, where λ is the wavelength of the light.[3] This formulation allows M² to predict beam behavior throughout propagation, with the beam radius at any position z given by w(z) = w0 √[1 + (M² z λ / (π w0²))²], extending the Gaussian beam propagation equation to real beams.[1][3]
Measurement of M² follows the ISO 11146 protocol, which requires profiling the beam at multiple positions along its propagation path—typically five locations in the near field, waist, and far field—using techniques such as scanning-slit or knife-edge profilers to determine width, waist location, and divergence.[2][1] Commercial systems automate this process by focusing the beam with a lens and scanning the profile over a range of propagation distances, often yielding results in seconds while ensuring accuracy for beams from continuous-wave to pulsed lasers.[3][2]
The significance of M² lies in its direct impact on laser performance in practical applications, as lower values enable tighter focusing for higher power density, reduced divergence for efficient beam delivery over distance, and better coupling into optical fibers or resonators.[1][3] In fields like materials processing, where focused spot size determines cutting or welding precision, or in medical lasers requiring minimal thermal spread, optimizing M² is essential for maximizing effective power and system efficiency.[3] It also serves as a key specification for laser manufacturers and system integrators to verify compliance with design goals and predict integration challenges.[3]
where P is the total optical power and λ is the wavelength; this relation shows that brightness scales inversely with M⁴, imposing a fundamental limit on the intensity that can be concentrated in the focal spot regardless of focusing optics.[6] Conceptually, the M² factor draws an analogy to light propagation in multimode optical fibers, where higher values correspond to the excitation and mixing of multiple transverse modes, increasing the effective numerical aperture and mimicking the reduced coherence and higher divergence observed in multimode fiber outputs compared to single-mode counterparts.[7]
Fundamentals
Definition
The beam quality factor , also known as the beam propagation factor, is a dimensionless parameter that quantifies the quality of a laser beam by measuring its deviation from the diffraction-limited performance of an ideal Gaussian beam.[4] It provides a single, propagation-invariant metric to characterize how effectively a beam can be focused and its overall coherence, with values closer to unity indicating higher quality.[4] For a fundamental Gaussian mode, which represents the ideal transverse electromagnetic mode with the lowest divergence, .[5] In contrast, real laser beams typically have due to contributions from higher-order multimodes, phase aberrations, or thermal distortions that degrade the beam's spatial profile and increase divergence.[5] The concept originated in the early 1990s, introduced by Anthony E. Siegman as a standardized, invariant measure of beam quality that remains constant through linear optical systems, building on earlier work in laser resonator theory.[4] A fundamental relation defining is where is the beam waist radius, is the far-field half-angle divergence, and is the wavelength in the propagation medium.[5] This expression scales the product of the beam's minimum width and divergence relative to the diffraction limit for a Gaussian beam.[5]Physical Interpretation
The M² factor provides a measure of how closely a laser beam approximates the ideal diffraction-limited behavior of a fundamental Gaussian mode, with values greater than 1 indicating deviations due to higher-order modes or aberrations that degrade performance. Physically, an M² > 1 results in a larger beam size at the focus and increased far-field divergence compared to an ideal Gaussian beam of the same input waist size and power, leading to a corresponding reduction in peak intensity and overall brightness. This degradation arises because the beam parameter product (waist radius times divergence angle) scales linearly with M², effectively spreading the energy over a larger phase-space volume.[5] For instance, a beam with M² = 2 will produce a focused spot whose area is four times that of an ideal Gaussian beam when propagated through the same focusing optics, quartering the achievable intensity for a given power level. Such effects limit the beam's utility in applications demanding high spatial resolution, as the excess divergence causes faster beam expansion over distance, reducing the effective range for maintaining collimation.[5] The impact on brightness further underscores these limitations, as M² governs an invariant quantity conserved through lossless paraxial optics. The beam brightness B, representing the maximum radiance, is expressed aswhere P is the total optical power and λ is the wavelength; this relation shows that brightness scales inversely with M⁴, imposing a fundamental limit on the intensity that can be concentrated in the focal spot regardless of focusing optics.[6] Conceptually, the M² factor draws an analogy to light propagation in multimode optical fibers, where higher values correspond to the excitation and mixing of multiple transverse modes, increasing the effective numerical aperture and mimicking the reduced coherence and higher divergence observed in multimode fiber outputs compared to single-mode counterparts.[7]