M^2 factor (original) (raw)

Definition: a parameter for quantifying the beam quality of laser beams

Alternative term: beam quality factor

Categories: general optics, optical metrology

• beam properties
  • beam divergence
  • beam parameter product
  • beam pointing fluctuations
  • beam quality
  • beam radius
  • beam waist
  • diameter–divergence product
  • _M_2 factor

Related: laser beam characterizationbeam qualitybeam parameter productbeam divergenceradiancebrightnessGaussian beamsBeam Quality Measurements Can Easily Go Wrong

Units: (dimensionless)

Formula symbol: ($M^2$)

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Contents

Definition of _M_2 Factor

The ($M^2$) factor (M squared factor), also called beam quality factor or beam propagation factor, is a common measure of the beam quality of a laser beam. According to ISO Standard 11146 [6], it is defined as the beam parameter product divided by ($\lambda / \pi$), the latter being the beam parameter product for a diffraction-limited Gaussian beam with the same wavelength. In other words, the half-angle beam divergence is \theta = {M^2}\frac{\lambda }{{\pi {w_0}}}$$

where ($w_0$) is the beam radius at the beam waist and ($\lambda$) the wavelength in the medium (e.g. air). A laser beam is often said to be “($M^2$) times diffraction-limited”.

_M_2 Factor of Hermite–Gaussian Beams

A diffraction-limited beam has an ($M^2$) factor of 1, and is a Gaussian beam. Smaller values of ($M^2$) are physically not possible. A Hermite–Gaussian beam, related to a TEMnm resonator mode, has an ($M^2$) factor of ($(2n + 1)$) in the ($x$) direction, and ($(2m + 1)$) in the ($y$) direction [1].

Focusability of a Beam

The ($M^2$) factor of a laser beam limits the degree to which the beam can be focused for a given beam divergence angle, which is often limited by the numerical aperture of the focusing lens. Together with the optical power, the beam quality factor determines the brightness (more precisely, the radiance) of a laser beam.

_M_2 Factor for Elliptical Beams

For beams that are not circularly symmetric, the ($M^2$) factor can be different for two directions orthogonal to the beam axis and to each other. This is particularly the case for the output of diode bars, where the ($M^2$) factor is fairly low for the fast axis and much higher for the slow axis.

ISO Standard 11146

According to ISO Standard 11146 [6], the ($M^2$) factor can be calculated from the measured evolution of the beam radius along the propagation direction (i.e. from the so-called caustic). See the article on beam quality for more details. A number of rules have to be observed, e.g. concerning the exact definition of the beam radius and details of the fitting procedure. Alternative methods are based on wavefront sensors, e.g. Shack–Hartmann wavefront sensors, which require the characterization of the beam only in a single plane.

Note that the ($M^2$) factor, being a single number, cannot be considered as a complete characterization of beam quality. The actual quality of a beam for a certain application can depend on details which are not captured with such a single number.

The concept of the ($M^2$) factor not only allows one to quantify the beam quality with a single number, but also to predict the evolution of the beam radius with a technically very simple extension of the Gaussian beam analysis: one simply has to replace the wavelength with ($M^2$) times the wavelength in all equations. This is very convenient for, e.g., designing the pump optics of diode-pumped lasers. Note, however, that this method works only when the D4σ method for obtaining the beam radius is used, which is suitable also for non-Gaussian beam shapes; see again ISO Standard 11146 [6] for details.

Errors in _M_2 Measurements

Unfortunately, essential details of the ISO 11146 standard are often not observed in ($M^2$) measurements, with the result that wrong ($M^2$) values are obtained or even published. Some frequently made mistakes in measurements based on the beam caustic are explained in the following:

• The beam radius is measured with a simple criterion, not based on the full intensity profile, although the beam profiles are not all close to Gaussian. Only for nearly Gaussian beam shapes, such simple measurement methods are allowed. For others, the D4σ method based on the second moment of the intensity distribution must be used.
• The beam is focused too tightly, so that the beam waist is too small to measure its beam radius precisely. For example, a CCD camera has a limited spatial resolution; it cannot be used for precise measurements if the beam diameter corresponds only to a few pixels.
• Background subtraction, a sensitive issue for the second-moment method, is not correctly done. Camera images can exhibit some background intensity level, which may either really belong to the laser beam (and should not be removed then) or is an artifact which must be removed. If such a background results from ambient light, the most reliable measure is to switch this off, or to carefully shield it with a black tube in front of the camera. (Subtracting a fixed level for all images is problematic since ambient light levels may change, e.g. when somebody moves in the room.) The background issue is particularly serious when the beam size is only a fraction of the camera's sensitive area.
• The beam intensity on a camera is too high or too low. If it is too high, the center pixels may be saturated, so that the beam intensity at the center is underestimated and the measured beam radius is too large. For too low intensities, intensity background issues may become more severe.
• The beam radii are not measured sufficiently far from the focus. In order to properly judge the beam divergence, the ISO 11146 standard demands that about half of the measurement points must be more than two effective Rayleigh lengths away from the beam focus (whereas the other half of the points is close to the focus, i.e., within one Rayleigh length). This may be difficult in practice when the beam waist is made relatively large, leading to a long Rayleigh length and correspondingly large space requirements for a correct measurement.

When different instruments deliver different ($M^2$) values, this may easily be caused by such errors, rather than by the instruments themselves.

Calculation of _M_2 Factor From Complex Field Distribution in a Plane

If the complex field distribution of a monochromatic field is known in one plane perpendicular to the beam direction, the field distribution in any other plane can be computed numerically, and the ($M^2$) could be obtained from that. As a technically simpler solution, one can directly compute ($M^2$) from the field distribution in one plane based on a few integrals [3].

Frequently Asked Questions

What is the ($M^2$) factor?

What is the ($M^2$) value of an ideal laser beam?

A perfectly diffraction-limited beam, which has a Gaussian intensity profile, has an ($M^2$) factor of 1. ($M^2$) values smaller than 1 are not physically possible.

How does the ($M^2$) factor affect focusing a laser beam?

The ($M^2$) factor limits how tightly a laser beam can be focused. For a given focusing lens and wavelength, a beam with a higher ($M^2$) value will result in a larger spot size at the focus and a lower brightness.

How is the ($M^2$) factor of a laser beam measured?

According to the ISO 11146 standard, the ($M^2$) factor is typically determined by measuring the beam radius at multiple positions along the beam's propagation path, particularly around the beam waist. This measurement of the beam's caustic allows for the calculation of ($M^2$).

Can the ($M^2$) factor be different for the horizontal and vertical directions?

Yes, for beams that are not circularly symmetric, the ($M^2$) factor can have different values for two orthogonal directions. This is common for diode bars, which have a low ($M^2$) for the fast axis and a much higher ($M^2$) for the slow axis.

How can one predict the propagation of a non-ideal laser beam?

The evolution of the beam radius for a beam with a known ($M^2$) factor can be predicted using the standard equations for Gaussian beam propagation. One simply has to replace the wavelength ($\lambda$) with the term ($M^2 \cdot \lambda$).

What are common errors in ($M^2$) measurements?

Common mistakes include using an incorrect definition for the beam radius for non-Gaussian beams, focusing the beam too tightly for the detector's resolution, improper background subtraction, and saturating the detector.

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Bibliography

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