Beam propagation and quality factors: A primer

Characterizing a laser system may seem like a daunting task, especially when just the laser source itself has many influencing factors, such as power stability, coherence, and wavelength, that govern which application it is most suitable for. However, there are a few key parameters concerning the beam profile itself that provide some insight into the overall system performance, whether it is a Gaussian beam in metrology, or the higher-order modes often produced by a diode laser system or by high-power materials-processing lasers.

In a single plane perpendicular to the propagation axis, a laser beam has a beam width, a radius, and a spatial intensity distribution. From the near field to the far field, each of these values will change. How much they change will affect the brightness of the beam and its ability to be focused to a spot.

This article will discuss beam propagation and quality factors, including the beam quality (M²) factor, beam parameter product (BPP), power in the bucket (PIB), and Strehl ratio, as well as which laser systems they are most suited for. Each beam-quality metric has its advantages and disadvantages, and as a result the correct metric for an individual use case is highly application-dependent. The ISO 11146 standard will be referenced here, as it provides an excellent starting place for definitions and test methods covering laser beam propagation.

M² factor

For laser beams for which the second-order moments of intensity (beam width) and divergence can be determined, the beam-propagation ratio M² is often used to characterize beam quality. The M² factor compares the actual shape of the beam to that of an ideal Gaussian beam.

The ISO Standard 11146¹ defines the M² factor as:

In Equation 1, w₀ is the beam waist, θ is the divergence angle of the laser, and λ is the lasing wavelength (see Fig. 1).

The divergence angle of a Gaussian beam is determined by Equation 2:

The resulting divergence angle can be inserted into Equation 2 to simplify the calculation of the M² factor of a Gaussian beam:

An M² factor of 1 corresponds to a diffraction-limited Gaussian beam, and M² factors larger than 1 correspond to a beam that deviates from the ideal Gaussian beam. It is impossible to achieve a value less than 1. Helium-neon gas lasers typically have an M² factor somewhere between 1 and 1.1.

It is important to know the M² factor of a laser beam for any application in which it is focused to a spot. For a given divergence angle, the M² factor can be a measure of how effectively the beam can be focused. The lower the M² factor, the tighter the focus and the more intense a beam spot can be achieved.

The Rayleigh range can also be shown to decrease with increasing M² factor:

Measuring M² is more involved than simply taking a measurement of the beam profile at a single plane along the propagation axis. According to the ISO 11146 standard, five separate beam-radius measurements must be recorded at various positions in both the near field and the far field along the propagation axis. The test equipment for the measurement, which may include a camera, moving pinhole, moving knife edge, moving slit, and/or variable aperture, is dependent on the beam itself, and ISO 11146 should be referenced for the best method to be used.

Even though the cross-section at a specific plane may look like a perfect Gaussian distribution, the beam may propagate very differently than a Gaussian beam and in that case will have a greater divergence angle. Figure 2 shows why multiple radius measurements at different planes are needed to calculate M².

FIGURE 2. This cross section of a laser beam doesn’t contain any of the Gaussian propagation mode TEM00, but the combination of other modes makes the profile appear to be Gaussian. While M2 is a standardized measurement and useful for tracing the beam as it propagates through complex optics, it is limited to second-moment measurements of near-Gaussian beams (single-mode, multimode, partially coherent, etc.). M2 is not an appropriate beam-quality metric for non-Gaussian beams, such as top hats, and is susceptible to detector background noise or nonlinearities.

One disadvantage of this metric is that it weighs the contributions from the “wings” (the low-power-density regions further away from the center of the beam) more than other methods, occasionally making the M² factor more useful in academic settings than industrial applications. Described below are a couple of other methods that can provide additional information about the beam quality and its propagation characteristics.

Beam-parameter product

The beam-parameter product (BPP) is another parameter often used to characterize the quality of a laser beam. According to the ISO 11146 standard, the BPP is given by the product of the beam radius (measured at the beam waist) and the half-angle beam divergence (measured in the far field). BPP is typically reported in units of mm-mrad, and it can be related to the M² factor by:

Because the M² factor is directly proportional to the BPP, a larger BPP indicates a beam of worse quality that is less similar to an ideal Gaussian beam. The minimum achievable BPP value is λ/π, which corresponds to an ideal Gaussian beam. For example, the minimum possible BPP of a beam with a 1064 nm wavelength is about 0.339 mm-mrad.

BPP is often used for multimode semiconductor or fiber lasers with large M² factors. Typical BPP values for high-power commercial products are 3 to 10 mm-rad.² Engineers looking for a simplified calculation to evaluate diode-laser fiber-coupled designs often look to BPP to help determine the amount of light that can be coupled into the fiber.³

Power in the bucket

Power in the bucket (PIB) is another parameter used for describing beam propagation and quality, and is commonly used in materials-processing and other high-power laser applications, as PIB has a direct bearing on the ability to form a small and intense spot on an object or a material being processed.

PIB refers to the fraction of total power that can be delivered into a cone, measured in angular units λ/D and called a “bucket,” where λ is the laser wavelength and D is a characteristic size of the initial beam. While there is no industry standard for defining and measuring PIB, the High Energy Laser Joint Tech office uses a two hard aperture geometry for PIB measurement, as shown in Figure 3.⁴

FIGURE 3. In this typical laboratory setup for measuring PIB in a two-hard-aperture configuration (a), the opening of the far-field aperture is varied to produce a curve of detected power vs. the aperture angular radius in units of λ/D (b).

A lower PIB value corresponds with a higher-quality beam, just as for the M² factor and BPP.

A laser beam’s vertical beam quality⁵ is given by the square root of the ratio of the fraction of power in the bucket of a perfect Gaussian beam. This ratio is unique for a given λ/D. The vertical beam quality corresponds to the plot’s vertical dimension and is most meaningful when the beam is nearly diffraction-limited.

The beam’s horizontal beam quality is given by the ratio of the λ/D value of a perfect Gaussian beam to that of the real beam at a given fraction of power in the bucket. This corresponds to the horizontal dimension of the plot and is useful when the beam is significantly worse than diffraction-limited.

PIB curves, such as in Figure 4, clearly display the amount of total beam power contained in the central beam lobe and what fraction of the beam is distributed in a weak but large area extending outside of the main portion of the beam. This can be more informative and useful than a complete beam-intensity profile.

Simply measuring several points along a PIB curve gives a substantially improved understanding of a laser beam’s quality of a laser beam.⁶

Strehl ratio

Similarly to how the M² factor allows the performance of a real laser to be compared to that of an ideal beam, the Strehl ratio of an optical system or component allows the optics’ true performance to be compared with that of an ideal version. Historically used in the astronomy community thanks to the work of Karl Strehl in 1894 and revisited by Virendra Mahajan in 1983,⁷ the Strehl ratio of an optic is approximately related to its root-mean-squared (RMS) transmitted wavefront error by Equation 8, where S is the optic’s Strehl ratio and σ is the optic’s RMS wavefront error in waves. This equation is valid for RMS wavefront error values less than or equal to λ/5.

Strehl ratio is often used for characterizing the performance of aspheric lenses. As measured by the Strehl ratio, one can attempt to quantify how much the wavefront error of a lens reduces the peak power of a focused spot. Physically, the lens wavefront error redistributes the power into the outer regions of the spot, thus reducing the peak value from its ideal diffraction limit (see Fig. 4).

FIGURE 4. The Strehl ratio of a lens is given by the ratio of the peak power of its focused spot to that of an ideal, diffraction-limited lens.

Strehl Ratio is also dependent on the f-number of the lens. In general, faster lenses, or lenses with small f-numbers, are more sensitive to wavefront error and thus can lead to more performance degradation.

For a Gaussian or aberrated-Gaussian beam,⁸ we can define the Strehl equivalent M² for the wavefront error of the lens or mirror as:⁹

For systems where thermal distortion is not a concern, the M² factors for the laser and the optics can be combined to produce an M² factor for the total system.

10.

One takeaway from this is that for high-power lasers with poor M² values, the effort to implement diffraction-limited optics is less critical, as small amounts of lens wavefront error will not significantly affect the beam divergence and image size.

Whether you are designing a laser system for fluorescence microscopy, materials processing, or rangefinding, it is important to understand your beam at the source and at the target. The factors and ratios described herein will help you on your way to an improved understanding of your overall laser system performance.