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# Flat-top Beams

Author: the photonics expert

Definition: a light beam with a flat intensity profile

Alternative term: top-hat beams

More general terms: light beams, laser beams

A flat-top beam (or top-hat beam) is a light beam (often a transformed laser beam) having an intensity profile which is flat over most of the covered area. This is in contrast to Gaussian beams, for example, where the intensity smoothly decays from its maximum on the beam axis to zero. Such beam profiles are required for some laser applications. For example, one requires a constant intensity over some area in some techniques for the processing of semiconductor wafers and other materials. Also, nonlinear frequency conversion at very high power levels can be more efficient when performed with flat-top beams.

Typically, however, a flat-top beam profile still has some smooth edges, so that it can be approximated with a supergaussian profile, rather than a rectangular profile. A supergaussian intensity profile of order <$n$> is defined by the following equation:

$$I(r) = {I_{\rm{p}}}\;\exp \left[ { - 2{{\left( {\frac{r}{w}} \right)}^n}} \right]$$

The higher the order, the steeper are the edges of the profile.

Flat-top beams can in principle be spatially coherent, having smooth phase profiles. However, flat-top beams made in practice (e.g. with certain beam homogenizers, or from multimode step-index fibers) are often spatially incoherent, having rather complicated phase profiles. In such cases, a flat intensity profile is achieved only with a superposition of many spectral components, each of which may have a quite structured intensity profile.

Figure 2 shows the beam quality factor <$M^2$> of coherent supergaussian beams with flat phase front as a function of the supergaussian order. The more we approximate a rectangular profile with a high beam order, the worse is the beam quality; the <$M^2$> factor rises with no limit (and eventually we leave the regime where we can apply the paraxial approximation).

## Propagation of Coherent Flat-top Beams

Note that in contrast to a Gaussian beam, a flat-top beam is not a free-space mode. This means that during propagation in free space, the shape of the intensity profile will change. The steeper the edges of the intensity profile are, the more rapidly will such changes occur. Figure 3 shows a simulated example for an initially supergaussian beam profile with supergaussian order 8 and flat wavefronts.

The beam profile first contracts and then expands again, now getting smooth edges. Note that the color scale of each profile is adjusted such that the same color saturation is achieved on the beam axis; in reality, the intensity decreases for the expanding beam.

Of course, that change of beam profile may be negligible within the distance to the application. For beams with larger diameter and not too steep edges of the intensity profile, the beam size and shape may stay approximately constant.

## Generation of Flat-top Beams

In many cases, a flat-top beam is obtained by first generating a Gaussian beam from a laser and then transforming its intensity profile with a suitable optical element. There are different kinds of beam homogenizers to do that transformation, using different operation principles; some are based on diffractive optics. Different types of beam shapers can differ a lot concerning spatially coherent or incoherent beam profiles, length of the usable top-hat region, sensitivity to input beam parameters etc.

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## Questions and Comments from Users

2020-09-04

I am wondering how to calculate the peak intensity value of a supergaussian beam with power <$P$>.

If the beam profile is close to rectangular, just take the power divided by the beam area.

2021-05-26

I would like to know how to calculate the peak fluence of a laser beam which is in x direction a top-hat, and in y direction a Gaussian beam.

You can calculate that for arbitrary intensity profiles based on an integral of the optical intensity over the full beam area.

2021-06-16

Is the coupling efficiency from free space into an optical fiber higher if the beam has a flat top or Gaussian profile before it enters the fiber?

If it is a single-mode fiber, the input profile should fit the mode profile, which is usually closer to Gaussian. For multimode fibers, flat-top beams may also be good for reaching a high coupling efficiency, as long as the edges of the profile are not too steep, leading to excessive beam divergence.

2021-10-16

Any guidance on how to simulate the supergaussian propagation such as shown in Figure 3?

That has been done with a simple numerical beam propagation algorithm based on Fourier optics. The original beam profile is Fourier-transformed, each plane wave component acquires a certain phase shift according to the propagation distance, and back-transforming that gives you the field in the other plane.

2023-04-03

How can be it claimed that a laser beam with <$M^2$> = 1.6 is far more effective in terms of paint ablation than a flat-top beam, even if the latter has a much higher pulse energy?

This is hard to judge without knowing more about that flat-top beam, but maybe it is one with far higher <$M^2$> value, which is thus more difficult to focus down to the required spot size.

2023-11-01

Can the mathematical equations developed for Gaussian beams (for beam waist, divergence and Rayleigh range) be applied to flat-top beams? How can one compute such parameters for non-Gaussian beams like flat-top?

The equations for Gaussian beams cannot be applied here. The situation is more difficult for non-Gaussian beams – partly, because these generally do not have a conserved shape.

2023-11-20

Is it impossible to have a flat top beam that extends forever in a perfect vacuum? Will it “decay” to a Gaussian beam? If so, what is the physical mechanism behind this?