# Importance of Beam Diameter on Laser Damage Threshold

Laser-Induced Damage Threshold (LIDT) values specify whether an optical component can be safely used with a laser of a given power. But, laser optics sometimes fail when illuminated by lasers with powers below the specified LIDT because the laser beam used for LIDT testing had too small of a beam diameter relative to defect density.

## Modeling Laser Damage Threshold with Probability

Laser damage under nanosecond pulse lasers is generally caused by defects on the optic surface. The probability of finding any particular number of defects (n) within any given area of the sample surface is a function of the defect density (D) and follows the Poisson distribution:

**(1)**

P = |
e ^{D} D ^{n}2 |

The defect density is unit less because it is the product of beam area (A) and the number density of defects per unit surface area (δ) on the optic. The probability (P) of finding a region free of defects can be determined by solving for n = 0. The maximum fraction of undamaged test sites (assuming a flat-top beam and uniform fluence) is equal to the probability of that region on the surface being free of defects. The damage probability is its compliment.

**(2)**

*P*= 1 - e^{-D}Increasing defect density should increase the damage probability, but increasing the beam area has the identical effect. This allows the damage probability to be normalized by the beam radius.

**(3)**

P = 1 - [1 - P _{0} ] |
(ω) ^{2}ω_{0} |

P_{0} is the damage probability using the tested beam diameter (ω_{0}) and P is the expected damage probability given the true application beam size (ω). Normally, the LIDT test is conducted using a relatively small diameter beam (200μm minimum according to ISO-21254).

In the case of a Gaussian beam, fluence is not uniform and varies as a function of distance from the beam center. For a Gaussian beam and a defect population following a normal distribution, the damage probability is approximated using a Burr distribution - a continuous probability distribution for a random variable that is non-negative. The cumulative distribution function (CDF) can be graphed using the following equation where F is fluence, µ is the mean and σ is the standard deviation of the defect distribution:

**(4)**

*P*= 1 - [1 + (*F/μ*)^{σ-1}]^{-σD}## Scaling of LIDT Value with Beam Diameter

Laser beam diameter directly impacts the probability of damage during testing. When beam size is significantly larger than defect areal density (w2» δ), unlikely events are detectable. When the beam size is too small, low defect densities are not always detectable and parts may appear better than they actually are. The scaling of laser damage with beam diameter is demonstrated in Figure 2. In this scenario a large number of defects have a threshold fluence of 10J, and a small number (1%) have a threshold fluence of 1J. Scaling the beam diameter from 0.2 to 20mm will drastically change the damage probability function and therefore change the LIDT value that would be produced from this test. With the 0.2mm beam, the chance that one of the 1J threshold defects will be detected is small. For this reason, the damage probability will remain very low until after a fluence of 10J, the fluence equaling that of the most common defect. Increasing the beam size from 0.2 to 2mm makes it much more likely that the 1J threshold defects will be detected, causing a sharp increase in damage probability at a fluence of 1J. By scaling the beam diameter to 20mm, the damage probability at 1J increases to almost certain probability of damage.

This illustrates the importance of using a laser beam with a large enough diameter to adequately sample the surface of the optic being tested. Using too small of a beam in testing LIDT will result in an inaccurate LIDT specification and possible failure during realworld applications. Published LIDT values can be misleading if the beam diameter used for testing is not reported. Talk to your optics manufacturer about testing protocols and their statistical implications for your laser application.

**Figure 1:** Profile of a Gaussian beam.

**Figure 2:** Scaling of laser damage probability with beam size

### References

• ISO 21254-1:2011 – Lasers and laser-related equipment