# Theoretical basis of the diagonal scan method for determining the laser ablation threshold for femtosecond vortex pulses

###### Abstract

In femtosecond laser micromachining, the ablation threshold is a key processing parameter that characterises the energy density required to cause ablation. Current techniques for measuring the ablation threshold such as the diameter regression and diagonal scan methods are based on the assumption of a Gaussian spatial profile, however no techniques currently exist for measuring the ablation threshold using a non-Gaussian beam shape.

Here we present a formalism of the diagonal scan method for determining the ablation threshold and pulse superposition for femtosecond vortex pulses. To the authors’ knowledge this is the first ablation threshold technique developed for pulses with non-Gaussian spatial profiles.

Using this method, the ablation threshold can be calculated using measurement of a single feature (the maximum damage radius ), which allows investigations of ablation threshold and incubation effects to be carried out quickly and easily. Extending this method to non-Gaussian beams will allow exploration of new avenues of research, enabling characterisation of the ablation threshold and incubation behaviour for a material when ablated with femtosecond vortex pulses.

## I Introduction

Femtosecond laser ablation is an advanced materials processing technique that enables microstructures to be fabricated in almost any material with very high accuracy and resolution Krueger04 ; Cheng13 . The ultrashort pulse duration leads to non-linear absorption, allowing materials to be ablated regardless of their linear absorption characteristics Perry99 . In addition, this ultrashort timescale limits energy transfer into the atomic lattice, greatly reducing heat effects, allowing materials to be ablated with little to no damage in the surrounding area Krueger97 . These advantages make femtosecond laser ablation an attractive prospect for industrial applications.

In all work involving femtosecond laser ablation, the ablation threshold (, in ) is a key parameter for characterising the interaction between laser and material, and is defined as the minimum energy density required to cause material removal. Therefore, it is of utmost importance that robust and useful methods of measuring the ablation threshold are available and widely applicable. Current methods for determining the ablation threshold are the diameter regression method Sanner09 and the diagonal scan method Samad06 . Both of these methods rely on the assumption that ablation is carried out using a laser beam with Gaussian spatial distribution. With rapid advances being made in the area of spatial beam shaping however, this assumption is not always valid. In particular, optical vortex beams have been investigated recently, indicating generation of different nanostructures to those obtained when using a Gaussian beam Hnatovsky10 ; Hnatovsky12 ; Anoop14a ; Anoop14b .

In this work we present a formalism of the diagonal scan method for measuring the femtosecond laser ablation threshold that is applicable to vortex beams. We derive an expression for calculating the ablation threshold based on measurement of the maximum damage radius, and also demonstrate a method for calculating the pulse superposition obtained during a diagonal scan experiment.

## Ii Damage Radius

To determine an expression for the ablation threshold, we consider a diagonal scan experiment where a sample is translated diagonally through the focal point of a focused laser beam, as shown in Figure 1.

The spatial fluence distribution of an optical vortex beam is given by Hnatovsky10 :

(1) |

where: is the topological charge, is the radial direction (in ), and is the radius of a Gaussian beam (in ) at , which is equal to:

(2) |

where is the propogation direction, is the wavelength, and is the beam waist (all in ).

(3) |

At Equation 1 simplifies to the equation for a Gaussian beam:

(4) |

We can set the damage radius to be equal to the radius at which the damage threshold is exceeded:

(5) |

(6) |

(7) |

We can set:

(8) |

(9) |

such that:

(10) |

(11) |

This is an equation of the form , which can be solved according to the Lambert Omega function Corless96 by .

(12) |

(13) |

The Lambert Omega function is multivalued (except at zero), therefore for any two radii can be defined, the inner and outer damage radius, which can be calculated using the principal and non-principal branches of the Lambert Omega function:

(14) |

(15) |

Equations 14 and 15 describe the damage radius as a function of the propogation direction , and are shown in Figure 2. For application in a diagonal scan ablation threshold measurement technique, we are only interested in the outer damage radius . For this reason we do not consider the inner damage radius further, and all references to refer to the outer damage radius .

For real numbers, the non-principal branch of the Lambert Omega function has the limits:

(16) |

The implications of this limit will be discussed further in Section V.

## Iii Finding the Maxima

To find we must set . Let:

(17) |

(18) |

(19) |

(20) |

(21) |

(22) |

(23) |

By applying the chain rule to these definitions, we can obtain an expression for

(24) |

(25) |

(26) |

(27) |

Defining these derivatives:

(28) |

(29) |

(30) |

(31) |

(32) |

(33) |

We now evaluate the derivatives. From equations 27, 28 and 29:

(34) |

(35) |

(36) |

Let:

(37) |

(38) |

From equations 26, 28, 31 and 33:

(39) |

(40) |

(41) |

(42) |

(43) |

(44) |

From equations 19, 23, 25, 38 and 44:

(45) |

(46) |

(47) |

(48) |

(49) |

(50) |

(51) |

(52) |

To find the maxima we set and rearrange for z, therefore:

(53) |

(54) |

(55) |

We now apply the definition of the Lambert Omega function once more, to carry out the reverse of the transform done previously (equations 11 and 12):

(56) |

Substituting in equation 21:

(57) |

The -value where the damage radius reaches its maximum is denoted , where for , therefore:

(58) |

(59) |

(60) |

(61) |

(62) |

This expression has a similar form to the equivalent expression below for a Gaussian beam Samad06 , and simplifies to this for .

(63) |

## Iv Isolating the Damage Threshold

To isolate the damage threshold, we substitute (equation 62) as the -value into the expression for damage radius (equation 15). Values of and are replicated from equations 8 and 9, with equation 3 substituted in for .

(64) |

(65) |

(66) |

When , , therefore:

(67) |

(68) |

(69) |

(70) |

We now need to substitute equation 62 into equation 70 in place of . Considering the left hand side only:

(71) |

(72) |

(73) |

(74) |

Now considering the right hand side:

(75) |

(76) |

(77) |

(78) |

(79) |

Setting :

(80) |

(81) |

This equation gives the damage threshold as a function of the maximum damage radius , for a given power and vortex charge , allowing calculation of the ablation threshold from measurement of .

Simplifications of this formula for vortex beams of order are shown below:

(82) |

(83) |

(84) |

(85) |

(86) |

These have a similar form to the equivalent expression for a Gaussian beam Samad06 :

(87) |

## V Limitations of the Lambert Omega Function

The non-principal branch of the Lambert Omega function used to calculate , is defined for real variables only when . Therefore to find the limits (in the -direction) of the solution defined in equation 15 we set:

(88) |

From the definitions of and in equations 8 and 9, it is clear that the inequality is true, as , , and are physical parameters with positive, real values. Now we consider only the inequality . Substituting in equations 8 and 9 for and we get: