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*Taking the X and Y projections of the focused beam and fitting them with a Gaussian distribution,  

*Taking the X and Y projections of the focused beam and fitting them with a Gaussian distribution,we are able to attain <math>\sigma_{X} = 0.63mm\,</math> and <math>\sigma_{Y} = 0.23mm \,</math>.

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*Assuming a Gaussian distribution at the waist of the beam, we now find the FWHM (full width at half maximum) by the following relation,  

*Assuming a Gaussian distribution at the waist of the beam, we now find the FWHM (full width at half maximum) by the following relation,  

:<math>\mathrm{FWHM} = 2 \sqrt{2 \ln 2}\ \sigma. </math>  

:<math>\mathrm{FWHM} = 2 \sqrt{2 \ln 2}\ \sigma. </math>  

*The smallest value of <math>\mathrm{FWHM_{X}}</math> and <math>\mathrm{FWHM_{Y}}</math> were 1.49mm and 0.552mm respectively.

*The smallest values of <math>\mathrm{FWHM_{X}}</math> and <math>\mathrm{FWHM_{Y}}</math> were 1.49mm and 0.552mm respectively.

*The Rayleigh Length, <math>\mathrm{Z_{R}}</math> is defined as the distance from the beam waist along the axis of propagation to the point where its cross section is doubled, <math>\mathrm\omega_{R}</math>. Taking <math>\mathrm\omega_{0}</math> as the beam waist, and using the <math>\mathrm{FWHM}</math> as its value we are looking for the point where  

*The Rayleigh Length, <math>\mathrm{Z_{R}}</math> is defined as the distance from the beam waist along the axis of propagation to the point where its cross section is doubled (<math>\mathrm\omega_{R}</math>). This value represents the "play" we will have when trying to focus the beam onto the diamond target for ablation. Taking <math>\mathrm\omega_{0}</math> as the beam waist, and using the <math>\mathrm{FWHM}</math> as its value we are looking for the point where,

:<math>\omega_{R} = \sqrt{2}\ \omega_{0}. </math>

:<math>\omega_{R} = \sqrt{2}\ \omega_{0}. </math>

[[Image:rayleigh.png|center|thumb|400px|Describes the Rayleigh Length of a beam waist <math>\omega_{0}</math>.]]

[[Image:rayleigh.png|center|thumb|400px|Describes the Rayleigh Length of a beam waist <math>\omega_{0}</math>.]]

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