Laser beam quality

<h2 id="m2-definitions">M2 definitions</h2>
Main article: <a href="/facts/M_squared/tjjRPSnN">M squared</a>

The <a href="/facts/Equation/pxFzLAVR">equation</a> for the <a href="/facts/Beam_divergence/7EQX7tY0">divergence</a> of a pure <a href="/facts/Gaussian_beam/Hvo1wL29">Gaussian</a> TEM00 unfocused beam propagating through space is given by

Θ
 
 00
 
 
 =
 
 
 
 4
 λ
 
 
 π
 
 D
 
 00
 
 
 
 
 
 
 
 {\displaystyle \Theta _{00}={4\lambda \over \pi D_{00}}}
 
, (1)
where D00 is the diameter of the <a href="/facts/Beam_waist/Hvo1wL29">beam waist</a>, and λ is the wavelength. Higher mode beams often start with a larger beam waist, D0, and/or have a faster divergence Θ0. In this case Equation (1) becomes

Θ
 
 0
 
 
 =
 
 M
 
 2
 
 
 
 
 
 4
 λ
 
 
 π
 
 D
 
 0
 
 
 
 
 
 
 
 {\displaystyle \Theta _{0}=M^{2}{4\lambda \over \pi D_{0}}}
 
, (2)
where Θ0 and D0 are the divergence and waist of a higher mode beam and M2 is greater than 1 and is named the "Beam Propagation <a href="/facts/Ratio/mFmhQt7T">Ratio</a>" per the ISO 11146 standard. When a Gaussian laser beam is focused, the focused spot <a href="/facts/Beam_diameter/rxCGx8q8">diameter</a> is defined by

d
 
 00
 
 
 =
 
 
 
 4
 λ
 f
 
 
 π
 
 D
 
 00
 
 
 
 
 
 
 
 {\displaystyle d_{00}={4\lambda f \over \pi D_{00}}}
 
, (3)
where d00 is the ideal focused spot diameter, f is the <a href="/facts/Focal_length/brGIUkEY">focal length</a> of the focusing lens, and D00 is the input beam waist and is placed one focal length from the lens as shown in the figure. However, when a <a href="/facts/Transverse_mode/nCIl5D6a">multimode</a> beam is focused, Equation (3) becomes

d
 
 0
 
 
 =
 
 M
 
 2
 
 
 
 
 
 4
 λ
 f
 
 
 π
 
 D
 
 0
 
 
 
 
 
 
 
 {\displaystyle d_{0}=M^{2}{4\lambda f \over \pi D_{0}}}
 
. (4)
<h2 id="m2-measurement">M2 measurement</h2>
M2 cannot be determined from a single beam profile measurement. The ISO/DIS 11146 define that M2 should be calculated from a series of measurements as shown in the figure below.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> M2 is measured on real beams by focusing the beam with a fixed position lens of known focal length, and then measuring the characteristics of the beam waist and divergence. These measurements can be taken with a <a href="/facts/Laser_beam_profiler/Nmc8GDyj">laser beam profiler</a>.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>

The multiple measurements ensure that the minimum beam diameter is found and enable a "curve fit" that improves the accuracy of the calculation by minimizing measurement error.

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Laser Beam Quality Metrics by T. Sean Ross <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Marshall, Larry, "Applications a la mode," Laser Focus, April 1971 <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Siegman, Anthony E. (February 5, 1993). Bhowmik, Anup (ed.). "Defining, measuring, and optimizing laser beam quality". Proc. SPIE 1868, Laser Resonators and Coherent Optics: Modeling, Technology, and Applications. Laser Resonators and Coherent Optics: Modeling, Technology, and Applications. 2: 2. Bibcode:1993SPIE.1868....2S. doi:10.1117/12.150601. S2CID 73623358. <a href="http://proceedings.spiedigitallibrary.org/proceeding.aspx?articleid=1006401" target="_blank">http://proceedings.spiedigitallibrary.org/proceeding.aspx?articleid=1006401</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Siegman, A. E. (October 1997). "How to (Maybe) Measure Laser Beam Quality". CiteSeerX 10.1.1.177.3400. Tutorial presentation at the Optical Society of America Annual Meeting, Long Beach, California <a href="/wiki/CiteSeerX_(identifier)" target="_blank">/wiki/CiteSeerX_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Borghi, R.; Santarsiero, M. (March 1, 1997). "M2 factor of Bessel–Gauss beams" (PDF). Optics Letters. 22 (5): 262–264. Bibcode:1997OptL...22..262B. doi:10.1364/ol.22.000262. PMID 18183169. <a href="http://webusers.fis.uniroma3.it/~ottica/sant/pubs/Max035.pdf" target="_blank">http://webusers.fis.uniroma3.it/~ottica/sant/pubs/Max035.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Fry, Constance L.; Faulkner., Alan R. (eds.). Current Concepts in Aesthetic and Reconstructive Oculoplastic Surgery. <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">ISO 11146:2005(E), "Lasers and laser-related equipment — Test methods for laser beam widths, divergence angles and beam propagation ratios". <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">G. Langer et al., A webcam in Bayer-mode as a light beam profiler for the near infra-red, Optics and Lasers in Engineering 51 (2013) 571–575 <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
</ol>

Laser beam quality open-in-new

Laser beam quality