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| In [[laser science]], the '''beam parameter product''' ('''BPP''') is the product of a [[laser]] beam's [[Beam divergence|divergence]] angle (half-angle) and the [[radius]] of the beam at its narrowest point (the [[beam waist]]).<ref name="RP">{{cite web |url=http://www.rp-photonics.com/beam_parameter_product.html |title=Beam parameter product |accessdate=2006-09-22 |work=Encyclopedia of Laser Physics and Technology | archiveurl= http://web.archive.org/web/20061018175856/http://www.rp-photonics.com/beam_parameter_product.html |publisher=RP Photonics |first=Rüdiger |last=Paschotta| archivedate= 18 October 2006 <!--DASHBot-->| deadurl= no}}</ref> The BPP quantifies the quality of a laser beam, and how well it can be focused to a small spot.
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| A [[Gaussian beam]] has the lowest possible BPP, <math>\lambda/\pi</math>, where <math>\lambda</math> is the [[wavelength]] of the light.<ref name="RP"/> The ratio of the BPP of an actual beam to that of an ideal Gaussian beam at the same wavelength is denoted '''M²''' ("'''[[M squared]]'''"). This parameter is a wavelength-independent measure of beam quality.
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| The quality of a beam is important for many applications. In [[fiber-optic communications]] beams with an M<sup>2</sup> close to 1 are required for coupling to [[single-mode optical fiber]]. Laser machine shops care a lot about the M<sup>2</sup> parameter of their lasers because the beams will focus to an area that is M<sup>2</sup> times larger than that of a Gaussian beam with the same wavelength and D4σ waist width; in other words, the [[fluence]] scales as 1/M<sup>2</sup>. The general rule of thumb is that M<sup>2</sup> increases as the laser power increases. It is difficult to obtain excellent beam quality and high average power (100 W to kWs) due to [[thermal lensing]] in the [[laser gain medium]].
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| == Measurement ==
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| There are several ways to define the width of a beam. When measuring the beam parameter product and M², one uses the [[Beam diameter#D4σ or second moment width|D4σ or "second moment" width]] of the beam to determine both the radius of the beam's waist and the divergence in the far field.<ref>A. E. Siegman, "[http://web.archive.org/web/20110604095354/http://www.stanford.edu/~siegman/beams_and_resonators/beam_quality_tutorial_osa.pdf How to (Maybe) Measure Laser Beam Quality]," Tutorial presentation at the Optical Society of America Annual Meeting, Long Beach, California, October 1997.</ref>
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| The BPP can be easily measured by placing an [[array detector]] or [[scanning-slit profiler]] at multiple positions within the beam after focusing it with a [[lens (optics)|lens]] of high optical quality and known [[focal length]]. To properly obtain the BPP and M² the following steps must be followed:<ref name="ISO11146-1">ISO 11146-1:2005(E), "Lasers and laser-related equipment — Test methods for laser beam widths, divergence angles and beam propagation ratios — Part 1: Stigmatic and simple astigmatic beams."</ref>
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| # Measure the D4σ widths at 5 axial positions near the beam waist (the location where the beam is narrowest).
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| # Measure the D4σ widths at 5 axial positions at least one [[Rayleigh length]] away from the waist.
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| # Fit the 10 measured data points to <math> \sigma^2(z) = \sigma_0^2 + M^4 \left(\frac{\lambda}{\pi\sigma_0}\right)^2(z-z_0)^2 </math>,<ref name="Siegman_p9">A. E. Siegman, "[http://web.archive.org/web/20110604095354/http://www.stanford.edu/~siegman/beams_and_resonators/beam_quality_tutorial_osa.pdf How to (Maybe) Measure Laser Beam Quality]," Tutorial presentation at the Optical Society of America Annual Meeting
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| Long Beach, California, October 1997, p.9. (Note that there is a typo in equation on page 3. Correct form comes from equations on page 9.)</ref> where <math> \sigma^2(z) </math> is the second moment of the distribution in the x or y direction (see section on D4σ beam width), and <math> z_0 </math> is the location of the beam waist with second moment width of <math> \sigma_0 </math>. Fitting the 10 data points yields M<sup>2</sup>, <math> z_0 </math>, and <math> \sigma_0 </math>. Siegman showed that all beam profiles — Gaussian, [[tophat beam|flat top]], [[Hermite-Gaussian mode|TEMxy]], or any shape — must follow the equation above provided that the beam radius uses the D4σ definition of the beam width. Using other definitions of beam width does not work.
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| In principle, one could use a single measurement at the waist to obtain the waist diameter, a single measurement in the far field to obtain the divergence, and then use these to calculate the BPP. The procedure above gives a more accurate result in practice, however.
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| ==See also==
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| *[[List of laser articles]]
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| ==References==
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| <references/>
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| ==Further reading==
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| *{{cite conference |first1=Zuolan |last1=Wang |first2=Simon |last2=Drovs |first3=Armin |last3=Segref |first4=Tobias |last4=Koenning |first5=Rajiv |last5=Pandey |url=http://www.dilas.com/gdresources/downloads/whitepapers/DILAS_PW11_7918-8_ZW.pdf |title=Fiber coupled diode laser beam parameter product calculation and Rules for optimized design |conference=SPIE Lase. Photonics West |year=2011 |location=San Francisco, CA, USA |others=Paper 7918-8}}
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| [[Category:Laser science]]
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