Appendix H Intensity and Power of Gaussian Beams H.1 Peak Intensity In order to determine the peak intensity of a beam from the total power in that beam, consider a elliptical Gaussian beam with an intensity profile given by, 2 2 I = I0 e−(x/a) e−(y/b) , (H.1) where, a is the long axis and b is the short axis. Make the substitutions x a du du = a u= =⇒ y , b dv d= . b : v= : (H.2) (H.3) The total power in the beam, P , is given by, P = !∞ !∞ I dx dy , = !∞ !∞ I0 a b e−(u 0 0 (H.4) 0 2 +v 2 ) du dv . (H.5) 0 In order to integrate equation H.5 , make the substitution r2 = u2 + v 2 and switch to circular co-ordinates. 185 Appendix H. Intensity and Power of Gaussian Beams 186 Thus, !2π !∞ P = I0 a b 2 e−r r dr dθ , 0 "0 e−r = 2πI0 a b − 2 2 #∞ . (H.6) (H.7) 0 Hence the peak intensity is related to the power within the beam by, I0 = H.2 P . πab (H.8) Mean Intensity In some circumstances it will be useful to consider a mean intensity, I, as opposed to the peak intensity, I0 . As Gaussian beams have an infinite extent obviously some limit on the extent of the beam will have to be considered. Suppose that the mean is calculated such that a fraction Z of the total power is taken into account, this will give an approximation to the mean, I, of IZ . Hence, PZ . π xZ y Z IZ = (H.9) For the purposes of this calculation consider a circularly symmetric beam such that, a = b = r0 , and xZ = yZ = rZ . Then, PZ = I0 !2π !rZ 0 2 e−(r/r0 ) r dr dθ , (H.10) 0 where rZ is the radius encompassing the power PZ . It follows that, PZ = −πr02 I0 = −πr02 I0 $ $ e−(r/r0 ) 2 %r Z e−(rZ /r0 ) , (H.11) % −1 . (H.12) 0 2 Appendix H. Intensity and Power of Gaussian Beams 187 But PZ = Z P , ∴ =⇒ PZ = πr02 I0 · Z , (H.13) 2 Z = 1 − e−(rZ /r0 ) . (H.14) Thus it follows that the beam radius encompassing a fraction, Z, of the total beam power is given by, rZ2 = r02 & & & 1 & & & . ln & 1−Z& (H.15) Now substituting equation H.15 back into equation H.9 , allows the mean intensity to be determined in terms of either the peak intensity or the total power. =⇒ IZ = ∴ IZ = πr02 ZP & 1 & , & ln & 1−Z I0 & . 1 & Z & ln & 1−Z (H.16) (H.17) Thus the mean intensity depends only on the peak intensity and the fraction of total power in the beam over which the mean is to be calculated.