Question

2. (Worth 6 points total.) (a) Find the expression for the total power in a Gaussian beam with beam waist wo and peak field amplitude at the center of the beam waist Ao, meaning A2 (b) Find the expression for the fraction of the power of a Gaussian beam that will be transmitted through a circular aperture of radius a if the aperture is centered around the optical axis of the Gaussian beam and placed at the beam waist (the focus) of the beam (c) What percentage of the power will be transmitted for apertures as described in part (b) if a1.0 How about for a 1.5wo? And for a 2.0wo? (d) In terms of the Rayleigh length LR, at what distance along the optical axis from the beam waist (i.e. at what value of z) will the peak intensity of a Gaussian beam be reduced by a factor of 10? What is the 1/e2 width of the beam at this location in terms of wo?

Answer 1

Total power of the Gaussian beam across an arbitrary Here at z is, (a) P = 1/2z_0 doubleintegral E middot E^2x dA = 1/2 E^2_0/z_0 wr^2/w^2 (z) integral^dA_0 integral^infinity_0 exp [- 2 r^2/w^2 (z)] r dr d theta = pi w^2/4 z_0 E^2_1 Vertical-bar (r radius g the Gaussian beam at z, w (t) = beam at z] The intensity of Gaussian beam at z = 0 where w(z) is the beam waist = w_0 I = I_0 e^- 2 r^2/w^2_0 = 2P/pi w^2_0 e^- 2 r^2/w^2_0 (b) If, r = a, I = I_0 e^-2a^2/w^2_0 I/I_0 = e^-2a^2/w^2_0 this is the fraction of the power will be transmitted. (e) a = 10w_0, z/z_0 = e^-z w^2_0/w^2_0 = e^-2 = 1/e^2 a = 1.5 w_0 I/I_0 = e^-2 (1.5)^2 w^2_0/w^2_0 = 1/e^4.5 a = 2.0 w_0, I/I_0 = e^-8 w^2_0/w^2_0 = 1/e^8 \r\n (d) Rayleigh length L_R = pi eta w^2_0/lambda where eta, lambda, w_0 constant I = I_0 e^-2 r^2