July 2003 Chuck DiMarzio, Northeastern University ECEG105 & ECEU646 Optics for Engineers Course Notes Part 8: Gaussian Beams Prof. Charles A. DiMarzio Northeastern University Fall 2003

July 2003 Chuck DiMarzio, Northeastern University Some Solutions to the Wave Equation Plane Waves –Fourier Optics Spherical Waves –Spherical Harmonics; eg. In Mie Scattering Gaussian Waves –Hermite- and Laguerre- Gaussian Waves

July 2003 Chuck DiMarzio, Northeastern University The Spherical-Gaussian Beam Gaussian Profile Rayleigh Range Diameter Radius of Curvature Axial Irradiance

July 2003 Chuck DiMarzio, Northeastern University Size Scales of Gaussian Beams P E P 0.86P 0.14E 0.95P 0.76P 0.5E 0.5P 0.21P 0.79E 0.5P d

July 2003 Chuck DiMarzio, Northeastern University Visualization of Gaussian Beam z=0 w  Center of Curvature

July 2003 Chuck DiMarzio, Northeastern University Parameters vs. Axial Distance z/b, Axial Distance d/d 0, Beam Diameter z/b, Axial Distance  /b, Radius of Curvature m4053

July 2003 Chuck DiMarzio, Northeastern University Complex Radius of Curvature Spherical Wave Gaussian Spherical Wave

July 2003 Chuck DiMarzio, Northeastern University Paraxial Approximation

July 2003 Chuck DiMarzio, Northeastern University Complex Radius of Curvature: Physical Results

July 2003 Chuck DiMarzio, Northeastern University Collins Chart z b

July 2003 Chuck DiMarzio, Northeastern University A Lens on the Collins Chart z b

July 2003 Chuck DiMarzio, Northeastern University Looking For Solutions on the Collins Chart (1) -z 1 You Can’t Focus a Beam of diameter d 1 any Further Away than z 1 b’=b’ 2 b’=b’ 1 You Can’t Keep a beam diameter less than d 2 over a distance greater than. zz

July 2003 Chuck DiMarzio, Northeastern University Looking For Solutions on the Collins Chart (2) b’=b’ 3 There may be 0, 1, or 2 solutions. Watch out for your tie! I want to put a beam waist at a distance z 3 from a starting diameter of d 3. z b

July 2003 Chuck DiMarzio, Northeastern University Making a Laser Cavity Make the Mirror Curvatures Match Those of the Beam You Want.

July 2003 Chuck DiMarzio, Northeastern University Hermite-Gaussian Beams (1) Expansion in Hermite Gaussian Functions –Orthogonal Functions Infinite x,y –Freedom to Choose w Use Best Fit for Lowest Mode Alternative –Laguerre Gaussians For Circular Symmetry

July 2003 Chuck DiMarzio, Northeastern University Hermite-Gaussian Beams (2) Possible Applications –Approximation to Real Beams Simple Propagation –Description of Modes of Real Lasers –Calculation of Losses at Square Apertures

July 2003 Chuck DiMarzio, Northeastern University Coefficients for HG Expansion

July 2003 Chuck DiMarzio, Northeastern University Propagation Problems

July 2003 Chuck DiMarzio, Northeastern University Uniform Circular Aperture Radial Distance Normalized Irradiance Original Function 1 term 8 terms 20 terms Radial Distance Normalized Irradiance Far Field Diffraction 1 term 8 terms 20 terms 1.22 /D

July 2003 Chuck DiMarzio, Northeastern University Sample Hermite Gaussian Beams 0:00:10:3 1:01:11:3 2:02:12:3 5:05:15:3 (0:1)+i(1:0) = “Donut Mode” Most lasers prefer rectangular modes because something breaks the circular symmetry. Note: Irradiance Images rendered with  =0.5 from matlab program m

July 2003 Chuck DiMarzio, Northeastern University Losses at an Aperture (1) g,Gain Aperture E1E1 r 1, mirror r 2, mirror E2E2 Straight-Line Layout E1E1 E2E2 E1E1 E 1 = E 1 gMr 2 gr 1 One round trip: What is M?

July 2003 Chuck DiMarzio, Northeastern University Losses at an Aperture (2) E1E1 E2E2 E1E1 C 1 = C 1 gMr 2 gr 1 One round trip: Now, g and M and maybe r are matrices. All but M are likely to be nearly diagonal. Large Apertures: M is diagonal Finite Apertures: Diagonal elements become smaller, and off-diagonal elements become non-zero