1. July 2003 Chuck DiMarzio, Northeastern University 10351-8-1 ECEG105 & ECEU646 Optics for Engineers Course Notes Part 8: Gaussian Beams Prof. Charles A. DiMarzio Northeastern University Fall 2003

2. July 2003 Chuck DiMarzio, Northeastern University 10351-8-2 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

3. July 2003 Chuck DiMarzio, Northeastern University 10351-8-3 The Spherical-Gaussian Beam Gaussian Profile Rayleigh Range Diameter Radius of Curvature Axial Irradiance

4. July 2003 Chuck DiMarzio, Northeastern University 10351-8-4 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

5. July 2003 Chuck DiMarzio, Northeastern University 10351-8-5 Visualization of Gaussian Beam z=0 w  Center of Curvature

6. July 2003 Chuck DiMarzio, Northeastern University 10351-8-6 Parameters vs. Axial Distance -505 0 1 2 3 4 5 z/b, Axial Distance d/d 0, Beam Diameter -505 0 5 z/b, Axial Distance  /b, Radius of Curvature m4053

7. July 2003 Chuck DiMarzio, Northeastern University 10351-8-7 Complex Radius of Curvature Spherical Wave Gaussian Spherical Wave

8. July 2003 Chuck DiMarzio, Northeastern University 10351-8-8 Paraxial Approximation

9. July 2003 Chuck DiMarzio, Northeastern University 10351-8-9 Complex Radius of Curvature: Physical Results

10. July 2003 Chuck DiMarzio, Northeastern University 10351-8-10 Collins Chart z b

11. July 2003 Chuck DiMarzio, Northeastern University 10351-8-11 A Lens on the Collins Chart z b

12. July 2003 Chuck DiMarzio, Northeastern University 10351-8-12 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

13. July 2003 Chuck DiMarzio, Northeastern University 10351-8-13 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

14. July 2003 Chuck DiMarzio, Northeastern University 10351-8-14 Making a Laser Cavity Make the Mirror Curvatures Match Those of the Beam You Want.

15. July 2003 Chuck DiMarzio, Northeastern University 10351-8-15 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

16. July 2003 Chuck DiMarzio, Northeastern University 10351-8-16 Hermite-Gaussian Beams (2) Possible Applications –Approximation to Real Beams Simple Propagation –Description of Modes of Real Lasers –Calculation of Losses at Square Apertures

17. July 2003 Chuck DiMarzio, Northeastern University 10351-8-17 Coefficients for HG Expansion

18. July 2003 Chuck DiMarzio, Northeastern University 10351-8-18 Propagation Problems

19. July 2003 Chuck DiMarzio, Northeastern University 10351-8-19 Uniform Circular Aperture 0123456 -60 -50 -40 -30 -20 -10 0 Radial Distance Normalized Irradiance Original Function 1 term 8 terms 20 terms 0123456 -60 -50 -40 -30 -20 -10 0 Radial Distance Normalized Irradiance Far Field Diffraction 1 term 8 terms 20 terms 1.22 /D

20. July 2003 Chuck DiMarzio, Northeastern University 10351-8-20 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 10021.m

21. July 2003 Chuck DiMarzio, Northeastern University 10351-8-21 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?

22. July 2003 Chuck DiMarzio, Northeastern University 10351-8-22 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