ECEG105 & ECEU646 Optics for Engineers Course Notes Part 8: Gaussian Beams

1. ECEG105 & ECEU646 Optics for EngineersCourse NotesPart 8: Gaussian Beams Prof. Charles A. DiMarzio Northeastern University Fall 2003 Chuck DiMarzio, Northeastern University

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 Chuck DiMarzio, Northeastern University

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

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

5. Visualization of Gaussian Beam w r z=0 Center of Curvature Chuck DiMarzio, Northeastern University

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

7. Complex Radius of Curvature • Spherical Wave • Gaussian Spherical Wave Chuck DiMarzio, Northeastern University

8. Paraxial Approximation Chuck DiMarzio, Northeastern University

9. Complex Radius of Curvature: Physical Results Chuck DiMarzio, Northeastern University

10. Collins Chart b z Chuck DiMarzio, Northeastern University

11. A Lens on the Collins Chart b z Chuck DiMarzio, Northeastern University

12. Looking For Solutions on the Collins Chart (1) You Can’t Focus a Beam of diameter d1 any Further Away than z1 You Can’t Keep a beam diameter less than d2 over a distance greater than. b’=b’2 b’=b’1 dz -z1 Chuck DiMarzio, Northeastern University

13. Looking For Solutions on the Collins Chart (2) I want to put a beam waist at a distance z3 from a starting diameter of d3. b b’=b’3 There may be 0, 1, or 2 solutions. Watch out for your tie! z Chuck DiMarzio, Northeastern University

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

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 Chuck DiMarzio, Northeastern University

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 Chuck DiMarzio, Northeastern University

17. Coefficients for HG Expansion Chuck DiMarzio, Northeastern University

18. Propagation Problems Chuck DiMarzio, Northeastern University

19. Uniform Circular Aperture Far Field Diffraction Original Function 0 0 1.22 l/D -10 -10 1 term 20 terms 1 term -20 -20 8 terms -30 -30 Normalized Irradiance Normalized Irradiance 20 terms -40 -40 8 terms -50 -50 -60 -60 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Radial Distance Radial Distance Chuck DiMarzio, Northeastern University

20. Sample Hermite Gaussian Beams 0:0 0:1 0:3 (0:1)+i(1:0) = “Donut Mode” 1:0 1:1 1:3 2:0 2:1 2:3 Most lasers prefer rectangular modes because something breaks the circular symmetry. 5:0 5:1 5:3 from matlab program 10021.m Note: Irradiance Images rendered with g=0.5 Chuck DiMarzio, Northeastern University

21. Losses at an Aperture (1) r2, mirror E1 E2 g,Gain r1, mirror Aperture Straight-Line Layout E1 E2 E1 One round trip: E1 = E1gMr2gr1 What is M? Chuck DiMarzio, Northeastern University

22. Losses at an Aperture (2) Large Apertures: M is diagonal Finite Apertures: Diagonal elements become smaller, and off-diagonal elements become non-zero E1 E2 E1 One round trip: C1 = C1gMr2gr1 Now, g and M and maybe r are matrices. All but M are likely to be nearly diagonal. Chuck DiMarzio, Northeastern University