Lasers PH 645/ OSE 645/ EE 613 Summer 2010 Section 1: T/Th 2:45-4:45 PM Engineering Building PDF Free Download

Lasers PH 645/ OSE 645/ EE 613 Summer 2010 Section 1: T/Th 2:45-4:45 PM Engineering Building 240 John D. Williams, Ph.D. Department of Electrical and Computer Engineering 406 Optics Building - UAHuntsville, Huntsville, AL 35899 Ph. (256) 824-2898 email: williams@eng.uah.edu Office Hours: Tues/Thurs 2-3PM JDW, ECE Summer 2010

Chapter 12: Stable Laser Resonators and Gaussian Beams Stable curved mirror cavities Properties of Gaussian beams Properties of real Laser beams Propagation of Gaussian beams using ABCD matrices Chapter 12 Homework: 3,6,7,9,11 Cambridge University Press, 2004 ISBN-13: 9780521541053 All figures presented from this point on were taken directly from (unless otherwise cited): W.T. Silfvast, laser Fundamentals 2 nd ed., Cambridge University Press, 2004.

Stable 2 Mirror Cavity Designs

ABCD Matrix Elements for Transmission and Reflection

ABCD Matrix Consider a beam traveling from r1 to r2. The beam travels at an angle θ. Assuming no index change or reflection, then the value θ2 = the incident angle This can be written in matrix form as Where the ABCD translation matrix is

ABCD Matrix for lenses A ray passing through two lenses of focal length f1 and f2 can be written as

Stability Criteria for Two Mirror Cavities One can visualize the optical conditions of two semispherical mirrors as that of two thin lenses bending light as it refracts through.

Graphical Conditions for Stability

Graphical Conditions for Stability For N successive applications present within a resonator cavity

Graphical Conditions for Stability

Conditions on the Verge of Stability

Gaussian Beams Consider a Gaussian beam with a geometric beam radius, w (beam waist) Where w o is the waist of the beam at the focal plane

Gaussian Beams Consider a Gaussian beam with a geometric beam radius, w Beam wavefront of curvature is Angular spread function for z > z R

Gaussian Properties of 2 Mirror Cavities

Plane Parallel and Long Radius Cavities Two plain mirrors parallel to one another Mode operation is on the edge of stability with g 1 g 2 =1 Just inside stability where the radius of curvature for each mirrors is significantly larger than the distance between the mirrors. g 1 g 2 1

Symmetrical Mirror Cavities Two curved mirrors of equal radius mirrors parallel to one another Minimum waist at center waist at mirror 1 flat mirror and another curved mirror focusing the beam waist at curved mirror waist at flat mirror = minimum waist

Concave-Convex Mirror Cavities Two curved mirrors One focusing and the other defocusing such that Minimum waist lies outside the cavity providing a large diameter beam throughout the gain medium with a relatively uniform volume

Near Centric (Spherical) Mirror Cavities Two curved mirrors focusing at a point d/2 Minimum waist at center waist at mirror g 1 =1+ d/r 1, g 2 =1 R 1 =d, R 2 = ω 2 0 = dλ d + d π 2d + d 1/ 2

Mode Volume of a Hermite-Gaussian Mode Mode volume of an electric field in a cavity Using the identity

Example of Laser Beam Power using Gaussian Mode Calculations

Example of Laser Beam Power using Gaussian Mode Calculations This is the maximum power available in the laser if all of the cavity power was converted to stimulated emission

Properties of Real Laser Beams Previous sections referred to the propagation of plane wave and Gaussian (diffraction limited) beams Higher order modes are not Gaussian shaped Thus, real world multimode lasers are neither of the shapes previously discussed There is however a parameter that does allow one to model such systems Lets start with a diffraction limited (Gaussian beam). The product of the waist and the angular spread is The minimum possible product would be In a real beam defined my multiples of different diffraction limited beams, on can define the divergence as And the minimum waist as Yielding: M 2 is the propagation constant

Properties of Real Laser Beams One can solve for the propagation constant in terms of geometric measurables as: And therefore the waist of the beam at any point z within the cavity as Notice that if M = 1 then we have the same equations used to represent a Gaussian beam Finally, one can determine the Raleigh range as With a concept as simple as M2, it is possible to align laser mirrors to generate the desired mode quality easily and effectively It is also possible to examine aberrations induced in the beam and measure the astigmatism of the beam with ease

Gaussian Beam Propagation using Matricies Gaussian beams are evaluated using a complex beam parameter, q, which evaluates the shape of the beam at the prescribed wavelength One can evaluate the beam parameter at any point 2 from a known point 1 using