Chapter II The Propagation of Rays and Beams Lecture 2 Chapter II The Propagation of Rays and Beams Highlights Optical ray propagation through a variety of optical media The propagation of Gaussian beam Homogeneous and isotropic Thin lenses Dielectric interface Curved mirrors Characteristics ABCD law 2 x 2 matrices Stability condition of periodic optical system

  §2.1 Lens Waveguide Ray Matrices +: Ray output direction above A paraxial ray passing through a thin lens of focal length f tanri’ ~ ri’ f Focal Plane z (ri, ri’) (ro, ro’) ri’f tan-1ri’ +: Ray output direction above z axis -: Ray output direction below   Output side Input side

§2.1 Lens Waveguide Exercises 1. Spherical Mirror: Radius of curvature R ri’ ro’ a b z R 2. Straight sections ( d ) and a thin lens ( f ) f z (ri, ri’) (ro, ro’) d (r2, r2’) z1 z2 z3

Most close to output side Most close to input side §2.1 Lens Waveguide Exercises Ray matrices for multi-element cascade system Most close to output side Most close to input side 3. Biperiodic lens sequence f2 z (rs+1, rs+1’) d (rs, rs’) f1 s s +1

§2.1 Lens Waveguide Stability condition

§2.2 Propagation of Rays Between Mirrors d (r1, r1’) (r2, r2’) (r3 r3’) (r4, r4’) (r5, r5’) z

§2.2 Propagation of Rays Between Mirrors Stability condition

§2.2 Propagation of Rays Between Mirrors Examples 1. Plane parallel cavity Rays parallel to z axis are trapped, otherwise escape. Trapped rays are closed after one times reflection. 2. Concentric cavity Rays pass through concentric point are trapped, otherwise escape. Trapped rays are closed after one times reflection. 3. Confocal cavity All rays as long as reflected back are trapped. Trapped rays are closed after two times reflection.

§2.2 Propagation of Rays Between Mirrors Stability condition for coaxial spherical cavity Unstability condition Metastability condition

§2.3 Rays in Lenslike Media z (r0, r0’) Ray Equation: For g2 > 0

§2.3 Rays in Lenslike Media For g2 < 0 Physical situations Spread 1. Absorption medium: Focus Spread 2. Optical Pumped SL rods: Focus 3. Dielectric waveguide: 4. Optical fiber: 5. Optical waveguide:

§2.4 Wave Equation in Quadratic Index Media The mostly encountered optical beam is Gaussian beam. Isotropic, charge-free medium & Neglect the right side

§2.4 Wave Equation in Quadratic Index Media We allow for a possible dependence of e on position r and the possibility of losses (s > 0) or gain (s < 0) in the medium k2 is some constant characteristic of the medium

§2.4 Wave Equation in Quadratic Index Media To obtain above equation, we neglect all second order derivate terms. If above equation holds for all r, we then have: This is a basic wave propagation equation for Gaussian beams.

§2.5 Gaussian Beams in a Homogenous Medium q0 is an arbitrary integration constant The choice of imaginary q0 will be found to lead to physically meaningful waves whose energy density is confined near the z axis.

§2.5 Gaussian Beams in a Homogenous Medium The first term of right side: Where we used: The second term of right side:

§2.5 Gaussian Beams in a Homogenous Medium We define the following parameters:

§2.5 Gaussian Beams in a Homogenous Medium 纵向相位 振幅因子 径向相位 上面的结果就是基摸 (fundamental mode)高斯光束的解。

§2.5 Gaussian Beams in a Homogenous Medium Z=0 w0 z z0 q R Spot size: distance which the amplitude is down by factor of 1/e Radius of curvature: Reyleigh length Beam spread angle

Homework 章节后习题2.1、2.3、2.6。