Download Optical Resonators - Quantum Electronics - Lecture Notes and more Study notes Quantum Physics in PDF only on Docsity! 1 Optical Resonators The microwave resonators are metal boxes or pipes (1) to build up large field intensity with moderate input power, (2) to act as a special and frequency filter selectively to fields, (3) to be used in spectral analyses. The number of modes with a frequency between ν and ν +dν per unit volume in a three dimensional resonator is νπν d c n N 3 328= (1) For a box of 1cm3, a microwave resonator contains at most one mode at a frequency of 109 Hz. However, at an optical frequency of 1014 Hz , the number of modes per unit volume can be as high as 109 for a bandwidth of 1010 Hz, unless the dimension of the resonator is reduced to the size of a wavelength. This problem can be overcome by using open resonators with a pair of mirrors with small areas. Only the mode propagating normal to the mirrors have high Q for meaningful resonance. In a resonator of distance l between mirrors, the beam from one of the mirrors of size a1 must be smaller than a2 to avoid losses. 1212 1 ≥≤ l aa oral a λ λ Fabry-Perot resonator with plane parallel mirrors) It can be proved, by considering multiple reflections between the mirrors, that, for an incoming electromagnetic radiation of wavelength k0, the transmission intensify is given by )(sin4)1( )1)(1( 0 22 21 22 21 4 21 00 0 lkneRReRR eRR T r lknlkn lkn ii i +− −−= (2) where the R’s are the reflectance of the mirrors and the index of refraction is a complex number. For ni =0, docsity.com 2 )(sin4)1( )1)(1( 2 21 2 21 21 klRRRR RR T +− −− = (4) Case I : R1=R2=R )(sin) 2 (1 1 )(sin )1( 4 1 1 )(sin4)1( )1( 222 2 1 22 2 1 F F kl R RklRR R T ν πν π + = − + = +− − = (5) Where the Finesse is defined by )1( R R F − = π The linewidth is given by F Fνδν = Meaning? Case II : R1=R2 and ni ≠0. Case III: R1 exp(2ni k0l)≈ 1 Finesse and Q-value Factors affecting linewidth: • Reflectivity of mirrors • Length of resonators • Parallelism of mirrors • Diffraction losses • Imperfections in optical materials • Coherence length Resonator modes: Plane-parallel mirrors of finite sizes cannot confine electromagnetic radiation. Stable confinement always involves spherical mirrors. Hermite Gaussian mode: From (33) of Lecture 1, the solution for propagating beam in a monogenous medium of index n are: docsity.com 5 Where MBCADF DA b det 2 2 =−= += (13) The beam position parameter after m round trips can be related to the initial by )sin( 0max ϕϕ += mFyy m m (14) where the parameters ymax, and ϕ0 are to be determined by the initial condition, and F b1cos−=ϕ (15) For the round trip effect is to reproduce the original position, then F=1 and 1≤b or 1)1)(1(0 21 ≤++≤ R d R d (16) It is customary to express the condition for stability as 10 21 ≤≤ gg where 1 1 1 R d g += 2 2 1 R d g += Concave mirrors: R<0. Discussion of resonator stability diagram. g1 g2 symmetrical confocal 1 1 Planar planar concentric Symmetric stable resonator docsity.com 6 Discussions Planar-planar Planar concave Unstable resonators Diffraction losses due to finite mirror sizes Homework: 1. Prove Eqs (2) and (4) and write a computer program to plot the transmitted power (E2 ) as function of distance l for a range of variation over a few wavelengths for the case of (4) for various value of R from 0.3 to 1. You may assume that the index of refraction between the mirror is 1. 2. In a symmetric stable resonator, R1 =R2, and g1 =g2 . The waist of the beam supported by the resonator is at the center. Express the beam waist using the wavelength, g, and the distance between the mirrors. 3. Find the ratio of the beam sizes at the waist and at mirrors in a symmetric confocal resonator. 4 . If two mirrors of radius of curvature of 10 cm and 20 cm, respectively, are used to construct a stable resonator, find the maximum distance between two mirrors. docsity.com
