6. Optical Resonators. (1) c

Transcription

1 6. Optical esonators The icrowave resonators are etal boxes or pipes to buil up large fiel intensit with oerate input power, to act as a special an frequenc filter selectivel to fiels, 3 to be use in spectral analses. The nuber of oes with a frequenc between ν an ν ν per unit volue in a three iensional resonator is 3 8πν n ν c N 3 or a box of c 3, a icrowave resonator contains at ost one oe at a frequenc of 9 H. However, at an optical frequenc of 4 H, the nuber of oes per unit volue can be as high as 9 for a banwith of H, unless the iension of the resonator is reuce to the sie of a wavelength. This proble can be overcoe b using open resonators with a pair of irrors with sall areas. Onl the oe propagating noral to the irrors have high Q for eaningful resonance. In a resonator of istance l between irrors, the bea fro one of the irrors of sie a ust be saller than a to avoi losses. λ aa l a or a λl abr-perot resonator with plane parallel irrors It can be prove, b consiering ultiple reflections between the irrors, that, for an incoing electroagnetic raiation of wavelength k, the transission intensif is given b T e 4 nikl n kl n kl i i e 4 e sin nrkl where the s are the reflectance of the irrors an the inex of refraction is a coplex nuber. or n i,

2 T 4 4 sin kl Case I : T 4 sin kl 4 sin here the inesse is efine b kl sin π 5 π The linewith is given b δν ν Meaning? Case II : an n i. Case III: expn i k l inesse an Q-value actors affecting linewith: eflectivit of irrors Length of resonators Parallelis of irrors Diffraction losses Iperfections in optical aterials Coherence length esonator oes: πν ν Plane-parallel irrors of finite sies cannot confine electroagnetic raiation. Stable confineent alwas involves spherical irrors. Herite Gaussian oe: ro 33 of Lecture, the solution for propagating bea in a onogenous eiu of inex n are:

3 3 ] [,, ζ ρ ρ l j jk jk l l e H x H A x U 7 where l,,n are the orer of the oes, the spot sie is / ] [ λ π The phase factor ζ is tan ζ. Stable resonators can be fore b place two irrors at an with their raii of curvature atching the curvature of the wavefront. Starting with a bea with a waist at, the irrors place at an have raii of curvature 8 9 A variet of cobination of concave an convex irrors can for a stabl resonator.

4 4 esonance frequenc: p 335 Unfoling a resonator Consier a resonator fore b two irrors of raii of curvature an separate b a istance This sste can be unfole into perioic lens sequence Using the ACD atrix to represent the position an slop after roun trips D C A Consier the relation between consecutive roun trips D C A eliinating the angles using the relation, A A e have b

5 here A D b AD C et M 3 The bea position paraeter after roun trips can be relate to the initial b sin ax ϕ ϕ 4 where the paraeters ax, an ϕ are to be eterine b the initial conition, an b ϕ cos 5 or the roun trip effect is to reprouce the original position, then an b or It is custoar to express the conition for stabilit as g g where g g Concave irrors: <. 6 Discussion of resonator stabilit iagra. g Planar planar concentric confocal Setric stable resonator g setrical 5

6 Discussions Planar-planar Planar concave Unstable resonators Diffraction losses ue to finite irror sies Hoework:. Prove Eqs an 4 an write a coputer progra to plot the transitte power E as function of istance l for a range of variation over a few wavelengths for the case of 4 for various value of fro.3 to. You a assue that the inex of refraction between the irror is.. In a setric stable resonator,, an g g. The waist of the bea supporte b the resonator is at the center. Express the bea waist using the wavelength, g, an the istance between the irrors. 3. in the ratio of the bea sies at the waist an at irrors in a setric confocal resonator. 4. If two irrors of raius of curvature of c an c, respectivel, are use to construct a stable resonator, fin the axiu istance between two irrors. 6