S. Blair September 27,

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1 S. Blair September 7, Optical Resonators With Spherical Mirrors Laser resonators have the same characteristics as Fabry-Perot etalons. A laser resonator supports longitudinal modes of a discrete set of frequencies ν m = m c nl Laser resonators also support a discrete set of transverse, or spatial, modes. These modes are given by the Hermite-Gaussian beams: ( ( w o E l,m (x, y, z = E o w(z H x y l H m e (x +y /w(z w(z w(z e jk(x +y /R(z e jkz e j(l+m+1η(z where and ( z w(z =w o 1+ R(z =z [ 1+ ( zo ] z η(z =tan 1 ( z AGaussianbeamisamodeofaresonatorcavitywhentheradiiofcurvatureofthemirrors equals the curvature of the beam wavefronts. At this condition, the direction of energy!! % &!!"#"$ propagation (or the ray is perpendicular to the surface of the mirror. Thus, the beam will retrace its path, resulting in a self-reproducing field. These mirrors are placed at positions z 1 and z, such that ] R 1 = z 1 [1+ ( zo is the radius of curvature of the first mirror, and z 1 R = z + z o z = z 1 + z o z 1 is the radius of curvature of the second mirror. Note: a positive radius of curvature is when the center of the curvature is to the left of the wavefront; therefore, R 1 < 0 and R > 0.

2 S. Blair September 7, If we have two mirrors with radii R 1 and R, and mirror separation (or cavity length l = z z 1, the Rayleigh distance of the Gaussian mode is ( πw zo = o n = l ( R 1 l(r l(r R 1 l λ (R R 1 l and the position with respect to the first and second mirror is z 1 = R 1 ± 1 R1 4zo z = R ± 1 R 4z o. The signs are chosen to make physical sense. The spot sizes at the mirrors are w 1 = w o 1+ w = w o 1+ For a symmetric resonator R = R 1 R, and z o = ( z1 ( z (R ll 4 The confocal resonator is a common symmetric geometry, and occurs when R = l, i. e. the '% '& two mirrors are separated by their foci. Now and z o = l 4 or = l, z 1 = l, z = l w 1 = w o, w = w o

3 S. Blair September 7, Mode Stability First, we will consider the special case of the symmetric resonator cavity, R = R 1 = R, with mirror spacing l. The minimum spot size is ( 1/4 ( λ l w o = R l 1/4 πn The spot size at each of the mirrors is given by [ λl R w 1, = πn l (R l/ Resonator losses are small when the spot size at each mirror is smaller than the size of the mirror. The cavity is said to be stable when w 1, remain finite; however, this stability criterion does not necessarily mean that the cavity is low loss. The minimum mirror spot size occurs under the confocal condition R = l, so that (w 1, conf = ( 1/ λl (w o conf = πn The ratio of mirror spot size of a general symmetric cavity to that of a confocal cavity is [ ] 1/4 w 1, 1 = (w 1, conf (l/r[ (l/r] ] 1/4 $(&! """""# $(& (*+ $ % $ & '' Therefore, the confocal resonator has the minimum mirror spot size and the minimum loss (note that higher-order TEM modes may also lase in this cavity.

4 S. Blair September 7, For the case when l R =0,thecavityconsistsofplanarmirrorsandisunstable For the case when l R =,thecavityisconcentricandunstable,' ' In section.1, we analyzed the propagation of rays between a periodic sequence of two lenses, and derived the stability condition ( 0 1 l ( 1 l 1 f 1 f which is the condition under which the ray height r remains finite. Making the substitution f = R/, the condition for a laser resonator becomes 0 (1 (1 lr1 lr 1 Note: This sign convention is different than the one we have been using for the radius of curvature. This convention has R>0whenthecenterofcurvatureofonemirrorisinthe direction of the other mirror. In the other sign convention, R>0whenthecenterofcurvatureistotheleftofthe wavefront, the stability condition becomes: 0 (1+ (1 lr1 lr 1 We will use the first sign convention when dealing with stability, so be careful! The stability criterion can be illustrated with a graph: 0 g 1 g 1 The confocal cavity is at the origin Generalized Resonator g 1 =1 l R 1 g =1 l R In a resonator, a stable mode is one that reproduces itself after one round trip. One round trip can be described by the appropriate ray transfer matrix. The q parameter then changes according to: q s+1 = Aq s + B Cq s + D

5 S. Blair September 7, & -*./013 -*./013 4 % -*./013 -*./013 but, for a stable mode, q s+1 = q s,whichgives 1 (D A ± (D A +4BC = q s B As before, AD BC =1,sothat 1 = D A q s B ± i 1 [(D + A/] B = D A B + i sin θ B where Since cos θ = D + A 1 = 1 q s R i λ πw n then w = Bλ πn sin θ In order for w to be real, we must have or and sin θ B < 0 1 [(D + A/] < 0 B<0 B D + A real < 1

6 S. Blair September 7, Resonator Frequencies The resonant frequencies are determined by the condition that the round trip phase is a multiple of π. Mathematically, for our two-mirror resonator, θ l,m (z θ l,m (z 1 =qπ where θ l,m is the phase of the TEM l,m Hermite Gaussian wave between z =0andamirror. θ l,m (z =kz (l + m +1tan 1 ( z The resonance condition is then k q d (l + m +1 [ ( ( ] tan 1 z tan 1 z1 = qπ where k q = πν qn and d = z z 1 c For a fixed l,m: ν q+1 ν q = c nd is the longitudinal mode spacing. All transverse modes l,m with l + m =const have the same frequency; i.e. they are degenerate in frequency. For a confocal cavity with R 1 = R, we have such that k q d (l + m +1 π = πν qn d (l + m +1 π c = qπ ν q = c [q +(l + m +1] 4nd c n d = ν q, l + m q, l + m + 1 q + 1, l + m or q, l + m + q + 1, l+m+1 q, l + m + 3 q +, l + m q + 1, l + m + q, l + m + 4 ν g