Theory of Laser Resonators and of the Beam Divergence

Unstable confocal resonator with external flat semi-transparent mirror for additional feedback

“…The term ∆ |γ| takes into account the additional field feedback. We see that |γ| → 1/M at R → 0 and |γ| → 1 at R → 0, which is consistent with the general ideas about the properties of the resonant field [6].…”

“…Such dependence of losses on ∆L is due to the interference of diverging and converging beams in the paraxial region of the resonator. It is well-known [6] that a similar property of quasi-periodicity (however, depending on the equivalent Fresnel number N eq = N(1 − 1/M)/(2M) is also characteristic of two-mirror unstable cavities (R = 0) and is, in this case, associated with radiation scattering at the edges of the counterreflector.…”

“…In (2), the second term in brackets takes into account the radiation that returns to the resonator upon reflection from the flat coupling mirror 3 with the reflection coefficient R in terms of the field; φ is determined by the distance ∆L (∆L ≪L) between the pole of the counterreflector and the reflecting plane of the mirror 3, φ = (4π /λ) • ∆L. If R = 0 then the kernel of (2) takes the well-known form [6] for a two-mirror resonator with an unlimited reflector 1. Note that, for a given Fresnel number N, the transparency of the resonator of the negative branch M < −1 is equal to the transparency of the resonator of the positive branch, with the effective Fresnel number N M = N • M 2 /(abs(M) − 1).…”

“…The field V Na (⃗ ρ) together with the homogeneous field forms the output radiation. In a lens line equivalent to the unstable resonator [6], the evolution of the field V 0 (⃗ ρ) in the Fresnel approximation and without account taken of radiation losses beyond the edges of the counterreflector is described by the following recursive relation:…”

“…The value |γ d | is the modulus of the eigenvalue of equation ( 1) at R = 0 in the diffraction approximation. For a threedimensional resonator, |γ d | ⩾ |γ g |, as a rule [6]. R eff is the effective reflection coefficient of the flat mirror, which takes into account the reflection coefficient R of the coupling mirror and the diffraction scattering of radiation at the edges of the counterreflector.…”