Fabry-Perot: lossless Fabry-Perot: lossy Spherical mirror 1 Resonator optics Fabry-Perot resonator • no losses / with losses • resonator modes • spectral width and finesse • resonator lifetime and quality factor Spherical mirror resonators • ray confinement • Gaussian modes Fundamentals of Photonics, Ch. 10
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Resonator optics - ULisboa · Exercise - Resonator Modes and Spectral Width! Calculate:! • frequency spacing ν F! • spectral width δν! of the modes of a Fabry-Perot resonator
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Consider a monochromatic wave U(r) that satisfies the Helmholtz eq.:!!!!The resonator modes (allowed U(r)) are the solutions under the boundary conditions defined by the geometry of the resonator.!Lossless mirrors U(r)=0 at the mirrors!�
u(r,t ) = Re U(r)exp(i2πνt ){ }
�
U(r) = 0
�
U(r) = A sin(kz ), kd = qπ (q = 1,2,…)
kq = q πd
Uq (r) = Aq sin(kqz )
The modes are standing waves and q=1,2… is the mode number!
Losses may affect the amplitude and the phase: they are accounted for by a complex roundtrip attenuation factor:!!We have for the consecutive amplitudes U1, U2…!
�
h = r e−iϕ
�
Un = hUn−1 = hnU0
U =U0 +U1 +U2 +
=U0(1+h +h2 +) =U0 /(1− h)
The phase shift φ corresponding to a full roundtrip is:!
We can write the losses in the medium as a distributed loss proportional to d!The losses in the mirrors are considered located losses at the mirrors 1 and 2!
�
exp(−2αsd )
R1 = r12, R2 = r2
2
The total losses are then:!
�
r 2 = R1R2 exp(−2αsd )= exp(−2αrd )
→αr = αs + 12d
ln 1R1R2
⎛ ⎝ ⎜
⎞ ⎠ ⎟
If we replace |r |2 in the expression for the finesse we obtain (approx. valid for αrd<<1):!
Calculate:!• frequency spacing νF!• spectral width δν!of the modes of a Fabry-Perot resonator whose mirrors have reflectances R1=0.98 and R1=0.99 and are separated by a distance d = 100 cm.!Assume that the medium has refractive index n = 1 and negligible losses.!Is the derived approximation appropriate in this case?!
Why do resonator losses cause spectral line broadening?!
Consider the expression for the spectral width:!!Note that cαr has dimensions of (time)-1. We define the characteristic decay time as:!!The relation between time width and spectral width has the form of an uncertainty product:!
Extra: Fabry-Perot interferometer!In this case a plane wave comes from outside and is transmitted through a mirror with amplitude transmittance t and amplitude reflectance r :!
By making calculations similar to the FP resonator, we obtain a similar expression for the transmitted intensity!!This depends on the!separation d betweenthe mirrors!
When mirror 1 is planar (R1 = ∞), determine as a function of d / |R2|:!• the confinement condition!• the depth of focus (= 2z0)!• beam width at the waist and at each of the mirrors!