5.2 Fabry–Pérot cavity

The schematic view of a Fabry–Pérot cavity with two movable mirrors is drawn in Figure 29View Image. This simple scheme is important for at least two reasons: (i) it is the most common element for more sophisticated interferometer configurations, which are considered below; and (ii) the analysis of real-life high-sensitivity interferometers devoted, in particular, to detection of GWs, can be reduced to a single Fabry–Pérot cavity by virtue of the ‘scaling law’ theorem [34Jump To The Next Citation Point], see Section 5.3.
View Image

Figure 29: Fabry–Pérot cavity

A Fabry–Pérot cavity consists of two movable mirrors that are separated by a distance L + x1 + x2, where L = cτ is the distance at rest with τ standing for a single pass light travel time, and x1 and x2 are the small deviations of the mirrors’ position from the equilibrium. Each of the mirrors is described by the transfer matrix 𝕄 1,2 with real coefficients of reflection ∘R---- 1,2 and transmission ∘T---- 1,2 according to Eq. (243View Equation). As indicated on the scheme, the outer faces of the mirrors are assumed to have negative reflectivities. While the intermediate equations depend on this choice, the final results are invariant to it. The cavity is pumped from both sides by two laser sources with the same optical frequency ωp.

The coupling equations for the ingoing and outgoing fields at each of the mirrors are absolutely the same as in Section 2.2.5. The only new thing is the free propagation of light between the mirrors that adds two more field continuity equations to the full set, describing the transformation of light in the Fabry–Pérot cavity. It is illuminating to write down input-output relations first in the time domain:

√--- √ --- √ --- √--- ˆEb1(t) = − R1 Eˆa1(t + 2x1∕c) + T1Eˆe1(t), ˆEb2(t) = − R2 ˆEa2(t + 2x2∕c) + T2 ˆEe2(t), ˆE (t) = √R--Eˆ (t − 2x ∕c) + √T-Eˆ (t), ˆE (t) = √R--ˆE (t − 2x ∕c) + √T--ˆE (t),(274 ) f1 1 e1 1 1 a1 f2 2 a2 2 2 e2 ˆEe1(t) = ˆEf2(t − L∕c), ˆEe2(t) = Eˆf1 (t − L ∕c).
Further we use notation τ = L∕c for the light travel time between the mirrors.

The frequency domain version of the above equations and their solutions are derived in Appendix A.1. We write these I/O-relations given in Eqs. (545View Equation) in terms of complex amplitudes instead of 2 photon amplitudes, for the expressions look much more compact in that representation. However, one can simplify them even more using the single-mode approximation.

Single-mode approximation.
We note that:

(i) in GW detection, rather high-finesse cavities are used, which implies low transmittance coefficients for the mirrors

T1,2 ≪ 1; (275 )
(ii) the cavities are relatively short, so their Free Spectral Range (FSR) fFSR = (2τ)−1 is much larger than the characteristic frequencies of the mirrors’ motion:
|Ω|τ ≪ 1; (276 )
and (iii) the detuning of the pump frequency from one of the cavity eigenfrequencies:
πn δ = ωp − --- (n is an integer) (277 ) τ
is also small in comparison with the FSR:
|δ|τ ≪ 1. (278 )
In this case, only this mode of the cavity can be taken into account, and the cavity can be treated as a single-mode lumped resonator. Note also that, while our intermediate equations below depend on whether n is even or odd, the final results do not. Therefore, we assume for simplicity that n is even.

Expanding the numerators and denominators of Eqs. (540View Equation, 545View Equation) into Taylor series in τ and keeping only the first non-vanishing terms, we obtain that

B1,2 = ℛ1,2(0)A1,2 + 𝒯 (0)A2,1, √ γ1A1 + √ γ2A2 E1,2 = F1,2 = E = --------√-------, (279 ) ℓ(0 ) τ
and
√ ---- ˆb (ω) = ℛ (Ω )ˆa (ω) + 𝒯 (Ω )ˆa (ω) + 2--γ1,2X-(Ω), (280 ) 1,2 1,2 1,2 2,1 ℓ(Ω) √ --- √ --- ˆ ˆe1,2(ω) = ˆf1,2(ω) = ˆe(ω) = --γ1ˆa1(ω-) +--γ2ˆa√2(ω-) +-X-(Ω-), (281 ) ℓ(Ω ) τ
where
T γ1,2 = -1,2, (282 ) 4τ γ = γ1 + γ2 (283 )
is the cavity half-bandwidth,
ℓ(Ω) = γ − i(δ + Ω), (284 )
√ ----- ℛ1,2(Ω) = 2-γ1,2 − 1, 𝒯 (Ω ) = 2--γ1γ2- (285 ) ℓ(Ω ) ℓ(Ω )
are the cavity left and right reflectivities and its transmittance,
ikpE ˆx(Ω) ˆX (Ω) = ---√-----, (286 ) τ
and
ˆx = ˆx + ˆx (287 ) 1 2
is the sum variation of the cavity length.

5.2.1 Optical transfer matrix for a Fabry–Pérot cavity

The above optical I/O-relations are obtained in terms of the complex amplitudes. In order to transform them to two-photon quadrature notations, one needs to employ the following linear transformations:

  1. change frequency ω → ωp ± Ω and rewrite the relations between the input αˆ(ω) and output operators ˆ β(ω) in the form:
    ˆ ˆ ˆ β(ω ) = f (Ω)ˆα (ω ) → β+ ≡ β(ωp + Ω) = f (ωp + Ω)ˆα (ωp + Ω ) ≡ f+αˆ+ and ˆβ†− ≡ ˆβ†(ωp − Ω) = f ∗(ωp − Ω )ˆα†(ωp − Ω ) ≡ f∗ˆα †− ; (288) −
    where f (Ω) is an arbitrary complex-valued function of sideband frequency Ω;
  2. use the definition (57View Equation) to get the following relations for two-photon quadrature operators:
    [ ] [ ∗ ∗ ] [ ] βˆc (Ω ) = 1- (f+ + f− ) i(f+ − f− ) ⋅ ˆαc(Ω) . (289) βˆs (Ω ) 2 − i(f+ − f∗− ) (f+ + f−∗) ˆαs(Ω)

Applying transformations (289View Equation) to Eqs. (280View Equation), we rewrite the I/O-relations for a Fabry–Pérot cavity in the two-photon quadratures notations:

ˆb1,2(Ω) = ℝ1,2(Ω)ˆa1,2(Ω) + 𝕋(Ω )ˆa2,1(Ω ) + 2 √ γ1,2𝕃(Ω )ˆX (Ω), (290 ) [ ] eˆ(Ω) = √1--𝕃(Ω ) √ γ1,2ˆa1,2(Ω) + √ γ2,1ˆa2,1(Ω) + ˆX (Ω) , (291 ) τ
where
[ ] ˆ − Es kpˆx(Ω)- X (Ω) = Ec √ τ , (292 ) √ -- √ -- Ec = 2Re E, Es = 2Im E, (293 )
[ ] 1 γ − iΩ − δ 𝕃 (Ω) = ------ δ γ − iΩ , (294 ) 𝒟(Ω ) 𝒟 (Ω ) = ℓ(Ω )ℓ∗(− Ω ) = (γ − iΩ)2 + δ2, (295 )
√ ----- ℝ1,2(Ω ) = 2γ1,2𝕃 (Ω ) − 𝕀, 𝕋 (Ω ) = 2 γ1γ2𝕃 (Ω). (296 )

Therefore, the I/O-relations in standard form read:

[ ] [ ] [ ] ˆb1(Ω) (0) ˆa1(Ω ) RFP1 (Ω ) ˆb (Ω) = 𝕄 FP ⋅ ˆa2(Ω ) + RFP (Ω ) xˆ(Ω) (297 ) 2 2
with optical transfer matrix defined as:
[ ] (0) ℝ1 (Ω) 𝕋 (Ω) 𝕄 FP(Ω ) = 𝕋(Ω ) ℝ (Ω ) (298 ) 2
and the response to the cavity elongation xˆ(Ω) defined as:
∘ --- [ ] ∘ --- [ ] FP γ1 − Es FP γ2 − Es R 1 (Ω ) = 2kp τ-𝕃(Ω ) ⋅ E and R 2 (Ω) = 2kp -τ 𝕃 (Ω) ⋅ E . (299 ) c c

Note that due to the fact that (𝕄 (0))† = (𝕄 (0))−1 FP FP the reflectivity and the transmission matrices ℝ1,2 and 𝕋 satisfy the following unitarity relations:

† † † † † † ℝ1ℝ 1 + 𝕋𝕋 = ℝ2ℝ 2 + 𝕋 𝕋 = 𝕀, ℝ1𝕋 + 𝕋 ℝ 2 = 0. (300 )

5.2.2 Mirror dynamics, radiation pressure forces and ponderomotive rigidity

The mechanical equations of motion of the Fabry–Pérot cavity mirrors, in spectral representation, are the following:

−1 fχ xx,i(Ω )ˆxi(Ω) = Fˆi(Ω ) + Gi (Ω) i = 1, 2, (301 )
where χxx,i are the mechanical susceptibilities of the mirrors, Gi stand for any external classical forces that could act on the mirrors (for example, a signal force to be detected),
ˆ ˆ ˆ ˆ ˆF = ℐei +-ℐfi −-ℐai −-ℐbi (302 ) i c
are the radiation pressure forces acting on the mirrors, and ˆâ„ ei, ˆâ„ fi, ˆâ„ ai, ℐˆ bi are the powers of the waves ei, fi, etc. The signs for all forces are chosen in such a way that the positive forces are oriented outwards from the cavity, increasing the corresponding mirror displacement x1,2.

In the spectral representation, using the quadrature amplitudes notation, the radiation pressure forces read:

ˆF (Ω ) = ℏkp-(ET E + FT F − AT A − BT B ) 1,2 2 [ 1,2 1,2 1,2 1,2 1,2 1,2 1,2 1,2 ] T T ˆ T T ˆ +ℏkp E 1,2ˆe1,2(Ω ) + F1,2f1,2(Ω) − A 1,2ˆa1,2(Ω) − B 1,2b1,2(Ω ) . (303 )
The first group, as we have already seen, describes the regular constant force; therefore, we omit it henceforth.

In the single-mode approximation, the radiation pressure forces acting on both mirrors are equal to each other:

T Fˆ1,2(Ω) ≡ Fˆb.a.(Ω ) = 2ℏkpE ˆe(Ω ), (304 )
and the optical field in the cavity is sensitive only to the elongation mechanical mode described by the coordinate x. Therefore, combining Eqs. (301View Equation), we obtain for this mode:
−1 χxx(Ω )xˆ(Ω) = Fˆb.a.(Ω ) + G (Ω ), (305 )
where
χxx(Ω ) = [χxx,1(Ω) + χxx,2(Ω)] (306 )
is the reduced mechanical susceptibility and
χxx,1(Ω )G1(Ω ) + χxx,2(Ω )G2 (Ω ) G (Ω ) = ------------χ--(Ω-)------------ (307 ) xx
is the effective external force.

In the simplest and at the same time the most important particular case of free mirrors:

1 χxx,i(Ω ) = −------ i = 1,2, (308 ) mi Ω2
the reduced mechanical susceptibility and the effective external force are equal to
1 χxx(Ω) = − ----, (309 ) μ Ω2
and
[ ] G (Ω ) = μ G1(Ω-)+ G2-(Ω-)- , (310 ) m1 m2
where
( ) 1 1 − 1 μ = m-- + m-- (311 ) 1 2
is the effective mass of the elongation mechanical mode.

It follows from Eqs. (291View Equation) and (304View Equation) that the radiation pressure force can be written as a sum of the random and dynamical back-action terms, similarly to the single mirror case:

Fˆb.a.(Ω ) = ˆF(b0.a).(Ω) − K (Ω)ˆx (Ω ), (312 )
with the random component equal to
T Fˆ(0)(Ω ) = 2ℏk√pE---𝕃(Ω )[√γ1-ˆa1(Ω) + √ γ2ˆa2(Ω )] (313 ) b.a. τ
and the ponderomotive rigidity that reads
K (Ω) = M--Jδ-. (314 ) 𝒟 (Ω )

We introduced here the normalized optical power

2 2 J = 4ℏkp|E|--= 4-ωpℐc (315 ) M τ M cL
with
ℐc = ℏωp|E |2 (316 )
standing for the mean optical power circulating inside the cavity, and M is some (in general, arbitrary) mass. Typically, it is convenient to make it equal to the reduced mass μ.

Substitution of the force (312View Equation) into Eq. (305View Equation) gives the following final equation of motion:

−1 ˆ(0) [χxx(Ω ) + K (Ω )]xˆ(Ω) = Fb.a.(Ω ) + G (Ω ). (317 )
Thus, the effective mechanical susceptibility χeffxx,FP(Ω) for a Fabry–Pérot cavity reads:
χeff,FP(Ω ) = χ−1(Ω ) + K (Ω ) =-----1------. (318 ) xx xx K (Ω) − μΩ2

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