Scattering losses are introduced in a FP cavity, if an off-center membrane is in- cluded. This is because the reflected beam of the membrane will not match the cavity mode exactly. In this section, we will discuss the scattering losses by projecting the reflection of the fundamental mode back on itself. We limit our attention on the fundamental mode for - in practice - this mode possesses the lowest dissipation rateκ.
Figure 3.4: The reflection from the membrane is not an eigenmode of the FP cavity in general (depicted for one side, blue). By projecting the reflection mode profile on the FP mode (red), we attempt to get a notion of the scattering losses induced by an off-center the membrane.
We have to assume that the full-cavity eigenmode does not attune signifi- cantly upon the inclusion of the membrane, if we are to project the membrane’s reflection on the FP eigenmodes, and interpreted this as a scattering correction. The frequency of the resonant eigenmode may shift, and the field amplitude may differ between the sub-cavities (see section 3.2.3), however, the Hermite- Gaussian eigenmode shapes must be unaltered. Note that this assumption does not imply that the power reflected from the membrane is insignificant, just that any reflected field that does not overlap with the cavity mode is lost almost instantly.
The adaptation of the full-cavity eigenmode is driven by the collection of field in the eigenmodes between the membrane and a cavity mirror. If this pro- cess is efficient, then the full-cavity eigenmode and the sub-cavity eigenmode may mediate their field difference to lower scattering losses between them, thereby increasing the finesse.
It is a reasonable assumption that the process of collecting field in a sub- cavity eigenmode from the full-cavity mode is inefficient. Simply, the chance of the sub-cavity to have a eigenmode at the same resonance frequency as the cavity mode should be negligible.
To further shift our interest from the implication of an additional mode in the system, let us investigate the finesse of the sub-cavities. The sub-cavity round trip losses (ρsc) are given by
ρsc=RmemRm ≈ Rmem
whereRmemandRmare the reflectance of the membrane and mirror respec- tively.
The above approximation will hold in general, for a typical FP cavity for optomechnics will achieve a finesse ofO(105), which translates to a reflectance 1-Rm ofO(10−5). We obtain for the finesse (Fsc) of the sub-cavity
Fsc= π 2 arcsin1− √ ρsc 2√4ρsc ≈ π 4 √ Rmem 1−√Rmem .
A SiN membrane with an reflective aluminum coating will achieve Rmem ≈ 0.9. This translates to a finesse ofO(102). Based on TMM, a bare 50 nm SiN membrane should have R ≈ 0.2 (section 3.15). In this case, the finesse is of O(100). Then, the field build-up in the eigenmode between the membrane and the cavity mirror - if on-resonance - will be at least three orders less efficient as opposed to the field that builds up in the full cavity mode.
It is implied that the higher order modes can couple back to the fundamental mode after reflecting on the membrane. Same as for the fundamental mode, the reflected beam of the higher-order modes from the membrane is no longer an eigenmode of the cavity, and thus the reflection must be comprised of a combi- nation of eigenmodes (Equation (3.17)). It is thus implied that field gets trans- mitted into the fundamental cavity mode upon this reflection.
The problem of how higher order modes couple back to the fundamental mode is anything but trivial. For one, the projection between higher-modes is not analytically solvable. Second and foremost, the phases between the differ- ent modes can not be compared directly, for they all have different complex eigenvalues upon a full cavity round trip. It might be interesting to study sim- ple numerical models in combination with experimental data in order look for valuable extensions from the simplest case studied here.
At this point we neglect higher order modes in the reflected beam. It is worth stressing that we are thus investigating the absolute worst case scenario for the scattering losses.
To determine the field overlap between the reflection and cavity mode we use the in-plane-normalized fundamental Gaussian modes (similar to Equation (3.11)) [12]
3.1 Transverse modes in a Fabry-P´erot cavity 57 |w,κi =φ(w,κ;x) = (2/π)1/4w−1/2exp −(x/w)2+ikκx 2 2 where κ ≡ 1 R
is the wavefront curvature.
For the overlap integral of these modes we have (Equation (3.17)) [12]†
ν=|hw¯, ¯κ|w,κi|=2 (w¯/w+w/ ¯w)2+π λww¯ 2 (κ¯−κ)2 −1 2 . (3.21)
Upon reflection on the membrane, the radius of curvature is inverted [20]. We must thus assess (Figure 3.5)
ν(w,−R,w,R).
The values for w and R are determined based on the cavity specifications and the membrane position by the utilization of Equations (3.12), (3.13), and (3.19).
The above expression forνcan be obtained by performing the inner product
over a plane perpendicular to the optical axis (3.17). The Gaussian eigenmodes profiles are orthogonal over a perpendicular plane (Equation (3.15)). Then, the field overlap determined from the inner product over a perpendicular plane must yield the same coefficients as the inner product assessed in complete space
‡.
†Note that the power projection term ’
τ’ introduced by Joyce and DeLoach [12] is the power
projection of a 2D Gaussian mode, which needs to be squared to obtain the power transmission of the 3D Gaussian beam. This means that the 3D field projectionνused here actually equalsτ.
-0.005 0.000 0.005 0.010 0.015 0.020z(m) 0.2 0.4 0.6 0.8 1.0ν L=85 mm -0.005 0.000 0.005 0.010 0.015 0.020z(m) 0.2 0.4 0.6 0.8 1.0ν L=95 mm -0.005 0.000 0.005 0.010 0.015 0.020z(m) 0.2 0.4 0.6 0.8 1.0ν L=98 mm -0.005 0.000 0.005 0.010 0.015 0.020z(m) 0.2 0.4 0.6 0.8 1.0ν L=99.9 mm
Figure 3.5:(orange) The field projectionνof the reflection of the membrane onto the FP
cavity fundamental eigenmode. This is the case that the membrane is not tilted (θ
2 =0, Figure 3.6). The scattering loss is indicated for different cavity lengths (L), and increases with an increased offset of the membrane position as opposed to the cavity center (z) (Equation (3.21)). This is because the wave front is increasingly curved away from the center. Theblueline marks the Rayleigh range (b). The Rayleigh range gives an indication of the range over which the wave front is flat. If the membrane is positioned outside of this range, overlap between the reflected beam and cavity mode is poor. Perfect mode overlap - time reversal viewed from the reflected beam - is obtained if the phase of the beam is homogeneous over the membrane upon reflection, this occurs if the membrane is positioned in the center of the FP cavity (we neglect here the finite size of the membrane). Note thatνis a symmetric function ofz, so that the scattering losses
are the same for both sub-cavities.
If the membrane is tilted as opposed the cavity axis, additional mismatch between the reflection and cavity field is induced. At the intersection of the optical axes, in the plane perpendicular to the cavity field, for small angles we may treat the effect as a phase modulation based on the wavelength of the mode (so assuming thatκandware constant in the x-range where there is significant
3.1 Transverse modes in a Fabry-P´erot cavity 59 |w,κ,θi = (2/π)1/4w−1/2exp −(x/w)2+ikκx 2 2
·exp[iktan(θ)x]
= (2/π)1/4w−1/2exp −(x/w)2+ik θx+κx 2 2 .
Figure 3.6: A membrane at an angle θ
2 (orange) will generate a Gaussian reflection at angle ≈ θ. In the small angle approximation, it is presumed that the projection of the
angled fundamental Gaussian on the original beam may be determined from the field overlap at the plane where their optical axes meet. This plane is then chosen to be perpendicular to the cavity mode (blue line).
From this expression, Joyce and DeLoach continue their studying by men- tioning that the field coupling between the modes in the small field approxima- tion may now be determined by (still evaluated at the plane at the intersection of the optical axes that is perpendicular to one mode) [12]
νθ(w¯, ¯R,w,R) = |hw¯, ¯κ, 0|w,κ,θi| =ν·exp[−(θ/θe)] with θe = 1 πν 2 (w¯/λ)2+ (w/λ)2 12 . The results of this projection are shown in Figure 3.7.
It is not explicitly mentioned why it is justified to assess the inner product between these mode at this plane. Indeed, any paraxial field profile may be pro- jected on the basis defined by the cavity eigenmodes in this manner. However, the paraxial wave equation explicitly marks a function space with beams that share the same k-vector oscillation (expression (3.5)). The tilted beam is thus not
contained in the solution space of the paraxial wave equation with ˆk = zˆ. One may check this for oneself by substituting ˜u→ ue˜ i(θkz−θkx) in (3.7). Then, likely
this projection scheme is invalid for our system, though note that no explicit assumptions about the system are made in by Joyce and DeLoach [12].
To demonstrate the problem with assessing the inner product at a plane, the result is not the same once we move the evaluation plane a distance ’d’ across the z-axis (still small angle approximation)
|w,κ,θid = (2/π)1/4w−1/2exp −((x+θd)/w)2+ik θ(x+θd) + κ(x+θd) 2 2 = (2/π)1/4w−1/2exp −(x/w)2+2xd/w2+ik x(θ−κθd) + κx 2 2 +O(θ2)
Note thatwandκchange based on the position of the plane with respect to
the z-axis, this is not written down explicitly. If we neglect the real contribution ofdin the exponent,
|hw¯, ¯κ, 0|w,κ,θi|d ≈ν·exp[−(θ−κθd)/θe]
6=ν·exp[−θ/θe] = νθ(w¯, ¯R,w,R)
It then seems arbitrary to evaluate the inner product strictly in the plane as shown in Figure 3.6.
One might consider that d could be small in a specific system, however, the Gaussian modes are defined for −∞ <z< ∞. This implies that the inner product should also be assessed throughout all of space.
3.1 Transverse modes in a Fabry-P´erot cavity 61
Figure 3.7: The field projection coefficientν of the membrane reflection on the funda-
mental cavity mode versus cavity lengthL(=2zm) and membrane positionz. The effects
of the tilt of the membrane (θ
2) are computed based on the expression determined for the coupling of two fundamental Gaussian modes by Joyce and DeLoach [12]. How- ever, we have not been able to justify the validity of this expression for our system. Cross sections over the L-axis (cavity length) for the upper left graph are displayed in Figure 3.5.
Figure 3.8: Rather than projecting the reflection of a tilted membrane (green) on the original cavity mode (horizontal), it might be more insightful to project the reflection on the cavity eigenmode set with the same k-vector. These eigenmodes are then not rotated with respect to the reflection, but there exist an offset∆x between them. Note that the cavity mirrors only define a reflective sphere if their separation matches the sum of their absolute radii of curvature. This means that this is the only configuration in which there are multiple sets of Hermite-Gaussian eigenmodes (with different k- vectors). However, the length of the cavity should be close to2Rm, and we are only
interested in results for small angles. Therefor, it might be reasonable to implement this scheme anyway.
We established that is problematic that the reflection of a tilted membrane possess a different k-vector as opposed to the original cavity mode. A logical next step would be to project the reflection on the cavity eigenmodes with the same k-vector instead (Figure 3.8), because a shifted mode is still a solution to the (same) paraxial wave equation (3.7).
We obtain the form of a fundamental gaussian beam with an offset, where we assume again the the tilt (offset) is small
|w,κ,∂xi = (2/π)1/4w−1/2exp −((x−∂x)/w)2+ikκ(x−∂x) 2 2 = (2/π)1/4w−1/2exp 2x∂x−(x/w)2 +ikκ x2 2 −x∂x +O(∂x2) .
We end our discussion here, for important results are already obtained from the configuration without tilt. However, this might be a good starting point for an extended study.
3.1 Transverse modes in a Fabry-P´erot cavity 63