6.6 Concluding Remarks
7.1.3 Optomechanical Coupling, “Effective Displacement”, and “Spatial Overlap” . 103
We now consider the situation illustrated at the top of Figure 7.4: a compliant end-mirror vibrating in modeφk(x, y, z) with amplitudebkis coupled to a Hermite-Gauss mode,ψmn(x, y, z), of a Fabry- Perot cavity.
Note: for notational simplicity we hereafter drop the longitudinal and transverse indices, qmn, in the parameterization of the optomechanical system. This is justified by the fact that in practice usually only one cavity mode is excited.
When bk = 0, the resonance frequency of the cavity is hωci. When bk > 0, then the cavity resonance frequency is displaced by an amount
δωc=ωc− hωci=g0ηkbk ≡g0δzk0. (7.16) Factor g0 = hωci/L here corresponds to the optomechanical coupling for a rigid mirror in a two-mirror cavity, factor ηk depends on the spatial overlap between of the mechanical and optical spatial modes and will be referred to as the“spatial overlap factor”, andδz0k ≡ηcbk is referred to as the“effective displacement” of modeφk.
We ask: how is ηk related to the spatial profile of the mechanical mode and the cavity mode?
To get a handle on the situation, we can compute the phase shift experienced by the circulating Hermite-Gaussian beam upon reflection from the vibrating mirror (relative to the mirror in its equilibrium position bk = 0). To simplify the problem, we assume that the intracavity wavefront is nearly planar (ˆk≈z) at the mirror surface. We then follow closely the arguments of Gillespie andˆ
Raab [52], who express the phase shift upon reflection as:
ψ+mn(x, y, z)→ψmn− (x, y, z)e2ikmnbk~φk(x,y,z)·ˆz (7.17a)
≈ψ−mn(x, y, z)(1 + 2ikmnbkφ~k(x, y, z)·z)ˆ (7.17b)
= X
m0n0
cm0n0ψ−m0n0(x, y, z) (7.17c)
≈ψ−mn(x, y, z) 1 + 2ikmnbk· R
S|ψmn− (x, y, z)|2~φk(x, y, z)·zdσˆ R
S|ψ−mn(x, y, z)|2dσ
!
(7.17d)
=ψ−mn(x, y, z) 1 + 2ikmnbk· R
S|ψmn(x, y, z)|2~φk(x, y, z)·zdσˆ R
S|ψmn(x, y, z)|2dσ
!
. (7.17e)
Here |ψ|2 here indicates ψ∗ψ for scalar quantities and S denotes the integral over the surface of the miror. The assumptions behind each step are: (7.17b) bk is much smaller thanλmnq, (7.17d) the Hermite-Gauss modes form a complete set, (7.17d) the Hermite-Gauss modes are orthogonal according to (7.15) and only the original unperturbed mode of the cavity is resonant with the incoming laser. Eq. 7.17d is obtained by applying Eq. 7.15 to both sides of Eq. 7.17c. Eq. 7.17e follows from the relationship between Hermite-Gaussian beams and modes given in Eqs. 7.12a and 7.12b.
Eq. 7.17e suggests that displacing vibrational mode φk from equilibrium by an amount bk is equivalent to displacing the entire mirror by an“effective displacement” magnitude:
δz0k= R
S|ψmn(x, y, z)|2~φk(x, y, z)·ˆzdσ R
S|ψmn(x, y, z)|2dσ ·bk ≡ηkbk (7.18) such that the cavity resonance frequency shift is given byδωc=g0δzk0. The spatial overlap factor, ηk, is the weighted averaged of spatial mode functionφk(x, y, z) over the normalized intensity profile of the cavity mode,|ψmn(x, y, z)|2/R
S|ψmn(x, y, z)|2dσ, whereS indicates evaluation at the surface of the mirror.
By direct analogy, we infer that a membrane in the “membrane-in-the-middle” system, vibrating in a single modeφij with amplitudebij, will produce a cavity resonance frequency shift given by δωc=gm(zm)ηijbij where
ηij= R
S|ψmn(x, y, z)|2φ~ij(x, y, z)·zdσˆ R
S|ψmn(x, y, z)|2dσ . (7.19)
Mode φij is then described by an effective displacement δzijm ≡ ηijbij; δzmij corresponds to the displacement of the membrane’s equilibrum position required to shift the cavity resonance frequency by gm(zm)ηijbij. ηij is referred to as the spatial overlap factor for membrane mode φij. The main assumption made in drawing the above analogy is that, to first order, the phase of the field
Figure 7.4: Reducing the optomechanical system to one dimension. Upper plot: in a typical situa- tion, Hermite-Gaussian modeψ00(x, y, z) (TEM00) is coupled to a vibrational mode of one of the end- mirror substrates, characterized by the product of mode-shape functionφk(x, y, z) and generalized amplitudebk << λ. Amplitudebkgives rise to a shift in the cavity resonance frequencyδωc =g0ηkbk, where g0 = ωc,mn/L is the canonical end-mirror optomechanical coupling and ηk is the weighted average ofφk(x, y, z0) over the normalized intensity profile|ψ00(x, y, z0)|2/R
S|ψ00(x, y, z0)|2dσ, eval- uated at the position of the mirror surface,z0. This is equivalent to displacing the entire mirror by
“effective” magnitudeδzk0 =ηkδbk. Lower plot: the same set of reasoning applies to the “membrane- in-the-middle”. Here we have represented the (6,6) mechanical mode of a square membrane as the product of mode shape function φ66(x, y, z) and generalized amplitude b66. Vibration amplitude b66 gives rise to a resonance frequency shift δωc = gmη66b66, where gm is the 1D dispersive op- tomechanical coupling for the MIM system derived in Section 3.3.1 and η66 is defined as above.
This is equivalent to a displacement of the equilibrium position of the membrane by an amount δzm66=η66b66.
transmitted through the membrane isArg[tm] for all modesφij (doesn’t depend on the vibrational mode-shape).
Note that we can also define an effective mass of the effective displacement — a useful expres- sion for more complicated structures like the end-mirrors, discussed in Section 7.3.2. For generic vibrational modes with index k(representing either the mirror or the membrane), this quantity is defined:
mkef f ≡ mk
ηk2 = hEki
Ω2khδzmk2i (7.20a)
=
1 V
R
V |φ~k(x, y, z)|2dxdydz×R
S|ψmn(x, y, z)|2dxdydz R
Szˆ·~φk(x, y, z)|ψ(x, y, z)|2dxdydz ·mphys, (7.20b) where recall |φ|~2=φ~∗·φ~ for vectors and |ψ|2 =ψ∗ψ for scalars. The expression formkef f is useful because it does not depend on the choice of normalization for eitherψmn orφk.
The common thread between all of these expressions is the spatial overlap factor, ηk. We now consider a relevant example of its computation.
7.1.3.1 Example: TEM00 and Square Membrane
We will usually be concerned with the spatial overlap between a square membrane (widthwm) and a TEM00 cavity mode (waistwc). Assume that the membrane is located near the geometric center of the cavity (zm<< L/2) and displaced by (x0, y0) from the cavity axis. In this case the expression for the spatial overlap at the equilibrium position of the membrane (z=zm) simplifies to:
~φij(x, y, zm) = sin
iπ(x+w2m) wm
sin
jπ(y+w2m) wm
ˆ
z (7.21a)
ψmn+ (x, y, zm)≈i|N00|e
−(x−x0 )2−(y−y0 )2 w2
c eikmnzm (7.21b)
ηij ≈ 2 πw2m
Z wm2
−wm2
Z wm2
−wm2
sin
iπ(x+w2m) wm
sin
jπ(y+w2m) wm
e
−2(x−x0 )2−2(y−y0 )2 w2
c dxdy. (7.21c)
In Figure 7.5 we compute the spatial overlap between a square membrane mode and a TEM00
optical mode for two sets of parameters, {wc, wm, x0, y0} = {35.7 µm,500 µm,0 µm,0 µm} and {35.7µ m,500 µ m,45 µm,120 µm}, the latter corresponding to the position used in the optome- chanical cooling experiment in Chapter 9. Note that in the first case, only odd-ordered modes are coupled to the cavity. In the latter case, the (6,6) mode that we would like to address has reduced optomechanical coupling characterized by a spatial overlap factor ofη66= 0.64.
frequency (MHz) -1.0
-0.5 0.0 0.5 1.0
9 8 7 6 5 4 3 2 1 0
overlapfactor
x( ) x( )
y( )
(1,1) (2,2)
(3,3)
(6,6) (5,5) (4,4)
200 100 0 -100 -200
200 100 0 -100 -200
�m
�m
200 100 0 -100 -200
200 100 0 -100 -200
�m
�my( )
Figure 7.5: Spatial overlap factor between the TEM00cavity mode (wc= 35.7µm) and the first 100 modes of a square membrane with dimensions{dm, wm}={50 nm,0.5 mm}. Red points correspond to the membrane centered on the optical mode, {x0, y0} ={0µm,0µm}. Blue points correspond to an offset of {x0, y0} = {45µm,120µm}. Circles indicate “diagonal” modes φii. Contours for φ11,33,66(x, y, zm) =±1/2 are shown in the upper- and lower-right plots, with a density plot of the optical modeψ00(x−x0, y−y0, zm) overlaid. zmis the the equilibrium position of the membrane.