Internal Modes of an Elastic Body: Displacement and Effective Mass

6.6 Concluding Remarks

7.1.1 Internal Modes of an Elastic Body: Displacement and Effective Mass

The internal modes of a three-dimensional elastic body are obtained by solving the elastic wave- equation (for example, see Section 4.2.1). Eigensolutions to the wave equation (“modes”) are vector fields, ~uk(x, y, z, t) which describe the displacement of each point in the body from its equilibrium position (x, y, z) at time t. We will use index k to denote a mode of a generic 3-dimensional body (like the membrane or mirror substrates) and indices (i, j) when referring specifically to a square membrane vibrational mode. We assume solutions of the form:

~uk(x, y, z, t) =bk(t)φ~k(x, y, z). (7.1) Here φ~k(x, y, z) is a real-valued, unitless mode-shape function and bk(t) is a function of time with units of length, representing the generalized amplitude ofφk. Hereafter will refer to thebk as the amplitude of “mode φk”, to emphasize that the magnitude of bk depends on the normalization of the mode-shape function. bk(t) is assumed to obey the equation of motion for an internally damped harmonic oscillator (see Section 2.2). In the Fourier domain (bk(Ω) =R∞

−∞bk(t)e^−iΩtdt):

(−Ω²+iΓk(Ω)Ω + Ω²_k)bk(Ω) =Fk(Ω)/mk (7.2)

whereF_kis a generalized external force andm_k is the “effective mass” of coordinateb_k, to be defined below.

We will also assume that the eigenmodes comprise a complete, orthogonal set, so that any vibration of the elastic body can be written in the form~u(x, y, z, t) =P

kbk(t)φ~k(x, y, z) and Z

V

φ~_k(x, y, z)·φ~_k⁰(x, y, z)dxdydz=N δ_k,k⁰, (7.3)

whereδk,k⁰ is the Kronecker-delta function,N is a normalization factor, andV is the volume of the body.

In the absence of dissipation (Γ_k = 0) each infinitesimal mass element in the body,ρ(x, y, z)dV, describes 1D harmonic motion around its equilibrium position with amplitudeb_k(t)φ~_k(x, y, z), fre- quency Ω_k, and potential energy ¹₂ρ(x, y, z)dV b²_k(t)|φ~_k(x, y, z)|²Ω²_k. The total energyE_kof the mode is given by summing the energy of each mass element. For a thermally excited damped harmonic oscillator, the average energy of modeφk is given by equipartition:

hEki=hb²_kiΩ²_k Z

V

ρ(x, y, z)|φ~_k(x, y, z)|²dV =k_BT, (7.4)

where h isignifies the time average and |φ|~² ≡~φ^∗·φ~ for vector quantities (for scalar quantities it means the square modulus).

The integral in Eq. 7.4 has units of mass and can be identified as the effective mass m_k of the generalized amplitude b_k relative to the energy normalization condition hE_ki = m_kΩ²_khb²_ki. The effective mass is related to the physical mass mphys ≡ R

V ρ(x, y, z)dxdydz by a purely geometric

“effective mass coefficient”, αk ≡mk/mphys. In the simple case of uniform density, ρ(x, y, z) =ρ, we find

α_k≡ mk

m_phys = 1 V

Z

V

|φ~_k(x, y, z)|²dV = N

V (7.5a)

hb²_ki= hEki

α_km_physΩ²_k. (7.5b)

Note that there remains an essential ambiguity in the definition ofb_kandm_kuntil a normalization N for~φk(x, y, z) is chosen. This choice is completely arbitrary, and the game is to make a convenient choice depending on the physical process being modeled. We will ultimately be interested in the displacement of a small patch of the membrane surface, defined by the size and location of the cavity mode piercing the membrane. Towards this end, it is convenient to normalize the mode- shape function by setting its maximum value to unity, i.e., max

|φ~k|

= 1. bk then describes the amplitude of the point of maximum displacement in modeφk.

For a square membrane with mode indices (i, j), this normalization gives |φ~_ij| = 1 at each antinode. b_ij then represents the displacement of an antinode from the equilibrium position of the membrane. The vibrational mode shapes and eigenfrequencies of a square membrane with width wmare given by (see Section 4.2.1):

φ~_ij(x, y, z) = sin i πx

wm

sin

j πy wm

ˆ

z; {i, j} ∈ {1,2,3...} (7.6a) Ω_ij= 1

wm

s T 2ρ

ri²+j²

2 . (7.6b)

The effective mass ofb_ij turns out to be the same for all modes of a square membrane:

mij =ρ t Z w_m

0

Z w_m 0

sin² i πx

wm

sin²

j πy wm

dxdy (7.7a)

=1

4ρ tw²_m= 1

4m_phys. (7.7b)

It’s interesting to contrast this against the case of a circular drum of diameterwm, whose modes are described by Bessel functionsJm(r) [49]. Using the same normalization convention:

~φij(r, θ, z) = Jj(2xijr/wm) cos(jθ)

max (|J_j(2x_ijr/w_m)|)zˆ (7.8a) i∈ {1,2,3...}, j∈ {0,1,2, ...};x_ij≡i^thzero of J_j(r) (7.8b)

Ω_ij/2π= 1 wm

s T 2ρ

2x_ij

π . (7.8c)

In contrast to square membrane modes, the effective mass of circular membrane modes varies widely with mode order. Of particular interest are the mode-shape functions of axisymmetric (j= 0,max (|J0(2xi0r/wm)|) = 1) modes. This subset of “confined” modes can have significantly reduced effective masses because most of their displacement is localized to a reduced diameter∼wm/xi0:

mi0=ρ t Z 2π

0

Z w_m/2 0

J₀²(2xi0r/wm)rdrdθ=mphysJ₁²(xi0). (7.9) In Figure 7.20, the inverse effective mass coefficient (m_phys/m_ij) of square and circular drum modes is compared. In particular, we compare the subset of odd-ordered square modes (φi,j; i= 1,3,5...; j = 1,3,5...) and axisymmetric circular (φi0; i = 1,2,3...) modes. These are modes that have an antinode at the geometric center of the membrane. Their significance is as follows:

if the cavity waist were infinitesimally small and located at the center of the membrane, then δωc = ωc − hωci = gmbij would be the cavity resonance frequency shift induced by vibrational amplitude bij. Circular membranes may indeed be an attractive alternative in the future, owing to their small effective mass and the reduced density of modes which have non-zero amplitude at

frequency (MHz) 40

30 20 10

0 0 2 4 6 8

(1,1) (3,3) (5,5) (7,7) (9,9)

(1,0) (2,0)

(3,0)

(4,0)

(5,0)

(6,0)

(7,0)

Figure 7.3: Inverse effective mass coefficient of odd-ordered square membrane modes (φ_i,j; i = 1,3,5...; j = 1,3,5...) and axisymmetric circular membrane modes (φ_i0, i= 1,2,3...) as a function of frequency for membrane diameterwm= 500µm and tensionT = 900 MPa. The choice of modes is relevant to the situation where the membrane is probed at its center by a TEM00 cavity mode.

In this case — as explained in Section 7.1.3 — all square and non-axisymmetric circular membrane modes have vanishing optomechanical coupling.

cavity center. At the time of this writing, collaboration with Richard Norte in the Painter group has produced some high quality preliminary devices (wm= 200− −300µm,Q10∼10⁶).

In the next two subsections we derive the effective optomechanical coupling for a cavity spatial mode with finite size and arbitrary location with respect to cavity center.