End-Mirror Coupling in a MIM Cavity

7.3 Multimode Thermal Noise Spectrum

7.3.2 Mirror Substrate Thermal Noise

7.3.2.3 End-Mirror Coupling in a MIM Cavity

It’s important to mention that taking an incoherent sum of the effective displacement noise of each end-mirror substratedoes not describe the effective cavity length noise due to the substrates when

Figure 7.10: Plots of membrane and single end-mirror coupling in the MIM cavity as a function of position of the membrane. Red, purple, and cyan curves correspond to the optomechanical coupling of the membrane, mirror 1, and mirror 2, respectively, for the 1-dimensional system sketched at the top. The left column and the right column correspond to the equilibrium membrane position, z_m located near cavity center and near mirror 2, respectively.

the membrane is present. This is because unlike a normal Fabry-Perot cavity, the MIM cavity is sensitive to common-mode translation of the two mirrors (indeed, translating the two mirrors in tandem is indistinguishable from translating the the membrane). To include this effect, we first numerically compute the optomechanical coupling of the entry (mirror 1) and exit (mirror 2) end mirrors in the MIM cavity as a function of membrane position, using the plane-wave model discussed in Section 3.3.1. The results are shown in Figure 7.10, with g_m/g₀, g₁/g₀, and g₂/g₀ representing the optomechanical coupling of the membrane (effective displacement δz_m), mirror 1 (effective displacementδz₁), and mirror 2 (effective displacementδz₂) relative to the canonical end-

mirror coupling,g₀=ω_c/L. All three couplings are functions of the membrane’s equilibrum position, z_m. By numerically adding and subtracting the curves shown in Figure 7.10, it can be shown that:

δωc =g1δz1+g2δz2+gmδzm (7.33a) 2g₀=g₁

1 +2zm

L

−g₂

1−2zm

L

(7.33b)

−g_m=g₁+g₂. (7.33c)

There are at least two interesting ways to write (7.33a): in terms of end mirror displacementsδz1,2, and in terms of symmetric/antisymmetric displacement: δz_± = (δz₁±δz₂)/2. Combining (7.33b) and (7.33c), we find:

δωc=g1δz1+g2δz2+gmδzm=g+δz++g−δz−+gmδzm (7.34a) g₁=g₀−g_m

2

1−2z_m L

(7.34b) g₂=−g₀−gm

2

1 + 2zm

L

(7.34c)

g₊=g₁+g₂=−g_m (7.34d)

g−=g1−g2= 2 g0+gm

zm

L

. (7.34e)

Evidently the optomechanical coupling ofδz1 andδz2are both functions of the membrane position zm. As expected, the optomechanical coupling (g+) to symmetric displacement (δz+) is equivalent to the optomechanical coupling of the membrane (gm) and vanishes when the membrane is at a node or an antinode of the intracavity field. The optomechanical coupling (g−) to antisymmetric (cavity-length-changing) displacement of the mirrors (δz−) is a complicated function of membrane position, but reduces to twice the canonical coupling (2g₀) when the membrane is located at the midpoint of the cavity.

Finally, using expressions for g₁ and g₂ and assuming z_m = 0, we can write down the total effective displacement noise of MIM cavity as:

SL(Ω) =

1− gm

2g0

2

Sz₁(Ω) +

1 + gm

2g0

2

Sz₂(Ω) + gm

g0

2

Sz_m(Ω), (7.35a) where superscripts{1,2, m} indicate displacement noise of mirror 1, mirror 2, and the membrane, respectively.

Chapter 8

Cavity-Based Thermal Noise Measurement

In the previous chapter we described the optomechanical coupling of an internally vibrating mirror or membrane in the MIM system. We then developed a model for the multimode thermal noise of the composite system. In this chapter we discuss the task of measuring thermal noise in lab, developing in detail several subjects that have been points of concern. In overview: (1) we describe an input- output model for the MIM cavity based on the canonical two-mirror optomechanical system. (2) We derive transfer function characterizing two techniques used to map membrane/mirror fluctuations into photocurrent fluctuations. The first technique consists of driving the cavity with a detuned input field and monitoring the transmitted power. The second technique consists of driving the cavity with a resonant input field and monitoring the phase of the reflected field, using the “Pound- Drever-Hall” method. (3) We predict the shot noise limited sensitivity of these measurements and compare to the predicted thermal noise of the membrane. (4) We detail a technique for calibration of the displacement measurement by phase modulating of the input field. (5) We walk through three important examples: (a) measurement of the temperature of a single membrane mode, (b) calibration of the spectrum of “spatial overlap” factors, {η_ij} (Section 7.1.3) using a multimode thermal noise measurement, and (c) characterization of the broadband displacement noise background due to laser frequency noise and thermal motion of the end-mirror substrates.

8.1 Basic Approach: Measurement Response Function

To treat mirror and membrane motion on equal footing, we will use the fact that in any optome- chanical system, displacement of the mechanical element manifests itself as a displacement of the cavity resonance frequency, δω_c(t) =gδz(t), where the definition of displacement coordinateδz(t) and optomechanical couplinggdepends on the geometry of the system (see, e.g., Section 7.1.3). In the lab, with help of the cavity transfer function, we transform cavity frequency fluctuations into

photocurrent fluctuations, δi(t) =i(t)− hii, by directing the field leaking out of the cavity onto a photodiode. The transformation constitutes a “measurement”, and we will assume that it is linear and described by a response functionGi,ω_c(Ω) (Section 2.1):

G_i,ωc(Ω) = δi(Ω)

δωc(Ω) A/Hz; {δi(Ω), δωc(Ω)}= Z ∞

−∞

{δi(t), δωc(t)}e^−iΩtdt. (8.1)

In the framework of linear response theory, measurement of noisy signals can be described using the transfer function |Gi,ω_c(Ω)|² (Eq. 2.12). By noise, we will mean a random signal, e.g., a noisy photocurrenti(t), described by a single-sided power spectral density,Si(Ω) (Eq. 2.9), with units of A²/Hz, normalized such that R

S_i(Ω)dΩ/2π=hi²(t)i. The mapping of cavity resonance frequency noise into photocurrent noise is characterized by:

Si(Ω) =|Gi,ω_c(Ω)|²Sω_c(Ω). (8.2) Using the nomenclature from Section 7.1.3, the generic relationship between photocurrent noise, cavity resonance frequency noise, and effective/actual displacement noise for multimode vibration of the membrane can be summarized as follows:

Si(Ω) =|Gi,ω_c(Ω)|²Sω_c(Ω) =|Gi,ω_c(Ω)|²g_m²Sz_m(Ω) =|Gi,ω_c(Ω)|²g_m² X

ij

η²_ijSb_ij(Ω). (8.3a)

Here S_zm(Ω) is the spectral density of effective membrane displacement, and S_bij(Ω) is spectral density of fluctuations in physical vibrational mode amplitudeb_ij.