The anatomy of coating thermal noise

2.4 Cross spectra of thermal noise components

2.4.3 The anatomy of coating thermal noise

Here let us assemble Eqs. (2.81), (2.86a) and (2.86b) from the previous sections, and write:

S_S^ij

zzS_zz(~x, z;~x⁰, z⁰) = 4kBT 3πf

(1 +σj)(1−2σj) Y_j(1−σ_j)²

h1 +σj

2 φBj + (1−2σj)φSj

i

δijδ⁽²⁾(~x−~x⁰)δ(z−z⁰), (2.87a) S_zszs(~x, ~x⁰) = 4k_BT

3πf

(1−σ_s−2σ_s²)² Y_s²

X

j

Y_jl_j (1−σj)²

h1−2σ_j

2 φ_Bj+1−σj+σ²_j 1 +σi

φ_Sji

δ⁽²⁾(~x−~x⁰), (2.87b) S_zsSzz(~x;~x⁰, z⁰) = 2kBT

3πf

(1−σs−2σ_s²)(1−σj−2σ_i²)

Y_s(1−σ_j)² [φ_Bj−φ_Sj]δ²(~x−~x⁰). (2.87c) Here we have assumed that z belongs to the i-th layer and that z⁰ belongs to the j-th layer, respectively. The thickness fluctuations of different layers are mutually independent [note the Kro- necker delta in Eq. (2.87a)], while the thickness fluctuation of each layer is correlated with the height fluctuation of the coating-substrate interface [Eq. (2.87c)].

Fluctuations in the strainSzz and the coating-substrate interfacezs, described by Eqs. (2.87a)–

(2.87b), can be represented alternatively as being driven by a number of independent fluctuating fields that exist throughout the coating. Such a representation allows us to better appreciate the origin and the magnitude of these fluctuations.

In order to do so, let us first define 3N thermal noise fields (i.e., 3 for each coating layer),n^B_j(x),

Thickness (δj) Surface height (zs) Bulk C_j^B=

r1 +σj

2 D_j^B= 1−σs−2σ²_s p2(1 +σj)

Yj

Ys

Shear

A C_j^SA=p

1−2σj D^S_j^A =−1−σs−2σ²_s 2p

1−2σ_j Yj

Y_s Shear

B (none) D_j^SB =

√3(1−σ_j)(1−σ_s−2σ_s²) 2p

1−2σj(1 +σj) Y_j Ys

Table 2.1: Transfer functions from bulk and shear noise fields to layer thickness and surface height.

n^S_j^A(x) andn^S_j^B(x), all independent from each other, with

S_nB

jn^B_k =4kBT(1−σj−2σ_j²)

3πf Yj(1−σj)² φ^j_Bδjkδ⁽³⁾(x−x⁰), (2.88a) S_nSA

j n^SA_k =S_nSB

j n^SB_k =4k_BT(1−σ_j−2σ_j²)

3πf Y_j(1−σ_j)² φ^j_Sδjkδ⁽³⁾(x−x⁰), (2.88b) and all other cross spectra vanishing. Herej labels coating layer, the superscriptB indicates bulk fluctuation, whileS_AandS_B label two types of shear fluctuations. The normalization of these fields are chosen such that each of these fields, when integrated over a length lj along z, have a noise spectrum that is roughly the same magnitude as a single-layer thermal noise.

Noise fieldsn^B_j(x),n^S_j^A(x) andn^S_j^B can be used to generate thickness fluctuations of the layers and the interface fluctuation (2.87a)–(2.87b) if we define

uzz(~x, z) =C_j^Bn^B_j(~x, z) +C_j^SAn^S_j^A(~x, z) (2.89) and

zs(~x) = X

j

Z L_j L_j+1

dz

D_j^Bn^B_j(~x, z) +D_j^SAn^S_j^A(~x, z) +D_j^SBn^S_j^B(~x, z)

. (2.90)

For each coating layer, C_j^B andD^B_j are transfer functions from the bulk noise field n^B_j to its own thickness δlj and to surface height zs, respectively; C_j^SA and D_j^SA are transfer functions from the first type of shear noise to thickness and surface height; finallyD_j^SB is the transfer function from the second type of shear noise to surface height (note that this noise field does not affect layer thickness).

Explicit forms of these transfer functions are listed in Table. 2.1.

Equations (2.89) and (2.90) owe their simple forms to the underlying physics of thermal fluctu- ations:

For bulk noise, i.e., terms involving n^B_j, the form of Eqs. (2.89) and (2.90) indicates that the interface fluctuation due to bulk dissipation is simply a sum of pieces that are directly proportional

substrate

coa ti ng l aye rs

_bulk

deformation

shear deformation

Figure 2.3: Illustration of the correlations between coating thicknessδl_jand the height of the coating- substrate interface, z_s. On the left, for a bulk deformation: when a coating element is expanding, its expansion along the x-y plane lifts the coating-substrate interface upwards, causing additional motion of the coating-air interface correlated to that caused by the increase in coating thickness. On the right, a particular shear mode: when a coating element is expanding, its contraction along thex- yplan pushes the coating-substrate interface downwards, causing addition motion of the coating-air interface anti-correlated to that caused by the increase in coating thickenss.

to the bulk-induced thickness fluctuations of each layer. This means the thermal bulk stress in a layer drive simultaneously the thickness fluctuation of that layer and a fluctuation of the coating-substrate interface. The fact that D^B_j and C_j^B have the same sign means that when thickness increases, the interface also rises (with intuitive explanation shown in Figure 2.3). This sign of correlation is generally unfavorable because the two noises add constructively towards the rise of the coating-air interface.

For shear noise, the situation is a little more complicated, because unlike bulk deformations, there are a total of five possible shear modes. From Eq. (2.73) and (2.74), it is clear thatf1, applied on opposites of Layer I (Figure 2.2), only drives thexx+yy−2zzshear mode and thexx+yy+zz bulk mode, while from Eq. (2.82) and (2.83), the force distributionfs drives three shear modes of xx−yy,xy+yx, andxx+yy−2zz. This means while thermal shear stresses in thexx+yy−2zz mode drives layer thickness and interface fluctuation simultaneously, there are additional modes of shear stress, xx−yy and xy+yx, that only drive the interface without driving layer thickness.

Our mode S_A, which drives both layer thickness and interface height, therefore corresponds to the physical shear mode ofxx+yy−2zz; our modeS_B, which only drives interface height, corresponds to the joint effect of the physical shear modesxx−yy andxy+yx. It is interesting to note that for S_A, its contributions toδl_j andz_s are anti correlated, because C^SA andD^SA have opposite signs.

This is intuitively explained in Fig. 2.3.

As an example application of Eqs. (2.89) and (2.90), if we ignore light penetration into the coating

layers, namely, when thermal noise is equal to

ξ^np≡ −zs−X

j

δlj (2.91)

we have

ξ^np=−X

j

Lj+1

Z

L_j

dzh

C_j^B+D_j^B n^B_j

+

C_j^SA+D_j^SA n^S_j^A

+D_j^SBn^S_j^Bi

(2.92) in which contributions from each layer have been divided into three mutually uncorrelated groups, each arising from a different type of fluctuation. Here we see explicitly that C^B and D^B sharing the same sign increases contributions fromn^B, andC^SA andD^SA having opposite signs suppresses contributions fromn^SA.

Finally, we note that in the spectral density ofξ^np, contributions directly from coating thickness will be proportional to|C_j^B|²and|C_j^SA|², and hence proportional to 1/Y_c, those from interface height will be|D^B_j|², |D_j^SA|² and |D_j^SB|2, and hence proportional toY_c/Y_s², while those from correlations will be proportional to C_j^BD_j^B and C_j^SAD^S_j^A, and hence proportional to 1/Y_s. This confirms our anticipation at the end of Sec. 2.3.4.