Optickle 2

M. Evans

June 12, 2014

Abstract

Optickle is a general model for the electro-opto-mechanical part of an interferometric GW detector. It ventures into mechanics only as far as is necessary to include radiation pressure effects, and into electronics only far enough to produce demodulation signals, and into optics only up to first order. There are many other tools that do all these things in greater detail. Optickle is for quick, but essentially complete interferometer design studies.

1 Introduction

As a general tool for opto-mechanical modeling Optickle can simulate any interferometer in which a single wavelength of light (with RF sidebands) is used. The construction of an Optickle model is discussed in detail in the “Optickle Function Reference” and several example systems are provided in the “Demo” area of the Optickle package. This document discusses the theory and implementation of the functions which compute the behavior of an Optickle system in terms of field amplitudes and transfer functions.

2 Example System

In the following sections, I will make frequent attempts to clarify by reference to the example optical system presented in this section. The example is a Fabry-Perot cavity, composed of two mirrors a source (the laser) and a sink (the photo-detector, see figure 1).

The source is responsible for illuminating the system. Let’s assume that it generates a carrier and 2 RF sidebands from RF phase modulation approximated to first order. Thus, the frequencies of the field components,

Figure 1: The example system: a Fabry-Perot cavity.

relative to the carrier field, are v⃗_{f_{RF}} = [−f_{mod}, 0, f_{mod}], where f_{mod} is the RF modulation frequency.

The field produced by the source propagates via Link 1 to the back input of Mirror A. At the end of each link is a ”field evaluation point”, or FEP. FEPs are represented by light red balls in figure 1, and are labeled E1, E2, etc. The field at E1, for example, is clearly just the source field multiplied by the propagation phase determined by the length of Link 1.

The field computation can be described in two parts: static (DC) fields, and audio frequency (AF) fields. The DC fields are present in the optical system when none of the optics are driven. The AF fields are the fields generated by driving the optics. The following sections describe these computations in detail.

3 DC Fields

The collection of DC fields is relatively small: one for each RF field component, at each field evaluation point (FEP). That is, N_{field} = N_{RF} N_{link}, where N_{RF} is the number of RF components and N_{link} the number of links, such that for our example system N_{RF} = 3, N_{link} = 4 ⇒ N_{field} = 12. Despite this not being a very big number, it is enough to make matrices unwieldy and complicated, so for the next few paragraphs I’ll use just one RF component v⃗_{f_{RF}} = [f_{mod}]. To compute the DC fields in the optical system Optickle assumes the steady state equation

v⃗_{DC} = M_{DC}v⃗_{DC} + v⃗_{source}  (1)

where M_{DC} is the optical propagation matrix between FEPs, v⃗_{DC} is the vector of fields at each FEP, and v⃗_{source} is the vector of injected fields (e.g., from the source in figure 1). Solving for v⃗_{DC}

v⃗_{DC} = (𝕀 − M_{DC})^{−1}v⃗_{source}  (2)

where 𝕀 is the identity matrix. The sources are given, so v⃗_{source} is known and the computation boils down to computing the inverse of 𝕀 − M_{DC}. The propagation matrix is built from the optics and links in the optical system. For example, the matrix element which takes E4 to E3 is determined by the reflectivity of Mirror A and the length of Link 3. Thus, this element^{1} of M_{DC} is

M_{DC}(3, 4) = −r_{A}e^{i2π f_{mod} l_{3}/c}.  (3)

More generally speaking, M_{DC} is the sum of all of the input to output transfer matrices of all optics, multiplied by the phase induced by the links,

M_{DC} = M_{ϕ}M_{opt}  (4)

with

M_{opt} = ∑_{n=1}^{N_{optic}} M_{out_{n}}M_{opt_{n}}M_{in_{n}}.  (5)

As shown in the above equation, the contribution of a given optic to the overall transfer matrix is made up of three parts, a matrix which maps FEPs onto the inputs of the optic, the optic’s own transfer matrix, and a matrix which maps the optic’s outputs back to FEPs.

Taking Mirror A again from our example system (figure 1), and noting that its index is 3, its front input is from FEP 4 and its back input from FEP 1, the corresponding input matrix is

M_{in_{3}} = \begin{aligned}[ \\ [ & 0 0 0 1 1 0 0 0 & ] \\ & & ] \end{aligned} .

The output matrix, sending the field from the front of Mirror A to FEP 3 and from the back to FEP 2 is

M_{out_{3}} = [ ; [ \begin{matrix}0 & 0 \\ 0 & 1 \\ 1 & 0 \\ 0 & 0 \end{matrix} ] ; ] .

1The indexing used in this document will reflect what is used in the Optickle code. Indices will start at 1, and the colon operator indicates a series of integers (both standard in Matlab).

Finally, assuming that Mirror A has an amplitude transmission coefficient of t_{A} and an amplitude reflectivity of r_{A} (which may be complex),

M_{opt_{3}} = \begin{aligned}[ \\ [ & \begin{matrix}−r_{A} & t_{A} \\ t_{A} & r_{A} \end{matrix} & ] \\ & & ] \end{aligned} .  (6)

The sum is used in equation 5 to combine the matrices of all optics, though it is expected that no element will contain contributions from more than one optic. That is, the transfer from one FEP to another happens only through the optic that connects them, which Optickle forces to be unique during construction.

Returning to the full set of RF components in our example system v⃗_{f_{RF}} = [−f_{mod}, 0, f_{mod}], the matrices for Mirror A are duplicated in block diagonal form with one block for each RF component, becoming

M_{in_{3}} = [ ; [ \begin{matrix}0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ & & & & 0 & 0 & 0 & 1 \\ & & & & 1 & 0 & 0 & 0 \\ & & & & & & & & 0 & 0 & 0 & 1 \\ & & & & & & & & 1 & 0 & 0 & 0 \end{matrix} ] ; ] ,

M_{out_{3}} = [ ; [ \begin{matrix}0 & 0 \\ 0 & 1 \\ 1 & 0 \\ 0 & 0 \\ & & 0 & 0 \\ & & 0 & 1 \\ & & 1 & 0 \\ & & 0 & 0 \\ & & & & 0 & 0 \\ & & & & 0 & 1 \\ & & & & 1 & 0 \\ & & & & 0 & 0 \end{matrix} ] ; ] ,

M_{opt_{3}} = [ ; [ \begin{matrix}−r_{A} & t_{A} \\ t_{A} & r_{A} \\ & & −r_{A} & t_{A} \\ & & t_{A} & r_{A} \\ & & & & −r_{A} & t_{A} \\ & & & & t_{A} & r_{A} \end{matrix} ] ; ] .

Note that the input and output transformations do not change the RF frequency of a field component, and in the case of a mirror, nor does the optic. However, some optics can convert one RF component to another (e.g., an RF modulator), and this will cause non-zero matrix elements appear in the normally empty off-diagonal areas of the optic’s transfer matrix.

Lastly, the propagation phase associated with moving from the input of a link to its output appears in the diagonal matrix

M_{ϕ}(k, k) = e^{i v⃗_{f_{RF}} (m) v⃗_{length}(n)}  (7)

with n ∈ 1 : N_{link}, m ∈ 1 : N_{RF} and k = n + N_{link}(m − 1).

v⃗_{length}(n) = 2π l_{n}/c  (8)

At this point we have constructed M_{DC} from the parameters of the Optickle model, so we need only put it into equation 2 to find the DC field vector v⃗_{DC}. The result is then used as the seed for computing the response of the system to excitation of one of the optic’s internal degrees of freedom, discussed in the following section.

4 AC Fields

The computation of AC fields is similar to that of DC fields, except that the optics now play a more active role. Moving Mirror A at 100Hz, for example, generates 2 audio frequency sidebands on each RF field component of the reflected fields (from E1 into Link 2 and from E4 to Link 3). AF sidebands, in turn, beat against static fields to produce forces on optics and signals on sensors.

In the AC case, the number of degrees of freedom to be considered more than doubles relative to the DC computation, and the matrices are no longer relationships just between fields, but are generalized to include the optics in the system. The AC equation analogous to equation 2 is

v⃗_{AC} = (𝕀 − M_{AC})^{−1}v⃗_{excitation}  (9)

where a given excitation vector produces a vector of audio sideband fields. A further transformation, determined by the probes place in the system, converts fields to signals

v⃗_{signal} = M_{prb}(𝕀 − M_{AC})^{−1}v⃗_{excitation}  (10)

The transfer matrix, from normalized excitations to signals, is thus just the core of the above equation,

M_{T F} = M_{prb}(𝕀 − M_{AC})^{−1}.  (11)

This matrix is one of the primary results produced by the Optickle compute function.

In Optickle the audio frequency transfer matrix, M_{AC}, is constructed in blocks

M_{AC} = \begin{aligned}[ \\ [ & \begin{matrix}M_{field−field} & M_{optic−field} \\ M_{field−optic} & M_{optic−optic} \end{matrix} & ] \\ & & ] \end{aligned} .  (12)

where M_{field−field} is a matrix similar to M_{DC} which represents transfers among fields, though in this case they are AF sideband fields. As the names indicate, M_{optic−field} and M_{field−optic} represent the relationships between the optics’ degrees of freedom and the AF sideband fields. The M_{optic−optic} matrix is diagonal, with non-zero terms resulting from the radiation reaction force which can be produced by static fields on an optic. The following sections will describe the construction of each of these matrices in detail.

4.1 Field to Field Transfer Matrix

This part of is almost identical to M_{DC}, the major differences being that for each RF component 2 AF sidebands must be computed, and that the propagation phase associated with the links must account for the sum of the RF and AF phases. Restating,

M_{field−field} = \begin{aligned}[ \\ [ & \begin{matrix}M_{ϕ_{+}}M_{opt} & 0 \\ 0 & M_{ϕ_{−}}^{∗}M_{opt}^{∗} \end{matrix} & ] \\ & & ] \end{aligned}  (13)

where in this case the link phase is

M_{ϕ_{±}}(k, k) = e^{i (v⃗_{f_{RF}} (m)±f_{AF}) v⃗_{length}(n)}  (14)

which is the same as equation 7, with the addition of the audio frequency component. The conjugation of the lower sideband component is necessary for the optic-field and field-optic calculations which follow, and allows for a direct mapping from the audio sideband fields in Optickle to the two-photon formalism.

4.2 Optic to Field Transfer Matrix

The optic to field matrix, much like the field to field matrix, must result in both upper and lower audio sidebands at FEPs, so it is constructed as

M_{optic−field} = \begin{aligned}[ \\ [ & \begin{matrix}M_{ϕ_{+}}M_{gen} \\ M_{ϕ_{−}}^{∗}M_{gen}^{∗} \end{matrix} & ] \\ & & ] \end{aligned}  (15)

The matrix M_{gen} is composed of an optic specified matrix and DC fields, and can be written as

M_{gen} = ∑_{n=1}^{N_{optic}} M_{out_{n}}M_{gen_{n}}M_{drv_{n}}^{T}.  (16)

where M_{drv_{n}} is the matrix which maps the internal degrees of freedom of the n^{th} optic onto the system wide degrees of freedom. The DC fields are present as source fields which the optics modulate to produce audio SBs,

M_{gen_{n}} = \begin{aligned}[ \\ [ & M_{gen_{n,1}}, M_{gen_{n,1}}, ⋯, M_{gen_{n,N_{dofn}}} & ] \\ & & ] \end{aligned} (17)

where

M_{gen_{n,m}} = M_{cpl_{n,m}}v⃗_{DC_{n,in}}  (18)

Here, M_{cpl_{n,m}} is the modulation produced by driving optic n, degree of freedom m. The indices are n ∈ 1 : N_{optic}, m ∈ 1 : N_{dof_{n}}.

The coupling matrix M_{cpl_{n,m}} can be written in terms of M_{opt_{n}}

M_{cpl_{n,m}} = \frac{1}{2} \frac{∂M_{opt_{n}}}{∂x_{n,m}}  (19)

where x_{n,m} the m^{th} degree of freedom of the n^{th} optic. The usual input and output transforms are in place to map the source fields in and the generated audio-sidebands out. After the output mapping, M_{ϕ} appears in equation 15 to carry the audio-sidebands from the output of the optics where they are generated, which are inputs of the associated links, to the FEPs at the outputs of the links.

As an example, let’s look at audio sideband generation by a mirror at normal incidence. Given an input field E_{in}, the field reflected from the front of the mirror is

E_{out} = −re^{−2ikx} E_{in}  (20)

since forward motion of the mirror by x causes a shortening the propagation path of the beam by 2x. We will use a sinusoidal motion for x

x = Re\left[{ \atop x̃}\right] cos(Ωt) + Im\left[{ \atop x̃}\right] sin(Ωt)  (21)
= \frac{1}{2}Re\left[{ \atop x̃}\right] (e^{iΩt} + e^{−iΩt}) − \frac{i}{2}Im\left[{ \atop x̃}\right] (e^{iΩt} − e^{−iΩt})  (22)
= \frac{1}{2} (x̃^{∗} e^{iΩt} + x̃ e^{−iΩt})  (23)

for some complex x̃.

For small x, such that kx ≪ 1, we can approximate the exponential to first order as

E_{out} = −r(1 − 2ikx)E_{in}  (24)
= −rE_{in} + ikr E_{in} (x̃^{∗} e^{iΩt} + x̃ e^{−iΩt})  (25)
= −rE_{in} + E_{+}e^{−iΩt} + E_{−}e^{iΩt}  (26)
where  E_{+} = ikr E_{in} x̃ E_{−} = ikr E_{in} x̃^{∗}  (27)

Thus, the production of audio sidebands in reflection from the mirror is given by

\begin{aligned}[ \\ [ & \begin{matrix}E_{+} \\ E_{−}^{∗} \end{matrix} & ] \\ & & ] \end{aligned} = \begin{aligned}[ \\ [ & \begin{matrix}+ikrE_{in} \\ −ikrE_{in}^{∗} \end{matrix} & ] \\ & & ] \end{aligned} x̃  (28)

which is just a single element version of equations 15, 17 and 19 in which E_{in} is one element of v⃗_{DC}, −re^{−2ikx} is one element of M_{opt}, and the propagation phase to the next FEP is not yet accounted for.

4.3 Field to Optic Transfer Matrix

This matrix is actually the product of the field-to-force matrix, and the forceto-degree of freedom matrix, where “degree of freedom” most often means “position”.

M_{field−optic} = M_{resp}M_{rad}  (29)

The response matrix M_{resp} is typically the mechanical response of the optics to radiation pressure induced force, summed over all optics in the system,

M_{resp} = ∑_{n=1}^{N_{optic}} M_{drv_{n}}M_{resp_{n}}M_{drv_{n}}^{T}  (30)

M_{rad} converts audio sideband fields to radiation forces on the optics, and the computation of this matrix is the topic of rest of this section.^{2}

2 In the following derivation, we will assume the optic in question is a mirror with incoming beams from the front and back at normal incidence. We will also consider only a single RF field component. The generalization of the result will follow.

To compute the force available for generating mechanical motion, we start by summing up the incoming and outgoing momenta

F = (P_{back_{in}} + P_{back_{out}} − P_{front_{in}} − P_{front_{out}}) /c  (31)

Note that this produces an easily recognized result in the special case of a perfect absorber: F_{a} = (P_{back_{in}} − P_{front_{in}})/c, since the beam on the back pushes forward and the one on the front pushes back. And also in the case of a perfect reflector: F_{r} = 2(P_{back_{in}} − P_{front_{in}})/c, since the outgoing beams have the same power as the incoming beams.

The modulation of each incoming and outgoing power by the co-propagating DC field and its audio SBs is given by

P_{X} = |E_{DC_{X}} + E_{+_{X}} e^{−iΩt} + E_{−_{X}} e^{iΩt}|^{2}  (32)

where X ∈ {back_{in,out}, front_{in,out}}, E_{±_{X}} are the upper and lower sideband amplitudes, and Ω = 2πf_{audio} is the audio angular frequency. Converting this expression to trigonometric form

P_{X} = Re\left[{ \atop P̃_{X}}\right] cos(Ωt) + Im\left[{ \atop P̃_{X}}\right] sin(Ωt)  (33)
P̃_{X} = 2 (E_{DC_{X}}^{∗} E_{+_{X}} + E_{DC_{X}} E_{−_{X}}^{∗} )  (34)

For the n^{th} optic we can compute E_{DC} at the outputs, given the inputs, with the field to field matrix M_{opt_{n}} as

v⃗_{DC_{n,out}} = M_{opt_{n}}v⃗_{DC_{n,in}} where v⃗_{DC_{n,in}} = M_{in_{n}}v⃗_{DC}  (35)

For the AF fields, the situation is somewhat more complicated since they can be produced by motion of the optic and thus appear at the output without coming in the input

v⃗_{AC_{n,in}} = \begin{aligned}[ \\ [ & \begin{matrix}M_{in_{n}} & 0 \\ 0 & M_{in_{n}}^{∗} \end{matrix} & ] \\ & & ] \end{aligned} v⃗_{AC_{fld}}  (36)
v⃗_{AC_{n,out}} = \begin{aligned}[ \\ [ & \begin{matrix}M_{opt_{n}} & 0 \\ 0 & M_{opt_{n}}^{∗} \end{matrix} & ] \\ & & ] \end{aligned} v⃗_{AC_{n,in}} + \begin{aligned}[ \\ [ & \begin{matrix}M_{gen_{n}} \\ M_{gen_{n}}^{∗} \end{matrix} & ] \\ & & ] \end{aligned} x̃_{n} (37)

where M_{gen_{n}} is a matrix representing the AF fields generated by the n^{th} optic (see equation 17).

To finish the translation to matrix math, we should convert the sum in equation 31 to a matrix multiplication. Before continuing with this math, we will, however, drop the optic index n; all of these matrices and vectors are for a particular optic.

F̃_{in} = \frac{2}{c} \begin{aligned}[ \\ [ & v_{DC_{in}}^{†}M_{dir_{in}}, v_{DC_{in}}^{T}M_{dir_{in}} & ] \\ & & ] \end{aligned} v⃗_{AC_{in}}  (38)
= \frac{2}{c} \begin{aligned}[ \\ [ & \begin{matrix}M_{dir_{in}}v⃗_{DC_{in}} \\ M_{dir_{in}}v_{DC_{in}}^{∗} \end{matrix} & ] \\ & & ] \end{aligned}^{†} v⃗_{AC_{in}}  (39)

where, M_{dir_{in}} is a diagonal matrix of real coefficients. For the example of a mirror with the first input on the front and the second on the back

M_{dir_{in}} = \begin{aligned}[ \\ [ & \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} & ] \\ & & ] \end{aligned}  (40)

The output is similar,

F̃_{out} = \frac{2}{c} \begin{aligned}[ \\ [ & v_{DC_{out}}^{†}M_{dir_{out}}, v_{DC_{out}}^{T}M_{dir_{out}}^{∗} & ] \\ & & ] \end{aligned} v⃗_{AC_{out}} (41)
= \frac{2}{c} \begin{aligned}[ \\ [ & \begin{matrix}M_{dir_{out}}M_{opt}v⃗_{DC_{in}} \\ M_{dir_{out}}M_{opt}^{∗}v_{DC_{in}}^{∗} \end{matrix} & ] \\ & & ] \end{aligned}^{†} v⃗_{AC_{out}}  (42)

such that the total force is

F̃ = \frac{2}{c} [ [ \begin{matrix}(M_{opt}^{†}M_{dir_{out}}M_{opt} + M_{dir_{in}})v⃗_{DC_{in}} \\ (M_{opt}^{T}M_{dir_{out}}M_{opt}^{∗} + M_{dir_{in}})v_{DC_{in}}^{∗} \end{matrix} ] ]^{†} v⃗_{AC_{in}} (43)

+\frac{2}{c} \begin{aligned}[ \\ [ & \begin{matrix}M_{dir_{out}}M_{opt}v⃗_{DC_{in}} \\ M_{dir_{out}}M_{opt}^{∗}v_{DC_{in}}^{∗} \end{matrix} & ] \\ & & ] \end{aligned}^{†} \begin{aligned}[ \\ [ & \begin{matrix}M_{gen} \\ M_{gen}^{∗} \end{matrix} & ] \\ & & ] \end{aligned} x̃

= \frac{2}{c} [ [ \begin{matrix}(M_{opt}^{†}M_{dir_{out}}M_{opt} + M_{dir_{in}})v⃗_{DC_{in}} \\ (M_{opt}^{T}M_{dir_{out}}M_{opt}^{∗} + M_{dir_{in}})v_{DC_{in}}^{∗} \end{matrix} ] ]^{†} v⃗_{AC_{in}} (44)

+\frac{4}{c}Re[v_{DC_{in}}^{†}M_{opt}^{†}M_{dir_{out}}M_{gen}] x̃

This can be reconnected to our starting point with the concrete example of a mirror using M_{opt} from equation 6, and a single incoming field on the front surface:

M_{opt} = \begin{aligned}[ \\ [ & \begin{matrix}−r & t \\ t & r \end{matrix} & ] \\ & & ] \end{aligned} , v⃗_{DC_{in}} = \begin{aligned}[ \\ [ & \begin{matrix}E_{DC} \\ 0 \end{matrix} & ] \\ & & ] \end{aligned} , v⃗_{AC_{in}} = [ ; [ \begin{matrix}E_{+} \\ 0 \\ E_{−}^{∗} \\ 0 \end{matrix} ] ; ] (45)

Thus,

M_{opt}^{†}M_{dir_{out}}M_{opt} = \begin{aligned}[ \\ [ & \begin{matrix}−r & t \\ t & r \end{matrix} & ] \\ & & ] \end{aligned}^{†} \begin{aligned}[ \\ [ & \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} & ] \\ & & ] \end{aligned} \begin{aligned}[ \\ [ & \begin{matrix}−r & t \\ t & r \end{matrix} & ] \\ & & ] \end{aligned} (46)
= \begin{aligned}[ \\ [ & \begin{matrix}|t|^{2} − |r|^{2} & tr^{∗} + rt^{∗} \\ tr^{∗} + rt^{∗} & |r|^{2} − |t|^{2} \end{matrix} & ] \\ & & ] \end{aligned}  (47)
= \begin{aligned}[ \\ [ & \begin{matrix}1 − 2R − L & tr^{∗} + rt^{∗} \\ tr^{∗} + rt^{∗} & 2R + L − 1 \end{matrix} & ] \\ & & ] \end{aligned}  (48)

where the power reflectivities and losses are R = |r|^{2}, and L = 1−|r|^{2}−|t|^{2}. This leads us to

M_{opt}^{†}M_{dir_{out}}M_{opt} + M_{dir_{in}} = \begin{aligned}[ \\ [ & \begin{matrix}−(2R + L) & tr^{∗} + rt^{∗} \\ tr^{∗} + rt^{∗} & 2R + L \end{matrix} & ] \\ & & ] \end{aligned} (49)

from which we conclude that the force due to incoming AF fields is

F̃_{AC} = \frac{2}{c} [ ; [ \begin{matrix}−(2R + L)E_{DC} \\ (tr^{∗} + rt^{∗})E_{DC} \\ −(2R + L)E_{DC}^{∗} \\ (tr^{∗} + rt^{∗})E_{DC}^{∗} \end{matrix} ] ; ]^{†} [ ; [ \begin{matrix}E_{+} \\ 0 \\ E_{−}^{∗} \\ 0 \end{matrix} ] ; ]  (50)

= −\frac{2}{c}(2R + L)(E_{DC}^{∗}E_{+} + E_{DC}E_{−}^{∗})  (51)
= −\frac{1}{c}(2R + L)P̃_{front_{in}}  (52)

This matches the naive result: the radiation force is P/c times the absorption coefficient L and twice the reflectivity R.

The radiation force which results from production of AF fields by the optic is given by the second term in equation 43. This term contains M_{cpl}, as shown in equation 19, and equation 28 for a mirror. For our simple mirror example,

M_{cpl} = \begin{aligned}[ \\ [ & \begin{matrix}ikr & 0 \\ 0 & ikr \end{matrix} & ] \\ & & ] \end{aligned}  (53)
⇒ M_{opt}^{†}M_{dir_{out}}M_{cpl} = ikr \begin{aligned}[ \\ [ & \begin{matrix}r & t \\ −t & r \end{matrix} & ] \\ & & ] \end{aligned}^{∗}  (54)
⇒ F̃_{x} = \frac{4}{c}Re[ik |rE_{DC}|^{2}] x̃ = 0  (55)

which again matches the naive expectation: phase modulation of a beam does not create a radiation reaction force. Note, however, that if fields were incident on both sides of the mirror the radiation reaction would be

F̃_{x2} = \frac{4}{c}Re[ikrt^{∗}(E_{DC1}^{∗}E_{DC2} − E_{DC1}E_{DC2}^{∗})] x̃  (56)
= \frac{8}{c}Re[krt^{∗}] Im[E_{DC1}E_{DC2}^{∗}] x̃  (57)

which may be non-zero if the two incoming beams are not phase matched.

Finally, we can return to the initial problem of finding M_{rad_{n}} for a given optic in equation 29. We can see from equation 43 that the force driven by incoming audio fields has a transfer matrix given by

M_{rad_{n,m}} = \frac{2}{c} [ [ \begin{matrix}(M_{opt}^{†}M_{dir_{out}}M_{opt} + M_{dir_{in}})_{n,m} M_{in_{n}}v⃗_{DC} \\ (M_{opt}^{T}M_{dir_{out}}M_{opt}^{∗} + M_{dir_{in}})_{n,m} M_{in_{n}}^{∗}v_{DC}^{∗} \end{matrix} ] ]^{†} (58)

such that the overall radiation reaction matrix is

M_{rad} = ∑_{n=1}^{N_{optic}} M_{drv_{n}}M_{rad_{n}} \begin{aligned}[ \\ [ & \begin{matrix}M_{in_{n}} & 0 \\ 0 & M_{in_{n}}^{∗} \end{matrix} & ] \\ & & ] \end{aligned}  (59)

where the M_{rad_{n}} for all drives of a given optic is a vertical concatenation of the radiation pressure applied to each individual drive

M_{rad_{n}} = [ ; [ M_{rad_{n,1}} ; ⋮ ; M_{rad_{n,N_{drive}}} ] ; ]  (60)

4.4 Optic to Optic Transfer Matrix

Similar to the field-optic matrix, the optic-optic matrix is driven by radiation forces.

M_{optic−optic} = M_{resp}M_{frc}  (61)

where M_{frc} comes from the radiation reaction force F̃_{x} derived in the previous section.

From equation 43, the force coefficient for a given drive is

M_{frc_{n,m}} = \frac{4}{c}Re[v_{DC}^{†}M_{in_{n}}^{T}M_{opt}^{†}M_{dir_{out}}M_{gen_{n,m}}]  (62)

and the force matrix for a given optic is diagonal

M_{frc_{n}} = [ ; [ \begin{matrix}M_{frc_{n,1}} & 0 & 0 & ⋯ \\ 0 & M_{frc_{n,2}} & 0 \\ 0 & 0 & ... \\ ⋮ & & & M_{frc_{n,N_{drive}}} \end{matrix} ] ; ]  (63)

The system wide matrix is made in the usual way, as a sum over all optics with the appropriate mapping matrices:

M_{frc} = ∑_{n=1}^{N_{optic}} M_{drv_{n}}M_{frc_{n}}M_{drv_{n}}^{T}  (64)

4.5 AC Matrix Inversion

While the inversion in equation 11 can be performed directly, and pervious versions of Optickle did just this, the result is somewhat difficult to interpret. The resulting transfer matrix M_{T F} is from drives, such as mirror displacements, to AC signals. The input drive to a mirror is not, however, the actual mirror displacement, but rather the displacement that you would get in the absence of radiation pressure effects. That is, the modification of the drives by the system’s opto-mechanics is included in M_{T F}. It is also true that much of v⃗_{excitation} is not used, since the excitations are generally always drives rather than direct audio sideband fiend injections.

To give a more easily interpreted result, and to reduce computation by addressing on the excitations which can actually be used, Optickle performs the computation suggested in equation 11 using the block form of M_{AC} shown in equation 12. The inverse of a matrix divided into 4 blocks is:

\begin{aligned}[ \\ [ & \begin{matrix}A & B \\ C & D \end{matrix} & ] \\ & & ] \end{aligned}^{−1} = \begin{aligned}[ \\ [ & \begin{matrix}K & L \\ M & N \end{matrix} & ] \\ & & ] \end{aligned}  (65)
where K = (A − BD^{−1}C)^{−1}  (66)
N = (D − CA^{−1}B)^{−1}  (67)
L = −A^{−1}BN = KBD^{−1}  (68)
M = −D^{−1}CK = NCA^{−1}  (69)

The part of M_{T F} which takes drives to signals is L in the above block inversion, and the corresponding piece of the inverse of 𝕀 − M_{AC} is

v⃗_{signal} = −M_{prb} M_{optT F} M_{optic−field} M_{mech} v⃗_{drv}  (70)
M_{optT F} = (𝕀 − M_{field−field})^{−1}  (71)
M_{mech} = (𝕀 − M_{optic−optic} − M_{field−optic} M_{optT F} M_{optic−field})^{−1} (72)

This has the advantage that the result can be returned to the user in 2 parts, one representing the drive-to-signal matrix

M_{drv−sig} = M_{prb} M_{optT F} M_{optic−field}  (73)

and the other representing the opto-mechanical modification of the drives, M_{mech}.

5 Signal Production

Converting DC and audio frequency fields to signals is the job of probes placed in the system. Probe signals result from power measured by the probe and is thus similar to the radiation pressure interaction described in M_{field−optic}. The major difference is that probes can demodulate signals at RF frequencies, thereby mixing RF components.

As with fields, there are two computations to perform: DC and AC. DC signals are the signals present given only the DC fields present, and are thus related to the current working point of the Optickle system. AC signals, on the other hand, are products between DC and audio frequency fields: they tell us about the response of the system around the current working point as seen in equation 10.

5.1 DC Signals

To compute the DC signals, we start with an expression for the intensity present at a given FEP,

I = |∑_{n=1}^{N_{RF}} E_{DC_{n}}e^{i2π[ν+v⃗_{f_{RF}} (n)]t}|^{2}  (74)

= ∑_{m=1}^{N_{RF}} ∑_{n=1}^{N_{RF}} E_{DC_{n}}^{∗}E_{DC_{m}}e^{i2π(v⃗_{f_{RF}} (m)−v⃗_{f_{RF}} (n))t}  (75)

where E_{DC_{n}} is the field amplitude of the n^{th} RF component, with the RF phase rotation factored out (much like equation ?? for field induced forces).

Requiring that the output be at the demodulation frequency f_{demod}, such that the integral

S_{DC} = ∫_{−∞}^{∞} I cos(2πf_{demod}t + ϕ_{demod}) dt  (76)

is non-zero removes most of the terms in the sum by introducing a delta function

S_{DC} = ∑_{m=1}^{N_{RF}} ∑_{n=1}^{N_{RF}} E_{DC_{n}}^{∗}E_{DC_{m}}δ_{n,m}  (77)

where

δ_{n,m} =\frac{1}{2} δ(v⃗_{f_{RF}} (m) − v⃗_{f_{RF}} (n) + f_{demod}) e^{iϕ_{demod}} +

\frac{1}{2} δ(v⃗_{f_{RF}} (m) − v⃗_{f_{RF}} (n) − f_{demod}) e^{−iϕ_{demod}}.

Translating this into the language of matrix manipulation, the DC signal from the k^{th} probe is

v⃗_{sig_{DC}}(k) = (M_{prb_{k}}M_{pin_{k}}v_{DC}^{∗})^{T} M_{pin_{k}}v⃗_{DC},  (78)

where the probe’s matrix expresses the delta function in equation 77 as the matrix

M_{prb_{k}}(m, n) = δ_{n,m},  (79)

and M_{pin_{k}} is the input map for the probe.

5.2 AC Signals

AC signal computation follows a very similar path. Intensity present at a given FEP is

I = |∑_{n=1}^{N_{RF}} (E_{DC_{n}}e^{i2π[ν+v⃗_{f_{RF}} (n)]t} + E_{+_{n}}e^{i2π[ν+v⃗_{f_{RF}} (n)+f_{audio}]t}
+E_{−_{n}}e^{i2π[ν+v⃗_{f_{RF}} (n)−f_{audio}]t})|^{2}  (80)
= ∑_{m=1}^{N_{RF}} ∑_{n=1}^{N_{RF}} Re [E_{DC_{n}}^{∗}E_{−_{m}}e^{i2π(v⃗_{f_{RF}} (m)−v⃗_{f_{RF}} (n)−f_{audio})t}
E_{DC_{n}}E_{+_{m}}^{∗}e^{i2π(−v⃗_{f_{RF}} (m)+v⃗_{f_{RF}} (n)−f_{audio})t}] .  (81)

The audio frequency signal present at a given FEP is

S_{AC} = ∑_{m=1}^{N_{RF}} ∑_{n=1}^{N_{RF}} Re[S̃_{AC}] cos (2πf_{audio}t) + Im[S̃_{AC}] sin (2πf_{audio}t)

(82)

where

S̃_{AC} = 2(E_{DC_{n}}^{∗}E_{−_{m}}δ_{n,m} + E_{DC_{n}}E_{+_{m}}^{∗}δ_{m,n})  (83)

(see, also, equation 34). The overall probe matrix is constructed from the individuals row at a time according to

M_{prb}(k,:) = \begin{aligned}[ \\ [ & \begin{matrix}M_{prb_{k}}M_{pin_{k}}v_{DC}^{∗} \\ M_{prb_{k}}^{T}M_{pin_{k}}v⃗_{DC} \end{matrix} & ] \\ & & ] \end{aligned}^{T} \begin{aligned}[ \\ [ & \begin{matrix}M_{pin_{k}} \\ M_{pin_{k}} \end{matrix} & ] \\ & & ] \end{aligned} ,  (84)

where k ∈ 1 : N_{probe}. Note that the probe matrix M_{prb_{k}} for the upper audio sideband is the transpose of that for the lower sideband, as m and n are exchanged for the upper sideband in equation 82. It may seem odd that the input map is used so may times: two instances are for the DC fields in v⃗_{DC} and v_{DC}^{∗}, other two bring in the correct the upper and lower audio frequency fields from v⃗_{AC} which is expected to arrive on the right (see equations 9 and 10).

6 Quantum Noise

The computation of quantum noise in Optickle follows the 2-photon formalism originally presented by H. Yuen and later by C. Caves. However, since Optickle was written before the author was aware of this formalism, there are some formal and notational differences in the computation. In particular, Optickle propagates audio SBs separately, but since the conjugate of the upper SB is propagated this is equivalent to propagating a linear combination of 2-photon states, and thus the quantum noise computation can follow the 2-photon formalism without modification.

Optickle takes a direct approach to quantum noise computation: each loss point, unconnected port, and source object is treated as a source of uncorrelated vacuum fluctuations. The output noise spectra are given by the incoherent sum of all of the injected vacuum fluctuations as seen by each probe.

7 Angular Transfer Functions

Optickle can compute transfer functions for the TEM01 mode produced, for example, by pitch of a mirror. This is done in almost complete analogy with the TEM00 computations described in the previous sections. The exceptions are the Gouy phase, and the dependence of audio sideband injection amplitude on the Hermite-Gaussian beam parameters.