Session 01 — Solving the detector by matrix inversion: cavities, loops, and transfer functions - GW101

Spine placement: Block 1 (The Interferometer) → optics/matrix_methods, session 1 of 3. This is the session where the curriculum’s organizing idea — “the detector is a hierarchy of linear systems” — becomes a concrete, repeatable computational recipe.

The recipe (one method, used everywhere below):
Draw the system as a directed graph of fields/signals (nodes) and couplings (edges).
Write the node-coupling (“adjacency”) matrix M by inspection of the drawing.
Invert: solve (I − M) x = x_in, i.e. x = (I − M)⁻¹ x_in. The inverse is the closed-loop response of the whole system.
A Fabry–Perot cavity, a SISO feedback loop, and (later) a full interferometer are the same calculation with a bigger matrix.

Commissioning question¶

“When I push on a mirror or step the laser frequency, how much does the field inside the arm cavity actually move — and how fast can my control loop correct it?”

Every answer in commissioning is some entry of an inverted matrix. This session teaches you to build that matrix from a drawing and read the answer off the inverse.

Learning objectives¶

Participants can:

Draw a two-mirror cavity as a field-node graph and write its coupling matrix M by inspection.
Solve the cavity by matrix inversion and show the result equals the analytic self-consistency solution E = t₁E_in /(1 − r₁r₂e^{2iφ}).
Recognize that a SISO feedback loop is the identical construction, and read the sensitivity S = 1/(1+PC) and complementary sensitivity T = PC/(1+PC) directly off (I − M)⁻¹.
Compute a frequency response in the audio-sideband (small-signal) approximation by evaluating the inverted system at s = iΩ, and extract the cavity pole.
Explain why “scaling up” to MIMO / a full interferometer changes only the size of M, not the method.

Retrieval warm-up¶

(Answers at the end of this note.)

In our convention (exp(−iωt), propagation exp(+ikL)), what is the one-way phase a field accumulates traversing a cavity of length L at frequency ω?
A cavity is “resonant” when the round-trip field reinforces the input. In one sentence, why is that a feedback condition rather than a static one?
For a single-pole low-pass response, what happens to gain and phase as frequency passes the pole?

Scope¶

Included

The draw → coupling-matrix → invert recipe.
Two-mirror Fabry–Perot cavity: circulating field, reflected field, buildup vs detuning.
The cavity ↔ feedback-loop equivalence (round-trip gain r₁r₂e^{2iφ} is an open-loop gain).
SISO closed-loop transfer functions S, T from the matrix inverse.
Frequency response via the audio-sideband (small-signal) approximation; the cavity pole.

Excluded (named here so you know where they live)

RF modulation / PDH error signals → optics/pdh_signals.
Michelson topology and the dark port → optics/ifo_topology.
Multi-DOF control design (loop shaping, stability margins) → controls/lsc, foundations/feedback_basics.
Higher-order spatial modes → optics/spatial_modes.

Core model¶

1. The system is a graph; the matrix is its adjacency¶

A linear optical/control system is a directed graph: nodes are complex amplitudes (fields or signals), edges carry a complex gain. Each node equals the sum of its incoming edges plus any external injection:

x_i = \sum_j M_{ij}\,x_j + (x_\text{in})_i \qquad\Longrightarrow\qquad (I-M)\,x = x_\text{in}.

(1)

$M_{ij}$ is the gain of the edge from node $j$ into node $i$ — read straight off the drawing; it is the (complex-weighted) adjacency matrix. Solving is one numpy.linalg.solve, and the inverse $(I-M)^{-1}$ is the closed-loop response: every node’s response to every injection at once.

Go deeper — the inverse is a sum over all physical paths

When the loop gain is below unity (spectral radius $\rho(M)<1$ ), the inverse is the Neumann series

(I-M)^{-1} = \sum_{k=0}^{\infty} M^{k}.

(2)

The $(i,j)$ entry of $M^{k}$ sums the gain products of every directed walk of length $k$ from $j$ to $i$ , so $[(I-M)^{-1}]_{ij}$ is the sum over all paths from $j$ to $i$ — physically every route the wave can take, including every number of round trips. For the cavity below, the $M^k$ terms are the $k$ -bounce contributions and $\sum_k$ is exactly the geometric buildup series $1/(1-r_1r_2e^{2i\phi})$ . “Invert the matrix” and “sum the multiple-reflection series” are the same statement.

2. Fabry–Perot cavity¶

Mirrors ITM $(r_1,t_1)$ , ETM $(r_2,t_2)$ ; one-way phase $\phi=kL=\omega L/c$ . Four nodes around the round trip, with one sign convention that matters: a field reflecting off the HR/cavity side of a mirror picks up $-r$ (off the substrate side, $+r$ ); transmission is real. (This is the unitary convention in shared/code/conventions.py.)

$E_1$ leaves the ITM toward the ETM; $E_2 = e^{i\phi}E_1$ reaches the ETM,
$E_3 = -r_2 E_2$ reflects back; $E_4 = e^{i\phi}E_3$ returns to the ITM,
the ITM closes the loop: $E_1 = t_1 E_\text{in} - r_1 E_4$ .

By inspection,

M=\begin{pmatrix} 0&0&0&-r_1\\ e^{i\phi}&0&0&0\\ 0&-r_2&0&0\\ 0&0&e^{i\phi}&0\end{pmatrix}, \qquad x_\text{in}=(t_1E_\text{in},\,0,\,0,\,0)^{\mathsf T}.

(3)

The drawing is the matrix: each directed edge below is one nonzero entry of $M$ (edge gain from node $j$ into node $i$ is $M_{ij}$ ), and the round trip carries the two HR-side reflection signs $-r_1$ (ITM) and $-r_2$ (ETM).

Caption: the Fabry–Perot field graph. Read $M$ straight off this drawing — the four loop edges $E_1\!\to\!E_2$ , $E_2\!\to\!E_3$ , $E_3\!\to\!E_4$ , $E_4\!\to\!E_1$ are exactly the four nonzero entries $e^{i\phi},\,-r_2,\,e^{i\phi},\,-r_1$ of $M$ above; the injection $t_1E_\text{in}$ into $E_1$ is $x_\text{in}$ , and the reflected field $E_\text{refl}=r_1E_\text{in}+t_1E_4$ leaves toward the laser.

Solving $(I-M)E=x_\text{in}$ reproduces the textbook result — but we never derive it by hand; we invert.

Go deeper — closed forms, buildup, and over/under/critical coupling

Eliminating nodes gives $E_4=-r_2e^{2i\phi}E_1$ , hence the circulating field

E_1=\frac{t_1}{1-r_1r_2e^{2i\phi}}\,E_\text{in},

(4)

and the field reflected back toward the laser, $E_\text{refl}=r_1E_\text{in}+t_1E_4$ , becomes

\frac{E_\text{refl}}{E_\text{in}}=r_1-\frac{t_1^2 r_2 e^{2i\phi}}{1-r_1r_2e^{2i\phi}} \;\xrightarrow{\ \text{resonance}\ }\;\frac{r_1-r_2}{1-r_1r_2}.

(5)

On resonance the power builds up by $|E_1/E_\text{in}|^2=T_1/(1-r_1r_2)^2\approx 2\mathcal F/\pi$ . The reflected amplitude $(r_1-r_2)/(1-r_1r_2)$ encodes the coupling:

under-coupled $r_1>r_2$ : $r_\text{cav}>0$ ;
critically coupled $r_1=r_2$ : $r_\text{cav}=0$ — all incident light enters, none reflects;
over-coupled $r_1<r_2$ (the LIGO arm case, ETM ≈ perfect): $r_\text{cav}<0$ , $|r_\text{cav}|\to1$ .

Energy is conserved: $|r_\text{cav}|^2+|t_2E_2|^2=1$ (the notebook asserts this). With a naive “ $+r$ on both sides” convention the mirror is non-unitary and $|r_\text{cav}|$ exceeds 1 — keep that as a sanity check on any field model you build.

Resonance is a feedback condition. The denominator $1-r_1r_2e^{2i\phi}$ is $1-(\text{round-trip gain})$ . As $r_1r_2e^{2i\phi}\to1$ the loop gain approaches unity and the field diverges — exactly a feedback loop approaching instability. That is the bridge to the next part.

3. SISO feedback loop — the same drawing¶

Plant $P$ , controller $C$ , error $e$ , actuation $u$ , output $y$ , reference $r$ , disturbance $d$ :

e=r-y,\qquad u=Ce,\qquad y=Pu+d.

(6)

Three nodes $(e,u,y)$ ; edges $y\to e$ (gain -1), $e\to u$ ( $C$ ), $u\to y$ ( $P$ ); injections $r$ into $e$ , $d$ into $y$ :

M=\begin{pmatrix}0&0&-1\\ C&0&0\\ 0&P&0\end{pmatrix},\qquad x_\text{in}=(r,0,d)^{\mathsf T}.

(7)

Drawn as a block diagram, this is the same graph: the comparison node forms $e=r-y$ , the controller $C$ and plant $P$ are the forward edges, and the output $y$ feeds back with gain -1 (negative feedback) — the optical analogue of the $-r_1,-r_2$ round-trip signs above.

Caption: the SISO feedback loop as a block diagram. The summing node computes the error $e=r-y$ ; $u=Ce$ drives the plant; the disturbance $d$ adds at the plant output to make $y=Pu+d$ ; and $y$ returns through the -1 feedback edge. This is the same three-node graph $(e,u,y)$ whose coupling matrix $M$ is written above.

The inverse hands you the closed-loop transfer functions with no extra algebra:

S=\frac{1}{1+PC}\ (\text{sensitivity}),\qquad T=\frac{PC}{1+PC}\ (\text{complementary}),\qquad S+T=1.

(8)

Go deeper — reading S and T off (I−M)⁻¹

Substituting, $y=PC(r-y)+d\Rightarrow y=\frac{PC}{1+PC}r+\frac{1}{1+PC}d=Tr+Sd$ , and $e=Sr-Sd$ . So the columns of $(I-M)^{-1}$ acting on $(r,d)$ are literally $S$ and $T$ . This is the same recipe as the cavity: the round-trip gain $r_1r_2e^{2i\phi}$ plays the role of the loop gain $PC$ , and $1-(\text{loop gain})$ is the common denominator. A resonant cavity is an optical feedback loop.

4. Transfer functions: the audio-sideband (small-signal) approximation¶

So far $M$ ’s entries were numbers (one frequency). For a frequency response we let a quantity wiggle at audio frequency $\Omega$ and ask how the steady state responds — the small-signal picture. A real modulation at $\Omega$ places audio sidebands on the carrier at $\pm\Omega$ ; in the linear regime each sideband propagates independently, so we re-evaluate the same matrix with frequency-dependent edges and read the response at offset $\Omega$ .

Loop: make $P,C$ functions of $s$ and evaluate at $s=i\Omega$ ⇒ $S(i\Omega),T(i\Omega)$ , the Bode response.
Cavity: the round-trip gain at sideband offset $\Omega$ is $r_1r_2e^{2i(\omega_0+\Omega)L/c}$ ; the circulating field rolls off as a single pole $H(\Omega)\approx 1/(1+i\Omega/\Omega_c)$ , with the cavity pole $f_c=\mathrm{FSR}/(2\mathcal F)$ .

Go deeper — where the cavity pole comes from

On resonance the carrier sees $g(0)=r_1r_2\equiv\rho$ ; a sideband at offset $\Omega$ sees $g(\Omega)=\rho\,e^{2i\Omega L/c}$ , so the normalized response is $H(\Omega)=\dfrac{1-\rho}{1-\rho e^{2i\Omega L/c}}$ . With $\psi=2\Omega L/c$ , the half-power point ( $|H|^2=\tfrac12$ ) is where $4\rho\sin^2(\psi/2)=(1-\rho)^2$ , i.e. $\sin(\psi/2)=\frac{1-\rho}{2\sqrt\rho}$ . For $\rho\to1$ this is $\Omega_c=\frac{(1-\rho)c}{2\sqrt\rho\,L}$ , i.e.

f_c=\frac{\mathrm{FSR}}{2\mathcal F},\qquad \mathrm{FSR}=\frac{c}{2L},\qquad \mathcal F=\frac{\pi\sqrt{r_1r_2}}{1-r_1r_2}.

(9)

The notebook extracts $f_c$ from the -3 dB point and matches this to ~0.00%. Higher finesse ⇒ more buildup but a lower $f_c$ ⇒ less control bandwidth — the commissioning trade-off.

Concept questions (ConcepTests)¶

You double the ITM transmissivity T₁ (cavity becomes “leakier”). Predict, before computing: does the cavity pole f_c go up or down, and does on-resonance buildup go up or down?
In the loop matrix, the y→e edge has gain −1 (negative feedback). What single sign change in M would turn this into positive feedback, and what would the inverse do near PC → 1? (Connect this back to cavity resonance.)

Common misconceptions¶

“Matrix inversion is just a fancier way to write the formula I already know.” The point is the opposite: you stop deriving formulas. You draw, fill in M, invert. The same code solves a cavity, a loop, and an interferometer.
“Resonance and feedback are different topics.” They are the same denominator 1 − (loop gain).
“Audio sidebands are an RF/PDH thing.” RF sidebands are added deliberately for sensing (later). Audio sidebands are just the small-signal content of any slow modulation — they are how a transfer function exists at all.
“A higher-finesse cavity is always better.” Higher finesse means more buildup but a lower pole — less control bandwidth. Commissioning lives in this trade-off.

Laptop block (30 min) — Predict → Model → Measure → Explain¶

Notebook: notebooks/optics/matrix_methods/session_01_matrix_inversion.ipynb. Imports the canonical conventions from shared/code/conventions.py. The notebook fades from a worked example → guided completion → an independent task; each part has an analytic check (assert) so you get private, immediate pass/fail feedback.

Predict — sketch circulating power vs detuning, and the reflected-field phase through resonance, before running anything.
Model (worked) — build the cavity coupling matrix M(φ) by inspection; solve (I − M)E = x_in.
Measure — sweep detuning; overlay the matrix solution on the analytic Airy curve (they must coincide to machine precision); compute FSR, finesse, and the cavity pole.
Guided — build the SISO loop matrix and confirm (I − M)⁻¹ reproduces S and T.
Independent — compute the cavity frequency response in the audio-sideband approximation, extract f_c numerically, and compare to FSR/(2𝓕).
Explain — write one or two sentences reconciling your prediction with the result.

Mandatory figures (produced by the notebook):

Circulating power vs detuning (Airy curve), matrix vs analytic overlaid.
Complex reflectivity vs detuning (magnitude and phase flip through resonance).
|H(Ω)| vs frequency showing the cavity pole.
Condition number of (I − M) vs detuning (why near-resonance is numerically delicate).

Exit prompt¶

In your own words (2–3 sentences): what is the same, and what is different, between solving a Fabry–Perot cavity and solving a feedback loop? If you can answer this, you can read the next two sessions as “the matrix just got bigger.”

Go deeper / refresher¶

Refresher — if exp(−iωt) / exp(+ikL) conventions feel shaky, skim shared/code/conventions.py (it is short and is the single source of truth).
Go deeper — session 2 turns (P, C) and the optical nodes into matrices (MIMO): the same (I − M)⁻¹ becomes a closed-loop response matrix. Session 3 grows the optical graph to a Michelson with arm cavities — a bigger M, identical recipe.
This session implements the field/cavity content of the optics seed lectures L01–L02; the audio-sideband viewpoint sets up the RF/PDH treatment in optics/pdh_signals.

Retrieval warm-up — answers¶

φ = kL = ωL/c one way; 2φ for a round trip.
The circulating field appears on both sides of its own equation (E₁ = t₁E_in + r₁r₂e^{2iφ}E₁): the output feeds back into the input, so the steady state is a self-consistency (fixed-point) condition, not a one-pass calculation.
At the pole the gain is down 3 dB (×1/√2) and the phase lag is 45°; well above it gain rolls off ∝ 1/Ω and phase approaches 90°.