# C3 — Critical coupling is a steady-state rule; short pulses want κ_e τ ≈ 2.51

**From:** E3-H183, rank 3/30 · **Status: DERIVATION + COMPUTATIONAL VERIFICATION**, including a
correction to my own first statement of the result.

---

## A. The claim, refined

For a single-mode cavity with intrinsic loss $\kappa_0$, external coupling $\kappa_e$, driven by a
square pulse of amplitude $s_0$ and duration $T$, the fraction of incident energy stored at the end
of the pulse is

$$\eta(\kappa_e)=\frac{4\kappa_e\left(1-e^{-\kappa T/2}\right)^2}{\kappa^2 T},
\qquad \kappa=\kappa_0+\kappa_e .$$

The maximiser obeys

$$\kappa_e^\* \to \kappa_0 \quad(\kappa_0T\gg1), \qquad
\kappa_e^\* T \to 2.51289\ldots \quad(\kappa_0T\ll1),$$

where $2.51289 = 2y$ and $y=1.25644$ solves $2y+1=e^{y}$.

**Both limits are in units of the coupling *rate*, and the two differ in kind.** In steady state the
optimum is set by a ratio to the intrinsic loss; in the short-pulse limit it is set by the pulse
duration alone, with $\kappa_0$ dropping out.

---

## B. Derivation

Cavity field from $t=0$:

$$\dot a=-\tfrac{\kappa}{2}a+\sqrt{\kappa_e}\,s_0
\ \Longrightarrow\
a(T)=\frac{2\sqrt{\kappa_e}s_0}{\kappa}\left(1-e^{-\kappa T/2}\right),$$

$$\eta=\frac{|a(T)|^2}{|s_0|^2T}
=\frac{4\kappa_e}{\kappa^{2}T}\left(1-e^{-\kappa T/2}\right)^2 .$$

**Steady state** ($\kappa T\gg1$): $\eta\to4\kappa_e/(\kappa^2T)$, maximised at
$\kappa_e=\kappa_0$ — the textbook critical-coupling rule, recovered exactly.

**Short pulse** ($\kappa_0T\ll1$): if in addition $\kappa_eT\ll1$,
$\eta\to\kappa_eT$, which increases with $\kappa_e$. That expansion is the trap: it holds only while
$\kappa_e T\ll1$, and the true optimum sits precisely where it fails. Setting
$\partial\eta/\partial\kappa_e=0$ with $u=\kappa_e/\kappa$ gives, in the limit $\kappa_0\to0$,

$$\frac{2y+1=e^{y}}{}\quad\text{with}\quad y=\frac{\kappa^\*T}{2},
\qquad \kappa^\*T = 2.51289\ldots$$

∎

**The correction.** My first version of this investigation reported an *unbounded* optimum,
extrapolating the small-$\kappa_eT$ expansion past its validity. The Phase II dossier D03,
written independently, derived the finite value $2.51286$. Recomputing confirmed D03: the product
$\kappa_e^\*T$ converges to $2.5132$ as $\kappa_0T\to10^{-3}$. The expansion was the error, not the
physics.

---

## C. Computational verification

Maximisation over $\log_{10}(\kappa_e/\kappa_0)$ on a $4\times10^{5}$-point grid.

| $\kappa_0T$ | $\kappa_e^*/\kappa_0$ | $\kappa_e^*T$ | $\eta_{\max}$ |
|---|---|---|---|
| 0.001 | 2513 | **2.5132** | — |
| 0.01 | 252 | 2.5161 | 0.811 |
| 0.1 | 25.5 | 2.5455 | 0.783 |
| 1.0 | 2.87 | 2.8750 | 0.561 |
| 10 | 1.000 | 10.0 | 0.1000 |
| 100 | 1.000 | 100 | 0.0100 |

The product $\kappa_e^\*T\to2.5129$ from above as $\kappa_0T\to0$, exactly as the implicit equation
requires. At $\kappa_0T=10$ the ratio is $1.000$ to three figures — critical coupling, recovered.

**What critical coupling costs:**

| $\kappa_0T$ | $\eta$ at critical | $\eta$ at optimum | shortfall |
|---|---|---|---|
| 0.05 | 0.0476 | 0.799 | **94%** |
| 0.10 | 0.0906 | 0.783 | **88%** |
| 0.30 | 0.224 | 0.725 | 69% |
| 1.00 | 0.432 | 0.561 | 23% |

**Independent check.** The closed form was compared against direct numerical integration of the
cavity ODE at $T=0.3$ and $T=2.0$; agreement to $10^{-8}$ relative.

---

## D. Stress tests

1. **"η can exceed 1 in the short-pulse expansion."** It can, and that is the signature of the
   invalid expansion: for $\kappa_eT\gtrsim1$ the driven-field model stops conserving energy in the
   way the formula assumes, and the true $\eta$ peaks near $0.81$ and falls as $4/(\kappa_eT)$.
   The finite optimum is what keeps the result physical.
2. **"The coefficient is of order one."** It is not; it is 2.513, and the difference matters — a
   factor-of-2.5 error in the coupling you should choose.
3. **Square pulse assumed.** A different pulse shape changes the constant but not the structure:
   in the short-pulse limit the optimum is $\kappa_e T=O(1)$, independent of $\kappa_0$.

---

## E. Status

| Statement | Status |
|---|---|
| Closed form for $\eta(\kappa_e)$ | **Derived**, verified against the ODE to $10^{-8}$ |
| Steady-state optimum at $\kappa_e=\kappa_0$ | **Derived and verified** |
| Short-pulse optimum $\kappa_e^\*T=2.51289$ | **Derived** (implicit equation) and **verified numerically** ($2.5132$ at $\kappa_0T=10^{-3}$) |
| Optimum is unbounded | **Refuted** — my own earlier claim, corrected here |
| Critical coupling loses up to 94% for short pulses | **Computational result** |
| Measured in a real cavity | **Open** |

## Reproducibility

```
cd 03_VOLUME_C/C3_pulse_coupling
python3 c3_pulse.py
```
Deterministic; numpy only; runtime < 40 s.