L18 · Calibrate the optical ruler

Optical Frequency Comb

Connect pulse-to-pulse carrier phase with tooth positions. Find an octave pair, acquire its RF beat, and reconstruct an unknown frequency with a known tooth index.

RF records acquired $\text{—}$

Pulse repetition period $\text{—}$

Uniform-band pulse FWHM reference $\text{—}$

Ideal CEO frequency $\text{—}$

Carrier phase slip per pulse $\text{—}$

Supported optical teeth $\text{—}$

Supported doubling pair $\text{—}$

Nearest target tooth index $\text{—}$

Signed ideal target beat $\text{—}$

Counted repetition frequency $\text{—}$

Counted CEO frequency $\text{—}$

Reconstructed target frequency $\text{—}$

Propagated counting error $\text{—}$

Acquisition state $\text{—}$

L18

What must be measured before the comb becomes a ruler?

Pulse repetition frequency

f_{\rm rep}

100\,\mathrm{MHz}

CEO fraction of repetition

\alpha

0.2

Geometric band center

\nu_c

250\,\mathrm{THz}

Maximum to minimum frequency ratio

R_\nu

1.5

Unknown target frequency

\nu_{\rm target}

250.000037\,\mathrm{THz}

Coarse prior half-width

\delta\nu

10\,\mathrm{MHz}

RF recording window

T_{\rm RF}

400\,\mathrm{ns}

The optical zoom shows exact tooth locations. A beat magnitude alone does not identify its sign or the tooth index; the ideal signed beat and coarse prior are explicit inputs here.

An ideal stationary comb has equal-power, flat-phase teeth inside a hard frequency support. The band center is geometric; the span is the ratio of maximum to minimum supported frequency. Repetition frequency is in megahertz, and optical frequencies are in terahertz. CEO is chosen as a fraction of repetition frequency in its fundamental interval. Every supported tooth has integer index plus this same offset. Local carrier snapshots use the exact finite sum in closed form; they share the displayed spectrum, rather than an independently drawn pulse. The pulse-width readout is the continuum sinc-squared FWHM reference for this very large tooth count. Frequency doubling uses an actually supported index and its doubled index; a merely one-octave geometric span can fail the discrete tooth test. The two arms overlap on a common ideal photodiode only when both teeth exist. Their difference gives CEO. The repetition pickoff and CEO signals are ideal selected RF components, sampled at 4097 points over the chosen record. Acquired RF values count interpolated positive-going zero crossings, requiring at least two crossings. Exactly zero CEO is treated as a known ideal DC branch; realistic zero-offset discrimination is not modeled. Acquisition also requires the target to lie inside support and a supplied coarse frequency prior narrower than half the tooth spacing. The displayed signed target beat and integer index are ideal known inputs; a photodiode magnitude alone would leave a sign ambiguity. Reconstruction propagates finite counting error through the large index and shows the resulting frequency error. The prior is a sufficient educational condition, not a Bayesian index estimator. Controls clear the capture. There is no servo, external clock error, stochastic noise, nonlinear propagation, phase-matching efficiency or laboratory accuracy claim. Optical layout and dimensions are schematic.

Start with the essentials

Focus question: What blocks an absolute frequency reconstruction?
One-sentence intuition: Doubling a supported low-frequency tooth and beating it with its doubled-index partner isolates the common offset.

Core mathematical model

The optical ruler has two coordinates

\begin{aligned}\nu_n&=nf_{\rm rep}+f_{\rm CEO}\\0&\le f_{\rm CEO}<f_{\rm rep}\end{aligned}

Spacing alone leaves the ruler origin unknown.

Phase slips between envelopes

\begin{aligned}T_R&=1/f_{\rm rep}\\\Delta\phi_{\rm CE}&=2\pi f_{\rm CEO}/f_{\rm rep}\end{aligned}

The envelope repeats while the carrier phase can advance.

A supported doubling pair

\begin{aligned}2\nu_n-\nu_{2n}&=2(nf_{\rm rep}+f_{\rm CEO})\\&\quad-(2nf_{\rm rep}+f_{\rm CEO})\\&=f_{\rm CEO}\end{aligned}

Both original teeth must lie in the support; doubling is idealized.

Count a sampled RF record

\widehat f=\frac{K-1}{t_K-t_1},\qquad K\ge2

Interpolate positive-going crossings and report unresolved short records.

Reconstruct with an identified tooth

\begin{aligned}\widehat\nu_{\rm target}&=n\widehat f_{\rm rep}+\widehat f_{\rm CEO}+b\\\delta\nu&=n\delta f_{\rm rep}+\delta f_{\rm CEO}\end{aligned}

The signed ideal beat is supplied here; large index amplifies counting error.

Finite equal-power comb field

\begin{aligned}A(\tau)&\propto\frac{\sin(N\pi f_{\rm rep}\tau)}{N\sin(\pi f_{\rm rep}\tau)}\\\Delta t_{1/2}&\simeq\frac{0.886}{\Delta\nu}\end{aligned}

The snapshots use the exact finite sum; pulse width is its many-tooth continuum reference.

Common difficulties

One coordinate is missing

Typical misconceptionKnowing the tooth spacing determines every optical frequency.

Better mental modelThe entire comb can shift while preserving spacing; CEO determines that shift.

Photodiode beats have a sign ambiguity

Typical misconceptionBeat magnitude identifies which side of the tooth the target lies on.

Better mental modelThe signed beat is explicitly known in this ideal model; real instruments need an additional discriminator.

Run the experiment

01
Watch carrier slip
Select the half-cycle preset and compare the three local pulse snapshots.
What to observe: Envelopes repeat while the carrier alternates sign.
02
Manufacture missing self-reference
Acquire the narrow preset, then use supported self-reference.
What to observe: No CEO beat is recorded without a supported tooth pair.
03
Count the actual RF record
Acquire, inspect the counted values, then choose record too short.
What to observe: Too few crossings leave the offset unresolved.
04
Test the integer ambiguity
Try the ambiguous-index preset and reduce prior half-width.
What to observe: A precise RF beat does not identify the optical tooth by itself.