L18 · Calibrate the optical ruler

Optical Frequency Comb

What must you measure before a frequency comb can work as a ruler for light? The tooth spacing and their common offset. The spacing equals the pulse repetition rate. Doubling a low-frequency tooth and beating it against the tooth with twice its index isolates the offset. An unknown laser also needs an identified tooth and the sign of its beat relative to that tooth.

Count repetition and offset signals, then reconstruct a target frequency. Compare missing overlap, a short record and an ambiguous 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 you measure before a frequency comb can work as a ruler for light?

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

2.2

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 the exact tooth positions. The size of a beat alone tells you neither its sign nor the tooth index, so this Lab supplies the ideal signed beat and a coarse prior as inputs.

Entry and physical parameter changes count a fresh ideal sampled RF record. Replay changes only channel markers. The RF plot shows the first fifty nanoseconds of the same 4097-point window; the counter uses the whole window. The arms ideally select the two specified teeth; a full multi-tooth photocurrent, optical powers and splitter losses are not computed. Even without the partner tooth, doubled light can reach the diode; the selected CEO beat is absent. Precision readouts retain extra digits to expose large-index error amplification.

The comb is ideal and stationary: its teeth have equal power and flat phase, and the spectrum stops sharply at its edges. The band center is the geometric mean of the edges, and the span is the ratio of the highest to the lowest frequency in the comb. Repetition frequency is in megahertz, and optical frequencies are in terahertz. The CEO frequency is set as a fraction of the repetition frequency, from zero up to but not including one repetition frequency. Every tooth sits at an integer multiple of the repetition frequency plus this same offset. The carrier snapshots use the exact finite sum in closed form, so they match the displayed spectrum. The pulse-width readout is the continuum sinc-squared FWHM, used as a reference for this very large number of teeth. Self-referencing needs a tooth that actually exists and a second tooth at twice its index. Because the teeth are discrete, a span of exactly one octave can still fail this test. The two arms meet on a shared ideal photodiode only when both teeth exist, and their difference frequency gives the CEO frequency. The repetition-rate pickoff and the CEO signal are ideal selected RF components, sampled at 4097 points over the chosen record. The counter uses interpolated upward zero crossings and needs at least two of them. A CEO of exactly zero is treated as a known ideal DC case; telling a real zero offset apart is not modeled. A reconstruction also needs the target inside the comb's spectrum and a coarse frequency prior narrower than half the tooth spacing. The signed target beat and the integer index are supplied as ideal known inputs; a photodiode alone measures only the size of the beat and leaves its sign ambiguous. The reconstruction carries the finite counting error through the large index and shows the resulting frequency error. The prior is a sufficient condition chosen for teaching, not a Bayesian estimate of the index. Changing a control replaces the old capture with a freshly counted ideal RF record. The model has no servo, external clock error, random noise, nonlinear propagation or phase-matching efficiency, and it makes no claim of laboratory accuracy. Optical layout and dimensions are schematic.

Start with the essentials

Focus question: What blocks an absolute frequency reconstruction?
One-sentence intuition: Double the frequency of a tooth near the low end of the comb and beat it against the tooth with twice its index: the repetition frequency cancels and only the offset is left.

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}

The spacing alone leaves the starting point of the ruler unknown.

Carrier phase slip from pulse to pulse

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

Each pulse envelope repeats the last one, but the carrier under it can shift forward in phase.

Self-referencing with a doubled tooth

\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 inside the comb's spectrum. The frequency doubling is treated as ideal.

Counting the RF frequency

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

The counter interpolates the upward zero crossings. If the record is too short, the frequency is reported as unresolved.

Reconstructing the target frequency

\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}

Here the beat and its sign are given as an ideal input. Because the tooth index is large, a small counting error in the repetition frequency becomes a large optical frequency error.

Field of a finite comb with equal-power teeth

\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 pulse snapshots use the exact finite sum. The pulse-width readout uses the continuum limit for many teeth.

Common difficulties

One coordinate is missing

Typical misconceptionKnowing the tooth spacing is enough to know every optical frequency.

Better mental modelThe whole comb can slide without changing its spacing. The CEO frequency tells you how far it has slid.

Photodiode beats have a sign ambiguity

Typical misconceptionThe size of the beat frequency tells you which side of the tooth the target is on.

Better mental modelA photodiode gives the same beat for a target the same distance above or below the tooth. This ideal model supplies the sign; a real instrument needs an extra discriminator to find it.

Run the experiment

01
Watch the carrier slip
Select Half-cycle phase slip and compare the three pulse snapshots.
What to observe: The envelopes repeat, but the carrier flips sign from one pulse to the next.
02
See self-referencing fail
Compare the loaded Supported self-reference record with Missing octave.
What to observe: Unless a tooth and its doubled-index partner both lie in the spectrum, no CEO beat is recorded.
03
Count the RF record
Read the default counted RF values, then choose Record too short. Use Acquire RF record to repeat the acquisition.
What to observe: With too few zero crossings, the offset cannot be resolved.
04
Test the tooth-index ambiguity
Try Ambiguous tooth index, then reduce the Coarse prior half-width.
What to observe: A precise RF beat alone does not tell you which optical tooth the target is next to.

Optical Frequency Comb

Measure both ruler coordinates

Core mathematical model

The optical ruler has two coordinates

Carrier phase slip from pulse to pulse

Self-referencing with a doubled tooth

Counting the RF frequency

Reconstructing the target frequency

Field of a finite comb with equal-power teeth

Common difficulties

One coordinate is missing

Photodiode beats have a sign ambiguity

Run the experiment

Watch the carrier slip

See self-referencing fail

Count the RF record

Test the tooth-index ambiguity

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