---
title: "Optical Frequency Comb Interactive Lab | Sandbox Physics"
description: "What must you measure before a frequency comb can work as a ruler for light? The tooth spacing and their common offset."
canonical_url: "https://sandboxphysics.com/labs/optical-frequency-comb/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/optical-frequency-comb/"
last_modified: "2026-10-28"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

Laser foundations · Light, gain and feedback

[L01 · Compare sources](https://sandboxphysics.com/labs/laser-source-comparison/) [L02 · Count transitions](https://sandboxphysics.com/labs/light-matter-transitions/) [L03 · Build inversion](https://sandboxphysics.com/labs/population-inversion/) [L04 · Amplify once](https://sandboxphysics.com/labs/single-pass-amplifier/) [L05 · Add feedback](https://sandboxphysics.com/labs/passive-optical-resonator/) [L06 · Start from noise](https://sandboxphysics.com/labs/laser-noise-startup/) [L07 · Build and debug](https://sandboxphysics.com/labs/build-debug-laser/) [L08 · Measure and diagnose](https://sandboxphysics.com/labs/laser-diagnostic-bench/) [L09 · Select axial modes](https://sandboxphysics.com/labs/gain-longitudinal-modes/) [L10 · Confine the beam](https://sandboxphysics.com/labs/stable-resonator-geometry/) [L11 · Select spatial modes](https://sandboxphysics.com/labs/transverse-laser-modes/) [L12 · Propagate and measure](https://sandboxphysics.com/labs/gaussian-beam-quality/) [L13 · Align and match](https://sandboxphysics.com/labs/alignment-mode-matching/) [L14 · Optimize extraction](https://sandboxphysics.com/labs/output-coupler-optimization/) [L15 · Drive the dynamics](https://sandboxphysics.com/labs/laser-dynamics-relaxation/) [L16 · Store and release](https://sandboxphysics.com/labs/q-switched-giant-pulse/) [L17 · Lock phases into pulses](https://sandboxphysics.com/labs/mode-locking-frequency-comb/) [L18 · Calibrate the comb](https://sandboxphysics.com/labs/optical-frequency-comb/) [L19 · Inject carriers](https://sandboxphysics.com/labs/semiconductor-laser-diode/) [L20 · Compare gain media](https://sandboxphysics.com/labs/laser-medium-zoo/) [L21 · Distributed gain](https://sandboxphysics.com/labs/fiber-fbg-laser/) [L22 · Choose feedback](https://sandboxphysics.com/labs/laser-feedback-architectures/) [L23 · Balance ultrafast pulses](https://sandboxphysics.com/labs/ti-sapphire-oscillator/) [L24 · Stretch and compress](https://sandboxphysics.com/labs/chirped-pulse-amplification/) [L25 · Trap and extract](https://sandboxphysics.com/labs/regenerative-multipass-amplifier/) [L26 · Convert frequencies](https://sandboxphysics.com/labs/nonlinear-frequency-conversion/) [L27 · Scatter and cascade](https://sandboxphysics.com/labs/raman-brillouin-lasers/) [L28 · Synchronize a slave](https://sandboxphysics.com/labs/injection-locking/) [L29 · Lock to a reference](https://sandboxphysics.com/labs/pdh-frequency-lock/) [L30 · Measure phase noise](https://sandboxphysics.com/labs/laser-linewidth-phase-noise/) [L31 · Scatter for feedback](https://sandboxphysics.com/labs/random-laser/) [L32 · Diagnose thermal scaling](https://sandboxphysics.com/labs/thermal-lensing/) [L33 · Combine coherent fields](https://sandboxphysics.com/labs/coherent-beam-combining/) [L34 · Sustain a Kerr pulse](https://sandboxphysics.com/labs/microresonator-kerr-comb/) [L35 · Return a delayed field](https://sandboxphysics.com/labs/external-feedback-chaos/) [L36 · Scan without mode hops](https://sandboxphysics.com/labs/tunable-laser-scan/)

Physical basis: [MIT: coherence and spectrum](https://ocw.mit.edu/courses/res-6-006-video-demonstrations-in-lasers-and-optics-spring-2008/resources/coherence-length-and-source-spectrum/) · [Rate equations](https://ocw.mit.edu/courses/6-977-ultrafast-optics-spring-2005/resources/chapter2/) · [Pumping and gain media (§7.2)](https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/bf3bdeb971819ddc9c68719d28bb2745_classnotes.pdf) · [Laser dynamics](https://ocw.mit.edu/courses/6-977-ultrafast-optics-spring-2005/resources/chapter4/)

Physics tutorial

## Measure both ruler coordinates

**Background** A regular train of pulses contains a comb of equally spaced optical frequencies. Knowing the spacing does not tell you where the comb starts.

**Why it matters** To use the comb as a ruler for light, you need the repetition frequency, the carrier-envelope offset (CEO) frequency and the integer index of the tooth you are using.

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 misconception** Knowing the tooth spacing is enough to know every optical frequency.

**Better mental model** The 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 misconception** The size of the beat frequency tells you which side of the tooth the target is on.

**Better mental model** A 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

1. 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.

2. 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.

3. 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.

4. 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.

Physical conclusion

Before you trust the counted RF values, check that a doubling pair exists in the spectrum, that the record is long enough and that the tooth index is not ambiguous.

## Related experiments and articles

Probes and measurement · Reading path

### Controlled light and precision measurement

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Frequency combs measure both the spacing and the offset**](https://sandboxphysics.com/stories/probe-evidence/frequency-comb-self-reference/)
2. [Lab **Mode Locking** Watch random phases lock into one pulse](https://sandboxphysics.com/labs/mode-locking-frequency-comb/)
3. [Lab **Microresonator Kerr Comb** Compare a circulating pulse with its comb](https://sandboxphysics.com/labs/microresonator-kerr-comb/)
4. [Lab **Nonlinear Frequency Conversion** See why the doubling crystal needs phase matching](https://sandboxphysics.com/labs/nonlinear-frequency-conversion/)

Waves & optics · Starter path

## From one ray to interference measurement

Start with refraction, find where the ray picture stops working, then use interference to measure tiny distances.

[Start here → **Optical Boundary Explorer** Turn the incident angle and watch reflected and transmitted power trade off. Then switch to the critical-angle scene. Observe · 11 min](https://sandboxphysics.com/labs/optical-boundary-explorer/)

1. [1 Optical Boundary Explorer](https://sandboxphysics.com/labs/optical-boundary-explorer/)
2. [2 Rainbow Caustic Observatory](https://sandboxphysics.com/labs/rainbow-laboratory/)
3. [3 Camera Obscura](https://sandboxphysics.com/labs/camera-obscura/)
4. [4 Coherent Wave Composer](https://sandboxphysics.com/labs/coherent-wave-composer/)
5. [5 Young Double-Slit Bench](https://sandboxphysics.com/labs/young-double-slit-bench/)
6. [6 Michelson Precision Lab](https://sandboxphysics.com/labs/michelson-precision-lab/)

[View the full path →](https://sandboxphysics.com/topics/waves-and-optics/#starter-path)