A train of short pulses becomes an optical ruler
A mode-locked laser emits a repeating train of brief light pulses. Its spectrum has evenly spaced narrow lines, called comb teeth. The pulse repetition rate fixes their spacing. Counting that spacing is like knowing the distance between marks on a ruler: the first mark still needs a position.
Inside each pulse, the optical field oscillates much faster than the pulse envelope rises and falls. The crests can slide relative to that envelope from one pulse to the next. This carrier-envelope phase slip shifts every comb tooth by the same offset frequency.
In the formula, the integer labels the tooth; the other two quantities are the repetition and offset frequencies.
Sources: [1]
The pulse needs its frequencies to stay in step
Many frequencies can reinforce one another during a brief interval and largely cancel between pulses. Mode locking keeps their relative timing repeatable. A wide spectrum supplies the frequencies needed for a short pulse, but those frequencies must also arrive together.
In one common source, a Ti:sapphire laser, intense light focuses itself more strongly in the crystal. With a suitably placed aperture, the strong central part of a pulse can lose less light than its weak wings. This Kerr-lens mechanism favors pulsed operation. The gain amplifies different frequencies by different amounts. Those frequencies acquire different delays, and the stronger parts acquire larger phase shifts. Together these effects determine the circulating shape. Other frequency-comb sources use other mechanisms.
Locking the phases turns noise into one pulse
In a linear cavity with mirror spacing L, light returns to the output mirror every 2L/c. The teeth are therefore c/(2L) apart, about 1 gigahertz for a 15-centimetre cavity; doubling the length halves the spacing. When the teeth start with random phases, they add up to noisy, nearly continuous light, and the peak is only a few times the average.
A modulator whose loss dips once per round trip, or an absorber that becomes more transparent in strong light, favours one time slot on every pass. Light arriving then survives best, the teeth settle to a common phase and the output becomes one pulse per round trip. In active locking the gain’s limited bandwidth broadens the pulse slightly on each pass while the modulator trims it. Their balance sets the width, which grows as the square root of the cavity length.
A passive ring can produce pulses too
A comb can also start with a continuous laser feeding a tiny passive ring. The material's refractive index changes slightly with intensity, an effect called the Kerr response. That change couples the ring's allowed frequencies. A localized pulse can circulate on top of a weak continuous background when the intensity-dependent phase change balances dispersion, the different delays acquired by different frequencies. The external pump supplies the energy that leaks away each round trip.
Herr and colleagues reported temporal solitons in a microresonator in work published in 2014. They compared the observed spectrum and pulse measurements with the circulating field calculated from a driven, lossy nonlinear model. A pulse present at the start and a pulse reached by tuning the laser are different preparations. Heating shifts the resonance during tuning, and several pulses or a repeating modulation can also produce comb lines. A broad comb spectrum therefore needs a time-domain and stability check before it is identified with one persistent pulse. Absolute frequency measurement still needs the spacing, offset and reference described below.
Sources: [6]
Doubling one tooth reveals the common offset
An octave-spanning comb reaches from one optical frequency to twice that frequency. Double a tooth at the low end, then beat it against the high-end tooth with twice its mode number. Doubling the low tooth doubles both its mode contribution and its offset. The high tooth has the same mode contribution but only one copy of the offset. Their difference leaves one offset frequency.
The two fields must overlap at a detector with enough coherence and power to produce a beat. Spectral broadening across the octave is part of the apparatus; a broad intensity spectrum alone does not guarantee that usable comparison. NIST describes fibre broadening and the contributions of the Hall/JILA and Hänsch teams.
The crystal changes the frequency, not the arithmetic
A nonlinear crystal can combine two photons from one input frequency into a photon at twice that frequency. The extra photon energy comes from the input beam. Light generated at successive depths builds up efficiently when its phase keeps pace with the generated field already present. This is phase matching.
Turning a birefringent crystal changes the refractive index seen by one polarization, so its phase matching can be tuned. Detuning the angle weakens the doubled light; it does not turn frequency doubling into an arbitrary frequency shift. Pump depletion and separation of the finite beams can limit the useful crystal length. Those details affect the beat strength, while the offset cancellation still follows the comb frequencies.
The unknown laser needs another beat
Combine the unknown laser with a nearby comb tooth on a photodetector. The beat measures the magnitude of their frequency difference. A coarse wavelength measurement identifies the nearby teeth; changing a known setting or using additional information determines which side of the tooth the laser lies on.
A wrong tooth number, a missed counted cycle or an unstable reference can produce a wrong result with many apparently precise digits. Absolute measurement therefore ties the repetition, offset and unknown-laser beat to a common reference linked to an established standard by documented comparisons with stated uncertainties. Electronic counters measure the radio-frequency signals, which determine the optical frequency through the tooth relation.
The large tooth number magnifies a repetition-rate error. As an arithmetic example, light near 250 terahertz with teeth 100 megahertz apart uses a tooth number near two and a half million. A one-hertz error in that spacing contributes about two and a half megahertz to the reconstructed optical frequency. Extra displayed digits cannot remove it. The offset count and target beat contribute their own errors, and a real measurement must include the reference clock and detector noise. A sufficiently long record lets the counter compare several complete cycles of a slow beat.
Independent combs test the measurement
NIST’s account reports comparisons between separate comb systems. Agreement between independent apparatuses demonstrates that the frequency-transfer method is reproducible. Frequency combs can compare optical clocks, connect optical frequencies to microwave signals and measure frequency ratios. Their accuracy comes from the realized reference and the systematic corrections of the devices being compared. The 2005 contribution of Hall and Hänsch spans precision laser spectroscopy and frequency-comb techniques, built through a broad metrology effort.