Carrier-envelope offset (CEO)

An ultrashort pulse is an oscillating electric field, the carrier, inside a smooth envelope. The carrier-envelope phase (CEP), $\varphi_{CE}$, is the offset between the peak of the envelope and the nearest crest of the carrier. In a mode-locked laser the envelope travels at the group velocity and the carrier at the phase velocity, which differ in any dispersive medium, so the CEP changes by a fixed amount $\Delta\varphi_{CE}$ on every round trip. In the frequency domain this slip shifts every line of the frequency comb by the carrier-envelope offset frequency

$$f_{CEO} = \frac{\Delta\varphi_{CE}}{2\pi}\,f_{rep},$$

taken modulo the repetition rate $f_{rep}$. A laser with $f_{rep} = 100$ MHz and $f_{CEO} = 20$ MHz slips by one fifth of a cycle, 1.26 rad, from pulse to pulse, and the pattern repeats every five pulses.

Comb lines and a worked mode number

Each line of the comb sits at

$$f_n = n\,f_{rep} + f_{CEO},$$

with $n$ an integer of order 10⁵ to 10⁶. For an erbium fiber comb with $f_{rep} = 250$ MHz and $f_{CEO} = 35$ MHz, the line nearest 1560 nm has $n = 768\,698$ and lies at 192.174535 THz. Knowing $f_{rep}$ and $f_{CEO}$ precisely fixes the frequency of every line; the integer $n$ is found from a wavemeter reading good to a fraction of $f_{rep}$, here to better than about 100 MHz.

Measuring f_CEO: the f-2f interferometer

$f_{CEO}$ cannot be read from the pulse train directly, since a photodiode detects only the envelope. The standard method is self-referencing with an f-2f interferometer. The spectrum is first broadened to span an octave, usually by supercontinuum generation in a highly nonlinear or microstructured fiber. Light from the low-frequency end, near $f_n$, is frequency-doubled by second-harmonic generation to $2f_n = 2n f_{rep} + 2f_{CEO}$ and overlapped on a photodiode with light from the high-frequency end near $f_{2n} = 2n f_{rep} + f_{CEO}$. Their beat note is

$$2f_n - f_{2n} = f_{CEO}.$$

In the example above, the doubled 1560 nm line at 384.349070 THz beats with the comb line at 384.349035 THz, near 780 nm, to give 35 MHz. Many pairs of lines contribute the same beat, which is why the signal is strong enough to lock. A servo then feeds back to the pump power, which changes the intracavity dispersion and nonlinear phase and thus $\Delta\varphi_{CE}$, or to an acousto-optic modulator outside the cavity. Telle and co-workers proposed the scheme in 1999, and Jones, Diddams and colleagues demonstrated CEP control and direct optical frequency synthesis with it in 2000. Variants such as 2f-3f need less than an octave of bandwidth.

Dispersion and the CEP of a single pulse

Material between the laser and the experiment changes the CEP of each pulse by

$$\Delta\varphi = \frac{2\pi L\,(n_g - n)}{\lambda}.$$

For fused silica at 800 nm, $n_g - n = 0.0138$, so 1 mm of glass shifts the CEP by about 109 rad, and a 58 µm change in thickness shifts it by $2\pi$. Pairs of thin glass wedges, translated into the beam, are the usual way to set the CEP of an amplified pulse, and air-path fluctuations and beam pointing are common sources of CEP noise.

Where it matters

For pulses many cycles long the CEP has no practical effect, because the envelope barely changes over one cycle. For few-cycle pulses, such as the 5 fs pulses from Kerr-lens mode-locked Ti:sapphire oscillators, fewer than two cycles lie under the envelope, and the peak field depends on the CEP. Strong-field processes such as high-harmonic generation, isolated attosecond pulse production and above-threshold ionization respond to the instantaneous electric field and show clear CEP dependence. In metrology, a stabilized $f_{CEO}$ and $f_{rep}$ turn the comb into a ruler linking microwave and optical frequencies, the basis for counting the output of an optical atomic clock; Hall and Hänsch shared half of the 2005 Nobel Prize in Physics for this work.

Common questions

What is the difference between CEP and f_CEO?

The CEP is the phase of a single pulse. $f_{CEO}$ describes how that phase changes from pulse to pulse. Locking $f_{CEO}$ to zero, or to a fraction of $f_{rep}$ such as a quarter, makes every pulse, or every fourth pulse, identical in CEP; the absolute CEP value still has to be measured separately, for example by stereo detection of photoelectrons.

Why does f_CEO need an octave-spanning spectrum?

The f-2f scheme compares a doubled low-frequency line with a line of exactly twice its frequency, and both must be present in the same spectrum.

References: H. R. Telle et al., Appl. Phys. B 69, 327 (1999); D. J. Jones et al., Science 288, 635 (2000); Th. Udem, R. Holzwarth, T. W. Hänsch, Nature 416, 233 (2002); J.-C. Diels, W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic Press, 2006); I. H. Malitson, J. Opt. Soc. Am. 55, 1205 (1965).