Carrier–envelope Offset

The carrier–envelope offset frequency (CEO frequency) of a mode-locked laser is

where Δφ_ceo is the change in the carrier–envelope offset phase (also called carrier–envelope phase, CEP) per resonator round trip and f_rep is the pulse repetition rate. The carrier–envelope offset frequency thus lies between zero and the repetition rate f_rep. The optical frequencies of the pulse train (which for simplicity is assumed to be noiseless) are

with integer values of the index j. This means that there is a so-called equidistant frequency comb, and all occurring optical frequencies are determined by the repetition frequency and the CEO frequency.

The carrier–envelope offset frequency is important in optical frequency metrology and also in high-intensity physics with few-cycle laser pulses, because it affects the oscillation pattern of the electric field and even the peak electric field strength.

The carrier envelope offset frequency can be detected e.g. with a so-called f−2f interferometer via a beat note between the higher-frequency end of the comb spectrum with the frequency-doubled lower-frequency end, if the optical spectrum covers an optical octave. (Such broad spectra can be achieved e.g. with supercontinuum generation in photonic crystal fibers, if the spectrum from the laser is not broad enough.) The article on frequency combs gives more details.

The CEO frequency of a laser can be influenced e.g. via the pump power, by slightly tilting a resonator mirror, or by inserting a glass wedge to a variable extent. (Various other control elements have also been developed.) The combination of detection and control of the CEO frequency allows the CEO frequency to be stabilized to a constant well-known value, so that all optical frequencies in the frequency comb are related to two RF or microwave frequencies [4]. Under such conditions, a laser is called CEO-stabilized or CEP-stabilized.

It is also possible to obtain frequency combs which naturally have a zero carrier–envelope offset frequency, i.e., an essentially constant carrier–envelope offset phase. For this purpose, it is necessary to arrange for difference frequency generation with both inputs from a single frequency comb [6]. This method leads to so-called self-phase-stabilized pulses.

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