Definition: the offset between the optical phase and the maximum of the wave envelope of an optical pulse
The time dependence of the electric field associated with a light pulse can be described as a fast sinusoidal oscillation (the carrier), multiplied by a more slowly varying envelope function. When the pulse propagates through a medium, the relative position between the carrier wave and envelope will in general change due to chromatic dispersion, causing a difference between phase velocity and group velocity, and possibly also due to optical nonlinearities. The carrier–envelope offset phase (or absolute phase) of a pulse is defined as the difference between the optical phase of the carrier wave and the envelope position, the latter being converted to a phase value. Figure 1 shows pulses with different values of the carrier–envelope offset phase.
Figure 1: Electric field (blue curves) of laser pulses with a 5-fs duration and a variable CEO phase. This phase changes by π/2 from one pulse to the next one, corresponding to a CEO frequency which is one quarter of the pulse repetition rate.
In a mode-locked laser, a pulse train is usually generated from a single pulse circulating in the laser resonator. Every time when this pulse hits the output coupler, an attenuated copy of it is emitted to the laser output. Typically, there is a certain change in the carrier–envelope offset phase in each round trip, which can be hundreds or thousands of radians. Therefore, each emitted pulse will have a different carrier–envelope phase. For the output pulse train, only the change in this phase value modulo 2 π is relevant. This can very sensitively depend on factors such as the laser power, resonator alignment, etc.
Carrier–Envelope Offset Frequency
The carrier–envelope offset frequency (CEO frequency) of a mode-locked laser is
where Δφceo is the change in the carrier–envelope offset phase (also called carrier–envelope phase, CEP) per resonator round trip and frep is the pulse repetition rate. The carrier–envelope offset frequency thus lies between zero and the repetition rate frep. 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.
CEO Measurement
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.
CEO Stabilization
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 in an automatic feedback system 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. In some cases, a stabilization to zero CEO frequency is achieved.
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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