Carrier envelope phase

M. Kremer, B. Fischer, B. Feuerstein, V.L.B. de Jesus, V. Sharma, C. Hofrichter, et al., Electron localization in molecular fragmentation of H2 by carrier-envelope phase stabilized laser pulses, Phys. Rev. Lett. 103 (21) (2009) 213003. 

Jones, D., Diddams, S., Ranka, J. et al.. Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical freqnency synthesis. Science, 288, 635, 2000. 

Fig. 8.7 Integrated asymmetry over several energy ranges versus carrier envelope phase and theoretical prediction. Adapted from [63, 80]...

Fig. 3. Time dependent field of the 3.9fs pulse reconstructed from the FROG trace. The solid curve is the field and the dotted curve is the envelope of the field. The central wavelength is 595nm corresponding to the oscillation period of 2 fs. The carrier envelope cannot be determined and arbitrary phase is used for the description.

Fig. 2. Consecutive pulses of a chirp free pulse train (A(t) real) and the corresponding spectrum. Because the carrier propagates with a different velocity within the laser cavity than the envelope (phase- and group velocity), the electric field does not repeat itself after one round trip. A pulse-to-pulse phase shift Aip results in an offset frequency...

Things are not quite as simple as they seem. In order for the constructive interference, which is at the core of wavepacket interferometry, to occur, not only must (t + At) = (t), but also the phases of apump and aprobe> which depend on the optical phase of the femtosecond laser rather than the molecular phase, must match. A rigorous treatment of the phase coherent pump/probe scheme using optically phase-locked pulse pairs is presented by Scherer, et al., [1990, 1991, 1992] and refined by Albrecht, et al., (1999), who discuss the distinction between and consequences of pulse envelope delays vs. carrier wave phase shifts (see Fig. 9.6). A simplified treatment, valid only for weak optical pulses is presented here. 

In an FM MMW spectrometer the spectral source frequency is modulated at a certain rate /, typically 1 kHz. This gives rise to sidebands of the spectral source frequency above and below the carrier frequency. The frequency modulated MMW carrier has in its modulation envelope phase and amplitude relationships to the carrier. Mixing in the non-linear junction of the detector yields the modulation signals altered by their interaction with the cavity and gas inside it, with their preserved amplitude and phase relationship to the original modulation signals. Those properties are measured by passing the heterodyne mixer output and the thermal noise contribution from the mixer, to a filtered phase-sensitive detection system, with the original modulation as reference. 

Fig. 11.5 Differential phase contrast detection of patterned protein in a 10 mm by 30 mm region, (a) Protein height signal showing ridges of protein in a checker board pattern, (b) Side band demodulated signal image in which the carrier frequency of the ridges is removed to show only the protein envelope. Reprinted from Ref. 21. with permission. 2008 Optical Society of America...

In polarization modulated ENDOR spectroscopy (PM-ENDOR)45, discussed in Sect. 4.7, the linearly polarized rf field B2 rotates in the laboratory xy-plane at a frequency fr fm, where fm denotes the modulation frequency of the rf carrier. In a PM-ENDOR experiment the same type of cavity, with two rf fields perpendicular to each other, and the same rf level and phase control units used in CP-ENDOR can be utilized. To obtain a rotating, linearly polarized rf field with a constant magnitude B2 and a constant angular velocity Q = 2 fr (fr typically 30-100 Hz), double sideband modulation with a suppressed carrier is applied to both rf signals. With this kind of modulation the phase of the carrier in each channel is switched by 180° for sinQt = 0. In addition, the phases of the two low-frequency envelopes have to be shifted by 90° with respect to each other. The coding of the two rf signals is shown in Fig. 8. 

The envelope function defines the pulse repetition time T = 27r/u>r by demanding A(t) = A(t — T). Inside the laser cavity the difference between the group velocity and the phase velocity shifts the carrier with respect to the envelope after each round trip. The electric field is therefore in general not periodic with T. To obtain the spectrum of E(t) the Fourier integral has to be calculated ... 

Jig. 2. A sine-wave carrier (frequency ojq) is assumed to be periodically amplitude modulated at frequency 2 with a Gaussian envelope. Exact frequency division (ioq/ 2 = integer) is obtained if the phase of the carrier is fixed to the phase of the modulation envelope. 

Using amplitude or frequency modulation of a carrier at frequency tog, we can achieve exact frequency division if we make sidebands of the carrier to such low frequency that we can force the condition wg-nQ = (n+2) 2-a>g = 2 so that tog/ 2 = n+1. For example, if we examine Fig. 2, we can achieve exact frequency division by any means which locks the phase of the carrier to the phase of the amplitude modulation that is, the undulations of the carrier do not "slip" under the envelope of the amplitude modulation. A divider based on these principles would be quite useful if 2 is in the microwave region (or below) where precise frequency synthesis is possible. Since 2 and n could be freely chosen, any value of uig could be measured in a single device. 

The optical phase of the carrier wave in a linearly polarized femtosecond pulse can be measured by the photoelectron rate (Fig. 6.59). If the electrons are produced by the nth harmonic of the visible femtosecond pulse, the rate is proportional to the 2nth power of the visible field amplimde. The amplitude depends strongly on the phase of the optical wave relative to the envelope maximum of the pulse. Measurements of this photo-electron rate as a function of the phase shift of the field amplitude against the pulse maximum allows the determination of the phase and the pulse width of the high-harmonic attosecond pulse [754]. There are many more applications of attosecond pulses these can be found, for example, in the publications of the groups of P. Corkum at the NRC in Ottawa [753] and F. Krausz at the MPI for Quantum Optics in Garching, who have pioneered this field [754]. 

The electric field of the femtosecond pulses is shown in Fig. 9.87. If there were no phase shifts, each pulse would be an exact replica of the previous pulse, i.e. E(t) = E(t — T). However, due to intracavity dispersion in the laser resonator, the group velocity and the phase velocity may be different. This causes a phase shift A< = T carrier wave E(t) with respect to the peak of the pulse envelope. 

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