Carrier-envelope phase of ultrashort pulses generated by optical rectification process

Passive stabilization of carrier-envelope phase (CEP) of few-cycle pulses by use of difference frequency generation (DFG) is a key technology for frequency comb and attosecond science [1,2]. Similarly, the CEP of the ultrashort terahertz pulse generated by optical rectification is also passively stabilized. The optical rectification process has been considered as just a special case of DFG, where the frequencies of the two input pulses are the same.In this contribution, we report how the CEP of the generated pulse is determined through the DFG and the optical rectification. We have found a clear difference of the CEP determination between these two processes. Assuming the two complex input electric fields are E1(t) = E (t)exp(iω1t + iφ1) and E2(t) = E (t)exp(iω2t + iφ2), the DFG field generated through the process ω1 - ω2 → ω0 can be described as ∂ E1(t)E 2(t) = ∂ E (t)E (t)expi(ω1 - ω2)t + i(φ1 - φ2) E0(t) o = E0(t)exp(iω0t + iΔφ), (1) ∂t ∂t where E0(t) = ∂ E (t)E (t) + iω0E (t)E (t), ω1 - ω2 = ω0, and φ1 - φ2 = Δφ. When ω0 ~ 0, namely, the ∂t process is called as optical rectification, the real field of the DFG reduces to ∂ E (t)E (t)cosΔφ. As a result, Δφ ∂t does not contribute to the phase but to the amplitude of the output field. On the other hand, when ω0 0, the real output field can be written as ω0E (t)E (t)cos(ω0t + Δφ + π/2). In this case, the CEP of the field is Δφ + π/2, which means the relative phase between the two input pulses directly affect the CEP of the output pulse. To experimentally investigate the CEP variation, we have generated phase-stable mid-infrared (MIR) pulses by using four-wave mix- ng (ω1 + ω1 - ω2 → ω0) through filamentation [3] and characterized the pulse including CEP information by measuring FROG and electro-optic sampling [4]. Figure 1(a) and (b) shows waveforms and spectral phases of the MIR pulses at different relative phases of the input pulses, respectively. Figure 1(c) shows phase change for each frequency components of the MIR pulses. The phase of the high frequency components (ω0 >3000 cm-1) changes continuously and linearly with respect to the relative phase. On the other hand, the phase of the low frequency components (ω0 <;3000 cm-1) changes by 0 or π like a step function, which means that Δφ basically affect only the amplitude. The π phase jump means that only the sign of the amplitude (the sign of cosΔφ) changes. The phase variation is understood with the above mentioned simple theory. At the conference, we plan to show the detail of the physics of the phase variation with numerical simulation results.