2. Generation of attosecond pulses by synthesis of cascaded harmonics

Fundamentally, an optical pulse train with a repetition rate of ωm can be viewed as the sum of a set of frequency components that form an arithmetic series [13]. The electric field of each component can be written in the following form:

$$E\_q(t) = A\_q e^{i\phi\_q} e^{i\omega\_q t},\tag{1}$$

where ω<sup>q</sup> = ω<sup>0</sup> + qωm, for q = 0, 1, 2, … To shape the pulse envelope, the phase term ϕ<sup>q</sup> and amplitude term Aq of each component are controlled. One can set the phase term ϕ<sup>q</sup> and rewrite it as ϕ<sup>q</sup> = ϕ<sup>0</sup> + qϕm. The synthesized pulse could then be expressed as:

$$E(t) = \sum\_{q} E\_{q}(t) = e^{j\left(a\_{0}t + \phi\_{0}\right)} \sum\_{q} A\_{q} e^{i q a\_{n} \left(t + \frac{\phi\_{n}}{a\_{m}}\right)} = e^{j\left(a\_{0}t + \phi\_{0}\right)} E\_{c}\left(t + \phi\_{n}/\_{w\_{n}}\right),\tag{2}$$

where EcðÞ� <sup>t</sup> <sup>P</sup> <sup>q</sup> Aqeiqωmt is a typical cosine pulse train and <sup>ω</sup>0<sup>t</sup> <sup>þ</sup> <sup>ϕ</sup><sup>0</sup> is the time-varying CEP with frequency of ω0. In the commensurate case, the CEP is equal to ϕ<sup>0</sup> for all ultrashort pulses belonging to the same attosecond pulse train or within the ns pulse envelope in the HSRS approach since ω<sup>0</sup> equals to zero. As a result, CEP will be randomly changing if ϕ<sup>0</sup> is random from 1 ns pulse to another. For instance, a 802 nm and a 602 nm laser with pulsewidth around ns and repetition rate of 30 Hz (corresponding to q = 3 and 4 of the Raman resonance of molecular hydrogen) were employed to stimulate the Raman sidebands in early work by one of the co-authors [14, 15]. Because the phases of the two driving lasers, denoted as ϕ<sup>3</sup> and ϕ4, are random and independent of each other in individual ns pulses, both ϕ<sup>m</sup> = ϕ<sup>4</sup> � ϕ<sup>3</sup> and ϕ<sup>0</sup> = ϕ<sup>3</sup> 3ϕ<sup>m</sup> = 4ϕ<sup>3</sup> 3ϕ<sup>4</sup> are random in time as well. Although CEP of generated pulse trains with 1 ns pulse every 33 ms is fixed. The CEP of attosecond pulses in different ns pulse envelopes varies randomly as well. This severely limits the application of this type of attosecond light source. Cross-correlation by four-wave-mixing interaction among attosecond pulses within the same ns pulse, which are commensurate. Therefore, correlation behavior could still be observed.

Alternatively, CEP will be fixed if all phase-controlled frequency components of the pulse train are optical harmonics from the same laser, rather than through Raman sideband generation. It is clear that the relative phase among generated higher-order harmonics and the lower ones are fixed. For example, relative phase among ϕ5, ϕ<sup>2</sup> and ϕ<sup>3</sup> of the fifth, second and third harmonic of the same laser will not be changing, if light of frequency ω<sup>5</sup> is generated from SFG of ω<sup>2</sup> and ω3.

At this junction, it is instructive to note that Hansch proposed that sub-femtosecond pulse could be synthesized by nonlinear phase locking of lasers nearly a decade ago [16]. Later, his group further demonstrated the feasibility of this approach with three cw phase-locked semiconductor lasers [17]. This approach, however, was not pursued since primarily because of the low power generated.

In the following, we summarize the potential advantages and unique features of attosecond pulse generation through pulse synthesis of harmonics of the same laser in contrast to the earlier Raman sideband approach:
