Three-dimensional carrier-envelope-phase map of focused few-cycle pulsed Gaussian beams

We derive an analytical expression that describes the complete three-dimensional carrier-envelope-phase (CEP) distribution in the focal volume of ultrashort pulsed Gaussian beams focused by spherical mirrors or lenses. The focal CEP map depends on the so-called factor $g$ specifying the frequency dependence of the beam width of the source few-cycle pulse, on its chirp, and on the small chromatic aberration introduced by a lens without appreciably distorting or broadening the few-cycle pulse. We show how to tailor the CEP map of mirror-focused and lens-focused few-cycle pulses in order to produce negligible transversal and axial CEP variations in specific regions of the focal volume for phase-sensitive interactions of light with matter taking place in a volume or on a surface. We propose a quasiachromatic doublet lens that can implement in practice these tailored CEP distributions.

Figure 1

For spherical mirror focusing, focal CEP maps (cut along the $y=0$ plane) for an input beam with $g=-0.5$, (a) transform-limited pulse $2C/\mathrm{\Delta }{T}^2=0$, and (b) chirped pulse $2C/\mathrm{\Delta }{T}^2=-0.571$. The CEP is transversally flat on the focal plane in (a) and at $Z=L_R/2$ in (b). (c) Axial variation of the CEP in the two above cases (solid curves) compared to Gouy's phase shift (dashed curve). (d) Needed relative chirp as a function of the position $Z_{\mathrm{frozen}}$ at which the CEP is constant transversally for a few values of $g$ of the input pulse. Chirps with opposite signs cause the same constant CEP transversally in the first half of the focal region.

Figure 2

(a) Needed relative chirp as a function of the position $Z_{\mathrm{frozen}}$ at which the effective CEP with $n=4$ is locally constant axially for a few values of $g$ of the input beam for which this chirp is relatively low. Chirps with opposite signs cause the same locally constant axial CEP variation in the first half of the focal region. (b) Needed relative chirp to freeze the effective CEP with $n=4$ both axially and transversally and the position within the focus at which this happens as functions of the values of $g$ for which this effect is possible. (c) On-axis and effective CEPs with $n=4$ (solid curves) for $g=1.26$ and $2C/\mathrm{\Delta }{T}^2=0.574$ and Gouy's phase (dashed curve), showing locally constant effective CEPs at $Z=L_R/2$. (d) CEP map in the focal region for $g=1.26$ and $2C/\mathrm{\Delta }{T}^2=0.574$, showing also a transversally constant CEP at $Z=L_R/2$.

Figure 3

(a) Chromatic aberration parameter $\gamma$ as a function of the relative chirp $2C/\mathrm{\Delta }{T}^2$ (a) to freeze axially the on-axis CEP at the focal point for different values of $g$ of the input pulse. (b) The same but to have a transversally flat CEP on the focal plane. (c) and (d) Chromatic aberration and chirp to produce on-axis and transversally frozen CEPs at the focus. Positive chromatic aberration requires negative chirp (solid curves) and vice versa (dashed curves).

Figure 4

CEP maps in the focal volume ($y=0$ sections) for $g=0$ and the indicated values of $\gamma$ and $2C/\mathrm{\Delta }{T}^2$. The CEP at $\zeta =0$ is axially constant in (a), transversally constant in (b), and axially and transversally constant in (c) and (d) with a maximum in (c) and a minimum in (d).

Figure 5

Upper plots: On-axis ($n\to \infty$) CEP shift for input pulses with the indicated values of $g$, focused by a lens with the indicated values of chromatic aberration $\gamma$ and chirp $2C/\mathrm{\Delta }{T}^2$ such that the CEP is approximately flat in the first half of the focus (red curves). The dashed curves represent Gouy's phase shift for reference, and the black curves represent the CEP shift in the case of focusing without chromatic aberration and without chirp. Lower plots: Corresponding CEP maps ($y=0$ sections) showing that the CEP is also transversally flat at positions about $-L_R/2$ as indicated by the vertical dashed lines in the upper plots.

Figure 6

(a) On-axis CEP shift about the focus for two different lens separation distances evaluated from Eq. (14) (curves) and from the numerical simulation described in the text (symbols). (b) The same but on the focal plane.