Reply to comments on: Line broadening by focusing; Gas ionization by focused laser beams

Abstract

In ref. 1 we were only concerned with the focused beam as such, not with the question by which device (lens, mirror, etc.) or by which mechanism the focusing is achieved. We mathematically showed that under certain, well-defined circumstances and to a well-defined accuracy the stronger the focusing (Ak,/ko etc.), the broader the energy spectrum (Ao/oo) , other parameters being equal. Actually, since the original laser beam in any of the considered cases is only weakly focused and quasi-monochromatic, the only thing we have to show is that the focused beam is less monochromatic even if some parameters change during the focusing. If one accepts the mathematical results one might object that our mathematics does not apply to the physical situation considered, in that the wave packets of the focused beam are, in reality, other than those described by our formulas. The mathematical form of our wave packets is, however, a very general one (formula [2], ref. 1). The form in our treatment enters only via the two parameters x and 5 (formula [16], ref. l ) , and the range of values that lead to some perceptible broadening is very wide, wider even than that considered in Table 1, ref. 1. The restrictions put on the form of our wave packets are the following: (i) only positive frequencies are considered, which is in accordance with the standard treatment by Born and Wolf (2); (ii) the form of the wave packet in momentum space $ ( k ) separates into a product (formula [12], ref. 1); (iii) the single-factor functions are symmetrical with respect to their respective maximum values; and (iv) they allow for the relations (k:) = x(AkJ4, (k:) = 5(AkJ6 etc. (eq. [16], ref. 1). The restrictions (ii)-(iv) are only introduced to obtain a handy final formula, and we do not see any reason why one should not obtain a line broadening without these restrictions, for example, when using a computer. In ref. 1 we only wanted to show that the effect exists. The application to the ionization of gas molecules, whose ionization energy is well above the mean photon energy, has been treated bv Dr. De Brito in ref. 3. There he does not claim that line broadening gives the explanation of the observed ionization, but only that it may contribute to it, and he shows that the contribution crucially depends on the line shape far from the line center, where little seems to be known experimentally