Comment on " Controlling the Spectral Shape of Nonlinear Thomson Scattering With Proper Laser Chirping "

This Response or Comment is brought to you for free and open access by the Physics at ODU Digital Commons. It has been accepted for inclusion in Physics Faculty Publications by an authorized administrator of ODU Digital Commons. For more information, please contact digitalcommons@odu.edu. Repository Citation Terzić, Balša and Krafft, Geoffrey A., "Comment on "Controlling the Spectral Shape of Nonlinear Thomson Scattering With Proper Laser Chirping"" (2016). Physics Faculty Publications. 41. http://digitalcommons.odu.edu/physics_fac_pubs/41


I. INTRODUCTION
In this comment we begin by noting that two important ideas used in Rykovanov, Geddes, Schroeder, Esarey and Leemans [1] (RGSEL) already appear in Terzić, Deitrick, Hofler and Krafft [2] (TDHK), their Ref.[47], for the linearly polarized case.In particular, the analytic requirement of constant lab-frame emission frequency and the use of a stationary phase argument prominent in TDHK are reused in RGSEL.The result is to derive essentially the same laser chirping prescription as appears in TDHK, but modified for circular polarization.It is the purpose of this comment to make clearer the appropriate connections between the results in TDHK and RGSEL.

II. DERIVATION OF THE EXACT LASER CHIRPING
First note how the key equation of RGSEL, their Eq.( 22), can be found from the exact chirping prescription reported in TDHK.The equivalent equation for the linearly polarized laser pulse is derived in TDHK as an unnumbered equation preceding their Eq.( 4): From here deriving the exact laser chirping function is just solving a first order differential equation with a boundary condition.We use fð0Þ ¼ 1 in TDHK.The solution for the exact modulation function in TDHK becomes, as reported in their Eq.( 4): The derivation of the corresponding exact modulation function for a circularly polarized pulse instead of the linearly polarized pulse from TDHK is straightforward.
One should remove factors of 1=2 from Eq. ( 1) for the circular polarization case to obtain For the boundary condition fð0Þ ¼ 1 (our preferred frequency normalization) one obtains For fðAE∞Þ ¼ 1 (as chosen in RGSEL), after relating the modulation function from TDHK, f, to that of RGSEL, ϕ, as ϕðξÞ ¼ ξfðξÞ, we obtain their Eq.( 21): Equation (19) of RGSEL, which led to their crucial Eq. ( 22), is derived using the stationary phase argument as in [3].TDHK previously used the stationary phase method to give an alternate derivation of the exact frequency modulation condition.Therefore, the chirping mechanism reported by RGSEL is exactly the same as the prescription of TDHK applied to circularly polarized pulses as opposed to linearly polarized pulses.

III. NUMBER OF SUBSIDIARY PEAKS IN THE UNCHIRPED SPECTRUM
Next, we note that RGSEL might have beneficially used the patently exact (to within the accuracy of the stationary phase approximation) expression for the number of subsidiary peaks in the unchirped spectrum, derived in TDHK and reported in their Eq.( 1), instead of their approximation.
Equation (3) of RGSEL approximates the number of oscillations in the unchirped spectrum as where Δω L is the FWHM bandwidth of the laser, ω L is the laser frequency and a 0 the laser pulse amplitude.The relationship to the Eq. ( 1) in TDHK is established if we recognize that the bandwidth of the laser pulse and the laser frequency are given as, respectively, which is to be compared to Eq. ( 1) of TDHK for the first harmonic n h ¼ 1: with ξ ≡ ξ=ð ffiffi ffi 2 p σÞ. RGSEL use a half-sine laser envelope profile: aðηÞ ¼ a 0 sin½ πη τ L for 0 < η < τ L .Substituting this expression for the laser profile into Eq.(8) yields We note is that the result of RGSEL depends on the amplitude a 0 in a different fashion than quadratic.Quadratic dependence of the number of subsidiary peaks on the amplitude a 0 was first empirically observed by Heinzl et al. [4] (N τ ≈ 0.24T½fsa 2 0 ), and later analytically explained and generalized to an arbitrary pulse shape and laser wavelength in THDK.
For the two examples reported in the left panel of In summary, we believe Rykovanov, Geddes, Schroeder, Esarey and Leemans have shown clearly and convincingly that our chirping prescription applies for circularly polarized lasers by accounting for the usual change in the field strength in going from linear to circular polarization.We have shown that our formula for the number of subsidiary peaks in the unchirped spectrum yields quantitative agreement when applied to cases presented in RGSEL.

Fig. 1 (
a 0 ¼ 0.4) and the right panel of Fig.(2) (a 0 ¼ 1) of RGSEL, their estimates are significantly different from those of TDHK.The TDHK expression correctly predicts 8 subsidiary peaks in the left panel of Fig.1and 50 subsidiary peaks in the right panel of Fig.2of RGSEL.RGSEL estimates these to be 9 and 32, respectively.