Signatures of the Carrier Envelope Phase in Nonlinear Thomson Scattering
, Vittoria Petrillo 2,3,†, Thomas C. Teter 4, Maksim Valialshchikov 5 and Sergey Rykovanov 5,†Round 1
Reviewer 1 Report
The manuscript presents a theoretical study of electron-photon scattering focused on the effect of the carrier envelope phase of the inonizing laser pulse. The problem is solved first for a single particle interacting with a plane wave, and then extended to a more realistic setting involving many electrons
as well as non-uniform laser beams.
In my view, the procedure of extracting the value of the carrier envelope phase from the spectrum is not explained clearly enough to be comprehensible for the readers and should be improved.
To be more specific, I would like to suggest that the quantity "f_{nu_k}" referred to at the beginning of the paragraph that leads to equation 7
is defined explicitly given that it is later plotted in figure 2 and also used in equation 8. Furthermore, I think that equation 8 and its consequences for extracting value of the carrier envelope phase should be given significantly more attention then just in caption of figure 3. In the present form, the claim that "the interval ... is an integer multiple between 0 and pi/2" is simply not meaningful to me. What interval? Integer multiple of what?
If the authors expand and clarify the presentation, I believe that the manuscript is suitable for publication.
Author Response
In my view, the procedure of extracting the value of the carrier envelope phase from the spectrum is not explained clearly enough to be comprehensible for the readers and should be improved.
1) To be more specific, I would like to suggest that the quantity "f_{nu_k}" referred to at the beginning of the paragraph that leads to equation 7 is defined explicitly given that it is later plotted in figure 2 and also used in equation 8.
In principle f_{nu_H} is explicitly defined in Eq. 5. However, we rewrote the paragraph to be more clear on the quantitative analysis.
2)
Furthermore, I think that equation 8 and its consequences for extracting value of the carrier envelope phase should be given significantly more attention then just in caption of figure 3.
We added a small paragraph under Eq. 8
3)
In the present form, the claim that "the interval ... is an integer multiple between 0 and pi/2" is simply not meaningful to me. What interval? Integer multiple of what?
The full sentence is
“The interval for which phi_cep can be determined through harmonic interference in the spectrum is an integer multiple between 0 and pi/2”
We meant to say that there is no unique solution of phi_cep to the shift of the harmonic peaks. We changed the sentence in the manuscript.
Best regards,
Marcel Ruijter
Author Response File: 
Author Response.docx
Reviewer 2 Report
Referee report for
Manuscript No: crystals-1202790
Authors: Marcel Ruijter, Vittoria Petrillo, Thomas Christopher Teter, Maksim Valialshchikov, Sergey Rykovanov
Title: Signatures of the Carrier Envelope Phase in nonlinear Thomson scattering
The manuscript under consideration studies the possibilities of using nonlinear Thomson scattering to probe the Carrier Envelope Phase of short laser pulses. To this end, the probe electrons are assumed to collide head-on with the laser pulse. Three different cases are considered: the collision of a single electron with a plane wave laser pulse, which can be solved analytically, is discussed in detail. Moreover, the interaction of an electron bunch with a plane wave laser pulse and a "beamed laser pulse" are briefly studied. The manuscript is rather well-written and in a good shape. The presentation can be easily followed.
However, I cannot give a positive recommendation for publication in Crystals before the authors have addressed the comments and questions given below.
1.) Above Eq. (5) the authors note that they identify theta=pi. However, the angle theta was not introduced before. I suggest to provide the explicit expression for the unit vector n right after Eq. (1) where it is first mentioned.
(A suggestion of a subleading optimization:) After Eq. (2) you note that F is the electromagnetic field tensor as function of position. In Eq. (2) you explicitly include the argument X^alpha for F but skip it for a. I would suggest to omit the argument here and instead note right below Eq. (2) that F is the field strength tensor, a the normalized vector potential, both
of which depend on the position X^alpha via A.
2.) Below Eq. (3) I suggest to mention that the normalization is such that E(0)=psi(vec{0})=1. The fact that E(0)=1 is for instance used in Eq. (7).
Concerning wording: below Eq. (3) and at various instances in the manuscript the authors refer to psi as transverse envelope function. I would prefer the word "spatial" instead of "transverse" here, because modifications of the transverse profile usually (when preserving Maxwell equations) also come with a localization in longitudinal direction; see the factor
of 1/q(z) in Eq. (10).
3.) I would consider it illustrative and helpful for the reader to explicitly show the field profiles for phi_cep = {0, pi/4, pi/2} in (the context of) Fig. 1. What is the choice for N_c here? In Eq. (5) you set sigma_l=1?
4.) I suppose the statement |f_v1|^2+|2 f_v1 f_v3|^2+|f_v3|^2+... above Eq. (8) is wrong and "|2 f_v1 f_v3|^2" should be replaced be "2 Re(f_v1 f_v3)".
5.) The last sentence in the caption of Fig. 3 (and a similar statement in the conclusions) is not clear to me. What is "an integer multiple between 0 and pi/2" supposed to mean? I guess that phi_cep is determined modulo pi and that due to symmetry even in the interval 0 to pi the solution is not unique as phi_cep=pi-phi_cep.
6.) What is the reason of the shift of the numerical peaks from the solid (dashed) vertical lines? Is this an effect due to the emittance? While you give the values for epsilon and sigma_gamma/gamma, you do not detail how they enter your simulations. Some general comments about the nature of the performed numerical simulations would be very helpful for the reader.
7.) It would be helpful if the notation Delta{a}_0 used in Fig. 6 would be briefly explained in the main text. To me the top left figure does not look particularly clear. The horizontal lines indicate peak values?
8.) In the Discussion section you note that Eq. (3) is no longer valid for shorter pulse durations. Do you mean here that Eq. (3) does no longer constitute a reasonably accurate approximation of a solution to the Maxwell equations?
9.) In the Discussion section you emphasize that the classical description becomes insufficient for an electron energy of gamma=80. The corresponding reference [21] has "Compton scattering" in its title. Moreover in the last sentence you mention "Compton experiments". As the focus of your work (especially your Sec. 2) is on Thomson scattering, I would consider it helpful to (once again) clarify the distinction between these regimes here.
Author Response
We would like to thank the referee for the extensive and constructive report.
1.) Above Eq. (5) the authors note that they identify theta=pi. However, the angle theta was not introduced before. I suggest to provide the explicit expression for the unit vector n right after Eq. (1) where it is first mentioned.
We added the definition of the unit vector and wrote after introducing the laser pulse the following:
“Given that with our definition the laser pulse travels in+ˆz, the angle for backscattered radiation is θ=π.”
(A suggestion of a subleading optimization:) After Eq. (2) you note that F is the electromagnetic field tensor as function of position. In Eq. (2) you explicitly include the argument X^alpha for F but skip it for a. I would suggest to omit the argument here and instead note right below Eq. (2) that F is the field strength tensor, a the normalized vector potential, both of which depend on the position X^alpha via A.
We adopted the consideration of the referee.
2.) Below Eq. (3) I suggest to mention that the normalization is such that E(0)=psi(vec{0})=1. The fact that E(0)=1 is for instance used in Eq. (7).
We adopted the consideration of the referee.
Concerning wording: below Eq. (3) and at various instances in the manuscript the authors refer to psi as transverse envelope function. I would prefer the word "spatial" instead of "transverse" here, because modifications of the transverse profile usually (when preserving Maxwell equations) also come with a localization in longitudinal direction; see the factor of 1/q(z) in Eq. (10).
We adopted the consideration of the referee.
3.) I would consider it illustrative and helpful for the reader to explicitly show the field profiles for phi_cep = {0, pi/4, pi/2} in (the context of) Fig. 1. What is the choice for N_c here?
We added the sentence to the caption “The laser pulse parameters are $N_c=5$ and $a_0=2$.”.
In Eq. (5) you set sigma_l=1?
No this is incorrect. To avoid confusion we rewrote \mathcal{E}(\frac{\zeta}{\sigma_l} → \mathcal{E}(\zeta) and only in its definition we use the length of the laser pulse, i.e. mathcal{E}(\zeta) = sech( \frac{\zeta}{\sigma_l}).
4.) I suppose the statement |f_v1|^2+|2 f_v1 f_v3|^2+|f_v3|^2+... above Eq. (8) is wrong and "|2 f_v1 f_v3|^2" should be replaced be "2 Re(f_v1 f_v3)".
We thank the referee for noticing the mistake.
5.) The last sentence in the caption of Fig. 3 (and a similar statement in the conclusions) is not clear to me. What is "an integer multiple between 0 and pi/2" supposed to mean? I guess that phi_cep is determined modulo pi and that due to symmetry even in the interval 0 to pi the solution is not unique as phi_cep=pi-phi_cep.
Your guess is correct. We rewrote the sentence in question to be more clear, as well as under Eq. 8.
6.) While you give the values for epsilon and sigma_gamma/gamma, you do not detail how they enter your simulations. Some general comments about the nature of the performed numerical simulations would be very helpful for the reader.
We added a sentence at the end of the general introduction of Thomson scattering: “To obtainnumerical results we use a particle tracker based on the Vay method [20] to obtain the trajectory of the electron which issubsequently used to calculate the spectrum according to Eq. 1.”
7.) It would be helpful if the notation Delta{a}_0 used in Fig. 6 would be briefly explained in the main text. To me the top left figure does not look particularly clear. The horizontal lines indicate peak values?
The idea was to to show in that panel what the difference is in intensity as an electron experiences at a transverse position as compared to an electron traveling on axis. Since in Eq. 8 we give the broadening contribution of this effect in the spectrum, we opted to change the figure.
8.) In the Discussion section you note that Eq. (3) is no longer valid for shorter pulse durations. Do you mean here that Eq. (3) does no longer constitute a reasonably accurate approximation of a solution to the Maxwell equations?
This is indeed what we meant and we implemented the suggestion.
9.) In the Discussion section you emphasize that the classical description becomes insufficient for an electron energy of gamma=80. The corresponding reference [21] has "Compton scattering" in its title. Moreover in the last sentence you mention "Compton experiments". As the focus of your work (especially your Sec. 2) is on Thomson scattering, I would consider it helpful to (once again) clarify the distinction between these regimes here.
First we would like to point out that Thomson and Compton scattering are used interchangably in literature. For example the experiments we refer to [23] “ … inverse-Compton-scattering sources (also referred to as Thomson sources) … ” and [24] “Therefore, the quantum recoil is negligible and can be treated as Thomson scattering. ” while retainging to Compton scattering as the description of the experiment.
Furthermore reference [21] has a dedicated section on Thomson scattering and compares the numerical solution of the classical and quantum spectrum in their figure 6.
However, we took your point into consideration and clarified it in the Discussion.
Best regards,
Marcel Ruijter
Author Response File: Author Response.docx
Round 2
Reviewer 2 Report
The authors have carefully revised their manuscript taking into account all the comments of the referee. I can now recommend the manuscript for publication in Crystals in its present form.
