Estimations of Low-Inertia Cubic Nonlinearity Featured by Electro-Optical Crystals in the THz Range
Round 1
Reviewer 1 Report
Zhukova et al. present a mini-review of the nonlinear refraction in a list of common THz materials. The paper is nice, but I miss its novelty. It seems to me the authors are just taking an equation (eq.1) from the literature, and use it to estimate n2 from other, previous works already published by others. Thus, in the present form, the paper seems to me more like a paragraph in a thesis rather than a standalone research paper. I am happy to engage the authors on this point and I retain an open mind but, in my opinion, the authors should either add at least one experimental proof or include this work into a larger, longer, full review. Conversely, if the authors already measured n2 themselves, this wasn't clear in the current version of the manuscript and it should be strongly improved before resubmission.
Author Response
We are grateful for the valuable comments which helped us to significantly improve the Letter. Below we provide the answers and the corresponding corrections in the text of the manuscript.
We have amended the manuscript text so that to highlight the article novelty. The substantive content of our article is that we have analyzed the dependences of the n2 coefficient magnitude on the media parameters for some widely used crystals. The emphasis on that is now introduced to all the parts of the article body. The analysis performed is a reference point for experimental teams to justify the choice of materials featuring high n2.
It seems to me the authors are just taking an equation (eq.1) from the literature, and use it to estimate n2 from other, previous works already published by others.
The equation (1) has been derived by professor Kozlov in collaboration with Dolgaleva and Boyd, and he is the coauthor of the paper [Dolgaleva et al. Phys. Rev. A 2015, 92, 023809] as well as current Letter. Professor Kozlov is the ideologist of the approach and this article is the sequel to the series of publications devoted to the issue. The calculations for the crystals have been performed as there are just few experimental works with the n2 values measured directly. The existing works are referenced in our article and prove that the theoretical model predicts the n2 value correctly.
I am happy to engage the authors on this point and I retain an open mind but, in my opinion, the authors should either add at least one experimental proof or include this work into a larger, longer, full review. Conversely, if the authors already measured n2 themselves, this wasn't clear in the current version of the manuscript and it should be strongly improved before resubmission.
The manuscript text was extended by the review of our recent work, devoted to both theoretical and experimental study of the n2 of water. Thus, it is demonstrated that the analysis performed for crystals is valid in case of water and so can be experimentally confirmed for liquids. The tendency for the n2 value increase for optimal parameters obtained for crystals logically correlates with the results for water.
This text was added to the Letter:
Moreover, we have tested this theoretical approach experimentally earlier for water [9]. The z-scan method was used to conduct the measurements [39]. The technique essence consists of the induced narrowing and broadening of an intense spherical light beam when a nonlinear medium moves along its propagation axis and passes through the focus. The nonlinear medium then acts as a thin lens and leads to a minimal change in the distribution of the beam field in the far field when placed in or near the focus. The resulting characteristic z-scan curve represents the peak and valley of the nonlinear medium transmission. The magnitude of the difference between the maximal and minimal values allows to calculate the nonlinear refractive index coefficient. Earlier we have shown that this technique is applicable for THz frequency range featuring a very broad spectrum with the correct ratio of the crystal thickness to the spatial size of the pulse [40]. Regarding water as the nonlinear medium we obtained theoretically value of n2= 5×10−10 cm2/W and experimentally n2= 7±5×10−10cm2/W. These values have a very good correspondence and show that for THz frequency range coefficient of nonlinear refractive index coefficient of water 6 order of magnitude higher than in NIR frequency range. In addition, these results confirm our medium parameter contribution analysis since water is characterized by required parameters correlation for n2 increase.
Reviewer 2 Report
The manuscript reports calculations of low-inertia nonlinear refractive index coefficient for several electro-optical crystals used in THz time-domain spectroscopy. The main concern is about the comparison with experimental data. First, the authors should clarify (in the abstract as well) that the reported experimental data have been taken from literature since they did not perform experiments at all. Moreover, being the experimental values of the THz refractive index reported only for two samples among five in Table II, a declared comparison between calculations and experiments may mislead the reader. Last but not least, the difference of one order of magnitude between
the two values does not improve the overall strength of the paper. The authors should consider this aspect.
A second smaller concern is for the title. The authors mention indeed a "cubic" nonlinearity, even though they do not use this word in the text anymore. They should clarify this point too.
Author Response
We are grateful for the valuable comments which helped us greatly improve our Letter. Below we provide the answers and the corresponding corrections in the text of the manuscript.The main concern is about the comparison with experimental data. First, the authors should clarify (in the abstract as well) that the reported experimental data have been taken from literature since they did not perform experiments at all.
Thank you for the comment. We have now switched the article focus to our results, addressed the known experimental results in the Discussion section, the work we performed for water included. It is clearly highlighted in the Abstract part also.
Moreover, being the experimental values of the THz refractive index reported only for two samples among five in Table II, a declared comparison between calculations and experiments may mislead the reader.
There are just few articles providing experimental estimations of the n2 value in THz spectral range. Our work will set the direction for the experimental teams to choose the materials featuring high n2 value to investigate.
This text was added to the Letter:
Comparison with other works for crystals:
The experimental confirmation is needed to prove the analytical model validity. However, there are just few experimental works conducted so far. For instance, in the paper [37] authors using the measurement of the angular dependence of the Kerr signal and the theoretical analysis of the experiment determined the nonzero tensor elements of the third-order response function for GaP and estimate its n2 parameter to be 1.2×10−13 cm2/W. For lithium niobate crystal experimental measurements based on the change of the transmitted THz pulse shape [38] gives the n2 value of 5.4×10−12 cm2/W. For the THz range results the values taken from other sources correlate with our estimations within about an order of magnitude, which is a tolerable error and a standard phenomenon for such a small n2 value. However, it should be mentioned, that these experimental results were indirectly estimated.
In addition, the review of our work is now included, providing the approbation of the theoretical model used for water.
Moreover, we have tested this theoretical approach experimentally earlier for water [9]. The z-scan method was used to conduct the measurements [39]. The technique essence consists of the induced narrowing and broadening of an intense spherical light beam when a nonlinear medium moves along its propagation axis and passes through the focus. The nonlinear medium then acts as a thin lens and leads to a minimal change in the distribution of the beam field in the far field when placed in or near the focus. The resulting characteristic z-scan curve represents the peak and valley of the nonlinear medium transmission. The magnitude of the difference between the maximal and minimal values allows to calculate the nonlinear refractive index coefficient. Earlier we have shown that this technique is applicable for THz frequency range featuring a very broad spectrum with the correct ratio of the crystal thickness to the spatial size of the pulse [40]. Regarding water as the nonlinear medium we obtained theoretically value of n2= 5×10−10 cm2/W and experimentally n2= 7±5×10−10cm2/W. These values have a very good correspondence and show that for THz frequency range coefficient of nonlinear refractive index coefficient of water 6 order of magnitude higher than in NIR frequency range. In addition, these results confirm our medium parameter contribution analysis since water is characterized by required parameters correlation for n2 increase.
Last but not least, the difference of one order of magnitude between the two values does not improve the overall strength of the paper. The authors should consider this aspect.
Thank you. We have now emphasized on our results and pointed out the article novelty in the article body. Indeed, some materials feature subtle difference of n2 value comparing to NIR range (around one order of magnitude). Our analysis illustrates and justifies it. The review of our article is another proof of the analysis performed as it presents the results of water n2 estimation, which is 6 order of magnitude higher.
Regarding the comparison between n2,THz values from theory and experiments discrepancy of one order of magnitude can be explained by two factors: the existing experiments only indirectly estimate presented value and due to the small value this parameter cannot be precisely measured.
A second smaller concern is for the title. The authors mention indeed a "cubic" nonlinearity, even though they do not use this word in the text anymore. They should clarify this point too.
Thank you for noticing. The notions “cubic nonlinearity” and “the third-order nonlinearity” are synonymous. We clarified it in the manuscript text.
Round 2
Reviewer 1 Report
I thank the Authors for the reply. I would only ask them to expand and clarify what they mean with: "these results confirm our medium parameter contribution analysis since water is characterized by required parameters correlation for n2 increase." What are these parameters? Moreover, water displays a well-documented list of anomalies, and comparing results on solid state crystals with an anomalous liquid needs to be done with caution. The paper should be accepted after such minor revisions.
Author Response
I would only ask them to expand and clarify what they mean with: "these results confirm our medium parameter contribution analysis since water is characterized by required parameters correlation for n2 increase." What are these parameters?
Thank you for the remark. Indeed, our explanation was a little bit vague. We’ve clarified it in the new version of this paragraph:
In addition, it can be seen that the n2 value for water in the THz range is several orders of magnitude higher on average than for crystals. Based on the above analysis of the contribution of medium parameters to cubic nonlinearity, it can be is easily explained. The water manifests higher fundamental vibrational frequency (100 THz), larger difference between linear refractive index in the range with non-resonant electronic contribution (1.33) and linear refractive index in the THz range (2.3), and smaller value of the numerical density of vibrations ($3.3\times10^{22}$). Furthermore, water exhibit 1000 times higher thermal expansion coefficient ($0.2\times10^{-3} ^{\circ}C^{-1}$) which can also contribute to the overall n2 value.
Moreover, water displays a well-documented list of anomalies, and comparing results on solid state crystals with an anomalous liquid needs to be done with caution. The paper should be accepted after such minor revisions.
Thank you for your question, it is an important comment. We have provided some justifications for this theoretical application in the case of water:
Obviously, the structure featured by liquids is different from the one of crystals, so it seems quite logical to doubt the idea to implement this theoretical model in case of water. But here the anharmonic oscillator vibration model is considered. Crystals being the case, the vibrations mentioned are the ones of lattice ions, whereas vibrations of a molecule as a separate structure are addressed regarding liquids. Both oscillation types have the same nature and, therefore, the approach is valid for liquids as well.
Reviewer 2 Report
I thank the authors for their effort in revising the manuscript. They addressed all the raised issues and they certainly improved the text.
I appreciate their work since the manuscript clearly and strongly provides a very well analysed and discussed topic now.
I am glad to recommend it for publication on Photonics.
Author Response
Thank you for your comment and positive decision
