Dynamic switching from coherent perfect absorption to parametric amplification in a nonlinear spoof plasmonic waveguide

Coherent perfect absorption (CPA) and amplification of electromagnetic waves are converse phenomena, where incoming radiations are coherently dissipated or amplified by structured incidences. Realizing such two phenomena simultaneously in a single device may benefit various applications such as biological sensing, photo detection, radar stealth, solar-thermal energy sharing, and wireless communications. However, previous experimental realizations of CPA and amplification generally require precise controls to the loss and gain of a system, making dynamic switching between the absorption and amplification states a challenge. To this end, we propose a nonlinear approach to realize CPA and parametric amplification (PA) simultaneously at the same frequency and demonstrate experimentally dynamic switch from the CPA to PA states in a judiciously designed nonlinear spoof plasmonic waveguide. The measured output signal gain can be continuously tuned from −33 dB to 22 dB in a propagation length of 9.2 wavelengths. Compared to the traditional linear CPA, our approach relaxes the stringent requirements on device dimensions and material losses, opening a new route to actively modulate the electromagnetic waves with giant amplification-to-absorption contrast in a compact platform. The proposed nonlinear plasmonic platform has potential applications in on-chip systems and wireless communications.

1) The concept of nonlinear CPA in a two-port system is not intuitively given.In this work, the signal gain is used as the exclusive parameter to define CPA.However, different from the one-port system (shown in Fig. 1(a)) where R+A=1, in the two-port system, zero transmission doesn't equal to unity absorption.As in Fig. 3(a), only S21 is given, yet S11 in such system is more likely to be non-zero.Thus, the authors are suggested to calculate and measure (if possible) the reflection for all cases (from Fig. 3 to Fig. 5).
2) It would be better to give the nonlinear CPA condition step by step, if possible.From Eq. (3), it is not intuitive that how to obtain the CPA condition of φ_p-2φ_s=90°.Is it only satisfied for some specific χ_eff^2 * |A_p |, or for arbitrary χ_eff^2* |A_p |?For example, if χ_eff^2 * |A_p | increases or decreases 10 times, will the gain at the CPA frequency significantly change?
3)Traditional linear CPA can only be achieved at some fixed resonance conditions, i.e. once the frequency is fixed, the CPA phenomenon can only be observed for some discrete cavity sizes.On the contrary, Fig. 5a shows that the nonlinear CPA is quite different as it is quite robust to the waveguide geometry once the waveguide length is larger than a critical value.I believe that this point is very interesting and hence deserves more discussions.For instance, what parameter(s) will affect the critical length of the waveguide, above which the gain/absorption coefficient approaches a constant.
4)The theoretical model developed to describe the PA and CPA is based on undepletion pump approximation, which requires the amplitude of the signal wave to be sufficiently small compared with that of the pump wave.I am wondering what the maximum input signal power the nonlinear waveguide is able to amplify/absorb.

5)
The maximum gain appears when φ_p-2φ_s=-90°, while the minimum gain emerges when φ_p-2φ_s=90°.By substituting the phase matching condition of the three-wave mixing process (φ_p=φ_s+φ_i) into them, one can get that φ_i=φ_s-90° and φ_i=φ_s+90°, respectively.Why does the constructive and destructive interference occur when the phase difference between the signal wave and the generated differential wave reaches −90° and °,instead of 0° and 180°?6)In the manuscript, the authors state that the increase of pump energy yields larger maximum gain and absorption rate of the signal wave.Please give more details (a quantitative study) on this point.
2) Fig. 4(a), the arrows of the signal wave and the pump wave seem to go backwards.Fig. 4(b), the dip and peaks are not clearly seen.
3) Fig. 5(a), it seems that perfect absorption occurs when the propagation length is larger than 7 wavelengths.However, on page 4, line 80, it claims that "it is insensitive to the material's loss tangent and dimensions of the device (e.g. the length of SSPP waveguide).The statement should be reconsidered.
4)The legend of the horizontal axis of Fig. 5a is "Propagation length (wavelength number)".The "wavelength number" here can easily get people confused with "wave number".It is better to rephrase the legend here, for example, to "Normalized waveguide length" or simply "L/wavelength " to avoid confusion.5)I find that a pioneering work in the field of CPA is missing in the reference: Y. D. Chong, Li Ge, Hui Cao, and A. D. Stone,Phys. Rev. Lett. 105,053901 Reviewer #2 (Remarks to the Author): In this manuscript Cui and co-workers present a very interesting design able to achieve both coherent perfect absorption and parametric amplification at the same frequency in the same device.The design is based on a corrugated thin waveguide that supports the propagation of spoof surface plasmons in which a varactor diode is introduced to act as a nonlinear generator.The manuscript is very well written, the results are interesting and relevant, and the main conclusions regarding the feasibility of using this type of dynamic switchers is supported by both the experimental findings and the background theory.Also, the agreement between the numerical simulations and the experimental results is indeed very good.I do not have any objection on the scientific quality and depth of the results.Therefore, in principle, I am inclined to support the publication of this manuscript in Nature Communications.
My only major suggestion is related to incorporating a more detailed discussion on the operation linewidth of the device, as shown in Figure 4. First, it would be instructive to analyse the physical mechanism behind the gain to understand why is maximum near the frequency of operation (4.43 GHz).Second, it would be great to know why there is only one point leading to complete absorption.In every resonant phenomenon, there is always a linewidth that is associated with a limiting physical magnitude.What is the linewidth and the associated limiting factor in this case?I think that this discussion could add more physical insight into this proposal.In addition, as in the title the word ``dynamic'' has been incorporated, it would be interesting to discuss the timescale of the dynamical switching present in this type of devices.
Reviewer #3 (Remarks to the Author): The authors proposed and realized a dynamic switching mechanism from coherent perfect absorption to parameter amplication by using a nonlinear spoof plasmonic waveguide.The idea is very interesting.The results are validated by both simulations and microwave experiments.The device is a compact structure that could find useful applications in the future.Therefore, I highly recommend publishing this paper on Nat.Commun., and I only have some minor suggestions for the authors to improve the manuscript.
1.The mechanism of nonlinear process is elaborated by Eqs.1-3.But can the authors describe briefly in a physical way why they choose this structure for nonlinear process and is it optimized?2. It seems from Fig. 2 that the amplification or attenuation is applied to the signal wave, instead of the pump wave.I wonder if it is possible to apply to the pump wave instead.In other words, what happens if the signal wave has a much larger amplitude than the pump wave?Or in general, is it possibel to control a wave of larger amplitude via a wave of smaller amplitude, like a triode.

What happens if the signal and pump waves have similar amplitudes.
Usually what is the criteria on the ratio between them? 4. Will the structure of effective nonlinear media induce reflection?I understand that here the input port may reflect the reflected wave back into the structure again, but in this way, the length between the port and the structure might be important and worth mentioning.Since the absorption is discussed, it is better to make a clearer clarification.

Replies to the Referees' Comments
We appreciate the referees' constructive comments and suggestions very much, which help improve the quality of the manuscript significantly.Based on these comments and suggestions, we have revised the manuscript carefully, and the correction parts are highlighted in the yellow background.Below are our point-by-point replies (in blue fonts) to the referees' comments (in black fonts).

To Referee #1 Comment:
This work demonstrates a new scheme for dynamic switching from parametric amplification (PA) to coherent perfect absorption (CPA) in a nonlinear spoof plasmonic waveguide by using three-wave mixing process.To my knowledge, such scheme is novel and has not been reported.The signal gain at the transmission side can be tuned from -33dB to 22dB by readily tailoring the phase difference between the pump and the signal.The signal gain is also expected to be tuned continuously when an analog phase shifter is employed.Experimental results show good agreements with the calculated results.Both the idea and the experimental realization are interesting, and the manuscript is well organized.I feel that it merits publication in Nature Communications after the authors address the following comments:

Our Reply:
Thank you very much for your positive comments.

Comment:
1) The concept of nonlinear CPA in a two-port system is not intuitively given.In this work, the signal gain is used as the exclusive parameter to define CPA.However, different from the one-port system (shown in Fig. 1(a)) where R+A=1, in the two-port system, zero transmission doesn't equal to unity absorption.As in Fig. 3(a), only S21 is given, yet S11 in such system is more likely to be non-zero.Thus, the authors are suggested to calculate and measure (if possible) the reflection for all cases (from Fig. 3 to Fig. 5).

Our Reply:
Thank you very much for raising this point.In fact, this paper focuses on two-port cases with good impedance matching and negligible reflection.To simplify the theoretical analysis, we suppose that the two-port system is non-reflective, where T+A=1.In our experiment, we designed an SSPP waveguide which has very small reflection.As shown in Figure R1, the measured reflection coefficient of the SSPP waveguide is smaller than -10 dB (meaning that the reflection is less than 10%) around the frequency of phase matching point 4.46 GHz, indicating that our case can be approximated as non-reflective, and the maximum absorption rate (when the measured gain is -33dB) is higher than 90%.Thus, the calculation results from our simplified theoretical model agree well with our experiments.To make this point clearer, we add the simulated and measured S11 parameters in Figure 3a (see Figure R1 below), and point out in the revised manuscript that our theoretical model applies only to ideal non-reflective cases.We would also like to highlight that, in contrast to a two-port linear system which is unable to realize CPA without reflection, our nonlinear CPA approach does not rely on the interference between the incident and reflected waves and can be achieved in a nonreflective system.

Comment:
2) It would be better to give the nonlinear CPA condition step by step, if possible.
From Eq. (3), it is not intuitive that how to obtain the CPA condition of φ_p-

Comment:
3) Traditional linear CPA can only be achieved at some fixed resonance conditions, i.e. once the frequency is fixed, the CPA phenomenon can only be observed for some discrete cavity sizes.On the contrary, Figure 5a shows that the nonlinear CPA is quite different as it is quite robust to the waveguide geometry once the waveguide length is larger than a critical value.I believe that this point is very interesting and hence deserves more discussions.For instance, what parameter(s) will affect the critical length of the waveguide, above which the gain/absorption coefficient approaches a constant.

Our Reply:
Thank you very much for your good suggestion.In an ideal lossless case, the gain of the signal wave will increase monotonically with the propagation length under the phase matching condition in the PA cases.However, in the presence of the loss, there is a critical length at which the gain reaches the peak value.Note that the gain remains relatively stable (within 2dB variation) around the peak in a certain range of propagation length, as highlighted in Figures R3(a), (c) and (e).As the propagation length further increases, the loss will dominate and the total gain will gradually decrease.So the loss tangent determined by the substrate material and nonlinear elements will affect s k and p k , and thus the critical length as well as the robustness.
In addition, the   2 eff  determined by the nonlinear elements also has a direct effect on the critical length and robustness.Here, we plot the signal gain as the function of the propagation length L, and investigate how s k , p k , and   2 eff  would affect the critical length, as shown in Figure R3.In the PA case, we can observe that the maximum signal gain as well as the critical length decrease with larger s k or p k , but increase Meanwhile, the attenuation increases monotonically with the propagation length in the CPA cases, as shown in Figure R3.Here, we mark the critical length at which the attenuation reaches 30 dB (corresponding to 99.9% attenuation).Note that the critical length for attenuation decreases with the increase of s k or   2 eff  , but increases with p k .We have added these discussions on Pages 6-7 of the revised supplementary information.

Comment:
4) The theoretical model developed to describe the PA and CPA is based on undepletion pump approximation, which requires the amplitude of the signal wave to be sufficiently small compared with that of the pump wave.I am wondering what the maximum input signal power the nonlinear waveguide is able to amplify/absorb.

Our Reply:
Thank you very much for your good question.From Eq. ( 3), we note that the signal power has no influence on the signal gain under the undepletion pump approximation.
However, as the signal power increases, the undepletion approximation is no longer valid, and the measured signal gain deviates from the calculated results, as shown in Figure R4.Here, we plot the calculated and measured signal gains as functions of the signal power at 2 90 arg( ) and -89° when the pump power is 21.59dBm.We observe that the signal gain will finally fall as the signal power reaches a certain value.In order to achieve the maximum signal gain, the signal power needs to be below -25dBm.In other words, for the undepleted pumped approximation to work, the signal wave should remain 4 order of magnitude smaller than the pumped one.However, we would like to highlight that beyond the undepleted pump approximation, our approach still works with a downgraded performance.These discussions have been added on Page 7-8 of the revised supplementary information.Comment: 5) The maximum gain appears when φ_p-2φ_s=-90°, while the minimum gain emerges when φ_p-2φ_s=90°.By substituting the phase matching condition of the three-wave mixing process (φ_p=φ_s+φ_i) into them, one can get that φ_i=φ_s-90° and φ_i=φ_s+90°, respectively.Why does the constructive and destructive interference occur when the phase difference between the signal wave and the generated differential wave reaches −90° and 90°, instead of 0° and 180°?
Our Reply: Thank you for your question.For the three-wave mixing process, the nonlinear coupled equation is given by: In our degenerate cases,

arg( )
Thus, ( 2 ) 90 arg( ) is    , and the idler wave generated by the DFG process interferes with the signal wave and amplifies/completely cancels it to realize PA/CPA.We have added these discussions on Page 9 of the revised manuscript.

Comment:
6)In the manuscript, the authors state that the increase of pump energy yields larger maximum gain and absorption rate of the signal wave.Please give more details (a quantitative study) on this point.

Our Reply:
Thank you for your suggestion.Here, we calculate and measure the signal gain versus 2 ps   with different pump power, as shown in Figure R5.We can observe that the signal gain is very sensitive to the pump power.The lower the pump power is, the smaller the maximum gain and absorption will be.Theoretically, a 2dB decrease in pump power can reduce both the maximum gain and attenuation by about 6dB.These discussions can be found on Page 8 of the revised supplementary information.

Our Reply:
Thank you for pointing it out.As discussed above, CPA and PA depend on

Comment:
2) Fig. 4(a), the arrows of the signal wave and the pump wave seem to go backwards.

Comment:
3) Fig. 5(a), it seems that perfect absorption occurs when the propagation length is larger than 7 wavelengths.However, on page 4, line 80, it claims that "it is insensitive to the material's loss tangent and dimensions of the device (e.g. the length of SSPP waveguide).The statement should be re-considered.

Our Reply:
Thank you for your correction.The statement here is not rigorous and we have revised it in the revised manuscript as: "it is insensitive to the material's loss tangent the CPA/PA effect can be achieved within a certain range of device dimensions".

Comment:
4)The legend of the horizontal axis of Fig. 5a is "Propagation length (wavelength number)".The "wavelength number" here can easily get people confused with "wave number".It is better to rephrase the legend here, for example, to "Normalized waveguide length" or simply "L/wavelength" to avoid confusion.

Our Reply:
Thank you very much for your kind advice.I have unified the legend of the horizontal axis of the relevant figures in the revised manuscript to / L  , as shown in Figure R8.

Comment:
5)I find that a pioneering work in the field of CPA is missing in the reference: Y. D.

Our Reply:
Thank you for reminding us about this very relevant reference.We have added this work into the reference list in the revised manuscript.

To Referee #2
Comment: In this manuscript Cui and co-workers present a very interesting design able to achieve both coherent perfect absorption and parametric amplification at the same frequency in the same device.The design is based on a corrugated thin waveguide that supports the propagation of spoof surface plasmons in which a varactor diode is introduced to act as a nonlinear generator.The manuscript is very well written, the results are interesting and relevant, and the main conclusions regarding the feasibility of using this type of dynamic switchers is supported by both the experimental findings and the background theory.Also, the agreement between the numerical simulations and the experimental results is indeed very good.I do not have any objection on the scientific quality and depth of the results.Therefore, in principle, I am inclined to support the publication of this manuscript in Nature Communications.

Our Reply:
Thank you very much for your positive comments.

Comment:
1. My only major suggestion is related to incorporating a more detailed discussion on the operation linewidth of the device, as shown in Figure 4. First, it would be instructive to analyse the physical mechanism behind the gain to understand why is maximum near the frequency of operation (4.43 GHz).Second, it would be great to know why there is only one point leading to complete absorption.

Our Reply:
Thank you very much for your good suggestion.As discussed in in the first paragraph of the theory section of the manuscript, our device is based on a three-wave-mixing process, where the generated wave (idler wave) has the same frequency of the signal wave.Thus, tuning the phase difference between the differential wave and the signal wave can result in destructive to instructive interference between them, leading to the control of signal wave from attenuation to amplification., the signal gain (and/or the absorption ratio) will change smoothly with the frequency.
Only in this case, we can define the operation linewidth.

Comment:
2. In every resonant phenomenon, there is always a linewidth that is associated with a limiting physical magnitude.What is the linewidth and the associated limiting factor in this case?I think that this discussion could add more physical insight into this proposal.

Our Reply:
Thank you very much for your good question.As discussed above, if we fix the pump frequency at the phase matching frequency point and vary the input signal frequency, we will only observe a single peak/dip of the signal gain.However, if we fix the  small and the optimal waveguide length short.We have added these discussions on pages 4-6 of the Supplementary Information.

Comment:
In addition, as in the title the word ``dynamic'' has been incorporated, it would be interesting to discuss the timescale of the dynamical switching present in this type of devices.

Our Reply:
Thank you for your suggestion.Three-wave mixing is a transient optical effect, so the timescale of the dynamical switching depends primarily on the phase shifter.The switching speed of most of the digital phase shifter models is 100ns.We found that the switching speed of the model HWQDPH2060-6 is only 35ns, but it can only be used to tune the signal wave due to its operating frequency.

To Referee #3
Comment: The authors proposed and realized a dynamic switching mechanism from coherent perfect absorption to parameter amplification by using a nonlinear spoof plasmonic waveguide.The idea is very interesting.The results are validated by both simulations and microwave experiments.The device is a compact structure that could find useful applications in the future.Therefore, I highly recommend publishing this paper on Nat.
Commun., and I only have some minor suggestions for the authors to improve the manuscript.

Our Reply:
Thank you very much for your positive comments.

Comment:
1.The mechanism of nonlinear process is elaborated by Eqs.1-3.But can the authors describe briefly in a physical way why they choose this structure for nonlinear process and is it optimized?
Our Reply: Thank you very much for your good question.Phase matching is critical to our approach, which requires precise control of the wave vectors of signal, pump, and idler waves.Spoof SPP platform give us more freedom to control the wave vectors and hence to design the phase matching.This is why we choose this platform.However, the operating linewidth of our structure has not been well optimized.To achieve a broad bandwidth, the dispersion curve should be well designed to keep the phase mismatch (out of the phase matching frequency) small and the optimal waveguide length short.Detailed discussions on the linewidth of our system are given on pages 4-6 of Supplementary Information.

Comment:
2. It seems from Fig. 2 The signal gain can be calculated as, A also has an effect on the gain.Thus, by making the input signal power small enough, a weak pump wave is also able to control the signal wave, which is not possible in 2 ps   case.The above discussions can be found on Pages 9-12 of the revised supplementary information.

Comment:
3. What happens if the signal and pump waves have similar amplitudes.Usually what is the criteria on the ratio between them?
Our Reply: Thank you very much for your good question.If the signal and pump powers have similar amplitudes, the amplification/absorption effect will be very poor.For the PA case, because the pump wave power is transferred to the signal wave through the nonlinear process, the pump power will drop drastically if the initial pump and signal waves have similar power amplitudes.Then the signal only has weak amplification due to the small pump power (As we have discussed above, the pump power has critical influence on the gain).In the CPA case, if the pump power is not high enough (e.g.comparable to the signal power), the generated differential wave which is out of phase with the signal wave will be very weak, so only a small portion of the signal wave will be absorbed.
In order to find the power ratio between the signal wave and the pump wave that gives our SSPP device the maximum modulation depth, we measured the gains at different signal input powers when the pump power is fixed at 21.59dBm.Note that as the signal power increases, the undepletion approximation fails, and the signal gain will fall.Figure R15 indicates that the higher gain requires lower signal power, and that the power ratio of the pump wave to the signal wave should be greater than 21.59dBm/-25dBm 45604  to achieve largest modulation depth of the device.These discussions have been added on Pages 7-8 of the revised supplementary information.

Comment:
Will the structure of effective nonlinear media induce reflection?I understand that here the input port may reflect the reflected wave back into the structure again, but in this way, the length between the port and the structure might be important and worth mentioning.Since the absorption is discussed, it is better to make a clearer clarification.

Our Reply:
Thank you very much for your good question.We have designed the reference impedance at the ports to be about 50Ω to match the port impedance of the coaxial lines, so no back and forth reflections occur.The S-parameter curves for our SSPP structure are shown in Figure R16.The measured reflection coefficient of the SSPP waveguide is smaller than -10 dB around the phase mating point 4.46 GHz (meaning that the reflected energy is less than 10%), indicating that our case can be approximated as non-reflective, and the maximum absorption rate (when the measured gain is -33dB) is higher than 90%.377, 995 (2022).I suggest the authors add them and discuss appropriately.

Our Reply:
Thank you very much for your kind reminder.We have added these important papers of CPA into the reference list and the related discussions in introduction section of the revised manuscript.

Figure R1 |
Figure R1 | The simulated and measured S-parameters of the SSPP waveguide.
2φ_s=90°.Is it only satisfied for some specific χ_eff^2 * |A_p |, or for arbitrary χ_eff^2* |A_p |?For example, if χ_eff^2 * |A_p | increases or decreases 10 times, will the gain at the CPA frequency significantly change?Our Reply: Thank you very much for your question.Our nonlinear CPA method is enlightened by the difference frequency generation (DFG) of three-wave mixing.The principle is that the idler wave generated by the DFG process interferes with the signal wave and completely cancels it at a specific phase, which requires 2 ps ff  .The detailed derivation of the nonlinear CPA is given in the supporting information.The phase relation of the three waves can be deduced from the nonlinear coupled equation:

sk
in the denominator is a complex number, the factor arg( ) s k also affects the phase.The phases on both sides of the equation are also equal, i.e.
signal and idler waves is   will change during propagation.The PA (CPA) requires the phase difference between the idler wave and the signal wave always equal to 0 ( ) so the nonlinear PA (CPA) process also requires the phase-

Figure
Figure R2 | The signal gains versus the phase difference

Figure R3 |
Figure R3 | The signal gains versus the propagation length ( / L  ) with different (a-b) s k , (c-d)

Figure R4 |
Figure R4 | The signal gains versus the signal power.
the phase.The phases on both sides of the equation are also equal, i.e.

Figure
Figure R5 | The signal gains versus premise of Figure 1(d) is that the input phase of the pump wave p  is a fixed value.Following your suggestion, we have changed the y- axis in Figure 1(d) to 2 ps   in the revised manuscript, as shown in Figure R6.

Figure R6 |
Figure R6 | Principles of linear and nonlinear CPAs.(a) of a linear coherent perfect absorber consisting of a gain-loss medium layer on top of a metallic ground plane.The destructive interference between the waves reflected at the first and second interfaces results in the enhancement of absorption.(b) Gain/absorption of the linear system in terms of the thickness Land loss tangent of the gain-loss medium layer.CPA can be realized only at some specific slab thicknesses and with a fixed loss tangent of the absorbing material (c) Schematic of the nonlinear CPA.By controlling the phase difference between the input signal and pump waves, the interference between the transmitted signal and generated differential waves from a nonlinear medium can be tuned from constructive to destructive to achievePA and CPA.(d)

Fig. 4
Fig. 4(b), the dip and peaks are not clearly seen.

Figure R7 |
Figure R7 | (a) The diagram of experimental setup.(b) The measured signal gains versus the

Figure R8 |
Figure R8 | The calculated and measured signal gains of the nonlinear SSPP waveguide.(a) The is indeed a linewidth (blue shaded region), as shown in FigureR9.The measured 3dB bandwidth is about 0

Figure
Figure R9 | The measured signal gains versus the signal frequencies at

Figure R10 |
Figure R10 | The calculated operation linewidth and Q factor versus the waveguide length / L 

Figure
Figure R11 | (a) The signal gains versus

Figure R12 |
Figure R12 | The signal gains versus the propagation length under distinct phase differences

Figure R13 |
Figure R13 | Counterplot of the signal gain/absorption in terms of the propagation length / L 

Figure R14 |
Figure R14 | The calculated and measured signal gains of the nonlinear SSPP waveguide.(a) The signal gains versus the signal power.(b) The signal gains versus the phase difference between the signal and pump waves.

Figure R15 |
Figure R15| The signal gains versus the signal power.

Figure R16 |
Figure R16 | The simulated and measured S-parameter curves of the SSPP waveguide.
that the amplification or attenuation is applied to the signal wave, instead of the pump wave.I wonder if it is possible to apply to the pump wave instead.In other words, what happens if the signal wave has a much larger amplitude than the pump wave?Or in general, is it possible to control a wave of larger amplitude via a wave of smaller amplitude, like a triode.Thank you for your question.If the signal wave has a much larger amplitude than the pump wave, we can use the signal wave to control the pump wave.In this case, it is equivalent to exchanging the frequency of the signal wave and the pump wave in the Comparison of the equation above with Eq. (3) in the manuscript shows that the biggest difference from the case of 2 p A , s