Abstract

We reply to the comment [Optica 8, 824 (2021) [CrossRef]  ] on our recent article [Optica 7, 323 (2020) [CrossRef]  ], where we exploited time-dependent metamaterials to achieve antireflection temporal coatings in the time domain. Clearly our approach has equivalences, but also differences, with its spatial analogue due to some fundamental differences in the physics behind temporal and spatial boundaries. In this context, we fully stand by our original claims and remark that a comparison and analogy between both worlds can be made with the understanding that each approach provides opportunities for different applications. Here, we provide a comprehensive and detailed reply in order to address the claims made in the comment.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

We thank Mai et al. [1] for their interest in our recently published article [2]. As we will show, the observations made by the authors of the comment [1] are fully consistent with the results provided in our published article [2] and Supplement 1 therein. Moreover, as we have discussed in our original manuscript [2] and Supplement 1 of [2], and further detailed below, a direct comparison between the spatial and temporal versions of antireflection coating should not be made on a one-to-one basis as carried out in the comment [1]. This is simply because the spatial and temporal versions rely on analogue yet fundamentally different physical phenomena.

First. To explain our technique, Mai et al. [1] said “By letting…$\Delta t = {t_2} - {t_1} = n{T_{\rm{eq}}}/{{4}}$, the two backward terms… will cancel each other out due to the 180 ˚ phase difference.”

We would like to clarify that this condition, $\Delta t = {t_2} - {t_1} = n{T_{\rm{eq}}}/{{4}}$, alone does not make the two backward terms cancel out. In fact, as we explained in detail in the supplementary materials of our work [2], in order to generate an antireflection temporal coating, not only should the condition of phase difference be fulfilled (requiring that $({{v_{\rm{eq}}}/{v_1}})\omega({{t_2} - {t_1}}) = n({\pi /2})$ with $n = {{1}}$, 3, 5), but also the condition that ${T_1}{R_2} = {R_1}{T_2}$ should hold. Hence, by working with these two conditions in our original manuscript [2], we demonstrated that the time duration of the intermediate step has to be $\Delta t = ({n{T_{\rm{eq}}}})/4$, with $n = {{1}}$, 3, 5 and the permittivity of the intermediate step should be ${\varepsilon _{\rm{eq}}} = \sqrt {{\varepsilon _1}{\varepsilon _3}}$. If only one of the two conditions is present, the backward (BW) wave will not be eliminated. As an example, we provide in Figs. 1(a) and 1(b) of this reply our analytical calculations (see Supplement 1 of Ref. [2]) of the forward (FW) and BW waves considering a monochromatic wave traveling in an unbounded medium where its relative $\varepsilon$ is rapidly changed in two steps from (a) ${\varepsilon _{1}} = {{1}}$ to ${\varepsilon _{2}} = {{3}}$ at ${\rm{t}} = {{\rm{t}}_1}$ and to ${\varepsilon _{3}} = {{4}}$ at ${\rm{t}} = {{\rm{t}}_2}$, and when it is changed from (b) ${\varepsilon _{1}} = {{1}}$ to ${\varepsilon _{2}} = {{2}}$ at ${\rm{t}} = {{\rm{t}}_1}$ and to ${\varepsilon _{3}} = {{4}}$ at ${\rm{t}} = {{\rm{t}}_2}$. The duration of the intermediate step for each case is calculated as mentioned above, $\Delta t = ({n{T_{\rm{eq}}}})/4,$ by using ${T_{\rm{eq}}} = {T_2}$, i.e., the values for the intermediate step. Note that from the results shown in Figs. 1(a) and 1(b), for both cases the FW (${{\rm{T}}_1}{{\rm{T}}_2}$, ${{\rm{R}}_1}{{\rm{R}}_2}$) and BW (${{\rm{T}}_1}{{\rm{R}}_2}$, ${{\rm{T}}_1}{{\rm{R}}_2}$) waves are always 180°, but the BW is eliminated only with one of the setups, demonstrating that not only is the condition of phase difference required to produce antireflection temporal coatings but also the value of the permittivity in the intermediate step is important.

 figure: Fig. 1.

Fig. 1. Analytical results of the two waves forming the FW wave (${{\rm{T}}_1}{{\rm{T}}_2}$ and ${{\rm{R}}_1}{{\rm{R}}_2}$, first row) and the two waves forming the BW wave (${{\rm{T}}_1}{{\rm{R}}_2}$ and ${{\rm{R}}_1}{{\rm{T}}_2}$, second row), along with the total FW and BW waves (third row) considering a monochromatic wave traveling in a spatially unbounded medium with a time-dependent relative permittivity that is rapidly changed from (a) ${\varepsilon _{1}} = {{1}}$ to ${\varepsilon _{2}} = {{3}}$ at ${\rm{t}} = {{\rm{t}}_1}$ and then to ${\varepsilon _{r3}} = {{4}}$ at ${\rm{t}} = {{\rm{t}}_2}$, and from (b) ${\varepsilon _{1}} = {{1}}$ to ${\varepsilon _{2}} = {{2}}$ at ${\rm{t}} = {{\rm{t}}_1}$ and then to ${\varepsilon _{3}} = {{4}}$ at ${\rm{t}} = {{\rm{t}}_2}$. The time duration of the intermediate step is calculated as $\Delta t = {t_2} - {t_1} = ({n{T_{\rm{eq}}}})/4$ with ${T_{\rm{eq}}} = {T_2}$ for each case. All plots represent a snapshot of the E-field at a time after inducing the second temporal change of permittivity.

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Second. Mai et al. mentioned that [1] “…the phase difference between the two forward terms… is also 180 ˚… Hence the total transmission will be reduced but cannot be completely cancelled.”

While we fully agree with the authors of the comment [1] on this issue, we would like to point out that this observation is indeed fully consistent with the results we already have in our manuscript [2] and supplementary materials therein. Specifically, Eqs. S4(a) and S4(b) in Supplement 1 of [2] clearly present the details of the FW signal, so all the details (including the phase differences) are there. Moreover, the amplitudes of the electric field provided in Figs. 3(c) and 3(g) of our manuscript [2] are about ${\sim}\;{6.5}$ and ${\sim}{5.6}$, respectively (consistent with our analytical results in the supplementary materials), noting that the amplitude of the transmitted wave with the antireflection temporal coating is less than that without such temporal coating. Incidentally, please note that this is also consistent with Fig. 1 in our reply below, where the amplitude of the FW is slightly different (${\sim}{\rm{FW}} = {0.36}$ and ${\rm{FW}} = {0.35}$) for both cases (a) and (b) (see third point for more details).

Third. In line with the previous point, the comment [1] also states that “… both the quarter-wavelength matching technique and the ATCs minimize the reflection, their transmission responses are quite opposite… their underlying physics is different…”

This point has also been extensively highlighted in our original manuscript [2] indicating the similarities and differences between the spatial and temporal versions of the antireflection coatings. Also, we would like to point out that such performance is obviously not unusual in the case of temporal coating because the total electromagnetic energy is not the same before and after the rapid temporal change in the medium $\varepsilon$ [35] due to the well-known fact that an abrupt change of the $\varepsilon$ of the medium involves an external action in the medium. Hence, if an extra step of $\varepsilon$ is added into the system to produce an antireflection temporal coating, the amplitude of the FW wave after the whole two-step process will indeed be different, in agreement with our results already included in [2].

Fourth. Mai et al. [1] then evaluated the propagation of a Gaussian temporal pulse, mentioning that “To compare with the result in [2], we firstly consider a similar sinusoidally modulated Gaussian pulse…”

We want to note that in our original manuscript [2], we considered a quasi-monochromatic Gaussian beam with 21 modulated cycles. However, in the comment [1], a few-cycle (i.e., three cycles) Gaussian pulse is used, which is rather broadband. It is important to note that our temporal coating is not intended to work for broadband signals. We provided an extensive study of this matter in Supplement 1 of [2], demonstrating how our temporal version is narrowband, in equivalence to the spatial version of the quarter-wave impedance transformer [6]. Based on this, the results and discussion provided in Fig. 1 of the comment [1] are expected and do not provide new physical insights on the performance of the antireflection temporal coating.

Fifth. In the comment [1] it is mentioned: “As a good compromise, we choose a 4-cycle quasi-monochromatic incidence.” And in Figs. 2 and 3 of the comment [1], Mai et al. evaluate the antireflection temporal coating considering that the relative permittivity of the medium is changed from ${\varepsilon _{1}} = {{1}}$ to ${\varepsilon _{2}} = {{10}}$ and then to ${\varepsilon _{3}} = {{100}}$ and then they mention “…Because of the 180 ˚ phase difference, the backward waves cancel each other except for the two non-overlapping half-cycles at both ends… the waveform of the total transmitted wave is also non-uniform…”.

The wave-packet considered in the comment [1] has only four cycles of a sinusoidal wave, resulting in a signal that is again not narrowband in the frequency domain, and consequently they observe a “non-uniform” field distribution in their results after introducing the antireflection temporal coating. We reiterate (as we made it clear in [2]) the antireflection temporal coating is an intrinsically narrowband phenomenon in an analogous way to its spatial equivalent (the quarter-wave impedance transformer). The study performed and included in the comment [1], however, corresponds to a rectangular-pulse modulated by a sinusoidal wave with a duration of only 4 cycles. This is not a “semi-monochromatic wave”, and not narrowband. Hence, it is expected that the antireflection temporal coating technique will not provide a uniform FW wave. This bandwidth point was detailed in depth in the Section H and Figs. S9-S10 of Supplement 1 in [2]. We should also mention that this is why in our simulations in [2] and Supplement 1 therein, we used the narrowband modulated Gaussian as the incident signal, and therefore in our results the two “non-overlapping” half cycles in the “front” and “back” ends of the Gaussian signal are negligibly small and thus not noticeable. It is also worth noting that in the comment [1] a temporal change of permittivity with huge contrast between the two values (i.e., from ${\varepsilon _{1}} = {{1}}$ to ${\varepsilon _3} = {{100}}$) is used, whereas in our paper we used much less contrast in our temporal $\varepsilon$, which is more realistic, i.e., from ${\varepsilon _{1}} = {{1}}$ to ${\varepsilon _3} = {{4}}$ (see Fig. 2 in [2]).

Sixth. Mai et al. [1] mentioned “The transmission reduction phenomenon represents a fundamental difference… of ATCs compared to their spatial analog, which maximizes the transmission”. Then it is mentioned that “Hence, their target applications would necessarily also be different.”

We fully agree with the authors. In [2], we made clear that the applications are different compared to the spatial version of antireflection coating. It is also important to note, as we did in [2], that our antireflection temporal coating can also provide frequency conversion.

Seventh. Linked to the previous point, the authors of the comment [1] then mention “One could observe those high-frequency effects from the small ripples in the reflected waveform shown in Fig. 3(b). A similar phenomenon can also be seen from Fig. 2(i) in Pacheco-Peña and Engheta’s ATC paper with a sinusoidally modulated Gaussian pulse.

The small ripples observed in Fig. 3(b) of the comment [1] have a different nature than what we reported in Fig. 2(i) in our original manuscript. In the case of Mai et al., the ripples are the direct effect of the broadband sinusoidally modulated square pulse with sharp rise time that is being used as the original pulse. Hence, Mai et al. observe “ripples” at the end/beginning of the BW wave. The high frequency content from our results in Fig. 2(i) from [2] comes from the direct consequence of using a narrowband sinusoidally modulated Gaussian pulse. In fact, if one compares the BW wave for the antireflection temporal coating (Fig. 2(i) from [2]) and the reflected temporal signal of the spatial version (Fig. 2(d) from [2]), it can be observed how they are similar. This is due to the fact that these frequencies fall outside the limits of the antireflection temporal/spatial coatings, respectively, as both approaches are intrinsically narrowband.

Eighth. In Fig. 4(b) of the comment [1], the authors calculate the amplitude of the FW using the antireflection temporal coating relative to the case without using it (i.e., only a single temporal boundary is applied). We agree that the change of amplitude of the FW wave is an important aspect of our antireflection temporal coating. However, as we have mentioned earlier in this reply, a change of amplitude of the FW wave is an expected response either with or without using the antireflection temporal coating, as each temporal change of $\varepsilon$ will involve an external action on the medium. However, it is worth mentioning that for that difference to be noticeable, one needs to have very high value of changes of $\varepsilon$ in time, as with those used in Fig. 4(b) of [1]; for instance, the relative change of amplitude of about 0.65 in Fig. 4(b) of [1] requires a final $\varepsilon$ of ${\varepsilon _3} = {{1000}}{\varepsilon _1}$. Moreover, we highlight the fact that for such high contrast in temporal variation of permittivity, the absolute values of the amplitude of the FW(BW) are extremely small compared to the amplitude of the “incident” wave before inducing the temporal boundaries. For instance, by using the analytical approach shown in our original manuscript [2] and considering a huge contrast of 1000 as used in the comment [1], the amplitudes of the FW(BW) waves are 0.0056(0) and 0.0163(0.0153) for the cases with and without the temporal coating, respectively. As observed, these values are almost negligible compared to the incident wave amplitude before inducing the temporal boundaries, which have an amplitude of 1. This demonstrates how extreme contrast in changes of $\varepsilon$ as those highlighted in the comment [1] are impractical. For completeness, in Fig. 2 in our reply here, we plot the analytical results of the amplitudes of the FW and BW waves as a function of the ratio between the initial and final values of $\varepsilon$ for cases with and without the antireflection temporal coating.

 figure: Fig. 2.

Fig. 2. Analytical results of the amplitude of the FW (first column) and BW (second column) waves as a function of the ratio between the initial and final values of $\varepsilon$ for a time-dependent medium with and without using the antireflection temporal coating.

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Also, it is important to note that the comment [1] has only focused on the case when ${\varepsilon}$ is changed from one value to a larger value. However, if the $\varepsilon$ is changed from a larger to a smaller value, e.g.,  ${\varepsilon _1}$ to ${\varepsilon _2}$ and then to ${\varepsilon _3}$ with ${\varepsilon _{1}} \gt {\varepsilon _2} \gt {\varepsilon _3}$, the amplitude of the FW wave will be increased to a value larger than the incident amplitude, which is well known from the literature based on the equations for the BW and FW waves [35,7]. In this context, antireflection temporal coatings may provide frequency conversion and an elimination of the BW wave, while the amplitude of the FW wave will be higher than the incident amplitude. Please note that, as we mentioned before, the total electromagnetic energy will be different before and after the temporal change of the medium due to the external action involved in the change of medium. This scenario is also demonstrated in our original manuscript in Fig. 3 [2]. Moreover, it is worth highlighting that in this scenario, the amplitude of the FW wave for cases with and without using our antireflection temporal coating will also be different, the difference being that now the amplitude of both FW waves will be increased compared to the incident signal (see Fig. 2 and Fig. 3 of this reply). Similarly, the BW wave will also have increased amplitude when not using our antireflection temporal coating technique. This was discussed in Fig. 3 in [2]. In this context, our technique will provide an extra protection to the source side, as it will eliminate BW waves with large amplitudes. We provide an example in Fig. 3 in this reply, where we calculated both analytically (using [2]) and numerically (COMSOL Multiphysics) the response of a system where the $\varepsilon$ is changed from ${\varepsilon _1} = {{4}}$ to ${\varepsilon _3} = {{1}}$ for cases with and without using the antireflection temporal coating. Again, note that we are using more realistic values rather than extreme differences between the different values of $\varepsilon$ used in the comment.

 figure: Fig. 3.

Fig. 3. Analytical (first row) and numerical (second row) results of the time snapshot of the electric field distribution for the incident (purple), FW (gray), and BW (green) waves considering an incident monochromatic wave traveling in an unbounded medium (analytical) or a sine-modulated narrow band Gaussian beam (simulations). The medium is considered time-dependent with $\varepsilon$ that is changed from ${\varepsilon _{1}} = {{4}}$ to ${\varepsilon _{3}} = {{1}}$ at ${\rm{t}} = {{\rm{t}}_1}$ (first column), and from ${\varepsilon _{1}} = {{4}}$ to ${\varepsilon _{2}} = {{2}}$ at ${\rm{t}} = {{\rm{t}}_1}$ and then to ${\varepsilon _{3}} = {{1}}$ at ${\rm{t}} = {{\rm{t}}_2}$ (second column), i.e.,  without and with the antireflection temporal coating.

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In conclusion, we have addressed the claims made in the comment [1] on our original paper [2]. While our antireflection temporal coating is the temporal analogue of the conventional antireflection coating, expectedly there are similarities and differences between the two phenomena, as explored in our original work [2] and Supplement 1 therein.

Funding

Office of Naval Research (N00014-16-1-2029); Newcastle University (Newcastle University Research Fellowship).

Disclosures

The authors declare that there are no conflicts of interest related to this paper.

REFERENCES

1. W. Mai, J. Xu, and D. H. Werner, “Antireflection temporal coatings: comment,” Optica 8, 824–825 (2021). [CrossRef]  

2. V. Pacheco-Peña and N. Engheta, “Antireflection temporal coatings,” Optica 7, 323–331 (2020). [CrossRef]  

3. F. Morgenthaler, “Velocity modulation of electromagnetic waves,” IEEE Trans. Microw. Theory Tech. 6, 167–172 (1958). [CrossRef]  

4. R. Fante, “Transmission of electromagnetic waves into time-varying media,” IEEE Trans. Antennas Propag. 19, 417–424 (1971). [CrossRef]  

5. V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016). [CrossRef]  

6. D. M. Pozar, Microwave Engineering, 4th ed. (Wiley, 1998).

7. Y. Xiao, D. N. Maywar, and G. P. Agrawal, “Reflection and transmission of electromagnetic waves at temporal boundary,” Opt. Lett. 39, 574–577 (2014). [CrossRef]  

References

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  1. W. Mai, J. Xu, and D. H. Werner, “Antireflection temporal coatings: comment,” Optica 8, 824–825 (2021).
    [Crossref]
  2. V. Pacheco-Peña and N. Engheta, “Antireflection temporal coatings,” Optica 7, 323–331 (2020).
    [Crossref]
  3. F. Morgenthaler, “Velocity modulation of electromagnetic waves,” IEEE Trans. Microw. Theory Tech. 6, 167–172 (1958).
    [Crossref]
  4. R. Fante, “Transmission of electromagnetic waves into time-varying media,” IEEE Trans. Antennas Propag. 19, 417–424 (1971).
    [Crossref]
  5. V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016).
    [Crossref]
  6. D. M. Pozar, Microwave Engineering, 4th ed. (Wiley, 1998).
  7. Y. Xiao, D. N. Maywar, and G. P. Agrawal, “Reflection and transmission of electromagnetic waves at temporal boundary,” Opt. Lett. 39, 574–577 (2014).
    [Crossref]

2021 (1)

2020 (1)

2016 (1)

V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016).
[Crossref]

2014 (1)

1971 (1)

R. Fante, “Transmission of electromagnetic waves into time-varying media,” IEEE Trans. Antennas Propag. 19, 417–424 (1971).
[Crossref]

1958 (1)

F. Morgenthaler, “Velocity modulation of electromagnetic waves,” IEEE Trans. Microw. Theory Tech. 6, 167–172 (1958).
[Crossref]

Agrawal, G. P.

Bacot, V.

V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016).
[Crossref]

Eddi, A.

V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016).
[Crossref]

Engheta, N.

Fante, R.

R. Fante, “Transmission of electromagnetic waves into time-varying media,” IEEE Trans. Antennas Propag. 19, 417–424 (1971).
[Crossref]

Fink, M.

V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016).
[Crossref]

Fort, E.

V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016).
[Crossref]

Labousse, M.

V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016).
[Crossref]

Mai, W.

Maywar, D. N.

Morgenthaler, F.

F. Morgenthaler, “Velocity modulation of electromagnetic waves,” IEEE Trans. Microw. Theory Tech. 6, 167–172 (1958).
[Crossref]

Pacheco-Peña, V.

Pozar, D. M.

D. M. Pozar, Microwave Engineering, 4th ed. (Wiley, 1998).

Werner, D. H.

Xiao, Y.

Xu, J.

IEEE Trans. Antennas Propag. (1)

R. Fante, “Transmission of electromagnetic waves into time-varying media,” IEEE Trans. Antennas Propag. 19, 417–424 (1971).
[Crossref]

IEEE Trans. Microw. Theory Tech. (1)

F. Morgenthaler, “Velocity modulation of electromagnetic waves,” IEEE Trans. Microw. Theory Tech. 6, 167–172 (1958).
[Crossref]

Nat. Phys. (1)

V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016).
[Crossref]

Opt. Lett. (1)

Optica (2)

Other (1)

D. M. Pozar, Microwave Engineering, 4th ed. (Wiley, 1998).

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Figures (3)

Fig. 1.
Fig. 1. Analytical results of the two waves forming the FW wave ( ${{\rm{T}}_1}{{\rm{T}}_2}$ and ${{\rm{R}}_1}{{\rm{R}}_2}$ , first row) and the two waves forming the BW wave ( ${{\rm{T}}_1}{{\rm{R}}_2}$ and ${{\rm{R}}_1}{{\rm{T}}_2}$ , second row), along with the total FW and BW waves (third row) considering a monochromatic wave traveling in a spatially unbounded medium with a time-dependent relative permittivity that is rapidly changed from (a)  ${\varepsilon _{1}} = {{1}}$ to ${\varepsilon _{2}} = {{3}}$ at ${\rm{t}} = {{\rm{t}}_1}$ and then to ${\varepsilon _{r3}} = {{4}}$ at ${\rm{t}} = {{\rm{t}}_2}$ , and from (b)  ${\varepsilon _{1}} = {{1}}$ to ${\varepsilon _{2}} = {{2}}$ at ${\rm{t}} = {{\rm{t}}_1}$ and then to ${\varepsilon _{3}} = {{4}}$ at ${\rm{t}} = {{\rm{t}}_2}$ . The time duration of the intermediate step is calculated as $\Delta t = {t_2} - {t_1} = ({n{T_{\rm{eq}}}})/4$ with ${T_{\rm{eq}}} = {T_2}$ for each case. All plots represent a snapshot of the E-field at a time after inducing the second temporal change of permittivity.
Fig. 2.
Fig. 2. Analytical results of the amplitude of the FW (first column) and BW (second column) waves as a function of the ratio between the initial and final values of $\varepsilon$ for a time-dependent medium with and without using the antireflection temporal coating.
Fig. 3.
Fig. 3. Analytical (first row) and numerical (second row) results of the time snapshot of the electric field distribution for the incident (purple), FW (gray), and BW (green) waves considering an incident monochromatic wave traveling in an unbounded medium (analytical) or a sine-modulated narrow band Gaussian beam (simulations). The medium is considered time-dependent with $\varepsilon$ that is changed from ${\varepsilon _{1}} = {{4}}$ to ${\varepsilon _{3}} = {{1}}$ at ${\rm{t}} = {{\rm{t}}_1}$ (first column), and from ${\varepsilon _{1}} = {{4}}$ to ${\varepsilon _{2}} = {{2}}$ at ${\rm{t}} = {{\rm{t}}_1}$ and then to ${\varepsilon _{3}} = {{1}}$ at ${\rm{t}} = {{\rm{t}}_2}$ (second column), i.e.,  without and with the antireflection temporal coating.