Abstract

The quarter-wavelength matching technique is widely used because it minimizes the reflection while it maximizes the transmission. The recently introduced antireflection temporal coatings (ATCs) [Optica 7, 323 (2020) [CrossRef]  ] have been considered as its temporal analog. However, our study shows that by introducing an ATC, not only will the reflection be reduced but also the transmission. This phenomenon is opposite its spatial counterpart, which indicates that ATCs are more than simply a temporal dual of quarter-wavelength matching. This is a direct consequence of the different physical phenomena that are manifested in the temporal and spatial domains.

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

In [1], Pacheco-Peña and Engheta investigated antireflection temporal coatings (ATCs), which are considered as an interesting time domain analog of the well-known quarter-wavelength matching technique. As derived in [1], an incident wave upon an ATC will be split into four components after encountering two temporal boundaries. By letting the time duration of the intermediate state be represented by $\Delta t = {t_2} - {t_1} = n{T_{\text{eq}}}/4$, the two backward terms (${T_1}{R_2} -$ and ${R_1}{T_2} -$) will cancel each other out due to the 180° phase difference.

It is not mentioned in [1], however, that the phase difference between the two forward terms (${T_1}{T_2} +$ and ${R_1}{R_2} +)$ is also 180°. The amplitudes can be calculated using Eqs. (4) and (5) in [2], which suggests that $| {{T_1}{T_2} +} | \gt | {{R_1}{R_2} +} |$. Hence, the total transmission will be reduced, but it cannot be completely canceled.

Although both the quarter-wavelength matching technique and the ATCs minimize the reflection, their transmission responses are quite opposite. This phenomenon indicates that their underlying physics is different. In the spatial domain, the total transmission of a quarter-wavelength matching layer can be represented as an infinite sum of partial transmission terms [3]. On the other hand, because of causality, the direction of time flow cannot be reversed. Consequently, there are no “back-and-forth bouncing waves” between the two temporal boundaries. The total transmission, when an ATC is used, is simply a sum of two terms. Therefore, the transmission performance of the ATCs is intrinsically different from that of quarter-wavelength matching.

To compare with the result in [1], we firstly consider a similar sinusoidally modulated Gaussian pulse:

$${A_0}(t ) = - \exp \left[{- \frac{{4{\pi}{{({t - {t_0}} )}^2}}}{{{t_m}}}} \right]{\cos}[{2{\pi}{f_m}t} ],$$
where ${t_0}$ denotes the center of the Gaussian pulse, ${t_m}$ is a parameter related to the spectral bandwidth, and ${f_m}$ is the frequency of the modulated signal when $\varepsilon (t) = {\varepsilon _1}$. In this example, we have chosen ${t_m} = 5.337 \times {10^{- 15}}$ s, ${t_0} = 3 \times {t_m}$, and ${f_m} = 4.9965 \times {10^{14}} \; {\rm Hz}$. The incident waveform at $t = t_1^ -$ is shown in Fig. 1(a). To demonstrate the transmission reduction phenomenon, we consider that the wave travels in a medium with permittivity, which changes from ${\varepsilon _1} = {\varepsilon _0}$ to ${\varepsilon _3} = 100{\varepsilon _0}$. The ATC is designed with ${\varepsilon _2} = \sqrt {{\varepsilon _1}{\varepsilon _3}} = 10{\varepsilon _0}$, as shown in Fig. 1(a). The simulated reflected and transmitted waveforms at $t = {t_{\rm s}} = 14.21{T_{\text{eq}}}$ are shown in Fig. 1(b).

It can be seen from Fig. 1(b) that the magnitude of both the transmitted and reflected waves are reduced with the existence of the ATCs. But, since the sinusoidally modulated Gaussian pulse has a broad bandwidth centered around the design frequency, the backward wave cannot be completely canceled. Therefore, we believe that a monochromatic incident wave can best reveal the underling physics of the ATCs. However, a monochromatic wave has infinite time duration, so the reflected and incident waves will overlap, making it difficult to differentiate between them.

As a good compromise, we choose a four-cycle quasi-monochromatic incidence. The excitation signal is defined as

$${E_z}({{\rm x} = 0} ) = \left\{{\begin{array}{l}{\sin ({2\pi\! {f_{{\varepsilon _1}}}t} ),0 \lt t \lt 4{T_{{\varepsilon _1}}} \lt {t_1}}\\{0,t \gt 4{T_{{\varepsilon _1}}}}\end{array}} \right.,$$
where ${f_{{\varepsilon _1}}} = {c_0}/\lambda = 4.9965 \times {10^{14}}$ Hz and ${T_{{\varepsilon _1}}} = 1/{f_0}$ are the frequency and period of the incident wave, respectively, when $\varepsilon (t) = {\varepsilon _1}$. The frequency and period of the wave will be transformed over the ATC duration (${t_1} \le t \le {t_2}$) because of the change of permittivity: ${f_{\text{eq}}} = {f_{{\varepsilon _2}}} = {f_{{\varepsilon _1}}}/\sqrt {10}$ and ${T_{\text{eq}}} = \sqrt {{\varepsilon _2}/{\varepsilon _1}} {T_{{\varepsilon _1}}} = \sqrt {10} {T_{{\varepsilon _1}}}$.

Theoretically, we analyzed the normalized four-cycle semi-monochromatic incident wave defined in Eq. (2). Because of the 180° phase difference, the backward waves cancel each other except for the two non-overlapping half-cycles at both ends, as shown in Fig. 2(a). Meanwhile, the waveform of the total transmitted wave in Fig. 2(b) is also non-uniform, which is due to the same reason. This unique waveform proves that the reason for the ATC transmission reduction is not just due to a reduction in the value of the Fresnel transmission coefficients, but is also because of the partial cancellation of the two forward propagating terms.

 figure: Fig. 1.

Fig. 1. (a) ATCs time setup (permittivity profile of the medium) and (b) the simulated spatial distribution of the reflected and transmitted waves with and without ATCs at $t = {t_{\rm s}}$.

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 figure: Fig. 2.

Fig. 2. Theoretically derived (a) reflection and (b) transmission with and without the ATC. With ATC, ${R_{\text{total}}} = ({T_1}{R_2} -) + ({R_1}{T_2} -$), ${T_{\text{total}}} = ({T_1}{T_2} +) + ({R_1}{R_2} +$).

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For validation, we performed simulations using a customized finite-difference time-domain (FDTD) code. The incident normalized wave is polarized in the $z$ direction. Figure 3(a) illustrates the ATC setup, while Fig. 3(b) shows the transmitted and reflected waves with and without the ATC. The simulated result matches very well with the analytical results shown in Fig. 2. The transmission reduction phenomenon represents a fundamental difference of ATCs compared to their spatial analog, which maximizes the transmission. Hence, their target applications would necessarily also be different.

 figure: Fig. 3.

Fig. 3. (a) ATC setup (permittivity profile of the medium) and (b) the simulated spatial distribution with and without the ATC (see Visualization 1 and Visualization 2).

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The abrupt discontinuities of the incident waveform will introduce high-frequency terms, but the effects are negligible. 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 [1] with a sinusoidally modulated Gaussian pulse.

The transmission reduction of ATCs is dependent on the permittivity contrast. In [1], Pacheco-Peña and Engheta investigated a moderate temporal discontinuity corresponding to $({{\varepsilon _3}/{\varepsilon _1}}) = 10$. The simulated transmission and reflection results of the ATC with the same moderate permittivity contrast are shown in Fig. 4(a). For this case, the transmission reduction becomes negligible. This might be the reason why this phenomenon went unnoticed in the Pacheco-Peña and Engheta paper [1]. We also investigated the relative difference of transmission with and without ATCs [$({T_{w/{o{\rm ATCs}}} - {T_{w/{\rm ATCs}}})/({T_{w/o{\rm ATCs}}}})]$ as a function of permittivity contrast. The simulated results are shown in Fig. 4(b), from which we can observe that the transmission reduction effect becomes increasingly manifest for extreme permittivity contrasts. Moreover, it needs to be pointed out that the transmission reduction, even though small with moderate permittivity contrast, is nontrivial. It represents a fundamental difference between ATCs and their spatial analog.

 figure: Fig. 4.

Fig. 4. (a) Simulated transmission and reflection for moderate temporal discontinuities $({{\varepsilon _3}/{\varepsilon _1}}) = 10$ at $t = {t_s}$ and (b) the relative difference between the transmission cases with and without the ATC, as a function of permittivity contrast. Notice that ${\varepsilon _2}$ changes accordingly to satisfy the ATC requirement: ${\varepsilon _2} = \sqrt {{\varepsilon _1}{\varepsilon _3}}$.

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Although both the ATC and quarter-wavelength matching minimize reflection, their underlying physics is not the same. In the latter case, waves bounce back and forth between spatial discontinuities, while, on the other hand, such bouncing waves do not exist in the temporal case because of the irreversibility of time. The unique transmission phenomenon of the ATC is rooted in causality, which is a fundamental property of the governing temporal domain physics.

Funding

Penn State MRSEC; National Science Foundation (DMR-1420620); Defense Advanced Research Projects Agency (HR00111720032).

Disclosures

The authors declare no conflicts of interest.

REFERENCES

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

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

3. D. Pozar, Microwave Engineering, 3rd ed. (Wiley, 2005).

References

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  1. V. Pacheco-Peña and N. Engheta, “Antireflection temporal coatings,” Optica 7, 323–331 (2020).
    [Crossref]
  2. Y. Xiao, D. N. Maywar, and G. P. Agrawal, “Reflection and transmission of electromagnetic waves at a temporal boundary,” Opt. Lett. 39, 574–577 (2014).
    [Crossref]
  3. D. Pozar, Microwave Engineering, 3rd ed. (Wiley, 2005).

2020 (1)

2014 (1)

Supplementary Material (2)

NameDescription
» Visualization 1       Transmission and reflection without ATCs.
» Visualization 2       Transmission and reflection with ATCs.

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Figures (4)

Fig. 1.
Fig. 1. (a) ATCs time setup (permittivity profile of the medium) and (b) the simulated spatial distribution of the reflected and transmitted waves with and without ATCs at $t = {t_{\rm s}}$ .
Fig. 2.
Fig. 2. Theoretically derived (a) reflection and (b) transmission with and without the ATC. With ATC, ${R_{\text{total}}} = ({T_1}{R_2} -) + ({R_1}{T_2} -$ ), ${T_{\text{total}}} = ({T_1}{T_2} +) + ({R_1}{R_2} +$ ).
Fig. 3.
Fig. 3. (a) ATC setup (permittivity profile of the medium) and (b) the simulated spatial distribution with and without the ATC (see Visualization 1 and Visualization 2).
Fig. 4.
Fig. 4. (a) Simulated transmission and reflection for moderate temporal discontinuities $({{\varepsilon _3}/{\varepsilon _1}}) = 10$ at $t = {t_s}$ and (b) the relative difference between the transmission cases with and without the ATC, as a function of permittivity contrast. Notice that ${\varepsilon _2}$ changes accordingly to satisfy the ATC requirement: ${\varepsilon _2} = \sqrt {{\varepsilon _1}{\varepsilon _3}}$ .

Equations (2)

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A 0 ( t ) = exp [ 4 π ( t t 0 ) 2 t m ] cos [ 2 π f m t ] ,
E z ( x = 0 ) = { sin ( 2 π f ε 1 t ) , 0 < t < 4 T ε 1 < t 1 0 , t > 4 T ε 1 ,