1. Introduction

Photonic crystals (PhCs) are periodic dielectric structures with a photonic band gap (PBG) originating from Bragg scattering, which can help control light wave propagation at certain frequency ranges [1–10]. As new photonic materials, PhCs open a window to design many novel devices which is beyond the traditional homogeneous materials. On the other hand, ultrafast all-optical nonlinear processes have received substantial attention as they help satisfy the requirements of picosecond (ps) or femtosecond (fs) tuning performance [11,12]. These ultrafast all-optical nonlinear processes illuminate the possibility to realize new photonic logical devices whose responding time could be several orders shorter than the electronic ones. In recent years, by combining the PhC with nonlinearity, several remarkable phenomena have been observed, and novel devices have been designed to realize ultrafast processes, such as nanocavities and optical switches [13–21]. With the help of PBG, some new phenomena have been observed in nonlinear PhCs. One of them is the considerable transmission increase at the gap-edge frequency range for the signal pulse if the PhCs are pumped by a strong pumping pulse. Traditionally, a band-gap shift mechanism (BSM) is employed to explain the considerable transmission increase at the gap-edge frequencies in ultrafast processes. According to the BSM, a pumping pulse leads to a variation in the dielectric constant of the PhC, such that the signal pulse experiences a shifted PBG. Therefore, the transmission of the signal pulse at the gap-edge frequencies considerably increases. Evidently, for such a static picture, the time length of the PhC nonlinear dielectric constant change should be longer than the dwelling time of the signal pulse inside the PhC. The PBG shifting can then be sustained. In other words, in the dwelling time the signal pulse in the PhC only can sense the shifted PBG. For the PhC switches with ps pulses [13–15], this condition is well satisfied because the time length of a pumping pulse (1 ps-level) is considerably longer than the dwelling time (∼60 fs-level) of a signal pulse in the PhC. Recent experiments conducted on such ultrafast switching processes utilized fs pulses [18–21] instead of ps pulses. The traditional BSM cannot explain the variation in transmission for cases with fs pulses, because the pumping process can be as short as 6 fs approximately [18–20], which is nearly one order shorter than the dwelling time (∼60 fs) of the signal pulse inside the PhCs. A new dynamical nonlinear mechanism [22] was proposed, in which the evident increase in the transmission at the gap edge is because the fs pumping process adds an extra phase to the field that is still inside the PhC. In the new mechanism, the low transmission at the gap-edge frequency for a linear finite PhC is due to the destructive interference between the transmitted fields, comprising the first-order pulse that is the remaining part of the incident pulse, and the higher-order pulses, generated by the Bragg scattering. Such an added extra phase can partially eliminate the destructive interference between the first-order pulse and the higher-order pulses, causing a considerable increase in transmission at the gap-edge frequency. The new dynamical nonlinear mechanism helps understand the complex trend of the transmission increase curve versus the different arriving times of the pumping pulse. This trend has been observed in both laboratory experiments and numerical experiments. Further, the extra-phase effect on the transmitted field explains the presence of a main peak and other peaks, which is beyond the explanation of the BSM. Evidently, the two mechanisms are different and are suitable for different regimes. However, the quantitative and detailed study of the new dynamical nonlinear mechanism has not yet been performed. Several essential questions should be addressed, such as “Which mechanism is more energy-efficient if a certain increase of transmission is required?” and “Can we use the new mechanism to indirectly detect the fs electronic processes?”. With knowledge of the new mechanism, we not only can obtain the deeper insights of the dynamical nonlinear processes in the PhC, but also can design new all-optical high-efficiency switches, photonic logical devices and ultrafast electronic process detectors.

In this study, we will quantitatively investigate the nonlinear efficiency and the possibility of detecting fs-level electronic processes based on the new dynamical nonlinear mechanism. First, we compare the energy efficiencies required to realize a certain increase in transmission at a certain gap-edge frequency for the conditions that satisfy the two mechanisms. The numerical results demonstrate that the new dynamical nonlinear mechanism is considerably more energy-efficient than the traditional BSM; thus, the new dynamical nonlinear mechanism can effectively enhance nonlinear efficiency. We also find that the patterns of transmission and reflection spectra of the dynamical nonlinear mechanism are considerably different from those of BSM, which imply the deep mechanism differences. Second, we demonstrate that the characteristics of the transmission curve for such fs-level switches depend on the ultrafast parameters of electronic systems, e.g., the electronic lifetime is on the higher level if the nonlinearity is from a two-level electronic system. Such correlation can be employed to indirectly detect ultrafast electronic processes in nonlinear materials.

The remainder of this paper is organized as follows: In Section 2, we propose two numerical experiments that help measure the energy efficiency of the traditional BSM and the new dynamical nonlinear mechanism required to realize a certain increase in transmission; in Section 3, we quantitatively study the characteristics of the increase in transmission for different higher-level lifetimes and demonstrate how our new mechanism helps detect the fs-level process of electronic systems; Section 4 concludes the paper.

2. Method to enhance nonlinear efficiency

The system in this study is a two-dimensional (2D) PhC [Fig. 1(a)] and is similar to the system studied in [22]. There are air holes on the polystyrene background material of the square lattice PhC, where the lattice constant is a = 273 nm and the hole radius is r = 0.2a. The dielectric constant of the background material is $\varepsilon = {\varepsilon _0} + {\chi ^{(3 )}}{|E |^2} + i{\varepsilon _i}$, where the linear dielectric constant is ɛ₀ = 2.5, the third-order nonlinear coefficient is χ⁽³⁾ = 10⁻¹² cm²/W, and E is the electric field of the external pumping pulse. Because the imaginary part of dielectric constant ɛ_i is extremely small (ɛ_i ≤ 0.0002) in numerical models, which is also the real condition in laboratory experiments, it is neglected in our simulation [18–20]. The size of the PhC is 40×30a². The absorbing boundary condition is set along both x- and y-directions. Numerical experiments are simulated using the commercial EastWave finite-difference time-domain (FDTD) software [23]. In our numerical simulation, the signal pulse is a Gaussian pulse with a central frequency of 3.75×10¹⁴ Hz and time width of 6 fs, as shown in Fig. 1(b). For the convenience of analysis, the time at which the incident signal pulse is sent out from the signal source is set as the starting point t = 0 fs for all numerical experiments. The signal pulse, which is supposed to have a spatial width of 28a along the y-direction, is incident along the x-direction and arrives at the left-side interface of the PhC at 29 fs, as shown in Fig. 1(a).

 

Fig. 1. (a) System of transmission measurement of linear and nonlinear 2D PhC. The blue part is the background material of the PhC with air holes. The thick blue arrows represent the signal pulse that is incident from left-side interface, and the transmitted field is detected at the right-side interface using detectors. The pumping pulse is supposed to illuminate the whole PhC homogeneously from the z-direction that is perpendicular to the PhC. (b) Temporal electric field of the incident signal pulse.

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In our numerical experiments, the pumping field that causes the nonlinear dielectric constant change will have different time lengths for different mechanisms. The third-order nonlinearity ${\varepsilon _{non}}\textrm{ = }{{\chi }^{(3)}}{|E |^2}$ of the background material is realized by the two-level electronic system that could be pumped by an external pumping field. Theoretically, it is easy to prove that the variation in the electronic population difference of such electronic systems corresponds to the Kerr-like nonlinear effect $\delta N \propto {|E |^2}$ [22]. We set the lifetime of the higher electronic level as τ₂₁ = 2.0 fs [20]. The pumping pulse is supposed to illuminate the whole PhC homogeneously along the z-direction, as shown in Fig. 1(a). There are two different mechanisms to interpret the increase in transmission caused by the pumping pulse in different regimes. For nonlinear PhCs in which the dwelling time is on approximately 100 fs timescales, the BSM is suitable for cases with the ps pumping pulse [13–15], while the new dynamical nonlinear mechanism is suitable for cases with the fs pumping pulse [18–20]. Next, from the perspective of total pumping energy, we will demonstrate that the new dynamical nonlinear mechanism is a considerably more efficient method to increase the transmission to a certain value. In other words, the new mechanism can enhance the nonlinear switching efficiency. To quantitatively study the ability of the new dynamical nonlinear mechanism to enhance the nonlinear switching efficiency, we define “nonlinear action” as the integral of the time-varying Poynting vector of the pumping pulse in the whole process $I = \int {|{{{\vec{S}}_{pump}}(t )} |\cdot dt}$, where ${\vec{S}_{pump}}(t )= {\vec{E}_{pump}}(t )\times {\vec{H}_{pump}}(t )$, and ${\vec{E}_{pump}}(t )$ and ${\vec{H}_{pump}}(t )$ are the electric and magnetic fields of the external pumping pulse, respectively. To determine the energy efficiency of different mechanisms, the nonlinear actions of numerical experiments are compared based on two different mechanisms to realize an equal increase in transmission at a certain gap-edge frequency.

For the linear case, where χ⁽³⁾ = 0 or the pumping power is zero, the spectra of the transmission, reflection, and their sum of the linear PhC are shown in Fig. 2. The figure shows that the first gap ranges from 3.516×10¹⁴ to 3.737×10¹⁴ Hz. The concerned frequency in this study is selected as f₀ = 3.735×10¹⁴ Hz that is inside the first gap and near the second band, as denoted by the dashed line in Fig. 2. The transmission at this frequency is approximately 12% in the linear case. To accurately compare the two different mechanisms, numerical experiments are conducted where the transmission to achieve is set as 41% at f₀.

 

Fig. 2. Spectra of transmission (red curve), reflection (blue curve), and their sum (green curve) obtained using FDTD numerical experiments for linear case. The vertical dotted line denotes the gap-edge frequency f₀.

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Our first numerical experiment is based on the BSM. According to this mechanism, the nonlinear dielectric constant change is maintained at a value when the signal pulse passes through the PhC, such that the signal pulse experiences a shifted PBG. Thus, we suppose that the power of the pumping pulse is constant in the range t₁ = 0 fs to t₂ = 200 fs, which can almost cover the total transmitting process of the signal pulse. Its effect on electron population change δN(t) = δ[N₀(t) – N₁(t)] is shown in Fig. 3(a). To increase the transmission at f₀ from 12% to 41%, the required nonlinear action ${I_1} = \int_{{t_1}}^{{t_2}} {|{{{\vec{S}}_{pump}}(t )} |\cdot dt}$ is approximately 1.40×10 J/m². The spectra of the transmission, reflection, and their sum with such pumping are shown in Fig. 3(b). Comparing the spectra in this figure to those in Fig. 2, we observe that the band-gap shift effect is evident.

 

Fig. 3. (a) Value of δN(t) versus time in the nonlinear case based on the BSM. The time length and height of the curve represent the lifetime and strength of the nonlinearity, respectively. (b) Spectra of transmission (red curve), reflection (blue curve), and their sum (green curve) for the nonlinear case based on the BSM.

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For numerical experiments based on the new dynamical nonlinear mechanism, the pumping pulse has the same temporal profile as the signal pulse, as shown in Fig. 1(b), similar to the laboratory experiment conditions [18–20]. The time length of the nonlinear dielectric constant change is then approximately several femtoseconds, considerably shorter than the dwelling time of the signal pulse. According to the new dynamical nonlinear mechanism, the transmitted field in linear or nonlinear cases can generally be separated into two parts: the first-order pulse, which is the remaining part of incident signal pulse, and the higher-order pulses, which are from the Bragg scattering. The low transmission at the gap-edge frequency f₀ for a finite linear PhC is due to the destructive interference between the first-order pulse and the higher-order pulses. For nonlinear cases, an ultrashort nonlinear dielectric constant change caused by the fs pumping pulse can generate an extra phase on the field that is still inside the PhC, and such an extra phase can partially eliminate the destructive interference between the first-order pulse and the higher-order pulses, such that the transmission at the gap edge could be increased. If the pumping pulse arrives at the instant the first-order pulse has just left the PhC, the extra phase is observed solely on the higher-order pulses that are still inside the PhC, and the effect of eliminating the destructive interference between the first-order pulse and the higher-order pulses reaches a maximum; hence, the transmission at our concerned frequency f₀ reaches its maximum as well [22].

In our numerical experiments based on the new dynamical nonlinear mechanism, we expect the pumping pulse to arrive at t = 98 fs, which is the instant at which the first-order pulse leaves the PhC, corresponding to the cases of the “first peak” on the transmission curve observed in experiments [18–20]. The effect of the Gaussian pumping pulse, whose time length is approximately 6 fs, on the electron population change δN(t) is shown in Fig. 4(a). To increase the transmission at f₀ from 12% to 41%, the required nonlinear action ${I_2} = \int_{{t_1}}^{{t_2}} {|{{{\vec{S}}_{pump}}(t )} |\cdot dt}$ is approximately 2.01×10 J/m² in our numerical experiment. The spectra of transmission, reflection, and their sum with such pumping are shown in Fig. 4(b).

 

Fig. 4. (a) Value of δN(t) versus time in the nonlinear case based on the new dynamical nonlinear mechanism. The time length and height of the curve represent the lifetime and strength of the nonlinearity, respectively. (b) Spectra of transmission (red curve), reflection (blue curve), and their sum (green curve) for nonlinear cases based on the new dynamical nonlinear mechanism.

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To compare the energy efficiencies of the two different mechanisms, we define the ratio of the total nonlinear actions of the two different mechanisms η = I₁/I₂ as the nonlinear enhancement factor with similar switching effects at the concerned frequency, i.e., an equal increase in transmission. Our numerical experiments indicate that η = I₁/I₂ ≈ 7 is required at the gap-edge frequency f₀ to achieve the increase in transmission of 41%. This result indicates that to obtain an equal increase in transmission, the new dynamical nonlinear mechanism case requires just 1/7 of the energy compared to the traditional BSM case. Evidently, the new dynamical nonlinear mechanism efficiently enhances the nonlinear switching efficiency.

The extensive physical differences between the BSM and the new dynamical nonlinear mechanism are emphasized in the evident pattern difference between Figs. 3(b) and 4(b), which is revealed for the first time. In Fig. 3(b), the band gap is slightly shifted; however, the patterns of spectra are moderately similar to the spectra of the linear case in Fig. 2. However, in Fig. 4(b), the patterns are considerably more complex, because the sum of the transmission and reflection at a few frequency ranges can be considerably larger or smaller than unity. Physically, these new patterns are observed because the dielectric constant of the PhC varies owing to the pumping pulse while the signal transmission process is still ongoing; hence, the PhC is no longer a conservative system for the signal pulse and the energy conservation at single frequency cannot be guaranteed. Further analysis of the new patterns will be published in [24].

In this section, we have demonstrated that the new dynamical nonlinear mechanism considerably and efficiently enhances the nonlinear efficiency. The pattern difference of transmission and reflection spectra also imply the deep difference between two mechanisms. This new dynamical nonlinear mechanism is universal and can be employed in different systems, independent of certain geometrical structures, pumping methods, and materials. The high efficiency of the new dynamical mechanism helps achieve a sufficient nonlinear effect with considerably lower energy consumption than the traditional BSM.

3. Method to detect ultrafast electronic process

Detecting ultrafast electronic processes is an important direction for ultrafast optics, in which the optical signals can carry the information of electronic systems in direct or indirect ways [25–30]. Therefore, it is natural to ask whether the new dynamical nonlinear mechanism can provide a new method to detect ultrafast electronic processes in nonlinear materials. In previous experimental and theoretical studies [18–20,22], the PhC transmission curves at a gap-edge frequency versus the arriving time of the pumping pulse were analyzed. From the perspective of the new dynamical nonlinear mechanism, the characteristics of such curves, particularly the positions and heights of the peaks, reveal the ultrafast nonlinearity effect caused by the fs pumping pulse. In this section, we will demonstrate that the characteristics of such curves can be related to the parameters of ultrafast electronic processes, leading to the nonlinearity in our system. The electronic ultrafast properties in the nonlinear material can be demonstrated if the rule of the variation in the characteristics of the curve with different electronic parameters is determined. In other words, a novel method to detect fs-level or even attosecond-level ultrafast electronic process can be set-up.

The nonlinearity in our system is generated by the electronic population difference between two electronic levels. The lifetime τ₂₁ of the higher level, which determines the time length for which an electron stays on the higher level before it spontaneously jumps down to the lower level, is a core parameter for our electronic systems. The longer the lifetime τ₂₁ of the higher level, the more electrons are accumulated on the higher level with an ultrashort pumping pulse whose length is set as 6 fs. Keeping the other parameters constant, the variations in population difference δN(t) versus time for different τ₂₁ are shown in Fig. 5(a). Evidently, when τ₂₁ is larger, the variations in population difference δN(t) increase, and the peak time and tail part are shifted.

 

Fig. 5. (a) Value of δN(t) versus time for different τ₂₁; (b) curves of transmission at the gap-edge frequency f₀ versus the different arriving times of the pumping pulse for different τ₂₁.

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For different τ₂₁, we investigate the curves of the transmission at the gap-edge frequency f₀ versus the different arriving times of the pumping pulse. The results are presented in Fig. 5(b) with τ₂₁ = 2, 2.5, 3, 3.5 fs. Figure 5(b) shows that for different lifetimes τ₂₁, the characteristics of the curves vary. The heights of the first peak and second peak versus different τ₂₁ are shown in Fig. 6 and the straight solid lines are fitted by the linear ones. The heights of first peak are almost proportional to the lifetime τ₂₁ in this range with a certain slope. Meanwhile, the heights of the second peak versus different τ₂₁ indicate that the second peak heights increase linearly with another slope. Thus, for different τ₂₁, the relative heights between the first peak and the second peak are different, which could be an important indication to determine τ₂₁. Figure 5(b) shows that not only the peaks but also the valleys on the curves are subtly different with different τ₂₁. The origins of these detailed differences will be further studied in [24].

 

Fig. 6. Heights of the first and second peaks of curves in Fig. 5(b) versus τ₂₁. The red and blue straight lines are the linear fitting results for the first and second peak, respectively.

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Further, we note that the new dynamical nonlinear mechanism can even detect the sub-fs-level or attosecond-level electronic processes. In our upper study, the working wavelength is chosen as 800 nm, the 0.5 fs difference of the fs-level electronic process can already be detected. According to the scaling rule of electromagnetic field, if we select a shorter wavelength (e.g., an ultraviolet wavelength 100 nm) as the working wavelength and the corresponding PhC structure, such variation in the characteristic curve can even help us to detect the attosecond-level difference of electronic systems.

We intend to emphasize that such variation in characteristics is universal and independent of the lattice type of the PhC, nonlinear materials, pulse type, and working frequency. Therefore, the new dynamical nonlinear mechanism is a novel method to indirectly detect the fs-level or even attosecond-level electronic processes of nonlinear materials in wide directions.

4. Conclusion

In conclusion, we quantitatively studied two special properties of the new dynamical nonlinear mechanism of all-optical nonlinear PhC switches. First, the traditional BSM was compared to the new mechanism, which indicated that the latter more efficiently enhanced the nonlinear efficiency, because it requires a pumping energy almost one order lower than the former. The pattern difference of transmission and reflection spectra, revealed in this work, also imply the deep difference between new mechanism and traditional BSM. Second, the new mechanism demonstrates that the nonlinear PhC switching is a novel method to indirectly detect ultrafast electronic processes at fs or attosecond levels. These two properties exhibit potential in numerous applications, such as ultrafast high-efficiency switches, all-optical logical devices, and ultrafast electronic process detectors. Further studies regarding the evident variation in transmission profile at the band region and the nonlinear physics of such variation will be published in a future paper.