1. Introduction

Metamaterial is a form of artificial electromagnetic material made up of a periodic array of subwavelength, which has an extraordinary electromagnetic response characteristics that natural materials do not have. It can achieve negative refraction [1], perfect prisms [2], gradient phase [3] and optical stealth [4], and achieve many applications such as broadband chirality [5,6], anomalous refraction [6,7], holography [8], optoelectronic switch [9], sensor [10–14], polarization control [15] and coherent modulation [16]. The coherent modulation is achieved by using two beams which incident vertically from both sides of the metamaterial as input signals, one as signal light and the other as control light. The superposition of the transmitted and reflected light produces the output signal. The phase difference between the two beams can be adjusted by altering the phase of the signal light due to light coherence. So that the interaction between metamaterial and light can be inhibited or strengthened [17]. Before the discovery of coherent modulation in metamaterials, wave interaction on nonlinear materials was the only technique that could dynamically adjust the wavefront of one beam of light to another, but this light-to-light modulation method was slow and required a large amount of optical power.

Moreover, the optical switching based on coherent modulation can overcome the above-mentioned drawbacks of the light-with-light modulation approach for nonlinear materials. An optical switching is a device with one or more optional transmission ports that can interconvert or logically operate optical signals in an optical transmission line or integrated optical path. It can be used in optical fiber communication systems, optical fiber network systems, optical fiber measurement systems or instruments, optical fiber sensor systems, optical switches. It has significant scientific and practical value for optical fiber communication networks. At present, a variety of new types of optical switches have been developed, which are mainly divided into electronic control optical switchings and all-optical switchings. Nowadays, there is an electronic bitrate bottleneck in the optoelectronic hybrid technology platform, and its energy consumption tends to increase, while the photon itself is not charged, so the all-optical switching without photoelectric conversion can overcome the proceeding issues. The concept of all-optical switching originated from integrated optoelectronics, which is a kind of "light-controlled light" switching in integrated optoelectronics. It does not need to go through opto-electric conversion, but only uses the interaction between photon and medium to achieve the "on" and "off" effect of photon communication device. However, there are still some problems such as slow response time and large device feature size. To overcome these problems, we proposed an ultrafast all-optical switching device based on the principle of coherent modulation.

Here, we demonstrate an ultrafast all-optical switching based on coherent modulation. It employs sub-wavelength thick plasma nanostructures as the primary body for the optical switching, which allows the transmission and absorption of one signal beam to be modulated by another coherent control beam. The switching device can not only realize the light-to-light modulation of continuous light, but also realize the control of pulse duration of 80 fs pulse with 12.5 THz bandwidth.

2. Theoretical model

The coherent control device is a four-port device with two input ports and two output ports [18]. As shown in the Fig. 1, there is a film with a thickness $d$ and a complex refractive index $n$. Two coherent lights A and B incident on the film, and the transmission of A and the reflection of B constitute the output light C, while the reflection of A and transmission of B constitute the output light D. The transmission matrix connects the output signals (C and D) to the input signals (A and B). Assuming that the optical properties of the flat material are linear and have a purely linear isotropic dipole response, the relationship between the input field and the output field can be expressed by a complex scattering matrix [18]:

 

Fig. 1. Thin film four-port device based on coherent control. The membrane is irradiated by two coherent beams A and B, travelling in opposite directions. Beams C and D are two output beams.

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Since the metamaterial is linear, when the two incident lights are scaled up or down by $\eta$ times, the output signals in both directions are scaled up or down by $\eta$ times:

Therefore, the general situation is as follows:

Assuming that the metamaterial is sufficiently thin, the delay over the entire thickness of the metamaterial is negligible, and we can assume that each cell is subjected to the same electric field, i.e., the combined field of the input signals A and B is equal to $E_{\rm A}+E_{\rm B}$. The metamaterial cell will re-radiate the absorbed energy equally in both positive and negative directions, and the efficiency of this energy depends on the excitation wavelength $\lambda$, resulting in a re-radiating field $S(\lambda )(E_{\rm A}+E_{\rm B})$, where $S(\lambda )$ is the scattering coefficient of the metamaterial under a single light. The amplitude and phase of $S(\lambda )$ correspond to the relative amplitude of the reradiation field and the phase lag between the reradiation field and the incident field, respectively. $S(\lambda )$ may include losses, so it can be assumed that the total input strength is not equal to the total output strength. Therefore, the expression of the scattering matrix for a metamaterial four-port device with total loss can be written as [18]:

According to the principle of coherent control, the above four-port device can be set as follows: two incident lights (A and B) propagate from opposite sides of the metamaterial and incident vertically to the surface of the material with the same amplitude, i.e., $|E_{\rm A}|=|E_{\rm B}|$. The two lights form standing waves along the propagation direction. When the metamaterial is placed vertically at the node of the standing wave, the phase difference between the two lights is $\alpha =180^{\circ }$, that is, $E_{\rm A}=-E_{\rm B}$, the effect of $S(\lambda )$ is ignored, or the real part of the scattering parameter $S(\lambda )$ is 1 and the imaginary part is 0, thus $E_{\rm C}=E_{\rm B}$, $E_{\rm D}=E_{\rm A}$. This phenomenon is called "coherent perfect transmission". The electric field components of the two coherent electromagnetic waves at this node are opposite, so they cancel each other, the combined electric field is zero, and the coherent electromagnetic waves do not interact with the metamaterial. When the metamaterial is placed vertically at the abdomen of the standing wave, the phase difference between the two lights is $\alpha =0^{\circ }$, that is, $E_{\rm A}=E_{\rm B}$, so that the real part of the scattering parameter $S(\lambda )$ is $-$0.5 and the imaginary part is 0, thus obtaining $E_{\rm D}=E_{\rm A}=0$. This phenomenon is called "coherent perfect absorption" [19,20].

3. Simulation design and analysis

According to the principle of coherent perfect absorption, the absorbent of deep subwavelength thickness should ideally exhibit the maximum possible level of 50 $\%$ traveling wave absorption. However, the traditional unstructured metal film is unable to satisfy this requirement. The traditional unstructured metal films have a high degree of transparency or reflection. Consequently, metamaterials are excellent for making such absorbers.

The metamaterial structure used in this paper is asymmetric split rings (ASR) structure. The structure can be realized either as metallic ASRs fabricated on a dielectric substrate, or as a complementary structure of ASR slits perforated in a metal film. The ASR slit structure, also known as the inverted Babinet metasurface, is designed in this paper. As shown in Fig. 2, the overall structure designed consists of a layer of gold film milled with asymmetrically split ring array and a layer of silica substrate. The metamaterials mentioned in this paper are made by depositing a 80 nm gold film on a silicon dioxide film (170 $\mathrm{\mu}$m thick) by thermal evaporation. A set of asymmetrical split ring (ASR) slits were milled through the gold layer by a focused ion beam (FIB) method. The right side of the Fig. 2 shows the structure of the metamaterial unit overlooking a larger view of figure, each unit corresponding to the same asymmetrically split ring and its upper part is a fan-shaped ring and the lower part of a U-shaped ring. In addition, Fig. 2 shows some key parameters of unit structure to determine the specific position of asymmetric split ring, $D=440$ nm, $b_1=5$ nm, $d=100$ nm, $R=190$ nm, $r=90$ nm, $b_2=20$ nm, $L-l=100$ nm, and the specific parameter values of $\theta$, $L$ and $l$ will be discussed in the following.

 

Fig. 2. The overall structure of an ultrafast coherent metamaterial modulator consists of a gold film with asymmetric split rings on the surface and a silica substrate.

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It has been proved that the structure of the slit milling on the surface of precious metal has been proved to generate an excitation mode of absorbing resonance, and the destruction of the structural symmetry can realize the excitation of Fano resonance in the narrow band range [21]. The Fano resonance in metamaterial resonance system is the result of the interaction of broad band bright mode and narrow band dark mode. In bright mode, the open ring is directly excited by incident light to produce a dipole response equivalent to a continuous line. On the contrary, the narrow-band dark resonance mode has very weak interaction with external electromagnetic field, that is to say, the dark mode cannot be directly excited by external electromagnetic field. When the bright mode indirectly induces the dark mode by means of near-field coupling, a weak annular current is induced inside the structure of the asymmetric open ring. The radiation loss is effectively inhibited by the dipole moment self-elimination effect caused by the left and right parts of the ring current. The dark mode is indirectly excited by the bright mode, and then reacts on the bright mode, and the interaction between the two leads to Fano resonance. Therefore, the excitation of dark mode and the coupling between light mode and dark mode can be controlled by adjusting the degree of structural asymmetry. In this paper, the design of the metamaterial breaks the symmetry of the structure, thus damaging the symmetry of the dark mode, which can be directly coupled with the external electromagnetic field. The resonance mode of high-quality factors in the asymmetric metamaterial is also known as the trapped mode. The physical mechanism of trapped mode can also be explained by the theory of Fano resonance. When excited by electromagnetic waves, the two inhomogeneous arcs of the open-loop structure support in-phase current oscillations except in the narrow frequency range where the antisymmetric current is established. Antisymmetric excitation forms magnetic dipole arrays that oscillate perpendicular to the metamaterial plane. This collective subradiative mode is only weakly coupled to free space by interacting with the wider superradiative dipole mode of the in-phase current, resulting in the classical Fano complex. By breaking the symmetry of the structure, different electromagnetic dipole resonance modes can be hybridized with each other, and the narrow line width and wide line width dipole resonance modes can be coupled to form Fano resonance. In addition, two dipole resonance modes with the same properties can be coupled by breaking the structural symmetry, and a mode with a wider linewidth and a mode with a narrower linewidth can be hybridized to act as the continuous state and discrete state of Fano resonance respectively.

Figure 3 shows the simulated vertical incident visible to near-infrared spectral response of the aperture of the asymmetric opening ring in the gold film and the normalized field distribution of the $z$-component of the magnetic field at the resonant wavelength. According to Babinet’s principle [22], $y$-polarized lights perpendicular to the annular aperture are used to excite the Fano resonance of the subsurface. As shown in Fig. 3(a), the reflection spectrum shows a Fano type band near 1500 nm, accompanied by two weak much resonance, corresponding to transmission peaks at about 1020 nm and 1640 nm respectively (see Fig. 3(b)). The $z$-component magnetic near field distribution at the reflection peak shows the characteristics of two oscillating magnetic dipoles in the metasurface plane shown in Fig. 3, and is therefore in magnetodark mode. The corresponding trapped mode is weakly coupled to free space, so most of the energy is confined around the slit (see Fig. 3(c)).

 

Fig. 3. Asymmetrically split ring metamaterial. (a) (b) Simulated normal incidence visible to near-infrared spectral response for asymmetric split ring apertures in a gold film. (c) Polarizations of incident waves and z-component magnetic near-field distributions at II.

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From the aspects of the results, ideally the metamaterial designed should behave the largest possible level of 50 $\%$ of the traveling wave absorption, and because of the existence of error, the actual situation can’t perfectly reached 50 $\%$ of maximum absorption level, we expect is to be closer to the 50 $\%$ level. The metamaterial designed in this paper exhibits a traveling wave absorption level up to 49.7 $\%$.

The structure of the designed metamaterial unit is shown in Fig. 4(a), which is an ASR with a thickness of 50 nm. The upper part is a fan-shaped ring and the lower part is a U-shaped half ring. This metamaterial is characterized by a Fano type plasmon mode with strong resonance absorption [23,24]. Simultaneously, the thickness of the designed metamaterial is only $\lambda /31$, which is the key factor to realize light-by-light modulation of two lights incident from opposite directions, where $\lambda$ is the working wavelength. Because the thickness is small enough, the metamaterial can be placed at the nodes and abdomens of standing waves to inhibit or enhance the interaction of light matter. The relative permittivity of gold is derived from the Drude-Lorentz model [25,26]. The two input signals used here are $y$-polarized continuous light. The target wavelength of this metamaterial is the communication band centered at 1550 nm, which provides "trapped mode" resonance for $y$-polarized light. The model’s overall size and the size of the slits are specifically designed to fit in the target wavelength. Based on prior research, we firstly calculated the overall size of the metamaterial unit to achieve absorption resonance at 1550 nm, and then modified the thickness of the gold film using a 5 nm step size to obtain maximum absorption while maintaining deep subwavelength thickness (see Fig. 4(b)). Then, the opening angle of the fan-shaped ring and the length of the two arms of the U-shaped ring were changed to adjust the drift of the absorption resonance peak (see Fig. 4(c), (d)). On the one hand, when the thickness of gold film is increased, the resonance spectrum will be redshifted. This is due to the fact that the strong electric field exists only in the localized region above the metasurface and weakens exponentially with the increasing distance [27]. On the other hand, when the thickness of the gold film is determined, the asymmetry of the metamaterial structure increases by changing the opening angle of the fan-shaped ring or the arm length of the U-shaped ring, which leads to the drift of the resonance spectrum. Based on the above analysis, we can obtain the optimal structural parameters of the independent gold film: the structural size of the metamaterial element $D=440$ ${\rm nm}$, the thickness of the gold film $t_1=50$ ${\rm nm}$, the opening angle of the fan-shaped ring $\theta =150$ $^ {\circ }$, and the length of the two arms of the U-shaped ring $L=180$ ${\rm nm}$.

 

Fig. 4. Independent metamaterial optimization process. (a) Self-contained metamaterial units. Here $\theta$ is the opening angle of the fan-shaped ring, $L$ is the height of the U-shaped half-ring, and $t_1$ is the thickness of the metamaterial element. (b) The opening angle of the fan-shaped ring $\theta$ and the height of the U-shaped half ring $L = 180$ ${\rm nm}$, and the thickness of the gold film $t_1=50$ ${\rm nm}$. (c) The absorption of the fan-shaped ring at different opening angles $\theta$ and $t_1=50$ ${\rm nm}$, $L = 180$ ${\rm nm}$ under single beam illumination was calculated numerically. (d) The absorption of U-shaped half ring at different heights $L$ and $t_1=50$ ${\rm nm}$, $\theta =150^{\circ }$ under single beam illumination was calculated numerically.

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As shown in Fig. 5(b) , the absorption, reflection and transmission spectra of the independent gold metamaterial were numerically simulated by using the finite difference time domain (FDTD) method when illuminated by continuous single light. The element structure of the model has periodic boundary conditions along the $x$ and $y$ axis (see Fig. 5(a)), while the incident lights propagate along the positive or negative $z$ axis. Here, the peak absorption of a single beam is about 49.7 $\%$. Ideally, the film absorbs only 50 $\%$ of the light from a single light, with the remaining light intensity transmitted and reflected. But in the case of two coherent lights, this structure can provide perfect coherently controlled transmission and absorption of the plasma. Our device is an optical four-port device consisting of two inputs ($E_1$ and $E_2$) and two outputs($S_1$ and $S_2$). We define the total output strength as $S=S_1+S_2$ and the total absorption as $A=1-(S_1+S_2)$, where the total input strength corresponds to 1. We used the parameter sweep function in FDTD software to sweep the phase of the control light from 0 to 2$\mathrm{\pi}$. In experiment, the phase difference is usually controlled by a piezoelectrically-driven optical delay line in the control light propagation path [17]. By changing the phase difference between the two incident lights, the relationship between the total output intensity and absorption intensity and the phase difference can be obtained at the working wavelength of 1550 nm and 1350 nm. The resonance is switched on when the coherent lights are illuminated at 1550 nm, which provides a total absorption level from 0.43 $\%$ to 99.99 $\%$ and a total output level from 0.05 $\%$ to 99.69 $\%$. The relatively broad nature of metamaterial resonance provides modulation with input intensity level exceeding 90 $\%$ in the whole spectral range of 1534 to 1592 nm, giving a bandwidth of 7.24 THz. Thus, the optical switch function can be implemented by controlling the phase difference between the two coherent lights at this working wavelength. In this case, the metamaterial performs a cross-error function similar to a semi-transparent mirror, transmitting input energy between the two output ports as a function of the relative phase of the incident beam, and performing the limited modulation for the total absorption or output intensity [17].

 

Fig. 5. Design of independent asymmetric split rings gold film. (a) Geometric design of metamaterial units. The thickness of the gold film is 50 nm. (b) Reflection, transmission, and absorption of a single beam vertically incident into a metamaterial. $T$, $R$ and $A$ are the normalized transmittance, reflectance and absorptivity respectively. (c,d) Light-by-light modulation output at 1550 nm and 1350 nm is a function of the phase difference between the two input beams: $S_1$ is the integral of the output intensity along the $E_1$ direction. $S_2$ is integral of the output strength along the $E_2$ direction. $S$ is total output strength. $A$ is total absorption strength.

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The next step is to consider the role of the substrate, since it is significantly easier to manufacture ASR structures supported by the substrate. In previous studies, it has been proved both silicon nitride [28] and silica [17] can be used as substrate materials of ASR to achieve coherent modulation function. According to the working wavelength as well as structure size, silica was selected as the substrate material of the gold film. The model works in the working wavelength by adjusting the thickness of the substrate and the gold film as well as the size of the slits. We used the above-mentioned independent gold film metamaterial’s cell and slit sizes, and then modified the gold film thickness in 5 nm increments to achieve maximum absorption (see Fig. 6(b)). Then, the opening angle of the fan-shaped ring and the length of the U-shaped ring’s two arms were changed to adjust the drift of the absorption resonance peak, and the absorption resonance at 1550 nm was obtained (see Fig. 6(c), (d)). At this point, we can obtain the optimal structured parameters: the structural size of metamaterial element $D=440$ ${\rm nm}$, the thickness of the gold film $t_1=80$ ${\rm nm}$, the thickness of the substrate $t_2=170$ $\mathrm{\mu}$m, the opening angle of the fan-shaped ring $\theta =140^{\circ }$, and the length of the arms of the U-shaped ring $L=160$ ${\rm nm}$.

 

Fig. 6. Optimization process of metamaterials with substrates. (a) Metamaterial units with substrates. Here $\theta$ is the opening angle of the fan-shaped ring, $L$ is the height of the U-shaped half-ring, $t_1$ is the thickness of the metamaterial unit, and $t_2$ is the thickness of the silica substrate. (b) The absorption of gold film with different thickness $t_1$ and $\theta =150^{\circ }$, $L=180$ ${\rm nm}$ and $t_2=170$ $\mathrm{\mu}$m under single beam illumination was calculated numerically. (c) The absorption of the fan-shaped ring at different opening angles $\theta$ and $t_1=80$ ${\rm nm}$, $t_2=170$ $\mathrm{\mu}$m, and $L=180$ ${\rm nm}$ under single beam illumination was calculated numerically. (d) The absorption of u-shaped half-rings at different heights $L$ and $t_1=80$ ${\rm nm}$, $t_2=170$ $\mathrm{\mu}$m, $\theta =140^ {\circ }$ under single beam illumination was calculated numerically.

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Figure 7(b) shows the numerical simulation spectra of a single beam incident from two different directions, the gold film and the substrate. It is shown that for the 80 nm thick gold sample on the 170 $\mathrm{\mu}$m thick silica substrate, the numerical simulation spectra of the single light irradiation in the two input directions are different due to the asymmetry introduced by the substrate in the two output directions along the propagation direction. (absorption rate: 58.2 $\%$/39.6 $\%$ light incident on the silica/gold side). This explains the asymmetry between $S_1$ and $S_2$ in Fig. 7(c), (d), but the phase dependence of the total output power remains similar [29]. In addition, the metamaterial will provide 4.44 $\%$ to 94.55 $\%$ of total phase control absorption and 95.69 $\%$ to 2.27 $\%$ of total output intensity modulation at 1550 nm (see Fig. 7(c)). The relatively broad nature of metamaterial resonance provides modulation with input intensity level exceeding 90 $\%$ in the whole spectral range of 1527 to 1577 nm, giving a bandwidth of 6.24 THz. Theoretically, the maximum or minimum value of $S$ and $A$ should be at zero phase difference, but due to the imbalance between the group velocity dispersion in the input optical path, it prevents the simultaneous phase matching of all wavelength components [28], thus results in the asymmetry of $S$ and $A$ in Fig. 7(c), (d).

 

Fig. 7. Gold film design with substrate asymmetric split rings. (a) Geometric design of metamaterial units. The thickness of gold film is 80 nm, and the thickness of silica substrate is 170 $\mathrm{\mu}$m. (b) Reflection, transmission, and absorption of a single beam vertically incident into a metamaterial. $T$, $R$ and $A$ are the normalized transmittance, reflectance and absorptivity respectively. (c,d) Light-by-light modulation output at 1550 nm and 1350 nm is a function of the phase difference between the two input beams: $S_1$ is the integral of the output intensity along the $E_1$ direction. $S_2$ is the integral of the output strength along the $E_2$ direction. $S$ is total output strength. $A$ is total absorption strength.

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According to the above simulation results, it is proved that the metamaterial structure designed in this paper can realize the optical switching function under the irradiation of coherent continuous light. In order to extend the application range of our designed metamaterial devices, short pulses with a pulse duration of 80 fs are selected as input signals. Ultra-fast switching time is one of the most important indicators to achieve the practical application level of the all-optical switch [30]. Figure 8 shows the total metamaterial modulation output as a function of the mutual delay between two coherent beams at a wavelength of 1550 nm. When the positive and negative delay is at a large level, the output of the system is at a constant level, which corresponds to the incoherent absorption of two incident beams by metamaterials (as shown in Fig. 8). The two incident pulses are independent in time until they reach the designed structure, so they interact independently with the metamaterial. When the two backward propagating pulses overlap in time on the metamaterials, the system oscillates back and forth between coherent perfect absorption and coherent transparency according to the principle of coherent modulation. The time-varying electric field in the metamaterial plane can be expressed as the sum of two inversely transmitted pulse fields, $E(t)+E(t-\tau )$, where $\tau$ is the time delay between the two pulses. The level of coherent absorption due to the applied field is [28]

 

Fig. 8. Spectral dispersion of ultrafast coherent modulation contrast. (a) The dependence of metamaterial coherent modulation output power on the time delay between two inversely transmitted input pulses at a wavelength of 1550 nm. Simulation data (red dots) overlap with the envelope curve (black line) and modulation contrast (blue line) given in Eq. (5). (b) Peak modulation contrast as a function of wavelength.

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In Fig. 8(a), the red dot is the simulated experimental result, and the black line is the relative output envelope simulated by Eq. (5). It can be seen from Fig. 8(a) that the simulation result is accurately fitted with the envelope, and the full-width half-maximum value of the envelope is 171 fs, and the value $A_1=0.269$, $A_2 =0.228$ [28]. The modulation contrast can be defined as the ratio of upper and lower envelope, which can reach 3 in the case of zero delay. As shown in Fig. 8(b), about 1400, 1450, 1500, 1550, 1600, 1650 and 1700 nm pulse center wavelength repeated simulation using the above method. Figure 8(b) gives the peak values of the modulation contrast at all the above wavelengths, which can be expected to reflect the metamaterial absorption resonance centered at 1550 nm. The 2.5 : 1 modulation contrast is sufficient for short-distance optical interconnection applications in data processing [32]. Therefore, it can be seen from the results in Fig. 8(b) that the designed all-optical switching can achieve a modulation contrast ratio of more than 2.5 : 1 at a bandwidth of about 12.5 THz.

Table 1 shows the comparison of key indicators of some metamaterial based optical switching devices. From the comparison in the table, we can see that the results obtained in this paper are the best for switching time, and the modulation contrast is within the range required for data processing in short-distance optical interconnection. Therefore, the metamaterial optical switching device based on coherent modulation designed in this paper has the characteristics of ultrafast response and ultra-high switching efficiency.

Table 1. A comparison of the various optical switching metrics discussed, obtained from some key example devices.

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4. Conclusion

In conclusion, we demonstrate a femtosecond ultrafast all-optical switching based on the principle of coherent modulation, which can be switched for continuous light or pulse light. The basic mechanism of coherent control is linear interference effect, so it can still be achieved under low intensity excitation, unlike the traditional interferometer through the output channel for light separation. Instead, the metamaterial device is designed to switch between high and low output states by manipulating the phase or intensity between the input beams, thus choosing between high and low absorption excitation states at the metamaterial plane. The switching function of the ultra-high speed all-optical modulator is proved in the communication band centered at 1550 nm. Through the special design of the metamaterial structure, the concept can be realized freely in a wide range of communication bands. In practice, manufacturing defects, pulse duration, and group velocity dispersion matching in the two beam channels will limit the operating frequency and modulation contrast that can be achieved [28]. The contrast ratio of 3 : 1 shown here can be effectively applied to short-distance optical interconnections for data processing, in which case only a ratio of 2.5 : 1 is sufficient to achieve the desired function [32].

Many optical functions can be effectively realized through the coherent regulation of light on metamaterial nanostructures, and absorption is only one of them [39]. Thus, in a coherent network environment where metamaterial devices can be easily interconnected and cascaded, the coherent control paradigm can provide a range of solutions, including logic gate capabilities for ultrafast all-optical data processing.