1. Introduction

Since the pioneering work on spatial invisibility cloaks for rendering any object in a specific region of space invisible [1,2], rapid developments in artificial composite materials has allowed different kinds of spatial invisibility cloaks to be designed theoretically and realized experimentally [3–8]. On the other hand, another kind of distant cloaking devices based on complementary media has been proposed and numerically demonstrated to not only hide distant objects [9] but also create illusions [10]. Recently, the concept of cloaking has even been theoretically and experimentally extended to cloaking in time, using a device that hides events occurring during a specific time window [11,12].

No matter whether the cloaks are for cloaking objects or for cloaking events, they are mainly designed based upon the popular transformation optics method [3,11,13]. Here, we introduce Fourier analysis method to constitute temporal cloaks to hide events in time domain, instead of designing spatial invisibility cloaks for concealing objects or creating optical illusions [14–17]. By synthesizing two linear time-invariant filters with different transfer functions, we can orderly create and close a temporal gap, in which any events occurring will not be detectable to the outside observers. Further analyses reveal that, analogue to distant spatial cloaks, hiding event outside the no-gap temporal cloaks is also feasible. In this case, the events can completely be compensated for by a conjugated temporal filter as nothing occurs. Theoretical analytical results are well confirmed by fast Fourier transformation numerical simulations.

2. Theoretical analysis

For an event (wave packet), it can be described as complex amplitude in time domain and frequency spectrum in frequency domain, respectively [18,19]

It is known from phase-shift theorem that a phase shift in frequency domain will lead to delay of an event in time domain [20,21]. For example, to give an incident event $u(t)$a time delay $\Delta t$ around the center time $t_0$, one can add a phase shift $\exp (-i2\pi \mu \Delta t)$ to its frequency spectrum $U(\mu )$. Therefore, to create a temporal gap, we can firstly synthesize a temporal filter with transfer function,

To close the opening time gap, we can synthesize another filter with transfer function$H_2(\mu )=H_1{}^{\ast }(\mu )$, where $\ast$ denotes the conjugation operator. In this case, the transfer function of the second filter can be written as

Comparing $H_2(\mu )$ and $H_1(\mu )$, we see that $H_2(\mu )$plays the role of closing the temporal gap opened by $H_1(\mu )$, making the output signal continuous again and looked as nothing occurs in time domain. Therefore, when an interferential event $u_i(t)$ occurs during the temporal gap$t_0-\Delta t<t<t_0+\Delta t$, from Eq. (5) we see that $u_i(t)u\text{'}(t)=0$, meaning that $u_i(t)$ shows no effect on the output signal after the time window. Thus, the output signal $u_{out}(t)$ is still observed as $u(t)$. However, when the interferential event $u_i(t)$ occurs outside the temporal gap, the output signal becomes into $u\text{'}(t)u_i(t)\ne 0$[see Eq. (5)], indicating that interferential $u_i(t)$ is detectable.

The above temporal filters can be implemented by using a classical pulse shaping configuration consisted of a pair of gratings, lenses, and masks [20,22], which was employed in a recent temporal cloak set-up [12] as shown in Fig. 1 . The role of the masks played is to modify the phase of frequency spectrum of the incident event through precise control over their thickness $d$ and/or refractive index $n$. From Eq. (3) we see that when optical phase change is $\pm 2\pi \mu \Delta t$, then $n$ and $d$ should satisfy $2\Delta (nd)=c\Delta t$, where $c$ is the speed of light in vacuum. By modulating $n$ and/or $d$of the masks, we can open a time window by the first grating, lens, and mask system and then close the time window by the second grating, lens, and mask system orderly. For instance, choosing vacuum ($n=1.0$, constant) as the medium of the masks but changing the thickness of the first mask $\Delta d=\pm 3\text{mm}$ while the second mask $\Delta d=\mp 3\text{mm}$, we can create and then close a $20\text{ps}$temporal gap in time domain in sequence. The thickness change of the masks can be realized by moving the two mirrors attached on the surfaces of the masks [12]: One mirror reflects the frequency spectrum before $t_0$ and the other reflects the frequency spectrum after $t_0$. In this configuration, the incident event occurs before the first filter and the interferential event occurs between these two filters separately. After the incident event passing through the first filter, it interacts with interferential event. Then the observer can detect output signal behind the second filter. So incident event occurs earlier than interferential event and output signal is detected after interaction of two events.

 

Fig. 1 Scheme of temporal filters consisted of a pair of gratings, lenses and mirrors.

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3. Numerical simulations

To verify the above analytical results, we employ fast Fourier transform method [21,23] to simulate an event occurs in the above filtering system. Since temporal filters act on a finite time slot, incident event is chosen occurring when filters are working. This avoids that staring or stopping filter will cause time gap not open. The incident event is set to be a Gauss beam $u(t)=1.0\sin ({\mu }_{0}t)\text{exp}[-{(t/40)}^2]$ with ${\mu }_0=0.5\text{THz}$[Fig. 2(a) ]. The interferential event, read as $u_i(t)=1.0+20exp\{-{[(t-500)/10]}^2\}$[Fig. 2(b)], occurs behind the first temporal filter during the time interval $200\text{ps}$ (from$400\text{ps}$to $600\text{ps}$), where the two events are mixed together. Behind the second filter, we get the total output signal.

 

Fig. 2 Time-dependent complex amplitude distributions of (a) an incident event, (b) an interferential event occurs during the temporal gap, and the output signal as the cloak is (c) off and (d) on. Time-dependent complex amplitude distributions of (e) an interferential event occurs during the edge of temporal gap and (f) corresponding output signal.

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Figure 2(c) shows the output signal as the cloak is off [setting $H_1(\mu )=H_2(\mu )=1$, meaning that two filters are removed and no time window opens]. Obviously, the incident event [Fig. 2(a)] is disturbed by the interferential event. However, when two filters are employed (the time gap is orderly opened and closed), a completely different effect happened. The output signal [Fig. 2(d)] is exactly the same as that of the incident event and no any information of the interferential event involved.

To know the effect of an interferential event occurring at the edge of temporal gap but instead of during the gap, we simulate the output signal [Fig. 2(f)] as the interferential event is assumed to be $u_i\text{'}(t)=1.0+20exp\{-{[(t-400)/10]}^2\}$[see Fig. 2(e)], which indicates that the interferential event occurs during the time interval of from$300\text{ps}$ to $500\text{ps}$but instead of from $400\text{ps}$to $600\text{ps}$. Comparing Fig. 2(f) and Fig. 2(a), we see that the output signal is also disturbed by the interferential event. Consequently, we conclude that only those interferential events occurred during the time gap can be canceled completely by such temporal filtering system.

Similar to distant spatial cloaks [9,10], which can cloak objects at a distance outside the cloaks and be regarded as a conjugated spatial filter [14–17], we can also construct a temporal cloak without time window to conceal events. For an interferential event$U_i(\mu )$in frequency domain, if there is a temporal filter with transfer function $H_c(\mu )=U_i{}^{\ast }(\mu )$, then behind the filter, the event$U_i(\mu )$ will completely be compensated for and not be detectable. Consequently, only incident event $U(\mu )$ is measurable. In addition, to conceal an interferential event $A(\mu )U_i(\mu )$ with amplitude change$A(\mu )$, we can employ a temporal filter with transfer function $H_c{}^{\prime }(\mu )=U_i{}^{\ast }(\mu )/A(\mu )$ to compensate both phase and amplitude simultaneously.

Such conjugated temporal filter with transfer function$H_c(\mu )$ can also be constructed with a grating-lens-mask system, where the mask can theoretically be used to tune the transmission coefficient of the interferential event so as to synthesize such $H_c(\mu )$ [20]. Here we introduce another technique to get a distant temporal cloak. For an interferential event occurs in a positive dispersive medium [right, Fig. 3(a) ], we can design a cloak constructed with a negative dispersive medium [left, Fig. 3(a)] to compensate for the effect of the interferential event. In this case, an interferential event occurs in a positive dispersive medium can be written as $U_i(\mu )=exp(\frac{1}{4{\pi }^2}i{\mu }^2\frac{d^{2}n}{d{\mu }^2}z)$ with $\frac{d^{2}n}{d{\mu }^2}>0$ in frequency domain, while the corresponding conjugated temporal cloak should have a transfer function $H_c(\mu )=exp(-\frac{1}{4{\pi }^2}i{\mu }^2\frac{d^{2}n}{d{\mu }^2}z)$, which can be realized by using a negative dispersive medium with$\frac{d^{2}n}{d{\mu }^2}<0$.

 

Fig. 3 Scheme of the dispersion curves of media (a) for creating interferential events (left) and constructing conjugated temporal cloaks (right). Time-dependent complex amplitude distributions of (b) an interferential event, (c) mixed output signal of the incident event and interferential event, (d) conjugated temporal filter, and (e) output signal after filter is implemented for compensating for the interferential event, respectively.

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In the simulation, supposing that during an incident event as that of Fig. 2(a) occurs, another interferential event with complex amplitude in frequency domain $U_i(\mu )=exp(1.0i{\mu }^2/3600)$ also occurs at $t=456\text{ps}$ [For comparison, we show in Fig. 3 (b) the amplitude of the interferential event in time domain]. Figure 3(c) presents the simulated output signal. We can see that the output signal distinguishes from the incident event [Fig. 2(a)], meaning that the incident event is obviously disturbed by interferential event. However, as a temporal filter with transfer function $H_c(\mu )=exp(-1.0i{\mu }^2/3600)$ [Fig. 3(d) shows its complex amplitude in time domain] at $t=544\text{ps}$ is used to compensate for the interferential event, we see that the output signal [Fig. 3(e)] is exactly recovered into the form of the incident event [Fig. 2(a)], indicating that interferential event is concealed completely as nothing occurs.

4. Conclusions and discussions

To conclude, we have introduced a Fourier analysis method for constituting temporal cloaks to hide events in time domain. The cloaks can either be constructed by synthesizing two linear time-invariant filters with different transfer functions to open and close a temporal gap orderly, so that any events occurring during the gap are not detectable outside, or be designed based upon temporal compensation to make event outside the cloaks not detectable. All the analytical results are verified by numerical simulations. Our results may provide another way of constructing time cloaks for interesting applications in such fields as temporal imaging and sensing etc.