## Abstract

The time-reversed second-harmonic generation in one-dimensional nonlinear photonic crystals has been theoretically studied without the undepleted pump approximation. A simple criterion has been deduced which determines the energy flow. Based on it, two kinds of structures with different symmetries are presented to realize the nonlinear time reversal effect. A completely reciprocal nonlinear response is also found in the same process. Furthermore, a multi-section-cascaded structure is proposed to realize the nonlinear time reversal at any given position.

© 2017 Optical Society of America

## 1. Introduction

Recent years have witnessed a growing tendency to study time-reversed symmetry in all kinds of areas, including acoustics [1, 2], plasmonics [3], telecommunications [4] and many other scientific topics [5, 6]. Time-reversed symmetry implies that a transmitting signal is allowed to reverse accurately in a backward direction of time [7]. In optics, time reversal is shown to be intimately linked with negative refraction [8] and reciprocity [9, 10]. It is well-known that the replication process can be demonstrated by the use of the four-wave nonlinearities [11]. In the previous researches, the nonlinear four-wave mixing process was widely utilized to achieve the time reversal of optical signals, which is helpful to compensate for chromatic dispersion and pulse distortion [12–15]. Another universal method to generate time-reversed waves is using time-modulated photonic structures [16–20]. Particularly in nonlinear optics, the time-reversed optical parametric oscillation has been proposed and demonstrated by Longhi to realize a coherent perfect absorber in a nonlinear medium [21]. The time-reversed symmetry for two other classical nonlinear processes—second-harmonic generation (SHG) and optical parametric amplification is considered and shown later by Y. Zheng et al., which has been utilized to observe the annihilation of coherent beams. The condition for time-reversed symmetry in an undepleted-pump scheme is obtained by adding a $\pi $ phase shift between two identical thin beta barium borate crystals and flipping the sign of phase-mismatching vector [22].

The above nonlinear optical processes have showed that using appropriate photonic structures is a good way to exactly reverse the energy conversion direction and phase variation of light waves. However, the previous approaches are mainly implemented based on the undepleted-pump scheme, which neglects pump depletion effects. In this article, a more comprehensive analysis for the nonlinear time-reversed SHG process with pump depletion has been conducted. It is found that the energy flow can be controlled by a criterion which is essential for the study of the reciprocal response. Based on this criterion, an integrated cascaded structure has been proposed that two sections of one-dimensional (1D) nonlinear photonic crystals (NPC) with equal length are linked by an embedded defect. Two kinds of NPC structures with different symmetries in the reciprocal space are presented to realize the nonlinear time reversal effect. By solving the nonlinear coupled equations analytically, it is found that that the nonlinear time reversal can be realized together with a completely reciprocal response for the SHG process as long as an appropriate phase shift is induced, which is verified by numerical simulations as well. The two-section-cascaded structure can also be extended to the multi-section-cascaded structures for the nonlinear time reversal at any given position.

## 2. Theoretical model

The schematic of the cascaded NPC structure to generate time-reversed SHG is shown in Fig. 1, where the up and down arrows represent the domain poling directions. Theoretically, the whole structure can be treated as three sections, where the first and third sections have the equal crystal length $L$ and the second section is the embedded defect of $\delta L$ in length. In this case, a phase shift between the fundamental wave (FW) and the second-harmonic wave (SHW) will be induced by the structure defect, thus allowing a more accurate and flexible time-reversed SHG process. Here in general, the time-reversed replication can be revealed by studying the SHG processes of the first and third sections separately, which can be treated analytically with the nonlinear coupled equations in a depleted-pump scheme [23].

The nonlinear coupled equations in the NPC can be written as:

For the first section of the NPC (0~$L$), we use the effective phase mismatch $\Delta k=\Delta {k}_{1}-G$ to describe the generalized phase mismatches involving the reciprocal vector $G$ provided by the periodical domain structures, which greatly influences the energy flow oscillation. The effective value of $K$ can be expressed as $K={K}_{1}\cdot {g}_{G}$, where ${g}_{G}$ is the Fourier coefficient of the reciprocal vector. Thus Eq. (1) can be simplified as:

For the second section ($L$~$L+\delta L$), a phase shift (denoted as $\Delta \phi $) is induced between the FW and SHW, which is proportional to the length of the defect. Strictly speaking, the wave mixing in this defect area will pose an impact on the overall reversal symmetry, but it is negligible in our situation since the length of the defect is rather short compared to the whole structure.

For the third section ($L+\delta L$~$2L+\delta L$), the reciprocal vector ${G}^{\text{'}}$ is adopted, thus the nonlinear coupled equations are similar with Eq. (2) except the phase mismatch $\Delta {k}^{\text{'}}=\Delta {k}_{1}-{G}^{\text{'}}$ and the corresponding coupling coefficient ${K}^{\text{'}}$, which can be expressed as:

Here ${K}^{\text{'}}={K}_{1}\cdot {g}_{{G}^{\text{'}}}\cdot \mathrm{exp}\left(i\Delta \phi \right)$ represents the effective coupling coefficient considering the reciprocal vector and the phase shift induced by the second section. If we concentrate on the variation of amplitudes and phases respectively, ${A}_{1}$,${A}_{2}$ will have their specific expressions:

Different from the usual analysis by elliptic function solution [24], here we employ a simpler method for the designing the NPC structure considering the pump depletion [25]. To achieve a time-reversed counterpart, the intensities of FW and SHW should be back converted in the third section in the same rate with that in the first section. Thus the derivative of wave intensity rather than the intensity itself is concerned. This means a simple criterion should be held in the whole SHG process, that is, the derivative of SHW intensity should be opposite at the symmetrical position on both sides of the axial plane and so is the derivative of FW intensity, which can be described as:

According to the energy conservation law, when Eq. (5) is satisfied the time-reversed requirement for FW will be satisfied automatically. Thus only the variation of SHW is needed to take into consideration. Here, $d$ represents the distance away from the axial plane and $d{I}_{2}/dx$ is defined as:

The detailed expressions of $\frac{d{A}_{2}}{dx}$ and its complex conjugation at the positions of $x=L-d$ and $x=L+d$ in different sections can be obtained from Eqs. (2) and (3), which can be expressed as:

Substituting Eq. (4), Eq. (7) and Eq. (8) into Eq. (5), it can be obtained as follows:

Appropriate values of phase shift $\Delta \phi $ and phase mismatches ($\Delta k$ and $\Delta {k}^{\text{'}}$) will make it possible that Eq. (9) always holds true no matter where the sample location is, thus the time-reversed SHG process is established.

There are two cases worth of serious consideration. One of the conditions is that the phase mismatches are in even symmetry: $\Delta {k}^{\text{'}}=\Delta k$. In this situation, the needed phase shift induced by the defect can be solved from Eq. (8) as:

Here ${\varphi}_{1}\left(L\right)$ and ${\varphi}_{2}\left(L\right)$ are the corresponding phases of the FW and SHW at the specific position $x=L$, which can be calculated by the analytical solutions of field distributions in the SHG process [24, 26].

The other structure condition is that the phase-mismatches in the first and third sections are in odd symmetry, that is $\Delta {k}^{\text{'}}=-\Delta k$. Under this circumstance, the needed phase shift is calculated as:

Both the two structure cases that the phase mismatches between the FW and the SHW are in odd and even symmetries can bring out a time-reversed counterpart without the undepleted pump approximation as long as the explicit phase shift is induced. When using undepleted pump approach, the needed phase shift is a constant $\pi $ [22], which can be considered as an approximation of Eq. (11). Thus the model with depleted pump will provide more possibilities to achieve a time-reversed SHG process.

## 3. Numerical results and discussions

Up to now, the possible conditions for the time-reversed and reciprocal SHG process in the 1D NPC structure with a defect have been obtained. We have also conducted numerical calculations to verify these analytical solutions. Lithium niobate is set for all the practical simulations under the room temperature in a defective NPC, the nonlinear coefficient ${d}_{33}=27pm/V$ and the numerical calculation mesh size is 1$\mu m$.

Figure 2 are the simulations of the propagating processes of FW and SHW under the two different phase mismatching conditions. The SHG process in the first section of the NPC structure is time-reversed in the third section due to the appropriate phase shift added in the second section. Figure 2(a) shows that time-reversal symmetry is obtained when the phase mismatches remain unchanged. The wavelength of FW is set to be 1064nm. The two cascaded crystal length is 5.5 mm respectively and the corresponding period is 6.7$\mu m$. The thickness of the defect is about 2.5$\mu m$. Figure 2(b) is simulated under the similar crystal parameters by flipping the sign of phase-mismatching vector. Phase shifts needed at the central axis are determined by Eqs. (10) and (11), respectively. For a spontaneously oscillating SHW (starting from zero) in the first section, its time-reversed process exhibits perfect SH annihilation in the third section when neglecting the wave mixing in the defect area. This kind of structure can be used as a dispersion compensation device in which optical pulse distortion due to dispersion can be reshaped [27].

In fact, not only the time-reversed symmetry but the reciprocal response for the SHG process is also worth noting in this cascaded structure with pump depletion. The reciprocal response focuses more on the FW/SHW outputs rather than the propagating process, thus the pump depletion is the essential condition so that the intensity of the pump waves is considered to change along with the propagating distance. Figure 3 exhibits the SHW output distribution after the NPC with $\Delta kL/2$ and $\Delta {k}^{\text{'}}L/2$ being controlled in the range of $\left(-\pi ,\pi \right)$. The initial input SHW intensity is zero. Figure 3(a) is obtained by numerical calculation when the phase shift induced by the defect is $\Delta \phi =2\left[\Delta kL+2{\varphi}_{1}\left(L\right)-{\varphi}_{2}\left(L\right)\right]$. It is found from the white dash dots that the output intensity will return to zero only when $\Delta {k}^{\text{'}}=\Delta k$. Figure 3(b) is similarly obtained when the phase shift induced is $\Delta \phi =\pi -2\Delta kL$ and the output SHW is back to zero when $\Delta {k}^{\text{'}}=-\Delta k$ is satisfied. Numerical Simulations have demonstrated that the output of the SHW propagating in the backward direction is the same with the one in the forward direction, which means the reciprocal response can be established under the above conditions to achieve time-reversed SHG as well. The reciprocity of the device has potential applications in the all-optical computing area, such as all-optical diodes, all-optical isolators and so on.

Actually, time-reversed symmetry can be realized at any given position by design of a multi-section-cascaded defective structure. Figure 4(a) shows the time-reversed SHG process in a multi-section structure being composed of four cascaded periodic structures and three defects, where the positions of the defects are $\frac{1}{10}{L}_{T}$,$\frac{3}{10}{L}_{T}$,$\frac{3}{5}{L}_{T}$ (described by the orange dash dots) and ${L}_{T}$ is the total length of the integrated NPC. In this situation, the phase mismatches are the same in different structure sections; The required phase shift induced by the defect is related to the length of the corresponding section and the phases of FW and SHW at the position of the defect. Similarly, the positions to realize time-reversed symmetry in Fig. 4(b) are $\frac{1}{10}{L}_{T}$,$\frac{1}{2}{L}_{T}$,$\frac{7}{10}{L}_{T}$, respectively. The sign of the phase mismatches is flipped every other section and the phase shift induced is determined only by the length of the corresponding section and the value of phase mismatches. However, the reciprocal response is no longer in existence in this multi- section-cascaded structure because of the random length of each section. The phase shift needed at the specific position in the backward propagating process may be different from the forward one, which is hardly to unify in the same structure.

## 4. Conclusion

In summary, analytical solutions and numerical calculations have demonstrated that the time-reversed and completely reciprocal SHG process can be realized in the 1D NPC structures. Starting from the nonlinear coupled equations with pump depletion we have proposed a simple criterion for energy flow control. It has been theoretically verified that there always exists a time-reversed symmetry and a reciprocal response for the SHG process as long as an appropriate phase shift is induced at the middle position of a symmetrical structure. Two phase mismatching conditions between the FW and the SHW are separately discussed in detail. In addition, the multi- section-cascaded NPC structures can also be designed for the nonlinear time reversal at any given position. We believe our discoveries may offer new avenues for flexible control in nonlinear time-reversed or reciprocal processes and lead to a variety of relevant applications.

## Funding

National Key R&D Program of China (2017YFA0303700); National Natural Science Foundation of China (NSFC) (Grant Nos. 11774165, 11504166, 11374150); Natural Science Foundation of Jiangsu Province (Grant No. BK20150563); The Priority Academic Program Development of Jiangsu Higher Education Institutions of China (PAPD).

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