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Abstract
It is known that phase conjugation technique is achieved by the nonlinearity of materials, and widely adopted in various applications, such as high-resolution imaging, signal amplification, and target detecting. Here, we have proposed a field-enhancement method for the degenerate four wave mixing (FWM) for phase conjugation purpose. In this method, a thin film waveguide with nonlinear property is utilized to confine and guide the pumps, achieving the enhanced FWM within a flexible structure. Compared to existing degenerate FWM methods, three merits are introduced by the proposed pump-guided nonlinear film. First, the pump is confined and guided in the nonlinear waveguide, and the pump energy is concentrated to achieve high power level of the phase-conjugated signals. Second, less pump energy leaks out from the thin film, with less interference to the phase-conjugated signals. The last one is that pump-guided film can be engineered into flexible shapes for different practical applications. Based on these advantages, the phase conjugation property is numerically verified for high-resolution image reconstruction, even with damping of waveguide or in the presence of the metallic particles and the dielectric blocks.
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1. Introduction
In recent year, the development of phase conjugation opens up an exciting field for light-matter interaction in nonlinear optics. Phase conjugation utilizes the nonlinear susceptibility of materials to general a “time reversal” wave with a correlated wavefront, by reversing the phase of the original wave [1–3]. In this process, when a propagating wave from a source impinges on a nonlinear medium, a back-propagating wave with complex conjugated phase is produced and retraced to the original source, forming a reconstructed source image [4]. Phase conjugation technology is widely studied and adopted in numerous literatures, including the field of high-resolution imaging [4–11], signal amplification [12], target detecting and positioning [13], optical frequency mixer [14], beam splitter [15], and so on. The image resolution can be enhanced by retracing its trajectory using a “phase conjugator”, mitigating the scatter and reflection in the propagating path, and widely adopted in the bio and plasmonic imaging [4–6]. Another heuristic to increase the image resolution is to design negative reflection using time reversal signals as “perfect lens”. Subwavelength imaging is achieved by using phase-conjugating slab or antenna arrays to overcome the diffraction limit [10,11,16].
It is known that the process of degenerate four wave mixing (FWM) is extensively utilized in the phase conjugation applications [17,18]. In the setup of FWM, the nonlinear material is excited by a monochromatic input wave with frequency ω, and two counter propagating pumps with kp1 + kp2 = 0 and operating at the same frequency ω. As a result, a phase conjugated wave is generated with the same frequency ω and index matching. The idea of degenerated FWM was first proposed by Hellwarth [17], and utilized to amplify the incoming wave by oscillating. The first observations of FWM phase conjugation were reported in [18], using the nonlinear bulk medium of CS2. There are two key study directions for the phase conjugation using degenerate FWM: (1) Thin surface-structured materials are preferred instead of bulky materials for the advantages of easy fabricating. (2) The third order nonlinear susceptibility is usually very small, resulting in the weak reconstructed wave. How to boost the magnitude of phase conjugated wave is a challenge [3]. With the design consideration of magnitude enhanced nonlinear metasurface, a series of feasible solutions are developed in recent years, such as extreme ultraviolet transient gratings [19], graphene metasurface [20–22], negative refractive metallic nanostructures [23–25], photonic integrated cavities [26], quantum-dot nanostructures [27], epsilon-near-zero mode [28], plasmonic metasurface [29], Bessel beam [30], plasmonic nanofocus [31], and space-time modulation [32], just named a few. In this paper, we investigated a new pump-guided nonlinear film for the degenerate FWM process. The nonlinear film waveguide not only provides phase conjugation ability, also confines the pumps without leaky out. Different from traditional guided-pump approaches using regular waveguides or optical fibers [33,34], the pump is concentrated in the film made of nonlinear materials for phase conjugation, to enhance the phase conjugated wave and decrease the interference. The thin film waveguide can be designed with arbitrary shape and surround the source for the largest phase conjugated wave energy. Excellent resolution and power level are achieved for the reconstructed image, even in the presence of the perfect electric conductor (PEC) particle and dielectric with turbid scattering, reflection and refraction.
2. General concept, configuration, and properties
For the degenerate FWM process, we assume that all the waves are linear polarized with electric field along z axis and the 2-D nonlinear material is in x-y plane. The electric fields of signal wave, pump 1 and pump 2 are expressed as following (the time convention is assumed as e-iωt):
where r = xx + yy, ks, kp1 and kp2 are wave vector of the signal wave, pump1 and pump 2, and As, A1 and A2 are the input field magnitude of the signal wave, pump1 and pump 2. The induced nonlinear polarization from the film waveguide is in the form of
Fig. 1. Generic concept of the phase conjugation using pump-guided film. TE1 mode pump propagates inside the film waveguide. The film is engineered in a loop shape. A point source (black solid) is at arbitrary position inside the film loop, and a reconstructive point source (orange dash) is achieved at the same spot as the original point source.
An intuitive method to excite the pump is to cut the waveguide loop, leaving two input ports for two counter propagating pumps, as illustrated in Fig. 2(a). Pump 1 (red color) and Pump 2 (blue color) are excited separately with equal magnitude and wave number of TE1 modes. The two input ports should be isolated with a distance more than one wavelength, avoiding the mutual coupling between two ports. However, in this structure, a gap of waveguide exists. And a part of radiating wave from the original point source propagates without the phase conjugating reflection. The power of reconstructed point source decreases, compared with closed loop of film waveguide. For this consideration, we have merged the two input ports into one, aiming to achieve a closed loop for film, as shown in Fig. 2(b). The pump is excited on the left with TE1 mode. The pump is split into two parts: pump 1 propagating along red arrow, and pump 2 propagating along blue arrow. Pump 1 and pump 2 are mainly concentrated in the film waveguide.
Fig. 2. Generic structures of the pump excitation methods for phase conjugation using pump-guided film. (a) Two pump input ports: pump 1 and pump 2 are excited from two isolated ends of the film waveguide. (b) Single pump input port: the pump is excited on the left end of the film waveguide. The input pump splits into two counter propagating pumps with equal magnitude and wave number. (c) Electric field distribution of the pump in the film waveguide, also with the dimension: w = 0.5 µm and d = 10 µm, the film is positioned in the far field of the point source. (d) Magnitude distribution of the power flow of the pump in the film waveguide.
Here, we use the example with the dimension shown in Fig. 2. The right part is a semi annulus, and the left part is an asymptotic curve, closed on the left side as the input port for pump. The degenerate FWM process is operating at the frequency of 100 THz (wavelength is 3 µm in free space). Some materials have been demonstrated to have third-order nonlinearity at this frequency, such as doped silicon [35], graphene [36], InSb [37], GaSb [38] and GaAs [3]. Without loss of generality, the film waveguide is with the permittivity of εms = 9 and the third order nonlinear susceptibility $\chi^{(3)}=1.4 \times 10^{-18} \cdot \mathrm{m}^2 / \mathrm{V}^2$ (the values of GaAs [3]). It is worth noting that the thickness of the film is only 0.5 µm, relatively thin compared to the wavelength in free space. The input electric field of the pump is TE1 mode, with the magnitude of 107 V/m on the left port, and is split into two components (pump 1 and pump 2) with equal magnitude and wave vector. We used the commercial simulator COMSOL Multiphysics to examine the pump performance in the phase conjugated film waveguide. The model is simulated in the time domain in a time range of [0, 800 fs], with a time step of 0.5 fs. All the field distributions shown in this paper is at the end of the time range (at t = 800 fs), when the simulations are converged and the field distribution is stable. Figure 2(c) shows the z-axis polarized electric fields (Ep1 and Ep2) distribution of the pump in the film waveguide. It can be clearly seen that TE1 mode appears inside the waveguide and evanescent field outside. The magnitude of the power flow of the guided pump is illustrated in Fig. 2(d). The power is uniformly distributed along the film waveguide with the peak intensity of 8 × 1011 W/m2 (realistic intensity for the FWM process [10]), and the intensity of the power flow is nearly zero outside the waveguide. That is to say, almost all the pump energy is concentrated in the film waveguide with little energy leak out as an interference. Therefore, high power level of the reflected phase-conjugated wave is achieved with the nonlinearity enhancement by using the proposed pump-guided film.
The process of phase conjugation often suffers from very poor third-order optical nonlinear effects, which limits their practical applications. A lot of methods for nonlinear enhancement have been proposed as shown in Table. 1 In [24], a hybrid metal-dielectric (BaTiO3-Au) nanodimer is fabricated to realize localized surface plasmon resonance, results in an up to 15-fold enhancement compared to single BaTiO3 nanoparticle. Another method is assisted by Epsilon-Near-Zero (ENZ) metamaterials: the ENZ mode is realized based on plasmonic gratings, efficiently boosting nonlinear effects up to 20 times compared with conventional resonances [28]. Besides, nanoantennas are properly designed with the nonlinear materials for field enhancement. Literature [39] shows that the nonlinear effects with nanoantennas is over 10 times stronger than the case without nanoantennas. In traditional FWM schemes, the pumps are directly impinging on the nonlinear materials, which makes the pump energy “everywhere”. As for our scheme, the pump signal is guided into the nonlinear film with TE1 mode. On the one hand, the energy is concentrated into film waveguide, leading to field enhancement with less energy leaking out. On the other hand, with the transverse resonance of TE1 mode in film waveguide, we can realize a stronger electric intensity for enhanced nonlinear effect. We have investigated a simple example base on the example of Fig. 2(c). Rather than the guided pumps, we use plane-wave pump with identical intensity from upside and downside to excite the nonlinear film, as the traditional FWM scheme. The electric field distribution of guided-pump scheme is much stronger than that of traditional scheme, which indicates an over-100-times field enhancement of the nonlinear effects.
3. Phase conjugation for high-resolution imaging
Next, we investigate the focusing performance of the phase conjugated wave using pump-guided film. A z-axis polarized point source is arbitrarily positioned inside the film loop of Fig. 2. The point source also operates at 100 THz and the input magnitude of electric field is 1 V/m. In Fig. 3(a), we only illustrate the field distribution of the radiated wave from the nonlinear dipole moment inside the pump-guided film, including the waves propagate inside and outside. The reflected phase-conjugated wave is focusing at exact the same spot as the original point source, treated as a reconstructive image. As discussed above, the power level of the phase conjugated wave is determined by the values of the pump magnitude and the third order nonlinear susceptibility. Phase conjugation is with the merit of turbidity suppression in the imaging applications. That is to say, the focusing effect of the phase conjugated wave maintains even in the presence of unexpected scattering, reflection, and refraction. Here, we have investigated two examples: one is with a circular perfect electric conductor (PEC) particle with the radius of 3 µm, and the other is with a cross-shaped dielectric made of glass (εr = 2.4) with the length of 3 µm for each arm. Firstly, in the presence of PEC particle, part of the electric field radiated from the point source are scattered by the PEC particle. However, due to the phase conjugation film waveguide, the reflected phase conjugated wave can retrace its trajectory and focus at the original spot of the point source, as illustrated in Fig. 3(b). The similar phenomenon also appears in the presence of the dielectric with arbitrary shape. Here, we use a cross-shaped glass dielectric near the original point source. Part of the electric field are interacted with glass dielectric, and its propagating trajectory is changed. However, phase conjugated wave refocusing still maintain at the original spot of the point source, as shown in Fig. 3(c). The magnitudes of the reconstructed point source images focused from the phase conjugated wave are also unchanged. The whole process of phase conjugation and signal reconstruction of Fig .3 can be seen in Visualization 1. We have proved that the same results can be observed in three-dimensional nonlinear waveguide. Detailed descriptions and results are presented in Supplement Section 1, Fig. S1 of Supplement 1.
Fig. 3. Electric field distributions of the reconstructed point sources at the time t = 800 fs: (a) is for the original source only, (b) is with PEC particle, (c) is with dielectric of εr = 2.4. The field distributions are only from the induced nonlinear dipole moment in the film. The phase conjugated wave focuses as a reconstructed point source. (d) and (e) show electric energy intensity distributions of the reconstructed spot using phase conjugated film: (d) along x axis, (e) along y axis.
What’s more, the resolution of reconstructed image using the pump-guided film is also studied and discussed. We set a coordinate in the x-y plane, and assume the spot of the origin point source as (0, 0). Figure 3(d) and Fig. 3(e) show the corresponding electric energy density along the cross lines of x axis and y axis. We record the results at the time of t = 800 fs. Three different cases are illustrated in Fig. 3(d) and Fig. 3(e): (1) original point source only (black solid line), (2) with PEC particle (red dashed line), and (3) with dielectric (blue dotted line), consisting with the electric field distributions illustrated in Fig. 3(a), Fig. 3(b) and Fig. 3(c). Subwavelength resolution, e.g., below half wavelength, can be achieved in all the three cases. That is to say, in the presence of PEC particle or dielectric, high resolution is maintained in the focusing image through the phase conjugation process using the proposed pump-guided nonlinear film approach.
As mentioned above, the proposed flexible pump-guided nonlinear film can also be engineered in different shapes, still with the effect of high-resolution imaging. Here, we have investigated three examples with different shapes, which are with the same pump excitation. First is a concave curve as the right part, Fig. 4(a) shows that the pumps are perfectly guided in the film waveguide. Thus, the phase conjugated wave can retrace its trajectory and focus at the original spot of the point source, as illustrated in Fig. 4(b). As to the second shape of semi-hexagon, some pump energy leaks out from the sharp corners, as shown in Fig. 4(c). However, seeing from Fig. 4(d), the phase conjugated wave still refocuses at the original spot of the point source but carries more noise at the sharp corners compared with the smoother shape in the first example. We can replace the sharp corners with rounded edges of polygon shapes. As an example, Fig. 4(e) illustrates a film waveguide of semi-rounded-rectangular shape with the pump guided perfectly in it. And as shown in Fig. 4(f), the phase conjugated wave focuses as a reconstructed point source. We can find that the imaging quality of phase conjugating effect is related to the guided pump. In Fig. 4(c) and Fig. 4(d), affected by the energy leakage at sharp corner, the pump energy is inhomogeneous along the film waveguide, leading to poor imaging accuracy and weak imaging intensity. As for the shapes without sharp corners, shown in Fig. 4(b) and Fig. 4(f), the focusing effect of the phase conjugated wave can realize same imaging resolution and intensity, independent from the shapes of film.
Fig. 4. Electric field distributions of pumps and reconstructed point sources with flexible pump-guided nonlinear film: (a) and (b) are with concave shape, (c) and (d) are with semi-hexagon shape, (e) and (f) are with semi-rounded-rectangle shape. In (a), (c) and (e), the field distributions are only from the guided pumps. In (b), (d) and (f), the field distributions are only from the induced nonlinear dipole moment in the film. The phase conjugated wave focuses as a reconstructed point source.
At last, the impact of damping of film waveguide on the phase conjugation process is illustrated and discussed. Here, we have investigated three examples with different values of damping, which are with the same structures and settings as Fig. 3(a). The damping is represented by the imaginary part of the permittivity εms′′. Thus, the permittivity of nonlinear waveguide is εms=εms′-j*εms′′. Figure 5 shows the electric field results with damping of waveguide. Separately, Fig. 5(a) and Fig. 5(b) are with εms′′=0.01, Fig. 5(c) and Fig. 5(d) are with εms′′=0.02, Fig. 5(e) and Fig. 5(f) are with εms′′=0.1. It can be illustrated that the energy in the waveguide decreases with the increase of damping, and the strength of the phase conjugated wave also weakens at the same time. However, phase conjugated wave still refocuses at the original spot of the point source with the same resolution. To be concluded, the damping of the nonlinear material only effects the amplitude of reconstructed signal instead of the resolution of imaging.
Fig. 5. Electric field distributions of pumps and reconstructed point sources with lossy pump-guided nonlinear film: (a) and (b) are with εms′′=0.01, (c) and (d) are with εms′′=0.02, (e) and (f) are with εms′′=0.1. In (a), (c) and (e), the field distributions are only from the guided pumps. In (b), (d) and (f), the field distributions are only from the induced nonlinear dipole moment in the film. The phase conjugated wave focuses as a reconstructed point source.
4. Conclusion
In this paper, we have investigated the pump-guided nonlinear film for phase conjugation application using degenerate FWM process. The pump is concentrated inside the film waveguide made of nonlinear materials, enhancing the phase-conjugated signal without leakage outside. The film can be designed in arbitrary shapes for different application requirements. High reconstructed resolution is achieved by utilizing the proposed pump-guided film, even with damping of waveguide or in the presence of the metallic particles and the dielectric blocks. Based on our understanding of the phase conjugation application, the proposed concept will pave the way of nonlinear thin film materials, which is easily fabricated and integrated. This configuration also can be connected with the enhancement with intrinsic nonlinear susceptibility of materials through local collective resonance, such as nanoantennas or nanocavities. The proposed pump-guided film based on phase conjugated nonlinear materials offers the possibility to enhance the nonlinearity by engineering the structure of the thin film materials.
Table 1. Several methods for nonlinear enhancement
Funding
National Natural Science Foundation of China (62022045); Tsinghua Initiative Scientific Research Program.
Acknowledgements
The authors thank Prof. Nader Engheta for the helpful discussions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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