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Abstract
Gain and loss balanced parity-time (PT) inversion symmetry has been achieved across multiple platforms including acoustics, electronics, and photonics. Tunable subwavelength asymmetric transmission based on PT symmetry breaking has attracted great interest. However, due to the diffraction limit, the geometric size of an optical PT symmetric system is much larger than the resonant wavelength, which limits the device miniaturization. Here, we theoretically studied a subwavelength optical PT symmetry breaking nanocircuit based on the similarity between a plasmonic system and an RLC circuit. Firstly, the asymmetric coupling of an input signal is observed by varying the coupling strength and gain-loss ratio between the nanocircuits. Furthermore, a subwavelength modulator is proposed by modulating the gain of the amplified nanocircuit. Notably, the modulation effect near the exceptional point is remarkable. Finally, we introduce a four-level atomic model modified by the Pauli exclusion principle to simulate the nonlinear dynamics of a PT symmetry broken laser. The asymmetric emission of a coherent laser is realized by full-wave simulation with a contrast of about 50. This subwavelength optical nanocircuit with broken PT symmetry is of great significance for realizing directional guided light, modulator and asymmetric-emission laser at subwavelength scales.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In a closed quantum system, quantum mechanics requires that all physical observables must be represented by Hermitian in Hilbert space, and the system energy eigenvalues are real. However, in many real situations, the quantum system can exchange energy with its environment to become a non-Hermitian system. In 1998, Bender and Boettcher theoretically proved that if PT symmetry is satisfied in a non-Hermitian system, i.e. $[{H\; PT} ]= 0$, the system can also have real eigenvalues [1,2], which requires the complex potential function in the Schrödinger equation to satisfy $V(\textrm{x} )= {V^\ast }({ - x} )$. Interestingly, in 2007, El-Ganainy and his colleagues utilized the mathematical equivalence between the single-particle Schrödinger equation and the paraxial approximate electromagnetic wave equation to replace the complex potential function with the complex refractive index, i.e. $n(\textrm{x} )= {n^\ast }({ - x} )$, and introduced the PT symmetry to optical system [3]. This leads to intensive research on the realization of PT symmetry on optoelectronic platforms. At present, optical PT symmetry has been realized in dielectric waveguides [4–8], photonic crystals [9–11], whispering gallery microcavities [12–20], atoms [21], metasurfaces [22–33], plasmonic systems [33–35], plasmon-exciton [36] and other structures. It has led to many interesting phenomena, such as unidirectional invisibility [7,22,33], coherent perfect absorbers [29–31], PT symmetry-induced bound state in the continuum [37], non-reciprocal light transmission [5,6,12,13], EP enhanced sensing [17–19,35], single-mode lasing [14–16], chiral light field [24–28], negative refraction [32], etc.
The PT symmetry RLC circuit based on Kirchhoff's equation theory has also been studied and verified [38–41], where subwavelength asymmetric transmission is demonstrated. However, there is a fundamental difference between electronic systems and optical systems: the element size of electronic systems is generally much smaller than the resonant wavelength, while that of optical system is just the opposite due to the diffraction limit, which limits the development of device miniaturization. Therefore, to achieve tunable subwavelength PT symmetry in optical system is extremely challenging. In 2005, Salandrino, Engheta and Alù proposed the similarity of plasmonic system to RLC circuit [42]. This result was experimentally verified later by electron-beam lithography (EBL) and nano manipulation [43,44]. Therefore, with the continuous development of nanofabrication technology, metal nanoparticles with gain-loss balance are an important platform for realizing subwavelength PT symmetric systems. We also note that S. Sanders and A. Manjavacas have found that the asymmetric scattering can be generated by the interaction between the artificial Lorentzian-gain and -loss nanoparticles but without including quantum effects [45].
In this work, based on the similarity between metal plasmonic system and RLC circuit, we include negative and positive resistance into the classical coupled mode theory of RLC resonance circuit to theoretically construct a PT symmetry optical nanocircuit. The nanorods structure has the following advantages: (1) Since the field is mainly concentrated at the end of the nanorod, the radiation loss is small; and it is conducive to the near-field coupling of the two localized surface plasmon resonance modes (LSPRs), which leads to a large coupling strength. The hybrid bonding and anti-bonding modes of the nanorods are beyond the Rayleigh criterion, making it suitable for the design of subwavelength PT symmetric structures. (2) Compared with nanospheres, nanorods are more conducive to couple with waveguides and to integrate with other devices. As the coupling strength and gain-loss ratio are tuned, the system undergoes from PT symmetry phase to PT symmetry broken phase, characterized by the far-field scattering spectrum and near-field strength distribution. The direction of signal coupling can be changed from bidirectional equal-intensity to asymmetric or even quasi unidirectional by modifying the gap between the two nanocircuits. In addition, based on the optical nanocircuit model, an optical modulator is proposed by gradually increasing the pump power with fixed coupling strength. Significantly, the signal intensity increases nonlinearly near the EP, resulting in a larger modulation depth. Finally, we introduce a four-level atomic model to simulate the nonlinear dynamics of the laser. When the pump power is further increased to the laser threshold, asymmetric single mode laser emission can be realized with a contrast near 50. The optical PT nanocircuit proposed here provides an excellent platform for realizing tunable subwavelength PT symmetry devices, which may find applications in integration optics and photonics.
2. Theory and model design
PT symmetric coupled RLC circuit model with amplification and attenuation is shown in Fig. 1(a). The two independent circuits are composed of the same resistor ${R_0}$, capacitor C and inductor L. The amplification circuit has an extra negative resistive element ${R_1}$. The resonance frequency of each individual nanocircuit is ${\omega _0} = 1/\sqrt {LC} $, and the two nanocircuits are coupled to each other. According to the similarity between plasmonic structure and RLC circuit: a non-metallic material, metal material (such as noble metals gold and silver) and gain material can be regarded as an effective positive “inductor”, effective positive “capacitor” and effective negative “resistor” in equivalent circuit elements respectively. The plasmon subwavelength PT symmetry nanocircuit based on this similarity is shown in Fig. 1(b). The Ag nanorods and two waveguides for far-field observation are placed on the Ag substrate. The optical parameter of silver is taken from Palik (0-2 μm) [46]. The gain nanorods with a length of ${l_1} = $ 180 nm and a radius of ${r_1} = \; $40 nm is covered with a layer of gain material (introduced by Lorentz function) with a thickness of $h = $ 5 nm. The center frequency of gain material is set to be the same as the resonance of the nanorod, and the imaginary part of the dielectric constant of the gain material is within the range reported previously [37,47]. In order to make the system PT symmetric, the radius of the loss nanorod is set to ${r_2} = \; $42.5 nm, and other parameters are consistent with the gain nanorod. The length of Ag waveguides at both ends is ${l_3} = {l_4} = 4$ µm, and the gap between the nanorod and the waveguide is set to ${s_3} = {s_4} = \; $30 nm. The impedance of a nanoparticle can be expressed as [48]:
Fig. 1. Design of PT symmetry subwavelength structure. (a) The active PT symmetry coupling RLC circuit model. (b) Schematic diagram of the plasmonic subwavelength PT symmetry nanocircuit corresponding to PT symmetry RLC circuit model. The Ag nanorods with a certain gap and two waveguides are placed on the Ag substrate, and one of the Ag nanorods is covered with gain material. The parameters are ${l_1} = {l_2} = 180\; $nm, $\textrm{}{l_3} = {l_4} = 4$ µm, ${r_1} = 40\; $nm, ${r_2} = 42.5$ nm, h = 5 nm, ${s_1} = {s_2} = 30\; $nm, ${s_0}$ is tuned from 30 nm to 140 nm. The polarization direction of signal light is along the x direction, and the gain material is pumped by another light. (c) The analytical solutions of the real and imaginary parts of the resonance frequency vary with the gain, where $\delta = {R_1}/2Lg$, black dot is EP, and the red dot indicates the laser threshold point of the amplification mode. (d) The black line and red line represent the scattering spectra of a single nanorod and two nanorods with a gap of 30 nm, respectively. The right panel shows the electric field distribution of bonding mode (red star) and anti-bonding mode (red triangle).
Therefore, $R/L$ represents the gain and loss of the system which has the dimension of frequency. A 2 × 2 matrix is proposed to describe the coupled non-Hermitian circuit:
The evolution of this analytical solution with varying gain is shown in Fig. 1(c), where $\delta = {R_1}/2Lg$, the optical nanocircuit is in the PT symmetry phase with two eigenvalues of different real parts and the same imaginary parts. Both the eigenvalues and the eigenvectors of the optical nanocircuits degenerate at EP (${R_1} = 2Lg$), which is different from the traditional diabolic points (DP) where only the eigenvalues degenerate [49]. The system is in the PT symmetry broken phase when ${R_1} > 2Lg$, the signal amplifier and signal absorber can be realized at the two ports of the coupling circuit due to the increase and decrease of the imaginary part of the eigenvalues.
Next, the subwavelength optical nanocircuit are characterized by full-wave simulation using finite difference time domain (FDTD) method. Single Ag nanorod is excited by a TM wave (along the x-direction) of a full-field scattering light source (signal light), and the first-order and third-order LSPRs can be observed in the scattering spectrum of the black solid line in Fig. 1(d) (mode analysis can be found in Supplement 1 Section S1). Here, only the first-order LSPR is considered. The resonance mode of the optical nanocircuit can be tuned in a wide range by modifying the length and radius of Ag nanorod (see more details in Supplement 1 Section S2). Strong coupling occurs when two optical nanocircuits are separated by an appropriate gap. The first-order LSPR of the nanocircuits is split into a bonding mode (low-frequency branch) and an anti-bonding mode (high-frequency branch), as shown by the red line in Fig. 1(d). Notice that to excite the anti-bonding mode, the light source is also asymmetric, but it is set to symmetric in the PT symmetry-broken phase (see more details in Supplement 1 Section Materials and Methods). The electric field of bonding mode is mainly concentrated between the two nanorods, while that of the anti-bonding mode is mainly distributed outside of the two nanorods as shown in the right panel of Fig. 1(d). In addition, the coupling strength decays exponentially with the increase of the gap between the nanocircuits (see more details in Supplement 1 Section S3). The magnitude of the gain is determined by the pump power, which also red-shifts the resonant frequency of the gain rod and breaks the EP. In order to make the system PT symmetric and heal the EP [50] (resonant at ω0 = 447.113 THz without detuning), the radius of the loss nanorod is set to 42.5 nm (Supplement 1 Section S4).
3. Results and analysis
3.1 Asymmetric signal coupling
By changing the coupling strength, the process of PT symmetry breaking is characterized and the modulation of signal propagation direction is demonstrated. The scattering spectra of the two nanocircuits at different coupling strength are shown in Fig. 2(a). The bonding and anti-bonding modes gradually degenerate as the coupling strength decreases. When the gap is set to 85 nm ($g = {R_1}/2L$= 9.36 THz), only one degenerate resonant mode can be seen in Fig. 2(a) marked by the red diamond, which is EP. After EP ($g < $9.36 THz), the system also has only one resonant frequency, which is actually the superposition of two modes with different losses. Therefore, the two output ports of the nanocircuit in Fig. 2(b) act as the signal amplifier and attenuator respectively. In Fig. 2(c), the blue and red solid curves show the analytical calculation of the real and imaginary parts of the resonance frequency of the PT system with different coupling strength, respectively. The full-wave simulation results (solid circle) are in good agreement with the analytical results. In the PT symmetry broken phase, because the real parts of the two broken modes are the same, it is difficult to separate the two modes in the scattering spectrum, so the imaginary part cannot be obtained by direct fitting. To demonstrate the process of PT symmetry breaking directly, the near-field electric and magnetic field distributions at EP (Fig. 2(a) red diamond) and PT symmetry broken phase (Fig. 2(a) yellow and green diamonds) are shown in Supplement 1 Section S5.
Fig. 2. Asymmetric signal coupling. (a) The scattering spectra at different gap when the gain is fixed. The system is at EP and the mode coalesces when the gap is 85 nm. (b) A coupled RLC circuit model with amplification and attenuation output ports. (c) The analytical calculation results of the real (blue) and imaginary (red) parts of the system resonance frequency with varying g, where $1/\delta = 2Lg/{R_1}$. The blue circle and red circle represent the simulation results of the real and imaginary parts of the resonance frequency, respectively. (d) The light intensity distributions and contours of the bonding mode (red star), anti-bonding mode (red triangle), and in the PT symmetry broken phase (green diamond).
In order to demonstrate the evolution of the asymmetric system in the far field, two waveguides are introduced at the end of each nanorod. The resonant frequency of the system red-shifts to 412 THz due to the introduction of waveguide. After that the light intensity distribution of x-y plane of bonding mode (red star in Fig. 2(a), anti-bonding mode (red triangle in Fig. 2(a) and PT symmetry broken phase (green diamond in Fig. 2(a) respectively are show in Fig. 2(d), the detailed information of contours can be found in Supplement 1 Section S6. Obviously, the light is transmitted to both ends with equal intensity if the system is PT symmetric. When the system is in the PT symmetry broken phase, one mode is amplified and the other mode is attenuated with the decrease of the coupling strength (the ratio of light transmission to the left and right is about 12:1), thus realizing asymmetric light coupling. In addition, the cross section of the light intensity distribution at the end of the gain and loss waveguide are shown in Supplement 1 Section S5, which also shows the process of PT symmetry breaking and asymmetric coupling.
3.2 Subwavelength modulator
According to Eq. (4), the PT symmetry breaking process can also be realized by tuning the gain/loss ratio, which is more feasible for many potential applications, such as modulator and unidirectional laser. In this section we discuss the possibility of using the above system as a subwavelength modulator.
We first analytically calculate the frequency modulation with different gain and coupling strength. The result is shown in Supplement 1 Section S7. The frequency modulation becomes more significant with increased coupling strength. Here, the coupling strength between the loss and gain nanocircuits is fixed at 16.24 THz. The scattering spectrum of the system under different gain is shown in Fig. 3(a) (more details can be found in the Materials and Methods of Supplement 1). Since the resonance frequency of a single nanorod red-shifts when the gain is increased (see more details in Supplement 1 Section S7), the system will be detuned in part of the parameter space, therefore, we use the three-dimensional diagram shown in Fig. 3(b) to describe its actual path in the parameter space precisely. By carefully optimizing the parameters, the detuning of the system near EP can be set to zero, at the cost of a small detuning of the two nanocircuits in the region away from EP. In Fig. 3(b), the black line and white line represent the paths of the real part of the eigenfrequency of the system with increasing gain under the ideal situation (Δ=0) and the actual situation (Δ≠0), and the white circle represents the corresponding simulation result. Our system enables greater frequency modulation because the coupling strength between plasmonic nanoparticles is much larger compared to many dielectric microcavities such as whispering galley microcavities. What’s more, we show the field electric and magnetic field distributions near the EP (red diamond in Fig. 3(a) and PT symmetry broken phase (yellow and green diamond in Fig. 3(a) (see more details in Supplement 1 Section S8).
Fig. 3. Subwavelength modulator. (a) The evolution of the scattering spectra with increasing gain. $g$=16.24 THz. (b) 3D analytic diagram of the real part of the eigenfrequency of the system. The black line and white line represent the paths of the real part of the eigenfrequency of the system with increasing gain under the ideal situation (Δ=0) and the actual situation (Δ≠0 due to Kramers-Kronig relations), and the white circle represents the corresponding simulation result. (c) The super exponential growth of scattering intensity at the center frequency (red dotted line in Fig. 3(a) varies with the gain, it can be fitted with two exponential functions. The intensity modulation depth of the system near EP is about 4.5 times than that of PT symmetry phase. (d) The light intensity distributions and contours of the bonding mode (red star), anti-bonding mode (red triangle), and in the PT symmetry broken phase (green diamond).
Especially, we find that the signal intensity near EP has a nonlinear modulation effect by monitoring scattering intensity at the center frequency (red dotted line in Fig. 3(a) under different gain, as shown in Fig. 3(c). That is to say, the relationship between scattering intensity and gain is super exponential. The data before EP can be fitted with: ${I_1} = $ 0.0343exp(16.67$\varepsilon$′′ + 0.16) + 0.274 (Fig. 3(c) black line); the data after EP can be fitted with: ${I_2} = $ 0.082exp(17.54$\varepsilon$′′) + 0.16 (Fig. 3(c) red line), where ε′′ is the imaginary part of dielectric constant of the gain material. The modulation depth (defined as ($({I_{\textrm{max}}} - {I_{\textrm{min}}})/({I_{\textrm{max}}} + {I_{\textrm{min}}}))$) of the system near the EP is about 4.5 times that of PT symmetric phase (unbroken, see more details in Supplement 1 Section S7). This result reveals the unique property of the PT symmetry modulator, which may be used to improve the performance of modulator in principle in the future.
In Fig. 3(d), we further show the light intensity distribution of bonding mode (red star in Fig. 3(a)), anti-bonding mode (red triangle in Fig. 3(a)) and PT symmetry broken phase (green diamond in Fig. 3(a)) in the x-y plane, the detailed information of contours can be found in Supplement 1 Section S9. Correspondingly, the cross section of the light intensity distribution at both the gain and loss ends of the optical nanocircuit waveguide are shown in Supplement 1 Section S8. Apparently, the modulator can also guide the light asymmetrically. The maximal contrast in Fig. 3 is 20:1. Therefore, the modal structure changes significantly before and after EP, for signal with a fixed frequency, the intensity can also be greatly modulated by changing the pump power on the gain material. Using this unique nature at EP, it is possible to improve the performance of the modulator.
3.3 Tunable subwavelength asymmetric-emission laser
According to the theoretical analysis, single mode lasing can be realized when the gain of amplification mode is further increased and completely compensates its loss in the PT symmetry broken phase. The total loss of the amplification mode is ${R_0}/L - {R_1}/2L - \sqrt {R_1^2/4{L^2} - {g^2}} $, the gain can completely compensate the loss if ${R_{1th}}/L = {R_0}/L + {g^2}L/{R_0}$, which is the laser threshold. Obviously, when coming close to the laser threshold, which is a nonlinear process, the simple linear gain used in section 3.1-3.2 does not hold anymore. For example, it cannot describe the gain saturation.
In the following, we introduce a four-level atomic model (inset in Fig. 4(a) as the gain to simulate the asymmetric response of the PT system laser dynamic by FDTD [50]. Here ${N_i}$ is the electron population density probability in level i and ${\tau _{ij}}$ is the decay time constant between levels i and j. Levels 1 and 2 are the lasing levels, correspond to dipole ${P_a}$. Levels 0 and 3 are the pumping levels, correspond to dipole ${P_b}$. The detailed gain material parameters used in our simulation are [51]: the pump light wavelength is ${\lambda _a} = $ 500 nm, the laser wavelength (cavity mode) is ${\lambda _b} = $ 652.6 nm, with damping coefficients ${\gamma _a} = {\gamma _b} = 1.0 \times {10^{13}}$; the lifetimes of the levels are ${\tau _{30}} = {\tau _{21}} = 300\; \textrm{ps};$ ${\tau _{32}} = {\tau _{10}} = 100\,\textrm{fs};$ the initial population density is ${N_0} = {N_1} = 5 \times {10^{23}}$. This parameter corresponds to many semiconductors gain materials, which have been used in multiple work [52–54]. The simulation is carried out by including the coupled rate equation and considering Pauli exclusion principle [51]. The calculated time evolution of the populations of the four levels and the inverted population density ($\Delta N = \; {N_2}\; $- ${N_1}$) is shown in Fig. 4(a) with the pump amplitude of 2.5×${10^6}\,\textrm{V}/$m. After the onset of pumping, the population of level 1 starts to decrease, and that of level 2 starts to increase. They tend to be stable after a short period of oscillation and then generate stable laser. Here, because the plasmonic system has a significant loss, more population inversion $\Delta N$ is needed to maintain laser emission. The time domain results of the pump and laser output signals (obtained by integration of electric field strength) at the gain and loss ends are shown in Fig. 4(b). At 7 ps, both gain and loss end start to generate spike pulse and then stabilize quickly, and the laser output signal at the gain end is much larger than that at the loss end.
To further obtain the laser threshold, we plot the laser output intensity (peak intensity) and its contrast intensity at the gain and loss ends under different pump light amplitudes in Fig. 4(c). The laser threshold is achieved when pump amplitude is 1.85×${10^6}$ V/m, corresponding to the intensity of 4.54×${10^5}\,\textrm{W}/\textrm{c}{\textrm{m}^2}$, which can be achieved by experiment. In addition, the laser emission spectra at the gain end under different pump amplitudes are shown in Fig. 4(d). The variation of the full width at half-maximum (FWHM) with the pump amplitude is shown in Supplement 1 Section S10. To further characterize the asymmetric emission, the emitted laser intensities at both ends are compared, and the emitted intensity at the gain end is about 47 times that at the loss end when the laser reaches stability (maximal contrast is 53 right after threshold). Obviously, a tunable highly asymmetric laser is realized through the analysis in both time and frequency domain. Here, because the shape of the laser emission spectrum at the loss end is the same as that at the gain end, and the temporal behaviors of the laser output signals at both ends are the same, therefore we think that the laser at the loss end also comes from the gain nanorod by, e.g., the guiding of the silver substrate. However, in actual situations, surface effects may increase the loss and the laser threshold. One biggest issue is the flatness of the surface. Firstly, single crystal silver plates as substrates have been reported in literatures, which have atomic-level surface smoothness. Secondly, the structure can be fabricated by EBL technology and thermal deposition (it can effectively solve the alignment problem of nanorods), or by focused ion beam (FIB) on commercial single crystal or twin crystal nanowire and nanorods. The surface can be further optimized by methods such as chemical polishing. Finally, the fluorescence of the gain material could be quenched by the metal in practice, and it can be solved by adding a 2-5 nm dielectric spacer between the silver nanorod and the gain material [55,56]. A possible fabrication process is also proposed (Supplement 1 Section S11).
Fig. 4. Subwavelength asymmetric-emission laser. (a) The temporal evolution results of the populations of the four levels and the inverted population density ($\Delta N = {N_2}\; $- ${N_1}$) when the pump amplitude is 2.5×${10^6}\textrm{}$ V$/$m. The inset is the four-level atomic model. (b) The time domain results of pumping and lasing output signals intensity (integration of electric field strength) of the gain and loss ends. (c) The laser output intensity (peak intensity) and its contrast intensity at the gain and loss ends under different pump amplitudes. The laser threshold is 4.54×${10^5}\,\textrm{W}/\textrm{c}{\textrm{m}^2}$, and the laser intensity at the gain end is about 47 times that at the loss end when the laser reaches stability (maximum contrast is 53 right after threshold). (d) The laser emission spectrum at the gain end under different pump amplitudes.
4. Conclusion
In summary, based on the similarity between plasmon and RLC circuit, we introduced negative and positive resistance into the classical coupled mode theory of RLC resonance circuit to construct PT symmetry optical nanocircuit theoretically. The PT symmetry broken phase transition is demonstrated on the subwavelength plasmon platform through full-wave simulation. By tuning the coupling strength between the nanocircuits, we observed the asymmetric signal coupling. Next, a subwavelength modulator was proposed by tuning the gain of the nano resonator. Notably, the modulation effect near the EP is super exponential and also asymmetric. Finally, a four-level atomic model modified by Pauli exclusion principle was introduced to simulate the asymmetric response of the PT system laser dynamic. The highly asymmetric laser emission is realized. The results of this work provide a valuable method for the design of devices that actively control optical transmission on the nanoscale. The tunable subwavelength optical PT nanocircuit is of great significance for the miniaturization and integration of on-chip PT devices, and paves the way to integrated circuit working at optical frequency.
Funding
National Natural Science Foundation of China (12174031, 91950108, 11774035).
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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