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Abstract
Multiple nonlinear effects with second-harmonic (SH) generation and optical bistability (OB) are highly desired but rather rare due to insufficient intrinsic nonlinearity in most nonlinear media. Here, a nonlinear microcavity coupled to a metallic nanoparticle (MNP) is suggested to realize the bistable SH generation. When two counterpropagating driving fields are injected into the cavity, a traditional SH pathway is constructed via the two-photon process of fundamental-harmonic (FH) photons. In addition, we report the coexistence between the SH and OB effects under the condition that the strong excitation and the detuning management for the driving fields boost system nonlinear responses. In the presence of a control field toward the MNP, our calculation finds that the traditional and plasmon-induced SH pathways in the present system allow enhancing the SH conversion efficiency and reducing the OB thresholds simultaneously. With the control field intensity increasing, a linear growth rate for the SH efficiency maximum is achieved. Furthermore, the relative phase between the driving and control fields has profound effects on modulating the bistable SH efficiency, the bistable interval and the thresholds.
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1. Introduction
Second-harmonic (SH) generation in on-chip optical microcavities has stimulated great interest in the field of nonlinear photonics due to its significant scientific importance and wide applications in biological imaging microscopies [1], frequency conversion laser sources [2], quantum information processing and quantum networking [3,4]. Two core issues in this subject are to pursue an ultralow pump threshold and ultrahigh conversion efficiency in the SH responses. Recent studies focus on synthesizing noncentrosymmetric bulk crystals, like lithium niobate LiNbO$_3$ [5] and antiperovskite Cs$_3$Cl(HC$_3$N$_3$S$_3$) [6], to explore second-order nonlinearity $\chi ^{(2)}$ that is the critical factor for the SH generation. Especially their composite structures have given rise to several merits including thermodynamic stability [7], the wide band gap and the high laser-induced damage threshold [8]. However, the available nonlinear coefficient in these crystals is so weak that it cannot meet the level of light sources and commercial applications.
Thus, to address the core issues of the SH generation, it is interesting to search for an optical environment, where the system can provide long nonlinear interaction time and the strong intensity of the fundamental waves [9–13]. To realize this goal, quantum interference in the high-Q microcavity is a promising approach. For example, under the specific illumination of two counterpropagating coherent beams, the interference-induced photon absorber not only collects a large amount of coherent light in the resonator [14–18] but also operates in the nonlinear regime [19–22]. Moreover, the utilization of the high-Q microcavity is conducive to further trapping long-lived coherent light with ultralow energy dissipation and holding the potential for various nonlinear parametric conversion [23–25]. These versatile performance of the quantum interference may open the possibility to host optimized SH processes.
On the other hand, localized surface plasmons (LSPs), originating from the coherent oscillation of conduction electrons near the surface of metal materials, have been suggested to manipulate local electromagnetic fields and tailor light-matter interactions. In particular, the coupling of light to the surface plasmons enables the strong localization of electromagnetic fields [26] and the robust ability for controlling light with light [27–29]. Also, the local-field enhancement naturally amplifies the nonlinear strength of both the metal itself and surrounding dielectric materials, forming the surface-enhanced SH and Raman scattering [30,31]. Nevertheless, inevitable nonradiative dissipation in the resonant plasmon limits its further development in nonlinear optics.
Very recently, the hybrid plasmon-photon cavity has emerged as a novel nanophotonic platform that can inherit the advantages of the low-loss microcavity and the large local-field enhancement [32–35]. Given these synergistic advantages in the hybrid system, a nonlinear microcavity coupled to a metallic nanoparticle (MNP) is designed in this paper. When the cavity is excited by two counterpropagating driving fields with the sufficient intensity and frequency detuning, the strong nonlinear responses of the system trigger the linear transmission and the SH spectra both operating in the regime of optical bistability (OB). Once the destructive interference between the transmitted and reflected fields exists in the system, an amount of system energy spontaneously flows into the SH spectrum, rather than the linear transmission spectrum. Then, we find that the SH efficiency maximum and the near-perfect absorption of the fundamental-harmonic (FH) mode simultaneously appear near the OB lower threshold. After implementing a control field on the MNP, the strong LSPs on the nanoparticle surface would carry out more plasmon polaritons and contribute to opening the plasmon-induced SH pathway. Under the cooperation of the traditional and plasmon-induced SH pathways, the strong plasmon-photon interaction allows us to enhance the SH conversion efficiency and reduce the OB thresholds. More importantly, the phase control technology for the incident lasers offers a different strategy for modulating the bistable SH efficiency, the bistable interval and the thresholds.
2. Physical system and theoretical framework model
Previous studies for plasmon-coupled microcavities have confirmed that the hybridization between plasmonics and photonics is beneficial for tailoring light-matter interactions [36–38] and designing sensing applications [39]. Similar to the hybrid plasmon-photon coupled structure, our proposed nonlinear microcavity coupled to a MNP is shown in Fig. 1(a). In this nonlinear microcavity, a second-order nonlinearity crystal is placed into an optical two-mode microcavity, which contains the FH mode (represented by $\hat a$) with the eigenfrequency $\omega _a$ and the SH mode (represented by $\hat b$) with the eigenfrequency $\omega _b$. In the SH generated process motivated by the $\chi ^{(2)}$ nonlinearity crystal, two FH photons convert to a SH photon via the parametric frequency conversion, in which the nonlinear coupling constant is marked by $\xi$ shown in Fig. 1(b). Additionally, a spherical nanoparticle inside the cavity is expected to generate LSP resonance and improve the SH conversion efficiency. The MNP dielectric response is governed by relative Drude permittivity dispersion ${\epsilon _{\rm {m}}(\omega )} = {\epsilon _\infty } - \omega _{sp}^2/(\omega ^2 + i{\omega }{\gamma _o})$, where ${\epsilon _{{\rm {m}}}}$ and ${\epsilon _\infty }$ are the permittivity and the ultraviolet permittivity of the metal. The surface plasma frequency and the Ohmic loss of the MNP mode are denoted as $\omega _{sp}$ and ${\gamma _o}$, respectively. Associating the Drude model and the Fröhlich condition $Re[{\epsilon _{\rm {m}}}(\omega )] = -2 {\epsilon _b}$, the resonant frequency of the metal plasmonic field is given as $\omega _p = \omega _{sp}/ \sqrt {(2 {\epsilon _b}+ {\epsilon _\infty })}$ with ${\epsilon _b}$ being the relative permittivity of the background environment. When this plasmonic resonant mode (represented by $\hat p$) matches with the FH mode of the cavity (i.e., $\omega _a = \omega _p$), the plasmons and the photons directly couple with each other, and their coupling strength is $g_{ap}$. Then, we apply the left- and right-driving fields $a^{r,l}_{in} =\varepsilon _d^{r,l} e^{- i \omega _d t + \phi _d}$ to excite the system. Meanwhile, the MNP is excited by the external control field $S_p = {\Omega _p}{e^{ - i{ {\omega _d}t + \phi _p}}}$. Here, $\varepsilon _d^{r,l}$ and ${\Omega _p}$ refer to the amplitudes of the corresponding lasers, which can be normalized to the input light powers $P_d^{r,l}$ and $P_p$ via the relationship of $\varepsilon _d^{r,l} = \sqrt {P_d^{r,l}/{\hbar }{\omega _d}}$ and $\Omega _p = \sqrt {P_p/{\hbar }{\omega _d}}$. For simplify, the left- and right-driving fields keep the same input amplitudes (i.e., $\varepsilon _d^r=\varepsilon _d^l=\varepsilon _d$) and all the input lasers have the same frequency $\omega _d$. The phases of the lasers are denoted as $\phi _{d,p}$.
In a rotating frame at the frequency $\omega _d$ of the driving fields, the model Hamiltonian of the hybrid plasmon-photon system can be expressed as ($\hbar =1$)
The first three terms in the right of the Hamiltonian represent the energy of the two cavity modes and the plasmonic resonant mode. $\hat {\cal O}$ and $\hat {\cal O}^{\dagger}$ (${\cal O} = a,b,p$) are the annihilation and creation operators of the corresponding modes. $\Delta _{a} = \omega _{ a} - \omega _d$, $\Delta _{ b} = \omega _{ b} - 2\omega _d$ and $\Delta _{ p} = \omega _{ p} - \omega _d$ denote the corresponding frequency detunings. If the frequency difference between the FH and SH modes is defined as $W = \omega _b - 2\omega _a$, one can obtain $\Delta _{ b} = 2\Delta _{ a} + W$. Only when the two cavity modes satisfy energy conservation (i.e., $\omega _b = 2\omega _a$), can the harmonic field be easily fulfilled in an experiment. In other words, the zero frequency difference $W=0$ is the precondition for capturing the SH signal [9]. The fourth term is the interplay between the cavity and the MNP. The coupling strength obeys the formula $g_{ap} = - 2{\pi }R^3 \sqrt {\frac {\omega _a \omega _p}{\epsilon _g V_a V_p}} |f(r_0)|$ [37] with the sphere radius $R$, the mode volume $V_a$ of the cavity and the mode volume $V_p$ of the plasmonic mode. Usually, we have the normalized mode distribution function $|f(r_0)| = 1$ and $\epsilon _g = 2 (\epsilon _{\infty } +\epsilon _b)$. The fifth term describes the process of the SH frequency conversion induced by the $\chi ^{(2)}$ nonlinearity. The coupling constant $\xi$ is determined by the intrinsic property of the nonlinear material [5–8]. The last term representing energy transfer from the external lasers to the cavity and MNP modes can be read as $\hat H_{dri} = i\sqrt {{\eta _r}{\kappa _a}} {{\hat a}^\dagger}\varepsilon _{d}^r + i\sqrt {{\eta _l}{\kappa _a}} {{\hat a}^\dagger} \varepsilon _{d}^l + i\sqrt {{\gamma _p}} {{\hat p}^\dagger}{\Omega _p}{e^{ - i{\phi _{pd}}}} + H.c.$. Due to the equivalent light-harvesting efficiency between the left and right input ports of the cavity, the light-cavity coupling rates have the same value, i.e., ${\eta _r}={\eta _l}={\eta _a}$. And the coupling rate ${\eta _a}$ for the FH mode satisfies ${\eta _a} = \kappa _a^e/(\kappa _a^i + \kappa _a^e) = \kappa _a^e/ \kappa _a$, where $\kappa _a^i$, $\kappa _a^e$ and ${\kappa _a}$ refer to the intrinsic, external, and total decay rates of the FH mode. Likewise, ${\eta _b} = \kappa _b^e/(\kappa _b^i + \kappa _b^e) = \kappa _b^e/ \kappa _b$ will be involved below this paper, where $\kappa _b^i$, $\kappa _b^e$ and ${\kappa _b}$ refer to the intrinsic, external, and total decay rates of the SH mode. The total decay rate of the dipolar plasmonic mode ${\gamma _p}$ includes the radiation rate $\gamma _r$ and the Ohmic loss rate $\gamma _o$ via the relation ${\gamma _p} = \gamma _r +\gamma _o$. At the end of the $\hat H_{dri}$ expression, ${\phi _{pd}} = \phi _{p} - \phi _{d}$ represents the relative phase between the driving and control fields.
Fig. 1. (a) Schematic diagram of a nonlinear microcavity coupled to a MNP. This nonlinear microcavity consists of a two-mode cavity and a second-order nonlinearity $\chi ^{(2)}$ crystal. The cavity is driven by two counterpropagating driving fields $a_{in}^l$ and $a_{in}^r$, while the MNP can be excited by the external control field $S_p$. (b) The coupling between the three modes in the system. Optical modes $\hat a$ and $\hat b$ belong to the two-mode cavity, while $\hat p$ is the plasmonic resonant mode.
With the mean-field approximation considered, all of the system operators can be reduced to their mean values. And the quantum noise operators, whose mean values are zero, can be safely dropped. Then, according to the Heisenberg-Langevin formalism $\frac {{d \hat {\cal O}}}{{d t}} = -i[\hat {\cal O}, \hat H]$, the motion dynamics of the system is fully written as
Obviously, the denominator on the right side of Eq. (5) possesses the intracavity photon intensity term $|a|^2$, which implies the nonlinear dependence of the intracavity field on the input-light fields. This nonlinear behavior has been used to study coherent photon absorption in the OB regime [19–22].
Subsequently, we are more concerned about the output fields at the frequencies of the FH and SH photons. By employing the input-output theory, one has the output amplitudes of the FH and SH modes from the left- and right-sides of the cavity
Considering that the left- and right-driving fields are assumed to be consistent in the present scheme (i.e., $a_{in}^{r}=a_{in}^{l}$), we can receive the same output signals from both sides of the cavity (i.e., $a_{out}^{r}=a_{out}^{l}$ and $b_{out}^{r}=b_{out}^{l}$). To intuitively assess the intensity of these output signals, we introduce the relative transmission intensities of the FH and SH modes with respect to the intensity of the driving fields, as follows
Here $T_a$ can be regarded as the transmission intensity of the driving fields, while $T_b$ can be treated as the SH conversion efficiency. Their values are two important indexes for evaluating linear and nonlinear responses of the hybrid system.
It is necessary to give a realistic parameter set in the hybrid plasmon-photon configuration. The related model and parameters can be found in recent experimental and theoretical works [9,37]. The combination of the high-Q cavity and the MNP has been reported in Ref. [37]. The parameters of the optical cavity are chosen as $(\omega _a, \kappa _a, Q_a)=(2.3 eV, 1 meV, 2300)$ and $(\omega _b, \kappa _b, Q_b)=(2 \omega _a, 3 \kappa _a, 2{Q_a}/3)$, in which these parameters represent the frequency, the decay rate and the quality factor of the FH and SH modes. The mode volume of the cavity is $V_a = 1 {\mu }m{^3}$. The MNP can be prepared by a gold material [37] and its related parameters are set as: $(\omega _p, \gamma _o, \gamma _r)=(2.3 eV, 0.2 eV, 2.45 meV)$ referring to the plasmonic resonant frequency, the Ohmic loss and the radiation rate. The mode volume of the plasmonic mode depends on the MNP radius. For example, we have $V_p = 6300 nm^3$ when $R=10 nm$. Under the phase-matching condition, intrinsic lithium niobate inside the optical cavity can support a strong second-order nonlinearity with the coupling strength $\xi =2 \kappa _a$ [9]. The outstanding nanofabrication technology allows integrating such a micromaterial into an optical cavity and maintains the compatibility between the cavity mode and $\chi ^{(2)}$-type nonlinearity. Besides, other parameters are selected as $\epsilon _\infty = 1$, $\epsilon _b = 1$, $\eta _a = \eta _b = 1/2$.
3. Results and discussions about plasmon-modulated bistable second-harmonic generation
Recent research advances about cavity-enhanced SH generation have reported meaningful progresses for ultralow thresholds [9,10], ultraviolet spectra [12] and SH frequency combs [40,41]. Among these advances, the SH formation mostly initiates from the stimulated up-conversion of two FH photons, whose reverse process is often used to prepare optical parametric amplifiers. Taking the optical properties of the second-order nonlinear crystal into consideration, the FH photons have an opportunity to yield the other nonlinear behavior, like the OB effect [21,42]. However, the coexistence between the SH and the OB remains unexplored. Exploring the coexistence of multiple nonlinear effects, especially for their cooperative or competitive relationships, is highly important to discover an unusual SH process with unconventional mechanisms.
To perform a proof-of-principle investigation for the nonlinear dynamics of the hybrid plasmon-photon cavity, Figs. 2(a) and 2(b) respectively show the $3D$ surface images of the FH transmission intensity $T_a$ and the normalized SH intensity $T_b$ as a function of the driving field intensity $P_d$ and its detuning $\Delta _a$. From the view of the surface shape of the two images, it is clear that both the FH transmission intensity $T_a$ and the SH intensity $T_b$ experience a higher flat area and a lower flat area, where the intensities of the FH and SH modes change smoothly. Inversely, the output intensities in the connected region of the two flat areas change drastically. Direct comparison of the two spectra finds that the low (high) FH transmission intensity and the high (low) SH conversion efficiency are synchronously located at the position of the same system parameters. This synchronous change in the linear and nonlinear spectra can be explained by the SH frequency conversion. Based on the principle of energy conservation, one deduces that the lower the FH transmission intensity is, the stronger the feature of this mode localization in the intracavity is. It means that the FH cavity mode carrying sufficient energy participates in the SH frequency conversion. After that, an amount of energy in the FH mode is continuously depleted and converted into the SH photon via the two-photon process, which is also called as the traditional SH pathway.
Fig. 2. The surface plots of (a) the transmission intensity $T_a$ of the FH mode and (b) the normalized SH intensity $T_b$ as a function of the driving field intensity $P_d$ and its detuning $\Delta _a$. (c) When the driving field detuning is fixed as $\Delta _a =\kappa _a, 4.5\kappa _a$, the transmission intensity $T_a$ and the normalized SH intensity $T_b$ varying with the driving field intensity $P_d$. (d) When the driving field intensity is fixed as $P_p =0.5 {\mu }W, 5 {\mu }W$, the transmission intensity $T_a$ and the normalized SH intensity $T_b$ varying with the driving field detuning $\Delta _a$. Other parameters are given in the text, expect for $W=0$, $\Delta _p=\Delta _a$, $P_p=0$ and $\phi _{pd}=0$.
Because the $2D$ plot can explicitly disclose the details of the FH and SH spectra, we depict their output intensities separately varying with the driving field intensity $P_d$ and detuning $\Delta _a$ in Figs. 2(c) and 2(d). In the case of the fixed driving field detuning $\Delta _a = \kappa _a$, the blue SH spectrum in Fig. 2(c) shows the standard lineshape of a new nonlinear signal. More specifically, with the driving field intensity increasing, the intensity of the SH generation grows fastly, up to the saturate point at the position of $(P_d, T_b) = (0.16 {\mu }W, 0.42)$, and then drops slowly. In the other case of $\Delta _a = 4.5 \kappa _a$, the black dash line in the SH spectrum exhibits a bistable behavior with respect to the input field intensity. Similar to the hysteresis curve, the common OB characteristic is composed of two stable branches (i.e., upper and lower branches) and an unstable branch. Note that the excellent nonlinear properties of the bistability are mainly determined by low thresholds and wide bistable interval, which are beneficial for designing optical switches and laser applications [43]. For example, in the present bistable spectra of Fig. 2(c), the lower and upper thresholds are respectively marked as $P_d^{LT}$ and $P_d^{UT}$, while its bistable interval is equal to $P_d^{UT} - P_d^{LT}$.
Combining Figs. 2(c) and 2(d), one finds that the bigger both the driving field intensity and detuning are, the more obvious the bistable behavior is. Essentially, there are two underlying mechanisms for explaining the coexistence of two kinds of nonlinear effects. First, since the nonlinear strength depends strongly on the medium polarization, increasing input field intensity is an effective method to improve nonlinear polarization responses. It is the reason that the coexistence between the SH and OB effects always occurs in the region of the strong input fields. Second, the detuning management for nonlinear responses plays an irreplaceable role in the modulation of the optical nonlinearity, which has an important effect on the bistable patterns. [44,45]. As a result of the collaboration of the two nonlinear enhanced mechanisms, the output spectra simultaneously possess the SH and OB effects. Nonetheless, whatever the intensity and the detuning of the driving fields are, the SH efficiency maximum (i.e., the normalized intensity) only reaches $T_b \approx 0.42$. Thus, it is necessary to amplify and control the SH signals, especially in the bistable operation domain.
Next, we start to coherently manipulate the SH signals when modifying the plasmon-photon interaction in reasonable ranges. In order to ensure that the present system operates in the bistable region as illustrated above, the driving field detuning is appointed as $\Delta _a=4.5 \kappa _a$. In the absence of the phase modulation, i.e., ${\phi }_{pd}=0$, Figs. 3(a-c) give the FH transmission intensity $T_a$ and the normalized SH intensity $T_b$ as a function of the driving field intensity $P_d$, when the control field intensity $P_p$ varies continuously or discretely. From the view of the whole spectra in Figs. 3(a-b), it is evident that the linear transmission intensity $T_a$ and the SH intensity $T_b$ still undergo the bistable behavior and the direct competition with each other. This is to say, under the premise of the energy conservation, the most energy of the system, except for those staying in the cavity modes, selectively flows into the linear spectra or the SH spectra. For example, due to the destructive interference between the transmitted and reflected fields, the nearly vanishing minimums in the transmission spectra of Fig. 3(c) are approximately located at the lower thresholds. Correspondingly, the parametric conversion makes use of the remaining energy and creates the efficiency maximums in the SH spectra. Interestingly enough, the bistable pattern of the FH transmission spectra evolves from a conventional lineshape to an unconventional loop shape, as $P_p$ increases from $1{\mu }W$ to $10{\mu }W$.
Fig. 3. The surface plots of (a) the transmission intensity $T_a$ of the FH mode and (b) the normalized SH intensity $T_b$ as a function of the driving field intensity $P_d$ and the control field intensity $P_p$. (c) With the control field intensity fixed as $P_p = 1{\mu }W, 10{\mu }W$, the transmission intensity $T_a$ and the normalized SH intensity $T_b$ versus the driving field intensity $P_d$. (d) The FH transmission minimum $T_a|_{min}$, the SH efficiency maximum $T_b|_{max}$ and the OB thresholds versus the control field intensity $P_p$. Other parameters are the same as in Fig. 2 expect for $\Delta _a=4.5 \kappa _a$ and $\phi _{pd}=0$.
To give a better insight into the bistable properties of these spectra, the FH transmission minimum $T_a|_{min}$, the SH efficiency maximum $T_b|_{max}$ and the OB thresholds versus the control field intensity $P_p$ are plotted in Fig. 3(d). When the control field intensity increases, there are two main characteristics for these physical quantities. First, the SH efficiency maximum $T_b|_{max}$ linearly increases close to $0.7$, while the transmission minimum $T_a|_{min}$ keeps nearly zero. As mentioned above, both the maximum $T_b|_{max}$ and the minimum $T_a|_{min}$ actually are located near the position of the lower thresholds. Second, with the increase of the control field intensity, the upper and lower thresholds $P_d^{LT/UT}$ decrease synchronously, whereas the bistable interval remains constant $P_d^{UT} - P_d^{LT} \approx 2.3 {\mu }W$. Physically, the MNP excited by the strong control field provides a well-established environment, where the LSP effect can be enhanced. Hence, the plasmon-photon interaction becomes more intense and boosts the energy of the localized plasmon mode into the inherently SH processes, ultimately resulting in the other plasmon-induced SH pathway. In comparison with the SH spectra only generated by the traditional SH pathway in Fig. 2(b), the present hybrid system containing the traditional and plasmon-induced pathways allows enhancing the SH conversion efficiency and reducing the OB thresholds.
It is well known that the phase control technology can offer another strategy for tailoring light-matter interactions to gain more efficient and accessible SH spectra. Since the relative phase is attached in the control field, the relatively large incident power with $P_p=4{\mu }W$ are applied to improve the phase sensitivity. Accordingly, Figs. 4(a) and 4(b) depict the linear output intensity $T_a$ and the normalized SH intensity $T_b$ as a function of the driving field intensity $P_d$ and the relative phase $\phi _{pd}$ of the incident lasers. It shows that the intensity and the bistable lineshapes in the output spectra have the sensitive dependence on the relative phase. Because the greatest spectral difference caused by phase-sensitive dependence occurs in the two cases of $\phi _{pd} = {\pi }/2, 3{\pi }/2$, we give the details of the two spectra in Fig. 4(c). Through the quantitative analysis, the blue dot line in the SH spectrum for the $\phi _{pd} = {\pi }/2$ case describes that the efficiency maximum exceeds $1$ and the minimum thresholds are $P_d^{LT} \approx 0.9{\mu }W$ and $P_d^{UT}\approx 2.5{\mu }W$. As expected, the dynamics behavior of the FH transmission intensity $T_a$ is related to the SH efficiency, in which the points of the near-perfect absorption and the SH efficiency maximum are located at the lower thresholds. This result verifies again that the nonlinear SH responses depend on whether the incident lasers are absorbed into the FH cavity mode via the destructive interference.
Fig. 4. The surface plots of (a) the transmission intensity $T_a$ of the FH mode and (b) the normalized SH intensity $T_b$ as a function of the driving field intensity $P_d$ and the relative phase $\phi _{pd}$ between the incident lasers. (c) With the relative phase fixed as $\phi _{pd} = {\pi }/2, 3{\pi }/2$, the transmission intensity $T_a$ and the normalized SH intensity $T_b$ varying with the driving field intensity $P_d$. (d) The FH transmission minimum $T_a|_{min}$, the SH efficiency maximum $T_b|_{max}$ and the OB thresholds versus the relative phase $\phi _{pd}$. Other parameters are the same as in Fig. 2 expect for $\Delta _a=4.5 \kappa _a$ and $P_p=4{\mu }W$.
Furthermore, we analyze the influence of the relative phase on the FH transmission minimum $T_a|_{min}$, the SH efficiency maximum $T_b|_{max}$, the lower and upper OB thresholds $P_d^{LT/UT}$, as shown in Fig. 4(d). It is readily seen that the four physical quantities are extremely sensitive to the relative phase. From the viewpoint of applications, this extraordinary sensitivity of phase dependence can be designed to control light with light [46]. Moreover, the SH efficiency maximum exists in the most narrow bistable interval (with $P_d^{UT}-P_d^{LT} \approx 1.6{\mu }W$) at the position $\phi _{pd} = {\pi }/2$, while the SH efficiency minimum exists in the widest bistable interval (with $P_d^{UT}-P_d^{LT} \approx 3.1{\mu }W$) at the position $\phi _{pd} = 3{\pi }/2$. Consequently, this phase control technology opens an alternative way to pursue the optimal coexistence between the SH and OB effects.
Based on the analysis for the plasmon-modulated bistable SH generation, we can predict that the size of the sphere nanoparticle plays a crucial role in the plasmon-photon interaction and the SH processes. The calculation results display the normalized SH intensity $T_b$ versus the driving field intensity $P_d$ for different MNP sphere radii in Fig. 5(a) and for different control fields acting on the MNP in Fig. 5(b). Because of the high-consistency in the FH transmission and SH spectral structures, we focus primarily on the SH spectral features. As the MNP radius increases from $5nm$ to $20nm$ in Fig. 5(a), the SH efficiency maximum in the bistable region has the remarkable enhancement. It also finds the larger the MNP radius is, the lower the bistable interval and the thresholds become. The physical reason for these SH spectral features can be understood as following. Under the fixed strong-excitation with $P_p =5{\mu }W$, the large surface area of the MNP would carry out more plasmon polaritons and enhance the coupling strength between the cavity and the MNP [47]. This result can be also derived from the expression of the coupling strength $g_{ap}$. Relying on this physical process, more energy in the MNP continuously converts into the FH mode via plasmon-photon interaction, then indirectly accelerates the bistable SH generation. Besides, when the large MNP with the $20{nm}$ radius is placed in this system, Fig. 5(b) illustrates that the SH conversion efficiency experiences the enhancement of more than four times as the control field intensity increases from $1{\mu }W$ to $6{\mu }W$. Although the bistable thresholds reduce, the bistable interval remains almost unchanged, which agrees with the result of Fig. 3(d).
Fig. 5. The normalized SH intensity $T_b$ versus the driving field intensity $P_d$ (a) for different sphere radii of the MNP with $P_p =5{\mu }W$ and (b) for different control fields acting on the MNP with $R =20{nm}$. Other parameters are the same as in Fig. 2 expect for $\Delta _a=4.5 \kappa _a$ and $\phi _{pd}=0$.
4. Conclusion
In summary, plasmon-modulated bistable SH generation has been studied in a nonlinear microcavity coupled to a MNP. In order to explore the multiple nonlinear effects, the strong excitation and the detuning management for the driving fields are employed to boost the nonlinear polarization of the medium. Different from the previous cavity-enhanced SH generation via the traditional two-photon process only [9–13], our proposed scheme involves several unconventional mechanisms. First, in the nonlinear microcavity the photon-photon and plasmon-photon interactions respectively lead to the traditional and plasmon-induced SH pathways, which contribute to enhancing the SH conversion efficiency and reducing the thresholds of the bistability. Second, once the destructive interference between the transmitted and reflected fields exists in the system, an amount of energy of the system participates in the SH processes, rather than linear transmission spectra. Then the SH efficiency maximum and the near-perfect absorption of the FH mode are located near the lower threshold of the bistability. Third, the phase control technology for the incident lasers offers a strategy for modulating the SH efficiency, the bistable interval and the thresholds. Fourth, considering the influence of the size of the MNP sphere on the plasmon-photon interaction, the SH conversion efficiency can be further enhanced by increasing the MNP surface area. These underlying mechanisms can work for the other parametric conversion, leading to more choices to pursue high-performance nonlinearity.
Funding
Natural Science Foundation of Universities of Jiangsu Province (19KJA180007); National Natural Science Foundation of China (11904176, 12104233, 61874060, 61911530220, 62005124, U1932159).
Disclosures
The authors declare no conflicts of interest.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
References
1. P. Campagnola, “Second harmonic generation imaging microscopy: applications to diseases diagnostics,” Anal. Chem. 83(9), 3224–3231 (2011). [CrossRef]
2. L. P. Gonzalez, D. C. Upchurch, P. G. Schunemann, et al., “Second-harmonic generation of a tunable continuous-wave CO2 laser in orientation-patterned GaAs,” Opt. Lett. 38(3), 320–322 (2013). [CrossRef]
3. G. Y. Wang, T. Li, Q. Ai, et al., “Faithful entanglement purification for high-capacity quantum communication with two-photon four-qubit systems,” Phys. Rev. Appl. 10(5), 054058 (2018). [CrossRef]
4. D. Poderini, I. Agresti, G. Marchese, et al., “Experimental violation of n-locality in a star quantum network,” Nat. Commun. 11(1), 2467 (2020). [CrossRef]
5. Y. Li, Z. Huang, Z. Sui, et al., “Optical anapole mode in nanostructured lithium niobate for enhancing second harmonic generation,” Nanophotonics 9(11), 3575–3585 (2020). [CrossRef]
6. M. Li, X. Zhang, Z. Xiong, et al., “A hybrid antiperovskite with strong linear and second-order nonlinear optical responses,” Angew. Chem. Int. Ed. 61(42), e202211151 (2022). [CrossRef]
7. G. H. Zou, H. Jo, S. Lim, et al., “Rb3VO(O2)2CO3: a four-in-one carbonatoperoxovanadate exhibiting an extremely strong second-harmonic generation response,” Angew. Chem. Int. Ed. 57(28), 8619–8622 (2018). [CrossRef]
8. J. Chen, C. L. Hu, F. Kong, et al., “High-performance second-harmonic-generation (SHG) materials: new developments and new strategies,” Acc. Chem. Res. 54(12), 2775–2783 (2021). [CrossRef]
9. X. Guo, C.-L. Zou, and H. X. Tang, “Second-harmonic generation in aluminum nitride microrings with 2500%W conversion efficiency,” Optica 3(10), 1126 (2016). [CrossRef]
10. J. Li, R. Yu, Y. Qu, et al., “Second-harmonic generation with ultralow-power pump thresholds in a dimer of two active-passive cavities,” Phys. Rev. A 96(1), 013815 (2017). [CrossRef]
11. Z. Lin, X. Liang, M. Lonćar, et al., “Cavity-enhanced second-harmonic generation via nonlinear-overlap optimization,” Optica 3(3), 233 (2016). [CrossRef]
12. S. Hannig, J. Mielke, J. A. Fenske, et al., “A highly stable monolithic enhancement cavity for second harmonic generation in the ultraviolet,” Rev. Sci. Instrum. 89(1), 013106 (2018). [CrossRef]
13. W. H. P. Pernice, C. Xiong, C. Schuck, et al., “Second harmonic generation in phase matched aluminum nitride waveguides and micro-ring resonators,” Appl. Phys. Lett. 100(22), 223501 (2012). [CrossRef]
14. G. S. Agarwal and Y. Zhu, “Photon trapping in cavity quantum electrodynamics,” Phys. Rev. A 92(2), 023824 (2015). [CrossRef]
15. L. Wang, K. Di, Y. Zhu, et al., “Interference control of perfect photon absorption in cavity quantum electrodynamics,” Phys. Rev. A 95(1), 013841 (2017). [CrossRef]
16. C. Wang, W. R. Sweeney, A. D. Stone, et al., “Coherent perfect absorption at an exception point,” Science 373(6560), 1261–1265 (2021). [CrossRef]
17. Y. D. Chong, L. Ge, H. Cao, et al., “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105(5), 053901 (2010). [CrossRef]
18. B. Midya and V. V. Konotop, “Coherent-perfect-absorber and laser for bound states in a continuum,” Opt. Lett. 43(3), 607 (2018). [CrossRef]
19. G. S. Agarwal, K. Di, L. Wang, et al., “Perfect photon absorption in the nonlinear regime of cavity quantum electrodynamics,” Phys. Rev. A 93(6), 063805 (2016). [CrossRef]
20. Y. Wei, W. Gu, G. Yang, et al., “Coherent perfect absorption in a quantum nonlinear regime of cavity quantum electrodynamics,” Phys. Rev. A 97(5), 053825 (2018). [CrossRef]
21. W. Xiong, J. Chen, B. Fang, et al., “Coherent perfect absorption in a weakly coupled atom-cavity system,” Phys. Rev. A 101(6), 063822 (2020). [CrossRef]
22. L. Wang, J. Hu, J. Du, et al., “Broadband coherent perfect absorption by cavity coupled to three-level atoms in linear and nonlinear regimes,” New J. Phys. 23(12), 123040 (2021). [CrossRef]
23. H. Xiong, L.-G. Si, A.-S. Zheng, et al., “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86(1), 013815 (2012). [CrossRef]
24. Q. Bin, Y. Wu, and X. Y. Lü, “Parity-symmetry-protected multiphoton bundle emission,” Phys. Rev. Lett. 127(7), 073602 (2021). [CrossRef]
25. S. Liu, W.-X. Yang, Z. Zhu, et al., “Quadrature squeezing of a higher-order sideband spectrum in cavity optomechanics,” Opt. Lett. 43(1), 9 (2018). [CrossRef]
26. J. H. Kim, J. Y. Lee, E. S. Kim, et al., “Electric field enhancement of coupled plasmonic nanostructures for optical amplification,” PhotoniX 4(1), 8 (2023). [CrossRef]
27. N. Liu, M. Hentschel, T. Weiss, et al., “Three-dimensional plasmon rulers,” Science 332(6036), 1407–1410 (2011). [CrossRef]
28. D. E. Gómez, X. Shi, T. Oshikiri, et al., “Near-perfect absorption of light by coherent plasmon-exciton states,” Nano Lett. 21(9), 3864–3870 (2021). [CrossRef]
29. S. A. Mann, D. L. Sounas, and A. Alu, “Nonreciprocal cavities and the time-bandwidth limit,” Optica 6(1), 104 (2019). [CrossRef]
30. M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012). [CrossRef]
31. H. Linnenbank, Y. Grynko, J. Förstner, et al., “Second harmonic generation spectroscopy on hybrid plasmonic/dielectric nanoantennas,” Light: Sci. Appl. 5(1), e16013 (2016). [CrossRef]
32. J. T. Hugall, A. Singh, and N. F. van Hulst, “Plasmonic cavity coupling,” ACS Photonics 5(1), 43–53 (2018). [CrossRef]
33. S. Zhang, S. R. Panikkanvalappil, S. Kang, et al., “Enhancing plasmonicphotonic hybrid cavity modes by coupling of individual plasmonic nanoparticles,” J. Phys. Chem. C 123(39), 24255–24262 (2019). [CrossRef]
34. F. Pan, K. C. Smith, H. L. Nguyen, et al., “Elucidating energy pathways through simultaneous measurement of absorption and transmission in a coupled plasmonic-photonic cavity,” Nano Lett. 20(1), 50–58 (2020). [CrossRef]
35. Y. W. Lu, W. J. Zhou, Y. Li, et al., “Unveiling atom-photon quasi-bound states in hybrid plasmonic-photonic cavity,” Nanophotonics 11(14), 3307–3317 (2022). [CrossRef]
36. Y.-F. Xiao, Y.-C. Liu, B.-B. Li, et al., “Strongly enhanced light-matter interaction in a hybrid photonic-plasmonic resonator,” Phys. Rev. A 85(3), 031805 (2012). [CrossRef]
37. P. Peng, Y. C. Liu, D. Xu, et al., “Enhancing coherent light-matter interactions through microcavity-engineered plasmonic resonances,” Phys. Rev. Lett. 119(23), 233901 (2017). [CrossRef]
38. E. Waks and D. Sridharan, “Cavity QED treatment of interactions between a metal nanoparticle and a dipole emitter,” Phys. Rev. A 82(4), 043845 (2010). [CrossRef]
39. L. Li, W.-X. Yang, T. Shui, et al., “Ultrasensitive sizing sensor for a single nanoparticle in a hybrid nonlinear microcavity,” IEEE Photonics J. 12(3), 1–8 (2020). [CrossRef]
40. T. Hansson, F. Leo, M. Erkintalo, et al., “Singly resonant second-harmonic-generation frequency combs,” Phys. Rev. A 95(1), 013805 (2017). [CrossRef]
41. F. Leo, T. Hansson, I. Ricciardi, et al., “Walk-off-induced modulation instability, temporal pattern formation, and frequency comb generation in cavity-enhanced second-harmonic generation,” Phys. Rev. Lett. 116(3), 033901 (2016). [CrossRef]
42. T. K. Fryett, C. M. Dodson, and A. Majumdar, “Cavity enhanced nonlinear optics for few photon optical bistability,” Opt. Express 23(12), 16246 (2015). [CrossRef]
43. B. Megyeri, G. Harvie, A. Lampis, et al., “Directional bistability and nonreciprocal lasing with cold atoms in a ring cavity,” Phys. Rev. Lett. 121(16), 163603 (2018). [CrossRef]
44. L. Pickup, K. Kalinin, A. Askitopoulos, et al., “Optical bistability under nonresonant excitation in spinor polariton condensates,” Phys. Rev. Lett. 120(22), 225301 (2018). [CrossRef]
45. J. H. Li, “Controllable optical bistability in a four-subband semiconductor quantum well system,” Phys. Rev. B 75(15), 155329 (2007). [CrossRef]
46. D. G. Baranov, A. Krasnok, T. Shegai, et al., “Coherent perfect absorbers: linear control of light with light,” Nat. Rev. Mater. 2(12), 17064 (2017). [CrossRef]
47. P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96(11), 113002 (2006). [CrossRef]
