For laser devices, laser oscillation is generated by the interaction of optical feedback in the cavity with the laser gain medium. Only a laser mode with lower loss can form a stable laser field distribution in the laser gain medium. There are two kinds of loss in the resonant cavity. One is the loss independent of the laser transverse mode, such as the internal loss of the laser working substance, the transmission loss of the cavity mirror, the absorption of the intracavity component, the scattering loss, etc. The other is the diffraction loss, which is closely related to the laser transverse mode. The diffraction loss has an important influence on the oscillation of the laser transverse modes with different orders. This characteristic is the physical basis for achieving laser transverse mode selection and obtaining high beam quality laser output. The ZnO microwire F-P resonant mode uses two mutually parallel reflecting surfaces as the cavity structure, as shown in Fig. 1(a). The F-P excitation mode in the ZnO microcavity can be observed in Fig. 1(b) and exhibits optical waveguide characteristics.This structure effectively limits the optical path between two mutually parallel end faces(M1 and M2) to realize gain amplification. The steady-state field of the optical wave of M1 on one of the side of the optical cavity is E1(x, y) and the steady-state field of the optical wave on the M2 plane is E2(x, y). The relationship between the two is as follows ：
图 1 (a) 类F-P激光腔结构示意图；(b)激光在ZnO微腔中传播的示意图
Figure 1. (a)Structure diagram of an F-P laser cavity; (b)schematic diagram of laser propagation in ZnO microcavity
Where ∂ is a complex constant factor defined as the change in the amplitude and the phase of the light field after a single pass propagation. The total energy loss Ψ of light during propagation can be expressed as[16-17]：
If the loss of transmission, absorption, and output is not being considered, α is defined as the diffraction loss of the laser cavity.
The dielectric constant of the noble metal material has a negative real part and a small positive imaginary part. And, the outer free electrons are easy to oscillate under the excitation of the incident light field. When the free electron oscillation frequency matches the frequency of the incident light field, the metal surface plasmon resonance is generated. According to the structural form of metal materials, this form of resonance is generally divided into two categories. The first is LSPR. The incident light field interacts with the metal nanoparticles. The free electrons on the conduction band collectively oscillate on the metal surface and the generated surface plasmon waves are localized near the nanostructure(Fig. 2(a)). In the resonant state, the energy of the electromagnetic field is effectively converted into the collective vibrational energy of the free electrons on the metal surface, which in turn produces LSPR[19-20]. The second is surface plasmon resonance(SPR) on the propagation surface, in which light field acts on the continuous metal surface to produce a surface plasmon resonance wave that propagates on the metal surface(Fig. 2(b)). In this paper, the LSPR resonance coupling effect is used to generate near-field local enhancement, which effectively improves the optical field confinement of ZnO micro-wire structure and then enhances F-P stimulated radiation.
In order to study the relationship between the ZnO micron-diameter and optical loss, the calculation and simulation are carried out by using three-dimensional space time domain finite-difference method for the hexahedral ZnO micro-wires. The effects of the electric field distribution of different cross-sections and the electric field intensity ratio inside and outside the cavity are used to analyze the effect of ZnO micron-diameter on optical loss. In the simulation, the ZnO microwires were placed vertically on the substrate. The height h of the microwires is set to 2.1 μm. Using a single-frequency light source and using a (n, k) material model for simulation, the refractive index of the ZnO material is set to n=2.06, k=1 by reference . The corresponding model is used to simulate the change of the mode number in the resonant cavity formed by the ZnO nanowires. The Perfectly Matched Layer(PML) is selected as the simulation boundary condition by setting a special dielectric layer at the truncation boundary in the FDTD region so that the wave impedance of the layer medium and the wave impedance of the adjacent medium are completely matched. Therefore, the incident wave does not reflect at the interface and passes through before it finally enters the PML layer. Although PML has only a finite thickness, it still has a good absorption for incident waves. Therefore, PML is also an absorbing boundary that is often used in actual calculations.
The electric field distributions of the ZnO micro-wires with diameters of 1.6, 2, 4, 6, and 8μm in the E(x, y) and E(x, z) sections were calculated using the established model, as shown in Fig. 3. The figure clearly shows that the ZnO microwire has optical waveguide characteristics and intracavity F-P mode oscillation. It is found by the electric field distribution that light is confined in the ZnO micron line and forms a standing wave propagating along the cavity. When the cavity diameter is 1.6 and 2 μm, a strong electric field distribution is observed outside the cavity, which indicates that the light will have a severe energy loss during the propagation in the cavity. As the cavity diameter increases, more energy is added to the cavity to achieve laser resonance. When the cavity diameter is increased to 6 and 8 μm, most of the light can be locked in the cavity and a perfect optical resonance mode can be observed. Through the electric field distribution of the cross-section of the ZnO micron line, it can be found that as the diameter of the ZnO micron line increases, its ability to confine the optical field is enhanced, and the optical loss is reduced. This phenomenon can be analyzed according to the theoretical formula for the laser cavity：
图 3 不同直径时的(x, y)，(x, z)面电场分布图
Figure 3. The electric field distributions of (x, y), (x, z) planes of the ZnO micro-wires with different diameters
where D is the lateral dimension of the microcavity(ie. the diameter of the microwire), h is the cavity length and λ is the wavelength. N is the Fresnel number, which is a characteristic parameter in the diffraction phenomenon. The larger the value, the smaller the diffraction loss. It can be seen from the formula (3) that, under conditions where the cavity length h is constant, the Fresnel number N gradually increases as the diameter D of the microwire increases at the same wavelength λ, thereby reducing the diffraction loss in the cavity.
Fig. 4 shows the distribution of light intensity and the efficiency of light confinement at different cavity diameters. The trend of the light field distribution in the cavity with different cavity diameters can be observed by Fig. 4(a). It can be seen as the cavity diameter increases, the intensity of the cavity increases gradually, while the intensity of the light field outside the cavity gradually decreases. Fig. 4(b) shows the light confinement efficiency curve for different cavity diameters. The light confinement efficiency here is defined as the percentage of the light field intensity area in the ZnO micron-line cavity to the total light field intensity area. It can be found that when the diameter is 1.6 μm, the optical field energy loss is at its highest and almost no light exists in the cavity. On the contrary, as the diameter of the microcavity increases, the light field confinement efficiency reaches about 99%, indicating that the light field energy is almost entirely limited to the intracavity propagation when the diameter reaches a certain level. Therefore, the F-P microcavity mode oscillation can be constructed by using the ZnO micro-wire structure light field confinement ability, and the optical loss characteristic can be effectively regulated by changing the micro-wire diameter.
In order to study the influence of ZnO micron-cavity diameter on the F-P oscillation mode, the resonance mode analysis group was used to simulate the resonance spectrum evolution process in the micro-wire, as shown in Fig. 5. It can be seen from the above figure that as the diameter of ZnO increases, multiple resonant modes appear gradually in the resonant cavity. When D=2 μm, there are three resonance peaks in the resonance spectrum, where the strongest peak appears at f=800 THz. When D is increased to 4 μm, three strong resonance peaks appear and the strongest peak position is substantially unchanged. As the diameter of the micron wire continues to increase to 6 and 8 μm, the multi-stage resonance peak increases significantly, and the resonance intensity increases accordingly. Studies have shown that as the diameter of ZnO increases, the number of intracavity resonance modes can also increase. Therefore, small-cavity ZnO micron-wire has the possibility of realizing a single-mode laser. The reason for the formation of the single-mode laser is mainly due to the side mode suppression caused by optical loss. It is thus crucial to study the small cavity optical loss. In order to improve the performance of the traditional optical microcavity system, an optical cavity mode is proposed using the microcavity structure modified by metal local surface plasmons to achieve the coupling of the evanescent field loss light and the metal local plasmon wave. It is expected that metal local surface plasmons can be used to enhance light interaction with matter and to provide a limited waveguide mode in the sub-wavelength range.
By using Ag nanoparticles to modify the ZnO micro-wire excitation to form the LSPR electric field, the local field enhancement and optical field confinement effect can be achieved on the surface of the micro-wire[24-25]. The problem of large optical loss in small-cavity ZnO micro-wires can be effectively suppressed by metal LSPR. To this end, Ag nanoparticles were used to modify the surface of ZnO microwires with a diameter of 2 μm and the electric field was recorded by an x-y section electromagnetic field monitor.
Fig. 6 is a structural diagram and a light field distribution diagram of 2, 4, and 6 surfaces of Ag-modified ZnO microwires. It can be easily seen from the electric field distribution diagram of the x-y cross-section that the optical field in the cavity is enhanced by the modification of the Ag nanoparticle, which proves the ability of the LSPR to limit the optical field. Since the work function of Ag equates to 4.35 eV and the electron affinity of ZnO is 4.26 eV, the electrons are easily transferred to each other at the interface of Ag/ZnO. In addition, the ZnO emission energy is close to the energy of the plasmon resonance of the Ag surface, so the enhancement in light field intensity is the result of the coupling between the local surface resonance plasmon of the Ag nanoparticle and the photons from the stimulated emission of the ZnO microrod.
图 6 Ag纳米颗粒修饰的LSPR微腔的模拟示意图和对应电场分布
Figure 6. Simulation diagram of LSPR microcavity modified by Ag nanoparticles and corresponding electric field distributions
Fig. 7 is an x-y plane electric field distribution diagram of the Ag particle modified ZnO microwire and its electric field intensity curve. It can be seen that there is a clear light field enhancement effect in the area where the Ag particles are in contact with the micron lines, corresponding to the two sharp enhancement peaks of the intensity curve of Fig. 7. The formation of these two enhancement peaks is the result of the resonant coupling of the local surface plasmon with the loss light in the microwire.
图 7 Ag纳米颗粒修饰微腔的电场强度分布图
Figure 7. Distribution of the electric field intensity of the Ag nanoparticle modified microcavity
Since the micron-scale ZnO micro-wire structure is much larger than that of the nano-scale Ag particles, the contact surface of ZnO and Ag is equivalent to a plane. A local simulation of the Ag/ZnO microwire interface was performed to monitor electric field in the x-y cross-section of the local field and the x-z end surface, respectively, as shown in Fig. 8. 5×5 equal-sized Ag nanoparticles were placed on the ZnO plane. Through the electric field distribution diagrams of the cross-section and the end surface, it can be found that there is coupling light enhancement at the edge of the Ag particle and that between the two metal particles, the electric field strength along the X-axis direction has a two-fold enhancement effect. This phenomenon is mainly due to the secondary coupling phenomenon between metal particles. First, the excitation of the plasmon and the micron-line photons create resonant coupling and amplification. Second, the oscillating coupling between the X-direction metal particles is generated by the resonance-enhanced energy.
图 8 Ag纳米颗粒装饰ZnO微米线的局部电场图
Figure 8. Local electric field diagrams of Ag nanoparticle modified ZnO microwire
In order to explain and quantify the effect of metal LSPR on the optical loss of the ZnO micron wire structure, Fig. 9 simulates the distribution curve and light confinement efficiency of ZnO micron lines with 2, 4 and 6 sides modified by Ag nanoparticles respectively. Fig. 9(a) is a graph of the light intensity change of different amount of Ag nanoparticle modification, and Fig. 9(b) is a graph of the corresponding light-confinment efficiency of ZnO micron line. It can be seen from the figure that as the decorative surface of Ag nanoparticles increases, the light field confinement efficiency improves significantly. Through calculation for the six-face-modified ZnO micro-wires with Ag nanoparticles, the light field-confinement efficiency of the ZnO micro-wires increased by 6.72% and the light field in the cavity is enhanced significantly.
图 9 (a) 随装饰面数目变化的光强曲线；(b)随装饰面数目变化的光限制效率图
Figure 9. (a)Light intensity curve as a function of the number of decorative surface; (b)light confinement efficiency map as a function of the number of decorative surfaces
This is because the evanescent wave field intersects with the plasma exciton localized on the surface of the ZnO, providing an environment in which the surface plasmon resonance mode is coupled with the traditional F-P resonant cavity mode. Thus, the energy distribution and oscillation process in the microcavity are regulated.
The resonance pattern of the decorative Ag particles modified micro-wires was also simulated by the resonance mode analysis group, as shown in Fig. 10. It can be seen that the multi-stage resonance peak in the ZnO micron-wire resonator modified by Ag nanoparticles becomes strong and increases from the original three resonance peaks to five, and that the stable resonance mode in the micro-wire increases significantly. This is mainly due to the surface plasmon resonance effect produced by Ag nanoparticles and the optical field confinement effect of the intersection region, allowing the ZnO cavity with same diameter to have a stronger light field. This also reduces the band edge loss, helps improve the performance of the laser, and achieves more resonant modes.
贵金属材料介电常数具有负实部和小的正虚部，且外层自由电子在入射光场的激发下易发生振荡，当自由电子振荡频率与入射光场的频率相匹配时产生金属表面等离激元共振。根据金属材料结构形式的不同，这种共振形式一般分为两类：第一种为LSPR，入射光场与金属纳米颗粒相互作用，导带上的自由电子在金属表面发生集体振荡, 产生的表面等离子激元波被局域在纳米结构附近(如图 2(a))。共振状态下, 电磁场的能量有效地转化为金属表面自由电子振动能，进而产生LSPR效应[19-20]。第二种是传播表面等离子共振(SPR)，光场作用在连续金属表面，从而在金属表面产生传输型的表面等离子激元共振波(图 2(b))。本文利用LSPR共振耦合效应产生近场局域增强，有效提高ZnO微米线结构光场限域能力，进而增强F-P受激辐射。
为了研究ZnO微米线腔径变化对其光学损耗造成的影响，本文对正六面体ZnO微米线进行三维空间时域有限差分法的计算模拟，利用不同横截面的电场分布及腔内腔外电场强度的比值，分析ZnO微米线腔径变化与光学损耗之间的联系。在仿真模型中，将ZnO微米线垂直放置在衬底上，微米线的高度h设置为2.1 μm，使用单频光源，并在仿真模拟时利用(n，k)材料模型创建ZnO材料，通过文献估算设置其折射率n=2.06，k=1的对应模型来模拟ZnO微米线形成的谐振腔里模式数量的变化。选择完美匹配层(Perfectly Matched Layer, PML)作为其模拟边界条件。方法是在FDTD区域内的截断边界处设置一种特殊介质层，使得该层介质的波阻抗与相邻介质波阻抗达到完全匹配，从而确保入射波在分界面上不发生反射，穿过分界面最后进入PML层，而且PML层存在有耗介质，可以使进入PML层的透射波迅速衰减，尽管PML的厚度有限，但它对入射波却有很好的吸收效果。因此，PML在实际计算中也是一种常被使用的吸收边界。
利用建立的模型分别计算直径为1.6、2、4、6和8 μm的ZnO微米线在E(x，y)截面及E(x，z)截面的电场分布，如图 3所示。图中可以清晰反映出ZnO微米线具有光波导特性，以及腔内F-P模式振荡。通过电场分布可以发现光被限制在ZnO微米线中并形成驻波沿腔体传播。当腔径为1.6和2 μm时，观察到腔外具有很强的电场分布，这说明光在腔内传播的过程中会产生严重的能量损失。随着腔径的逐步增加，腔内会加入更多的能量来实现激光的共振。当腔径增加到6和8 μm时，绝大部分光可以被锁定在腔中并可观察到完美的光学共振模式。通过ZnO微米线横截面上的电场分布可以发现，随着ZnO微米线腔径的增加，其对光场的限域能力增强，光学损耗降低。这一现象可以根据激光谐振腔理论公式来分析：
其中, D为微腔横向尺寸(即微米线的直径), h为腔长, λ为波长。N为菲涅耳数，它是衍射现象中的一个特征参数, 其数值越大表征衍射损耗越小。通过公式(3)可知，在腔长h不变的情况下，在同一波长λ下，随着微米线直径D的增加，菲涅耳数N逐步增大，进而导致谐振腔内的衍射损耗降低。
图 4为不同腔径下光强的分布曲线及光限制效率。通过图 4(a)可以观察到不同腔径下腔内光场分布的变化趋势。随着腔径的增加，腔内光强逐渐增强，而腔外的光场强度逐渐减弱。图 4(b)给出不同腔径对应的光限制效率变化曲线。这里的光限制效率定义为：ZnO微米线腔内光场强度面积占总的光场强度面积的百分比。可以发现当直径为1.6 μm时，光场能量损耗最大，腔内几乎没有光场存在。相反，随着微腔直径的增加，光场限制效率达到了99%左右，说明直径达到一定程度时，光场能量几乎全部限制在腔内传播。因此利用ZnO微米线结构的光场限域能力可以构建F-P微腔模式振荡，并且通过改变微米线直径可以有效调控光学损耗特性。
通过图 5可以看出，随着ZnO直径的增加，谐振腔中逐渐出现多个谐振模式。当D=2 μm时，谐振谱中共有3个谐振峰，其中f=800 THz处表现出一个最强峰。当D增加到4 μm时，出现3个较强的谐振峰，且最强峰位置基本不变。随着微米线直径继续增加到6和8 μm时，多级谐振峰明显增多，谐振强度也相应增加。有研究表明随着ZnO直径的增加可以使腔内谐振模式的数量增加，而小腔径ZnO微米线具备实现单模激光的可能性。形成单模激光的机理主要是光学损耗引起的边模抑制，因此对小腔光学损耗的研究至关重要。为了改善传统光学微腔系统的性能，提出利用金属局域表面等离子激元修饰微腔结构，来实现微腔侧面倏逝场损耗光与金属局域等离子激元波耦合的一种光学腔模式。这将有望利用金属局域表面等离子激元加强光与物质相互作用并提供一个亚波长范围的限制波导模式。
利用Ag纳米颗粒修饰ZnO微米线激发形成LSPR电场，可以在微米线表面实现局域场增强及光场限域效应[24-25]。对于小腔径ZnO微米线光学损耗较大的问题，可以通过金属LSPR进行有效抑制。为此，应用Ag纳米颗粒修饰直径为2 μm的ZnO微米线表面，并通过x-y截面电磁场监视器记录电场的变化。图 6为Ag修饰ZnO微米线2、4、6个面的结构示意图及光场分布图。通过x-y截面的电场分布图可以看出，经Ag纳米颗粒修饰后腔内光场明显增强，这证明LSPR对光场具有限域能力。由于Ag的功函数为4.35 eV，ZnO的电子亲和能为4.26 eV，在Ag/ZnO交界面上电子容易相互转移。此外ZnO发射能量接近于Ag表面等离子激元共振的能量，因此光场强度的增强是Ag纳米颗粒的局域表面等离激元共振与ZnO微米棒受激发射的光子相耦合的结果。
为了解释及量化金属LSPR对ZnO微米线结构光学损耗的影响，图 9仿真计算了Ag纳米颗粒分别修饰ZnO微米线2、4、6个侧面下光强的分布曲线及光限制效率，如图 9所示。图 9(a)为不同数量Ag纳米颗粒修饰面对应的光场强度变化曲线图；图 9(b)为对应的ZnO微米线光限制效率的曲线图。通过图 9可以看出，随着Ag纳米颗粒修饰面的增加，光场限制效率有了明显的提高。经过6面修饰Ag纳米颗粒后，ZnO微米线的光场限制效率提高了6.72%，并且腔内的光场有了明显的增强。这是因为倏逝波光场与局域在ZnO表面的等离子激元形成交叉区，提供了一个实现表面等离子激元共振模式与传统的F-P谐振腔模式耦合的环境，从而调节微腔中的能量分布和振荡过程。
Resonant mode of Fabry-Perot microcavity regulated by metal surface plasmons
摘要: 目前，利用氧化锌（ZnO）微纳米线结构形成具有自然谐振腔的紫外激光器件引起国内外广泛关注。针对ZnO本征缺陷导致器件发光及稳定性不足等问题，开展金属局域等离子激元局域场发光增强方面的研究，对ZnO基紫外激光器件的应用具有十分重要的意义。本文通过理论仿真构建氧化锌微米线结构模型，对微腔光学损耗及Fabry-Perot（F-P）谐振腔模式演化进行了理论分析。得到ZnO微腔直径变化与F-P谐振模式演化、光学损耗和光强分布的关系。在此基础上通过金属Ag纳米颗粒对ZnO微米线6个表面进行修饰，发现金属局域表面等离子激元共振耦合效应对微腔周围的损耗光有明显的抑制作用，并且在金属与微腔的交叉区通过共振耦合效应实现局域场增强。模拟结果表明，在损耗较大的微腔表面修饰Ag纳米颗粒以后，光场限域能力提高6.72%，而在金属颗粒之间沿X轴方向产生二次耦合现象，其电场强度更有2倍的增强效果。Abstract: At present, the use of zinc oxide(ZnO) micro-nanowire structures in ultraviolet laser devices with natural resonant cavities has attracted wide attention at home and abroad. Aiming at the problems of the luminescence and stability caused by ZnO intrinsic defects, research on the local field luminescence enhancement of plasmons is very important for the application of ZnO-based UV laser devices. In this paper, ZnO micro-wire structure model is constructed by theoretical simulation and the micro-cavity optical loss and Fabry-Perot(F-P) resonant cavity mode evolution are theoretically analyzed. The relationship between the diameter change of ZnO microcavity and the evolution of the F-P resonance mode, optical loss and light intensity distribution is obtained. On this basis, the six surfaces of ZnO microwires are modified by metal Ag nanoparticles. It is found that the resonance coupling effect of metal local surface plasmons significantly inhibited the loss of light around the microcavity and the local field enhancement is realized by the resonance coupling effect at the intersection of the metal and the microcavity. The simulation results show that after the surface of the microcavity is modified with Ag nanoparticles, the confinement ability of the optical field increased by 6.72%, while the secondary coupling occurs along the X axis between the metal particles, and the electric field intensity is enhanced by 2 times.
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