WO2025035986A1 - 补偿波传播中的损耗的方法 - Google Patents

补偿波传播中的损耗的方法 Download PDF

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WO2025035986A1
WO2025035986A1 PCT/CN2024/103061 CN2024103061W WO2025035986A1 WO 2025035986 A1 WO2025035986 A1 WO 2025035986A1 CN 2024103061 W CN2024103061 W CN 2024103061W WO 2025035986 A1 WO2025035986 A1 WO 2025035986A1
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frequency
wave
wave propagation
losses
compensating
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French (fr)
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张霜
管福鑫
戴庆
郭相东
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National Center for Nanosccience and Technology China
University of Hong Kong HKU
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National Center for Nanosccience and Technology China
University of Hong Kong HKU
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    • H—ELECTRICITY
    • H01—ELECTRIC ELEMENTS
    • H01P—WAVEGUIDES; RESONATORS, LINES, OR OTHER DEVICES OF THE WAVEGUIDE TYPE
    • H01P3/00—Waveguides; Transmission lines of the waveguide type
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04B—TRANSMISSION
    • H04B11/00—Transmission systems employing ultrasonic, sonic or infrasonic waves
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04B—TRANSMISSION
    • H04B15/00—Suppression or limitation of noise or interference

Definitions

  • the present invention relates to the field of wave propagation, and in particular to a method for compensating for loss in wave propagation by synthesizing complex frequency waves.
  • Polaritons are a type of surface electromagnetic wave formed by the interaction between particles on the surface of a medium and the incident light field. They include surface plasmons based on electron-photon hybridization and phonon polaritons based on phonon-photon hybridization. The energy density of this polariton electromagnetic wave is higher than that of the incident light field, so it can produce a strong electric field enhancement effect on the surface of the medium. Polaritons can compress and focus the light field to a very small scale, thus enabling optical information transmission and processing at the nanoscale.
  • the present invention proposes a method for compensating for the loss in wave propagation by synthesizing complex frequency waves, comprising:
  • the complex frequency is selected based on the frequency of the wave and the dielectric constant of the medium to at least partially cancel the imaginary part of the in-plane wave vector in the wave propagation;
  • a complex frequency response is obtained based on the plurality of real frequency responses and the complex frequency to compensate for losses in wave propagation.
  • the step of obtaining a complex frequency response based on a plurality of real frequency responses and the complex frequency to compensate for the loss in wave propagation comprises calculating the complex frequency response by the following formula:
  • F( ⁇ ′ i ,r) is the real frequency response
  • ⁇ ′ i is the real frequency
  • i is a positive integer
  • t is the time
  • ⁇ is the interval frequency of the real frequency ⁇ ′ i
  • r is the position of the wave.
  • the wave is a polariton and the measurement result of the wave propagation is the electric field strength of the wave.
  • the complex frequency ⁇ is the center frequency and ⁇ is the imaginary gain, which is chosen so that the imaginary part of the in-plane wave vector is zero.
  • the dielectric constant of the medium in which the wave propagates is described by the Drude model as ⁇ m is the dielectric constant of the medium, ⁇ r is the dielectric constant when the frequency ⁇ tends to infinity, ⁇ p is the plasmon frequency of the dielectric layer, and ⁇ is the damping rate of the dielectric layer.
  • the present invention also provides a computer-readable storage medium, which contains a computer program, and the computer program can be executed by a processor to implement the steps of the above method.
  • the present invention also provides an electronic device, comprising:
  • a memory for storing one or more executable instructions
  • the one or more processors are configured to implement the steps of the above method by executing the one or more executable instructions.
  • the method of compensating the loss in wave propagation by synthesizing complex frequency waves of the present invention can compensate the loss in wave propagation and realize lossless wave propagation by synthesizing complex frequency waves with complex frequencies having virtual gain and multiple real frequency responses.
  • FIG. 1 shows a flow chart of a method for compensating for losses in wave propagation by synthesizing complex frequency waves according to an embodiment of the present invention.
  • FIG. 2A shows the excitation field distribution of plasmon polaritons (SPPs).
  • FIG2B shows the corresponding Fourier distribution of SPPs.
  • FIG2C shows the dispersion under complex frequency excitation.
  • FIG2D shows Dynamic evolution of the complex frequency excitation at different times.
  • FIG3A shows an experimental setup based on the scattering scanning near-field optical microscopy technique.
  • FIG3B shows the field distribution for two real frequencies.
  • FIG3C shows the field distributions for two complex frequencies.
  • Figure 4 shows atomic force microscopy images showing the near-field distribution at the MoO3 interface measured at different frequencies.
  • FIG5A shows the resultant electric field distribution of the complex frequency (910-6.5i) cm -1 at different times, with a time step of 0.53 ps.
  • FIG5B shows imaging patterns with different synthetic frequency numbers, wherein the center frequency is fixed at 910 cm ⁇ 1 and the frequency interval is fixed at 1 cm ⁇ 1 .
  • FIG6A shows an atomic force microscope image of an experimental sample.
  • Polaritons including surface plasmon polaritons (SPPs) and phonon polaritons (PhPs), have become the first choice for building nanophotonic circuits, thereby promoting the development of ultracompact and high-speed optical devices.
  • Utilizing polaritons in nanophotonics provides a way to overcome the diffraction limit of light, thereby manipulating light at the nanoscale.
  • intrinsic losses hinder many loss-sensitive polariton-based applications, including waveguides, biosensing, subdiffraction-limited imaging, and plasmonic structured illumination microscopy. Intrinsic losses negatively affect polaritons in two main ways: (1) the propagation distance is significantly shortened; (2) the dispersion curve of polaritons far away from the light cone is significantly blurred, which seriously affects the subwavelength applications of polaritons.
  • Intrinsic losses are caused by the imaginary part of the dielectric in the material, such as Ohmic losses in plasmon systems and lattice vibration relaxation processes in phonon-plasmon systems.
  • the method is to add an external gain medium.
  • it is very challenging to completely offset the plasma loss with gain, and gain compensation is susceptible to noise and instability.
  • the present invention proposes a method for achieving lossless propagation of waves by synthesizing complex frequency waves, which can achieve almost lossless propagation of highly confined polaritons.
  • Typical materials supporting plasmon/phonon polaritons can be described by the Drude model, the Lorentz model, or the multi-Lorentz model, which includes loss terms that lead to the imaginary part of the in-plane wave vector.
  • the resulting complex frequency It is possible to ensure that the imaginary part of the in-plane wave vector k is zero, which means that there is no spatial growth or attenuation of the SPP along the interface. Therefore, in order to achieve lossless propagation, it is necessary to find a suitable value for the imaginary part of the complex frequency.
  • ⁇ m ⁇ r - ⁇ p 2 /( ⁇ 2 +i ⁇ - ⁇ 0 2 ), where ⁇ 0 is the resonant frequency of the model.
  • the complex frequency can be obtained Virtual The ⁇ is:
  • the complex frequency makes the imaginary part of the in-plane wave vector equal to 0, which can offset the intrinsic loss and achieve lossless propagation of the wave.
  • the complex frequency The value of the imaginary part ⁇ is set so that the imaginary part of the in-plane wave vector approaches 0, so as to at least partially compensate for the loss and allow the wave to propagate farther.
  • the truncated complex frequency wave can be expanded to the real frequency domain It shows a Lorentz line shape.
  • the final electric field in the time domain can be expressed as: in are the Fourier coefficients.
  • any response in a system excited by a truncated complex frequency wave (such as electric field strength, magnetic field strength, etc.) can be expressed as the integral of the real frequency response in the quasi-steady state
  • the above integral can be discretized as:
  • ⁇ ′ i is the real frequency
  • F( ⁇ ′ i ,r) is the real frequency response under the real frequency ⁇ ′ i , where i is a positive integer and r is the position of the wave, for example, a one-dimensional or two-dimensional coordinate. It can be calculated based on the measurement and/or calculation results F( ⁇ ′ i , ⁇ ) at multiple real frequencies ⁇ ′ i
  • the discrete frequencies ⁇ ′ 1 , ⁇ ′ 2 , ..., ⁇ ′ i are equally spaced, with an interval of ⁇ .
  • the decay length of highly confined polaritons can be significantly enhanced by complex frequency methods, restoring nearly lossless propagation limited only by the noise level.
  • the method of increasing the decay length by synthesizing complex frequency waves can also be applied to other frequency ranges and other types of waves, including but not limited to acoustic waves and elastic waves.
  • FIG1 shows a flow chart of the method, and the method comprises:
  • Step S101 Obtaining a real frequency response based on the measurement result of wave propagation.
  • the measurement results of wave propagation can be the strength of the wave at a certain location, such as the amplitude of the wave, the electric field strength, the magnetic field strength, etc.
  • a plurality of real frequency responses are obtained based on the measurement results of a plurality of wave propagations over a period of time.
  • Step S102 Selecting a complex frequency based on the frequency of the wave and the dielectric constant of the medium to at least partially cancel out the imaginary part of the in-plane wave vector in wave propagation.
  • the complex frequency is selected so that the imaginary part of the in-plane wave vector is zero or approaches zero.
  • the value of the imaginary part of the complex frequency can be calculated by the above formula (5) and formula (6).
  • Step S103 obtaining a complex frequency response based on a plurality of real frequency responses and the complex frequency to compensate for the loss in wave propagation.
  • the complex frequency response may be calculated based on the above formula (6).
  • the complex frequency response may be calculated based on an integral formula, which may be, for example, The lower limit of integration a is less than the complex frequency The real part of the integral, the upper limit b is greater than the complex frequency The real part of .
  • the above formula (6) and the above integral formula can be modified or altered to achieve the effect of offsetting or compensating the loss in wave propagation, so the above modifications or alterations all belong to the protection scope of the present invention. In other words, any method of obtaining a complex frequency response based on a combination of multiple real frequency responses is within the protection scope of the present invention.
  • the genus supports SPPs below the plasmon frequency.
  • An infinitely long antenna placed on a planar plasmonic metal acts as an SPP source under light irradiation.
  • Figure 2A shows the excitation field distribution of the SPP. As the frequency increases, the propagation distance of the SPP decreases, which means that higher frequency SPPs have a shorter propagation length due to stronger confinement to the interface.
  • Figure 2B shows the corresponding Fourier distribution of the SPP. The results show that the high-frequency mode becomes blurred and eventually becomes invisible. Therefore, it is necessary to balance the plasma loss by utilizing the complex frequency to achieve lossless propagation.
  • the complex frequency field distribution can be synthesized by a linear combination of the real frequency field distribution,
  • the complex frequency The value of ⁇ is obtained by formula (4), where E is the electric field strength of the wave and r is the position of the wave, such as a one-dimensional or two-dimensional coordinate.
  • Figure 2C shows the dispersion under complex frequency excitation, which completely restores the dispersion under higher frequencies.
  • Figure 2D shows The dynamic evolution of the complex frequency excitation at different times (the real part of which is represented by the bottom dashed line in Figure 2C). As time increases, the propagation distance extends linearly, but the amplitude remains uniform at different locations. This provides a visualization of the lossless propagation of surface plasmon polaritons at complex frequencies, which is in sharp contrast to the propagation at real frequencies.
  • the loss compensation is applicable not only to metals described by the Drude model, but also to materials with more complex dielectric functions, such as van der Waals materials supporting PhP.
  • materials with more complex dielectric functions such as van der Waals materials supporting PhP.
  • a hexagonal boron nitride (hBN) film is used, which supports in-plane isotropic PhPs and whose intrinsic losses can be compensated by complex frequencies to observe the lossless propagation of phonon polaritons.
  • FIG3A shows the experimental setup based on the scattering scanning near-field optical microscopy (s-SNOM) technique.
  • s-SNOM scattering scanning near-field optical microscopy
  • a long gold antenna placed on the hBN film is used to emit the 1D PhP.
  • the electric field distribution is measured in the frequency range of 1421 cm -1 to 1503 cm -1 with a step size of 2 cm -1 .
  • Two field distributions with real frequencies of 1451 cm -1 and 1477 cm -1 are selected as the center frequencies, and the synthetic and Complex frequency response at .
  • a complex frequency approach is used to study the temporal evolution of a more complex field distribution supported by a van der Waals crystalline ⁇ -MoO 3 (MoO 3 ) film, which is highly anisotropic and supports natural in-plane hyperbolic polaritons.
  • a gold antenna is placed on the MoO 3 film to excite the PhP.
  • Figure 4 shows an atomic force microscope (AFM) image showing the near-field distribution at the MoO 3 interface measured by SNOM at different frequencies.
  • the overall frequency range is 891-943 cm -1 with an interval of 1 cm -1 .
  • the field distribution changes show a characteristic hyperbolic propagation behavior with a concave wave front. As the frequency increases, the wavelength decreases, the field confinement is stronger, and the propagation attenuation is stronger. At all measured frequencies, the decay length of the polaritons is less than two wavelengths due to the significant intrinsic losses of the material.
  • Figure 5A shows the synthetic electric field distribution at different times for the complex frequency (910-6.5i) cm -1 , with a time step of 0.53ps and a total number of frequencies of 53.
  • the noise is largely smoothed by summing the multi-frequency signals.
  • the two-dimensional field diagram reaches an optimum at around 4.24ps, with a propagation length of 8.7 ⁇ m (in the vertical direction), while the propagation length corresponding to the center real frequency is only 1.6 ⁇ m. Therefore, the propagation length under this complex frequency method is increased by more than four times compared to the center real frequency.
  • FIG5B shows imaging patterns with different numbers of synthetic frequencies, where the center frequency is fixed at 910 cm -1 , the frequency interval is fixed at 1 cm -1 , and the time snapshot is fixed at 4.24 ps. It is obvious that the field diagram of 9 frequency points has been significantly improved compared with the real frequency. As the number of frequencies increases to 21, the attenuation length increases significantly. However, further increasing to 39 does not bring significant improvement.
  • the complex frequency method is used to study the interference behavior of PhP.
  • Two circular antennas with different diameters (0.8 ⁇ m and 3 ⁇ m) are made on the MoO 3 film to excite phonon polaritons.
  • the PhP emitted from the two antennas propagate in opposite directions, but due to attenuation, they do not form discernible interference fringes.
  • the new technology could be used in the development of photonic integrated circuits, where the ability to transmit signals over long distances is critical to ensuring circuit performance and reliability, and could also benefit applications including photonic circuits, waveguides, and plasmon/phononic structure illumination microscopy.
  • the method of compensating the loss in wave propagation by synthesizing complex frequency waves of the present invention can compensate the loss in wave propagation by synthesizing complex frequency waves with complex frequencies having virtual gain and multiple real frequency responses to achieve lossless wave propagation.

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Abstract

提供一种补偿波传播中的损耗的方法，包括：基于波传播的测量结果获得实频率响应；基于波的频率和介质的介电常数选择复频率，以至少部分抵消波传播中的面内波矢的虚部；基于多个实频率响应和所述复频率获得复频率响应以补偿波传播中的损耗。

Description

补偿波传播中的损耗的方法 技术领域
本发明涉及波传播领域，尤其涉及一种通过合成复频波补偿波传播中的损耗的方法。
背景技术
极化激元是由于介质表面上的粒子与入射光场相互作用，形成的一种表面电磁波。其中包含基于电子-光子杂化的表面等离激元和基于声子-光子杂化的声子极化激元。这种极化激元电磁波的能量密度高于入射光场，因此可以在介质表面上产生强烈的电场增强效应。极化激元能够将光场压缩聚焦至很小尺度，因而可以实现在纳米尺度上的光信息传输和处理。受益于亚波长光限制，表面等离极化激元和声子极化激元提供了一种超越传统光学衍射极限的方法，并有助于高效能量存储、局部场增强以及高灵敏度。但是损耗限制了极化激元的传播长度，从而限制了极化激元的实际使用。虽然优化制造技术有助于避免不完美结构造成的散射损耗，但导致热量产生的本征损耗无法消除。因此，如何克服本征损耗以实现波的无损传播是本领域亟需解决的问题。
发明内容
基于现有技术的上述问题，本发明提出一种通过合成复频波补偿波传播中的损耗的方法，包括：
基于波传播的测量结果获得实频率响应；
基于波的频率和介质的介电常数选择复频率，以至少部分抵消波传播中的面内波矢的虚部；
基于多个实频率响应和所述复频率获得复频率响应以补偿波传播中的损耗。
在一个实施例中，基于多个实频率响应和所述复频率获得复频率响应以补偿波传播中的损耗的步骤包括通过以下公式计算复频率响应：
其中，F(ω′i,r)为实频率响应，为复频率响应，ω′i为实频率，i为正整数，为复频率，t为时间，Δω为实频率ω′i的间隔频率，r为波的位置。
在一个实施例中，所述波为极化激元，波传播的测量结果为波的电场强度。
在一个实施例中，所述复频率ω为中心频率，β为虚拟增益，其被选择为使得面内波矢的虚部为零。
在一个实施例中，波在其中传播的介质的介电常数由德鲁德模型描述为εm为介质的介电常数，εr为频率ω趋向于无穷时的介电常数，ωp为介质层的等离激元频率，γ为介质层的阻尼率。
在一个实施例中，
在一个实施例中，波在其中传播的介质的介电常数由洛伦兹模型描述为εm＝εr-ωp 2/(ω2+iωγ-ω0 2)，εm为介质的介电常数，εr为频率ω趋向于无穷时的介电常数，ωp为介质层的等离激元频率，γ为介质层的阻尼率，ω0为模型的共振频率。
在一个实施例中，
本发明还提供一种计算机可读存储介质，其上包含有计算机程序，所述计算机程序能够被处理器执行以实现上述方法的步骤。
本发明还提供一种电子设备，其包括：
一个或多个处理器；以及
存储器，用于存储一个或多个可执行指令；
所述一个或多个处理器被配置为经由执行所述一个或多个可执行指令以实现上述方法的步骤。
本发明的通过合成复频波补偿波传播中的损耗的方法，通过具有虚拟增益的复频率以及多个实频率响应合成复频波，能够补偿波传播中的损耗，实现波的无损传播。
附图说明
图1示出了根据本发明一个实施例的通过合成复频波补偿波传播中的损耗的方法的流程图。
图2A示出了等离极化激元(SPP)的激发场分布。
图2B示出了SPP相应的傅里叶分布。
图2C示出了复频率激励下的色散。
图2D示出了处的复频率激励在不同时刻的动态演化。
图2E示出了在t＝5.5×10-2ps处相应的空间场分布。
图3A示出了基于散射式扫描近场光学显微镜技术的实验装置。
图3B示出了两个实频率的场分布。
图3C示出了两个复频率的场分布。
图4示出了原子力显微镜图像，其示出了在不同频率下测量的MoO3界面上的近场分布。
图5A示出了复频率(910-6.5i)cm-1在不同时间的合成电场分布，时间步长为0.53ps。
图5B示出了不同合成频率数的成像图案，其中，中心频率固定为910cm-1，频率间隔固定为1cm-1。
图6A示出了实验样品的原子力显微镜图像。
图6B示出了频率为f＝990cm-1时电场分布的幅度和实部。
图6C示出了频率为f＝(990-2i)cm-1时天线之间的干涉图案。
具体实施方式
为了使本发明的目的、技术方案以及优点更加清楚明白，下面将结合附图通过具体实施例对本发明作进一步详细说明。应当注意，本发明给出的实施例仅用于说明，而不限制本发明的保护范围。
极化激元，包括表面等离极化激元(SPP)和声子极化激元(PhP)，已成为构建纳米光子电路的首选，从而促进超紧凑和高速光学器件的开发。在纳米光子学中利用极化激元提供了克服光衍射极限的途径，从而可以在纳米尺度上操纵光。然而，本征损耗阻碍了许多对损耗敏感的基于极化激元的应用，包括波导、生物传感、亚衍射极限成像和等离子体结构照明显微镜。本征损耗主要通过两个方面对极化激元产生负面影响：(1)传播距离显著缩短；(2)远离光锥的极化激元的色散曲线显著模糊，严重影响极化激元的亚波长应用。
本征损耗是由材料中电介质的虚部引起的，例如等离激元系统中的欧姆损耗以及声子-等离激元系统中的晶格振动弛豫过程。抵消损耗最常用的 方法是加入外部增益介质。然而，用增益完全抵消等离子体损耗是非常具有挑战性的，并且增益补偿容易受到噪声和不稳定的影响。经过研究，本发明提出了一种通过合成复频波实现波的无损传播的方法，能够实现高度受限的极化激元的几乎无损耗传播。
支持等离/声子极化激元的典型材料可以通过德鲁德模型(Drude model)、洛伦兹模型(Lorentz model)或多洛伦兹模型(multi-Lorentz model)进行描述，其中包含导致面内波矢虚部的损耗项。
为简单起见，以界面处的SPP为例进行说明，面内波矢k的数学解为其中，ω为波频率，c为光速，εm为介质的介电常数。在一个实施例中，假设等离子体金属的介电常数εm由德鲁德模型描述，即其中，εr为频率ω趋向于无穷时的介电常数，ωp为介质层的等离激元频率，γ为介质层的阻尼率。其中，分母中的耗散项引入了SPP传播的衰减。通常认为通过用特定的复频率补偿德鲁德模型中的损耗，就可以实现SPP的无损传播。然而，这并不完全准确，将复频率代入面内波矢k的公式中后，面内波矢k中仍然存在一个与频率相关的项其会导致波矢k中出现虚部。因此，波无损传播的条件应为面内波矢k的虚部为0，即亦即得到：
其中，ω对应于中心频率，β表示虚拟增益，ω>>β，ωp>>γ。通过忽略二阶小量，得到：
将等式(2)代入等式(1)得到：
由此得到：
所得到的复频率能够确保面内波矢k的虚部为零，这意味着SPP沿界面没有空间增长或衰减。因此，为了实现无损传播，需要找到合适的复频率虚部值。
在另一实施例中，假设等离子体金属的介电常数εm由洛伦兹模型描述，即εm＝εr-ωp 2/(ω2+iωγ-ω0 2)，ω0为模型的共振频率。可以得到复频率的虚 部β为：
通过上述公式可以看出，通过选择复频率的虚部β的值，使得面内波矢的虚部为0，可以抵消本征损耗，实现波的无损传播。在另一实施例中也可以选择复频率的虚部β的值以使得面内波矢的虚部趋近于0，以至少部分补偿损耗，使得波可以传播得更远。
本领域技术人员应当理解，上述公式(4)和公式(5)仅仅是示例性的，并不构成对本发明的限制。也可以使用其他的模型描述介电常数εm，从而得到不同的复频率的虚部的值。因此，在实际应用中，可以使得面内波矢的虚部为0或趋近于0，计算得到的复频率能够至少部分抵消面内波矢的虚部，从而至少部分抵消本征损耗，减少波的传播损耗。
在确定了选择复频率的虚部β的值之后，下面将说明如何利用该值生成复频波，以实现波的无损传播或者使其传播得更远。
具有时间衰减的复频波在时间接近负无穷时是发散的。由于直接生成复频波非常困难，因此本发明人使用一种新的方法来合成截断的复频波，表示为其中ET(t)为截断的复频波的强度；θ(t)是避免能量发散的时间截断函数，其中对于时间t≥0，θ(t)＝1，对于时间t<0，θ(t)＝0。
通过使用傅里叶变换，截断的复频波可以展开至实频域其表现为洛伦兹线型。在时间域的最终电场可以表示为：其中为傅里叶系数。自然地，由截断的复频波激发的系统中的任何响应(例如电场强度、磁场强度等)都可以表示为处于准稳态的实频响应的积分实际上，对于足够宽的光谱范围，上述积分可以离散化为：
其中，可以是表征波传播的任何物理量，例如电场强度、磁场强度等。ω′i为实频率，F(ω′i,r)是实频率ω′i下的实频率响应，其中i为正整数，r 为波的位置，例如一维或二维坐标。可以基于多个实频率ω′i下的测量和/或计算结果F(ω′i,β)来计算离散频率ω′1，ω′2，……，ω′i是等间隔的，其间隔为Δω。复频率将复频率代入上述公式(6)中，即可至少部分抵消本征损耗。
基于上述内容，可以通过复频率方法显著增强高度受限的极化激元的衰减长度，恢复仅受噪声水平限制的近乎无损的传播。然而本领域技术人员应当理解，通过合成复频波增加衰减长度的方法也可以应用到其他频率范围和其他类型的波，包括但不限于声波和弹性波。
基于上述内容，本发明提供了一种通过合成复频波补偿波在介质传播中的损耗的方法，图1示出了该方法的流程图，该方法包括：
步骤S101：基于波传播的测量结果获得实频率响应。
波传播的测量结果可以是在一定位置处波的强度，例如波的振幅，电场强度，磁场强度等。
其中，基于一段时间内多个波传播的测量结果获得多个实频率响应。
步骤S102：基于波的频率和介质的介电常数选择复频率，以至少部分抵消波传播中的面内波矢的虚部。
在一个实施例中，复频率被选择为使得面内波矢的虚部为零或趋近于零。在一个实施例中，可以通过上述公式(5)和公式(6)计算得到复频率的虚部的值。
步骤S103：基于多个实频率响应和所述复频率获得复频率响应以补偿波传播中的损耗。
在一个实施例中，可以基于上述公式(6)计算复频率响应。在另一个实施例中，可以基于积分公式计算复频率响应，积分公式可以例如是其中积分下限a小于复频率的实部，积分上限b大于复频率的实部。本领域技术人员应当理解，可以对上述公式(6)以及上述积分公式进行修改或改动，均能达到抵消或补偿波传播中的损耗的效果，因此上述修改或改动均属于本发明的保护范围。也就是说，凡是基于多个实频率响应的组合获得复频率响应的方法，均在本发明的保护范围内。
在一个实施例中，假设由德鲁德模型描述的真实等离子体金属的介电常数为其中ωp＝1.442×1016Hz，γ＝3×1014Hz。该金 属支持低于等离激元频率的SPP。放置在平面等离子体金属上的无限长天线在光照射下充当SPP源。图2A示出了SPP的激发场分布。随着频率增加，SPP的传播距离减小，这意味着较高频率的SPP由于对界面的限制更强，因此具有更短的传播长度。图2B示出了SPP相应的傅里叶分布。结果表明，高频模式变得模糊并最终变得不可见。因此，需要通过利用复频率来平衡等离子体损耗，以实现无损传播。可以通过实频率场分布的线性组合来合成复频率场分布，
其中复频率β的值由公式(4)获得，E为波的电场强度，r为波的位置，例如一维或者二维坐标。图2C示出了复频率激励下的色散，完全恢复了较高频率下的色散。图2D示出了(其实部由图2C中底部虚线表示)处的复频率激励在不同时刻的动态演化。随着时间的增加，传播距离线性延伸，但幅度在不同位置保持均匀。这可视化地提供了表面等离极化激元在复频率下的无损传播，其与实频率下的传播形成鲜明对比。图2E示出了在t＝5.5×10-2ps处相应的空间场分布，图2E的上半部分示出了的空间场分布，图2E的下半部分示出了(其实部由图2C中上部虚线表示)的空间场分布。如图2E所示，与实频率情况相比，衰减要慢得多。
损耗的补偿不仅适用于德鲁德模型描述的金属，还适用于具有更复杂介电功能的材料，例如支持PhP的范德华材料。对于PhP，也会存在满足Im(k)＝0的特定复频率解。
在一个实施例中，使用六方氮化硼(hBN)薄膜，其支持面内各向同性PhP，并且可以利用复频率来补偿其本征损耗，以观察声子极化激元的无损传播。图3A示出了基于散射式扫描近场光学显微镜(s-SNOM)技术的实验装置。放置在hBN薄膜上的长金天线用于发射1D PhP。在1421cm-1至1503cm-1的频率范围内测量电场分布，步长为2cm-1。选择实频率为1451cm-1和1477cm-1的两个场分布作为中心频率，合成 和处的复频率响应。相应的虚部表示针对每个频率的优化值。图3B示出了两个实频率的场分布，图3C示出了两个复频率的场分布。实验结果表明，实频率下的传播衰减很强，复频率下极化激元的传播几乎是非耗散的。
在一个实施例中，使用复频率方法来研究由范德华晶体α-MoO3(MoO3)薄膜支持的更复杂场分布的时间演化，该薄膜具有高度各向异性并支持自然面内双曲极化激元。将金天线放置在MoO3薄膜上以激发PhP。图4示出了原子力显微镜(AFM)图像，其示出了在不同频率下SNOM测量的MoO3界面上的近场分布。总体频率范围为891-943cm-1，间隔为1cm-1。场分布变化表现出具有凹波前的特征双曲线传播行为。随着频率的增加，波长减小，场限制更强，同时传播衰减更强。在所有测量的频率下，由于材料显著的本征损耗，极化激元的衰减长度小于两个波长。
图5A示出了复频率(910-6.5i)cm-1在不同时间的合成电场分布，时间步长为0.53ps，总频率数为53。对于较小的t，噪声在很大程度上通过多频率信号的求和而被平滑。随着时间的推移，波向上传播得越来越远，远远超出了实频率下两个波长的衰减长度。二维场图在4.24ps左右达到最佳，传播长度为8.7μm(沿垂直方向)，而中心实频率对应的传播长度仅为1.6μm。因此，与中心实频率相比，这种复频率方法下的传播长度增加了四倍以上。通过进一步增加t，随着信号在时域中继续衰减，噪声开始占主导地位。当时间超过4.77ps时，场分布开始表现出混乱。因此，在决定在复频域中构造波的最佳时间时，需要在较长的传播和信噪比之间进行权衡。
本发明人还研究了频率点数量(即离散频率ω′i的数量)的影响，图5B示出了不同合成频率数的成像图案，其中，中心频率固定为910cm-1，频率间隔固定为1cm-1，时间快照固定为4.24ps。很明显，9个频率点的场图与实频率相比已经有了显著的改善。随着频率数量增加到21，衰减长度显著增加。然而，进一步增加到39个并没有带来明显的改善。
在一个实施例中，使用复频率方法来研究PhP的干涉行为。在MoO3薄膜上制作两个不同直径(0.8μm和3μm)的圆形天线以激发声子极化激元，图6A示出了实验样品的原子力显微镜图像。使用s-SNOM探针扫描白色虚线框内的场分布，频率为f＝990cm-1时电场分布的幅度和实部分别如图6B的上图和下图所示。从两个天线发出的PhP相向传播，但由于衰减，它们不会形成可辨别的干涉条纹。通过合成复频率下的场图，MoO3薄膜中的本征损耗得到补偿，从而使PhP的传播时间更长。这有利于天线之间形成清晰的干涉图案，如图6C所示。
实现极化激元近乎无损的传播具有重要意义。极化激元的无损传播可 以用于光子集成电路的开发，其中长距离传输信号的能力对于确保电路的性能和可靠性至关重要，同时也有益于包括光子电路、波导和等离子体/声子结构照明显微镜在内的应用。
本发明的通过合成复频波补偿波传播中的损耗的方法，通过具有虚拟增益的复频率以及多个实频率响应合成复频波，能够补偿波传播中的损耗，以实现波的无损传播。
虽然本发明已经通过优选实施例进行了描述，然而本发明并非局限于这里所描述的实施例，在不脱离本发明范围的情况下还包括所作出的各种改变以及变化。

Claims (10)

  1. 一种补偿波传播中的损耗的方法，包括：
    基于波传播的测量结果获得实频率响应；
    基于波的频率和介质的介电常数选择复频率，以至少部分抵消波传播中的面内波矢的虚部；
    基于多个实频率响应和所述复频率获得复频率响应以补偿波传播中的损耗。
  2. 根据权利要求1所述的补偿波传播中的损耗的方法，其中，基于多个实频率响应和所述复频率获得复频率响应以补偿波传播中的损耗的步骤包括通过以下公式计算复频率响应：
    其中，F(ω′i,r)为实频率响应，为复频率响应，ω′i为实频率，i为正整数，为复频率，t为时间，Δω为实频率ω′i的间隔频率，r为波的位置。
  3. 根据权利要求1或2所述的补偿波传播中的损耗的方法，其中，所述波为极化激元，波传播的测量结果为波的电场强度。
  4. 根据权利要求3所述的补偿波传播中的损耗的方法，其中，复频率ω为中心频率，β为虚拟增益，其被选择为使得面内波矢的虚部为零。
  5. 根据权利要求4所述的补偿波传播中的损耗的方法，其中，波在其中传播的介质的介电常数由德鲁德模型描述为εm为介质的介电常数，ωr为频率ω趋向于无穷时的介电常数，ωp为介质层的等离激元频率，γ为介质层的阻尼率。
  6. 根据权利要求5所述的补偿波传播中的损耗的方法，其中，
  7. 根据权利要求4所述的补偿波传播中的损耗的方法，其中，波在其中传播的介质的介电常数由洛伦兹模型描述为εm＝εr-ωp 2/(ω2+iωγ-ω0 2)，εm为介质的介电常数，εr为频率ω趋向于无穷时的介电常数，ωp为介质层的等离激元频率，γ为介质层的阻尼率，ω0为模型的共振频率。
  8. 根据权利要求7所述的补偿波传播中的损耗的方法，其中，
  9. 一种计算机可读存储介质，其特征在于，其上包含有计算机程序，所述计算机程序能够被处理器执行以实现权利要求1-8之一所述方法的步骤。
  10. 一种电子设备，其特征在于，包括：
    一个或多个处理器；以及
    存储器，用于存储一个或多个可执行指令；
    所述一个或多个处理器被配置为经由执行所述一个或多个可执行指令以实现权利要求1-8之一所述方法的步骤。
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