Method for compensating for loss in wave propagation

Abstract

Provided is a method for compensating for loss in wave propagation, comprising: obtaining real frequency responses on the basis of a measurement result of wave propagation; selecting a complex frequency on the basis of the frequency of a wave and the dielectric constant of a medium so as to at least partially counteract the imaginary part of an in-plane wave vector in the wave propagation; and on the basis of a plurality of real frequency responses and the complex frequency, obtaining a complex frequency response to compensate for loss in the wave propagation.

Description

Methods for compensating for losses in wave propagation Technical Field

[0001] 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. Background Art

[0002] Polaritons are surface electromagnetic waves formed by the interaction between particles on a medium's surface and the incident light field. These include surface plasmons based on electron-photon hybridization and phonon polaritons based on phonon-photon hybridization. These polariton electromagnetic waves have a higher energy density than the incident light field, resulting in a strong electric field enhancement effect on the surface of the medium. Polaritons can compress and focus the light field to very small dimensions, enabling optical information transmission and processing at the nanoscale. Benefiting from subwavelength light confinement, surface plasmon polaritons and phonon polaritons offer a way to surpass the diffraction limit of traditional optics and contribute to efficient energy storage, local field enhancement, and high sensitivity. However, losses limit the propagation length of polaritons, thus restricting their practical application. While optimizing manufacturing techniques can help avoid scattering losses caused by structural imperfections, intrinsic losses that generate heat cannot be eliminated. Therefore, overcoming intrinsic losses to achieve lossless wave propagation is an urgent challenge in this field.

[0003] Summary of the Invention

[0004] In view of the above problems in the prior art, the present invention proposes a method for compensating for losses in wave propagation by synthesizing complex frequency waves, comprising:

[0005] Obtaining real frequency response based on wave propagation measurement results;

[0006] Selecting a complex frequency based on the frequency of the wave and the dielectric constant of the medium to at least partially cancel an imaginary part of an in-plane wave vector in wave propagation;

[0007] 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.

[0008] In one embodiment, 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:

[0009] Among them, F(ω′ i ,r) is the real frequency response, is the complex frequency response, ω′ i is the real frequency, i is a positive integer, is the complex frequency, t is the time, Δω is the real frequency ω′ i The interval frequency is , and r is the position of the wave.

[0010] In one embodiment, the wave is a polariton and the measurement result of the wave propagation is the electric field strength of the wave.

[0011] In one embodiment, the complex frequency ω is the center frequency and β is the imaginary gain, which is chosen so that the imaginary part of the in-plane wave vector is zero.

[0012] In one embodiment, the dielectric constant of the medium in which the wave propagates is described by the Drude model as ε m is the dielectric constant of the medium, ε r is the dielectric constant when the frequency ω tends to infinity, ω p is the plasmon frequency of the dielectric layer, and γ is the damping rate of the dielectric layer.

[0013] In one embodiment,

[0014] In one embodiment, the dielectric constant of the medium in which the wave propagates is described by the Lorentz model as ε m =ε r -ω p 2 / (ω 2 +iωγ-ω0 2 ), ε m is the dielectric constant of the medium, ε r is the dielectric constant when the frequency ω tends to infinity, ω p is the plasmon frequency of the dielectric layer, γ is the damping rate of the dielectric layer, and ω0 is the resonant frequency of the model.

[0015] In one embodiment,

[0016] The present invention also provides a computer-readable storage medium containing a computer program, wherein the computer program can be executed by a processor to implement the steps of the above method.

[0017] The present invention also provides an electronic device, comprising:

[0018] one or more processors; and

[0019] a memory for storing one or more executable instructions;

[0020] The one or more processors are configured to implement the steps of the above method by executing the one or more executable instructions.

[0021] The method of compensating for loss in wave propagation by synthesizing complex frequency waves of the present invention can compensate for 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. BRIEF DESCRIPTION OF THE DRAWINGS

[0022] FIG1 shows a flow chart of a method for compensating for loss in wave propagation by synthesizing complex frequency waves according to one embodiment of the present invention.

[0023] FIG. 2A shows the excitation field distribution of plasmon polaritons (SPPs).

[0024] FIG2B shows the corresponding Fourier distribution of SPPs.

[0025] FIG2C shows the dispersion under complex frequency excitation.

[0026] FIG2D shows Dynamic evolution of the complex frequency excitation at different times.

[0027] FIG2E shows that at t=5.5×10 -2 The corresponding spatial field distribution at ps.

[0028] FIG3A shows an experimental setup based on scattering scanning near-field optical microscopy technology.

[0029] FIG3B shows the field distribution for two real frequencies.

[0030] FIG3C shows the field distributions for two complex frequencies.

[0031] FIG4 shows atomic force microscopy images illustrating the near-field distribution at the MoO 3 interface measured at different frequencies.

[0032] FIG5A shows the complex frequency (910-6.5i) cm -1 The synthetic electric field distribution at different times, with a time step of 0.53 ps.

[0033] FIG5B shows imaging patterns with different synthetic frequencies, where the center frequency is fixed at 910 cm -1 , the frequency interval is fixed at 1cm -1 .

[0034] FIG6A shows an atomic force microscope image of the experimental sample.

[0035] FIG6B shows a frequency of f=990 cm -1 The amplitude and real part of the electric field distribution when .

[0036] FIG. 6C shows a frequency of f=(990-2i)cm -1 The interference pattern between the antennas. DETAILED DESCRIPTION

[0037] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention will be further described in detail below through specific embodiments in conjunction with the accompanying drawings. It should be noted that the embodiments given in the present invention are only for illustration and do not limit the scope of protection of the present invention.

[0038] Polaritons, including surface plasmon polaritons (SPPs) and phonon polaritons (PhPs), have become the preferred 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. However, intrinsic loss hinders many loss-sensitive polariton-based applications, including waveguides, biosensing, subdiffraction-limited imaging, and plasmonic structure illumination microscopy. Intrinsic loss has a negative impact on 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.

[0039] 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 most common way to offset losses is to add an external gain medium. However, it is very challenging to completely offset plasma losses with gain, and gain compensation is easily affected by noise and instability. After research, the present invention proposes a method for achieving lossless wave propagation by synthesizing complex frequency waves, which can achieve almost lossless propagation of highly confined polaritons.

[0040] 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.

[0041] For simplicity, we take the SPP at the interface as an example. The mathematical solution of the in-plane wave vector k is Where ω is the wave frequency, c is the speed of light, and ε m is the dielectric constant of the medium. In one embodiment, it is assumed that the dielectric constant ε of the plasma metal m Described by the Drude model, that is Among them, ε r is the dielectric constant when the frequency ω tends to infinity, ω p is the plasmon frequency of the dielectric layer, and γ is the damping rate of the dielectric layer. The dissipation term in the denominator introduces the attenuation of SPP propagation. It is generally believed that by using a specific complex frequency By compensating for the loss in the Drude model, lossless propagation of SPP can be achieved. However, this is not entirely accurate. After substituting into the formula of the in-plane wave vector k, there is still a frequency-dependent term in the in-plane wave vector k This will result in an imaginary part in the wave vector k. Therefore, the condition for lossless wave propagation should be that the imaginary part of the in-plane wave vector k is 0, that is, that is get:

[0042] in, ω corresponds to the center frequency, β represents the virtual gain, ω>>β, ω p >>γ. By neglecting the second-order small quantities, we obtain:

[0043] Substituting equation (2) into equation (1) yields:

[0044] From this we get:

[0045] 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.

[0046] In another embodiment, assuming that the dielectric constant ε of the plasma metal m Described by the Lorentz model, that is, ε m =ε r -ω p 2 / (ω 2 +iωγ-ω0 2 ), ω0 is the resonant frequency of the model. The complex frequency can be obtained The imaginary part β is:

[0047] It can be seen from the above formula that by choosing the complex frequency The value of the imaginary part β 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. In another embodiment, the complex frequency can also be selected The value of the imaginary part β is adjusted 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.

[0048] It should be understood by those skilled in the art that the above formulas (4) and (5) are merely exemplary and do not constitute a limitation to the present invention. Other models can also be used to describe the dielectric constant ε m, thus obtaining different values ​​of the imaginary part of the complex frequency. Therefore, in practical applications, the imaginary part of the in-plane wave vector can be made 0 or close to 0, and the calculated complex frequency can at least partially offset the imaginary part of the in-plane wave vector, thereby at least partially offsetting the intrinsic loss and reducing the wave propagation loss.

[0049] After determining the selected complex frequency After determining the value of the imaginary part β, the following will explain how to use this value to generate a complex frequency wave to achieve lossless wave propagation or make it propagate farther.

[0050] Complex frequency wave with time decay It diverges when time approaches negative infinity. Since it is very difficult to directly generate complex frequency waves, the inventors use a new method to synthesize truncated complex frequency waves, which is expressed as Among them E T (t) is the intensity of the truncated complex frequency wave; θ(t) is the time truncation function to avoid energy divergence, where for time t≥0, θ(t)=1, for time t<0, θ(t)=0.

[0051] By using Fourier transform, 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. Naturally, 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 In fact, for a sufficiently wide spectral range, the above integral can be discretized as:

[0052] in, It can be any physical quantity that characterizes wave propagation, such as electric field strength, magnetic field strength, etc. i is the real frequency, F(ω′ i ,r) is the real frequency ω′ i The real frequency response under , where i is a positive integer and r is the position of the wave, such as a one-dimensional or two-dimensional coordinate. It can be based on multiple real frequencies ω′ i The measurement and / or calculation results F(ω′ i ,β) to calculate Discrete frequencies ω′1, ω′2, …, ω′ i are equally spaced, with an interval of Δω. Complex frequency The complex frequency Substituting this into the above formula (6), the intrinsic loss can be at least partially offset.

[0053] Based on the above, complex frequency methods can significantly enhance the decay length of highly confined polaritons, restoring near-lossless propagation limited only by the noise level. However, those skilled in the art will appreciate that this method of increasing 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 and elastic waves.

[0054] Based on the above, the present invention provides a method for compensating for wave loss in medium propagation by synthesizing complex frequency waves. FIG1 shows a flow chart of the method, which includes:

[0055] Step S101: Obtaining a real frequency response based on the measurement result of wave propagation.

[0056] The measurement results of wave propagation can be the intensity of the wave at a certain location, such as the amplitude of the wave, the electric field strength, the magnetic field strength, etc.

[0057] Therein, a plurality of real frequency responses are obtained based on measurement results of a plurality of wave propagations over a period of time.

[0058] 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 the imaginary part of the in-plane wave vector in wave propagation.

[0059] In one embodiment, the complex frequency is selected so that the imaginary part of the in-plane wave vector is zero or approaches zero. In one embodiment, the value of the imaginary part of the complex frequency can be calculated by the above formula (5) and formula (6).

[0060] Step S103: obtaining a complex frequency response based on a plurality of real frequency responses and the complex frequency to compensate for loss in wave propagation.

[0061] In one embodiment, the complex frequency response can be calculated based on the above formula (6). In another embodiment, the complex frequency response can be calculated based on an integral formula, which can be, for example, The lower limit of integration a is less than the complex frequency The real part of the integral upper limit b is greater than the complex frequency The real part of . Those skilled in the art will appreciate that modifications or alterations to Formula (6) and the integral formula can be made to offset or compensate for losses in wave propagation, and therefore, such modifications or alterations fall within the scope of the present invention. In other words, any method for obtaining a complex frequency response based on a combination of multiple real frequency responses falls within the scope of the present invention.

[0062] In one embodiment, the dielectric constant of a real plasmonic metal described by the Drude model is assumed to be where ω p =1.442×10 16Hz, γ=3×10 14 Hz. The metal 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 complex frequency to achieve lossless propagation. The complex frequency field distribution can be synthesized by a linear combination of the real frequency field distribution,

[0063] The complex frequency The value of β 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 fully recovers the dispersion at 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. Figure 2E shows the dynamic evolution of the complex frequency excitation at t = 5.5 × 10 -2 The corresponding spatial field distribution at ps is shown in the upper part of Figure 2E. The spatial field distribution of the (The real part is shown by the upper dashed line in FIG2C ) As shown in FIG2E , the decay is much slower than that in the real frequency case.

[0064] 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 that support PhP. For PhP, there will also be a specific complex frequency solution that satisfies Im(k) = 0.

[0065] In one embodiment, a hexagonal boron nitride (hBN) thin film is used, which supports isotropic PhP in the plane and can use complex frequency to compensate its intrinsic loss to observe the lossless propagation of phonon polaritons. Figure 3A shows the experimental setup based on the scattering scanning near-field optical microscopy (s-SNOM) technique. A long gold antenna placed on the hBN film is used to emit 1D PhP. At 1421 cm -1 to 1503cm -1 The electric field distribution is measured within a frequency range with a step size of 2 cm -1 . Select the real frequency as 1451cm -1and 1477cm -1 The two field distributions of are used as the center frequency, synthesizing and The complex frequency response at . The corresponding imaginary part represents the optimized value for each frequency. Figure 3B shows the field distribution at two real frequencies, and Figure 3C shows the field distribution at two complex frequencies. The experimental results show that the propagation attenuation is strong at real frequencies, while the propagation of polaritons at complex frequencies is almost non-dissipative.

[0066] In one embodiment, a complex frequency method is used to study the temporal evolution of a more complex field distribution supported by a van der Waals crystalline α-MoO3 (MoO3) film, which is highly anisotropic and supports natural in-plane hyperbolic polaritons. A gold antenna is placed on the MoO3 film to excite the PhP. Figure 4 shows atomic force microscopy (AFM) images showing the near-field distribution at the MoO3 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 exhibits a characteristic hyperbolic propagation behavior with a concave wavefront. As the frequency increases, the wavelength decreases, the field confinement becomes stronger, and the propagation attenuation becomes stronger. At all measured frequencies, the decay length of the polaritons is less than two wavelengths due to the significant intrinsic losses in the material.

[0067] FIG5A shows the complex frequency (910-6.5i) cm -1 The synthetic electric field distribution at different times, with a time step of 0.53 ps and a total number of frequencies of 53. For smaller t, the noise is largely smoothed by summing the multi-frequency signals. As time passes, the wave propagates farther and farther upward, far exceeding the decay length of two wavelengths at real frequencies. The two-dimensional field pattern reaches an optimum at around 4.24 ps, with a propagation length of 8.7 μm (along the vertical direction), while the propagation length corresponding to the center real frequency is only 1.6 μm. Therefore, the propagation length under this complex frequency method is increased by more than a factor of four compared to the center real frequency. By further increasing t, as the signal continues to decay in the time domain, noise begins to dominate. When the time exceeds 4.77 ps, the field distribution begins to show chaos. Therefore, when determining the optimal time to construct the wave in the complex frequency domain, a trade-off needs to be made between longer propagation and signal-to-noise ratio.

[0068] The inventors also studied the number of frequency points (i.e. discrete frequency ω′ i FIG5B shows the imaging patterns of different synthetic frequency numbers, where the center frequency is fixed at 910 cm -1 , the frequency interval is fixed at 1cm -1, with the time snapshot fixed at 4.24 ps. Clearly, the field pattern at nine frequencies shows a significant improvement compared to the real frequency. Increasing the number of frequencies to 21 significantly increases the decay length. However, further increasing the number to 39 does not bring significant improvement.

[0069] In one embodiment, the complex frequency method is used to study the interference behavior of PhP. Two circular antennas with different diameters (0.8 μm and 3 μm) are made on the MoO3 film to excite phonon polaritons. Figure 6A shows the atomic force microscope image of the experimental sample. The field distribution within the white dotted box is scanned using an s-SNOM probe with a frequency of f = 990 cm -1 The amplitude and real part of the electric field distribution are shown in the upper and lower figures of Figure 6B, respectively. The PhP emitted from the two antennas propagate in opposite directions, but due to attenuation, they do not form discernible interference fringes. By synthesizing the complex frequency In the field diagram below, the intrinsic loss in the MoO3 film is compensated, resulting in a longer propagation time for the PhP. This facilitates the formation of a clear interference pattern between the antennas, as shown in Figure 6C.

[0070] Achieving near-lossless propagation of polaritons is of great significance. This can be used in the development of photonic integrated circuits, where the ability to transmit signals over long distances is crucial to ensuring circuit performance and reliability. It also benefits applications including photonic circuits, waveguides, and plasmon / phononic structure illumination microscopy.

[0071] The method of compensating for loss in wave propagation by synthesizing complex frequency waves of the present invention can compensate for loss in wave propagation by synthesizing complex frequency waves with a complex frequency having a virtual gain and multiple real frequency responses to achieve lossless wave propagation.

[0072] Although the present invention has been described through preferred embodiments, the present invention is not limited to the embodiments described herein but includes various changes and modifications that may be made without departing from the scope of the present invention.

Claims

1. A method for compensating for losses in wave propagation, comprising: Obtaining real frequency response based on the measurement results of wave propagation; 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.

2. A method for compensating for losses in wave propagation according to claim 1, wherein: 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: Among them, F(ω′ i ,r) is the real frequency response, is the complex frequency response, ω′ i is the real frequency, i is a positive integer, is the complex frequency, t is the time, Δω is the real frequency ω′ i The interval frequency is , and r is the position of the wave.

3. A method for compensating for losses in wave propagation according to claim 1 or 2, wherein: The wave is a polariton, and the measurement result of the wave propagation is the electric field strength of the wave.

4. A method for compensating for losses in wave propagation according to claim 3, wherein: Complex frequency ω is the center frequency and β is the imaginary gain, which is chosen so that the imaginary part of the in-plane wave vector is zero.

5. A method for compensating for losses in wave propagation according to claim 4, wherein: The dielectric constant of the medium in which the wave propagates is described by the Drude model as ε m is the dielectric constant of the medium, ω r is the dielectric constant when the frequency ω tends to infinity, ω p is the plasmon frequency of the dielectric layer, and γ is the damping rate of the dielectric layer.

6. A method for compensating for losses in wave propagation according to claim 5, wherein:

7. The method of compensating for losses in wave propagation according to claim 4, wherein: The dielectric constant of the medium in which the wave propagates is described by the Lorentz model as ε m =ε r -ω p 2 / (ω 2 +iωγ-ω0 2 ), ε m is the dielectric constant of the medium, ε r is the dielectric constant when the frequency ω tends to infinity, ω p is the plasmon frequency of the dielectric layer, γ is the damping rate of the dielectric layer, and ω0 is the resonant frequency of the model.

8. The method of compensating for losses in wave propagation according to claim 7, wherein:

9. A computer-readable storage medium, characterized in that: A computer program is included thereon, and the computer program can be executed by a processor to implement the steps of the method according to any one of claims 1 to 8.

10. An electronic device, characterized in that: include: one or more processors; as well as A memory for storing one or more executable instructions; The one or more processors are configured to implement the steps of the method of any one of claims 1-8 by executing the one or more executable instructions.

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