Method for recovering output spectrum distortion caused by fast linear frequency modulation continuous waves

Abstract

The invention relates to the technical field of optical microcavity sensing, in particular to a method for recovering output spectrum distortion caused by fast linear frequency modulation continuous waves. By introducing a phase compensation spectrum recovery algorithm, any distorted read-out spectrum is recovered into a steady-state spectrum, which benefits from long photon lifetime and photon local characteristics, and the optical microcavity has already become a promising high-performance sensing tool. A ringing effect is derived from a secondary phase of interference between light emitted by a microcavity and waveguide input and output, the secondary phase depends on the chirp rate of an input linear frequency modulation continuous wave and is irrelevant to the type of the microcavity, and after the introduced phase is compensated, any steady-state spectrum can be recovered from a distorted read-out spectrum; a bridge is erected between a transient ringing spectrum and a steady-state spectrum, and a new way is opened up for high-speed sensing, optical communication and advanced integrated chip technologies.

Description

Technical Field

[0001] The invention relates to the technical field of optical microcavity sensing, and in particular to a method for restoring output spectrum distortion caused by fast linear frequency modulation continuous wave. Background Art

[0002] An optical microresonator is an optical device that can confine light in a very small area. Due to the resonance of light, the light intensity in the microresonator will be continuously enhanced, which can enhance the interaction between light and matter. Optical microcavities have greatly promoted the research of basic physics due to their small size, high quality factor, and strong confinement effect on the light field, and are widely used in the field of optoelectronic devices. Optical microcavities are known for their ability to capture photons in a small mode volume, making the confined light lifetime longer. The accumulation of photons in the microcavity produces a huge decay field, making it an important platform for studying the interaction between light and matter. Therefore, microcavities have promoted the extensive development of basic physics, including research on parity-time (anti)symmetry, optomechanics, photon blockade, magnetomechanical systems, and the generation of optical parameters. In addition, they are also widely used in microlasers, chaotic sources, filters, and frequency combs.

[0003] Suppose we use a whispering gallery mode microcavity coupled with a straight waveguide. The straight waveguide has two ends, one end is equipped with a photodetector as the output end, and the other end is used as the input end to input light waves. When we use a linear frequency modulation continuous wave with a slower chirp rate to scan the resonance of the whispering gallery mode microcavity, a certain spectrum can be obtained at the output end. Due to the coupling effect, the light wave of a specific frequency enters the whispering gallery mode microcavity and is confined therein, which appears as a resonance peak centered on the specific frequency in the spectrum. Due to the high sensitivity of the whispering gallery mode microcavity to external factors, such as pressure, temperature, humidity, electromagnetic field and impurities, slight changes in the geometry of the whispering gallery mode microcavity or the refractive index of the surrounding environment will cause a significant change in the position of the resonance peak in the spectrum, making it very suitable for sensing applications.

[0004] Mainstream microcavity sensors are usually based on detecting steady-state transmission spectra or reflection spectra. This method usually uses a linear frequency modulated continuous wave to scan the resonance of the microcavity at a slow speed. Because when the frequency change rate (chirp rate) of the input linear frequency modulated continuous wave exceeds the square of the Lorentz line width, the ringing phenomenon will distort the readout spectrum. Since the Lorentz line width of the high-quality factor microcavity is narrower, this distortion becomes more obvious in the microcavity with a high Q value, which makes it impossible to obtain a high-resolution readout spectrum by simply increasing the scanning speed. So far, scientists have proposed two methods to utilize the ringing phenomenon to meet the growing demand for transient response. The first method is to develop a new sensing protocol for ringing spectroscopy. In the research of this method, scientists have published articles to explain the basic physical principles, and based on these studies, various characteristics of the ringing spectrum are used to achieve sensing purposes. However, it is still a challenge to understand the elusive data characteristics given by the ringing spectrum, which hinders the application of optical microcavities in transient sensing. The second method focuses on reconstructing the spectrum using the ringing tail. For example, the counterpart to the well-known “ring-down” spectroscopy technique is “ring-up” spectroscopy, which uses a pulsed laser at a fixed wavelength to monitor the transient response of a microcavity. However, the spectral resolution is limited by the time scale of the ringing tail, making it ineffective in the case of fast decays.

[0005] Therefore, the present application proposes a method for restoring output spectrum distortion caused by fast linear frequency modulation continuous wave. Summary of the invention

[0006] In order to make up for the deficiencies of the prior art and solve the technical problems existing in the background technology, the present invention proposes a method for restoring the output spectrum distortion caused by a fast linear frequency modulation continuous wave.

[0007] The present invention is achieved through the following technical solutions:

[0008] A method for recovering output spectrum distortion caused by fast linear frequency modulated continuous wave, wherein the secondary phase depends on the chirp rate of the input linear frequency modulated continuous wave and has nothing to do with the type of microcavity, comprising the following steps:

[0009] Step 1: Based on the linear time-invariant system, the response of the microcavity system is expressed by the following formula:

[0010] α out (f)=H(f)×α in (f)

[0011] Where H(f) is the Green function of the system, α in (f) and α out (f) Input and output respectively;

[0012] Step 2: Introduce the conditions of waveguide-coupled microcavity, which consists of the input signal and the reflection spectrum of the microcavity system. The Green function of the system is expressed by the following formula:

[0013] H(f)=1-R(f)

[0014] And the reflection spectrum describes the microcavity absorption, and the dispersion caused by absorption is expressed by the following formula:

[0015]

[0016] Among them, γ(τ) represents the amplitude, represents the phase, τ represents the time delay;

[0017] Step 3: Input the linear frequency modulated continuous light into the microcavity system, and its output is expressed by the following formula:

[0018]

[0019] Among them, α0e iθ ( t )=α in (t),θ(t)=2π(f S +kt / 2), α0 is the amplitude of the input linear frequency modulated continuous wave, f S is the starting frequency of the input linear frequency modulated continuous wave, k is the chirp rate of the input linear frequency modulated continuous wave;

[0020] Step 4: Based on steps 1 to 3, the intensity read out in the time domain is expressed by the following formula:

[0021]

[0022] Preferably, s(t) is represented by the following formula:

[0023]

[0024] Where θ1 = -2πf S τ is the starting frequency phase, θ2=πkτ 2 is a phase shift with quadratic function characteristics.

[0025] Preferably, when not affected by θ1 and θ2, the term s(t) is able to map R(t), where θ1 brings a linear shift and θ2 produces a ringing oscillation, causing distortion in the spectral readout.

[0026] Preferably, when the ringing effect occurs, the frequency change rate of the linear frequency modulated continuous wave input into the system exceeds the square of the Lorentz line width of the optical microcavity, so that the readout spectrum produces a ringing effect.

[0027] Preferably, the secondary phase is compensated to restore the steady-state spectrum from the read spectrum with any ringing distortion, which specifically includes the following steps:

[0028] S1: The removal of θ1 and θ2 is achieved in the frequency domain through Fourier transform, as shown below:

[0029]

[0030] If f = ±kτ is a term of δ(f = ±kτ), then the equation is well defined;

[0031] S2: S(f) is expressed as follows:

[0032]

[0033] That is the result of the phases of θ1 and θ2;

[0034] S3: Applied to the spectrum, the readout distortion will be eliminated;

[0035] Finally, an inverse Fourier transform is performed to obtain the recovered readout in the time domain

[0036]

[0037] Among them, τ k = |f / k| represents the spectral resolution, which depends only on the chirp rate, due to the terms γ and This approach is independent of the spectrum of the microcavity system and can therefore recover any readout spectrum with arbitrary linear frequency modulated continuous wave chirp rate.

[0038] The beneficial effects of the present invention are:

[0039] The present invention simulates complex spectra by cascading three different Fano-like line shapes and generates steady-state spectra, which proves that the proposed method is applicable to various line shapes; in addition, by introducing a phase-compensated spectrum recovery algorithm, an arbitrarily distorted readout spectrum is restored to its steady-state spectrum. Thanks to the long photon lifetime and photon localization characteristics, optical microcavities have become a promising high-performance sensing tool. By proposing that the ringing effect originates from the secondary phase of interference between the light emitted by the microcavity and the waveguide input and output, this secondary phase depends on the chirp rate of the input linear frequency modulated continuous wave and is independent of the type of microcavity, after compensating for the introduced phase, any steady-state spectrum can be restored from the distorted readout spectrum, thus building a bridge between the transient ringing spectrum and the steady-state spectrum, and opening up a new path for high-speed sensing, optical communications and advanced integrated chip technology. BRIEF DESCRIPTION OF THE DRAWINGS

[0040] Figure 1Several typical configuration diagrams of microcavity systems are shown;

[0041] Figure 2 The spectrum restoration effect diagram of the present invention under complex line shape is shown;

[0042] in, Figure 2 (a) shows three cascaded Fano-like line shapes simulating complex steady-state spectra and readout results at different chirp rates; (b) shows the comparison of steady-state and recovery readout at different chirp rates;

[0043] Figure 3 It is a schematic diagram of the experimental verification system of the present invention and a diagram of the experimental results obtained;

[0044] in, Figure 3 (a) is a schematic diagram of the experimental verification system; (b) is the ringing distortion spectrum read out experimentally; (c) is the ringing distortion spectrum obtained by theoretical simulation under the same parameters; (d) is the restored spectrum obtained in the experiment by the scheme of the present invention; (e) is the restored spectrum obtained theoretically using the scheme of the present invention under the same parameters. DETAILED DESCRIPTION

[0045] The present invention will be further described below in conjunction with specific examples. It should be understood that these examples are only used to illustrate the present invention and are not intended to limit the scope of the present invention. The experimental methods in the following examples without specifying specific conditions are usually carried out under conventional conditions or under conditions recommended by the manufacturer.

[0046] Unless otherwise defined, all professional and scientific terms used herein have the same meanings as those familiar to those skilled in the art. The reagents or raw materials used in the present invention can be purchased through conventional channels. If not otherwise specified, the reagents or raw materials used in the present invention are used in a conventional manner in the art or in accordance with the product instructions. In addition, any method and material similar to or equivalent to the described content can be applied to the method of the present invention. The present invention is further described according to the accompanying drawings and specific embodiments of the present invention, and the preferred implementation methods and materials described in the present invention are only for demonstration purposes.

[0047] The present invention proposes for the first time that the ringing effect originates from the secondary phase of the interference between the light emitted by the microcavity and the waveguide input and output. This secondary phase depends on the chirp rate of the input linear frequency modulated continuous wave and has nothing to do with the type of microcavity. In view of this discovery, after compensating for the introduced phase, the method proposed by the present invention can recover any steady-state spectrum from the distorted readout spectrum.

[0048] We first start with the factors that cause output distortion. Figure 1Several typical configurations of microcavity systems are demonstrated, including a single microcavity, coupled microcavities with parity-time (PT) symmetry, electromagnetic induced transparency (EIT)-like, and Fano-like spectral features.

[0049] The following technical solution is used for verification. The specific steps are as follows:

[0050] Make a system that can read the ringing distortion spectrum through an optical microcavity, as shown in the attached manual. Figure 3 (a). Where AWG stands for arbitrary waveform generator; VOA stands for variable optical attenuator; PC stands for polarization controller; PD stands for photodetector; OSC stands for oscilloscope. The inset on the right shows the optical image of the optical microcavity. In this example, a structure of a whispering gallery microcavity and a fiber taper coupling is used. The arbitrary waveform generator outputs a sawtooth wave, which controls the laser to output a continuous wave with linear frequency modulation within a certain range. After being filtered by the variable optical attenuator and polarization controller, it is incident on the coupling structure of the whispering gallery microcavity and the fiber taper. The readout spectrum of the system is obtained at the output end of the structure through the photodetector and oscilloscope. When the frequency of the input sawtooth wave is fast enough so that the chirp rate of the linear frequency modulated continuous wave exceeds the square of the Lorentz line width of the optical microcavity, a readout spectrum with ringing distortion is obtained.

[0051] First, let’s start with the factors that cause output distortion. Considering the linear time-invariant system, the response of the microcavity system should be expressed as:

[0052] α out (f)=H(f)×α in (f)

[0053] Where H(f) is the Green’s function of the system; α in (f) and α out (f) are the input and output respectively. Considering the conditions of waveguide coupled microcavity, the Green function of the system can be written as:

[0054] H(f)=1-R(f)

[0055] It consists of the input signal and the reflection spectrum of the microcavity system. In general, the reflection spectrum describes the microcavity absorption, and the dispersion caused by absorption can be written as:

[0056]

[0057] The amplitude γ(τ), phase and time delay τ. In order to obtain the microcavity spectrum, we input linear frequency modulated continuous light into the microcavity system. Therefore, the output should be written as:

[0058]

[0059] where αin (t) = α0e iθ(t) ,θ(t)=2π(f s +kt / 2), and α0, f s and k are the amplitude, starting frequency and chirp rate of the input linear frequency modulated continuous wave, respectively. So the intensity read out in the time domain can be written as

[0060]

[0061] in θ1=-2πf s τ and θ2 = πkr 2 are the starting frequency phase and the phase shift with quadratic function characteristics. When not affected by θ1 and θ2, the s(t) term can map R(f); conversely, θ1 brings a linear shift and θ2 produces ringing oscillation, which distorts the spectral readout.

[0062] In order to remove the introduced θ1 and θ2 phases, a Fourier transform is implemented on the readout device, which is realized in the frequency domain.

[0063]

[0064] if is the term of δ(f±kτ), then the equation is well defined. Therefore, S(f) can be simplified to

[0065]

[0066] Found this item is the result of the phases of θ1 and θ2. Therefore, if

[0067] Applied to the spectrum, the readout distortion will be eliminated. Finally, an inverse Fourier transform is performed to obtain the recovered readout in the time domain

[0068]

[0069] Among them, τ k = |f / k| represents the spectral resolution, which depends only on the chirp rate. This approach is independent of the spectrum of the microcavity system and can therefore recover any readout spectrum with arbitrary linear frequency modulated continuous wave chirp rate.

[0070] The results obtained by the embodiments of the present disclosure are shown in the accompanying drawings. Figure 3 In theory, Figure 3 (c) The ringing distortion spectrum was read out at a chirp rate of 3 MH / μs and the Figure 3 (e) shows the recovered spectrum. Experimentally, Figure 3(b) Using the same parameters as the theory, we obtain the distorted spectrum and obtain Figure 3 (d) shows the recovered spectrum. The recovered readout results in an identified steady-state spectrum, allowing the application to obtain parameters that are obscured by distortion.

[0071] Finally: The technical means disclosed in the technical solution of the present invention are not limited to the technical means disclosed in the above embodiments, but also include technical solutions composed of any combination of the above technical features. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principle of the present invention, and these improvements and modifications are also regarded as the protection scope of the present invention.

[0072] Any matters not described in the present invention are applicable to the prior art.

Claims

1. A method for restoring output spectrum distortion caused by fast linear frequency modulation continuous wave, characterized in that: The secondary phase depends on the chirp rate of the input linear frequency modulated continuous wave and has nothing to do with the type of microcavity. It includes the following steps: Step 1: Based on the linear time-invariant system, the response of the microcavity system is expressed by the following formula: α out (f)=H(f)×α in (f) Where H(f) is the Green function of the system, α in (f) and α out (f) Input and output respectively; Step 2: Introduce the conditions of waveguide-coupled microcavity, which consists of the input signal and the reflection spectrum of the microcavity system. The Green function of the system is expressed by the following formula: H(f)=1-R(f) And the reflection spectrum describes the microcavity absorption, and the dispersion caused by absorption is expressed by the following formula: Among them, γ(τ) represents the amplitude, represents the phase, τ represents the time delay; Step 3: Input the linear frequency modulated continuous light into the microcavity system, and its output is expressed by the following formula: Among them, α0e iθ(t) =α in (t),θ(t)=2π(f S +kt / 2), α0 is the amplitude of the input linear frequency modulated continuous wave, f S is the starting frequency of the input linear frequency modulated continuous wave, k is the chirp rate of the input linear frequency modulated continuous wave; Step 4: Based on steps 1 to 3, the intensity read out in the time domain is expressed by the following formula:

2. The method for restoring output spectrum distortion caused by fast linear frequency modulation continuous wave according to claim 1, characterized in that: s(t) is expressed by the following formula: Where θ1 = -2πf S τ is the starting frequency phase, θ2=πkτ 2 is a phase shift with quadratic function characteristics.

3. The method for restoring output spectrum distortion caused by fast linear frequency modulation continuous wave according to claim 2, characterized in that: When not affected by θ1 and θ2, the term s(t) can map R(t), with θ1 causing a linear shift and θ2 producing a ringing oscillation, which distorts the spectral readout.

4. The method for restoring output spectrum distortion caused by fast linear frequency modulation continuous wave according to claim 3, characterized in that: When the ringing effect occurs, the frequency change rate of the linear frequency modulated continuous wave input to the system exceeds the square of the Lorentz line width of the optical microcavity, causing a ringing effect in the readout spectrum.

5. The method for restoring output spectrum distortion caused by fast linear frequency modulation continuous wave according to claim 1, characterized in that: The secondary phase is compensated to restore the steady-state spectrum from the read spectrum with any ringing distortion, which specifically includes the following steps: S1: The removal of θ1 and θ2 is achieved in the frequency domain through Fourier transform, as shown below: If f = ±kτ is a term of δ(f = ±kτ), then the equation is well defined; S2: S(f) is expressed as follows: That is the result of the phases of θ1 and θ2; S3: Applied to the spectrum, the readout distortion will be eliminated; Finally, an inverse Fourier transform is performed to obtain the recovered readout in the time domain Among them, τ k = |f / k| represents the spectral resolution, which depends only on the chirp rate, due to the terms γ and This approach is independent of the spectrum of the microcavity system and can therefore recover any readout spectrum with arbitrary linear frequency modulated continuous wave chirp rate.

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