A method for recovering the output spectrum distortion caused by fast chirped continuous wave

Abstract

The present invention relates to the field of optical microcavity sensing technology, and particularly to a method for recovering the output spectral distortion caused by fast linear frequency modulation continuous wave; by introducing a spectral recovery algorithm with phase compensation, any distorted readout spectrum can be 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 quadratic phase of the interference between the light emitted from the microcavity and the input / output of the waveguide, this quadratic phase depends on the chirp rate of the input linear frequency modulation 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, bridging the gap between the transient ringing spectrum and the steady-state spectrum, and opening up a new path for high-speed sensing, optical communication, and advanced integrated chip technology.

Description

Technical Field

[0001] The present invention relates to the technical field of optical microcavity sensing, and particularly to a method for recovering the output spectrum distortion caused by fast chirped continuous wave. Background Art

[0002] An optical microresonator is an optical device that can confine light within a very small area. Moreover, due to the resonance effect of light, the light intensity in the microcavity will be continuously enhanced, which can enhance the interaction between light and matter. Benefiting from its characteristics such as small volume, high quality factor, and strong confinement effect on the optical field, the optical microcavity has greatly promoted the research of basic physics and has been widely applied in the field of optoelectronic devices. The optical microcavity is famous for capturing photons in a small mode volume and making the confined light have a longer lifetime. Photon accumulation in the microcavity generates 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 in aspects such as parity-time (anti) symmetry, optomechanics, photon blockade, magnetomechanical systems, and optical parametric generation. In addition, they are widely used in fields such as micro-lasers, 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, with a photodetector installed at one end as the output end and the other end as the input end for inputting light waves. When we use a linearly chirped continuous wave with a slower chirp rate to scan the resonance of the whispering gallery mode microcavity, a definite spectrogram can be obtained at the output end. Light waves of a specific frequency enter the whispering gallery mode microcavity due to the coupling effect and are confined therein, which is manifested as a resonance peak centered at this specific frequency in the spectrogram. Due to the high sensitivity of the whispering gallery mode microcavity to external factors, for example, small changes in pressure, temperature, humidity, electromagnetic field, and impurities affect the geometry of the whispering gallery mode microcavity or the refractive index of the surrounding environment, which will cause an obvious change in the position of the resonance peak in the spectrum. Therefore, it is very suitable for sensing applications.

[0004] Mainstream microcavity sensors are typically based on detecting steady-state transmission spectra or reflection spectra. This method usually uses a chirped continuous wave to scan the resonance of the microcavity at a relatively slow speed. Because when the frequency change rate (chirp rate) of the input chirped continuous wave exceeds the square of the Lorentz linewidth, the ringing phenomenon will distort the readout spectrum. Due to the narrower Lorentz linewidth of high-quality factor microcavities, this distortion becomes more obvious in high-Q microcavities, which makes it impossible to obtain a high-resolution readout spectrum simply by 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 the ringing spectrum. In the research of this method, scientists have published papers explaining the basic physical principles, and based on these studies, various characteristics of the ringing spectrum are used to achieve the sensing purpose. 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 using the ringing tail to reconstruct the spectrum. For example, the "ring-up" spectroscopy technique corresponding to the well-known "ring-down" spectroscopy technique uses a pulsed laser with a fixed wavelength to monitor the transient response of the microcavity. However, the spectral resolution is limited by the time scale of the ringing tail, making it ineffective in the case of rapid decay.

[0005] Therefore, this application proposes a method for recovering the distortion of the output spectrum caused by a fast chirped continuous wave. Summary of the Invention

[0006] To make up for the deficiencies of the prior art and solve the technical problems existing in the background art, this invention proposes a method for recovering the distortion of the output spectrum caused by a fast chirped continuous wave.

[0007] This invention is realized through the following technical solutions:

[0008] A method for recovering the distortion of the output spectrum caused by a fast chirped continuous wave, where the quadratic phase depends on the chirp rate of the input chirped continuous wave and is independent of the type of microcavity, and it includes the following steps:

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

[0010]

[0011] Where, is the Green's function of the system, and are the input and output respectively;

[0012] Step 2: Introduce the conditions for waveguide-coupled microcavities, which consist of the input signal and the reflection spectrum of the microcavity system. The Green's function of the system is expressed as follows:

[0013]

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

[0015]

[0016] Wherein, represents the amplitude, represents the phase, represents the time delay;

[0017] Step 3: Input the chirped continuous wave into the microcavity system, and its output is expressed as follows:

[0018]

[0019] Wherein, , , is the amplitude of the input chirped continuous wave, is the starting frequency of the input chirped continuous wave, is the chirp rate of the input chirped continuous wave;

[0020] Step 4: Based on Steps 1 to 3, the readout intensity is expressed as follows in the time domain:

[0021] .

[0022] Preferably, is expressed as follows:

[0023]

[0024] Wherein, is the starting frequency phase, is the phase shift with quadratic function characteristics.

[0025] Preferably, when not affected by and , term can map , and its brings a linear offset, generates ringing oscillations, causing spectral readout distortion.

[0026] Preferably, when generating the ringing effect, the frequency change rate of the chirped continuous wave input to the system exceeds the square of the Lorentz linewidth of the optical microcavity, causing the readout spectrum to generate the ringing effect.

[0027] Preferably, the quadratic phase is compensated to recover the steady-state spectrum from any ringing-distorted readout spectrum, which specifically includes the following steps:

[0028] S1: Remove and in the frequency domain through Fourier transform, as follows:

[0029]

[0030]

[0031] If is the term of , the equation is well-defined;

[0032] S2: Express by the following formula:

[0033] where is and caused by the phase;

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

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

[0036] where represents the spectral resolution, which only depends on the chirp rate. Since the terms and are independent of the spectrum of the microcavity system, this method can recover any readout spectrum with any linear frequency modulation continuous wave chirp rate.

[0037] The beneficial effects of the present invention are:

[0038] The present invention generates a steady-state spectrum by cascading three different Fano-like line shapes to simulate a complex spectrum, verifying that the proposed method is applicable to various line shapes. Additionally, by introducing a spectral recovery algorithm with phase compensation, any distorted readout spectrum can be restored to its steady-state spectrum. Benefiting from 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 quadratic phase of the interference between the light emitted from the microcavity and the input / output of the waveguide, this quadratic 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, bridging the gap between the transient ringing spectrum and the steady-state spectrum, and opening up a new approach for high-speed sensing, optical communication, and advanced integrated chip technology. Brief Description of the Drawings

[0039] Figure 1 Several typical configuration diagrams of the microcavity system are shown;

[0040] Figure 2 The spectral recovery effect diagram of the present invention under a complex line shape is shown;

[0041] Among them, Figure 2 In (a), it shows that three cascaded Fano-like line shapes simulate the complex steady-state spectrum and the readout results at different chirp rates; (b) shows the comparison between the steady state and the restored readout at different chirp rates;

[0042] Figure 3 It is the schematic diagram of the experimental verification system of the present invention and the obtained experimental result diagram;

[0043] Among them, Figure 3 In (a), it is the schematic diagram of the experimental verification system; (b) is the ringing distorted spectrum read out experimentally; (c) is the ringing distorted spectrum obtained by theoretical simulation under the same parameters; (d) is the restored spectrum obtained by the present invention in the experiment; (e) is the restored spectrum theoretically obtained by using the present invention under the same parameters. Detailed Embodiment

[0044] The present invention will be further described below in conjunction with specific embodiments. It should be understood that these embodiments are only used to illustrate the present invention and not to limit the scope of the present invention. The experimental methods without specific conditions noted in the following embodiments are usually carried out under conventional conditions or according to the conditions recommended by the manufacturer.

[0045] Unless otherwise defined, all professional and scientific terms used herein have the same meaning as those familiar to persons skilled in the art. The reagents or raw materials used in the present invention can all be obtained by conventional means. Unless otherwise specified, the reagents or raw materials used in the present invention are used in accordance with the conventional methods in the art or in accordance with the product instructions. In addition, any methods and materials similar or equivalent to the described content can be applied to the method of the present invention. The present invention will now be further described with reference to the accompanying drawings and specific embodiments. The preferred implementation methods and materials described in the present invention are for illustrative purposes only.

[0046] The present invention first proposes that the ringing effect originates from the second-order phase of the interference between the light emitted from the microcavity and the input and output of the waveguide. This second-order phase depends on the chirp rate of the input chirped continuous wave and is independent of 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.

[0047] We first start with the factors that cause output distortion. As shown in the accompanying drawings of the specification Figure 1 shows several typical configurations of the microcavity system, including a single microcavity, coupled microcavities with parity-time (PT) symmetry, electromagnetically induced transparency (EIT)-like, and Fano-spectroscopy-like characteristics.

[0048] It is verified through the following technical solutions, and the specific steps are as follows:

[0049] Fabricate a system that can read the ringing distortion spectrum through an optical microcavity, as shown in the accompanying drawings of the specification Figure 3 as shown in (a). Where AWG represents an arbitrary waveform generator; VOA represents a variable optical attenuator; PC represents a polarization controller; PD represents a photodetector; OSC represents an oscilloscope. The right inset shows the optical image of the optical microcavity. In this example, a structure of a whispering-gallery microcavity coupled with an optical fiber taper is used. A sawtooth wave is output by the arbitrary waveform generator to control the laser to output a chirped continuous wave within a certain range. After filtering through the variable optical attenuator and the polarization controller, it is incident into the structure of the whispering-gallery microcavity coupled with the optical fiber taper. The readout spectrum of the system is obtained at the output end of this structure through the photodetector and the oscilloscope. When the frequency of the input sawtooth wave is fast enough so that the chirp rate of the chirped continuous wave exceeds the square of the Lorentz linewidth of the optical microcavity, a ringing-distorted readout spectrum will be obtained.

[0050] First, start with the factors that cause output distortion. Considering a linear time-invariant system, the response of the microcavity system can be expressed as:

[0051] where is the Green's function of the system; and They are the input and output respectively. Considering the conditions of the waveguide-coupled microcavity, the Green's function of the system can be written as:

[0052] It consists of the input signal and the reflection spectrum of the microcavity system. Generally speaking, the reflection spectrum describes the microcavity absorption, and the dispersion caused by absorption can be written as:

[0053] where the amplitude , the phase and the time delay . To obtain the microcavity spectrum, we input the chirped continuous wave into the microcavity system. So the output should be written as:

[0054] where , and, , and are the amplitude, the starting frequency and the chirp rate of the input chirped continuous wave respectively. So the intensity read out in the time domain can be written as

[0055]

[0056] where , and are the starting frequency phase and the phase shift with quadratic function characteristics respectively. When not affected by and , the term can map out ; conversely, brings a linear shift, generates ringing oscillations, distorting the spectral readout.

[0057] To remove the introduced and phases, a Fourier transform is implemented on the readout device, and

[0058]

[0059]

[0060] If is the term of , the equation is well-defined. Therefore can be simplified to

[0061] It is found that this term is the result of and phases. Therefore, if Applied to the spectrum, the readout distortion will be eliminated. Finally, an inverse Fourier transform is performed to obtain the restored readout in the time domain.

[0062]

[0063] Among them, represents the spectral resolution and depends only on the chirp rate. Since the terms and are independent of the spectrum of the microcavity system, this method can restore any readout spectrum with an arbitrary linear frequency modulation continuous wave chirp rate.

[0064] The results obtained in the embodiments of the present disclosure are as shown in the accompanying drawings of the specification. Figure 3 Theoretically, Figure 3 (c) Read out the ringing distortion spectrum at a chirp rate of 3 kHz / µs and obtained the restored spectrum shown in Figure 3 (e). Experimentally, Figure 3 (b) Obtained the distorted spectrum using exactly the same parameters as in the theory and obtained the restored spectrum shown in Figure 3 (d). The restored readout results produced an identified steady-state spectrum, enabling this application to obtain the parameters masked by the distortion.

Claims

1. A method for recovering the output spectrum distortion caused by fast linear frequency modulation continuous wave, characterized in that, The quadratic phase depends on the chirp rate of the input linear frequency modulated continuous wave and is independent of 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 represented by the following formula: α out (f) = H(f) × α in (f) where H(f) is the Green's function of the system, and α in (f) and α out (f) are the input and output respectively; Step 2: Introduce the conditions of the waveguide-coupled microcavity. The Green's function of the system consists of the input signal and the reflection spectrum of the microcavity system. The Green's function of the system is represented by the following formula: H(f) = 1 - R(f) And the reflection spectrum describes the microcavity absorption, and the dispersion caused by the absorption is represented by the following formula: Among them, γ(τ) represents the amplitude, represents the phase, and τ represents the time delay; Step 3: Input the linear frequency modulated continuous wave into the microcavity system, and the output is represented 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, and 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 by the system is represented by the following formula in the time domain: s(t) is represented by the following formula: where θ1 = -2πf S τ is the starting frequency phase, and θ2 = πkτ 2 is a phase shift with quadratic function characteristics Compensate the quadratic phase to recover its steady-state spectrum from the readout spectrum with any ringing distortion. It specifically includes the following steps: S1: Remove θ1 and θ2 in the frequency domain through Fourier transform, as shown below: If the term for f = ±kτ is δ(f = ±kτ), the equation is well-defined; S2: After simplifying S(f), it is represented by the following formula: wherein, is the result caused by the phases θ1 and θ2; S3: Apply to the spectrum, and the readout distortion will be eliminated; Finally, perform the inverse Fourier transform to obtain the recovered readout in the time domain; where τ k = |f / k| represents the spectral resolution, which depends only on the chirp rate. Since the terms γ and are independent of the spectrum of the microcavity system, this method can recover any readout spectrum with an arbitrary linear frequency modulation continuous wave chirp rate.

2. A method for restoring the output spectrum distortion caused by a fast linear frequency modulation continuous wave according to claim 1, characterized in that, When not affected by θ1 and θ2, the s(t) term can map R(t). θ1 brings a linear offset, and θ2 generates a ringing oscillation, causing the spectral readout to be distorted.

3. A method for recovering output spectral distortion caused by fast linear frequency modulation continuous wave according to claim 2, characterized in that, 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 linewidth of the optical microcavity, causing the readout spectrum to produce the ringing effect.

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