Inter-mode beat ranging noise suppression device based on cascaded delay superposition filtering

By filtering out redundant inter-mode beat signals generated by the optical comb through a cascaded delay superposition filter optical path, the problem of noise suppression in optical comb inter-mode beating ranging was solved, and a higher accuracy ranging effect was achieved.

CN121613467APending Publication Date: 2026-03-06HARBIN INST OF TECH
View PDF 0 Cites 0 Cited by

Patent Information

Application Number
CN202511910292.7
Authority / Receiving Office
CN · China
Patent Type
Applications(China)
Current Assignee / Owner
Filing Date
2025-12-17
Publication Date
2026-03-06

AI Technical Summary

Technical Problem

Existing technologies struggle to effectively suppress inter-mode beating ranging noise, especially in the context of the need for μm-level ranging accuracy during the flight of micro, nano, or even picosatellites, where effective noise suppression methods are lacking.

Method used

A cascaded delay superposition filtering inter-mode beat ranging noise suppression device is used to optically filter the redundant inter-mode beat signals generated by the optical comb through a cascaded optical comb delay superposition filtering optical path, retaining the ranging signal and filtering out the redundant signal.

Benefits of technology

It significantly suppresses inter-mode ranging noise, reduces the bandwidth requirements and cost of the ranging system, and improves ranging accuracy.

✦ Generated by Eureka AI based on patent content.

Smart Images

  • Figure CN121613467A_ABST
    Figure CN121613467A_ABST
Patent Text Reader

Abstract

The invention belongs to the technical field of laser ranging, and provides an inter-mode beat ranging noise suppression device based on cascade type delay superposition filtering, which is characterized in that laser emitted by an optical comb light source firstly performs optical filtering on redundant inter-mode beat signals generated by an optical comb through a cascade type optical comb delay superposition filtering optical path; and then the light enters an optical comb inter-mode beat signal detection optical path to obtain an optical comb inter-mode beat signal for distance measurement, and finally the optical comb inter-mode beat signal is transmitted to an optical comb inter-mode beat signal processing, acquisition and calculation module to obtain a distance measurement result. Because the redundant inter-mode beat signal generated by the optical comb is filtered by the cascaded optical comb delay superposition filtering optical path, the ranging noise introduced by the redundant inter-mode beat signal to the ranging result is suppressed. According to the method, the suppression effect on the inter-mode beat ranging noise of the optical comb is obvious, and the cost is low. According to the method, the redundant inter-mode beat signals generated by the optical comb are filtered, and meanwhile, the inter-mode beat signals used for distance measurement are reserved, so that the distance measurement noise introduced by the redundant inter-mode beat signals is remarkably suppressed.
Need to check novelty before this filing date? Find Prior Art

Description

Technical Field

[0001] This invention belongs to the field of laser ranging technology, specifically relating to an intermodal beat ranging noise suppression device based on cascaded delay superposition filtering. Background Technology

[0002] Inter-mode beat ranging is a high-precision laser ranging technique. This technique utilizes the characteristic that the phase of the beat signal generated by the beat frequencies of different longitudinal modes in an optical comb varies with the propagation distance of the comb to achieve ranging. This beat signal is called the inter-mode beat signal, hence the name inter-mode beat ranging.

[0003] Ranging noise is one of the important indicators for evaluating ranging accuracy. For the inter-comb beat ranging technique, the averaging time of the ranging results is 10. ms Allan bias analysis was performed, and the results were used as an evaluation index for inter-mode beat ranging noise. Currently, the noise level of the inter-mode beat ranging prototype is in the tens of... μm Quantity (@10) ms However, as research into the formation flight of tiny, nano, and even pico satellites continues to deepen, inter-satellite ranging technology has also emerged accordingly. μm The need for higher precision, or even greater precision, necessitates the suppression of inter-module ranging noise.

[0004] The main method for suppressing inter-mode beat noise in ranging is to boost the frequency of the ranging signal. The ranging signal is a first-order inter-mode beat signal generated by the optical comb; the propagation distance of the optical comb is calculated by extracting the phase of the ranging signal. Existing technologies use 1026... MHz When the ranging signal is received, the ranging noise reaches 30. μm @10 ms However, because this method relies too heavily on the bandwidth of the ranging system (such as the response bandwidth of the photodetector and the spectral bandwidth of the optical comb), it is difficult to further improve the ranging noise suppression effect.

[0005] Furthermore, existing technologies have revealed that the intermodal beat signals generated by the optical comb introduce significant noise into each other during photoelectric detection. Therefore, for intermodal beat ranging systems, redundant intermodal beat signals (other than the ranging signal) become one of the main sources of intermodal beat ranging noise. However, there is currently a lack of effective methods to filter redundant intermodal beat signals before photoelectric detection, making it difficult to suppress intermodal beat ranging noise.

[0006] In summary, current inter-module ranging technology lacks a device for processing ranging noise. Summary of the Invention

[0007] To address the aforementioned technical problems, this invention provides an inter-mode beat ranging noise suppression device based on cascaded delay superposition filtering, thereby resolving the issues in the prior art. The technical solution adopted by this invention is as follows: The noise suppression device for intermodal beat ranging based on cascaded delay superposition filtering includes a comb light source, a cascaded comb delay superposition filtering optical path, a comb intermodal beat signal detection optical path, and a comb intermodal beat signal processing, acquisition, and calculation module connected in sequence. In this process, the laser emitted by the optical comb source first passes through a cascaded optical comb delay superposition filtering optical path to perform optical filtering on the redundant inter-mode beat signals generated by the optical comb. Then, it enters the optical comb inter-mode beat signal detection optical path to acquire the optical comb inter-mode beat signal used for ranging. Finally, the optical comb inter-mode beat signal is transmitted to the optical comb inter-mode beat signal processing, acquisition and calculation module to obtain the ranging result.

[0008] Furthermore, the cascaded optical comb delay superposition filtering optical path includes a 1×2 fiber coupler and a cascaded filtering unit; The cascaded filtering unit includes an upper channel fiber delay line, a lower channel fiber delay line, and a 2×2 fiber coupler; the upper channel fiber delay line and the lower channel fiber delay line together form a parallel optical path structure, and the output ends of the upper channel fiber delay line and the lower channel fiber delay line are connected to the 2×2 fiber coupler. The cascaded filter units are arranged in a series of interconnected units. The output of the 2×2 fiber coupler of the upper-level cascaded filter unit is connected to the input of the upper channel fiber delay line and the lower channel fiber delay line of the lower-level cascaded filter unit. The input ends of the upper and lower channel fiber delay lines of the cascaded filter unit located in the first stage are connected to the output end of the 1×2 fiber coupler, and the input end of the 1×2 fiber coupler is connected to the optical comb light source. The output of the 2×2 fiber coupler in the last stage of the cascaded filter unit is connected to the optical path for detecting the intermodal beat signal of the optical comb.

[0009] Furthermore, the repetition frequency of the optical comb light source is locked to [value missing]. fr ; Extract the first generated by the optical comb N 2 X Intermodal beat signals are used for distance measurement.

[0010] Furthermore, the first X The optical path difference between the upper and lower fiber delay lines of the cascaded filter unit is c / fr / 2 X ,in c It is the speed of light in a vacuum.

[0011] Furthermore, the optical comb signal emitted by the optical comb light source is represented as follows: ; In the formula fr , f 0 represents the repetition frequency and offset frequency of the optical comb, respectively; z This indicates the order of each longitudinal mode in the optical comb. A ( z ) indicates the first z The amplitude of the longitudinal mode; l Indicates the optical path length of the optical comb; t Indicates time; c It represents the speed of light in a vacuum.

[0012] Furthermore, the laser emitted from the optical comb source passes through a cascaded optical comb delay superposition filtering path before entering the optical comb intermodal beat signal detection path. At this point, the optical comb signal under the same optical power is represented as: ; In the formula x for X The ordinal number of each channel; fr , f 0 represents the repetition frequency and offset frequency of the optical comb, respectively; z This indicates the order of each longitudinal mode in the optical comb. A ( z ) indicates the first z The amplitude of the longitudinal mode; l Indicates the optical path length of the optical comb; t Indicates time; c This represents the speed of light in a vacuum; at this time, each pair of longitudinal modes in the optical comb beats with each other, and the generated intermodal beat signal model is: ; In the formula, X The number of fiber delay lines with increasing optical path length; k It is a positive integer; fr Indicates the repetition frequency of the optical comb; z This indicates the order of each longitudinal mode in the optical comb. A ( z ) indicates the first z The amplitude of the longitudinal mode; l Indicates the optical path length of the optical comb; t Indicates time; c It represents the speed of light in a vacuum.

[0013] The present invention has the following beneficial effects: This invention offers significant and cost-effective suppression of inter-mode beat noise in optical comb ranging. It filters out redundant inter-mode beat signals generated by the optical comb while retaining the inter-mode beat signals used for ranging, thus significantly suppressing the ranging noise introduced by redundant inter-mode beat signals. Furthermore, it does not rely on increasing the frequency of the ranging signal, reducing the bandwidth requirements and cost of the ranging system. Attached Figure Description

[0014] Figure 1 This is an overall structural diagram of the present invention; Figure 2 The intermodal beat signal spectrum before and after applying the present invention; Figure 3 The inter-module ranging results before and after applying this invention; Figure 4 The results of Allen's deviation analysis are shown for the inter-modal shooting distance measurement results before and after applying the present invention. Detailed Implementation

[0015] The following will be based on embodiments of the present invention. Figures 1-4 The technical solutions in the embodiments of the present invention will be clearly and completely described. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Unless otherwise specified, the technical means used in the embodiments are conventional means well known to those skilled in the art.

[0016] like Figure 1 This invention proposes an inter-mode beat ranging noise suppression device based on a cascaded optical comb delay superposition filtering optical path, including an optical comb light source 1, a cascaded optical comb delay superposition filtering optical path 2, an optical comb inter-mode beat signal detection optical path 3, and an optical comb inter-mode beat signal processing, acquisition, and calculation module 4.

[0017] The laser emitted by the optical comb source 1 first passes through the cascaded optical comb delay superposition filter optical path 2 to optically filter the redundant inter-mode beat signal generated by the optical comb. Then, it enters the optical comb inter-mode beat signal detection optical path 3 to acquire the optical comb inter-mode beat signal used for ranging. Finally, the optical comb inter-mode beat signal is transmitted to the optical comb inter-mode beat signal processing, acquisition, and calculation module 4 to obtain the ranging result. Since the redundant inter-mode beat signal generated by the optical comb is filtered out by the cascaded optical comb delay superposition filter optical path 2, the ranging noise introduced by the redundant inter-mode beat signal to the ranging result is suppressed.

[0018] In this invention, both the optical comb inter-mode beat signal detection optical path 3 and the optical comb inter-mode beat signal processing, acquisition, and calculation module 4 are existing technologies; specifically, the optical comb inter-mode beat signal detection optical path 3 can be found in the reference "High-precision absolute distance measurement based on femtosecond laser longitudinal mode beat signal", Optics and Optoelectronics Technology, 2017; whereby... Figure 3The optical path section, namely the typical structure of optical comb intermode beat signal detection optical path 3, is described. The optical comb intermode beat signal processing, acquisition, and calculation module 4 can be referenced in the literature "Real-time suppression of random phase drift for laser ranging with high-frequency intermode beats," Chinese Optics Letters, 2023; among which... Figure 2 The circuit part, namely the typical structure of the optical comb inter-mode beat signal processing, acquisition and calculation module 4.

[0019] In this invention, the cascaded optical comb delay superposition filter optical path 2 includes a 1×2 fiber coupler 5 and a cascaded filter unit; The cascaded filtering unit includes an upper channel fiber delay line, a lower channel fiber delay line, and a 2×2 fiber coupler; the upper channel fiber delay line and the lower channel fiber delay line together form a parallel optical path structure, and the output ends of the upper channel fiber delay line and the lower channel fiber delay line are connected to the 2×2 fiber coupler. The cascaded filter units are arranged in a series of interconnected units. The output of the 2×2 fiber coupler of the upper-level cascaded filter unit is connected to the input of the upper channel fiber delay line and the lower channel fiber delay line of the lower-level cascaded filter unit. The input ends of the upper and lower channel fiber delay lines of the cascaded filter unit located in the first stage are connected to the output end of the 1×2 fiber coupler 5, and the input end of the 1×2 fiber coupler 5 is connected to the optical comb light source 1. The output of the 2×2 fiber coupler in the last stage of the cascaded filter unit is connected to the optical comb intermodal beat signal detection optical path 3.

[0020] Specifically, such as Figure 1 The first-stage cascaded filter unit includes a first-stage upper channel fiber delay line 6, a first-stage lower channel fiber delay line 7, and a first-stage 2×2 fiber coupler; the second-stage cascaded filter unit includes a second-stage upper channel fiber delay line 9, a second-stage lower channel fiber delay line 10, and a second-stage 2×2 fiber coupler 11; X Cascaded filter units include the first X Level 12 Fiber Delay Line X Level 13 fiber delay line X 14-stage 2×2 fiber optic couplers, in which X It is a positive integer.

[0021] The light beam emitted from the optical comb source 1 is split into two beams by the 1×2 fiber coupler 5. The two beams then enter the first-stage upper channel fiber delay line 6 and the first-stage lower channel fiber delay line 7, respectively, and are then combined and split again by the first-stage 2×2 fiber coupler 8. The resulting two beams then enter the second-stage upper channel fiber delay line 9 and the second-stage lower channel fiber delay line 10, respectively, and are then combined and split again by the second-stage 2×2 fiber coupler 11. This process continues until the beams enter the first-stage upper channel fiber delay line 9 and the second-stage lower channel fiber delay line 10, respectively. X The 12th and 1st stage upper channel fiber delay line X After the first-stage lower channel fiber delay line 13, through the first... X The 2×2 fiber coupler 14 combines and splits the light into two beams again. Either of the two beams enters the intermodal beat signal detection optical path 3 of the optical comb.

[0022] In this invention, the repetition frequency of the optical comb light source 1 is locked to [value missing]. fr The ranging system extracts the ( ) generated by the optical comb. N 2 X Intermodal beat signals are used for ranging. N (The value is a positive integer). The optical path difference between the first-stage upper channel fiber delay line 6 and the first-stage lower channel fiber delay line 7 is... c / fr / 2 1 The optical path difference between the second-stage upper channel fiber delay line 9 and the second-stage lower channel fiber delay line 10 is... c / fr / 2 2 And so on, the first X The 12th and 1st stage upper channel fiber delay line X The optical path difference of the lower channel fiber delay line 13 is c / fr / 2 X ,in c It is the speed of light in a vacuum.

[0023] Alternatively, the final stage 2×2 fiber coupler can be replaced with a 2×1 fiber coupler, and its output beam can be fed into the intermodal beat signal detection optical path 3 of the optical comb.

[0024] The technical principle of this invention is as follows: An optical comb has multiple longitudinal modes in its spectrum, and the signal generated by the beat frequencies between these modes is called the inter-mode beat signal. The inter-mode beat signal contains multiple signal components of different frequencies, of which one or more frequencies are used by the ranging system for distance measurement, i.e., as the ranging signal, while the other useless frequencies are called redundant inter-mode beat signals. This invention achieves ranging noise suppression by filtering out the redundant inter-mode beat signals generated by the optical comb. The optical filtering of the redundant inter-mode beat signals is based on the principle of destructive interference, i.e., for... Two signals with equal amplitude and frequency, when this The phases of the signals form an arithmetic sequence with intervals of . At that time, this The superposition of these signals can be expressed as follows: ; The formula shows that The signals are canceled out together. Based on this principle, we first analyze the inter-mode beat signals before applying this invention: The optical comb signal emitted by optical comb source 1 can be represented as: ; In the formula fr , f 0 represents the repetition frequency and offset frequency of the optical comb, respectively; z This indicates the order of each longitudinal mode in the optical comb. A ( z ) indicates the first z The amplitude of the longitudinal mode; l Indicates the optical path length of the optical comb; t Indicates time; c It represents the speed of light in a vacuum.

[0025] Before applying this invention, in the optical comb where each order of longitudinal modes directly enters the optical comb intermodal beat signal detection optical path 3 from the optical comb light source 1, mutual beat frequencies occur between each pair of modes, and the generated intermodal beat signal model is as follows: ; In the formula y A positive integer, representing the difference in order. y The two-order longitudinal mode beat frequency occurs; fr Indicates the repetition frequency of the optical comb; z This indicates the order of each longitudinal mode in the optical comb. A ( z ) indicates the first z The amplitude of the longitudinal mode; l Indicates the optical path length of the optical comb; t Indicates time; c It represents the speed of light in a vacuum.

[0026] According to the above formula, the intermodal beat signal generated by the optical comb includes multiple frequency signal components, where the frequency of each signal component is a repetition frequency. fr An integer multiple of the order of the inter-modal beat signal is used, which represents the order of the inter-modal beat signal. Since multiple orders of inter-modal beat signals exist simultaneously, the order is needed to distinguish the individual signal components. When a ranging system selects an inter-modal beat signal of a certain order as the ranging signal, inter-modal beat signals of other orders become redundant inter-modal beat signals.

[0027] After applying this invention, the optical comb signal originating from the optical comb source 1 passes through the cascaded optical comb delay superposition filter optical path 2 and then enters the optical comb intermodal beat signal detection optical path 3. At this time, the optical comb signal under the same optical power can be expressed as: ; In the formula x for X The ordinal number of each channel; fr , f 0 represents the repetition frequency and offset frequency of the optical comb, respectively; z This indicates the order of each longitudinal mode in the optical comb. A ( z ) indicates the first z The amplitude of the longitudinal mode; l Indicates the optical path length of the optical comb; t Indicates time; c This represents the speed of light in a vacuum. At this point, each pair of longitudinal modes in the optical comb experiences a beat frequency exchange, and the resulting intermodal beat signal model is: ; In the formula k It is a positive integer; fr Indicates the repetition frequency of the optical comb; z This indicates the order of each longitudinal mode in the optical comb. A ( z ) indicates the first z The amplitude of the longitudinal mode; l Indicates the optical path length of the optical comb; t Indicates time; c It represents the speed of light in a vacuum.

[0028] A comparison was made between the models of inter-mode beat signals detected by optical comb inter-mode beat signal detection optical path 3 before and after applying this invention. It was found that after applying this method, the () kX Intermodal signal of order (i.e., order of) X Intermodal beat signals that are integer multiples of each other are preserved, while other orders are not. X Integer multiples of the intermodal beat signal are filtered out. When the optical comb generates the ( NX When the intermodal beat signal of order 1 is used as the ranging signal, signals of other orders are used as redundant intermodal beat signals. In this case, the ranging signal is retained, while the redundant intermodal beat signals are filtered out. As described in the background art, redundant intermodal beat signals introduce noise into the ranging signal measurement. Therefore, this invention can suppress intermodal beat ranging noise.

[0029] In a specific implementation of this invention, the repetition frequency of the optical comb light source 1 is locked to [value missing]. fr =31.25 MHzThe ranging system extracts the 32nd-order intermodal beat signal generated by the optical comb for ranging. The optical path difference between the first-stage upper channel fiber delay line 6 and the first-stage lower channel fiber delay line 7 is 4.8. m The optical path difference between the second-stage upper channel fiber delay line 9 and the second-stage lower channel fiber delay line 10 is 2.4. m , No. X The 12th and 1st stage upper channel fiber delay line X The optical path difference of the lower-level channel fiber delay line 13 is 0.6. m .

[0030] Figure 2 The image shows the inter-mode beat signal spectrum before and after applying this invention. Referring to the power of the first-order inter-mode beat signal before applying this invention, the power of each signal in the spectrum is standardized. Since the total number of stages in the cascaded optical comb delay superposition filter optical path 2 is... X =4, therefore the order is 2. 4 Inter-modal beat signals that are integer multiples of each other are retained (e.g., the 16th and 32nd orders), while inter-modal beat signals of other orders are filtered out. The 32nd order inter-modal beat signal is used for ranging; therefore, the other signals are redundant inter-modal beat signals. In summary, this invention filters out redundant inter-modal beat signals.

[0031] Figure 3 The figures show the inter-module shot distance measurement results before and after applying the present invention. By comparing the distance measurement results before and after applying the present invention, it was found that the fluctuation range of the distance measurement results was reduced after applying the present invention, indicating that the present invention effectively suppresses the noise in inter-module shot distance measurement.

[0032] Figure 4 This document presents the Allan bias analysis results for inter-modal beat ranging before and after applying this invention. Referring to the inter-modal beat ranging noise evaluation index proposed by research teams from the Korea Advanced Institute of Science and Technology (KAIST) and ETH Zurich, the results before and after applying this invention are compared within a 10... ms The results of the Allen bias analysis over the mean time period, from 10 μm Reduced to 3 μm This indicates that the noise in the inter-mode ranging is significantly suppressed.

[0033] The above embodiments are merely descriptions of preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Any modifications, alterations, alterations, or substitutions made by those skilled in the art to the technical solutions of the present invention without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.

Claims

1. A device for intermodal beat ranging noise suppression based on cascaded delay-and-sum filtering, characterized by, The light comb light source (1), the cascaded light comb delay superposition filter light path (2), the light comb intermodal beat signal detection light path (3) and the light comb intermodal beat signal processing, acquisition and calculation module (4) are sequentially connected. The laser emitted by the light comb light source (1) first passes through the cascaded light comb delay superposition filter light path (2) to filter the redundant intermodal beat signals generated by the light comb, then enters the light comb intermodal beat signal detection light path (3) to obtain the light comb intermodal beat signals used for distance measurement, and finally transmits the light comb intermodal beat signals to the light comb intermodal beat signal processing, acquisition and calculation module (4) to obtain the distance measurement result.

2. The inter- modal beat noise rejection apparatus based on cascaded delay-and- sum filtering according to claim 1, wherein, The cascaded light comb delay superposition filter light path (2) comprises a 1*2 optical fiber coupler (5) and a cascaded filter unit. The cascaded filter unit comprises an upper channel optical fiber delay line, a lower channel optical fiber delay line and a 2*2 optical fiber coupler; the upper channel optical fiber delay line and the lower channel optical fiber delay line jointly constitute a parallel light path structure, and the output ends of the upper channel optical fiber delay line and the lower channel optical fiber delay line are connected to the 2*2 optical fiber coupler; A plurality of cascaded filter units are sequentially connected in the front and back, and the output end of the 2*2 optical fiber coupler of the upper cascaded filter unit is connected to the input ends of the upper channel optical fiber delay line and the lower channel optical fiber delay line of the lower cascaded filter unit. The input ends of the upper channel optical fiber delay line and the lower channel optical fiber delay line of the cascaded filter unit at the first stage are connected to the output end of the 1*2 optical fiber coupler (5), and the input end of the 1*2 optical fiber coupler (5) is connected to the light comb light source (1). The output end of the 2*2 optical fiber coupler of the cascaded filter unit at the last stage is connected to the light comb intermodal beat signal detection light path (3).

3. The inter- modal beat noise rejection apparatus based on cascaded delay-and- sum filtering according to claim 1 or 2, characterized in that, The repetition frequency of the optical comb light source (1) is locked to fr ; an inter-2 N 2 X order mode beat signal generated by the optical comb is used for ranging.

4. The inter- modal beat noise rejection apparatus based on cascaded delay-and- sum filtering according to claim 3, wherein, The optical path difference between the upper pass optical fiber delay line and the lower pass optical fiber delay line of the stage cascade filter unit is X c / 2 c / fr / 2 X wherein c c is the vacuum light speed.

5. The inter- modal beat noise suppression apparatus based on cascaded delay-and- sum filtering according to claim 1 or 2, characterized in that, The light comb signal emitted by the light comb light source (1) is represented as: ; wherein fr , f 0 represent the repetition frequency, the offset frequency of the optical comb, respectively; z represents the order of the longitudinal mode in the optical comb, A z represents the amplitude of the longitudinal mode of order z .​ l represents an optical path of the optical comb propagation; t represents time; c represents the vacuum light speed.

6. The inter- modal beat noise rejection apparatus based on cascaded delay-and- sum filtering according to claim 5, wherein, After the laser emitted by the light comb light source (1) passes through the cascaded light comb delay superposition filter light path (2) and then enters the light comb intermodal beat signal detection light path (3), the light comb signal under the same optical power is represented as: ; wherein x is X the ordinal number of the channel; fr , f 0 denote the repetition frequency, the offset frequency of the optical comb, respectively; z denotes the order of the longitudinal mode in the optical comb, A ( z ) denotes the amplitude of the longitudinal mode of order z . l represents an optical path of the optical comb propagation; t represents time; c represents the vacuum light speed; at this time, the mutual beat frequencies occur between each two longitudinal modes in the optical comb, and the mode inter-beat signal model generated is: ; wherein X is the number of optical fiber delay lines with increasing optical path lengths; k is a positive integer; fr denotes the repetition frequency of the optical comb; z denotes the order of the longitudinal modes in the optical comb, A ( z ) denotes the amplitude of the longitudinal mode of order z . l represents an optical path of the optical comb propagation; t represents time; c represents the vacuum light speed.