Light-controlled multi-beam forming network chip and network

By introducing Sagnac rings and delay line structures into the optical beamforming network, the problems of cross-connection and limited scalability in the optical beamforming network are solved, achieving chip area reduction and performance improvement, which is suitable for scenarios such as phased array radar.

CN114217293BActive Publication Date: 2026-01-02UNITED MICROELECTRONICS CENT CO LTD
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Patent Information

Application Number
CN202111502957.2
Authority / Receiving Office
CN · China
Patent Type
Patents(China)
Current Assignee / Owner
Filing Date
2021-12-09
Publication Date
2026-01-02
Estimated Expiration
2041-12-09

AI Technical Summary

Technical Problem

Existing optically controlled beamforming networks suffer from crossover issues and limited scalability. In particular, under the requirement of multi-beamforming, the number of waveguide crossovers increases, leading to system performance degradation and making integration and expansion difficult.

Method used

A Sagnac ring is used to realize clockwise and counterclockwise transmission loops of optical signals. Combined with first-level and second-level delay lines, the combination of wavelength division multiplexer and delay lines reduces on-chip waveguide crossings and reuses the first-level wavelength division multiplexer and first-level delay line to design an optically controlled multi-beamforming network chip.

Benefits of technology

It effectively reduces chip area, lowers crosstalk and loss between channels, and has good scalability, meeting the requirements of multi-element, multi-beam beamforming systems.

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Abstract

The application provides an optical control multi-beam forming network chip and network, the chip comprising a first wavelength division multiplexer and MxN delay lines, each delay line being connected with the first wavelength division multiplexer, each delay line comprising a first delay line and a Sagnac ring, one end of the first delay line being connected with the first wavelength division multiplexer; the first wavelength division multiplexer is used for dividing MxN optical signals of different wavelengths into MxN paths and inputting each optical signal into a corresponding first delay line; the other end of the first delay line is connected with the Sagnac ring of the delay line, the Sagnac ring is used for realizing clockwise and counterclockwise positive and negative two-way transmission loops of the optical signals and transmitting the optical signals back to the first delay line; the first delay line is also used for transmitting the optical signals transmitted back by the Sagnac ring to the first wavelength division multiplexer after receiving the optical signals, and the first wavelength division multiplexer is also used for combining the MxN optical signals with time delay difference. The application effectively avoids the problem of on-chip waveguide intersection, has small chip area size and is easy to expand.
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Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of photonic integrated delay chip, and particularly relates to an optical control multi-beam forming network chip and network. BACKGROUND

[0002] The optical control beam forming network based on photonic true delay technology has important application value in the field of phased array radar, can effectively solve the problems of beam tilting and the like existing in the traditional phased array based on electrical phase shifter, and has technical advantages of low loss, large bandwidth, small volume and weight, and anti-electromagnetic interference.

[0003] At present, the implementation of photonic true delay technology gradually transits from a system based on discrete devices such as optical fibers to a system based on on-chip devices such as optical switches, micro rings and optical gratings, and the adjustment of delay amount is realized by tuning the on-chip photonic devices such as optical switches, micro rings and optical gratings. However, when the optical control beam forming network is realized based on the above method, the crossing of waveguides in the network will be inevitably introduced on the chip (such as the Rotman lens type optical control multi-beam forming network chip shown in Figure 1 and the adjustable optical delay line type optical control multi-beam forming network chip shown in Figure 2 ), which brings certain influence on the loss and crosstalk of the beam forming network. Especially under the demand of multi-beam forming, multiple beams are generally realized by the way of scale replication, and under the condition of increasing the number of array elements and beams, the number of waveguide crossings is greatly increased, which deteriorates the system performance and greatly limits the expansibility, and is not easy to integrate and expand. SUMMARY

[0004] Therefore, the embodiments of the present application provide an optical control multi-beam forming network chip and network to solve the problems of crossing and expansion limitation existing in the multi-beam forming network in the prior art.

[0005] In a first aspect of the present application, an optical control multi-beam forming network chip is provided, comprising: a first wavelength division multiplexer and MxN delay lines, each delay line is connected with the first wavelength division multiplexer,

[0006] Each delay line comprises a first delay line and a Sagnac ring, one end of the first delay line is connected with the first wavelength division multiplexer; the first wavelength division multiplexer is used for dividing the optical signals of MxN different wavelengths into MxN paths, and inputting each optical signal into a corresponding first delay line; the other end of the first delay line is connected with the Sagnac ring of the delay line, the Sagnac ring is used for realizing clockwise and counterclockwise transmission loops of the optical signals, and transmitting the optical signals back to the first delay line; the first delay line is also used for transmitting the optical signals transmitted back by the Sagnac ring to the first wavelength division multiplexer after receiving the optical signals, and the first wavelength division multiplexer is also used for combining the optical signals of MxN paths with time delay difference into a bundle.

[0007] Wherein M and N are natural numbers, corresponding to the number of beams and the number of phased array antenna elements respectively.

[0008] Further, the Sagnac ring comprises a 1x2 coupler and a second delay line, and the 1x2 coupler is connected with the first delay line and the second delay line respectively.

[0009] Further, the time delay of each delay line is:

[0010]

[0011] Wherein τ i represents the time delay, θ represents the beam pointing angle, d represents the antenna element spacing, and c represents the speed of light.

[0012] Further, the first delay line is a switch-type optical delay line or a micro-ring type optical delay line.

[0013] Further, the second delay line is a switch-type optical delay line or a micro-ring type optical delay line.

[0014] Further, the first wavelength division multiplexer is a waveguide array grating or a cascaded unequal arm Mach-Zehnder type interferometer.

[0015] Further, the 1x2 coupler is an MMI coupler or a directional coupler.

[0016] In the second aspect of the present application, an optical control multi-beam forming network is provided, comprising a multi-wavelength laser, an electro-optical modulator, a circulator, a second wavelength division multiplexer, a photoelectric detector, a phased array antenna, a chip control circuit and the optical control multi-beam forming network chip of the first aspect,

[0017] The multi-wavelength laser is used for generating optical signals of MxN different wavelengths,

[0018] The electro-optical modulator is used for modulating a microwave signal onto the optical signal, and transmitting the modulated optical signal to the circulator,

[0019] The circulator is configured to transmit the optical signal to the optical-controlled multi-beam forming network chip, and transmit the optical signals of MxN different wavelengths with time delay difference processed by the optical-controlled multi-beam forming network chip to the second wavelength division multiplexer,

[0020] The chip control circuit is connected with the optical-controlled multi-beam forming network chip,

[0021] The second wavelength division multiplexer is configured to divide the optical signals of MxN different wavelengths into N paths and transmit to the corresponding photodetectors respectively,

[0022] The photodetector is configured to convert the optical signal into an electrical signal and transmit to the phased array antenna,

[0023] The phased array antenna is configured to emit the electrical signal.

[0024] Further, an optical fiber amplifier is connected between the circulator and the second wavelength division multiplexer.

[0025] In a third aspect, the application provides an optical-controlled multi-beam forming network, comprising N phased array antenna elements, N multi-wavelength lasers, N electro-optical modulators, a third wavelength division multiplexer, a circulator, a fourth wavelength division multiplexer, N photodetectors, a data processing unit, a chip control circuit and the optical-controlled multi-beam forming network chip of the first aspect,

[0026] The phased array antenna is configured to receive the electrical signal and transmit N different electrical signals to the corresponding electro-optical modulators,

[0027] The multi-wavelength laser is configured to generate N paths of optical signals containing M wavelengths,

[0028] The electro-optical modulator is configured to modulate the information of the electrical signal onto the optical signal, and transmit the optical signal to the third wavelength division multiplexer,

[0029] The third wavelength division multiplexer is configured to combine the modulated optical signals and transmit to the circulator,

[0030] The circulator transmits the combined optical signal to the optical-controlled multi-beam forming network chip, and transmits the optical signals of MxN different wavelengths with time delay difference processed by the optical-controlled multi-beam forming network chip to the fourth wavelength division multiplexer,

[0031] The optical-controlled multi-beam forming network chip is connected with the chip control circuit,

[0032] The fourth wavelength division multiplexer is configured to divide the optical signals of MxN different wavelengths with time delay difference into N paths and transmit to the corresponding photodetectors respectively;

[0033] The photodetector is used to convert the optical signal into an electrical signal and transmit the electrical signal to a digital processing unit.

[0034] Compared with the prior art, the present application has the following advantages:

[0035] The present application realizes the transmission loop by the Sagnac ring, reuses the first wavelength division multiplexer and the first delay line, effectively reduces the layout size, and makes the chip area size smaller; meanwhile, the chip architecture based on the wavelength division multiplexer can effectively avoid the problem of on-chip waveguide crossing, reduce the inter-channel crosstalk and loss, meet the demand of the beam forming system of multiple array elements and multiple beams, and has good scalability, especially in the case of a large number of array elements and channels. BRIEF DESCRIPTION OF DRAWINGS

[0036] More details, features and advantages of the present disclosure are disclosed in the following description of exemplary embodiments in conjunction with the accompanying drawings, in which:

[0037] Figure 1 A structure schematic diagram of an existing Rotman lens type optical control multi-beam forming network chip is shown;

[0038] Figure 2 A structure schematic diagram of an existing adjustable optical delay line type optical control multi-beam forming network chip is shown;

[0039] Figure 3 A structure schematic diagram of an optical control multi-beam forming network chip according to an exemplary embodiment of the present disclosure is shown;

[0040] Figure 4 A structure schematic diagram of a switch type optical delay line according to an exemplary embodiment of the present disclosure is shown;

[0041] Figure 5 A structure schematic diagram of a micro-ring type optical delay line according to an exemplary embodiment of the present disclosure is shown;

[0042] Figure 6 A structure schematic diagram of an optical control multi-beam forming network according to an exemplary embodiment of the present disclosure is shown;

[0043] Figure 7 A structure schematic diagram of an optical control multi-beam forming network according to another exemplary embodiment of the present disclosure is shown. DETAILED DESCRIPTION

[0044] Embodiments of the present disclosure will be described below in greater detail with reference to the accompanying drawings. While certain embodiments of the present disclosure are shown in the drawings, it is understood that the present disclosure can be embodied in various forms and should not be interpreted as being limited to the embodiments set forth herein; rather, these embodiments are provided so as to more completely and thoroughly understand the present disclosure. It is understood that the drawings and embodiments of the present disclosure are for exemplary purposes only and are not intended to limit the scope of protection of the present disclosure.

[0045] It should be understood that each of the steps recited in the method embodiments of the present disclosure can be performed in different orders and / or in parallel. In addition, the method embodiments can include additional steps and / or omit the steps shown. The scope of the present disclosure is not limited in this respect.

[0046] The term "comprising" and variations thereof as used herein are open-ended, and mean "including but not limited to". The term "based on" means "based, at least in part, on". The term "one embodiment" means "at least one embodiment"; the term "another embodiment" means "at least one additional embodiment"; the term "some embodiments" means "at least some embodiments". Related terms are defined in the description that follows. It should be noted that reference to a "first", "second", etc. concept in the present disclosure is merely for differentiating between different devices, modules, or units, and does not imply a sequence or interdependence of the functions performed by these devices, modules, or units.

[0047] It should be noted that the terms "one", "multiple" mentioned in the present disclosure are illustrative rather than restrictive, and those skilled in the art should understand that, unless otherwise explicitly indicated in the context, it should be understood as "one or more".

[0048] The names of the messages or information exchanged between the devices in the embodiments of the present disclosure are only for illustrative purposes, and are not intended to limit the scope of the messages or information.

[0049] The embodiments of the present disclosure provide an optical-controlled multi-beam forming network chip for the beam forming requirements of N array elements and M beams, such as Figure 3As shown, the chip includes a first wavelength division multiplexer and MxN delay lines, each of the delay lines is connected with the first wavelength division multiplexer, each of the delay lines includes a first delay line and a Sagnac loop, one end of the first delay line is connected with the first wavelength division multiplexer, the first wavelength division multiplexer is used to divide the MxN optical signals of different wavelengths into MxN paths, and input each path of the optical signals into a corresponding first delay line, the other end of the first delay line is connected with the Sagnac loop of the delay line, the Sagnac loop is used to realize a positive and negative transmission loop of the optical signals clockwise and counterclockwise, and the clockwise and counterclockwise transmission light fields are generated in the Sagnac loop, so that on-chip cross can be effectively avoided. The Sagnac loop also transmits the optical signals back to the first delay line, and the first delay line is also used to transmit the optical signals transmitted back by the Sagnac loop to the first wavelength division multiplexer after receiving the optical signals, and the first wavelength division multiplexer is also used to combine the MxN optical signals with a delay difference and then transmit the combined optical signals out of the chip. Wherein M and N are natural numbers, M and N correspond to the number of beams and the number of phased array antenna elements respectively. The values of N and M can be set according to specific needs, which can meet the requirements of the beam forming system of multiple elements and multiple beams, especially in the case of a large number of elements and channels, which has good scalability.

[0050] The first wavelength division multiplexer and the first delay line are used when receiving and transmitting optical signals, the first wavelength division multiplexer and the first delay line are repeatedly used, the layout size is effectively reduced, and the chip area size is smaller.

[0051] In the embodiment, the Sagnac loop includes a 1x2 coupler and a second delay line, and the 1x2 coupler is connected with the first delay line and the second delay line respectively. After receiving the optical signals delayed by the first delay line, the 1x2 coupler divides one path into two paths, so that the two paths are transmitted clockwise and counterclockwise through the second delay line for delay, and then transmitted back to the first delay line through the 1x2 coupler. The first delay line is responsible for realizing a large true delay, and the second delay line is responsible for realizing a small true delay, which can effectively reduce the layout area and further reduce the chip area.

[0052] The delay amount of each delay line is:

[0053]

[0054] Wherein, τ i represents the delay amount, θ represents the beam pointing angle, d represents the antenna element spacing, and c represents the speed of light.

[0055] In some optional embodiments, the first delay line is a switch type optical delay line or a micro-ring type optical delay line, and can also be other delay line structures with reciprocity. The second delay line is a switch type optical delay line or a micro-ring type optical delay line, and can also be other delay line structures with reciprocity. The structure of the switch type optical delay line is as follows:Figure 4 As shown, the structure of the micro-ring optical delay line is as follows: Figure 5 As shown.

[0056] In some alternative embodiments, the first wavelength division multiplexer is a waveguide array grating or a cascaded unequal-arm Mach-Zehnder interferometer, which can realize the multiplexing of wavelengths generated by multi-wavelength light sources. 11 ,λ 12 ,…,λ 1M ,λ 21 ,λ 22 ,…,λ 2M ,…,λ NM MxN optical signals of different wavelengths are divided into MxN paths and input into different first-stage delay lines.

[0057] In some alternative embodiments, the 1x2 coupler is an MMI coupler (Multi-Mode Interference Coupler, MM1Coupler) or a directional coupler.

[0058] This invention also provides an optically controlled multi-beamforming network, which is suitable for the transmission system of phased array radar, such as... Figure 6 As shown, it includes a multi-wavelength laser, an electro-optic modulator, a circulator, a second wavelength division multiplexer, a photodetector, a phased array antenna, a chip control circuit, and the optically controlled multibeamforming network chip described in the above embodiments.

[0059] Multi-wavelength lasers are used to generate λ 11 ,λ 12 ,…,λ 1M ,λ 21 ,λ 22 ,…,λ 2M ,…,λ NM M x N optical signals of different wavelengths.

[0060] Electro-optic modulators are used to modulate microwave signals onto optical signals and transmit the modulated optical signals to circulators. Intensity modulation and other methods can be used.

[0061] The circulator is used to transmit optical signals to the optically controlled multibeamforming network chip and transmit the MxN optical signals with different wavelengths with time delays processed by the optically controlled multibeamforming network chip to the second wavelength division multiplexer.

[0062] The chip control circuit is connected to the optically controlled multibeamforming network chip, λ 11 ,λ 21 ,λ 31 ,…,λ N1 The N array elements corresponding to the first beam have their delay amounts controlled by the delay lines of their respective paths. The delay amount for each path is... Thus, the corresponding directed beamforming is obtained; λ 12 ,λ 22 ,λ 32 ,…,λ N2 correspond to the N elements of the second beam, and so on, λ 1M ,λ 2M ,λ 3M ,…,λ NM correspond to the N elements of the Mth beam.

[0063] The second wavelength division multiplexer is used to divide the MxN different wavelength optical signals into N paths and transmit them to the corresponding photodetectors. The second wavelength division multiplexer can be a waveguide array grating or a cascade unequal arm Mach-Zehnder interferometer. The second wavelength division multiplexer can divide MxN different wavelength optical signals into N paths, the first path containing M wavelengths of optical signals λ 11 ,λ 12 ,…,λ 1M ,λ 21 ,λ 22 ,…,λ 2M ,…,λ NM , and so on. 11 ,λ 12 ,…,λ 1M , and the second path containing M wavelengths of optical signals λ 21 ,λ 22 ,…,λ 2M , and so on.

[0064] The photodetector is used to convert the optical signal into an electrical signal and transmit it to the phased array antenna. The phased array antenna is used to emit the electrical signal.

[0065] In some optional embodiments, an optical fiber amplifier is connected between the circulator and the second wavelength division multiplexer, and an amplifier can also be connected between the phased array antenna and the photodetector to amplify the signal.

[0066] The embodiment of the application also provides an optical control multi-beam forming network, which is suitable for receiving systems, such as Figure 7 As shown in the figure, the network includes N phased array antenna elements, N multi-wavelength lasers, N electro-optical modulators, a third wavelength division multiplexer, a circulator, a fourth wavelength division multiplexer, N photodetectors, a data processing unit, a chip control circuit, and the optical control multi-beam forming network chip described in the above embodiment.

[0067] The phased array antenna is used to receive the electrical signal and transmit N different electrical signals to the corresponding electro-optical modulator.

[0068] The multi-wavelength laser is used to generate N paths of M wavelength optical signals, the first path being λ 11 ,λ12 ,…,λ 1M ,second road is λ 21 ,λ 22 ,…,λ 2M ,etc.

[0069] The electro-optical modulator is used for modulating the information of the electrical signal onto the optical signal, and transmitting the optical signal to the third wavelength division multiplexer.

[0070] The third wavelength division multiplexer is used for transmitting the modulated optical signal to the circulator after beam combination, and the third wavelength division multiplexer can be a waveguide array grating or a cascaded unequal arm Mach-Zehnder interferometer.

[0071] The circulator transmits the combined optical signal to the optical multi-beam forming network chip, and transmits the MxN different wavelengths of optical signals with delay difference processed by the optical multi-beam forming network chip to the fourth wavelength division multiplexer.

[0072] The optical multi-beam forming network chip is connected with the chip control circuit.

[0073] The fourth wavelength division multiplexer is used for dividing the MxN different wavelengths of optical signals with delay difference into N paths and transmitting them to corresponding photodetectors respectively; the fourth wavelength division multiplexer can be a waveguide array grating or a cascaded unequal arm Mach-Zehnder interferometer. The fourth wavelength division multiplexer can realize the division of λ 11 ,λ 12 ,…,λ 1M ,λ 21 ,λ 22 ,…,λ 2M ,…,λ NM MxN different wavelengths of optical signals into N paths, the first path contains λ 11 ,λ 12 ,…,λ 1M M wavelengths of optical signals, the second path contains λ 21 ,λ 22 ,…,λ 2M M wavelengths of optical signals, and so on.

[0074] The photodetector is used for converting the optical signal into an electrical signal and transmitting it to the digital processing unit, and N photodetectors transmit N beam-formed electrical signals to the digital processing unit, and the digital processing unit receives and processes the signals to complete the reception.

[0075] In summary, the embodiment of the present application proposes an optical control multi-beam forming network chip, compared with the traditional optical control multi-beam forming chip, the architecture of the chip effectively avoids on-chip cross, reduces inter-channel crosstalk and loss, and has smaller area size. The chip can be used in the beam forming component of a transmitting system or a receiving system, and can meet the demand of a multi-element multi-beam beam forming system. It mainly faces phased array radar, electronic countermeasure system and other scene applications.

[0076] Although the embodiments of the present application are described in conjunction with the drawings, various modifications and changes can be made by those skilled in the art without departing from the spirit and scope of the present application, and such modifications and changes fall within the scope defined by the appended claims.

Claims

1. A light controlled multi-beam forming network chip, characterized in that, Comprising: a first wavelength division multiplexer and MxN a plurality of delay lines, each of the plurality of delay lines being connected to the first wavelength division multiplexer, Each delay line includes a first-stage delay line and a Sagnac loop, with one end of the first-stage delay line connected to the first wavelength division multiplexer; the first wavelength division multiplexer is used to... MxN Optical signals of different wavelengths are divided into MxN The optical signal is input into a corresponding first-stage delay line. The other end of each first-stage delay line is connected to a Sagnac loop, which is used to implement clockwise and counterclockwise transmission loops for the optical signal and to transmit the optical signal back to the first-stage delay line. The first-stage delay line also receives the optical signal returned from the Sagnac loop and transmits it to a first wavelength division multiplexer. The first wavelength division multiplexer is further used to... MxN The path combines optical signals with time delay. wherein M, N are natural numbers, respectively corresponding to the number of beams and the number of array elements of the phased array antenna; The Sagnac ring comprises a 1x2 coupler and a second-order delay line, and the 1x2 coupler is connected with the first-order delay line and the second-order delay line respectively. The delay time of each path delay line is: wherein, τ i denotes a delay amount, θ denotes a beam pointing angle, d denotes an antenna element spacing, c denotes the speed of light; The first-order delay line is a switch-type optical delay line or a micro-ring type optical delay line, and the second-order delay line is a switch-type optical delay line or a micro-ring type optical delay line.

2. The light-controlled multi-beam forming network chip of claim 1, wherein, The first wavelength division multiplexer is a waveguide array grating or a cascaded unequal arm Mach-Zehnder type interferometer.

3. The light control multi-beam forming network chip of claim 1, wherein, The 1x2 coupler is an MMI coupler or a directional coupler.

4. A light controlled multi-beam forming network characterized by, Comprising a multi-wavelength laser, an electro-optical modulator, a circulator, a second wavelength division multiplexer, a photoelectric detector, a phased array antenna, a chip control circuit and the optical-controlled multi-beam forming network chip of any one of claims 1-3, The multi-wavelength laser is used to generate MxN light signals of different wavelengths, The electro-optical modulator is used for modulating a microwave signal onto an optical signal, and transmitting the modulated optical signal to the circulator, The circulator is used to transmit the optical signal to the optical control multi-beam forming network chip, and transmit the optical signal with delay difference of different wavelengths processed by the optical control multi-beam forming network chip to the second wavelength division multiplexer. MxN The circulator is used to transmit the optical signal to the optical control multi-beam forming network chip, and transmit the optical signal with delay difference of different wavelengths processed by the optical control multi-beam forming network chip to the second wavelength division multiplexer. The chip control circuit is connected with the optical-controlled multi-beam forming network chip, The second wavelength division multiplexer is used to divide the optical signals of different wavelengths into MxN N different paths and transmit to corresponding photodetectors respectively.​ The photoelectric detector is used for converting the optical signal into an electrical signal and then transmitting the electrical signal to the phased array antenna, The phased array antenna is used for emitting the electrical signal.

5. The optically controlled multi-beam forming network of claim 4, wherein, An optical fiber amplifier is connected between the circulator and the second wavelength division multiplexer.

6. A light controlled multi-beam forming network characterized by, comprising N a phased array antenna element, N a multi-wavelength laser, N an electro-optical modulator, a third wavelength division multiplexer, a circulator, a fourth wavelength division multiplexer, N a photodetector, a data processing unit, a chip control circuit and the optical control multi-beam forming network chip of any one of claims 1-3, The phased array antenna is used to receive electrical signals and transmit N a different electrical signal to a corresponding electro-optical modulator, The multi-wavelength laser is used to generate N The route comprises M a light signal of one wavelength, The electro-optical modulator is used for modulating information of an electrical signal onto an optical signal, and transmitting the optical signal to a third wavelength division multiplexer, The third wavelength division multiplexer is used for combining the modulated optical signal and then transmitting the combined optical signal to the circulator, The circulator transmits the combined light signal to the optical control multi-beam forming network chip, and transmits the light signal with delay difference of different wavelengths processed by the optical control multi-beam forming network chip to the fourth wavelength division multiplexer. MxN The circulator transmits the combined light signal to the optical control multi-beam forming network chip, and transmits the light signal with delay difference of different wavelengths processed by the optical control multi-beam forming network chip to the fourth wavelength division multiplexer. The optical-controlled multi-beam forming network chip is connected with the chip control circuit, The fourth wavelength division multiplexer is used to divide the optical signals with delay difference of different wavelengths into MxN N paths and transmit to corresponding photodetectors respectively;​ The photoelectric detector is used for converting the optical signal into an electrical signal and then transmitting the electrical signal to the digital processing unit.

Citation Information

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