A dual-input dual-output mode converter and its design method

The dual-input dual-output mode converter designed using reverse engineering algorithms solves the single-port limitation of existing mode converters, realizes multi-path conversion and high integration of mode converters, and improves the flexibility of the modular multiplexing system.

CN116088096BActive Publication Date: 2026-01-30ZHEJIANG LAB
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Patent Information

Application Number
CN202211582802.9
Authority / Receiving Office
CN · China
Patent Type
Patents(China)
Current Assignee / Owner
Filing Date
2022-12-09
Publication Date
2026-01-30
Estimated Expiration
2042-12-09

AI Technical Summary

Technical Problem

Existing mode converters are mostly single-port input and single-port output, and the design process relies on manual adjustment, making it difficult to implement more complex and highly integrated devices.

Method used

A dual-input dual-output mode converter is designed using a reverse design algorithm. By utilizing a silicon dioxide substrate, a top silicon layer, and a low-refractive-index cladding material, and by setting the etching state of rectangular cells, the mode conversion region can be flexibly controlled, supporting bidirectional conversion between TE0 and TE1 modes.

Benefits of technology

It achieves multi-path mode conversion in a compact size, improving the flexibility and integration of the modulus multiplexing system and enhancing the functional complexity of the mode converter.

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Abstract

This invention discloses a dual-input dual-output mode converter and its design method, relating to the field of micro-nano optoelectronic device technology. The device includes a silicon dioxide substrate, a top silicon layer, and a low-refractive-index cladding. The top silicon layer includes a first input waveguide, a second input waveguide, a mode conversion region, a first output waveguide, and a second output waveguide. The first and second input waveguides are connected to the first and second output waveguides respectively through the mode conversion region. The mode conversion region is divided into M×N rectangular units, which can be etched or unetched, with the etching depth being the same as the thickness of the top silicon layer. The state of the rectangular units is set by a quality factor. Based on reverse design, the device increases the mode conversion path within a limited size through reasonable division and calculation of the mode conversion region, achieving dual-input dual-output mode conversion functionality. This is of great significance for further improving the flexibility and integration of mode-division multiplexing systems.
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Description

Technical Field

[0001] This application relates to the field of micro-nano optoelectronic components technology, and in particular to a dual-input dual-output mode converter and its design method. Background Technology

[0002] With the continuous development of information technology in modern society, the transmission capacity of communication systems based on single-mode fiber has reached its limit due to the effects of optical nonlinearity and fiber fusion effect, making it increasingly difficult to meet the ever-increasing demand for communication traffic. To address this issue, various multiplexing technologies have been proposed, among which Mode Division Multiplexing (MDM) technology, which can provide more data multiplexing capabilities within a limited space, has attracted widespread attention from researchers. The main components involved in an MDM system include mode divider / demultiplexers, mode switches, and mode converters. The mode converter, used to convert between different modes at the receiving and transmitting ends of the MDM system, is one of the key components for improving the flexibility of the MDM system.

[0003] Current mode converters are based on various structures, such as asymmetric directional couplers, Mach-Zehnder interferometers, multimode interferometers, and Bragg gratings. Most of these mode converters are single-port input and single-port output; that is, an m-order mode input at the input port is converted to an n-order mode and output from the input port (where m and n are non-negative integers, and m ≠ n). This offers limited improvement in the flexibility of mode-division multiplexing systems. Furthermore, these structures have regular shapes, and the design process is basically based on existing analytical theory, manually setting device parameters, using optical simulation software for simulation calculations or experiments, and adjusting and optimizing parameters based on the obtained data until satisfactory results are achieved. Implementing more complex functions, increasing integration, and designing irregularly shaped devices presents significant challenges. Summary of the Invention

[0004] To address the problems existing in the prior art, the purpose of this application is to provide a dual-input dual-output mode converter and its design method, so as to provide a mode converter with low loss, ultra-compact size and higher flexibility.

[0005] This application is achieved through the following technical solution:

[0006] A dual-input dual-output mode converter and its design method include a silicon dioxide substrate, a top silicon layer, and a low-refractive-index cladding layer. The top silicon layer includes a first input waveguide, a second input waveguide, a mode conversion region, a first output waveguide, and a second output waveguide. The first input waveguide is connected to the first output waveguide through the mode conversion region, and the second input waveguide is connected to the second output waveguide through the mode conversion region. The mode conversion region is divided into M×N rectangular units. The rectangular units are either etched or unetched, with the etching depth being the same as the thickness of the top silicon layer. When the rectangular unit is etched, the internal material is a low-refractive-index material; when the rectangular unit is unetched, the internal material is silicon. The state of the rectangular units in the mode conversion region is set by a quality factor.

[0007] Furthermore, the wavelength range of the first input waveguide and the second input waveguide is from 1520nm to 1580nm.

[0008] Furthermore, the first and second input waveguides only support TE0 mode passage, while the first and second output waveguides simultaneously allow TE0 and TE1 modes to pass through.

[0009] Furthermore, the thickness of the silicon dioxide substrate is 3 μm, and the thickness of the top silicon layer is 220 nm.

[0010] Furthermore, the rectangular unit is a square with a side length of 100 nm.

[0011] Furthermore, the rectangular unit achieves the etching state through single-step etching.

[0012] Furthermore, setting the state of the rectangular cells in the mode transition region through the quality factor includes:

[0013] (1) Set all rectangular units to be in an unetched state as the initial state, and calculate the initial quality factor of the dual-input dual-output mode converter in the initial state.

[0014] (2) Starting from the first rectangular cell, perform the following operations on all rectangular cells in sequence: change the state of the rectangular cell and calculate the new quality factor, compare the new quality factor with the initial quality factor, if the new quality factor is greater than the initial quality factor, keep the state of the rectangular cell as the changed state and assign the new quality factor to the initial quality factor; otherwise, set the state of the rectangular cell to the state before the change.

[0015] (3) Repeat step (2) until the quality factor no longer increases.

[0016] Furthermore, the quality factor Where M is the number of wavelength channels of the input light, t1 is the transmittance of the TE1 mode detected in the first output waveguide when the TE0 mode is input from the first input waveguide at the input light wavelength, and t2 is the transmittance of the TE1 mode detected in the second output waveguide when the TE0 mode is input from the second input waveguide at the input light wavelength.

[0017] Furthermore, the quality factor

[0018] Where M is the number of wavelength channels, the calculation of FOM consists of two parts. The part before the "+" is composed of these parameters: at a certain wavelength, when the TE0 mode is input from the first input waveguide, the transmittance of the TE0 mode detected in the first output waveguide is x. 11 The transmittance of TE1 mode is t 11 , representing the crosstalk of the TE0 mode and the slew rate of the TE1 mode at the first output waveguide, respectively; the transmittance of the TE0 mode detected at the second output waveguide is x. 12 The transmittance of TE1 mode is t 12 , representing the crosstalk between the TE0 and TE1 modes at the second output waveguide, respectively; the transmittance of the TE0 mode monitored at the second input waveguide is r. 12 This represents the crosstalk of the TE0 mode at the second input waveguide. Since the second input waveguide is designed to support only the TE0 mode, the crosstalk of the TE1 mode at the second input waveguide is not considered. The part after the "+" consists of these parameters: at a certain wavelength, when the TE0 mode is input from the second input waveguide, the transmittance of the TE0 mode detected in the second output waveguide is x. 22 The transmittance of TE1 mode is t 22 , representing the crosstalk of the TE0 mode and the slew rate of the TE1 mode at the second output waveguide, respectively; the transmittance of the TE0 mode detected at the first output waveguide is x. 21 The transmittance of TE1 mode is t 21 , representing the crosstalk between the TE0 and TE1 modes at the first output waveguide, respectively; the transmittance of the TE0 mode monitored at the first input waveguide is r. 21 , representing the crosstalk of the TE0 mode at the first input waveguide. Since the first input waveguide is designed to only support the TE0 mode, the crosstalk of the TE1 mode at the first input waveguide is not considered.

[0019] The technical solutions provided by the embodiments of this application may include the following beneficial effects:

[0020] As demonstrated by the above embodiments, the reverse design algorithm employed in this application is significant in addressing the problems inherent in traditional "forward design," designing photonic devices with irregular structures, and further improving the integration density of integrated photonic circuits. The final designed dual-input dual-output mode converter is feasible and can be used in mode-division multiplexing systems to achieve mode switching. Based on the reverse design concept, the device design achieves dual-input dual-output mode switching functionality within a very compact size. Compared to existing work, the mode converter proposed in this invention increases the mode switching path within a limited size, which is of great significance for further improving the flexibility of mode-division multiplexing systems and enhancing system integration.

[0021] It should be understood that the above general description and the following detailed description are exemplary and explanatory only, and do not limit this application. Attached Figure Description

[0022] The accompanying drawings, which are incorporated in and form part of this specification, illustrate embodiments consistent with this application and, together with the description, serve to explain the principles of this application.

[0023] Figure 1 This is a schematic diagram of the top silicon structure of a dual-input dual-output mode converter according to Embodiment 1.

[0024] Figure 2 The diagram shows the field distribution simulation results of the dual-input dual-output mode converter according to Embodiment 1, where (a) is the field distribution simulation result of the dual-input dual-output mode converter in Embodiment 1 when the TE0 mode is input from the first input waveguide, and (b) is the field distribution simulation result of the dual-input dual-output mode converter in Embodiment 1 when the TE0 mode is input from the second input waveguide.

[0025] Figure 3 This is a schematic diagram of the top silicon structure of a dual-input dual-output mode converter according to Embodiment 2.

[0026] Figure 4 The diagram shows the field distribution simulation results of the dual-input dual-output mode converter according to Embodiment 2, where (a) is the field distribution simulation result of the dual-input dual-output mode converter in Embodiment 1 when the TE0 mode is input from the first input waveguide, and (b) is the field distribution simulation result of the dual-input dual-output mode converter in Embodiment 1 when the TE0 mode is input from the second input waveguide.

[0027] Figure label:

[0028] 1. First input waveguide; 2. Second input waveguide; 3. Mode conversion region; 4. First output waveguide; 5. Second output waveguide. Detailed Implementation

[0029] Exemplary embodiments will now be described in detail, examples of which are illustrated in the accompanying drawings. When the following description relates to the drawings, unless otherwise indicated, the same numbers in different drawings represent the same or similar elements. The embodiments described in the following exemplary embodiments do not represent all embodiments consistent with this application.

[0030] The terminology used in this application is for the purpose of describing particular embodiments only and is not intended to be limiting of the application. The singular forms “a,” “the,” and “the” used in this application and the appended claims are also intended to include the plural forms unless the context clearly indicates otherwise. It should also be understood that the term “and / or” as used herein refers to and includes any or all possible combinations of one or more of the associated listed items.

[0031] It should be understood that although the terms first, second, third, etc., may be used in this application to describe various information, such information should not be limited to these terms. These terms are only used to distinguish information of the same type from one another. For example, without departing from the scope of this application, first information may also be referred to as second information, and similarly, second information may also be referred to as first information. Depending on the context, the word "if" as used herein may be interpreted as "when," "when," or "in response to determination."

[0032] Example 1:

[0033] Figure 1 This is a schematic diagram of the top silicon structure of a dual-input dual-output mode converter according to Embodiment 1, as shown below. Figure 1 As shown, the dual-input dual-output mode converter may include a silicon dioxide substrate, a top silicon layer, and an upper cladding layer formed of a material with a refractive index lower than silicon (hereinafter referred to as a low-refractive-index material). The top silicon layer includes a first input waveguide 1, a second input waveguide 2, a mode conversion region 3, a first output waveguide 4, and a second output waveguide 5. The first input waveguide 1 is connected to the first output waveguide 4 through the mode conversion region 3, and the second input waveguide 2 is connected to the second output waveguide 5 through the mode conversion region 3. The mode conversion region 3 is divided into M×N rectangular units. The rectangular units are either etched or unetched, and the etching depth is the same as the thickness of the top silicon layer. When the rectangular unit is etched, the internal material is a low-refractive-index material; when the rectangular unit is unetched, the internal material is silicon. The state of the rectangular units in the mode conversion region 3 is set by a quality factor.

[0034] As can be seen from the above embodiments, this application's use of reverse design algorithms is significant in addressing the problem that existing mode converters are primarily single-input, single-output. This application increases the mode conversion path within a limited size, which is crucial for further improving the flexibility and system integration of modular multiplexing systems. The device design, based on the reverse design concept, achieves dual-input, dual-output mode conversion functionality within a very compact size. Compared to existing work, the mode converter proposed in this invention increases the mode conversion path within a limited size, which is significant for further improving the flexibility and system integration of modular multiplexing systems.

[0035] Specifically, the wavelength range of the input light in the first input waveguide 1 and the second input waveguide 2 is 1520nm to 1580nm, covering the C-band commonly used in the field of optical communication, ensuring the bandwidth characteristics of the device, and the wavelength range of the input light can also be adjusted according to the actual situation.

[0036] Specifically, the first input waveguide 1 and the second input waveguide 2 are designed to support only the TE0 mode, and the waveguide width is 500nm; the first output waveguide 4 and the second output waveguide 5 are designed to allow both the TE0 and TE1 modes to pass through, and the waveguide width is 800nm. The above waveguide widths can be adjusted according to the actual situation.

[0037] Specifically, the silicon dioxide substrate has a thickness of 3 μm, the top silicon layer has a thickness of 220 nm, the mode conversion region 3 is a square with a side length of 2 μm, and the rectangular unit is a square with a side length of 100 nm. It should be noted that the above dimensions can be adjusted according to actual conditions and requirements, and this application does not limit them. In this embodiment, the mode conversion region 3 is divided into 20×20 rectangular units. The rectangular units are either etched or not etched, and the etching depth is the same as the thickness of the top silicon layer. That is, when the rectangular unit is etched, the material is air; when the rectangular unit is not etched, the material is silicon. Device fabrication can be achieved through single-step etching.

[0038] Specifically, based on the desired device function (i.e., converting from the TE0 mode input to the first input waveguide 1 to the TE1 mode output to the first output waveguide 4, and from the TE0 mode input to the second input waveguide 2 to the TE1 mode output to the second output waveguide 5), a suitable figure of merit (FOM) is defined to evaluate the device performance. Setting the state of the rectangular cells in the mode conversion region 3 using the FOM includes:

[0039] (1) Set all rectangular cells to the unetched state as the initial state, and calculate the initial quality factor of the dual-input dual-output mode converter in the initial state, denoted as FOM. max ;

[0040] (2) Since the function of the device is symmetrical about the center line L of the mode conversion region 3 along the y-axis, the operation of the rectangular cells is also symmetrical about the left and right when performing simulation optimization. Starting from the first rectangular cell and the rectangular cells that are symmetrical about the first rectangular cell about the center line L of the mode conversion region 3 along the y-axis, the following operations are performed on all rectangular cells in sequence: change the state of the rectangular cell (that is, if the rectangular cell was originally in the non-etched state, change it to the etched state; otherwise, if the rectangular cell was originally in the etched state, change it to the non-etched state) and calculate the new quality factor (denoted as temp). The new quality factor temp is compared with the initial quality factor FOM. max If the new quality factor temp is greater than the initial quality factor FOM, then... max Then, the state of the rectangular unit remains the changed state, and the new quality factor temp is assigned to the initial quality factor FOM. max Conversely, the state of the rectangular unit is set back to its original state.

[0041] (3) Repeat step (2) until FOMmax no longer increases.

[0042] Specifically, after the calculation of 400 rectangular units is completed, it is called the end of one iteration. Then, step (2) is repeated for multiple iterations. When FOMmax no longer increases, the output device pattern is as follows. Figure 1 As shown.

[0043] Specifically, the state transformation of the rectangular element, the calculation and comparison of the FOM are all achieved using the programming language Python and the three-dimensional finite-time difference (3D-FDTD) function in the optical simulation software Lumerical.

[0044] In this embodiment, the quality factor Where M is the number of input optical wavelength channels, t1 is the transmittance of TE1 mode detected in the first output waveguide 4 when TE0 mode is input from the first input waveguide 1 at the input optical wavelength, and t2 is the transmittance of TE1 mode detected in the second output waveguide 5 when TE0 mode is input from the second input waveguide 2 at the input optical wavelength. FOM increases when the mode conversion efficiency increases and decreases when the mode conversion efficiency decreases.

[0045] At a wavelength of 1550nm, when the TE0 mode is input from the first input waveguide 1, the field distribution simulation results of the dual-input dual-output mode converter are shown in the appendix. Figure 2As shown in (a) above, the TE0 mode input from the first input waveguide 1 propagates along the positive y-axis. After passing through the mode conversion region 3, it is converted into the TE1 mode and output from the first output waveguide 4 along the positive x-axis. This achieves the conversion from TE0 to TE1 mode and the conversion of the mode propagation direction.

[0046] At a wavelength of 1550nm, when the TE0 mode is input from the second input waveguide 2, the field distribution simulation results of the dual-input dual-output mode converter are shown in the appendix. Figure 2 As shown in (b) above, the TE0 mode input from the second input waveguide 2 propagates along the positive y-axis. After passing through the mode conversion region 3, it is converted into the TE1 mode and output from the second output waveguide 5 along the negative x-axis, realizing the conversion from TE0 to TE1 mode and the conversion of mode propagation direction. Simulation results show that the designed mode converter achieves the expected dual-input dual-output mode conversion function, proving the feasibility of the design.

[0047] Example 2:

[0048] This embodiment is basically the same as the technical solution provided in Embodiment 1, except that: the mode conversion region 3 is a square with a side length of 2.6 μm, and the mode conversion region 3 is divided into 26×26 rectangular units. The definition of the quality factor FOM is as follows:

[0049]

[0050] Where M is the number of wavelength channels. The calculation of FOM consists of two parts. The part before the "+" is composed of these parameters: at a certain wavelength, when the TE0 mode is input from the first input waveguide 1, the transmittance of the TE0 mode detected in the first output waveguide 4 is x. 11 The transmittance of TE1 mode is t 11 , representing the crosstalk of the TE0 mode and the conversion rate of the TE1 mode at point 4 of the first output waveguide, respectively; the transmittance of the TE0 mode detected in the second output waveguide 5 is x. 12 The transmittance of TE1 mode is t 12 , representing the crosstalk of TE0 and TE1 modes at point 5 of the second output waveguide, respectively; the transmittance of the TE0 mode monitored at point 2 of the second input waveguide is r. 12 , representing the crosstalk of the TE0 mode at the second input waveguide 2. Since the second input waveguide 2 is designed to only support the TE0 mode, the crosstalk of the TE1 mode at the second input waveguide 2 is not considered.

[0051] The part after the "+" sign consists of these parameters: at a certain wavelength, when the TE0 mode is input from the second input waveguide 2, the transmittance of the TE0 mode monitored in the second output waveguide 5 is x. 22 The transmittance of TE1 mode is t22 , representing the crosstalk of the TE0 mode and the conversion rate of the TE1 mode at point 5 of the second output waveguide, respectively; the transmittance of the TE0 mode detected at point 4 of the first output waveguide is x. 21 The transmittance of TE1 mode is t 21 , representing the crosstalk of TE0 and TE1 modes at point 4 of the first output waveguide, respectively; the transmittance of the TE0 mode monitored at point 1 of the first input waveguide is r. 21 , represents the crosstalk of the TE0 mode at the first input waveguide 1. Since the first input waveguide 1 is designed to only support the TE0 mode, the crosstalk of the TE1 mode at the first input waveguide 1 is not considered.

[0052] As defined above, crosstalk in each waveguide of the mode converter also affects the FOM parameter value: the higher the mode conversion rate and the lower the crosstalk in each waveguide, the larger the FOM; conversely, the lower the mode conversion rate and the higher the crosstalk in each waveguide, the smaller the FOM. This definition of FOM takes into account both the insertion loss of the mode converter and the crosstalk performance of the device.

[0053] The mode converter finally calculated in this embodiment (such as...) Figure 3 (As shown) It can also achieve the same function as the dual-input dual-output mode converter described in Embodiment 1. The simulation results are as follows. Figure 4 (a) and Figure 4 As shown in (b):

[0054] At a wavelength of 1550nm, when the TE0 mode is input from the first input waveguide 1, the field distribution simulation results of the dual-input dual-output mode converter are shown in the appendix. Figure 4 As shown in (a) above, the TE0 mode input from the first input waveguide 1 propagates along the positive y-axis. After passing through the mode conversion region 3, it is converted into the TE1 mode and output from the first output waveguide 4 along the positive x-axis. This achieves the conversion from TE0 to TE1 mode and the conversion of the mode propagation direction.

[0055] At a wavelength of 1550nm, when the TE0 mode is input from the second input waveguide 2, the field distribution simulation results of the dual-input dual-output mode converter are shown in the appendix. Figure 4 As shown in (b) above, the TE0 mode input from the second input waveguide 2 propagates along the positive y-axis. After passing through the mode conversion region 3, it is converted into the TE1 mode and output from the second output waveguide 5 along the negative x-axis, thus realizing the conversion from TE0 to TE1 mode and the conversion of the mode propagation direction.

[0056] Other embodiments of this application will readily occur to those skilled in the art upon consideration of the specification and practice of the disclosure herein. This application is intended to cover any variations, uses, or adaptations of this application that follow the general principles of this application and include common knowledge or customary techniques in the art not disclosed herein.

[0057] It should be understood that this application is not limited to the precise structure described above and shown in the accompanying drawings, and various modifications and changes can be made without departing from its scope.

Claims

1. A dual-in dual-out mode converter and design method, characterized by, A silicon-on-insulator waveguide device includes a silicon dioxide substrate, a top layer of silicon, and a cladding layer of low refractive index material, the top layer of silicon including a first input waveguide, a second input waveguide, a mode conversion region, a first output waveguide, and a second output waveguide, wherein the first input waveguide is connected to the first output waveguide through the mode conversion region, the second input waveguide is connected to the second output waveguide through the mode conversion region, the mode conversion region is divided into M × N a plurality of rectangular cells, the state of the rectangular cells being etched or not etched, the etching depth being the same as the thickness of the top layer of silicon, the internal material being the low refractive index material when the rectangular cells are in the etched state, the internal material being silicon when the rectangular cells are in the not etched state, wherein the state of the rectangular cells in the mode conversion region is set by a quality factor. The first input waveguide and the second input waveguide only support TE0 mode passing, and the first output waveguide and the second output waveguide allow TE0 and TE1 modes to pass simultaneously.

2. The dual-input dual-output mode converter and design method of claim 1, wherein, The wavelength range input by the first input waveguide and the second input waveguide is 1520 nm to 1580 nm.

3. The dual input dual output mode converter and design method of claim 1, wherein, The thickness of the silicon dioxide substrate is 3 μm, and the thickness of the top layer of silicon is 220 nm.

4. The dual input dual output mode converter and design method of claim 1, wherein, The rectangular unit is a square with a side length of 100 nm.

5. The dual input dual output mode converter and design method of claim 1, wherein, The rectangular unit is realized by single-step etching.

6. The dual input dual output mode converter and design method of claim 1, wherein, Setting the state of the rectangular unit in the mode conversion region by the quality factor includes: (1) setting all the rectangular units in the non-etching state as the initial state, calculating the initial quality factor of the double-in double-out mode converter in the initial state; (2) starting from the first rectangular unit, sequentially performing the following operations on all the rectangular units: changing the state of the rectangular unit and calculating the new quality factor, comparing the new quality factor with the initial quality factor, if the new quality factor is greater than the initial quality factor, keeping the state of the rectangular unit as the changed state, and assigning the new quality factor to the initial quality factor; otherwise, setting the state of the rectangular unit to the state before the change; (3) repeating step (2) until the quality factor no longer increases.

7. The dual-2-input dual-2-output mode converter and design method of claim 6, wherein, the quality factor wherein, M is the number of wavelength channels of the input light, t 1 is the transmittance of TE1 mode monitored at the first output waveguide when TE0 mode is input from the first input waveguide at the input light wavelength; t 2 is the transmittance of TE1 mode monitored at the second output waveguide when TE0 mode is input from the second input waveguide at the input light wavelength.

8. The dual-2-input dual-2-output mode converter and design method of claim 6, wherein, the quality factor , wherein, M is the number of wavelength channels, the calculation of FOM contains two parts, the part before "+" is composed of these parameters: the transmittance of TE0 mode monitored at the first output waveguide when TE0 mode is input from the first input waveguide at a certain wavelength is x 11 the transmittance of TE1 mode is t 11 respectively represent the crosstalk of TE0 mode and the conversion rate of TE1 mode at the first output waveguide; the transmittance of TE0 mode monitored at the second output waveguide when TE0 mode is input from the first input waveguide at a certain wavelength is x 12 the transmittance of TE1 mode is t 12 respectively represent the crosstalk of TE0 mode and TE1 mode at the second output waveguide; the transmittance of TE0 mode monitored at the second input waveguide is r 12 representing the crosstalk of TE0 mode at the second input waveguide, since the second input waveguide is designed to support only TE0 mode, the crosstalk of TE1 mode at the second input waveguide is not considered, the part after "+" is composed of these parameters: the transmittance of TE0 mode monitored at the second output waveguide when TE0 mode is input from the second input waveguide at a certain wavelength is x 22 the transmittance of TE1 mode is t 22 respectively represent the crosstalk of TE0 mode and the conversion rate of TE1 mode at the second output waveguide; the transmittance of TE0 mode monitored at the first output waveguide when TE0 mode is input from the second input waveguide at a certain wavelength is x 21 the transmittance of TE1 mode is t 21 respectively represent the crosstalk of TE0 mode and TE1 mode at the first output waveguide; the transmittance of TE0 mode monitored at the first input waveguide is r 21 representing the crosstalk of TE0 mode at the first input waveguide, since the first input waveguide is designed to support only TE0 mode, the crosstalk of TE1 mode at the first input waveguide is not considered.

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