CN116088096A - Dual-input dual-output mode converter and design method - Google Patents
Dual-input dual-output mode converter and design method Download PDFInfo
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- G—PHYSICS
- G02—OPTICS
- G02B—OPTICAL ELEMENTS, SYSTEMS OR APPARATUS
- G02B6/00—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings
- G02B6/10—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings of the optical waveguide type
- G02B6/14—Mode converters
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- G—PHYSICS
- G02—OPTICS
- G02B—OPTICAL ELEMENTS, SYSTEMS OR APPARATUS
- G02B6/00—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings
- G02B6/10—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings of the optical waveguide type
- G02B6/12—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings of the optical waveguide type of the integrated circuit kind
- G02B2006/12035—Materials
- G02B2006/12061—Silicon
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- G—PHYSICS
- G02—OPTICS
- G02B—OPTICAL ELEMENTS, SYSTEMS OR APPARATUS
- G02B6/00—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings
- G02B6/10—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings of the optical waveguide type
- G02B6/12—Light guides; Structural details of arrangements comprising light guides and other optical elements, e.g. couplings of the optical waveguide type of the integrated circuit kind
- G02B2006/12133—Functions
- G02B2006/12152—Mode converter
Abstract
The invention discloses a double-input double-output mode converter and a design method, and relates to the technical field of micro-nano optoelectronic components. The device increases the path of mode conversion on the limited size by reasonable division and calculation of the mode conversion area based on reverse design, and realizes the double-in and double-out mode conversion function. Has important significance for further improving the flexibility and the integration level of the mode division multiplexing system.
Description
Technical Field
The application relates to the technical field of micro-nano optoelectronic components, in particular to a double-input double-output mode converter and a design method.
Background
With the continuous development of information technology in modern society, the transmission capacity of a communication system based on a single-mode fiber is approaching a limit due to the influence of optical nonlinearity and fiber fusion effect, and it is increasingly difficult to meet the increasing communication traffic demand. To solve this problem, various multiplexing techniques have been proposed successively, in which a mode division multiplexing (Mode Division Multiplexing, MDM) technique can provide multiplexing functions of more data in a limited space, and thus has received a great deal of attention from researchers. The main devices involved in the MDM system include a mode division multiplexer/demultiplexer, a mode switch, a mode converter, etc., where the mode converter is used to implement conversion between different modes at a receiving end and an emitting end in the MDM system, and is one of the key devices for improving flexibility of the MDM system.
Mode converters have been proposed today based on various structures such as asymmetric directional couplers, mach-zehnder interferometers, multimode interferometers, bragg gratings, etc. Most of the mode converters are single-port input and single-port output, namely, m-order modes input from an input port are converted into n-order modes through the mode converters and output from the input port (m and n are non-negative integers, and m is not equal to n), so that the flexibility of the mode division multiplexing system is limited. The shape rule of the structures is basically based on the existing analysis theory, the design flow is to manually set the parameters of the devices, perform simulation calculation or experiments by using optical simulation software, and adjust and optimize each parameter according to the obtained data result until a satisfactory result is obtained. The device with more complex functions, further improved integration and irregular design shape is very difficult to realize.
Disclosure of Invention
Aiming at the problems existing in the prior art, an object of the embodiments of the present application is to provide a dual-input dual-output mode converter and a design method thereof, so as to provide a mode converter with low loss, ultra-compact size and higher flexibility.
The application is realized by the following technical scheme:
the double-in and double-out mode converter comprises a silicon dioxide substrate, top silicon and a low refractive index material cladding layer, wherein the top silicon comprises a first input waveguide, a second input waveguide, a mode conversion area, a first output waveguide and a second output waveguide, the first input waveguide is connected with the first output waveguide through the mode conversion area, the second input waveguide is connected with the second output waveguide through the mode conversion area, the mode conversion area is divided into M multiplied by N rectangular units, the rectangular units are etched or not etched, the etching depth is the same as the thickness of the top silicon, and when the rectangular units are in an etching state, the internal material is the low refractive index material; when the rectangular unit is in a non-etching state, the internal material is silicon; wherein the states of the rectangular cells in the mode transition area are set by a quality factor.
Further, the first input waveguide and the second input waveguide input wavelengths in the range 1520nm to 1580nm.
Further, the first and second input waveguides support only TE 0 Mode pass, the first output waveguide and the second output waveguide simultaneously allow TE 0 And TE (TE) 1 The mode passes.
Further, the thickness of the silicon dioxide substrate is 3 μm, and the thickness of the top silicon layer is 220nm.
Further, the rectangular unit is square with a side length of 100 nm.
Further, the rectangular unit realizes an etching state through single-step etching.
Further, setting the state of the rectangular unit in the mode conversion area by the quality factor includes:
(1) Setting all rectangular units in a non-etching state as an initial state, and calculating an initial quality factor of the double-input double-output mode converter in the initial state;
(2) Starting from the first rectangular unit, the following operations are sequentially performed on all the rectangular units: changing the state of a rectangular unit, calculating a new quality factor, comparing the new quality factor with the initial quality factor, if the new quality factor is larger 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 as the state before changing;
(3) And (5) repeatedly executing the step (2) until the quality factor is not increased any more.
Further, the quality factor
Wherein M is the number of wavelength channels of the input light, t 1 Inputting TE from a first input waveguide for an input light wavelength 0 In mode, TE is monitored at the first output waveguide 1 Transmittance of the mode; t is t 2 Inputting TE from the second input waveguide for the input light wavelength 0 In mode, TE is monitored at the second output waveguide 1 Transmittance of the mode.
Further, the quality factor
Where M is the number of wavelength channels, the calculation of FOM comprises two parts, the part preceding "+" being composed of these parameters: at a certain wavelength, TE is input from a first input waveguide 0 In mode, TE is monitored at the first output waveguide 0 The transmittance of the mode is x 11 ，TE 1 The transmittance of the mode is t 11 Respectively represent TE at the first output waveguide 0 Crosstalk and TE of modes 1 A conversion rate of the mode; TE monitored at the second output waveguide 0 The transmittance of the mode is x 12 ，TE 1 The transmittance of the mode is t 12 Respectively represent TE at the second output waveguide 0 Mode and TE 1 Crosstalk of modes; TE monitored at the second input waveguide 0 Mode transmittance r 12 Representing TE at the second input waveguide 0 Crosstalk of modes, since the second input waveguide is designed to support only TE 0 Mode, therefore, does not consider TE at the second input waveguide 1 Crosstalk of modes. The part following "+" is made up of these parameters: at a certain wavelength, TE is input from the second input waveguide 0 In mode, TE is monitored at the second output waveguide 0 The transmittance of the mode is x 22 ，TE 1 The transmittance of the mode is t 22 Respectively represent TE at the second output waveguide 0 Strings of patternsScrambling and TE 1 A conversion rate of the mode; TE monitored at a first output waveguide 0 The transmittance of the mode is x 21 ，TE 1 The transmittance of the mode is t 21 Respectively represent TE at the first output waveguide 0 Mode and TE 1 Crosstalk of modes; TE monitored at first input waveguide 0 Mode transmittance r 21 Representing TE at the first input waveguide 0 Crosstalk of modes, since the first input waveguide is designed to support only TE 0 Mode, therefore, does not consider TE at the first input waveguide 1 Crosstalk of modes.
The technical scheme provided by the embodiment of the application can comprise the following beneficial effects:
according to the embodiment, the reverse design algorithm is used for solving the problem of the traditional forward design, and designing the photonic device with irregular structure, so that the integration level of the integrated photonic circuit is improved. The finally designed double-input and double-output mode converter can be realized and can be used for realizing mode conversion in an analog-to-digital multiplexing system. The device design is based on the design concept of reverse design, and the double-in and double-out mode conversion function is realized on a very compact size. Compared with the prior art, the mode converter provided by the invention increases the mode conversion path on the limited size, and has important significance for further improving the flexibility of the mode division multiplexing system and improving the system integration level.
It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory only and are not restrictive of the application.
Drawings
The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments consistent with the application and together with the description, serve to explain the principles of the application.
Fig. 1 is a schematic diagram of a top silicon structure of a dual-in dual-out mode converter according to embodiment 1.
FIG. 2 is a schematic diagram showing the result of field distribution simulation in a dual-input dual-output mode converter according to embodiment 1, where (a) is when TE 0 The field distribution simulation result in the dual-input dual-output mode converter in the first embodiment when the mode is input from the first input waveguide, (b) is the TE 0 The field distribution simulation results in the dual-input dual-output mode converter in the first embodiment when the mode is input from the second input waveguide.
Fig. 3 is a schematic diagram of a top silicon structure of a dual-in dual-out mode converter according to embodiment 2.
FIG. 4 is a schematic diagram showing the result of field distribution simulation in a dual-input dual-output mode converter according to embodiment 2, where (a) is when TE 0 The field distribution simulation result in the dual-input dual-output mode converter in the first embodiment when the mode is input from the first input waveguide, (b) is the TE 0 The field distribution simulation results in the dual-input dual-output mode converter in the first embodiment when the mode is input from the second input waveguide.
Reference numerals:
1. a first input waveguide; 2. a second input waveguide; 3. a mode conversion region; 4. a first output waveguide; 5. and a second output waveguide.
Detailed Description
Reference will now be made in detail to exemplary embodiments, examples of which are illustrated in the accompanying drawings. When the following description refers to the accompanying drawings, the same numbers in different drawings refer to the same or similar elements, unless otherwise indicated. The implementations described in the following exemplary examples are not representative of all implementations consistent with the present application.
The terminology used in the present application is for the purpose of describing particular embodiments only and is not intended to be limiting of the present application. As used in this application and the appended claims, the singular forms "a," "an," and "the" are intended to include the plural forms as well, unless the context clearly indicates otherwise. It should also be understood that the term "and/or" as used herein refers to and encompasses any or all possible combinations of one or more of the associated listed items.
It should be understood that although the terms first, second, third, etc. may be used herein to describe various information, these information should not be limited by these terms. These terms are only used to distinguish one type of information from another. For example, a first message may also be referred to as a second message, and similarly, a second message may also be referred to as a first message, without departing from the scope of the present application. The word "if" as used herein may be interpreted as "at … …" or "at … …" or "responsive to a determination", depending on the context.
Example 1:
fig. 1 is a schematic diagram of a top silicon structure of a dual-input dual-output mode converter according to embodiment 1, and as shown in fig. 1, the dual-input dual-output mode converter may include a silicon dioxide substrate, a top silicon and an upper cladding layer formed of a material having a lower refractive index than silicon (hereinafter referred to as a low refractive index material), the top silicon including 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, wherein the first input waveguide 1 is connected to the first output waveguide 4 through the mode conversion region 3, 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 etched or not etched, the etching depth is the same as the thickness of the top silicon, and the internal material is a low refractive index material when the rectangular units are in an etched state; when the rectangular unit is in a non-etching state, the internal material is silicon; wherein the states of the rectangular cells in the mode transition area 3 are set by a quality factor.
The above embodiments can be known that the application of the reverse design algorithm has an important meaning for solving the problem that the working types of the existing mode converter are basically single-input and single-output. The method increases the path of mode conversion on the limited size, and has important significance for further improving the flexibility of the mode division multiplexing system and improving the system integration level. The device design is based on the design concept of reverse design, and the double-in and double-out mode conversion function is realized on a very compact size. Compared with the prior art, the mode converter provided by the invention increases the mode conversion path on the limited size, and has important significance for further improving the flexibility of the mode division multiplexing system and improving the system integration level.
Specifically, the wavelength ranges of the input light of the first input waveguide 1 and the second input waveguide 2 are 1520nm to 1580nm, which covers the C-band commonly used in the field of optical communication, ensures the bandwidth characteristic of the device, and can also adjust the wavelength range of the input light according to practical situations.
In particular, the first input waveguide 1 and the second input waveguide 2 are designed to support only TE 0 The mode passes through, and the waveguide width is 500nm; the first output waveguide 4 and the second output waveguide 5 are designed to allow TE simultaneously 0 And TE (TE) 1 The mode passes through, the waveguide width is 800nm, wherein the waveguide width can be adjusted according to the actual situation.
Specifically, the thickness of the silicon dioxide substrate is 3 μm, the thickness of the top layer silicon is 220nm, the mode conversion area 3 is a square with a side length of 2 μm, and the rectangular unit is a square with a side length of 100nm, and it should be noted that the above dimensions can be adjusted according to practical situations and requirements, and the application is not limited thereto. In this embodiment, the mode conversion area 3 is divided into 20×20 rectangular units, where the rectangular units are etched or not etched, and the etching depth is the same as the thickness of the top silicon, that is, when the rectangular units are etched, the material is air; when the rectangular unit is not etched, wherein the material is silicon, the device can be manufactured by single-step etching.
Specifically, according to the device function to be realized (i.e., TE input from the first input waveguide 1 0 Mode conversion to TE output by the first output waveguide 4 1 Mode, TE input from the second input waveguide 2 0 Mode conversion to TE output from the second output waveguide 5 1 Mode) to define an appropriate figure of merit (FOM) for evaluating the performance of the device. Setting the state of the rectangular unit in the mode conversion area 3 by the quality factor includes:
(1) Setting all rectangular units in a non-etching state as an initial state, calculating an initial quality factor of the double-input double-output mode converter in the initial state, and recording the initial quality factor as FOM max ；
(2) Due toThe function of the device is to be symmetrical left and right with the center line L of the mode conversion area 3 along the y-axis direction as a symmetry axis, so that the operation of the rectangular unit is also adopted in a left and right symmetrical manner when simulation optimization is performed. Starting from the first rectangular unit and rectangular units which are symmetrical to the first rectangular unit left and right by taking the central line L of the mode conversion area 3 along the y-axis direction as a symmetry axis, the following operations are sequentially carried out on all the rectangular units: changing the state of the rectangular unit (i.e. changing to an etching state if the rectangular unit is originally in a non-etching state; conversely, changing to a non-etching state if the rectangular unit is originally in an etching state) and calculating a new quality factor (denoted temp), and combining the new quality factor temp with the initial quality factor FOM max Comparing if the new quality factor temp is greater than the initial quality factor FOM max Maintaining the state of the rectangular unit as changed state, and assigning the new quality factor temp to the initial quality factor FOM max The method comprises the steps of carrying out a first treatment on the surface of the Otherwise, setting the state of the rectangular unit as the state before changing;
(3) Step (2) is repeatedly performed until FOMmax is no longer increased.
Specifically, when the calculation of 400 rectangular units is finished, called one iteration is finished, and then the step (2) is repeated, and after a plurality of iterations, the device graph output when FOMmax is no longer increased is shown in fig. 1.
Specifically, the state transformation of the rectangular unit, the calculation and the comparison of the FOM are realized by means of a programming language Python and a three-dimensional finite time domain difference (3D-FDTD) function in optical simulation software Lumerical.
In this embodiment, the quality factor
Wherein M is the number of input optical wavelength channels, t 1 For inputting TE from the first input waveguide 1 at the input light wavelength 0 In mode, TE is monitored at the first output waveguide 4 1 Transmittance of the mode; t is t 2 For inputting TE from the second input waveguide 2 at the input light wavelength 0 In the mode, at the second output waveGuide 5 TE monitored 1 Transmittance of the mode. FOM increases as mode conversion efficiency increases; and vice versa.
At 1550nm wavelength, when TE 0 The simulation result of the field distribution in the dual-input dual-output mode converter when the mode is input from the first input waveguide 1 is shown in fig. 2 (a). It can be seen that TE is input from the first input waveguide 1 0 The mode propagates in the positive y-axis direction, passes through the mode conversion region 3, and is converted into TE 1 Mode is output from the first output waveguide 4 along the positive direction of the x-axis, TE is realized 0 To TE 1 The switching of modes and the switching of the propagation direction of modes,
at 1550nm wavelength, when TE 0 The field distribution simulation result in the dual-input dual-output mode converter when the mode is input from the second input waveguide 2 is shown in fig. 2 (b). The TE input from the second input waveguide 2 can be seen 0 The mode propagates in the positive y-axis direction, passes through the mode conversion region 3, and is converted into TE 1 Mode is output from the second output waveguide 5 along the negative x-axis direction, TE is realized 0 To TE 1 Mode conversion and mode propagation direction conversion. Simulation results show that the designed mode converter achieves the expected double-input double-output mode conversion function, and the feasibility of design is proved.
Example 2:
the technical scheme provided in this embodiment is basically the same as that provided in embodiment 1, except that: the mode conversion area 3 is square with a side length of 2.6 mu m, the mode conversion area 3 is divided into 26×26 rectangular units, and the definition of the quality factor FOM is as follows:
where M is the number of wavelength channels. The calculation of FOM consists of two parts, the part preceding "+" consisting of these parameters: at a certain wavelength, TE is input from the first input waveguide 1 0 In mode, TE is monitored at the first output waveguide 4 0 The transmittance of the mode is x 11 ，TE 1 The transmittance of the mode is t 11 Respectively represent TE at the first output waveguide 4 0 Crosstalk and TE of modes 1 A conversion rate of the mode; TE monitored at the second output waveguide 5 0 The transmittance of the mode is x 12 ，TE 1 The transmittance of the mode is t 12 Respectively represent TE at the second output waveguide 5 0 Mode and TE 1 Crosstalk of modes; TE monitored at the second input waveguide 2 0 Mode transmittance r 12 Representing TE at the second input waveguide 2 0 Crosstalk of modes, since the second input waveguide 2 is designed to support only TE 0 Mode, therefore, does not take into account TE at the second input waveguide 2 1 Crosstalk of modes.
The part following "+" is made up of these parameters: at a certain wavelength, TE is input from the second input waveguide 2 0 In mode, TE is monitored at the second output waveguide 5 0 The transmittance of the mode is x 22 ，TE 1 The transmittance of the mode is t 22 Respectively represent TE at the second output waveguide 5 0 Crosstalk and TE of modes 1 A conversion rate of the mode; TE monitored at the first output waveguide 4 0 The transmittance of the mode is x 21 ，TE 1 The transmittance of the mode is t 21 Respectively represent TE at the first output waveguide 4 0 Mode and TE 1 Crosstalk of modes; TE monitored at the first input waveguide 1 0 Mode transmittance r 21 Representing TE at the first input waveguide 1 0 Crosstalk of modes, since the first input waveguide 1 is designed to support only TE 0 Mode, therefore, does not take into account TE at the first input waveguide 1 1 Crosstalk of modes.
As can be seen from the above parameter definitions, crosstalk in each waveguide of the mode converter also affects the value of the FOM parameter: the greater the mode conversion rate, the smaller the crosstalk of the modes in each waveguide, the greater the FOM; conversely, the smaller the mode conversion ratio, the greater the crosstalk of modes in each waveguide, and the smaller the FOM. The definition of such FOM takes into account both the insertion loss of the mode converter and the crosstalk performance of the device.
The mode converter (shown in fig. 3) finally calculated in this embodiment can also achieve the same functions as the dual-input dual-output mode converter described in the first embodiment, and the simulation results are shown in fig. 4 (a) and fig. 4 (b):
at 1550nm wavelength, when TE 0 The simulation result of the field distribution in the dual-input dual-output mode converter when the mode is input from the first input waveguide 1 is shown in fig. 4 (a). It can be seen that TE is input from the first input waveguide 1 0 The mode propagates in the positive y-axis direction, passes through the mode conversion region 3, and is converted into TE 1 Mode is output from the first output waveguide 4 along the positive direction of the x-axis, TE is realized 0 To TE 1 The switching of modes and the switching of the propagation direction of modes,
at 1550nm wavelength, when TE 0 The field distribution simulation result in the dual-in dual-out mode converter when the mode is input from the second input waveguide 2 is shown in fig. 4 (b). The TE input from the second input waveguide 2 can be seen 0 The mode propagates in the positive y-axis direction, passes through the mode conversion region 3, and is converted into TE 1 Mode is output from the second output waveguide 5 along the negative x-axis direction, TE is realized 0 To TE 1 Mode conversion and mode propagation direction conversion.
Other embodiments of the present application will be apparent to those skilled in the art from consideration of the specification and practice of the disclosure herein. This application is intended to cover any variations, uses, or adaptations of the application following, in general, the principles of the application and including such departures from the present disclosure as come within known or customary practice within the art to which the application pertains.
It is to be understood that the present application is not limited to the precise arrangements and instrumentalities shown in the drawings, which have been described above, and that various modifications and changes may be effected without departing from the scope thereof.
Claims (9)
1. The double-in and double-out mode converter is characterized by comprising a silicon dioxide substrate, top silicon and a low-refractive-index material cladding layer, wherein the top silicon comprises a first input waveguide, a second input waveguide, a mode conversion area, a first output waveguide and a second output waveguide, the first input waveguide is connected with the first output waveguide through the mode conversion area, the second input waveguide is connected with the second output waveguide through the mode conversion area, the mode conversion area is divided into M multiplied by N rectangular units, the rectangular units are etched or not etched, the etching depth is the same as the thickness of the top silicon, and when the rectangular units are in an etching state, the internal materials are low-refractive-index materials; when the rectangular unit is in a non-etching state, the internal material is silicon; wherein the states of the rectangular cells in the mode transition area are set by a quality factor.
2. The dual-in and dual-out mode converter and the design method according to claim 1, wherein the first input waveguide and the second input waveguide input a wavelength range of 1520nm to 1580nm.
3. The dual-in and dual-out mode converter and design method as set forth in claim 1, wherein said first and second input waveguides support only TE 0 Mode pass, the first output waveguide and the second output waveguide simultaneously allow TE 0 And TE (TE) 1 The mode passes.
4. The dual-in and dual-out mode converter and the design method as claimed in claim 1, wherein the thickness of the silicon dioxide substrate is 3 μm and the thickness of the top silicon layer is 220nm.
5. The dual-in and dual-out mode converter and the design method according to claim 1, wherein the rectangular unit is square with a side length of 100 nm.
6. The dual-in and dual-out mode converter and the design method according to claim 1, wherein the rectangular unit realizes an etching state by single-step etching.
7. The dual-in and dual-out mode converter and the design method according to claim 1, wherein setting the state of the rectangular cells in the mode conversion region by the quality factor comprises:
(1) Setting all rectangular units in a non-etching state as an initial state, and calculating an initial quality factor of the double-input double-output mode converter in the initial state;
(2) Starting from the first rectangular unit, the following operations are sequentially performed on all the rectangular units: changing the state of a rectangular unit, calculating a new quality factor, comparing the new quality factor with the initial quality factor, if the new quality factor is larger 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 as the state before changing;
(3) And (5) repeatedly executing the step (2) until the quality factor is not increased any more.
8. The dual-in and dual-out mode converter and design method according to claim 5, wherein the quality factor
Wherein M is the number of wavelength channels of the input light, t 1 Inputting TE from a first input waveguide for an input light wavelength 0 In mode, TE is monitored at the first output waveguide 1 Transmittance of the mode; t is t 2 Inputting TE from the second input waveguide for the input light wavelength 0 In mode, TE is monitored at the second output waveguide 1 Transmittance of the mode.
9. The dual-in and dual-out mode converter and design method according to claim 5, wherein the quality factor
Wherein M is a wavelength channelThe calculation of FOM consists of two parts, the part preceding "+" consisting of these parameters: at a certain wavelength, TE is input from a first input waveguide 0 In mode, TE is monitored at the first output waveguide 0 The transmittance of the mode is x 11 ，TE 1 The transmittance of the mode is t 11 Respectively represent TE at the first output waveguide 0 Crosstalk and TE of modes 1 A conversion rate of the mode; TE monitored at the second output waveguide 0 The transmittance of the mode is x 12 ，TE 1 The transmittance of the mode is t 12 Respectively represent TE at the second output waveguide 0 Mode and TE 1 Crosstalk of modes; TE monitored at the second input waveguide 0 Mode transmittance r 12 Representing TE at the second input waveguide 0 Crosstalk of modes, since the second input waveguide is designed to support only TE 0 Mode, therefore, does not consider TE at the second input waveguide 1 Crosstalk of modes. The part following "+" is made up of these parameters: at a certain wavelength, TE is input from the second input waveguide 0 In mode, TE is monitored at the second output waveguide 0 The transmittance of the mode is x 22 ，TE 1 The transmittance of the mode is t 22 Respectively represent TE at the second output waveguide 0 Crosstalk and TE of modes 1 A conversion rate of the mode; TE monitored at a first output waveguide 0 The transmittance of the mode is x 21 ，TE 1 The transmittance of the mode is t 21 Respectively represent TE at the first output waveguide 0 Mode and TE 1 Crosstalk of modes; TE monitored at first input waveguide 0 Mode transmittance r 21 Representing TE at the first input waveguide 0 Crosstalk of modes, since the first input waveguide is designed to support only TE 0 Mode, therefore, does not consider TE at the first input waveguide 1 Crosstalk of modes.
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