A differential equation solver based on Sagnac ring and its design method

Abstract

The present invention provides a differential equation solver based on a Sagnac ring and a design method thereof, comprising the following steps: Step A: Based on the transfer matrix method, mathematical operations are performed on the constructed differential equation solver to obtain a transfer expression for the differential equation solver; Step B: Amplitude and phase diagrams of the differential equation solver are obtained through simulation and compared with the spectrum and phase diagrams of an ideal first-order differential equation; Step C: Implementing the principle of adjustable first-order constant coefficient differential equations; Step D: Inputting a Gaussian pulse to the input end and observing the output time domain waveform at the output end. The present invention can achieve second-order differential equation solving with adjustable constant coefficients based on the first-order differential equation through further cascading, further expanding the practicality of the present structure and having broad application prospects in the field of silicon photonic signal processing.

Description

Technical Field

[0001] The present invention belongs to the field of communication technology, and in particular relates to a structure for solving optical constant coefficient differential equations based on Sagnac rings. Background Art

[0002] With the rapid development of computers and information communications, signal rates are constantly increasing. Electronics-based communication networks have almost reached their speed limits, and signal processing in the electrical domain can no longer meet the required speed requirements. Compared to electrical signal processing, optical signal processing can better meet the high bandwidth and high speed requirements of current information transmission and storage. In recent years, many optical signal processing device structures have been proposed, such as optical differentiators, optical integrators, optical Hilbert transformers, and optical differential equation solvers.

[0003] Differential equations, a key branch of mathematics, were originally applied to solving problems in fields like physics, astronomy, and geometry. Today, numerous practical problems, such as population development and urban traffic flow, require differential equations to establish models for analysis and prediction. They are widely used in nearly all natural sciences and engineering fields.

[0004] Devices used to solve optical differential equations are often fiber Bragg gratings, FP cavities, and microring resonators. However, most existing technologies can only solve first-order differential equations with a fixed k coefficient. However, different k coefficients represent different differential equation systems. Whether the k coefficient is adjustable and the adjustment range of the k coefficient are particularly important. A more critical performance parameter is whether second-order or even higher-order differential equations can be achieved through cascading. This solution can solve first-order differential equations with constant coefficients by cascading two Sagnac rings. The value of k can be adjusted by adjusting the thermal power on one arm. On this basis, cascading two more Sagnac rings can easily solve second-order differential equations. Due to its advantages such as small footprint, easy integration, and ability to solve first-order and second-order differential equations with an adjustable constant coefficient k, this solution has broad application prospects in the field of optics. Summary of the Invention

[0005] In order to solve the above technical problems, the present invention proposes a structure for solving optical constant coefficient differential equations based on Sagnac rings and a design method thereof.

[0006] A differential equation solver based on a Sagnac ring adopts silicon waveguide material. The length of the ring waveguide L2 in each SLR is 10-50 μm, the lengths of the MZI arm L1 with a heater and the MZI arm L3 without a heater are 10-50 μm, the length of the MZI arm L4 without a heater is 10-50 μm, and the LC coupling length is 10-50 μm.

[0007] Among them, the Mach-Zehnder coupler is a special coupler in optics. Its structure mainly consists of three parts: two directional couplers on the left and right, and a waveguide in the middle consisting of two arms (divided into upper and lower arms). Generally, a heater is placed on one of the arms. The specific function depends on the situation. The detailed structure is as follows Figure 1 The MZI part.

[0008] The present invention adopts the above technical solution, which has the advantage that, unlike the differential equation solving structure of traditional electronic devices, this differential equation solver can overcome the rate limit brought about by electronic devices solving differential equations, and can realize first-order differential equations with adjustable k coefficient.

[0009] This method uses a cascaded structure of two simple silicon-based Saganc ring reflectors, replacing the directional coupler (DC) in the Sagnac ring with a tunable coupler. By varying the input thermal power of the phase shifter in the tunable coupler, and thus its refractive index, the reflectivity of each Sagnac ring can be changed, thereby enabling the solution of a first-order differential equation with an adjustable k-factor. Compared to other structures, this structure offers advantages such as adjustable k-factor, a simple structure, and the need to control only a single parameter: input thermal power. This structure has broad application prospects in the field of integrated silicon photonics.

[0010] The present invention also provides a design method for a differential equation solver based on a Sagnac ring, comprising the following steps:

[0011] Step A: Based on the transfer matrix method, mathematical operations are performed on the constructed differential equation solver to obtain a transfer expression of the differential equation solver;

[0012] Step B: obtaining an amplitude diagram and a phase diagram of the differential equation solver through simulation, and comparing them with the spectrum diagram and phase diagram of an ideal first-order differential equation;

[0013] Step C: Implement the principle of adjustable first-order constant coefficient differential equation;

[0014] Step D: Input a Gaussian pulse into the input terminal and observe the output time domain waveform at the output terminal.

[0015] In the field of optical signal processing, in order to verify whether a certain operation is realized or not, the most common Gaussian pulse is usually input as the input pulse. By using the Gaussian pulse as the input light source, the light source passes through this structure, and its output waveform can be observed at the output end using an optical oscilloscope, and then compared with the output waveform under the ideal differential equation.

[0016] The present invention adopts the above technical solution, which has the advantages of being easy to cascade and able to achieve the function of solving high-order optical differential equations that most solutions cannot achieve. It also has a simple and clear structure and can be widely used in the field of silicon photonic device integration.

[0017] Preferably, in step A, according to the transfer matrix method, the transfer function of the differential equation solver is:

[0018]

[0019]

[0020] r s =2ja2(a1+a3)(a1kt 3 -a3k 3 t) (1)

[0021] Where ts and rs are the transmission function and reflection function of the SLR with MZI coupler, t and k are the transmission coefficient and coupling coefficient of the directional coupler, respectively; a i =exp(-αl i -jβl i )(i=1, 2, 3, 4) is the transmission factor of the waveguide, l i (i = 1, 2, 3, 4) represent the length of the MZI arm with heater, the length of the ring waveguide, the length of the MZI arm without heater, and the length of the waveguide connecting the two MZIs, respectively. α represents the loss factor, and β is the propagation constant of the silicon waveguide. Ignoring waveguide loss, i.e., when α = 0, we have:

[0022]

[0023] Substituting (2) into (1) we get:

[0024]

[0025] Since βl = wτ, where w represents the angular frequency of the input light, τ = ngl4 / c represents the time delay of a single pass through l4, ng is the group refractive index of the waveguide, and c is the speed of light in vacuum, substituting this equation into (3) yields:

[0026]

[0027] When w approaches the resonant frequency w0 of the structure, we have:

[0028]

[0029] Formula (5) is the transfer function when the angular frequency of the input light approaches the resonant frequency w0;

[0030] The frequency domain expression of an ideal first-order ordinary differential equation is:

[0031]

[0032] It can be seen that the transfer function expression of this differential equation solver has the same form as the transfer function expression of the ideal first-order constant coefficient differential equation, and the constant coefficient k of this differential equation solver is:

[0033]

[0034] Therefore, in theory, this differential equation solver can fully realize the first-order differential equation with adjustable constant coefficients.

[0035] Preferably, step B includes: after building the differential equation solver on Interconnnect software, measuring the amplitude diagram and phase diagram of the differential equation solver and the ideal first-order differential equation system, and obtaining the amplitude-frequency response and phase response curves that are consistent with the amplitude-frequency response and phase response curves of the ideal first-order ordinary differential equation.

[0036] Through simulation comparison, we can check whether the curve of this structure and the ideal graph are consistent. If they are almost consistent, it further proves theoretically that this structure can solve the differential equation function; if the difference is large, then theoretically it is almost impossible for this structure to be used to realize the differential equation function.

[0037] Preferably, the step C comprises:

[0038] From equation (7), we can see that the value of k is related to rs and τ. Since τ = ngl4 / c is fixed, but rs is equal to:

[0039] r s =2ja2(a1+a3)(a1kt 3 -a3k 3 t) (8)

[0040] When the directional coupler is a 3dB coupler, that is, k2 = t2 = 0.5, equation (8) is simplified to:

[0041]

[0042] And for a1, the following relationship holds:

[0043]

[0044] Therefore, the corresponding ng group refractive index on l1 can be changed by adjusting the thermal power on the l1 arm of the MZI on each SLR. Obviously, a1 will also change accordingly, which will naturally lead to a change in rs, thereby changing the value of the first-order constant coefficient k, realizing the function of solving first-order ordinary differential equations with adjustable k coefficient.

[0045] The present invention brings the following effects:

[0046] 1. Compared with traditional large-scale active devices, this differential equation solver based on Sagnac ring uses simple passive silicon waveguide as material, and the system structure is simple and clear, and easy to integrate.

[0047] 2. Most traditional solutions can only solve first-order ordinary differential equations with a fixed coefficient k. However, this differential equation solver structure based on the Sagnac ring can solve first-order differential equations with an adjustable constant coefficient k. And only the heating power parameter needs to be changed.

[0048] 3. By cascading the differential equation solver structure based on the Sagnac ring, the high-order optical differential equation solving function that most solutions cannot achieve can be achieved.

[0049] 4. This invention develops a novel structure that, compared to traditional differential equation solvers, offers advantages such as a smaller footprint and the elimination of the need for additional active media, which would complicate the system. Simply adjusting the thermal power parameter on the MZI allows for solving first-order differential equations with adjustable constant coefficients. Furthermore, through further cascading, this invention can solve second-order differential equations with adjustable constant coefficients, building on the first-order differential equations. This further expands the practicality of this differential equation solver structure and offers broad application prospects in silicon photonics signal processing. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] The technical solution of the present application is further described below with reference to the accompanying drawings and embodiments.

[0051] Figure 1 It is a structural diagram of a differential equation solver according to an embodiment of the present application.

[0052] Figure 2 1 is an amplitude-frequency response and phase response diagram of an ideal ODE and differential equation solver according to an embodiment of the present application.

[0053] Figure 3 1 is an output waveform diagram of an ideal ODE and a differential equation solver when k=0.035 / ps in an embodiment of the present application.

[0054] Figure 41 is an output waveform diagram of an ideal ODE and a differential equation solver when k=0.012 / ps in an embodiment of the present application. DETAILED DESCRIPTION

[0055] The technical solution of the present application will be described in detail below with reference to the accompanying drawings and in combination with embodiments.

[0056] Step 1: Build the structure using silicon waveguides on a silicon-on-insulator (SOI) platform.

[0057] Step 2: Based on the transfer matrix method, perform mathematical calculations on the constructed structure to obtain the transfer expression of this structure;

[0058] Step 3: Obtain the amplitude diagram and phase diagram of the structure through software simulation, and compare them with the spectrum diagram and phase diagram of the ideal first-order differential equation;

[0059] Step 4: Implement the principle of adjustable first-order constant coefficient differential equation;

[0060] Step 5: Input a Gaussian pulse into the input terminal and observe the output time domain waveform at the output terminal.

[0061] Step 1: Build the structure with silicon waveguide on silicon-on-insulator (SOI) platform. Figure 1 As shown in the figure, the design is a cascaded Sagnac ring structure based on a silicon-on-insulator platform. Silicon waveguide materials are used. 2 SLRs are formed by designing Mach-Zehnder couplers instead of traditional directional couplers. The overall structure consists of two SLRs, where the length of l2 in each SLR is 20μm, the lengths of l1 and l3 are 30μm, and the length of l4 is 10μm. c The coupling length is 10 μm.

[0062] Among them, step 2: Based on the transfer matrix method, the expression of the structure is calculated

[0063] like Figure 1 The proposed structure shown in the figure consists of two SLRs forming a Fabry-Perot (FP) cavity. According to the transfer matrix method, the transfer function of the structure is

[0064]

[0065]

[0066] r s =2ja2(a1+a3)(a1kt 3 -a3k 3 t) (1)

[0067] Where ts and rs are the transmission function and reflection function of the SLR with MZI coupler, respectively, and t and k are the transmission coefficient and coupling coefficient of the directional coupler, respectively. i =exp(-αl i -jβl i )(i=1, 2, 3, 4) is the transmission factor of the waveguide, l i (i = 1, 2, 3, 4) represent the length of the MZI arm with heater, the length of the ring waveguide, the length of the MZI arm without heater, and the length of the waveguide connecting the two MZIs, respectively. α represents the loss factor, and β is the propagation constant of the silicon waveguide. Assuming that waveguide loss is ignored, that is, when α = 0, we have:

[0068]

[0069] Substituting (2) into (1) we get:

[0070]

[0071] Since βl = wτ, where w represents the angular frequency of the input light, τ = ngl4 / c represents the time delay of a single pass through l4, ng is the group refractive index of the waveguide, and c is the speed of light in vacuum, substituting this equation into (3) yields:

[0072]

[0073] When w approaches the resonant frequency w0 of the structure, we have:

[0074]

[0075] Equation (5) is the transfer function when the angular frequency of the input light approaches the resonant frequency w0.

[0076] The frequency domain expression of an ideal first-order ordinary differential equation is:

[0077]

[0078] Therefore, it can be seen that the transfer function expression of this structure has the same form as the transfer function expression of the ideal first-order constant coefficient differential equation, and the constant coefficient k of this structure is:

[0079]

[0080] Therefore, in theory, this structure can fully realize the first-order differential equation with adjustable constant coefficients.

[0081] Step 3: Analyze the amplitude and phase diagrams of this structure

[0082] After building this structure on Interconnnect software, the amplitude and phase diagrams of this structure and the ideal first-order differential equation system were measured, as shown in Figure 2 As shown, the blue and red dashed lines represent the amplitude and phase of the ideal first-order ordinary differential equation, while the black and yellow dashed lines represent the amplitude and phase at the output of this system. We can see that the amplitude-frequency response and phase response curves of this structure are very consistent with those of the ideal first-order ordinary differential equation. This further verifies that this system can be used to solve first-order ordinary differential equations.

[0083] Step 4: Principle of implementing a first-order differential equation with adjustable constant coefficient k

[0084] From equation (7), we can see that the value of k is related to rs and τ. Since τ = ngl4 / c is fixed, but rs is equal to:

[0085] r s =2ja2(a1+a3)(a1kt 3 -a3k 3 t) (8)

[0086] when Figure 1 When the directional coupler in is a 3dB coupler, that is, k2=t2=0.5, then equation (8) is simplified to:

[0087]

[0088] And for a1, the following relationship holds:

[0089]

[0090] Therefore, the corresponding ng group refractive index on l1 can be changed by adjusting the thermal power on the l1 arm of the MZI on each SLR. Obviously, a1 will also change accordingly, which will naturally lead to a change in rs, thereby changing the value of the first-order constant coefficient k, realizing the function of solving first-order ordinary differential equations with adjustable k coefficient.

[0091] Step 5: Input a Gaussian pulse into the input terminal and observe the output time domain waveform at the output terminal.

[0092] When the input power of both arms of the SLR MZI is 7 mW, an ideal Gaussian pulse with a full width at half maximum (FWHM) of 80 ps is input as the light source. Figure 3 The output waveform measured at the output terminal at k = 0.035 / ps is compared with the ideal ODE. The solid blue line represents the solution to the ideal ODE, while the dashed red line represents the solution of this system. It can be seen that the error is very small. Therefore, it is feasible to use this system to solve first-order linear differential equations with constant coefficients.

[0093] When the input power of both arms of the SLR MZI is 30 mW, an ideal Gaussian pulse with a full width at half maximum (FWHM) of 80 ps is input to the in port as the light source. Figure 4 The output waveform measured at the output terminal at k = 0.012 / ps is compared to the ideal ODE. The solid blue line represents the solution to the ideal ODE, while the dashed red line represents the solution to this system. Therefore, by adjusting the thermal power of each MZI arm, a first-order optical differential equation with adjustable constant coefficients can be implemented.

[0094] Moreover, in the field of optical signal processing, if you want to verify whether a structure can realize a certain function, you usually build the structure first, then derive an expression based on the built structure, and then use the expression to study and prove the principle that can realize the function. Finally, the final result diagram of the structure is compared with the ideal diagram to judge. If the error is large, the structure is not feasible, and if the error is small, it is feasible.

[0095] Based on the above-mentioned ideal embodiments of this application, and in accordance with the above description, relevant personnel can make various changes and modifications without departing from the scope of the technical concept of this application. The technical scope of this application is not limited to the contents of the specification, but must be determined according to the scope of the claims.

Claims

1. A design method for a differential equation solver based on a Sagnac ring, characterized in that: The following steps are involved: Step A: Based on the transfer matrix method, mathematical operations are performed on the constructed differential equation solver to obtain a transfer expression of the differential equation solver; Step B: obtaining an amplitude diagram and a phase diagram of the differential equation solver through simulation, and comparing them with the spectrum diagram and phase diagram of an ideal first-order differential equation; Step C: Implement the principle of adjustable first-order constant coefficient differential equation; Step D: Input a Gaussian pulse to the input terminal and observe the output time domain waveform at the output terminal; In step A, according to the transfer matrix method, the transfer function of the differential equation solver is <h2 style=";text-align:left;direction:ltr">r<h2 style=";text-align:left;direction:ltr"> s <h2 style=";text-align:left;direction:ltr"> = 2ja2(a1+a3)(a1kt)<h2 style=";text-align:left;direction:ltr"> 3 <h2 style=";text-align:left;direction:ltr"> -a3k<h2 style=";text-align:left;direction:ltr"> 3 <h2 style=";text-align:left;direction:ltr"> t) (1) Where ts and rs are the transmission function and reflection function of the SLR with MZI coupler, t and k are the transmission coefficient and coupling coefficient of the directional coupler, respectively; a i =exp(-αl i -jβl i )i=1, 2, 3, 4 are the transmission factors of the waveguide, l i i = 1, 2, 3, 4, respectively, representing the length of the MZI arm with heater, the length of the ring waveguide, the length of the MZI arm without heater, and the length of the waveguide connecting the two MZIs; α is the loss factor, and β is the propagation constant of the silicon waveguide. Under the premise of ignoring the waveguide loss, that is, when α = 0, we have: Substituting (2) into (1) we get: Since βl = wτ, where w represents the angular frequency of the input light, τ = ngl4 / c represents the time delay of a single pass through l4, ng is the group refractive index of the waveguide, and c is the speed of light in vacuum; substituting the equation βl = wτ into (3) yields: When w approaches the resonant frequency w0 of the structure of the differential equation solver based on the Sagnac ring, we have: Formula (5) is the transfer function when the angular frequency of the input light approaches the resonant frequency w0; The frequency domain expression of an ideal first-order ordinary differential equation is: It can be seen that the transfer function expression of this differential equation solver has the same form as the transfer function expression of the ideal first-order constant coefficient differential equation, and the constant coefficient k of this differential equation solver is: Therefore, in theory, this differential equation solver can fully realize the differential equation with adjustable first-order constant coefficients; The differential equation solver of the Sagnac ring of the design method adopts silicon waveguide material and a Mach-Zehnder coupler to form an SLR; the length of the ring waveguide L2 in each SLR is 10-50 μm, the length of the MZI arm L1 with a heater and the MZI arm L3 without a heater are 10-50 μm, the length of the MZI arm L4 without a heater is 10-50 μm, and the LC coupling length is 10-50 μm.

2. The method for designing a differential equation solver based on a Sagnac ring according to claim 1, wherein: The step B includes: after building the differential equation solver on the Interconnnect software, measuring the amplitude diagram and phase diagram of the differential equation solver and the ideal first-order differential equation system, and obtaining an amplitude-frequency response and phase response curve that are consistent with the amplitude-frequency response and phase response curve of the ideal first-order ordinary differential equation.

3. The method for designing a differential equation solver based on a Sagnac ring according to claim 1, wherein: The step C comprises: From equation (7), we can see that the value of k is related to rs and τ. Since τ = ngl4 / c is fixed, but rs is equal to: <h2 style=";text-align:left;direction:ltr">r<h2 style=";text-align:left;direction:ltr"> s <h2 style=";text-align:left;direction:ltr"> = 2ja2(a1+a3)(a1kt)<h2 style=";text-align:left;direction:ltr"> 3 <h2 style=";text-align:left;direction:ltr"> -a3k<h2 style=";text-align:left;direction:ltr"> 3 <h2 style=";text-align:left;direction:ltr"> t) (8) When the directional coupler is a 3dB coupler, that is, k2 = t2 = 0.5, equation (8) is simplified to: And for a1, the following relationship holds:

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