Coaxial waveguide high power microwave mode component analysis method and system based on four-frame difference sampling

Abstract

The application discloses a coaxial waveguide high-power microwave mode component analysis method and system based on four-frame difference sampling, and belongs to the technical field of high-power microwave device diagnosis and signal processing. The method obtains transverse electric field distribution data at four specific time points (t1, t2=t1+T / 4, t3=t1+T / 2, t4=t1+3T / 4) in one microwave base frequency period at the same axial position of the output end face of the coaxial waveguide; four integral values Q j1 , Q j2 , Q j3 , Q j4 are obtained by carrying out cross-sectional integration on electric fields at different time points and normalized magnetic fields of target modes; and difference quantities Q j1 -Q j3 , Q j2 -Q j4 are constructed to eliminate second harmonic interference, and then mode amplitude and phase are accurately solved. The application effectively solves problems of large mode power calculation error and inaccurate mode purity evaluation caused by harmonic influence in a traditional two-frame orthogonal sampling method, and has high analysis precision, strong anti-harmonic interference capability and simple algorithm.

Description

Technical Field

[0001] This invention relates to the field of high-power microwave technology, and more specifically to a method and system for coaxial waveguide high-power microwave mode composition analysis based on four-frame differential sampling. Background Technology

[0002] High-power microwave (HPM) devices (such as relativistic backward-wave oscillators and relativistic klystron amplifiers) often exhibit significant harmonic components (especially second harmonics) in their output signals during beam-wave interactions due to strong field nonlinearity and relativistic modulation of the electron beam. During device design and optimization, engineers typically rely on particle simulation (PIC) software to analyze the mode composition of the output microwaves to evaluate the device's efficiency and mode purity.

[0003] Currently, the most mainstream method for mode component analysis based on PIC simulation post-processing is the "two-frame orthogonal sampling method." Its basic principle is: within a fundamental frequency period T, two moments with a phase difference of π / 2 (i.e., an interval of T / 4) are selected to extract the transverse electric field. Utilizing the orthogonality of the coaxial waveguide eigenmodes, the electric field is projected onto each target mode to extract amplitude and phase. However, this method implicitly assumes a stringent prerequisite—the signal to be analyzed must be an ideal single-frequency signal. When there are non-negligible harmonic components in the output waveguide, the two-frame orthogonal sampling method faces serious fundamental flaws: First, mathematically, two-frame sampling can only construct one set of orthogonal bases (first-order sine and first-order cosine), but it cannot distinguish the coupling contributions of the fundamental frequency and even-order harmonics at the sampling points. Specifically, the projection values ​​of the second harmonic at the two sampling points with an interval of T / 4 do not cancel each other out due to the orthogonality; instead, they are superimposed into the integral result of the fundamental frequency component, causing the mode projection value to deviate from the true value. Ultimately, this error is directly propagated to the power calculation stage, causing the mode power to be systematically overestimated or underestimated, with errors reaching 5% to 10%, and in severe cases even exceeding 7% (e.g., in strong harmonic scenarios of X-band coaxial relativistic backward wave oscillators). This leads to deviations in the judgment of the device's optimal operating point and mode competition. While existing technologies have attempted to address these shortcomings by employing time-domain filtering or spectral windowing for preprocessing, these methods either destroy the phase information of the original signal or significantly increase post-processing time due to iterative calculations, making it difficult to balance accuracy and efficiency.

[0004] Therefore, there is an urgent need for a coaxial waveguide high-power microwave mode composition analysis method based on four-frame differential sampling, which can eliminate even-order harmonic interference at its source without destroying the original signal phase. Summary of the Invention

[0005] To address the aforementioned problems, this invention provides a method and system for coaxial waveguide high-power microwave mode component analysis based on four-frame differential sampling. By selecting four specific symmetrical moments within a fundamental frequency period for sampling and constructing a differential linear combination, even-order harmonic interference is directly eliminated during mode decomposition, achieving high-precision mode parameter extraction.

[0006] To achieve the above objectives, this invention provides a method for coaxial waveguide high-power microwave mode composition analysis based on four-frame differential sampling, comprising the following complete steps: S1: At the same axial position z0 on the output end face of the coaxial waveguide, within one microwave fundamental frequency period T, acquire the transverse electric field distribution data E at four times t1, t2 = t1 + T / 4, t3 = t1 + T / 2, and t4 = t1 + 3T / 4. T (z0, r, φ, t k ), where k=1,2,3,4; the electric field data are derived from particle simulation results.

[0007] The selection of the four sampling times is as follows: using T / 4 as the basic interval, four sampling points are evenly distributed within a complete fundamental frequency cycle, forming a symmetrical sampling sequence. The special characteristic of this sequence is that any two sampling points differing by T / 2 (i.e., t1 and t3, t2 and t4) are time-separated by half a fundamental frequency cycle. This causes the fundamental frequency component to have opposite amplitudes at these two sampling points, while the even harmonic components (taking the second harmonic as an example) have the same amplitude at these two sampling points. It is this rule of "fundamental frequency out of phase, even harmonics in phase" that lays the foundation for subsequent differential elimination of even harmonics. Simultaneously, the T / 4 phase difference between t1 and t2, and between t3 and t4, corresponds to a π / 2 phase difference in the fundamental frequency, thus constructing orthogonal components.

[0008] In specific implementation, the selection of t1 should satisfy the following: within the steady-state output stage of the PIC simulation, any time point can be selected as the starting sampling point; preferably, the selection of t1 can ensure that all four sampling times are within the time window when the PIC simulation has reached a steady state, so as to ensure that the acquired electric field data has stable amplitude and frequency characteristics.

[0009] S2: Transverse electric field distribution E at each time k T (z0, r, φ, t k The normalized transverse magnetic field h of the target mode j Tj Performing cross-sectional integration on (r, φ) yields four integral values. Qjk : (1) Where S is the cross-section of the coaxial waveguide, h Tj(r, φ) is the normalized transverse magnetic field distribution function of the target mode j, which satisfies the coaxial waveguide eigenmode equation and the corresponding boundary conditions.

[0010] The physical meaning of this step is to utilize the orthogonality between the eigenmodes of the coaxial waveguide to decompose the actual transverse electric field distribution onto each target mode. Normalized transverse magnetic field h Tj The specific form of (r, φ) is determined by the geometric parameters of the coaxial waveguide (inner radius R). in Outer radius R out The transverse magnetic field is determined by both the TEM mode and the mode type. For the TEM mode, the transverse magnetic field has only an angular component, and its amplitude is uniformly distributed across the cross-section; for the TM mode... 0n The transverse magnetic field has only an angular component, and its radial distribution is described by a Bessel function. The specific functional forms mentioned above are common knowledge to those skilled in the art and will not be elaborated here.

[0011] S3: For each target pattern j, calculate the difference component: (2) (3) Through the above differential operation, the contributions of even harmonics (including second and fourth harmonics, etc.) are completely canceled out, while the fundamental frequency component is preserved and enhanced. Specifically, for any even harmonic order m = 2, 4, 6, ..., its phase difference at times t1 and t3 = t1 + T / 2 is mπ. Since m is an even number and mπ is an integer multiple of 2π, the instantaneous values ​​of the even harmonics at times t1 and t3 are the same. Therefore, in the differential operation... The even harmonics are completely canceled out; similarly, even harmonics are completely canceled out. The even harmonics are also completely canceled out. This principle also applies to all even harmonics of the fourth order and above, achieving even harmonic suppression over a wide frequency range.

[0012] S4: Calculate the amplitude V of target mode j based on the difference component. j and phase φ j : (4) (5) in Let ω be the longitudinal wavenumber of target mode j, ω = 2π / T be the microwave angular frequency, and atan2 be the arctangent function in the four quadrants.

[0013] It should be noted that in the above formulas for calculating amplitude and phase... The term is used to compensate for the initial phase shift introduced by the initial sampling time t1, so that the phase obtained in different modes... Phase has comparable physical meaning.

[0014] S5: Based on the amplitude Calculate target pattern j power (6) in Let j be the wave impedance of the target mode. For the TEM mode, For TM in the propagation state 0n mold, .

[0015] S6: Calculate the power percentage of each target mode: (7) This allows us to obtain information on the power distribution and mode purity of each mode in the coaxial waveguide.

[0016] It should be noted that the core inventive idea of ​​this invention lies in: selecting four symmetrical sampling times with intervals of T / 4, T / 2, and 3T / 4 within a fundamental frequency period, and utilizing differential combination... This allows even-order harmonic components to be automatically canceled out during differential operations, while the fundamental frequency component is fully preserved. Four-frame sampling is merely a preferred implementation of this core idea. Those skilled in the art can extend it to even-numbered frame symmetrical sampling schemes such as six-frame or eight-frame sampling based on the same principle, which also fall within the protection scope of this invention.

[0017] The present invention also provides a coaxial waveguide high-power microwave mode composition analysis system based on four-frame differential sampling, comprising: The data acquisition module is used to acquire transverse electric field distribution data at four specified times within one microwave fundamental frequency period T at the same axial position z0 on the output end face of the coaxial waveguide. The integration calculation module is used to perform cross-sectional integration of the transverse electric field distribution and the normalized transverse magnetic field of the target mode at each time point to obtain four integral values. The harmonic cancellation module is used to construct a difference component based on the four integral values ​​to eliminate even harmonic components in the transverse electric field distribution data. The parameter solving module is used to calculate the amplitude and phase of the target mode based on the difference components; A power calculation module is used to calculate the power of the target mode based on the amplitude; The mode composition analysis module is used to calculate the power proportion of each target mode.

[0018] The above modules can be implemented in software and integrated into the post-processing module of particle simulation software, or they can be implemented in real time using dedicated hardware circuits.

[0019] The beneficial effects of this invention are: 1. Strong harmonic suppression capability: By selecting four specific symmetrical moments within a fundamental frequency cycle for sampling and constructing a differential combination, the power error caused by the second harmonic can be reduced from 5%~10% in the traditional method to below 0.5%, which significantly improves the accuracy of mode component analysis. At the same time, this differential mechanism has an automatic cancellation effect on all even harmonics such as the fourth and sixth harmonics, realizing broadband even harmonic suppression.

[0020] 2. High computational efficiency: Only two sets of transverse electric field distribution data at different times need to be added to the traditional two-frame method to achieve an order-of-magnitude improvement in accuracy without the need for complex iterative calculations or filtering; the computational complexity remains at the order of O(N), where N is the number of grid points in the cross section, which is comparable to the two-frame method.

[0021] 3. Wide applicability: Applicable to various HPM devices that generate strong harmonics, including relativistic backward wave tubes, relativistic klystrons, virtual cathode oscillators, etc., and maintains high accuracy in both single-mode and multi-mode propagation scenarios; also applicable to TEM and TM modes in coaxial waveguides. 0n Hybrid mode analysis of the model.

[0022] 4. Simple algorithm: No additional filtering is required, the phase integrity of the original signal is maintained, and it is easy to integrate into existing PIC analog post-processing workflows; it does not depend on a specific PIC software platform and has good versatility and portability. Attached Figure Description

[0023] Figure 1 This is a schematic diagram of the device structure of the X-band relativistic backward wave oscillator (RBWO) in an embodiment of the present invention; Figure 2 Spectral analysis diagram of the microwave output of the X-band relativistic backward wave oscillator; Figure 3 A graph showing the time-varying output microwave power of an X-band relativistic backward wave oscillator; Figure 4 The image shows the microwave output power obtained by numerically calculating the X-band RBWO using the four-frame harmonic elimination method of this invention. Figure 5 This is a schematic diagram of the device structure of the Q-band relativistic backward wave oscillator (RBWO) in an embodiment of the present invention; Figure 6 Spectral analysis diagram of the output microwave of a Q-band relativistic backward wave oscillator; Figure 7 A graph showing the time-varying output microwave power of a Q-band relativistic backward wave oscillator; Figure 8 The image shows the microwave output power obtained by numerically calculating the Q-band RBWO using the four-frame harmonic elimination method of this invention. Detailed Implementation

[0024] The embodiments of the present invention will be further described below with reference to the accompanying drawings.

[0025] Example 1: Theoretical Derivation and Principle Explanation Suppose that the transverse electric field distribution on the cross section at point z of the coaxial waveguide output segment can be represented as the superposition of various propagation modes: (8) in Let j be the longitudinal wavenumber of mode j. Let j be the transverse field amplitude. Let ω be the initial phase of mode j, and ω be the microwave angular frequency. This is the normalized transverse electric field distribution function.

[0026] The normalized transverse electromagnetic field distribution function satisfies the orthogonality relation: (9) Where S is the waveguide cross-section. and These represent the normalized transverse electric field of mode i and the normalized transverse magnetic field of mode j, respectively.

[0027] To calculate the power flow in mode j, the field distribution data at the cross section at point z is extracted from the PIC simulation results. Regarding its relationship with the normalized transverse magnetic field h Tj By performing cross-sectional integration, and applying the orthogonality relation, we can obtain: (10) However, the electric field output by the actual PIC simulation contains a significant second harmonic component: (11) in Let be the projection amplitude of the second harmonic component onto mode j. Its initial phase.

[0028] To eliminate the interference of the second harmonic, this invention selects four moments within one microwave cycle T: (12) The following derivation is performed for a fixed pattern j. For simplicity, Q is... jk Abbreviated as Q k (k=1,2,3,4).

[0029] Perform cross-sectional integration at z0 over the four time points, and define: (13) (14) (15) (16) Substituting the electric field expression containing harmonics into the expression, let... The four integral values ​​are then expanded as follows: (17) (18) (19) (20) In the above expansion, the fundamental frequency component has opposite amplitudes at times t1 and t3. and The opposite is also true at times t2 and t4. and The second harmonic component has the same amplitude at times t1 and t3 (both are...). The same applies at times t2 and t4 (both are...). ).

[0030] right and Making a difference: (twenty one) right and Making a difference: (twenty two) As can be seen from the above two equations, the second harmonic term is completely eliminated, leaving only the fundamental frequency component. Similarly, it can be verified that all even harmonics, such as the fourth and sixth harmonics, are canceled out in this differential operation. Therefore, the amplitude and phase of mode j can be obtained: (twenty three) (twenty four) Find the amplitude of mode j Then, its power flow is given by the integral of the Poynting vector over the cross section: (25) in Let be the wave impedance of mode j. Wave impedance of the TEM mode. TM in the propagation state 0n Mode impedance .

[0031] The power percentage of each mode is defined as follows: (26) The above theoretical derivation shows that the present invention accurately eliminates even-order harmonic interference without filtering by using four-frame differential sampling, thus achieving accurate extraction of fundamental frequency mode parameters.

[0032] Example 2: Sampling Time Selection Principles This embodiment illustrates the selection principle of the four-frame sampling time t1 and its impact on the analysis accuracy.

[0033] In engineering implementation, the selection of t1 directly affects the phase position of the four sampling points relative to the microwave signal waveform. Theoretically, any selection of t1, as long as the four sampling points are distributed at equal intervals of T / 4, will ensure the effectiveness of the differential harmonic elimination mechanism. However, in actual PIC simulations, the output signal may exhibit slow amplitude drift or frequency chirp effects. Therefore, the selection of t1 should follow these principles: (1) All four sampling times should be within the steady-state output time window of the PIC simulation, and the selection should be avoided during the start-up or cut-off phases to ensure the stability of the amplitude and frequency of the electric field data; (2) The selection of t1 should ensure that the electric field data corresponding to the four sampling times are available in the simulation output file, that is, it does not exceed the total time range of the PIC simulation. (3) Under the premise of satisfying the above conditions, the selection of t1 has no substantial impact on the final analysis results, which reflects the robustness of the present invention.

[0034] As a preferred implementation, the following automated selection strategy can be adopted: During the steady-state output phase of the PIC simulation, using the time step Δt of the PIC simulation as the basic unit, select any value that satisfies t1 + T / 2 ≤ t end The time t is taken as the starting sampling time, where t end This is the termination time of the PIC simulation. More preferably, t1 can be selected in the middle region of the steady-state output stage to minimize its distance from the start-up and cutoff transient processes.

[0035] Example 3: Numerical Verification of X-band Relativistic Backward Wave Oscillator The verification was performed using actual PIC simulation data of an X-band relativistic backward wave oscillator. The basic structure of the device is as follows: Figure 1 As shown. The spectrum analysis diagram of this device is as follows. Figure 2 As shown, the device operates at a fundamental frequency of 9.380 GHz and generates a significant second harmonic component in the output coaxial waveguide. The amplitude of the second harmonic is about 9% of the fundamental frequency amplitude, which belongs to the strong harmonic scenario.

[0036] The specific verification parameters are as follows: Fundamental frequency: f0 = 9.380 GHz, period T = 106.61 ps The four sampling times are: t1 = 13.000 ns, t2 = 13.027 ns, t3 = 13.053 ns, and t4 = 13.080 ns. Coaxial waveguide dimensions: Inner radius R in =27.46mm, outer radius R out =40.75mm Analysis modes: TEM mode (propagation state), TM 01 TM 02 TM03 Modulus (all cutoff) Simulated microwave output such as Figure 3 As shown, the total output power is approximately 413 MW. Substituting the transverse electric field detected in the simulation, the numerical calculation results are as follows. Figure 4 As shown. Under this strong harmonic condition, the analysis results of the four-frame harmonic elimination method and the traditional two-frame method are compared: Table 1: Comparison of Power Analysis of X-band Coaxial RBWO Mode

[0037] Experimental results show that: The traditional two-frame method overestimates the TEM mode power by 7.20% due to the influence of the second harmonic, resulting in a significant error. The four-frame harmonic elimination method of this invention effectively suppresses second harmonic interference, with a power deviation of only -0.05%, which is far superior to the two-frame method.

[0038] Example 4: Numerical Verification of Q-band Multimode The verification was performed using actual PIC simulation data of a Q-band relativistic backward wave oscillator. The basic structure of the device is as follows: Figure 5 As shown. The spectrum analysis diagram of this device is as follows. Figure 6 As shown, the device operates at a fundamental frequency of 46.10 GHz, with virtually no higher harmonics (the second harmonic amplitude is less than 1% of the fundamental frequency). The TEM mode is the dominant operating mode in this device, but the TM mode... 01 The model can also propagate, and is mainly used to verify the correctness of model component analysis under multi-mode propagation conditions.

[0039] The specific verification parameters are as follows: Fundamental frequency: f0 = 46.10 GHz, period T = 21.68 ps The four sampling times are: t1 = 20.000ns, t2 = 20.0054ns, t3 = 20.0108ns, and t4 = 20.0162ns. Coaxial waveguide dimensions: Inner radius R in =15.00mm, outer radius R out =19.40mm Analysis modes: TEM mode (propagation state), TM 01 (Propagation state), TM 02 TM 03 Modulus (all cutoff) Simulated microwave output such as Figure 7 As shown, the total output power is approximately 363MW. Substituting the transverse electric field detected in the simulation, the numerical calculation results are as follows. Figure 8 As shown. Due to the small size of the second harmonic component, the analysis results of the four-frame harmonic elimination method and the traditional two-frame method are compared: Table 2: Comparison of Q-band coaxial RBWO mode power analysis

[0040] Experimental results show that: When the higher harmonic components are small, the results of the four-frame method are not significantly different from those of the two-frame method, with a power deviation of only 0.03%, which is better than the 1.9% of the two-frame method. Under multimode propagation conditions, the four-frame method can accurately separate the TEM mode and the TM mode. 01 The power components of the modes were analyzed to verify the universality and reliability of the method of the present invention in multi-mode coexistence scenarios.

[0041] Example 5: Algorithm Flow and System Implementation This invention also provides a coaxial waveguide high-power microwave mode composition analysis system based on four-frame differential sampling, comprising the following modules: Data acquisition module: Used to acquire transverse electric field distribution data E at the same axial position z0 on the output end face of the coaxial waveguide at four specified times within one microwave fundamental frequency period T. T (z0, r, φ, t k (k=1,2,3,4). This module can directly read the field distribution file output by the PIC software, or collect field data in real time during the simulation process through the data interface.

[0042] Integration module: used to calculate the electric field distribution at each time k and the normalized transverse magnetic field of the target mode j. Performing cross-sectional integration yields four integral values ​​Q. jk The core algorithm of this module is numerical integration. For field data on discrete grids in PIC simulations, appropriate numerical integration methods (such as the trapezoidal method or Simpson's method) are used to calculate the surface integral on the cross-section.

[0043] Harmonic cancellation module: used to construct differential components , This is to eliminate even-order harmonic interference in the electric field data.

[0044] Parameter solving module: used to calculate the amplitude of mode j based on the difference component. and phase Numerical calculations are performed using formulas (4) and (5).

[0045] Power calculation module: used for amplitude-based calculations Calculation mode power

[0046] Mode composition analysis module: used to calculate the power proportion of each target mode. It also outputs a list of power distributions for each mode and a mode purity index.

[0047] The above modules can be implemented in software and integrated into the post-processing module of particle simulation software, or they can be implemented in real time using dedicated hardware circuits. In software implementation, programming languages ​​such as Python, MATLAB, and C++ can be used as post-processing plugins or standalone tools for PIC simulation software (such as UNIPIC, CHIPIC, CST Particle Studio, etc.).

[0048] This invention can be widely applied to post-processing analysis of particle simulations for high-power microwave devices, and is particularly suitable for the output mode composition analysis of HPM devices such as relativistic backward wave oscillators, relativistic klystron amplifiers, and virtual cathode oscillators. This invention has significant engineering application value in improving the design efficiency and mode purity evaluation accuracy of HPM devices.

[0049] The embodiments described above are merely illustrative of specific implementations of the present invention, and while the descriptions are detailed, they should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the scope of protection of the present invention.

Claims

1. A method for high-power microwave mode composition analysis of coaxial waveguides based on four-frame differential sampling, characterized in that, Includes the following steps: S1: At the same axial position z0 on the output end face of the coaxial waveguide, within one microwave fundamental frequency period T, acquire the transverse electric field distribution data E at four times t1, t2 = t1 + T / 4, t3 = t1 + T / 2, and t4 = t1 + 3T / 4. T (z0,r, φ, t k ), where k=1,2,3,4; S2: Transverse electric field distribution E at each time k T (z0, r, φ, t k The normalized transverse magnetic field h of the target mode j Tj Performing cross-sectional integration on (r, φ) yields four integral values. Qjk : ; Where S is the cross section of the coaxial waveguide; S3: Calculate the difference component , The even harmonic components in the transverse electric field distribution data are eliminated by the difference component. S4: Calculate the amplitude V of target mode j based on the difference component. j and phase φ j : ; in Let ω be the longitudinal wavenumber of target mode j, and ω = 2π / T be the microwave angular frequency. S5: Based on the amplitude Calculate target pattern j power ,in Let be the wave impedance of target mode j; S6: Calculate the power percentage of each target mode. This allows us to obtain information on the power distribution and mode purity of each mode in the coaxial waveguide.

2. The method according to claim 1, characterized in that, Before step S1, the method also includes: based on the inner radius R of the coaxial waveguide. in Outer radius R out And the operating frequency ω, to determine the target mode j to be analyzed, the target mode j including TEM mode and TM mode. 0n Model; and calculate the normalized transverse magnetic field distribution of each target model. .

3. The method according to claim 1, characterized in that, The wave impedance mentioned in step S5 For the TEM mode, For TM in the propagation state 0n mold, .

4. The method according to claim 1, characterized in that, The transverse electric field distribution data The output comes from particle simulation software.

5. The method according to claim 1, characterized in that, The coaxial waveguide is used for output microwave mode analysis of relativistic backward wave oscillators, virtual cathode oscillators, or Cherenkov generators.

6. The method according to claim 1, characterized in that, The four times are selected from at least four symmetrical sampling times within a fundamental frequency period.

7. The method according to claim 1, characterized in that, The power ratio Used to evaluate the mode purity of target mode j in the output microwave.

8. A coaxial waveguide high-power microwave mode composition analysis system based on four-frame differential sampling, characterized in that, include: The data acquisition module is used to acquire transverse electric field distribution data at four specified times within one microwave fundamental frequency period T at the same axial position z0 on the output end face of the coaxial waveguide. The integration calculation module is used to perform cross-sectional integration of the transverse electric field distribution and the normalized transverse magnetic field of the target mode at each time point to obtain four integral values. The harmonic cancellation module is used to construct a difference component based on the four integral values ​​to eliminate even harmonic components in the transverse electric field distribution data. The parameter solving module is used to calculate the amplitude and phase of the target mode based on the difference components; A power calculation module is used to calculate the power of the target mode based on the amplitude; The mode composition analysis module is used to calculate the power proportion of each target mode.

9. The system according to claim 8, characterized in that, The data acquisition module extracts the transverse electric field distribution data from the output of the particle simulation software.

10. The system according to claim 8, characterized in that, The four specified times are t1, t1 + T / 4, t1 + T / 2, and t1 + 3T / 4.

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