A multi-frequency-point half-wave voltage parallel measurement system of an electro-optic phase modulator

Abstract

The application provides an electro-optic phase modulator multi-frequency point half-wave voltage parallel measurement system, which is sequentially provided with a narrow linewidth laser, a first coupler, a polarization controller, a phase modulator, a delay optical fiber, a frequency shifter, a second coupler, a dispersion optical device, a photodetector array, a multi-channel signal acquisition module and a data processing module; the input end of the phase modulator is further sequentially connected with a first microwave signal source, a first power amplifier and a first low-pass filter; and the input end of the frequency shifter is sequentially connected with a second microwave signal source, a second power amplifier and a second low-pass filter. The scheme can improve the measurement efficiency and accuracy of the half-wave voltage.

Description

Technical Field

[0001] This invention relates to the field of voltage measurement technology, and in particular to a multi-frequency half-wave voltage parallel measurement system using an electro-optic phase modulator. Background Technology

[0002] Half-wave voltage (V) of electro-optic phase modulator π The voltage required to induce a π-radian change in the phase of an optical wave is a key parameter characterizing the performance of electro-optic modulation. It is defined as the voltage required to induce a π-radian change in the phase of an optical wave. This parameter directly affects the efficiency, power consumption, and linearity of the modulator and is of great significance in fields such as optical communication, fiber optic sensing, lidar, and quantum optics.

[0003] Electro-optic modulation technology is based on the electro-optic effect. When an external electric field is applied to an electro-optic crystal (such as lithium niobate LiNbO3), it causes a change in the crystal's refractive index, thereby altering the phase of the light wave passing through the crystal. The half-wave voltage is a core indicator for evaluating the performance of electro-optic modulators, and its accurate measurement is crucial for device design, performance evaluation, and application optimization.

[0004] Common methods for measuring half-wave voltage include:

[0005] Scanning DC bias voltage method: By measuring the change in DC bias voltage and output light intensity, the maximum and minimum points are determined from the transfer function curve. The voltage difference between adjacent extreme points is the half-wave voltage.

[0006] Frequency doubling modulation method: DC voltage and AC signal are applied simultaneously. When the DC voltage is adjusted to the point where the output light intensity reaches an extreme value, the AC signal will exhibit frequency doubling distortion. The difference in DC voltage corresponding to the occurrence of frequency doubling distortion is the half-wave voltage.

[0007] Spectral analysis method: The half-wave voltage is estimated by analyzing the changes in the spectral characteristics of the modulated optical signal;

[0008] Polarization interferometry: Measurement is performed based on the interference principle of two polarization modes;

[0009] Sawtooth wave modulation method: The half-wave voltage is determined by applying a sawtooth wave voltage signal and observing the change in output light intensity;

[0010] Zero-crossing detection method: By adjusting the bias voltage, the modulated light passing through the phase modulator and the local oscillator light will have a coherent zero point, and the half-wave voltage value can be calculated from this.

[0011] The current half-wave voltage measurement method has the following problems:

[0012] Measurement accuracy is limited: the extreme value method and the frequency doubling method are easily affected by environmental disturbances and system noise during the measurement process, which leads to inaccurate judgment of extreme points. For example, temperature fluctuations can cause phase drift in the interferometric system, resulting in measurement errors of up to 5%-10%. The spectral analysis method requires high resolution of the spectrometer, and the data processing introduces approximate calculations, making it difficult to achieve high-precision measurement.

[0013] The system is complex and inconvenient to operate: polarization interferometry requires precise optical path alignment and a stable interferometry system. The system is complex and has extremely high requirements for environmental stability, making it unsuitable for online measurement or engineering field applications; variable amplitude sine wave modulation method requires dedicated modulation and demodulation circuits, which increases the system complexity and cost; some methods (such as the laser phased array principle method) require complex optical component configuration and precision mechanical adjustment, making operation cumbersome.

[0014] These technical limitations restrict the accuracy, efficiency, and applicability of half-wave voltage measurement. There is an urgent need to develop simple and practical new measurement methods to solve these problems and provide more reliable technical support for the design optimization and performance evaluation of electro-optic phase modulators. Summary of the Invention

[0015] To address the aforementioned problems, this invention proposes a parallel measurement system for half-wave voltage at multiple frequencies using an electro-optic phase modulator. By employing a frequency shifter, a dispersive beam splitter, a photodetector array, a multi-channel signal acquisition module, and a data processing module, the system improves the measurement efficiency and accuracy of half-wave voltage.

[0016] A multi-frequency half-wave voltage parallel measurement system for an electro-optic phase modulator, the system comprising, in sequence, a narrow linewidth laser, a first coupler, a polarization controller, a phase modulator, a delay fiber, a frequency shifter, a second coupler, a dispersive beam splitter, a photodetector array, a multi-channel signal acquisition module, and a data processing module; the input terminal of the phase modulator is further connected in sequence to a first microwave signal source, a first power amplifier, and a first low-pass filter; the input terminal of the frequency shifter is connected in sequence to a second microwave signal source, a second power amplifier, and a second low-pass filter.

[0017] A narrow-linewidth laser emits a single-frequency laser source, which is split into two beams by a 50:50 first coupler and used as optical inputs to a phase modulator and a frequency shifter, respectively. Before the optical carrier enters the frequency shifter, a delay fiber is used to delay the two optical signals to remove coherence. Before the optical carrier enters the phase modulator, a polarization controller is used to adjust the polarization state to ensure that the polarization state remains consistent when the two optical signals are combined.

[0018] The optical signals output from the phase modulator and frequency shifter are combined by the second coupler and then enter the dispersive beam splitter to spatially separate the frequency-shifted optical carrier, first-order sideband, second-order sideband, and each higher-order sideband.

[0019] The separated optical carrier and each sideband are fed into a photodetector array composed of different photodetectors and converted into electrical signals. The multi-channel signal acquisition module acquires the current signal output by the photodetector and transmits the data to the data processing module.

[0020] Furthermore, the data processing module controls the first microwave signal source to generate a first single-tone sine wave signal, setting the frequency of the first single-tone sine wave signal to ω1 and the power to be... Therefore, the amplitude of the corresponding first single-tone sine wave signal is calculated to be... The relationship between amplitude and power is as follows:

[0021]

[0022] in, For matching resistors.

[0023] Furthermore, the data processing module controls the second microwave signal source to generate a second single-tone sine wave signal, and sets the frequency of the second single-tone sine wave signal to ω2.

[0024] Furthermore, the electrical signal output by the photodetector is sent to the multi-channel signal acquisition module and data processing module to obtain the interference signal with a center wavelength of ω2 and a sideband spectrum of ω2±nω1 after beam combining. The power of ω2 corresponds to the zero-order Bessel function, and the power of ω2±nω1 corresponds to the higher-order Bessel function.

[0025] Furthermore, the data processing module's data processing procedure includes:

[0026] Step 1: Record the power P2 of the signal at ω2, and convert the power P2 into the amplitude of a sinusoidal signal. Record the power P2' of the signal at ω2±nω1, and convert the power P2' into the amplitude of a sinusoidal signal. ;

[0027] Step 2: Fit using spline interpolation , The function curve is used to obtain the fitted curve. ;

[0028] Step 3: Fit using spline interpolation , The function curve is used to obtain the fitted curve. ;

[0029] Thus, the fitted curve is obtained. The local minimum points: x1, x2, x3, x4, ..., x NGiven that the local minima of the zero-order Bessel function are 3.832, 7.016, 10.173, 13.324, ..., and the half-wave voltage values ​​calculated from each local minimum are V... π1 =x1 / 3.832*π, V π2 =x² / 7.016*π, V π3 =x³ / 10.173*π, V π4 =x4 / 13.324*π…;Then the final calculated half-wave voltage value is V π =(V π1 + V π2 + V π3 + V π4 +…) / N, where N is the number of local minima used;

[0030] Step 4: Based on the fitted curve The local minimum values ​​of the curve are obtained, and the half-wave voltage at ω2±nω1 is calculated from them. For different values ​​of n, the Bessel function of the corresponding order needs to be used. That is, when n=1, the first-order Bessel function is needed to measure the half-wave voltage at ω1; when n=2, the second-order Bessel function is needed to measure the half-wave voltage at 2ω1, and so on.

[0031] Step 5: Change the frequency ω1 and repeat the operations of steps 1 to 4 above to complete the half-wave voltage measurement at different frequencies.

[0032] This invention provides a parallel measurement system for multi-frequency half-wave voltage of an electro-optic phase modulator, achieving low hardware complexity and high reliability in half-wave voltage testing. By adjusting the power value of the sinusoidal signal output from the signal generator, the beat frequency output of the phase modulator output spectrum corresponding to different RF input powers is obtained. Through FFT calculation, the center wavelength and higher-order sideband power values ​​can be obtained. The sinusoidal signal power is numerically fitted with the power values ​​of the corresponding sidebands of the spectrum to obtain the corresponding input-output function curve, and then multiple local minima of the function curve are obtained. By averaging the half-wave voltage values ​​obtained from multiple local minima, an accurate half-wave voltage measurement result can be obtained. By changing the frequency of the RF signal, the half-wave voltage at different frequency points can be measured. The beneficial technical effects of this invention are as follows:

[0033] 1) By using a photodetector array, digital signal processing (FFT operation) is implemented, which can calculate the half-wave voltage at DC and nω1 in parallel, thus improving the measurement efficiency of the half-wave voltage;

[0034] 2) By using frequency shifting, the measurement of half-wave voltage near DC can be transferred to the measurement of half-wave voltage at intermediate frequency, avoiding the influence of environmental noise such as power frequency interference near zero frequency, and improving the accuracy of half-wave voltage measurement near zero frequency and at low frequency.

[0035] 3) It reduces the requirements for measurement points and can measure any radio frequency power and the corresponding modulation output power, avoiding the problems of requiring complex optical instruments for zero-crossing detection, making it difficult to measure accurately and the radio frequency power difficult to adjust accurately to the optical zero-crossing point.

[0036] 4) By using curve fitting of the Bessel function, more measurement data is utilized, which improves the accuracy of feature point finding and avoids the problem that it is not easy to accurately modulate the zero-order or higher-order sidebands to 0 in zero-crossing detection (affected by the noise of the measuring instrument and optical device).

[0037] 5) By averaging the half-wave voltage values ​​obtained from multiple extreme points, the accuracy of half-wave voltage measurement can be further improved.

[0038] 6) The signal acquisition module and data processing module in this invention can be replaced by general-purpose equipment. The signal acquisition module can be a spectrum analyzer or oscilloscope, and the data processing module can be a personal computer, tablet computer, or other device with simple calculation functions, which is relatively simple to implement and has a low cost. Attached Figure Description

[0039] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0040] Figure 1 This is a schematic diagram of the structure of a multi-frequency half-wave voltage parallel measurement system for an electro-optic phase modulator according to an embodiment of the present invention;

[0041] Figure 2 This is a schematic diagram of the spectral composition before and after phase modulation provided in an embodiment of the present invention;

[0042] Figure 3 This is a theoretical curve diagram of the 0th to 4th order Bessel functions provided in the embodiments of the present invention;

[0043] Figure 4 This is a schematic diagram of the fitting result of the zero-order Bessel function at ω2 provided in an embodiment of the present invention.

[0044] Explanation of reference numerals in the attached figures: 1-First coupler, 2-Second coupler, 3-First microwave signal source, 4-First power amplifier, 5-First low-pass filter, 6-Second microwave signal source, 7-Second power amplifier, 8-Second low-pass filter. Detailed Implementation

[0045] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0046] A multi-frequency half-wave voltage parallel measurement system using an electro-optic phase modulator, such as... Figure 1 As shown, the system sequentially includes a narrow linewidth laser, a first coupler 1, a polarization controller, a phase modulator, a time-delay fiber, a frequency shifter, a second coupler 2, a dispersive beam splitter, a photodetector array, a multi-channel signal acquisition module, and a data processing module. The input of the phase modulator is also sequentially connected to a first microwave signal source 3, a first power amplifier 4, and a first low-pass filter 5. The input of the frequency shifter is sequentially connected to a second microwave signal source 6, a second power amplifier 7, and a second low-pass filter 8.

[0047] A high-stability, narrow-linewidth laser emits a single-frequency laser source. After being split by a 50:50 first coupler 1, the beams are used as optical inputs to the phase modulator and frequency shifter, respectively. Before the optical carrier enters the frequency shifter, a delay fiber is used to delay the two optical signals to remove coherence. Before the optical carrier enters the phase modulator, a polarization controller is used to adjust the polarization state to ensure that the polarization state remains consistent when the two optical signals are combined.

[0048] A frequency shifter shifts the signal frequency, while a phase modulator modulates the signal phase. The principle of phase modulation is as follows:

[0049] Suppose that the single-frequency laser outputs a narrow-linewidth single-frequency laser:

[0050]

[0051] Where ω0 is the frequency of the laser, A0 is the amplitude of the light wave, and E0 represents the light field that changes with time t.

[0052] The laser beam output from a single-frequency laser is split by a coupler and then enters a phase modulator. A sinusoidal signal is used for phase modulation, resulting in the following output light:

[0053]

[0054] in, It is the frequency of the modulation signal. It is the modulation index. and These are the amplitude of the modulation voltage and the half-wave voltage of the phase modulator, respectively. Expanding the above equation using a Fourier series, we get:

[0055]

[0056] in, It is a Bessel function of the first kind. As can be seen from the above equation, the spectrum after phase modulation is as follows: Figure 2 As shown, the signal is transformed from a single tone into a comb pattern, with the frequency of each comb tooth being... The corresponding optical amplitude is That is, the nth order Bézier function, where the curves of each order Bézier function are as follows: Figure 3 As shown in the figure, five colors are used to represent Bessel functions of orders 0 to 4, J. v In (z), the subscript v indicates the order of the Bessel function, and the horizontal axis z is the independent variable, which is positively correlated with the amplitude of the single-tone signal input to the phase modulator in this invention. v (z) represents the variation of Bessel functions of different orders with respect to the independent variable.

[0057] The optical signals output from the phase modulator and frequency shifter are combined by the second coupler 2 and then enter the dispersive beam splitter to spatially separate the frequency-shifted optical carrier, first-order sideband, second-order sideband, and each higher-order sideband.

[0058] The separated optical carrier and each order sideband are sent to different photodetectors (i.e., photodetector arrays) to be converted into electrical signals. The multi-channel signal acquisition module acquires the current signal output by the photodetector and transmits the data to the data processing module.

[0059] The data processing module controls the first microwave signal source 3 to generate a first single tone (sine wave signal). The frequency of this single tone signal is the frequency to be measured and its harmonics. The data processing module can generate single-tone signals with arbitrary frequency precision by adjusting the parameters of the DDS (Direct Digital Synthesizer) in the FPGA (Field Programmable Gate Array). The single-tone signal is amplified by a fixed-gain power amplifier, with the signal power adjusted by the FPGA program and calibrated using a power meter. Alternatively, a standard signal source can be used to generate the single tone (sine wave signal), and its frequency and amplitude (power) can be set by the instrument.

[0060] Set the frequency of the first single-tone (sine modulation) signal to ω1, and the power to... The amplitude of the corresponding sinusoidal signal can be calculated from this. For a sinusoidal signal, the relationship between its amplitude and power is as follows:

[0061]

[0062] The radio frequency circuit typically uses a matching resistor of R=50Ω, and then the first power amplifier 4 and the first low-pass filter 5 amplify and filter the first single-tone signal.

[0063] Using the same method, a second single-tone sine wave signal is generated by a second microwave signal source 6. The frequency of the second single-tone sine wave modulation signal is set to ω2. The power of the second microwave signal source 6 is determined by the manufacturer's specifications of the frequency shifter. The second single tone is only used for frequency shifting of the light wave and does not participate in the half-wave voltage calculation. This invention uses an SGTF800-1064-1P(H) polarization-maintaining fiber optic acousto-optic modulator, which can achieve 800MHz frequency shifting. A recommended RF input power of 1W is suggested. The second power amplifier 7 amplifies the power of the second single-tone sine wave signal. Considering that the power amplifier usually generates high-order harmonics under high power conditions, a second low-pass filter 8 with a cutoff frequency of 1.5 times ω2 is placed after the second microwave signal source 6. For the 800MHz single tone used in this invention, a low-pass filter with a cutoff frequency of 1.2GHz is used.

[0064] The electrical signal output from the photodetector is sent to the multi-channel signal acquisition module and data processing module to obtain an interference signal with a center wavelength of ω2 (power corresponding to the zero-order Bessel function) and a sideband spectrum of ω2±nω1 (power corresponding to the higher-order Bessel function) after beam combining.

[0065] Adjust the polarization controller to maximize the power of the interference signal. At this point, the amplitude of the single tone at ω2 measured by the spectrum analyzer is stable and at its maximum value. Fix the state of the polarization controller at this point.

[0066] The data processing module also performs FFT on the current signal to obtain the power P2 and P2' of the signal at ω2 (power corresponding to the zero-order Bessel function) and ω2±nω1 (power corresponding to the higher-order Bessel function), and converts them into sinusoidal signal amplitudes V2 and V2'.

[0067] Considering that the ADC (Analog-to-Digital Converter) used in the signal acquisition module has a fixed sampling rate f s (Usually determined by the device itself and the driving circuit), it cannot be guaranteed that for every frequency point, f sAll are integer multiples of ω2 and ω2±nω1; therefore, when performing FFT operations, in order to avoid spectral leakage (also known as spectral "diffuse"), windowed FFT is used; common window functions include Hamming window, Hanning window, triangular window, etc. By selecting an appropriate window function, spectral leakage can be reduced and the realism and accuracy of the calculated spectrum can be improved.

[0068] The data processing module adjusts the power of the first microwave signal source 3 in a certain step. To avoid signal distortion during the fitting process, it should ensure that sampling is performed at least twice within a half-wave voltage variation range. For example, for a phase modulator of a bulk waveguide, the half-wave voltage range at low frequencies is approximately 1V~4V. Therefore, when adjusting the power, the amplitude step should be less than 0.5V. Different power values ​​are denoted as P1, and the amplitude value is calculated. .

[0069] The data processing module processes the data, specifically as follows:

[0070] (1) Record the power P2 of the signal at the corresponding ω2 and convert it into the amplitude of a sinusoidal signal. Record the power P2' of the signal at ω2±nω1 (the power corresponds to the higher-order Bessel function) and convert it into the amplitude of a sinusoidal signal. ;

[0071] (2) Use spline interpolation to fit the data. , The function curve is used to obtain the fitted curve. The result of a certain calculation is as follows Figure 4 As shown in the figure, the horizontal axis V1 represents the amplitude of the single-tone signal applied to the phase modulator. The amplitude of the interference signal at ω2 is obtained after FFT calculation, and x1 and x2 are local minima;

[0072] (3) Use spline interpolation to fit the data. , The function curve is used to obtain the fitted curve. ;

[0073] Thus, the fitted curve is obtained. The local minimum points: x1, x2, x3, x 4、 …、x N Given that the local minima of the zero-order Bessel function are 3.832, 7.016, 10.173, 13.324, ..., the half-wave voltage values ​​calculated from each local minimum are V. π1 =x1 / 3.832*π, V π2 =x² / 7.016*π, V π3 =x³ / 10.173*π, Vπ4 =x4 / 13.324*π…;Then the final calculated half-wave voltage value is V π =(V π1 + V π2 + V π3 + V π4 +…) / N, where N is the number of local minima used.

[0074] The algorithm principle of this invention is to use actual measurement data to perform Bessel function fitting to obtain feature points (local minimum points), and then use these feature points to calculate the half-wave voltage value. Figure 3 As can be seen from the curves of the 0th to 4th order Bessel functions, each order Bessel function exhibits oscillatory convergence characteristics as the independent variable (positively correlated with the amplitude of the first single tone in this invention) increases, that is, the amplitude becomes smaller and smaller. In actual measurement, due to the limitations of device and instrument conversion and measurement accuracy, only a finite number of local minimum points can be obtained.

[0075] (4) Similarly, based on the fitted curve By obtaining the local minimum values ​​of the curve, the half-wave voltage at ω2±nω1 can be calculated.

[0076] It should be noted that for different values ​​of n, the corresponding order of the Bessel function needs to be used. That is, when n=1, a first-order Bessel function is needed to measure the half-wave voltage at ω1; when n=2, a second-order Bessel function is needed to measure the half-wave voltage at 2ω1; and so on.

[0077] (5) Change the frequency ω1 and repeat the above (1) to (4) steps to complete the half-wave voltage measurement at different frequencies.

[0078] Since both the photodetector and the signal acquisition module have DC bias, meaning the reference level is not zero, therefore, relative to Figure 3 , Figure 4 The curve shifts upwards overall, and the curve does not cross zero. That is, Figure 4 The middle curve is ,in, The DC bias is introduced by the photodetector and signal acquisition module. Therefore, in this invention, the vertical axis only needs to consider local minima, while the periodicity of the horizontal axis reflects the value of the half-wave voltage. Thus, by calculating the periodicity of the horizontal axis variable, the fitted value of the half-wave voltage can be obtained; the vertical axis only needs to find local minima, without concern for the absolute value of the measured voltage, therefore the introduction of the vertical axis bias voltage does not affect the calculation accuracy of the half-wave voltage.

[0079] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A parallel measurement system for multi-frequency half-wave voltage of an electro-optic phase modulator, characterized in that, The system is provided with a narrow linewidth laser, a first coupler, a polarization controller, a phase modulator, a time-delay fiber, a frequency shifter, a second coupler, a dispersive beam splitter, a photodetector array, a multi-channel signal acquisition module, and a data processing module in sequence. The input of the phase modulator is also connected to a first microwave signal source, a first power amplifier, and a first low-pass filter in sequence. The input of the frequency shifter is connected to a second microwave signal source, a second power amplifier, and a second low-pass filter in sequence. A narrow-linewidth laser emits a single-frequency laser source, which is split by a 50:50 first coupler and used as the optical inputs of a phase modulator and a frequency shifter, respectively. Before the optical carrier enters the frequency shifter, a delay fiber is used to delay the two optical signals to remove the coherence. Before the optical carrier enters the phase modulator, the polarization state is adjusted by a polarization controller to ensure that the polarization state remains consistent when the two optical signals are combined. The optical signals output from the phase modulator and frequency shifter are combined by the second coupler and then enter the dispersive beam splitter to spatially separate the frequency-shifted optical carrier, first-order sideband, second-order sideband, and each higher-order sideband. The separated optical carrier and each order sideband are respectively sent to a photodetector array composed of different photodetectors to be converted into electrical signals. The multi-channel signal acquisition module acquires the current signal output by the photodetector and transmits the data to the data processing module. The data processing module controls the first microwave signal source to generate a first single-tone sine wave signal, setting the frequency of the first single-tone sine wave signal to ω1 and the power to be... Therefore, the amplitude of the corresponding first single-tone sine wave signal is calculated to be... The relationship between amplitude and power is as follows: ,in, To match the resistors, the first power amplifier and the first low-pass filter then amplify and filter the first single-tone signal. The data processing module also controls the second microwave signal source to generate a second single-tone sine wave signal. The frequency of the second single-tone sine wave signal is set to ω2. The second single-tone sine wave signal is only used to shift the frequency of the light wave and does not participate in the half-wave voltage calculation. The second power amplifier amplifies the power of the second single-tone sine wave signal and places a second low-pass filter with a cutoff frequency of 1.5 times ω2 after the second microwave signal source. The electrical signal output by the photodetector is sent to the multi-channel signal acquisition module and data processing module to obtain the interference signal with a center wavelength of ω2 and a sideband spectrum of ω2±nω1 after beam combining. The power of ω2 corresponds to the zero-order Bessel function, and the power of ω2±nω1 corresponds to the higher-order Bessel function. The data processing module adjusts the power of the first microwave signal source to ensure that more than two samples are taken within a half-wave voltage variation range; The data processing module processes the data in the following ways: Step 1: Record the power P2 of the signal at ω2, and convert the power P2 into the amplitude of a sinusoidal signal. Record the power P2' of the signal at ω2±nω1, and convert the power P2' into the amplitude of a sinusoidal signal. ; Step 2: Fit using spline interpolation , The function curve is used to obtain the fitted curve. ; Step 3: Fit using spline interpolation , The function curve is used to obtain the fitted curve. ; Thus, the fitted curve is obtained. The local minimum points: x1, x2, x3, x4, ..., x N Given that the local minima of the zero-order Bessel function are 3.832, 7.016, 10.173, 13.324, ..., and the half-wave voltage values ​​calculated from each local minimum are V... π1 =x1 / 3.832*π, V π2 =x² / 7.016*π, V π3 =x³ / 10.173*π, V π4 =x4 / 13.324*π…;Then the final calculated half-wave voltage value is V π =(V π1 + V π2 + V π3 + V π4 +…) / N, where N is the number of local minima used; Step 4: Based on the fitted curve The local minimum values ​​of the curve are obtained, and the half-wave voltage at ω2±nω1 is calculated from them. For different n values, the Bessel function of the corresponding order is used. That is, when n=1, the first-order Bessel function is used to measure the half-wave voltage at ω1; when n=2, the second-order Bessel function is used to measure the half-wave voltage at 2ω1, and so on. Step 5: Change the frequency ω1 and repeat the operations of steps 1 to 4 above to complete the half-wave voltage measurement at different frequencies.

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