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Improved heterodyne system using double-passed acousto-optic frequency shifters for measuring the frequency response of photodetectors in ultrasonic applications

Xiujuan Feng, Ping Yang, Longbiao He, Guangzhen Xing, Min Wang, and Wei Ke

Open Access Open Access

Abstract

The improved heterodyne system is presented to measure the frequency response of photodetectors (PDs) with bandwidth about 100 MHz utilized in ultrasonic applications. In order to eliminate the periodic oscillation following the frequency tuning of the acousto-optic frequency shifters (AOFSs) in the measurement results, the AOFSs’ double-passed scheme in Michelson interferometer is used. Compared with the AOFSs’ single-passed configuration in Mach-Zehnder interferometer, the double-passed scheme can avoid additional optical alignment during the process of the frequency tuning and then reduce its induced measurement uncertainty. With two double-passed AOFSs connected in parallel, the experimental setup is designed to demonstrate the feasibility of the improved heterodyne system in the frequency range from 500 kHz to 135 MHz. Experimental results with measurement uncertainty are provided and discussed.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The optical method based on homodyne or heterodyne interferometer has been well-established as the primary standard of hydrophone calibration instead of the reciprocity method for ultrasonic metrology [1–4]. The laser interferometer based primary standard can extend the calibration frequency to more than 40 MHz with the sound pressure realized directly and absolutely by the measured acoustic displacement and the known acoustic impedance. With ultrasonic waves imposed on an acoustically transparent and optically reflecting pellicle to follow its motion, the laser interferometer is used to measure the acoustic displacement from the change in the phase of the interference signal generated by a reference beam and the reflected beam from the pellicle. The photodetector (PD), usually including a photodiode and the trans-impedance amplifier, is the crucial component in the laser interferometer used to convert the optical interference signal to the electronic signal mostly depending on its frequency response, from which the acoustic displacement is demodulated. In order to acquire the adequate signal-to-noise ratio, the amplitude of the sound pressure where the pellicle is positioned must be large enough and the PD with large gain and thus limited bandwidth is required to be used. Previous studies demonstrate that the non-flat magnitude frequency response of the utilized PD is the main uncertainty component in the homodyne interferometer based ultrasonic sound pressure standard, especially in the high-frequency range for medical ultrasound applications [1–2]. Our recent studies show that both the non-ideal magnitude and phase frequency response of the PD can also significantly influence the measurement results of the heterodyne interferometer based sound pressure standard, such as the high intensity focused ultrasound (HIFU) application [5–6]. So, it is necessary to measure and correct the frequency response of the utilized PDs in the interested frequency range of the optical sound pressure standard, which is also applicable to other optical measurement systems for ultrasonic applications, such as fiber-optic hydrophones and optical multilayer hydrophones [7–10].

In contrast to the measurement requirements with frequency up to tens of gigahertz for optical communications, it requires much lower measurement uncertainty and higher frequency resolution for ultrasonic applications. The methods can be used include opto-mechanical calibrator [11, 12], heterodyne system with two tunable lasers [13] and twice-modulated light intensity scheme [14]. The opto-mechanical calibrator consists of three parts to cover the frequency range from 1 Hz to 115 MHz, and the measurement stability is greatly influenced by the environmental vibration at low frequency and mechanical resonance effects at high frequency [11]. The expanded uncertainty of less than 5% (k = 2) is achieved and it can be less than 3% (k = 2) from 1 kHz to 25 MHz. The improved heterodyne system using two distributed feedback lasers can cover the frequency range from 1 MHz to 450 MHz with measurement uncertainty 5% (k = 2). But the difference-frequency servo control is necessary to suppress the frequency jitter of the optical excitation signal and ensure the frequency resolution better than 1 MHz with the required stability. In the twice-modulated light intensity scheme, the measurement frequency range from 1 MHz to 20 MHz is reported, which is restricted by the modulation bandwidth of acousto-optic modulators.

Alternative heterodyne system using acousto-optic frequency modulation in Mach-Zehnder interferometer is also presented with verification experiments in the frequency range from 500 kHz to 150 MHz, which can be easily expanded by choosing acousto-optic frequency shifters (AOFSs) with suitable frequency tuning range, number and combination configurations [15]. Compared with the above three methods, the heterodyne method using acousto-optic frequency modulation offers the advantages of simple configuration, easily expanding in the high-frequency range, frequency resolution better than 500 kHz. However, it needs much careful optical alignment to make the two laser beams of the interferometer coincide with each other during the process of the frequency tuning; otherwise the frequency response of the interference signal used as the optical excitation signal in the measurement will vary with the misalignment angle of the two laser beams. The deflection angle of one laser beam in the interferometer is too small to be re-aligned unless the re-alignment works again with relative larger frequency shift of AOFSs. So, obvious periodic oscillation can be seen in the frequency response measurement results of PDs.

In order to avoid the optical alignment difficulties and the periodic oscillation in the measurement results of the heterodyne system using AOFSs based on Mach-Zehnder interferometer, the improved scheme based on Michelson interferometer using AOFSs’ double-passed configuration is presented in this paper. Experiments are implemented to demonstrate the performance of the improved scheme in the frequency range from 500 kHz to 135 MHz. Experimental results with measurement uncertainty are provided and discussed.

2. Measurement scheme

Figure 1 shows the schematic diagram of the heterodyne system using acousto-optic frequency modulation with different connection configurations of AOFSs. The AOFS with tunable frequency shift is used in one arm of Mach-Zehnder interferometer. Due to acousto-optic interaction, the frequency difference is generated between the first-order diffracted beam and the input laser beam of the AOFS. And the magnitude of the frequency difference depends on the frequency of the built-in ultrasonic transducer controlled by the DC voltage for the RF driver of AOFS. The first-order diffracted beam is selected as the output of AOFS and then combine with the laser beam in the other arm of the interferometer at the second polarized beam splitter (PBS 2). The interference signal of the combined two laser beams after the following polarizer and the non-polarizing beam splitter (BS) is used as the optical excitation signal to measure the frequency response of PDs. The output voltage of the measured PD in response to the optical excitation signal can be expressed as:

$${V_{out}} = \eta ({I_s} + {I_r})\left[ {1 + M(f)\frac{{2\sqrt {{I_s}{I_r}} }}{{{I_s} + {I_r}}}\cos (2\pi ft)} \right],$$
where η is defined as the ratio of the PD’s output voltage to the input optical power and is the function of the optical wavelength, Is and Ir are the optical power of the two interfering beams, f is the measurement frequency equaling to the frequency difference of the two interfering beams, M(f) represents the frequency response of the PD to be measured.

 figure: Fig. 1.

Fig. 1. (a) Schematic diagram of the heterodyne system based on Mach-Zehnder interferometer using acousto-optic frequency modulation. (b) Connection configurations of AOFSs in the heterodyne interferometer. AOFS is acousto-optic frequency shifter. PBS is polarizing beam splitter. BS is non-polarizing beam splitter.

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The frequency response of the PD to be measured can be obtained by the ratio of the AC component to the DC component from the spectrum of its output voltage, which requires the frequency response of the optical excitation signal to be flat in the measurement. That means, the fluctuation of the optical power ratio between the two interfering beams before the PD to be measured, the modulation efficiency of AOFSs and other attenuation in the two arms of the interferometer must be controlled, which can be realized by the adjustment of the two half waveplates before and after PBS 1.

The measurement frequency range of the PD in ultrasonic applications is usually from 500 kHz to 100 MHz or above for practical requirements, which can be covered by choosing AOFSs with suitable frequency tuning range, number and combination configurations. For the heterodyne system based on Mach-Zehnder interferometer, three commercial AOFSs with the configuration shown in Fig. 1(b) are usually necessary to cover the frequency range from 500 kHz to 150 MHz.

The frequency resolution of the heterodyne system depends on the frequency tuning interval, the linewidth and the frequency stability of the optical excitation signal. The frequency interval in the measurement is adjusted continuously by the frequency shift difference of AOFSs in two arms of the interferometer controlled by the DC voltage for their RF drivers. Experimental results show that there is no frequency spectrum broadening in the output of the PD in response to the optical excitation signal and the measured linewidth is less than 50 kHz [15]. Considering the frequency shift stability of AOFSs, the measured maximum frequency fluctuation of the optical excitation signal is 51 kHz within 10 minutes. So, it is obvious that the frequency resolution better than 500 kHz can be realized in the measurement, which is enough for most of ultrasonic applications [13].

However, the question must to be improved is the difficult re-alignment of the AOFS output beams for the dependence of their deflection angles on the frequency shift in the measurement. Slight deviation between the two interfering beams before the PD to be measured will significantly change the optical excitation signal and then the frequency response measurement results. When the frequency shift of any one of AOFSs is changed to achieve the desired measurement frequency, the propagation direction of the first-order diffraction beam of the AOFS’s output will also changes. If the increment of the frequency shift is too small, it is difficult to distinguish the deflection angle of the AOFS’s output beam unless it accumulates to be large enough. That is why obvious periodic oscillation can be seen in the frequency response measurement results in previous work [15].

In order to make the optical re-alignment easier to eliminate the periodic oscillation in the measurement results, the optimized heterodyne system using double-passed AOFSs in Michelson interferometer is presented as shown in Fig. 2. The half waveplate and the PBS 1 after the laser are used to adjust the optical power inserted to the interferometer. Then the following half waveplate and the PBS 2 are used to control the optical power ratio between the two arms of the interferometer. On each arm of the interferometer, the focal points of the two convex lenses must coincide with each other and the AOFS is placed at the overlapped focal point. Then, when the selected first-order diffraction beams of AOFS with different deflection angles pass the second convex lens, the propagation directions of the output beams are all parallel to the principle axis of the convex lens. The reflected beam from the following mirror can return back to PBS 2 along the same path. The quarter waveplate is used to convert the polarization direction of the retuned laser beam from vertical to horizontal or from horizontal to vertical so as to pass PBS 2 and arrive at the PD to be measured. The laser beam passes the AOFS two times and the frequency shift also doubles [16–17]. The returned two laser beams combine in PBS 2 and generate the optical excitation signal after the polarizer and the BS before the PD to be measured.

 figure: Fig. 2.

Fig. 2. Schematic diagram of the improved heterodyne system based on Michelson interferometer using AOFSs’ double-passed scheme.

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In the improved scheme, when the optical system is lined up, additional re-alignment is not needed during the frequency tuning in the measurement. Also, for the double-passed configuration of AOFSs used, the frequency modulation efficiency is also doubled. Compared with the heterodyne system based on Mach-Zehnder interferometer, the same frequency range can be covered with less AOFSs and simpler combination configurations in the improved system, which will offer better efficiency and stability in the measurement. For example, only two variable AOFSs (MT80-B30A1, MT110-B50A1, AA Opto Electronic, France) with one combination configuration as shown in Fig. 2 are needed to cover the frequency range from 500 kHz to 135 MHz in the improved heterodyne system and the same two AOFSs with three combination configurations as shown in Fig. 1(b) are needed to cover the same frequency range before.

3. Experimental results and discussion

The experimental setup shown in Fig. 3 is designed to demonstrate the feasibility of the improvement in the frequency range from 500 kHz to 135 MHz. The operation wavelength of the used laser (LMX-532S-50-COL-PP, Oxxius, France) is 532 nm. Two variable AOFSs (MT80-B30A1, MT110-B50A1, AA Opto Electronic, France) are used as shown in Fig. 2. The frequency shift ranges of the two AOFSs are 65 MHz ∼ 95 MHz and 85 MHz ∼ 135 MHz respectively, numbered as No. 1 and No. 2. There are many frequency shift settings to cover the designed measurement frequency range. In the following experiments, the frequency shift of the No. 1 AOFS is set to be fixed 85 MHz to cover the range from 500 kHz to 100 MHz and then is set to be fixed 65 MHz to cover the range from 100 MHz to 135 MHz. All the four convex lenses have the same focal length 100 mm. The output of the measured PD in response to the optical excitation signal is detected by the oscilloscope (HDO6104, LeCroy, USA). And then, the DC and AC components of the PD’s output are obtained by the spectrum analysis with the frequency resolution 25 kHz.

 figure: Fig. 3.

Fig. 3. Experimental setup for the improved heterodyne system based on Michelson interferometer using AOFSs’ double-passed scheme

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Two types of PDs commonly used in our experiments are measured in this paper, including a Si trans-impedance amplified detector (PDA10A, Thorlabs, USA) and a Si avalanche detector (APD120A/M, Thorlabs, USA). The measurement results using the heterodyne system based on Mach-Zehnder interferometer and the improved scheme based on Michelson interferometer for the PD with type PDA10A and the PD with type APD120A/M are shown in Fig. 4 and Fig. 5. The measurement results have been normalized to the data at the frequency about 500 kHz.

 figure: Fig. 4.

Fig. 4. Measurement results comparison between the heterodyne system shown in Fig. 1 and the improved scheme shown in Fig. 2 for the PD with type PDA10A.

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 figure: Fig. 5.

Fig. 5. Measurement results comparison between the heterodyne system shown in Fig. 1 and the improved scheme shown in Fig. 2 for the PD with type APD120A/M.

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As is shown in Fig. 4 and Fig. 5, the measurement results of the improved heterodyne system based on Michelson interferometer has good agreement with that obtained by the scheme based on Mach-Zehnder interferometer. And the periodic oscillation has been eliminated in the measurement results of the improved heterodyne system. Also, the measurement stability of the heterodyne system and its improved system are compared by the standard deviation of six measurements for the PD with type PDA10A. It can be seen obviously from Fig. 6 that the improved heterodyne system offers better measurement stability with smaller standard deviation of measurement results. However, it can be seen from Fig. 4 that there is a bump around 100 MHz in the frequency response measurement results obtained by the improved heterodyne system, which is not detected by the heterodyne system before. According to the analysis of the typical equivalent circuit model of the PD to be measured, the open-loop gain of the built-in trans-impedance amplifier depends on the frequency and decreases with increasing frequency. So, the transfer function of the trans-impedance amplifier is a high-order linear system and then the bump is generated in the high-frequency part of its frequency response curve.

 figure: Fig. 6.

Fig. 6. Measurement stability comparison between the heterodyne system and its improved scheme for the PD with type PDA10A.

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4. Uncertainty analysis

The main uncertainty component is the spatial mismatch between the two interfering beams before the PD to be measured, which will lead to the frequency response fluctuation of the optical excitation signal during the frequency tuning in the measurement. Other components include the fluctuation of the optical power ratio between the two interfering beams, the frequency response of the oscilloscope, data fitting and so on.

4.1 Spatial mismatch between the two interfering beams

The spatial mismatch induced by the optical misalignment means that there is an intersection angle between the propagation directions of the two interfering beams received by the PD to be measured, and for the general case, the two light spots on the active region of the PD don't coincide with each other, as shown in Fig. 7. As a result, the visibility of the optical excitation signal and then the measurement results of the PD will be significantly influenced during the frequency tuning process in the measurement. Assuming that the two interfering beams are plane waves and ignoring the diffraction effect induced by the apertures in the interferometer to simplify the calculations, the uncertainty is calculated as follows.

 figure: Fig. 7.

Fig. 7. Illustration of the spatial mismatch between the two interfering beams before the PD to be measured.

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The complex amplitude of the two interfering beams can be expressed as:

$${\vec{E}_S}(\vec{r},t) = {U_S}(\vec{r})\exp [{i({{{\vec{K}}_S} \cdot \vec{r} - {\omega_S}t + {\varphi_{S0}}} )} ],$$
$${\vec{E}_R}(\vec{r},t) = {U_R}(\vec{r})\exp [{i({{{\vec{K}}_R} \cdot \vec{r} - {\omega_R}t + {\varphi_{R0}}} )} ],$$
where ${U_S}(\vec{r})$ and ${U_R}(\vec{r})$ are the amplitude, ${\vec{K}_S}$ and ${\vec{K}_R}$ are the wave vector, ${\omega _S}$ and ${\omega _R}$ are the angular frequency, ${\varphi _{S0}}$ and ${\varphi _{R0}}$ are the initial phase.

The DC and AC component of the PD's output in response to the optical excitation signal can be expressed as:

$${V_{ou{t_ - }DC}}(f) = \eta \int_0^{2\pi } {\int_0^{{r_s}} {({U_S^2(\vec{r}) + U_R^2(\vec{r})} )rdrd\varphi } } ,$$
$${V_{ou{t_ - }AC}}(f) = 2\eta \sqrt {{A^2} + {B^2}} \cos ({\Delta \omega t - \Delta {\varphi_0} + \alpha } ),$$
where Δω is the angular frequency difference of the two interfering beams, Δφ0 is the initial phase difference of the two interfering beams, rs is the radius of the integral area on the active region of the PD, (r, φ) is the radius and azimuth of an arbitrary point on the integral surface, θ is the intersection angle between the propagation directions of the two interfering beams, and A, B, α are denoted as:
$$\tan \alpha = \frac{B}{A},$$
$${\vec{K}_R} \cdot \vec{r} - {\vec{K}_S} \cdot \vec{r} = {K_R}r\sin \theta \cos \varphi ,$$
$$A = \int_0^{2\pi } {\int_0^{{r_s}} {{U_S}(\vec{r}){U_R}(\vec{r})\cos ({{K_R}r\sin \theta \cos \varphi } )rdrd\varphi } } ,$$
$$B = \int_0^{2\pi } {\int_0^{{r_s}} {{U_S}(\vec{r}){U_R}(\vec{r})\sin ({{K_R}r\sin \theta \cos \varphi } )rdrd\varphi } } .$$
In the measurement, two interfering beams incident to the PD after a convex lens. The focused two light spots are usually much less than the active region of the PD. As shown in Fig. 7, the wave vector of one laser beam coincides with z axis, and the intersection angle between z axis and the other laser beam represents as θ, the distance between the centers of the two light spots on the active region of the PD is denoted as r0. The simulation result of the relationship among the intersection angle θ, the distance r0 and the ratio M of the AC component Vout_AC to the DC component Vout_DC is shown in Fig. 8. In the simulation, the radius of the two light spots is set to be 0.1 mm and the integral radius on the active region of the PD is 0.3 mm. The optical power ratio of the two interfering beams is set to be the typical value 0.95.

It can be seen from Fig. 8, the ratio of the AC component to the DC component of the PD’s output is significantly influenced by the intersection angle between the propagation directions of the two interfering beams and the center distance of the two light spots on the active region of the PD, which is the main uncertainty component in the frequency response measurement of the PD.

 figure: Fig. 8.

Fig. 8. The relationship among the intersection angle θ, the center distance r0 of the two light spots on the active region of the PD and the ratio M of the AC component Vout_AC to the DC component Vout_DC of the PD’s output.

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There is a screen 2 m away from the BS to monitor the spatial mismatch in the measurement. If the two light spots coincide with each other on the active region of the PD and the intersection angle between the propagation directions of the two interfering beams is 0.020°, the distance between the two light spots on the screen is 0.70 mm, which can be distinguished by human eyes. So, it is reasonable to set 0.020° as the typical value of the intersection angle. A high precision linear translation stage with the PD mounted can be used to reduce the center distance of the two light spots on its active region, the typical value of which is then set to 8 μm. In the measurement, the change range of the intersection angle can be controlled within 0.020°±0.010°. So, the relative standard uncertainty induced by the spatial mismatch ur(H1) is calculated as 2.43%.

4.2 Fluctuation of the optical power ratio between the two interfering beams

In order to make the optical excitation signal invariable, the fluctuation of the optical power ratio between the two interfering beams should be monitored and controlled. The optical power of the two interfering beams can be monitored directly by the DC component from the output of the PD to be measured. The optical power of the two interfering beams is measured alternatively with one of them blocked. If the PD to be measured can’t detect the DC component, the optical power meter placed after the BS can be used. The uncertainty component related to the optical power ratio can be expressed as:

$${H_2} = \frac{{2\sqrt {{R_I}} }}{{1 + {R_I}}},$$
where RI is the optical power ratio of the two interfering beams, which can be controlled from 0.9 to 1.0 with typical value 0.95 in the measurement. So, the standard uncertainty of the optical power ratio is 0.029. And then the calculated relative standard uncertainty ur(H2) induced by the fluctuation of the optical power ratio is 0.04%.

4.3 Frequency response of the oscilloscope

The oscilloscope with type HDO6104 and bandwidth 1 GHz is used for the data acquisition and the spectrum analysis. The used sampling rate is set to be 2.5 GS/s. The DC coupling mode with the input impedance 1MΩ is used. The relative standard uncertainty induced by the frequency response during the measurement frequency range is estimated as 0.50%.

4.4 Data fitting

The measurement can only be performed at discrete frequencies using the heterodyne system presented in this paper. The polynomial fitting is used to obtain the frequency response value at continuous frequencies and reduce the influence of noise. The uncertainty induced by the fitting procedure is calculated from the standard deviation by the method presented in [13], which can be expressed:

$$sd = \sqrt {\frac{{\sum\limits_{i = 1}^N {{{[{H({f_i}) - {H_{Fit}}({f_i})} ]}^2}} }}{{({N - {N_f}} )H_2({f_i})}}} ,$$
where H(fi) and HFit(fi) are the experimental and fitting data respectively, N is the total measurement point, Nf is the degree of freedom in the fitting procedure. So, the calculated relative standard uncertainty induced by the fitting procedure is about 0.05%.

4.5 Measurement repeatability

The standard uncertainty induced by the measurement repeatability is calculated by the standard deviation of six measurements at each frequency point. And the maximum value of the calculated standard deviation at all the measurement points is used to determine the relative standard uncertainty, which is less than 1.00%.

4.6 Other components

Other uncertainty components include the frequency measurement error, data rounding and so on. The relative standard uncertainty induced by these components is estimated as 0.20%.

4.7 Combined standard uncertainty and expanded uncertainty

The combined relative standard uncertainty is 2.7% and the expanded uncertainty is 5.4% (k = 2) as shown in Table 1.

Tables Icon

Table 1. Uncertainty budget

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5. Conclusion

The improved heterodyne system based on Michelson interferometer using double-passed AOFS configuration is presented for the frequency response measurement of PDs in ultrasonic applications. For the double-passed AOFS configuration, the same frequency range can be covered with less AOFSs and simpler combination configurations in the improved system. The demonstrated experiments are implemented in the frequency range from 500 kHz to 135 MHz. The frequency resolution 500 kHz can be obtained without any servo frequency control system. Compared with the heterodyne system and its improved scheme, the experimental results have good agreement for the same PD. And there is no periodic oscillation in the measurement results of the improved heterodyne system. Also, the measurement stability has been obviously improved for the double-passed configuration of AOFSs used. The expanded uncertainty of the improved heterodyne system is about 5.4% (k = 2).

Funding

Young Elite Scientists Sponsorship Program by China Association for Science and Technology (2017QNRC001); National Natural Science Foundation of China (NFSC) (51805506, 51505453, 51575502); National Key R&D Program of China (2017YFF0205004).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

References

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Figures (8)
Fig. 1.
Fig. 1. (a) Schematic diagram of the heterodyne system based on Mach-Zehnder interferometer using acousto-optic frequency modulation. (b) Connection configurations of AOFSs in the heterodyne interferometer. AOFS is acousto-optic frequency shifter. PBS is polarizing beam splitter. BS is non-polarizing beam splitter.

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Fig. 2.
Fig. 2. Schematic diagram of the improved heterodyne system based on Michelson interferometer using AOFSs’ double-passed scheme.

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Fig. 3.
Fig. 3. Experimental setup for the improved heterodyne system based on Michelson interferometer using AOFSs’ double-passed scheme

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Fig. 4.
Fig. 4. Measurement results comparison between the heterodyne system shown in Fig. 1 and the improved scheme shown in Fig. 2 for the PD with type PDA10A.

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Fig. 5.
Fig. 5. Measurement results comparison between the heterodyne system shown in Fig. 1 and the improved scheme shown in Fig. 2 for the PD with type APD120A/M.

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Fig. 6.
Fig. 6. Measurement stability comparison between the heterodyne system and its improved scheme for the PD with type PDA10A.

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Fig. 7.
Fig. 7. Illustration of the spatial mismatch between the two interfering beams before the PD to be measured.

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Fig. 8.
Fig. 8. The relationship among the intersection angle θ, the center distance r0 of the two light spots on the active region of the PD and the ratio M of the AC component Vout_AC to the DC component Vout_DC of the PD’s output.

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Tables (1)

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Table 1. Uncertainty budget

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Equations (11)

Equations on this page are rendered with MathJax. Learn more.

V o u t = η ( I s + I r ) [ 1 + M ( f ) 2 I s I r I s + I r cos ⁡ ( 2 π f t ) ] ,
E → S ( r → , t ) = U S ( r → ) exp ⁡ [ i ( K → S ⋅ r → − ω S t + φ S 0 ) ] ,
E → R ( r → , t ) = U R ( r → ) exp ⁡ [ i ( K → R ⋅ r → − ω R t + φ R 0 ) ] ,
V o u t − D C ( f ) = η ∫ 0 2 π ∫ 0 r s ( U S 2 ( r → ) + U R 2 ( r → ) ) r d r d φ ,
V o u t − A C ( f ) = 2 η A 2 + B 2 cos ⁡ ( Δ ω t − Δ φ 0 + α ) ,
tan ⁡ α = B A ,
K → R ⋅ r → − K → S ⋅ r → = K R r sin ⁡ θ cos ⁡ φ ,
A = ∫ 0 2 π ∫ 0 r s U S ( r → ) U R ( r → ) cos ⁡ ( K R r sin ⁡ θ cos ⁡ φ ) r d r d φ ,
B = ∫ 0 2 π ∫ 0 r s U S ( r → ) U R ( r → ) sin ⁡ ( K R r sin ⁡ θ cos ⁡ φ ) r d r d φ .
H 2 = 2 R I 1 + R I ,
s d = ∑ i = 1 N [ H ( f i ) − H F i t ( f i ) ] 2 ( N − N f ) H 2 ( f i ) ,
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