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Micro-newton strain force and temperature synchronous fiber sensor with a high Q-factor based on the quartz microbubble integrated in the capillary-taper structure

Chongbin Sun, Yi Liu, Yan Li, and Shiliang Qu

Open Access Open Access

Abstract

A micro-newton strain force and temperature synchronous fiber sensor with a high Q-factor is proposed. The sensor is based on a commercial quartz microbubble (QMB, the diameter is less than 80 µm) that is attached to the end surface of the suspending taper integrated in the hollow core fiber. The multi-beam interference and long-active-length make the sensor show both high sensitivity (0.150 nm/mN) and Q-factor (1470 based on the 3dB-bandwidth). The actual detection limit of the strain force reaches about 50 µN. The UV-cured polymer between the QMB and taper improves the temperature sensitivity. The strain force and temperature can be demodulated synchronously by using band-pass filtering and sensing matrix. The sensor can have actual application in micro-newton strain force detection as its low cost and flexible structure.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Optical fiber structures such as Mach-Zehnder interferometers (MZIs) [1,2], Fabry-Perot interferometers (FPIs) [3,4], fiber brag gratings (FBGs) [5,6] have been widely used for strain force sensing. Among these kinds of structures, sensors based on FPIs are most popular because of their advantages such as small size, good durability, and ease to fabrication. Air cavities based on different shapes such as cylinder [7,8], oblate spheroid [9,10], cuboid [11] and expanded bubble [12–14] have been fabricated between single mode fibers (SMFs) to form fiber in-line FPIs. However, their strain force sensitivities are generally less than 0.04 nm/mN due to the limitation of the interference principle [9,11].

To further improve the sensitivity, a kind of long-active-length FP strain force sensor based on the suspending taper in the fiber has been reported [15–18]. The interference length Li is separate from the cavity active-length La, and the strain force sensitivity is proportional to the ratio of La-to-Li. When the active-length is increased to 1360 µm by using capillary-taper structure, the strain force sensitivity reaches as high as 0.841 nm/mN [18]. However, as the FP interference length of these structures is only less than 10µm, the free spectrum range (FSR) is more than 200 nm. Meanwhile, as the large reflection loss at the taper end face, the interference quality degenerates. As a result, the Q-factor of the interference valley based on 3dB-bandwidth is generally less than 70 for this kind of sensor. In fact, the Q-factor and sensitivity both determinate the detection limit (DL) [19], which is the key parameter to evaluate the actual performance of the sensor. The sensitivities of the above reported sensors are improved by sacrificing the Q-factor values, which influences the actual DL. There needs to be an effective way to improve the sensitivity and Q-factor at the same time. For common fiber in-line FPIs, multi-beam interferences based on the cascading cavities are often employed to increase the Q-factor values. For example, Deng et al. proposed a high-temperature sensor by use of cascaded FPI based on four in-fiber mirrors. The Q-factor of their sensor is more than 1800, which effectively improves the DL value [20]. Therefore, the mothed of multi-beam interference can be introduced to the strain force sensor with a long-active-length.

Besides, the strain force sensor is usually used in the environment with a temperature variation, where the strain force and temperature are needed to be detected at the same time. A temperature sensor such as FBG can be combined with the long-active-length strain force sensor [21]. However, the reported FBGs and long-active-length strain force sensors show low temperature sensitivities (generally less than 30 pm/°C) [21] due to the low thermo-optical coefficient (TOC) and thermal expansion coefficient (TEC) of quartz material.

In this paper, we proposed a long-active-length fiber strain force sensor with both high sensitivity and high Q-factor. A commercial quartz microbubble (QMB, the diameter is less than 80 µm) is attached to the end surface of the suspending taper by the UV polymer. The suspending taper is fabricated by arc discharge fiber tapering, and its length can be more than 3000 µm. The taper is then plugged into a section of HCF that is spliced to the SMF in advance. The joint between the taper and HCF is sealed by arc discharge in a Fiber Fusion Splicer. The end face of SMF and two side faces of the QMB form three reflection mirrors. Owing to the multi-beam interference, the Q-factor of the interference valley based on the 3dB-bandwidth is improved to be 1470. Meanwhile, owing to the long active length, the strain force sensitivity reaches 0.150 nm/mN. As a result, the actual DL of the strain force is about 50 µN. Besides, two different interference fringes can be extracted individually by using band-pass filtering. They are both sensitive to strain force and temperature. The two parameters can be demodulated synchronously by using sensing matrix with the maximum sensitivities of 0.667 nm/mN and −0.251 nm/°C.

2. Theory

The schematic diagram of the sensor is shown in Figs. 1(a) and 1(b). The length between the SMF end face and left-face of the QMB is L1. The diameter of the QMB is L2. The length of HCF is L. The side wall thickness of the QMB is about 200nm, which can be ignored. There is a section of UV-cured polymer whose length is L3 between the QMB and taper.

 figure: Fig. 1.

Fig. 1. (a)Three-dimensional model diagram of the sensor. (b) The diagram of the sectional view of the sensor. (c)Simulated spectrum of two-beam FPI. (d)Simulated spectrum of three-beam FPI.

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The previously reported strain sensors are usually based on two-beam FPI. We assume that the first reflection face is the flat end-face of the SMF and the second reflection face is the interface between air and quartz. The length of the interference length is assumed to be L1. According to the FP interference theory [22], the reflection spectrum can be expressed as

$$R = {\left|{\sqrt {{R_1}} + (1 - \alpha )(1 - {R_1})\sqrt {1 - C} \sqrt {{R_2}} {e^{ - j4\pi \frac{{{L_1}}}{\lambda } + j\pi }}} \right|^2},$$
where α is the transmission loss of the light spreading in the FP cavity, R1 and R2 are the reflectivity of two reflection faces, and C is the reflection loss at the second reflection face. By putting α=0.01, R1 = 0.0358, R2 = 0.0331, C = 0.7, and L1 = 14 µm into Eq. (1), we get the simulated reflection pattern as shown in Fig. 1(c). The Q-factor based on the 3dB-bandwidth is about 74.

For our structure, as the refractive index (RI) of the UV polymer is close to the quartz material, the reflections at the interfaces QMB-UV polymer and UV polymer-taper can be ignored. The end-face of SMF and two side faces of the QMB form three reflection mirrors. The sensor in this paper is mainly based on three-beam FPI. According to the theory of three-beam FPI [22], the reflection spectrum can be expressed as

$$R = {\left|\begin{array}{l} \sqrt {{R_1}} + \sqrt {1 - {C_1}} \sqrt {{R_2}} (1 - \alpha )(1 - {R_1}){e^{ - \frac{{j{L_1}n4\pi }}{\lambda } + j\pi }}\\ + \sqrt {1 - {C_2}} \sqrt {{R_3}} (1 - \alpha )(1 - {R_1})(1 - {R_2}){e^{ - j({L_1}n + {L_2}n)4\pi /\lambda + j\pi }} \end{array} \right|^2},$$
where R1, R2 and R3 are the reflectivity of the three reflection mirrors, respectively, and R1 = 0.0358, R2= R3 = 0.0331. C1 and C2 are the reflection loss at two side faces of the QMB due to the curved shape. n is the RI of air, and n = 1.00. It is assumed that α=0.01, C1 = 0.7, C2 = 0.9, L1 = 14 µm and L2 = 75 µm, the simulated reflection pattern of three-beam FPI is shown in Fig. 1(d) according to Eq. (2). The Q-factor of three-beam FPI is as high as about 770, which is about ten times higher than that of two-beam FPI.

There are generally three types of interference fringes superimposed in the reflection spectrum of the common three-beam FPI [23], which corresponds to three FP cavities formed by two of three reflection mirrors. As the reflectivity of the two side faces of the QMB is much smaller than that of the SMF due to the transmission and reflection losses, the interference fringe formed by the cavity between the two side faces of the QMB can be ignored. Therefore, there are mainly two types of interference fringes superimposed in the reflection spectrum corresponds to the cavity length L1 and L1 + L2. The central wavelength values of the mth-order interference valley for the first-type interference fringe and m´th-order interference valley for the second-type interference fringe can be expressed as

$${\lambda _m} = \frac{{2n{L_1}}}{m},$$
$${\lambda _{m^{\prime}}} = \frac{{2n({{L_1} + {L_2}} )}}{{m^{\prime}}}.$$
As the two types of interference fringes have different FSR values, they can be extracted individually by using band-pass filtering.

When the sensor is stretched, the central wavelength shift sensitivity of the first-type interference fringe can be expressed as

$$\frac{{\partial {\lambda _m}}}{{\partial F}} = \frac{{{\lambda _m}}}{{{L_1}}}\frac{{\partial {L_1}}}{{\partial F}}.$$
The Young’s modulus E is expressed as
$$E = \frac{{\partial F/A}}{{\partial L/L}},$$
where, A is the cross-sectional area of optical fiber. According to the expression of Young’s modulus, Eq. (5) can be rewritten as
$$\frac{{\partial {\lambda _m}}}{{\partial F}} = \frac{{{\lambda _m}}}{{AE}}\frac{L}{{{L_1}}}.$$
In Eq. (7), A and E are about 0. 01 mm2 and 72. 9 GPa respectively, so that the sensitivity is in direct proportion to the value of L/L1. To obtain higher sensitivity, we can increase L and decrease L1.

In the same way, the central wavelength shift sensitivity of the second-type interference fringe can be expressed as

$$\frac{{\partial {\lambda _{m^{\prime}}}}}{{\partial F}} = \frac{{{\lambda _{m^{\prime}}}}}{{AE}}\frac{L}{{{L_1} + {L_2}}}.$$
Comparing Eq. (8) with Eq. (7) we can see, the central wavelength shift sensitivity of the second-type interference fringe is smaller than that of the first-type interference fringe. When L1 = 14 µm, L2 = 75 µm, and L = 4500 µm, the sensitivities of fringe I and fringe II can be calculated to be 0.683 nm/mN and 0.107 nm/mN, respectively. As the final interference fringe in the spectrum is the superposition of two types of interference fringes, the actual sensitivity value is between the two sensitivity values calculated by Eq. (7) and Eq. (8) respectively.

When the temperature changes, the UV polymer between the QMB and taper will expand or shrink, causing L1 and L1 + L2 changing. The temperature sensitivity for the first-type interference fringe can be expressed as

$$\frac{{\partial {\lambda _m}}}{{\partial T}} ={-} {\lambda _m}\alpha \frac{{{L_3}}}{{{L_1}}},$$
where, α=∂L3/(L3∂T) is the TEC of the UV polymer. In the same way, the temperature sensitivity for the second-type interference fringe can be expressed as
$$\frac{{\partial {\lambda _{m^{\prime}}}}}{{\partial T}} ={-} {\lambda _{m^{\prime}}}\alpha \frac{{{L_3}}}{{{L_1} + {L_2}}}.$$
From Eqs. (9) and (10), we can see that the temperature sensitivity depends on the thickness and the TEC of UV polymer. α of the UV polymer adopted in the sensor is about 2.4×10−4/°C [24]. The calculated sensitivity of fringe I and fringe II is −0.266 nm/°C and −0.042nm/°C when L3 = 10 µm.

3. Fabrication

The fabrication process includes three steps as shown in Fig. 2. In the first step, the structure consisting of a section of SMF and a short section of HCF (SMF-HCF) was used to catch a QMB as shown in Fig. 2(a). The HCF is made by drawing the quartz capillary by Rui-feng chromatographic device company. A few QMBs purchased from 3M Corporation with a type of GB-K1 were put on a glass slide. SMF-HCF was controlled by a five-dimensional displacement platform that consists of three translation axes and two rotary axes driven by five manual spiral micrometers. The opening of the SMF-HCF was moved to reach the QMB with a proper size by the adjusting three translation axes, and then the SMF-HCF was controlled to be perpendicular to the upper surface of the glass slide by adjusting two rotary axes. The QMB was caught in the HCF and it would attach on the inner wall of the HCF due to Van Der Waals force. In the second step, a fiber taper was prepared beforehand by using the two-step arc discharge fiber tapering method [14], and then some UV polymer (NOA81) was dipped on the end surface of the fiber taper. The fiber taper was placed on a displacement platform and it was adjusted to make the end surface of fiber taper touch QMB as shown in Fig. 2(b). Once the location of the fiber taper and QMB was adjusted well, the structure was exposed in UV light (365nm) until the UV polymer was absolutely solidified. In the last step, the taper was then plugged into a section of HCF that was spliced to the SMF in advance as Fig. 2(c). This step was operated in a Fiber Fusion Splicer, and the joint between the taper and HCF is sealed by arc discharge. It’s worth noting that the QMB will be destroyed by high temperature if it is too close from the discharging point. Therefore, the length of the fiber taper should be more than 3mm. The image of the final fabricated fiber sensor is shown in Figs. 2(d) and 2(e). The diameter of the QMB is about 75 µm and the diameter of the fiber taper is about 60 µm. The fiber taper is strong enough to support the QMB. The thickness of the polymer is about 10 µm. The inner and outer diameter of HCF is 80 µm and 125 µm respectively. The length of HCF is about 4500 µm and the length of cavity 1 is about 14 µm.

 figure: Fig. 2.

Fig. 2. (a) The QMB is caught in HCF. (b)The QMB is fixed on the end of the taper. (c) The joint between the taper and HCF is sealed by arc discharge. (d) The micrograph of the QMB in the sensor. (e) The micrograph of the splicing position.

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The measured reflection spectrum of the sensor is shown in Fig. 3(a), and the fringe pattern is similar with the simulation result shown in Fig. 1(d). The Q-factor value of the interference valley near the wavelength 1430 nm based on the 3dB-bandwidth is as high as 622 owing to the superposition of interference fringes. Figure 3 (b) is the fast Fourier transform (FFT) result of the spectrum. There are two main peaks in the frequency spectrum, which corresponds to the interference length L1 and L1 + L2. Their spatial frequencies are 0.015/nm and 0.080/nm. The two types of interference fringes are extracted individually by using band-pass filtering. For the second dominant peak (Peak II), there is a small peak near its left, which corresponds to the interference accrued in the QMB. It will influence the filtering result if the bandwidth for the band-pass filtering is larger than 0.01/nm. Therefore, the bandwidth is set as 0.009/nm. The spectrum results are shown in Figs. 3(c) and 3(d). The fringe I with larger FSR results from the interference cavity with the length of L1, and the fringe II with smaller FSR results from the interference cavity with the length of L1 + L2. The two kinds of fringes can also be achieved by using Inverse Fourier Transform (IFT) method based on the frequency values of the two dominant peaks, which can avoid the influence of the bandwidth selection on band-pass filtering.

 figure: Fig. 3.

Fig. 3. (a) Original spectrum of the sensor. (b) FFT result of the spectrum. (c) Spectrum after band-pass filtering at peak I. (d) Spectrum after band-pass filtering at peak II.

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4. Experimental results and discussions

Strain experiments were carried out with the sensor. Strain force was applied by loading objects on the sensor. The value of strain force is increased by increasing the mass of the objects that is measured by an Electronic Scale. At each strain force, the reflection spectrum was recorded, and two types of interference fringes (I and II) were extracted from the original spectrum by using band-pass filtering. The original spectrum and the extracted two types of interference fringes at different strain force conditions are shown in Figs. 4(a)–4(c). Three spectra all redshift with the increase of the strain force. The interference valley near wavelength 1430 nm was chosen to show its wavelength shift as the variation of the strain force for three spectra. The linear fitted results are shown in Fig. 4(d). The sensitivities of fringe I and II are 0.667 nm/mN and 0.092 nm/mN, which are basically consistent with the calculated theory values (0.683 nm/mN and 0.107 nm/mN). The little difference between the calculated and experimental sensitivity values may result from the error of the measurement of the interference length. The sensitivity of the superimposed spectrum is 0.124 nm/mN, which is between the sensitivities of fringe I and II agreeing with the theoretical analysis. The three spectra all show good linearities with values of 99.9%, 99.2% and 98.5%, respectively.

 figure: Fig. 4.

Fig. 4. (a) The change of the fringe I with strain force. (b) The change of the fringe II with strain force. (c) The change of the original spectrum with strain force. (d) The linear relations between the wavelength shift and strain force.

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Temperature experiment was also carried out by heating the sensor in the water bath. The sensor was heated from 30°C to 70°C with a step of 5℃. The original spectrum and the extracted fringes I and II all redshift with the increase of the temperature as shown in Figs. 5(a)–5(c). The linear fitted results between the wavelength shift and temperature are shown in Fig. 5(d). The temperature sensitivity of the fringe I is −0.251 nm/℃ with a linearity of 99.5%, and the temperature sensitivity of the fringe II is −0.038 nm/℃ with a linearity of 98.3%. The calculated theory values for fringes I and II are −0.266 nm/°C and −0.042nm/°C. The calculated and experimental sensitivity values also coincide with each other. The sensitivity of the superimposed spectrum is −0.049 nm/°C with a linearity of 98.2%. Owing to the long active length and large TEC of the UV polymer, the sensor shows both high sensitivities in strain force and temperature sensing. As fringes I and II can be extracted from the single sensor at the same time, sensitive synchronous detection of strain force and temperature can be achieved by using the follow sensing matrix based on the fringes I and II:

$$\left[ {\begin{array}{c} {\varDelta F}\\ {\varDelta T} \end{array}} \right] = {\left[ {\begin{array}{cc} {{S_{1 - F}}}&{{S_{1 - T}}}\\ {{S_{2 - F}}}&{{S_{2 - T}}} \end{array}} \right]^{ - 1}}\left[ {\begin{array}{c} {\varDelta {\lambda^\textrm{{I} }}}\\ {\varDelta {\lambda^\textrm{{II} }}} \end{array}} \right],$$
where, S1-F and S1-T are strain force and temperature sensitivities of the fringe I, and S2-F and S2-T are strain force and temperature sensitivities of the fringe II. The binary linear equations are effective only if these four sensitivity values are always constant, which means that these four sensitivity values should be all independent on the strain force and temperature. As the experimental test range in this paper is small, the influence of the strain force and temperature on these four sensitivities can be ignored. Then we discuss the detection limit (DL) based on the sensing matrix. We assume that the smallest wavelength shift value that can be distinguished by OSA is R (unit nm), the small strain force and temperature variations that cannot be distinguished by OSA satisfy the inequalities
$$\left\{ {\begin{array}{c} {|{{S_{1 - F}}\varDelta F + {S_{1 - T}}\varDelta T} |< R}\\ {|{{S_{2 - F}}\varDelta F + {S_{2 - T}}\varDelta T} |< R} \end{array}} \right.$$

 figure: Fig. 5.

Fig. 5. (a) The change of the fringe I with temperature. (b) The change of the fringe II with temperature. (c) The change of the original spectrum with temperature. (d) The linear relations between the wavelength shift and temperature.

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Here, S1-F =0.667 nm/mN, S1-T =−0.251 nm/°C, S2-F =0.092 nm/mN, and S2-T =−0.038 nm/°C. The solutions of the inequalities can be shown in the coordinate system by using the linear programming. The result is shown in Fig. 6. The variations of strain force and temperature can only be detected when they are beyond the area enclosed by the red lines. Smaller area shown by using the linear programming results in smaller value of DL. From Fig. 6 we can also see that, the DL of the strain force is influenced by that of the temperature. If the environmental temperature fluctuates in a large range, the DL of the strain force will degenerate. If the values of R and temperature fluctuation is known, the DL value can be obtained from Fig. 6.

 figure: Fig. 6.

Fig. 6. The linear programming region to show the DL.

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As the Q-factor value of the interference valley in the original superimposed spectrum near the wavelength 1430nm is 622, the sensor is expected to show better wavelength demodulation resolution and smaller wavelength shift noise. Therefore, it has additional advantages in small strain force detection. If the synchronous detection of strain force and temperature is not required, the QMB can be fixed on the end face of the taper by using CO2 laser heating [25]. As the UV polymer is not used, the influence of the temperature crosstalk on strain force detection can be ignored due to the small TEC (10−7/°C) of quartz material. To show the advantage of the sensor in small strain force detection, we fabricated another sensor (L1 = 8 µm, L = 3500 mm) to further improve the strain force sensitivity and the Q-factor. The measured reflection spectrum is shown in Fig. 7(a). As the transmission and reflection losses are reduced as the decrease of L1, the fringe contrast of the spectrum near the wavelength 1470 nm reaches about 19 dB. As a result, the Q-factor based on the 3dB-bandwidth reaches as high as 1633. The strain force response of the sensor was also tested, the sensitivity results are shown in Fig. 7(b). The strain force sensitivities of fringe I and II are 0.991 nm/mN and 0.089 nm/mN. The strain force sensitivity of the chosen valley with high Q-factor in the actual reflection spectrum is 0.150 nm/mN.

 figure: Fig. 7.

Fig. 7. (a) The spectrum of another sensor. (b) The strain force sensitivities of the fringe I, fringe II and the original spectrum.

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To test the actual DL in small strain force detection, an extremely light object with the mass of 0. 02g (corresponding to the strain force of 196 µN) was loaded on the sensor and then removed for three cycles in constant room temperature. In each step, the reflection spectrum of the sensor was recorded repeatedly for 20 times at an interval of 2 s. The central wavelength value of the chosen valley with high Q-factor is measured by using Lorentz fitting of the valley. The wavelength shift values in the whole test process are shown in Fig. 8(a). From the result we can see, the sensor can easily show the small strain force change of 196 µN. The measured wavelength errors at fixed strain force for three times are shown in Fig. 8(b). The wavelength variation is about 0.008 nm owing to the large Q-factor value [19]. By putting this value and sensitivity 0.150 nm/mN into DL = R/S, DL can be measured to be 53 µN.

 figure: Fig. 8.

Fig. 8. (a) The “ladder-mode” experiments: the sensor was loaded with 0 g and 0.02 g for three cycles. (b) The measured wavelength errors at fixed strain force for three times.

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The Q-factor and sensitivity of the proposed sensor is compared with other reported strain force sensors as shown in Table 1. Our sensor shows large Q-factor and high sensitivity at the same time. Besides, synchronous measurement of strain force and temperature can be achieved by using the single sensor, which further increases the competitiveness of the sensor in actual application.

Tables Icon

Table 1. Compared with Other strain force Sensors

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5. Conclusion

A suspending taper and QMB was integrated in the HCF and a long-active-length FP strain force sensor is achieved. The suspending taper structure can effectively improve the strain fore sensitivity. Meanwhile, the three reflection faces increase the Q-factor value of the interference fringe, and the DL of about 50 µN is realized for the proposed strain force sensor. Though the QMB can be fixed on the end-face of the taper by using CO2 laser heating, UV-cured polymer can be used between the QMB and taper to achieve sensitive temperature detection at the same time. The strain force and temperature can be demodulated synchronously by using band-pass filtering and sensing matrix.

Funding

National Natural Science Foundation of China (11874010, 11874133); Natural Science Foundation of Shandong Province (ZR2021MF111).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (8)
Fig. 1.
Fig. 1. (a)Three-dimensional model diagram of the sensor. (b) The diagram of the sectional view of the sensor. (c)Simulated spectrum of two-beam FPI. (d)Simulated spectrum of three-beam FPI.

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Fig. 2.
Fig. 2. (a) The QMB is caught in HCF. (b)The QMB is fixed on the end of the taper. (c) The joint between the taper and HCF is sealed by arc discharge. (d) The micrograph of the QMB in the sensor. (e) The micrograph of the splicing position.

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Fig. 3.
Fig. 3. (a) Original spectrum of the sensor. (b) FFT result of the spectrum. (c) Spectrum after band-pass filtering at peak I. (d) Spectrum after band-pass filtering at peak II.

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Fig. 4.
Fig. 4. (a) The change of the fringe I with strain force. (b) The change of the fringe II with strain force. (c) The change of the original spectrum with strain force. (d) The linear relations between the wavelength shift and strain force.

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Fig. 5.
Fig. 5. (a) The change of the fringe I with temperature. (b) The change of the fringe II with temperature. (c) The change of the original spectrum with temperature. (d) The linear relations between the wavelength shift and temperature.

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Fig. 6.
Fig. 6. The linear programming region to show the DL.

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Fig. 7.
Fig. 7. (a) The spectrum of another sensor. (b) The strain force sensitivities of the fringe I, fringe II and the original spectrum.

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Fig. 8.
Fig. 8. (a) The “ladder-mode” experiments: the sensor was loaded with 0 g and 0.02 g for three cycles. (b) The measured wavelength errors at fixed strain force for three times.

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Tables (1)

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Table 1. Compared with Other strain force Sensors

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Equations (12)

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

R = | R 1 + ( 1 − α ) ( 1 − R 1 ) 1 − C R 2 e − j 4 π L 1 λ + j π | 2 ,
R = | R 1 + 1 − C 1 R 2 ( 1 − α ) ( 1 − R 1 ) e − j L 1 n 4 π λ + j π + 1 − C 2 R 3 ( 1 − α ) ( 1 − R 1 ) ( 1 − R 2 ) e − j ( L 1 n + L 2 n ) 4 π / λ + j π | 2 ,
λ m = 2 n L 1 m ,
λ m ′ = 2 n ( L 1 + L 2 ) m ′ .
∂ λ m ∂ F = λ m L 1 ∂ L 1 ∂ F .
E = ∂ F / A ∂ L / L ,
∂ λ m ∂ F = λ m A E L L 1 .
∂ λ m ′ ∂ F = λ m ′ A E L L 1 + L 2 .
∂ λ m ∂ T = − λ m α L 3 L 1 ,
∂ λ m ′ ∂ T = − λ m ′ α L 3 L 1 + L 2 .
[ Δ F Δ T ] = [ S 1 − F S 1 − T S 2 − F S 2 − T ] − 1 [ Δ λ {I}  Δ λ {II}  ] ,
{ | S 1 − F Δ F + S 1 − T Δ T | < R | S 2 − F Δ F + S 2 − T Δ T | < R
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