Axial clearance measurement method based on wavelength division multiplexing with all-fiber microwave photonic mixing

Zhenxin Yu; Fajie Duan; Xiao Fu; Guangyue Niu; Ruijia Bao; Jingxin Wu

doi:10.1364/OE.516498

1. Introduction

The operating parameters of large rotating machinery such as aero-engines, gas turbines, and steam turbines significantly influence the performance of major equipment such as aircraft and ship [1–8]. Notably, the axial clearance between the rotating and stationary parts of rotating machinery is critical, impacting engine efficiency and operational safety [4–8]. Minimizing axial clearance is essential for enhancing engine operation efficiency. However, excessively small axial clearances can raise the likelihood of safety incidents. In order to realize the effective monitoring of axial clearance, high accuracy online measurement of axial clearance is indispensable, which also provides data support for further realizing active clearance control of large rotating machinery [9,10].

However, the measurement of axial clearance necessitates installing the sensor probe inside rotating machinery, presenting significant challenges. Firstly, the narrow space inside the machinery restricts the size of the sensor probe, while a large axial clearance measurement range is often required, leading to an inherent contradiction. Secondly, the complex inner mechanical structure results in a lengthy and convoluted path for the sensing signals, requiring the use of long, thin, and flexible transmission lines. Furthermore, the high and variable internal temperatures during machinery operation demand careful consideration of temperature-induced drift in the long-distance transmission of measurement signals. These challenges have rendered many previously proposed methods for clearance measurement ineffective. These methods include the discharge probe method, the eddy current method, the capacitance method, and the optical fiber reflection intensity method. For instance, the discharge probe method [11] has a limited maximum measurement range, which is considered unsatisfactory, and it is not suitable for narrow measurement space. The eddy current method [12] is constrained to operate below the Curie temperature, making it inapplicable in high-temperature environments. The capacitance method [13,14], when tailored for axial clearance measurement, results in an excessively large sensor probe. Moreover, long-distance transmission cables increase parasitic capacitance, undermining its effectiveness. Lastly, the optical fiber reflection intensity method [15,16] is limited by its narrow measurement range and the measurement signal is prone to fluctuations in light intensity.

In recent years, to overcome the aforementioned challenges in axial clearance measurement, researchers have developed some new methods. From 2020 to 2023, some research has focused on frequency-swept interferometry-based axial clearance measurement methods [17–21]. These methods utilize all-fiber optical path structures, featuring small-sized probes that are suitable for the narrow space within rotating machinery. However, the measurement speed is constrained by the scanning rate of the light source. Furthermore, Doppler error significantly affects this method, necessitating intricate compensation paths or algorithms to correct this error [22,23]. This necessity not only adds to the system complexity but also further restricts the measurement speed. To address the limitations, an axial clearance measurement method based on dispersion interferometry has been proposed [24]. Essentially, like the frequency-swept interferometry approach, this method also belongs to the coherent interferometric measurement techniques. The difference is the use of a pulse-type broadband light source that emits the broadband light onto the axial end face of the rotor, without the need for periodic scanning, thus eliminating the impact of Doppler errors during scanning periods. However, both coherent interferometric methods share a common limitation: the intensity and contrast of the interferometric signal are heavily influenced by the ratio of intensities between the measurement and reference light. As the rotor end face rotates, the intensity of the diffusely reflected measurement light varies, often being weak, which substantially affects the interferometric signal's intensity and contrast. Consequently, both methods impose stringent requirements on the smoothness of the axial surface. It is crucial to precisely adjust the angle of the probe to align its end face as parallel as possible to the rotor's end face. This limitation becomes more pronounced when attempting to achieve a large range of axial clearance measurements.

In 2021, A heterodyne-based microwave axial clearance measurement method was proposed [25]. The measurement and reference signal are independent, highlighting the significant advantages in achieving large measurement range. It can measure axial clearances up to 18.5 mm with a sensor diameter of 9 mm. However, this method encounters limitations due to nonlinear responses in the microwave components. Additionally, a notable issue is the large divergence angle of the microwave sensing probe, which diminishes the signal-to-noise ratio for remote axial clearance measurements. As a result, the method's accuracy is constrained. In early 2023, an axial clearance measurement method based on all-fiber microwave photonic mixing was introduced [26]. This technique effectively merges the extensive measurement range characteristic of microwave methods with the high accuracy of optical methods. A key feature is its ability to perform down-conversion in the optical domain, thereby eliminating the need for processing microwave electrical signals. This advancement significantly reduces electromagnetic interference among microwave signals and circumvents the impact of nonlinear microwave components on the system [27–30], enhancing measurement accuracy. It achieves a measurement accuracy of better than 10.5µm within a 20 mm range. Furthermore, the method employs an all-fiber optical path structure with a probe diameter of 2.78 mm, which is suitable for narrow space. However, the measurement and reference path of the method are different, making it difficult to compensate for the drift in the measurement signal transmission path caused by the internal temperature change in rotating machinery.

In this paper, a large measurement range, high accuracy and low drift axial clearance measurement method based on wavelength division multiplexing (WDM) with all-fiber microwave photonic mixing is proposed. The measurement system employs light of different wavelengths for the measurement and reference light. Through WDM, the measurement and reference light are transmitted, modulated, and down-converted along a common fiber optical path. This approach effectively mitigates axial clearance drift induced by temperature changes on the measurement light transmission path. Based on the thermo-optic effect of optical fiber material, a comprehensive theoretical model of axial clearance drift determined by wavelength and temperature is developed. In addition, the optical design for a compact optical bandpass filter (OBPF) fiber probe is proposed, aimed at efficiently separating the measurement and reference light at the probe. The performance of the measurement system is validated through simulations and experiments. The effectiveness of this method in mitigating temperature-induced drift is further substantiated through high-temperature drift and comparative experiments.

2. Principle

2.1 Principle of axial clearance measurement based on WDM with microwave photonic mixing

The schematic diagram of the axial clearance measurement system based on WDM with All-fiber microwave photonic mixing is shown in Fig. 1.

Fig. 1. Schematic diagram of the measurement system structure. PFL: polarization-maintaining fiber-coupled laser, CWDM: coarse wavelength division multiplexer, EOM: Mach-Zehnder electro-optic intensity modulator, HP EDFA: high-power erbium-doped fiber amplifier, OBPF probe: optical bandpass filter fiber probe, EDFA: small signal erbium-doped fiber amplifier, PD: photodetector, DAQ: data acquisition card, RF: microwave signal, MBC: modulator bias controller, PLL: phase-locked loop frequency synthesizer with integrated VCO, PA: microwave power amplifier, PC: personal computer.

Download Full Size | PDF

RF1 and RF2, microwave signals generated by microwave signal synthesis module, with an intermediate frequency difference, are applied to EOM #1 and EOM #2, respectively. Light emitted by PFL #1 and PFL #2 enters the two corresponding input terminals of the coarse wavelength division multiplexer (CWDM #1), followed by intensity modulation via RF1 through EOM #1. The initially modulated light, amplified by HP EDFA #1, traverses a circulator to reach the optical bandpass filter fiber probe (OBPF probe) inside the rotating machinery. The OBPF probe incorporates an optical fused-silica window at its front end, coated with an optical bandpass filter film as depicted in Fig. 4. This probe distinctively interacts with light of two different wavelengths: transmitting one as measurement light and reflecting the other as reference light. Emitted from the OBPF probe, the measurement light, focused by a lens within the probe, travels through the axial clearance space, illuminating the rotor's axial end face. The diffusely reflected measurement light, along with directly reflected reference light, is recaptured by the OBPF probe and re-enters the same optical fiber. This combined light, passing through the circulator, is amplified by EDFA #2 and undergoes a second intensity modulation via RF2 through EOM #2. The doubly modulated light, upon reaching common input of CWDM #2, is separated into its component wavelengths. Finally, distinct wavelengths of measurement and reference light are received and photoelectrically converted into corresponding signals by PD #1 and PD #2.

The wavelengths of light emitted by PFL #1 and PFL #2 are denoted as ${\lambda _1}$ and ${\lambda _2}$. Light of wavelength ${\lambda _1}$ serves as the measurement light, while ${\lambda _2}$ functions as the reference light. Upon modulation by EOM #1, the initial phases of the microwave signals carried by the measurement and reference light are represented as ${\varphi _{m0}}$ and ${\varphi _{r0}}$. As both the measurement and reference light travel through the same optical fiber path post-modulation and prior to reaching CWDM #2, their distinct wavelengths result in different refractive indices within the optical fiber. This discrepancy leads to variations in optical path lengths, thereby inducing distinct phase changes for each type of light. Consequently, the overall phase differences for the microwave signals carried by the measurement and reference light in the optical fiber path are recorded as ${\varphi _m}$ and ${\varphi _r}$. Additionally, the phase difference attributed to the round-trip propagation of the measurement light-carried microwave signal within the axial clearance space, situated between the OBPF probe and the rotor end face, is denoted as ${\varphi _d}$.

The phase difference $\mathrm{\Delta }\varphi $ between the measurement signal and the reference signal can be expressed as

(1)$$\Delta \varphi = \Delta {\varphi _0} + \Delta {\varphi _d} = ({\varphi _{m0}} + {\varphi _m} - {\varphi _{r0}} - {\varphi _r}) + {\varphi _d}$$

Therefore, utilizing the principle of microwave photonic mixing phase difference measurement, the value of axial clearance can be mathematically represented as

(2)$$d = \frac{{[{\Delta \varphi - ({\varphi_{m0}} + {\varphi_m} - {\varphi_{r0}} - {\varphi_r})} ]c}}{{4\pi {n_{air}}{f_M}}}$$

where ${n_{air}}$ is the refractive index of air. c denotes the speed of light in a vacuum. ${f_M}$ signifies the frequency of light intensity modulation.

In order to solve the phase difference of the microwave signals, a down-conversion process is required, which is executed through all-fiber microwave photonic mixing within the optical domain [31,32]. Leveraging the attributes of the MZM and applying the Jacobi-Anger expansion, the post-modulation intensities of the measurement light and reference light through EOM #1 can be mathematically formulated as

(3)$$\scalebox{0.95}{$\displaystyle \left\{ \begin{array}{@{}l@{}} {I_{m0}}(t )= {A_{m0}} + {A_{m0}}\left\{ \begin{array}{@{}l@{}} \cos \left( {\frac{{\pi {U_{DC1}}}}{{{V_{\pi m1,{\kern 1pt} {\kern 1pt} DC}}}}} \right)\left\{ {{J_0}\left( {\frac{{\pi {A_{M1}}}}{{{V_{\pi m1,{\kern 1pt} {\kern 1pt} RF{\kern 1pt} }}}}} \right) + 2\sum\limits_{m = 1}^\infty {{J_{2m}}\left( {\frac{{\pi {A_{M1}}}}{{{V_{\pi m1,{\kern 1pt} {\kern 1pt} RF{\kern 1pt} }}}}} \right)\cos [{2m({2\pi {f_M}t + {\varphi_1}} )} ]} } \right\}\\ + \sin \left( {\frac{{\pi {U_{DC1}}}}{{{V_{\pi m1,{\kern 1pt} {\kern 1pt} DC}}}}} \right)\left\{ {2\sum\limits_{m = 1}^\infty {{J_{2m - 1}}\left( {\frac{{\pi {A_{M1}}}}{{{V_{\pi m1,{\kern 1pt} {\kern 1pt} RF{\kern 1pt} }}}}} \right)\sin [{({2m - 1} )\cdot ({2\pi {f_M}t + {\varphi_1}} )} ]} } \right\} \end{array} \right\}{\kern 1pt} {\kern 1pt} \\ {I_{r0}}(t )= {A_{r0}} + {A_{r0}}\left\{ \begin{array}{@{}l@{}} \cos \left( {\frac{{\pi {U_{DC1}}}}{{{V_{\pi r1,{\kern 1pt} {\kern 1pt} DC}}}}} \right)\left\{ {{J_0}\left( {\frac{{\pi {A_{M1}}}}{{{V_{\pi r1,{\kern 1pt} {\kern 1pt} RF{\kern 1pt} }}}}} \right) + 2\sum\limits_{m = 1}^\infty {{J_{2m}}\left( {\frac{{\pi {A_{M1}}}}{{{V_{\pi r1,{\kern 1pt} {\kern 1pt} RF{\kern 1pt} }}}}} \right)\cos [{2m({2\pi {f_M}t + {\varphi_1}} )} ]} } \right\}\\ + \sin \left( {\frac{{\pi {U_{DC1}}}}{{{V_{\pi r1,{\kern 1pt} {\kern 1pt} DC}}}}} \right)\left\{ {2\sum\limits_{m = 1}^\infty {{J_{2m - 1}}\left( {\frac{{\pi {A_{M1}}}}{{{V_{\pi r1,{\kern 1pt} {\kern 1pt} RF{\kern 1pt} }}}}} \right)\sin [{({2m - 1} )\cdot ({2\pi {f_M}t + {\varphi_1}} )} ]} } \right\} \end{array} \right\} \end{array} \right.$}$$

where ${J_n}(x )$ is the $n$th-order Bessel function of the first kind. ${\varphi _1}$ is the phase value of RF1, and ${\varphi _{m0}} = {\varphi _{r0}} = {\varphi _1}$. ${A_{M1}}$ is the amplitude of microwave signal RF1. ${A_{m0}}$ and ${A_{r0}}$ are defined as the intensity transmission constants for the measurement and reference light, which are determined by the input intensity of the light of two wavelengths and the power loss coefficients of EOM #1. ${U_{DC1}}$ indicates the DC bias voltage supplied by MBC #1 to EOM #1. ${V_{\pi m1,RF}}$ and ${V_{\pi r1,RF}}$ represent the RF half-wave voltage of EOM #1 with respect to the measurement and reference light, whereas ${V_{\pi m1,DC}}$ and ${V_{\pi r1,DC}}$ represent the DC half-wave voltage for the same. Given the relationship between the half-wave voltage and the wavelength of the MZM [33], it is evident that the RF and DC half-wave voltages for the measurement and reference light of different wavelengths vary.

From the expanded signal expression, it is clear that the signal comprises a rich spectrum of harmonic components. Among these, the fundamental part, characterized by the frequency ${f_M}$, is designated as the effective signal for axial clearance measurement. Aiming to maximize the coefficient ${J_1}({\pi {A_{M1}}/{V_{\pi m1,RF}}} )$ of the fundamental part, the DC bias voltage ${U_{DC1}}$, output by MBC #1, and the amplitude ${A_{M1}}$ of the microwave signal are tuned. In the subsequent analytical derivation, focus is maintained solely on this fundamental part.

The phase differences induced by the measurement and reference signals within the optical fiber path prior to reaching EOM #2 are denoted by ${\varphi _{m1}}$ and ${\varphi _{r1}}$. When reaching EOM #2, the intensities of the measurement and reference light can be expressed as

(4)$$\left\{ \begin{array}{l} {I_{m1}}(t )= {A_{m1}}\left[ {1 + 2\sin \left( {\frac{{\pi {U_{DC1}}}}{{{V_{\pi m1,{\kern 1pt} {\kern 1pt} DC}}}}} \right){J_1}\left( {\frac{{\pi {A_{M1}}}}{{{V_{\pi m1,{\kern 1pt} {\kern 1pt} RF{\kern 1pt} }}}}} \right)\sin ({2\pi {f_M}t + {\varphi_1} + {\varphi_{m1}} + {\varphi_d}} )} \right]\\ {I_{r1}}(t )= {A_{r1}}\left[ {1 + 2\sin \left( {\frac{{\pi {U_{DC1}}}}{{{V_{\pi r1,{\kern 1pt} {\kern 1pt} DC}}}}} \right){J_1}\left( {\frac{{\pi {A_{M1}}}}{{{V_{\pi r1,{\kern 1pt} {\kern 1pt} RF{\kern 1pt} }}}}} \right)\sin ({2\pi {f_M}t + {\varphi_1} + {\varphi_{r1}}} )} \right] \end{array} \right.$$

where ${A_{m1}}$ and ${A_{r1}}$ are the intensity transmission constants of the measurement and reference light during the propagation process between EOM #1 and EOM #2. ${\varphi _{r1}}$, ${\varphi _{m1}}$ are the phase differences of the microwave signals carried by the reference and measurement light introduced by the propagation in the optical paths between EOM #1 and EOM #2.

After undergoing a second intensity modulation by the microwave signal RF2 through EOM #2, an intermediate frequency component emerges in the output signal. The intensity of the intermediate frequency component can be expressed as

(5)$$\scalebox{0.9}{$\displaystyle\left\{ \begin{array}{@{}l@{}} {I_{m2}}(t )= 2{A_{m2}}\sin \left( {\frac{{\pi {U_{DC1}}}}{{{V_{\pi m1,{\kern 1pt} {\kern 1pt} DC}}}}} \right)\sin \left( {\frac{{\pi {U_{DC2}}}}{{{V_{\pi m2,{\kern 1pt} {\kern 1pt} DC}}}}} \right){J_1}\left( {\frac{{\pi {A_{M1}}}}{{{V_{\pi m1,{\kern 1pt} {\kern 1pt} RF{\kern 1pt} }}}}} \right){J_1}\left( {\frac{{\pi {A_{M2}}}}{{{V_{\pi m2,{\kern 1pt} {\kern 1pt} RF{\kern 1pt} }}}}} \right)\cos ({2\pi {f_{IM}}t + {\varphi_1} + {\varphi_{m1}} + {\varphi_d} - {\varphi_2}} )\\ {I_{r2}}(t )= 2{A_{r2}}\sin \left( {\frac{{\pi {U_{DC1}}}}{{{V_{\pi r1,{\kern 1pt} {\kern 1pt} DC}}}}} \right)\sin \left( {\frac{{\pi {U_{DC2}}}}{{{V_{\pi r2,{\kern 1pt} {\kern 1pt} DC}}}}} \right){J_1}\left( {\frac{{\pi {A_{M1}}}}{{{V_{\pi r1,{\kern 1pt} {\kern 1pt} RF{\kern 1pt} }}}}} \right){J_1}\left( {\frac{{\pi {A_{M2}}}}{{{V_{\pi r2,{\kern 1pt} {\kern 1pt} RF{\kern 1pt} }}}}} \right)\cos ({2\pi {f_{IM}}t + {\varphi_1} + {\varphi_{r1}} - {\varphi_2}} )\end{array} \right.$}$$

where ${A_{m2}}$ and ${A_{r2}}$ represent the intensity transmission constants of the measurement and reference light during their propagation through EOM #2. ${\varphi _2}$ is the phase value of signal RF2. ${A_{M2}}$ is the amplitude value of signal RF2. ${U_{DC2}}$ represents the DC bias voltage input from MBC #2 to EOM #2. ${V_{\pi m2,RF}}$ and ${V_{\pi r2,RF}}$ represent the RF half-wave voltage of EOM #2 with respect to the measurement and reference light, whereas ${V_{\pi m2,DC}}$ and ${V_{\pi r2,DC}}$ represent the DC half-wave voltage for the same.

Subsequently, the total phase difference between the measurement signal and the reference signal received by the photodetector can be expressed as

(6)$$\Delta \varphi = ({\varphi _{m1}} - {\varphi _{r1}}) + {\varphi _d}$$

The process of down-conversion in microwave photonic mixing does not alter the phase difference between the measurement signal and the reference signal. Then, the phase difference between the measurement and reference signals can be determined using phase identification methods such as quadrature demodulation [34], Hilbert transform [35], all-phase Fourier transform [36]. The phase demodulation method chosen in this paper is Hilbert transform method. Finally, the axial clearance value can be calculated by Eq. (2).

2.2 Theoretical model of axial clearance temperature drift for WDM-based common optical path structure

There exists a critical challenge in measuring axial clearance within rotating machinery. As illustrated in Fig. 2, this process necessitates the extension of the sensor probe and signal transmission line into the machinery's interior. Consequently, a segment of the measurement path must operate in a high-temperature environment. During online measurement of the axial gap, significant internal temperature variations within the rotating machinery can lead to changes in the measurement path length. This results in an unbalanced phase difference between the measurement signal and the reference signal, subsequently introducing measurement errors. In this study, the measurement error attributable to temperature variations is termed ‘temperature drift.’ The calculation of axial clearance temperature drift for the microwave photonic mixing method without compensation, hereinafter referred to as the ‘single-wavelength single-path structure,’ is outlined as follows.

Fig. 2. Schematic diagram of the optical path structure of microwave photonic mixing method without compensation (single-wavelength single-path structure).

Download Full Size | PDF

The axial clearance measurement utilizing the microwave photonic mixing method primarily hinges on quantifying the phase difference between the microwave-modulated measurement signal and the reference signal. This phase difference is dictated by the time-of-flight (TOF) values of the microwave-modulated light, where TOF values are contingent upon the total optical path length (OPL) encountered during light transmission.

When the measurement optical path is subjected to a high-temperature environment with fluctuating temperatures, both the length and refractive index of the optical fiber in the measurement path undergo changes due to thermal expansion and the thermo-optic effect. For simplification, it is initially assumed that the thermal expansion coefficient of the optical fiber remains constant with temperature variations. The analysis utilizes the average temperature and average refractive index of the optical fiber path. Consequently, the fiber length and refractive index in the measurement path can be mathematically expressed as

(7)$$\left\{ \begin{array}{l} {l_m}^{\prime} = {l_m} + \Delta {l_m} = {l_m} + \alpha \cdot {l_m} \cdot \Delta T\\ {n_m}^{\prime} = {n_m}\textrm{ + }\Delta {n_m} = {n_m}\textrm{ + }\beta \cdot \Delta T \end{array} \right.$$

where $l_m^{\prime}$ represents the length of the measurement optical fiber in high-temperature environment. $n_m^{\prime}$ represents the average refractive index of the measuring optical fiber in high-temperature environment. $\mathrm{\Delta }{l_m}$ represents the change in optical fiber length caused by temperature changes. $\mathrm{\Delta }{n_m}$ represents the change in the measured light refractive index caused by temperature changes. ${l_m}$ indicates the length of fiber of the measurement path at room temperature. ${n_m}$ indicates the refractive index of the measurement light in fiber at room temperature. $\alpha $ represents the thermal expansion coefficient of the optical fiber. $\beta $ represents the thermo-optic coefficient of the measurement light with wavelength $\lambda $ in the optical fiber. $\mathrm{\Delta }T$ represents the average temperature change of the high-temperature environment with temperature T relative to the room temperature environment with temperature ${T_0}$, i.e., $\mathrm{\Delta }T = T - {T_0}$.

Variations in both the length of the measurement fiber and the average refractive index of the measurement light lead to changes in OPL. These alterations in OPL induce a phase difference, which in turn introduces a temperature drift in axial clearance measurement. The temperature drift can be mathematically articulated as

(8)$$\Delta {d_a} = \Delta {L_m} = \alpha \cdot {n_m} \cdot {l_m} \cdot \Delta T + \beta \cdot {l_m} \cdot \Delta T + \alpha \cdot \beta \cdot {l_m} \cdot \Delta {T^2}$$

where $\mathrm{\Delta }{d_a}$ represents the axial clearance temperature drift. $\mathrm{\Delta }{L_m}$ indicates the change in OPL of the measurement light.

In this study, to mitigate the axial clearance temperature drift, a WDM-based common optical path structure is proposed. As depicted in Fig. 1 and Fig. 3, both the measurement light with wavelength ${\lambda _1}$ and the reference light with wavelength ${\lambda _2}$ traverse the same optical fiber path within the rotating machinery. Consequently, both the measurement and reference light simultaneously experience the temperature changes in the high-temperature environment inside the machinery. This arrangement ensures that the OPL changes for both types of light are nearly identical, thereby effectively offsetting the temperature drift. The subsequent analysis is focused on the temperature drift due to the differences in wavelengths in high-temperature environment for the WDM-based common optical path structure.

Fig. 3. Schematic diagram of the WDM-based common optical path structure.

Download Full Size | PDF

Fig. 4. Schematic diagram of the structure of OBPF probe.

Download Full Size | PDF

When the internal temperature of the rotating machinery shifts to T, the OPLs of both the measurement light and the reference light undergo the following changes.

Firstly, as a result of the thermal expansion effect, the length $l(T )$ of the fiber undergoes changes. Given the substantial range of internal temperature variations in rotating machinery, this model incorporates the second-order effect of the thermal expansion coefficient of fused silica for enhanced accuracy, as referenced in [37,38]. The fiber length can be expressed as

(9)$$l(T )= {l_{T0}} \cdot ({1 + {\alpha_1} \cdot \Delta T + {\alpha_2} \cdot \Delta {T^2}} )$$

where ${l_{T0}}$ represents the fiber length of the measurement path at room temperature. ${\alpha _1}$ and ${\alpha _2}$ represent the first-order coefficient and the second-order coefficient of thermal expansion coefficient.

Secondly, as a consequence of the thermo-optic effect, the refractive indices of the measurement and reference light in the optical fiber are concurrently influenced by the wavelength and temperature. The thermo-optic coefficient $\beta ({\lambda ,T} )$ varies with both the wavelength and temperature. According to Ghosh’s model [39–42], the dispersion expression of thermo-optic coefficient is

(10)$$2n\frac{{dn}}{{dT}} = GR + H{R^2}$$

where $dn/dT$ is the thermo-optic coefficient $\beta ({\lambda ,T} )$, R can be expressed as

(11)$$R(\lambda )= \frac{{{\lambda ^2}}}{{{\lambda ^2} - \lambda _{ig}^2}}$$

where ${\lambda _{ig}}$ represents the correspondent wavelength of the fused silica intrinsic bandgap, which is a constant.

Parameter G in Eq. (10) is determined as

(12)$$G ={-} 3\alpha {K^2}$$

where ${K^2} = n_\infty ^2 - 1$, and ${n_\infty }$ is the low-frequency refractive index in the infrared region, and is considered a constant. Consequently, the parameter G is primarily influenced by the thermal expansion coefficient of the material. However, it is important to note that Ghosh's definition of G accounts only for the first-order coefficient of the fused silica thermal expansion coefficient. Thus, the thermal expansion coefficient and G are regarded as constants, independent of temperature. Nonetheless, in scenarios where the range of temperature change is significant, the second-order coefficient of the thermal expansion coefficient becomes relevant. This necessitates the inclusion of temperature as a variable in the formula for G. In light of this, an optimized expression for G can be articulated as

(13)$$G^{\prime}(T )={-} 3({{\alpha_1} + {\alpha_2}\Delta T} ){K^2}$$

Parameter H in Eq. (10) is determined as

(14)$$H ={-} \frac{1}{{{E_g}}}\left( {\frac{{d{E_g}}}{{dT}}} \right){K^2}$$

where ${E_g}$ represents the silica excitonic bandgap, which is also dependent on temperature. Here, temperature T is included as a variable in the formula of H. According to Gaspar’s work [41,43], for silica glass, the relationship between ${E_g}$ and temperature T can be expressed as

(15)$${E_g}(T )= {H_0} + {H_1}T + {H_2}{T^2} + {H_3}{T^3}$$

where ${H_0}$, ${H_1}$, ${H_2}$, ${H_3}$ are the corresponding coefficient constants of each order. H can be optimized as

(16)$$H^{\prime}(T )={-} \frac{1}{{{E_g}(T )}}\left( {\frac{{d{E_g}(T )}}{{dT}}} \right){K^2}$$

Then the dispersion equation of the thermo-optic coefficient after optimization can be obtained, which is represented by the following partial differential equation.

(17)$$2n({\lambda ,T} )\frac{{\partial n({\lambda ,T} )}}{{\partial T}} = G^{\prime}(T )R(\lambda )+ H^{\prime}(T )R{(\lambda )^2}$$

Substituting Eq. (11), Eq. (13) and Eq. (16) into the above formula, we can get

(18)$$\scalebox{0.86}{$\displaystyle 2\left[ {n({\lambda ,{T_0}} )+ \int_{{T_0}}^T {\beta ({\lambda ,T} )dT} } \right]\beta ({\lambda ,T} )= [{ - 3({{\alpha_1} + {\alpha_2}\Delta T} ){K^2}} ]\frac{{{\lambda ^2}}}{{{\lambda ^2} - \lambda _{ig}^2}} + \left[ { - \frac{1}{{{E_g}(T )}}\left( {\frac{{d{E_g}(T )}}{{dT}}} \right){K^2}} \right]{\left( {\frac{{{\lambda^2}}}{{{\lambda^2} - \lambda_{ig}^2}}} \right)^2}$}$$

where $n({\lambda ,{T_0}} )$ is the refractive index at room temperature, which can be determined by the Sellmeier formula [44].

By solving the partial differential equation, the expression of the thermo-optic coefficient $\beta ({\lambda ,T} )$ at high temperature T ($T > {T_0}$) can be determined as

(19)$$\scalebox{0.9}{$\displaystyle \beta ({\lambda ,T} )= \frac{1}{{2\Delta T}}\left\{ \begin{array}{l} - \sqrt {A + \frac{{B{\lambda^2}}}{{({{\lambda^2} - C} )}} + \frac{{D{\lambda^2}}}{{({{\lambda^2} - E} )}}} + \\ \sqrt {A + \frac{{B{\lambda^2}}}{{({{\lambda^2} - C} )}} + \frac{{D{\lambda^2}}}{{({{\lambda^2} - E} )}} - 2\Delta T\frac{{{K^2}{\lambda^2}}}{{{\lambda^2} - \lambda_{ig}^2}}\left[ {3{\alpha_1} + 3{\alpha_2}\Delta T + \frac{{{\lambda^2}({{H_1} + 2{H_2}T + 3{H_3}{T^2}} )}}{{({{\lambda^2} - \lambda_{ig}^2} )({{H_0} + {H_1}T + {H_2}{T^2} + {H_3}{T^3}} )}}} \right]} \end{array} \right\}$}$$

where A, B, C, D, E are Sellmeier coefficients of fused silica, which can be determined in handbook of optical constants of solids [44].

Then the refractive index $n({\lambda ,T} )$ can be expressed as

(20)$$n({\lambda ,T} )= n({\lambda ,{T_0}} )+ \int_{{T_0}}^T {\beta ({\lambda ,T} )dT} $$

The OPLs of measurement light and reference light can be expressed as

(21)$$\left\{ \begin{array}{l} L({{\lambda_1},T} )= n({{\lambda_1},T} )\cdot l(T )\\ L({{\lambda_2},T} )= n({{\lambda_2},T} )\cdot l(T )\end{array} \right.$$

Therefore, the axial clearance temperature drift can be expressed as

(22)$$\scalebox{0.9}{$\displaystyle\begin{array}{l} \Delta {d_b} = l(T )[{n({{\lambda_1},T} )- n({{\lambda_2},T} )} ]- \Delta {L_0}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = {l_{T0}} \cdot ({1 + {\alpha_1} \cdot \Delta T + {\alpha_2} \cdot \Delta {T^2}} )\left[ {n({{\lambda_1},{T_0}} )+ \int_{{T_0}}^T {\beta ({{\lambda_1},T} )dT} - n({{\lambda_2},{T_0}} )- \int_{{T_0}}^T {\beta ({{\lambda_2},T} )dT} } \right] - \Delta {L_0} \end{array}$}$$

where $\mathrm{\Delta }{L_0}$ represents the initial OPD between measurement light and reference light.

The above formula depicts comprehensive theoretical model of axial clearance drift determined by wavelength and temperature for WDM-based common optical path structure.

2.3 Theoretical design for OBPF probe

A pivotal aspect of the axial clearance measurement method based on WDM all-fiber common optical path is to separate the measurement light and the reference light with different wavelengths at the fiber probe. As shown in Fig. 4, the optical structure for OBPF probe is proposed, designed for efficient separation of the measurement and reference light.

In an ideal scenario, the measurement light would be fully transmitted through the filter window, while the reference light would be entirely reflected. However, within the confines of current technology and our understanding, optical bandpass filter films exhibit limited transmission band isolation ($is{o_t}$) and reflection band isolation ($is{o_r}$) for both wavelengths of light. On the one hand, the light passing through the filter window includes not only the measurement light of wavelength ${\lambda _1}$, but also some reference light of wavelength ${\lambda _2}$. The impact of this component is typically negligible because transmission band isolation is generally higher and the intensity diminishes a lot after diffuse reflection at the axial end surface. On the other hand, the light reflected by the filter window contains both the reference light of wavelength ${\lambda _2}$ and some measurement light of wavelength ${\lambda _1}$. Due to the weak intensity of the diffuse reflection measurement light received from the rotor axial end face by the OBPF probe, the relative intensity of the ${\lambda _1}$ measurement light reflected by the filter window is comparatively large. This factor significantly influences the measurement, potentially reducing the signal-to-noise ratio to a level that might render accurate measurement challenging.

To optimize the signal-to-noise ratio of both the measurement and reference signals as much as possible, it is crucial to maintain a low return loss (RL) for the reference light at the filter window, while simultaneously maximizing the return loss for the measurement light. This approach ensures that the intensity of reflected reference light is maximized, whereas the intensity of the measurement light reflected from the filter window is minimized. In the OBPF probe design method proposed in this paper, a balance in the return loss for both the measurement and reference light is achieved by adjusting the axial position of the filter window and altering the angle between the normal line of the filter window and the optical axis. The theoretical model for this approach is analyzed as follows.

When evaluating the reflection of the filter coating on the fused-silica window in the OBPF probe, this coating can be conceptualized as a reflective mirror. The reflectivity to the reference light and measurement light are denoted as ${\eta _1}$ and ${\eta _2}$. Consequently, the calculation of the OBPF probe's return loss can be equated to the computation of the coupling efficiency between the Gaussian beam transmitted by the OBPF probe and that received by a mirror OBPF probe, as illustrated in Fig. 5. The angle between the plane where the filter coating is located and the vertical plane of the optical axis is $\psi $, then the angle between the optical axes of the probe and the mirror probe is $2\psi $. Assuming that the distance along the optical axis between the beam waist of the Gaussian beam and the beam waist of the mirrored Gaussian beam is ${Z_0}$, and the distance between the center of the left end of reflected lens image and the optical axis is ${X_0}$. There are the following relationships.

(23)$$\begin{array}{l} {X_0} = 2({{l_{WD}} - {l_2}} )\tan 2\psi \\ {Z_0} ={-} 2({{l_{WD}} - {l_2}} )= 2{l_2} - 2{l_{WD}} \end{array}$$

where ${l_{WD}}$ is the working distance of the OBPF optical fiber probe, determined by two-stage ray tracing method [26]. ${l_2}$ represents the distance from the right end surface of the lens to the coating.

Fig. 5. Schematic diagram of specular reflection model of OBPF probe optical filter window. $\psi $: the angle between the plane where the coating is located and the vertical plane of the optical axis, ${l_1}$: the distance from the fiber end face to the left end face of the lens, ${l_c}$: the thickness of the lens, ${l_2}$: the distance from the right end face of the lens to the coating, ${X_0}$: the vertical distance between the center of the left end of the reflected lens image and the optical axis.

Download Full Size | PDF

This paper establishes a return loss analysis model for the OBPF probe based on ray-transfer matrix theory and overlapping integration method [45–47]. The transmission matrix of the OBPF probe can be expressed as

(24)$${M_T} = \left[ {\begin{array}{{cc}} {{A_0}}&{{B_0}}\\ {{C_0}}&{{D_0}} \end{array}} \right] = \left[ {\begin{array}{{cc}} {1 + \frac{{{l_{WD}}({{n_1} - {n_c}} )}}{{{n_1}{R_c}}}}&{{l_1} + \frac{{{n_1}{l_c}}}{{{n_c}}} + {l_{WD}}\left[ {1 + \frac{{({{n_1} - {n_c}} )}}{{{R_c}}}\left( {\frac{{{l_c}}}{{{n_c}}} - \frac{{{l_1}}}{{{n_1}}}} \right)} \right]}\\ {\frac{{{n_1} - {n_c}}}{{{n_1}{R_c}}}}&{1 + \frac{{({{n_1} - {n_c}} )}}{{{R_c}}}\left( {\frac{{{l_c}}}{{{n_c}}} - \frac{{{l_1}}}{{{n_1}}}} \right)} \end{array}} \right]$$

where ${n_1}$ denotes the refractive index of the medium filling the gap in the optical fiber probe. ${n_c}$ indicates the refractive index of the lens material. ${R_c}$ indicates the curvature radius of the right surface of the lens. ${l_1}$ indicates the distance from the fiber end face to the left end face of the lens. ${l_c}$ indicates the thickness of the lens.

The light field emitted from a single-mode fiber can be approximated as Gaussian beam [45]. The beam waist is located at the end face of the fiber. The beam waist radius (${\omega _0}$) is the mode field radius (${r_{MFD}}$) of the single-mode fiber, and the radius (${R_0}$) of the equiphasic surface is infinite.

Use the beam radius ${\omega _i}$ and the curvature radius ${R_i}$ of the equiphasic surface to describe the fundamental mode of the Gaussian beam. The complex curvature parameter ${q_i}$ is defined as

(25)$$\frac{1}{{{q_i}}} = \frac{1}{{{R_i}}} - j\frac{\lambda }{{\pi {n_1}{\omega _i}^2}}$$

Then the complex parameter at the end face of the single-mode fiber Gaussian beam can be expressed as

(26)$${q_0} = j\frac{{\pi {n_1}{\omega _0}^2}}{\lambda }$$

Gaussian beam transmission follows the ABCD law, which can be expressed as

(27)$${q_{i + 1}} = \frac{{{A_i}{q_i} + {B_i}}}{{{C_i}{q_i} + {D_i}}}$$

When the Gaussian beam passes through the OBPF probe, its beam waist position is at a distance ${l_{WD}}$ from the right end face of the lens. Utilizing Eq. (24), Eq. (26), and Eq. (27), and setting $\rho = \pi {n_1}\omega _0^2/\lambda $, the beam waist radius can be expressed as

(28)$${\omega _1} = {\omega _0}\sqrt {\frac{{{\rho ^2}A_0^2 + B_0^2}}{{{\rho ^2}({{A_0}{D_0} - {B_0}{C_0}} )}}}$$

Based on overlapping integration method, the coupling efficiency between two Gaussian beams can be expressed as

(29)$${\eta _c} = \frac{2}{{\pi {E_1}^2{\omega _1}^2}}\int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } {{E_x}|{_{z^{\prime} = 0}} } } \cdot {E_{x^{\prime}}}^ \ast |{_{z^{\prime} = 0}\textrm{d}x^{\prime}\textrm{d}y^{\prime}} $$

where ${E_1}$ represents the electric field amplitude of the light field output from the lens at the center of the beam waist position ($x = 0,\; y = 0,\; z = 0$). ${E_x}$ and ${E_{x^{\prime}}}$ represent the components of the Gaussian beam light field and the mirrored Gaussian beam light field in the x direction.

By substituting the expression for the Gaussian beam light field and performing integration, the expression for the coupling efficiency of the two Gaussian beams can be determined. This enables the calculation of the power coupling efficiency, which can be mathematically expressed as

(30)$${\eta _p} = {\eta _c} \cdot \eta _c^ \ast{=} \frac{{{k^2}\omega _1^4}}{{Z_0^2 + {k^2}\omega _1^4}}exp \left\{ { - \frac{{{k^2}\omega_1^2[{4X_0^2 + 4{X_0}{Z_0}\sin ({2\psi } )+ ({2Z_0^2 + {k^2}\omega_1^4} ){{\sin }^2}({2\psi } )} ]}}{{4{k^2}\omega_1^4 + 4Z_0^2}}} \right\}$$

where k is the wave number, expressed as $2\pi {n_1}/\lambda $.

When combined with Eq. (23), the parameter ${\eta _p}$ is determined by the following factors: the wavelength $\lambda $ of the light, the deflection angle $\psi $ of the window, and the distance ${l_2}$ between the window and the lens.

Then the return loss of the measurement and reference light can be expressed as

(31)$$\begin{array}{l} {R_L}({{\lambda_1},\psi ,{l_2}} )={-} 10\lg ({{\eta_1} \cdot {\eta_{p1}}} )\\ {R_L}({{\lambda_2},\psi ,{l_2}} )={-} 10\lg ({{\eta_2} \cdot {\eta_{p2}}} )\end{array}$$

During the design phase of the OBPF probe, a key objective is to maximize the return loss for the measurement light while minimizing the return loss for the reference light. Achieving this balance involves adjustments of the window declination angle and the distance between the lens and the window.

3. Simulations and results

3.1 Simulations of temperature-drift in axial clearance measurement

In the axial clearance measurement system based on WDM all-fiber microwave photonic mixing, a high-temperature resistant optical fiber is used. In this study, a polyimide-coated fiber (PI-coated fiber) has been chosen for its high-temperature resistance. This fiber comprises a polyimide coating and a core made of fused silica, making it suitable for long-term operation at temperatures up to 300°C. The relevant thermo-optic parameters for both PI-coated fiber and fused silica material are comprehensively detailed in Table 1.

Table 1. Parameters of fiber and fused silica material

View Table | View all tables in this article

Initially, the temperature drift of the single-wavelength single-path structure, as depicted in Fig. 2, was simulated. For this simulation, a measurement light wavelength of 1550 nm was selected. Utilizing Eq. (8) and MATLAB for the simulation, the results for the axial clearance temperature drift were obtained, as illustrated in Fig. 6. The simulation indicates that the temperature drift in the single-wavelength single-path structure is primarily influenced by the optical path difference arising from the thermal expansion of the optical fiber. Notably, with temperature variations, the axial clearance temperature drift exhibits significant changes. For instance, when the temperature rises from 20°C to 280°C, the axial clearance temperature drift value escalates to 2.87 mm. This magnitude of drift is substantially larger than the acceptable limit for axial clearance measurement drift, potentially leading to the failure of the measurement system.

Fig. 6. Simulation results of temperature drift of single-wavelength single-path structure.

Download Full Size | PDF

Secondly, for comparative analysis, the axial clearance temperature drift of a single-wavelength dual-path structure, an enhancement over the single-wavelength single-path structure, was simulated. In this improved design, the first microwave modulated light is bifurcated into two paths using an optical fiber coupler. These light, after traversing circulators, enter two distinct optical fiber paths. One of these serves as the measurement optical path, while the other functions as the reference optical path. Both optical fibers extend into the rotating machinery concurrently, sensing the temperature changes within. Although this dual-path structure incorporates a reference optical path to mitigate temperature drift in axial clearance measurement, the measurement and reference paths are inherently different. Owing to the limited space and intricate structure of the rotating machinery, the paths are convoluted and complex, leading to two primary disparities between them. Firstly, maintaining identical absolute lengths for both the measurement and reference paths inside machinery like an aeroengine is challenging. Secondly, the temperature fields impacting the two optical fibers differ. Consequently, despite the dual-path approach, the temperature drift in axial clearance remains significant due to these path differences.

Assuming a discrepancy of 20 cm in the absolute lengths of the two optical fibers that sense the changing temperature field within the rotating machinery, simulation results for axial clearance temperature drift were obtained, as illustrated in Fig. 7. The simulation indicates that when the temperature shifts from 20°C to 280°C, the axial clearance temperature drift reaches 573µm. This drift, although relatively smaller, is still significant when compared to the precision requirements of several microns to tens of microns typically necessary for axial clearance measurements. Furthermore, the temperature drift is expected to increase with larger temperature variation ranges or more uneven heating of the two optical fibers.

Fig. 7. Simulation results of temperature drift of single-wavelength dual-path structure.

Download Full Size | PDF

Finally, the temperature drift of the WDM-based common optical path structure, as depicted in Fig. 3, was simulated. Drawing from the theoretical analysis presented in part 2.2, the axial clearance temperature drift can be represented by Eq. (22). Considering the parameters of the optical fiber device selected for the actual measurement system design, the wavelengths of the measurement and reference light were chosen to fall within the C-band range (1525 nm to 1565 nm). During the simulation, the initial wavelength of the reference light was set at 1550 nm. The wavelength variation range for the reference light during the simulation was set between 1525 nm and 1570 nm, with a temperature variation range from 20°C to 300°C. Utilizing MATLAB, the simulation results for the axial clearance measurement temperature drift of the WDM-based common optical path structure were obtained, as shown in Fig. 8. The results indicate that when the reference light wavelength varies across the entire C-band range and the temperature fluctuates between 20°C and 300°C, the axial clearance measurement temperature drift value changes within the range of -0.9µm to 1.3µm. It is observed that the closer the reference light wavelength is to the measurement light wavelength, the smaller the temperature drift value. Specifically, when the measurement light wavelength is set at 1530 nm and the temperature changes from 20°C to 280°C, the temperature drift value is 0.91µm. Additionally, when the measurement light wavelength is 1565 nm and the temperature shifts from 20°C to 280°C, the temperature drift value is -0.63µm.

Fig. 8. Simulation results of temperature drift of WDM-based common optical path structure.

Download Full Size | PDF

Comparing the simulation results of the aforementioned three optical path structures, it is evident that the WDM-based common optical path structure proposed in this paper significantly reduces the axial clearance temperature drift value to less than 1µm. This efficacy makes it particularly suitable for axial clearance measurement in environments with narrow space, where there are wide temperature variation range and complex internal structure.

3.2 Simulations of return loss of OBPF probe

The parameters for the OBPF probe are detailed in Table 2.

Table 2. Parameters of the OBPF probe components

View Table | View all tables in this article

For the simulation, we assumed the wavelengths of the measurement and reference light to be 1530 nm and 1550 nm, respectively. Utilizing Eq. (31) and the parameters from Table 2, and employing MATLAB, we derived the simulation results for the OBPF probe's return loss, which is influenced by the window declination angle $\psi $ and the distance ${l_2}$ between the lens and the window, as shown in Fig. 9. The optical bandpass filter film exhibits a reflection band isolation of 25 dB, and the return loss of the OBPF probe to the measurement light, excluding the filter window, is 65 dB. Therefore, it is crucial to adjust the window declination angle and the distance between the lens and the window to achieve a return loss of 40 dB for the optical filter window with respect to the reference light, as indicated by the red line in Fig. 9. The optimal scenario involves adjusting the window declination and distance such that it aligns with the red line, thereby maximizing the signal-to-noise ratio of both the measurement and reference signals. It is important to note that the adjustment of OBPF probe parameters, specifically the window deflection angle $\psi $ and distance ${l_2}$, should be done in conjunction with the differences in actual probes and various other influencing factors. This implies that adjustments are based on real-time feedback from return loss measurements, utilizing a six-degree-of-freedom displacement stage.

Fig. 9. Simulation results of return loss of OBPF probe determined by the window declination angle and the distance between the lens and the window.

Download Full Size | PDF

4. Experiments, results and discussion

4.1 Experimental setup

In order to verify the performance of the method proposed in this paper, an experimental setup was built as shown in Fig. 10.

Fig. 10. Experimental setup for axial clearance measurement.

Download Full Size | PDF

In the all-fiber optical module, two polarization-maintaining laser diodes (PL-1530-20-T-PM-M and PL-1550-20-T-PM-M, Max-ray Photonics) were used, the central wavelength of which were 1530 nm and 1550 nm. Two Mach-Zehnder electro-optic modulators (MXAN-LN-10, EXAIL Technologies) were used for the first microwave modulation and the second microwave modulation, respectively. Microwave signal synthesis module or microwave synthesizer (N5172B and N5222A from Keysight Technologies used in this paper) was utilized to generate microwave signals. The frequencies of microwave signals RF1 and RF2 are 6.0025 GHz and 6 GHz, respectively, and the corresponding ambiguity range is about 25 mm. A high-power erbium-doped fiber amplifier (EYDFA-HP-C-BA-23-PM-M, Max-ray Photonics) and a small-signal erbium-doped fiber amplifier (EDFA-C-PA-35-PM-M, Max- ray Photonics) were used as preamplification and small signal amplification in the optical path. Two coarse wavelength division multiplexers were used respectively for multiplexing and demultiplexing the measurement light and the reference light in the optical path. Two photodetectors (UPD-15-IR2-FC, ALPHALAS GMBH) were used for reception and photoelectric conversion of the second modulated light signals. The photoelectrically converted intermediate frequency signal was collected by a dual-channel high-speed acquisition card. The frequency of the intermediate frequency signal is 2.5 MHz, and the sampling rate of the acquisition card is set to 125MSps. In order to simulate the change of axial clearance, the OBPF probe was fixed on the designed mechanism and installed on a motorized precision translation stage (PA300 and SC300-3B, Zolix Instruments CO., LTD.), which could make axial displacement. The screw lead of the motorized precision translation stage is 5 mm, the step angle of the drive stepper motor is 1.8 degrees, and the subdivision is set to 8. In addition, a metal disc was fixed on a motor with its end face facing the OBPF probe to simulate the rotor. Install the motor equipped with the disc on a lifting platform to facilitate height adjustment to align with the OBPF probe. In order to test the temperature drift, a high-temperature cabinet (101-0S, Shanghai Han Yu Technology Co., Ltd.) was used to simulate the high-temperature environment. The high-temperature-resistant optical fiber used in this experimental system was PI coated fiber (SM1550P, Thorlabs Inc.), which has a long-term working temperature resistance of up to 300°C. Finally, a laser interferometer (SJ6000, CHOTEST) was used in this experimental system. The distance measurement results were used as standard values in this experimental system and for comparison.

Additionally, Fig. 11 presents the transmission and reflection spectra of the Optical Bandpass Filter (OBPF) utilized in the experiment. Analyzing the spectral data reveals that the transmission isolation ($is{o_t}$) at wavelengths of 1530 nm and 1550 nm reaches approximately 66.5 dB, while the reflection isolation ($is{o_r}$) is around 26.5 dB. This value of reflection isolation closely aligns with the 25 dB figure employed in the simulations.

Fig. 11. The transmission and reflection spectra of the optical bandpass filter (OBPF).

Download Full Size | PDF

4.2 Measurement performance evaluation experiments

The experiments were designed to verify measurement performances, including resolution, range, repeatability, accuracy, and temperature drift of the system. To enhance accuracy, a laser interferometer (SJ6000, CHOTEST) was employed to calibrate the measurement system over a range of 0-23.5 mm. And the results obtained from the laser interferometer served as reference values, against which the measurement results obtained by the proposed system were compared. Following calibration, a series of verification experiments were conducted.

4.2.1 Measurement resolution, repeatability and accuracy evaluation

Initially, experiments to verify the measurement resolution were conducted. The motorized precision translation stage (PA300) was either controlled by the driver (SC300-3B) or manually operated to attain any position within the 0-23.5 mm measurement range. Subsequently, the stepper motor of the motorized precision translation stage was controlled by the driver (SC300-3B) to advance with incremental steps, each constituting a 3µm displacement. The phase difference measurement results for each position were recorded, as depicted in Fig. 12. At each position, the measurement frequency of the system is set to 25 kHz. Analysis of these results reveals that the axial clearance measurement resolution of the proposed system is better than 3µm. And from the measurement results, it can be calculated that the sensitivity is approximately 0.28 rad/mm.

Fig. 12. The results of the measurement resolution verification experiments.

Download Full Size | PDF

Secondly, experiments to verify the measurement repeatability were carried out. Commencing from the zero position, the motorized precision translation stage was advanced in 0.5 mm increments within the 0-23.5 mm range, facilitating axial clearance measurements at a total of 48 distinct positions. At each of these positions, the total measurement duration was set to 60 seconds, with the axial clearance measurement values being recorded every 20 milliseconds. Following this, the standard deviation of the 60-second measurement results at each position was calculated. The results of these standard deviation calculations are presented in Fig. 13. As indicated in the figure, the measurement standard deviation within the entire range consistently remained better than 5.8µm.

Fig. 13. The results of the measurement repeatability verification experiments.

Download Full Size | PDF

Furthermore, experiments to verify measurement accuracy were conducted. Within the measurement range of 0-23.5 mm, the motorized precision translation stage was moved in 0.5 mm increments, and axial clearance was measured at each step. This procedure was repeated to perform measurements at 48 reference value positions. The measurement errors were determined by subtracting the reference values from the axial clearance measurement values obtained by the measurement system at each point. Three sets of such measurement experiments were carried out. The error results from the three groups of axial clearance measurements are illustrated in Fig. 14. Analysis of these results indicates that within the 23.5 mm range, the accuracy of the measurement system is maintained at better than 2.8µm.

Fig. 14. The results of the measurement accuracy verification experiments.

Download Full Size | PDF

4.2.2 Temperature drift evaluation

To assess the measurement stability of the axial clearance measurement system in a room temperature environment, and to facilitate a subsequent comparison with the experimental results of axial clearance temperature drift in high-temperature conditions, an extended axial clearance measurement was conducted over a duration of approximately 10 minutes. The results of this measurement are depicted in Fig. 15. Analysis of these results indicates that the axial clearance measurement values remained stable over the 10-minute period, exhibiting no significant drift and fluctuating around the mean value.

Fig. 15. 10-minute measurement results of the axial clearance measurement system.

Download Full Size | PDF

Subsequently, temperature drift measurement experiments were conducted. The internal environment of the high-temperature cabinet was set to six different temperatures: 50°C, 80°C, 130°C, 180°C, 230°C, and 260°C. At each of these temperature settings, axial clearance measurements were conducted for a duration of 60 seconds. These measurements were performed at axial clearance positions of 5 mm, 17 mm, and 23 mm. The results for each of these positions are presented in Fig. 16, Fig. 17, and Fig. 18, respectively. To provide a more intuitive understanding of the axial clearance measurement drift under different temperature conditions, the average axial clearance measurement results at each temperature were compiled and compared. From these results, it can be inferred that with a temperature change of 210°C, the drift measurements at the axial clearance positions of 5 mm, 17 mm, and 23 mm are 8.7µm, 11.1µm, and 3.2µm, respectively.

Fig. 16. Measurement results of temperature drift at 5 mm axial clearance.

Download Full Size | PDF

Fig. 17. Measurement results of temperature drift at 17 mm axial clearance.

Download Full Size | PDF

Fig. 18. Measurement results of temperature drift at 23 mm axial clearance.

Download Full Size | PDF

Based on the measurement results obtained at each location, it is observed that the measured temperature drift values exceed those predicted by the simulations. Moreover, the drift varies across different axial clearance positions. This discrepancy suggests that the drift may not be solely attributable to temperature changes but could also be influenced by other factors, such as system noise. Additionally, during the temperature drift experiments, there was a need to allow time for the high-temperature cabinet to heat up and stabilize the temperature. Conducting a complete experiment is time-consuming, and such prolonged measurements might introduce additional drifts. These include thermal drifts in devices like the EDFA, photoelectric conversion and receiving circuits within the measurement system, and mechanical drifts in the system’s mechanical components.

To more effectively demonstrate the efficacy of the axial clearance measurement method proposed in this paper in mitigating temperature drift, experiments were conducted with both single-wavelength single-path and dual-wavelength dual-path structures at the 17 mm axial clearance position. These experiments correspond to the simulation results depicted in Fig. 6 and Fig. 7. The measurement outcomes are presented in Fig. 19 and Fig. 20. From these results, it is deduced that with a temperature change of 210°C, the drift measurement values for the single-wavelength single-path and the dual-wavelength dual-path structures are approximately 2.169 mm and 495.5µm, respectively. During the experiment, due to the need to lead out the optical fiber interface from the side of the high-temperature cabinet and to prevent high-temperature conduction to room-temperature devices, a considerable length of optical fiber was left at room temperature to facilitate cooling. Consequently, the actual effective length of the optical fiber subjected to heating might have been less than 1 m. Additionally, the optical fiber undergoes a temperature transition from high to room temperature, which could account for why the single-wavelength single-path structure's measurement results were smaller than the simulation predictions. Furthermore, for the single-wavelength dual-path structure, the variation in heating uniformity between the two optical fiber bundles presents a challenge and contributes to the observed discrepancies between the experimental and simulation results.

Fig. 19. Measurement results of temperature drift of single-wavelength single-path structure.

Download Full Size | PDF

Fig. 20. Measurement results of temperature drift of single-wavelength dual-path structure.

Download Full Size | PDF

The comparative chart displaying the temperature drift measurement results for the three optical path structures is illustrated in Fig. 21. An analysis of these comparative results clearly demonstrates that the temperature drifts observed in both the single-wavelength single-path and the single-wavelength dual-path structures are significantly higher than those measured using the method proposed in this paper. This stark contrast underscores the effectiveness of the proposed method in reducing temperature-induced measurement drift.

Fig. 21. Comparison of temperature drift experimental measurement results.

Download Full Size | PDF

5. Conclusions and prospect

Due to narrow space, long signal transmission path and high temperature inside rotating machinery, it remains a great challenge to realize large measurement range, high accuracy, and low drift in axial clearance measurement. To overcome the above challenge, this paper proposes an axial clearance measurement method based on wavelength division multiplexing (WDM) with all-fiber microwave photonic mixing. Down-conversion is achieved in the optical domain, improving the measurement accuracy. Based on WDM, the measurement and reference light with different wavelengths are transmitted, modulated and down-converted through the common optical path, suppressing the axial clearance measurement drift. Based on the thermo-optic effect of optical fiber materials, a comprehensive theoretical model for axial clearance measurement drift determined by wavelength and temperature is derived. To achieve efficient separation of measurement and reference light at the probe, the structure for optical bandpass filter (OBPF) fiber probe is presented, aiming to optimize the signal-to-noise ratio. Finally, the measurement system is built and a series of experiments are conducted to verify the performance of the measurement system. The results show that the resolution of the measurement system is better than 3µm. The axial clearance measurement range can reach 23.5 mm, a performance not previously reported in the literature. Within the measurement range, the accuracy is better than 2.8µm. These performances have all been improved compared to previous all-fiber microwave photonic axial clearance measurement system. The diameter of the OBPF probe used is 3 mm, which is suitable for axial clearance measurement in narrow space. In addition, high-temperature drift and comparative experiments were conducted. The experimental results from the high-temperature drift verification experiments show that when the temperature changes by 210°C, the drift at the 23 mm axial clearance is only 3.2µm. Further, comparative experiments with two other optical path structures confirmed the effectiveness of the proposed method in suppressing temperature drift.

In summary, the proposed method has great application potential for measuring axial clearance in rotating machinery characterized by narrow space, complex internal structure, and high-temperature environment. Additionally, the WDM-based common optical path structure proposed in this paper also has promising application prospects in other fields of absolute distance measurement, such as microwave photonic absolute distance measurement [31,32]. This approach not only suppresses distance measurement drift but also simplifies the optical path of the measurement system, thereby saving the need for a Mach-Zehnder intensity modulator and its ancillary components, reducing the overall cost of the measurement system.

Funding

National Natural Science Foundation of China (52205573, 61971307, 62231011, 92360306, U2241265); National Science and Technology Major Project (J2022-V-0005-0031); Chinese Aeronautical Establishment (2022Z060048001); China Postdoctoral Science Foundation (2022M720106); Young Teacher Research Initiation Project of State Key Laboratory (Pilq2304); Joint Fund of Ministry of Education for Equipment Pre-research (8091B022144); National Defense Science and Technology Innovation Fund of the Chinese Academy of Sciences (6142212210304); Special Project for Research and Development in Key areas of Guangdong Province (2020B0404030001); Fok Ying Tung Education Foundation (171055); Young Elite Scientists Sponsorship Program by CAST (2021QNRC001).

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.

References

1. H. Cheng, X. Lu, S. Zhao, et al., “Effect of tip clearance variation in the transonic axial compressor of a miniature gas turbine at different Reynolds numbers,” Aerosp. Sci. Technol. 128, 107793 (2022). [CrossRef]

2. Q. Xu, J. Wu, L. Wu, et al., “Pressure and velocity fluctuations characteristics of the tip clearance flow in an axial compressor stage at the near-stall condition,” Aerosp. Sci. Technol. 129, 107796 (2022). [CrossRef]

3. M. Zhang, X. Dong, J. Li, et al., “Effect of differential tip clearance on the performance and noise of an axial compressor,” Aerosp. Sci. Technol. 132, 108070 (2023). [CrossRef]

4. C. Yang, L. Fan, Z. Zhong, et al., “Numerical and experimental investigation on flow field of the turbine stage under different axial gaps,” Processes 11(7), 2138 (2023). [CrossRef]

5. S. N. Danish, S. R. Qureshi, M. M. Imran, et al., “Effect of tip clearance and rotor-stator axial gap on the efficiency of a multistage compressor,” Appl. Therm. Eng. 99, 988–995 (2016). [CrossRef]

6. M. G. Choi and J. Ryu, “Numerical study of the axial gap and hot streak effects on thermal and flow characteristics in two-stage high pressure gas turbine,” Energies 11(10), 2654 (2018). [CrossRef]

7. H. Li, Q. Zheng, and B. Jiang, “Influence of rotor-stator axial clearance on compressor rotating stall characteristics,” Aerosp. Sci. Technol. 139, 108373 (2023). [CrossRef]

8. P. Li and L. Cao, “Effect of rotor-stator axial gap on the tip seal leakage flow of a steam turbine,” J. Therm. Sci. 30(3), 973–982 (2021). [CrossRef]

9. B. Gao, W. Shen, H. Guan, et al., “Review and comparison of clearance control strategies,” Machines 10(6), 492 (2022). [CrossRef]

10. B. Jia and X. Zhang, “An optical fiber blade tip clearance sensor for active clearance control applications,” Procedia Eng. 15, 984–988 (2011). [CrossRef]

11. B. Yu, T. Zhang, H. Ke, et al., “Research on the tip clearance measuring method based on AC discharge,” IEEE Access 8, 60355–60363 (2020). [CrossRef]

12. Q. Yang, L. Li, Y. Zhao, et al., “Experimental measurement of axial clearance in scroll compressor using eddy current displacement sensor,” Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 222(7), 1315–1320 (2008). [CrossRef]

13. T. Addabbo, F. Bertocci, A. Fort, et al., “A clearance measurement system based on on-component multilayer tri-axial capacitive probe,” Measurement 124(01), 575–581 (2018). [CrossRef]

14. S. Lavagnoli, G. Paniagua, M. Tulkens, et al., “High-fidelity rotor gap measurements in a short-duration turbine rig,” Mech. Syst. Signal Process. 27(1), 590–603 (2012). [CrossRef]

15. J. Binghui and H. Lei, “An optical fiber measurement system for blade tip clearance of engine,” Int. J. Aerosp. Eng. 2017, 1 (2017). [CrossRef]

16. G. Durana, J. Amorebieta, R. Fernandez, et al., “Design, fabrication and testing of a high-sensitive fibre sensor for tip clearance measurements,” Sensors 18(8), 2610 (2018). [CrossRef]

17. Y. Ren, P. Zhang, B. Shao, et al., “Correlation-changed-EMD algorithm for single frequency-sweep interferometry signal of high-speed rotating structure clearance measurement,” IEEE Sens. J. 23(23), 29366–29375 (2023). [CrossRef]

18. H. Liu, W. Zhang, B. Shao, et al., “Algorithm of Doppler error suppression in frequency-swept interferometry for the dynamic axial clearance measurement of high-speed rotating machinery,” Opt. Express 29(26), 42471 (2021). [CrossRef]

19. Y. Ren, L. Hao, Z. Peng, et al., “Fast algorithm of single frequency-swept interferometry for the dynamic axial clearance measurement of high-speed rotating machinery,” Opt. Express 31(3), 4253 (2023). [CrossRef]

20. R. Bao, F. Duan, X. Fu, et al., “Frequency-scanning interferometry for axial clearance of rotating machinery based on speed synchronization and extended Kalman filter,” Opt. Lasers Eng. 164(01), 107515 (2023). [CrossRef]

21. B. Shao, W. Zhang, P. Zhang, et al., “Dynamic Clearance Measurement Using Fiber-Optic Frequency-Swept and Frequency-Fixed Interferometry,” IEEE Photonics Technol. Lett. 32(20), 1331–1334 (2020). [CrossRef]

22. Y. Shang, J. Lin, L. Yang, et al., “Precision improvement in frequency scanning interferometry based on suppression of the magnification effect,” Opt. Express 28(4), 5822 (2020). [CrossRef]

23. L. Tao, Z. Liu, W. Zhang, et al., “Frequency-scanning interferometry for dynamic absolute distance measurement using Kalman filter,” Opt. Lett. 39(24), 6997 (2014). [CrossRef]

24. F. Duan, X. Li, R. Bao, et al., “High precision dynamic measurement method for axial clearance based on time division multiplexing with dispersive interferometry,” Opt. Express 31(17), 28549 (2023). [CrossRef]

25. G. Niu, F. Duan, Z. Liu, et al., “A high-accuracy non-contact online measurement method of the rotor-stator axial gap based on the microwave heterodyne structure,” Mech. Syst. Signal Process. 150, 107320 (2021). [CrossRef]

26. Z. Yu, W. Liu, F. Duan, et al., “High-accuracy measurement system for rotor-stator axial clearance in narrow spaces based on all-fiber microwave photonic mixing,” Opt. Express 31(13), 20994 (2023). [CrossRef]

27. X. Zou, B. Lu, W. Pan, et al., “Photonics for microwave measurements,” Laser Photonics Rev. 10(5), 711–734 (2016). [CrossRef]

28. Y. Gao, F. Shi, J. Zhang, et al., “Reconfigurable microwave photonic mixer for hybrid macro-micro cellular systems,” Opt. Express 31(4), 5314 (2023). [CrossRef]

29. S. Pan and J. Yao, “Photonics-based broadband microwave measurement,” J. Lightwave Technol. 35(16), 3498–3513 (2017). [CrossRef]

30. G. Wang, Q. Meng, Y. J. Li, et al., “Photonic-assisted multiple microwave frequency measurement with improved robustness,” Opt. Lett. 48(5), 1172 (2023). [CrossRef]

31. Y.-S. Jang, J. Park, and J. Jin, “Sub-100-nm precision distance measurement by means of all-fiber photonic microwave mixing,” Opt. Express 29(8), 12229 (2021). [CrossRef]

32. Y. S. Jang, J. Park, and J. Jin, “Periodic-error-free all-fiber distance measurement method with photonic microwave modulation toward on-chip-based devices,” IEEE Trans. Instrum. Meas. 71, 1–7 (2022). [CrossRef]

33. J. Li, D. Lu, and Z. Qi, “Miniature Fourier transform spectrometer based on wavelength dependence of half-wave voltage of a LiNbO_3 waveguide interferometer,” Opt. Lett. 39(13), 3923 (2014). [CrossRef]

34. F. Xi, S. Chen, and Z. Liu, “Quadrature compressive sampling for radar signals,” IEEE Trans. Signal Process. 62(11), 2787–2802 (2014). [CrossRef]

35. A. Matsuki, H. Kori, and R. Kobayashi, “An extended Hilbert transform method for reconstructing the phase from an oscillatory signal,” Sci. Rep. 13(1), 3535 (2023). [CrossRef]

36. H. Li, J. Kang, L. Feng, et al., “Phase extraction of optical carrier-based microwave interferometry with all-phase fast Fourier transform for distance measurement,” Opt. Lasers Eng. 156(April), 107090 (2022). [CrossRef]

37. H. Gao, Y. Jiang, Y. Cui, et al., “Investigation on the thermo-optic coefficient of silica fiber within a wide temperature range,” J. Lightwave Technol. 36(24), 5881–5886 (2018). [CrossRef]

38. L. Li, D. Lv, M. Yang, et al., “A IR-femtosecond laser hybrid sensor to measure the thermal expansion and thermo-optical coefficient of silica-based FBG at high temperatures,” Sensors 18(2), 3535 (2018). [CrossRef]

39. G. Ghosh, Handbook of Optical Constants of Solids: Thermo-Optic Coefficients, E. D. Palik, ed. (Academic, 1997), pp. 115–261.

40. J. D. Musgraves, J. Hu, and L. Calvez, Springer Handbook of Glass (Springer, 2019), Chap 29.

41. G. Rego, “Temperature dependence of the thermo-optic coefficient of SiO2 glass,” Sensors 23(13), 3535 (2023). [CrossRef]

42. G. Ghosh, “Model for the thermo-optic coefficients of some standard optical glasses,” J. Non. Cryst. Solids 189(1-2), 191–196 (1995). [CrossRef]

43. F. Messina, E. Vella, M. Cannas, et al., “Evidence of delocalized excitons in amorphous solids,” Phys. Rev. Lett. 105(11), 116401 (2010). [CrossRef]

44. G. Ghosh, Handbook of Optical Constants of Solids: Refractive Index, E. D. Palik, ed. (Academic, 1997), pp. 5–114.

45. S. Yuan and N. A. Riza, “General formula for coupling-loss characterization of single-mode fiber collimators by use of gradient-index rod lenses,” Appl. Opt. 38(15), 3214 (1999). [CrossRef]

46. Z. Zhao, F. Cao, S. Wang, et al., “Study on optical coupling characteristics of a high-radial-tolerance thick-lens beam expanding fiber collimator,” Infrared Phys. Technol. 133(June), 104846 (2023). [CrossRef]

47. S. D. Le, P. Rochard, J. B. Briand, et al., “Coupling efficiency and reflectance analysis of graded index expanded beam connectors,” J. Lightwave Technol. 34(9), 2092–2099 (2016). [CrossRef]

Object	Parameter	Symbol	Value
PI coated fiber	Room temperature length	${l_{T0}}$	1m
Fused silica	Sellmeier Coefficients^a	A	1.3091
		B	0.7951
		C	1.0944 $\times $ 10⁻²
		D	0.9211
		E	100
	The 1^st and 2^nd order of thermal expansion coefficients^b	${\alpha _1}$	5.36 $\times $ 10⁻⁷/°C
	The 1^st and 2^nd order of thermal expansion coefficients^b	${\alpha _2}$	6.2$\times $ 10⁻¹¹/°C²
	Low-frequency refractive index^a	${n_\infty }$	1.44
	Temperature coefficient of excitation bandgap^c	${H_0}$	10.415
		${H_1}$	4.0929 $\times $ 10⁻⁵
		${H_2}$	-6.6159 $\times $ 10⁻⁷
		${H_3}$	4.0092 $\times $ 10⁻¹⁰

Object	Parameter	Symbol	Value
OBPF Probe	Working Distance^a	${l_{WD}}$	12mm
	Return loss of the probe itself (Except optical window)	${\eta _{p0}}$	65dB
	Interstitial refractive index (Approximate value)	${n_1}$	1
OBPF Window	Isolation of reflection band (1530 nm and 1550 nm)	$is{o_r}$	25dB
Lens	Diameter	/	1.8mm
	Thickness	${l_c}$	3mm
	Refractive index	${n_c}$	1.4585
	Radius of curvature	${R_c}$	1.419mm
Fiber core	Mode field diameter	${r_{MFD}}$	5µm

Object	Parameter	Symbol	Value
PI coated fiber	Room temperature length	${l_{T0}}$	1m
Fused silica	Sellmeier Coefficients^a	A	1.3091
		B	0.7951
		C	1.0944 $\times $ 10⁻²
		D	0.9211
		E	100
	The 1^st and 2^nd order of thermal expansion coefficients^b	${\alpha _1}$	5.36 $\times $ 10⁻⁷/°C
	The 1^st and 2^nd order of thermal expansion coefficients^b	${\alpha _2}$	6.2$\times $ 10⁻¹¹/°C²
	Low-frequency refractive index^a	${n_\infty }$	1.44
	Temperature coefficient of excitation bandgap^c	${H_0}$	10.415
		${H_1}$	4.0929 $\times $ 10⁻⁵
		${H_2}$	-6.6159 $\times $ 10⁻⁷
		${H_3}$	4.0092 $\times $ 10⁻¹⁰

Object	Parameter	Symbol	Value
OBPF Probe	Working Distance^a	${l_{WD}}$	12mm
	Return loss of the probe itself (Except optical window)	${\eta _{p0}}$	65dB
	Interstitial refractive index (Approximate value)	${n_1}$	1
OBPF Window	Isolation of reflection band (1530 nm and 1550 nm)	$is{o_r}$	25dB
Lens	Diameter	/	1.8mm
	Thickness	${l_c}$	3mm
	Refractive index	${n_c}$	1.4585
	Radius of curvature	${R_c}$	1.419mm
Fiber core	Mode field diameter	${r_{MFD}}$	5µm

Axial clearance measurement method based on wavelength division multiplexing with all-fiber microwave photonic mixing

Abstract

1. Introduction

2. Principle

2.1 Principle of axial clearance measurement based on WDM with microwave photonic mixing

2.2 Theoretical model of axial clearance temperature drift for WDM-based common optical path structure

2.3 Theoretical design for OBPF probe

3. Simulations and results

3.1 Simulations of temperature-drift in axial clearance measurement

3.2 Simulations of return loss of OBPF probe

4. Experiments, results and discussion

4.1 Experimental setup

4.2 Measurement performance evaluation experiments

4.2.1 Measurement resolution, repeatability and accuracy evaluation

4.2.2 Temperature drift evaluation

5. Conclusions and prospect

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (21)

Tables (2)

Equations (31)

Optics Express