Investigation of Temperature Sensitivity of a Polymer-Overlaid Microfiber Mach-Zehnder Interferometer

The temperature sensitivity of the free spectral range (FSR) for a polymer-overlaid microfiber Mach-Zehnder interferometer (MZI) is investigated both theoretically and experimentally. The waist diameter of the optical microfiber can be controlled to alter the thermal expansion and optic properties of the polymer-coated MZI. Inserting an optical microfiber with a strong evanescent field into the MZI, a low index polymer with high thermal characteristics is deposited on the surface of the microfibers to realize a polymer-overlaid microfiber MZI. It was found that the thermal expansion factor in the proposed MZI plays an important role in the temperature sensitivity of the FSR. The temperature sensitivity of the polymer-overlaid microfiber MZI is improved, which is measured to be −8.29 nm/°C at 25 °C. The optical transmission spectrum of the polymer-overlaid microfiber MZI is converted to the spatial frequency spectrum via fast Fourier transform. The temperature sensitivity of the spatial frequency in the proposed polymer-overlaid MZI is estimated to be 18.31 pm−1 °C−1, which is 17 times higher than that of the microfiber MZI without polymer coating (1.04 pm−1 °C−1).

In this paper, a polymer-overlaid microfiber MZI is fabricated by coating the microfiber with low index polymer. By considering the waist diameter of the microfiber, the thermal expansion, and the thermo-optic factors in the polymer-overlaid MZI, the temperature sensitivity of the proposed microfiber MZI is investigated. To initially configure a microfiber MZI, a microfiber with a waist diameter of 10 µm is fabricated. Then, low index polymer with high thermal quantiles is deposited on the surface of the microfiber, and consequently the polymer-overlaid microfiber MZI is realized. By considering the contributions of the thermal properties of polymer-coated microfibers to the temperature sensitivity of the proposed microfiber MZI, it was determined that the thermal expansion factor is higher than the thermo-optic factor. Further, the temperature sensitivity of the free spectral range (FSR) in the polymer-overlaid microfiber MZI is improved, and is measured to be −8.29 nm/ • C at 25 • C. The transmission spectra of the proposed polymer-overlaid microfiber MZI is transformed to the spatial frequency spectra using fast Fourier transform (FFT). Compared with the temperature sensitivity of the microfiber MZI without the low index polymer overlay (1.04 pm −1 • C −1 ), the temperature sensitivity of the spatial frequency is effectively increased via the proposed polymer-overlaid microfiber MZI, and is measured to be 18.31 pm −1 • C −1 in the temperature range of 25 to 80 • C. Figure 1 depicts the experimental configuration of the polymer-overlaid microfiber MZI, which includes two 3-dB couplers, as well as reference and sensing arms. The reference arm (Region 3) is composed of a conventional single-mode fiber (SMF) with a diameter of 125 µm and a length of 46.8 cm (l 3 ). The sensing arm contains a polymer-overlaid microfiber (Region 1) between two SMFs (Region 2) with a total length of 40.6 cm (l 2 ). A flame brushing technique was exploited to adiabatically taper an SMF to gradually reduce the waist diameter [25]. The SMF fixed on the motorized translation stage was elongated by exposure to a computer-controlled heater. High-order modes in the microfiber can be excited from the fundamental core mode by non-adiabatically tapering the SMF, resulting in an in-line modal interferometer depending on the waist diameter of the microfiber [25][26][27]. The waist diameter and the length (l 1 ) of the fabricated microfiber were measured to be~10 µm and~6.3 cm, respectively. Then, the microfiber was spin-coated using a UV-curable polymer with a low refractive index (PC-373, SSCP, n = 1.375) and, consequently, the polymer-overlaid microfiber MZI was realized. Figure 2 shows the experimental results of the transmission spectra of the microfiber MZI (black line) and the polymer-overlaid microfiber MZI (red line). The free spectral range (FSR) of the microfiber MZI at room temperature (25 • C) changed from 47.5 to 27.8 nm before and after polymer coating, respectively. Since the polymer coating overlay increased the effective refractive index of the microfiber, the optical path length difference of the arms in the proposed microfiber MZI would also be increased. Consequently, the FSR should be diminished by coating the microfiber MZI with low index polymer. By considering the contributions of the thermal properties of polymer-coated microfibers to the temperature sensitivity of the proposed microfiber MZI, it was determined that the thermal expansion factor is higher than the thermo-optic factor. Further, the temperature sensitivity of the free spectral range (FSR) in the polymer-overlaid microfiber MZI is improved, and is measured to be −8.29 nm/°C at 25 °C. The transmission spectra of the proposed polymer-overlaid microfiber MZI is transformed to the spatial frequency spectra using fast Fourier transform (FFT). Compared with the temperature sensitivity of the microfiber MZI without the low index polymer overlay (1.04 pm −1 °C −1 ), the temperature sensitivity of the spatial frequency is effectively increased via the proposed polymer-overlaid microfiber MZI, and is measured to be 18.31 pm −1 °C −1 in the temperature range of 25 to 80 °C. Figure 1 depicts the experimental configuration of the polymer-overlaid microfiber MZI, which includes two 3-dB couplers, as well as reference and sensing arms. The reference arm (Region 3) is composed of a conventional single-mode fiber (SMF) with a diameter of 125 μm and a length of 46.8 cm (l3). The sensing arm contains a polymer-overlaid microfiber (Region 1) between two SMFs (Region 2) with a total length of 40.6 cm (l2). A flame brushing technique was exploited to adiabatically taper an SMF to gradually reduce the waist diameter [25]. The SMF fixed on the motorized translation stage was elongated by exposure to a computer-controlled heater. High-order modes in the microfiber can be excited from the fundamental core mode by non-adiabatically tapering the SMF, resulting in an in-line modal interferometer depending on the waist diameter of the microfiber [25][26][27]. The waist diameter and the length (l1) of the fabricated microfiber were measured to be ~10 μm and ~6.3 cm, respectively. Then, the microfiber was spin-coated using a UV-curable polymer with a low refractive index (PC-373, SSCP, n = 1.375) and, consequently, the polymer-overlaid microfiber MZI was realized. Figure 2 shows the experimental results of the transmission spectra of the microfiber MZI (black line) and the polymer-overlaid microfiber MZI (red line). The free spectral range (FSR) of the microfiber MZI at room temperature (25 °C) changed from 47.5 to 27.8 nm before and after polymer coating, respectively. Since the polymer coating overlay increased the effective refractive index of the microfiber, the optical path length difference of the arms in the proposed microfiber MZI would also be increased. Consequently, the FSR should be diminished by coating the microfiber MZI with low index polymer.

Theoretical Analysis of the Temperature Sensitivity of the FSR in the Polymer-Overlaid Microfiber MZI
In general, the FSR (Δλ) of the MZI can be written as [7]: where λ1 and λ2 represent the two adjacent maximum and minimum wavelengths, respectively, and Δφ is the optical path length difference between the two arms in the MZI. From Equation (1), the dependence of the FSR on the temperature (T) can be described by: The proposed microfiber MZI has two different optical sections that consist of polymer-overlaid microfibers and SMFs. The optical path length difference (Δφ) between the sensing and the reference arms in the polymer-overlaid MZI can be described as where φ1, φ2, and φ3 are the optical path lengths in Regions 1, 2, and 3, respectively, and n1, n2, n3 and l1, l2, l3 are the effective refractive indices and the physical lengths of Regions 1, 2, 3, respectively. Since Region 1 (microfiber), when viewed as a sensing element, is only exposed to different temperatures via a heating oven, the optical quantities of Regions 2 and 3 (n2, n3, l2, and l3) cannot be changed by changing the temperature. The derivative of the optical path length difference with respect to temperature can be written as: By substituting Equation (4) into Equation (2), we obtain: In Equation (5), it is evident that two dominant factors, the thermal expansion (TE) and the thermo-optic factors (TO), play an important role in determining the temperature sensitivity of the FSR in the polymer-overlaid microfiber MZI. By definition of the thermal expansion coefficient (α = ∂l/l∂T), the thermal expansion (TE) term in Equation (5) can be modified as:

Theoretical Analysis of the Temperature Sensitivity of the FSR in the Polymer-Overlaid Microfiber MZI
In general, the FSR (∆λ) of the MZI can be written as [7]: where λ 1 and λ 2 represent the two adjacent maximum and minimum wavelengths, respectively, and ∆φ is the optical path length difference between the two arms in the MZI. From Equation (1), the dependence of the FSR on the temperature (T) can be described by: The proposed microfiber MZI has two different optical sections that consist of polymer-overlaid microfibers and SMFs. The optical path length difference (∆φ) between the sensing and the reference arms in the polymer-overlaid MZI can be described as where φ 1 , φ 2 , and φ 3 are the optical path lengths in Regions 1, 2, and 3, respectively, and n 1 , n 2 , n 3 and l 1 , l 2 , l 3 are the effective refractive indices and the physical lengths of Regions 1, 2, 3, respectively. Since Region 1 (microfiber), when viewed as a sensing element, is only exposed to different temperatures via a heating oven, the optical quantities of Regions 2 and 3 (n 2 , n 3 , l 2 , and l 3 ) cannot be changed by changing the temperature. The derivative of the optical path length difference with respect to temperature can be written as: By substituting Equation (4) into Equation (2), we obtain: In Equation (5), it is evident that two dominant factors, the thermal expansion (TE) and the thermo-optic factors (TO), play an important role in determining the temperature sensitivity of the FSR in the polymer-overlaid microfiber MZI. By definition of the thermal expansion coefficient (α = ∂l/l∂T), the thermal expansion (TE) term in Equation (5) can be modified as: where α 1 is the thermal expansion coefficient of the polymer-coated microfiber (Region 1), which consists of two terms, α Silica and α Coating . Microfibers with a small waist diameter of less than 10 µm can be largely considered as a fused silica waveguide [28,29]. To improve the thermal expansion factor in Equations (5) and (6), we exploited the low index coating polymer with a high thermal expansion coefficient (α Coating = 400 × 10 −6 / • C), which is 800 times higher than that of fused silica (α Silica = 0.55 × 10 −6 / • C). Consequently, we approximated α 1 in Equation (6) and found it to be mostly determined by the thermal expansion coefficient of the low index polymer (α Coating ); this is because the thermal expansion coefficient of silica (α Silica ) is negligible. The thermo-optic factor (TO) in Equation (5) can be written as [7,28,29]: where σ Silica (= 1.1 × 10 −5 / • C) and σ Coating (= −3.4 × 10 −4 / • C) are the thermo-optic coefficients of fused silica and the coating polymer, respectively, and n Coating is the refractive index of the coating polymer around the microfiber. By substituting Equations (6) and (7) into Equation (5), the temperature sensitivity of the FSR in the polymer-overlaid microfiber MZI can be derived as: where the first and second terms in the parentheses correspond to the thermal expansion (TE) and thermo-optic (TO) terms, respectively, in the polymer-overlaid microfiber MZI. By using the finite element method (FEM), we analyzed the effect of the thermal expansion and the thermo-optic factors on the FSR variation in the proposed polymer-overlaid microfiber MZI as the external temperature increased. In Equation (8), it is clear that the thermal expansion factor (TE) can be readily achieved by considering α Coating and the effective index (n 1 ) of the polymer-overlaid microfiber. After theoretically evaluating the variation of the effective refractive index (n 1 ) for the microfiber relative to variations in the refractive index of the low index polymer (n Coating ), we obtained the value of ∂n 1 /∂n Coating in order to analyze the total thermo-optic factor (TO) in Equation (8). Since the effective index of the microfiber with a strong evanescent field is highly sensitive to ambient index change, the value of ∂n 1 /∂n Coating depends on the waist diameter of the microfiber [30]. Figure 3a presents the FSR variation in the polymer-overlaid microfiber MZI as a function of temperature when considering the thermal expansion and thermo-optic factors. It is evident that the temperature sensitivity of the FSR in the polymer-overlaid microfiber MZI is eventually negative, which is predominantly due to the thermal expansion factor (black square), as seen in Figure 3a. Increasing the external temperature consequently reduces the FSR of the polymer-overlaid MZI. Figure 3b shows the theoretical results of the spatial frequency shift as a function of temperature after applying FFT. Since the spatial frequency (∆f ) is inversely proportional to the FSR, the spatial frequency shifts to higher frequencies with increasing temperature. The temperature sensitivities of the spatial frequencies (∆f ) affected by the thermal expansion and the thermo-optic terms were theoretically estimated to bẽ 17.2 and~1.05 pm −1 • C −1 , respectively. It is notable that the contribution of the thermal expansion factor to the temperature sensitivity of the FSR in the polymer-overlaid microfiber MZI is~16 times higher than that of the thermo-optic term.  Figure 4 depicts the theoretical (black line) and experimental results (red circles and blue triangles) on the variation of the FSR in the proposed polymer-overlaid microfiber MZI as a function of temperature. The temperature dependence of the FSR shows a strong nonmonotonic behavior because the FSR of the polymer-overlaid microfiber MZI is inversely proportional to Δn⋅l. As seen in Figure 4 (specifically the red circles), increasing the external temperature reduced the FSR of the polymer-overlaid microfiber MZI due to the enhanced thermal expansion and thermos-optic terms of the polymer-overlaid microfiber MZI. In contrast, decreasing the temperature increased the FSR of the polymer-overlaid microfiber MZI, as seen in Figure 4 (blue triangle). No differences in the FSR variations via increasing or decreasing the temperature were observed. The temperature sensitivity of the FSR was experimentally measured to be −8.29 nm/°C at a temperature of 25 °C, as shown in Figure 4. The theoretical results were in good agreement with the experimental ones.  Figure 5 shows the spatial frequency spectra of the polymer-overlaid microfiber MZI (transformed using FFT) as the applied temperature increases. The zero padding method was applied to enhance the frequency resolution and analysis accuracy [31]. Figures 6a,b show the spatial frequency shift (Δf) with and without polymer overlay, respectively, as a function of temperature. The nonmonotonic behavior of the temperature dependence of the FSR was effectively eliminated by considering the spatial frequency after applying the FFT, as seen in Figures 6a,b. As seen in Figure 6,    Figure 4 depicts the theoretical (black line) and experimental results (red circles and blue triangles) on the variation of the FSR in the proposed polymer-overlaid microfiber MZI as a function of temperature. The temperature dependence of the FSR shows a strong nonmonotonic behavior because the FSR of the polymer-overlaid microfiber MZI is inversely proportional to ∆n·l. As seen in Figure 4 (specifically the red circles), increasing the external temperature reduced the FSR of the polymer-overlaid microfiber MZI due to the enhanced thermal expansion and thermos-optic terms of the polymer-overlaid microfiber MZI. In contrast, decreasing the temperature increased the FSR of the polymer-overlaid microfiber MZI, as seen in Figure 4 (blue triangle). No differences in the FSR variations via increasing or decreasing the temperature were observed. The temperature sensitivity of the FSR was experimentally measured to be −8.29 nm/ • C at a temperature of 25 • C, as shown in Figure 4. The theoretical results were in good agreement with the experimental ones.  Figure 4 depicts the theoretical (black line) and experimental results (red circles and blue triangles) on the variation of the FSR in the proposed polymer-overlaid microfiber MZI as a function of temperature. The temperature dependence of the FSR shows a strong nonmonotonic behavior because the FSR of the polymer-overlaid microfiber MZI is inversely proportional to Δn⋅l. As seen in Figure 4 (specifically the red circles), increasing the external temperature reduced the FSR of the polymer-overlaid microfiber MZI due to the enhanced thermal expansion and thermos-optic terms of the polymer-overlaid microfiber MZI. In contrast, decreasing the temperature increased the FSR of the polymer-overlaid microfiber MZI, as seen in Figure 4 (blue triangle). No differences in the FSR variations via increasing or decreasing the temperature were observed. The temperature sensitivity of the FSR was experimentally measured to be −8.29 nm/°C at a temperature of 25 °C, as shown in Figure 4. The theoretical results were in good agreement with the experimental ones.  Figure 5 shows the spatial frequency spectra of the polymer-overlaid microfiber MZI (transformed using FFT) as the applied temperature increases. The zero padding method was applied to enhance the frequency resolution and analysis accuracy [31]. Figures 6a,b show the spatial frequency shift (Δf) with and without polymer overlay, respectively, as a function of temperature. The nonmonotonic behavior of the temperature dependence of the FSR was effectively eliminated by considering the spatial frequency after applying the FFT, as seen in Figures 6a,b. As seen in Figure 6,    Figure 5 shows the spatial frequency spectra of the polymer-overlaid microfiber MZI (transformed using FFT) as the applied temperature increases. The zero padding method was applied to enhance the frequency resolution and analysis accuracy [31]. Figure 6a,b show the spatial frequency shift (∆f ) with and without polymer overlay, respectively, as a function of temperature. The nonmonotonic behavior of Sensors 2017, 17, 2403 6 of 8 the temperature dependence of the FSR was effectively eliminated by considering the spatial frequency after applying the FFT, as seen in Figure 6a,b. As seen in Figure 6, increasing the temperature shifted the spatial frequency peak to higher frequencies (red circles) because of the decreased FSR in the polymer-overlaid microfiber MZI. Conversely, decreasing the temperature shifted the spatial frequency peak to lower frequencies (blue squares). However, no severe differences in the spatial frequency shifts when increasing or decreasing the temperature were observed. The temperature sensitivity of the spatial frequency peak of the polymer-overlaid microfiber MZI was higher than that without a polymer overlay. The temperature sensitivity of the spatial frequency in the polymer-overlaid microfiber MZI was measured to be 18.3 pm −1 • C −1 , which was~17 times higher than that of the microfiber MZI without polymer coating (1.04 pm −1 • C −1 ). increasing the temperature shifted the spatial frequency peak to higher frequencies (red circles) because of the decreased FSR in the polymer-overlaid microfiber MZI. Conversely, decreasing the temperature shifted the spatial frequency peak to lower frequencies (blue squares). However, no severe differences in the spatial frequency shifts when increasing or decreasing the temperature were observed. The temperature sensitivity of the spatial frequency peak of the polymer-overlaid microfiber MZI was higher than that without a polymer overlay. The temperature sensitivity of the spatial frequency in the polymer-overlaid microfiber MZI was measured to be 18.3 pm −1 °C −1 , which was ~17 times higher than that of the microfiber MZI without polymer coating (1.04 pm −1 °C −1 ).

Conclusions
The temperature sensitivity of the FSR in a polymer-overlaid microfiber MZI was investigated by considering the thermal expansion and thermo-optic factors. After fabricating the microfiber with a waist diameter of ~10 μm, the microfiber was spin-coated with a low index polymer and eventually the polymer-overlaid microfiber MZI was configured. Since the microfiber has a strong evanescent field and a small waist diameter, the temperature sensitivity of the microfiber MZI was inevitably affected by the thermal properties of the low index polymer overlay. The effects of the thermal expansion and the thermo-optic factors on the temperature sensitivity of the FSR in the proposed polymer-overlaid microfiber MZI were theoretically analyzed. It was determined that the   increasing the temperature shifted the spatial frequency peak to higher frequencies (red circles) because of the decreased FSR in the polymer-overlaid microfiber MZI. Conversely, decreasing the temperature shifted the spatial frequency peak to lower frequencies (blue squares). However, no severe differences in the spatial frequency shifts when increasing or decreasing the temperature were observed. The temperature sensitivity of the spatial frequency peak of the polymer-overlaid microfiber MZI was higher than that without a polymer overlay. The temperature sensitivity of the spatial frequency in the polymer-overlaid microfiber MZI was measured to be 18.3 pm −1 °C −1 , which was ~17 times higher than that of the microfiber MZI without polymer coating (1.04 pm −1 °C −1 ).

Conclusions
The temperature sensitivity of the FSR in a polymer-overlaid microfiber MZI was investigated by considering the thermal expansion and thermo-optic factors. After fabricating the microfiber with a waist diameter of ~10 μm, the microfiber was spin-coated with a low index polymer and eventually the polymer-overlaid microfiber MZI was configured. Since the microfiber has a strong evanescent field and a small waist diameter, the temperature sensitivity of the microfiber MZI was inevitably affected by the thermal properties of the low index polymer overlay. The effects of the thermal expansion and the thermo-optic factors on the temperature sensitivity of the FSR in the proposed polymer-overlaid microfiber MZI were theoretically analyzed. It was determined that the

Conclusions
The temperature sensitivity of the FSR in a polymer-overlaid microfiber MZI was investigated by considering the thermal expansion and thermo-optic factors. After fabricating the microfiber with a waist diameter of~10 µm, the microfiber was spin-coated with a low index polymer and eventually the polymer-overlaid microfiber MZI was configured. Since the microfiber has a strong evanescent field and a small waist diameter, the temperature sensitivity of the microfiber MZI was inevitably affected by the thermal properties of the low index polymer overlay. The effects of the thermal expansion and the thermo-optic factors on the temperature sensitivity of the FSR in the proposed polymer-overlaid microfiber MZI were theoretically analyzed. It was determined that the thermal expansion contribution to the temperature sensitivity of the FSR for the polymer-overlaid microfiber MZI was 16 times higher than that of the thermo-optic factor. The temperature sensitivity of the FSR in the polymer-overlaid microfiber MZI was successfully improved to −8.29 nm/ • C at a temperature of 25 • C. After applying the FFT and zero padding methods, the optical spectra of the polymer-overlaid microfiber MZI was converted to spatial frequency spectra. The temperature sensitivity of the spatial frequency in the proposed polymer-overlaid MZI was found to be 18.31 pm −1 • C −1 in the temperature range of 25 to 80 • C, which was 17 times higher than that of the microfiber MZI without polymer coating (1.04 pm −1 • C −1 ).