1 Introduction

Optical all-fiber interferometric comb filters have attracted much research interest because of their potential applications in wavelength-division-multiplexed (WDM) fiber communication systems, compact optical fiber sensors, multiwavelength laser sources, etc. Independent of their alternative topological configurations, such as Michelson, Mach–Zehnder, Fabry–Perot, and Sagnac, different implementations of these filters have been proposed and experimentally demonstrated [1–6]. However, these kinds of optical filters do not satisfy most of the requirements of a multiwavelength fiber laser source as well as of a temperature sensor, such as tunable center wavelengths, tunable wavelength spacings, switchable wavelength channels, and good temperature sensitivity. Thus, optical comb filters that can satisfy these requirements are desired for the development of multiwavelength fiber laser sources and temperature sensors for future applications. Several architectures have been proposed for tunable comb filters, in order to enhance their spacing tunability, such as a combination of a sampled chirped fiber Bragg grating with a beam bending technique [6, 7] and multisegment polarization-maintaining fiber (PMF) comb filters [8, 9]. A Mach–Zehnder interferometer (MZI) filter is usually considered as a nontunable optical filter unless an optical variable delay line is employed to change the path difference [10]. However, in a recent work on a tunable polarization-controlled all-fiber comb filter based on a modified dual-pass MZI structure [11], single-pass and double-pass characteristics of a conventional MZI were demonstrated using a polarization controller (PC) in the interferometric arm. A polarization-dependent isolator and another PC were employed at the input to control the polarization, which allowed two discrete spectral spacing tuning for single-pass and double-pass configurations. However, none of the proposed designs were able to realize filters with continuous tunability [5–12]. Spectral spacing can be selected by adjusting the polarization state of the light beam inside the loop. Normally, it is performed using a PC [13, 14]. However, a PMF is made of different glasses having different thermal expansion coefficients; therefore, the propagation behavior of the wave in the fiber may change with varying temperature [15]. In addition, the temperature-dependent birefringence effect has high potential for application to temperature sensors [16]. Fiber optic sensors have high durability to harsh environments, high sensitivity, stability, high resolution, fast response, and immunity to electromagnetic interference [17]. Fiber optic temperature sensors are particularly useful in environments in which the optical signal is not affected by external factors. They also provide the advantage of facilitating remote sensing without much effort. In addition, fiber optic temperature sensors are of interest for many industrial applications; they produce minimal disturbance in the measurement environment, are passive and immune to electrical interference, and may act as their own telemetric channel [18–21]. Recently, long-period fiber gratings have been proposed [22] for temperature and strain sensing. These sensors exhibit higher sensitivity and low back-reflections and can be implemented with economical signal demodulation schemes [22]. However, the high bend sensitivity of the long-period gratings may cause undesirable changes in the spectral response. Birefringent fiber interferometers and wavelength-division multiplexers [23, 24], in which two different polarization modes exhibit different responses to temperature, can also be regarded as temperature sensors. However, such interferometric sensors are polarization-sensitive and the consequent need for polarization control elements complicates the scheme and adds to the cost of the measurement system. A thermally controlled fiber Sagnac loop mirror has been proposed previously [25], but its extinction ratio (ER) is low and the change in spectral spacing is not continuous. This mirror is also polarization-sensitive. Birefringence as a function of temperature can be used to manipulate the spectral spacing as well as the shift in the peak wavelength position. In the present work, we aim to address these issues by proposing a simple thermally controlled polarization-independent tunable comb filter with flexible and continuous tunability of its spectral spacing. The filter combines the advantages of an MZI and a Sagnac interferometer with the unique filtering and temperature sensing properties of fiber birefringence. The theoretical analyses governing the filter design are presented together with experimental results in order to validate the feasibility of the developed filter.

2 Theoretical analysis

Both the temperature dependence of the spectral spacing and the phase shifting of the transmission spectrum are functions of two parameters: birefringence and the length of the PMF segment. The shift of the peak wavelength position of the transmission spectrum is a result of the phase shifting of two incoming light waves. As the variation in birefringence with temperature is three orders of magnitude larger than that in the length [26], it is reasonable to neglect the length variation for simplicity of analysis. In this proposed configuration, the PMF segment consists of two different lengths that are spliced at an offset rotation of 90° relative to the principal axes of the PMF, which results in a wavelength spacing ∆λ of the peaks:

$$\varDelta \lambda = \frac{{\lambda^{2} }}{{\varDelta n_{\text{eo}} L_{\text{eff}} }},$$
(1)

where ∆n eo is the effective birefringence between the two orthogonal polarization modes, L eff is the effective length of the two PMF segments, and λ is the center wavelength of the light source.

Phase difference ϕ occurs between the two fields propagating along the fast and slow axes and can be expressed as

$$\phi = \frac{{2\pi \varDelta n_{\text{eo}} L_{\text{eff}} }}{\lambda }.$$
(2)

Suppose that the PMF segment of the filter is placed in a furnace of temperature T (°C); then, the spectral spacing and phase difference can be expressed as a function of T as follows:

$$\varDelta \lambda (T) = \frac{{\lambda^{2} }}{{\varDelta n_{\text{eo}} (T)L_{\text{eff}} }}$$
(3)
$$\varphi (T) = \frac{{2\pi \varDelta n_{\text{eo}} (T)L_{\text{eff}} }}{\lambda }.$$
(4)

In the analysis, the temperature response of birefringence can well be characterized by the second-order polynomial model expressed in Eq. (5) and the higher-order terms are considered as negligibly small.

$$\varDelta n_{\text{eo}} (T) = \varDelta n_{\text{eo}} (T_{0} ) - b_{1} (T - T_{0} ) - b_{2} (T - T_{0} )^{2} ,$$
(5)

where b 1, b 2 > 0. The variation is well modeled by the polynomial expression in Eq. (5) for b 1 = 2.55 × 10−7 °C−1, b 2 = 1.5 × 10−9 °C−2, and Δn eo (T 0 = 30 °C) = 4.02 × 10−4. The birefringence is calculated by Eq. (1) based on the spectral spacing obtained experimentally.

3 Experimental setup

The general function of the proposed filter is described by referring to its schematic shown in Fig. 1. The filter was designed with two couplers (3 dB and non −3 dB): one PMF segment and one PC. The employed PC consisted of two quarter-wave plates (QWPs) and one half-wave plate (HWP).

Fig. 1
figure 1

Schematic of proposed thermally tunable all-fiber comb filter based on dual-pass MZI

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The PMF segment comprises two PMF lengths L 1 and L 2 that were set at 6 and 3 m, respectively. Both fibers were spliced to each other by rotating the principal axis of L 2 at 90° relative to that of L 1. The rationale behind selecting such a configuration was to allow variation in the effective length of the PMF segment, which is the sum of L 1 and L 2 depending on the alignment of the principal axes of the fibers and enables the two PMFs to act as a single fiber length. Through tuning of the HWP of the PC, the axes can be aligned in the same or the opposite direction, thus realizing the corresponding maximum |L 1 + L 2| or minimum |L 1 − L 2| effective length. The sinusoidal output spectrum of the filter is the product of interference due to the phase difference created during the propagation of the two waves in the slow and fast axes. Changes in the effective length of the PMF segment affect both the phase difference and the birefringence properties of the fiber, since the phase difference, φ = 2πΔn eo L eff/λ, is dependent on both the effective length and Δn eo (i.e., the fiber birefringence). The effective length of the PMF also affects the spectral spacing, which is given by Δλ = λ 2/ΔnL eff. Therefore, variation in the effective length of the PMF causes a variation in the spectral spacing. The PMF segment is placed inside the furnace. Further, to ensure a consistent and uniform distribution of temperature, the fiber segment is placed on the metal plate, which in turn is placed at the center of the furnace. The temperature of the furnace is set to range from 30 to 100 °C in 10 °C steps, and the output transmission spectrum is recorded. The temperature is increased in 10 °C steps at a time interval of 10 min. In this experiment, light propagation is handled in the concatenated L 1 and L 2 segments, leading to the two length extremes L 1 + L 2 and L 1 − L 2. During the experiment, the PC is not used for tuning the spectral spacing or the shifting of the peak wavelength position. For both the experiments, an amplified stimulated emission (ASE) source is used along with an optical spectrum analyzer (OSA) as the input source to obtain the transmission spectrum.

4 Results and discussion

Birefringence is a function of temperature: It decreases with increasing temperature. Figure 2 shows the variation in birefringence with temperature for the PMF segment with lengths L 1 = 6 m and L 2 = 3 m.

Fig. 2
figure 2

Variation in birefringence in PMF with temperature

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The variation is well modeled by the polynomial expression in Eq. (5). The experimental results for the variation in birefringence with temperature are almost similar to the theoretical calculation results. The change in birefringence causes a small change in the length of the fiber. This small change is responsible for creating a phase shift among the propagating light waves [27]. Therefore, a continuous shift of the peak wavelength position occurs with a small change in temperature. The temperature dependence of the spectral spacing is investigated by considering two important related parameters: the birefringence and the effective length of the PMF segments. As mentioned earlier, the variation in birefringence with temperature is three orders of magnitude larger than that in the length. Therefore, the temperature dependence property of the birefringence of the PMF material is employed to change the spectral spacing. In this experiment, temperature is varied from 30 to 100 °C, and the output transmission at different spectral spacings is recorded at intervals of 10 °C. The experiment is separately performed for the maximum and minimum effective lengths of the PMF segment. The spectral spacing is different at different temperatures. The output transmission at different spectral spacings for the maximum and minimum effective lengths is shown in Fig. 3a, b, respectively. From Fig. 3a, it is seen that the spectral spacing changes from 0.72 to 1.48 nm with a change in temperature from 30 to 100 °C. Thermal excitement of the PMF material leads to a change in the birefringence magnitude as well as the effective length of the PMF segment. To observe the stability of the filter, the temperature is kept constant at every increment step for 30 min during the experiment, and the output transmission during this period is confirmed to be stable.

Fig. 3
figure 3

Output transmission with different spectral spacings at different temperatures for effective lengths of a |L 1 + L 2| and b |L 1 − L 2|

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The maximum spectral spacing of 1.48 nm is observed at 100 °C for the maximum effective length (i.e., |L 1 + L 2|). Owing to the limitations of our thermal equipment and surrounding environment, the temperature was not increased beyond 100 °C during the experiment. However, according to the theoretical calculation, the maximum spectral spacing occurs at 190 °C, as shown in Fig. 4. At the minimum effective length, the spectral spacing varies from 2.00 to 2.90 nm for the same temperature range. Therefore, the tunable range depends on the effective length of the PMF segment. Depending on the intended application, the tunable range can be adjusted by selecting an appropriate length of the PMF segment. The main objective of this experiment is to present the technique for changing the birefringence to tune the spectral spacing as well as the peak wavelength position continuously by exploiting the thermal effect. According to the theoretical analysis, it is desirable to obtain different spectral spacings for both the effective lengths above the temperature of 100 °C. The ER is measured to be 21 dB, and it does not degrade during this experiment.

Fig. 4
figure 4

Variation in spectral spacing with temperature

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A nonlinear variation in birefringence with temperature results in a nonlinear relation between the spectral spacing and the temperature. The variation in spectral spacing with temperature is shown in Fig. 4.

Based on these findings, we can see that the change in spectral spacing is minimal for changes in temperature from 30 to 60 °C. However, the spectral spacing varies nonlinearly with a linear change in temperature from 60 to 100 °C. The gradient of change in the spectral spacing within a 10 °C band is calculated and presented in Table 1.

Table 1 The gradient of spectral spacing
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At temperatures lower than 60 °C, the change in spectral spacing with temperature variation is less than 10 pm/°C for both effective lengths. It is observed from Table 1 that above 60 °C, the gradient of the spectral spacing changes considerably. This results in an increase in the sensitivity of the comb filter to temperature. There is also a slight difference between the experimental and theoretical observations for both lengths. A common scenario encountered for a birefringent material is that there are two distinct indices of refraction. The birefringence property is related to anisotropy in the binding forces between the atoms to form a material. When the birefringent material is stimulated with temperature, the binding forces weaken, which causes a change in the birefringence property [28]. The vibration of atoms at low temperatures is smaller than that at high temperatures; an increase in temperature lends additional energy to an atom, so it vibrates more vigorously. Owing to this, an abrupt change in the vibration of atoms causes random interaction among them. The random interaction, in turn, induces a nonlinear change in both the birefringence and the spectral spacing at high temperatures. Theoretically, atoms in a birefringent material are considered to have a uniform arrangement, but practically, a few atomic dislocations have been reported [28]. During stimulation with temperature, these atomic dislocations affect the experimental observation, as compared to the theoretical calculations. During the experiment, the PMF segment is initially in a relaxation state (i.e., free from any external pressure) and the strain condition is assumed to be constant. Referring to Table 1, the spectral spacing remains almost constant in the temperature range of 10–60 °C. Our observations suggest that in this temperature range, a 1 °C rise in temperature leads to a change of less than 10 pm in the spectral spacing. As mentioned above, to check the stability of the filter, the temperature is kept constant for 30 min at each step of the temperature increment. In this period, the peak wavelength does not shift at all. Even though the variations in spectral spacing are almost negligible, the peak wavelength can be shifted continuously within the free spectral range (FSR), as depicted in Fig. 5.

Fig. 5
figure 5

Variation in peak wavelength position with temperature

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The experimental results for the variation in peak wavelength position with temperature for the maximum effective length of the PMF segment are obtained. Initially, the temperature is set to 40 °C, and the transmission spectrum is recorded at 1 °C increments up to 45 °C. Figure 5 shows that within this 5 °C variation range of temperature, the peak wavelength position shifts from one peak to a consecutive peak. The same results are recorded for any reference temperature within the range of 30–60 °C. Here, the filter response time is limited by the thermal response of the heating device. Figure 6 shows a comparison between the theoretical calculation and the experimental observation of the shift in the peak wavelength position with temperature. In the experiment, the gradients of the shift in the peak wavelength position are found to be 0.15 and 0.41 nm/°C, respectively, for the maximum and minimum effective lengths, whereas the corresponding theoretical gradients are 0.18 and 0.35 nm/°C, respectively. The shift in the peak wavelength position has a linear relationship with the variation in temperature. For the maximum effective length, the experimental observations and theoretical calculations are found to differ slightly. An atom of a material with lower volume absorbs more energy than that of a material with higher volume when the materials are stimulated at the same temperature [29]. As the temperature increases, the vibration of atoms increases, which causes random interaction among them after a certain temperature [29].

Fig. 6
figure 6

Comparison between theoretical calculation and experimental observation of variation in peak wavelength position with temperature

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The increased vibration of the atoms causes a phase change. The volume of the fiber material at the minimum effective length is smaller than that at the maximum effective length. Therefore, vibration increases faster in the case of the minimum effective length than in the case of the maximum effective length. These behaviors are responsible for the significant discrepancy between the experimental observations and theoretical calculations. Another factor responsible for this discrepancy is an elongation in the original length due to the thermal effect. Theoretically, the length of the PMF segment is assumed as constant throughout the experiment; in other words, thermal expansion of the PMF segment is not taken into account in the theoretical calculation.

5 Conclusion

We have successfully demonstrated a newly developed thermally tunable all-fiber comb filter that is based on the MZI. By adjusting the temperature, we could tune the spectral spacings of the filter from 0.72 to 1.48 nm (maximum effective length) and from 2.00 to 2.90 nm (minimum effective length), which are restricted by the effective PMF length. The range of spectral spacing can be designed and controlled easily by changing the length of the PMF segment. Further, the shifting of the peak wavelength position can be achieved by adjusting the temperature, and the peak wavelength position can be tuned continuously within the entire FSR. The developed filter also has advantages of having a simple implementation and being cost-effective. This comb filter can be used as a multiwavelength laser source, temperature sensor, and a self-interference interferometer, the last of which has high potential for use in miniaturized sensors for some parametric sensing applications.