Fiber Bragg grating Fabry–Perot cavities (FBG-FPs) are desirable for narrow-band filters and higher accuracy wavelength measurement, due to the fact that the linewidth of their ultra-narrow resonance peaks in the stopband of the fiber Bragg grating (FBGs) could easily be reduced to several MHz. Hence FBG-FP is one of the most important fiber components for lasers and suppressing noise,^[1–5] tunable filters,^[6–8] wavelength division multiplexing,^[9] and fiber sensing.^[10–12] Wei et al. designed an external cavity diode laser that realized a noise suppression of over 70 dB with the Fourier frequencies between 5 Hz and 1 kHz by an optical feedback ring based on FBG-FP cavity.^[1] A short-linear-cavity (SLC) single-frequency fiber laser based on the slow-light effect of FBG-FP filter exhibited a several-hundred-Hz linewidth.^[4] A minimum static strain resolution of was achieved by a static-strain sensor based on FBG-FP interferometers.^[12]

However, it was found that FBG-FPs transmission spectra show some loss characteristics in some practical applications. For example, the intrinsic fiber loss and the additional loss induced by ultraviolet light (UV) seriously limit the improvement of output power and linewidth of the distributed feedback (DFB) fiber laser.^[13] The resonance transmission peaks of FBG-FP with high Q-factor are demanded for noise suppression demonstrated in Ref. ^[1]. However, it is difficult to achieve a higher Q-factor because of intra-cavity loss, which limits the performance of this configure for suppressing noise. Although the fact that the loss of FBG-FP is a significant limiting factor for its further development is well known, there are few reports with in-depth analyses on the loss characteristics in FBG-FP.

In this work, the loss characteristics of FBG-FPs are analyzed in detail. A theoretical model of the FBG-FP transmission spectrum considering FBG and intra-cavity fiber loss is established. Several types of FBG-FPs are inscribed experimentally, and fitting the measured spectrum with the theoretical model, the results show that the theoretical model is in good agreement with the experimental results. It is further demonstrated that the weak loss in FBG-FP results in ann observable reduction of resonance transmission peaks transmissivity, which is called the loss enhancement effect of resonance transmission peak in this paper. The theoretical model can be used to accurately estimate FBG loss and the loss of intra-cavity pure fiber region caused by the UV exposure. Hence it provides a new idea for the measurement of UV-induced loss.

The typical structure of FBG-FP consisting of two uniform single-mode FBGs with a certain distance is shown in Fig. 1. According to the FBG formation mechanism,^[14] hydrogen-loading and UV exposure will induce loss to the FBG during the FBG-FP inscription process. In hydrogen-loaded fiber, H₂ molecules react at normal Si–O–Ge sites, and form Ge–OH, Si–OH, Ge–H, Si–H, and oxygen deficient Ge defects, resulting in loss increase in fiber.^{[15, 16]} FBG inscription could introduce additional extrinsic defects, such as mechanical damage, contamination, inhomogeneous index profile in the fiber core cross section,^[17] and so on. These defects will cause absorption, scattering, and radiation-mode coupling in FBG.^[18] In addition, post-exposing the intra-cavity pure fiber region in FBG-FP is a common method to make the transmission peak wavelength consistent with the design wavelength,^[19] which would generate additional loss in the intra-cavity pure fiber region. Due to the existence of the FP effect, the weak loss will be amplified and more serious loss will be introduced at transmission peaks. In general, the main source of intra-cavity loss of FBG-FPs contains two aspects, one is the loss of FBG, and another is the loss of the intra-cavity pure fiber region.

Fig. 1.

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	Fig. 1. (color online) Schematic diagram of FBG-FP.

The coupled mode theory is a commonly used tool for treating interactions between confined modes with refractive index perturbation, and the coupled mode equations of FBG in single fibers can be expressed as^[20]

It is assumed that an FBG-FP is composed of two FBGs with lengths l₁ and l₂, and the distance of d between these two FBGs. According to the multi-beam interference theory,^[21] the transmission coefficient of the FBG-FP shown in Fig. 1 is given by

Here we divided the transmission spectrum of the FBG-FP into two parts for discussions, that is, the wavelength range within the stop bandwidth of the FBGs and the wavelength range outside of the stop bandwidth of the FBGs. If the wavelength is out of the stop bandwidth of the FBGs, regarding as 0, equation (9) can be simplified as . This is the transmissivity of the light one-pass flight FBG-FP. If the wavelength is within the stop bandwidth of the FBGs, is a function of λ. When the phase factor of Eq. (5) satisfies the resonance condition, that is, , the transmissivity of the FBG-FP can be given by

Firstly, the FBG-FPs are inscribed by the phase mask method as shown in Fig. 2. One diaphragm with two rectangular windows with the same length is installed in front of the phase mask. There are two sections of the same length of the UV beam exposing the phase mask and the fiber for writing the two FBG reflectors of the FBG-FP. The length of rectangular windows in the diaphragm and the distance between them are determined by the length and distance of the FBG reflectors in FBG-FPs that we designed. Two groups of FBG-FP, called group A and group B, are manufactured in SMF-28e using a KrF excimer laser with the output wavelength of 248 nm, the pulse rate of 12 Hz and single pulse energy of 80 mJ. The transmission spectra of samples in group A contain a single transmission peak and in group B contain multi transmission peaks. Each group has two samples, where A(B)-2 is obtained by post-exposing the intra-cavity pure fiber region of A(B)-1. The parameters of FBG-FP inscription are described in detail in Table 1.

Fig. 2.

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	Fig. 2. (color online) Schematic diagram of FBG-FP inscription (a) A(B)-1; (b) A(B)-2.

Table 1.

Table 1.

Table 1. Parameters of FBG-FP inscription. . l/mm d/mm Pulse counts of UV uniform exposure A-1 8.5 2 0 A-2 8.5 2 4000 B-1 3 13 0 B-2 3 13 3000

Table 1. Parameters of FBG-FP inscription. .

The spectra of FBG-FPs are monitored using an optical spectrum analyzer (OSA) with a resolution of 0.04 pm (APEX Technologies, AP-2041B). The probe light is provided by the OSA’s affiliated tunable laser with a linewidth of 500 kHz. The spectra of FBG-FPs of two groups are shown in Fig. 3 and Fig. 4, respectively, where blue solid curves are measured results and red dashed curves are results fitting with Eq. (9) in section 2. Fitting parameters of A-1 and A-2 in Fig. 3 are , , , and , , , respectively. The fitted coefficient of determination R² are 0.9932 and 0.9859, respectively. It is seen that the theoretical model is in good agreement with the experimental results, and confirms that the theoretical model is valid for calculating the FBG-FP spectrum. In the ideal case, the transmissivity of the transmission peak should be 100%, but if there is a weak intra-cavity loss, the transmissivity of the resonance transmission peak is reduced obviously. The loss of the resonance transmission peak in samples A-1 and A-2 are 2.07 dB and 3.76 dB, respectively. These loss values are much greater than UV- induced loss in fiber, which indicates that the resonance effect enhances the loss of the resonance transmission peak. This phenomenon is called the loss enhancement effect of the resonance transmission peak in this paper.

Fig. 3.

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	Fig. 3. (color online) Spectra of FBG-FP in group A, where blue solid curves are measured results and red dash curves are fitting results (a) A-1; (b) A-2.

Fig. 4.

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	Fig. 4. (color online) Spectra of FBG-FP in group B, where blue solid curves are measured results and red dash curves are fitting results (a) B-1; (b) B-2.

Fitting parameters of B-1 and B-2 in Fig. 4 are , , , and , , , respectively. The coefficient of the determination R² are 0.9604 and 0.9662, respectively. The results illustrate that loss values are related with the wavelength of light. For example, in Fig. 4(b), the loss of peak 1 is 4.13 dB, and the loss of peak 2 is 9.10 dB. This phenomenon is attributed to the FP resonant effect. The light components at the transmission wavelength repeatedly oscillate, resulting in experiencing longer optical paths before it transmits out. Thus it suffers much more loss than a single path. The number of round trips, or photon lifetime, is related with the reflection of FBG at the transmission peak wavelength. Hence, the higher the reflectivity is, the more significant the loss enhancement effect of the resonance transmission peak is exhibited. Figure 5 shows the FBG-FP transmission spectrum and its envelope calculated with Eq. (10), the parameters are the same as the fitted values of FBG-FP in group B. The pink dotted curve, blue solid line, and red dashed line present the reflection spectrum of single FBG, transmission spectrum of FBG-FP, and envelope of the transmission spectrum respectively. It can be seen that the transmission wavelength is closer to the Bragg wavelength, then the single FBG reflectivity is higher, and the loss is more remarkable.

Fig. 5.

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	Fig. 5. (color online) Simulation of the FBG-FP using Group B’s parameter with , , .

The post-exposure induced loss for FBG-FP is shown in Fig. 3(b) and Fig. 4(b). According to the fitting parameters, we estimate that loss of about 0.0014 mm⁻¹ is induced by 4000-pulse counts uniform UV-exposure, and 3000-pulse counts UV-exposure results in loss of 0.00085 mm⁻¹. It provides a new way to measure UV-induced loss. In the past, in order to avoid some disturbing factors, such as fiber splicing loss, measuring a spectrum as reference before FBG is written and monitoring the spectrum during FBG writing were used to estimate UV-induced loss.^[22] This method is limited by the accuracy of OSA, whose accuracy generally is 0.01 dB. Besides it is useless after annealing. We demonstrated that the photothermal effect in phase shift FBG can be used to estimate the loss coefficient of FBG in previous work.^[23] However, the measurement error is large by this method, due to the fact that the heat dissipation condition have a strong impact on the transmission peak drift of the phase shift FBG. So it is expected to be a new method to measure UV-induced loss by using the loss enhancement effect of the transmission peak, which can increase the accuracy by two orders of magnitude without reference light.

In this paper we analyzed the effect of the UV-induced loss on the spectral response of FBG-FPs. The theoretical study is achieved by extending the standard coupled-mode theory including the effect of periodic loss and the multi-beam interference theory including the intra-cavity fiber loss. Several FBG-FPs with a single resonance peak or multi-resonance peaks are manufactured in experiments, and the measured spectrum is fitted with the theoretical model in Section 2; the theoretical result is in good agreement with the experimental results. It is demonstrated that the weak loss in FBG-FP results in serious reduction of resonance transmission peaks transmissivity. For FBG-FP with multi-resonance peaks, the resonance peak wavelength is closer to the Bragg wavelength, the more significant the loss effect of the resonance transmission peak is exhibited. This study can be applied to measure FBG loss more accurately, without reference light, and it can play an important role in FBG and FBG-FP writing process optimization and application parameter optimization.