Optical cavities with a high $Q$-factor and/or large group delays have many potential and actual applications, including sensing, communications, nonlinear optical interactions, and as optical frequency combs [1,2]. Record $Q$-factors of $8\times {10}^9$ have been reported in a microsphere [3,4]. Although such high performance holds great promise, achieving stable and mechanically robust coupling into such cavities is challenging. This problem can be potentially solved by developing high $Q$-factor cavities in all fiber devices. A promising avenue is a fiber Fabry–Perot (FP) interferometer consisting of two fiber Bragg gratings (FBGs) spaced by a short length of fiber. Experimental prototypes have been shown to exhibit $Q$-factors as high as $1.9\times {10}^6$ (a group index $n_g\approx 47.8$) [5], $2.6\times {10}^7$ ($n_g\approx 102$) [6], and $2\times {10}^7$ ($n_g\approx 90$) [7]. Achieving higher group indices is important in some applications; for example, when a large field enhancement is required (the field enhancement is proportional to the group index). Also for the same $Q$-factor, a higher group index means the same group delay in a shorter length.

Another alternative is the narrow slow-light resonances that exist in FBGs of suitable design [8–10], a simpler and more compact device. Slow-light peaks in uniform FBGs were predicted theoretically [8,9] and observed experimentally [10–13]. Just outside its bandgap, a uniform FBG acts as a partial reflector; light penetrates it, is partially reflected, and the reflected light traveling back through the FBG is reflected again [9]. The FBG therefore acts as an FP interferometer. At wavelengths where these multiple reflections are in phase in the forward direction, constructive interference produces a transmission peak. At these wavelengths, light has traveled multiple times through the device and experienced a large group delay. Because this phenomenon occurs only at the edges of the bandgap, reflections are typically weak, and the group delay is moderately large.

Larger group delays can be obtained in $\pi$-shifted FBGs [11]. The FBG then acts as two shorter FBGs separated by a half-wavelength, i.e., an FP with an ultrashort cavity that supports only one slow-light transmission peak in the middle of the bandgap. The same effect can occur in an apodized FBG [8,9]. The Bragg wavelength is then a function of position $z$ along the FBG. When the spatial profile of the mean effective mode index $\mathrm{\Delta }{n}_{\mathrm{dc}}(z)$ is single-humped, such as a Gaussian profile, there are pairs of positions along the grating, where the Bragg wavelengths are equal within a pair. In the space between these positions, the bandgap is shifted, and light is essentially transmitted, i.e., an FP is formed [8,9]. This FP can support much stronger slow-light resonances than a uniform FBG because resonances occur inside the bandgap where the FBG reflectivity is high and the $Q$-factor of the interferometer consequently sizeable.

To exploit this potential, we recently demonstrated a group delay of 5.1 ns ($n_g=127$) in an apodized FBG with a length of 1.2 cm and an index modulation $\mathrm{\Delta }{n}_{\mathrm{ac}}=\mathrm{\Delta }{n}_{\mathrm{dc}}=\mathrm{\Delta }n\approx {10}^{-3}$ fabricated using conventional UV writing [14]. The inferred power loss of this grating was $1.16{\mathrm{m}}^{-1}$. To increase the group delay, we evaluated FBGs fabricated using a femtosecond laser [15], a technique that produces significantly lower internal losses [16]. In a femtosecond FBG with a slight apodization, a longer length of 2 cm and a $\mathrm{\Delta }n$ of also $\sim {10}^{-3}$, we observed a group index of 58 (a group delay of 3.9 ns) [17]. This value was not as high as expected given the very low loss of the grating ($0.1{\mathrm{m}}^{-1}$), which is largely because the FBG length was not optimized [17]. However, this grating had an exceedingly high transmission (89%), and, when used as a strain sensor, it produced a record resolution of $280\mathrm{f\epsilon }/\surd \mathrm{Hz}$ [17].

In this Letter, we report significant improvements over these results in femtosecond FBGs with an even lower loss coefficient, a larger index modulation, and a stronger apodization. This FBG exhibits a new measured record-high group delay of 19.5 ns (a group index of 292, or a $Q$-factor of $1.5\times {10}^7$) and a very low power loss coefficient of only $0.12{\mathrm{m}}^{-1}$. The amplitude, location, and birefringence of the six slow-light peaks in this FBG’s measured group-index spectrum are in good agreement with predictions.

To illustrate the importance of apodization, we show in Figs. 1–3 the simulated group-index and transmission spectra of FBGs with three different index profiles. These numerical simulations were performed by solving the well-known differential equations that rule the evolution of the light’s electric field inside an FBG [8] using the transfer matrix method [8]. All three gratings have the same length $L=2\mathrm{cm}$, peak index modulation $\mathrm{\Delta }{n}_{\mathrm{ac}}\approx {10}^{-3}$, period $\mathrm{\Lambda }=532.98\mathrm{nm}$, and power loss coefficient $\alpha =0.4{\mathrm{m}}^{-1}$, which is close to the loss measured in FBGs of this strength [11].

 

Fig. 1. (a) Index profile of a uniform FBG (period $\mathrm{\Lambda }$ not to scale). (b) Simulated transmission and group index spectra of this FBG. $\mathrm{\Lambda }$ is the FBG period, and $n_0$ the refractive index of the unperturbed fiber mode.

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Fig. 2. (a) Index profile of a Gaussian-apodized FBG. (b) Simulated transmission and group index spectra of this FBG.

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Fig. 3. (a) Index profile of an FBG written using the femtosecond writing technique (period $\mathrm{\Lambda }$ not to scale). (b) Simulated transmission and group index spectra of this FBG.

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Figure 1(a) shows the index profile of a uniform FBG, and Fig. 1(b) shows its simulated spectra. The slow-light peaks occur just outside the bandgap, as described earlier. The strongest peak is the one closest to the band edge. Its peak group index is 146.9. In a uniform profile, $\mathrm{\Delta }{n}_{\mathrm{ac}}=\mathrm{\Delta }{n}_{\mathrm{dc}}$, and, as discussed above, the FP has relatively low finesse, which explains this relatively low group index.

Figure 2(a) shows the profile of a Gaussian-apodized FBG with a peak value for both $\mathrm{\Delta }{n}_{\mathrm{ac}}$ and $\mathrm{\Delta }{n}_{\mathrm{dc}}$ of ${10}^{-3}$, and an FWHM equal to twice the length. Both the transmission and the group index spectra [Fig. 2(b)] exhibit sharp resonances, centered on the same wavelengths, on the short-wavelength side of the bandgap. The strongest slow-light resonance is the one closest to the bandgap; its peak group index is as large as 361.5, i.e., $\sim 2.5$ times slower than in the uniform grating of Fig. 1. When the power loss coefficient is reduced from $0.4$ to $0.12{\mathrm{m}}^{-1}$, the experimental value inferred for the measurements reported below, the peak group index increases to 1204, which illustrates the critical importance of reducing the loss.

To fabricate the FBG characterized in this Letter, the beam of a femtosecond laser was scanned across the length of the fiber, and an aperture was used to determine the length of the FBG. For the first order, the profile was therefore the convolution of the laser beam’s intensity profile with a square aperture. The index profile expected for this grating [Fig. 3(a)] was calculated by taking the convolution of the measured laser-beam intensity profile (a Gaussian with an FWHM of 8 mm) with a 10 mm aperture. With this apodization, the slowest peak has a predicted group index of 353 [see Fig. 3(b)]. The small reduction in group index compared with the case of Fig. 2 is caused by the steeper slope of the $\mathrm{\Delta }{n}_{\mathrm{dc}}$ profile at the edges of the FBG, which results in a shorter reflective region at each of the slow-light peaks and, hence, lower finesse. These simulations demonstrate that the most critical parameters to achieve a large group index are the loss and the presence of a strong $z$-dependent $\mathrm{\Delta }{n}_{\mathrm{dc}}$, not so much the shape of the apodization.

The loss of an FBG depends on several parameters, including the fabrication technique and the index modulation. Measurements using a technique described in [16] show that in FBGs written in a conventional SMF-28 fiber with a femtosecond laser, the loss coefficient ranged from $\sim 0.02$ to $0.9{\mathrm{m}}^{-1}$ for a slightly apodized $\mathrm{\Delta }{n}_{\mathrm{ac}}$ in the range of ${10}^{-3}-3.5\times {10}^{-3}$. In contrast, FBGs written using the conventional UV-writing technique had a significantly higher estimated loss, greater than $1{\mathrm{m}}^{-1}$ for similar values of index modulation [16,18]. Although a larger sample is needed to establish a more precise quantitative comparison, it is readily apparent that the femtosecond-laser technique produces much lower losses by an estimated ratio in the vicinity of 4–5. This significant loss reduction, combined with the ability to produce large index modulations, are the primary reasons for evaluating femtosecond FBGs to produce large group delays in FBGs.

To verify these predictions experimentally and demonstrate a much larger group delay than reported so far with slightly apodized gratings (a FWHM of $2L$) written with the same technique, we fabricated a strong FBG with a large $\mathrm{\Delta }{n}_{\mathrm{dc}}$ Gaussian apodization utilizing the technique reported in [15]. The FBG reported here, fabricated in SMF-28 fiber, had a length of 2 cm and a peak index modulation $\mathrm{\Delta }{n}_{\mathrm{ac}}=\mathrm{\Delta }{n}_{\mathrm{dc}}\approx 1.69\times {10}^{-3}$, calculated from the measured FWHM of the bandgap ($\sim 1.8\mathrm{nm}$).

The experimental setup used to measure the group delay is shown in Fig. 4 [11]. Light from a fiber-coupled tunable 1550 nm laser was amplitude modulated at a frequency $f_m$ of 10 MHz. The modulated light was passed through a fiber polarization controller then coupled into the FBG. The output signal from the FBG was sent to a detector followed by a lock-in amplifier to measure the phase of the output signal. The laser was first tuned to a wavelength far away from the bandgap (1555 nm), which provided a point of reference where the output phase (unaffected by the FBG dispersion) was set to zero, and the transmission was measured to be close to 100%. The laser wavelength was then scanned, and the measurement repeated at individual wavelengths across the slow-light region. The measured spectrum of the phase delay $\phi$ provided the spectrum of the group delay $t_g$, which is related to the phase delay by $t_g=\phi /(2\pi f_m)$. Figure 5 shows the measured group delay spectrum, and the numerically calculated group delay spectrum for this particular FBG. The measured spectrum exhibits six slow-light peaks, aperiodically spaced between $\sim 1536.48$ and 1536.55 nm. For comparison, the measured 10 dB band edge of the FBG was 1536.474 nm, so all of these peaks are within the bandgap, as predicted for an apodized FBG (see Figs. 2 and 3). The highest measured group delay is 19.5 ns, which corresponds to a group index of 292. This group delay is significantly higher than the previous record established in a slightly apodized FBG [11,14] by a factor of 3.8, which demonstrates the significant benefits of apodization and loss in generating strong field confinements in a periodic structure. This group index is lower than the highest value predicted in Fig. 3(b) because this is the tenth closest peak to the bandgap (the closest peak was too weak to be observed). The peak transmission for this slow-light peak is about $-16\mathrm{dB}$. The $Q$-factor for this resonance, calculated from its measured linewidth (0.1 pm), is $1.5\times {10}^7$, which is close to the value calculated from the group delay ($1.2\times {10}^7$) [19].

 

Fig. 4. Experimental setup used to measure the group delay and transmission of the FBG (see text for details).

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Fig. 5. Group index measurement (solid line), theoretical prediction (dashed lines).

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The simulated spectrum in Fig. 5, predicted using the transfer matrix method described earlier, was fitted to the measured spectrum by adjusting four fitting parameters: (1) the peak value of the two index spectra, assumed to be equal ($\mathrm{\Delta }{n}_{\mathrm{ac}}(L/2)=\mathrm{\Delta }{n}_{\mathrm{dc}}(L/2)$); (2) the peak value of the loss profile; (3) the FBG length; and (4) the FWHM of the Gaussian beam. $\mathrm{\Delta }{n}_{\mathrm{ac}}(z)$ and $\mathrm{\Delta }{n}_{\mathrm{dc}}(z)$ were calculated in the same manner and had the same general shape as the profiles shown in Fig. 3(a). The period was taken to be 529.61 nm, corresponding to the measured Bragg wavelength of 1536.87 nm. This is consistent with the fact that during fabrication the target period was 534.3 nm and the fiber was under tension. To represent the fact that the loss increases with index modulation [16], we assumed that the loss profile had the same shape as the index profiles.

The measured group-index spectrum (see Fig. 5) is aperiodic because it consists of two independent series of slow-light peaks, one for each eigenpolarization of the FBG. The separation between the peaks in a pair is $\sim 7.5\mathrm{pm}$, a value that has been observed in many of the femtosecond gratings that we have characterized. This was confirmed experimentally by adjusting the state of polarization (SOP) of the light launched into the FBG with the polarization controller (see Fig. 4). For some input SOPs, one of the series of peaks was extinguished, while for other input SOPs it was the other series that vanished. As discussed in [11,20], this behavior arises from birefringence induced in the FBG’s baseline index $n_0$ [see Fig. 1(a)] by the asymmetric irradiation of the fiber during fabrication. To explain a peak separation of $\sim 7.5\mathrm{pm}$ in our simulations, we used a birefringence of $\mathrm{\Delta }{n}_0\approx 1.7\times {10}^{-5}$ for both $\mathrm{\Delta }{n}_{\mathrm{ac}}$ and $\mathrm{\Delta }{n}_{\mathrm{dc}}$. This value is close to what has been reported by others in femtosecond gratings with a $\mathrm{\Delta }{n}_{\mathrm{ac}}$ of similar magnitude [21].

To reproduce numerically all the peaks in the measured spectrum, it was necessary to add birefringence to the model. This was done by superposing two spectra: one calculated with $n_0$, and the other with $n_0+\mathrm{\Delta }{n}_0$. Because of the way the grating was written, $\mathrm{\Delta }{n}_{\mathrm{dc}}$ and $\mathrm{\Delta }{n}_{\mathrm{ac}}$ were expected, and thus assumed, to have the same birefringence. The simulated spectrum generated by this process that best fits the measured spectrum, giving priority to fitting the peaks’ amplitudes best, is shown in Fig. 5 (solid line), which predicts the locations and amplitudes of the six observed slow-light peaks fairly well. This general agreement confirms (1) the mechanism behind the formation of the slow-light peaks; (2) that the birefringence is responsible for the two sets of peaks; and (3) that the magnitude of the birefringence is correct. The slight disagreement in the peaks’ locations is due partly to temperature drifts during the long measurement (a few hours), which induce a variable relative shift between the peaks of up to 0.2 pm. More slow-light peaks are predicted than observed experimentally (see Fig. 5) because these additional peaks were too weak to be detected. The peak value of $\mathrm{\Delta }{n}_{\mathrm{ac}}$ (and $\mathrm{\Delta }{n}_{\mathrm{dc}}$) that produced this best fit is $1.69\times {10}^{-3}$, which is in agreement with the value inferred from the measured width of the bandgap. The fitted values of the FBG length (18.8 mm) and of the laser-beam FWHM (8 mm) are also in good agreement with nominal experimental values (20 and 8 mm, respectively). The fitted peak value of the power loss profile is $0.12{\mathrm{m}}^{-1}$, which is close to the value reported in an earlier femtosecond FBG [17].

In conclusion, we have shown through numerical simulations that FBGs with strong Gaussian-like dc apodizations can support much stronger slow-light resonances than can uniform FBGs of comparable index modulation. This enhancement arises from the creation of Fabry–Perot resonances due to the apodization. To exploit this prediction, we fabricated a strongly apodized 2 cm FBG using a femtosecond laser technique. Its measured group-index spectrum exhibits several slow-light resonances. The strongest one has a group delay of 19.5 ns (group index of 292), which is about four times (two times) higher than the previous record. The existence of twin peaks in the spectrum indicates the presence of birefringence. The measured spectrum is in good agreement with the spectrum predicted by a model that takes into account the index profile estimated from fabrication conditions, the expected loss profile of the FBG, and this birefringence. The fitted values of the birefringence ($1.7\times {10}^{-5}$) and of the peak ac and dc index modulations ($1.69\times {10}^{-3}$) agree well with previous reports and expectations. The peak power loss inferred from this fit is $0.12{\mathrm{m}}^{-1}$, which is one of the lowest values reported for an FBG.