1. Introduction and motivation

Optical filters, which filter out noise or unwanted signals, or flatten gain after amplification are one of the most important optical components in optical fiber networks. Fiber based filters are desirable because they can be spliced to yield low insertion loss and very stable coupling to the input and output fibers. Several fiber bandpass filter designs, both narrow and broadband, have been proposed and demonstrated [1-9]. Tunable bandpass filters based on photonic bandgap fibers [10, 11] have also been demonstrated. However, no efforts have been made to design a fiber bandpass filter based on the leakage loss properties of the fiber by enhancing the leakage loss of the fundamental mode before and after a certain cut-off wavelength. In this work, photonic crystal fibers (PCFs) with a specially designed core are employed for achieving bandpass filtering characteristics. PCFs, for which the cladding is an array of tiny air-holes, have gained significant interest due to their superior optical properties such as wide-band single mode operation, scalable effective mode areas, and controllable dispersion characteristics [12, 13]. The proposed PCF bandpass filter design is mainly based on achieving short- and long-wavelength cut-offs (where the index of the fundamental guided mode becomes smaller than the average cladding index). The short-wavelength cut-off is achieved by doping the core by low index material [14] such as fluorine, while the long-wavelength cut-off is obtained by enlarging the air-holes surrounding the doped core region. We compute the short- and long-wavelength cut-offs for various PCF design parameters using a modal analysis based on a fully-vectorial finite element method (V-FEM) [15], and accordingly the transmission characteristics are obtained. The bandpass characteristics of the proposed PCF filter are also evaluated through a fully-vectorial finite-element beam propagation (BPM) [16] simulation. We show that the bandpass window can be shifted by tuning the pitch constant. Numerical simulations reveal that for low doping concentration and small air-hole diameters in the first ring, the bandpass window is broader; however, the bandpass window can be made narrow by using high doping concentration and large air-holes. We find that a 5-cm long PCF with a heterostructured, down-doped core and eleven rings of air-holes can result in ~440 nm 3-dB bandwidth with ~90% of transmission. A longer fiber results in reduced transmission and a narrower 3-dB bandwidth, e.g. for a length of 30 cm, the transmission drops to 65% while the 3-dB bandwidth reduces to ~280 nm. This suggests that the bandpass characteristics of the PCF filter can be tuned by varying the fiber length, as the operation of the proposed bandpass filter depends on the leakage loss of the fiber.

2. Fiber structure

Figure 1 depicts the transverse cross-section of the proposed PCF bandpass filter. The background material is pure silica with a refractive index of 1.45. The PCF is characterized through four geometrical parameters, namely, doping size D, hole-diameters d′ and d (d′> d), and pitch constant Λ. The core is doped with low index material [14], such as fluorine and is surrounded by six enlarged air-holes. The doping concentration is defined by the difference between the pure silica index (n _si) and fluorine doping index (n _f) i.e. Δ=n _f-n _si. The air-holes beyond the first ring, which form the cladding of the proposed fiber filter, have uniform diameters. The key point to enlarge the air-holes and down-doping the core is to obtain long and short wavelength cut-offs, so that the fiber can behave as short- and long-wavelength pass filter, thus providing a band over which the fiber passes certain range of the wavelengths. This passband can be scaled by varying the pitch constant of the fiber. In Fig. 2 (a), we plot the refractive index variation of the cladding (solid blue curve), the fundamental mode (dotted red curve with open circles), and doped core (dotted black curve) of a PCF with d′/Λ=0.34, d/Λ=0.30, Δ=-0.004, and D=Λ. It can be seen from the graph that the index of the fundamental mode (n ^fm _eff) is smaller than the cladding index (n _cl) i.e. n ^fm _eff<n _cl for certain wavelengths. As the wavelength increases the n ^fm _eff gets closer to the cladding index and intersects at a particular wavelength. We refer this wavelength as the short wavelength cut-off λ_S. After λ_S, the modal field is well confined into the doped core. The n ^fm _eff again becomes smaller than the cladding index due to enlarged inner air-holes and the long wavelength cut-off λ_L is obtained. The bandpass window is determined by all wavelengths which fall between λ_S and λ_L i.e. λ_S<λ< λ_L. Figure 2(b) shows the magnified image of the effective index variation of the cladding index and the guided mode before the short wavelength cut-off. The antiguiding region, where the index of the fundamental mode becomes smaller than the cladding index, is clearly seen. Note that the wavelength dependency of the silica refractive index has been ignored to compute the cut-off wavelengths as well as transmission characteristics. As the device length is small, one can ignore the material dispersion of the silica for simplifying the calculations. However, one can estimate the wavelength cut-offs and thereof the transmission characteristics by taking into account the wavelength dependency of the refractive index of the silica through Sellmeier’s equation. In appendix A, we have shown the effect of silica’s material dispersion on the wavelength cut-offs and the transmission characteristics for its practical application.

We can qualitatively estimate how the bandpass characteristics depend on the PCF geometrical parameters by considering how the curves in Fig. 2 shift by changing the geometry. When the cladding hole-diameter is small (or low cladding air-filling fraction), the cladding index curve shifts up. One therefore expects narrow passband. At long wavelengths, we can treat the low index rod (i.e. fluorine-doped rod) and first ring of holes as a composite core; increasing the inner hole size or decreasing the fluorine-doped rod index thus lowers the effective core index at a given wavelength and decreases the long wavelength cut-off. Increasing the inner hole-size has a stronger effect on the long wavelength cut-off relative to the short wavelength cut-off because the latter is dominated by stronger mode overlap with the low index rod.

 

Fig. 1. Schematic view of the proposed PCF band-pass filter.

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Fig. 2. (a). Effective index variation for a fluorine-doped PCF with d’/Λ=0.34, d/Λ=0.30, and Δ=-0.004, (b) magnified view of effective index variation before the short wavelength cut-off.

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3. Design strategies and numerical results

In this section, we explain the design methodology to achieve broad bandpass PCF filter. As mentioned before, the PCF bandpass filter design is characterized through the doping concentration, d/Λ, and d′/Λ. The doping radius is fixed at 0.5Λ in all numerical simulations. We use V-FEM to evaluate the short- and long-wavelengths cut-off for different values of cladding hole-diameter, namely d/Λ=0.40, 0.35, 0.30, and 0.25 by varying the doping concentration Δ. In Fig. 3, we plot the short- and long-wavelengths cut-offs as a function of enlarged air-hole size for different cladding hole-diameter and doping concentrations. The solid curves depict the short wavelength cut-off whereas the dotted curves correspond to the long wavelength cut-off. The colors blue, red, black, and green stand for doping concentration of Δ=-0.001, -0.002, -0.003, and -0.004, respectively.

 

Fig. 3. Variation of short- and long-wavelength cut-offs in (a) d/Λ=0.40, (b) d/Λ=0.35, (c) d/Λ=0.30, (d) d/Λ=0.25 for various doping levels, Δ=-0.001, -0.002, -0.003, and -0.004. The solid and dotted curves stand for short- and long-wavelength cut-offs, respectively.

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From Fig. 3, several conclusions can be made. The long wavelength cut-off decreases as d′/Λ increases, whereas the short wavelength cut-off increases as d′/Λ increases. This is because enlarging the air-holes surrounding the doped core decreases the effective index of the fundamental mode at all wavelengths, which pushes the short-wavelength cut-off towards longer wavelengths and long-wavelength cut-off to shorter wavelengths. As we increase the doping concentration, the gap between the short- and long-wavelength cut-off curves becomes small, suggesting narrow bandpass filtering characteristics. The short- and long-wavelength cut-off curves stop after a particular value of d′/Λ. If we further increase the d′/Λ the PCF may not guide the fundamental mode at all. If we look to curves in Fig. 3, any x-cut parallel to y-axis gives a set of short- and long-wavelength cut-offs whose difference can gives a quantitative idea of the passband. We can see that the curves terminate at a particular d′/Λ and we define this point as end-tips of the curve (towards large d′/Λ). It is clear that the passband is narrower at these end-tips than at any x-cut points on the cut-off curves. Decreasing the cladding hole-diameter d/Λ shifts the short- and long-wavelength cut-offs. As we decrease the cladding hole-diameter d/Λ, the cladding index curve approaches to silica index, making a red-shift in the antiguiding wavelength [14] and hence short wavelength cut-off is red-shifted. However, the long wavelength cut-off is blue-shifted. These red- or blue-shifts in the cut-offs can be clearly seen in Fig. 4(a), where cut-off wavelengths are plotted for two different cladding hole-diameters, namely d/Λ=0.35 and 0.40. In Fig. 4(b), we plot the wavelength cut-offs for several cladding hole-diameters at a particular doping concentration of -0.004. The solid curves correspond to the short-wavelength cut-off, whereas the dotted curves stand for the long-wavelength cut-off. From Fig. 4(b), we can ascertain that the gap between the end tips of both short- and long-wavelength cut-offs widens as the cladding hole diameter increases. As a design rule for the PCF band-pass filter, we can fairly conclude from the cut-off analysis that (i) to achieve broad bandpass filtering characteristics, low doping concentration and small d′/Λ should be used, and (ii) however to get narrow filtering characteristics, high doping concentration with large d′/Λ should be used.

Based on the above analysis, we select one particular PCF as an example, with structural parameters d/Λ=0.35, d′/Λ=0.475, Λ=3.2 µm, and Δ=-0.004 and air-hole rings N as a parameter. We obtain the leakage loss and transmission characteristics through V-FEM. In Fig. 5, mode-field distributions of the PCF bandpass filter have been plotted at different wavelengths, namely 1.0 µm (smaller than short-wavelength cut-off, 5(a)), 1.35 µm (short-wavelength cut-off λ_S, 5(b)), 1.50 µm (center wavelength of the band, estimated through λ_S+λ_L/2, 5(c)), 1.66 µm (long-wavelength cut-off λ_L, 5(d)), and 2.0 µm (larger than long-wavelength cut-off, 5(e)). If we look to Fig. 5(a), we can see that the mode is not confined into the core and radiates into the cladding, as the operating wavelength is below the short-wavelength cut-off. At the short wavelength cut-off (Fig. 5(b)), there is partial confinement of the fundamental mode, but it is too leaky to propagate. Therefore, it’s difficult to observe it at the end of the fiber. As we further increase the wavelength, the mode is confined and guided in the core of the fiber (Fig. 5(c)). Near the long-wavelength cut-off (Fig. 5(d)), the index of the fundamental mode again approaches to the cladding index and the fiber leakage loss is enhanced. In Fig. 5(e), we depict the mode field after the long-wavelength cut-off, where one can see the radiated field in the outer cladding.

 

Fig. 4. Variation of short- and long-wavelength cut-offs (a) for two different cladding air-hole diameters namely, d/Λ=0.40 and 0.35, and (b) for three different cladding air-hole diameters viz. d/Λ=0.40, 0.35, 0.30 at a fixed doping concentration Δ=-0.004. The solid and dotted curves stand for short- and long-wavelength cut-offs, respectively.

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Fig. 5. Modal field distributions in PCF (d′/Λ=0.475, d/Λ=0.35, Λ=3.2 µm, Δ=-0.004, N=11) at (a) 1.0 µm, (b) 1.35 µm, short-wavelength cut-off, (c) 1.5 µm central wavelength, (d) 1.66 µm long-wavelength cut-off, and (e) 2.0 µm beyond long-wavelength cut-off.

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Next, we evaluate the transmission characteristics of the proposed PCF bandpass filter for several rings of air-holes in the cladding. Figure 6 shows the transmission characteristics of a 5-cm long PCF with d/Λ=0.35, d′/Λ=0.475, Λ=3.2 µm, and Δ=-0.004. From Fig. 6, we predict that more than 80% of transmission can be obtained when there are more than eight air-hole rings. The 3-dB bandwidth increases as air-hole rings increase. The fiber bandpass filter gives ~440 nm 3-dB bandwidth for eleven air-hole rings. In Fig. 7(a), we show the transmission characteristics of the PCF bandpass filter (N=11) as a function of the fiber length. The transmission varies because the filter operation depends on the leakage loss of the fiber. Even when the fiber length is large, viz. 30 cm, 65% of transmission can be obtained with a reduced 3-dB bandwidth of ~280 nm and an out of band suppression of more than 40 dB. The out-of-band suppression decreases as the fiber length decreases. Note that the side-lobes before the passband may not be observed at the output after connecting the proposed PCF bandpass filter to a conventional optical fiber, as the side-lobes correspond to the modal field radiated in the cladding as shown by Fig. 5(a).

 

Fig. 6. Transmission characteristics of the PCF band-pass filter (d′/Λ=0.475, d/Λ=0.35, Λ=3.2 µm, Δ=-0.004) with air-hole rings N as a parameter.

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Finally, we perform the BPM simulation of the PCF bandpass filter with eleven air-hole rings, d/Λ=0.35, d′/Λ=0.475, Λ=3.2 µm, Δ=-0.004, and L=5-cm. The modal field from a PCF without down-doped core and enlarged air-holes is launched into the PCF band-pass filter. This gives rise to the insertion loss between both launching fiber and the PCF bandpass filter. The results from the BPM (solid red curve) and modal analysis (solid blue curve) are exhibited in Fig. 7(b). We obtain an insertion loss of 0.49 dB from the BPM simulation, which compares favorably with 0.19 dB insertion loss computed from the modal analysis. This validates our design rules established from the modal analysis to achieve tunable PCF bandpass filters. The kink appearing in the BPM transmission curve at short wavelength is due to the excitation of the cladding mode at short wavelength which is 1.20 µm. We further confirm the appearance of the kink in BPM simulation by running the BPM simulation for different fiber lengths. We find that the kink appears at the same wavelength whereas the transmission magnitude corresponding to the kink increases or decreases as we decrease or increase the fiber length. We also observe a beating between the core mode and the cladding mode at 1.20 µm wavelength. We confirm from BPM simulations that the total power at 1.20 µm wavelength decreases non-monotonically over the propagation distance with a beating between the core mode and the cladding mode. The beating occurs at an every step of ~5 mm.

 

Fig. 7. (a). Transmission characteristics of the PCF bandpass filter evaluated through modal analysis as a fiber length parameter and (b) BPM and FEM simulated transmission characteristics.

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3.1 Tolerance analysis

In this section, we study the effect of the fiber tolerances on the transmission characteristics of the PCF bandpass filter (d′/Λ=0.475, d/Λ=0.35, Δ=-0.004, and N=11) and show how the transmission changes for a ±1% variation in d or d′. Figure 8 depicts the peak transmission values plotted on both left and right hand y-axis of the graph for tolerances varying steps of 0.5%. The red and black curves correspond to the tolerances in inner hole-diameter d′ and cladding hole-diameter d, respectively. As the cladding hole-diameter decreases or increases, the transmission drops or rises accordingly. It can be seen from the tolerance analysis that if the cladding hole-diameter d decreases by 1%, the transmission falls to 20% from 93%; however, it rises to 99.5% with broader passband when d increases by 1%. A reverse effect is observed for tolerances in inner hole-diameter d′. The peak transmission decreases to 76% for a 1% change in d′, and it increases to 98.4% for a -1% variation in the d′. From this analysis, we may predict that the performance of the PCF bandpass filter can degrade if a ±1% tolerance present in the cladding hole-diameter; however, for a ±1% tolerance in inner holediameter d′, the transmission is still more than 75%.

Note that, variation in d or d′ also affects the 3-dB bandwidth of the PCF bandpass filter. The bandwidth increases as the cladding hole-diameter d increases and decreases as d decreases, whereas an increment in the inner-hole diameter d′ reduces the bandwidth and decrement in d′ increases the bandwidth. It can be seen from Fig. 8 that the decrement in the cladding hole-diameter d severely degrades the transmission of the PCF bandpass filter, but one can enhance the transmission by tuning the fiber length as demonstrated in Fig. 7(a). We can use shorter length of the fiber to raise the transmission. In such a case, the bandwidth also changes as there is a trade-off between the fiber length, transmission, and the bandwidth.

 

Fig. 8. Impact of tolerances in d or d′ on the transmission characteristics of the PCF bandpass filter (d′/Λ=0.475, d/Λ=0.35, Δ=-0.004, and N=11). The peak transmission drops drastically for a -1% change in the cladding hole-diameter d. However, for a 1% change in the inner holediameter d′, the transmission decreases by 17% from the peak transmission value of 93.3%. The decrement in the cladding hole-diameter d severely affects the transmission characteristics. However, we can use shorter length of the fiber to raise the transmission at the cost of large bandwidth.

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4. Summary

In summary, we have presented the design rules and the methodology to achieve narrow or broad PCF bandpass filters. The PCF bandpass filter design was based on a very simple air-hole arrangement in the cladding, similar to W-type index profile with a depressed core. The filtering characteristics were computed by accurate V-FEM and BPM solvers. It was found that by controlling the doping concentration, d′/Λ, and d/Λ, the bandpass window can be tuned and moreover, the passband can be scaled by varying the pitch-constant. The effect of air-hole rings and fiber length has also been demonstrated. By varying the fiber length, the transmission as well bandpass window can be tuned as the operation of the proposed filter relies on the leakage loss characteristics. The cut-off wavelength curves which give the transmission characteristics of the PCF bandpass filter are also evaluated by taking into account the material dispersion of the silica. Additionally, we have performed a tolerance analysis. We found that the performance of the PCF bandpass filter degrades for a -1% variation in cladding hole-diameter d, but the transmission doesn’t drop below 75% for ±1% tolerances in the inner hole-diameter d′. PCFs with fluorine-doped core have already been fabricated [14] and enlargement of few air-holes in the final design will not add significant complexity in the fabrication of PCF bandpass filters. Therefore, we expect the realization of the device in near future.

Appendix A:

The cut-off wavelength curves and transmission characteristics of the PCF bandpass filter (d′/Λ=0.475, d/Λ=0.35, Δ=-0.004, and N=11) have been computed by taking into account the wavelength dependent refractive index of the silica through Sellmeier’s formula. The results are shown in Fig. 9. The dashed curves with filled circles represent the computation done by considering wavelength-dependent refractive index of silica. It can be seen from Fig. 9(a) that the long wavelength cut-off decreases, while the short wavelength cut-off almost remains the same. The transmission characteristics calculated for a 5-cm long PCF bandpass filter are exhibited in Fig. 9(b). The transmission magnitude doesn’t change, only a shift in the passband is seen which is due to the shift in the long wavelength cut-off. Therefore, in practice we expect the material dispersion to affect the center frequency and width of the passband, but not its transmission.

 

Fig. 9. The effect of wavelength dependency of the refractive index of silica on (a) cut-off wavelengths and (b) the transmission characteristics of the PCF bandpass filter (d′/Λ=0.475, d/Λ=0.35, Δ=-0.004, and N=11). The cut-off wavelength curves and transmission curve calculated by taking into account the wavelength-dependent refractive index of silica are shown by dotted curves with filled circles.

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Acknowledgments

S. K. Varshney kindly acknowledges to Japan Society for Promotion of Science (JSPS) for their support in carrying out this work.

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