Abstract

We propose and demonstrate a pre-compensation mechanism to account for the writing-beam profile which when applied to the design of advanced fibre Bragg gratings helps to achieve a desired design spectral response. We use the example of a complex multi-channel grating as an example to demonstrate the improvement achievable using the pre-compensation and find good agreement between experimental results and numerical calculations.

© 2015 Optical Society of America

1. Introduction

Advanced multi-channel fibre Bragg gratings (FBG) are receiving much attention recently with applications in WDM (wave division multiplexing) telecommunication systems [1], multi-wavelength fibre lasers [2], and astrophotonics [3]. Demands for larger channel counts and dispersion control across the channels increases the complexity of the designed FBGs and the challenges of fabricating them.

There have been many different methods for designing multichannel gratings investigated over the past decade. Examples include the inverse layer peeling (LP) algorithm [4], sampling methods (phase only, amplitude only, and phase-amplitude sampling) [57], and Moiré grating structures [8]. The layer peeling algorithm is the most general approach. However as the number of channels increases, the maximum required refractive index modulation of these designs increases too, such that they become impractical. Combining layer peeling with optimisation methods one can reduce the required index modulation to a realisable level [9].

A common approach to fabricating complex FBG devices is to use a uniform phase mask and gradually move the writing beam and phase mask relative to the fibre while modulating the beam to enable apodisation and wavelength and phase shifts in the fabricate device e.g. [10,11]. While this approach is flexible, there are limitations, due to the writing beam size, on the rate at which the grating phase can vary with distance [10, 12]. Previous work [10] has accounted for this assuming a rectangular beam profile but there has been little attention on how to redesign the gratings to mitigate the effect. To overcome this problem, Li and Li in [12] apply pre-compensation to the target reflectivity spectrum and then obtain the coupling coefficient with the layer peeling algorithm. However, their approach can lead to unphysical pre-compensated target spectra with reflectivities greater than 1.

In this paper we propose and demonstrate a simple mechanism of pre-compensation applied in the coupling coefficient design process that is more generally applicable and doesn’t lead to unphysical target spectra. It enables a significant improvement in the performance of the fabricated gratings bringing them closer to the target design parameters. The technique uses the experimentally measured beam profile which may not be rectangular. We use the example of multichannel FBG-based comb filters to demonstrate both the effect and its mitigation taking account of the measured Gaussian writing beam profile.

2. Design of a multi-channel dispersion free grating

We consider a 13 channel FBG with 0.2 nm bandwidth of each channel and 0.5 nm channel separation designed for application in a mode-locked fibre laser. The target reflectivity is described by

|r(λ)|=0.75j=113exp((λλjb)4)×exp(i2πneff(1λ1λj)dj)

Here b = 0.2 nm is the bandwidth of each channel, λj = 1545 + 0.5 × j nm is the central wavelength for channel j, neff the effective refractive index, and dj is a tailored group delay profile for each channel. The group delay term helps to avoid the concentration of the index modulation in the middle of the grating, which would occur if each channel had the identical group delay [9,13]. The central wavelength of the full spectrum is 1550 nm and the total length of the grating is 6 cm. Given Eq. (1), the LP algorithm is used to reconstruct the grating structure and obtain the corresponding coupling coefficient q(z). Optimisation of group delay values dj is used to smoothen the grating apodisation profile and to reduce the period change variation and chirp along its length. Figure 1 illustrates the corresponding reconstructed grating in terms of the complex coupling coefficient q(z) and its spectral response. Here z represents the grating position along the fibre. Using the same approach we also modelled 5, 9, 11 and 13 peak gratings with the same channel separation and channel bandwidth for comparison.

 

Fig. 1 A 13 peak FBG design profile: (a) coupling coefficient, (b) grating phase. Resulting calculated spectral response: (c) reflectivity; (d) group delay.

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3. Spectral degradation due to the writing beam size

The fabrication of fibre Bragg gratings with complex and precise profiles requires elaborate equipment and a detailed consideration of the constraints and the trade-offs of the fabrication process. An ideal fabrication system would be one that can support a wide range of phase and index modulation change with high precision. Initial demonstrations used phase mask dithering to perform apodisation and phase change [14]. The approach used here involves micro-stitching together a large number of short homogenous FBGs while moving the fibre continuously in a fringe pattern formed by a phase mask. The illuminating UV beam is modulated based on the position of the fibre and the required apodisation and localised wavelength of the FBG. The modulation is controlled such that successive FBGs overlay each other coherently. Chirp in the grating is achieved by allowing a small phase offset between successive exposures. If the overall shift reaches π over the beam width the grating will be erased. Thus the beam size (number of fringes) has an impact on the achievable change in wavelength and, at a specific wavelength offset, has an affect on the overall grating strength. This is a well known problem previously discussed in [10], where the authors considered a rectangular laser beam profile and made a prediction of the impact of beam size on FBG reflectivity but did not compensate for it as we do here.

Here we consider a UV writing-beam with a Gaussian beam profile expressed as

b(z)=exp((zw/2)2)
Here z is the position along the fibre and w is the beam width parameter. The index-modulation profile formed in a fibre core qconv is the result of convolution of the ideal grating q(z) and this Gaussian beam b(z).
qconv(z)=q(z)*b(z)

The values of beam width used in the calculations were taken from measurements of the beam from our Coherent I300 244 nm laser taken using a Thorlabs UV106-UV beam profiler. They are quoted as full width at the 1/e2 level of the measured Gaussian profile.

Considering various values for the beam width, we calculated the resulting grating profile and its corresponding spectral response. Figure 2(a) shows the numerically simulated resulting apodisation for the gratings after the convolution with the beams of 265 μm and 430 μm width. The apodisation results in gradually reduced the coupling coefficient at the grating edges. The phase of the grating wasn’t affected by the convolution and its profile remained the same as shown in Fig. 1(b). These changes in apodisation profile inevitably lead to degradation of the resulting spectral response with increasing beam width (shown in Fig. 2(b)), which was also confirmed experimentally with the results illustrated in Fig. 3.

 

Fig. 2 Effect of beam size: (a) on the apodisation of the grating, (b) on its spectral response.

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Fig. 3 Experimental gratings spectra for differing beam widths without compensation.

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We compare the spectral response of fabricated 5 and 11 channel gratings with beam sizes of 265 μm and 430 μm. The gratings were fabricated using a computer controller high precision fabrication setup using a 244 nm frequency doubled CW argon ion laser. A phase mask was used to produce the fringes, with the wavelength centred at 1550 nm. The fibre was mounted on a high accuracy air-bearing stage and scanned across the incident beam at a constant speed. Numerically modelled coupling coefficients were used for generating their apodisation and chirp that were controlled during fabrication process by an acousto-optic modulator put in the path of the beam. Varying beam sizes were achieved using focusing optics in the path of the beam. The exposed fibre was hydrogen loaded. The gratings were measured with an optical vector analyser with high resolution post annealing at 80°C for 48 hrs after fabrication.

4. Pre-compensation

There have been a few attempts at solving the problem of adjusting the designed grating profile to reduce the effect of the beam size. Li and Li in [12] apply pre-compensation to the target reflectivity spectrum and then obtain the coupling coefficient with the layer peeling algorithm. However, their approach can lead to unphysical pre-compensated target spectra with reflectivities greater than 1. The authors showed that even for a beam size as small as 65 μm, they had to reduce the target spectrum reflectivity to 0.75.

Here we propose to apply pre-compensation to the designed “ideal” coupling coefficient. By performing the Fourier transform to Eq. (3), we obtain

(qconv(z))=(q(z))(b(z))
where represents the Fourier transform. Now we assume that our ideal coupling coefficient q(z) is the result of convolution of some pre-compensated coupling coefficient qprecomp(z) and the beam b(z):
q(z)=qprecomp(z)*b(z).
To retrieve qprecomp(z) from Eq. (5) one has to perform deconvolution via the Fourier and inverse Fourier transform:
qprecomp(z)=1[(q)/(b)]

Figure 4 shows the comparison of the apodisation profile of the original grating and numerically simulated pre-compensated (using Eq. (6)) gratings for the beam sizes of 265 μm and 430 μm. We see the increase in the required index modulation at the edges of the grating. It is also seen that for a large beam of 430 μm the maximum index modulation is 4.5 times larger than for smaller beam of 265 μm which makes its fabrication more challenging.

 

Fig. 4 Original and pre-compensated FBG designs.

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If we fabricate the grating using the pre-compensated coupling coefficient profile qprecomp(z), we would expect to achieve the target spectrum, because the convolution of the coupling coefficient qprecomp(z) with the beam b(z) will result in the original (ideal) coupling coefficient q(z). The spectra of the experimentally fabricated gratings (Fig. 5) show a significant improvement, compared to the original (without pre-compensation) gratings as shown in Fig. 3.

 

Fig. 5 Spectra of fabricated 5 and 11 channels gratings both uncompensated and compensated for the 265 μm beam size used.

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Figure 6 shows the experimentally measured relative channel positions for fabricated gratings using pre-compensated and uncompensated designs and using two different writing beam sizes. The improvement brought about by the pre-compensation can be seen in the flattening of the relative position curves.

 

Fig. 6 Channel positions for fabricated 5, 9, 11 and 13-channels gratings. U stands for uncompensated, C for compensated grating designs, 265 and 430 correspond to the beam size in μm.

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5. Discussion

The demonstrated mechanism of pre-compensation uses a very simple and well-known deconvolution principle. One can easily apply such method to some gratings profiles and achieve the target design spectrum. It is not very efficient regarding the index modulation contrast, but it does offer a solution to the discussed problem for gratings with limited refractive index modulations.

The maximum beam width, which may be used to fabricate a given structure, is limited by the maximum index change available to use in the compensated design. Using a 100 mW Ar+ laser and hydrogen loaded standard telecommunications fibre we reliably are able to achieve a maximum coupling coefficient of |q(z)| = 2500 m−1. For our 11 and 5 peak compensated grating designs the maximum usable beam sizes are 460 μm and 1250 μm respectively. With 21 channels, the resulting compensated coupling coefficient becomes larger, and the maximum usable beam size reduces to 350 μm.

The beam size has an impact on the relative positions of the channels of an order of 0.05 nm in addition to their strength (Fig. 6). We suggest, this is due to diffusion of the loaded hydrogen in the fibre, and is the focus of on-going research.

6. Conclusions

In this work we demonstrate a mechanism of pre-compensation of designed fibre Bragg gratings in order to reduce the impact of writing beam size on the resulting grating. We apply a deconvolution of the Gaussian profiled writing laser beam to the designed coupling coefficient. Inevitably, it requires larger values of coupling coefficient to compensate for lower coupling coefficient contrast at the edges. We show that for large beam sizes it is unfeasible to apply enough pre-compensation to achieve the desired spectral response, whereas for sufficiently small beam sizes this mechanism significantly improves the performance of the resulting grating. Numerical simulations show a good match with experimental results.

References and links

1. F. Ouellette, “Dispersion cancellation using linearly chirped Bragg grating filters in optical waveguides,” Opt. Lett. 12(10), 847–849 (1987). [CrossRef]   [PubMed]  

2. X. Shu, S. Jiang, and D. Huang, “Fiber grating Sagnac loop and its multiwavelength laser application,” IEEE Photon. Technol. Lett. 12, 980–982 (2000). [CrossRef]  

3. J. Bland-Hawthorn, A. Buryak, and K. Kolossovski, “Optimization algorithm for ultrabroadband multichannel aperiodic fiber Bragg grating filters,” J. Opt. Soc. Am. A 25(1), 153–158 (2008). [CrossRef]  

4. J. Skaar and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37(2), 165–173 (2001). [CrossRef]  

5. Y. Dai and J. P. Yao, “Multi-channel Bragg gratings, Opt. Express 16(15), 11216–11223 (2008). [CrossRef]   [PubMed]  

6. X. Chen, J. Hayashi, and H. Li, “Simultaneous dispersion and dispersion-slope compensator based on a doubly sampled ultrahigh-channel-count fiber Bragg grating,” Appl. Opt. 49(5), 823–828 (2010). [CrossRef]   [PubMed]  

7. H. Li, M. Li, Y. Sheng, and J. E. Rothenberg, “Advances in the design and fabrication of high-channel-count fiber Bragg gratings,” J. Lightwave Technol. 25(9), 2739–2750 (2007). [CrossRef]  

8. R. Kashyap, Fiber Bragg Gratings (Academic, 1999).

9. A. V. Buryak, K. Y. Kolossovski, and D. Y. Stepanov, “Optimization of refractive index sampling for multichannel fiber Bragg gratings,” IEEE J. Quantum Electron. 39(1), 91–98 (2003). [CrossRef]  

10. M. J. Cole, W. H. Loh, R. I. Laming, M. N. Zervas, and S. Barcelos, “Moving fibre/phase mask-scanning beam technique for enhanced flexibility in producing fibre gratings with uniform phase mask,” Electron. Lett. 31(17), 1488–1489 (1995). [CrossRef]  

11. W. H. Loh, M. J. Cole, M. N. Zervas, S. Barcelos, and R. I. Laming, “Complex grating structures with uniform phase masks based on the moving fiber-scanning beam technique,” Opt. Lett. 20, 2051–2053 (1995). [CrossRef]   [PubMed]  

12. M. Li and H. Li, “Influences of writing-beam size on the performances of dispersion-free multi-channel fiber Bragg grating,” Opt. Fiber Technol. 15(1), 33–38 (2009). [CrossRef]  

13. H. Cao, J. Atai, X. Shu, and G. Chen, “Direct design of high channel-count fiber Bragg grating filters with low index modulation,” Opt. Express 20(11), 12095–12110 (2012). [CrossRef]   [PubMed]  

14. J. Williams, “Method of fabricating an optical waveguide grating and apparatus for implementing the method,” U.S. Patent EP1447691A1 (August 18, 2004).

References

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  1. F. Ouellette, “Dispersion cancellation using linearly chirped Bragg grating filters in optical waveguides,” Opt. Lett. 12(10), 847–849 (1987).
    [Crossref] [PubMed]
  2. X. Shu, S. Jiang, and D. Huang, “Fiber grating Sagnac loop and its multiwavelength laser application,” IEEE Photon. Technol. Lett. 12, 980–982 (2000).
    [Crossref]
  3. J. Bland-Hawthorn, A. Buryak, and K. Kolossovski, “Optimization algorithm for ultrabroadband multichannel aperiodic fiber Bragg grating filters,” J. Opt. Soc. Am. A 25(1), 153–158 (2008).
    [Crossref]
  4. J. Skaar and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37(2), 165–173 (2001).
    [Crossref]
  5. Y. Dai and J. P. Yao, “Multi-channel Bragg gratings, Opt. Express 16(15), 11216–11223 (2008).
    [Crossref] [PubMed]
  6. X. Chen, J. Hayashi, and H. Li, “Simultaneous dispersion and dispersion-slope compensator based on a doubly sampled ultrahigh-channel-count fiber Bragg grating,” Appl. Opt. 49(5), 823–828 (2010).
    [Crossref] [PubMed]
  7. H. Li, M. Li, Y. Sheng, and J. E. Rothenberg, “Advances in the design and fabrication of high-channel-count fiber Bragg gratings,” J. Lightwave Technol. 25(9), 2739–2750 (2007).
    [Crossref]
  8. R. Kashyap, Fiber Bragg Gratings (Academic, 1999).
  9. A. V. Buryak, K. Y. Kolossovski, and D. Y. Stepanov, “Optimization of refractive index sampling for multichannel fiber Bragg gratings,” IEEE J. Quantum Electron. 39(1), 91–98 (2003).
    [Crossref]
  10. M. J. Cole, W. H. Loh, R. I. Laming, M. N. Zervas, and S. Barcelos, “Moving fibre/phase mask-scanning beam technique for enhanced flexibility in producing fibre gratings with uniform phase mask,” Electron. Lett. 31(17), 1488–1489 (1995).
    [Crossref]
  11. W. H. Loh, M. J. Cole, M. N. Zervas, S. Barcelos, and R. I. Laming, “Complex grating structures with uniform phase masks based on the moving fiber-scanning beam technique,” Opt. Lett. 20, 2051–2053 (1995).
    [Crossref] [PubMed]
  12. M. Li and H. Li, “Influences of writing-beam size on the performances of dispersion-free multi-channel fiber Bragg grating,” Opt. Fiber Technol. 15(1), 33–38 (2009).
    [Crossref]
  13. H. Cao, J. Atai, X. Shu, and G. Chen, “Direct design of high channel-count fiber Bragg grating filters with low index modulation,” Opt. Express 20(11), 12095–12110 (2012).
    [Crossref] [PubMed]
  14. J. Williams, “Method of fabricating an optical waveguide grating and apparatus for implementing the method,” U.S. Patent EP1447691A1 (August18, 2004).

2012 (1)

2010 (1)

2009 (1)

M. Li and H. Li, “Influences of writing-beam size on the performances of dispersion-free multi-channel fiber Bragg grating,” Opt. Fiber Technol. 15(1), 33–38 (2009).
[Crossref]

2008 (2)

2007 (1)

2003 (1)

A. V. Buryak, K. Y. Kolossovski, and D. Y. Stepanov, “Optimization of refractive index sampling for multichannel fiber Bragg gratings,” IEEE J. Quantum Electron. 39(1), 91–98 (2003).
[Crossref]

2001 (1)

J. Skaar and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37(2), 165–173 (2001).
[Crossref]

2000 (1)

X. Shu, S. Jiang, and D. Huang, “Fiber grating Sagnac loop and its multiwavelength laser application,” IEEE Photon. Technol. Lett. 12, 980–982 (2000).
[Crossref]

1995 (2)

M. J. Cole, W. H. Loh, R. I. Laming, M. N. Zervas, and S. Barcelos, “Moving fibre/phase mask-scanning beam technique for enhanced flexibility in producing fibre gratings with uniform phase mask,” Electron. Lett. 31(17), 1488–1489 (1995).
[Crossref]

W. H. Loh, M. J. Cole, M. N. Zervas, S. Barcelos, and R. I. Laming, “Complex grating structures with uniform phase masks based on the moving fiber-scanning beam technique,” Opt. Lett. 20, 2051–2053 (1995).
[Crossref] [PubMed]

1987 (1)

Atai, J.

Barcelos, S.

W. H. Loh, M. J. Cole, M. N. Zervas, S. Barcelos, and R. I. Laming, “Complex grating structures with uniform phase masks based on the moving fiber-scanning beam technique,” Opt. Lett. 20, 2051–2053 (1995).
[Crossref] [PubMed]

M. J. Cole, W. H. Loh, R. I. Laming, M. N. Zervas, and S. Barcelos, “Moving fibre/phase mask-scanning beam technique for enhanced flexibility in producing fibre gratings with uniform phase mask,” Electron. Lett. 31(17), 1488–1489 (1995).
[Crossref]

Bland-Hawthorn, J.

Buryak, A.

Buryak, A. V.

A. V. Buryak, K. Y. Kolossovski, and D. Y. Stepanov, “Optimization of refractive index sampling for multichannel fiber Bragg gratings,” IEEE J. Quantum Electron. 39(1), 91–98 (2003).
[Crossref]

Cao, H.

Chen, G.

Chen, X.

Cole, M. J.

M. J. Cole, W. H. Loh, R. I. Laming, M. N. Zervas, and S. Barcelos, “Moving fibre/phase mask-scanning beam technique for enhanced flexibility in producing fibre gratings with uniform phase mask,” Electron. Lett. 31(17), 1488–1489 (1995).
[Crossref]

W. H. Loh, M. J. Cole, M. N. Zervas, S. Barcelos, and R. I. Laming, “Complex grating structures with uniform phase masks based on the moving fiber-scanning beam technique,” Opt. Lett. 20, 2051–2053 (1995).
[Crossref] [PubMed]

Dai, Y.

Erdogan, T.

J. Skaar and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37(2), 165–173 (2001).
[Crossref]

Hayashi, J.

Huang, D.

X. Shu, S. Jiang, and D. Huang, “Fiber grating Sagnac loop and its multiwavelength laser application,” IEEE Photon. Technol. Lett. 12, 980–982 (2000).
[Crossref]

Jiang, S.

X. Shu, S. Jiang, and D. Huang, “Fiber grating Sagnac loop and its multiwavelength laser application,” IEEE Photon. Technol. Lett. 12, 980–982 (2000).
[Crossref]

Kashyap, R.

R. Kashyap, Fiber Bragg Gratings (Academic, 1999).

Kolossovski, K.

Kolossovski, K. Y.

A. V. Buryak, K. Y. Kolossovski, and D. Y. Stepanov, “Optimization of refractive index sampling for multichannel fiber Bragg gratings,” IEEE J. Quantum Electron. 39(1), 91–98 (2003).
[Crossref]

Laming, R. I.

M. J. Cole, W. H. Loh, R. I. Laming, M. N. Zervas, and S. Barcelos, “Moving fibre/phase mask-scanning beam technique for enhanced flexibility in producing fibre gratings with uniform phase mask,” Electron. Lett. 31(17), 1488–1489 (1995).
[Crossref]

W. H. Loh, M. J. Cole, M. N. Zervas, S. Barcelos, and R. I. Laming, “Complex grating structures with uniform phase masks based on the moving fiber-scanning beam technique,” Opt. Lett. 20, 2051–2053 (1995).
[Crossref] [PubMed]

Li, H.

Li, M.

M. Li and H. Li, “Influences of writing-beam size on the performances of dispersion-free multi-channel fiber Bragg grating,” Opt. Fiber Technol. 15(1), 33–38 (2009).
[Crossref]

H. Li, M. Li, Y. Sheng, and J. E. Rothenberg, “Advances in the design and fabrication of high-channel-count fiber Bragg gratings,” J. Lightwave Technol. 25(9), 2739–2750 (2007).
[Crossref]

Loh, W. H.

W. H. Loh, M. J. Cole, M. N. Zervas, S. Barcelos, and R. I. Laming, “Complex grating structures with uniform phase masks based on the moving fiber-scanning beam technique,” Opt. Lett. 20, 2051–2053 (1995).
[Crossref] [PubMed]

M. J. Cole, W. H. Loh, R. I. Laming, M. N. Zervas, and S. Barcelos, “Moving fibre/phase mask-scanning beam technique for enhanced flexibility in producing fibre gratings with uniform phase mask,” Electron. Lett. 31(17), 1488–1489 (1995).
[Crossref]

Ouellette, F.

Rothenberg, J. E.

Sheng, Y.

Shu, X.

H. Cao, J. Atai, X. Shu, and G. Chen, “Direct design of high channel-count fiber Bragg grating filters with low index modulation,” Opt. Express 20(11), 12095–12110 (2012).
[Crossref] [PubMed]

X. Shu, S. Jiang, and D. Huang, “Fiber grating Sagnac loop and its multiwavelength laser application,” IEEE Photon. Technol. Lett. 12, 980–982 (2000).
[Crossref]

Skaar, J.

J. Skaar and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37(2), 165–173 (2001).
[Crossref]

Stepanov, D. Y.

A. V. Buryak, K. Y. Kolossovski, and D. Y. Stepanov, “Optimization of refractive index sampling for multichannel fiber Bragg gratings,” IEEE J. Quantum Electron. 39(1), 91–98 (2003).
[Crossref]

Williams, J.

J. Williams, “Method of fabricating an optical waveguide grating and apparatus for implementing the method,” U.S. Patent EP1447691A1 (August18, 2004).

Yao, J. P.

Zervas, M. N.

M. J. Cole, W. H. Loh, R. I. Laming, M. N. Zervas, and S. Barcelos, “Moving fibre/phase mask-scanning beam technique for enhanced flexibility in producing fibre gratings with uniform phase mask,” Electron. Lett. 31(17), 1488–1489 (1995).
[Crossref]

W. H. Loh, M. J. Cole, M. N. Zervas, S. Barcelos, and R. I. Laming, “Complex grating structures with uniform phase masks based on the moving fiber-scanning beam technique,” Opt. Lett. 20, 2051–2053 (1995).
[Crossref] [PubMed]

Appl. Opt. (1)

Electron. Lett. (1)

M. J. Cole, W. H. Loh, R. I. Laming, M. N. Zervas, and S. Barcelos, “Moving fibre/phase mask-scanning beam technique for enhanced flexibility in producing fibre gratings with uniform phase mask,” Electron. Lett. 31(17), 1488–1489 (1995).
[Crossref]

IEEE J. Quantum Electron. (2)

A. V. Buryak, K. Y. Kolossovski, and D. Y. Stepanov, “Optimization of refractive index sampling for multichannel fiber Bragg gratings,” IEEE J. Quantum Electron. 39(1), 91–98 (2003).
[Crossref]

J. Skaar and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37(2), 165–173 (2001).
[Crossref]

IEEE Photon. Technol. Lett. (1)

X. Shu, S. Jiang, and D. Huang, “Fiber grating Sagnac loop and its multiwavelength laser application,” IEEE Photon. Technol. Lett. 12, 980–982 (2000).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. A (1)

Opt. Express (2)

Opt. Fiber Technol. (1)

M. Li and H. Li, “Influences of writing-beam size on the performances of dispersion-free multi-channel fiber Bragg grating,” Opt. Fiber Technol. 15(1), 33–38 (2009).
[Crossref]

Opt. Lett. (2)

Other (2)

R. Kashyap, Fiber Bragg Gratings (Academic, 1999).

J. Williams, “Method of fabricating an optical waveguide grating and apparatus for implementing the method,” U.S. Patent EP1447691A1 (August18, 2004).

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Figures (6)

Fig. 1
Fig. 1 A 13 peak FBG design profile: (a) coupling coefficient, (b) grating phase. Resulting calculated spectral response: (c) reflectivity; (d) group delay.
Fig. 2
Fig. 2 Effect of beam size: (a) on the apodisation of the grating, (b) on its spectral response.
Fig. 3
Fig. 3 Experimental gratings spectra for differing beam widths without compensation.
Fig. 4
Fig. 4 Original and pre-compensated FBG designs.
Fig. 5
Fig. 5 Spectra of fabricated 5 and 11 channels gratings both uncompensated and compensated for the 265 μm beam size used.
Fig. 6
Fig. 6 Channel positions for fabricated 5, 9, 11 and 13-channels gratings. U stands for uncompensated, C for compensated grating designs, 265 and 430 correspond to the beam size in μm.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

| r ( λ ) | = 0.75 j = 1 13 exp ( ( λ λ j b ) 4 ) × exp ( i 2 π n eff ( 1 λ 1 λ j ) d j )
b ( z ) = exp ( ( z w / 2 ) 2 )
q conv ( z ) = q ( z ) * b ( z )
( q conv ( z ) ) = ( q ( z ) ) ( b ( z ) )
q ( z ) = q precomp ( z ) * b ( z ) .
q precomp ( z ) = 1 [ ( q ) / ( b ) ]

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