## Gratings superposed spatially by writing in laterally separated regions

Optics Express, Vol. 13, Issue 4, pp. 1161-1171 (2005)

http://dx.doi.org/10.1364/OPEX.13.001161

Acrobat PDF (252 KB)

### Abstract

A new type of gratings superposed spatially in laterally separated areas of the guide is introduced and analyzed using coupled-mode theory. The guided mode overlaps the constituent gratings and sees the superposition of them. Also, special characteristics that the structure might synthesize are considered, including one example where a phase-only sampled split grating provides zero response for out-of-band channels. A conventional grating requires both phase- and amplitude- sampling for zero out-of-band channels. The split grating, however, requires alignment of the constituent gratings in addition to requirements on the accuracy of the amplitude and pitch structures.

© 2005 Optical Society of America

## 1. Introduction

8. J.E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R.B. Wilcox, and J. Zweiback, “Dammann Fiber Bragg Gratings and Phase-Only Sampling for High Channel Counts,” IEEE Photon. Technol. Lett. **14**, 1309–1311 (2002). [CrossRef]

10. K. Y. Kolossovski, R. A. Sammut, A. V. Buryak, and D. Yu. Stepanov, “Three-step design optimization for multi-channel fibre Bragg gratings,” Opt. Express **11**, 1029–1038 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1029. [CrossRef] [PubMed]

11. A. Martinez, M. Dubov, I. Y. Khrushchev, and I. Bennion, “Femtosecond Inscription of Superimposed, Non-Overlapping Fibre Bragg Gratings,” presented at the 30th European Conference on Optical Communication, Stockholm, Sweden, 5–9 Sept. 2004, http://www.aston.ac.uk/~khrushci/Research/ECOC_04_FBG.pdf.

## 2. Basic features of the structure

*f*(

_{ri}*z*) and

*ϕ*(

_{ri}*z*). Equation (3) is written assuming the “dc” index change, the index change spatially averaged over one grating period, is proportional to the “ac” index change. Note however that a fabrication method might also provide a constant or zero “dc” index change.

## 3. Coupled-mode analysis

12. T. Erdogan, “Fiber Grating Spectra,” J. Lightwave Technol. **15**, 1277–1294 (1997). [CrossRef]

*ψ*

_{1}(

*z*) and

*ψ*

_{2}(

*z*) are slowly varying amplitudes of the forward and backward traveling modes respectively, and

*C*;

_{km}*k*,

*m*=1,2, are the coupling coefficients given by

*β*

_{1})=sgn(

*β*)=1 and sgn(

*β*

_{2})=sgn(-

*β*)=-1. The integral in Eq. (5) is over the cross-section of the waveguide (xy- plane) and

**e**

*and*

_{k}**e**

_{m}are such that

**E**=

**e**

_{1}(

*x*)exp(

*jβz*) and

**E**-=

**e**

_{2}(

*x*)exp(-

*jβz*) represent the normalized fields of the forward and backward traveling modes, respectively. With normalized fields, the power of each mode is unity. Inserting Eq. (3) into the following approximation,

*b*=(

_{i}*i*-1)

*b*/4,

*i*=1,…4. Under the 2-D guide assumption the width of the guide,

*b*, cancels out of the equations since the integral over

*dy*cancels with 1/

*b*that occurs with normalizing the fields. Though Eq. (7a) is for

*C*

_{11}(

*z*) it can be shown that

*β*) in Eq. (5). Eq. (8) is valid if the modes correspond to TE

_{k}_{0}because

**e**

_{1}=

**e**

_{2}. Alternatively, if they correspond to TM

_{0}and the guide is weakly guiding, it still is true that

**e**

_{1}≈

**e**

_{2}since

**e**

_{1z}≪

**e**

_{1x}in

**e**

_{2}=

**e**

_{1x}-

**e**

_{1z}.

*n*

*;*

_{dci}*i*=1,2,3,4, are constants. Eqs. (4) through (8) still apply except for (7c) which is replaced by

*ψ*

_{1}(

*z*) and

*ψ*

_{2}(

*z*) gives

12. T. Erdogan, “Fiber Grating Spectra,” J. Lightwave Technol. **15**, 1277–1294 (1997). [CrossRef]

*π*phase-shifts. For both examples, since the intention is to reproduce the responses of a conventional grating, the split grating should reproduce its “ac” coupling coefficient. If the split grating has symmetry where it comprises only 2 constituent gratings, and the boundary between them is the centerline of the guide, the integral over the fields incorporated in the “ac” coupling coefficient cancels with the one for the conventional grating. Thus we choose this symmetric split grating for the examples. With this symmetric split grating, setting the “ac” coupling coefficient equal to that of the conventional grating leads to

*f*(

_{u}*z*) and

*ϕ*(

_{u}*z*) are the index modulation and phase, where

*u*stands for unsplit. The right hand side is one-half the sum of the “ac” index changes in the constituent gratings. Whether the index change of the split grating has the form of Eq. (3) or (9), the “ac” part has the form seen in the right-hand side of Eq. (12). Eq. (12) was found by setting the “ac” coupling coefficient of the conventional grating, given by

## 4. Example: phase-only sampled split grating

8. J.E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R.B. Wilcox, and J. Zweiback, “Dammann Fiber Bragg Gratings and Phase-Only Sampling for High Channel Counts,” IEEE Photon. Technol. Lett. **14**, 1309–1311 (2002). [CrossRef]

9. A. V. Buryak, K. Y. Kolossovski, and D. Yu. Stepanov, “Optimization of Refractive Index Sampling for Multichannel Fiber Bragg Gratings,” IEEE J. Quantum Electron. **39**, 91–98 (2003). [CrossRef]

*ϕ*. Putting the computed

_{m}*ϕ*in Eq. (14) the magnitude and phase are shown in Fig. 2. The computed

_{m}*ϕ*are only moderately different than the initial estimate, indicating the closeness of that estimate to a mini-max solution. The theoretical limit on the peak value of the sampling function is √

_{m}*N*[9

9. A. V. Buryak, K. Y. Kolossovski, and D. Yu. Stepanov, “Optimization of Refractive Index Sampling for Multichannel Fiber Bragg Gratings,” IEEE J. Quantum Electron. **39**, 91–98 (2003). [CrossRef]

15. H. Li, Y. Sheng, Y. Li, and J. E. Rothenberg, “Phased-Only Sampled Fiber Bragg Gratings for High-Channel-Count Chromatic Dispersion Compensation,” J. Lightwave Technol. **21**, 2074–2083 (2003). [CrossRef]

8. J.E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R.B. Wilcox, and J. Zweiback, “Dammann Fiber Bragg Gratings and Phase-Only Sampling for High Channel Counts,” IEEE Photon. Technol. Lett. **14**, 1309–1311 (2002). [CrossRef]

15. H. Li, Y. Sheng, Y. Li, and J. E. Rothenberg, “Phased-Only Sampled Fiber Bragg Gratings for High-Channel-Count Chromatic Dispersion Compensation,” J. Lightwave Technol. **21**, 2074–2083 (2003). [CrossRef]

*f*(

*z*), the central pitch Λ

*, and the phase*

_{u}*ϕ*(

_{g}*z*).

*s*(

*z*)=

*a*(

*z*)exp[

*iϕ*(

*z*)] where the amplitude

*a*(

*z*) has a maximum value

*A*, the terms are (as with LINC)

*f*(

*z*) and

*ϕ*(

_{g}*z*). The factors of 2 are for counteracting the factor 0.5 in (12). Thus the left- and right-hand sides of Eq. (12) are equal, and the conventional and split gratings have the same reflectivity responses. Due to the factors of 2 the maximum index change equals, though at least it does not exceed, the maximum index change of the conventional grating. Figure 2 shows the reflectivity, assuming a seed grating with a bell-shaped apodizing function and

*ϕ*(

_{g}*z*)=0. The slab waveguide has a

_{1}=0.15 µm, n

_{1}=1.4579, and n

_{2}=n

_{SiO2}. Finally, note that the reflectivity for out-of-band channels is zero.

## 5. Example: amplitude-only sampled split grating

*ϕ*. The one difference is this time a symmetry for the channel phases will be enforced in order to produce a sampling function that’s purely real. Splitting this purely real sampling function into a pair of purely real terms will then be trivial, and thus the constituent gratings will be amplitude-only modulated which is the goal. The symmetry enforced is

_{m}*s*

_{2}(

*z*). The sum of these index changes equals 2 times the perturbation of the conventional grating. Thus Eq. (12) holds, and thus the split grating provides the same responses as the conventional grating.

## 6. Conclusions

## References and links

1. | W. H. Loh, F. Q. Zhou, and J. J. Pan, “Sampled Fiber Grating Based-Dispersion Slope Compensator,” IEEE Photon. Technol. Lett. |

2. | Y. Painchaud, A. Mailloux, H. Chotard, E. Pelletier, and M. Guy, “Multi-channel fiber Bragg gratings for dispersion and slope compensation,” in |

3. | A. Othonos, X. Lee, and R. M. Measures, “Superimposed multiple Bragg gratings,” Electron. Lett. |

4. | K. Zhou, A. G. Simpson, X. Chen, L. Zhang, and I. Bennion, “Fiber Bragg Grating Sensor Interrogation System Using a CCD Side Detection Method With Superimposed Blazed Gratings,” IEEE Photon. Technol. Lett. |

5. | M. G. Xu, J. L. Archambault, L. Reekie, and J. P. Dakin, “Discrimination between strain and temperature effects using dual-wavelength fibre grating sensors,” Electron. Lett. |

6. | J. Azana, R. Slavik, P. Kockaert, L. R. Chen, and S. LaRochelle, “Generation of Customized Ultrahigh Repetition Rate Pulse Sequences Using Superimposed Fiber Bragg Gratings,” J. Lightwave Technol. |

7. | A. Safaai-Jazi and T. L. Gradishar, “Gratings with Independently Apodized Layers,” presented at the Southeast Regional Meeting on Optoelectronics, Photonics, and Imaging, Charlotte, North Carolina, 18–19 Sept. 2000. |

8. | J.E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R.B. Wilcox, and J. Zweiback, “Dammann Fiber Bragg Gratings and Phase-Only Sampling for High Channel Counts,” IEEE Photon. Technol. Lett. |

9. | A. V. Buryak, K. Y. Kolossovski, and D. Yu. Stepanov, “Optimization of Refractive Index Sampling for Multichannel Fiber Bragg Gratings,” IEEE J. Quantum Electron. |

10. | K. Y. Kolossovski, R. A. Sammut, A. V. Buryak, and D. Yu. Stepanov, “Three-step design optimization for multi-channel fibre Bragg gratings,” Opt. Express |

11. | A. Martinez, M. Dubov, I. Y. Khrushchev, and I. Bennion, “Femtosecond Inscription of Superimposed, Non-Overlapping Fibre Bragg Gratings,” presented at the 30th European Conference on Optical Communication, Stockholm, Sweden, 5–9 Sept. 2004, http://www.aston.ac.uk/~khrushci/Research/ECOC_04_FBG.pdf. |

12. | T. Erdogan, “Fiber Grating Spectra,” J. Lightwave Technol. |

13. | M. Ibsen, M. K. Durkin, M. J. Cole, and R. I. Laming, “Sinc-Sampled Fiber Bragg Gratings for Identical Multiple Wavelength Operation,” IEEE Photon. Technol. Lett. |

14. | M. Guy, “Recent Advances in Fiber Bragg Grating Technology Enable Cost-Effective Fabrication of High-Performance Optical Components,” Physics in Canada60, 2004. |

15. | H. Li, Y. Sheng, Y. Li, and J. E. Rothenberg, “Phased-Only Sampled Fiber Bragg Gratings for High-Channel-Count Chromatic Dispersion Compensation,” J. Lightwave Technol. |

16. | S. A. Hetzel, A. Bateman, and J. P. McGeehan, “LINC Transmitter,” Electron. Lett. |

**OCIS Codes**

(060.2340) Fiber optics and optical communications : Fiber optics components

(230.1480) Optical devices : Bragg reflectors

(230.7390) Optical devices : Waveguides, planar

(230.7400) Optical devices : Waveguides, slab

**ToC Category:**

Research Papers

**History**

Original Manuscript: January 5, 2005

Revised Manuscript: January 2, 2005

Published: February 21, 2005

**Citation**

Thomas Gradishar and A. Safaai-Jazi, "Gratings superposed spatially by writing in laterally separated regions," Opt. Express **13**, 1161-1171 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-4-1161

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### References

- W. H. Loh, F. Q. Zhou, and J. J. Pan, “Sampled Fiber Grating Based-Dispersion Slope Compensator ,” IEEE Photon. Technol. Lett. 11, 1280-1282 (1999). [CrossRef]
- Y. Painchaud,A. Mailloux, H. Chotard, E. Pelletier, and M. Guy, “Multi-channel fiber Bragg gratings for dispersion and slope compensation,” in Proc. Optical Fiber Communication Conference (Optical Society of America, Washington, DC, 2002) paper ThAA5.
- A. Othonos, X. Lee, and R. M. Measures, “Superimposed multiple Bragg gratings,” Electron. Lett. 30, 1972-1974 (1994). [CrossRef]
- K. Zhou, A. G. Simpson, X. Chen, L. Zhang, and I. Bennion, “Fiber Bragg Grating Sensor Interrogation System Using a CCD Side Detection Method With Superimposed Blazed Gratings,” IEEE Photon. Technol. Lett. 16, 1549-1551 (2004). [CrossRef]
- M. G. Xu, J. L. Archambault, L. Reekie, and J. P. Dakin, “Discrimination between strain and temperature effects using dual-wavelength fibre grating sensors,” Electron. Lett. 30, 1085-1087 (1994). [CrossRef]
- J. Azana, R. Slavik, P. Kockaert, L. R. Chen, S. LaRochelle, “Generation of Customized Ultrahigh Repetition Rate Pulse Sequences Using Superimposed Fiber Bragg Gratings,” J. Lightwave Technol. 21, 1490-1498 (2003). [CrossRef]
- A. Safaai-Jazi and T. L. Gradishar, “Gratings with Independently Apodized Layers,” presented at the Southeast Regional Meeting on Optoelectronics, Photonics, and Imaging, Charlotte, North Carolina, 18-19 Sept. 2000.
- J.E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R.B. Wilcox, and J. Zweiback, “Dammann Fiber Bragg Gratings and Phase-Only Sampling for High Channel Counts,” IEEE Photon. Technol. Lett. 14, 1309-1311 (2002). [CrossRef]
- A. V. Buryak, K. Y. Kolossovski, and D. Yu. Stepanov, “Optimization of Refractive Index Sampling for Multichannel Fiber Bragg Gratings,” IEEE J. Quantum Electron. 39, 91-98 (2003). [CrossRef]
- K. Y. Kolossovski, R. A. Sammut, A. V. Buryak, D. Yu. Stepanov, “Three-step design optimization for multi-channel fibre Bragg gratings,” Opt. Express 11, 1029-1038 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1029.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1029</a> [CrossRef] [PubMed]
- A. Martinez, M. Dubov, I. Y. Khrushchev, I. Bennion, “Femtosecond Inscription of Superimposed, Non-Overlapping Fibre Bragg Gratings,” presented at the 30th European Conference on Optical Communication, Stockholm, Sweden, 5-9 Sept. 2004, <a href="http://www.aston.ac.uk/~khrushci/Research/ECOC_04_FBG.pdf.">http://www.aston.ac.uk/~khrushci/Research/ECOC_04_FBG.pdf.</a>
- T. Erdogan, “Fiber Grating Spectra,” J. Lightwave Technol. 15, 1277-1294 (1997). [CrossRef]
- M. Ibsen, M. K. Durkin, M. J. Cole, and R. I. Laming, “Sinc-Sampled Fiber Bragg Gratings for Identical Multiple Wavelength Operation,” IEEE Photon. Technol. Lett. 10, 842-844 (1998). [CrossRef]
- M. Guy, “Recent Advances in Fiber Bragg Grating Technology Enable Cost-Effective Fabrication of High-Performance Optical Components,” Physics in Canada 60, 2004.
- H. Li, Y. Sheng, Y. Li, J. E. Rothenberg, “Phased-Only Sampled Fiber Bragg Gratings for High-Channel-Count Chromatic Dispersion Compensation,” J. Lightwave Technol. 21, 2074-2083 (2003). [CrossRef]
- S. A. Hetzel, A. Bateman, and J. P. McGeehan, “LINC Transmitter,” Electron. Lett. 27, 844-846 (1991). [CrossRef]

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