## Design of discrete, nearly-uniform Bragg gratings in planar waveguides

Optics Express, Vol. 13, Issue 4, pp. 1098-1106 (2005)

http://dx.doi.org/10.1364/OPEX.13.001098

Acrobat PDF (152 KB)

### Abstract

In this paper we present an efficient method for designing discrete, nearly-uniform Bragg gratings in generic planar waveguides. Various schemes have already been proposed to design continuous Bragg gratings in optical fibers, but a general scheme for creating their discrete counterpart is still lacking. Taking a continuous Bragg grating as our starting point, we show that the same grating functionalities can also be realized in any planar waveguide by discretizing it into a series of air holes. The relationship between the two gratings is established in terms of grating strength and local grating period.

© 2005 Optical Society of America

## 1. Introduction

1. J. Zhang, P. Shum, and S. Y. Li, et al., “Design and fabrication of flat-band long-period grating,” IEEE Photonics Technol. Lett. **15**, 1558–1560 (2003). [CrossRef]

5. M. Ibsen and R. Feced, “Fiber Bragg gratings for pure dispersion-slope compensation,” Opt. Lett. **28**, 980–982 (2003). [CrossRef] [PubMed]

6. A. Yariv, “Coupled-wave formalism for optical waveguiding by transverse Bragg reflection,” Opt. Lett. **27**, 936–938 (2002). [CrossRef]

7. W. Kuang and J.D. O’Brien, “Reducing the out-of-plane radiation loss of photonic crystal waveguides on high-index substrates,” Opt. Lett. **29**, 860–862 (2004). [CrossRef] [PubMed]

*et al*. first proposed a simple discretization procedure in the microwave regime, in a similar effort to set up an equivalence relation between an FBG and a discrete air-hole grating in a microstrip waveguide [8

8. M. A. G. Laso et al., “Analysis and design of 1-D photonic bandgap microstrip structures using a fiber grating model,” Microwave Opt. Tech. Lett. **22**, 223–226 (1999). [CrossRef]

## 2. Background

*n*

_{0}is the average effective refractive index and

*K*

_{0}=(=2

*π*/Λ), Δ

*n*(

*z*), and

*θ*(

*z*) specify the grating parameters: Λ is the reference period, Δ

*n*(

*z*) accounts for the local grating strength (apodization), and

*θ*(

*z*) determines its phase variation and local period.

## 3. Grating design

9. R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. **35**, 1105–1115 (1999). [CrossRef]

*ñ*

_{0}=the effective refractive index of the unperturbed waveguide,

*a*=(area of the mth air hole)×(1-

_{m}*n*),

_{substrate}*b*=deviation of the mth air hole from

_{m}*z*=

*m*Λ,

*W*(

*y*) is a windowing function that is nonzero only within the thickness of the waveguide. Also embedded in Eqs. (3-a) and (3-b) is the implicit assumption that the areas of the air holes, hence the

*a*’s, are small, for otherwise we would not have been able to approximate the index perturbation by

_{m}*δ*functions. Additionally the deviation of the air holes from their respective reference locations, i.e., the

*b*’s, are also taken to be small since the grating is assumed to be “nearly uniform”.

_{m}*n*(

*z*) in (1) and

*ñ*(

*x,y,z*) in Eq. (3) we arrive at the following intuition:

*θ′*(

*z*) is the first order derivative of

*θ*(

*z*).

*n*(

*z*) and

*ñ*(

*x,y,z*) give the same filter response we go back to the original wave equation. Assuming linearly polarized electric field, say

*E*⃗=

*E*, the perturbed wave equation can be written [10]:

_{x}*P*(

_{pert}*r⃗,t*) is the only term in the wave equation that provides coupling between the forward and backward propagating waves,

*n*(

*z*) and

*ñ*(

*x,y,z*) are equivalent as long as they give equivalent perturbative coupling term

*P*(

_{pert}*r,t*).

*n*(

*z*)≪

*n*

_{0},

*P*(

_{pert}*r⃗,t*) is given by (1):

*n*(

*z*) and

*θ*(

*z*), Δ(

*n*

^{2}) consists of two distinct Fourier components centered respectively at ±

*K*

_{0}(see Fig. 2). Without loss of generality we filter out the positive frequency component and apply the sampling theorem on its slowly-varying envelope (including the phase factor

*e*. The end result is an alternative but equivalent expression for the perturbative component at +

^{iθ(z)}*K*

_{0}:

*ñ*(

*x, y, z*)≪

*ñ*

_{0}and from Eqs. (3-a) and (3-b) we get:

*δ*(

*x*) and

*W*(

*y*) for now and once again assuming

*a*’s and

_{m}*b*’s are sufficiently slowly varying, we plot the Fourier spectrum of Δ(

_{m}*ñ*

^{2}) in Fig. 3. Note only the two components located at

*K*

_{0}and -

*K*

_{0}can provide coupling between the forward and backward propagating waves and hence are relevant in our analysis.

*K*

_{0}with a pass-band bandwidth of 2Γ :

*b*’s are small, we obtain the following relations:

_{m}*a*is proportional to the area of the mth air hole, Eq. (11-a) confirms our first intuition in Eq. (4-a).

_{m}_{1}and Λ

_{2}, and recall

*K*

_{0}=2

*π*/Λ, we get:

*b*and

_{m+1}*b*

_{m-1}and made the slowly-varying assumption on

*θ*(

*z*). (12) indeed confirms our second intuition in Eq. (4-b).

*δ*(

*x*) and

*W*(

*y*) functions that we have deliberately left out of Δ(

*ñ*

^{2}) in our derivation so far. We show in the following that the inclusion of the two functions requires the introduction of an extra scaling factor

*γ*into Δ

*ñ*(

*x, y, z*), i.e.,

*γ*is presumably highly dependent on the geometry of the waveguide.

*γ*by matching coupled mode equations. In the Appendix we derive the coupled mode equations for wave propagation in FBG’s (i.e., the continuous grating), and so long as the goal is to generate the same filter response via a discrete grating in a planar waveguide, we need to reproduce the same set of coupled mode equations for the new waveguide as well. Following through the derivation in the Appendix, we realize that we need to make the following substitution for the discrete grating:

*k*is the propagation constant of the mth mode.

_{m}*∂*operator in Eq. (15) by -

^{2}/∂t^{2}*ω*

^{2}, multiply both sides by E*

_{B}(

*x,y*)=E*

_{F}(

*x,y*), integrate over the entire transverse plane, and then apply the orthonormality relation (16). Collecting resonant terms in the resultant equation such as the ones that vary as

*t*is the thickness of the planar waveguide and E

_{m}=E

_{B}=E

_{F}. Or in other words,

*ω*by

*ω*

_{0}, the center frequency, and

*k*by

_{m}*K*

_{0}/2.

*c*is the speed of light in vacuum and

*ñ*is the effective modal index given by Eq. (3-a).

## 4. Conclusion

## Appendix

*b*(

_{B}*z,β*) (backward propagating) and

*b*(

_{F}*z,β*) (forward propagating) [9

9. R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. **35**, 1105–1115 (1999). [CrossRef]

*β*is the detuning parameter

*n*(

*z*) and

*θ*(

*z*) have been defined in (1).

*E*⃗=

*E*, and then expand the total field as:

_{x}*b*and

_{B}*b*:

_{F}*b*″ and

_{B}*b*″

_{F}have been omitted from (A-7).

*k=ω/c*and

*n*

_{0}is the effective modal refractive index of the unperturbed waveguide.

*kn*

_{0})

^{2}as

*β≪K*

_{0}. Plugging (A-9) and (A-10) into (A-7), we find the first term in the square brackets becomes

_{B}(

*x,y*)=E

_{F}(

*x,y*). So we can cross them out on both sides of (A-11). Also if we replace the ∂

^{2}/

*∂t*

^{2}operator by -

*ω*

^{2}and collect the resonant terms, say the ones that vary as

*q*(

*z*) has been defined in (A-4). Thus (A-14) gives back the original mode equation (A-1). In a similar fashion, by collecting resonant terms with

## Acknowledgments

## References and links

1. | J. Zhang, P. Shum, and S. Y. Li, et al., “Design and fabrication of flat-band long-period grating,” IEEE Photonics Technol. Lett. |

2. | J. Skaar and O. H. Waagaard, “Design and characterization of finite-length fiber gratings,” IEEE J. Quantum Electron. |

3. | H. P. Li, Y. L. Sheng, and Y. Li, et al. “Phased-only sampled fiber Bragg gratings for high-channel-count chromatic dispersion compensation,” J. Lightwave Technol. |

4. | L. G. Sheu, K. P. Chuang, and Y.C. Lai, “Fiber bragg grating dispersion compensator by single-period overlap-step-scan exposure,” IEEE Photonics Technol. Lett. |

5. | M. Ibsen and R. Feced, “Fiber Bragg gratings for pure dispersion-slope compensation,” Opt. Lett. |

6. | A. Yariv, “Coupled-wave formalism for optical waveguiding by transverse Bragg reflection,” Opt. Lett. |

7. | W. Kuang and J.D. O’Brien, “Reducing the out-of-plane radiation loss of photonic crystal waveguides on high-index substrates,” Opt. Lett. |

8. | M. A. G. Laso et al., “Analysis and design of 1-D photonic bandgap microstrip structures using a fiber grating model,” Microwave Opt. Tech. Lett. |

9. | R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. |

10. | A. Yariv, |

**OCIS Codes**

(050.2770) Diffraction and gratings : Gratings

(060.2330) Fiber optics and optical communications : Fiber optics communications

(230.1480) Optical devices : Bragg reflectors

**ToC Category:**

Research Papers

**History**

Original Manuscript: November 19, 2004

Revised Manuscript: November 18, 2004

Published: February 21, 2005

**Citation**

George Ouyang and Amnon Yariv, "Design of discrete, nearly-uniform Bragg gratings in planar waveguides," Opt. Express **13**, 1098-1106 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-4-1098

Sort: Journal | Reset

### References

- J. Zhang, P. Shum, S. Y. Li, et al., “Design and fabrication of flat-band long-period grating,” IEEE Photonics Technol. Lett. 15, 1558-1560 (2003). [CrossRef]
- J. Skaar and O. H. Waagaard, “Design and characterization of finite-length fiber gratings,” IEEE J. Quantum Electron. 39, 1238-1245 (2003). [CrossRef]
- H. P. Li, Y. L. Sheng, Y. Li, et al. “Phased-only sampled fiber Bragg gratings for high-channel-count chromatic dispersion compensation,” J. Lightwave Technol. 21, 2074–2083 (2003). [CrossRef]
- L. G. Sheu, K. P. Chuang, and Y.C. Lai, “Fiber bragg grating dispersion compensator by single-period overlap-step-scan exposure,” IEEE Photonics Technol. Lett. 15, 939-941 (2003). [CrossRef]
- M. Ibsen and R. Feced, “Fiber Bragg gratings for pure dispersion-slope compensation,” Opt. Lett. 28, 980-982 (2003). [CrossRef] [PubMed]
- A. Yariv, “Coupled-wave formalism for optical waveguiding by transverse Bragg reflection,” Opt. Lett. 27, 936-938 (2002). [CrossRef]
- W. Kuang and J.D. O’Brien, “Reducing the out-of-plane radiation loss of photonic crystal waveguides on high-index substrates,” Opt. Lett. 29, 860-862 (2004). [CrossRef] [PubMed]
- M. A. G. Laso et al., “Analysis and design of 1-D photonic bandgap microstrip structures using a fiber grating model,” Microwave Opt. Tech. Lett. 22, 223-226 (1999). [CrossRef]
- R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105-1115 (1999). [CrossRef]
- A. Yariv, Optics Electronics (4th edition, Saunders College Publishing, 1991).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.