## Simple model of errors in chirped fiber gratings

Optics Express, Vol. 12, Issue 1, pp. 189-197 (2004)

http://dx.doi.org/10.1364/OPEX.12.000189

Acrobat PDF (133 KB)

### Abstract

A simple etalon based model is presented to show the origin of the wavelength-dependent ripples in the group delay and phase, and in the intensity of optical signals reflected from chirped fiber gratings. The simplicity of the model allows intuitive understanding of the effects, and quantitative predictions. We derive accurate scaling laws that allow the experimenter to make quantitative connections between the grating writing process parameters and grating performance.

© 2004 Optical Society of America

## 1. Introduction

1. R. Kashyap and M. deLacerda-Rocha, “On the group delay of chirped fibre Bragg gratings,” Opt. Commun. **153**, 19–22 (1998). [CrossRef]

2. L. Poladian, “Understanding profile-induced group-delay ripple in Bragg gratings,” Appl. Opt. **39**, 1920–1923 (2000). [CrossRef]

4. M. Sumetsky, B.J. Eggleton, and C. Martijn de Sterke, “Theory of group delay ripple generated by chirped fiber gratings,” Opt. Express **10**, 332–340, 2002. http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-332. [CrossRef] [PubMed]

5. D. Garthe, G. Milner, and Y. Cai, “System performance of broadband dispersion compensating gratings,” Electron. Lett. **19**, 582–583 (1998). [CrossRef]

2. L. Poladian, “Understanding profile-induced group-delay ripple in Bragg gratings,” Appl. Opt. **39**, 1920–1923 (2000). [CrossRef]

6. L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E **48**, 4758–4767 (1993). [CrossRef]

## 2. Sideband model

*n*

_{0}. In general, amplitude and phase modulations contain all information about the apodization and imposed chirp, as well as errors due to non-ideal fabrication. The refractive index pattern in a grating may be described as

*m*(

*z*) is an amplitude perturbation of the oscillating refractive index, and

*ϕ*(

*z*) is a perturbation of the grating period Λ(

*z*)=2π/

*p*(

*z*). A grating of period Λ

_{0}reflects light of wavelength

*λ*

_{0}=2

*n*

_{0}Λ

_{0}, and a linearly chirped grating has a position dependent period characterized by the chirp parameter

*C*

_{0}, such that Λ(

*z*)=Λ

_{0}+

*C*

_{0}

*z*.

*n*(

*z*) is the index modulation envelope, and

*p*(

*z*)≈

*p*

_{0}(1-

*C*

_{0}

*z*Λ

_{0}) is the chirp profile. We refer to these as “bulk” grating properties, considered here to be varying spatially with much slower rates than the nominal grating frequency

*p*

_{0}. Throughout we make the approximation that the grating bandwidth is much smaller than the center wavelength (

*C*

_{0}

*z*≪Λ

_{0}), which is well justified in current telecommunication systems centered near 1550 nm with bandwidth 30 nm.

*W*,

*X*,

*Y*and

*Z*are the amplitudes of the perturbation at the spatial frequency

*g*. In the limit that

*m*,

*ϕ*≪1,

*n*(

*z*) is rewritten in terms of these sidebands in the spatial domain,

*N*

^{±}≡

*W*

^{∓}

*iX*+

*iY*±

*Z*. The important result is that spectrally distinct periodicities are present with periods 2

*π*/

*p*, 2

*π*/(

*p*+

*g*) and 2

*π*/(

*p*-

*g*). These will form the main band and sidebands respectively, where the spectral separation of the side bands from the main band is Δ

*λ*≈±

*λ*

_{0}

*g*/

*p*when

*g*≪

*p*. All three bands are identical in their bulk phase and amplitude response functions

*p*(

*z*) and Δ

*n*(

*z*) respectively, but differ in the scaling and wavelength offsets.

*C*

_{0}=0.079 nm/cm and perturbation period 2

*π*/

*g*=1.1 mm, yielding spectral sidebands separated by ~0.72 nm. The grating was apodized with a super-Gaussian profile expected to yield <5 ps peak-to-peak GDR. Note that according Eq. (3), the sidebands have the same bandwidth as and overlap the main band. In the overlap region the sidebands are indicated by the effect on the GDR and phase ripple. Phase ripple in Fig. 1(b) is defined as the departure from the ideal phase of a linearly chirped grating, and is calculated by integrating the full group delay and taking the residual of a second order polynomial fit.

*k*. The spectral separation of these bands from Eq. (3) gives rise to a spatial separation of the reflection point for a particular wavelength, i.e. when a grating is written at a particular point

*z*, the three separate wavelengths are written simultaneously. Upon chirping, a particular wavelength will then be resonant with each band at a different value of

*z*, forming etalons of relatively simple structure. The reflected field has four contributions of first order in sideband reflectivity or larger, as shown in Fig. 2. In the time domain, an incoming pulse will produce both early and late echoes according to the travel time.

*i*(

*ωt*-

*kx*) has been dropped,

*φ*is the ideal phase response and

*τ*=2

*n*

_{0}

*L*/

*c*is the optical propagation delay due to the path length

*L*between the bands. The path length is determined by the period chirp

*C*

_{0}through

*L*=Δ

*λ*/2

*n*

_{0}

*C*

_{0}≈

*λ*

_{0}

*g*/2

*n*

_{0}

*p*

*C*

_{0}.

*A*,

*B*and

*C*are the field reflection coefficients for each band: they are a function

*f*of the index contrast Δ

*n*(

*k*) from Eq. (3), where a change of variable has been used from Eq. (1); Δ

*n*(

*z*→

*π*/

*n*

_{0}

*C*

_{0}

*k*). In this formalism, the function

*f*[] is difficult to calculate analytically, and is normally found numerically. However, the actual magnitude of the reflections is not necessary to make useful predictions as we will show later. Note that the negative sign in front of the third term of

*E*

_{r}is a consequence of reflection at

*A*from the opposite side as the first term. Physically, a single grating fringe has width

*λ*/4

*n*

_{0}, causing a

*π*phase shift.

*E*

_{0}is coherent and

*E*

_{0}(

*k*,

*t*±

*τ*)=

*E*

_{0}exp(±

*iϕ*) with

*ϕ*=2

*kn*

_{0}

*L*. Equation (4) is then easily solved for the intensity

*I*and the phase

*α*of the reflected field, using the approximation that

*A*,

*C*≪1;

*c*

^{-1}(

*∂α*/

*∂k*), and the GDR Δ

*τ*, is commonly defined as the departure from the ideal group delay

*c*

^{-1}(

*∂φ*/

*∂k*) ;

*φ*.

*A*and

*C*above. It is therefore straightforward to include additional discrete or continuous sideband spectra. Equations (5) and (6) show that the bulk properties are determined by the reflectivity

*B*and phase

*φ*of the main band. As shown previously [4

4. M. Sumetsky, B.J. Eggleton, and C. Martijn de Sterke, “Theory of group delay ripple generated by chirped fiber gratings,” Opt. Express **10**, 332–340, 2002. http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-332. [CrossRef] [PubMed]

*A*,

*B*and

*C*. This minimum period observable in Eq. (7) is caused by the longest etalon supported by the grating, or simply when

*L*equals the grating length.

8. M. Yamada and K. Sakuda, “Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach,” Appl. Opt. **26**, 3474 (1987). [CrossRef] [PubMed]

9. T. Erdogan, “Fiber grating spectra,” J. Lightwave Techno. **15**, 1277–1294 (1997). [CrossRef]

*sin*term in Eq. (6)) compared to coupled mode (CM) simulation, versus the size of the sideband. For clarity, we take only one sideband (

*C*=0) and the ripple is determined only where there is spectral overlap between the side band and main band.

*B*=0.95). Since the phase ripple is not commonly shown in the literature, we show the GDR on the right axis for a particular grating. However, phase ripple is much more universal since it is independent of grating parameters such as chirp and, in general, perturbation frequency g. We also show the comparison for an uncommonly weak main band

*B*=0.2, (equivalent to a reflection power loss of 14dB). For a stronger sideband the approximation that the perturbation is small begins to break down and the addition of more terms in Eq. (4) would restore the accuracy.

*A*and

*C*from that of

*B*, in the limit of small side bands (we refer to

*A*and

*C*interchangeably). The field reflectivity is

*R*=tanh(

*κL*

_{eff}), where

*κ*∝Δ

*n*is the coupling strength [9

9. T. Erdogan, “Fiber grating spectra,” J. Lightwave Techno. **15**, 1277–1294 (1997). [CrossRef]

*L*

_{eff}, proportional to

6. L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E **48**, 4758–4767 (1993). [CrossRef]

*A*using only the sideband to main band ratios from Eq. (3), and the reflectivity

*B*of the main band, so we are only concerned with the relative magnitudes of the reflections. Under the assumption of weak coupling of the sidebands, we define the index contrast ratio between the side band and main band;

*n*

_{sb,mb}is the coefficient of the appropriate band in Eq. (3). The reflectivity

*A*may be calculated without using CM simulations, as long as one knows the reflectivity of the main band,

*B*and the ratio

*γ*. Calculation of

*γ*must be done through understanding the effect of specific errors on grating writing.

## 3. Example

*η*(

*z*) that are near the spatial frequency

*p*since they are expected to be very small in any practical system (even if they are not, this model can be used by considering gratings written by

*η*(

*z*) alone in place of the sidebands). Instead, a slow index variation must be thought of as affecting the local reflective wavelength of the grating. An effective spatial frequency of the index

*p*

_{eff}, can be defined to contain the index perturbation since

*p*=2

*π*2

*n*/

*λ*;

*p*

_{eff}to obtain the phase, in which slow components escape integration, gives the effective index

*p*(

*z*)≈

*p*

_{0}has been used for purposes of scaling the perturbation. Equation (11) has the form of a classical frequency modulation and is easily cast into the sideband form when

*η*(

*x*) is written in terms of its Fourier components. Again we specialize to a single component such that

*η*(

*x*)=

*δn*sin(

*gx*), noting that other components can be simply added in the final result. Integrating

*η*(

*x*) and rewriting Eq. (11) gives

*π*/

*g*=1mm, slow compared to the grating period of 2

*π*/

*p*

_{0}=532 nm. For a grating with 10dB loss in transmission (

*B*=0.95) and

*C*

_{0}=0.079 nm/cm (~420 ps/nm dispersion), consider a perturbation to the overall index

*δn*required to give the -28 dB sideband seen in Fig.1(a). Using

*n*

_{0}=1.45 gives

*A*=0.039 and the required

*δn*is 3.3×10

^{-5}. Using Eqs. (5–7), the phase ripple is ±78 mrad and the GDR is ±24 ps, in agreement with Fig.1b. Further, the power penalty can be calculated from reference [11] to be 0.4 dB for a 10Gb/s NRZ signal. By rewriting the index in sideband formulation, all types of perturbation may be considered in this way.

## 4. Discussion

*A*is proportional to

*L*∝1/

*C*

_{0}, we see from Eq. (7) that the GDR Δτ∝

4. M. Sumetsky, B.J. Eggleton, and C. Martijn de Sterke, “Theory of group delay ripple generated by chirped fiber gratings,” Opt. Express **10**, 332–340, 2002. http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-332. [CrossRef] [PubMed]

*A*(when

*B*~1 for simplicity). Therefore, upon comparing gratings of different chirp but similar strength, grating performance is independent of chirp.

*B*≈1), the combination of the two reflections from sideband

*A*cancel, and the reflectivity variation is small. This behavior is exactly that of a Gires-Tournois etalon, and provides justification for neglecting reflectivity variations in previous models [11–13]. For weaker gratings (

*B*<1) light penetrates to the reflection at sideband

*C*, contributing to the fine structure when there is spectral overlap between the bands, and enhancing reflectivity variations.

*A*=

*C*(1-

*B*

^{2})/(1+

*B*

^{2}), Eqs. (5,7) demonstrate that there is no phase ripple, but only amplitude ripple where all three bands overlap spectrally. This is extremely important when attempting to assign performance levels based upon grating measurements, as many techniques focus on the phase or GDR. However, we maintain that the sideband amplitude is the relevant quantity, and fortuitous cancellations of phase or GDR cannot be used to increase grating performance for incoherent signals.

*C*, occurring at time

*T*+2

*nL*/

*c*cannot interfere with the early reflection from sideband

*A*at time

*T*-2

*nL*/

*c*, as can occur when making coherent measurements. Since the penalty of a dispersion compensating grating can be stated in terms of the energy removed from the main pulse [11], it is more meaningful to consider the amplitude of the sidebands than the phase ripple alone, especially for weaker gratings. This can present measurement issues, since determination of the sideband size requires both the reflected phase and amplitude, as well as the relative phase between periodic ripples. Generally, the sideband cannot be measured directly as in Fig. 1 since the relevant wavelength range is obscured by the main band. In some cases, it may be useful to obtain better data about the far sideband

*C*by measuring the transmitted amplitude and phase, repeating the analysis above. Alternatively, the grating may be measured from the other direction.

*l*

_{tr}for the perturbation increases. Particular perturbations that are very long may make the transition from the chromatic dispersion regime (intra-symbol) into the phase ripple regime (inter-symbol), changing their contribution to the system penalty. This is not a fundamental issue, but a subtlety of application to a particular communication format.

## 5. Conclusion

## Acknowledgments

## References and links

1. | R. Kashyap and M. deLacerda-Rocha, “On the group delay of chirped fibre Bragg gratings,” Opt. Commun. |

2. | L. Poladian, “Understanding profile-induced group-delay ripple in Bragg gratings,” Appl. Opt. |

3. | R. Feced and M.N. Zervas, “Effects of Random Phase and Amplitude Errors in Optical Fiber Bragg Gratings,” J. Lightwave Technol. |

4. | M. Sumetsky, B.J. Eggleton, and C. Martijn de Sterke, “Theory of group delay ripple generated by chirped fiber gratings,” Opt. Express |

5. | D. Garthe, G. Milner, and Y. Cai, “System performance of broadband dispersion compensating gratings,” Electron. Lett. |

6. | L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E |

7. | J.F. Brennan and D.L. LaBrake, “Realization of >10-m-long chirped fiber Bragg gratings,” OSA, Bragg Gratings, Photosensitivity, and Poling, Stuart, FL, ThD2, pp. 35–37 (September 1999). |

8. | M. Yamada and K. Sakuda, “Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach,” Appl. Opt. |

9. | T. Erdogan, “Fiber grating spectra,” J. Lightwave Techno. |

10. | T. Erdogan, private communication. We have also verified this scaling against coupled-mode simulations. |

11. | X. Fan, D.L. LaBrake, and J.F. Brennan, “Chirped fiber grating characterization with phase ripples,” OSA Optical Fiber Communications (Optical Society of America, Washington, D.C., 2003), FC2. |

12. | M. Eiselt, C.B. Clausen, and R.W. Tkach, “Performance characterization of components with group delay fluctuations,” Symposium on Optical Fiber Measurements (NIST, Boulder, Colorado, 2002), Session III. |

13. | C. Sheerer, C. Glingener, G. Fisher, M. Bohn, and W. Rosenkranz, “Influence of filter group delay ripples on system performance,” European Conf. Opt. Commun. (Nice, France, 1999), I-410. |

14. | M. Sumetsky, P.I. Reyes, P.S. Westbrook, N.M. Litchinitser, and B.J. Eggleton, “Group-delay ripple correction in chirped fiber Bragg gratings,” Opt. Lett. |

**OCIS Codes**

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

(230.1480) Optical devices : Bragg reflectors

**ToC Category:**

Research Papers

**History**

Original Manuscript: November 19, 2003

Revised Manuscript: December 30, 2003

Published: January 12, 2004

**Citation**

Michael Matthews, J. Porque, C. Hoyle, M. Vos, and T. Smith, "Simple model of errors in chirped fiber gratings," Opt. Express **12**, 189-197 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-1-189

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### References

- R. Kashyap, M. deLacerda-Rocha, �??On the group delay of chirped fibre Bragg gratings,�?? Opt. Commun. 153, 19-22 (1998). [CrossRef]
- L. Poladian, �??Understanding profile-induced group-delay ripple in Bragg gratings,�?? Appl. Opt. 39, 1920-1923 (2000). [CrossRef]
- R. Feced, M.N. Zervas, �??Effects of Random Phase and Amplitude Errors in Optical Fiber Bragg Gratings,�?? J. Lightwave Technol. 18, 90-101 (2000). [CrossRef]
- M. Sumetsky, B.J. Eggleton, C. Martijn de Sterke, �??Theory of group delay ripple generated by chirped fiber gratings,�?? Opt. Express 10, 332-340, 2002. <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-332">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-332</a>. [CrossRef] [PubMed]
- D. Garthe, G. Milner, Y. Cai, �??System performance of broadband dispersion compensating gratings,�?? Electron. Lett. 19, 582-583 (1998). [CrossRef]
- L. Poladian, �??Graphical and WKB analysis of nonuniform Bragg gratings,�?? Phys. Rev. E 48, 4758-4767 (1993). [CrossRef]
- J.F. Brennan, D.L. LaBrake, �??Realization of >10-m-long chirped fiber Bragg gratings,�?? OSA, Bragg Gratings, Photosensitivity, and Poling, Stuart, FL, ThD2, pp. 35-37 (September 1999).
- M. Yamada, K. Sakuda, �??Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach,�?? Appl. Opt. 26, 3474 (1987). [CrossRef] [PubMed]
- T. Erdogan, �??Fiber grating spectra,�?? J. Lightwave Technol. 15, 1277-1294 (1997). [CrossRef]
- T. Erdogan, private communication. We have also verified this scaling against coupled-mode simulations.
- X. Fan, D.L. LaBrake, and J.F. Brennan, �??Chirped fiber grating characterization with phase ripples,�?? OSA Optical Fiber Communications (Optical Society of America, Washington, D.C., 2003), FC2.
- M. Eiselt, C.B. Clausen, and R.W. Tkach, �??Performance characterization of components with group delay fluctuations,�?? Symposium on Optical Fiber Measurements (NIST, Boulder, Colorado, 2002), Session III.
- C. Sheerer, C. Glingener, G. Fisher, M. Bohn, and W. Rosenkranz, �??Influence of filter group delay ripples on system performance,�?? European Conf. Opt. Commun. (Nice, France, 1999), I-410
- M. Sumetsky, P.I. Reyes, P.S. Westbrook, N.M. Litchinitser and B.J. Eggleton, �??Group-delay ripple correction in chirped fiber Bragg gratings,�?? Opt. Lett. 28, 777-779, (2003). [CrossRef] [PubMed]

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