## Theory of group delay ripple generated by chirped fiber gratings

Optics Express, Vol. 10, Issue 7, pp. 332-340 (2002)

http://dx.doi.org/10.1364/OE.10.000332

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### Abstract

The theory of the group delay ripple generated by apodized chirped fiber gratings is developed using the analogy between noisy gratings and superstructure Bragg gratings. It predicts the fundamental cutoff of the high frequency spatial noise of grating parameters in excellent agreement with the experimental data. We find simple general relationship between the high-frequency ripple in the grating period and the group delay ripple. In particular, we show that the amplitude of a single-frequency group delay ripple component changes with grating period chirp, *C*, as *C*^{-3/2} and is proportional to the grating index modulation, while its phase shift and period changes as *C*^{-1} .

© 2002 Optical Society of America

1. B. J. Eggleton, A. Ahuja, P. S. Westbrook, J. A. Rogers, P. Kuo, T. N. Nielsen, and B. Mikkelsen, “Integrated tunable fiber gratings for dispersion management in high-bit rate systems,” J. Lightwave Technol. **18**, 1418–1432 (2000) [CrossRef]

1. B. J. Eggleton, A. Ahuja, P. S. Westbrook, J. A. Rogers, P. Kuo, T. N. Nielsen, and B. Mikkelsen, “Integrated tunable fiber gratings for dispersion management in high-bit rate systems,” J. Lightwave Technol. **18**, 1418–1432 (2000) [CrossRef]

3. M. Ibsen, M. K. Durkin, R. Feced, M.J. Cole, M.N. Zervas, and R.I. Laming, “Dispersion compensating fibre Bragg gratings,” in *Active and Passive Optical Components for WDM Communication*, Proc. SPIE **4532**, 540–551 (2001). [CrossRef]

5. R. Feced and M. N. Zervas, “Effect of random phase and amplitude errors in optical fiber gratings,” J. Lightwave Technol. **18**, 90–101 (2000). [CrossRef]

*C*, is proportional to

*C*

^{-3/2}rather than

*C*

^{-1}as predicted by the classical ray approximation.

*z*),

*z*). The spatial ripple of the other grating parameters, like the amplitude of index modulation and of the DC component of refractive index, can be described similarly.

*z*

_{t}, and the grating period noise elsewhere does not matter (see Fig. 1). Simple calculation using Eq. (1) defines GDR in the form demonstrating its inverse proportionality to the chirp

*C*:

*n*

_{0}is the effective refractive index and

*c*

_{0}is the speed of light.

*q*(see Eq. (1)) this reflection is nothing but the secondary Bragg reflection governed by the coupled wave equations [2] with (2

*π*/

*q*)-periodic coefficients.

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

8. N. G. R. Broderick and C. M. de Sterke, “Theory of grating superstructures,” Phys. Rev. E **55**, 3634–3646 (1997). [CrossRef]

*z*) defined by Eq. (1). This modulation produces narrow and weak reflection sidebands, or photonic bandgaps, shaded in Fig. 2, spaced by interval ∆

*λ*

_{q}∝

*q*(superstructure effect [8

8. N. G. R. Broderick and C. M. de Sterke, “Theory of grating superstructures,” Phys. Rev. E **55**, 3634–3646 (1997). [CrossRef]

*a*.(

*above cutoff*): If the light is reflected from the main reflection band only, without intersecting the sidebands then no GDR is observed.

*b*.(

*below cutoff*): If the light also crosses the sideband then the interference of the wave reflected from the main band and the sideband causes GDR.

*a*, the grating will exhibit GDR because of interference with the wave reflected from the upper sideband. Below we assume that the grating is strongly reflective and the GDR produced by this interference is negligible. Also, it is assumed that the grating is apodized so that the reflections near edge

*z*

_{0}are small. As it was mentioned, the corresponding noise in amplitude of grating modulation can be considered similarly.

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

*z*

_{st}, corresponding to the noise component with frequency

*q*(see Fig. 2) is defined by equation:

*k*

_{eff}(

*z*) is the effective (envelope) wavenumber, ∆

*λ*is the detuning of wavelength of incident light, which is calculated from the high-wavelength edge of the reflection band, and ∆

*n*(

*z*) is the index modulation. It is seen from Fig. 2 that whenever

*z*

_{st}is outside the interval (

*z*

_{0},

*z*

_{t}) the component with frequency

*q*does not generate GDR. Eq. (3) defines the cutoff spatial frequency,

*q*

_{c}, and the cutoff GDR frequency,

*ν*

_{c}, in the form:

*q*~

*π*∆

*λ*

_{b}/(2

*n*

_{0}

*λ*

_{b}is the reflection bandwidth, which can be expressed through the grating length

*L*by the equation ∆

*λ*

_{b}= 2

*n*

_{0}

*CL*. For example, for the bandwidth

*λ*

_{b}= 1 nm our theory is valid for the spatial noise frequencies of the order 4 mm

^{-1}and, respectively, for the periods of the order 1 mm. This high-frequency interval is, in fact, the interval of our interest where the following expression for the GDR is found:

*τ*

_{q}|, given by this equation with numerical calculation for

*L*=100 mm, Λ

_{0}= 540 nm,

*C*=0.05 nm/cm, and ∆

*n*

_{0}= 2·10

^{-4}. The agreement is very good in the interval of the spatial ripple period between the cut-off period equal to 0.8 mm and up to 1cm. This figure also shows that the GDR value calculated from Eq. (2) is wrong. This approximation is, obviously, inadequate when the spatial frequency

*q*and effective wavenumber

*k*

_{eff}are of the same order of magnitude. Interestingly, according to Eq. (5) a single harmonic of GDR depends on chirp as |∆

*τ*

_{q}|~

*C*

^{-3/2}but not as |∆

*τ*

_{q}|~

*C*

^{-1}, which follows from Eq. (2). Fig. 3b demonstrates that the

*C*

^{-3/2}dependence is very accurate for the periods of spatial ripple up to 2 cm. The linear dependence on the index modulation, ∆

*n*, follows from Eq. (5) and can be understood from the linear dependence of the GDR on the amplitude of reflection coefficient from the sideband, which is proportional to ∆

*n*.

*λ*

_{q}=

*n*

*q*/

*π*in Fig. 4 explains the gradual suppression of the high frequency GDR, which is usually observed experimentally [1

1. B. J. Eggleton, A. Ahuja, P. S. Westbrook, J. A. Rogers, P. Kuo, T. N. Nielsen, and B. Mikkelsen, “Integrated tunable fiber gratings for dispersion management in high-bit rate systems,” J. Lightwave Technol. **18**, 1418–1432 (2000) [CrossRef]

*λ*measured from the high-wavelength edge of the reflection band. It is seen that the experimental cutoff frequencies are in excellent agreement with the ones predicted by Eq. (4). Note, that the adiabatically small GDR generated by imperfect apodization [2] has the same cut-off frequencies. However, its numerically estimated value has the 0.1 picosecond order and is negligible compared to the effect of spatial noise considered.

## APPENDIX

## 1. Solution of coupled wave equations by perturbation theory

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

*n*is the amplitude of index modulation,

*z*) is defined by Eq. (1).

*z*) (see Eq. (1)) can be calculated by solving Eq. (A.2) by perturbation theory. To the first order in power of ∆Λ(

*z*), we find GDR in the form:

*r*

_{0}is the reflection coefficient in the absence of noise and functions

*z*), together with

*z*) = (

*z*))

^{*}, are the solutions of the coupled wave equation in the absence of noise which satisfy the boundary conditions

*u*

^{+}(

*z*) = exp(

*iβz*), and

*u*

^{-}(

*z*) = 0 for

*z*<

*z*

_{0}.

## 2. WKB solution in the absence of noise

*δ*(

*z*)=

*δ*

_{0}(

*z*) and

*κ*(

*z*) are slowly varying functions, we can solve Eqs. (A.2) using the WKB approximation [7

**48**, 4758–4767 (1993). [CrossRef]

*z*

_{t}, defined by equation

*k*

_{eff}(

*z*

_{t}) = 0, where |

*z*-

*z*

_{t}| is less than or of order Λ

_{0}(

*n*

_{0}/ ∆

*nC*)

^{1/3}. In this neighborhood, which is of order 5 mm for the grating parameters considered in the main text, Eqs. (A.6) should be substituted by the accurate solution of the couple wave equations. The noise term in Eq.(1) can contain fast varying components with frequencies

*q*which are comparable or greater than the value of the wave-number

*k*

_{eff}in Eqs. (A.5). For these components the WKB approach fails and ∆Λ(

*z*) cannot be directly incorporated into Eqs. (A.6) by substitution

*δ*(

*z*) for

*δ*

_{0}(

*z*).

## 3. Group delay ripple

*q*, wherever WKB approximation for

*z*) is valid, one can substitute Eqs. (A.6) and Eq. (1) into Eqs. (A.5), which yields for the GDR:

*τ*

_{1}, does not describe reflection from grating period ripple and can be obtained in WKB approximation, when the noise ∆Λ(

*z*) is directly inserted into Eqs. (A.6). This term dominates for the components of grating period noise with low frequencies

*q*≪

*k*

_{eff}. When the amplitude of grating modulation, ∆

*n*, is small we have

*k*

_{eff}(

*z*) ≈

*δ*

_{0}(

*z*) and ∆

*τ*

_{1}coincides with the classical ray approximation of Eq. (2). For low frequency components, the second item, ∆

*τ*

_{2}, is small, because it contains the fast oscillating exponents with frequency

*k*

^{eff}≫

*q*under the integral. For higher frequencies, WKB approximation for ∆

*τ*

_{1}fails and the total contribution into integral of Eq. (A.4) of the turning point

*z*

_{t}becomes exponentially small for big

*q*. Exponential dependence on

*q*follows from the general property of Fourier integral of an analytical function

*f*(

*z*) vanishing for

*z*→±∞. For example, if

*d*

_{p}is the distance between the real axis

*z*and the nearest complex pole of

*f*(

*z*) then

*dze*

^{iqz}

*f*(

*z*) ~

*e*

^{-qdp}[9]. On the contrary, for high frequencies, when

*q*~

*k*

_{eff}, the term ∆

*τ*

_{2}, which describes the Bragg reflection from “long-period” noise frequency components, becomes dominant. The main integral in the expression for ∆

*τ*

_{2}in Eqs. (A.7) can be calculated by the stationary phase method. While the first term in this integral is strongly oscillating and has negligible contribution, the second one may have a stationary point

*z*

_{st}defined by equation

*κ*(

*z*) (or ∆

*n*) in the Eq. (A.6) for

*k*

_{eff}(

*z*) we find for the turning point and the stationary point, respectively:

*z*

_{t}-

*z*

_{st}cannot exceed the grating length

*L*defined through the bandwidth ∆

*λ*

_{b}as

*L*= ∆

*λ*

_{b}/

*C*the stationary point can only exist if the frequency

*q*is less than the cutoff frequency defined by Eqs. (4). Whenever

*z*

_{st}becomes less than

*z*

_{0}the second integral in Eq. (A.7) vanishes and the resulting GDR experiences the cutoff effect. Otherwise, calculation of ∆

*τ*

_{2}by the stationary point method yields the result of Eq. (5).

## References

1. | B. J. Eggleton, A. Ahuja, P. S. Westbrook, J. A. Rogers, P. Kuo, T. N. Nielsen, and B. Mikkelsen, “Integrated tunable fiber gratings for dispersion management in high-bit rate systems,” J. Lightwave Technol. |

2. | R. Kashyap, |

3. | M. Ibsen, M. K. Durkin, R. Feced, M.J. Cole, M.N. Zervas, and R.I. Laming, “Dispersion compensating fibre Bragg gratings,” in |

4. | F. Ouellette, “The effect of profile noise on the spectral response of fiber gratings,” in |

5. | R. Feced and M. N. Zervas, “Effect of random phase and amplitude errors in optical fiber gratings,” J. Lightwave Technol. |

6. | R. Feced, J. A. J. Fells, S. E. Kanellopoulos, P. J. Bennett, and H. F. M. Priddle, “Impact of random phase errors on the performance of fiber grating dispersion compensators,” in |

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

8. | N. G. R. Broderick and C. M. de Sterke, “Theory of grating superstructures,” Phys. Rev. E |

9. | A. B. Migdal, |

**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: March 4, 2002

Revised Manuscript: March 27, 2002

Published: April 8, 2002

**Citation**

Michael Sumetsky, Benjamin Eggleton, and C. de Sterke, "Theory of group delay ripple generated by chirped fiber gratings," Opt. Express **10**, 332-340 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-7-332

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### References

- B. J. Eggleton, A. Ahuja, P. S. Westbrook, J. A. Rogers, P. Kuo, T. N. Nielsen, and B. Mikkelsen, �Integrated tunable fiber gratings for dispersion management in high-bit rate systems,� J. Lightwave Technol. 18, 1418-1432 (2000) [CrossRef]
- R. Kashyap, Fiber Bragg Gratings, (Academic Press, 1999).
- M. Ibsen, M. K. Durkin, R. Feced, M. J. Cole, M. N. Zervas, and R.I. Laming, �Dispersion compensating fibre Bragg gratings,� in Active and Passive Optical Components for WDM Communication, Proc. SPIE 4532, 540-551 (2001). [CrossRef]
- F. Ouellette, �The effect of profile noise on the spectral response of fiber gratings,� in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, (Optical Society of America Williamsburg 1997) Paper BMG13-2.
- R. Feced, M. N. Zervas, �Effect of random phase and amplitude errors in optical fiber gratings,� J. Lightwave Technol. 18, 90-101 (2000). [CrossRef]
- R. Feced, J. A. J. Fells, S. E. Kanellopoulos, P. J. Bennett, and H. F. M. Priddle, �Impact of random phase errors on the performance of fiber grating dispersion compensators,� in Optical Fiber Communication Conference, (Optical Sosiety of America, Washington, D.C., 2001) Paper WDD89.
- L. Poladian, �Graphical andWKB analysis of nonuniform Bragg gratings,� Phys. Rev. E 48, 4758-4767 (1993). [CrossRef]
- N. G. R. Broderick and C. M. de Sterke, �Theory of grating superstructures,� Phys. Rev. E 55, 3634-3646 (1997). [CrossRef]
- A. B. Migdal, Qualitative Methods in Quantum Theory, (W.A. Benjamin, Inc, 1977).

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