OSA's Digital Library

Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 10, Iss. 7 — Apr. 8, 2002
  • pp: 332–340
« Show journal navigation

Theory of group delay ripple generated by chirped fiber gratings

Michael Sumetsky, Benjamin J. Eggleton, and C. Martijn de Sterke  »View Author Affiliations


Optics Express, Vol. 10, Issue 7, pp. 332-340 (2002)
http://dx.doi.org/10.1364/OE.10.000332


View Full Text Article

Acrobat PDF (166 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

Fig 1. Reflection of light from chirped Bragg fiber grating

Without the loss of generality of our approach, it is assumed that there is no spatial ripple in the grating parameters other than in the period Λ(z),

Λ(z)=Λ0+C(zz0)+ΔΛ(z),
ΔΛ(z)=q>0ΔΛqexp[iq(zz0)]+c.c.,
(1)

where, for convenience, we use the spectral representation of the grating period noise, ∆Λ(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.

It is instructive to consider first the description of the GDR based on the classical ray approximation, which, as it is shown below, is inadequate for the gratings of our concern. In this approximation, GDR appears only due to the ripple in the position of turning point zt , 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:

Δτ(λ)=2n0c0CΔΛ(zt),zt=z0+(λ2n0Λ0)2n0C,
(2)

where n 0 is the effective refractive index and c 0 is the speed of light.

By contrast, for the GDR of gratings considered in this paper the contribution of the turning point ripple is irrelevant. Rather, the GDR is predominantly generated by the interference of the waves reflected from the weak non-homogeneities of grating parameters along the whole grating length traveled by the light. In terms of noise component with frequency q (see Eq. (1)) this reflection is nothing but the secondary Bragg reflection governed by the coupled wave equations [2

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

] with (2π/q)-periodic coefficients.

A simple explanation of the properties of GDR is given in Fig. 2, which is a band diagram [7

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

,8

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

] of a CFBG with a weak long-period modulation representing a frequency component of the spatial noise ∆Λ(z) defined by Eq. (1). This modulation produces narrow and weak reflection sidebands, or photonic bandgaps, shaded in Fig. 2, spaced by interval ∆λqq (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]

]). The GDR arises due to the interference of the light reflected from the main band and its sideband. There are two different wavelength regimes:

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.

Fig. 2. Physical picture of the GDR cutoff effect

Thus, the formation of sidebands is accompanied by GDR if the ripple period is below the cutoff magnitude. It is also clear from Fig. 2 that if the main band is partly transparent then, even in region 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.

We calculate the GDR using the WKB approximation for the unperturbed solution of the coupled wave equations [7

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

], and the first order perturbation theory for the noise. The developed theory is described in the Appendix where one can find derivation of the following Eqs. (3)–(5). The secondary Bragg reflection point from the sideband, zst , corresponding to the noise component with frequency q (see Fig. 2) is defined by equation:

2keff(zst)=q,keff(z)=π2n0Λ02[Δλ2n0C(zz0)]2Λ02Δn2(z).
(3)

Here keff (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 zst is outside the interval (z 0, zt ) the component with frequency q does not generate GDR. Eq. (3) defines the cutoff spatial frequency, qc , and the cutoff GDR frequency, νc , in the form:

qc=πΔλn0Λ02,νc=πΔλ2Cn02Λ02.
(4)

Eqs. (3) and (4) determine the frequency range, q ~ πλb /(2n 0Λ02), where the developed theory is valid. Here ∆λb is the reflection bandwidth, which can be expressed through the grating length L by the equation ∆λb = 2n 0 CL . For example, for the bandwidth λb = 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:

Δτ(Δλ)=0<q<qcΔτqexp[iqΔλ2n0C]+c.c.,
Δτq=2iπΔnc0(2C)3/2exp(iq2Λ024πC)ΔΛq.
(5)
Fig 3. a – comparison of the single-harmonic GDR amplitude calculated from Eq. (4) with numerical calculations and with classical ray approximation Eq. (2) for the amplitude of spatial ripple 0.0025 nm; b – fitting the numerically calculated GDR amplitude vs. chirp dependence (squares) by C -3/2 power law (solid lines) for the spatial period ripple 5 mm and 20 mm.

Fig. 3a shows the comparison of the absolute value of GDR harmonic amplitude, 2| ∆τq |, given by this equation with numerical calculation for L=100 mm, Λ0= 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 keff are of the same order of magnitude. Interestingly, according to Eq. (5) a single harmonic of GDR depends on chirp as |∆τq |~C -3/2 but not as |∆τq |~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.

Fig. 4. Ripple in grating parameters and corresponding GDR

The predicted cutoff effect is confirmed by the numerical solution of the coupled wave equation for CFBG with the parameters indicated above. Fig. 4.a demonstrates the cutoff effect for GDR generated by spatial harmonic ripples having different periods. The sum of these harmonics creates GDR, which has gradually disappearing high frequency components (Fig. 4.b). In this example the noise components with the periods less than 0.8 mm do not affect the GDR at all. Eq. (4) defines the GDR-free bandwidth in the form ∆λq = nΛ02 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]

,2

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

] and, in particular, is seen from Fig.5. Fig. 5a demonstrates typical group delay and GDR measurements for one of the gratings written from a phase mask fabricated at OFS Labs. Fig. 5b shows the Fourier spectrums of this GDR calculated for different bandwidths ∆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

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

] 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.

Fig.5. a - Experimentally measured group delay and GDR of a typical CFBG; b - Fourier spectrum of this GDR calculated for different bandwidths ∆λ measured from the high-wavelength edge of the reflection band and demonstrating the cutoff frequencies coincident with the ones predicted by Eq. (4).

In summary, we have investigated the effect of spatial noise of the apodized CFBGs on the GDR and found that the high frequency noise is cutting off. The developed theory is based on the analogy between the noisy gratings and the superstructure Bragg gratings and on the isolation of the high-frequency components in the spectral representation of the GDR. The cutoff effect is demonstrated here for the spatial noise in the grating period. It can be similarly shown that Eq. (4) holds for the spatial noise in any grating parameter. The results obtained are important both for basic understanding and for the further improvement of the grating performance.

APPENDIX

1. Solution of coupled wave equations by perturbation theory

Propagation of light in fiber grating defined by the refractive index variation

n(z)=n0+Δncos(2πΛ0z+2πΛ02Zdz(Λ(z))Λ0)
(A.1)

is governed by the coupled wave equation [2

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

,7

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

]:

u(z)=+i[δ(z)u(z)+κ(z)v(z)]
v(z)=i[δ(z)v(z)+κ(z)u(z)]
(A.2)

where

δ(z)=βπΛ02(Λ(z)Λ0),κ(z)=πΔn(z)2n0Λ0,
(A.3)

n is the amplitude of index modulation,

β=2πn0λπΛ0πΔλ2n0Λ02
(A.4)

is the detuning parameter, and grating period variation Λ(z) is defined by Eq. (1).

The effect of the weak spatial grating period noise ∆Λ(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:

Δτ=Δτ1+Δτ2,Δτi=2πn0c0Λ02ddβRe[z0dxΔΛ(z)Gi(z)]
G1=u0+(u0+)*+u0(u0)*,G2=r01u0+(u0)*+r0u0(u0+)*
(A.5)

where r 0 is the reflection coefficient in the absence of noise and functions u0±(z), together with v0±(z) = (u0 (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

In the absence of noise, assuming that δ(z)=δ 0(z) and κ(z) are slowly varying functions, we can solve Eqs. (A.2) using the WKB approximation [7

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

]:

(u0±(z)v0±(z))=e±iz0zkeff(z)dz,2Q(z)(Q(z)±1Q(z)±1),
Q(z)=δ0(z)κ(z)δ0(z)+κ(z),keff(z)=δ02(z)κ2(z)
(A.6)

Eqs. (A.6) have a singularity and fail in close proximity of the turning point zt , defined by equation keff (zt ) = 0, where |z-zt | is less than or of order Λ0(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 keff 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

However, independently of the value of q, wherever WKB approximation for u0±(z) is valid, one can substitute Eqs. (A.6) and Eq. (1) into Eqs. (A.5), which yields for the GDR:

Δτ=Δτ1+Δτ2
Δτ1=πn0c0Λ02q>0ΔΛqddβz0ztdzδ0(z)keff(z)eiq(zz0)+c.c.
Δτ2=πn02c0Λ02q>0ΔΛqddβz0ztdzκ(z)keff(z)(r01eiq(zz0)+2iz0zkeff(z)dz+r0eiq(zz0)2iz0zkeff(z)dz)+c.c.
(A.7)

The first item of 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 qkeff . When the amplitude of grating modulation, ∆n, is small we have keff (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 keffq 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 zt 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 dp is the distance between the real axis z and the nearest complex pole of f(z) then + dze iqz f(z) ~ e -qdp [9

9. A. B. Migdal, Qualitative Methods in Quantum Theory, (W.A. Benjamin, Inc, 1977).

]. On the contrary, for high frequencies, when q~keff , 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 zst defined by equation

2keff(zst)=q.
(A.8)

This is Eq. (3) of the main text. Neglecting κ(z) (or ∆n) in the Eq. (A.6) for keff (z) we find for the turning point and the stationary point, respectively:

zt=Λ02βπC,zst=Λ02πC(βq2).
(A.9)

It is now easy to show that because the difference zt - zst cannot exceed the grating length L defined through the bandwidth ∆λb as L = ∆λb / C the stationary point can only exist if the frequency q is less than the cutoff frequency defined by Eqs. (4). Whenever zst 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. 18, 1418–1432 (2000) [CrossRef]

2.

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

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]

4.

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 Sosiety of America Williamsburg1997) Paper BMG13-2.

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]

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 Opcal Fiber Communication Conference, (Optical Sosiety of America, Washington, D.C., 2001) Paper WDD89.

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]

9.

A. B. Migdal, Qualitative Methods in Quantum Theory, (W.A. Benjamin, Inc, 1977).

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


Sort:  Journal  |  Reset  

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. 18, 1418-1432 (2000) [CrossRef]
  2. R. Kashyap, Fiber Bragg Gratings, (Academic Press, 1999).
  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]
  4. 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.
  5. R. Feced, M. N. Zervas, �Effect of random phase and amplitude errors in optical fiber gratings,� J. Lightwave Technol. 18, 90-101 (2000). [CrossRef]
  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 Optical Fiber Communication Conference, (Optical Sosiety of America, Washington, D.C., 2001) Paper WDD89.
  7. L. Poladian, �Graphical andWKB 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]
  9. A. B. Migdal, Qualitative Methods in Quantum Theory, (W.A. Benjamin, Inc, 1977).

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.

CrossCheck Deposited