1. Introduction

Chirped fiber Bragg gratings have been the subject of considerable investigation for use as dispersion compensators in optical networks. Much of the focus has been on non-ideal behavior, specifically group delay ripple (GDR) and reflection ripple. Numerous calculations have been done to design optimized index profiles, which reduce the ringing associated with finite bandwidth devices to acceptable levels [1,2]. When the optimal index profile is known, the critical issue then becomes how to generate this profile in a real grating manufacturing process. To date, understanding of the relationship between fabrication tolerances and grating performance has been inadequate. One goal of this paper is to make a connection between fabrication parameters and grating quality using a simple optical model.

Due to the extreme number of multi-path reflections possible in fiber Bragg gratings, solutions for the optical phase and amplitude response in non-ideal gratings have generally been accomplished numerically [3,4 and references therein]. Few analytical models have emerged, leaving little room for intuitive understanding. It has been suggested that perturbations to the Bragg period form spectral sidebands, interfering with the ideal optical properties of the grating [4,5]. For a chirped grating, the sidebands play the role of spatial etalons, creating the phase and intensity ripples often seen. Recently the sideband model has been successfully used to qualitatively explain a host of commonly observed grating features, and has also achieved great quantitative success. Still, the complexity inherent in these analyses provides little intuition into the behavior and origin of analytic dependencies.

Here we present an extremely simple model that is both very satisfying intuitively, and also quantitatively accurate. We represent the grating response as the sum of two parts. The first part is the idealized response that results from the spatially-slowly-varying chirp and apodization of the index modulation that are intentionally designed into the grating. This part of the response includes the grating bandwidth and average dispersion, and we will refer to properties resulting from this first part as “bulk” properties of the grating. This part of the spectrum is commonly calculated by numerical solution of the coupled mode equations. The second part of the response is the result of smaller, more rapidly varying perturbations on the slowly varying grating profile. By assuming the perturbations, and therefore the resultant sidebands, are small, we can effectively decouple the bulk properties of a grating such as chirp and apodization from the perturbations. Then we can use the sideband picture to calculate the effects of these perturbations on chirped gratings. Only a simple algebraic model is required to deduce the fine structure on the phase and amplitude that ultimately determine the quality of the grating. Furthermore, for practical applications, the model leads to simple scaling laws that can be substituted for more intuitive insight.

The model presented below treats the perturbed grating as a simple optical etalon with discrete reflections [2]. A fiber Bragg grating is not truly a discrete reflector however, as each index fringe contributes very little to the overall reflection. If the etalons are shorter than an effective interaction length [6], the reflection is distributed over a length larger than the etalon spacing, and our model of discrete reflections does not apply. We therefore consider only relatively rapid variations in grating properties, often termed “dense GDR”. Under the assumption of small perturbations, we find very good agreement between this model and coupled-mode simulations. In addition, the analytic nature of the derivation elucidates many of the dependencies, and even reveals some new effects.

2. Sideband model

To begin, we consider the index profile of a Bragg grating written in a fiber with nominal refractive index 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

where 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 Λ₀ reflects light of wavelength λ ₀=2n ₀Λ₀, and a linearly chirped grating has a position dependent period characterized by the chirp parameter C ₀, such that Λ(z)=Λ₀+C ₀ z.

In order to separate any intentional profile from error induced modulations, the index has been written with terms clearly separated; Δn(z) is the index modulation envelope, and p(z)≈p ₀(1-C ₀ zΛ₀) 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≪Λ₀), which is well justified in current telecommunication systems centered near 1550 nm with bandwidth 30 nm.

The creation of sidebands is illustrated by rewriting Eq. (1) using the spatial Fourier components of each perturbation;

in which 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,

Here N ^±≡W ^∓ iX+iY±Z. The important result is that spectrally distinct periodicities are present with periods 2π/p, 2π/(p+g) and 2π/(p-g). These will form the main band and sidebands respectively, where the spectral separation of the side bands from the main band is Δλ≈±λ ₀ g/p when g≪p. All three bands are identical in their bulk phase and amplitude response functions p(z) and Δn(z) respectively, but differ in the scaling and wavelength offsets.

Generally sidebands caused by fabrication errors consist of many periodicities, and are not obvious as replicas of the main band in the spectral domain. To emphasize the sideband picture, we wrote gratings using the contolled flow method [7], but with an intentional perturbation at a periodicity chosen to illustrate the sideband structure and the effect on the main band’s response (GDR) (Fig 1). The grating was ~7 cm long, with chirp C ₀=0.079 nm/cm and perturbation period 2π/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.

 

Fig. 1. (a) Reflection spectrum of a grating written with intentional periodic errors to generate sidebands. (b) The GDR and phase ripple of the same grating, measured from the long wavelength side.

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To find the detailed optical response within the main band using the index profile in Eq. (3), we determine the combined reflectivity of these three bands as a function of wave number 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.

 

Fig. 2. (left) Diagram of reflections to first order in sideband strength. Light enters from the right (E ₀) and is reflected from the grating at various points. The highlighted regions of the fiber grating A,B,C represent the near sideband, main band, and far sideband respectively, for a particular wavelength. (right) Time domain representation of a single pulse after reflection from the band structure, showing early and late echoes.

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To make Eq. (3) more illustrative, we specialize to a single perturbation period as in Figs. 1(a) and (b). The reflected field can be modeled from four discrete reflections in Fig. 2;

in which the phase expi(ωt-kx) has been dropped, φ is the ideal phase response and τ=2n 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=Δλ/2n ₀ C ₀≈λ ₀ g/2n 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 Δn(k) from Eq. (3), where a change of variable has been used from Eq. (1); Δn(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/4n ₀, causing a π phase shift.

For many measurement methods E ₀ is coherent and E ₀(k, t±τ)=E ₀exp(±iϕ) with ϕ=2kn 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;

The group delay may be calculated as c ^-1(∂α/∂k), and the GDR Δτ, is commonly defined as the departure from the ideal group delay c ^-1(∂φ/∂k) ;

In analogy with this, we define the phase ripple as the perturbative terms in Eq. (6) not associated with the ideal phase φ.

Although we have specialized to a single perturbation period, the assumption of small sidebands means additional contributions simply add to both the phase, power and group delay in the same manner as the terms containing 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], there is a cutoff period associated with phase ripple, stemming from the spectral overlap of 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.

To show the quantitative accuracy of the approximations used here, we compared the results with numerical simulations of the coupled-mode model, based upon the transfer matrix method [8,9]. Fig. 3 shows the phase ripple from the model here (the coefficient of the 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.

 

Fig. 3. Peak to peak phase ripple versus relative sideband size for two different grating strengths. Lines show sideband model and points show CM simulation. Right axis is peak-to-peak GDR for the given phase ripple based on a grating with C ₀=0.079 nm/cm and sideband spacing Δλ=0.72 nm.

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Very good agreement is seen when the main band is strong (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.

In order to gain some insight over numerical models, simple scaling laws can be used to infer the strength of 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∝Δn is the coupling strength [9] and L_eff , proportional to $1⁄\sqrt{C_0}$, is the effective interaction length of light in a chirped grating [6,10]. The goal is to find 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;

where Δn_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.

To validate this scaling method and the etalon model, we fabricated gratings of different strengths with an intentional perturbation as in Fig 1. The group delay was measured using the standard modulation phase-shift method at 125 MHz. We attempted to make the measurement of the GDR magnitude model independent by measuring only the peak-to-peak (p-p) value and not specializing to the single period we expected given our perturbation. We applied a band pass filter to the GDR data from 0.25 to 2 times the period due to our perturbation. This allowed us to include the possibility of higher order terms not present in our model, at the cost of some additional noise due to the residual GDR from uncontrolled noise sources and non-ideal apodization. Then, the p-p GDR was determined as the average peak value minus the average valley value over 0.5 nm of the low wavelength side of the grating spectrum.

 

Fig. 4. Peak to peak GDR versus grating strength. Line is from Eq. (7). Solid points are data from gratings as shown in Fig.1. Open points are the same gratings after annealing.

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3. Example

As an example of the utility of this model, consider variations in the average index of the fiber. This can be due to changes in fiber chemistry or dimension, or specific changes in the UV exposure. With only this perturbation present, the index has the form

We ignore variations of 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=2π2n/λ ;

Integrating p_eff to obtain the phase, in which slow components escape integration, gives the effective index

where 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)=δnsin(gx), noting that other components can be simply added in the final result. Integrating η(x) and rewriting Eq. (11) gives

Using Eq. (8), the sidebands become

As an example a single tone modulation is taken with a spatial period of 2π/g=1mm, slow compared to the grating period of 2π/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

The analytic nature of this analysis enables a very intuitive look at grating behavior. Using the scaling above, whereby A is proportional to $1⁄\sqrt{C_0}$ and L∝1/C ₀, we see from Eq. (7) that the GDR Δτ∝$C_0^{-3/2}$, in agreement with [4]. However, it is noteworthy that this scaling is not meaningful when considering the performance of the grating used to compensate dispersion. Recently it has been established that the phase ripple amplitude is a more direct measure of grating performance than the GDR [11,12], in light of its direct relationship to the magnitude of the reflected echoes (Eq. (6)). From Eq. (3), phase ripple amplitude shows no dependence on the chirp other than through the sideband strength A (when B~1 for simplicity). Therefore, upon comparing gratings of different chirp but similar strength, grating performance is independent of chirp.

When the main reflection is strong (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.

Due to the number of possible sources of perturbation when writing a grating, many combinations of amplitude and phase modulation are possible. For example, if 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.

For high bandwidth communication in which signals are not coherent, the full time domain pulse description must be used in Eq. (4). A complete treatment of this is data format dependent and is beyond the scope of this paper, but some qualitative observations are clear. First, if the separation between echoes (Fig. 2) is greater than the pulse width, then there is no interference possible except between the two delayed echoes. This is the regime that causes dense GDR in coherent measurements. It is obvious that the reflection from sideband C, occurring at time T+2nL/c cannot interfere with the early reflection from sideband A at time T-2nL/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.

Others have shown that there is a distinct difference in how fast and slow phase or group delay ripples must be handled in calculating grating performance [13]. The slow regime, in which the ripple period is larger than the signal bandwidth, is often thought of as chromatic dispersion, and the penalty is calculated by treating the residual dispersion value. The fast regime is more commonly thought of in terms of dense GDR and has been the subject of much more work.

The designation of fast and slow is related to the difference between inter-symbol and intra-symbol interference, respectively. Penalty in a communication system is determined differently in each regime. The transition between regimes occurs when the echo time becomes larger than the data pulse width τ. Therefore, the fast regime occurs when the sidebands are spaced greater than cτ/2n. Since the sideband spacing is L_sb =λ ₀ g 2np ₀ C ₀ we find there is a subtle mechanism to link chirp to grating performance, contrary to our earlier assertion. Explicitly, for a given pulse length we can define the transition between regimes such that L_sb =cτ/2n. The perturbation period at this transition is

As the chirp becomes slower (and dispersion increases), the transition length scale 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

Using a simple etalon picture we have derived an accurate analytic model for noise on chirped fiber gratings. The assumption of small perturbations is required to separate large-scale effects like chirp and apodization, from much faster noise related contributions. We have shown that this approximation is accurate, even for grating quality much worse than current capabilities. Slow errors not considered here are also important, and become more detrimental as the push for faster data rates increases signal bandwidth [14].

Simple scaling laws emerged from this model that can be used for all types of grating writing systems. For very long gratings, ultra-violet exposure induced loss can add complexity to this model since the effective strength of a sideband reflection will change as its distance from the main band increases. Cladding modes provide a wavelength dependent loss that will change the scaling of sideband to main band size in a more complex way than presented here. Despite this added richness, our analytic solution provides a simple understanding for the origin of common grating effects and even shows the limitation of current analysis methods.