1. Introduction

Optical communications system designers have wanted to use chirped fiber Bragg gratings (FBGs) as dispersion compensators since they were first proposed over a decade ago. In principle, chirped gratings have ideal characteristics compared with other solutions, such as low insertion loss, small package size, and significantly reduced nonlinear optical interactions. Fiber birefringence causes polarization-dependent variations in the system performance of chirped gratings. A portion of these variations may be due to polarization mode dispersion (PMD), but fluctuations also occur because different regions of the grating phase ripple, with corresponding changes in the system performance, are being sampled in a birefringent grating as the polarization space is explored.

Many effects may cause birefringence in fiber gratings, but generally the largest effect is due to the side-write fabrication process [1–4]. As shown herein, PMD measurements are complicated by group delay ripple (GDR), which may cause the measured differential group delay (DGD) between polarizations at a given wavelength to imply nonsensically large PMD values if common definitions are used [1, 5–6], i.e., the measured DGD implies complete eye-closure, yet little measurable difference in system performance is seen as the polarization space is explored in system testbed measurements [7].

While it is common, and sometimes desirable, to measure both DGD and GDR with high resolution [8], we demonstrate that this information is redundant in the case of chirped FBGs. Although multimeter length gratings are used as examples in the work, the methods and results are equally applicable to much shorter length gratings. Typically, long length gratings have larger GDR amplitudes than that of shorter gratings, but as discussed in Section 3, broadband measurements extract the useful DGD information regardless of the GDR magnitudes.

2. Theory

DGD measurements of chirped fiber gratings typically yield very fine structure in the presence of GDR, making it difficult to assign a ‘PMD’ value to a grating with standard characterization methods. We clarify the relationship between DGD and GDR and derive two different, but equivalent, representations of these grating imperfections. One representation requires one to measure only the birefringence of the fiber and the phase ripple, and the other involves measurement of the full PMD vector. For simplicity, we don’t include polarization dependent loss in this analysis, but this methodology can be extended to include it. We discuss these two methods of representing grating imperfections and illustrate how to characterize a fiber grating with them.

The optical phase shift induced by a grating along a birefringent axis ‘x’ is

where the mode at optical frequency ω travels at a speed c/n_x within the chirped grating for a total length L_x that varies with frequency. The phase shift along the other axis ‘y’ can be expressed in a similar manner, but here the effective index is different by Δn from that along the other axis. Since the grating pitch Λ(ω) is the same for both axes, the index change and phase matching conditions force the mode to be reflected at a different location along the grating length, which can be calculated ed as L_x (ω+Δω), where Δω is the apparent frequency shift of one grating axis relative to the other. The wavelength shift Δλ between the two birefringent axes of the grating is related to the birefringence Δn as Δλ=2·Δn·Λ, where Λ is the grating pitch. Phase matching conditions yield

which may be approximated as

This formalism can be used to express the phase shift along the y-axis in terms of the x-axis parameters as

where the propagation length is expanded about the frequency of interest and the difference in phase shift Δϕ between the two axes is

The first term $\Delta n·\frac{\omega }{c}·L_x\left(\omega \right)$ is due to the optical path length difference within the grating between the two axes arising from the birefringence, and the second term (with the bracketed terms defining ΔL) arises in chirped gratings since the mode is reflected from a different location. The bracketed terms account for the dispersion and dispersion slope of the chirped grating, which are slightly different between the axes due to the birefringence. By comparing Eq. (1) and Eq. (4), we obtain

when n_x ≫Δn, indicating that the optical phase along one axis is approximately that along the other shifted by Δω. The birefringence Δn is typically less than ~10^-5 that of the effective index of the propagating mode [2]. The group delay of a propagating wave, t_g , after traversing a length of fiber L at velocity v_g is defined as

The difference between the group delays of the two axes is thus obtained from Eq. (1) and Eq. (4) as

where the material and waveguide dispersions of the fiber are assumed negligible compared to the other terms.

As an example, consider a grating fabricated for dispersion correction in a telecommunications system having a 1360-ps/nm-dispersion with a bandwidth covering wavelengths between 1530 nm and 1560 nm. Assume this ~4-meter-long grating has a birefringence of 10^-6 and a pitch varying linearly with frequency to yield L_x (ω)=K·(ω ₀-ω)/2π, where ω ₀/2π=196.078 0 THz (1530 nm) and K=2.194 mm/GHz. Note that L is twice the length from the grating edge to the reflection point, since that length is double-passed in reflection. The birefringence will induce a frequency shift of Δω/2π=~129 MHz (Δλ=~1 pm) at 193.548 THz (1550 nm). The differential group delay at 1550 nm calculated with Eq. (8) is

$\mathrm{Example}\Delta t\approx \frac{\Delta n}{c}·\left(5.55m\right)+\frac{n_x}{c}·283um$

$\approx 0.019\mathrm{ps}+1.42\mathrm{ps}.$.

As demonstrated, the first term of Eq. (8) is usually much smaller than the grating dispersion contributions and is often neglected to approximate the differential group delay as Δt≈K·Δω·n/2πc, which may be re-written as Δt≈D·Δλ with D≡-K·n_x /λ ² as the grating dispersion.

Polarization mode dispersion is typically expressed in vector form as Ω⃗=Δτ·q̂, which points in the direction of the local fast principle state of polarization of the fiber system. The 2^nd-order PMD is expressed by differentiating this vector with respect to frequency to yield $\frac{\partial \overrightarrow{\Omega }}{\partial \omega }=\frac{\partial \Delta \tau }{\partial \omega }·\hat{q}+\Delta \tau ·\frac{\partial \hat{q}}{\partial \omega }$ [e.g., 9]. The second component is a depolarization term, related to ∂q̂/∂ω, and the first component is a polarization-dependent chromatic dispersion (PCD) term, which can be obtained for a chirped grating by differentiating Eq. (8) as

The first PCD term arises since the group delay dispersions along the grating axes are slightly different due to Δn, and the second PCD term occurs in nonlinearly chirped gratings when ΔL varies with optical frequency. Since the birefringence axis of a fiber grating is determined during the inscription process by the incident writing laser beam, variations in the vector direction over the bandwidths of interest are typically negligible, i.e., ∂q̂/∂ω≈0 as shown in later sections, making $\frac{\partial \overrightarrow{\Omega }}{\partial \omega }=\frac{\partial \Delta \tau }{\partial \omega }·\hat{q}·$. The PCD for the example grating given above is estimated by Eq. (9) to be <1 fs/nm.

Phase ripple is generally taken as the rapidly varying, residual phase information remaining after subtraction of the slowly varying dispersion-related terms within the spectral bandwidth of interest [10, 11]. Optical phase ripple ϕ^R incorporates readily into this formalism by adding a perturbation function to the length function L(ω) and with similar arguments obtaining ${\varphi }_y^R$ (ω)≈${\varphi }_x^R$ (ω+Δω).

2.1 Representation A

The Jones matrix for a chirped grating in reflection is then expressed as

where the grating reflectivity is set to unity. The first matrix represents what is considered the 1^st-order PMD and provides a relative time delay between the axes, while the second is the phase ripple measured along each axis. Note that the phase ripple is needed along only one axis to characterize a grating, since the ripple along an axis is that along the other axis shifted by Δω. High-resolution measurement of the ripple is required, but only the magnitude of the fiber birefringence need be obtained for DGD determination.

2.2 Representation B

Another method of characterizing chirped gratings can be found by rewriting Eq. (10) as

where the differential phase shift due to ripple between the birefringent axes at a given wavelength is Δϕ^R (ω)=${\varphi }_y^R$ (ω)-${\varphi }_x^R$ (ω). The first term of Eq. (11), which applies to both polarizations, is the full phase shift measured along an axis, and the second term (the matrix) represents a ‘low-bandwidth’ DGD measurement, where the vector PMD is measured. We denote this method as a low-bandwidth technique, since the spectral bandwidth of each point in the measurement is much less than that of the system signal for which the grating device is intended, while a ‘high-bandwidth’ technique has a bandwidth comparable to or greater than that of the signal.

For a linearly chirped grating, the differential group delay can be obtained from Eq. (7) and the matrix in Eq. (11) as

where GDR_x (ω) is the group delay ripple along the x-axis. Typically the contribution of group delay ripple to DGD is much larger than that of dispersion at a given wavelength.

As illustrated, low-bandwidth measurements of both DGD and GDR give redundant information and are not both needed to completely characterize a chirped grating. For Eq. (11) to be used correctly, the phase ripple and DGD measurements must be very accurate in wavelength and magnitude. In general, it is preferable to use the method of Eq. (10) since the birefringence and phase ripple measurements may be taken with standard techniques and are less susceptible to measurement errors.

These two representations of a grating suggest different interpretations of how ripple and birefringence will impact a device performance in a communications system, yet they are mathematically equivalent. The birefringence delays the signal and its ripple-produced distortions between the axes, which is evident in Eq. (10) upon inspection. But an incomplete interpretation of Eq. (12) may suggest that a rapidly-varying high-order PMD is further corrupting the signal, in addition to the ripple distortions. As Eq. (11) illustrates, one must track the direction and magnitude of the PMD vector with measurements that are well calibrated with the ripple measurements to correctly characterize the device and interpret its impact on a communications system.

3. Chirped grating characterization

As discussed, the PMD and GDR of a grating can be characterized completely by measuring the grating phase ripple and fiber birefringence. Careful selection of measurement system parameters are needed to avoid double counting the group delay ripple while determining these values. For instance as detailed in Eq. (12), low-bandwidth DGD measurements across a chirped grating bandwidth may vary rapidly as a function of wavelength and confound birefringence measurements.

The differential group delay of a 1360-ps/nm-dispersion chirped FBG is shown in Fig 1(a) that was measured in reflection by launching 4 distinct polarizations into it and determining their respective delays with the modulation-phase shift (MPS) method at a modulation frequency of 50 MHz [12]. For reference, this data was also averaged with a 20 GHz moving window, which yielded the ~40 ps line that is also displayed.

The directions in Stokes space of Ω are shown in Figs. 1(b) and 2, illustrating that the vector changes direction quickly by π within the signal spectrum bandwidth. The PMD vector or Ω⃗=Δτ·q̂ is chosen traditionally to point in the direction of the local fast principle state of polarization of the fiber system. An alternative method of handling the rapidly changing direction of the vector is to allow a negative Δτ from an average delay and align q̂ along a birefringence axis of the fiber grating. To illustrate this point, we measured directly the group delay along each birefringent axis of the grating and calculated their difference, which yielded both positive and negative DGD values shown in Fig. 3. The GDR along an axis is that along the other axis shifted in frequency, as predicted in Section II. Note that Fig. 3 illustrates the characterization method detailed in Eq. (12), and also note that the spectral shift between the phase ripple measurements can be used to calculate the birefringence, as will be shown in Fig. 4.

 

Fig. 1. (a) Low-bandwidth measurement of DGD across the bandwidth of a chirped fiber grating in reflection. The thick line corresponds to a 20 GHz moving-window average. (b) The corresponding position of the PMD vector in Poincaré space. The solid diamond is the angle θ on the sphere horizontal plane; the open square, the azimuth angle φ. (c) Q-penalty due to the grating in a 10 Gb/s communications system. Error bars indicate the penalty variation with launch polarization at each wavelength.

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Fig. 2. Orientation of the PMD vector across the bandwidth of the grating in Poincaré space.

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Fig. 3. GDR along each birefringent axes of a grating (upper curves), and the lower curve is the difference between them, i.e. DGD with sign.

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Contained within the low-bandwidth measurements in Fig. 1 are both the phase ripple and birefringence information of the grating, but one has difficulty assigning values for PMD and phase ripple by examining these results. To disentangle this information, high-bandwidth DGD measurement techniques, such as the Jones matrix eigen-analysis or the 4-point MPS method with high-modulation frequencies, can be used to obtain 1^st-order PMD values. Increasing the modulation frequency in a MPS measurement has the effect of decreasing the GDR and averaging over bandwidth [13], which reduces the effect of GDR on the DGD measurement, i.e., |GDR_x -GDR_y |<|D·Δλ in Eq. (12). For a 10 Gb/s communications system, modulation frequencies >5 GHz are appropriate to average over the spectral extent of the signal. The direction and magnitude of the PMD vector of the grating in Fig. 1 were measured with the 4-point MPS method at 10 GHz and are shown in Fig. 4. As expected, the vector direction is relatively constant over the signal bandwidth, and the DGD fluctuates much less than in Fig. 1. This measured differential group delay agrees with values calculated with Eq. (12) as ~D·Δλ=1.36 ps with Δλ=1 pm (Δn~10^-6), denoted as the diamond line in Fig. 4(b).

 

Fig. 4. The angles θ and φ (a) and magnitude (b) of the 1^st-order PMD vector in Poincaré space measured with high-frequency MPS methods. The thick line in (b) is the 1^st-order PMD calculated with the fiber birefringence and grating dispersion.

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Accurate birefringence measurements in chirped gratings may also be obtained with Jones matrix eigen-analysis methods, provided the step-size between measurements is comparable to the spectral bandwidth of the given signal. Jones matrix methods provide information similar to high-bandwidth MPS methods as illustrated in Fig. 5 for another grating region [8], where adjacent wavelength measurements for the Jones matrix calculations were spaced at 160 pm to match the 20 GHz sampling spacing set of the MPS method at 10 GHz. The slight wavelength shift for the calculated DGD between the MPS and Jones matrix measurements is due likely to environmental temperature fluctuations between measurements.

 

Fig. 5. Comparison of PMD measurements made with MPS (black solid line) and Jones eigen-analysis (grey dashed line) methods when care is taken to use equivalent bandwidths.

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High-bandwidth results may also be obtained with low-bandwidth measurements (such as in Fig. 1) by recording the PMD vector magnitude and direction across the device bandwidth and then calculating the average vector over a spectral width Δω comparable to that of the input signal as

where Δτ_{x, y, z} are the three projections of Ω onto Cartesian coordinates and 〈〉is the average over bandwidth Δω. The data in Figure 1 was processed with Eq. (13) over a 20 GHz bandwidth and the results are shown in Fig. 6. Here the diamond-demarcated line is the PMD measured with a high-bandwidth MPS method (also shown in Fig. 3(b)), and the gray line is that calculated with Eq. (13) based on the data from Fig. 1. Our implementation of this measurement method was slow and thus made this method difficult to implement. Low-frequency measurements of the PMD vector direction were corrupted easily by many factors, such as environmental changes, measurement noise, and varying birefringence in the measurement system, as illustrated in Fig. 2 by the data spread in Stokes space.

 

Fig. 6. Grating DGD measured with the MPS method at a 10 GHz modulation frequency (diamond demarcated line) compared with that calculated by taking the vector average of the data in Fig. 1(a)

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4. Chirped grating performance in optical communications systems

Low-bandwidth DGD measurements of a chirped grating may suggest that a rapidly-varying high-order PMD is further corrupting an optical communications signal in addition to ripple distortions, but one must track the direction of the PMD vector, as well as its magnitude, to correctly characterize the device and interpret its impact on a communications system. Here we explore the impact of these imperfections on device performance as dispersion compensators in optical communications systems with system testbed simulations and measurements.

4.1 Testbed measurements

We spliced the grating characterized in Figure 1 to an optical circulator and used it in reflection for dispersion compensation in an optical communications test system. The Q-penalties were calculated with the same testbed system and methods as discussed in [14] and are shown in Fig. 1c. Penalty variations across the grating bandwidth can be attributed to imperfect dispersion correction and signal distortion due to phase ripple, but polarization-dependent effects also appear. The error bars in Fig. 1(c) show that the Q penalty varies <0.25 dB as the launch polarization space is explored, which is a similar variation as seen in our measurement repeatability experiments. But if the DGD measurements of Fig. 1(a) are interpreted as PMD, the Q-penalty estimates will be greatly exaggerated. The DGD values peaking at >150 ps imply significant eye-closure, and the average DGD of ~40 ps in Fig 1(a) predicts a penalty of ~1.6 dB. This prediction is based on an experiment where we launched 10 Gb/s signals into a PMD Emulator [JDSU, Model # PE4] with a 40 ps DGD. Clearly the measured penalty variation with launch polarization is much smaller than what would be predicted by the DGD measurements of Fig. 1(a).

4.2 Testbed simulations

We modeled the system degradations induced by ripple and rapidly varying DGD to understand the measured system penalties [15, 16], and found that optical pulse distortion due to PMD can be exaggerated if the direction of the vector in low-bandwidth measurements is neglected. In our numerical model, the Q factor value of the system at the receiver was 19 dB before grating imperfections were added to the simulation, and we used optical-noise-dominated 10 Gb/s nonreturn-to-zero system signals. To facilitate comparisons between the ripple and the signal, we express the ripple periodicity in terms of optical frequency units.

 

Fig. 7. Simulations of pulse distortion when the PMD is modeled as a vector, the input pulse (solid line) is not distorted after propagating through the grating (open circles), but the pulse is distorted significantly when the scalar average is used (filled circles).

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We used Eq. (11) to simulate a Gaussian pulse passing through a component with Ω⃗=Δτ·q̂=[τ ₀·sin(ω/f_r )+D·Δλ]·q̂, with the input polarization aligned to split the power equally between the fast and slow axes. The PMD vector q̂ was fixed to point along a birefringence axis. To simulate characteristics similar to those of the grating of Fig. 1, the ripple frequency f_r was taken as 156 MHz; the amplitude of the differential delay τ ₀, as 50 ps; and the 1^st-order PMD D·Δλ, as 1.2 ps. As shown in Fig. 7, there is little difference between the input (line) and output pulse (empty circles), but if the PMD is treated as Δτ=|τ ₀·sin(ω/f_r )+D·Δλ| where the vector direction information is lost, then significant distortion occurs (solid circles), equivalent to that caused by a constant PMD equal to the average over one ripple cycle. The simulated pulses are surrounded by echo pulses caused by the phase ripple [17,18], which are too small in magnitude to be seen in the figure. Optical system designers account for the echo pulses with ripple specifications and do not need to consider these in PMD specifications.

To better understand how DGD ripple is related to pulse distortion, we varied the ripple frequency and the fiber birefringence during simulations and calculated the pulse width, taken as its standard deviation within the original time slot, as shown in Fig. 8. Since the pulse distortion depends on the fiber birefringence, the wavelength shift between axes was varied from 1 pm to 50 pm, corresponding to a PMD of 1 ps to 50 ps for a grating with a 1000 ps/nm dispersion. In the figure, the ripple frequency was normalized to that of the standard deviation of the input pulse. Although the ripple-caused DGD fluctuations are very large in amplitude, their effect on the pulse width is not felt until the ripple frequency approaches the signal spectral bandwidth becomes a 1^st-order PMD. These simulations support the previous assertion that high-bandwidth measurements of the DGD, which smooth these rapid fluctuations, yield PMD magnitudes that effect pulse broadening and should be used in designing optical communications system.

 

Fig. 8. Pulse width within the original time slot as a function of the ripple frequency normalized to the standard deviation of the pulse spectrum.

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The system penalty at a given wavelength will vary with the input launch polarization, since different portions of the phase ripple will be sampled as the polarization space is explored. The frequency of this variation will depend on the periodicity of the phase ripple. Here we simulate a grating with a periodicity closer to that of a 10 GHz signal to enhance this effect and take the ripple frequency as f_r =600 MHz and Δω=260 MHz. When the launch polarization is along a birefringent axis and the carrier frequency is varied, the system penalty is periodic with the same period as the phase ripple (Fig. 9(a)). Similar results, with a 2 pm relative shift, are obtained for launches along the other axis. As the polarization alignment of the input signal varies from one axis to the other at a fixed wavelength, the penalty variation is as shown in Fig. 9(b) for an example wavelength, demonstrating how the system penalty variation with polarization may have occurred as measured in Fig. 1(c).

 

Fig. 9. (a) Simulations of Q-penalty induced by rapidly varying DGD when the input signal polarization is aligned with X (solid line) or Y birefringence axis (dotted line). (b) Penalty variation when the polarization alignment of the input signal changes gradually from the X to the Y birefringence axis. The penalty offset is caused by noise added to the simulations.

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5. Conclusion

The phase ripple of chirped fiber gratings causes the DGD to vary rapidly with wavelength and makes PMD measurements difficult to interpret. The value that system designers desire for their PMD budget is related to the underlying birefringence of the device and is masked by the ripple-dominated DGD measurements made with low-bandwidth techniques. Since many commercial instruments utilize low-frequency modulations for PMD measurements, one must record the PMD vector direction and magnitude while making measurements to calculate 1^st-order PMD values.

To summarize, low-bandwidth measurements of both PMD and phase ripple provide redundant information and are not both needed to completely characterize a chirped grating. Measurements of the DGD of chirped fiber gratings typically yield very fine structure in the presence of ripple, making it difficult to assign a ‘PMD’ value to a grating with low-bandwidth characterization methods. In general, it is preferable to use high-bandwidth measurement methods to determine the PMD and low-bandwidth methods for the phase ripple, since one can then use common definitions of PMD and phase ripple (or GDR) in optical communications system designs.