Chromatic dispersion of optical fibers is a major consideration and must be accounted for when developing or deploying optical equipment for use in high-speed optical fiber communications and microwave photonics links ^[1]. A variety of techniques have been demonstrated for measuring the chromatic dispersion of optical fibers, including the time-of-flight method ^[2], phase shift method ^[3], interferometric method ^[4,5], and transfer function method ^[6,7]. Among these methods, the widely used phase shift method and transfer function method are suitable for characterization of large dispersions with temporal resolution on the order of picoseconds ^[8]. However, the phase shift method needs complicated systems to compensate the fiber length variation caused by the thermal fluctuation with a reference light ^[4]. The traditional transfer function method requires chirp-free operation of intensity modulators, or at least two notches for correcting the chirp error, which leads to a limited measurement range ^[8,9].

The chirp parameter varies with the applied bias voltage of the modulator. In the case of chirped intensity modulation, the fiber transfer function will depend on both the fiber dispersion and the chirp parameter; that is to say, the transfer function can be tuned by operating the modulator at different chirp parameters through adjusting the applied bias voltages. In this paper, we propose a novel method that achieves high accuracy over a large range for measuring the fiber dispersion by using chirped intensity-modulated optical signals. In contrast to the traditional method, the proposed method requires one notch rather than two notches, which not only eliminates the chirp error but also achieves an improved measurement range through adjusting the chirp parameters (bias voltages) of intensity modulators. Moreover, it allows fast and accurate measurement of the magnitude and sign of chromatic dispersion at different operating wavelengths for any kind of optical fibers simply by using a vector network analyzer (VNA).

As is shown in Fig. 1, an optical carrier at the wavelength λ is intensity modulated by a microwave sinusoid signal at the frequency f. The optical field of the intensity-modulated signal can be expressed as ^[10,11] with α being the chirp parameter, m the modulation index, and I0 the average optical power. After fiber propagation, the chromatic dispersion will introduce an additional phase shift to each optical sideband depending on its frequency offset relative to the optical carrier, and result in the partial fading of the intensity-modulated signal. Under the small-signal approximation, the transfer function, relating the instantaneous optical intensity after and before fiber propagation, can simply be given by ^[7,10,11] where the total fiber dispersion DT is related to the group velocity dispersion parameter β2 by DT=−2πβ2Lc/λ2 with the fiber length L. We did not take into account the absolute phase and group delay, because both terms produced only phase delay of the optical carrier signal. The transfer function H acts as a function of the total dispersion DT and the chirp parameter α, which periodically achieves null (zero) at the notch frequencies corresponding to with u being the order of the notch. It is obvious that the notch frequency depends on not only the total dispersion of optical fibers but also the chirp parameter of the intensity modulators.

Conventionally, there are two methods of eliminating the chirp error when extracting the chromatic dispersion from the measured transfer function. The chirp-free method employs zero chirp intensity modulation and determines the chromatic dispersion with one notch frequency as in which the minimum measurable dispersion DTmin is restricted to be with fmax being the maximum modulation frequency. Alternatively, the double-notch method is to extract the chromatic dispersion by subtracting two notch frequencies as in which the minimum measurable dispersion DTmin is as follows: One can see from Eqs. (5) and (7) that the chirp-free method holds much larger measurement range but requires chirp-free intensity modulation; meanwhile, the double-notch method alleviates the chirp error at the expense of a smaller measurement range.

To take care of both the chirp error and the measurement range, we develop a single-notch method by introducing an associated frequency of the first-order notch. As we know, the transfer function will achieve a null at the notch frequency, corresponding to The transfer function can be expressed with the notch frequency as Here, we choose an associated frequency of the first-order notch given by and substitute Eq. (10) into Eq. (9), leading to which allows us to extract the fiber dispersion with a very simple expression given by Since only one notch is required, the minimum measurable dispersion DTmin meets Figures 2(a) and 2(b) show the minimum measurable dispersion as a function of the chirp parameter for the normal dispersion (DT>0) case and for the anomalous dispersion (DT<0) case, respectively, where λ=1550nm and fmax=20GHz are set. Similar results can also be obtained for other wavelengths and frequency ranges. In fact, one can conclude from Eqs. (5), (7), and (13) that our method has the smallest minimum measurable dispersion and the largest measurement range in both the normal and anomalous dispersion regimes regardless of any chirp parameter.

The measurement range can be further improved by optimizing the chirp parameter of the optical modulators. From Fig. 2, it is a favorable characteristic for the measurement range to have a positive chirp parameter that is as large as possible in the normal dispersion regime and a negative chirp parameter that is as large as possible in the anomalous dispersion regime. For example, the minimum measurable dispersion in the normal dispersion regime will be DTmin>19.53ps/nm in the case of α=5, corresponding to a 1.15 km length of standard single-mode fiber (SMF) at 1550 nm. In comparison, the measurable dispersion of the double-notch method is DTmin>331.72ps/nm, corresponding to a 19.51 km length of SMF at 1550 nm. A similar conclusion can also be drawn in the anomalous dispersion regime with a negative chirp parameter. In practice, a positive or negative chirp parameter can be achieved by changing the applied bias voltage of an optical external modulator, such as a Mach–Zehnder modulator ^[11] or electro-absorption modulator ^[12]. It has been demonstrated that the chirp parameter can be tuned from −10 to 10 as a function of the bias voltage for a typical electro-absorption modulator ^[12].

In our measurement, an electro-absorption modulator (OKI OM5653C) is used as the optical modulator with a tunable chirp parameter. As is shown in Fig. 1, the optical carrier at 1550.20 nm from a tunable laser diode is intensity modulated by a sinusoidal microwave signal and sent to a length of SMF under test. With a full two-port calibration, an HP8720D VNA is used to generate the microwave swept frequency signal within 20 GHz and to measure the fiber transfer function of the chirped intensity-modulated signal. At first, port 1 and port 2 are directly connected to determine the response contributions of the modulator and receiver as an optical-to-optical reference. Then, the dispersive fiber is connected between port 1 and port 2 to measure the total response including the modulator, dispersive medium, and receiver. The fiber transfer function is therefore obtained by dividing the response with the optical fiber by the response without the optical fiber, in order to exclude the influence of the modulator and receiver. It should be noted that the fiber loss is normalized from the measured transfer function. A similar measurement procedure can also be found in Refs. ^[7,9].

Figure 3 shows the measured transfer function of a 25.4 km SMF at different bias voltages of the intensity modulator. The transfer function, in the case of a bias voltage of −1.4V, holds two notches at 6.215 and 18.335 GHz, respectively, from which the fiber dispersion can be extracted to be DT=419.27ps/nm with the double-notch method based on Eq. (6). Meanwhile, the associated frequency of the first-order notch is fh=4.394GHz and H(fh)=−5.838dB, and the fiber dispersion can also be determined to be DT=420.06ps/nm with our method based on Eq. (12). When the applied bias voltage is tuned to −2.2V, the two notch frequencies are accordingly changed to 8.665 and 19.305 GHz, respectively, and the associated frequency of the first-order notch is fh=6.127GHz and H(fh)=−5.32dB. In this case, the fiber dispersion is measured to be DT=418.31ps/nm with the double-notch method and DT=419.19ps/nm with our method. Agreeable results can also be obtained at other bias voltages, indicating the efficient error correction of the chirp parameter. For a further validation, we adopt a least-squares fitting of the measured transfer function based on Eq. (9), in which the first-order notch frequencies f1 are set as 6.215 and 8.665 GHz, respectively. The determined dispersions from the curve fitting are 419.7 and 420.3ps/nm, respectively. The fitted curves are illustrated in the same figure together with the measured transfer functions for comparison, where both agreeable results verify the accuracy of our method.

In order to further verify our method, a 6.1 km length of SMF is chosen to repeat the measurement. As is shown in Fig. 4, the measured transfer function achieves only the first-order notch, and all the higher-order notches are beyond a 20 GHz frequency range even if different chirp parameters are tried by changing the bias voltage of the intensity modulator, which results in difficulty determining the fiber dispersion with the double-notch method according to Eq. (6). With our method, the fiber dispersion can be determined to be DT=98.54ps/nm according to the associated frequency fh=10.02GHz and H(fh)=−5.749dB of the first-order notch at 14.168 GHz in the case of a bias voltage of −1.8V. The measured result in the case of a bias voltage of −2.1V is DT=98.60ps/nm according to the associated frequency fh=12.385GHz and H(fh)=−5.375dB of the first-order notch at 17.518 GHz. For accuracy, we check our results by fitting the measured transfer function using Eq. (9) within 20 GHz, where the first-order notch frequencies f1 are set as 14.168 and 17.518 GHz, respectively. The least-square curve fittings lead to the determined dispersions 99.55 and 99.83ps/nm, respectively, which are also included in Fig. 4 for comparison.

Furthermore, we also measure the dispersion of the 81.5 km fiber at different operating wavelengths. Figure 5 shows the measured fiber dispersion as a function of operating wavelength. The measured results with the double-notch method are also included in Fig. 5 for comparison. Both agreeable results verify that our method works over a wide wavelength range with high accuracy.

If the sign of the fiber dispersion also needs to be tested, we use a length of standard SMF as the reference fiber to cascade with the fiber under test. If the notch frequencies in the cascade case become larger than those of the reference fiber alone, then the fiber under test will have the same sign of chromatic dispersion as the reference fiber; otherwise, it holds the opposite sign of chromatic dispersion of the reference fiber.

Since both the fiber dispersion and the notch frequency are determined from the transfer function, the uncertainty of the notch frequency will be delivered to the fiber dispersion through the error transfer factor Fwhich is deduced from the total differential of Eq. (2). An error factor of 2 means that the relative error of the notch frequency will be transferred to the fiber dispersion two times. In our experiment, the uncertainty of the notch frequency is within 50 MHz, corresponding to a relative error of less than 0.25%, and the maximum uncertainty of 0.5% might be transferred from the notch frequency to the fiber dispersion.

The accuracy of the extracted fiber dispersion will also be impacted by the measured transfer function. The error dependence of fiber dispersion on the uncertainty of the transfer function can be investigated with the help of the total derivative of Eq. (2), given by According to Eqs. (8) and (10), we have Substituting Eq. (16) into Eq. (15), the error transfer factor at the associated frequency can be expressed by As is shown in Fig. 6, the error transfer factor at the associated frequency is plotted as a function of chirp parameter. According to the VNA specification, the measurement uncertainty is less than 0.2 dB, corresponding to a relative error of less than 2.33%=(10.^(0.2/20)−1)*100%. The uncertainties of the fiber dispersion resulting from the measured dispersion are listed in Table 1 for reference. In the worst case, the maximum relative error of the measured dispersion is about 3.46%(=1.27*2.33%+0.5%), which is contributed by both the notch frequency and the transfer function.

To summarize, we have demonstrated the use of a chirped intensity-modulated signal for accurately extracting the chromatic dispersion of optical fibers based on fiber transfer function measurement with a VNA. Our method allows a wide measurement range with a simple configuration and can achieve fast measurement with high accuracy, which enhances the characteristics of optical fibers, dispersive devices, and optical transmission paths.