Measurements of Group Delay and Chromatic Dispersion of Hollow-Core Fiber Using a Frequency Domain Method

Abstract

Although hollow-core fibers are intended to be single-mode, they can potentially carry slightly higher-order mode content depending on the specific fiber structure. The presence of one or more higher-order modes makes the measurement of group delay and chromatic dispersion difficult if one relies on an instrument that is designed to work with single-mode fiber, in particular, a commercial instrument. In this work, we present the measurements of hollow-core fibers using a frequency domain method by acquiring the complex transfer function over a range of modulation frequencies. The measurement technique is immune to the higher-order mode nature presented by some of the hollow-core fibers. We measured hollow-core fibers with a five-capillary structure and a six-capillary structure. We obtained the absolute group delay as well as the chromatic dispersion information. In particular, we were able to measure one hollow-core fiber with at least two modes. The measured chromatic dispersion values are consistent with the modeling and those reported in the literature.

1. Introduction

Anti-resonant hollow-core fibers (HCFs), which utilize a cladding structure composed of thin capillaries to confine light within a hollow core, have garnered significant research interest recently due to their relatively simple structures and low confinement loss (CL). Among these, the nested antiresonant nodeless fiber (NANF) designs, which incorporate features of nested capillaries with non-touching to prevent node formation, have demonstrated record-low fiber loss over a broad bandwidth [1,2,3], with the lowest loss reported at 0.22 dB/km at 1625 nm [3]. Subsequently, the double-nested design (DNANF) was proposed to achieve even better performance with lower loss [4,5,6], such as below 0.11 dB/km at 1550 nm [5].

The measurement of HCF is crucial for understanding its fundamental properties, such as group delay (GD) and chromatic dispersion (CD). Although HCFs are primarily designed to support single-mode propagation, they are not strictly single-mode in the traditional sense and may support higher-order modes (HOMs), albeit with significantly high attenuation. Existing measurement techniques for optical fibers are largely focused on single-mode fibers and strongly rely on the fiber’s single-mode nature [7,8]. The quality of these measurements can quickly degrade in the presence of HOMs, and the measurements themselves may fail when the HOM content is significant. When dealing with few-mode optical fibers, different measurement techniques specifically designed for each mode are often required [9]. Therefore, the measurement of HCF is not a trivial matter. In this paper, we intend to introduce a measurement method that can overcome the measurement challenge.

Recently, a frequency domain method for measuring various types of optical fibers from single-mode fiber, polarization maintaining fiber, few-mode fiber, and multimode fiber to multicore fiber using a vector network analyzer (VNA) has been presented [10,11]. Through proper electrical–optical (E-O) conversion to launch frequency sweeping signals into the fiber and optical–electrical (O-E) conversion at the receiving side, the VNA measures the complex transfer function (CTF) of the fiber transmission. The GD information can be calculated from the inverse Fourier transform of the CTF with proper mathematical treatments to de-alias the CTF to recover the full group delay of each mode. Since the group delay can be measured over a range of wavelengths, the CD can also be obtained through the first-order derivative of GD vs. wavelength, and dispersion of different modes can be obtained over a range of wavelengths.

The frequency domain measurement method is suitable for measuring the HCFs. The measurement technique works even when HOM is present by measuring the GD for each mode. In this paper, we apply the frequency domain measurement to HCF by measuring its CTFs over a range of wavelengths, regardless of whether the fiber is single-mode or few-mode. Three measurement examples are presented, with one case specifically demonstrating the measurement of an HCF that exhibits both the fundamental mode and a HOM.

2. Measurement Principles

In this section, we detail the experimental setup and the measurement principles involved. Figure 1 illustrates the experimental arrangement. A narrow linewidth tunable laser, which operates in the 1520 to 1580 nm range, is intensity modulated by a modulator. The primary measurement is performed with a VNA, an RF instrument that assesses the electrical response of a device under test over various driving frequencies. Specifically, we utilized an Agilent N5230C PNA-L Network Analyzer for our experiments. Our main focus is on examining the transmission characteristics of the fiber under test (FUT), also referred to as the S₂₁ parameter or CTF. To ascertain the optical properties of the FUT, we implemented an E-O conversion at the launching end using a continuous wave (CW) laser, where an intensity modulator (IM) converts sweeping-frequency electrical signals into optical signals within the FUT. At the receiving end, an O-E conversion is performed using a photodetector (PD) or a linear optical receiver to transform the optical signals back into electrical signals.

The CTF, referred to as S₂₁, can be measured through a VNA over a frequency range for the FUT at a given launch condition [10]. The CTF takes the form,

$$\mathit{CTF}\left(f\right)=S_{21}\left(f\right)={{\displaystyle \sum }}_{j=1}^n{a}_j·\exp \left(-i·2\pi f{\tau }_j\right)$$

(1)

where $a_j$ is the relative optical power in mode $j$, and ${\tau }_j$ is the GD of mode $j$. The CTF is directly related to the time domain picture. When one input pulse, $P_{in}\left(t\right)$ is launched into the FUT with $n$ modes, the output pulse consists of $n$ pulses resulting from different group delays, ${\tau }_j$ of the $n$ modes,

$$P_{\mathit{out}}\left(t\right)={{\displaystyle \sum }}_{j=1}^n{a}_j·P_{\mathrm{in}}\left(t-{\tau }_j\right)$$

(2)

The time domain information can be extracted from the frequency domain measurement by performing an inverse Fourier transform of $CTF\left(f\right)$ on either the real or imaginary part. This allows for the determination of the GD of each mode,${\tau }_j$. To fully resolve the time domain information without causing aliasing or ambiguity of the group delay time ${\tau }_j$, the sampling frequency step $df$ needs to meet the condition as set by Nyquist theorem, $df\le 1/\left(2·\max \left({\tau }_1,\dots {\tau }_n\right)\right)$, which results in unreasonable large numbers of points that need to be sampled. In case we have chosen the sample points below the Nyquist requirement, the frequency content of the CTF can be ‘folded’ differently based on the frequency step of sampling [12]. This problem can be solved if we apply a de-aliasing procedure. We transform the CTF into a local time frame centered at the peak location $t_p$ in the inversion Fourier transform,

$$CT{F}^{\prime }\left(f\right)=e^{i2\pi {\tau }_{0}f}·CTF\left(f\right), \mathrm{where}{\tau }_0=\left({\displaystyle \frac{k}{df}}\right)\pm t_p$$

(3)

where k is an integer. Depending on the sign before $t_p$, the time sequence from inverse Fourier transform can either have the same time sequence or the opposite one compared to the actual propagation times for each mode. With a proper choice of k, which can be chosen by searching around estimated group delay and the sign in Equation (3), one can de-alias the signal to obtain the full group delay ${\tau }_j$ for each mode [10]. The output pulse $P_{out}\left(t-{\tau }_0\right)$ from the fiber can then be obtained based on an assumed input pulse $P_{in}\left(t\right)$,

$$P_{out}\left(t-{\tau }_0\right)={\mathcal{F}}^{-1}(CT{F}^{\prime }\left(f\right)*\mathcal{F}\left(P_{in}\left(t\right)\right)$$

(4)

where $\mathcal{F}()$ and ${\mathcal{F}}^{-1}()$ denote the Fourier and inverse Fourier transform.

Once the group delays are obtained over a wavelength range, the chromatic dispersion, $D\left(\lambda \right),$ can be calculated as the first-order derivative of the GD [11],

$$D\left(\lambda \right)={\displaystyle \frac{1}{L}}{\displaystyle \frac{d\tau }{d\lambda }}$$

(5)

where L is the length of the fiber and τ is the total group delay of a particular mode.

3. Measurements of Three HCFs

The HCF samples we experimentally studied are either six-capillary or five-capillary HCFs, as illustrated in Figure 2. The light coming from a standard single-mode fiber jumper is spliced with a multimode fiber-based Graded-index (GRIN) lens with 50 μm core diameter and 270 μm length before coupling into the HCF so that the fundamental mode of HCF is dominantly launched [13]. Through proper calibration to take out the system contribution, the VNA measures the CTF contributed from the fiber.

We selected three experimental HCFs to demonstrate the measurement technique. One is based on a six-capillary structure shown in Figure 2a with a length of approximately 500 m. The capillaries have diameters of 30 and 16 μm, respectively. The core diameter is about 39 μm. The mode field diameter (MFD) of the fundamental mode at 1550 nm is around 34 μm. The fiber has a transmission window from 1300 to 1600 nm. As is known, six-capillary HCF tends to have higher-order modes [14]. This fiber can support HOM, as shown by the measurements at 1550 nm in Figure 3. The frequency range for the VNA sampling is from 10 MHz to 10 GHz with an NOP of 999. In Figure 3a, we present the real part of the CTF, which exhibits very fast oscillations that may be challenging to discern due to the resolution limitations of the display. By using Equation (3) with the choice of $k$ to be 16, the value of $t_p$ to be −30.0007 ns, we can obtain the modified CTF to transform the CTF to a local time frame, as shown in Figure 3b. Further, using Equation (4), we can obtain the output pulse with an assumed input pulse, as shown in Figure 3c. With the launch condition in the experiment, we can clearly see two modes. To measure multiple modes simultaneously, it is important to ensure that the output pulses associated with different modes are well separated. If they are not adequately separated, one can opt to use a longer fiber for the measurement or use a higher frequency sweeping range for the VNA. It can be found that the measurement technique can resolve the absolute GD change at the level of a few picoseconds (ps). The accuracy of GD as scaled in unit length is primarily determined by the accuracy of the fiber length at around 1 m. The measurements were performed over a sequence of wavelengths, and a second-order polynomial fitting was conducted to achieve an R-squared of 0.99933/0.9965 for the GDs of fundamental/HOM modes, as shown in Figure 4b. The GD is comparable to the time of flight of light in vacuum or air, which is significantly lower than that of conventional solid-core optical fiber with an approximate GD of 4920 ns/km. The CD as a first-order derivative of GD is then calculated using Equation (5) using a fitted GD curve, as shown in Figure 4b. The use of curve fitting can further enhance the reliability of the GD measurements to yield more stable CD values with measurement repeatability around 0.1–0.2 ps/(km·nm). The CD values are in the range of 3.26–3.64 ps/(km·nm) for the fundamental mode and in the range of 5.08–5.53 ps/(km·nm) for the HOM. The CD of the HOM is higher than that of the fundamental mode.

The second HCF is a fiber based on a six-capillary structure shown in Figure 2a with a length of approximately 1700 m. The capillaries have diameters of approximately 32.8 μm and 17.2 μm, respectively. The core diameter is about 47.7 μm. The fundamental mode MFD at 1550 nm is around 37.1 μm. This fiber is similar to the first fiber studied in Figure 3 and Figure 4, but it differs slightly in that it predominantly exhibits only one mode. As shown in Figure 5a, small amounts of HOMs are illustrated as the ripple pulses following the main pulse. For the fundamental mode, we have measured the GD over a sequence of wavelengths from 1520 nm to 1580 nm. Once again, it can be observed that the variation in GD is minimal, with only a 0.19 ns change over the 60 nm wavelength range. Despite the small variation in GD, we conducted a second-order polynomial fitting and achieved an R-squared value of 0.999976, as illustrated in Figure 5b. The CD is illustrated in Figure 5c, with values ranging from 1.61 to 2.06 ps/(km·nm) across the wavelength range of 1520 nm to 1580 nm.

The third fiber has a nested five-capillary structure shown in Figure 2b with a length of approximately 1300 m, which has better HOM suppression [14]. This fiber has a core diameter of 33.2 µm, an outer capillary average diameter of 37.7 µm, and a nested capillary average diameter of 19.9 µm. The fundamental mode MFD around 1550 nm is 28.5 μm. The fiber has a transmission window from 1300 to 1600 nm. The measurements were conducted from 1520 to 1580 nm with a 10 nm wavelength increment. Figure 6a shows the calculated output pulse by processing CTF following the procedure in [10] and Section 2. Figure 6b shows that the fundamental mode delay is around 3348.02 ns/km, and it increases slightly over the wavelength. The GD data were fitted using a second-order polynomial, resulting in an R-squared value of 0.999989. Additionally, the CD of the fiber was obtained and is shown in Figure 6c, indicating a range of 2.76–3.45 ps/(nm.km) across the wavelength range of 1520–1580 nm.

4. Discussion

In the current work, we presented a frequency domain method for measuring the HCFs for their GD and CD. The subsequent discussions explore several aspects that provide insights into this method, including comparisons to other techniques and CD results expected from HCFs.

• Capable of measuring HCFs that are not strictly single-mode: Matured measurement techniques of optical fibers are largely for single-mode fibers, strongly relying on the fiber’s single-mode nature [7,8]. If the FUT is not strictly single-mode, the quality of these measurements can quickly degrade in the presence of HOMs, and the measurements themselves may fail when the HOM content is significant. For two of the HCFs, the second and third one, we have attempted to use a commercial instrument designed for measuring single-mode fibers. In neither case do the measurements yield GDs that are close to expectation. Since the frequency domain method can detect each mode without being interfered with by another mode, the integrity of the measurement is not affected by the presence of another mode. This highlights the benefits of the method in measuring HCFs.
• High resolution of detecting GD changes: For HCFs, the GD change over the wavelength is very small in the order of 100–250 ps over the entire wavelength range for the three fibers measured. Despite the small changes in GD, the frequency domain measurements were able to detect these changes. The quality of the data is demonstrated by the R-squared value exceeding 0.9999. In comparison, a typical CD measurement instrument would require a fiber length of at least a few kilometers. However, in this study, we were able to measure the HCFs with lengths as short as around 500 m.
• Comparing the measured CD to the semi-analytical model: For comparison, we extracted the main fiber parameters from SEM images of the second and third HCFs and calculated the group index $n_g$ using a semi-analytical model [15]. From the group index versus wavelength results, we then calculated the dispersion using the following equation,

  $$D\left(\lambda \right)={\displaystyle \frac{1}{c}}{\displaystyle \frac{d{n}_g}{d\lambda }}$$

  (6)

A comparison of the experimental results and modeled CD is shown in Figure 7 for the second and third HCF measured in Section 4. As can be seen, reasonable agreements between measurement and modeling have been achieved. In addition, in Refs. [6,16], 5-capillary DNANFs have also been reported as having low single-digit CD at the C-band with a positive CD slope, consistent with the findings in the current work.

5. Conclusions

HCFs hold the promise for future optical communications with low loss and low latency. Although HCFs are intended to be single-mode, they can potentially carry slightly higher-order mode content depending on the specific fiber structure. The presence of a higher-order mode makes the measurement of GD and CD difficult if one relies on an instrument that is designed to work with single-mode fiber. In this work, we present the measurements of HCFs using a frequency domain method through acquiring CTF over a range of modulation frequencies. The measurement technique is immune to the higher-order mode nature as presented by some of the HCFs. We measured HCFs both with five-capillary structures and six-capillary structures. We obtained the absolute GD as well as the CD information. In particular, we were able to measure one HCF with at least two modes. The measured CD values are consistent with the modeling and those reported in the literature. The frequency domain measurement technique also has high time resolution for resolving the small GD changes over wavelength, in which case, we were able to conduct the measurements using a fiber as short as around 500 m. With the work presented in this paper, the frequency domain measurement technique has been illustrated to be capable of measuring essentially all types of optical fibers, from single-mode fiber, polarization maintaining fiber, few-mode fiber, multimode fiber, multi-core fiber, and now HCF. This method is highly robust and versatile, making it suitable for measuring both GD and CD of optical fibers in general.