Few-mode fiber true time delay lines for distributed radiofrequency signal processing

We report, for the first time to our knowledge, distributed radiofrequency signal processing built upon true time delay operation on a step-index few-mode fiber. Two 3sample configurations with different time delay properties are implemented over the same 60meter 4-LP-mode fiber link. The inscription of a long period grating at a specific fiber position converts part of the LP01 mode into the LP02, permitting sample time delay engineering. Delay line performance is experimentally demonstrated when applied to radiofrequency signal filtering, example of fiber-distributed processing functionality exhibiting one order or magnitude gain in terms of compactness. © 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

In this paper, we propose and experimentally demonstrate, for the first time to our knowledge, sampled TTDL operation with constant basic differential delay along a SI FMF. Two different TTDLs, which are characterized by different basic differential delays, are implemented over the same 60-m FMF that propagates 4 LP modes (LP 01 , LP 11 , LP 21 and LP 02 ). A long period grating (LPG) inscribed at a specific position along the fiber allows the modification of the time delay of the sample linked to the LP 02 mode. Delay line selection is realized by choosing a proper set of 3 modes at the fiber output. As a proof of concept, we apply the implemented sampled TTDL to a typical RF signal processing functionality, RF signal filtering, obtaining a good agreement between theoretical and experimental results.

True time delay line operation based on a few-mode fiber
We propose to implement sampled TTDL operation on a step-index FMF, provided by Prysmian, which supports four LP modes (LP 01 , LP 11 , LP 21 and LP 02 ). The fiber is characterized by a 15-μm core diameter, a 125-μm cladding diameter and a refractive index contrast of 1.1%. At a wavelength of 1550 nm, the differential group delays (DGDs) relative to LP 01 mode are 4.4, 8.9 and 7.9 ps/m, respectively for LP 11 , LP 21 and LP 02 modes, and the chromatic dispersion values are 21, 26, 19 and 8 ps/km/nm, respectively for LP 01 , LP 11 , LP 21 and LP 02 modes, [6]. The group delay per unit length τ n for a fiber mode n can be expanded as a first-order Taylor series [4] around an anchor wavelength λ 0 as: where D n is the chromatic dispersion parameter of mode n at λ 0 . Higher-order dispersion effects are assumed to be similar in all mode groups and, therefore, can be neglected in terms of DGD calculation. Sampled TTDL operability requires identical basic differential delays between adjacent samples Δt, [5]. From the above fiber group delay and dispersion specifications, we see that the first three modes (i.e., modes LP 01 , LP 11 and LP 21 ) almost satisfy this condition, and thus one can expect that it will occur at some point nearby 1550 nm. By solving the equation system formed by Eq. (1) for these modes, we found that the wavelength λ T = 1558.3 nm forces the DGD between LP 21 and LP 01 modes to double the DGD between LP 11 and LP 01 modes. If the first, second and third samples correspond, respectively, to the LP 01 , LP 11 and LP 21 modes, then the total time delay of the samples t k (where k = 1, 2, 3) fulfil: (2) As illustrated in Fig. 1(b), one can find, for any fiber length, three equally temporary spaced optical samples when operating at the optical wavelength λ T . The processing radiofrequency depends on the inverse of the basic differential delay Δt, which is determined by the fiber length. As further described in section 4, a fiber length of L = 60 m leads to Δt 1 = 285.7 ps, so that the processing radiofrequency falls within the frequency range of our measurement system.
We must take into account that we cannot implement a 4-sample TTDL with all the propagating modes since the LP 02 mode does not fulfil the constant basic differential delay condition. Therefore, we use this mode for implementing, at the same time, another 3-sample delay line in a way that one can manually switch between the two delay lines by simply changing the set of modes extracted at the output of the FMF link.
The first, second and third samples featuring this second delay line will be given, respectively, by LP 01 , LP 02 and LP 11 modes at the fiber output as long as we assure a constant basic differential delay. To achieve this, we propose to realize a mode conversion by the inscription of an LPG at a specific longitudinal position along the fiber so that the total delay of one sample is modified accordingly. Figure 1(a) illustrates this concept. Since each mode is characterized by a different group delay, the new sample's delay will then be determined by a ponderation between the group delays of both modes involved. In our case, the time delay of the second sample is adjusted by combining the propagation characteristics of the LP 01 and LP 02 modes. This sample will propagate firstly through LP 01 along a certain distance L' of the FMF link. At this point, part of the energy propagating through LP 01 will couple to LP 02 mode. This second sample will then propagate through LP 02 mode along the remaining distance L -L', as shown in Fig. 1(a). The time delay of the first and third samples are obtained, respectively, from the modal groups delays as t 1 = τ 01 L and t 2 = τ 11 L. For the second sample, the total time delay t 2 is given by: Equations (2) and (3) yield L' = 0.285L for λ = 1558.3 nm. Thus, by introducing the LPG at 0.285 times the total length, TTDL operation is possible for the second set of three fiber modes, as shown in Fig. 1(b) lower. In particular, for a fiber link length of 60 m, the LPG must be placed at the position L' = 17.1 m and the TTDL basic differential delay is Δt 2 = 137 ps. As mentioned before, we will evaluate the performance of both TTDLs when they are applied to three-tap RF photonic filtering. By choosing a given set of samples, (either the ones extracted from LP 01, LP 11 and LP 21 modes, or those extracted from LP 01, LP 02 and LP 11 modes at the FMF output), we can select a different microwave photonic filter.

Long period grating inscription
With grating periods Λ typically ranging from 100 μm up to 1 mm, LPGs can induce coupling from a propagating core mode to another forward-propagating core mode if the grating period equals the beat length of the modes. This energy transfer takes place at the central wavelength of the LPG that depends on the period Λ and the difference between the effective refractive indices of the modes [7]. In our case, for mode conversion between LP 01 and LP 02 , we have: LPGs have found application in band rejection filters [8], gain flattering filters [9], temperature sensors [10], curvature sensors [11], refractive index sensors [12], and mode conversion in FMFs [13]. They are low loss and have low reflection and high design flexibility. They can be created either as a mechanical perturbation over the fiber or inscribed as a permanent perturbation with a UV laser.
We inscribed the LPG with direct point-by-point UV laser inscription. Prior to the inscription, the FMF was hydrogen-loaded at ambient temperature for two weeks at a pressure of 50 bar to increase its photosensitivity. The grating was inscribed with a frequency-doubled argon-ion laser emitting at 244 nm, with an output power of 50 mW and an exposure time of 20 s. Photonic lanterns were spliced at both ends in order to monitor the optical output powers of all modes during the inscription. The inscribed period was 263 μm, which was selected to get the mode change at 1550-nm wavelength for the Prysmian fiber, while the LPG length was set to 29 mm to achieve the complete mode change. After the inscription, a heat annealing was done to assure the stabilization of the LPG by heating the fiber up to 200 °C for five hours and thus accelerating the degradation effects associated with the hydrogen diffusion and the refractive index thermal decay. Prior to heat annealing, we observed both short-time and long-time variations, due to the hydrogen content. Short-time variations are produced after inscription when hydrogen is redistributed from non-affected sections of fiber to irradiated sections where the grating is inscribed, and long-time variations are produced when hydrogen leaks out of the fiber [8,14,15]. Figure 2(a) shows the optical spectra for the two modes 16 hours after the inscription. Spectrum deformations can be observed at the mode-conversion wavelength due to energy leaking from LP 02 mode to higher-order cladding modes. After five hours of heat annealing, the optical spectrum is properly recovered and an efficient mode conversion between LP 01 and LP 02 modes is achieved and maintained over time at a wavelength of 1550 nm (see Fig.  2(b)). The output power of both modes can be controlled by tuning the wavelength in the vicinity of the LPG response. In particular, we are interested in working at 1558.3 nm, where the conversion efficiency of the LPG translates into output optical powers of −4.8 and −2.   Figure 3 shows the experimental setup for RF signal filtering assembled with the implemented TTDL. We use a laser emitting at 1558.3 nm, an external electro-optic intensity modulator, a photodetector and a vector network analyzer (VNA). After the modulator, the light is split in three paths and injected into the FMF using a photonic lantern [16], so that the modes LP 01 , LP 11 and LP 21 are excited. After the FMF link, the signals propagated by the four LP modes are recovered using the second photonic lantern. The photonics lanterns were fabricated by Olkin Optics explicitly for the Prysmian FMF [17]. We characterized their performance by measuring the insertion losses (gathered in Table 1) and the average intermodal crosstalk between modes at 1550 nm (gathered in Table 2) when both photonic lanterns are connected directly. One can observe that the insertion losses are not constant due to several phenomena that affect each mode differently, as fiber offset induced splice misalignments and possible manufacturer errors. We can see in addition higher levels of crosstalk for any mode coupling combination that involves the LP 11 mode, what may have important repercussions on the overall behavior of the system. In general, the effect of modal crosstalk must be carefully tackled since it can degrade the response of the microwave photonic filter.  At the output of the photonic lantern, we can choose between the two possible signal filters by selecting the appropriate set of modes. We must take into account that the RF filter must arise from the TTDL implemented by the FMF itself, so we compensated any other influence over the group delay introduced by external components by adjusting three variable delay lines (VDLs) introduced before the FMF. The basic differential delay between samples was therefore adjusted to zero when the FMF was not present in the system. We have measured the differential delay of each mode by using an interferometric-based technique [18], which allows us to compensate the setup with a precision of 2 ps.

Experimental demonstration and application to RF photonic signal filtering
The basic differential delays of both TTDL schemes (i.e., Δt 1 = 285.7 and Δt 2 = 137.0 ps), generate microwave photonic filters with expected free spectral ranges (FSRs) of 3.75 and 7.50 GHz, respectively. These FSR values allow us to measure multiple filter resonances in the vector network analyzer.
We work only with the spatial LP lma modes to avoid any additional system required to combine the a and b spatial field components. In the context of SDM distribution of parallel channels, the signals propagated through degenerate modes should be in principle combined using an optical or electrical diversity combiner stage before or after photodetection if direct detection is used, [19]. The application of these techniques to MWP signal processing must be further investigated since (1) the use of electrical diversity combining is not compatible, for instance, with RF signal filtering where all the samples must be combined optically before detection, and (2) the use of optical combining requires applying adaptive co-phasing before adding both modes together in the optical domain that will significantly increase the complexity of the system.  Figure 4 shows the normalized radiofrequency response of the microwave photonic filter implemented with the first TTDL, which is given by the set of modes LP 01 , LP 11a and LP 21a . The maroon solid line corresponds to the experimental result, while the black dotted line corresponds to the theoretical result for an ideal uniform three-tap filter, obtained as the timeto-frequency Fourier transform of three Gaussian samples spaced 285.7 ps. We can see that the theoretical and experimental results agree very well in terms of frequency periodicity, but there exists some degradation for increasing frequencies in terms of main-to-secondary sidelobe ratio. This degradation may be caused mainly by small variations in the differential delay between adjacent samples. Hence, we computed the frequency response where the time periodicity of the samples is varied a 5% around the theoretical Δt value, that is, a 7-ps variation (blue dashed line). As observed, this adjustment leads to a better agreement for high frequencies. This time variation can be understood as an accumulation of precision errors. Actually, the systematic error of our setup for the time delay measurement of one sample is 2 ps. The 7-ps time deviation corresponds to 3.5 times this systematic error, which falls close to the measurement uncertainty we can expect.
Additional effects that might contribute to the time delay variation include: (1) fluctuations in the groups delays of the FMF modes that arise from variations in the laboratory environmental conditions; (2) mismatching between the modal DGD values provided in the FMF specifications and the actual values, what in addition translates into a deviation from both the theoretical optimal wavelength λ T and the longitudinal position of the LPG along the fiber L'; and (3) slight variations in the fiber lengths L and L' due to fiber cutting imprecision.
Furthermore, we must take into account that the intermodal crosstalk arisen in both the FMF link and the photonic lanterns introduces some distortion in the RF frequency response as the amplitude windowing of the time samples is altered. In particular, the central TTDL sample, which is carried by the LP 11 mode, is the one contributing the most to this effect, as corroborated by Table 2.  Figure 5 illustrates the normalized radiofrequency response of the microwave photonic filter implemented with the second TTDL, which is produced by the set of modes LP 01 , LP 02 and LP 11a . The maroon solid line corresponds to the experimental results while the blue dashed line represents the theoretical response for an ideal apodized three-tap filter, obtained as the time-to-frequency Fourier transform of three Gaussian samples, where the power of central tap is 20% higher than the rest and the basic differential delay is Δt 2 = 137 ps. As we can see, the experimental results show a very good agreement with the apodized theoretical simulation. The different attenuation level introduced by the LPG at 1558.3 nm, which is −4.8 dB for LP 01 mode and −2.2 dB for LP 02 mode (as shown in Fig. 2(b)), is the main responsible for this amplitude apodization.

Conclusions
We have proposed and experimentally demonstrated the first parallel TTDL for RF signals developed on a few-mode fiber link, where the 4 LP modes act as the optical carriers for the delay line samples. The introduction of an LPG to partially transform the LP 01 mode into the LP 02 mode at the proper link distance enables participation of the LP 02 mode in the TDDL operation. Two different configurations with different time delay properties are implemented on the same fiber, both ensuring a constant differential delay between samples, which is an essential requirement for incoherent discrete-time signal processing. The performance of the TTDL has been evaluated in the context of microwave signal filtering where we successfully demonstrated two filters where the FSR depends on the set of modes recovered.
This work demonstrates the feasible implementation of parallel distributed signal processing along a single FMF without resorting to parallel singlemode fiber links. This solution can be applied to a variety of MWP functionalities that will especially demanded in next-generation fiber-wireless communications, including not only signal filtering but also optical beamforming in phased-array antennas and arbitrary waveform generation among others.