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We present a novel method for distortion elimination in phase-modulated analog optical links. A small part of the phase modulated signal seeds a four-wave mixing comb source, which generates lightwaves with integer multiples of the phase modulation of the original signal. These lightwaves are scaled and re-combined with the original phase-modulated signal to cancel the distortion generated in the interferometric phase-to-amplitude conversion process. Experimentally, we demonstrate full cancelation of the third-order distortion of the receiver and achieve a 19-dB improvement in the link’s SFDR at a 1-Hz bandwidth. This approach is readily extendable to eliminate all relevant higher-order distortion products or synthesize arbitrary phase-to-amplitude transfer functions.
© 2014 Optical Society of America
1. Introduction
Analog optical links have found extensive application in the RF and microwave domain primarily due to the benefits offered by optical fiber, including low propagation loss, high bandwidth, low size, low cost, low dispersion, and immunity to electromagnetic interference. In particular, antenna remoting, radio-over-fiber, and phased-array radar applications have benefited from advances in the field [1–4]. To spur further adoption, research efforts in this area have focused on improving the performance metrics of these links including reducing link loss, noise figure, and distortion and thereby increasing spurious-free dynamic range (SFDR) [5–9].
Due to receiver simplicity, the most commonly investigated links are intensity-modulated direct-detection (IMDD) links and therefore, most linearization techniques have been directed toward reducing distortion in the IMDD architecture [10–20]. More recently phase-modulated links have received increased attention due to advantages they offer, including the high degree of linearity in the phase modulation process, no need for active bias control at the transmit end, and the ability to implement balanced detection with a single fiber span [21–24]. In a phase-modulated interferometrically detected (ΦMID) link, the transmitter is a simple phase modulator and the receiver is an interferometer followed by a photodetectors (or two photodetectors for balanced detection). While the phase modulation process is linear, the interferometric phase-to-amplitude conversion is not and thus adds distortion. Previous linearization schemes for ΦMID links include digital post-processing [24] and parallel interferometric detection [25], which demonstrated 12-dB and 16-dB improvement in the link SFDR at 1-Hz bandwidth, respectively.
In this paper we investigate a novel linearization method for eliminating distortion in ΦMID links by utilizing the nonlinear optical process of cascaded four-wave mixing (FWM). Using a Fourier approach, the FWM generated lightwaves are scaled and combined with the signal prior to a single interferometric receiver such that distortion is eliminated [26]. We experimentally demonstrate elimination of the third-order distortion products in a ΦMID link and achieve a 19-dB improvement in SFDR at 1-Hz bandwidth. Most significantly, this linearization method readily offers a path to eliminating any relevant higher order distortion products or, more generally, the synthesis of arbitrary phase-to-amplitude transfer functions in the receiver.
2. Linearization background
Linearization of analog optical links involves modifying the transfer function (input voltage to output voltage) of the link to reduce one or more distortion products. In most analog optical links, the primary distortions are introduced by the transfer function of an interferometer. Specifically, in IMDD links employing Mach-Zehnder intensity modulators this occurs at the transmit end in the modulator whereas in ΦMID links it occurs at the receive end in the interferometric detector. The simplest linearization method involves biasing the interferometer at quadrature where the Taylor expansion of the transfer function is an odd function and thus all even-order distortions are eliminated. However, in links with less than an octave of bandwidth, the odd-order distortions are the most detrimental as their intermodulation products fall in band and thus cannot be filtered out. Thus in sub-octave links, third-order distortion poses the primary limit to the link’s SFDR. Therefore reducing or eliminating third-order distortion is highly desirable. The simultaneous elimination of both the second-order and third-order distortions is known as broadband linearization and typically comes at the expense of higher noise Fig [6].
There are three commonly employed methods of distortion elimination in analog photonic links: analog electronic (feedback/feedforward) [10–13], digital signal processing [14, 15, 23, 24], and electro-optic [16–20, 22, 25]. One example of electronic distortion elimination is predistortion, in which an RF signal with equal and opposite nonlinearity to the transfer function is fed into the modulator [10, 11]. Another example is feed-forward, in which part of the optical output is detected and compared with the input RF signal to generate an error signal [12, 13]. This error signal is inverted and sent to a second electro-optic modulator whose output is added to the first output to produce a more linear output. The second type of distortion elimination involves detecting the output, sending it through an analog-to-digital converter, and using digital signal processing (DSP) to electronically correct the signal [14, 15]. The third type involves connecting multiple modulators or multiple interferometric detectors in either series or parallel to produce an output that is more linear than either of the devices individually [19, 20]. For example, this usually means driving one modulator with a high optical power and low RF power (low distortion) and the other with low optical power and high RF power (high distortion), then combining the signals such that the distortion products cancel while the signal does not [16–20], however, due to device constraints this linearization generally comes at the cost of significantly reduced link gain.
3. Linearization using FWM comb source
Here we make use of cascaded four-wave mixing (FWM) of the signal with an unmodulatedcontinuous-wave laser to generate multiple lightwaves of predictable distortion. Due to the phase transfer of the FWM process, the cascaded FWM process that we employ generates an array of lightwaves that possess integer multiples of the signal’s phase-modulation, as depicted in Fig. 1 [27]. While the efficiency of the FWM process is polarization dependent, by implementing a polarization diversity technique it is possible to eliminate this dependency with increased receiver complexity if necessary [28]. Due to the sinusoidal response of the interferometric detection process, each of the generated lightwaves corresponds to a Fourier component of the link’s overall voltage transfer function. Therefore, as depicted in Fig. 2, these lightwaves can be scaled and combined with the original signal to linearize the overall transfer function of the link. Beyond linearization, these Fourier components can also be scaled and combined to yield any custom-made transfer function through a Fourier synthesis approach. Such arbitrary transfer functions could have many signal processing applications that could potentially eliminate the need for electronic components before or after the link. For example, by introducing the inverse nonlinearity of a microwave component after the link on top of a linear transfer function, we could make the link act as a predistorter that would enhance the performance of the overall system.
Fig. 1 A cascaded four-wave mixing process between the signal and a CW pump laser is employed to produce idlers with integer multiples of the phase modulation of the original signal at frequencies which are integer multiples of Δω = ω1–ω0 away from the two pumps.
Fig. 2 By appropriately choosing, scaling, and combining lightwaves with the correct phase multiples of the original signals, any transfer function can be synthesized.
Here we focus on distortion elimination in a ΦMID link. Distortion enters the link through the sinusoidal transfer function of the interferometric receiver. When biased at quadrature, we can readily calculate the response by performing a Taylor expansion of a sine function aboutits inflection point (see Eq. (1)),
where is the analog phase-encoded temporal signal. In addition to the desired linear term, the expansion also exhibits higher odd-order terms. This leads to large third-order distortion as shown in Fig. 3 (blue curve). By scaling and combining a lightwave of triple the phase-modulation of the original signal, the third-order distortion can be eliminated entirely, while suffering only a slight reduction in link gain (see Eqs. (2)-(3) and the red curve in Fig. 3) and leaving the fifth-order distortion as the primary limit. Adding additional lightwaves and following a similar process, it is possible to further eliminate higher-order distortion products, which also increases the quadrature bias tolerance of the interferometer (see blue curve in Fig. 3). The scale coefficients of the added lightwaves are readily found using a matrix inversion as detailed in the Appendix. Furthermore, in the limit as the number of terms , the scale coefficients asymptotically approach those of the Fourier expansion of a triangle wave.Fig. 3 (a) The normalized transfer function with various Fourier terms (solid lines) and ideal (dotted); (b) 1st derivative of transfer function or normalized link gain; (c) 2nd derivative or second-order distortion; (d) 3rd derivative or third-order distortion.
To investigate the impact of this linearization on the SFDR of the ΦMID link, we calculated the SFDR resulting from the fundamental tone and the intermodulation productgenerated by a two-tone test using the coefficients found by the method described above. To best compare with our experimental results, we use our experimental link parameters in this calculation. Namely, we assume a modulator of 5.9 V and a photocurrent of 5 mA for each detector in balanced configuration. In Fig. 4 we plot the received fundamental and intermodulation power as a function of microwave power input to the link. Here for SFDR calculation we assume a shot noise limited link. For these link parameters Fig. 5 shows the SFDR at 1-Hz bandwidth as a function of the number of linearization terms. The link SFDR also depends on signal bandwidth and we plot the SFDR versus bandwidth as a function of the number of linearization terms in Fig. 6.
Fig. 4 Theoretical SFDR performance calculated for the ΦMID link as an increasing number of linearization terms are added and assuming a modulator Vπ of 5.9 V and 5-mA per photodetector with balanced detection.
Fig. 5 The calculated SFDR as an increasing number of linearization terms are added. The dashed line represents the SFDR for a perfectly linear link when the maximum peak to peak voltage equals the modulator Vπ for reference.
4. Experiment
To experimentally investigate the linearization performance of this approach we construct a conventional ΦMID link and a linearized ΦMID link as depicted in Figs. 7 and 8. The conventional ΦMID link consists of a 20-mW laser operated at 1558.98 nm and a 20-GHz electrooptic phase modulator at the transmit end. At the receive end, we have an EDFA that receives about 10 mW and outputs 85 mW. A bandpass filter aligned with our laser wavelength removes the out of band amplified spontaneous emission noise from the signal. Finally, the phase modulated signal is detected using an asymmetric Mach-Zehnder interferometer (a-MZI), and a Discovery Semiconductors DSC740 balanced photodetector with 0.62 A/W responsivity and 26 GHz bandwidth (see Fig. 7). In the linearized ΦMID link, we use the same components, wavelengths, and power levels as in the conventional link, but we add several more components. After amplifying the phase-modulated signal, we tap off 10% and combine it with another CW laser at 1554.79 nm and 17 mW to seed the cascaded FWM comb generation process. This comb source consists of two cascaded spools of highly nonlinear fiber (HNLF), the first of which is 88-m long and is staircase-tensioned in ten steps from low to high tension to suppress Brillouin scattering, and the second of which is 100 m long and has uniform low tension [29]. We operate the EDFA prior to the comb generation at 2 W. After comb generation we isolate the −3 comb line by using an optical bandpassfilter at 1542.94 nm. We introduce the scaling factor by attenuating the −3 signal with the variable optical attenuator (VOA) such that the intermodulation product is minimized. Our experimentally measured scale factor is about 1/24. Note the slight deviation from the theoretical value of 1/27 most likely results from compensation for additional sources of distortion in the link (e.g. the photodetector). Finally we combine this lightwave with the original signal using a WDM filter and use the a-MZI and balanced photodetector to receive the combined signal (see Fig. 8).
Fig. 7 Experimental block diagram of conventional phase-modulated analog optical link. ϕM: phase modulator, MZI: asymmetric Mach-Zehnder interferometer, PD: photodetector.
Fig. 8 Experimental block diagram of the linearized link. EDFA: Erbium doped fiber amplifier, HNLF: highly nonlinear optical fiber, TDL: tunable delay line, −3φ filter: optical bandpass filter, VOA: variable optical attenuator, MZI: asymmetric Mach-Zehnder interferometer, PD: photodetector. Optical spectra at various points in the block diagram. A: Phase-modulated signal; B: Optical comb source output; C: Combined phase-modulated signal with −3φ component filtered from comb line.
For both links, we use a 100-ps path length difference in the a-MZI, which produces periodic dips spaced 10 GHz apart in the frequency response. Here we choose our center frequency to be the first peak near 5 GHz and our bandwidth is expected to be about 8 GHz based on Fig. 6 of [21]. The peak gain of the link is determined primarily by the Vπ of the phase modulator and the current generated at the photodetector. For both links, we use the same 5.9-V Vπ phase modulator and operate with 5-mA on each photodetector in the balanced receiver. We achieve a peak gain of −10.56 dB at 5 GHz in the linearized link and observe asmall 1-dB reduction in the linearized link gain over the conventional link gain, as expected. We use a tunable delay line in the upper branch of the linearized ΦMID link in order to accurately path match it with the lower branch. With matched path lengths in the two branches the link bandwidth is unaffected by the linearization approach.
The noise figure of a phase-modulated link with balanced detection is primarily impacted by thermal noise in the detector, shot noise, and laser phase noise since laser relative intensity noise (RIN) is canceled through balanced detection [21]. We experimentally measure the noise figure of our link to be 39 dB for both the conventional as well as the linearized link. Thus the linearization approach adds negligible noise to the link. This is expected since the lightwave represents the majority of the received power (upper path in Fig. 8) and is not involved in the cascaded FWM interaction. In comparison, the −3 comb line represents only 1/24 of the received power relative to the lightwave and thus any increase in the phase noise on the −3 line minimally impacts the noise figure of the link. Although balanced detection cancels the contribution of RIN to the link noise, the link is sensitive to the phase noise of the laser used. In our setup we use two New Focus 6700 series External Cavity Tunable Diode Lasers with < 200 kHz linewidth. However, diode lasers exhibit relatively large phase noise in the GHz range thus we expect to be able to reduce the noise figure to near the shot noise limited level of 22.5 dB in future work by incorporating fiber lasers into the architecture [21].
We measure the SFDR using a two-tone test in which two microwave tones, one at 5.35 GHz the other at 5.45 GHz, are combined and applied to the input of the phase modulator. For sufficiently high microwave drive powers, spurious intermodulation tones at 5.25 GHz and 5.55 GHz are produced. By sweeping the powers of the input tones, we can determine the link SFDR. Figure 9 shows the experimentally measured fundamental power and intermodulation power as a function of input power for both the conventional link and the linearized link. For the conventional link we obtain an SFDR of 103 dB based on our experimentally measured intermodulation characterization and experimentally measured noise level. With linearization we measure an SFDR of 122 dB, which represents a 19-dB improvement in SFDR for a 1-Hz noise bandwidth. Furthermore, we find that the intermodulation power now scales with the input power to the fifth indicating that the impact of third-order distortion is minimized over the measured power range. The slight increase in slope of the intermodulation power at the lower power levels is due to interference between residual third-order and fifth-order intermodulation products. This interference can be fully eliminated with more precise scaling of the power of the linearization lightwave. Due to the change in slope of the intermodulation power from 3 to 5, the SFDR improvement over the conventional link depends on the signal bandwidth. Figure 10 shows the link SFDR and SFDR improvement as a function of signal bandwidth based on the experimental characterizations.
Fig. 9 SFDR two-tone test (fundamental tones: 5.35 and 5.45 GHz, spurious tone: 5.25 and 5.55 GHz) comparison between conventional (red) and linearized (blue) phase-modulated analog optical link.
Fig. 10 Experimental SFDR versus signal bandwidth for the experimentally characterized conventional link and linearized link. The SFDR improvement is also plotted (dashed line).
5. Summary and conclusion
We have demonstrated a novel receiver-based method for linearizing phase-modulated analog optical links. In this method, we tap off a small part of the modulated signal and use it along with an additional CW laser to seed a cascaded FWM comb source. This source provides lightwaves with integer multiples of the original signal’s phase modulation, which can be scaled and combined with the original signal to linearize the link using a Fourier approach. Experimentally we demonstrate full cancelation of third-order distortion and achieve a 19-dB improvement in link SFDR for a 1-Hz bandwidth. Furthermore, with this method, there is a straightforward path to eliminating higher order distortions as well as generating arbitrary link transfer functions.
Appendix
Here we show how to derive the scaling factors for the Fourier coefficients of a linear transfer function, assuming quadrature bias and thus no even order distortion. Starting with two lightwaves and and using the first two terms of the Taylor expansion, we can readily solve for the scaling factor required for the second lightwave to eliminate the third order distortion and leave only the linear term:
Now, if we add another lightwave, 5 and find the first three terms of the Taylor series, we can eliminate both the third and fifth order distortions. To do this, we must find the scaling factors for both the 3 lightwave (because the scaling factors change as we change the number of lightwaves) as well as 5 lightwave. We can do this by performing the following matrix inversion:
In general, we can remove the first distortions by adding lightwaves, as follows:
Acknowledgments
This work was supported by the Office of Naval Research under Grant N000141210730.
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