1. Introduction

With the inherent wide bandwidth of photonic components, radio frequency (RF) photonics links have been attracting significant interest to military and commercial applications, such as aeronautics and astronautics, radio over fiber (ROF), 5G and beyond systems [1–4]. Spurious-free dynamic range (SFDR) is one of the most important parameters to evaluate the performance of a link. In high-carrier-frequency applications, SFDR is usually limited by the so-called third-order intermodulation distortion (IMD3) component, a nonlinear spur which results from the intrinsical nonlinear transfer function of an external electro-optic modulator [5]. Elimination of such same-frequency interfering is essential for high-quality links, which has been reported in previous literatures. Recently, new trend towards broadband processing capacity has been valued by the drive for future multi-band and multi-purpose remote sensing system, where the carrier frequency should be switched broadly and rapidly from 10 GHz to as large as 40 GHz [1]. Attention should then be paid in order to keep high SFDR while the carrier spans multiple octaves.

In recent proposals, IMD3 can be removed well either in the analog domain or digitally after down-conversion [6–20]. In almost all of the approaches, a second “nonlinear link” is built which is in parallel or series combined with the primary RF photonic link [7, 8]. When an arbitrary RF signal is input, a new-generated nonlinear spur is expected to cancel out exactly the original one. Electronic pre-distortion circuit is low-cost, with however limited SFDR performance in broad bandwidth [9–11]. Combination of electro-optic modulators has the potential to obtain much higher linearization, and SFDR within 1-Hz bandwidth over 130 dB has been reported. The corresponding implementations may show difficulties, though, when the carrier is tuned broadly. In schemes where the second link is driven directly by the input RF signal, e.g. typical setups with two Mach-Zehnder modulators (MZMs) in parallel or series, perfect IMD3 cancelation requires a precise power splitting of RF signals applied to the two modulators, as well as a stable phase relationship. Such signal division depends on half-wave voltage $(V_{\pi })$ or other modulator’s specifications, which are relevant to RF carrier. Preserving a constant power split ratio, loss, and phase retardation under widely changed RF carrier frequency range is a great challenge for current RF devices and cables. Since IMD3 cancelation depends on coherent subtraction of nonlinear spurs from primary and second links, tiny deviation results in large IMD3 remains. That is, though broadband IMD3 cancelation is supported, the second nonlinear link has to be adjusted accordingly when the carrier is tuned; this is very difficult in a real world when the incoming signal is unknown. Therefore, an adjustment-free broadband linearization approach that is totally insensitive to the frequency of the input carrier, is desired in practical scenarios.

A promising way is to perform linearization after nonlinear electro-optic modulation. For example [12], and [13] propose that IMD3 components can be reduced if one properly sets the attenuation and phase shift of the so-called optical carrier band of the modulated lightwave. The above adjustment is independent from input, provided a special all-pass optical filter is achievable. Afterwards digital linearization also works, since the second nonlinear link is software defined with arbitrary flexibility and accuracy [14–19]. Note the digital output limits its application in the receiving channel only. Broadband linearization has also been demonstrated by modulation-combination-based analog processing. For example [20], proposed an inherently broadband linearized modulator where the electronic and optic integration avoids the frequency-dependent parts such as RF power splitter and delay line. In this paper, we propose and experimentally demonstrate an adjustment-free linearization method compatible to large RF carrier tuning. Employing of-the-shelf microwave photonics parts, we use two MZMs in series to obtain feedforward IMD3 elimination in a simple intensity-modulated, direct-detection (IMDD) link. Instead of using multiple modulators, each of which is driven by a RF signal, here the full input is applied on the first modulator. Then by a baseband optical receiver, the key nonlinearity is extracted from the distorted optical intensity, which is used to correct the lightwave distortion through the second modulator. We show in theory that the proposed IMD3 suppression is free from carrier-dependent specifications such as $V_{\pi }$ of modulators, responsivity of photo detector (PD), etc. Experimentally, adjustment-free SFDR is always around 125 dB within 1-Hz bandwidth while the input carrier frequency is tuned from 4 GHz to 12 GHz.

2. Operation principle

Since the reported linearization approaches rely on coherent subtraction of nonlinear spurs from original RF photonics link and the specially designed second nonlinear link, the phase and power deviation of the second link from the optimized value will result in significant IMD3 remains. The relationship between the reduced IMD3 suppression ratio and the phase-power deviation is theoretically estimated and shown in Fig. 1.

 

Fig. 1 The residual IMD3 spurs caused by phase and power deviation.

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The above is simply evaluated by subtracting a unit sinusoid wave from the other one under different power and phase, i.e. $10{\text{log}}_{10}{\left|\sqrt{{\left(1+{\delta }_{\text{power}}\right)}^3}{e}^{i\left(2\pi f_{\text{IMD}3}t+{\delta }_{\text{phase}}\right)}-e^{i\left(2\pi f_{\text{IMD}3}t\right)}\right|}^2$. Note that the IMD3 component is stimulated by fundamental signal and its phase fluctuation follows that of the fundamental, but its power change in decibel is three times that of the fundamental, as shown in the above inline equation. This has been considered in Fig. 1, and the deviations correspond to the input signal. It can be seen that phase and power variation should be kept in a very small range if high IMD3 suppression is desired. For example, in order to get 25 dB suppression, the maximum phase deviation is as small as 3 degrees, while the power change should be kept within about −14 dB. Such performance is hard for current commercial microwave parts, e.g. power divider and cables, if the signal carrier frequency is tuned over multiple octave spans.

Figure 2 shows the proposed broadband adjustment-free feedforward linearization scheme, where two cascaded MZMs are employed, but the second one is driven by the output from the first one, rather than by part of the input RF signal directly.

 

Fig. 2 System configuration of the proposed linearization scheme.

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A continuous-wave (CW) light is firstly modulated by an RF signal, then part of the modulated light is opto-electronically converted and its baseband is amplified, which modulates the other part lightwave after the first MZM. After two intensity modulations, the RF signal is recovered by the second high-speed PD. The operation principle is explained as follows. Assume that the input RF signal is bandpass with tuned carrier, $x\left(t\right)=A\left(t\right)\cos \left[{\omega }_{\text{RF}}\left(t\right)+\phi \left(t\right)\right]$, where ${\omega }_{\text{RF}}$ is the RF angle frequency and $A\left(t\right)$ and $\phi \left(t\right)$ are the amplitude and phase modulation, respectively. Due to the nonlinear transmission curve of the first MZM, the output lightwave intensity involves all orders of harmonics of,$x\left(t\right)$ which can be expressed as $I\left(t\right)=a_0+a_{1}x+a_2{x}^2+a_3{x}^3$, where $a_i$ ($i=0, 1, 2, 3$) are the coefficients of the power series and are determined by the specific parameters of the link. Higher order harmonics than three are ignored due to their minor contributions. In many applications the harmonic components can be simply distinguished in the frequency domain, so here the fundamental RF band around ${\omega }_{\text{RF}}$, denoted as $I_1\left(t\right)$ is considered, as well as the baseband,$I_0\left(t\right)$. According to [16], they are

At a fixed RF carrier frequency, the linearization based on Eq. (2) has been demonstrated in [16] and [21], where the multiplying is realized in the digital domain and in the final high-speed bias-modulated PD, respectively. The digital method involves large quantization noise (e.g. −137.6 dBm/Hz [21]) and unavoidable down-conversion, while the specially-designed PD has limited modulation bandwidth (less than 100 MHz in our previous report). Here, the multiplication is physically realized by opto-electronically converting and low-noise amplifying $I_0\left(t\right)$, which modulates synchronized $I_1\left(t\right)$ by quadrature-biased MZM₂. Since the IMD3 components is usually low-power, the input of low-noise amplifier (LNA) and the modulation depth of MZM₂ is quite small, so that the corresponding nonlinearity is neglected. The linearization operation is the same: in MZM₂ as the second link, the nonlinear part, $A^2\left(t\right)$, in the baseband, $I_0\left(t\right)$, modulates the fundamental output of $I_1\left(t\right)$, which generates new IMD3 spurs; as long as MZM₁ is low-biased, and MZM₂ works on its positive electronic-optic transmission curve, the new-generated intermodulation has inverse phase with the original one; as a result, under an optimized input optical power or link gain of the second link, the new IMD3 cancels out exactly that from MZM₁. Mathematically, the output voltage of baseband receiver, after a DC block, is $\sqrt{G_{\text{LNA}}}{R}_{{\text{PD}}_1}\kappa a_2{A}^2\left(t\right)Z_{{\text{PD}}_1}/4$, where $G_{\text{LNA}}$ is the gain of LNA, $R_{{\text{PD}}_1}$ is responsivity of PD₁, $\kappa$ is loss of optical splitter from the output port of MZM₁ to PD₁, and $Z_{{\text{PD}}_1}$ is the load. Assume a linear intensity modulation of MZM₂, after MZM₂ $I_1\left(t\right)$ is multiplied by $1+\pi /V_{\pi }\sqrt{G_{\text{LNA}}}{R}_{{\text{PD}}_1}\kappa a_2{A}^2\left(t\right)Z_{{\text{PD}}_1}/4$. Compared with Eq. (1), $a_0$ and $a_2$ are updated accordingly, and the linearization condition is changed to

Differently, with off-the-shelf devices, the new scheme linearizes the optically carried analog signal with broadband and adjustment-free operation. Firstly, at a fixed carrier, the second baseband link is easy to have uniform frequency response (i.e. the $S_{21}$ parameter, both in amplitude and in phase) within a few GHz, that is, the power and phase fluctuation is limited under certain tolerance defined by Fig. 1. Such signal bandwidth is sufficient for many sensing radars. Secondly, when the carrier frequency scans, Eq. (2) stands always without adjusting link parameters. According to [16], $a_i$ depends on many link parameters. With a uniform $S_{21}$ of link two, only $V_{\pi }$ of MZM₁ is carrier-dependent. Note that $a_i$ ($i=0, 1, 2, 3$) is in proportional to $1/V_{\pi }^i$, so that (2) will stand for all carriers once it stands for one case. Physically, the correction term, $A^2\left(t\right)$, is generated with IMD3 meanwhile in MZM₁, and their powers are strictly associated according to $V_{\pi }$; different from typical MZM combination, the second link is baseband independent from carrier. As a result, the broadband adjustment-free operation is possible.

Note the above analysis ignores the higher-order nonlinearity. When IMD3 is highly suppressed, others, especially the fifth-order one, become dominant nonlinear spurs. By simulation, we illustrate the comprehensive in Fig. 3. Parameters are listed below. The power of CW light is 20 dBm, with relative intensity noise (RIN) of −155 dBc/Hz. MZM₁ is biased at asin(1/3) (0 accords to the quadrature point), and its insertion loss is 4 dB. In the baseband receiver, the PD responsivity is 1 A/W, the 3-dB bandwidth of PD is 1 GHz, the gain of LNA is 15 dB, and its noise figure is 3 dB. The $V_{\pi }$ of MZM₂ is 3 V, and its insertion loss is 2 dB. The responsivity of the final PD is 1 A/W. When the optical power splitter after MZM₁ is 7:3 where 70% is fed into MZM₂, we find the optimized IMD3 suppression. We assume $V_{\pi }$ of MZM₁ changes with input RF carrier, from 4 V to 8 V, while a dual-sinusoidal RF with 5 dBm per tone simulates the input.

 

Fig. 3 The output fundamental and IMD3 powers (a) before and (b) after the feedforward linearization. (c) and (d) are two linearized RF spectrums when $V_{\pi }$ is 4.5 V and 7.5 V, respectively. The resolution bandwidth (RBW) in simulation is 100 kHz.

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Figure 3(a) shows significantly changed IMD3 powers under different $V_{\pi }$ before the linearization, which is calculated by tuning off LNA. Broadband adjustment-free linearization requires automatic adaption of such variation, which is achieved by our proposal: in Fig. 3(b) where the LNA is ON, the power ratio of the fundamental to IMD3 ratio is kept more than 60 dB during $V_{\pi }$ spanning from 4 V to 8 V. Figure 3(c) and 3(d) show two specific examples when $V_{\pi }$ is 4.5 V and 7.5 V, respectively. It can be seen that when $V_{\pi }$ decreases so that modulation depth gets larger, higher-order nonlinearity gets remarkable. The linearization may fail under further increased modulation depth, since the small-signal approximation used above is no longer correct. However, such modulation depth is usually beyond the upper limit of the predicted dynamic range, and is out of consideration. We here note the compromise for IMD3 correction by the proposed scheme. The signal bandwidth is limited within one octave span, since the second-order production has to be clearly extracted from the baseband receiver. The frequency tuning range is limited by the bandwidth of either MZM₁ or PD₂. Additional power is consumed in the baseband link, especially in its LNA, and additional optical loss is induced by the power splitting after MZM₁ as well as the modulation from MZM₂. The optical loss could be compensated by increasing laser power.

3. Experiment and result

A proof of concept experiment based on the proposed linearization scheme is carried out according to Fig. 1. The CW light is from commercial Koheras AdjustiK Benchtop Fiber Laser, which operates at 1550 nm. Both MZMs are from EOSpace, and their bandwidth is around 20 GHz. Our proposal requires stable working point for both MZMs: low-bias angle for MZM₁ and quadrature point for MZM₂, which is achieved by two commercial bias controllers. The bias angle stability requirement can be estimated according to [16], which shares the similar mathematical equation with the proposed one here. IMD3 can be suppressed more than 40 dB as long as the bias angle drift away the best value is less than ± 2 degrees. Such stability can be easily achieved by our bias controller [22]. The PD as well as the following LNA in the baseband receiver is from New Focus (Model 1611) with a bandwidth of around 1 GHz. Since the power of the baseband signal has a dramatic effect on the elimination of distortion, we use a tunable optical power splitter (from New Focus) after MZM₁ to optimize the baseband output. In practice an optical coupler with fixed splitting ratio and a tunable optical attenuator can be used, which are common fiber devices and much cheaper. Note that splitting ratio should be optimized so that the loss of tunable attenuator could be as small as possible, and the lightwave power in the RF receiver path could be highest under a fixed total optical power. Such optimization is achieved by a tunable power splitter in our proof-of-concept demonstration. In order to get a stable IMD3 correction, the fundamental power should be stabilized according to a certain IMD3 suppression tolerance, as shown in Fig. 1. Such stability is demonstrated by employing a tunable optical coupler. We believe an optical power splitter with fixed ratio and the variable optical attenuator is more mature than tunable coupler, and such stability is then easier to achieve. A matched fiber delay after power splitter is used to synchronize the baseband and distorted signal before their multiplication at MZM₂. We achieve such synchronization by a tunable optical delay line, which could also be obtained in the baseband link by an electronic one with bandwidth compatible with the baseband receiver. Note the optical link for IMD3 correction is in a baseband rather than a high-carrier-frequency band as in previous modulator combination schemes, so that the above electronic part is feasible even if the adjustment-free operation is required. According to Fig. 1, under the given suppression ratio, the phase deviation should be less than 3 degrees, which is around 0.42 ps if the RF frequency is 20 GHz. Such precision could be realized by commercial optical components and by electronic ones. A two-tone RF signal with variable center carrier frequency is used to evaluate the linearization performance.

Firstly, at 10 GHz the linearization is illustrated. When the RF power of each tone is 5 dBm and baseband receiver is OFF, the electrical spectrum (measured by Agilent N9010A with RBW of 100 Hz) after final PD is shown in Fig. 4(a), where considerable IMD3 spurs are observed due to the intrinsic nonlinearity of MZM₁.

 

Fig. 4 The output RF spectrums (a) before and (b) after the proposed linearization. The IMD3 nonlinear spurs are suppressed significantly.

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The fundamental to IMD3 power ratio is 35.1 dB. When the baseband receiver is ON and the ratio of power splitter is optimized, the proposed linearization works so that the fundamental to IMD3 power ratio reaches 59.6 dB, as shown in Fig. 4(b), which is 24.5 dB better. While the input RF power is tuned from 5 to 10 dBm, output powers of both the fundamental and IMD3 are recorded and plotted in Fig. 5, where (a) and (b) are without and with linearization, respectively. By comparing the two plots, one can find that the slope of the measured inter-modulation component which is 3 before the linearization, become to be 5 after the second modulation, which indicates that the third-order nonlinearity is completely suppressed. The mean optical power hitting on the final PD is 0 dBm, and the measured noise floor is −165.8 dBm/Hz. Note that the noise floor does not change after the second baseband modulation. As a result, the SFDR increases from 104.4 to 125.5 dB within 1-Hz bandwidth, leading to an enhancement as large as 21.1 dB.

 

Fig. 5 The measured RF powers of fundamental and intermodulation component versus input RF power (a) before and (b) after the proposed linearization.

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Secondly, the input carrier frequency is tuned from 4 GHz to 12 GHz in order to illustrate the adjustment-free operation. During the carrier tuning, all the link parameters, especially the power splitting ratio of the tunable optical coupler, are unchanged. When the input RF power of each tone is 5 dBm, the output power ratios of fundamental to IMD3 sidebands are measured and calculated as above. It can be seen from Fig. 6(a) that across the 8-GHz bandwidth, the power ratios are kept around 60 dB with IMD3 compensation, which is improved about 25 dB comparing with link without linearization.

 

Fig. 6 When the input RF carrier frequency is tuned from 4 GHz to 12 GHz, the measured (a) fundamental to IMD3 ratios and (b) SFDR when the linearization is ON and OFF, respectively.

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Such uniform IMD3 suppression is consistent with simulation. Without further link optimization, the SFDR performance under varied carrier frequency is also measured and compared with un-linearized link. Over 8-GHz bandwidth, the SFDR is kept as high as 125 dB within 1-Hz bandwidth. The result shows that our proposed linearization approach is capable of adjustment-free uniform and high dynamic range when carrier frequency is broadly tuned.

4. Conclusion

In conclusion, we have proposed and experimentally demonstrated a novel feedforward linearization approach for wideband-working RF photonic link, which is capable of realizing uniform high dynamic range during RF carrier tuning. Though the scheme is based on cascaded MZMs, the distortion is extracted after the first modulator in baseband and then is used to linearize the signal in the optically-carried RF band. Such an implementation ensures an adjustment-free intermodulation distortion cancellation when carrier is changing. Experimentally we achieved a uniform 25-dB IMD3 suppression and 125 dB SFDR in 1-Hz bandwidth when the RF carrier was changed from 4 GHz to 8 GHz, and no link optimization was performed during the tuning. Our approach may benefit future applications in frequency-agile scenarios, e.g. multi-band and multi-purpose radar.