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SECTION I

Analog microwave photonic links are used in various applications including broadband wireless access networks, sensor networks, radar, satellite communications, instrumentation, and warfare systems [1], due to the advantages of large bandwidth, reduced attenuation, size, weight, and immunity to electromagnetic interference. In most of these links, the RF signal is externally modulated with a Mach–Zehnder modulator (MZM) based on different modulation schemes including optical double-sideband (ODSB), optical carrier suppression (OCS), and optical single-sideband (OSSB) [2], [3]. Among them, OSSB modulation is more preferable in analog microwave photonic links since it can effectively eliminate the fading effect [4] and produce higher bandwidth efficiency [5].

In the conventional OSSB modulation scheme, nonlinear distortion such as harmonic distortion and intermodulation distortion will be generated due to the nonlinear transfer function of the MZM, which limits the overall link spurious-free dynamic range (SFDR). The third-order intermodulation distortion (IMD3) is the most determining cause of distortion since it is in close proximity to the fundamental signal and difficult to filter out. In order to eliminate the IMD3 and enhance the SFDR of the link, many schemes have been presented. The general idea is to introduce certain predistortion to compensate for the existing ones using complex modulators, such as dual-drive MZM (DD-MZM) with fiber Bragg grating (FBG) [6], integrated dual MZM [7], mixed polarization DD-MZM [8], dual-parallel MZM (DP-MZM) [9], and electro-optic polymeric DP-MZM [10]. However, those methods are difficult to control at arbitrary operating points and increase the complexity of transmitter. Up to now, a little attention has been paid on postcompensation for the nonlinearity of modulators directly in the optical fields. However, these approaches are limited to ODSB modulation [11], [12], phase modulation [13], [14], or frequency modulation [15].

In this paper, a postcompensation scheme for nonlinearity of OSSB modulation analog microwave photonic link is proposed and demonstrated. At the transmit side of the link, OSSB modulation is realized by using a DD-MZM [16]. At the receive end, IMD3 is suppressed by utilizing two parallel interferometers. This architecture greatly simplifies the transmitter. The theoretical derivation of SFDR for our linearized link is presented and the comparison between our scheme and conventional OSSB modulation is carried out. Finally, the bandwidth over which our linearized link still out-performs a conventional OSSB modulation link is discussed.

SECTION II

Fig. 1 shows the proposed architecture for linearization of an OSSB modulation microwave photonic link. Two RF signals are optically modulated by a DFB laser with a DD-MZM, but they are 90° phase shift from each other and the MZM is biased at quadrature. Thus, an OSSB modulation signal is generated. Postmodulator the OSSB signal is equally divided into two parts by a 3 dB optical splitter, and drives two parallel interferometers with different differential delays. The differencing operation is performed with a balanced detector to suppress the IMD3. In order to make phase matching of IMD3 between two paths, a variable fiber delay line is used before each photodiode (PD) on the two demodulation paths.

According to the transfer function of DD-MZM, the optical field at the output of modulator can be quantified by TeX Source $$E_{out}(t) = {E_{c}e^{jw_{c}t} \over 2}\left[e^{j\varphi (t)} + je^{j\varphi^{\prime}(t)}\right]\eqno{\hbox{(1)}}$$ where TeX Source $$\eqalignno{\varphi(t) = &\, {\sqrt{2} \over 2}m \left[\cos(w_{1}t) + \cos(w_{2}t)\right]&\hbox{(2)} \cr \varphi^{\prime}(t) = &\, {\sqrt{2} \over 2}m\left[\sin (w_{1}t) + \sin(w_{2}t)\right]&\hbox{(3)}}$$ are the optical phase shift applied to each arms of modulator for a two-tone measurement, $E_{c}$ and $w_{c}$ are the lightwave amplitude and angular frequency, respectively, $m = \pi V/V_{\pi}$ is the modulation index of MZM, $V = \sqrt{2P_{rf}R_{i}}$ is the amplitude of RF input signal, $P_{rf}$ is the RF input power, $R_{i}$ is the input resistance of the modulator, $V_{\pi}$ is the half-wave voltage of modulator, $w_{1}$ and $w_{2}$ are the angular frequencies of RF input signals.

By using the Jacobi-Auger expansions $e^{jm\cos(wt)} = \sum_{n = - \infty}^{\infty}j^{n}J_{n}(m)e^{jnwt}$ and $e^{jm\sin(wt)} = \sum_{n = - \infty}^{\infty}J_{n}(m)e^{jnwt}$, (1) can be further expanded as TeX Source $$\eqalignno{E_{out}(t) = &\, {E_{c}e^{jw_{c}t} \over 2}\left[\sum_{n = - \infty}^{\infty} j^{n}J_{n}\left({\sqrt{2} \over 2}m\right)e^{jnw_{1}t} \sum_{k = - \infty}^{\infty}j^{k}J_{k}\left({\sqrt{2} \over 2}m\right)e^{jkw_{2}t}\right.\cr&{\hskip45pt} + \left. j\sum_{n = - \infty}^{\infty}J_{n}\left({\sqrt{2} \over 2}m\right)e^{jnw_{1}t}\sum_{k = - \infty}^{\infty}J_{k}\left({\sqrt{2} \over 2}m\right)e^{jkw_{2}t}\right]\cr = &\, {E_{c}e^{jw_{c}t} \over 2}\sum_{n = - \infty}^{\infty}\sum_{k = - \infty}^{\infty}(j^{n + k} + j)J_{n}\left({\sqrt{2} \over 2}m\right)J_{k}\left({\sqrt{2} \over 2}m\right)e^{j(nw_{1} + kw_{2})t}.&\hbox{(4)}}$$ It is shown from (4)that only the upper first-order sideband exists. Thus, an OSSB modulation signal is generated.

After interferometric demodulation, the output photocurrent from the up-path can be obtained as TeX Source $$\eqalignno{I_{1}(t) = &\, R{\sqrt{2} \over 8}E_{c}e^{jw_{c}t} \left[e^{- jw_{c}\tau_{1}} \left(e^{j\varphi(t - \tau_{1})} + je^{j\varphi^{\prime}(t - \tau_{1})}\right) - (e^{j \varphi (t)} + je^{j\varphi^{\prime}(t)})\right]\cr\noalign{\vskip-1pt} &\times {\sqrt{2} \over 8}E_{c}e^{- jw_{c}t}\left[e^{jw_{c}\tau_{1}}\left(e^{- j\varphi(t - \tau_{1})} - je^{- j\varphi^{\prime}(t - \tau_{1})}\right) - \left(e^{- j\varphi(t)} - je^{- j\varphi^{\prime}(t)}\right)\right]\cr\noalign{\vskip-1pt} = &\, {RE_{c}^{2} \over 16}\left\{2 + J_{0}(m)\sum_{n = 1}^{\infty}\left[ - 1 + (- 1)^{n}\right]j^{n + 1}J_{n}(m)\cos\left[nw_{1}(t - \tau_{1}) + {n\pi \over 4}\right]\right.\cr\noalign{\vskip-1pt}&{\hskip32pt} + J_{0}(m)\sum_{k = 1}^{\infty} \left[ - 1 + (- 1)^{k}\right]j^{k + 1}J_{k}(m)\cos \left[kw_{2}(t - \tau_{1}) + {k\pi \over 4}\right]\cr\noalign{\vskip-1pt}&{\hskip32pt} + 2\sum_{n = 1}^{\infty}\sum_{k = 1}^{\infty}\left[ - 1 + (- 1)^{n + k}\right]j^{n + k + 1}J_{n}(m)J_{k}(m)\cr\noalign{\vskip-1pt}&{\hskip32pt}\times\cos\left[nw_{1}(t - \tau_{1}) + {n\pi \over 4} \right]\cos\left[kw_{2}(t - \tau_{1}) + {k\pi \over 4}\right]\cr\noalign{\vskip-1pt} &{\hskip32pt} - \sum_{n = - \infty}^{\infty}\sum_{k = - \infty}^{\infty}J_{n}\left(\sqrt{2}m\sin\left({w_{1}\tau_{1} \over 2}\right)\right) J_{k}\left(\sqrt{2}m\sin\left({w_{2}\tau_{1} \over 2}\right)\right)\cr\noalign{\vskip-1pt}&{\hskip32pt}\times \cos\left[n \left(w_{1}t - {w_{1}\tau_{1} \over 2}\right) + k\left(w_{2}t - {w_{2}\tau_{1} \over 2}\right) - w_{c} \tau_{1}\right]\cr\noalign{\vskip-1pt} &{\hskip32pt} - \sum_{n = - \infty}^{\infty}\sum_{n = - \infty}^{\infty}J_{n}\left(\sqrt{2}m\sin\left(- {w_{1}\tau_{1} \over 2} + {\pi \over 4}\right)\right)J_{k}\left(\sqrt{2}m \sin\left(- {w_{2}\tau_{1} \over 2} + {\pi \over 4}\right)\right)\cr&{\hskip32pt}\times\sin\left[n\left(w_{1}t - {w_{1}\tau_{1} \over 2} + {3\pi \over 4}\right) - k \left(w_{2}t - {w_{2}\tau_{1} \over 2} - {\pi \over 4}\right) - w_{c}\tau_{1}\right]\cr &{\hskip32pt} + \sum_{n = - \infty}^{\infty}\sum_{n = - \infty}^{\infty}J_{n}\left(\sqrt{2}m\cos\left(- {w_{1}\tau_{1} \over 2} + {\pi \over 4}\right)\right)J_{k}\left(\sqrt{2}m\cos\left(- {w_{2} \tau_{1} \over 2} + {\pi \over 4}\right)\right)\cr&{\hskip32pt} \times \sin\left[n\left(w_{1}t - {w_{1}\tau_{1} \over 2} - {\pi \over 4}\right) + k \left(w_{2}t - {w_{2}\tau_{1} \over 2} - {\pi \over 4}\right) - w_{c}\tau_{1} \right]\cr &{\hskip32pt} - J_{0}\left(\sqrt{2}m\sin\left({w_{1}\tau_{1} \over 2}\right)\right)J_{0} \left(\sqrt{2}m\sin\left({w_{2}\tau_{1} \over 2}\right)\right) \cos w_{c}\tau_{1}\cr&{\hskip32pt} - 2J_{0}\left(\sqrt{2}m\sin\left({w_{1}\tau_{1} \over 2}\right)\right)\sum_{k = 1}^{\infty}J_{k}\left(\sqrt{2}m\sin\left({w_{2} \tau_{1} \over 2}\right)\right)\cr \times \cos \left[k\left(w_{2}t - {w_{2}\tau_{1} \over 2}\right)\right] \cos \left(w_{c}\tau_{1} + {k\pi \over 2}\right)\cr &{\hskip32pt} - 2J_{0}\left(\sqrt{2}m\sin \left({w_{2}\tau_{1} \over 2}\right)\right)\sum_{n = 1}^{\infty}J_{n}\left(\sqrt{2}m\sin\left({w_{1}\tau_{1} \over 2}\right)\right) \cr&{\hskip32pt} \times \cos \left[n\left(w_{1}t - {w_{1}\tau_{1} \over 2}\right)\right]\cos \left(w_{c}\tau_{1} + {n\pi \over 2}\right)\cr &{\hskip32pt} - 4\sum_{n = 1}^{\infty}\sum_{k = 1}^{\infty}(- j)^{n + k}J_{n}\left(\sqrt{2}m\sin\left({w_{1}\tau_{1} \over 2}\right)\right)J_{k} \left(\sqrt{2}m\sin\left({w_{2}\tau_{1} \over 2} \right)\right)\cr&{\hskip32pt} \times \cos\left[n\left(w_{1}t - {w_{1}\tau_{1} \over 2}\right)\right] \cos \left[k\left(w_{2}t - {w_{2}\tau_{1} \over 2}\right)\right]\cos\left[w_{c}\tau_{1} + {(k + n)\pi \over 2}\right]\cr &{\hskip32pt} + J_{0}(m)\sum_{k = 1}^{\infty}\left[ - 1 + (- 1)^{k}\right]j^{k + 1}J_{k}(m) \cos\left[kw_{2}t + {k\pi \over 4}\right]\cr&{\hskip32pt} + J_{0}(m)\sum_{n = 1}^{\infty}\left[ - 1 + (- 1)^{n}\right]j^{n + 1}J_{n}(m)\cos\left[nw_{1}t + {n\pi \over 4}\right]\cr &{\hskip32pt} + \left.2\sum_{n = 1}^{\infty}\sum_{k = 1}^{\infty}\left[ - 1 + (- 1)^{n + k}\right] j^{n + k + 1}J_{n}(m)J_{k}(m)\cos\left[nw_{1}t + {n\pi \over 4}\right] \cos \left[kw_{2}t + {k\pi \over 4}\right]\right\}\quad&\hbox{(5)}}$$ where $R$ is the responsivity of photodiode, $\tau_{1}$ is the differential delay of the MZI1, $J_{n}$ and $J_{k}$ are Bessel functions of the first kind. From (5), the amplitude of the fundamental response for up-path can be obtained as TeX Source $$\eqalignno{I_{1, w1} = &\, {R \over 8}E_{c}^{2}\left\{2J_{0}(m)J_{1}(m)\cos\left({w_{1}\tau_{1} \over 2}\right) + \sqrt{2}J_{0}\left(\sqrt{2}m\sin{w_{2}\tau_{1} \over 2}\right)J_{1}\left(\sqrt{2}m\sin{w_{1}\tau_{1} \over 2} \right)\sin(w_{c}\tau_{1})\right.\cr&{\hskip30pt} - J_{0}\left[\sqrt{2}m\sin\left(- {w_{2}\tau_{1} \over 2} + {\pi \over 4}\right)\right]J_{1}\left[\sqrt{2}m\sin\left(- {w_{1}\tau_{1} \over 2} + {\pi \over 4}\right)\right] \cos(w_{c}\tau_{1})\cr&{\hskip30pt} - \left. J_{0}\left[\sqrt{2}m\cos\left(- {w_{2}\tau_{1} \over 2} + {\pi \over 4}\right)\right]J_{1}\left[\sqrt{2}m\cos\left(- {w_{1}\tau_{1} \over 2} + {\pi \over 4}\right)\right] \cos(w_{c}\tau_{1})\right\}\cr&{\hskip30pt} \times \cos\left(w_{1}t - {w_{1}\tau_{1} \over 2} + {\pi \over 4} \right).&\hbox{(6)}}$$the in-band third-order intermodulation products are the limiting distortion components, the amplitude of this distortion term can be obtained as TeX Source $$\eqalignno{I_{1, 2w_{1} - w_{2}} = &\, {R \over 8}E_{c}^{2} \left\{- 2J_{1}(m)J_{2}(m)\cos\left(- w_{1}\tau_{1} + {w_{2}\tau_{1} \over 2}\right)\right.\cr&{\hskip32pt} - \sqrt{2}J_{1}\left[\sqrt{2}m\sin \left({w_{2}\tau_{1} \over 2}\right)\right]J_{2} \left[\sqrt{2}m\sin\left({w_{1} \tau_{1} \over 2}\right)\right]\sin(w_{c}\tau_{1})\cr&{\hskip32pt} + J_{1}\left[\sqrt{2}m\sin\left(- {w_{2} \tau_{1} \over 2} + {\pi \over 4}\right)\right]J_{2}\left[\sqrt{2}m\sin\left(- {w_{1}\tau_{1} \over 2} + {\pi \over 4}\right)\right] \cos (w_{c}\tau_{1})\cr&{\hskip32pt} + \left. J_{1}\left[\sqrt{2}m\cos \left(- {w_{2}\tau_{1} \over 2} + {\pi \over 4}\right)\right]J_{2}\left[\sqrt{2}m\cos \left(- {w_{1}\tau_{1} \over 2} + {\pi \over 4} \right)\right]\cos (w_{c}\tau_{1})\right\}\cr&{\hskip32pt} \times \cos\left(2w_{1}t - w_{2}t - w_{1}\tau_{1} + {w_{2}\tau_{1} \over 2} + {\pi \over 4}\right).&\hbox{(7)}}$$ In order to better understand the principle of our linearization scheme, a small-signal approximation $m \ll 1$ is considered. Thus, equation (6) can be further simplified as TeX Source $$\eqalignno{I_{1, w1} = &\, {R \over 8}E_{c}^{2}\left\{\cos \left({w_{1}\tau_{1} \over 2}\right)\left[1 - \cos(w_{c}\tau_{1})\right] + \sin \left({w_{1}\tau_{1} \over 2}\right) \sin (w_{c}\tau_{1})\right\}m\cos\left(w_{1}t - {w_{1}\tau_{1} \over 2} + {\pi \over 4}\right)\cr = &\, A_{1, 1}m\cos \left(w_{1}t - {w_{1}\tau_{1} \over 2} + {\pi \over 4}\right).&\hbox{(8)}}$$ Following the same method, equation (7) can be further simplified as: TeX Source $$\eqalignno{I_{1, 2w_{1} - w_{2}} = &\, {R \over 8}E_{c}^{2} \left\{\left[ - {\cos \left(- w_{1}\tau_{1} + {w_{2}\tau_{1} \over 2}\right)\over 8} - {\sin\left({w_{2}\tau_{1} \over 2}\right)\sin^{2}\left({w_{1}\tau_{1} \over 2}\right) \over 4}\sin(w_{c}\tau_{1})\right.\right.\cr\noalign{\vskip3pt}&{\hskip40pt} + {\sqrt{2}\sin \left(- {w_{2}\tau_{1} \over 2} + {\pi \over 4}\right)\sin^{2}\left(- {w_{1}\tau_{1} \over 2} + {\pi \over 4}\right) \over 8}\cos (w_{c}\tau_{1})\cr\noalign{\vskip3pt}&{\hskip40pt} + \left.{\sqrt{2}\cos \left(- {w_{2}\tau_{1} \over 2} + {\pi \over 4}\right)\cos^{2}\left(- {w_{1}\tau_{1} \over 2} + {\pi \over 4}\right) \over 8}\cos (w_{c}\tau_{1})\right]m^{3}\cr\noalign{\vskip3pt}&{\hskip31pt} + \left[{5 \over 192}\cos \left(- w_{1}\tau_{1} + {w_{2}\tau_{1} \over 2}\right) + {\sin \left({w_{2}\tau_{1} \over 2}\right) \sin^{4} \left({w_{1}\tau_{1} \over 2}\right) \over 24}\sin (w_{c}\tau_{1})\right.\cr\noalign{\vskip3pt}&{\hskip49pt} + {\sin^{3}\left({w_{2}\tau_{1} \over 2}\right)\sin^{2}\left({w_{1}\tau_{1} \over 2}\right) \over 16} \sin(w_{c}\tau_{1})\cr\noalign{\vskip3pt}&{\hskip49pt} - {\sqrt{2}\sin \left(- {w_{2}\tau_{1} \over 2} + {\pi \over 4}\right)\sin^{4}\left(- {w_{1}\tau_{1} \over 2} + {\pi \over 4}\right) \over 48}\cos (w_{c}\tau_{1})\cr \noalign{\vskip3pt}&{\hskip49pt} - {\sqrt{2}\sin^{3}\left(- {w_{2}\tau_{1} \over 2} + {\pi \over 4}\right) \sin^{2}\left(- {w_{1}\tau_{1} \over 2} + {\pi \over 4}\right) \over 32}\cos (w_{c}\tau_{1})\cr\noalign{\vskip3pt}&{\hskip49pt} - {\sqrt{2}\cos \left(- {w_{2}\tau_{1} \over 2} + {\pi \over 4}\right)\cos^{4}\left(- {w_{1}\tau_{1} \over 2} + {\pi \over 4}\right) \over 48}\cos (w_{c}\tau_{1})\cr\noalign{\vskip3pt}&{\hskip49pt} - \left.\left.{\sqrt{2}\cos^{3}\left(- {w_{2}\tau_{1} \over 2} + {\pi \over 4}\right)\cos^{2}\left(- {w_{1} \tau_{1} \over 2} + {\pi \over 4}\right) \over 32}\cos (w_{c}\tau_{1})\right]m^{5}\right\}\cr\noalign{\vskip3pt}&{\hskip49pt} \times \cos \left(2w_{1}t - w_{2}t - w_{1}\tau_{1} + {w_{2}\tau_{1} \over 2} + {\pi \over 4}\right)\cr\noalign{\vskip3pt} = &\, (B_{1, 3}m^{3} + B_{1, 5}m^{5}) \times \cos \left(2w_{1}t - w_{2}t - w_{1}\tau_{1} + {w_{2}\tau_{1} \over 2} + {\pi \over 4}\right).&\hbox{(9)}}$$ In this paper, the third-order term in equation (9) will be suppressed, which is the dominant distortion.

The output photocurrents from the two arms of MZI are $\pi$ out of phase. Thus, following the same method, the small-signal fundamental photocurrent for bottom-path can be obtained as: TeX Source $$\eqalignno{I_{2, w1} = &\, {R \over 8}E_{c}^{2} \left\{\cos\left({w_{1}\tau_{2} \over 2}\right)\left[1 + \cos(w_{c}\tau_{2})\right] - \sin\left({w_{1}\tau_{2} \over 2}\right)\sin(w_{c}\tau_{2})\right\}m\cos\left(w_{1}t - {w_{1}\tau_{2} \over 2} + {\pi \over 4}\right)\cr\noalign{\vskip3pt} = &\, A_{2, 1}m\cos\left(w_{1}t - {w_{1}\tau_{2} \over 2} + {\pi \over 4}\right)&\hbox{(10)}}$$ where $\tau_{2}$ is the differential delay of the MZI2.

The small-signal third-order intermodulation distortion (IMD3) photocurrent for bottom-path can be obtained as TeX Source $$\eqalignno{I_{2, 2w_{1} - w_{2}} = &\, {R \over 8}E_{c}^{2} \left\{\left[ - {\cos\left(- w_{1}\tau_{2} + {w_{2}\tau_{2} \over 2}\right) \over 8} + {\sin\left({w_{2}\tau_{2} \over 2}\right)\sin^{2}\left({w_{1}\tau_{2} \over 2}\right) \over 4}\sin(w_{c}\tau_{2})\right.\right.\cr\noalign{\vskip3pt}&{\hskip40pt} - {\sqrt{2}\sin \left(- {w_{2}\tau_{2} \over 2} + {\pi \over 4}\right)\sin^{2}\left(- {w_{1}\tau_{2} \over 2} + {\pi \over 4}\right) \over 8}\cos \left(w_{c}\tau_{2}\right)\cr\noalign{\vskip3pt}& {\hskip40pt} - \left.{\sqrt{2}\cos \left(- {w_{2}\tau_{2} \over 2} + {\pi \over 4}\right)\cos^{2}\left(- {w_{1} \tau_{2} \over 2} + {\pi \over 4}\right) \over 8}\cos (w_{c}\tau_{2})\right]m^{3}\cr &+ \left[{5 \over 192}\cos \left(- w_{1}\tau_{2} + {w_{2}\tau_{2} \over 2}\right) - {\sin\left({w_{2}\tau_{2} \over 2}\right)\sin^{4} \left({w_{1}\tau_{2} \over 2}\right) \over 24}\sin(w_{c}\tau_{2})\right.\cr\noalign{\vskip-3pt}&{\hskip53pt} - {\sin^{3}\left({w_{2}\tau_{2} \over 2}\right)\sin^{2}\left({w_{1}\tau_{2} \over 2}\right) \over 16}\sin(w_{c} \tau_{2})\cr\noalign{\vskip-3pt}&{\hskip53pt} + {\sqrt{2}\sin\left(- {w_{2}\tau_{2} \over 2} + {\pi \over 4} \right)\sin^{4}\left(- {w_{1}\tau_{2} \over 2} + {\pi \over 4}\right) \over 48}\cos(w_{c}\tau_{2})\cr\noalign{\vskip-3pt}&{\hskip53pt} + {\sqrt{2}\sin^{3}\left(- {w_{2}\tau_{2} \over 2} + {\pi \over 4}\right)\sin^{2} \left(- {w_{1}\tau_{2} \over 2} + {\pi \over 4}\right) \over 32}\cos (w_{c}\tau_{2})\cr\noalign{\vskip-3pt}&{\hskip53pt} + {\sqrt{2}\cos\left(- {w_{2}\tau_{2} \over 2} + {\pi \over 4}\right)\cos^{4}\left(- {w_{1}\tau_{2} \over 2} + {\pi \over 4}\right) \over 48}\cos (w_{c}\tau_{2})\cr\noalign{\vskip-3pt}&{\hskip53pt} + \left. \left.{\sqrt{2}\cos^{3}\left(- {w_{2}\tau_{2} \over 2} + {\pi \over 4}\right)\cos^{2}\left(- {w_{1}\tau_{2} \over 2} + {\pi \over 4}\right) \over 32}\cos (w_{c}\tau_{2})\right]m^{5}\right\}\cr\noalign{\vskip-3pt}&{\hskip53pt} \times \cos \left(2w_{1}t - w_{2}t - w_{1}\tau_{2} + {w_{2}\tau_{2} \over 2} + {\pi \over 4}\right)\cr\noalign{\vskip-3pt} = &\, (B_{2, 3}m^{3} + B_{2, 5}m^{5})\cos \left(2w_{1}t - w_{2}t - w_{1}\tau_{2} + {w_{2}\tau_{2} \over 2} + {\pi \over 4}\right).&\hbox{(11)}}$$

From (9) and (11), in order to make phase matching of IMD3 between two paths, the time delay of fiber delay line on the bottom-path should satisfy the condition TeX Source $$\tau = {1 \over 2}(\tau_{1} - \tau_{2}).\eqno{\hbox{(12)}}$$

Furthermore, the third-order terms have equal amplitude when the coefficients $B_{1, 3}$ and $B_{2, 3}$ satisfy the condition TeX Source $$10^{- {\alpha \over 10}}B_{1, 3} = B_{2, 3}\eqno{\hbox{(13)}}$$ where $\alpha$ is the power attenuation for up-path.

Thus, the third-order terms of IMD3 have equal intensity and phase. After the differencing operation is performed, these two kinds of IMD3 terms will cancel each other out while the fundamental term still exists. In the circumstances, the fifth-order terms at frequencies $2w_{1} - w_{2}$ and $2w_{2} - w_{1}$ are the dominant distortion components.

The spurious-free dynamic range of link is first determined by the output N th order intercept point, which can be written as TeX Source $$OIP_{N} = \left({P_{1}^{N} \over P_{N}}\right)^{1/(N - 1)}\eqno{\hbox{(14)}}$$ where $P_{1}$ and $P_{N}$ are the fundamental and $N$th order distortion output power, respectively.Then, the $N$th order limited SFDR can be expressed as TeX Source $$SFDR_{N} = \left({OIP_{N} \over N_{o}}\right)^{(N - 1)/N}\eqno{\hbox{(15)}}$$ where $N_{o}$ is the output noise power spectral density per unit bandwidth (W/Hz).

Taking expressions of output fundamental and distortion power into equation (14), the fifth-order intercept point of our linearization link can be written as TeX Source $$OIP_{5}^{(2w_{1} - w_{2})} = {\left(10^{- {\alpha \over 10}}A_{1, 1} - A_{2, 1}\right)^{5/2} \over 2 \left(10^{- {\alpha \over 10}}B_{1, 5} - B_{2, 5}\right)^{1/2}}R_{o}\eqno{\hbox{(16)}}$$ where $R_{o}$ is the photodiode output resistance.

Substituting (16) into (15) and assuming shot noise is the dominant noise contributions, the fifth order limited SFDR is obtained as TeX Source $$SFDR_{5}^{(2w_{1} - w_{2})} = \left[{\left(10^{- {\alpha \over 10}}A_{1, 1} - A_{2, 1}\right)^{5/2} \over 4qR\left(10^{- {\alpha \over 10}}B_{1, 5} - B_{2, 5}\right)^{1/2}P_{r}} \right]^{4/5}\eqno{\hbox{(17)}}$$ where TeX Source $$P_{r} = {E_{c}^{2} \over 8}\left\{10^{- {\alpha \over 10}}\left[1 - \cos (w_{c}\tau_{1})\right] + 1 + \cos (w_{c}\tau_{2}) \right\}\eqno{\hbox{(18)}}$$ is the total optical power at the photodiode, $q = 1.6 \times 10^{- 19}\ \hbox{C}$ is the electron charge.

If the link does not operate at the desired frequency, the coefficients ratio given by (13) will not be satisfied. In this case, the third order contribution to the IMD3 becomes dominant. Following the approach presented before, the third-order intercept point and SFDR can be expressed as TeX Source $$\eqalignno{OIP_{3}^{(2w_{1} - w_{2})} = &\, {\left(10^{- {\alpha \over 10}}A_{1, 1} - A_{2, 1}\right)^{3} \over 2\left(10^{- {\alpha \over 10}}B_{1, 3} - B_{2, 3}\right)}R_{o}&\hbox{(19)}\cr SFDR_{3}^{(2w_{1} - w_{2})} = &\, \left[{\left(10^{- {\alpha \over 10}}A_{1, 1} - A_{2, 1}\right)^{3} \over 4qR \left(10^{- {\alpha \over 10}}B_{1, 3} - B_{2, 3}\right)P_{r}} \right]^{2/3}.&\hbox{(20)}}$$

Since our linearized architecture requires phase matching to achieve the desired IMD3 suppression, any slight relative delay introduced between the two parallel interferometers or small changes in the time delays of the MZIs will make SFDR converted from fifth-order-limited to third-order-limited. In this case, the third-order intercept point and third-order-limited SFDR can be written as TeX Source $$\eqalignno{OIP_{3}^{(2w_{1} - w_{2})} = &\, \left\{{\left[\left(10^{- {\alpha \over 10}}A_{1, 1}\right)^{2} + A_{_{2, 1}}^{2}\right]^{3} \over 4\left[\left(10^{- {\alpha \over 10}}B_{1, 3}\right)^{2} + B_{_{2, 3}}^{2}\right]} \right\}^{1 \over 2}R_{o}&\hbox{(21)}\cr SFDR_{3}^{(2w_{1} - w_{2})} = &\, {\left(10^{- {\alpha \over 10}}A_{1, 1}\right)^{2} + A_{_{2, 1}}^{2} \over \left\{(4qRP_{r})^{2}\left[\left(10^{- {\alpha \over 10}}B_{1, 3} \right)^{2} + B_{_{2, 3}}^{2}\right]\right\}^{1 \over 3}}.&\hbox{(22)}}$$

SECTION III

To demonstrate the operation for linearization of an OSSB modulation microwave photonic link, a proof-of-concept simulation is constructed with the commercial software OptiSystem, for a two-tone investigation [see Fig. 1]. The simulation parameters for the setup are given in Table 1.

In order to explain the principle of our linearized scheme, the fundamental and IMD3 responses for both up and bottom paths are simulated independently, as shown in Fig. 2. Here, the simulated responses of the link employing $\tau_{1} = 100\ \hbox{ps}$ MZI1 are shown by the squares and those of the link constructed with the $\tau_{2} = 75\ \hbox{ps}$ MZI2 are shown by the circles. It can be seen that the IMD3 power of up and bottom paths are identical, while the fundamental power of bottom path is higher than that of the up path. When the third-order terms of IMD3 have identical phase, they will be eliminated and the fundamental power of bottom path will become dominant after differenced with a balanced detector.

In order to show the suppression of IMD3 by using our proposed scheme over the conventional OSSB modulation scheme, a microwave photonic link using the conventional OSSB modulation is also built for comparison, and the main simulation parameters are the same as mentioned in Table 1. Fig. 3 shows the RF output spectrum for conventional OSSB modulation link and our highly linearized link. It can be seen from Fig. 3(a)that the fundamental-to-IMD3 ratio (FIR) is only 25 dB using the conventional OSSB modulation scheme. However, it is obviously seen from Fig. 3(b)that the FIR is about 49 dB and other distortion components are greatly suppressed.

To further demonstrate the superiority of our linearized link, we compare its SFDR to that of traditional OSSB link for the identical received optical power, as shown in Fig. 4. It can be seen that our linearized link is obviously limited by the fifth-order term of IMD3. The simulated fifth-order intercept of approximately $OIP_{5}^{(2w_{1} - w_{2})} = 3.9\ \hbox{dBm}$ and shot noise level per unit bandwidth of $N_{o} = - 161.1\ \hbox{dBm/Hz}$ yield a fifth-order-limited spurious-free dynamic range of $SFDR_{5}^{(2w_{1} - w_{2})} =132\ \hbox{dB}$ (extrapolated to 1 Hz bandwidth). Both the intercept point and SFDR match very well with the theoretical results shown in (16) and (17) which are $OIP_{5}^{(2w_{1} - w_{2})} = 3.4\ \hbox{dBm}$ and $SFDR_{5}^{(2w_{1} - w_{2})} = 131.5\ \hbox{dB}$, respectively. For comparison, the simulated third-order intercept and spurious-free dynamic range of the conventional OSSB link operating at the same total received optical power are $OIP_{3}^{(2w_{1} - w_{2})} =7.2\ \hbox{dBm}$ and $SFDR_{3}^{(2w_{1} - w_{2})} = 112.1\ \hbox{dB}$ for a bandwidth of 1 Hz. Thus, the SFDR of our linearized link is approximately 20 dB higher than that of the traditional OSSB link at the identical received optical power.

The linearized scheme relies upon differencing two properly adjusted third-order terms of IMD3 resulting from two frequency-dependent MZIs. In order to explain the frequency-dependence of our scheme, the power attenuation for up-path is fixed so that the link is linearized at $w/2\pi = 1.01\ \hbox{GHz}$ and alter the frequencies of two RF input signals between 200 MHz and 2 GHz (maintain the power and frequency spacing of RF input signals). Circles in Fig. 5 represent the simulated SFDR as a function of the offset frequency from $w/2\pi = 1.01\ \hbox{GHz}$. Solid line and dashed line represent the calculated third-order [see (20)] and fifth-order [see (17)] limited SFDR of our linearized scheme, respectively. While dotted line represents the third-order limited SFDR of a conventional OSSB link. It can be seen obviously that the SFDR is third-order limited for almost all frequencies, except for the desired RF input signal frequency where it arrives at maximal value of $SFDR_{5}^{(2w_{1} - w_{2})} = 132\ \hbox{dB}$. However, the SFDR is greater than the $SFDR_{3}^{(2w_{1} - w_{2})} = 112.1\ \hbox{dB}$ of a traditional OSSB scheme in a frequency range of $\sim$8.8 GHz.

In real systems, the time delays of MZI1 and MZI2 may deviate from 100 ps and 75 ps, respectively. These conditions will result in the degradation of SFDR. Fig. 6 shows the simulated SFDR versus time delay deviation and theoretical calculation results using equation (22), the simulation results match well with the theoretical ones. It can be seen that the SFDR peaks at both $\Delta \tau_{1} = 0$ and $\Delta \tau_{2} = 0$. The proposed highly linear scheme improves the SFDR if the time delay variation for MZI1 or MZI2 is within $- 4\ \hbox{ps}\ <\ \Delta \tau_{1}\ <\ 42\ \hbox{ps}$ or $- 2\ \hbox{ps}\ <\ \Delta \tau_{2}\ <\ 34\ \hbox{ps}$, respectively. This indicates that $\Delta \tau_{1}$ has larger operation ranges, in which the SFDRs are still higher than that of the conventional OSSB link.

Finally, the advantages of our approach relative to the state of the art are discussed. First, our linearization approach could be readily realized by integrating the interferometers into a single planar lightwave circuit and easily scaled to achieve even higher linearity. Second, compared to the transmitter-based linearization OSSB links [6], [8], our link greatly reduces the complexity on the transmit side since linearization is achieved at the receiver end. Third, compared to the transmitter-based linearization OSSB scheme in [8], where a 13 dB improvement in SFDR is obtained, an approximately 20 dB improvement in SFDR is achieved using our OSSB linearization approach.

SECTION IV

We have proposed and investigated a linearized analog microwave photonic link that utilizes a DD-MZM to obtain OSSB modulation signal and two parallel interferometers to suppress IMD3. It is shown by theoretical derivation and simulation that fundamental-to-IMD3 ratio (FIR) is 49 dB for a RF input signal power of 10 dBm using our proposed linearized link, which is 24 dB more than a conventional OSSB modulation link, and SFDR is 132 dB for a bandwidth of 1 Hz at the received optical power of 8 dBm assuming shot noise is the dominant noise contributions, which is improved approximately 20 dB. Simulation results show that even if the time delays of MZIs deviate from the ideal values to a certain degree, the performance is still acceptable. Our scheme extremely simplifies the transmit end of link since the linearization is realized on the receive end.

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