The schematic diagram of proposed photonic link where all IMDs along Harmonics Distortions which are based on two D-DPMZM – have been eliminated by using balanced-photo-detectors, shown in the Fig. 1. In this configuration, we have used two-tone microwave frequency of 17 GHz and 17.5 GHz using RF shifters by 90 degrees and 270 degrees, respectively. The two D-DPMZM consist of two sub-dual electrode MZMs and two balance photo-detector combined by using a 3 dB power combiner.
Two RF signals are combined by RF combiner and then are shifted by 90 degrees for upper and lower arm of MZM1 (D-DPMZM1) while frequency one is shifted by 270 degrees whereas frequency two is shifted by 90 degrees for upper and lower arm of \({\mathrm{MZM}}_{2}\) (D-DPMZM1). In D-DPMZM2 the RF frequencies are combined and then shifted by 270 degrees for upper and lower arm of MZM1 (D-DPMZM2), frequency one is shifted by 90 degrees and frequency two is shifted by 270 degrees for upper and lower arm of \({\mathrm{MZM}}_{2}\) (D-DPMZM2). External DC bias is set to maximum for MZM1 and to quadrature for MZM2 for both D-DPMZMs. In addition, frequencies than are combined by 3 dB power combiner and detected by balanced photo-detector. Results show that proposed configuration allows elimination of all IMDs and all even harmonic distortions.
The drive voltage with DC biases of the D-DPMZM1 for the schematic configuration illustrated in Fig. 1 can be expressed as:
$$ Q_{11} (t) = V_{m} \left\{ {\cos (\omega_{1} t + \frac{\pi }{2}) + \cos (\omega_{2} t + \frac{\pi }{2})} \right\} + V_{\pi } $$
(1)
$$ Q_{12} (t) = V_{m} \left\{ {\cos (\omega_{1} t + \frac{\pi }{2}) + \cos (\omega_{2} t + \frac{\pi }{2})} \right\} $$
(2)
$$ \begin{gathered} Q_{21} (t) = V_{m} \left\{ {\cos (\omega_{1} t - \frac{\pi }{2}) + \cos (\omega_{2} t + \frac{\pi }{2})} \right\} + \frac{{V_{\pi } }}{2} \hfill \\ \hfill \\ \end{gathered} $$
(3)
$$ Q_{22} (t) = V_{m} \left\{ {\cos (\omega_{1} t - \frac{\pi }{2}) + \cos (\omega_{2} t + \frac{\pi }{2})} \right\} $$
(4)
where \({Q}_{11}\left(\mathrm{t}\right)\mathrm{ and }{Q}_{12}\left(\mathrm{t}\right)\) are drive voltages on two electrodes of MZM1 (D-DPMZM1); \({Q}_{21}\left(\mathrm{t}\right)\mathrm{ and }{Q}_{12}\left(\mathrm{t}\right)\) are drive voltages on two electrodes of MZM2 (D-DPMZM1); \({\mathrm{V}}_{\mathrm{m}}\) represent the amplitude of the RF input signals. The laser power is expressed as: \({E}_{\mathrm{in}}\left(t\right)={E}_{c}{\mathrm{e}}^{\mathrm{j}{\omega }_{\mathrm{c}}t}\) where the Ec is the input power and ωc is the angular frequency of the laser, consequently the output optical power in MZM1 (D-DPMZM1) can be expressed as:
$$ E_{{out1D - DPMZ_{1} }} (t) = E_{in} (t)\left\{ {\exp (j\pi \frac{{Q_{11} (t)}}{{V_{\pi } }}) + \exp ( - j\pi \frac{{Q_{12} (t)}}{{V_{\pi } }})} \right\} $$
(5)
The output optical power in MZM2 (D-DPMZM1) can be expressed as:
$$ E_{{out2D - DPMZ_{1} }} (t) = E_{in} (t)\left\{ {\exp (j\pi \frac{{Q_{21} (t)}}{{V_{\pi } }}) + \exp ( - j\pi \frac{{Q_{22} (t)}}{{V_{\pi } }})} \right\} $$
(6)
If \(m=\frac{\pi {V}_{m}}{{V}_{\pi }}\) than by substituting Eq. (1) and (2) into Eq. (5) we obtain:
$$ E_{{out1D - DPMZ_{1} }} (t) = E_{in} (t)\left\{ \begin{gathered} \exp (jm\left\{ {\sin (\omega_{1} t) + \sin (\omega_{2} t)} \right\} + j\pi ) \hfill \\ + \exp ( - jm\left\{ {\sin (\omega_{1} t) + \sin (\omega_{2} t)} \right\}) \hfill \\ \end{gathered} \right\} $$
(7)
Applying a Jacobi-Anger Expansion in Eq. (7), we obtain:
$$ E_{{out2D - DPMZM_{1} }} (t) = E_{in} (t)\sum\limits_{n,m = - \infty }^{\infty } {J_{n} (m)} J_{m} (m)e^{{j(n\omega_{1} t + \omega_{2} t)}} \left[ {( - 1)^{n + m} e^{j\pi } + 1} \right] $$
(8)
Similarly, by substituting Eq. (3) and (4) into Eq. (6), we can derive the output optical power in MZM2 (D-DPMZM1):
$$ E_{{out2D - DPMZM_{1} (t)}} = E_{in} (t)\left\{ \begin{gathered} \exp (jm\left\{ {\sin (\omega_{1} t) - \sin (\omega_{2} t)} \right\}) + j\frac{\pi }{2} \hfill \\ + \exp ( - jm\left\{ {\sin (\omega_{1} t) - \sin (\omega_{2} t)} \right\}) \hfill \\ \end{gathered} \right\} $$
(9)
Applying a Jacobi-Anger Expansion in Eq. (9), we find:
$$ E_{{out2D - DPMZM_{1} }} (t) = E_{in} (t)\sum\limits_{n,m = - \infty }^{\infty } {j_{n} (m)} j_{m} (m)e^{{j(n\omega_{1} t + \omega_{2} t)}} \left[ \begin{gathered} \left\{ {( - 1)^{m} + 1} \right\}e^{{j\frac{\pi }{2}}} \hfill \\ + ( - 1)^{n} + 1 \hfill \\ \end{gathered} \right] $$
(10)
Combined power of the two MZMs will represent the optical power for D-DPMZM1. The signal after 3 dB power combiner can be expressed as:
$$ E_{{1D - DPMZM_{1} }} (t) = \frac{{E_{{out1D - DPMZM_{1} }} (t) + E_{{out2D - DPMZM_{1} }} (t)}}{\sqrt 2 } $$
(11)
$$ E_{{1D - DPMZM_{1} }} (t) = \frac{{E_{{out1D - DPMZM_{1} }} (t) - E_{{out2D - DPMZM_{1} }} (t)}}{\sqrt 2 } $$
(12)
The generated photocurrent I(t) after the balance-photodetector is:
$$ I_{{PDD - DPMZM_{1} }} (t) = R\left[ \begin{gathered} E_{{out1D - DPMZM_{1} }} (t).E_{{out1D - DPMZM_{1} }} (t)^{*} - \hfill \\ E_{{out2D - DPMZM_{1} }} (t).E_{{out2D - DPMZM_{1} }} (t)^{*} \hfill \\ \end{gathered} \right] $$
(13)
where \(\mathcal{R}\) is responsivity of the photodetector. By deploying Tayler series expansion to the third order in m, following expression can be derived.
$$ I_{{PDD - DPMZM_{1} }} (t) = - \frac{1}{2}RP_{in} \left\{ \begin{gathered} 8m(\cos (\omega_{1} t) + \sin (\omega_{2} t)) + \hfill \\ 4m^{2} (\cos (2\omega_{1} t) - \cos (2\omega_{2} t)) \hfill \\ + 4m^{3} \left( \begin{gathered} \sin (3\omega_{1} t) + \sin (3\omega_{2} t) \hfill \\ - \sin (\omega_{1} t) - \sin (\omega_{2} t) \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} \right\} + o(m)^{4} $$
(14)
Similarly, we can derive equations for D-DPMZM2 as follows:
$$ Q_{11} (t) = V_{m} \left\{ {\cos (\omega_{1} t - \frac{\pi }{2}) + \cos (\omega_{2} t - \frac{\pi }{2})} \right\} + V_{\pi } $$
(15)
$$ Q_{12} (t) = V_{m} \left\{ {\cos (\omega_{1} t + \frac{\pi }{2}) + \cos (\omega_{2} t + \frac{\pi }{2})} \right\} $$
(16)
$$ Q_{21} (t) = V_{m} \left\{ {\cos (\omega_{1} t + \frac{\pi }{2}) + \cos (\omega_{2} t - \frac{\pi }{2})} \right\} + \frac{{V_{\pi } }}{2} $$
(17)
$$ Q_{22} (t) = V_{m} \left\{ {\cos (\omega_{1} t + \frac{\pi }{2}) + \cos (\omega_{2} t - \frac{\pi }{2})} \right\} $$
(18)
where \({Q}_{11}\left(t\right)\mathrm{ and }{Q}_{12}\left(t\right)\) are drive voltages on two electrodes of MZM1 (D-DPMZM2); \({Q}_{21}\left(\mathrm{t}\right)\mathrm{ and }{Q}_{12}\left(\mathrm{t}\right)\) are drive voltages on two electrodes of MZM2 (D-DPMZM2); \({\mathrm{V}}_{\mathrm{m}}\) represent the amplitude of the RF input signals. The laser power is expressed as: \({E}_{\mathrm{in}}\left(t\right)={E}_{\mathrm{c}}{\mathrm{e}}^{\mathrm{j}{\omega }_{\mathrm{c}}\mathrm{t}}\) where the Ec is the input power and ωc is the angular frequency of the laser, consequently the output optical in MZM1 (D-DPMZM2) can be expressed as:
$$ E_{out1D - DPMZ2} (t) = E_{in} (t)\left\{ {\exp (j\pi \frac{{Q_{11} (t)}}{{V_{\pi } }}) + \exp ( - j\pi \frac{{Q_{12} (t)}}{{V_{\pi } }})} \right\} $$
(19)
And the output optical power in MZM2 (D-DPMZM2) can be expressed as:
$$ E_{{out2D - DPMZ_{2} }} (t) = E_{in} (t)\left\{ {\exp (j\pi \frac{{Q_{21} (t)}}{{V_{\pi } }}) + \exp ( - j\pi \frac{{Q_{22} (t)}}{{V_{\pi } }})} \right\} $$
(20)
If \(m=\frac{\pi {V}_{m}}{{V}_{\pi }}\) than by substituting Eq. (15) and (16) into Eq. (19), we obtain:
$$ E_{out1D - DPMZ2} (t) = E_{in} (t)\left\{ \begin{gathered} \exp (jm\left\{ {\sin (\omega_{1} t) + \sin (\omega_{2} t)} \right\} + j\pi ) \hfill \\ + \exp (jm\left\{ {\sin (\omega_{1} t) + \sin (\omega_{2} t)} \right\}) \hfill \\ \end{gathered} \right\} $$
(21)
Applying a Jacobi-Anger Expansion in Eq. 20 we get:
$$ E_{out1D - DPMZM2} (t) = E_{in} (t)\sum\limits_{n,m = - \infty }^{\infty } {J_{n} (m)} J_{m} (m)e^{{j(n\omega_{1} t + \omega_{2} t)}} \left[ {( - 1)^{n + m} e^{j\pi } + 1} \right] $$
(22)
Similarly, by substituting Eq. (17) and (18) into Eq. (20), we can derive the output optical power in MZM2 (D-DPMZM2):
$$ E_{{out2D - DPMZM_{2} (t)}} = E_{in} (t)\left\{ \begin{gathered} \exp (jm\left\{ { - \sin (\omega_{1} t) + \sin (\omega_{2} t)} \right\}) + j\frac{\pi }{2} \hfill \\ + \exp ( - jm\left\{ { - \sin (\omega_{1} t) + \sin (\omega_{2} t)} \right\}) \hfill \\ \end{gathered} \right\} $$
(23)
Applying a Jacobi-Anger Expansion in Eq. (22), we obtain:
$$ E_{out2D - DPMZM2} (t) = E_{in} (t)\sum\limits_{n,m = - \infty }^{\infty } {j_{n} (m)} j_{m} (m)e^{{j(n\omega_{1} t + \omega_{2} t)}} \left[ \begin{gathered} \left\{ {( - 1)^{m} + 1} \right\}e^{{j\frac{\pi }{2}}} \hfill \\ + ( - 1)^{n} + 1 \hfill \\ \end{gathered} \right] $$
(24)
Combined power of the two MZMs will represent the optical power for D-DPMZM1. The signal after 3 dB power combiner can be expressed as:
$$ E_{{2D - DPMZM_{2} }} (t) = \frac{{E_{{out1D - DPMZM_{2} }} (t) + E_{out2D - DPMZM2} (t)}}{\sqrt 2 } $$
(25)
$$ E_{2D - DPMZM2} (t) = \frac{{E_{out1D - DPMZM2} (t) - E_{{out2D - DPMZM_{2} }} (t)}}{\sqrt 2 } $$
(26)
The generated photocurrent \({\mathrm{I}}_{\mathrm{PD}\_{\mathrm{D}-\mathrm{DPMZM}}_{2}}\left(\mathrm{t}\right)\) after the balance-photodetector is:
$$ I_{PDD - DPMZM2} (t) = R\left[ \begin{gathered} E_{{out1D - DPMZM_{2} }} (t).E_{{out1D - DPMZM_{2} }} (t)^{*} - \hfill \\ E_{out2D - DPMZM2} (t).E_{out2D - DPMZM2} (t)^{*} \hfill \\ \end{gathered} \right] $$
(27)
where \(\mathcal{R}\) is responsivity of photodetector. By using Tayler series expansion to the third order in m, following expression can be derived:
$$ I_{PDD - DPMZM2} (t) = - \frac{1}{2}RP_{in} \left\{ \begin{gathered} 8m(\cos (\omega_{1} t) + \sin (\omega_{2} t)) + \hfill \\ - 4m^{2} (\cos (2\omega_{1} t) - \cos (2\omega_{2} t)) \hfill \\ + 4m^{3} \left( \begin{gathered} \sin (3\omega_{1} t) + \sin (3\omega_{2} t) \hfill \\ - \sin (\omega_{1} t) - \sin (\omega_{2} t) \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} \right\} + o(m)^{4} $$
(28)
Combining power after two balance-photodetectors
$$ I_{PD} (t) = I_{PDD - DPMZM1} (t) + I_{PDD - DPMZM2} (t) $$
(29)
Then we obtain:
$$ I_{PD} (t) = RP_{in} \left\{ \begin{gathered} 16m(\cos (\omega_{1} t) + \sin (\omega_{2} t)) + \hfill \\ + 8m^{3} \left( \begin{gathered} \sin (3\omega_{1} t) + \sin (3\omega_{2} t) \hfill \\ - \sin (\omega_{1} t) - \sin (\omega_{2} t) \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} \right\} + o(m)^{4} $$
(30)
From results in Eq. (28), it can be seen that the third order Intermodulation Distortion of frequency \({2\omega }_{2}{-\omega }_{1}\) and \(2{\omega }_{1}-{\omega }_{2}\) is eliminated. Second order harmonic and intermodulation of frequency is \({\upomega }_{2}{-\upomega }_{1}\),\({\upomega }_{1}-{\upomega }_{2}\), and \({2\upomega }_{2}{-2\upomega }_{1}\),\(2{\upomega }_{1}-2{\upomega }_{2}\). We have used the Tayler series to higher order (up to ninth order) and to all intermodulation distortions and even harmonic distortions does not exist which means that the modulation index increases the IMDs and even harmonic distortions will not exist in this model.