Figure 1 shows that Simulative model of the proposed RoF Link based on DD-DPMZM. The RoF link consist of two applied RF sources, indicated as Tone-1 & Tone-2, Laser diode as a carrier source, three electrical phase shifters, a PIN photo diode, conventional MZM structure and one DD-DPMZM structure. The used Linearized RoF link consists of two MZMs, as MZM-1(Upper MZM) and MZM-2 (Lower MZM), and both modulators are represented as an intensity modulated sections and they are arranged in a parallel form to get its desired outcome. Each phase shifters have 900 phase shifts to get desired outcome in the suppression of IMD3 spurious components.

The Optical Intensity of the laser diode as a carrier source can be represented as:

$${E}_{in}\left(t\right)= {A}_{LD}\text{exp}\left(j{\varOmega }_{0}t\right)$$

1

Where \({\varOmega }_{0}\) & \({A}_{LD}\) are represented as angular frequency and amplitude of the transmitted optical wave respectively. The RF source are used to generate, Two RF tones as Tone-1 & Tone-2 having frequency \({\varOmega }_{1}\) & \({\varOmega }_{2}\) respectively, with same amplitude. MZM-1 is biased with zero voltage while MZM-2 is biased at \({V}_{\pi }\) voltage, here \({V}_{\pi }\) is representing switching voltage.

To get the linearized modulation & large suppression of the IMD3, the RF driving voltages at four electrodes of DD-DPMZM with proper biasing can be arranged as:

$${V}_{L1}\left(t\right)= {V}_{m}\left\{\text{cos}{(\varOmega }_{1}t\right) +\text{s}\text{i}\text{n}\left({\varOmega }_{2} \text{t}\right)\}$$

2

$${V}_{L2}\left(t\right)= {V}_{m}\left\{\text{sin}{(\varOmega }_{1}t\right) +\text{c}\text{o}\text{s}\left({\varOmega }_{2} \text{t}\right)\}$$

3

$${V}_{U1}\left(t\right)= {V}_{m}\left\{\text{sin}{(\varOmega }_{1}t\right) +\text{sin}\left({\varOmega }_{2} t\right)+ \frac{{V}_{\pi }}{2}\}$$

4

$${V}_{U2}\left(t\right)= {V}_{m}\left\{\text{cos}{(\varOmega }_{1}t\right) +\text{sin}\left({\varOmega }_{2} t\right)- \frac{{V}_{\pi }}{2}\}$$

5

Where \({V}_{m}\), is the amplitude of the applied RF tones. \({V}_{L1}\left(t\right)\) & \({V}_{L1}\left(t\right)\) are showing driving voltages of MZM-1 (Upper MZM); & \({V}_{U1}\left(t\right) \& {V}_{U2}\left(t\right)\) are showing driving voltages of MZM-2 (Lower MZM). Assume that the MZM-1 & MZM-2 have same half wave switching voltages \({V}_{\pi }\). The output of MZM-1 can be expressed as:

$${E}_{MZM-1}\left(t\right)= \frac{1}{2\sqrt{2}} {\alpha E}_{in}\left(t\right)\left[\text{exp} \frac{j\pi {V}_{L1}\left(t\right)}{{V}_{\pi }}+\text{exp} \frac{j\pi {V}_{L2}\left(t\right)}{{V}_{\pi }} \right]$$

6

$${E}_{MZM-1}\left(t\right)= \frac{1}{2\sqrt{2}} {\alpha E}_{in}\left(t\right)\left[\text{exp}\left\{\frac{j\pi {V}_{m}}{{V}_{\pi }}\left[\text{cos}{\varOmega }_{1}t+\text{sin}{\varOmega }_{2}t\right]\right\}+\text{exp}\left\{\frac{j\pi {V}_{m}}{{V}_{\pi }} \left[\text{sin}{\varOmega }_{1}t+\text{cos}{\varOmega }_{2}t\right]\right\} \right]$$

7

Modulation Index or Depth of Modulation is expressed as:

$$m= \frac{\pi {V}_{m}}{{V}_{\pi }}$$

8

In term of depth of modulation, the outcome of MZM-1 can be modified as:

$${E}_{MZM-1}\left(t\right)= \frac{1}{2\sqrt{2}} {\alpha E}_{in}\left(t\right)\left[\text{exp}\left\{jm \left[\text{cos}{\varOmega }_{1}t+\text{sin}{\varOmega }_{2}t\right]\right\}+\text{exp}\left\{jm \left[\text{sin}{\varOmega }_{1}t+\text{cos}{\varOmega }_{2}t\right]\right\} \right]$$

9

The inner terms in the above equations assume that

\(A=\text{exp}\left(\frac{j\pi }{{V}_{\pi }}{V}_{L1}\left(t\right)\right)\) , and \(B=\text{exp}\left(\frac{j\pi }{{V}_{\pi }}{V}_{L2}\left(t\right)\right)\) (10)

\(A= \text{exp}\left\{jm \left[\text{cos}{\varOmega }_{1}t+\text{sin}{\varOmega }_{2}t\right]\right\}\) , and \(\text{B}=\text{ exp}\left\{jm \left[\text{sin}{\varOmega }_{1}t+\text{cos}{\varOmega }_{2}t\right]\right\}\) (11)

Both terms A & B can be expressed using Taylor series expansions as:

$$A=[1+jm \left(\text{cos}{\varOmega }_{1}t+\text{sin}{\varOmega }_{2}t\right)- \frac{{m}^{2}}{2!} {\left(\text{cos}{\varOmega }_{1}t+\text{sin}{\varOmega }_{2}t\right)}^{2}- \frac{j{m}^{3}}{3!} {\left(\text{cos}{\varOmega }_{1}t+\text{sin}{\varOmega }_{2}t\right)}^{3}+ \frac{{m}^{4}}{4!} {\left(\text{cos}{\varOmega }_{1}t+\text{sin}{\varOmega }_{2}t\right)}^{4}+ \frac{{jm}^{5}}{5!} {\left(\text{cos}{\varOmega }_{1}t+\text{sin}{\varOmega }_{2}t\right)}^{5}- \frac{{m}^{6}}{6!} {\left(\text{cos}{\varOmega }_{1}t+\text{sin}{\varOmega }_{2}t\right)}^{6}- \frac{{jm}^{7}}{7!} {\left(\text{cos}{\varOmega }_{1}t+\text{sin}{\varOmega }_{2}t\right)}^{7}+\dots ]$$

12

$$B=[1+jm \left(\text{sin}{\varOmega }_{1}t+\text{cos}{\varOmega }_{2}t\right)- \frac{{m}^{2}}{2!} {\left(\text{sin}{\varOmega }_{1}t+\text{cos}{\varOmega }_{2}t\right)}^{2}- \frac{j{m}^{3}}{3!} {\left(\text{sin}{\varOmega }_{1}t+\text{cos}{\varOmega }_{2}t\right)}^{3}+ \frac{{m}^{4}}{4!} {\left(\text{sin}{\varOmega }_{1}t+\text{cos}{\varOmega }_{2}t\right)}^{4}+ \frac{{jm}^{5}}{5!} {\left(\text{sin}{\varOmega }_{1}t+\text{cos}{\varOmega }_{2}t\right)}^{5}- \frac{{m}^{6}}{6!} {\left(\text{sin}{\varOmega }_{1}t+\text{cos}{\varOmega }_{2}t\right)}^{6}- \frac{{jm}^{7}}{7!} {\left(\text{sin}{\varOmega }_{1}t+\text{cos}{\varOmega }_{2}t\right)}^{7}+\dots ]$$

13

After the simplification and using Taylor series expansions in the defined terms, the outcome of MZM-1 is given as:

$${E}_{MZM-1}\left(t\right)= \frac{1}{2\sqrt{2}} {\alpha E}_{in}\left(t\right)\left[A+B\right]$$

14

$${E}_{MZM-1}\left(t\right)= \frac{1}{2\sqrt{2}}{ \alpha E}_{in}\left(t\right)\left[\sqrt{2}+ \frac{m}{2}\text{exp}\left(\frac{j\pi }{4}\right)\left\{\text{exp }j\left({\varOmega }_{1}t\right)+ \text{exp} j\left({\varOmega }_{2}t\right)\right\}+ \frac{{m}^{3}}{16}\text{exp}\left(\frac{j\pi }{4}\right)\left\{\text{exp} j\left({2\varOmega }_{1}-{\varOmega }_{2}\right)t+ \text{exp}\left({-\varOmega }_{1}+2{\varOmega }_{2}\right)t\right\}+ \frac{{m}^{4}}{64\sqrt{2}}\text{exp}\left(\frac{j\pi }{4}\right)\left\{\text{exp} j\left({2\varOmega }_{1}-{2\varOmega }_{2}\right)t+ \text{exp}\left({-2\varOmega }_{1}+2{\varOmega }_{2}\right)t\right\}+ \frac{{m}^{5}}{384}\text{exp}\left(\frac{j5\pi }{4}\right)\left\{\text{exp} j\left({3\varOmega }_{1}-2{\varOmega }_{2}\right)t+ \text{exp}\left({-2\varOmega }_{1}+3{\varOmega }_{2}\right)t\right\}+ \frac{{m}^{7}}{18432}\text{exp}\left(\frac{j5\pi }{4}\right)\left\{\text{exp} j\left({4\varOmega }_{1}-3{\varOmega }_{2}\right)t+ \text{exp}\left({-3\varOmega }_{1}+4{\varOmega }_{2}\right)t\right\}\right]$$

15

The outcome of the equations (15) shows that sub-MZM-1 does not originates the second order component (\({\varOmega }_{1}-{\varOmega }_{2} and {\varOmega }_{2}-{\varOmega }_{1}\)), which indicated that the one of the major Spurious components of IMD3 has been suppressed.

Optical output at MZM-2 is given as:

$${E}_{MZM-2}\left(t\right)= \frac{1}{2\sqrt{2}} {\alpha E}_{in}\left(t\right) \left\{\text{exp} \frac{j\pi {V}_{U1}\left(t\right)}{{V}_{\pi }}+\text{exp} \frac{j\pi {V}_{U2}\left(t\right)}{{V}_{\pi }} \right\}$$

16

The inner terms in the above equations assume that

\(C=\text{exp}\left(\frac{j\pi }{{V}_{\pi }}{V}_{U1}\left(t\right)\right)\) , and \(D=\text{exp}\left(\frac{j\pi }{{V}_{\pi }}{V}_{U2}\left(t\right)\right)\)

$$C=\text{exp}\frac{j\pi }{{V}_{\pi }}\left[{V}_{m} \left\{\text{sin}{\varOmega }_{1}t+\text{sin}{\varOmega }_{2}t\right\}+ \frac{{V}_{\pi }}{2}\right]$$

17

$$D=\text{exp}\frac{j\pi }{{V}_{\pi }}\left[{V}_{m} \left\{\text{cos}{\varOmega }_{1}t+\text{cos}{\varOmega }_{2}t\right\}- \frac{{V}_{\pi }}{2}\right]$$

18

Using all mentioned values of C and D in the equation number 16, then the outcome of MZM-2 is given as:

$${E}_{MZM-2}\left(t\right)= \frac{1}{2\sqrt{2}} {\alpha E}_{in}\left(t\right)\left[C+D\right]$$

19

Using Taylor series expansions, the mentioned terms C and D, is expressed as:

$$C=j\left[1+jm \left(\text{sin}{\varOmega }_{1}t+\text{sin}{\varOmega }_{2}t\right)- \frac{{m}^{2}}{2!} {\left(\text{sin}{\varOmega }_{1}t+\text{sin}{\varOmega }_{2}t\right)}^{2}- \frac{j{m}^{3}}{3!} {\left(\text{sin}{\varOmega }_{1}t+\text{sin}{w}_{2}t\right)}^{3}+ \frac{{m}^{4}}{4!} {\left(\text{sin}{\varOmega }_{1}t+\text{sin}{\varOmega }_{2}t\right)}^{4}+ \frac{{jm}^{5}}{5!} {\left(\text{sin}{\varOmega }_{1}t+\text{sin}{\varOmega }_{2}t\right)}^{5}- \frac{{m}^{6}}{6!} {\left(\text{sin}{\varOmega }_{1}t+\text{sin}{\varOmega }_{2}t\right)}^{6}- \frac{{jm}^{7}}{7!} {\left(\text{sin}{\varOmega }_{1}t+\text{sin}{\varOmega }_{2}t\right)}^{7}+\dots \right]$$

20

$$D=(-j)\left[1+jm \left(\text{cos}{\varOmega }_{1}t+\text{cos}{\varOmega }_{2}t\right)- \frac{{m}^{2}}{2!} {\left(\text{cos}{\varOmega }_{1}t+\text{cos}{\varOmega }_{2}t\right)}^{2}- \frac{j{m}^{3}}{3!} {\left(\text{cos}{\varOmega }_{1}t+\text{cos}{\varOmega }_{2}t\right)}^{3}+ \frac{{m}^{4}}{4!} {\left(\text{cos}{\varOmega }_{1}t+\text{cos}{\varOmega }_{2}t\right)}^{4}+ \frac{{jm}^{5}}{5!} {\left(\text{cos}{\varOmega }_{1}t+\text{cos}{\varOmega }_{2}t\right)}^{5}- \frac{{m}^{6}}{6!} {\left(\text{cos}{\varOmega }_{1}t+\text{cos}{\varOmega }_{2}t\right)}^{6}- \frac{{jm}^{7}}{7!} {\left(\text{cos}{\varOmega }_{1}t+\text{cos}{\varOmega }_{2}t\right)}^{7}+\dots \right]$$

21

After the simplification and using Taylor series expansions in the defined terms, the outcome of MZM-2 is given as:

$${E}_{MZM-2}\left(t\right)= \frac{1}{2\sqrt{2}}{ \alpha E}_{in}\left(t\right)\left[\frac{m}{2}\text{exp}\left(\frac{j\pi }{4}\right)\left\{\text{exp }j\left({\varOmega }_{1}t\right)+ \text{exp} j\left({\varOmega }_{2}t\right)\right\}+ \frac{{m}^{3}}{16}\text{exp}\left(\frac{j5\pi }{4}\right)\left\{\text{exp} j\left({2\varOmega }_{1}-{\varOmega }_{2}\right)t+ \text{exp}\left({-\varOmega }_{1}+2{\varOmega }_{2}\right)t\right\}+ \frac{{m}^{5}}{384}\text{exp}\left(\frac{j\pi }{4}\right)\left\{\text{exp} j\left({3\varOmega }_{1}-2{\varOmega }_{2}\right)t+ \text{exp}\left({-2\varOmega }_{1}+3{\varOmega }_{2}\right)t\right\}+ \frac{{m}^{7}}{18432}\text{exp}\left(\frac{j5\pi }{4}\right)\left\{\text{exp} j\left({4\varOmega }_{1}-3{\varOmega }_{2}\right)t+ \text{exp}\left({-3\varOmega }_{1}+4{\varOmega }_{2}\right)t\right\}\right]$$

22

The outcome of the equations (22) shows that MZM-2 does not originates the second order component (\({\varOmega }_{1}-{\varOmega }_{2} and {\varOmega }_{2}-{\varOmega }_{1}\)), which indicated that the one of the major components of IMD3 has been supressed. The outcome of the equations (15) and (22) shoes that the third order component \({\left\{\right(2\varOmega }_{1}-{\varOmega }_{2}) and {(-\varOmega }_{1}+{2\varOmega }_{2})\}\) from MZM-1 and MZM-2 respectively, having the same magnitude (\(\frac{1}{2\sqrt{2}} \alpha {A}_{LD} \frac{{m}^{3}}{16}\)), but having opposite phases. As MZM-1 having phase \(\frac{\pi }{4}\), while MZM-2 having phase \(\frac{5\pi }{4}\), so once they are combining at the outcome stage of the linearized RoF link, both terms are cancelled out completely. Also the outcome of the equations (15) and (22) shoes that the fifth order component \({\left\{\right(3\varOmega }_{1}-2{\varOmega }_{2}) and {(-2\varOmega }_{1}+{3\varOmega }_{2})\}\) from MZM-1 and MZM-2 respectively, having the same magnitude (\(\frac{1}{2\sqrt{2}} \alpha {A}_{LD} \frac{{m}^{5}}{384}\)), but having opposite phases. As MZM-1 having phase \(\frac{5\pi }{4}\), while MZM-2 having phase \(\frac{\pi }{4}\) for the mentioned fifth order component, so once they are combining at the outcome stage of the linearized RoF link, both terms are cancelled out completely. Thus, the proposed RoF link shows that the combined outcome of the DD-DPMZM is free from the second, third and fifth order components, that one is the major resources of the IMD3.

The combined outcome of linearized RoF link is given as:

$${E}_{out}\left(t\right)= \left[{E}_{MZM-1}\left(t\right)+{E}_{MZM-2}\left(t\right)\right]/2$$

After simplifications, the combined outcome of the linearized RoF link can be expressed as:

$${E}_{out}\left(t\right)= \frac{1}{2\sqrt{2}}{ \alpha E}_{in}\left(t\right)\left[1 + \frac{m}{2}\text{exp}\left(\frac{j\pi }{4}\right)\left\{\text{exp }j\left({\varOmega }_{1}t\right)+ \text{exp} j\left({\varOmega }_{2}t\right)\right\}+ \frac{{m}^{4}}{64}\text{exp}\left(j\pi \right)\left\{\text{exp} j\left({2\varOmega }_{1}-{2\varOmega }_{2}\right)t+ \text{exp}\left({-2\varOmega }_{1}+2{\varOmega }_{2}\right)t\right\}+ \frac{{m}^{7}}{18432}\text{exp}\left(\frac{j\pi }{4}\right)\left\{\text{exp} j\left({4\varOmega }_{1}-3{\varOmega }_{2}\right)t+ \text{exp}\left({-3\varOmega }_{1}+4{\varOmega }_{2}\right)t\right\}+\dots \right]$$

23

The equations (23) shoes that the optical outcome of linearized RoF link, it contains only the fundamental component, fourth order component & seventh order component. The fundamental component having the same magnitude \(\frac{1}{\sqrt{2}} \alpha {A}_{LD} \frac{m}{2}\), from both sub-MZMs with the same phase \(\frac{\pi }{4}\), thus the proposed RoF link enhancing the magnitude of the fundamental component, that’s good indication about the linearization. The fourth and seventh order spurious components having very little magnitude as compared to the desired fundamental components, so they will impact a little effect on the proposed model performance.

Photo detector current is given as:

$${I}_{PD}\left(t\right)=R {E}_{out}\left(t\right){E}_{out}^{*}\left(t\right)$$

$$=R \frac{{\alpha }^{2}}{2}{E}_{in}^{2}\left(t\right)\left\{1 + m\sqrt{2} \left[\text{cos}\left({\varOmega }_{1}t+ \frac{\pi }{4}\right)t+ \text{cos}\left({\varOmega }_{2}t+ \frac{\pi }{4}\right)t\right]+{m}^{2}\left[1+ \text{cos}\left({\varOmega }_{1}-{\varOmega }_{2}\right)t\right]+\frac{{m}^{5}}{32\sqrt{2}} \left[\text{cos}\left(-{\varOmega }_{1}t+{2\varOmega }_{2}t- \frac{3\pi }{4}\right)t+ \text{cos}\left({2\varOmega }_{1}t-{\varOmega }_{2}t- \frac{3\pi }{4}\right)t\right]\right\}$$

24

Where R is the responsivity of PIN photodetector (PD). Eq. (24) shows the measured fundamental component with enhanced magnitude while the IMD3 spurious components \({\left\{\right(2\varOmega }_{1}-{\varOmega }_{2}) and {(-\varOmega }_{1}+{2\varOmega }_{2})\}\) having negligible magnitude compared with the fundamental components. Thus, the linearized RoF link with optimized performance can be obtained with large suppression of IMD3 spurious components in the proposed DD-DPMZM model.

From Eq. (24) the fundamental component power and IMD3 spurious component power in a simplified form is given as:

$${P}_{\varOmega 1}= \frac{{\alpha }^{4}{R}^{2}{P}_{LD}^{2}{m}^{2}{R}_{Load}}{4}$$

25

$${P}_{{2\varOmega }_{1}-{\varOmega }_{2}}= \frac{{\alpha }^{4}{R}^{2}{P}_{LD}^{2}{m}^{2}{R}_{Load}}{4*{8}^{4}}$$

26

Where \({P}_{LD}\) is the optical power and \({R}_{Load}\) is the load resistance at PIN PD. load resistance is setup to one ohm and considered attenuation coefficient is taken − 0.21 dB with permissible conditions.