Photon-Assisted Generation of Frequency 16-Tupling Millimeter Waves by Mach–Zehnder Modulators Without Filtering and Transmission over Fibers Without the Bit Walk-Off Effect

Abstract

A novel system to generate and transmit frequency 16-tupling millimeter waves (MMWs) without the bit walk-off effect created by the Mach–Zehnder modulator (MZM) is proposed. The ±8th-order sidebands are generated by four MZMs connected in parallel with the data carried only on the +8th-order sideband by adopting the composite radio frequency (RF) for the MZMs. The frequency 16-tupling MMW with data is generated by injecting the ±8th order sidebands into the photodetector. The system’s principle of operation is given. The feasibility of our designed schemes is verified by the simulation experiment. The effect on the system’s BER is discussed when the main parameters diverge from their theoretical values or default values. Our designed system can effectively eliminate the bit walk-off effect caused by fiber dispersion and greatly raise the fiber distribution distance.

1. Introduction

With the rapid development of modern wireless communications, such as artificial intelligence, virtual reality, 5G, 6G, etc., the frequency of carriers has extended to the frequency range of millimeter waves (MMWs) and terahertz waves [1,2]. To generate MMWs above 100 GHz in the electric domain, the frequency response of the electronics limits the MMW frequency [3]. Distributing MMWs in free space creates a great loss. The generation and distribution of MMWs by photonic technology is known as radio-on-fiber (ROF) technology. It can overcome the abovementioned problems, and it has become the key technology for modern wireless communication systems [4,5,6].

The two coherent light waves with frequency spacing are called optical MMWs. MMWs are produced by beating optical MMWs in a photodetector (PD). Many schemes to produce optical MMWs have been put forward, such as photonic heterodyning schemes [7], optoelectronic oscillator schemes [8], mode-locked laser schemes [9], external modulator schemes [10], optical nonlinear effect schemes [11], etc. Among those proposed schemes, the external schemes offer several advantages, such as a larger frequency multiplication factor (FMF), larger tunability, higher reliability, higher power efficiency, and higher receiver sensitivity [10]. The optical phase modulator (PM), Mach–Zehnder modulator (MZM), and polarization modulator (PoIM) are the main external modulators. The generated MMW’s purity is low when using PM-based methods [11]. High polarization sensitivity is the main drawback of PoIM-based methods [12,13]. The DC drift and limited extinction are the main drawbacks of MZM-based methods [13]. Adopting PoIM, the FMF of the generated MMW can be obtained as follows: 6 [14,15], 8 [16], 12 [17], 18 [18], 24 [19,20], and 32 [21]. MZM-based methods have the advantage of a high stable performance at a low cost. These are the main methods used to generate high-FMF MMWs. The FMF of generated MMWs can be obtained as follows: 12 [22,23], 16 [24,25,26], 18 [27], 24 [28], and 32 [29].

One of the most significant problems faced by the MMWE ROF system is fiber dispersion [30,31,32]. The data modulation format is an important factor affecting the size of the dispersion. In the MZM-based MMW generation method, the generation optical MMWs often are the ±nth-order sidebands with optical carrier (oth sideband). If the data are modulated on all the sidebands, this is called double sideband modulation (DSB). It has the effects of MMW signal periodic decay and bit walk-off. If the oth sideband is impressed and the data are modulated on the ±nth sideband, this is called carrier-suppressed DSB (CS-DSB) [33,34,35,36,37]. In this case, the MMWs’ signal periodic decay effects can be eliminated, but the bit walk-off effect remains. In the case of CS-DSB, if the data are modulated on one of the ±nth sidebands, this is called single sideband modulation (SSB). In this case, the MMW periodic decay effect and the bit walk-off effect can both be eliminated. In the MMW ROF system, the easiest way to implement SSB is by using the filter [38,39]. The main drawbacks of the filtering method are that the insertion loss is increased and the system’s tunability is limited. How to realize the SSB without using a filter has been a hot topic within research of the MMW ROF system recently. A novel method to modulate the data only on the −1th sideband without using a filter by a dual parallel MZM (DPMZM), which is formed by three MZMs (MZMa, MZMb and MZMc), was devised [40]. In this scheme, an SSB signal is generated by MZMa, and another SSB signal with data is modulated on the +1th-order sideband is generated by MZMb. By combing the output of MZMa and MZMb in MZMc, the optical carrier is eliminated. The main drawbacks of this scheme are the complex structure and low FMF. To overcome the limitations of the abovementioned method, Zhu et al. put forward three structures for realizing that the data are modulated only on the −1th, −2th, and −6th sidebands, respectively, and that the MMW with FMF of 2, 4, 12, are produced, respectively [41,42,43] The problem with those schemes is that the FMF is still not high enough. To obtain higher-frequency MMWs, and to decrease the frequency of the RF LO, the FMF needs to be further increased. To increase the fiber transmission distance, the bit walk-off effect needs to be conquered.

In this paper, a new method to produce 16-tupling frequency MMW without a filter to overcome the dispersion by MZMs is present. Its operation principle is theoretically analyzed, and its validity is verified with simulation experiments. A comparison of our designed scheme with other schemes designed in the related literature is given in

Table 1. Comparison of the methods to eliminate the bit walk-off effect.

Reference	FMF	Method	Characteristic
[30,31,32]	4, 6, 12	filter method	high loss, poor tunability
[40]	2	two MZM	Complex, FMF low, filterless
[41,42,43]	2, 4, 12	composite RF signal	FMF low, filterless
Our scheme	16	composite RF signal as the drive signal of the MZM	FMF high, filterless

.

The system structure is presented in Section 2. The operation principle is analyzed in Section 3. The results of the simulation experiments are presented in Section 4. The discussion is given in Section 5. Section 6 is devoted to the conclusion.

2. System Design

The diagrammatic sketch of our designed filterless frequency 16-tupling MMW ROF system is shown in Figure 1. The main photonic devices are a continuous wave laser (CW laser), six 3 dB optical couplers (signed as OC1~OC6), four MZMs (signed as MZM1~MZM4), an optical amplifier (OA), and a photodetector (PD). The main electrical devices are two RF signal local oscillators (LO) (signed as RF LO1, RF LO2), seven electrical phase shifters (EPSs) (signed as EPS1~EPS7), an electrical phase modulator (PM), and an electrical gainer (EG). MZM1 and MZM2 construct DPMZM1. MZM3 and MZM4 are used to construct DPMZM2. The photonic and electrical paths are expressed with solid and dashed lines.

3. Operation Principle

When the EPSi (i = 1, 3, 5, 7) is set at 180°, each MZM is operated at the maximum transmission point (MATP). The output signal from each MZM is a ±2n (n is an integer)-order optical sideband. Setting the initial phase difference of the RF driving signals loaded on the two MZMs in every DPMZM as 90°, the sidebands of the nth order taking an odd number are inversely equal, and those sidebands cancel each other out when they are added together. The output of each DPMZM is the 4m (m is an integer)-order sidebands. Setting the initial phase difference of the RF driving signal loaded on the two DPMZMs as 45°, the sidebands of m taking the odd number are inversely equal, and those sidebands cancel each other out when they are added together. The composite optical signal from the four MZMs is the 8k (k is an integer)-order sidebands. By setting appropriate modulation coefficients for the MZM, the optical sidebands except k = 1 are suppressed as much as possible, and the composite output signal of four MZMs are the ±8th-order sidebands. Beating the ±8th-order optical sidebands in the PD for optoelectrical conversion, the photocurrent from the PD contains the frequency 16-tupling MMW signal.

3.1. Output Signal from MZMs

The photonic carrier from the CW laser is expressed as $E_0\left(t\right)=8E_0\exp \left(j{\omega }_{0}t\right)$, where $8E_0$, ${\omega }_0$ are the amplitude and angular frequency of the photonic carrier, respectively.

The data are expressed as $s\left(t\right)={\displaystyle {\sum }_n{I}_n}g\left(t-nT\right)$, where $I_n$ is the sequences of binary numbers, $g\left(t\right)$ is the codeword function, and $T$ is the codeword period.

The composite RF driving signal is expressed as follows:

$$V_i\left(t\right)=V_{RF}\cos \left[{\omega }_{RF}t+P·s\left(t\right)+{\phi }_i\right]+G·s\left(t\right)$$

(1)

where $V_{RF}$, ${\omega }_{RF}$ and ${\phi }_i$ are the amplitude, angular frequency, and initial phase of the RF driving signal, respectively. The $G$ and $P$ are the EG’ gain and the PM’s modulation index, respectively. $V_{\pi }$ is MZM’s half-wave voltage.

Setting $P=3\pi /16$, $G=6$, $V_{\pi }=4V$,

${\phi }_i=0, \pi , \pi /2, 3\pi /2, \pi /4, 5\pi /4, 3\pi /4, 7\pi /4 \left(i=1, 2, 3, 4, 5, 6, 7, 8\right)$. The form of Equation (1) is converted as given below:

$$V_i\left(t\right)=V_{RF}\cos \left[{\omega }_{RF}t+{\displaystyle \frac{3\pi }{16}}s\left(t\right)+{\phi }_i\right]+6s\left(t\right)$$

(2)

The output signals of MZM1 and MZM2 are expressed as given below:

$$\begin{array}{c}{E}_{MZM1}=E_0{e}^{j\left\{{\omega }_{0}t+m\cos \left[{\omega }_{RF}t+\frac{3\pi }{16}s\left(t\right)\right]+\frac{6\pi }{4}s\left(t\right)\right\}}+E_0{e}^{j\left\{{\omega }_{0}t+m\cos \left[{\omega }_{RF}t+\frac{3\pi }{16}s\left(t\right)+\pi \right]+\frac{6\pi }{4}s\left(t\right)\right\}}\\ =2E_0{e}^{j{\omega }_{0}t}{\displaystyle \sum _{n=-\infty }^{\infty }{\left(-1\right)}^n{J}_{2n}(m)e^{j2n\left[{\omega }_{RF}t+\frac{3\pi }{16}s\left(t\right)\right]+j\frac{3\pi }{2}s\left(t\right)}}\end{array}$$

(3)

$$E_{MZM2}=2E_0{e}^{j{\omega }_{0}t}{\displaystyle \sum _{n=-\infty }^{\infty }{\left(-1\right)}^n{J}_{2n}(m)e^{j2n\left[{\omega }_{RF}t+\frac{3\pi }{16}s\left(t\right)+\frac{\pi }{2}\right]+j\frac{3\pi }{2}s\left(t\right)}}$$

(4)

where $m$ is the modulation index of MZM. Here, the Jacoby–Angell constant equation is utilized.

Combing the output signals of MZM1 and MZM2, the output signal of DPMZM1 is obtained as given below:

$$\begin{array}{cc}\hfill E_{DPMZM1}& =E_{MZM1}+E_{MZM2}\hfill \\ & =E_0{e}^{j{\omega }_{0}t}\left\{\begin{array}{l}{\displaystyle \sum _{n=-\infty }^{\infty }2{\left(-1\right)}^n{J}_{2n}(m)e^{j2n\left[{\omega }_{RF}t+\frac{3\pi }{16}s\left(t\right)\right]+j\frac{3\pi }{2}s\left(t\right)}}\\ +{\displaystyle \sum _{n=-\infty }^{\infty }2{\left(-1\right)}^n{J}_{2n}(m)e^{j2n\left[{\omega }_{RF}t+\frac{3\pi }{16}s\left(t\right)+\frac{\pi }{2}\right]+j\frac{3\pi }{2}s\left(t\right)}}\end{array}\right\}\hfill \\ & =E_0{e}^{j{\omega }_{0}t}\left\{{\displaystyle \sum _{n=-\infty }^{\infty }4J_{4n}(m)e^{j4n\left[{\omega }_{RF}t+\frac{3\pi }{16}s\left(t\right)\right]+j\frac{3\pi }{2}s\left(t\right)}}\right\}\hfill \end{array}$$

(5)

DPMZM2 has the same structure as DPMZM1. The output of DPMZM2 is obtained as given below:

$$E_{DPMZM2}=E_0{e}^{j{\omega }_{0}t}{\displaystyle \sum _{n=-\infty }^{\infty }4J_{4n}(m)e^{j4n\left[{\omega }_{RF}t+\frac{3\pi }{16}s\left(t\right)+\frac{\pi }{4}\right]+j\frac{3\pi }{2}s\left(t\right)}}$$

(6)

Adding the output signals of the two DPMZMs, the composite optical field is obtained as follows:

$$\begin{array}{cc}\hfill E_{out}& =E_{DPMZM1}+E_{DPMZM2}=E_0{e}^{j{\omega }_{0}t}\left\{\begin{array}{l}{\displaystyle \sum _{n=-\infty }^{\infty }4J_{4n}(m)e^{j4n\left[{\omega }_{RF}t+\frac{3\pi }{16}s\left(t\right)\right]+j\frac{3\pi }{2}s\left(t\right)}}\\ +{\displaystyle \sum _{n=-\infty }^{\infty }4J_{4n}(m)e^{j4n\left[{\omega }_{RF}t+\frac{3\pi }{16}s\left(t\right)+\frac{\pi }{4}\right]+j\frac{3\pi }{2}s\left(t\right)}}\end{array}\right\}\hfill \\ & =E_0{e}^{j{\omega }_{0}t}{\displaystyle \sum _{n=-\infty }^{\infty }8J_{8n}(m)e^{j8n\left[{\omega }_{RF}t+\frac{3\pi }{16}s\left(t\right)\right]+j\frac{3\pi }{2}s\left(t\right)}}\hfill \end{array}$$

(7)

It can be seen from Equation (7) that $E_{out}$ are ±8n-order sidebands signals, whose amplitudes are proportional to $J_n(m)$.

3.2. Selection of the Modulation Index of MZM

To generate a frequency 16-tupling MMW, it needs to beat the ±8th-order sidebands in the PD. Meanwhile, it is needed to restrain the other sidebands except the ±8th-order sidebands.

When m = 5.52, $J_0(5.52)=-2.65\times {10}^{-5}$, $J_8(5.52)=0.0344$, and $J_{16}(5.52)=3.4421\times {10}^{-7}$. The optical carrier suppression ratio (OCSR) is $OCSR=10\lg \left(P_8/P_0\right)=20\lg \left[J_8(5.52)/J_0(5.52)\right]=62.27 \mathrm{dB}$. The photonic spurious suppression ratio (OSSR) is $OSSR=10\lg \left(P_8/P_{16}\right)=20\lg \left[J_8(5.52)/J_{16}(5.52)\right]=100 \mathrm{dB}$.

According to the values of $J_{8n}(5.52)$, the amplitude of the 8n-order sidebands except n = ±1 is small and can be ignored. Neglecting all terms except for n = ±1, Equation (5) is simplified as given below:

$$E_{out}\left(0,t\right)=E_0\left\{J_8(m)e^{j\left({\omega }_{0}t+8{\omega }_{RF}t\right)+j\pi s\left(t\right)}+J_{-8}(m)e^{j\left({\omega }_{0}t-8{\omega }_{RF}t\right)}\right\}$$

(8)

As we see from Equation (8), the downlink data are carried only on the +8th-order sidebands.

3.3. The Dispersion Effect on the Generated Frequency 16-Tupling MMW

In the back-to-back (BTB) system, the photocurrent of the PD is obtained and expressed as shown below:

$$I\left(0,t\right)=\mu {\left|E_{out}\left(0,t\right)\right|}^2=2\mu E_0^2{J}_8^2\left(m\right)\left\{1+\cos \left[16{\omega }_{RF}+\pi s\left(t\right)\right]\right\}$$

(9)

where $\mu$ is the responsibility of the PD.

In the fiber transmission system, the dispersion effect is caused by the different velocities of the ±8th-order sideband. The composite signal of the four MZMs is expressed as shown below:

$$E_{out}\left(z,t\right)=e^{-\gamma z}{E}_0{J}_8(m)e^{j\left[{\omega }_{0}t+8{\omega }_{RF}t-\beta \left({\omega }_0+8{\omega }_{RF}\right)z\right]+j\pi s\left(t-t^{\prime }\right)}+e^{-\gamma z}{E}_0{J}_{-8}(m)e^{j\left[{\omega }_{0}t-8{\omega }_{RF}t-\beta \left({\omega }_0-8{\omega }_{RF}\right)z\right]}$$

(10)

where $\gamma$, $\beta \left(\omega \right)$ is the loss coefficient and transmission constant of fiber, respectively, $t^{\prime }={\left({\omega }_0+8{\omega }_{RF}\right)}^{-1}\beta \left({\omega }_0+8{\omega }_{RF}\right)z$ is the delay time of the codeword, and z is the length of the fiber. Here, nonlinear effects are not considered.

Comparing Equation (10) with Equation (8), it can be seen that their spectra are the same.

Expanding the term of $\beta \left({\omega }_0\pm 8{\omega }_{RF}\right)$ in Equation (10) with Taylor expansion and neglecting the terms ${\beta }^n\left({\omega }_0\right)\left(n\ge 3\right)$ (on account of their small amplitudes), we obtain the following:

$$\beta \left({\omega }_0\pm 8{\omega }_{RF}\right)=\beta \left({\omega }_0\right)\pm 8{\omega }_{RF}{\beta }^{\prime }\left({\omega }_0\right)+24{{\omega }^2}_{RF}{\beta }^{″}\left({\omega }_0\right)$$

(11)

The optoelectrical current of the PD is obtained and expressed as given below:

$$\begin{array}{cc}\hfill I\left(z,t\right)& =\mu {\left|E_{out}\left(z,t\right)\right|}^2\hfill \\ & =2e^{-2\gamma z}\mu E_0^2{J}_8^2(m)\left\{1+\cos \left[16{\omega }_{RF}t+\beta \left({\omega }_0+8{\omega }_{RF}\right)z-\beta \left({\omega }_0-8{\omega }_{RF}\right)z-\pi s\left(t-t^{\prime }\right)\right]\right\}\hfill \\ & =2e^{-2\gamma z}\mu E_0^2{J}_8^2(m)\left\{1+\cos \left[16{\omega }_{RF}t+16{\omega }_{RF}{\beta }^{\prime }\left({\omega }_0\right)z-\pi s\left(t-t^{\prime }\right)\right]\right\}\hfill \end{array}$$

(12)

In the derivation of Equation (12), Equation (11) is substituted into Equation (10).

From Equation (12), we can see that after the transmission distance of z, the bit walk-off effect is eliminated; the effect on codewords is delayed.

4. Simulation Experiments

4.1. Transmission Test

4.1.1. Simulation Circuit with OptiSystem

According to Figure 1, the simulation circuit of the 16-tupling ROF system based on four MZMs connected in parallel to overcome the bit walk-off effect is built using the OptiSystem simulation tool, as shown in Figure 2. Figure 2a is the circuit of the system. Figure 2b is the circuit of the subsystem; this is labeled as “DPMZM” in Figure 2. Figure 2c is the circuit of the subsystem labeled as “composite RF signal” in Figure 2. The meaning of each abbreviated word in Figure 2 is the same as that in Figure 1.

4.1.2. Simulation Parameter

The key parameters in the simulation circuit are listed in

Table 2. The main parameters in the simulation circuit.

Device Name	Parameter	Value
CW	Frequency	193.1 THz
	Linewidth	10 MHz
	Power	0 dBm
data	rate	Gbps
	Modulation mode	NRZ
RF-LO	Frequency	10 GHz
	Amplitude	7.0284 V
phase modulator	P	3π/16
electrical gainer	G	6
MZM	Vπ	4 V
	Insertion loss	5 dB
	Extinction ratio	40 dB
Fiber	Dispersion Loss	16.75 ps/nm/km 0.2 dB/km
OA	Gain	25 dB
	Noise figure	4 dB
PD	Responsivity	1 A/W
BPF	Frequency	160 GHz
	Bandwidth	1.5 bit rate
LPF	Cutoff frequency	2.5 GHz

.

4.1.3. Simulation Experiment Results

The optical spectrum at point A in Figure 1 is shown in Figure 3. We can see from Figure 3 that the main components of the composite signal of four MZMs are the ideal −8th-order optical sidebands and the +8th-order optical sideband with data.

The spectrum of the optoelectrical current from the PD is shown in Figure 4. In Figure 4, the signal situated at 160GHz is the frequency 16-tupling MMW with data.

The relation curve between the Q value and the transmission distance is shown in Figure 5. From Figure 5, it can be seen that even if the transmission distance exceeds 45 km, the Q value is greater than 6, and it still meets the requirements of a digital transmission system.

Figure 6 shows the relation curve BER with the accepted power in the PD with different fiber lengths. In Figure 6, “BTB” means the back-to-back system. It can be seen from Figure 6, corresponding to a BER of 10⁻⁹, that after the optical signal transmitting across distances of 10 km, 20 km, and 30 km, the power penalty is 0.5 dB, 0.9 dB, and 1.7 dB, respectively.

4.2. Impact of Irrational Parameters on System Performance

In our simulation experiments, most parameters are chosen as the default values of the OptiSystem, the key parameters are obtained by theoretical analysis. The parameters’ values may deviate from their default or analysis values in the engineering application. It will inevitably reduce the performance and increase the system’s BER. This needs to be analyzed.

In practical systems, the main parameters affecting the BER are as follows: (a) the MZM’s modulation index m, (b) the PM’s modulation index P, and (c) the EG’s gain G. The m, which is decided by the RF signal voltage, influences the suppression of unwanted sidebands. The values of the P and G directly determine that the achievement of the data is modulated only on the +8th-order optical sideband.

4.2.1. Impact of Modulation Index of MZM

According to the analysis given in Section 3.2, the m’s theoretical value is 5.52. According to $m=\left(V_{RF}/V_{\pi }\right)\pi$ ($V_{\pi }=4 \mathrm{V}$), the $V_{RF}$’s theoretical value is 7.0284 V. When the $V_{RF}$ changes in the range of ±0.2 V around 7.028 V, i.e., in the range of 6.8284 V < V_RF < 7.2284 V, the corresponding m value changes in the scope of 5.363 < m < 5.677.

Figure 7 shows the relationship between the $V_{RF}$ and the BER for different transmission distances. From Figure 7, we can see that when $V_{RF}$ is less than 7.0284 V, the system’s BER is more susceptible to deviations in $V_{RF}$.

4.2.2. Effect of Phase Modulation Index P of PM

The theoretical value of PM’s P is 33.75. When P deviates from ±3 around 33.75, the deviation of P versus BER is shown in Figure 8, from which we can see that, for the BTB system, the system’s BER is more susceptible to the deviation of P when it is greater or smaller than 33.75.

4.2.3. The Effect of the Gain G of EG

The theoretical value of EG’s G is 6. When G deflects from 6, the relationship of the BER versus G is shown in Figure 9, from which it can be seen that the system performance degrades more quickly when G is less than 6.

5. Discussion

(1)

Transmission distance

For the conventional OCS system, the maximum transmission distance for the system’s eye diagram closure can be obtained by Equation (13) [41,42,43]

$$z<{\displaystyle \frac{\eta \tau c}{{\lambda }_c^{2}D{f}_{RF}}}$$

(13)

where $\eta$ is the duty cycle, $D$ is the fiber’s dispersion parameter, $\lambda$ is the central optical carrier wavelength, and $\tau$ is the codeword period.

In our simulation experiments, the default values of $\eta$, $D$, $\lambda$ and $\tau$ are $\eta =1$, $D=16.75 \mathrm{ps}/\mathrm{nm}·\mathrm{km}$, $\lambda =1552.52 \mathrm{nm}$ and $\tau =0.4 \mathrm{ns}$, respectively. Substituting those values into Equation (13), we obtain the maximum transmission distance of 18.25 km. For our devised system, we can see from Figure 4 that when at a length of 45 km, the BER is greater than 10⁻⁹.

(2)

Scope of application

Our devised system is only suitable for the optical MMW generation schemes, which do not need to adopt methods to suppress the central carrier. The origin can be explained as follows: if the output composite signal contains the central carrier, the central carrier will be modulated by the data. In this case, the conventional carrier canceling methods could not suppress the carrier completely.

(3)

Carrier reuse

In our designed scheme, due to the limitation on the length of the manuscript, we do not devise the uplink. To cut down the BS’s cost of ROF, carrier reuse technology is often used in the BS.

In our devised system, to effectively cut down the cost of the BS, part of the power of the ideal −8th-order sideband can be reflected out by the FBG and used for the uplink carrier for carrier reuse. One of the main problems faced when adopting this approach is how much the reflectivity of the FGB is chosen. The larger the reflectivity, the better the performance for the uplink and the worse the performance for the downlink. To ensure the best performance for both the uplink and downlink, the reflectivity of the FBG is needed for optimization.

(4)

Nonlinear

Due to the low power in our designed system, we did not consider the fibers’ nonlinear effects. If we apply our method to a WDM ROF system, the power of the system is high, and we need to consider the nonlinear effect of the fiber.

(5)

The basis to choose the values of the PM’s P and the EG’s G.

According to Equation (1), the composite RF signal loaded onto the MZM is $V_i\left(t\right)=V_{RF}\cos \left[{\omega }_{RF}t+P·s\left(t\right)+{\phi }_i\right]+G·s\left(t\right)$, For the sake of simplicity, here we have ${\phi }_i=0$. When $m$ is chosen 5.52, according to Equation (7), the output signal of four MZMs in parallel is obtained as follows:

$$E_{out}=8E_0{e}^{j{\omega }_{0}t}\left\{J_8(m)e^{j8\left[{\omega }_{RF}t+Ps\left(t\right)\right]+jGs(t)\pi /V_{\pi }}+J_{-8}(m)e^{-j8\left[{\omega }_{RF}t+Ps\left(t\right)\right]+jGs(t)\pi /V_{\pi }}\right\}$$

(14)

To modulate the data on the +8th-order sideband only, the following relation needs to be satisfied.

$$\left\{\begin{array}{c}8P+G\pi /V_{\pi }=\left(2m-1\right)\pi \\ -8P+G\pi /V_{\pi }=2n\pi \end{array}\right.m,n=0, \pm 1, \pm 2, \pm 3, \cdots$$

(15)

From Equation (15), we obtain the following:

$$G=\left(m+n-1/2\right)V_{\pi }, P=\left(2m-2n-1\right)\pi /16$$

(16)

In Table 2, we see that $V_{\pi }=4$. For the setting $m=2,n=0$, we receive $G=6$, $P=3\pi /16$; then, Equation (14) is changed as given below:

$$e^{j8\left[{\omega }_{RF}t+\frac{3\pi }{16}s\left(t\right)\right]+j\frac{3\pi }{2}s\left(t\right)}+e^{-j8\left[{\omega }_{RF}t+\frac{3\pi }{16}s\left(t\right)\right]+j\frac{3\pi }{2}s\left(t\right)}=e^{j\left[8{\omega }_{RF}t+3\pi s\left(t\right)\right]}+e^{j\left(8{\omega }_{RF}t\right)}$$

(17)

As seen in Equation (17), the data signal is modulated only on the +8th-order sideband.

It should be noted that m and n can take any integer. Therefore, there can be more than one value of P and G. For example, if $m=n=1$, according to Equation (16), we obtain $G=6$, $P=-\pi /16$. Then, we have the following:

$$e^{j8\left[{\omega }_{RF}t-\frac{\pi }{16}s\left(t\right)\right]+j\frac{3\pi }{2}s\left(t\right)}+e^{-j8\left[{\omega }_{RF}t-\frac{\pi }{16}s\left(t\right)\right]+j\frac{3\pi }{2}s\left(t\right)}=e^{j\left[8{\omega }_{RF}t+\pi s\left(t\right)\right]}+e^{j\left(8{\omega }_{RF}t\right)}$$

(18)

As seen in Equation (18), the data signal is modulated only on the +8th-order sideband.

(6)

Optical energy efficiency

In this paper, four MZMs are connected in parallel to produce 8n-order sideband signals, where ±8th-order sidebands are useful sidebands and the rest of the sidebands are useless sidebands; thus, the energy efficiency of the light is a question of the power of the ±8th-order sidebands as a percentage of the power of the input optical carrier. This can be calculated using the following equation:

$$Energy efficiency={\displaystyle \frac{E_8^2+E_{-8}^2}{{\displaystyle \sum _{-\infty }^{\infty }{E}_{8n}^2}}}$$

(19)

where $E_{8n}$ is the amplitude of the optical sideband.

From Section 3.1, it follows that $E_0\left(t\right)=8E_0\exp \left(j{\omega }_{0}t\right)$, and $E_8(m)=E_{0}8J_8(m)$. Neglecting the insertion loss of MZM and the 3 dB coupler, we receive ${\displaystyle \sum _{-\infty }^{\infty }{E}_{8n}^2}=1$. In our manuscript, $m=5.52$ we see that $J_8(5.52)=0.0344$, according to Equation (19), and the optical energy efficiency is obtained as $Energy efficiency=2{\left[8\times J_8(5.52)\right]}^2=15.14\%$.

It can be seen that the optical energy efficiency in ROF is still relatively low, and how to improve the energy efficiency of ROF remains an important research topic.

6. Conclusions

In this paper, a new scheme is proposed in order to produce frequency 16-tupling without a filter and transmitting fibers without the bit walk-off effect by using four MZM connected in parallel.

In the case of the absence of data modulation, by setting the proper amplitude and initial phase of the RF driving signal, the composite signal from the four MZMs comes from the ±8th-order sidebands.

In the case of data modulation, the data are divided into two beams: one of them modulates the RF driving signal with the PM, and the other is magnified by the EG; then, the two beams are added together to form the composite RF driving signal. By setting the proper modulation index of the PM and gain of the EG, the data are carried only on the +8th-order sideband. When the values of P and G are taken as 3π/16 and 6; for the case of BER = 10⁻⁹, the maximum transmission distance can reach up to 45 km.

We also studied the effect on the Q value or BER when the amplitude of the RF driving signal, the EP’s P, and the EG’s G deviate from their theoretical values.

Our devised system can effectively eliminate the bit walk-off effect, significantly raise the fiber distribution length, and have potential application values in the MMW ROF.