© The Author(s), published by EDP Sciences, 2025

1 Introduction

With the increasing bandwidth demand by users of modern wireless communication systems, such as smartphones, HDTV, virtual reality, 5G, 6G, and so on, the frequency of carriers has been extended to the millimeter waves (MMW) and terahertz waves [1, 2]. To generate the MMWs above 100 GHz in the electric domain, the frequency is limited by the frequency response of the electronics [3]. Distributing the MMW in free space results in a great loss problem. To generate and distribute the MMW in the optical domain can overcome those problems, and it has become the key technology in the radio over fiber (ROF) system for wireless communication systems [4, 5]. In general, many methods are used to generate two coherent optical carriers whose frequency spacing is the desired MMW. The MMW is generated by beating the two coherent optical carriers in the photodetector (PD). The two coherent optical carriers are called optical MMW. Many methods to generate optical MMW have been proposed, such as optical heterodyning methods, optoelectronic oscillator methods [6], mode-locked laser methods [7], external modulator methods [8], optical nonlinear effect methods [9], and so on. Among them, the external method based on Mach-Zehnder modulators (MZMs) offers several advantages, such as a high-frequency multiplication factor (FMF), larger tunability, higher reliability, higher conversion efficiency, and higher receiver sensitivity [8]. The main external modulators include phase modulators (PM), Mach-Zehnder modulators (MZM), and polarization modulators (PoIM). The generation of MMW by PM suffers from the low purity of the resulting MMW. The generation of MMW by MZM suffers from problems such as DC drift and a limited extinction ratio. The generation of MMW based on PoIM does not have the problems of DC drift and limited extinction ratio, and it has been a hot research topic in recent years. Using PoIM, frequency quadrupling [10, 11], sixtupling [12, 13], octupling [14], 12-tupling [15], 16-tupling [16], 18-tupling [17], 24-tupling [18, 19], 32-tupling [20, 21] MMW could be obtained.

In the MMW ROF system, the optical wave is used as the carrier of the MMW, and the MMW is used as the carrier for the data signal. Compared with the optical carrier, the MMW is a subcarrier. The data modulation format directly affects the performance of the MMW ROF. The dispersion of optical fibers seriously affects the performance of ROF systems [22, 23]. There are three modulation formats in the MM ROF system, they are single-sideband (SSB) modulation, double-sideband (DSB) modulation, and optical carrier suppression double-sideband modulation (OCS) [24]. In the MMW ROF system based on external modulators, (a) when the generated main sidebands are the two ±nth-order sidebands with a carrier, the system is called DSB modulation system; and (b) when the generated main sidebands are ±nth order sidebands with carrier suppression, the system is called an OCS modulation system; (c) when the generated main sidebands is nth (or −nth) order sidebands with carrier, the system is called an SSB modulation system. For the DSB modulation system, the fiber chromatic dispersion not only leads to periodic attenuation effects but also to bit walk-off effects [24, 25], which will greatly limit the transmission distance of the system. When the digital signal is modulated on two coherent optical carriers, the code word of the digital signal is separated in the time domain caused by optical fiber dispersion. This effect is known as the bit walk-off effect. For the OCS and SSB modulation formats, the systems can overcome the periodic fading effect, but the bit walk-off effect cannot be eliminated [24, 26, 27]. If the data are modulated onto one of the two coherent optical sidebands, the periodic fading effect and the bit walk-off effect can be eliminated. Some researchers have proposed methods to modulate the data signal onto only one sideband using an optical filter [28–30]. The main problem associated with those methods is that the use of filters not only increases the insertion loss and the system cost, but also limits the tunability of the system. How to modulate the data signal on one of two sidebands without using a filter is currently a hot topic. Some research scholars have proposed a filterless method to modulate the data signal only to the 1st order sideband using dual parallel MZM (DPMZM) [31], and the main problems of this method are complex structure, high cost, and low-FMF. To simplify the system structure and improve FMF, Zihang Zhu et al. proposed three kinds of MMW ROF systems without a filter by MZM to modulate the data only on the −2nd, −4th, and −6th order sidebands with frequency quadrupling, octupling, 12-tupling [32–34]. Since all of the above schemes are realized with MZM, these schemes still suffer from the limitation extinction ratio and the DC drift of MZM. For the MMW ROF system based on PoIM, if the data signal is modulated only on one sideband without adopting a filter, such a system can overcome the extinction ratio limitation and DC drift problem caused by MZM, and also can eliminate the bit walk-off effect.

To reduce the cost of the base station (BS) in MMW ROF systems, carrier reuse is needed in the BS. The most common method for carrier reuse is to reflect a portion of the nth (or −nth) order sideband without data by a fiber Bragg grating (FBG) and use it as the uplink carrier [35, 36]. This approach leads to two problems, one is that the operating frequency point of the FBG must be located in the nth (or −nth) order optical sideband, it will lead to the tunability of the system is poor; the other problem is that since the FBG reflects part of the power of the nth order sideband, this will inevitably lead to the power imbalance for the ±nth order sidebands, it will degrade the performance of the downlink.

In this paper, we propose a novel frequency octupling inserting pilot MMW ROF system based on DP-PoIM. In our designed system, the bit walk-off effect can be overcome, and the carrier reuse is realized by inserting a pilot signal. We theoretically analyze its operation principle, design simulation links to verify its feasibility, and analyze the impact of irrational parameters in the system Bit Error Rate (BER).

This paper is organized as follows: in Section 2, a schematic diagram of the system structure is provided; in Section 3, the operation principle of the system is analyzed; in Section 4, the simulation experiments are designed and the simulation results are given; in Section 5, the system performance is studied with simulation when the main devices parameters deviate from their theoretical or design values; in Section 6, the discussion is given; and in Section 7, a conclusion is given.

2 System design

Figure 1 shows the schematic structure of the system we designed. The main optical components in the system include a continuous wave laser diode (CW-LD), two polarization modulators (PolM1 and PolM2), a polarization beam splitter (PBS), a polarization beam combiner (PBC), a polarizer (Pol), a fiber Bragg grating (FBG), an optical amplifier (OA), and two photodetectors (PD1 and PD2). The main electrical components in the system include two radio frequency (RF) local oscillator signal generators (RF-LO1, RF-LO2), an electrical amplifier (EA), an electrical phase modulator (PM), an electrical gainer (EG), a bandpass filter (BPF), three electrical phase shifter (EPS1, EPS2, EPS3), two low-pass filters (LPF1, LPF2), a mixer (Mixer), and an intensity modulator (IM). The dashed line and the solid line in Figure 1 indicate the electrical signal link and the optical signal link, respectively.

Fig. 1 Schematic of the frequency octupling MMW ROF without bit walk-off effect by PolMs.

The “composite RF signal”, “DP-PolM” and “recovery” modules are shown at the bottom in Figure 1, respectively. The “composite RF signal” module is constructed by the EG, PM and RF-LO1. It is used to form the composite RF signal. The “DP-PolM” module is formed by connecting PolM1 and PolM2 in parallel by the PBS and PBC. The “Recovery” module, which is used to recover the data, is constructed by the BPF, RF-LO2, Mixer, and LPF1.

To compare our scheme with the conventional frequency octupling MMW ROF system, we added an electrical switch (ES) to the module of the “composite RF signal” and an optical switch (OS) to the central station (CS). When the ES and OS are open, the system is the conventional frequency octupling MMW ROF system. The ES and OS are not necessary in the real system.

3 Operation principle

3.1 In the case of no data loading

The linearly polarized optical carrier from the CW-LD is expressed as E₀(t) = E₀exp(jω₀t), where E₀ and ω₀ are the amplitude and angular frequency of optical carrier, respectively. The RF drive signal output from RF-LOi (i = 1, 2) is expressed as V_i(t) = V_RFcos[ω_RFt + φ_i], where V_RF, ω_RF, and φ_i are the amplitude, the angular frequency, and the initial phase of the RF drive signal, respectively. The RF drive signals loaded onto PolMi (i = 1, 2) are V_i(t).

The optical carrier from the CW LD is split into two beams; one is injected into the DP-PolM, and the other is used as the pilot signal. The beam injected into the DP-PoIM is divided into upper and lower paths by the PBS first and then fed into PolM1 and PolM2, respectively. Let the polarization direction of the input optical carriers of PolM1 and PolM2 be at an angle of α = −45°, 45° with the main axis of PolM1 and PolM2, respectively.

The outputs of PolM1 and PolM2 are denoted respectively as follows [16]:$$\left[\begin{array}{c}{E}_{1x}\\ E_{1y}\end{array}\right]=\frac{1}{2}{E}_0{e}^{j{\omega }_{0}t}\left[\begin{array}{c}{e}^{\mathrm{jm}\cos \left({\omega }_{\mathrm{RF}}t+{\phi }_1\right)}\\ -e^{-\mathrm{jm}\cos \left({\omega }_{\mathrm{RF}}t+{\phi }_1\right)}\end{array}\right]$$(1) $$\left[\begin{array}{c}{E}_{2x}\\ E_{2y}\end{array}\right]=\frac{1}{2}{E}_0{e}^{j{\omega }_{0}t}\left[\begin{array}{c}{e}^{\mathrm{jm}\cos \left({\omega }_{\mathrm{RF}}t+{\phi }_2\right)}\\ e^{-\mathrm{jm}\cos \left({\omega }_{\mathrm{RF}}t+{\phi }_2\right)}\end{array}\right]$$(2)where m = π(V_RF/V_π) is the modulation index and V_π is the half-wave voltage of the PoIMi.

Combing the output optical signals from PolM1 and PolM2 by the PBC, the output signals of the PBC is the output signal of the DP-PoIM, it can be expressed as follows [17]:$$\left[\begin{array}{c}{E}_x\\ E_y\end{array}\right]=\frac{1}{2}\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]\left[\begin{array}{c}{E}_{1x}\\ E_{1y}\end{array}\right]+\frac{1}{2}\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]\left[\begin{array}{c}{E}_{2x}\\ E_{2y}\end{array}\right]=\left[\begin{array}{c}\frac{E_{1x}}{2}-\frac{E_{1y}}{2}+\frac{E_{2x}}{2}+\frac{E_{2y}}{2}\\ -\frac{E_{1x}}{2}+\frac{E_{1y}}{2}+\frac{E_{2x}}{2}+\frac{E_{2y}}{2}\end{array}\right].$$(3)

The input signal of the Pol is the output signal of the DP-PolM. When the polarization angle of the Pol is set to 0°, the output signal of the Pol can be expressed as follows [17]:$$E_{\mathrm{Pol}}=\left[\begin{array}{c}1\\ 0\end{array}\right]\left[\begin{array}{c}{E}_x\\ E_y\end{array}\right]=\frac{E_{1x}}{2}-\frac{E_{1y}}{2}+\frac{E_{2x}}{2}+\frac{E_{2y}}{2}.$$(4)

The φ₂ is introduced by EPS2. Setting φ₁ = 0, φ₂ = φ = π/2, bringing equation (1) and equation (2) into equation (4), and applying the $e^{\mathrm{jx}\cos \theta }=\sum _{n=-\infty }^{\infty }\mathrm{}{j}^n{J}_n\left(x\right)e^{\mathrm{jn\theta }}$ constant equation [17], we get$$E_{\mathrm{Pol}}=E_0{e}^{j{\omega }_{0}t}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{}{J}_{4n}\left(m\right)e^{j4n{\omega }_{\mathrm{RF}}t}.$$(5)

Setting m = 2.405, there are J₀(2.405) = −9.05566 × 10⁻⁵, J₄(2.405) = 0.06476, J₈(2.405) = 9.2217 × 10⁻⁵, J₁₂(2.405) = 1.7068 × 10⁻⁸. From that we can see that the optical carrier (0th order sideband) is well suppressed, ±4th and ±8th order sidebands are the dominant sidebands and the largest spurious sidebands, respectively. The sideband suppression ratio (OSSR) is OSSR = 20lg[J₄ (2.405)/J₈ (2.405)] = 56.93 dB. The amplitudes of the ±4nth (n > 1) sidebands are too small and can be neglected. After neglecting the terms of the ±4nth (n = 0, n > 1) order sidebands, equation (5) can be simplified as follows:$$E_{\mathrm{Pol}}=E_0\left[J_4\left(m\right)e^{j\left({\omega }_{0}t+4{\omega }_{\mathrm{RF}}t\right)}+J_{-4}\left(m\right)e^{j\left({\omega }_{0}t-4{\omega }_{\mathrm{RF}}t\right)}\right]$$(6)

We can see from equation (6) that the output signal of DP-PolM is ±4th order sidebands signal.

Combining the output signal of the DP-PoIM and the pilot signal by the OC2, the output signal of the CS2 can be expressed as follows:$$E_{\mathrm{out}}=E_{\mathrm{Pol}}+E_{\mathrm{os}}=E_0\left[J_4\left(m\right)e^{j\left({\omega }_{0}t+4{\omega }_{\mathrm{RF}}t\right)}+J_{-4}\left(m\right)e^{j\left({\omega }_{0}t-4{\omega }_{\mathrm{RF}}t\right)}\right]+E_0{e}^{j{\omega }_{0}t}.$$(7)

3.2 In the case of loading data signals

3.2.1 Back-to-back system

The data signal generated from the data source is expressed as s(t) = ∑_nI_ng(t − nT). where I_n ∈ {0, 1} is the binary sequence, g(t) is the code form function, and T is the code word duration. The RF drive signal $V_i^{\mathrm{\prime }}\left(t\right)$ of the PolMi are composite RF drive signal. The composite RF signal is formed by combining two signals. One is the RF driving signal, which is generated by the RF-LO1 first and then modulated by the data s(t). The other is the amplified s(t) which is amplified by the EG. Let ${V\text{'}}_i\left(t\right)$ be denoted as $V_i^{\mathrm{\prime }}\left(t\right)=V_{\mathrm{RF}}\cos \left[{\omega }_{\mathrm{RF}}t+{\phi }_i+\mathrm{Ps}\left(t\right)\right]+\mathrm{Gs}\left(t\right)$, where P and G is the phase modulation index of the PM and the amplification of the EG, respectively. Taking P = 3π/8, G = 1.5, then $V_i^{\mathrm{\prime }}\left(t\right)$ can be expressed as follows$${V_i}^{\mathrm{\prime }}\left(t\right)=V_{\mathrm{RF}}\cos \left[{\omega }_{\mathrm{RF}}t+{\phi }_i+\frac{3\pi }{8}s\left(t\right)\right]+\frac{3}{2}s\left(t\right).$$(8)

Replacing V_i(t) with ${V\text{'}}_i\left(t\right)$ and using the same process of deriving equation (5) to equation (7), the output signal of CS is obtained as follows:$${E\mathrm{\prime }}_{\mathrm{out}}\left(0,t\right)=E_0\left[J_4\left(m\right)e^{j\left({\omega }_{0}t+4{\omega }_{\mathrm{RF}}t\right)+j3\pi s\left(t\right)}+J_{-4}\left(m\right)e^{j\left({\omega }_{0}t-4{\omega }_{\mathrm{RF}}t\right)}\right]+E_0{e}^{j{\omega }_{0}t}.$$(9)

According to the previous analysis, from equation (9), we can see that the data signal s(t) is modulated on the +4th sideband only.

In the base station (BS), after filtering out the pilot signal (0th order sideband) in the input signal by the FBG, the output of the FBG is only the ±4th order sidebands. After beating the ±4th order sidebands in the PD1, the photocurrent signal from PD1 is expressed as follows:$$I\left(0,t\right)=\mathfrak{R}{\left|{E\mathrm{\prime }}_{\mathrm{out}}\left(0,t\right)\right|}^2=2\mathfrak{R}{E}_0^2{J}_4^2\left(m\right)\left\{1+\cos \left[8{\omega }_{\mathrm{RF}}t+3\pi s\left(t\right)\right]\right\}$$(10)where ℜ is the responsivity of the PD1.

We can see from equation (10) that the frequency octupling MMW signal with s(t) is generated.

3.2.2 Fiber transmission system

In the optical fiber transmission system, the ±4th order sidebands have different group velocities due to the fiber dispersion. Let the fiber transmission distance be represented as z, the input signal of the BS is expressed as follows:$$\begin{array}{c}{E\mathrm{\prime }}_{\mathrm{out}}\left(z,t\right)=e^{-\gamma z}{E}_0{J}_4\left(m\right)e^{j\left[{\omega }_{0}t+4{\omega }_{\mathrm{RF}}t-\beta \left({\omega }_0+4{\omega }_{\mathrm{RF}}\right)z\right]+j3\pi s\left(t-t\mathrm{\prime }\right)}+\\ e^{-\gamma z}{E}_0{J}_{-4}\left(m\right)e^{j\left[{\omega }_{0}t-4{\omega }_{\mathrm{RF}}t-\beta \left({\omega }_0-4{\omega }_{\mathrm{RF}}\right)z\right]}+E_0{e}^{j\left({\omega }_{0}t-\beta {\omega }_{0}z\right)}\end{array}$$(11)where, γ and β(ω) is the loss coefficient and the transmission constant of the fiber, and t′ = (ω₀ + 4ω_RF)⁻¹ β(ω₀ + 4ω_RF)z is the time delay of the code word. The effect of fiber nonlinearity is neglected here. The reasons can be found in Section 6.2.

Comparing equation (11) with equation (9), we find that the spectrum of the signal is not changed when the fiber transmission distance is z.

Expanding the term of β(ω₀ ± 4ω_RF) in equation (11) with Taylor expansion, and ignoring the term β ⁽ⁿ⁾(ω₀)(n > 2) for their very small amplitude, we get$$\beta \left({\omega }_0\pm 4{\omega }_{\mathrm{RF}}\right)=\beta \left({\omega }_0\right)\pm 4{\omega }_{\mathrm{RF}}{\beta }^{\mathrm{\prime }\left({\omega }_0\right)}+8{\omega }_{{ }_{\mathrm{RF}}}^2{\beta }^{\mathrm{\text{'}\text{'}}\left({\omega }_0\right)}.$$(12)

After filtering out the pilot signal via the FBG, injecting the $E_{\mathrm{out}}^{\mathrm{\prime }}\left(z,t\right)$ into the PD1 for photoelectric conversion, the photocurrent from PD1 can be expressed as follows:$$I\left(z,t\right)=\mathfrak{R}{\left|E_{\mathrm{out}}^{\mathrm{\prime }}\left(z,t\right)\right|}^2$$(13)

Substituting equations (11) and (12) into equation (13), we obtain that$$I\left(z,t\right)=2e^{-2\gamma z}\mathfrak{R}{E}_0^2{J}_4^2\left(m\right)\left\{1+\cos \left[8{\omega }_{\mathrm{RF}}t+8{\omega }_{\mathrm{RF}}{\beta }^{\mathrm{\prime }\left({\omega }_0\right)}z-3\pi s\left(t-t^{\mathrm{\prime }}\right)\right]\right\}.$$(14)

We can see from equation (14) that the data signal is not subject to the bit walk-off effect, except for the delay.

4. Simulation experiment

4.1 Simulation parameter setting

According to Figure 1, the simulation link is set up by the OptiSystem simulation software. Table 1 shows the settings of the main parameters in the simulation link.

4.2 Results of the simulation experiment

Figure 2a is the optical spectrum of point A in Figure 1, from which we can see that: all the sidebands are well suppressed except the ±4th order sidebands; the peak power of the +4th order sideband is lower than that of the −4th order; the data is only modulated on the +4th order sideband.

Fig. 2 Optical spectrum of points A–D in Figure 1.

Figure 2b is the optical spectrum of point B in Figure 1, from which we can see that the 0th order sideband is the inserted pilot signal.

Figure 2c is the optical spectrum of point C in Figure 1, is the optical spectrum after filtering out the pilot signal. Figure 2c is similar to Figure 2a.

Figure 2d is the optical spectrum of point D in Figure 1, from which we can see that the uplink data has been modulated onto the pilot signal.

Figure 3a is the spectrum of point E in Figure 1, from which we can see that the output of the PD1 contains the frequency octupling MMW signal with downlink data.

Fig. 3 Spectrum of points E and F in Figure 1.

Figure 3b is the spectrum of point F in Figure 1, is the baseband signal of the uplink data.

Figure 4a shows the time-domain waveform of the data signal output by the NRZ generator in an optical transmitter. Figure 4b shows the waveform of the data signal obtained by recovering the signal through the data recovery circuit of the receiver. Comparing Figures 4a and 4b, it can be seen that the data signal has been fully recovered.

Fig. 4 Time domain waveform of data.

In Figure 5, the relation between the Q-factor with the distance of fiber transmission z is shown. The eye diagrams of the data signal at those distances of 0 km (BTB), 20 km, 60 km, and 80 km are marked in Figure 4. From Figure 4, we can see that, in the case of the Q >6, the transmission distance z can be reached up to 80 km.

Fig. 5 The relationship of the Q factor with the fiber transmission distance.

Figures 6a and 6b show the relationship between the BER and the received power of PD1 and PD2 for downlink and uplink, respectively. From Figure 6a, we can see that the power penalty in the downlink is 0.29 dB and 0.72 dB for 20 km and 40 km transmission distances, respectively. From Figure 6b, we can see that the power penalty in the uplink is 0.1 dB and 0.22 dB for 20 km and 40 km transmission distances, respectively.

Fig. 6 The relationship between the BER and the received power of PD1 and PD2. (a) Downlink; (b) Uplink.

4.3 Compare the conventional frequency octupling ROF system with our scheme

When the OS and ES in Figure 1 are open, the system becomes to a conventional frequency octupling MMW ROF system in which the downlink data is simultaneously modulated onto the ±4th order sidebands.

For the conventional ROF system, the maximum transmission distance satisfies the following relationship [21]:$$z<\frac{\tau c}{8{\lambda }_{\mathrm{c}}^{2}D{f}_{\mathrm{RF}}}$$(15)where η is the duty cycle of code word, τ is code word period, D is the dispersion constant of fiber, f_RF is the frequency of RF driving signal, and λ_c is the center wavelength of the optical carrier wave, respectively. Setting η = 1, τ = 0.4 nsf_RF = 10 GHz, D = 16.75 ps/nm/km, λ = 1552.52 nm the maximum transmission distance is obtained to be 37 km by equation (15).

Figure 7 is the eye diagram obtained at different transmission distances in the conventional frequency octupling ROF system. From Figure 7, we can see that the width of the code word becomes narrower and narrower as the transmission distance increases. The eye diagram is almost closed when the distance is 37 km. These results are in agreement with equation (15).

Fig. 7 Eye diagram at a different transmission distance. (a) BTB; (b) 10 km; (c) 20 km; (d) 37 km.

To compare our scheme with the conventional scheme, we give the eye diagrams at different fiber transmission distances for our designed system in Figure 8. From Figure 8, we can see that the eye diagram is still open even the transmission distance reaches 80 km.

Fig. 8 Eye diagrams at different transmission distances in our scheme. (a) BTB; (b) 20 km; (c) 60 km; (d) 80 km.

4.4 Compare the conventional carrier reuse system with our scheme

When the OS is on and the ES is off, the system is the conventional carrier reuse system. The central frequency of the FBG is shifted to 193.06 THz, which is the frequency of the −4th order optical sideband.

Figure 8 is the relationship between BER and the received powers of the PD1 in the conventional carrier reuse system and our system.

We can see from Figure 9 that the improvements in receiver sensitivity are increased with the fiber transmission distances. Under the condition of BER = 10⁻⁹, the improvement of the sensitivity of receivers is listed in Table 2.

Fig. 9 The relationship of the BER of the downlink with the received power of PD1.

5 System stability

The values of the parameters of the main devices are chosen based on the values of the theoretical analysis or the default values of the simulation system. In practice, the parameters affecting the BER (or Q-factor) mostly are, (1) the amplitude of the RF drive signal, which determines the modulation index of PolM and thus affects the amplitude of the sidebands; (2) the modulation index P of the PM and the amplification factor G of the EG, they are the key parameters which realize the data is modulated to the +4th order sideband only; (3) the polarization angle of the Pol; (4) the value of the 90° EPS, which is the key parameter to determine whether the odd terms in the 2n-order sidebands in the output of the PolM1 and PolM2 can be canceled each other. When their values deviate from their theoretical values or default values, the BER is bound to increase. To ensure the BER < 10⁻⁹, it is necessary to investigate the range within which these parameters are allowed to deviate from their theoretical or default values.

5.1 The influence of V_RF on the Q-factor

According to the previous theoretical analysis, the theoretical value of the m is 2.405. When m varies in the range of 2.268 < m < 2.551, according to m = (V_RF/V_π)π (the default value of V_π is 1 V in the simulation system), the corresponding range of the V_RF is 0.7155 < V_RF < 0.8155. When the V_RF varies in this range, the relationship curve between the Q-factor and V_RF is obtained by simulation and shown in Figure 10. From Figure 10, we can see that Q > 6 is satisfied when V_RF varying in the range of 0.722~0.812.

Fig. 10 The relationship of the Q factor with VRF.

5.2 The influence of PM’s P on the Q factor

According to the previous theoretical analysis, the theoretical value of the PM’s phasing index P is 3π/8 = 67.5°. When P varies within the range of 67.5° ± 2°, the relationship curve between the Q-factor of the downlink and P is obtained by simulation experiments and shown in Figure 11.

Fig. 11 The relationship of the Q factor of the system with the P of PM.

We can see from Figure 11 that, when the P varies in the range of 66.25°~69.25°, the Q is greater than 6.

5.3 The influence the G on the Q-factor

According to the previous analysis, the theoretical value of the EG’s gain coefficient G is 1.5. When G is varied within the range of 1.5 ± 0.5, the relationship between the Q-factor of the downlink and G is obtained by simulation experiments and shown in Figure 12.

Fig. 12 The relationship of the G with the Q factor.

We can see from Figure 12 that, when G is varied in the range of 1.275 to 1.65, Q > 6.

5.4 The influence of the polarization angle of the Pol on Q-factor

The theoretical value of the polarization angle of the Pol is 0°. When the angle of the Pol is varied within the range of 0° ± 10°, the relationship between the Q factor and the Pol angle is obtained by simulation experiments and shown in Figure 13.

Fig. 13 The relationship of the Pol angle with the Q factor.

We can see from Figure 13 that, when the Pol angle is varied within the range of −7.1° to 7.4°, Q > 6.

5.5 The influence of the 90° EPS on Q-factor

The output signals of the two PolMs are ±2n order optical sidebands. To ensure that the output signal of the DP-PolM is ±4n order optical sidebands, it is necessary to let the output optical sidebands of the two PolMs have the opposite sign when n takes odd number. It can be realized by the 90° EPS. When the phase of the 90° EPS deviates from 90°, the terms in the ±2n (n is an odd number) order optical sidebands could not be effectively suppressed, and the Q-factor will be decreased. When the offset of the phase of the 90° EPS is within the range of 90° ± 5°, the relationship between the system Q-factor and the phase offset of the 90° EPS is obtained by simulation experiments and shown in Figure 14.

Fig. 14 The relationship of the Q factor with the phase of the 90° EPS.

We can see from Figure 14 that, when the phase of the 90° EPS is varied within the range of 87.1°~94.2°, Q > 6.

The summary of the system’s stability is listed in Table 3.

6 Discussion

6.1 Extension of methods

In our designed scheme, the modulation of the data only on the +4th order sideband is realized by appropriately setting the P of PM and the G of EG. This method can be extended to generate the frequency multiplication ROF with FMF is 2nf_RF based on external modulators. In this external modulator-based system, the main sidebands are the ±n order optical sidebands. To modulate the data signal only on the +n (or −n) order sideband is realized by using the composite RF drive signal as the drive RF signal of the external modulator. The composite RF drive signal is formed by combining the two signals. One signal is formed by modulating the data signal on the RF drive signal by the PM, and the other is formed by amplifying the data signal by the EG.

For a binary data signal, according to the analysis above, the amplitudes of the ±nth order sidebands are proportional to $e^{j\left[n{\omega }_{\mathrm{RF}}t+\mathrm{nPs}\left(t\right)\right]+\mathrm{j\pi }\mathrm{Gs}\left(t\right)/V_{\pi }}$ and $e^{-j\left[n{\omega }_{\mathrm{RF}}t+\mathrm{nPs}\left(t\right)\right]+\mathrm{j\pi }\mathrm{Gs}\left(t\right)/V_{\pi }}$, respectively.

To modulate the data signal on the +nth order optical sideband only, the following equations need to be satisfied.$$\{\begin{array}{c}\mathrm{nP}+\pi G/V_{\pi }=\left(2k_1+1\right)\pi \\ -\mathrm{nP}+\pi G/V_{\pi }=2k_2\pi \end{array}$$(16)

where the n is the order of the sidebands, k₁ and k₂ take integers.

According to equation (16), we can get$$G=\left(k+1/2\right)V_{\pi },P=\left(k\mathrm{\prime }+1/2\right)\pi /n$$(17)where k = k₁ + k₂ and k′ = k₁ − k₁ take integers.

6.2 Fiber nonlinear effect

(1) Analysis of nonlinear effects

In our designed scheme, the transmission of light waves in optical fibers mainly consists of three optical waves. They are the ±4th order optical sidebands and the 0th order optical sideband (optical carrier). The optical power injected into the fiber exceeds milliwatts, and the transmission distance exceeds 80 km. Therefore, the nonlinear effects of the fiber need to be considered. The nonlinear effects that affect the transmission performance of optical fibers are mostly self-phase modulation (SPM) and four-wave mixing (FWM). The nonlinear analysis of optical fibers below refers to reference [37].

1) SPM effect

Considering nonlinear effects, the refractive indices of the fiber core and cladding are expressed as$$n_j^{\mathrm{\prime }}=n_j+n_2\left(P/A_{\mathrm{eff}}\right)\left(j=\mathrm{1,2}\right).$$(18)

Among them, n₂ is the nonlinear refractive index, P is the optical power, A_eff is the effective area, and j = 1, 2 represent the core wave cladding, respectively.

The transmission constant of the optical wave is$$\beta \text{'}=\beta +k_0{n}_2\left(P/A_{\mathrm{eff}}\right)=\beta +\gamma P$$(19)where, γ = 2πn₂/(A_effλ) is a nonlinear parameter.

The non-linear phase shift generated under the item γ is$${\varphi }_{\mathrm{NL}}={\int }_0^L\mathrm{}\left(\beta \text{'}-\beta \right)\mathrm{d}z=\gamma P_{\mathrm{in}}{L}_{\mathrm{eff}}$$(20)P_in(t) produce ϕ_NL(t) which result in frequency chirp. This is the SPM effect.

When ϕ_NL ⩽ 0.1, SPM has little impact on the system’s BER. For long optical fibers, there are L_eff = 1/α, where α is the attenuation coefficient of the fiber. In our simulation experiment, there are γ = 2W⁻¹/km and α = 0.2 dB/km. For ϕ_NL ⩽ 0.1, there are P_in < 22 mW = 13.42 dBm.

In our simulation experiment, the input power of the PoIMs is 20 dBm. The optical power injected into the fiber is less than 5 dBm. So the SPM in our schemes has a relatively small impact.

2) FWM effect

FWM originated from third-order nonlinear polarizability χ⁽³⁾. When three optical waves with angular frequencies of ω₁, ω₂, and ω₃, are transmitted in an optical fiber, a fourth optical field with frequency ω₄ (ω₄ = ω₁ ± ω₂ ± ω₃) is generated by χ⁽³⁾. This is the FWM effect. In this paper, let ω₂ = ω₀, ω₃ = ω₀ + Ω, ω₁ = ω₀ − Ω, where ω₀ is the center carrier (0th optical sideband) angular frequencies, ω₃ and ω₁ are the angular frequencies of ±4th order optical sidebands, respectively, Ω = 4 × 2πf_RF and f_RF is the frequency of the RF drive signal loaded on PoIMs.

On the one hand, for multi-channel transmission, the optical wave with frequency ω₄ = ω₁ + ω₂ – ω₃ has the most severe impact on the system. In our scheme, there are ω₄ = ω₀ ± 2Ω, which are located in ±8th order sidebands. Their amplitudes are very small. At the base station (BS), the central carrier is first filtered out and used as the uplink carrier. The remaining light waves are mainly ±4th and ±8th order sidebands. The ±4th order sidebands are the main sidebands, and the ±8th order sidebands are the scattered sidebands. When those light waves are beat in PD, the resulting signal mainly consists of 4th, 8th, 2nd, and 16th harmonics f_RF. The amplitude of the 8th harmonic waves is the highest, followed by the 4th and 12th harmonic waves, and the 16th harmonic wave is the smallest. The 8th harmonic wave signal, which is the frequency octupling MMW, is filtered out by a bandpass filter. It can be seen that FWM has little impact on the system designed in our scheme.

On the other hand, when three light waves are transmitted in an optical fiber, a fourth light wave is generated due to FWM. When these four waves are transmitted in the same direction, phase adaptation can be written as$$\Delta =\beta \left({\omega }_3\right)+\beta \left({\omega }_4\right)-\beta \left({\omega }_1\right)-\beta \left({\omega }_2\right)$$(21)where β(ω_i) (i = 1, 2, 3, 4) is the transmission constant of the light wave.

Phase adaptation can be expressed as$$\Delta ={\beta }_2{\mathrm{\Omega }}^2.$$(22)

The phase-matching condition of FWM is that the greater the adaptability, the less likely FWM is to occur. When β₂ = 0, the phases are perfectly matched; When β₂ < 1 ps²/km, Ω < 100 GHz, it is easy to generate FWM.

In our simulation experiment, when β₂ and f_RF are chosen as 16.75 ps/km · nm, there are Ω = 40 GHz and Δ = 1.07 × 10⁵. So it is not easy to generate FWM. The influence of FWM in this article is relatively small.

According to our analysis, due to the relatively small impact of SPM and FWM on our designed system. For the sake of simplicity in theoretical analysis, we neglected the nonlinear effects of optical fibers.

(2) Design of nonlinear parameters of optical fiber in simulation system

In Optisystem, there is an option to consider the optical fiber nonlinear effects with “SPM” and “n₂”. When “SPM” and “n₂” are checked, the system considers the nonlinear effects of “SPM” and “FMW”. When these two options are not selected, the system does not account for the nonlinear effects of the optical fiber.

In the simulation experiment, we selected “SPM” and “n₂”, so the obtained Q-factor and BER have considered the nonlinear effects.To compare the impact of nonlinear effects on the system, we designed a new simulation experiment, in which we did not select “SPM” and “n₂” and kept other parameters unchanged. The obtained Q curve graph is shown in Figure 15. For comparison, we simultaneously present the curve considering “SPM” and “n₂” in Figure 15.

Fig. 15 Q factor variation with transmission distance.

From Figure 15, it can be seen that nonlinear effects reduce the Q value of the system, but the magnitude of the reduction is small and can be ignored in practical engineering applications.

6.3 Data modulation format

In the MMW ROF system, data signals are modulated onto MMWs, which are then modulated onto optical carriers. The MMW is the subcarrier, and the optical is the carrier.

The principle of modulating data signals only onto the +4th order sidebands in this article is as follows. The RF drive signal loaded on PoIM is a composite RF signal, as shown in formula (8), where the data signal is phase modulated onto the RF carrier.

The output signal of DP PoIM is a ±4th order sideband signal, where the phases of the ±4th order sideband signals are,$${\omega }_{0}t+4\left[{\omega }_{\mathrm{RF}}t+\frac{3}{8}\pi s\left(t\right)\right]+\frac{3}{8}\pi s\left(t\right)={\omega }_{0}t+{\omega }_{\mathrm{RF}}t+3\pi s\left(t\right)$$(23) $${\omega }_{0}t-4\left[{\omega }_{\mathrm{RF}}t+\frac{3}{8}\pi s\left(t\right)\right]+\frac{3}{8}\pi s\left(t\right)={\omega }_{0}t+4{\omega }_{\mathrm{RF}}t.$$(24)

From the above analysis, it can be seen that to modulate the data signal only to the +4th order sideband, the data signal in our proposed scheme is limited to phase modulation and is not suitable for amplitude modulation. Therefore, it is not suitable for advanced modulation formats such as QAM.

6.4 Experimental validation

In our paper, the experimental links are established using the OptiSystem simulation software, which is widely utilized in engineering design and scientific research for optical communication. The parameters used in this software are based on actual components. The practicality and reliability of the software have been verified by both academia and industry.

Due to the limitation of experimental conditions, among the schemes to generate the frequency multiplication MMW based on PoIMs, for the FMF less than 6, there are several schemes verified with physical experiments; for the FMF greater than 6, most schemes are generally verified by simulation experiments.

Due to the limitations of the conditions in our laboratory, it is difficult to verify our proposed scheme with physical experiments currently. In the future, if the experimental conditions are available, we will continue the related physical verification experiment.

7 Conclusion

A novel MMW ROF system to overcome the bit walk-off effect and realize the carrier reuse by inserting a pilot is proposed. At the CS, in the case of no data transmission, the main sidebands in the output signal of the DP-PolM are the ±4th order sidebands by adjusting the amplitude and initial phase of the RF driving signal of the PolMs. In the case of data transmission, using the composite RF driving signal as the RF driving signal for the PoIM, the downlink data can be modulated on the +4th order sideband. The composite RF driving signal is formed by two signals. One signal is the modulated RF driving signal, which is formed by modulate the data signal on the RF driving signal with the PM; the other signal is the amplified data signal by the EG. By appropriately adjusting the P of the PM and the gain G of the EG, the data signal is modulated only to the +4th order sideband. By splitting part of the power from the CW-LD as the inserting pilot in the CS and filtering out the pilot by an FBG as the uplink optical carrier in the BS, the carrier reuse is realized. In the BS, by injecting the ±4th sidebands into the PD1, the frequency octupling MMW with downlink data is generated.

The operation principle of the system was theoretically analyzed, and a simulation system was built using Optisystem tool to verify the feasibility of the system. At the same time, the influence of the main parameters of the system on the system Q when deviating from the theoretical or design values was studied. In order to highlight the advantages of our system, we also designed a traditional frequency octupling MMW ROF system with the same parameters except for the part of the data modulated, and compared their performances. For the transmission distance of 80 km with a data of 2.5 Gpbs, under the condition of Q-factor is greater than 6, the power penalty of our designed scheme is 1 dB, the Q-factor of the conventional ROF system is less than 6.

The MMW ROF system designed in this paper can effectively solve the bit walk-off effect caused by fiber chromatic dispersion, greatly increase the fiber transmission distance, significantly improve the downlink performance, completely avoid the DC drift and limitation extinction ratio issues by MZM. It has important application prospects in MMW ROF.

Funding

National Key R&D Program of China (2021YFB2500903).

Conflicts of interest

The authors declare that there are no conflicts of interest.

Data availability statement

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Author contribution statement

Fengjun Zhou: Writing – original draft, Investigation; Nana Zhou: Software, Validation; Qun Wang: Resources; Hengji Hu: Supervision; Xinqiao Chen: Project administration. All authors have read and agreed to the published version of the manuscript.

References

All Tables

All Figures

	Fig. 1 Schematic of the frequency octupling MMW ROF without bit walk-off effect by PolMs.
In the text

	Fig. 2 Optical spectrum of points A–D in Figure 1.
In the text

	Fig. 3 Spectrum of points E and F in Figure 1.
In the text

	Fig. 4 Time domain waveform of data.
In the text

	Fig. 5 The relationship of the Q factor with the fiber transmission distance.
In the text

	Fig. 6 The relationship between the BER and the received power of PD1 and PD2. (a) Downlink; (b) Uplink.
In the text

	Fig. 7 Eye diagram at a different transmission distance. (a) BTB; (b) 10 km; (c) 20 km; (d) 37 km.
In the text

	Fig. 8 Eye diagrams at different transmission distances in our scheme. (a) BTB; (b) 20 km; (c) 60 km; (d) 80 km.
In the text

	Fig. 9 The relationship of the BER of the downlink with the received power of PD1.
In the text

	Fig. 10 The relationship of the Q factor with VRF.
In the text

	Fig. 11 The relationship of the Q factor of the system with the P of PM.
In the text

	Fig. 12 The relationship of the G with the Q factor.
In the text

	Fig. 13 The relationship of the Pol angle with the Q factor.
In the text

	Fig. 14 The relationship of the Q factor with the phase of the 90° EPS.
In the text

	Fig. 15 Q factor variation with transmission distance.
In the text