A New Photonic Filterless Scheme for the Generation of Frequency 16-Tupling Millimeter Wave Signals Utilizing Cascading Polarization Modulators

Abstract

A new photonic scheme is proposed for the generation of a frequency 16-tupling millimeter wave (mm-wave) signal, based on four cascaded polarization modulators (PolMs). Firstly, through the precise control of these four PolMs, odd harmonics can be effectively suppressed sequentially. Next, combined with the modulation index of the PolMs, only the pure ± 8th-order optical harmonics are retained and enter the photodetector to beat the frequency. After multiple rounds of detailed, mathematical formula derivation and computer simulation, the accuracy and advantages of the proposed solution can be demonstrated. As a result, a 160 GHz mm-wave signal is obtained, which is produced from a 10 GHz radio frequency (RF) signal. By properly setting the modulation index and the angle of three phase differences, the optical suppression ratio (OSSR) and the radio frequency suppression ratio (RFSSR) of the acquired signal are 64.21 dB and 60.99 dB, respectively. Further, the impact of the variables on the OSSR and the RFSSR are analyzed and discussed.

1. Introduction

With the explosive growth in the volume of the flow of the wireless Internet, fifth-generation mobile communication technology (5G) has become the new generation of broadband mobile communication technology due to its higher speed, lower latency and greater connectivity [1,2]. Fifth-generation technology has been applied in various industries, such as energy, education, healthcare, etc. Mm-wave is a promising technology for future 5G cellular systems. Mm-waves have already been used in radar detection, satellite remote sensing and aerospace fields [3,4]. Therefore, more and more scholars are researching how to obtain higher-frequency mm-wave signals.

Compared to the traditional means of generating mm-wave signals in the electrical domain, the method of generating mm-wave signals in the optical domain is well received because of its lower cost and simpler structure [5,6,7]. The optical generation technology of mm-waves can be broadly classified into direct modulation [8], photoelectric oscillation [9], four-wave mixing [10], optical heterodyne [11] and external modulation [12,13,14,15,16,17,18,19,20,21,22,23,24,25]. Except for external modulation, these methods all have certain drawbacks, so external modulation is the most reliable. The most commonly used modulators in external modulation technology are the phase modulator (PM), the Mach–Zehnder modulator (MZM) and the polarization modulator (PolM). Moreover, there are many papers that have presented attempts to generate high-frequency (frequency quadrupling, frequency octupling, frequency 12-tupling, frequency 16-tupling, etc.) mm-wave signals using these modulators. In the published articles, most of them use MZMs but use different combinations of methods to generate frequency 16-tupling mm-wave signals. For instance, in 2013, Chen Xiaogang, et al. [17] proposed a novel scheme for frequency 16-tupling millimeter wave generation based on two cascaded integrated dual-parallel Mach–Zehnder modulators (DPMZMs) without optical filtering. There, the optical suppression ratio (OSSR) is about 49 dB and the radio frequency suppression ratio (RFSSR) is about 32 dB. Coincidentally, in 2015, Zhu Zihang et al. [18] also proposed a scheme to generate 16-tupling millimeter wave generation based on two cascaded DPMZMs without an optical filter. In that case, the OSSR and RFSSR of the generated signal were 21.5 dB and 38 dB, respectively. In 2020, Wu Zengyan et al. [19] proposed a new scheme to generate an 80 GHz mm-wave from a 5 GHz radio frequency (RF) driving signal using two Mach–Zehnder modulators (MZMs) with a 23.48 dB OSSR and a 26.38 dB RFSSR. The three schemes for generating frequency 16-tupling mm-wave signals mentioned above have been listed along with some others in Table 1. As has been described, the previous schemes for generating the frequency 16-tupling mm-wave signal were all obtained using MZMs in cascade or in parallel. However, there are bias points in the MZM, as the bias points may drift when the temperature changes or when vibration occurs in the external environment; this requires additional direct current (DC) bias voltage as a control, which increases the complexity of the system. We also maintain that the OSSR and RFSSR values of the obtained signals are not very high in the previous research.

In this study, a new photonic scheme is proposed for generating a frequency 16-tupling mm-wave signal based on four cascaded polarization modulators (PolMs). On the one hand, there is no bias point in the PolM, so there will not be any bias point drift caused by external environmental factors. On the other hand, no additional DC bias voltage needs to be introduced via the PolM. Both theoretical calculations and simulation results support that the frequency 16-tupling scheme can generate a higher quality and a purer signal, attaining an OSSR of 64.21 dB and an RFSSR of 60.99 dB.

2. Principles and Methods

In this work, there are four cascaded PolMs used to generate a 16-tupling mm-wave signal, and the schematic diagram is shown in Figure 1. As we can see in Figure 1, there are four subsystems, which consist of a polarization controller (PC), a PolM and a polarizer (Pol). The main process of the proposed scheme is as follows. To obtain the 16-tupling mm-wave signal, it is necessary to generate ±8th-order optical sidebands. This is carried out by adjusting the polarization controller and the polarizer for each subsystem to suppress odd-order optical sidebands so that even-order optical sidebands are generated. Next, the phase differences driven by the RF signal are loaded on the four PolMs to eliminate the even-order sidebands, except for ±(8n)th-order ones. At last, the modulation index is set to zero to make the amplitude of the 0th-order sideband. By means of the above steps, it is possible to attain only ±8th-order harmonics as the final output.

The signal output from the first polarization modulator (PolM1) can be expressed as

$$E_{PolM-1}(t)=\frac{E_{in}(t)}{2}\left[\begin{array}{c}\cos {\theta }_1{e}^{jm\cos {\omega }_{m}t}\\ \sin {\theta }_1{e}^{jm\cos ({\omega }_{m}t+\pi )}\end{array}\right]=\frac{E_c{e}^{j{\omega }_{c}t}}{2}\left[\begin{array}{c}\cos {\theta }_1{e}^{jm\cos {\omega }_{m}t}\\ \sin {\theta }_1{e}^{-jm\cos {\omega }_{m}t}\end{array}\right]$$

(1)

where $E_c$ and ${\omega }_c$ are the amplified and angular frequencies of the input signal, respectively. ${\theta }_1$ is the azimuth of the polarization controller (PC1). $m$ is the modulation index, which can be defined as $m=\pi V_m/V_{\pi }$. ${\omega }_m$ is the angular frequency of the RF signal.

Then, the signal is transmitted into the first polarizer (Pol1), so the signal output from Pol1—which is the output from Sub A—can be written as

$$E_A(t)=\frac{E_c{e}^{j{\omega }_{c}t}}{2}\left[\begin{array}{cc}\cos {\theta }_2& 0\\ 0& \sin {\theta }_2\end{array}\right]\left[\begin{array}{c}\cos {\theta }_1{e}^{jm\cos {\omega }_{m}t}\\ \sin {\theta }_1{e}^{-jm\cos {\omega }_{m}t}\end{array}\right]$$

(2)

where ${\theta }_2$ is the polarization angle of the Pol1.

In order to suppress odd-order optical sidebands, the angle of both the PC1 and the Pol1 is set to π/4 rad. Therefore, Equation (2) can be simplified as

$$E_A(t)=\frac{E_c{e}^{j{\omega }_{c}t}}{2}\left(e^{jm\cos {\omega }_{m}t}+e^{-jm\cos {\omega }_{m}t}\right)$$

(3)

Then, the Bessel function is used to expand Equation (3) and can be obtained as

$$\begin{array}{ll}{E}_A(t)& =\frac{E_c{e}^{j{\omega }_{c}t}}{2}\left\{{\displaystyle \sum _{n=-\infty }^{+\infty }\left[1+{(-1)}^n\right]\cdot j^n\cdot J_n(m)\cdot e^{jn{\omega }_{m}t}}\right\}\\ & =E_c{e}^{j{\omega }_{c}t}\left\{J_0(m)-2J_2(m)\cos (2{\omega }_{m}t)+2J_4(m)\cos (4{\omega }_{m}t)-2J_6(m)\cos (6{\omega }_{m}t)+\dots \right\}\end{array}$$

(4)

where $J_n(m)$ is the Bessel function of the first kind of n-th order. Moreover, it is observed clearly in Equation (4) that there are only even-order sidebands; all the odd-order sidebands are suppressed.

Similarly, the signal output from Sub B can be written as

$$\begin{array}{ll}{E}_B(t)& =\frac{E_A(t)}{2}\left[e^{jm\cos ({\omega }_{m}t+{\phi }_1)}+e^{-jm\cos ({\omega }_{m}t+{\phi }_1)}\right]\\ & =\frac{E_A(t)}{2}\left\{{\displaystyle \sum _{n=-\infty }^{+\infty }\left[1+{(-1)}^n\right]\cdot j^n\cdot J_n(m)\cdot e^{jn{\omega }_{m}t}\cdot e^{jn{\phi }_1}}\right\}\\ & =\frac{E_c{e}^{j{\omega }_{c}t}}{4}\left\{{\displaystyle \sum _{n=-\infty }^{+\infty }\left[1+{(-1)}^n\right]\cdot j^n\cdot J_n(m)\cdot e^{jn{\omega }_{m}t}}\right\}\\ & \times \left\{{\displaystyle \sum _{n=-\infty }^{+\infty }\left[1+{(-1)}^n\right]\cdot j^n\cdot J_n(m)\cdot e^{jn{\omega }_{m}t}\cdot e^{jn{\phi }_1}}\right\}\\ & =E_c{e}^{j{\omega }_{c}t}\left\{\begin{array}{l}{J}_0(m)-2J_2(m)\cos (2{\omega }_{m}t)+2J_4(m)\cos (4{\omega }_{m}t)\\ -2J_6(m)\cos (6{\omega }_{m}t)+\dots \end{array}\right\}\\ & \times \left\{\begin{array}{l}{J}_0(m)-2J_2(m)\cos (2{\omega }_{m}t+2{\phi }_1)+2J_4(m)\cos (4{\omega }_{m}t+4{\phi }_1)\\ -2J_6(m)\cos (6{\omega }_{m}t+6{\phi }_1)+\dots \end{array}\right\}\\ & =E_c{e}^{j{\omega }_{c}t}\left\{\begin{array}{l}{S}_1+T_1\cos (2{\omega }_{m}t)+U_1\cos (4{\omega }_{m}t)+V_1\cos (6{\omega }_{m}t)+W_1\cos (8{\omega }_{m}t)\\ +X_1\cos (10{\omega }_{m}t)+Y_1\cos (12{\omega }_{m}t)+\dots \end{array}\right\}\end{array}$$

(5)

where

$$\begin{array}{ll}{S}_1& =J_0^2(m)+2J_2^2(m)\cos (2{\phi }_1)+2J_4^2(m)\cos (4{\phi }_1)+2J_6^2(m)\cos (6{\phi }_1),\\ T_1& =-\{4J_0(m)J_2(m)[\cos (2{\phi }_1)+1]+4J_2(m)J_4(m)[\cos (2{\phi }_1)+\cos (4{\phi }_1)]\\ & +4J_4(m)J_6(m)[\cos (4{\phi }_1)+\cos (6{\phi }_1)]\},\\ U_1& =2J_2^2(m)\cos (2{\phi }_1)+4J_0(m)J_4(m)[\cos (4{\phi }_1)+1]+4J_2(m)J_6(m)[\cos (2{\phi }_1)+\cos (6{\phi }_1)],\\ V_1& =-\{4J_0(m)J_6(m)[\cos (6{\phi }_1)+1]+4J_2(m)J_4(m)[\cos (2{\phi }_1)+\cos (4{\phi }_1)]\},\\ W_1& =2J_4^2(m)\cos (4{\phi }_1)+4J_2(m)J_6(m)[\cos (2{\phi }_1)+\cos (6{\phi }_1)],\\ X_1& =-4J_4(m)J_6(m)[\cos (4{\phi }_1)+\cos (6{\phi }_1)],\\ Y_1& =2J_6^2(m)\cos (6{\phi }_1),\end{array}$$

$S_1$, $T_1$, $U_1$, $V_1$, $W_1$, $X_1$ and $Y_1$ represent the amplitude values of 0th, ±2nd, ±4th, ±6th, ±8th, ±10th and ±12th orders in Equation (5), respectively, and ${\phi }_1$ is the phase difference between the PolM1 and the PolM2 driven by the RF signal.

Next, the signal output from Sub C can be written as

$$\begin{array}{ll}{E}_C(t)& =\frac{E_B(t)}{2}\left[e^{jm\cos ({\omega }_{m}t+{\phi }_2)}+e^{-jm\cos ({\omega }_{m}t+{\phi }_2)}\right]\\ & =\frac{E_B(t)}{2}\left\{{\displaystyle \sum _{n=-\infty }^{+\infty }\left[1+{(-1)}^n\right]\cdot j^n\cdot J_n(m)\cdot e^{jn{\omega }_{m}t}\cdot e^{jn{\phi }_2}}\right\}\\ & =\frac{E_c{e}^{j{\omega }_{c}t}}{8}\left\{{\displaystyle \sum _{n=-\infty }^{+\infty }\left[1+{(-1)}^n\right]\cdot j^n\cdot J_n(m)\cdot e^{jn{\omega }_{m}t}}\right\}\\ & \times \left\{{\displaystyle \sum _{n=-\infty }^{+\infty }\left[1+{(-1)}^n\right]\cdot j^n\cdot J_n(m)\cdot e^{jn{\omega }_{m}t}\cdot e^{jn{\phi }_1}}\right\}\\ & \times \left\{{\displaystyle \sum _{n=-\infty }^{+\infty }\left[1+{(-1)}^n\right]\cdot j^n\cdot J_n(m)\cdot e^{jn{\omega }_{m}t}\cdot e^{jn{\phi }_2}}\right\}\\ & =E_c{e}^{j{\omega }_{c}t}\left\{\begin{array}{l}{J}_0(m)-2J_2(m)\cos (2{\omega }_{m}t)+2J_4(m)\cos (4{\omega }_{m}t)\\ -2J_6(m)\cos (6{\omega }_{m}t)+\dots \end{array}\right\}\\ & \times \left\{\begin{array}{l}{J}_0(m)-2J_2(m)\cos (2{\omega }_{m}t+2{\phi }_1)+2J_4(m)\cos (4{\omega }_{m}t+4{\phi }_1)\\ -2J_6(m)\cos (6{\omega }_{m}t+6{\phi }_1)+\dots \end{array}\right\}\\ & \times \left\{\begin{array}{l}{J}_0(m)-2J_2(m)\cos (2{\omega }_{m}t+2{\phi }_2)+2J_4(m)\cos (4{\omega }_{m}t+4{\phi }_2)\\ -2J_6(m)\cos (6{\omega }_{m}t+6{\phi }_2)+\dots \end{array}\right\}\\ & =E_c{e}^{j{\omega }_{c}t}\left\{\begin{array}{l}{S}_2+T_2\cos (2{\omega }_{m}t)+U_2\cos (4{\omega }_{m}t)+V_2\cos (6{\omega }_{m}t)+W_2\cos (8{\omega }_{m}t)\\ +X_2\cos (10{\omega }_{m}t)+Y_2\cos (12{\omega }_{m}t)+P_1\cos (14{\omega }_{m}t)+Q_1\cos (16{\omega }_{m}t)\\ +R_1\cos (18{\omega }_{m}t)+\dots \end{array}\right\}\end{array}$$

(6)

where

$$\begin{array}{ll}{S}_2& =S_1{J}_0(m)+2T_1{J}_2(m)\cos (2{\phi }_2)+2U_1{J}_4(m)\cos (4{\phi }_2)+2V_1{J}_6(m)\cos (6{\phi }_2),\\ T_2& =T_1(J_0(m)+2J_4(m)\cos (4{\phi }_2))+2S_1{J}_2(m)\cos (2{\phi }_2)+2U_1(J_2(m)\cos (2{\phi }_2)+J_6(m)\cos (6{\phi }_2))\\ & +2V_1{J}_4(m)\cos (4{\phi }_2)+2W_1{J}_6(m)\cos (6{\phi }_2),\\ U_2& =U_1{J}_0(m)+2T_1(J_2(m)\cos (2{\phi }_2)+J_6(m)\cos (6{\phi }_2))+2V_1{J}_2(m)\cos (2{\phi }_2)+2(S_1+W_1)J_4(m)\cos (4{\phi }_2)\\ & +2X_1{J}_6(m)\cos (6{\phi }_2),\\ V_2& =V_1{J}_0(m)+2(U_1+W_1)J_2(m)\cos (2{\phi }_2)+2(T_1+X_1)J_4(m)\cos (4{\phi }_2)+2(S_1+Y_1)J_6(m)\cos (6{\phi }_2),\\ W_2& =W_1{J}_0(m)+2(V_1+X_1)J_2(m)\cos (2{\phi }_2)+2(U_1+Y_1)J_4(m)\cos (4{\phi }_2)+2T_1{J}_6(m)\cos (6{\phi }_2),\\ X_2& =X_1{J}_0(m)+2(W_1+Y_1)J_2(m)\cos (2{\phi }_2)+2V_1{J}_4(m)\cos (4{\phi }_2)+2U_1{J}_6(m)\cos (6{\phi }_2),\\ Y_2& =Y_1{J}_0(m)+2X_1{J}_2(m)\cos (2{\phi }_2)+2W_1{J}_4(m)\cos (4{\phi }_2)+2V_1{J}_6(m)\cos (6{\phi }_2),\\ P_1& =2Y_1{J}_2(m)\cos (2{\phi }_2)+2X_1{J}_4(m)\cos (4{\phi }_2)+2W_1{J}_6(m)\cos (6{\phi }_2),\\ Q_1& =2Y_1{J}_4(m)\cos (4{\phi }_2)+2X_1{J}_6(m)\cos (6{\phi }_2),\\ R_1& =2Y_1{J}_6(m)\cos (6{\phi }_2),\end{array}$$

where $S_2$, $T_2$, $U_2$, $V_2$, $W_2$, $X_2$, $Y_2$, $P_2$, $Q_1$ and $R_1$ represent the amplitude values of 0th, ±2nd, ±4th, ±6th, ±8th, ±10th, ±12th, ±14th, ±16th and ±18th orders in Equation (6), respectively, and ${\phi }_2$ is the phase difference between the PolM2 and the PolM3 driven by the RF signal.

At last, the signal output from Sub D, which is the final output signal, can be written as

$$\begin{array}{ll}{E}_{out}(t)& =E_D(t)\\ & =\frac{E_C(t)}{2}\left[e^{jm\cos ({\omega }_{m}t+{\phi }_3)}+e^{-jm\cos ({\omega }_{m}t+{\phi }_3)}\right]\\ & =\frac{E_C(t)}{2}\left\{{\displaystyle \sum _{n=-\infty }^{+\infty }\left[1+{(-1)}^n\right]\cdot j^n\cdot J_n(m)\cdot e^{jn{\omega }_{m}t}\cdot e^{jn{\phi }_3}}\right\}\\ & =\frac{E_c{e}^{j{\omega }_{c}t}}{16}\left\{{\displaystyle \sum _{n=-\infty }^{+\infty }\left[1+{(-1)}^n\right]\cdot j^n\cdot J_n(m)\cdot e^{jn{\omega }_{m}t}}\right\}\\ & \times \left\{{\displaystyle \sum _{n=-\infty }^{+\infty }\left[1+{(-1)}^n\right]\cdot j^n\cdot J_n(m)\cdot e^{jn{\omega }_{m}t}\cdot e^{jn{\phi }_1}}\right\}\\ & \times \left\{{\displaystyle \sum _{n=-\infty }^{+\infty }\left[1+{(-1)}^n\right]\cdot j^n\cdot J_n(m)\cdot e^{jn{\omega }_{m}t}\cdot e^{jn{\phi }_2}}\right\}\\ & \times \left\{{\displaystyle \sum _{n=-\infty }^{+\infty }\left[1+{(-1)}^n\right]\cdot j^n\cdot J_n(m)\cdot e^{jn{\omega }_{m}t}\cdot e^{jn{\phi }_3}}\right\}\\ & =E_c{e}^{j{\omega }_{c}t}\left\{\begin{array}{l}{J}_0(m)-2J_2(m)\cos (2{\omega }_{m}t)+2J_4(m)\cos (4{\omega }_{m}t)\\ -2J_6(m)\cos (6{\omega }_{m}t)+\dots \end{array}\right\}\\ & \times \left\{\begin{array}{l}{J}_0(m)-2J_2(m)\cos (2{\omega }_{m}t+2{\phi }_1)+2J_4(m)\cos (4{\omega }_{m}t+4{\phi }_1)\\ -2J_6(m)\cos (6{\omega }_{m}t+6{\phi }_1)+\dots \end{array}\right\}\\ & \times \left\{\begin{array}{l}{J}_0(m)-2J_2(m)\cos (2{\omega }_{m}t+2{\phi }_2)+2J_4(m)\cos (4{\omega }_{m}t+4{\phi }_2)\\ -2J_6(m)\cos (6{\omega }_{m}t+6{\phi }_2)+\dots \end{array}\right\}\\ & \times \left\{\begin{array}{l}{J}_0(m)-2J_2(m)\cos (2{\omega }_{m}t+2{\phi }_3)+2J_4(m)\cos (4{\omega }_{m}t+4{\phi }_3)\\ -2J_6(m)\cos (6{\omega }_{m}t+6{\phi }_3)+\dots \end{array}\right\}\\ & =E_c{e}^{j{\omega }_{c}t}\left\{\begin{array}{l}{S}_3+T_3\cos (2{\omega }_{m}t)+U_3\cos (4{\omega }_{m}t)+V_3\cos (6{\omega }_{m}t)+W_3\cos (8{\omega }_{m}t)\\ +X_3\cos (10{\omega }_{m}t)+Y_3\cos (12{\omega }_{m}t)+P_2\cos (14{\omega }_{m}t)+Q_2\cos (16{\omega }_{m}t)\\ +R_2\cos (18{\omega }_{m}t)+M\cos (20{\omega }_{m}t)+N\cos (22{\omega }_{m}t)+O\cos (24{\omega }_{m}t)\\ +\dots \end{array}\right\}\end{array}$$

(7)

where

$$\begin{array}{ll}{S}_3& =S_2{J}_0(m)+2T_2{J}_2(m)\cos (2{\phi }_3)+2U_2{J}_4(m)\cos (4{\phi }_3)+2V_2{J}_6(m)\cos (6{\phi }_3),\\ T_3& =T_1(J_0(m)+2J_4(m)\cos (4{\phi }_3))+2S_2{J}_2(m)\cos (2{\phi }_3)+2U_2(J_2(m)\cos (2{\phi }_3)+J_6(m)\cos (6{\phi }_3))\\ & +2V_2{J}_4(m)\cos (4{\phi }_3)+2W_2{J}_6(m)\cos (6{\phi }_3),\\ U_3& =U_2{J}_0(m)+2T_2(J_2(m)\cos (2{\phi }_3)+J_6(m)\cos (6{\phi }_3))+2V_2{J}_2(m)\cos (2{\phi }_3)+2(S_2+W_2)J_4(m)\cos (4{\phi }_3)\\ & +2X_2{J}_2(m)\cos (6{\phi }_3),\\ V_3& =V_2{J}_0(m)+2(U_2+W_2)J_2(m)\cos (2{\phi }_3)+2(T_2+X_2)J_4(m)\cos (4{\phi }_3)+2(S_2+Y_2)J_6(m)\cos (6{\phi }_3),\\ W_3& =W_2{J}_0(m)+2(V_2+X_2)J_2(m)\cos (2{\phi }_3)+2(U_2+Y_2)J_4(m)\cos (4{\phi }_3)+2(T_2+P_1)J_6(m)\cos (6{\phi }_3),\\ X_3& =X_2{J}_0(m)+2(W_2+Y_2)J_2(m)\cos (2{\phi }_3)+2(V_2+P_1)J_4(m)\cos (4{\phi }_3)+2(U_2+Q_1)J_6(m)\cos (6{\phi }_3),\\ Y_3& =Y_2{J}_0(m)+2(X_2+P_1)J_2(m)\cos (2{\phi }_3)+2(W_2+Q_1)J_4(m)\cos (4{\phi }_3)+2(V_2+R_1)J_6(m)\cos (6{\phi }_3),\\ P_2& =P_1{J}_0(m)+2(Y_2+Q_1)J_2(m)\cos (2{\phi }_3)+2(X_2+R_1)J_4(m)\cos (4{\phi }_3)+2W_2{J}_6(m)\cos (6{\phi }_3),\\ Q_2& =Q_1{J}_0(m)+2(P_1+R_1)J_2(m)\cos (2{\phi }_3)+2Y_2{J}_4(m)\cos (4{\phi }_3)+2X_2{J}_6(m)\cos (6{\phi }_3),\\ R_2& =R_1{J}_0(m)+2Q_1{J}_2(m)\cos (2{\phi }_3)+2P_1{J}_4(m)\cos (4{\phi }_3)+2Y_2{J}_6(m)\cos (6{\phi }_3),\\ M& =2R_1{J}_2(m)\cos (2{\phi }_3)+2Q_1{J}_4(m)\cos (4{\phi }_3)+2P_1{J}_6(m)\cos (6{\phi }_3),\\ N& =2R_1{J}_4(m)\cos (4{\phi }_3)+2Q_1{J}_6(m)\cos (6{\phi }_3),\\ O& =2R_1{J}_6(m)\cos (6{\phi }_3),\end{array}$$

$S_3$, $T_3$, $U_3$, $V_3$, $W_3$, $X_3$, $Y_3$, $P_2$, $Q_2$, $R_2$, $M$, $N$ and $O$ represent the amplitude values of 0th, ±2nd, ±4th, ±6th, ±8th, ±10th, ±12th, ±14th, ±16th, ±18th, ±20th, ±22th and ±24th orders in Equation (7), respectively, and ${\phi }_3$ is the phase difference between the PolM3 and the PolM4 driven by the RF signal.

In the scheme, the phase differences are set to π/4, π/2 and 3π/4, respectively, in order to suppress the other even-order sidebands, except for the 0th and ±8th orders. Moreover, by adjusting the modulation index to 2.827, the 0th-order optical sideband can be completely eliminated. Then, Equation (7) can be simplified as

$$\begin{array}{ll}{E}_{out}(t)& =E_c{e}^{j{\omega }_{c}t}\{[J_2^4(m)+4J_4^4(m)+6J_0^2(m)J_4^2(m)+12J_2^2(m)J_4^2(m)+6J_2^2(m)J_6^2(m)\\ & +12J_4^2(m)J_6^2(m)+12J_2^3(m)J_6(m)+12J_2(m)J_6^3(m)+12J_0^2(m)J_2(m)J_6(m)\\ & +12J_0(m)J_2^2(m)J_4(m)+12J_0(m)J_4(m)J_6^2(m)+24J_2(m)J_4^2(m)J_6(m)\\ & +24J_0(m)J_2(m)J_4(m)J_6(m)]\cos (8{\omega }_{m}t)\\ & =E_c{e}^{j{\omega }_{c}t}\cdot A\cdot \cos (8{\omega }_{m}t)\end{array}$$

(8)

where $A$ is the amplitude value of ±8th-order optical sidebands in Equation (8).

Thus, the final output signal contains only ±8th-order sidebands and the photocurrent obtained from the photodiode (PD) can be written as

$$\begin{array}{ll}I(t)& \propto \Re {\left|G{E}_4(t)\right|}^2\\ & =\Re {(G{E}_c{e}^{j{\omega }_{c}t})}^2\cdot A_8^2\cdot \left[\frac{\cos (16{\omega }_{m}t)+1}{2}\right]\end{array}$$

(9)

where $\Re$ and $G$ refer to the responsivity and amplification gain of PD, respectively.

3. Simulation Results and Discussions

The simulation results have been verified using the simulation software optisystem 15. A continuous wave (CW) laser with a frequency of 193.1 THz, a power of 20 dBm and a linewidth of 10 MHz was employed to transmit the optical carrier signal, and then, the optical carrier signal was injected into the PolMs. The phase shifts driven by RF signals were set to π/4, π/2 and 3π/4, respectively. A sinusoidal signal with a frequency of 10 GHz acted as the RF signal to drive the four PolMs. Finally, the output sidebands consisted of ±8th-order sidebands, which were placed at 193.02 THz and 193.18 THz, respectively. All these parameter settings are summarized in Table 2.

The optical spectrum output graph, as observed via the optical spectrum analyzer (OSA) after each subsystem, is shown in Figure 2. As can be seen in Figure 2a, at the first polarizer only even-order sidebands are obtained. Then, the optical spectrum observed when the signal is transmitted to the second polarizer is shown in Figure 2b. The highest order is expanded to the ±10th order. The signal is next transmitted to the third polarizer; at this time, the highest order can approach the ±12th order, and the optical spectrum for this is shown in Figure 2c. Eventually, when the signal is transmitted to the last polarizer, only ±8th-order harmonics remain because of the cooperation of three phase differences, as shown in Figure 2d.

The output RF spectrum graph as observed via an electrical spectrum analyzer (ESA) is shown in Figure 3. It can be concluded that the frequency of the output signal is 160 GHz, which is 16 times that of the RF signal.

It can be observed from Figure 2 and Figure 3 that the OSSR and the RFSSR of the generated signal in ideal conditions can reach 64.21 dB and 60.99 dB, respectively. Then, in order to make the comparison more intuitive, the OSSRs and the RFSSRs of the previous works mentioned in the paper are summarized in Table 3. It can be seen in Table 3 that such high OSSRs and RFSSRs have not been achieved in the previous works of frequency 16-tupling mm-wave signal generation.

As described, the resulting simulation diagrams of the optical spectrum and the RF spectrum, when the values of all parameters are set to the desired values, are shown in Figure 2 and Figure 3. However, in practice, the data are so inaccurate that they may lead to errors in experiments. The deviation of the modulation index brings about an increase in the 0th-order sideband’s power, which further causes a decrease in OSSR and RFSSR. For example, Figure 4 and Figure 5 show the optical spectrum and the RF spectrum of the signal when the modulation index has drifted by ±0.005 and ±0.01, respectively. Therefore, only when the modulation index is set to 2.827 can the 0th-order sideband be completely eliminated and the OSSR and the RFSSR reach the maximum, as shown in Figure 2 and Figure 3. The effect of the modulation index on the OSSR and the RFSSR at different values is then shown in Figure 6. It can be seen in Figure 6 that when the modulation index fluctuates in the range of 2.787 to 2.867, the OSSR and the RFSSR can be maintained at more than 15 dB and 10 dB, respectively.

Another parameter that may affect the OSSR and the RFSSR in the experiment is the phase differences loaded on the PolMs driven by the RF signal. There are three electrical phase shifters (EPSs) in the proposed scheme; the three EPSs may shift individually or together, so the two cases are analyzed in turn. The three EPSs are used to eliminate even-order sidebands except for those of the ±(8n)th order. Therefore, once the EPS has drifted, the other even-order sidebands, such as ±2nd, ±4th order and so on, are not eliminated. It is then found that when the three EPSs drift individually, the impacts of their corresponding phase shifts on the OSSR and the RFSSR are the same; so, we summarize them in one figure, as shown in Figure 7. Moreover, the optical spectrum diagram and the RF spectrum diagram when one of three EPSs deviates from ±0.25 deg and ±0.5 deg are also shown in Figure 8 and Figure 9, respectively. It can be clearly seen from Figure 8 and Figure 9 that when the angle of EPS has drifted, the other even-order optical sidebands such as ±2nd, ±4th order optical sidebands and so on are all residue, leading to a reduced OSSR and RFSSR. Only when the phase differences are adjusted to π/4, π/2 and 3π/4, respectively, can even order optical sidebands be completely canceled and can the OSSR and the RFSSR reach the highest values shown in Figure 2 and Figure 3.

Moreover, when the three EPSs shift simultaneously, the effect on the OSSR and the RFSSR is the same as when they drift individually, as shown in Figure 10. In addition, the optical spectrum diagram, and the RF spectrum diagram when three EPSs deviate simultaneously from ±0.25 deg and ±0.5 deg are also the same as in Figure 8 and Figure 9, the results of which are shown in Figure 11 and Figure 12, respectively.

The function of each subsystem’s PC and Pol is to make PolMs produce even-order sidebands by adjusting their angles. Therefore, when each PC shifts or when four PCs simultaneously shift, odd-order optical sidebands will exist. For example, when the angle of PC1 deviates from ±1 deg, the optical spectrum diagram and the RF spectrum diagram are shown in Figure 13. It can be clearly seen that the spurious optical sidebands at this time are all odd-order sidebands and that the values of the OSSRs and the RFSSRs decrease as the offset angle increases. If the angle drifts only 1 degree, the OSSR drops sharply from 64.21 dB to 30.78 dB and the RFSSR drops from 60.99 dB to 52.92 dB. So, the impact of the offset PC angle on the OSSR and the RFSSR is shown in Figure 14. It can be seen that when the drift degree is within the range of ±5, the impact on the OSSR and the RFSSR is acceptable. When four PCs drift simultaneously, the influence on the OSSR and the RFSSR is the same as in Figure 14, as shown in Figure 15.

Similarly, when each Pol shifts or when four Pols shift simultaneously, the results obtained are the same as in the analysis of the PC. Therefore, the figures showing the same optical spectrum, RF spectrum and impact on the OSSR and the RFSSR are not shown repeatedly.

4. Conclusions

The paper proposed a new scheme to generate a frequency 16-tupling mm-wave signal using four cascaded PolMs and showed that the OSSR and the RFSSR of the generated signal can reach 64.21 dB and 60.99 dB, respectively. In addition, such a high OSSR or RFSSR has not been achieved in the previous works of frequency 16-tupling mm-wave signal generation. The mathematical calculations and simulation results are shown above. So, we can observe through the calculation process that as long as the angles of PC and Pol in each subsystem are both set to π/4, the three phase differences are set to π/4, π/2 and 3π/4 and the modulation index is set to 2.827, all the unwanted optical sidebands except for those of the ±8th order are eliminated and the final output signal only contains ±8th-order harmonics. Therefore, there are no extra spurious sidebands in the generated signal, which is also the reason for the high OSSR and RFSSR of the generated signal. In conclusion, when these parameters that play a key role are shifted, the obtained signal will be impure. Thus, it is necessary to analyze the influence of these offset parameters on the OSSR and the RFSSR. Based on our analysis, we can conclude that the OSSR and the RFSSR are tolerable when the PC’s angle drifts not more than ±5 deg, the Pol’s angle drifts within the range of ±5 deg, the EPS’s phase drifts within ±1 deg and the modulation index is controlled within the range of 2.817 to 2.837.