Nowadays, phase shifters are important in phased array radar, satellite communication, broadband wireless communication, modern radar systems, electrical warfare, and phased-array antennas, et al^[1]. Modern phase shifters are expected for high operating frequency and large instantaneous bandwidth, so traditional electric techniques suffer a serious electronic bottleneck^[2-4]. Microwave photonic (MWP) novel technology has attracted significant attention due to its significant advantages over electronic technology such as broad bandwidth, low loss, slight frequency-dependent loss, small size, and immunity to electromagnetic interference^[5-10]. In the past few years, numerous MPPSs have been proposed based on Brillouin scattering^[11], outlier mixing techniques^{[6, 12-14]}, non-linear optical effects^[15-16] and Fiber Bragg Gratings (FBGs)^[17]. However, the generated signals possess the same frequency as the driven RF signal frequency which has difficulty reaching an mm-wave frequency of at least 30 GHz.

To obtain a high-frequency phase-shifted signal with low-frequency devices, frequency multiplication is highly desired while performing the phase shift. In recent years, many schemes combining the phase shift and frequency multiplication functions into one system have been proposed, promising for future applications^[18-19]. However, these demonstrations use optical filters, which increase complexity and limit frequency tuning speed and tunable range. Recently, a filterless frequency-doubling phase shifter has been demonstrated using a Dual-Parallel Polarization Modulator (DP-PolM), a Polarization Controller (PC) and a PM^[20]. Moreover, a filterless frequency-doubled MPPS was proposed using a Polarization Modulator (PolM) with two modes of opposite-phase modulation without an optical filter^[21-22]. The systems possessed good frequency tunability when applied to the multi-wavelength operation. However, most of these realizations have been demonstrated for a single Frequency Multiplication Factor (FMF) value, hence lacking FMF tunability, which hardly fulfills the frequency selectivity of future systems. Although some approaches have been proposed to achieve an adjustable FMF^[23-25], the system is complicated due to using polarizers or multiple DPMZMs. Moreover, there are only even FMFs, but no odd ones, which cannot fulfill the future millimeter wave radar communications.

This paper presents a full-range tunable MPPS with adjustable FMF to obtain a high-frequency phase-shifted signal with a low-frequency device. In the proposed scheme, two 2^nd-order sidebands with a 90° relative phase were generated by two parallel MZMs and separated by an OC based on coherent superimposition. Then, the two separated 2^nd-order sidebands are further single-sideband modulated via DPMZMs with different sideband configurations, and a relative phase shift is introduced between the two optical tones by a PM in the down arm. After combination with the tones in the upper arm by an OC, the phase-shifted optical signal is converted to the electrical domain by a PD with a continuously tunable 360 deg phase shift and FMF from two to eight. The comparison between the previous schemes and our proposed scheme is shown in Table 1. Since the proposed MPPS doesn’t use any optical filter, it possesses better frequency tunability than the scheme proposed in Refs. [18-20]. And, the proposed system has stronger stability than the scheme proposed in Refs. [21-22] because no wavelength-sensitive polarizer is used. It is worth mentioning that the proposed system can have a larger working bandwidth due to adjustable FMF. Since our proposed MPPS does not use an optical filter, the bandwidth is limited only by the DPMZM bandwidth. The DPMZM is implemented using electro-optic technology with a bandwidth of 75 GHz^[26], and the MPPS can be designed to operate up to the V band (40–75 GHz). The phase shift can be tuned by the DC voltage applied to the PM and the FMF can be tuned by the bias voltages and relative phase shift of DPMZM. Therefore, this solution can be tuned to a specific transmission frequency according to the different functions that the radar needs to complete, to meet the frequency requirements of modern multi-function radars for transmission signals. Simulations on the proposed scheme are conducted. Desirable frequency tunability and multi-wavelength compatibility can be obtained without filters or wavelength-dependent devices. The proposed MPPS has relatively good phase and frequency tunability, low amplitude fluctuation, and large ESSR.

A schematic diagram of the proposed microwave photonic Frequency Multiplication Phase Shifter (FMPS) is shown in Fig. 1. The proposed setup consists of a continuous-wave light from the laser (CW laser), an optical splitter (OS), two parallel MZMs and DPMZMs, two 2×2 OCs with a coupling ratio of 1, and a PM. The DPMZM consists of three MZMs, including two parallel MZMs with a push-pull pattern (MZM_a and MZM_b) embedded in the main modulator (MZM_c).

A lightwave from the CW laser source expressed as ${{E}}_{0}{\mathrm{e}}^{{{{\rm{j}}}{\omega }}_{{c}}{t}}$ is split equally and fed into two parallel MZMs. Two MZMs, biased at the maximum transmission point, were driven by the RF signal at ω_m with a push-pull pattern to generate even-order sidebands. The lightwave in MZM₁ is modulated by the RF driving signal V_RFcos(ω_mt) and can be written as:

$ E_{\mathrm{MZM}_{1}}(t)=\frac{1}{2} {E}_{0} {\rm{e}}^{{\rm{j}} \omega_{c} t}\left[{\rm{e}}^{{\rm{j}} \beta_{1} \cos \omega_{m} t}+{\rm{e}}^{-{\rm{j}} \beta_{1} \cos \omega_{m} t}\right]\quad, $ (1)

here, β₁=πV_RF/V_π is the modulation depth of MZM₁. E₀ and ω_c are the amplitude and angular frequency of the optical carrier, respectively; V_RF and ω_m are the peak voltage and angular frequency of the RF driving signal, respectively; and V_π is the half-wave voltage of each MZM. Applying the Jacobi-Anger expansion to Eq. (1)

$ \begin{split} &E_{\mathrm{MZM}_{1}}(t)= \\ & \frac{1}{2} {E}_{0} {\rm{e}}^{{\rm{j}} \omega_{c} t}\left[\sum_{n=-\infty}^{+\infty} j^{n} {\rm{J}}_{n}\left(\beta_{1}\right) {\rm{e}}^{{\rm{j}} n \omega_{n} t}+\sum_{n=-\infty}^{+\infty} j^{n} {\rm{J}}_{n}\left(-\beta_{1}\right) {\rm{e}}^{{\rm{j}} n \omega_{n} t}\right]= \\ & {E}_{0} {\rm{e}}^{{\rm{j}} \omega_{c} t} \sum_{n=-\infty}^{+\infty}(-1)^{n} {\rm{J}}_{2 n}\left(\beta_{1}\right) {\rm{e}}^{2 {\rm{j}} n \omega_{m} t}\quad, \end{split} $ (2)

here J_n (·) represents the nth-order Bessel function of the first kind. It can be seen from Eq. (2) that the odd-order sidebands are suppressed, leaving the even-order sidebands. When the modulation index β₁=2.405, J₀(2.405)=0, J₂(2.405)=0.431, and J₄(2.405)=0.064, the ±2^nd-order sidebands will be dominant, which are 17.43 dB larger than the ±4^th order ones, and can be written as:

$ E_{\mathrm{MZM1}}(t)={E}_{0} \mathrm{e}^{{\rm{j}} \omega_{c} t}\left[\begin{array}{l} {\rm{J}}_{4}\left(\beta_{1}\right) \mathrm{e}^{-4 {\rm{j}} \omega_{m} t}-{\rm{J}}_{2}\left(\beta_{1}\right) \mathrm{e}^{-2 {\rm{j}} \omega_{m} t} \\ -{\rm{J}}_{2}\left(\beta_{1}\right) \mathrm{e}^{2 {\rm{j}} \omega_{m} t}+{\rm{J}}_{4}\left(\beta_{1}\right) \mathrm{e}^{4 {\rm{j}} \omega_{m} t} \end{array}\right]. $ (3)

Correspondingly, MZM₂ is driven by the RF driving signal V_RFcos(ω_mt+π/4), and the output lightwave can be written as:

$ \begin{split} E_{\mathrm{MZM}_{2}}(t)=&{E}_{0} {\rm{e}}^{{\rm{j}} \omega_{c} t} \displaystyle\sum_{n=-\infty}^{+\infty} {\rm{J}}_{2 n}\left(\beta_{1}\right) {\rm{e}}^{2 {\rm{j}} n \omega_{m^{t}} t+\tfrac{n {\text{π}} {\rm{j}}}{2}} =\\ &{E}_{0} \mathrm{e}^{{\rm{j}} \omega_{c} t}\left[\begin{array}{l} -{\rm{J}}_{4}\left(\beta_{1}\right) \mathrm{e}^{-4 {\rm{j}} \omega_{m} t}+{\rm{j}} {\rm{J}}_{2}\left(\beta_{1}\right) \mathrm{e}^{-2 {\rm{j}} \omega_{m} t} \\ -{\rm{j}} {\rm{J}}_{2}\left(\beta_{1}\right) \mathrm{e}^{2 {\rm{j}} \omega_{m} t}-{\rm{J}}_{4}\left(\beta_{1}\right) \mathrm{e}^{4 {\rm{j}} \omega_{m} t} \end{array}\right] .\\ \end{split} $ (4)

MZM₂ has the same modulation index b₁ as MZM₁. Then, the output signals of MZM₁ and MZM₂ are recombined using an OC (OC₁)^[27] with the transmission matrix

$ \boldsymbol{T}=\frac{\sqrt{2}}{2}\left[\begin{array}{ll} 1 & {\rm{j}} \\ {\rm{j}} & 1 \end{array}\right]\quad. $ (5)

The output lightwave from OC₁ can be expressed as:

$ \begin{aligned} {\left[\begin{array}{c} E_{\mathrm{c}}(t) \\ E_{\mathrm{d}}(t) \end{array}\right] } & ={E}_{0} {\rm{e}}^{{\rm{j}} \omega_{c} t} \frac{\sqrt{2}}{2}\left[\begin{array}{ll} 1 & {\rm{j}} \\ {\rm{j}} & 1 \end{array}\right]\left[\begin{array}{c} {\rm{J}}_{4}\left(\beta_{1}\right) {\rm{e}}^{-4 {\rm{j}} \omega_{m} t}-{\rm{J}}_{2}\left(\beta_{1}\right) {\rm{e}}^{-2 {\rm{j}} \omega_{m} t}-{\rm{J}}_{2}\left(\beta_{1}\right) {\rm{e}}^{2 {\rm{j}} \omega_{m} t}+{\rm{J}}_{4}\left(\beta_{1}\right) {\rm{e}}^{4 {\rm{j}} \omega_{m} t} \\ -{\rm{J}}_{4}\left(\beta_{1}\right) {\rm{e}}^{-4 {\rm{j}} \omega_{m} t}+{\rm{j}} {\rm{J}}_{2}\left(\beta_{1}\right) {\rm{e}}^{-2 {\rm{j}} \omega_{m} t}-{\rm{j}} {\rm{J}}_{2}\left(\beta_{1}\right) {\rm{e}}^{2 {\rm{j}} \omega_{m} t}-{\rm{J}}_{4}\left(\beta_{1}\right) {\rm{e}}^{4 {\rm{j}} \omega_{m} t} \end{array}\right] \\ & =\frac{\sqrt{2}}{2} {E}_{0} {\rm{e}}^{{\rm{j}} \omega_{c} t}\left[\begin{array}{l} -2 {\rm{J}}_{2}\left(\beta_{1}\right) {\rm{e}}^{-2 {\rm{j}} \omega_{m} t}+(1-{\rm{j}}) {\rm{J}}_{4}\left(\beta_{1}\right)\left(\mathrm{e}^{-4 {\rm{j}} \omega_{m} t}+{\rm{e}}^{4 {\rm{j}} \omega_{m} t}\right) \\ -2 {\rm{j}} {\rm{J}}_{2}\left(\beta_{1}\right) {\rm{e}}^{2 {\rm{j}} \omega_{m} t}+({\rm{j}}-1) {\rm{J}}_{4}\left(\beta_{1}\right)\left(\mathrm{e}^{-4 {\rm{j}} \omega_{m} t}+{\rm{e}}^{4 {\rm{j}} \omega_{m} t}\right) \end{array}\right]\quad. \end{aligned} $ (6)

It is observed from Eq. (6) that the ±2^nd -order sidebands are separated by OC₁ while OSSR is increased after OC_1. The ±4^th -order sidebands are suppressed by more than 19.58 dB, which is small enough to be ignored. Therefore, mainly the -2^nd -order sideband is output from the upper port, and the +2^nd-order one is output from the lower port. The OC₁ output optical field can be written in Jones matrix as:

$ \left[\begin{array}{l} E_{{\rm{c}}}(t) \\ E_{{\rm{d}}}(t) \end{array}\right]=-\sqrt{2} {E}_{0} {\rm{e}}^{{\rm{j}} w_ct}\left[\begin{array}{l} {\rm{J}}_{2}\left(\beta_{1}\right) {\rm{e}}^{-2 {\rm{j}} \omega_mt} \\ {\rm{j}} {\rm{J}}_{2}\left(\beta_{1}\right) {\rm{e}}^{2 {\rm{j}} \omega_mt} \end{array}\right].$ (7)

Then, the generated ±2^nd order sidebands are fed into DPMZM₁ and DPMZM₂, respectively, for single-sideband modulation shown in Fig.1.The optical field at the output of the DPMZM₁ with an ideal extinction can be written as:

$ \begin{split} E_{\mathrm{DPMZM}_{1}}(t)=&\frac{\sqrt{2}}{2} E_{\mathrm{MZM}_{\mathrm{a}}}(t)+\frac{\sqrt{2}}{2} E_{\mathrm{MZM}_{\mathrm{b}}}(t) {\rm{e}}^{\mathrm{j} {\text{π}} \frac{V_{\min }}{{V}_{{\text{π}}}}} =\\ &-{E}_{0} {\rm{e}}^{{\rm{j}}\left(\omega_{c}-2 \omega_{m}\right) t} {\rm{J}}_{2}\left(\beta_{1}\right)\left[\left(\frac{1}{2} {\rm{e}}^{{\rm{j}} \beta_{2} \sin \omega_{m} t}+\frac{1}{2} {\rm{e}}^{-{\rm{j}} \beta_{2} \sin \omega_{m} t} {\rm{e}}^{{\rm{j}} \frac{{\text{π}} V_{{\rm{D C}}}}{V_{{\text{π}}}}}\right)+{\rm{e}}^{{\rm{j}} {\text{π}} \frac{V_{{\rm{main}} }}{V_{{\text{π}}}}}\left(\frac{1}{2} {\rm{e}}^{{\rm{j}} \beta_{2} \sin \left(\omega_{m} t+\theta_{m}\right)}+\frac{1}{2} {\rm{e}}^{-{\rm{j}} \beta_{2} \sin \left(\omega_{m} t+\theta_{m}\right)} {\rm{e}}^{{\rm{j}} \frac{{\text{π}} V_{D C}}{V_{{\text{π}}}}}\right)\right]= \\ &-{E}_{0} {\rm{e}}^{{\rm{j}}\left(\omega_{c}-2 \omega_{m}\right) t} {\rm{J}}_{2}\left(\beta_{1}\right)\left\{\left(\frac{1}{2} \sum_{n=-\infty}^{+\infty} {\rm{J}}_{n}\left(\beta_{2}\right) {\rm{e}}^{n {\rm{j}} \omega_{\mathrm{m}} t}\left[1+(-1)^{n} {\rm{e}}^{{\rm{j}} \frac{{\text{π}} V_{D C}}{V_{{\text{π}}}}}\right]\left[1+{\rm{e}}^{\mathrm{j} {\text{π}} \frac{V_{{\rm{main}} }}{V_{{\text{π}}}}+n {\rm{j}} \theta_{m}}\right]\right\}\right. \quad,\\ \end{split} $ (8)

here E_DPMZM1(t),E_MZMa(t), and E_MZMb(t) are the output signals of the DPMZM, MZM_a, and MZM_b. β₂=πV_RF2/V_π is the modulation depth of DPMZM₁.V_RF2 is the peak voltage of the RF driving signal. V_main is the main bias voltage of DPMZM₁.V_DC is the DC bias voltage of MZM_a and MZM_b. θ_m is the optical relative phase shift introduced between the two RF driving signals. By adjusting the bias voltages of the modulator and the relative phase shift of the RF driving signals, a carrier-suppressed single-sideband RF modulated optical signal is generated.

When the two MZMs of DPMZM₁ are biased at the minimum transmission point by V_DC=V_π to suppress the optical carrier, and the RF signals satisfy the small-signal modulation condition, the lightwaves of their output mainly consist of the two first-order sidebands. As the RF driving signals to the two MZMs have a relative phase shift θ_m=π/2, and the main MZM has a bias voltage of V_main=−V_π/2, only the +1^st-order sideband at ω_c−ω_m is output from DPMZM₁ while the −1^st-order sideband at ω_c−3ω_m is suppressed,

$ E_{\mathrm{DPMZM}_{1}}=-2 {E}_{0} {\rm{J}}_{2}\left(\beta_{1}\right) {\rm{J}}_{1}\left(\beta_{2}\right) {\rm{e}}^{{\rm{j}}\left(\omega_{c}-\omega_{m}\right)t}\quad. $ (9)

In the same way, when V_main=V_π/2, only the −1^st-order sideband at ω_c-3ω_m appears at the output port of DPMZM₁ while the +1^st-order sideband is suppressed at ω_c-ω_m, which can be expressed as:

$ E_{\mathrm{DPMZM}_{1}}=2 {E}_{0} {\rm{J}}_{2}\left(\beta_{1}\right) {\rm{J}}_{1}\left(\beta_{2}\right) {\rm{e}}^{{\rm{j}}\left(\omega_{c}-3\omega_{m}\right)t} \quad.$ (10)

WhenV_DC=0,θ_m=π/4, and V_main=−V_π/2, the two MZMs in the DPMZM₁ are biased at the maximum transmission point, and odd-order signals are suppressed. Consistent with the analysis of Eq. (7), the high-order ones are small enough to be ignored; only the +2^nd sideband at ω_c appears at the output of the DPMZM₁ while the −2^nd sideband at ω_c−4ω_m is suppressed, which can be written as,

$ E_{\text {DPMZM }_{1}}(t)=-2 E_{0} {\rm{J}}_{2}\left(\beta_{1}\right) {\rm{J}}_{2}\left(\beta_{2}\right) {\rm{e}}^{{\rm{j}}\omega_{c}t} \quad.$ (11)

In the same way for DPMZM₂, when V_main=V_π/2, it can be written as:

$ E_{\text {DPMZM }_{1}}(t)=-2 {E}_{0} {\rm{J}}_{2}(\beta_1) {\rm{J}}_{2}\left(\beta_{2}\right) {\rm{e}}^{{\rm{j}}\left(\omega_{c}-4 \omega_{m}\right) t} \quad.$ (12)

Therefore, the lightwaves output from the two DPMZMs, as shown in Fig.1, can be summarized as

$ E_{\text {DPMZM }_{1}}(t)==-2 {E}_{0} {\rm{e}}^{{\rm{j}}\left(\omega_c-2\omega_m\right)t} {\rm{J}}_{2}(\beta_1)\left[{\rm{J}}_{p}(\beta_2) {\rm{e}}^{p{\rm{j}}\omega_mt}\right], $ (13)

$ E_{\text {DPMZM }_{2}}(t)==-2{\rm{ j}} {E}_{0} {\rm{e}}^{{\rm{j}}\left(\omega_c+2\omega_m\right)t} {\rm{J}}_{2}(\beta_1)\left[{\rm{J}}_{q}(\beta_3) {\rm{e}}^{q{\rm{j}}\omega_mt}\right] ,$ (14)

with p, q=±1, ±2. β₃=πV_RF3/V_π is the modulation depth of DPMZM₂. We used p, q to denote the orders of DPMZM₁ and DPMZM₂ modulated signals, respectively, which can be adjusted by setting the modulator bias voltages V_DC, V_main, and relative phase shiftθ_m of the RF driving signals applied to DPMZM₁ and DPMZM₂, respectively, as shown in Table 2.

When the relative phase shift θ_m=π/2, the modulator bias voltages V_DC=V_π andV_main=±V_π/2, p, q can be set as ±1, while as the relative phase shift θ_m=π/4 and the modulator bias voltages V_DC=0,V_main=±V_π/2, there are p, q=±2. The lightwave E_DPMZM2(t) experiences a phase shift introduced by a PM after DPMZM₂ and then combined with E_DPMZM1(t) via the OC₂ and becomes

$ \begin{aligned} {E}_{\text {coupler 2}}(t) =& \frac{\sqrt{2}}{2} {E}_{\mathrm{DPMZM}_{1}}(t)+\mathrm{j} {\rm{e}}^{\mathrm{j} \frac{{\text{π}} V_{\mathrm{PM}}}{V_{{\text{π}}}}} \frac{\sqrt{2}}{2} {E}_{\mathrm{DPMZM}_{2}}(t) =\\ & - \sqrt{2} {E}_{0} \mathrm{e}^{\mathrm{j} \omega_{c} t} \mathrm{J}_{2}\left(\beta_{1}\right)\left[\begin{array}{l} \mathrm{J}_{p}\left(\beta_{2}\right) \mathrm{e}^{(-2+p) \omega_{n t} t} \\ +\mathrm{J}_{q}\left(\beta_{3}\right) \mathrm{e}^{(2+q) \mathrm{j} \omega_{m} t} \mathrm{e}^{\mathrm{j} \psi} \end{array}\right], \end{aligned} $ (15)

here ψ is the phase shift introduced by the PM,

$ \psi ={\frac{{\text{π}} V_{{\rm{P M}}}}{V_{{\text{π}}}}}\quad, $ (16)

where V_PM is the DC voltage of the PM. The phase-shifted optical signal is then sent to the PD for square-law detection, and the microwave signal in the photocurrent can be expressed as:

$ {I_{{\rm{output}}}} \propto {{\rm{J}}_2}^2({\beta _1}){{{\rm{J}}} _P}({\beta _2}){{{\rm{J}}} _q}({\beta _3})\cos ((4 + q - p){\omega _m}t + \psi ). $ (17)

It can be seen from Eq. (17) that an RF signal with a frequency of (4+q−p)ω_m and a phase of ψ is generated in the photocurrent. The generated microwave signal has an adjustable FMF from 2 to 8, which can be set by the parameters of the two DPMZMs according to Table2. The phase shift ψ can be continuously tuned by adjusting the DC voltage of the PM.

To demonstrate the feasibility of the proposed frequency-multiplying phase shifter, a simulation was carried out based on the Optisystem platform as shown in Fig. 2. The frequency of the optical carrier from the LD is 193.1 THz, and the linewidth is 10 MHz. The CW lightwave from the laser source is split into two beams by an OS and then launched into two parallel MZMs (MZM1 and MZM2) with a half-wave voltage (V_π) of 4 V and extinction ratio (ER) of 40 dB. They are driven by the RF signals at a frequency of 10 GHz but with a 45-deg phase difference.

The RF driving signals have a voltage amplitude of 3.06 V to make the modulation index β₁=2.405. So, two Carrier-Suppressed Double-Side-Band (CS-DSB) modulated optical signals are generated with an OSSR larger than 16.5 dB, as shown by the optical spectra in Fig. 3(a)~3(b), which agrees with the theoretical analysis. After being coupled by a 2×2 Optical Coupler (OC₁), they are coherently superposed and converted from the CS-DSB optical signal to the CS-SSB ones with the spectra shown in Fig. 3(c)~3(d). It is observed that either of the ±2^nd -order optical sidebands are generated respectively, and the optical carrier and the other optical sidebands are further suppressed with an OSSR of more than 20 dB. Then, the separated ±2^nd -order sidebands are fed into DPMZM₁ or DPMZM₂ for further OCS-SSB modulation. A PM driven by the DC voltage is used to introduce a relative phase between the two tones from the two DPMZMs. The two optical tones are recombined by OC₂ and optoelectrical conversion by a high-speed PD to form a frequency multiplication microwave signal with full range phase shift over 360 deg and adjustable FMF from 2 to 8, as detailed below.

The main advantage of this FMPS is its highly tunable operation. Two optical sidebands with a frequency interval of 20 GHz corresponding to the FMF of 2 can be generated with an OSSR of around 20 dB at the output of OC₂ by setting V_DC=V_π, V_main=−V_π/2, and θ_m=π/2 according to Table 2, and the spectrum is shown in Fig. 4(a). After the PM introduced phase shift, the optical signal was sent to the PD for square-law detection, and the measured electrical spectrum of the photocurrent is shown in Fig.4 (b). It is observed that a 2^nd-order harmonic at 20 GHz is dominant, while the main spurious are 4^th and 6^th-order harmonics, which come mainly from the beating between the ±3^th-order sidebands and the ±1^st-order sidebands. The ESSR, defined as the power ratio of the primary signal to the most prominent spurious, is more than 18.51 dB.

As expected, other frequency multiplication signals with a step of 10 GHz are obtained at 30−80 GHz, corresponding to FMF of 3−8, by adjusting the parameters of the RF driving signal and DC bias voltages of the optical modulators. Their optical and electrical spectra of microwave signals in the photocurrent are shown in Fig. 5. Two optical sidebands are generated, and the optical carrier and other optical sidebands are deeply suppressed with a suppression ratio exceeding 18.95 dB. The different frequency components are found in the electrical spectrum due to the beating between residual sidebands. The OSSR and ESSR versus FMF are shown in Fig. 6. It can be seen that they all have similar OSSR and ESSR values greater than 15 dB.

Moreover, it can be seen from Fig. 6 that the 8^th-order harmonic performs better than the others due to the coherent superposition of the optical sidebands during modulation. Correspondingly the ESSR increases up to 41.06 dB at FMF=8. The spurious suppression ratio of the generated 80 GHz signal was measured for the input microwave signal with different powers, as shown in Fig. 7(a). The OSSR and ESSR of the generated microwave signal are more significant than 25 dB when the power of the CW lightwave is beyond 0 dBm. Fig. 7(b) shows that ESSR increases with ERs, while the OSSR is almost constant at about 35 dB when the ER increases above 30 dB. This indicates that excellent ESSR can obtain when the ER is above 30 dB.

To demonstrate the full phase shift tunability of the proposed FMPS over 360deg besides the frequency multiplication, we take a case in a 10 GHz RF sinusoidal signal becoming an 80 GHz signal. The waveforms of the generated 80 GHz microwave signals are depicted in Fig. 8(a−e) as the DC voltage of the PM with a half-wave voltage of 1.0 V is adjusted from 0 to 2.0 V with a step of 0.5 V.

For different applied DC voltages, the generated microwave signals have linearly increasing initial phases with a step of 90 deg. The phase shifts versus DC voltages have good linearity, shown in Fig. 8(f). This is consistent with the analytical prediction in Eq. (17). For the other cases with the FMF of 2−8, the generated microwave signals also have full tunable phases with the DC bias voltages, as shown in Fig. 8(f) (color online).

Stable operation is essential for an MPPS. To illustrate this property, simulations are carried out for the optical carrier with the wavelength varying from 1522 to 1586 nm. The stable power and the phase shift of the generated frequency multiplication phase shift microwave signals at different optical carrier frequencies are shown in Fig. 9 (a) and 9(b). The generated signals have nearly identical power at different wavelengths, and their power differences for different FMFs are below 7.0 dB. It is also observed from Fig. 9 (b) that the initial phases of the generated signals at different optical carrier frequencies have similar performance with some fluctuations smaller than 13.6 deg. The feature would allow the application of multichannel signal processing based on wavelength division multiplexing technologies.

The powers of the generated microwave signals are plotted in Fig. 9(c) when the frequency of the input signal is swept from 3 GHz to 21 GHz. It can be seen that the power of the generated signal remains constant although the frequency of the input signals varies, which means the power of the generated microwave signals is insensitive to the frequency of the RF driving signal. To better illustrate the phase-shifting capability, the simulated phase response of the MPPS for different frequencies of RF driving signal is shown in Fig. 9(d). As can be seen from Fig. 9(d), the phase shift of the generated microwave signal is independent of the frequency of the RF driving signal.

The analytical and simulation results in previous sections were obtained assuming the situation. However, in the real world, the accurate adjustment of the parameters is impossible, which affects the performance of the proposed MPPS. So, the performance of the proposed MPPS under the effect of the modulation index deviation and the RF signals’ phase deviation are investigated. We firstly fix the phase difference between the two input RF signals of MZM₁ and MZM₂ to be 45° and the modulation index of MZM₁ to be 2.408, but vary the modulation index of MZM₂ from 1.7 to 3.3 by varying the amplitude of the RF driving signal from 3.0 to 6.0 V. The effects of the deviation of the modulation indices on the ESSR and OSSR are shown in Fig. 10(a) which illustrates that the largest suppression ratio is obtained when β₁=2.3, which is the optimal modulation index. As the modulation index of MZM₂ has a value between 2.22 and 2.38, the ESSR is larger than 20 dB due to its better tolerance to modulation index deviation.

On the other hand, when the phase difference between the two input RF signals of MZM₁ and MZM₂ deviated from the ideal case of 45° because of imperfections of the electrical 45° hybrid, the performance of the proposed MPPS is also degraded. As the phase difference is varying from 16° to 75° when the modulation indexes of MZM₁ and MZM₂ are 2.3, the OSSR and ESSR are plotted in Fig. 9(b). It can be seen that to assure an ESSR larger than 20 dB, the phase difference should be controlled within the range of 36°–54°.

We have proposed and analyzed a photonic scheme to realize MPPS with tunable FMF based on a filterless architecture. The frequency of the generated microwave phase-shifted signal can be tuned over a wide frequency range since no optical filter is used in the system. The RF phase-shift operation can be implemented simply by tuning the DC bias voltage of the PM, and the phase shift has a linear relationship with the DC voltage and can be tuned over a 360 deg range. The 360 deg phase shift and frequency multiplication capabilities are verified by simulation, and the wavelength of the optical carrier and the frequency of the RF driving signal have little influence on the two features. Moreover, the excellent tolerance to the modulation index deviation and phase difference was demonstrated. All the above advantages such as very compact configuration, easy phase tuning, and wide operation bandwidth overcome electronic technology bottlenecks and make the proposed system have promising applications in many directions like future 5G systems with high efficiency and adaptability and compact high-resolution radars.