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Photonic-assisted frequency downconverter with self-interference cancellation and fiber dispersion elimination based on stimulated Brillouin scattering

Xiaojie Fan, Yinfang Chen, Xuhua Cao, Sha Zhu, Ming Li, Ninghua Zhu, and Wei Li

Open Access Open Access

Abstract

A photonic-assisted frequency downconverter with self-interference cancellation and fiber dispersion elimination is proposed for in-band full-duplex (IBFD) radio-over-fiber (ROF) systems based on stimulated Brillouin scattering. In this work, a dual-polarization Mach-Zehnder modulator (DPol-MZM) is employed to cancel the self-interference signal in the optical domain, thus the proposed system has a large operation bandwidth without the limitation of the electrical bottleneck. Meanwhile, a widely tunable microwave photonic filter (MPF) based on stimulated Brillouin scattering (SBS) is constructed to perform carrier-suppressed single-sideband modulation of LO. Therefore, the proposed photonic frequency downconverter is naturally free from fiber dispersion. Furthermore, thanks to the characteristics of the SBS-based MPF, such as large out-of-band rejection ratio, high resolution and wideband tunability, the proposed system has a high flexibility and downconversion efficiency, which is of great significance for IBFD ROF systems.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

In recent years, radio-over-fiber (ROF) systems have been a hot research topic for its advantages in wireless communications such as large bandwidth, low loss, high flexibility and low cost [1–4]. For conventional ROF systems, the uplink and downlink signals are in different frequency bands, so the interferences between them can be easily avoided. To meet the ever-increasing demand for higher data rate, in-band full-duplex (IBFD) ROF systems are proposed to improve spectral efficiency [5,6]. However, since the uplink and downlink signals share the same frequency band in IBFD ROF systems, the self-interference (SI) signal cannot be simply removed by optical or electrical filters. Therefore, it is essential for IBFD ROF systems to eliminate the SI signal.

Originally, the self-interference cancellation (SIC) based on electrical methods has been researched [7,8]. However, due to the limitation of the electronic bottleneck, the frequency of SI signal is usually low and the operation bandwidth is generally narrow in electrical SIC systems. To overcome the electronic bottleneck, photonic-assisted SIC systems have been widely investigated thanks to the advantages of microwave photonics, such as large bandwidth, low loss, high carrier frequency, electromagnetic inference elimination and so on [9]. With a balanced photodetector (BPD), two electro-absorption modulators (EAMs) [10], or two directly modulated lasers [11], or two electro-absorption modulated lasers (EMLs) [12] were utilized for RF SIC. In [13], instead of a BPD, a single-port PD in conjunction with two polarization modulators was used to cancel the SI signal for IBFD ROF system. However, the forementioned photonic schemes [10–13] all employed two separate optical paths, which is complicated and sensitive to the environmental changes. Therefore, some RF SIC schemes [14–16] based on one optical path were proposed. In [14], an integrated dual-parallel MZM was used to realize RF SIC in the optical domain by biasing at the quadrature and the minimum point. A dual-drive Mach-Zehnder modulator (DDMZM) and a fiber Bragg grating (FBG) were utilized for RF SIC in [15]. Recently, another photonic technique based on dual phase modulation in Sagnac loop has been proposed for RF SIC in [16].

As we know, for IBFD ROF systems, the received signals usually need to be transmitted from base stations (BSs) to central office (CO) for signal processing. Therefore, RF frequency downconversion and fiber transmission capability are also significant for IBFD ROF system applications. In recent years, some photonic-assisted schemes have been proposed for RF SIC and frequency downconversion with immunity to dispersion. In [17], a dual-polarization dual-parallel MZM (DPol-DPMZM) was used to realize RF SIC and frequency downconversion. The chromatic dispersion-induced power fading (CDIP) was neglected because the downconverted signal was at a low frequency. The SIC depths of single-frequency and wideband signals were up to 57 and 26 dB. However, if the fiber transmission distance is long enough or the frequency of downconverted signal increases, the system would suffer from severe power fading. In order to be free from fiber dispersion, other approaches [18–20] have been investigated based on single sideband (SSB) modulation or phase compensation. However, in [18], LO signal was modulated to generate multiple optical harmonics for frequency downconversion, which might lead to a large spurious. Two 90° electrical hybrid couplers were employed in [19], which would affect the response consistency of SI and self-reference (SR) signals. Due to phase compensation, the technique in [20] was not suitable for ROF systems with multiple BSs. In addition, the SIC cancellation depths of single-frequency and wideband signals were around 40 and 20 dB in [18–20], which were lower than that in [17] and mainly caused by the degeneration of the response consistency of SR and SI signals and the performance of modulator. Therefore, it is desirable to propose a photonic technique for IBFD ROF systems, which can simultaneously realize RF SIC and frequency downconversion with immunity to dispersion and large cancellation depth.

In this work, we propose a photonic-assisted frequency downconverter with self-interference cancellation and fiber dispersion elimination based on a DPol-MZM, which is promising for multi-functional or multi-scenarios IBFD ROF systems. The upper MZM is used to generate the optical carrier-suppressed double sidebands (CS-DSBs) of the signal of interest (SOI) and SI signal, while the lower MZM is used to generate the optical CS-DSBs of the local oscillator (LO) and self-reference (SR) signal. By matching the phases and powers of the applied SI and SR signals and introducing a π phase difference between SI- and SR-modulated optical signals, the SI signal can be eliminated well in the optical domain, meaning that the system has a large operation bandwidth. Meanwhile, a special microwave photonic filter (MPF) is constructed based on stimulated Brillouin scattering (SBS) to realize carrier-suppressed single sideband (SSB) modulation of the LO. After photodetection, a frequency downconverter is achieved. Thanks to the advantages of SBS-based MPF [21–23], such as large out-of-band rejection ratio, high resolution and wideband tunability, the proposed frequency downconverter can perform high flexibility and good fiber transmission capacity without dispersion. In addition, high-power downconverted signals can be achieved based on SBS-based MPF instead of increasing the power of LO signal itself, which improves the downconversion efficiency. Both theoretical analysis and experimental demonstration have been carried out in our work. The SIC depths of the single-frequency and wideband signals are up to around 55 and 28 dB, which has obvious improvements compared to the values in [18–20].

2. Principle

2.1 RF SIC in the optical domain

The schematic diagram of the proposed system is shown in Fig. 1(a). A linearly polarized optical carrier from a laser diode (LD) is split into two paths via a 50:50 optical coupler (OC). In the upper path, the optical carrier is injected into a dual-polarization Mach-Zehnder modulator (DPol-MZM), which consists of two sub-MZMs (x-MZM and y-MZM) at two orthogonal polarization state, a 90° polarization rotator (90° PR) and a polarization beam combiner (PBC). The x-MZM is driven by the SOI and SI signals while the y-MZM is driven by the LO and SR signals. Assuming that the SOI, SI, SR and LO signals are expressed as

$$\begin{array}{l} {U_{SOI}}(t) = {V_{SOI}}\cos [{\omega _1}t + {\theta _{SOI}}(t)],\\ {U_{SI}}(t) = {V_{SI}}\cos [{\omega _1}t + {\theta _{SI}}(t)],\\ {U_{SR}}(t) = {V_{SR}}\cos [{\omega _1}t + {\theta _{SR}}(t)],\\ {U_{LO}}(t) = {V_{LO}}\cos [{\omega _2}t + {\theta _{LO}}(t)]. \end{array}$$
where VSOI, VSI, VSR and VLO are the amplitudes of the SOI, SI, SR and LO signals, respectively; ω1 is the angular frequency of the SOI, SI and SR signals, ω2 is the angular frequency of the LO signal; θSOI(t), θSI(t), θSR(t) and θLO(t) are the phases of the SOI, SI, SR and LO signals, respectively. The x- and y-MZM are both biased at the minimum transmission point (MITP) to realize the CS-DSB modulation. Considering the small signal modulation and applying Jacobi-Anger expansion, the optical field at the output of the DPol-MZM can be expressed as
$$\begin{aligned}{E_{DPol - MZM}}(t)& = \frac{1}{4}\left[ {\begin{array}{{c}} {{E_{x - MZM}}(t)}\\ {{E_{y - MZM}}(t)} \end{array}} \right]\\& = \frac{1}{8}{E_0}{e^{j{\omega _0}t}}\left[ {\begin{array}{{c}} {{e^{j\frac{\pi }{{{V_\pi }}}[{U_{SOI}}(t) + {U_{SI}}(t)]}} - {e^{ - j\frac{\pi }{{{V_\pi }}}[{U_{SOI}}(t) + {U_{SI}}(t)]}}}\\ {{e^{j\frac{\pi }{{{V_\pi }}}[{U_{LO}}(t) + {U_{SR}}(t)]}} - {e^{ - j\frac{\pi }{{{V_\pi }}}[{U_{LO}}(t) + {U_{SR}}(t)]}}} \end{array}} \right]\\& = \frac{1}{4}{E_0}{e^{j{\omega _0}t + j\frac{\pi }{2}}}\left[ {\begin{array}{{c}} \begin{array}{l} \{ {J_1}({\beta_{SOI}}){J_0}({\beta_{SI}})[{e^{j({\omega_1}t + {\theta_{SOI}}(t))}} + {e^{ - j({\omega_1}t + {\theta_{SOI}}(t))}}]\\ + {J_0}({\beta_{SOI}}){J_1}({\beta_{SI}})[{e^{j({\omega_1}t + {\theta_{SI}}(t))}} + {e^{ - j({\omega_1}t + {\theta_{SI}}(t))}}]\} \end{array}\\ \begin{array}{l} \{ {J_0}({\beta_{LO}}){J_1}({\beta_{SR}})[{e^{j({\omega_1}t + {\theta_{SR}}(t))}} + {e^{ - j({\omega_1}t + {\theta_{SR}}(t))}}]\\ + {J_1}({\beta_{LO}}){J_0}({\beta_{SR}})[{e^{j({\omega_2}t + {\theta_{LO}}(t))}} + {e^{ - j({\omega_2}t + {\theta_{LO}}(t))}}]\} \end{array} \end{array}} \right] \end{aligned}$$
where E0 and ω0 are the amplitude and angular frequency of the optical carrier, Vπ is the half-wave voltage of the x- and y-MZM, βi=πVi/Vπ is the modulation indices of SOI, SI, SR and LO signals (i = SOI, SI, SR and LO), J0 and J1 are the 0th- and 1st-order Bessel function of the first kind. According to Eq. (2), it can be seen that the optical CS-DSBs of the SOI, SI, SR and LO are generated successfully, as shown in Figs. 1(b-i)-1(b-ii).

 figure: Fig. 1.

Fig. 1. (a) The schematic diagram of the proposed photonic-assisted frequency downconverter with self-interference cancellation and fiber dispersion elimination based on stimulated Brillouin scattering. LD, laser diode; OC, optical coupler; DPol-MZM, dual-polarization Mach-Zehnder modulator; 90° PR, 90° polarization rotator; PBC, polarization beam combiner; PC, polarization controller; Pol., polarizer; EDFA, erbium-doped fiber amplifier; HNLF, high nonlinear fiber; EATT, electrical attenuator; TTDL, tunable time delay line; TOBPF, tunable optical bandpass filter; SMF, single mode fiber; PD, photodetector. (b-i)-(b-vi) The simplified optical spectra corresponding to nodes (i)-(vi) in (a).

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Following the DPol-MZM, a PC is utilized to adjust the polarization states of the optical signals. The transfer function of the PC is given by [24]

$${P_{PC}} = \left[ {\begin{array}{{cc}} {\cos \alpha }&{ - \sin \alpha }\\ {\sin \alpha }&{\cos \alpha } \end{array}} \right]\left[ {\begin{array}{{cc}} {{e^{j\frac{\delta }{2}}}}&0\\ 0&{{e^{ - \frac{\delta }{2}}}} \end{array}} \right],$$
where α is the rotation angle, and δ is the phase difference between the two orthogonal polarized components introduced by the birefringence in a PC. When the state of PC is adjusted to satisfy α=π/4 and δ=π, the optical field after Pol. can be written as
$$\begin{aligned} {E_{Pol.}} &= \frac{1}{4}(\sin \alpha \cdot {E_{x - MZM}}(t)\cdot {e^{j\frac{\delta }{2}}} + \cos \alpha \cdot {E_{y - MZM}}(t)\cdot {e^{ - j\frac{\delta }{2}}})\\& ={-} \frac{{\sqrt 2 }}{8}{E_0}{e^{j{\omega _0}t}}\left[ {\begin{array}{{c}} \begin{array}{l} {J_1}({\beta_{SOI}}){J_0}({\beta_{SI}})[{e^{j({\omega_1}t + {\theta_{SOI}}(t))}} + {e^{ - j({\omega_1}t + {\theta_{SOI}}(t))}}]\\ + {J_0}({\beta_{SOI}}){J_1}({\beta_{SI}})[{e^{j({\omega_1}t + {\theta_{SI}}(t))}} + {e^{ - j({\omega_1}t + {\theta_{SI}}(t))}}]\\ - {J_0}({\beta_{LO}}){J_1}({\beta_{SR}})[{e^{j({\omega_1}t + {\theta_{SR}}(t))}} + {e^{ - j({\omega_1}t + {\theta_{SR}}(t))}}] \end{array}\\ { - {J_1}({\beta_{LO}}){J_0}({\beta_{SR}})[{e^{j({\omega_2}t + {\theta_{LO}}(t))}} + {e^{ - j({\omega_2}t + {\theta_{LO}}(t))}}]} \end{array}} \right] \end{aligned}$$

Under the small signal modulation ((βSOI, βSI, βLO, βSR<<1; J0(βSOI), J0(βSI), J0(βLO), J0(βSR) ≈1; J1(βSOI), J1(βSI), J1(βLO), J1(βSR) <<1), it is easy to satisfy the conditions J0(βSOI)J1(βSI)= J0(βLO)J1(βSR) and θSI(t)= θSR(t) by adjusting the amplitudes and phases of the applied electrical signals. Therefore, the Eq. (4) can be re-expressed as

$${E_{Pol.}} ={-} \frac{{\sqrt 2 }}{8}{E_0}{e^{j{\omega _0}t}}\left[ {\begin{array}{{c}} {{J_1}({\beta_{SOI}}){J_0}({\beta_{SI}})[{e^{j({\omega_1}t + {\theta_{SOI}}(t))}} + {e^{ - j({\omega_1}t + {\theta_{SOI}}(t))}}]}\\ { - {J_1}({\beta_{LO}}){J_0}({\beta_{SR}})[{e^{j({\omega_2}t + {\theta_{LO}}(t))}} + {e^{ - j({\omega_2}t + {\theta_{LO}}(t))}}]} \end{array}} \right]$$

According to Eq. (5), we can find that the SI-modulated optical CS-DSBs are cancelled successfully by SR-modulated optical CS-DSBs after Pol., as shown in Fig. 1(b-iii).

 figure: Fig. 2.

Fig. 2. Experimental setup of the proposed photonic-assisted frequency downconverter with improved self-interference cancellation and fiber dispersion elimination based on stimulated Brillouin scattering. LD, laser diode; OC, optical coupler; DPol-MZM, dual-polarization Mach-Zehnder modulator; 90° PR, 90° polarization rotator; PBC, polarization beam combiner; PC, polarization controller; Pol., polarizer; ISO, optical isolator; MZM, Mach-Zehnder modulator; EDFA, erbium-doped fiber amplifier; HNLF, high nonlinear fiber; EATT, electrical attenuator; TTDL, tunable time delay line; TOBPF, tunable optical bandpass filter; SMF, single mode fiber; PD, photodetector; MSG, microwave signal generator; VSG, vector signal generator; ESA, electrical spectrum analyzer.

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2.2 RF frequency downconversion with SIC and fiber dispersion elimination

Usually, in the IBFD ROF systems, the received signals need to be transmitted from BSs to CO via fiber, so the CDIP is a dominant factor which is need to be considered. Assuming that the optical signals at the output of Pol. are directly sent to CO by single mode fiber (SMF), the optical field at the output of SMF can be written as

$${E_{SMF}} ={-} \frac{{\sqrt 2 }}{8}{E_0}{e^{j{\omega _0}t}}\left[ {\begin{array}{{c}} {{J_1}({\beta_{SOI}}){J_0}({\beta_{SI}})[{e^{j({\omega_1}t + {\theta_{SOI}}(t) + {\theta_{ + 1}})}} + {e^{ - j({\omega_1}t + {\theta_{SOI}}(t) - {\theta_{ - 1}})}}]}\\ { - {J_1}({\beta_{LO}}){J_0}({\beta_{SR}})[{e^{j({\omega_2}t + {\theta_{LO}}(t) + \theta_{ + 1}^{\prime})}} + {e^{ - j({\omega_2}t + {\theta_{LO}}(t) - \theta_{ - 1}^{\prime})}}]} \end{array}} \right]$$
where θ+1, θ-1, θ’+1 and θ’-1 are respectively the dispersion-induced phase shifts of ±1st-order sidebands of SOI and LO. Expanding the propagation constant β in Talar series, we have [9]
$$\begin{array}{l} {\theta _{ + 1}}(\omega ) = z\beta ({\omega _0}) + z{\beta ^{\prime}}({\omega _0}){\omega _1} + \frac{1}{2}z{\beta ^{^{\prime\prime}}}({\omega _0})\omega _1^2\\ {\theta _{ - 1}}(\omega ) = z\beta ({\omega _0}) - z{\beta ^{\prime}}({\omega _0}){\omega _1} + \frac{1}{2}z{\beta ^{^{\prime\prime}}}({\omega _0})\omega _1^2\\ \theta _{ + 1}^{\prime}(\omega ) = z\beta ({\omega _0}) + z{\beta ^{\prime}}({\omega _0}){\omega _2} + \frac{1}{2}z{\beta ^{^{\prime\prime}}}({\omega _0})\omega _2^2\\ \theta _{ - 1}^{\prime}(\omega ) = z\beta ({\omega _0}) - z{\beta ^{\prime}}({\omega _0}){\omega _2} + \frac{1}{2}z{\beta ^{^{\prime\prime}}}({\omega _0})\omega _2^2 \end{array}$$
where z is the length of the SMF, β(ω0), β’(ω0), and β’’(ω0) are respectively the 0th-, 1st-, and 2nd- order derivatives of β. Ignoring the DC and double frequency components, the photocurrent after photodetection can be expressed as
$$\begin{array}{c} i(t) \propto E_0^2{J_0}({\beta _{SI}}){J_0}({\beta _{SR}}){J_1}({\beta _{SOI}}){J_1}({\beta _{LO}})\cos [\frac{{zD({\omega _0}){\lambda ^2}}}{{4\pi c}}(\omega _1^2 - \omega _2^2)]\\ \times \{ \cos [{\omega _1}t + {\omega _2}t + {\theta _{SOI}}(t) + {\theta _{LO}}(t) + z{\beta ^{\prime}}({\omega _0})({\omega _2} + {\omega _1})]\\ + \cos [{\omega _1}t - {\omega _2}t + {\theta _{SOI}}(t) - {\theta _{LO}}(t) + z{\beta ^{\prime}}({\omega _0})({\omega _1} - {\omega _2})\} \end{array}$$
where D(ω0) =-2πcβ’’(ω0)/λ2 is the dispersion coefficient of the fiber, c is the velocity of light and λ is the wavelength of the optical carrier. According to Eq. (8), we can find that the RF frequency up/down conversion can be realized simultaneously based on our system. However, the amplitudes of the generated frequency up/downconverted signals are proportional to the coefficient cos(zD(ω0)λ2(ω12-ω22)/4πc), which will bring about power fading. Therefore, to eliminate the power fading, the optical SSB modulation of LO is a promising candidate. In addition, considering that high conversion efficiency is a key performance for frequency downconverter in IBFD ROF systems, an MPF based on SBS is constructed to realize optical carrier-suppressed SSB modulation of LO and improve the conversion efficiency.

Firstly, as a “probe beam”, the optical signals after Pol. are coupled into a spool of high nonlinear fiber (HNLF). At the same time, the optical carrier in the lower path of OC is injected into another MZM, which is biased at MITP to generate optical CS-DSBs of the pump wave as a “control beam” shown in Fig. 1(b-iv). After amplified by an EDFA (erbium-doped fiber amplifier), the “control beam” are fed into the HNLF via a circulator in the opposite direction of the “probe beam”. At this moment, SBS process occurs in the HNLF, which introduces SBS gain and loss spectra at the both sides of the “control beam” shown in Fig. 1(b-iv). By adjusting the power and polarization state of the “control beam”, the performance of the SBS can be optimized. If the frequency of the pump wave is set as

$${f_{pump}} = {f_B} + {f_{LO}}$$
where fB is the Brillouin frequency shift of the HNLF. Defining that gB and αB are the SBS gain and loss factor, we have [25]
$$\begin{array}{l} {g_B}(\varDelta f) = \frac{{{g_0}}}{2}\frac{{{{(\Delta {\nu _B}/2)}^2}}}{{\Delta {f^2} + {{(\Delta {\nu _B}/2)}^2}}} + j\frac{{{g_0}}}{4}\frac{{\Delta {\nu _B}\Delta f}}{{\Delta {f^2} + {{(\Delta {\nu _B}/2)}^2}}},\\ {\alpha _B}(\varDelta f) ={-} \frac{{{g_0}}}{2}\frac{{{{(\Delta {\nu _B}/2)}^2}}}{{\Delta {f^2} + {{(\Delta {\nu _B}/2)}^2}}} - j\frac{{{g_0}}}{4}\frac{{\Delta {\nu _B}\Delta f}}{{\Delta {f^2} + {{(\Delta {\nu _B}/2)}^2}}} \end{array}$$
where g0 and ΔνB represent the line-center gain factor and Brillouin linewidth of the HNLF, Δf is the frequency deviation from the SBS gain/loss spectrum center. Therefore, the optical field after the circular can be written as
$${E_{circular}} ={-} \frac{{\sqrt 2 }}{8}{E_0}{e^{j{\omega _0}t}}\left[ {\begin{array}{{c}} {{J_1}({\beta_{SOI}}){J_0}({\beta_{SI}})[{e^{j({\omega_1}t + {\theta_{SOI}}(t))}} + {e^{ - j({\omega_1}t + {\theta_{SOI}}(t))}}]}\\ \begin{array}{l} - {J_1}({\beta_{LO}}){J_0}({\beta_{SR}})[G(\Delta f){e^{j({\omega_2}t + {\theta_{LO}}(t))}}{e^{j\varphi (\Delta f)}}\\ + A(\Delta f){e^{ - j({\omega_2}t + {\theta_{LO}}(t))}}{e^{ - j\varphi (\Delta f)}}]\\ - \sqrt 2 {J_1}({\beta_{pump}}){e^{j{\omega_{pump}}t + j\frac{\pi }{2}}} - \sqrt 2 {J_1}({\beta_{pump}}){e^{ - j{\omega_{pump}}t + j\frac{\pi }{2}}} \end{array} \end{array}} \right]$$

In details,

$$\begin{array}{l} G(\Delta f) = {e^{Re [{g_B}(\Delta f)]{I_p}L}} = {e^{\frac{{{g_0}{I_p}L}}{2}\frac{{{{(\Delta {\nu _B}/2)}^2}}}{{\Delta {f^2} + {{(\Delta {\nu _B}/2)}^2}}}}},\\ A(\Delta f) = {e^{Re [{\alpha _B}(\Delta f)]{I_p}L}} = {e^{ - \frac{{{g_0}{I_p}L}}{2}\frac{{{{(\Delta {\nu _B}/2)}^2}}}{{\Delta {f^2} + {{(\Delta {\nu _B}/2)}^2}}}}},\\ \varphi (\Delta f) = {\mathop{\rm Im}\nolimits} [{g_B}(\Delta f)]{I_p}L = \frac{{{g_0}{I_p}L}}{4}\frac{{\Delta {\nu _B}\Delta f}}{{\Delta {f^2} + {{(\Delta {\nu _B}/2)}^2}}} \end{array}$$
where Vpump and ωpump are respectively the amplitude and angular frequency of the pump wave, βpump=πVpump/Vπ is modulation index of pump wave, Ip is the optical power of pump-modulated signals, and L is the length of the HNLF. Due to Δf = 0 and considering the ideal performance of the SBS, we can get G(Δf)>>1, A(Δf)≈0 and φ(Δf) = 0. Therefore, the +1st-order optical sideband of the LO is amplified while the -1st-order optical sideband suffers from a significant fading, as shown in Fig. 1(b-v). In fact, if the condition fLO = fpump+fB is satisfied, we can also achieve the SSB modulation of LO by amplifying the -1st-order optical sideband and suppressing the +1st-order optical sideband of LO based on a low-frequency pump signal. A tunable optical bandpass filter (TOBPF) is cascade behind the circular to filter the optical CS-DSBs of the pump wave. Therefore, the optical field after the TOBPF can be written as
$${E_{TOBPF}} ={-} \frac{{\sqrt 2 }}{8}{E_0}{e^{j{\omega _0}t}}\left[ {\begin{array}{{c}} {{J_1}({\beta_{SOI}}){J_0}({\beta_{SI}})[{e^{j({\omega_1}t + {\theta_{SOI}}(t))}} + {e^{ - j({\omega_1}t + {\theta_{SOI}}(t))}}]}\\ { - {J_1}({\beta_{LO}}){J_0}({\beta_{SR}})G(\Delta f){e^{j({\omega_2}t + {\theta_{LO}}(t))}}} \end{array}} \right]$$

The simplified optical spectrum is shown in Fig. 1(b-vi), consisting of the optical CS-DSBs of the SOI signal and the optical CS-SSBs of LO signal. After transmitted by SMF, the optical signal is

$${E_{SMF}} ={-} \frac{{\sqrt 2 }}{8}{E_0}{e^{j{\omega _0}t}}\left[ {\begin{array}{{c}} {{J_1}({\beta_{SOI}}){J_0}({\beta_{SI}})[{e^{j({\omega_1}t + {\theta_{SOI}}(t) + {\theta_{ + 1}})}} + {e^{ - j({\omega_1}t + {\theta_{SOI}}(t) - {\theta_{ - 1}})}}]}\\ { - {J_1}({\beta_{LO}}){J_0}({\beta_{SR}})G(\Delta f){e^{j({\omega_2}t + {\theta_{LO}}(t) + \theta_{ + 1}^{\prime})}}} \end{array}} \right]$$

After detection by PD in BSs, the frequency conversion is realized. The photocurrent is

$$\begin{aligned} i(t) &\propto E_0^2{J_0}({\beta _{SI}}){J_0}({\beta _{SR}}){J_1}({\beta _{SOI}}){J_1}({\beta _{LO}})G(\Delta f)\\& \times \{ \cos [{\omega _1}t + {\omega _2}t + {\theta _{SOI}}(t) + {\theta _{LO}}(t) + \theta _{ + 1}^{\prime} - {\theta _{ - 1}}]\\& + \cos [{\omega _1}t - {\omega _2}t + {\theta _{SOI}}(t) - {\theta _{LO}}(t) + {\theta _{ + 1}} - \theta _{ + 1}^{\prime}]\} \end{aligned}$$

As can be seen from Eq. (15), the fiber dispersion only affects the phases of the RF frequency downconverted signal rather than its amplitude. In other words, the fiber dispersion-induced power fading is eliminated successfully regardless of the transmission frequency and distance. In addition, due to the gain coefficient G(Δf)>>1, the power of the generated downconverted signal increases, meaning that the conversion efficiency is improved. Furthermore, if necessary, the frequency upconverted signal with SIC and dispersion suppression can also be achieved by employing a large-bandwidth high-speed PD. Therefore, the RF SIC and frequency up/down conversion can be realized simultaneously with immunity to power fading based on the proposed system, which is potential for multi-functional ROF systems.

3. Results

3.1 RF SIC in the optical domain

A proof-of-concept experiment was carried out based on the setup shown in Fig. 2. An optical carrier at 1550.008 nm from an LD was divided into two paths via an optical coupler (OC). At the upper path, the optical carrier was injected into a DPol-MZM. An RF signal from a microwave signal generator (MSG1) was set to the SOI with a power of 8 dBm, whereas the LO with a power of 8 dBm was provided by another MSG2. An RF signal from a vector signal generator (VSG) was split into two parts via a 3-dB electrical coupler (EC1), as the SI and SR signals with powers of 10 dBm. The SOI and SI signals were combined by a 3-dB EC2 and sent to x-MZM via a tunable electrical attenuator, while the LO and SR signals were coupled by a 3-dB EC3 and applied to y-MZM via a tunable time delay line. To erase SI signal, the time delay and power consumption of the SR signal from VSG to y-MZM should be consistent with that of the SI signal from VSG to x-MZM. Therefore, it was necessary to carefully select the cables and connectors and precisely adjust the electrical attenuator and time delay line in the two paths. By setting the DC biases of x- and y-MZM at MITP, the CS-DSBs of the SOI, SI, LO and SR signals were generated successfully at the output of DPol-MZM. First, to observe the SIC in the optical domain, the SOI and LO were off in the experiment. The SI and SR signals were set to 11 GHz. An optical spectrum analyzer was employed to observe the optical signals at the output of the polarizer. Adjusting the PC, when the SI- and SR-modulated optical signals were suppressed to the maximum, the state of PC satisfied α=π/4 and δ=π according to the principle of vector superposition. Figure 3(a) shows the measured optical spectra at the output of Pol. The blue line represents the SI-modulated optical CS-DSBs from x-MZM, in which the optical carrier is not suppressed well because of the poor extinction ratio of the x-MZM. The orange line represents the SR-modulated optical CS-DSBs from y-MZM, where the optical carrier is also not completely suppressed due to finite extinction ratio of the y-MZM. The green dash line depicts the optical CS-DSBs of the SI and SR signals without SIC, whereas the red line shows the optical spectrum when the SIC is attained. The residual optical carrier and ±1 sidebands of the SI signal can be still observed after SIC, which does not fit well with the ideal simplified spectrum in Fig. 1(b-iii). This is mainly caused by the nonideal extinction ratio and half-wave voltage of the DPol-MZM. According to Fig. 3, the SI-modulated optical CS-DSBs are suppressed well by about 30 dB. To verify the frequency tunability of the proposed system, the SI and SR were reset to 13, 15 and 17 GHz. Figures 3(b)–3(d) disclose the measured corresponding optical spectra at the output of Pol. The optical sidebands of the SI signal are all cancelled successfully, indicating that the proposed system has a large operation bandwidth. After Pol., a spool of HNLF was cascaded to perform SBS.

 figure: Fig. 3.

Fig. 3. The measured optical spectra at the output of polarizer when the SI was set to (a) 11 GHz, (b) 13 GHz, (c) 15 GHz and (d) 17 GHz.

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3.2 RF frequency downconversion with SIC in the electrical domain

To further demonstrate the SIC in the electrical domain, the LO signal was set to 10 GHz and turned on to perform the frequency downconversion. As described in the principal section, an SBS-based MPF was built at the lower path of the OC. Specifically, an MZM was employed to modulate the optical carrier to generate the CS-DSBs of pump wave. To achieve the carrier-suppressed SSB modulation of the LO, the frequency of pump wave needs to satisfy fp = fLO+fB. Due to fB = 9.42 GHz, the pump wave was set to 19.42 GHz. After amplified by an EDFA, a PC2 was assigned to adjust the polarization state of the CS-DSBs of pump wave, thus optimizing the gain and loss of the SBS. With a circulator, the CS-DSBs of pump wave were injected into the HNLF, in which the SBS occurred and the carrier-suppressed SSB modulation of the LO was achieved. Thereafter, a tunable optical bandpass filter was inserted to remove the CS-DSBs of pump wave. In order to clearly observe the influence of gain and loss of SBS on ±1st-order optical sidebands of LO signal, the optical spectra of LO-modulated optical signals before and after SBS happened were measured under the SIC condition. Taking the case of 10-GHz LO signal and 15-GHz SI signal as an example, the measured optical spectra are shown in Fig. 4. The red line represents the optical spectrum at the output of the polarizer before the SBS happened, while the blue line represents the optical spectrum at the output of the TOBPF after SBS happened. It can be observed that the +1st -order optical sideband of the LO was amplified by 22 dB while the -1st -order optical sideband of the LO was suppressed by 26 dB under the influence of gain and loss of SBS, indicating that the SSB modulation was realized. After detection in a PD, the RF frequency downconversion was attained. Figure 5(a) denotes the measured electrical spectrum of the frequency downconverted to 1 GHz IF signal. As can be seen, the SIC depth is up to 55.4 dB, which has an obvious improvement compared to the SIC depths in [18–20]. Figures 5(b)–5(d) display the measured electrical spectra of the frequency downconverted to 3, 5 and 7 GHz. The SIC depths in Figs. 5(b)–5(d) are 53.5, 54.2 and 51.3 dB, indicating that the proposed system has a large operation bandwidth.

 figure: Fig. 4.

Fig. 4. the measure optical spectra of the LO-modulated optical signals without and with SBS under the SIC condition when LO signal is 10 GHz and SI signal is 15 GHz.

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 figure: Fig. 5.

Fig. 5. The measured electrical spectra of the frequency downconverted signals with and without RF SIC when the SI was set to (a) 11 GHz, (b) 13 GHz, (c) 15 GHz and (d) 17 GHz.

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To the best of our knowledge, the wideband RF signal plays an important role in IBFD ROF systems. Therefore, the wideband RF SIC performance needs to be evaluated based on the proposed system. To better observe the wideband RF SIC, the SOI was applied and set to a cosine signal, whereas the SI and SR signals from VSG were reset to a quadrature phase shift keying (QPSK) signal with a symbol rate of 20 Msys/s. In practical application, the SOI and SI signals should be the same RF signal. However, if the SOI is set to the same as SI signal in the experiment, it will be difficult to distinguish whether the residual signal after SIC is caused by the poor RF SIC performance or the SOI itself. The center frequencies of the SOI, SI and SR signals were 11, 13, 15 and 17 GHz, respectively. The LO was unchanged, still at 10 GHz. Figure 6 demonstrates the measured electrical spectra of the wideband RF frequency downconverted signals. It can be seen that if the SIC is not considered, the desired frequency downconverted signals are overlapped with the wideband interference signals. If the SIC is fulfilled, the wideband interference signals are suppressed well by around 28 dB, whereas the desired frequency downconverted signals have no significant decadences. Similarly, the wideband RF SIC depths are also promoted. Therefore, the SIC performance for the single-frequency and wideband signals downconverted to different frequencies has been successfully validated in our experiment.

 figure: Fig. 6.

Fig. 6. The measured electrical spectra of the wideband frequency downconverted signals with and without RF SIC when the single-frequency SOI and wideband SI signal were set to (a) 11 GHz, (b) 13 GHz, (c) 15 GHz and (d) 17 GHz.

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In addition, to further confirm the flexibility and tunability of the proposed system, the frequency of the LO was reset to 1 GHz lower than that of the SOI. Figure 7 illustrates the measured SIC depths versus the SOI frequencies when the downconverted signals are always 1 GHz. The blue line represents the SIC depths of the single-frequency downconverted signals, which is around 54 dB. The orange line indicates the SIC depths of the wideband downconverted signals, which is around 28 dB. Compared with the SIC depths in Figs. 5 and 6, the SIC performance is essentially the same. This means that the SIC capability of the proposed system is unaffected whether the frequency of the LO is fixed or varies with the frequency of the SOI. Therefore, the proposed system has a large tunability and flexibility.

 figure: Fig. 7.

Fig. 7. The measured SIC depths of the single-frequency and wideband downconverted to 1 GHz IF signals versus the frequencies of the SOI/SI signals.

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3.3 Fiber dispersion elimination

Furthermore, we evaluated the fiber transmission capacity of the proposed system by simulation. The dispersion coefficient of single-mode fiber (SMF) is -17 ps/km/nm, the wavelength of the optical carrier in the experiment is 1550.008 nm, and the velocity of light is 3×108 m/s. According to Eq. (8), the CDIP of the generated frequency downconverted signal based on DSB modulation is related to transmission distance and the frequencies of SOI and LO. Firstly, we fixed the LO frequency at 10 GHz to simulate the power of the downconverted signal versus the SOI frequency after SMF transmission. The Fig. 8(a) demonstrates the simulated results of 0-, 20-, 40-, 60-, 80- and 100-km SMF transmission, where the manifest power fading can be observed. This indicates that as the SOI frequency increases, the frequency of the downconverted signal is getting higher and higher, and its power will be affected by chromatic dispersion after SMF transmission. And the longer the transmission distance, the more the power is affected by chromatic dispersion. Then, we changed the frequencies of LO and SOI simultaneously to ensure that the frequency of the downconverted signal is always 1 GHz. Figure 8(b) shows the power of 1-GHz downconverted signal versus the SOI frequency after 0-, 20-, 40-, 60-, 80- and 100-km SMF transmission. It can be seen that as the transmission distance increases, the power of 1-GHz downconverted signal will still be affected by dispersion. And the higher the frequencies of LO and SOI, the more the power is susceptible to chromatic dispersion. Finally, according to Eq. (15), we simulated the power of the downconverted signal after SMF transmission based on SSB modulation, as shown in Fig. 8(c). We can find that the power of the downconverted signal is not affected by chromatic dispersion regardless of the transmission distance and the frequencies of SOI and LO.

 figure: Fig. 8.

Fig. 8. The simulated power of downconverted signal versus the SOI frequency after 0-, 20-, 40-, 60-, 80- and 100-km SMF transmission (a) at a fixed LO frequency of 10 GHz based on DSB modulation, (b) when the downconverted signal is always 1 GHz based on DSB modulation and (c) based on the proposed system.

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 figure: Fig. 9.

Fig. 9. The measured powers of single-frequency downconverted signals versus the SOI frequency before and after 25-km SMF transmission based on (a) DSB modulation and (b) SSB modulation; (c) The measured and simulated power fading based on DSB and SSB modulation after 25-km SMF.

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Meanwhile, we also verified the feasibility of anti-dispersion transmission of the proposed system by experiment. A spool of 25-km single mode fiber (SMF) was employed to introduce the dispersion in the experiment. Figure 9(a) displays the measured powers of DSB-modulated downconverted signals before and after 25-km SMF transmission when not applying SBS-based MPF. It can be observed that there is a significant power fading at around 15 GHz after 25-km SMF transmission. Figure 9(b) shows the measured powers of SSB-modulated downconverted signals before and after 25-km SMF transmission when applying SBS-based MPF, in which there is no significant fading after transmission. The 10-dB power attenuation after transmission in Fig. 9(b) is mainly caused by the loss of SMF. The blue line in Fig. 9(c) exhibits the simulated power fading of the downconverted signals after 25-km SMF transmission based on DSB modulation. The measured power differences of downconverted signals before and after transmission were normalized to remove the influence of fiber attenuation and the responses of the electrical and optical devices. The stars represent the normalized results of SSB modulation and the circles show the normalized results of DSB modulation in Fig. 9(c). As can be seen, the simulated and measured power fading based on DSB modulation are consistent and has a significant power fading, but the power fading based on SSB modulation is almost a constant. Therefore, the proposed system can perform fiber anti-dispersion transmission.

4. Discussion

In fact, we can simplify the proposed system by replacing the DPol-MZM with a DPMZM, as shown in Fig. 10. The PC1 and Pol. after DPol-MZM in Fig. 1(a) are omitted, which can reduce the polarization sensitivity of the proposed system. The sub-MZM1 and sub-MZM2 are still biased at MITP. Hence, the optical signals at the outputs of sub-MZM1 and sub-MZM2 be written as

$$\begin{array}{c} {E_{sub - MZM1}}(t) = \frac{{\sqrt 2 }}{4}{E_0}{e^{j{\omega _0}t + j\frac{\pi }{2}}}\{ {J_1}({\beta _{SOI}}){J_0}({\beta _{SI}})[{e^{j({\omega _1}t + {\theta _{SOI}}(t))}} + {e^{ - j({\omega _1}t + {\theta _{SOI}}(t))}}]\\ + {J_0}({\beta _{SOI}}){J_1}({\beta _{SI}})[{e^{j({\omega _1}t + {\theta _{SI}}(t))}} + {e^{ - j({\omega _1}t + {\theta _{SI}}(t))}}]\} \end{array}$$
$$\begin{array}{c} {E_{sub - MZM2}}(t) = \frac{{\sqrt 2 }}{4}{E_0}{e^{j{\omega _0}t + j\frac{\pi }{2}}}\{ {J_0}({\beta _{LO}}){J_1}({\beta _{SR}})[{e^{j({\omega _1}t + {\theta _{SR}}(t))}} + {e^{ - j({\omega _1}t + {\theta _{SR}}(t))}}]\\ + {J_1}({\beta _{LO}}){J_0}({\beta _{SR}})[{e^{j({\omega _2}t + {\theta _{LO}}(t))}} + {e^{ - j({\omega _2}t + {\theta _{LO}}(t))}}]\} \end{array}$$
which are the same as the optical field at the outputs of x- and y-MZM. By setting the main DC bias of DPMZM at Vπ and adjusting the amplitudes and phases of the applied electrical signals, the optical field at the output of the DPMZM can be expressed
$$\begin{aligned} {E_{DPMZM}}(t) &= \frac{{\sqrt 2 }}{2}{E_{sub - MZM1}}(t) + \frac{{\sqrt 2 }}{2}{E_{sub - MZM1}}(t){e^{j\pi }}\\& = \frac{1}{4}{E_0}{e^{j{\omega _0}t + j\frac{\pi }{2}}}\left[ {\begin{array}{{c}} \begin{array}{l} {J_1}({\beta_{SOI}}){J_0}({\beta_{SI}})[{e^{j({\omega_1}t + {\theta_{SOI}}(t))}} + {e^{ - j({\omega_1}t + {\theta_{SOI}}(t))}}]\\ + {J_0}({\beta_{SOI}}){J_1}({\beta_{SI}})[{e^{j({\omega_1}t + {\theta_{SI}}(t))}} + {e^{ - j({\omega_1}t + {\theta_{SI}}(t))}}] \end{array}\\ \begin{array}{l} - {J_1}({\beta_{LO}}){J_0}({\beta_{SR}})[{e^{j({\omega_2}t + {\theta_{LO}}(t))}} + {e^{ - j({\omega_2}t + {\theta_{LO}}(t))}}]\\ - {J_0}({\beta_{LO}}){J_1}({\beta_{SR}})[{e^{j({\omega_1}t + {\theta_{SR}}(t))}} + {e^{ - j({\omega_1}t + {\theta_{SR}}(t))}}] \end{array} \end{array}} \right]\\& = \frac{1}{4}{E_0}{e^{j{\omega _0}t + j\frac{\pi }{2}}}\left[ {\begin{array}{{c}} {{J_1}({\beta_{SOI}}){J_0}({\beta_{SI}})[{e^{j({\omega_1}t + {\theta_{SOI}}(t))}} + {e^{ - j({\omega_1}t + {\theta_{SOI}}(t))}}]}\\ { - {J_1}({\beta_{LO}}){J_0}({\beta_{SR}})[{e^{j({\omega_2}t + {\theta_{LO}}(t))}} + {e^{ - j({\omega_2}t + {\theta_{LO}}(t))}}]} \end{array}} \right] \end{aligned}$$
which is similar to the Eq. (5). It indicates that the photonic-assisted SIC can be directly achieved by adjusting the main DC bias of the DPMZM instead of tuning the PC1 and Pol. Hence, the experimental operation based on the simplified system is easier than that based on the practical system. However, the main bias Vπ drift of DPMZM is not a negligible issue that will affect the RF SIC performance, so a commercially available bias controller needs to be employed to stabilize the bias Vπ. Similar optical RF SIC by using a DPMZM has been reported in Ref. [14], indicating that the proposed simplified system based on DPMZM is feasible. Unfortunately, an effective DPMZM is not available in our lab, so we chose the DPol-MZM in conjunction with a PC and Pol. to carried out the scheme for RF SIC and frequency downconversion.

 figure: Fig. 10.

Fig. 10. The simplified schematic diagram of the proposed photonic-assisted frequency downconverter with self-interference cancellation and fiber dispersion elimination based on stimulated Brillouin scattering.

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5. Conclusion

In this work, we theoretically and experimentally analyzed a photonic-assisted frequency downconverter with large RF SIC depths and immunity to power fading. The SI signal is eliminated in the optical domain, implying that the proposed system has a large operation bandwidth. By means of a tunable SBS-based MPF, the carrier-suppressed SSB modulation of LO signal can be attained, so the proposed system has an ability to up/downconvert the signal and suppress the fiber dispersion during the transmission. In addition, due to the characteristic of SBS-based MPF, conversion efficiency can be improved based on the proposed technique.

Funding

National Key Research and Development Program of China (2019YFB2205302, 2019YFB2203200); National Natural Science Foundation of China (62075210, 61620106013, 61835010).

Disclosures

The authors declare no conflicts of interest.

Data availability

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.

References

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Data availability

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.

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Figures (10)
Fig. 1.
Fig. 1. (a) The schematic diagram of the proposed photonic-assisted frequency downconverter with self-interference cancellation and fiber dispersion elimination based on stimulated Brillouin scattering. LD, laser diode; OC, optical coupler; DPol-MZM, dual-polarization Mach-Zehnder modulator; 90° PR, 90° polarization rotator; PBC, polarization beam combiner; PC, polarization controller; Pol., polarizer; EDFA, erbium-doped fiber amplifier; HNLF, high nonlinear fiber; EATT, electrical attenuator; TTDL, tunable time delay line; TOBPF, tunable optical bandpass filter; SMF, single mode fiber; PD, photodetector. (b-i)-(b-vi) The simplified optical spectra corresponding to nodes (i)-(vi) in (a).

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Fig. 2.
Fig. 2. Experimental setup of the proposed photonic-assisted frequency downconverter with improved self-interference cancellation and fiber dispersion elimination based on stimulated Brillouin scattering. LD, laser diode; OC, optical coupler; DPol-MZM, dual-polarization Mach-Zehnder modulator; 90° PR, 90° polarization rotator; PBC, polarization beam combiner; PC, polarization controller; Pol., polarizer; ISO, optical isolator; MZM, Mach-Zehnder modulator; EDFA, erbium-doped fiber amplifier; HNLF, high nonlinear fiber; EATT, electrical attenuator; TTDL, tunable time delay line; TOBPF, tunable optical bandpass filter; SMF, single mode fiber; PD, photodetector; MSG, microwave signal generator; VSG, vector signal generator; ESA, electrical spectrum analyzer.

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Fig. 3.
Fig. 3. The measured optical spectra at the output of polarizer when the SI was set to (a) 11 GHz, (b) 13 GHz, (c) 15 GHz and (d) 17 GHz.

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Fig. 4.
Fig. 4. the measure optical spectra of the LO-modulated optical signals without and with SBS under the SIC condition when LO signal is 10 GHz and SI signal is 15 GHz.

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Fig. 5.
Fig. 5. The measured electrical spectra of the frequency downconverted signals with and without RF SIC when the SI was set to (a) 11 GHz, (b) 13 GHz, (c) 15 GHz and (d) 17 GHz.

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Fig. 6.
Fig. 6. The measured electrical spectra of the wideband frequency downconverted signals with and without RF SIC when the single-frequency SOI and wideband SI signal were set to (a) 11 GHz, (b) 13 GHz, (c) 15 GHz and (d) 17 GHz.

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Fig. 7.
Fig. 7. The measured SIC depths of the single-frequency and wideband downconverted to 1 GHz IF signals versus the frequencies of the SOI/SI signals.

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Fig. 8.
Fig. 8. The simulated power of downconverted signal versus the SOI frequency after 0-, 20-, 40-, 60-, 80- and 100-km SMF transmission (a) at a fixed LO frequency of 10 GHz based on DSB modulation, (b) when the downconverted signal is always 1 GHz based on DSB modulation and (c) based on the proposed system.

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Fig. 9.
Fig. 9. The measured powers of single-frequency downconverted signals versus the SOI frequency before and after 25-km SMF transmission based on (a) DSB modulation and (b) SSB modulation; (c) The measured and simulated power fading based on DSB and SSB modulation after 25-km SMF.

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Fig. 10.
Fig. 10. The simplified schematic diagram of the proposed photonic-assisted frequency downconverter with self-interference cancellation and fiber dispersion elimination based on stimulated Brillouin scattering.

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Equations (18)

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U S O I ( t ) = V S O I cos ⁡ [ ω 1 t + θ S O I ( t ) ] , U S I ( t ) = V S I cos ⁡ [ ω 1 t + θ S I ( t ) ] , U S R ( t ) = V S R cos ⁡ [ ω 1 t + θ S R ( t ) ] , U L O ( t ) = V L O cos ⁡ [ ω 2 t + θ L O ( t ) ] .
E D P o l − M Z M ( t ) = 1 4 [ E x − M Z M ( t ) E y − M Z M ( t ) ] = 1 8 E 0 e j ω 0 t [ e j π V π [ U S O I ( t ) + U S I ( t ) ] − e − j π V π [ U S O I ( t ) + U S I ( t ) ] e j π V π [ U L O ( t ) + U S R ( t ) ] − e − j π V π [ U L O ( t ) + U S R ( t ) ] ] = 1 4 E 0 e j ω 0 t + j π 2 [ { J 1 ( β S O I ) J 0 ( β S I ) [ e j ( ω 1 t + θ S O I ( t ) ) + e − j ( ω 1 t + θ S O I ( t ) ) ] + J 0 ( β S O I ) J 1 ( β S I ) [ e j ( ω 1 t + θ S I ( t ) ) + e − j ( ω 1 t + θ S I ( t ) ) ] } { J 0 ( β L O ) J 1 ( β S R ) [ e j ( ω 1 t + θ S R ( t ) ) + e − j ( ω 1 t + θ S R ( t ) ) ] + J 1 ( β L O ) J 0 ( β S R ) [ e j ( ω 2 t + θ L O ( t ) ) + e − j ( ω 2 t + θ L O ( t ) ) ] } ]
P P C = [ cos ⁡ α − sin ⁡ α sin ⁡ α cos ⁡ α ] [ e j δ 2 0 0 e − δ 2 ] ,
E P o l . = 1 4 ( sin ⁡ α ⋅ E x − M Z M ( t ) ⋅ e j δ 2 + cos ⁡ α ⋅ E y − M Z M ( t ) ⋅ e − j δ 2 ) = − 2 8 E 0 e j ω 0 t [ J 1 ( β S O I ) J 0 ( β S I ) [ e j ( ω 1 t + θ S O I ( t ) ) + e − j ( ω 1 t + θ S O I ( t ) ) ] + J 0 ( β S O I ) J 1 ( β S I ) [ e j ( ω 1 t + θ S I ( t ) ) + e − j ( ω 1 t + θ S I ( t ) ) ] − J 0 ( β L O ) J 1 ( β S R ) [ e j ( ω 1 t + θ S R ( t ) ) + e − j ( ω 1 t + θ S R ( t ) ) ] − J 1 ( β L O ) J 0 ( β S R ) [ e j ( ω 2 t + θ L O ( t ) ) + e − j ( ω 2 t + θ L O ( t ) ) ] ]
E P o l . = − 2 8 E 0 e j ω 0 t [ J 1 ( β S O I ) J 0 ( β S I ) [ e j ( ω 1 t + θ S O I ( t ) ) + e − j ( ω 1 t + θ S O I ( t ) ) ] − J 1 ( β L O ) J 0 ( β S R ) [ e j ( ω 2 t + θ L O ( t ) ) + e − j ( ω 2 t + θ L O ( t ) ) ] ]
E S M F = − 2 8 E 0 e j ω 0 t [ J 1 ( β S O I ) J 0 ( β S I ) [ e j ( ω 1 t + θ S O I ( t ) + θ + 1 ) + e − j ( ω 1 t + θ S O I ( t ) − θ − 1 ) ] − J 1 ( β L O ) J 0 ( β S R ) [ e j ( ω 2 t + θ L O ( t ) + θ + 1 ′ ) + e − j ( ω 2 t + θ L O ( t ) − θ − 1 ′ ) ] ]
θ + 1 ( ω ) = z β ( ω 0 ) + z β ′ ( ω 0 ) ω 1 + 1 2 z β ′ ′ ( ω 0 ) ω 1 2 θ − 1 ( ω ) = z β ( ω 0 ) − z β ′ ( ω 0 ) ω 1 + 1 2 z β ′ ′ ( ω 0 ) ω 1 2 θ + 1 ′ ( ω ) = z β ( ω 0 ) + z β ′ ( ω 0 ) ω 2 + 1 2 z β ′ ′ ( ω 0 ) ω 2 2 θ − 1 ′ ( ω ) = z β ( ω 0 ) − z β ′ ( ω 0 ) ω 2 + 1 2 z β ′ ′ ( ω 0 ) ω 2 2
i ( t ) ∝ E 0 2 J 0 ( β S I ) J 0 ( β S R ) J 1 ( β S O I ) J 1 ( β L O ) cos ⁡ [ z D ( ω 0 ) λ 2 4 π c ( ω 1 2 − ω 2 2 ) ] × { cos ⁡ [ ω 1 t + ω 2 t + θ S O I ( t ) + θ L O ( t ) + z β ′ ( ω 0 ) ( ω 2 + ω 1 ) ] + cos ⁡ [ ω 1 t − ω 2 t + θ S O I ( t ) − θ L O ( t ) + z β ′ ( ω 0 ) ( ω 1 − ω 2 ) }
f p u m p = f B + f L O
g B ( Δ f ) = g 0 2 ( Δ ν B / 2 ) 2 Δ f 2 + ( Δ ν B / 2 ) 2 + j g 0 4 Δ ν B Δ f Δ f 2 + ( Δ ν B / 2 ) 2 , α B ( Δ f ) = − g 0 2 ( Δ ν B / 2 ) 2 Δ f 2 + ( Δ ν B / 2 ) 2 − j g 0 4 Δ ν B Δ f Δ f 2 + ( Δ ν B / 2 ) 2
E c i r c u l a r = − 2 8 E 0 e j ω 0 t [ J 1 ( β S O I ) J 0 ( β S I ) [ e j ( ω 1 t + θ S O I ( t ) ) + e − j ( ω 1 t + θ S O I ( t ) ) ] − J 1 ( β L O ) J 0 ( β S R ) [ G ( Δ f ) e j ( ω 2 t + θ L O ( t ) ) e j φ ( Δ f ) + A ( Δ f ) e − j ( ω 2 t + θ L O ( t ) ) e − j φ ( Δ f ) ] − 2 J 1 ( β p u m p ) e j ω p u m p t + j π 2 − 2 J 1 ( β p u m p ) e − j ω p u m p t + j π 2 ]
G ( Δ f ) = e R e [ g B ( Δ f ) ] I p L = e g 0 I p L 2 ( Δ ν B / 2 ) 2 Δ f 2 + ( Δ ν B / 2 ) 2 , A ( Δ f ) = e R e [ α B ( Δ f ) ] I p L = e − g 0 I p L 2 ( Δ ν B / 2 ) 2 Δ f 2 + ( Δ ν B / 2 ) 2 , φ ( Δ f ) = Im [ g B ( Δ f ) ] I p L = g 0 I p L 4 Δ ν B Δ f Δ f 2 + ( Δ ν B / 2 ) 2
E T O B P F = − 2 8 E 0 e j ω 0 t [ J 1 ( β S O I ) J 0 ( β S I ) [ e j ( ω 1 t + θ S O I ( t ) ) + e − j ( ω 1 t + θ S O I ( t ) ) ] − J 1 ( β L O ) J 0 ( β S R ) G ( Δ f ) e j ( ω 2 t + θ L O ( t ) ) ]
E S M F = − 2 8 E 0 e j ω 0 t [ J 1 ( β S O I ) J 0 ( β S I ) [ e j ( ω 1 t + θ S O I ( t ) + θ + 1 ) + e − j ( ω 1 t + θ S O I ( t ) − θ − 1 ) ] − J 1 ( β L O ) J 0 ( β S R ) G ( Δ f ) e j ( ω 2 t + θ L O ( t ) + θ + 1 ′ ) ]
i ( t ) ∝ E 0 2 J 0 ( β S I ) J 0 ( β S R ) J 1 ( β S O I ) J 1 ( β L O ) G ( Δ f ) × { cos ⁡ [ ω 1 t + ω 2 t + θ S O I ( t ) + θ L O ( t ) + θ + 1 ′ − θ − 1 ] + cos ⁡ [ ω 1 t − ω 2 t + θ S O I ( t ) − θ L O ( t ) + θ + 1 − θ + 1 ′ ] }
E s u b − M Z M 1 ( t ) = 2 4 E 0 e j ω 0 t + j π 2 { J 1 ( β S O I ) J 0 ( β S I ) [ e j ( ω 1 t + θ S O I ( t ) ) + e − j ( ω 1 t + θ S O I ( t ) ) ] + J 0 ( β S O I ) J 1 ( β S I ) [ e j ( ω 1 t + θ S I ( t ) ) + e − j ( ω 1 t + θ S I ( t ) ) ] }
E s u b − M Z M 2 ( t ) = 2 4 E 0 e j ω 0 t + j π 2 { J 0 ( β L O ) J 1 ( β S R ) [ e j ( ω 1 t + θ S R ( t ) ) + e − j ( ω 1 t + θ S R ( t ) ) ] + J 1 ( β L O ) J 0 ( β S R ) [ e j ( ω 2 t + θ L O ( t ) ) + e − j ( ω 2 t + θ L O ( t ) ) ] }
E D P M Z M ( t ) = 2 2 E s u b − M Z M 1 ( t ) + 2 2 E s u b − M Z M 1 ( t ) e j π = 1 4 E 0 e j ω 0 t + j π 2 [ J 1 ( β S O I ) J 0 ( β S I ) [ e j ( ω 1 t + θ S O I ( t ) ) + e − j ( ω 1 t + θ S O I ( t ) ) ] + J 0 ( β S O I ) J 1 ( β S I ) [ e j ( ω 1 t + θ S I ( t ) ) + e − j ( ω 1 t + θ S I ( t ) ) ] − J 1 ( β L O ) J 0 ( β S R ) [ e j ( ω 2 t + θ L O ( t ) ) + e − j ( ω 2 t + θ L O ( t ) ) ] − J 0 ( β L O ) J 1 ( β S R ) [ e j ( ω 1 t + θ S R ( t ) ) + e − j ( ω 1 t + θ S R ( t ) ) ] ] = 1 4 E 0 e j ω 0 t + j π 2 [ J 1 ( β S O I ) J 0 ( β S I ) [ e j ( ω 1 t + θ S O I ( t ) ) + e − j ( ω 1 t + θ S O I ( t ) ) ] − J 1 ( β L O ) J 0 ( β S R ) [ e j ( ω 2 t + θ L O ( t ) ) + e − j ( ω 2 t + θ L O ( t ) ) ] ]
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