A radio frequency (RF) signal source with high quality is critical in applications including communication systems, radar systems, aerospace, and measurement systems [1–3]. Optoelectronic oscillators (OEOs) have attracted the attention of researchers throughout the last two decades for their ability to generate RF signals at high frequencies with ultra-low phase noise [4]. Benefiting from the low loss and the high Q factor of the long fiber, excellent phase noise is exhibited by integrating the long fiber into an OEO loop as an energy storage cavity. However, the long fiber used in an OEO loop is rather sensitive to environmental perturbations. The quality of the generating RF signals is deteriorated when the fiber parameters vary with the environment [5]. To overcome this problem, various methods of frequency stabilization have been explored, such as thermal stabilization [5], temperature-insensitive fiber [6], injection locking [7–10], and phase-locked loop (PLL) [11–14]. For schemes using PLL or injection locking, a stable external frequency source is required to stabilize the output frequency, which increases the size and cost of an OEO. In [11], the phase variations of the RF signals are compensated by controlling the bias voltage of the Mach-Zehnder modulator, which simultaneously results in loop gain variations of the OEO. In [12–14], an electrical RF phase shifter is incorporated into an OEO to compensate for the phase variations. However, the electrical RF phase shifter has some limitations such as phase shift range and bandwidth [15]. Moreover, the optical delay line is used to compensate for fiber length variations, which also faces the problem of limited compensation length in a wide temperature range [16, 17].

In addition, frequency tunability is essential for an RF signal source. A microwave photonic bandpass filter (MPBF) is a promising solution due to its broadband tunable characteristic. Therefore, a great variety of the MPBFs have been proposed to implement the selection of the OEO oscillation modes [18–22]. In [18], a phase modulator (PM) in conjunction with a linearly chirped fiber Bragg grating (LCFBG) form a tunable MPBF and are further incorporated in an OEO. In [19–22], stimulated Brillouin scattering (SBS) [19, 20] or a phase-shifted fiber Bragg grating (PS-FBG) [21, 22] is used to break the amplitude balance of the phase modulated sidebands to realize an MPBF. However, the mentioned tunable MPBF-based OEOs still face a frequency drift problem due to the long fiber in the loop.

In this article, a frequency stable and tunable OEO is designed and analyzed. A phase-shifted fiber Bragg grating (PS-FBG) is designed in combination with a PM to build a tunable MPBF. Furthermore, the phase variations of the RF signal can be simultaneously compensated by the designed optical phase compensation loop. Therefore, the oscillation frequency can be stabilized and shifted by tuning the wavelength of the tunable laser (TL).

Figure 1 shows a schematic of the designed OEO. It is composed of a single oscillation loop and an optical phase compensation loop. An output light of a TL is injected into a polarization controller (PC) and split into two beams by an optical coupler (OC1). In the upper path, an optical phase shifter based on piezoelectric translator (PZT) is used to shift the phase of the optical carrier. In the lower path, the optical carrier is modulated by a PM, then passed through an isolator (ISO) and a PS-FBG before being sent to the OC2. After the optical signals of the upper and lower paths are combined by the OC2, the signals at the two outputs of the OC2 are detected separately by different photodetectors (PDs). By a single mode fiber (SMF), the input optical signal of the PD1 is introduced with a time delay. The PD1 output signal is amplified and then divided into two parts by an RF power divider (DIV1). One part from the DIV1 is split into two parts by the DIV2, and one part is sent back to the PM to establish the single oscillation loop, and the other part is the output RF signal. To stabilize the output of the OEO, another part from the DIV1 is compared with the output signal of the electrical amplifier (EA2) by a mixer to obtain the phase variations. The signal from the mixer is sent to a control module, which is used to adjust the PZT-based optical phase shifter to form the optical phase compensation loop.

Figure 1. Schematic of the designed optoelectronic oscillator (OEO). PC, polarization controller; OC, optical coupler; PM, phase modulator; PS-FBG, phase-shifted fiber Bragg grating; ISO, isolator; PZT, piezoelectric translator; PD, photodetector; SMF, single mode fiber; EA, electrical amplifier; DIV, radio frequency power divider.

For the upper path, the optical carrier with a phase shift φ introduced by the PZT-based optical phase shifter is given by

(1)$$E_1\left(t\right)=\frac{E_c}{2}\exp \left(j{\omega }_{c}t+j\phi \right),$$

where E_c and ω_c are the electrical amplitude and frequency of the optical carrier. For the lower path, the amplitude of the modulated sidebands is modified through the PS-FBG transmission window. The power transmission coefficient of the PS-FBG T(ω) is given by [23]

(2)$$T\left(\omega \right)=1-{\left|\frac{{}^1{F}_{21}{}^2{F}_{11}{+}^1{F}_{22}{}^2{F}_{21}\exp \left(j{\phi }_0\right)}{{}^1{F}_{11}{}^2{F}_{11}{+}^1{F}_{12}{}^2{F}_{21}\exp \left(j{\phi }_0\right)}\right|}^2$$

(3)$${}^i{F}_{11}{=}^i{F}_{22}^{\ast }=\cosh \left(\gamma L_i\right)-j\left(\frac{\sigma }{\gamma }\right)\sinh \left(\gamma L_i\right),$$

(4)$${}^i{F}_{12}{=}^i{F}_{21}^{\ast }=-j\left(\frac{\kappa }{\gamma }\right)\sinh \left(\gamma L_i\right),$$

where i = 1, 2 is the sub-grating label, * denotes complex conjugation, φ⁰ is phase shift variable, L_i is the sub-grating length, γ² = κ² − σ², κ = ωΔn/(2c) is the coupling coefficient and σ = n₀(ω − ω_B)/c is the detuning, Δn is the depth of index modulation, ω_B = 2πc/λ_B, λ_B = 2n₀Λ and n₀ are the Bragg wavelength and the refractive index of the sub-grating, Λ is the grating period, c is the light speed in vacuum, and ω is the frequency of the input optical signal.

Under a small signal modulation condition, only a lower modulated sideband is passed by using the PS-FBG. Thus, the optical signal from the PS-FBG can be described as

(5)$$E_2\left(t\right)=\frac{E_c}{2}\sqrt{T\left({\omega }_c+{\omega }_e\right)}{J}_1\left(\beta \right)\exp \left[j\left({\omega }_c+{\omega }_e\right)t+j\frac{\pi }{2}\right],$$

where J₁(·) is the first-order Bessel function, β = πV_e/V_π and V_π are the modulation index and the half-wave voltage of the PM, and ω_e and V_e are the frequency and amplitude of the RF signal.

E₁(t) and E₂(t) are combined by the OC2 and then passed through the SMF, and the output optical signal can be written as

(6)$$\begin{array}{c}{E}_{\text{out}}\left(t\right)=\frac{E_c}{4}\sqrt{T\left({\omega }_c+{\omega }_e\right)}{J}_1\left(\beta \right)\\ \cdot \exp \left[j\left({\omega }_c+{\omega }_e\right)\left(t+{\tau }_s\right)+j\frac{\text{π}}{2}\right]\\ +\frac{E_c}{4}\exp \left[j{\omega }_c\left(t+{\tau }_s\right)+j\phi \right],\end{array}$$

where τ_s = n_sL_s/c is the time delay of the SMF, and L_s and n_s are the length and the refractive index of the SMF. The RF signal is recovered by the PD1, ignoring the higher harmonics and DC components, and it can be expressed as

(7)$$\begin{array}{c}{V}_e\left(t\right)\propto A{\left|E_{\text{out}}\left(t\right)\right|}^{\text{2}}\\ \approx A\sqrt{T\left({\omega }_c+{\omega }_e\right)}{J}_1\left(\beta \right)\sin \left({\omega }_{e}t+{\omega }_e{\tau }_s-\phi \right),\end{array}$$

where A is a constant that is determined by the responsivity and the load impedance of the PD1. From Eq. (7), the phase shift caused by the PZT-based optical phase shifter is brought into the RF signal. In that case, the phase of the RF signal can be shifted by adjusting the optical phase shifter. Thus, the phase variations can be compensated. Also, the RF phase shift range is equal to the phase shift range of the optical phase shifter. Given that the phase shift range of the PZT-based optical phase shifter is more than 8π, the phase variations can still be compensated in a wide temperature range.

According to Eq. (7), the power of the RF signal from the PD1 can be obtained as

(8)$$\begin{array}{c}{P}_e\left({\omega }_e\right)\propto {\left|V_e\left(t\right)\right|}^2\\ =A^{2}T\left({\omega }_c+{\omega }_e\right)J_1{}^2\left(\beta \right).\end{array}$$

From Eq. (8), T(ω_c + ω_e) is a frequency-dependent function and the peak of P_e(ω_e) appears when T(ω_c + ω_e) = 1, which means that the PS-FBG transmission window corresponds to a passband in the RF domain. Also, the peak of the passband can be changed by adjusting ω_c. Accordingly, a tunable MPBF is realized in the oscillation loop by tuning the output wavelength of the TL.

After oscillation, the output of the EA1 is equal to the sum of the RF signal of each loop circulation, which can be expressed as

(9)$$\begin{array}{c}{V}_{\text{out}}\left(t\right)=\exp \left(j{\omega }_{e}t\right){{\displaystyle \sum _{m=0}^{\infty }\left\{\underset{G_{\text{eff}}}{\underbrace{AG{J}_1\left(\beta \right)}}\underset{T}{\underbrace{\sqrt{T\left({\omega }_c+{\omega }_e\right)}}}\right\}}}^m\\ \cdot \exp \left[jm\left({\omega }_e{\tau }_s-\phi +\frac{\pi }{2}\right)\right],\end{array}$$

where G is amplitude gain from the EA1. Correspondingly, the RF power can be expressed as

(10)$$P\left({\omega }_e\right)\propto \frac{1}{1+{\left(G_{\text{eff}}T\right)}^2-2G_{\text{eff}}T\cos \left({\omega }_e\tau -\phi +\frac{\pi }{2}\right)},$$

where τ = τ_s + τ_e is the sum of the time delays of the SMF and the electrical link, and τ_e is the delay of the electrical link. From Eq. (10), when ω_eτ – φ + π/2 = 2kπ (k = 0, 1, 2, …), k is mode number, and T(ω_c + ω_e) = 1, the oscillation frequency is generated. Furthermore, the OEO phase noise based on the Yao-Maleki model is given by [4]

(11)$$S_{\text{RF}}\left(f^{\prime }\right)=\frac{\delta }{{\left(\delta /2\tau \right)}^2+{\left(2\text{π}\right)}^2{\left(f^{\prime }\tau \right)}^2}$$

where δ = ρ_NG²/P_osc, ρ_N is equivalent input noise density, and P_osc and f ′ are the power and offset frequency of the oscillation peak, respectively.

The time delay varies with ambient temperature because the refractive index and thermal expansion coefficient of the SMF are sensitive to temperature. The time delay variation can be expressed as [24]

(12)$$\Delta {\tau }_s=\left(\frac{L_s}{c}{C}_N+\frac{n_s}{c}{C}_L\right)\Delta T,$$

where ΔT is the ambient temperature variation, C_N is the refractive index variation with temperature, and C_L is the thermal expansion coefficient. For the SMF, C_N and C_L are equal to 1 × 10⁻⁵/℃ and 5.5 × 10⁻⁷/℃, respectively [25].

Considering the time delay variations in the SMF, the RF power of the OEO can be rewritten as

(13)$$P^{\prime }\left({\omega }_e\right)\propto \frac{1}{1+{\left(G_{\text{eff}}T\right)}^2-2G_{\text{eff}}T\cos \left({\omega }_e\tau +{\omega }_e\Delta {\tau }_s-\phi +\frac{\pi }{2}\right)}.$$

From Eq. (13), when φ is adjusted to make ω_eΔτ_s − φ = 0, the loop phase of the OEO will be stabilized. To obtain the time delay variations, the output signals of the EA1 and EA2 are compared by the mixer. The output of the mixer can be expressed as [26]

(14)$$V_{pv}\approx K_d\sin \left[2\text{π}\frac{k\Delta {\tau }_s\left(\tau -{\tau }_s\right)}{\tau \left(\tau -\Delta {\tau }_s\right)}\right].$$

where K_d is the gain of the mixer. From Eq. (14), the time delay variations cannot be extracted when mode-hopping occurs. Then, the output of the mixer is used to adjust the PZT-based optical phase shifter by the control module.

For the phase compensation and frequency tuning of the designed OEO, the PS-FBG is the key device to be analyzed.

The reflection spectrum of the designed PS-FBG can be calculated by using Eq. (2) based on the parameters shown in Table 1, as shown in Fig. 2. In Fig. 2, in the wavelength range from 1,549.838 nm to 1,550.359 nm, the center wavelength of the transmission window is 1,549.938 nm and 3-dB bandwidth is 21.43 MHz. To discuss the MPBF, the original wavelength λ is set more than the center wavelength of the PS-FBG transmission window and the smallest tuning step is 1 pm. The frequency responses of the MPBF can be calculated by combining Eq. (8), as shown in Fig. 3. In Fig. 3(a), when λ is tuned from 1,549.939 nm to 1,550.147 nm, the center frequency of the MPBF is changed from 118 MHz to 26.090 GHz. However, the magnitude of the MPBF is decreased with λ increased in the range from 1,550.132 nm (24.217 GHz) to 1,550.147 nm (26.090 GHz). When λ is tuned at 1,550.147 nm, the magnitude of the MPBF is less than −3 dB. Therefore, the MPBF can be tuned from 118 MHz to 25.965 GHz. Furthermore, to clearly show the 3-dB bandwidth and the tuning resolution, Fig. 3(b) shows the frequency responses of the MPBF with center frequencies of 16.101 GHz and 16.226 GHz. In Fig. 3(b), the average value of the 3-dB bandwidths of the MPBF is 21.32 MHz and the tuning resolution is 125 MHz.

TABLE 1. Parameters of the designed phase-shifted fiber Bragg grating (PS-FBG).

Λ (nm)	Li (mm)	Δn	n0	φ0
528	6	4.4 × 10−4	1.4679	π/4

Figure 2. Reflection spectrum of the PS-FBG.

Figure 3. Frequency responses of the microwave photonic bandpass filter (MPBF). (a) The wavelength is tuned from 1,549.939 nm to 1,550.147 nm. (b) The center frequencies are 16.101 GHz and 16.226 GHz.

The MPBF is incorporated into the oscillation loop to perform oscillation mode selection. The output RF signal power of the OEO can be calculated by using Eq. (10), as shown in Fig. 4. In the calculation, G_eff = 1−10⁻⁷, n_s = 1.46, L_s = 203 m. Considering that the magnitude of the MPBF is decreased with λ increased over 1,550.131 nm, λ is set in the range from 1,549.939 nm to 1,550.131 nm to obtain enough gain for the OEO. Figure 4(a) shows the output RF signals from 118 MHz to 12.106 GHz with a frequency step of approximately 1 GHz when λ is tuned from 1,549.939 nm to 1,550.035 nm. Figure 4(b) shows the output RF signals from 13.104 GHz to 24.092 GHz with a frequency step of approximately 1 GHz when λ is tuned from 1,550.043 nm to 1,550.131 nm. By fine tuning λ, the output RF signal from 14.228 GHz to 14.853 GHz is tuned in frequency steps of 125 MHz, as shown in Fig. 4(c). To clearly observe the side-mode suppression ratios (SMSRs) of the output RF signal, Fig. 4(d) shows a zoomed-in view of the 18.099 GHz signal. In Fig. 4(d), the SMSR of the signal is 75.64 dB. Therefore, the designed OEO can be tuned from 118 MHz to 24.092 GHz with a frequency tuning resolution of 125 MHz, and the SMSR is more than 73 dB.

Figure 4. Output radio frequency (RF) signals of the optoelectronic oscillator (OEO) at different frequency. (a) The frequency tuning in approximately 1 GHz steps from 118 MHz to 12.106 GHz. (b) The frequency tuning in approximately 1 GHz steps from 13.104 GHz to 24.092 GHz. (c) The fine frequency tuning in 125 MHz steps from 14.228 GHz to 14.853 GHz. (d) Zoomed-in view of the 18.099 GHz signal.

To study the compensation results of the phase variations, as an example, 18.099 GHz is selected as the initial oscillation signal. According to Eq. (12), mode-hopping of the OEO will appear when the temperature fluctuation is more than ±7.8 ℃. Considering that the loop phase condition of the OEO is multiples of 2π, the temperature variations are set within ±3.9 ℃. Figures 5(a) and 5(b) show the calculated frequency drifts and phase noise of the output RF signals under different temperatures. In the calculation, ρ_N = 10⁻¹⁷ mW/Hz and G = 1,000. In Fig. 5, when the temperature variations are 0.1 ℃, 1.5 ℃, 3.9 ℃ and −3.9 ℃, the frequency drifts of the output RF signals are −12.40 kHz, −186.02 kHz, −483.65 kHz and 483.68 kHz, respectively. The phase noise at 10 kHz offset frequency is −75.58 dBc/Hz, −49.00 dBc/Hz, −34.93 dBc/Hz and −35.97 dBc/Hz, and the SMSRs are 53.14 dB, 24.44 dB, 0.64 dB and 2.25 dB.

Figure 5. Output radio frequency (RF) signals of the optoelectronic oscillator (OEO) under different temperatures. (a) The frequency drifts and (b) the phase noise.

Correspondingly, the compensated output RF signals are shown in Fig 6. In Fig. 6, the frequency drifts are −55 Hz, 15 Hz, 68 Hz, and −12 Hz. The phase noise at 10 kHz offset frequency is −95.29 dBc/Hz, −92.49 dBc/Hz, −96.40 dBc/Hz and −96.11 dBc/Hz, and the SMSRs are 72.17 dB, 69.39 dB, 73.29 dB and 73.41 dB. The detailed compensation results are shown in Table 2. In Table 2, with optical phase compensation, the frequency drifts are less than 68 Hz, the SMSRs are more than 69.39 dB, and the phase noise is less than −92.49 dBc/Hz at 10 kHz offset frequency.

TABLE 2. Compensation results of the output radio frequency (RF) signals.

ΔT (℃)	Without Compensation			With Compensation
	Frequency Drift (kHz)	Phase Noise (dBc/Hz@10 kHz)	SMSR (dB)	Frequency Drift (Hz)	Phase Noise (dBc/Hz@10 kHz)	SMSR (dB)
0.1	−12.40	−75.58	53.14	−55	−95.29	72.17
1.5	−186.02	−49.00	24.44	15	−92.49	69.39
3.9	−483.65	−34.93	0.64	68	−96.40	73.29
−3.9	483.68	−35.97	2.25	−12	−96.11	73.41

Figure 6. Output radio frequency (RF) signals of the optoelectronic oscillator (OEO) with optical phase compensation. (a) The frequency drifts and (b) the phase noise.

The designed OEO is compared with the previously reported OEOs, as shown in Table 3. In Table 3, the OEO has a wider tuning range, higher SMSR, and phase variation compensation in comparison to [18] and [21]. However, the phase noise of the OEO is higher due to the SMF of 203 m used in the loop. Considering the phase variation compensation, a shorter SMF is employed. This is because the use of a longer SMF causes a denser oscillation mode, and the temperature range of extractable time delay variation is reduced.

TABLE 3. Comparison of the designed optoelectronic oscillator (OEO) with previously reported OEOs.

Architecture	Tunability	Phase Noise (dBc/Hz@10 kHz)	SMSR (dB)	Phase Variation Compensation
LCFBG-MPBF (4 km) [18]	3.3–7.3 GHz	−110	63	Without
SBS-MPBF (1 km) [19]	DC–40 GHz	−113	55	Without
PS-FBG-MPBF (747 m) [21]	2.483–12.571 GHz	−101.2	61	Without
Based on PLL (3 km) [12]	Fixed	−117	80	With
This Work (203 m)	118 MHz–24.092 GHz	−96.40	73.29	With

In conclusion, a stable OEO with a wider frequency tuning range is designed and analyzed. The tunable MPBF is achieved through the combined operation of the designed PS-FBG and the PM. By tuning the output wavelength of the TL, the MPBF passband can be shifted to select the oscillation mode. With the optical phase compensation loop, the phase variations of the RF signal caused by the temperature variations are compensated to stabilize the output frequency. The output RF signals are tuned from 118 MHz to 24.092 GHz when the output wavelength of the TL is adjusted from 1,549.939 nm to 1,550.131 nm. When the ambient temperature fluctuates within ±3.9 ℃, the frequency drifts of the output RF signals are less than 68 Hz, the SMSRs are more than 69.39 dB, and the phase noise is less than −92.49 dBc/Hz at 10 kHz offset frequency. All the results reveal that the designed OEO has advantages of wider frequency tunable range and better temperature stability.

The authors declare no conflicts of interest.

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

National Natural Science Foundation of China (NSFC 62162034); the General Program of the Basic Research Program of Yunnan Province (202201AT070189).