High spectral purity electromagnetically induced transparency-based microwave optoelectronic oscillator with a quasi-cylindrical microcavity

Xueying Jin; Mengyu Wang; Keyi Wang; Yongchao Dong; Liandong Yu

doi:10.1364/OE.27.000150

1. Introduction

High spectral purity microwave sources have a major impact on scientific and industrial areas. Driven by emerging applications comprising, for example, communication, frequency metrology, robotics, radar, and aerospace engineering, there is a growing demand for compact low-phase-noise oscillator systems. In microwave oscillator, there are three critical parameters: maximum frequency, spectral purity (i.e., phase noise), and frequency stability. The maximum frequency determines the amount of data that can be transmitted, while the spectral purity characterizes the number of data channels that can be carried. Among them, spectral purity is particularly important for an oscillation system. These general constraints apply to communication links, radar, navigation or any other systems consisting of a sender and a receiver.

Electronic oscillator depends on frequency multiplication of signals created by low noise quartz or surface acoustic wave oscillator. The quality factor and thus the noise accompanied with the multiplication process, unfortunately, degrades for high frequency signals beyond the levels required for high-end applications [1]. A major milestone has been achieved in the early 1990s, when Yao and Maleki [2,3] introduced a long-distance fiber delay to microwave oscillation system to achieve spectrally pure radio frequency (RF) signals at high frequency, which provides an elegant and reliable solution using the so-called optoelectronic oscillator (OEO). A narrow-band RF filter need to be inserted into the delay line optoelectronic oscillator (DL-OEO) loop to achieve stable single-mode operation. The oscillation frequency is defined by the center frequency of this filter. This DL-OEO provides a remarkable progress on low-phase-noise as well as long storage time with no frequency-dependent attenuation [4,5]. Yet, despite the excellent accomplishments, the DL-OEO platform still face some severe limitations. First, the fiber-ring cavity modes created by delay line leads to strong spurious peaks close to the carrier in the generated phase noise spectrum, which need to be reduced using complex configurations [6,7]. Second, a few kilometers long fiber delay is bulky and not transportable because of its weight, size (few dm³), and an indispensable temperature-stabilized box.

An alternative approach to resolve this problem is to replace the long fiber with a high-Q whispering gallery mode resonator (WGMR-OEO). The oscillation occurs at the free spectral range (FSR) of the WGMR, its spectral purity and stability are determined by the spectral width of WGM resonances, which has the narrowest characteristic linewidth. In contrast to DL-OEOs, microcavity-based OEOs suppress spurious modes significantly, eliminate the need for narrow-band RF filter, and at the same time, achieve large storage time in a compact space (few μm³ to few mm³). Many novel implementations have been developed to create highly spectrally pure microwave signals, including OEO with a high-Q crystalline WGMR in add-pass [8,9] or add-drop [10,11] configurations, or crystalline WGMR-based Kerr/Raman/Brillion combs [12–14].

The motivation for electromagnetically induced transparency (EIT)-based OEOs stems from the extremely narrow spectral linewidth exhibited by EIT resonances. EIT is a quantum interference effect originating from different transition pathways of optical fields. This effect has been applied to many applications such as ultraslow group velocity of light, sensing, optical switching, narrowband filtering, and optical storage [15–17]. Those results take the advantage of steep frequency dispersion property around the narrow EIT resonance. More recently, we have demonstrated a tunable EIT in a single quasi-cylindrical microresonator (QCMR) [18], which provides a stable platform for controlling the EIT and Fano resonances in a compact volume. We show here that EIT resonances can also be exploited for OEO generation and stabilization.

In this work, we report the application of tunable EIT effect generated within a single QCMR to a microwave optoelectronic oscillator system. Although WGMR coupling system have been studied for microwave generation, the present device is the first EIT-based microwave source with a single QCMR. In this case, the QCMR performs as a large energy storage element as well as a high selective optical band-pass filter. Moreover, the QCMR is exploited to stabilize the laser frequency onto one of its resonances. This simplifies the system and eliminates the need for narrow-band RF filter, two microfibers or prisms, which is an indispensable requirement in traditional OEOs [10,13]. The microwave oscillation frequency is defined by axial free spectral range (Axial-FSR) of QCMR, and the device has a low phase noise of −123 dBc/Hz at 10 kHz offset from the carrier, and −135 dBc/Hz at 100 kHz. Furthermore, we study the effect of laser-mode locking state, quality factor as well as spectral lineshapes of microcavity, laser coupling efficiency, and three configurations of optical energy storage elements on the phase noise performance of the device. The EIT-based OEO is theoretically described using the modified stochastic dynamic model, which shows a good agreement with our experimental results. Finally, we investigate the origin of noise peaks, as well as to improve its phase noise and stability performances. Our approach extends the advantages of EIT to microwave OEO device based on a single QCMR, opening a route for compact high spectrally pure microwave source.

2. Experimental setup

The experimental setup, sketched in Fig. 1(a), is used for EIT-based microwave OEO with a single QCMR platform. The proposed OEO is similar to the WGMR-OEO containing two coupling fibers [10,11], yet here only one microfiber is included and the optical filtering function is achieved by tunable EIT resonances within a QCMR. Importantly, a high-Q QCMR is exploited for tunable EIT [18] and microwave generation, which acts as an energy storage element, narrow bandwidth optical filter, and a stabilization element. The pump field is amplified by erbium-doped fiber amplifier (EDFA), and then injected into a Mach-Zehnder modulator (MZM), which generates a broadband optical comb. This broadband optical spectrum is next coupled to the QCMR by evanescent field through one microfiber in add-pass configuration. Two polarization controllers (PC) are adopted in this setup, the first one is before the MZM for linear polarization, and another one is for optimizing the WGM excitation. The coupling condition and hence the EIT lineshapes as well as the coupling efficiency are controlled by vertically or horizontally scanning the QCMR, while keeping it in physical contact with microfiber. The QCMR proposed in our lab [19,20] has a nanoscale parabolic profile, with a ~125 μm diameter corresponding to a ~538 GHz azimuthal FSR and a ~5 GHz axial FSR. Once a desired EIT lineshape is obtained for a selected WGM resonance, the pump wavelength is then stabilized onto this mode using a feedback loop as described in Fig. 1. The servo controller (LB1005, New Focus) holds the frequency detuning of the pump relative to the microcavity resonant frequency. Afterwards, the output light from the microfiber is a series of equidistant combs, separated by axial FSR of QCMR (~5 GHz), and subsequently routed to a fast PD (EOT ET-3500F, bandwidth 12.5 GHz) for heterodyne detection. The fast PD detects the various beatings kΩ_M (k is an integer), whereas filters the high-order and direct current (DC) components owing to its limited bandwidth. The generated RF signal is then filtered and amplified (32 dB gain), and finally fed back to the driving port of MZM to close the oscillation loop. The produced RF signal and its phase noise spectrum are monitored using an electrical signal and spectrum analyzer (ESA, Keysight N9030A). Figure 1(b) shows the observed three kinds of tunable transmission lineshapes, namely EIT peak, Fano, and Lorentzian dip. Also, an optical spectrum analyzer (OSA, Yokogawa AQ6370D) enables us to monitor the optical spectra, as displayed in Fig. 1(d). We implemented a wideband electrical filter (central frequency 5 GHz, bandwidth 500 MHz) to reject RF noise outside the frequency band of interest, whose bandwidth is an order of magnitude larger than the bandwidth of a WGM (Q ~3 × 10⁷, bandwidth 6.45 MHz). This is intrinsically different from the narrowband RF filter in a traditional DL-OEO system, which is used to select the microwave oscillation frequency. A phase shifter is also employed to adjust the microwave round-trip phase to the desired value for self-sustained oscillations.

Fig. 1 Experimental setup and performance characterization. (a) Experimental setup illustrating the EIT-based microwave oscillator device. The tunable EIT is generated within a single QCMR. EDFA, erbium-doped fiber amplifier; MZM, Mach-Zehnder modulator; PC, polarization controller; PD, photodetector; BPF, wideband bandpass filter; OSA, optical spectrum analyzer; ESA, electrical signal and spectrum analyzer. (b) Experimental transmission spectra of the QCMR-microfiber coupling system, showing three types of controlled lineshapes: EIT peak, Fano, and Lorentzian dip. (c) RF signal and the corresponding phase noise performance generated by the device. (Resolution Bandwidth (RBW) is 50 kHz). (d) Optical spectrum of the generated signal.

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3. Theoretical model and results

3.1 Tunable EIT within a single QCMR

We model the microfiber as a multimode waveguide in the near-waist region as displayed in Fig. 2(a). When two or more fiber modes are coupled with one WGM simultaneously, there exists a phase shift that produces the EIT/Fano lineshape. For simplicity, here we only consider two fiber modes, that is the fundamental mode (fiber mode 1) and the high-order mode (fiber mode 2). The dynamic field inside the QCMR a can be expressed in the language of coupled-mode theory [21]

\frac{d a}{d t} = - (i σ + \frac{κ_{0} + κ_{e x 1} + κ_{e x 2}}{2}) a + \sqrt{κ_{e x 1}} E_{1} + \sqrt{κ_{e x 2}} e^{i Δ φ_{0}} E_{2},

where

σ = ω - ω_{0}

designates the frequency detuning between the input light (ω) and the resonance frequency (ω₀).

κ_{0}

and

κ_{e x 1} (κ_{e x 2})

denote the intrinsic decay rate and external coupling decay rate for fiber mode 1 (mode 2). E₁ and E₂ are the input amplitudes of mode 1 (mode 2).

Δ φ_{0}

indicates the phase shift between the two fiber modes. In this case, only mode 1 can be detected at the output

E_{o u t} = E_{1} - \sqrt{κ_{e x 1}} a

. The normalized transmission with the steady state is given by

Fig. 2 (a) Schematic of EIT generation in a single QCMR: two fiber modes coupled with one WGM. (b) Four kinds of transmission lineshapes with different phase shifts Δφ₀. Other parameters are set: $κ_{0} =2 κ_{e x 1} = κ_{e x 2}$ , E₂/E₁ = 1. (b) Four kinds of transmission lineshapes with different distribution radios E₂/E₁. Other parameters are set: $κ_{0} =2 κ_{e x 1} = κ_{e x 2}$ , Δφ₀ = π.

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T_{0} = {| \frac{i σ + \frac{κ_{0} - κ_{e x 1} + κ_{e x 2}}{2} - \sqrt{κ_{e x 1} κ_{e x 2}} e^{i Δ φ_{0}} \cdot E_{2} / E_{1}}{i σ + \frac{κ_{0} + κ_{e x 1} + κ_{e x 2}}{2}} |}^{2} .

In essence, EIT/Fano resonance originates from interference between two related discrete states (a single WGM excited by two fiber modes) and a continuum (fiber mode 1), analogized to the classical EIT/Fano mechanism. This equation shows that, phase shift Δφ₀ characterizes the degree of asymmetry and hence, dominates the lineshape variation, and the field distribution ratio E₂/E₁ determines the maximum degree of asymmetry. Figures 2(b)-2(c) simulate the dependence of output spectra on these parameters. It is seen that, for phase shift Δφ₀ = Nπ (N is an integer), the lineshape exhibits a symmetric Lorentz [Fig. 2(b), green curve and red curve], and its shape can be tuned between an EIT peak and a Lorentzian dip continuously by modifying the distribution radio E₂/E₁. In other cases, the lineshape behaves as an asymmetric Fano-like lineshape [Fig. 2(b), orange curve and blue curve]. Moreover, a larger E₂/E₁ leads to a stronger EIT/Fano resonance, as displayed in Fig. 2(c). Thus, we here provide a simple and effective method for tunable EIT/Fano generation.

3.2 EIT-based optoelectronic oscillator model

The dynamics of this EIT-based oscillation system is defined by two slowly-varying amplitude variables. The first is the complex electric field $G_{n} (t) = G_{n} (t) e^{i Ψ_{n} (t)}$ output from different WGMR modes, and the second is the microwave amplitude $A (t) = A (t) e^{i φ (t)}$ [see Fig. 1(a)]. They obey the following dynamic equations in the time domain [11,22,23]

\frac{d G_{n}}{d t} = - [i σ + \frac{κ}{2}] \cdot G_{n} + \sqrt{2 κ_{e x 1}} ζ_{n} (t) + Γ_{n} (ϕ) [1 + η_{n} (t)] J_{n} (A (t - T)) e^{i n (φ (t - T) - ω_{L} T)},

A (t) = 2 β e^{i υ} \sum_{n = - \infty}^{+ \infty} G_{n + 1} G_{n}^{*},

with

Γ_{n} (ϕ) = E \cdot [e^{i ϕ} + {(- 1)}^{n} e^{- i ϕ}] i^{n} / 2,

E = i σ + \frac{κ_{0} - κ_{e x 1} + κ_{e x 2}}{2} - \sqrt{κ_{e x 1} κ_{e x 2}} e^{i Δ φ_{0}} \cdot E_{2} / E_{1} .

where E indicates the EIT-modified coefficient. The cavity overall decay rate κ is defined by

κ = κ_{0} + κ_{e x 1} + κ_{e x 2}

. n is an integer and represents n-th order for

G_{n}

,

Γ_{n}

,

J_{n}

, and

η_{n}

[11,22,23]. J_n denotes the n-th order Bessel functions of the first kind, and υ corresponds to the microwave round-trip phase shift. T refers to the time delay in the optical path when the effect of WGMR is not account for, and

ω_{L}

is the pump laser frequency.

β = π η G_{0} P_{0} S / 2 V_{π RF}

corresponds to the optoelectronic gain without regard to the effect of QCMR, with G₀, η, and S being the amplifier gain, losses, and the responsibility of PD, respectively.

V_{π RF}

and

V_{π}

are respectively the RF and DC half-wave voltages of MZM.

ϕ = π V_{B} / (2 V_{π})

represents the static phase of the MZM, with

V_{B}

being the bias input voltage of MZM. The dimensionless amplitude

A (t)

and the real voltage

V (t)

are simply related by

V (t) = (2 V_{π RF} / π) A (t) c o s [Ω_{M} t + φ (t)]

, with the microwave output angular frequency of

Ω_{M}

.

It is worth noting that, the microwave signal suffers from noise resulting from multiple sources, for example, laser, resonator, and PD. Theoretically, this noise produces perturbations around the steady state solution, which can be modeled by adding Langevin noise terms to the dynamic stochastic model, as shown in Eq. (3). Generally speaking, the noise in the system can be decomposed into two contributions, namely the multiplicative noise and the addictive noise. The multiplicative noise, resulting from the overall optical gain fluctuations, is assumed as Gaussian white noise with density power spectrum ${| η_{n} (ω) |}^{2} = 2 D_{m}$ for simplicity. The second contribution addictive noise, is originated from environmental fluctuations. This addictive noise is also assumed to be spectrally white, which corresponds to noise power density ${| ζ_{n} (ω) |}^{2} = 2 D_{a}$ . Typically, the non-trivial fixed point are obtained from Eqs. (3)-(4) by setting all the derivatives to zero, thereby yielding the stationary solution

G_{n}^{s t} = \frac{2 Γ_{n} (ϕ) J_{n} (A^{s t})}{\sqrt{4 σ^{2} + κ^{2}}},

A^{s t} = Γ J_{1} (2 A^{s t}),

with

Γ = β | \sin (2 ϕ) | T_{0} .

Γ is the overall loop gain, which is the product of power transmission factor of QCMR-coupled system [T₀ in Eq. (2)] and the optoelectronic gain. This stationary solution is of interest to us, since it determines the microwave amplitude. Note that with power transmission factor involved, the overall gain and hence the stationary solution is modified by EIT factor. Furthermore, by geometrical interpretation [11], the non-trivial fixed point is unconditionally stable for 1 < Γ < 15.52, in both of the QCMR-OEO and QCMR-DL OEO systems.

3.3 Phase noise performance

To characterize the purity of a microwave signal, an effective way is to measure its phase noise. Here, we extract the stochastic phase terms from Eqs. (3)-(4), disregard the higher-order terms and make use of the steady state solution. One can finally obtain

\frac{d Ψ_{n}}{d t} = σ η_{n} (t) + \frac{κ}{2} [n φ (t - T) - Ψ_{n}] + \sqrt{2 κ_{e x 1}} \frac{ζ_{n, Ψ} (t)}{G_{n}^{s t}},

0 = \sum_{n = - \infty}^{+ \infty} G_{n + 1}^{s t} G_{n}^{s t} [Ψ_{n + 1} - Ψ {}_{n}- φ] .

where

ζ_{n, Ψ} (t) = ζ_{n, Re} (t) \sin (- Ψ_{n}) + ζ_{n, Im} (t) \cos (- Ψ_{n})

are independent Gaussian white noises. Considering that Eq. (10) is linear, and all noise sources are uncorrelated, the phase noise spectrum of this microwave oscillator can be found from the squared modulus of Fourier transform of Eq. (10), which yields

{| Φ (ω) |}^{2} = \frac{2 σ^{2} α_{1}^{2} {| η (ω) |}^{2} + 4 D_{a} κ_{e x 1} α_{2}^{2}}{{| i ω + \frac{κ}{2} (1 - e^{- i ω T}) |}^{2}},

α_{1}^{2} = \frac{2 \sum_{n = 0}^{+ \infty} J_{n}^{2} (A^{s t}) J_{n + 1}^{2} (A^{s t})}{J_{1}^{2} (2 A^{s t})},

α_{2}^{2} = \frac{1}{J_{1}^{2} (2 A^{s t}) \sin^{2} (2 ϕ)} .

Equations (12)-(14) reveals that the multiplicative noise in the oscillation system is governed by the detuning between the pump laser and the microcavity mode. Also, the multiplicative noise will ultimately vanish provided that the pump laser frequency matches the center of the resonance, namely σ = 0. This may be attributed to the fact that this detuning is an intensity-to-phase noise conversion mechanism. Note that by setting T = 0, one can obtain the phase noise spectrum formula for a QCMR-OEO without delay.

We next implement multiple simulations using the above equations for such an EIT-based oscillator to theoretically investigate the phase noise properties. In our study, we consider the following parameters unless otherwise stated: κ_ex0 = 2.47 × 10⁶, κ_ex1 = 1.24 × 10⁶, κ_ex2 = 2.47 × 10⁶, which leads to overall decay rate κ = 6.18 × 10⁶, corresponding to cavity linewidth Δω ≈6.18 MHz. Essentially, the linewidth of EIT is much smaller than RF filter bandwidth (500 MHz) and therefore, it allows for filtering function in optical domain only. Other parameters are set: ϕ = −π/4, D_m = 1 × 10⁻¹⁷ rad²/Hz, D_a = 3 × 10⁻¹⁶ rad²/Hz, σ /2π = 40.85 MHz, β = 24.16, and A_st = 1.7590. Note that the value of A_st is calculated from Eqs. (7)-(9).

Figure 3(a) displays the microwave phase noise of an EIT-based OEO with different mode detuning. When the pump laser frequency is shifted away from the center of the optical resonance, we find that the phase noise increases and its spectral purity degrades. This confirms the importance of laser stabilization in an EIT-based oscillator. Regarding the influence of quality factor, a series of the phase noise spectra using QCMR of different Q factors (Q = 3.13 × 10⁷, 1.95 × 10⁷, 9.19 × 10⁶, and 4.88 × 10⁶) are shown in Fig. 3(b). It is seen that, for higher quality factor, the theory evidences a significant reduction of spurious modes but a degradation on phase noise floor. In particular, an improvement in quality factor by a factor of two will degrade the phase noise floor by ~4 dB, whereas reduce the spurious mode by ~4 dB. Thus, there is a trade-off between the spurious modes reduction and phase noise performance when choosing a mode with appropriate quality factor. Furthermore, we predict that, for a mode with enough high quality factor, the spur may ultimately vanish in the phase noise spectrum, whereas the phase noise performance will inevitably degrades.

Fig. 3 (a) Theoretical phase noise spectra for different mode detuning σ/2π = 40.85 MHz, 70.98 MHz, and 95.12 MHz. (b) Theoretical phase noise spectra for different mode quality factor Q = 3.13 × 10⁷, 1.95 × 10⁷, 9.19 × 10⁶, and 4.88 × 10⁶.

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Figure 4(a) compares the phase noise spectrum for T = 20 μs (corresponds to a 4 km fiber length) with that attained with extremely small delay (T = 2 ns). We find that longer delay produces a significant improvement on the phase noise performance by 36 dB. To demonstrate the role of different optical energy storage elements, we then compare the phase noise performance for three different configurations: QCMR-OEO, DL-OEO, QCMR-DL-OEO. Figure 4(b) displays the phase noise spectra for QCMR-OEO and QCMR-DL-OEO calculated from Eqs. (12)-(14), and DL-OEO calculated from Eq. (15). The EIT resonances are created inside the QCMR coupling system. For the common DL-OEO without a microcavity, the theoretical phase noise spectrum can be written as [24]

{| Φ (ω) |}^{2} = \frac{\frac{μ^{2}}{4 Q_{f}^{2}} {| η (ω) |}^{2} + \frac{2 μ^{2}}{{| A^{s t} |}^{2}} D_{a}}{{| i ω + μ (1 - e^{- i ω T}) |}^{2}} .

Where Q_f ≈Ω₀/ΔΩ denotes the quality factor determined by RF filter of half-bandwidth μ/π ≈ΔΩ. For comparison with our experiments, we employ D_m = 1 × 10⁻⁸ rad²/Hz, D_a = 3 × 10⁻¹⁴ rad²/Hz, Ω₀ = 5 GHz, Q_f ≃ 10, μ/π = 500 MHz. This comparison shows that, the configuration with both DL and QCMR achieves a better phase noise performance than the other two cases. More precisely, the QCMR enables a reduction for the first-neighboring spur of ~42 dB, and a 4 km long DL improve the phase noise performance by ~36 dB. This clearly evidences the role of a QCMR with EIT resonances in suppressing the spurious modes and thus producing a high spectral purity RF signal.

Fig. 4 (a) Comparison between the theoretical phase noise performance of QCMR-DL-OEO for different delay line: T = 2 μs and T = 2 ns. (b) Comparison of the theoretical phase noise performance of three different configurations of OEOs: QCMR-OEO, DL-OEO, and QCMR-DL-OEO. The DL is a 4 km long fiber delay line. The spurious peaks are highly rejected in a QCMR-DL-OEO system attributed to the narrow bandwidth filtering effect of EIT resonance.

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4. Generating a tunable EIT with a QCMR

Experimentally, for tunable EIT generation, we exploit a multimode microfiber-QCMR add-pass coupling configuration [see Fig. 1(a)]. The QCMR (with the length of ~450 μm and height of only ~36 nm, measured by MF scanning method [25]) was fabricated using a fiber fusion splicer via our previously developed method, which enables high-Q QCMR (quality factor up to 3 × 10⁷) with clean and regular spectrum [19,20]. The tunable CW laser (New Focus Velocity 6728, linewidth < 200 kHz) in 1550 nm band was injected into the microfiber and then coupled to the QCMR. The output light was split twice and 5% of the signal was sent to a slow PD followed by a 6 GHz real-time sampling oscilloscope (Keysight MSOS604A) to monitor the transmission spectrum of QCMR-microfiber coupling system. The multimode microfiber was kept in contact with the QCMR, which can be moved horizontally or vertically through a high-precision three-axis stage. In contrast to the coupling gap tuning method reported elsewhere [26,27], this leads to a stable tuning process.

The tunable EIT created within a QCMR is characterized in two perspectives. First, we characterize the dependence of EIT/Fano lineshapes on the coupling position of microfiber by horizontally scanning the QCMR. To achieve a stable tuning, the microfiber keeps in touch with the QCMR surface. We started from the Lorentian dip and slowly scanned the QCMR along x direction (see Fig. 5 inset), where the microfiber diameter decreased. The lineshapes were recorded every 2 μm, as shown in Fig. 5(a). It is seen that, the transmission lineshape evolved from a Lorentzian dip to an asymmetric Fano-like lineshape, and then became an EIT peak. This evolvement of lineshape can be attributed to the fact that the phase shift and the field distribution ratio between the two fiber modes depends on coupling position. With the decrease of the coupling diameter, the transmitted background becomes weak, which is originated from the increase of the insertion loss during the tuning process. Figure 5(b) compares the experimental lineshapes with the simulated results obtained using the theoretical model (Eq. (2)), where the correspondence between the coupling position of microfiber and the value of Δφ₀ and E₂/E₁ is given. It is evident that, the phase shift between the two fiber modes Δφ₀ dominates the lineshape variation, which makes the spectra evolve periodically by 2π, while field distribution ratio E₂/E₁ determines the degree of asymmetry. More specifically, for phase shift Δφ₀ = Nπ (N is an integer), the lineshape behaves as a symmetric Lorentz, and further, it can be engineered between a dip and an EIT peak by changing the distribution radio E₂/E₁. A large E₂/E₁ value leads to an enhanced EIT, which is also in good agreement with the theoretical result illustrated in Fig. 2(c). In other cases, for phase shift Δφ₀≠Nπ, the asymmetric Fano lineshape is always exhibited. In addition, one may note that with the decrease of the coupling diameter, the distribution radio E₂/E₁ increases. This further confirms the energy conversion between the two fiber modes in the tuning process. We found that the simulated lineshapes capture the major properties of the measured lineshapes. The slight deviation may be attributed to the involvement of other WGMs as well as the small variations of coupling parameters when scanning the microfiber. Therefore, the theoretical model is credible to explain the dependence of transmission lineshape on the coupling position of microfiber.

Fig. 5 Comparison between the experimental lineshapes (a) and the theoretical lineshapes (b). Normalized transmission spectra with six different coupling positions ∆x = 0, 2 μm, 4 μm, 6 μm, 8 μm, and 10 μm. ∆x = 0 corresponds to an assigned relative zero point in the taper waist region. The simulation parameters are κ₀ = 8 × 10⁷, κ_ex1 = κ₀/2, κ_ex2 = κ₀, Δφ₀ = 0, 0.76π, 0.82π, 0.88π, 0.90π, π, and E₂/E₁ = 0.6, 0.7, 0.8, 0.9, 1.0, 1.2. (c) Optical characterization of EIT resonances with different coupling efficiencies. The inset on the left sketches the tuning process in the experiment.

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Second, the tunability of coupling efficiency of EIT resonance is characterized by changing the relative vertical position between the microfiber and the QCMR. Figure 5(c) shows five typical EIT spectra with different Δz position for mode A. We observed that, as we vertically moving the QCMR, the coupling efficiency of EIT peak can be tuned from 0.73 to 0.35 continuously, and finally disappeared (not shown). In fact, mode A is a q = 19 axial modes (q denotes the axial mode quantum number, and q = 0 indicates the fundamental axial mode), that would disappear and reappear periodically and sequentially 20 times during the whole tuning process. This phenomenon can be attributed to the variations of spatial overlap between the WGM field and the microfiber mode field, which has been explained in detail in our previous work [19,28]. It is well known that, the mode selection and the strength of mode excitation is governed by the degree of phase matching as well as spatial overlap between the microfiber and QCMR mode fields. Here, we show that, this rule equally applies to the EIT resonances case. The accurate and flexible tuning of EIT resonances shows the dependence of transmission lineshape on the phase shift Δφ₀, field distribution radio E₂/E₁, and the coupling parameters of the system, which in turn, provides an approach for achieving the best performance of an EIT-based microwave oscillator.

5. EIT-based Optoelectronic oscillator characterization

We characterized the RF signal generated by this EIT-based OEO system and observed an exceptionally high spectral purity RF signal. The observed RF tone confirms that EIT-based OEO is promising for generation of RF signals with high spectrally pure. This type of photonic microwave generator is attractive, because it can be miniaturized and further packaged with chip-scale hybrid integration [20,29]. To find what limits the performance of the EIT-based oscillator, we conduct a series of experiments to investigate the impact of laser-mode locking state, quality factor as well as spectral lineshapes of microcavity, laser coupling efficiency, and three configurations of optical energy storage elements on the phase noise performance of the device. Therefore, different experimental measurements are implemented on this EIT-based OEO, especially to characterize its optical spectra, RF signal, and phase noise spectra. Finally, we characterize the stability performance of generated RF signals. From these interesting results, the phase noise limitations of the device can be obtained, which helps us to improve its phase noise and stability performances.

In a first experiment, we analyze the impact of laser-mode locking state on the phase noise performance by a detailed comparison of the results obtained from five different locking positions. These experimental results of phase noise spectra are those of microwave generated at ~5 GHz. Various patterns of RF signals [Fig. 6(a), left panel] and phase noise spectra [Fig. 6(a), right panel] were generated and identified when laser was locked on different positions of a selected optical EIT resonance, owing to the very good dynamic of ESA employed in our experiment. Note that we did not use any WGMR temperature stabilization setup. Figure 6(b) display the corresponding normalized EIT spectrum recorded on the OSC connected to a slow PD. To further confirm the dependence of phase noise level relying on laser-mode locking position, we provided the phase noise value for five locking state at 10 kHz offset and 100 kHz offset respectively, as depicted in Fig. 6(c). We observe that, when the locking position of pump laser is shifted towards the center [from S₅ to S₁ locking state, see Fig. 6(b)], the phase noise at 1 kHz−1 MHz frequency range experiences a significant improvement. More specifically, the phase noise at 10 kHz offset is significantly improved from the level of −100 dBc/Hz to −115 dBc/Hz, and for the case of 100 kHz offset, the phase noise performance is also promoted from the level of −120 dBc/Hz to −134 dBc/Hz. Here, the phase noise spectrum in the interested 1 kHz−1 MHz frequency range has a complex quasi-resonant frequency dependence [13], called ‘noise plateau’ (see Section 6). We attribute this noise plateau to the conversion of laser frequency noise to the amplitude noise inside the QCMR, which greatly contributes to the degradation the phase noise performance of generated RF signal. We conclude that, the laser-mode locking state noticeably influences the phase noise performance in noise plateau frequency range (1 kHz−1 MHz offset), particularly for smaller frequency offset. Therefore, high spectrally pure RF signal can be achieved by locking the laser near the center of the EIT resonance.

Fig. 6 (a) Three typical measured RF signal (left panel) and their corresponding phase noise spectra (right panel) at ~5 GHz for different laser-mode locking states (see Fig. 6(b)). (b) Normalized EIT spectrum recorded at a slow PD monitored by an oscilloscope for a silica QCMR, where multiple optical modes are excited within ~0.3 nm wavelength range. The inset shows five different laser-mode locking state (S₁, S₂,…S₅) in our experiment using a servo controller (LB1005, New Focus). (c) Extracted phase noise data versus laser-mode locking position for 10 kHz and 100 kHz frequency offset at ~5 GHz carrier RF signal.

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To investigate on effect of quality factors of EIT mode on the phase noise performance, we chose five EIT resonances with different Q factors, locked the pump laser to the same position of these modes respectively, and measured their RF and corresponding phase noise spectra[Figs. 7(a)-7(b)]. The RF signal spectra in Fig. 7(a) shows four stable RF signals with slightly different frequencies. The frequency of generated RF signal here is dominated by axial FSR of nearby EIT peaks. This further confirms the nearly equidistant axial modes exhibited by our QCMR [19,20], since the variations of axial FSR for different axial modes is very small. Figure 7(b) gives the corresponding phase noise spectra for EIT peaks with different Q factors. It is seen that, increase of the Q factor of EIT peak by a factor of two will lead to a reduction on spurious peaks by ~3 dB, while the phase noise floor may become higher. The measured results exhibit good agreement with the theoretical results in Fig. 3(b) regarding the impact of quality factors (~4 dB spur reduction) on phase noise levels. Deviations are attributed to the fact that the measurements on different Q factors are implemented independently and might hence not have locked to the exact same position. Moreover, we find that, higher Q factor corresponds to slightly higher noise bumps in the noise plateau frequency range (1 kHz−1 MHz). This further supports the prediction that this noise originates from transfer of laser frequency noise to noise inside the WGMR. Therefore, a trade-off has to be considered between the spur rejection and an improvement on phase noise floor when selecting an EIT mode with proper quality factor.

Fig. 7 Limitations on oscillator RF signal and phase noise. (a)−(b) Influence of WGMR quality factor on the generated RF signal (a) and phase noise (b). (c)−(d) Influence of coupling efficiency on the generated RF signal (c) and phase noise (d). (e)−(f) Influence of transmission lineshape on the phase noise (f). (e) Optical characterization of EIT resonances with three typical lineshapes, namely EIT, Fano, and Lorentzian dip.

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Figures 7(c)-7(d) study the impact of coupling efficiency on the phase noise performance. We kept the position of x unchanged and varied the coupling efficiency by vertically moving the QCMR along its axis. The RF signal and optical spectra are depicted in Fig. 7(c) and Fig. 5(c) respectively, showing stable microwave signals and decreasing coupling efficiency in optical characterization. The mechanism for this tunable coupling efficiency has been provided in Section 4. Figure 7(d) displays the phase noise performance for different coupling efficiencies. Here, the result shows clearly better close-in phase noise (frequency offset < 1 kHz) for a smaller coupling efficiency (under-coupling region), about ~16 dB improvement on phase noise level. However, for 1 kHz – 1 MHz offset frequency range (noise plateau), the impact is not obvious, and the phase noise floor for a smaller coupling efficiency may be higher [blue curve in Fig. 7(d)]. This interesting result can be explained by the fact that the close-in noise may come from the thermal noise within WGMR (see Section 6), which can be lowered down by decreasing the optical energy coupled in QCMR. Moreover, a smaller coupling efficiency corresponds to less thermal noise in the QCMR. As a result, a smaller coupling efficiency leads to improvement on close-in noise. Indeed, when the coupling efficiency becomes very small [pink curve in Fig. 5(c)], no stable oscillation can be obtained.

In a forth experiment, we investigate the impact of output lineshapes of QCMR on the phase noise level, and demonstrate that the EIT lineshape should be adjusted to optimize the phase noise performance. In this case, we kept the position of z fixed and horizontally translated the QCMR along the microfiber. The transmission spectrum gradually evolved from an EIT peak (Δx = 0, Δx = 156 μm) to asymmetric Fano-like lineshape (Δx = 196 μm), and finally changed to a symmetric Lorentzian dip (Δx = 313 μm) with the increase of microfiber diameter, as displayed on Fig. 7(e). This phenomenon in optical domain has been further discussed in Section 4. Figure 7(f) presents the phase noise spectra for these three cases: EIT, Fano, and Lorentzian dip respectively. For Fano-like (yellow curve) or a Lorentzian dip (blue curve) lineshape, one can notice an unstable RF signal accompanied with higher spurious peaks in phase noise spectrum, which is undesirable for a high spectrally pure OEO. The dip-OEO features a phase noise level of −98 dBc/Hz at 10 kHz offset from the carrier, and a Fano-OEO shows a phase noise level of −100 dBc/Hz. On the contrary, EIT-OEO can generate significantly high spectral purity RF signals (red curve and green curve), which exhibits a phase noise level of −110 dBc/Hz at 10 kHz offset. We attribute this to the excellent filtering function exhibited by an EIT lineshape transmission spectrum and argue that, by adjusting the lineshapes of transmission spectrum, the manipulation and optimization of the phase noise of microwave signals can be achieved in our device.

We next investigate the effect of three different optical energy storage elements on the phase noise performance. For comparison purposes, the DL-based OEO (4 km long delay line) and the QCMR-based OEO RF signal and phase-noise spectra are presented in Fig. 8 together with the phase noise of QCMR-DL OEO. The presented phase noise measurements in Fig. 8(b) give the phase noise levels, at 10 kHz offset from ~5 GHz carriers as follows: below −99 dBc/Hz for QCMR-OEO, below −108 dBc/Hz for DL-OEO, and below −123 dBc/Hz for QCMR-DL-OEO.

Fig. 8 Comparison of oscillation RF signals (a) and phase noise spectra (b) of three types of OEOs based on different optical storage element: a silica QCMR (QCMR-OEO, yellow curve), a 4 km long delay line (DL-OEO, blue curve), and a silica QCMR combined with a delay line (DL-QCMR-OEO, red curve).

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From the aspect of a filtering, EIT-based OEO seems to be very effective. In DL-OEO (blue curve), a various of spurious peaks emerge owing to the lack of a narrow-band RF filter. One can notice a strong phase noise reduction by ~40 dB when the QCMR is added to the oscillation loop (red curve). The spur mode rejection value is quite high compared to the conventional dual-loop OEO [6,30], which achieves ~30 dB spurious mode rejection by employing an 8.4 km long delay line combined with a 2.2 km delay line. However, this data still lower than the one obtained using a MgF₂ WGMR-DL OEO in add-drop configuration [10], ~53 dB spur rejection can be achieved, since the crystalline MgF₂ resonator shows a higher quality factor of ~6.1 × 10⁸. Aside from that, the phase noise curves for DL-OEO and DL-QCMR-OEO almost overlap in close-in noise frequency range, showing that the spur rejection effect actually does not work in this region.

On the other hand, it is seen that the measured phase noise spectra including a QCMR feature several noise bumps in the offset frequency range extending from 1 kHz to 30 kHz. Typically, the noise bump around 20 kHz offset frequency is not shown in the phase noise spectrum of DL-OEO, so that the WGMR (nonlinear scattering noise generated inside the WGMR) is responsible for the noise bumps in this region. Nevertheless, the noise bumps in the offset frequency range exceeding 30 kHz seem to result from other noise contributions, as these noise bumps are stronger in the DL-OEO phase noise spectrum where no WGMR is exploited. This may be attributed to the residual noise generated by RF amplifiers included in the OEO loop, it converts through nonlinearity of OEO loop and is finally added to the phase noise of output microwave signal. For comparison between the QCMR-OEO (orange curve) and DL-QCMR-OEO (red curve), we observe that DL is a good candidate for reducing the phase noise floor. Hence, we here adopt an EIT-generated QCMR combined with a DL as the optical energy storage element in our setup. The EIT-based QCMR performs as a narrow-band filter, while the DL for reduction in phase noise floor.

Finally, we evaluated the stability performance of generated microwave signals. The oscillation system was operated in a room environment for 30 min with a sampling period of 1 min, and multiple acquisitions are displayed in Fig. 9. The measured maximum oscillation frequency fluctuation and maximum power drift are lower than 4.8 kHz and 0.08 dBm, respectively. On another front, the variations of phase noise level are lower than 7.8 dB at 10 kHz offset frequency and lower than 6.3 dB at 100 kHz offset frequency. The frequency fluctuations may be attributed to the thermal instability in the WGMR, DL, or locking process here. We did not observe any mode-hopping during the process, demonstrating a stable oscillation signal in our EIT-based device.

Fig. 9 (a) Stability of generated oscillation signal recorded over half an hour. (b) Stability of phase noise at 10 kHz and 100 kHz offset frequency.

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6. Investigations on origin of excess noise peaks

Regarding the performance of our EIT-based OEO, an interesting question is what limits it. To identify the origin of noise peaks in phase noise spectra, we performed dedicated measurements using an assembled open-loop test set-up, as shown in Fig. 10. Here, we first measured the RF spectrum of a RF synthesizer with the same frequency as axial FSR of QCMR (~5 GHz), and then measured its RF spectra when the signal was transmitted through different optical links. In this way, various RF signal as well as phase noise spectra in different configurations can be obtained exploiting the ESA: after one or two RF amplifier, after the DL (~4 km), after a silica QCMR, and after a QCMR combined with DL. Note that the EIT peak is always exhibited by the QCMR. For the DL-QCMR case, we measured the transmitted RF spectra in three different locking states: lock on the FWHM, lock on the center, and lock on the edge.

Fig. 10 Origin of excess noise peaks. RF signal (a) and phase noise spectra (b) of a RF synthesizer signal at 5 GHz: directly measured and then measured after the signal was transmitted through different optical links. The OEO setup is designed in open loop configuration.

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Figure 10(b) shows that, when the QCMR is included in the optical link, multiple noise peaks arise, suggesting that these noise peaks are generated within the QCMR. In particular, the noise peaks in close-in frequency range (< 1 kHz offset frequency) seem to be very strong and remain almost the same for the last four cases containing the QCMR [green, red, orange, and light blue curves in Fig. 10(b)]. Note that the close-in phase noise is significantly improved provided that the RF signal does not pass through the optical link including a QCMR. Thus, we deduce that the noise peaks in this region may result from thermal noise of the QCMR associated with the laser relative intensity noise [13]. In addition, this noise will be weakened by adjusting the coupling condition to under-coupled region, as demonstrated in Figs. 7(c)-7(e), since less optical power is coupled in WGMR in this case. On the other hand, several noise bumps also emerge in ‘noise plateau’ frequency range (1 kHz – 1 MHz offset frequency). We find that, these noise bumps becomes stronger when the pump laser is locked on the edge of the EIT resonance, indicating that the noise peaks in the noise plateau region may result from conversion of laser frequency noise to amplitude noise inside the QCMR. Figure 6 in Section 5 further confirms the above result.

Involvement of a long DL (dark blue curve) will increase phase noise floor in noise plateau offset frequency range. We may attribute it to the laser frequency fluctuations converted into delay fluctuations through in-phase condition, which finally converts into a microwave phase noise [31]. This noise floor can be significantly improved by introducing the QCMR, as seen in Fig. 10(b), owing to the good filtering function of EIT. Subsequently, we compare the RF spectra obtained without a RF amplifier, with one amplifier, and with two amplifiers (black, purple, and yellow curves). One can notice a slightly higher phase noise floor in noise plateau region and typically a noise peak at 100 kHz offset frequency emerge and increase, when more RF amplifiers are included in the OEO loop. These phase noise floor and noise peak are mainly due to the residual noise generated by RF amplifiers, since in this case, the loop did not have other optical links such as a WGMR or a DL.

7. Discussion and conclusion

In a traditional DL-based OEO, a narrow-band RF filter is important for single-mode operation, which is not a necessity in our EIT-based OEO. Compared to the crystalline WGMR-based OEO in add-drop configuration [10,13], our device simplifies the system by eliminating the need for two microfibers or prisms. Moreover, our device allows manipulation and optimization of the phase noise performance in a stable manner, which can be attributed to unique mode field distribution and stable tuning approach of QCMR. However, the dual-fiber configuration combined with high quality crystalline WGMR can achieve slightly higher spur rejection. Hence, there is a trade-off between the EIT-based OEO and crystalline add-drop WGMR-based OEO.

The phase noise of our EIT-based OEO can be improved by adjusting the laser-mode locking state, quality factor as well as spectral lineshapes of the EIT peak, and coupling efficiency. Generally speaking, a typical phase noise spectrum has three distinct regions: close-in noise in < 1 kHz offset frequency range; ‘noise plateau’ in 1 kHz−1 MHz frequency range that features complex quasi-resonant frequency dependence; and the noise floor in > 10 MHz frequency range. The close-in noise is due to the thermal noise of QCMR with the ultimate limitation resulting from intensity noise of pump laser. From the above experiment results, we conclude that this close-in noise can be improved by adjusting the coupling condition to ‘under-coupled’ region through vertical displacement tuning. The noise plateau can be attributed to transfer of laser frequency noise to amplitude noise inside the QCMR. The noise in this region is related to the laser-mode locking state, namely the frequency detuning between the pump laser and the resonance. The noise floor is limited by the detector shot noise and relies on the RF signal power at the output of photodiode (not shown).

Regarding the optical energy storage element, the EIT-based QCMR combined with a DL seems to be very effective. This implementation exhibits strong rejection rate for spurious peaks and improved phase noise floor at the same time. Moreover, one may notice that a better phase noise performance can be achieved with an EIT peak exhibiting lower Q factor, yet in turn, this leads to stronger spurious peaks. As a result, a good compromise should be reached between employing a QCMR with high-Q factor and that with large linewidth. One may expect significantly improved phase noise performance by identifying and engineering the optical and RF noise contributions in the loop, for example, the noise accompanied with the RF amplifier. Note that our experiments were performed in a general laboratory environment, exposing to external thermal and mechanical perturbations, which may degrade the phase noise level and the stability of the OEO system. Besides that, further optimization of coupling condition may also improve the phase noise level. Thanks to the physical contact between the QCMR and microfiber in the tuning process, the coupling configuration in our device is stable, which makes the EIT-based OEO attractive in practical applications especially in strong vibration environments such as aircraft, rocket, and missile.

In summary, we apply the tunable EIT effect generated within a single QCMR to an important class of hybrid optoelectronic device, WGMR-based OEO. A prototype of the developed EIT-based OEO has been fabricated to test the applicability of this concept in microwave photonics field. We achieve single-mode operation in an EIT-based OEO without the use of narrow-band RF filters and eliminating the need for two microfibers or prisms. Indeed, the high-Q QCMR that create EIT resonances here not only serves as an energy storage element and a narrow bandwidth optical filter but also allows stabilization using a servo control. High spectral purity operation at ~5 GHz with a phase noise level of −123 dBc/Hz at 10 kHz from the carrier, and −135 dBc/Hz at 100 kHz is reported. Furthermore, we demonstrate that manipulation of spectral purity of microwaves by adjusting the laser-mode locking state, quality factor as well as spectral lineshapes of the EIT peak, and coupling efficiency is feasible and reliable. Based on this finding, EIT-based OEO may be an interesting example that can be exploited in further applications in microwave photonics area. Particularly, future work will explore the applications of this concept to achieve high spectrally pure RF signal generators with enhanced performance for communication, signal processing, radar, navigation, and other related fields.

Funding

National Natural Science Foundation of China (NSFC) (51875165, 61775209 and 61801129).

Acknowledgments

We would like to thank Dr. Zhen Shen for helpful discussions.

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High spectral purity electromagnetically induced transparency-based microwave optoelectronic oscillator with a quasi-cylindrical microcavity

Abstract

1. Introduction

2. Experimental setup

3. Theoretical model and results

3.1 Tunable EIT within a single QCMR

3.2 EIT-based optoelectronic oscillator model

3.3 Phase noise performance

4. Generating a tunable EIT with a QCMR

5. EIT-based Optoelectronic oscillator characterization

6. Investigations on origin of excess noise peaks

7. Discussion and conclusion

Funding

Acknowledgments

References

Cited By

Figures (10)

Equations (15)

Optics Express