Schematic representation of our analog PLL discriminator. The control voltage V(t) of the VCO constitutes the output signal that replicates the input frequency variations δν(t). This signal is analyzed using an oscilloscope, a lock-in amplifier or a FFT analyzer.
Difference between the ideal (dashed line) and actual (light thick curve) response of the digital phase detector. (a) DC analog output voltage as a function of the phase difference between the two inputs; some nonlinearities are visible at the encircled points. (b) Highlight of the nonlinearities of the detector occurring roughly every 2π phase difference. This curve has been obtained by applying a frequency-modulated carrier at one input of the device and performing lock-in detection at the output to determine the discriminator sensitivity. The dashed line corresponds to the average slope of the DC curve (a).
Amplitude (a) and phase (b) of the normalized transfer function of the different discriminators, measured by applying a frequency-modulated input carrier and performing lock-in detection of the discriminator demodulated signal. Each transfer function has been normalized by the discriminator sensitivity measured at 1 kHz modulation frequency (D ν = 7 × 10−7 [V/Hz] for PLL, D ν = 1.25 × 10−6 [V/Hz] for Miteq, D ν = 10−3 [V/Hz] for HF2PLL, D ν = 1.8 × 10‑5 [V/Hz] or D φ = 1.8 × 10−2 [V/rad] for DXD200). The amplitude response of the digital phase detector DXD200 is represented both in terms of response to frequency and phase modulation.
Normalized sensitivity of the frequency discriminators (measured for 1 kHz modulation frequency) as a function of the carrier frequency detuning. The gray area indicates the linear frequency range Δf lin of each discriminator, defined as the frequency interval for which the discriminator response differs by less than ±10% (±0.9 dB) from its nominal sensitivity. The frequency range of the HF2PLL is inversely proportional to the software-selected sensitivity D ν (Δf lin = ±10V/D ν) and is shown here for two particular cases (D ν = 100 μV/Hz and D ν = 5 μV/Hz) for illustration.
Noise floor of the different discriminators. The noise floor of the analog PLL depends on the PI gain and is presented here in an optimized configuration. The white frequency noise of the HF2PLL (at low frequency) results from white noise at the analog output and thus scales as 1/D ν 2 for sensitivities up to D ν = 10 mV/Hz. It is displayed here for two cases, D ν = 100 μV/Hz and D ν = 10 mV/Hz. The dashed lines represent an approximation of the noise floor of each discriminator in terms of a power series of f ( f −2, f −1, f 0, f 1, and f 2).
Cross-sensitivity of the discriminators to amplitude modulation (a) and to amplitude noise (b), expressed in terms of AM-to-FM (AN-to-FN) conversion factor (in Hz/%). The dashed lines represent an approximation of the AM–FM (AN–FN) conversion factor as a constant level (for Miteq) or proportional to f (for the other discriminators), obtained in the high frequency range where the measurements are out of the noise floor of each discriminator. These trend lines are used to extract numerical values for the AN–FN cross-sensitivity of each discriminator as listed in Table I.
Examples of a graphical representation of a frequency/phase discriminator with different bandwidths ( f BW = 100 Hz, 1 kHz, 10 kHz, 100 kHz, 1 MHz): (a) frequency discriminator in the plane ( f, S δν ), (b) phase discriminator in the plane ( f, S ϕ ), and (c) phase discriminator in the plane ( f, S δν ). The frequency discriminator has a range of Δf = 100 kHz and a noise floor S min = 0.01 Hz2/Hz; the phase discriminator has a range Δϕ = 2π and a noise floor S ϕ min = 10−9 rad2/Hz. The dashed line represents the β-separation line (S δν = (8Ln(2)/π2) · f) in the frequency noise spectrum and its correspondent (S ϕ = (8Ln(2)/π2)/f) in the phase noise spectrum.
Graphical comparison of the characteristics of the different discriminators. Each discriminator is represented by a surface delimited by its noise floor S min, its bandwidth f BW and the maximum measureable frequency noise PSD S max. The situation for HF2PLL depends on the selected discriminator value D ν and is shown here for two cases, D ν = 100 μV/Hz and D ν = 10 mV/Hz. The dashed line represents the β-separation line S δν(f) = (8Ln(2)/π2) · f.9
Frequency noise PSD of the CEO-beat in our frequency comb measured with the different discriminators; (a) free-running CEO and (b) CEO phase-stabilized to a 20-MHz reference signal. For HF2PLL, the discriminator value is 100 μV/Hz. The β-separation line that is relevant for the determination of the CEO-beat linewidth is also shown as a dashed line.9
Summary of the main properties of the frequency (phase) discriminators. The noise floor is approximated by a power series in f ( f α ) with up to three different exponents corresponding to flicker frequency noise (range 1, −2 < α < −1), white frequency noise (range 2, α = 0) and flicker phase noise or white phase noise (range 3, 1 < α < 2).
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