^{1,a)}, Nikola Bucalovic

^{1}, Lionel Tombez

^{1}, Vladimir Dolgovskiy

^{1}, Christian Schori

^{1}, Gianni Di Domenico

^{1}, Michele Zaffalon

^{2}and Pierre Thomann

^{1}

### Abstract

We describe a radio-frequency (RF) discriminator, or frequency-to-voltage converter, based on a voltage-controlled oscillator phase-locked to the signal under test, which has been developed to analyze the frequency noise properties of an RF signal, e.g., a heterodyne optical beat signal between two lasers or between a laser and an optical frequency comb. We present a detailed characterization of the properties of this discriminator and we compare it to three other commercially available discriminators. Owing to its large linear frequency range of 7 MHz, its bandwidth of 200 kHz and its noise floor below 0.01 Hz^{2}/Hz in a significant part of the spectrum, our frequency discriminator is able to fully characterize the frequency noise of a beat signal with a linewidth ranging from a couple of megahertz down to a few hertz. As an example of application, we present measurements of the frequency noise of the carrier envelope offset beat in a low-noise optical frequency comb.

The authors would like to thank R. Scholten for fruitful discussions about the analog PLL discriminator and Niels Haandbaek for the HF2PLL. The authors are also very grateful to Professor Ursula Keller, ETH Zurich, for making available the optical frequency comb developed in her Laboratory and used here for assessing the frequency discriminators. This work was financed by the Swiss National Science Foundation (SNSF) and by the Swiss Confederation Program Nano-Tera.ch, scientifically evaluated by the SNSF.

I. INTRODUCTION

II. FREQUENCY DISCRIMINATORS

A. Description of the RF frequency discriminators

1. Analog phase-locked loop discriminator

2. Miteq RF discriminator

3. Numerical phase-locked loop HF2PLL discriminator

4. Digital phase detector DXD200

B. Characterization of the frequency discriminators

1. Sensitivity and bandwidth

2. Frequency range

3. Noise floor

4. AM/AN cross-sensitivity

C. Comparison of the frequency discriminators

III. EXAMPLE OF APPLICATION OF THE FREQUENCY DISCRIMINATORS

IV. DISCUSSION AND CONCLUSION

### Key Topics

- Discriminators
- 156.0
- Linewidths
- 27.0
- Position sensitive detectors
- 20.0
- 1/f noise
- 17.0
- Frequency measurement
- 15.0

## Figures

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.

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).

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.

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 Δ*f* _{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 } (Δ*f* _{lin} = ±10V/*D* _{ν}) and is shown here for two particular cases (*D* _{ ν } = 100 *μ*V/Hz and *D* _{ ν } = 5 *μ*V/Hz) for illustration.

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 Δ*f* _{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 } (Δ*f* _{lin} = ±10V/*D* _{ν}) and is shown here for two particular cases (*D* _{ ν } = 100 *μ*V/Hz and *D* _{ ν } = 5 *μ*V/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 *μ*V/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}).

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 *μ*V/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.

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 Δ*f* = 100 kHz and a noise floor *S* _{min} = 0.01 Hz^{2}/Hz; the phase discriminator has a range Δ*ϕ* = 2π and a noise floor *S* _{ ϕ } _{min} = 10^{−9} rad^{2}/Hz. The dashed line represents the *β*-separation line (*S* _{δν} = (8Ln(2)/π^{2}) · *f*) in the frequency noise spectrum and its correspondent (*S* _{ϕ} = (8Ln(2)/π^{2})/*f*) in the phase noise spectrum.

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 Δ*f* = 100 kHz and a noise floor *S* _{min} = 0.01 Hz^{2}/Hz; the phase discriminator has a range Δ*ϕ* = 2π and a noise floor *S* _{ ϕ } _{min} = 10^{−9} rad^{2}/Hz. The dashed line represents the *β*-separation line (*S* _{δν} = (8Ln(2)/π^{2}) · *f*) in the frequency noise spectrum and its correspondent (*S* _{ϕ} = (8Ln(2)/π^{2})/*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 *μ*V/Hz and *D* _{ ν } = 10 mV/Hz. The dashed line represents the *β*-separation line *S* _{δν}(*f*) = (8Ln(2)/π^{2}) · *f*.^{9}

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 *μ*V/Hz and *D* _{ ν } = 10 mV/Hz. The dashed line represents the *β*-separation line *S* _{δν}(*f*) = (8Ln(2)/π^{2}) · *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 *μ*V/Hz. The *β*-separation line that is relevant for the determination of the CEO-beat linewidth is also shown as a dashed line.^{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 *μ*V/Hz. The *β*-separation line that is relevant for the determination of the CEO-beat linewidth is also shown as a dashed line.^{9}

## Tables

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).

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).

Article metrics loading...

Full text loading...

Commenting has been disabled for this content