3.3 Optical-to-electrical conversion
3.3.4 Operation at a zero crossing of the AM-PM conversion
The dependence of the AM-PM conversion coefficient on the optical power per pulse has been studied extensively [89, 90]. It varies between positive and negative values and exhibits vanishing points which can be used to reduce the AM-PM conversion. Some experiments have used power stabilisation techniques to demonstrate a reduction of the AM-PM conversion by reducing the RIN of the laser without operating at a vanishing point [92, 93, 94]. By stabilising the optical power of their combs via feedback to the pump current controls, Zhanget al.[94] demonstrated a residual optical-to-electrical phase
noise performance of the11.55 GHzharmonic of−120 dBc Hz−1 at1 Hz, and a residual fractional frequency instability of1.1×10−16at1 sand1.5×10−19 at 1000 s.
Other experiments have focussed on operating close to a vanishing point of α [89, 90, 91]. Zhang et al. have carried out a thorough analysis ofα in
commercial photodiodes operated at 1550 nm [89]. From their characteri-
sation of α and measurements of the long term power fluctuations of their
commercial fibre comb, they calculate values for α. For initial tuning to a
vanishing point, they predict an αof 0.003 radafter three days due to drifts
in the amplitude of their laser by 0.1 percent. Allowing for up to 1 per- cent of amplitude drifts they predict an α of0.03 rad for longer time scales.
These values ofα would correspond to an AM-PM suppression of30 dBand 50 dB respectively, which is a significant improvement compared to normal
photodetection.
locations of the zero crossings of α differ from device to device and strongly
depend on the optical energy per pulse and the bias voltage as well as the temperature. For some applications such as large-scale experiments it would be impractical to characterise every single photodiode and carefully control the environmental conditions to ensure operation near the vanishing point. However, since operation at a vanishing point ofαwas shown to significantly
reduce the AM-PM conversion, almost all state-of-the-art low-noise photonic microwave synthesisers operate at a vanishing point ofα. To avoid degrada-
tion of the performance, the optical power of the comb and the temperature of the photodiode are usually stabilised.
3.4
State of the art performance
Photonic microwave synthesis
In 2013, Fortieret al.[79] demonstrated state-of-the-art low-noise photonic-
microwave synthesis. A combination of several techniques was employed in order to achieve this performance. They compared two individual pho- tonic microwave synthesisers, locked to individual optical references (cavity- stabilised laser), and comprised of individual frequency combs (1 GHzTi:sap-
phire) and optical-to-electrical conversion systems (MUTC photodiodes). Since illumination of the MUTC photodiodes with the pulse train resulted in saturation, the repetition rate of the comb was doubled using free-space MZI pulse interleavers in order to increase the available rf power. The two photodiodes were furthermore operated at nulls of α in order to suppress
AM-PM conversion. This resulted in a phase noise of the 10 GHz harmonic
of −104 dBc Hz−1 at 1 Hz and −177 dBc Hz−1 at 2 MHz. At low offset fre- quencies the phase noise was limited by the optical cavity, and at high offset frequencies the phase noise was limited by an rf amplifier used in the phase noise measurement system. These are the best results in terms of short term frequency stability and close to carrier phase noise for any room-temperature
10 GHz oscillator, comparable to only the very best cryogenic dielectric os-
3.5 Conclusions and summary
Optical-to-electrical conversion
In 2014, Baynes et al. [95] demonstrated the lowest residual phase noise for
optical-to-electrical conversion reported to date. Two MUTC photodiodes were used to generate 10 GHz signals from a common fibre comb. The pho-
todiodes were operated at a vanishing point ofα, and the power of the comb
as well as the temperature of the photodiodes were stabilised. For increased linearity, the repetition rate of the fibre comb was multiplied from 208 MHz
to3.3 GHz using cascaded fibre MZIs. They achieved a residual phase noise
of −131 dBc Hz−1 at 1 Hz and −170 dBc Hz−1 at 10 kHz and a frequency stability of 1.4×10−17 at1 s and 5.5×10−20 at 1000 s. The phase noise at high offset frequencies was limited by uncorrelated noise of an EDFA, and for lower offset frequencies the phase noise was limited by the flicker noise of the photodiodes to −135f−1dBc Hz−1.
3.5
Conclusions and summary
Significant progress has been made over the last decade in the generation of low-noise and high-stability photonic microwave signals. As can be seen in figure 3.4, the residual stability of a frequency comb, acting as an opti- cal frequency divider, is sufficient to support state-of-the-art optical clocks. Furthermore, the optical-to-electrical conversion can also preserve the sta- bility of the best optical clocks. The residual stability of the best optical-to- electrical conversion reported to date by Baynes et al.is more than an order
of magnitude below the performance of the best optical clocks. This opens up a route for the generation of the most-stable microwave signals generated by any source, with long term stabilities orders of magnitude better than those achieved with the best cryogenic sapphire oscillators.
For shorter time-scales, the highest stability frequency sources are pro- vided by ultra-stable optical Fabry Pérot cavities. The phase noise con- tribution of thermal noise limited cavities to a 10 GHz signal is shown in
figure 3.5. Thermal noise limited cavities exhibit flicker frequency noise which has a slope of f−3 in the phase noise PSD. The dashed green lines
Figure 3.4: Fractional frequency stability contributions of several components of a pho- tonic microwave generator. Green: residual frequency stability of a frequency comb, Nicolodi et al. [64]; blue dashed: state-of-the-art optical lattice clock
stability; Nicholson et al. [4]; brown: state-of-the-art ultra-stable silicon
Fabry-Pérot cavity, Kessler et al. [36]; red: residual stability of state-of-the-
art optical-to-electrical conversion (MBW:10 Hz), Baynes et al.[95].
show the phase noise contribution of thermal noise limited cavities with fractional frequency stabilities of 1×10−15 and 1×10−16. With state-of- the-art cavities approaching a thermal noise limited fractional frequency stabilities of 1×10−16 [30], photonic microwave synthesis with phase noise of −124 dBc Hz−1 at 1 Hzappears feasible.
Advances in photonic microwave generation have already produced mi- crowave signals with one of the lowest phase noise levels available from any source, comparable to only the very best sapphire oscillators. The orange line in figure 3.5 shows the phase noise of state-of-the-art photonic microwave generation which was limited by the optical reference cavity at low offset frequencies [61].
As described in this chapter, the noise contribution of frequency combs and the optical-to-electrical conversion have been significantly reduced. At
3.5 Conclusions and summary
Figure 3.5: SSB phase noise contributions of different components in the photonic mi- crowave generation scaled to 10 GHz. Grey: typical contribution of direct photodetection and rf amplification; orange: State-of-the-art photonic mi- crowave generation, Fortier et al. [61]; solid black: state-of-the-art optical-
to-electrical conversion, Baynes et al. [95]; dashed black: flicker noise of the
photodiode used in [95]; dashed green: phase noise of thermal noise limited cavities with fractional stabilities of1×10−15 and1×10−16at1 s, assuming perfect optical frequency division.
high offset frequencies, repetition rate multiplication, operation below the normal shot noise limit and high-power MUTC photodiodes have greatly reduced the noise floor up to −177 dBc Hz−1 [61]. At low offset frequencies,
the use of photodiodes operated at a zero crossing of α have reduced the
phase noise to the flicker noise floor of photodiodes. The best optical-to- electrical conversion reported to date by Bayneset al.approaches this flicker
noise limit between 1 Hz and 10 kHz [95]. An alternative approach for the
reduction of the AM-PM conversion which does not require operation at a zero crossing ofαis based on the balanced optical-microwave phase detector
(BOM-PD). This method will be described in the following chapter.
The next generation of optical cavities have projected thermal noise lim- ited frequency stabilities below1×10−16[36], corresponding to a phase noise
contribution of−124f−3dBc Hz−1. As a result, the phase noise contribution of the cavity surpasses that of the flicker noise of the photodiodes at only a few Hz, making it the main limitation in photonic microwave synthesis at low offset frequencies. The reduction of flicker noise in semiconductor devices remains a difficult technical challenge, yet to be overcome.
4
Low noise optical-to-electrical
conversion using BOM-PDs
As discussed in the previous chapter (section 3.3.3), the AM-PM conversion in the optical-to-electrical conversion process is one of the key challenges in photonic microwave synthesis. It can be overcome by operating at a zero crossing of the AM-PM conversion coefficient (section 3.3.4). In this chap- ter, the balanced optical-microwave phase detector (BOM-PD), which offers an alternative approach for the suppression of AM-PM in the optical-to- electrical conversion, will be described. The BOM-PD was originally devel- oped by Kim and Kärtner at MIT [96]. In the first section of this chapter, the working principle of the balanced optical-microwave phase detector is described, and an analysis of noise processes is given. In the second section, results from free-space BOM-PD set-ups are presented. The third section presents results on low-noise optical-to-electrical conversion using improved fibre BOM-PDs. While designed to suppress the AM-PM conversion, the BOM-PD from Kimet al.only had an AM-PM conversion coefficient similar
to that found in a saturated photodiode [97]. The improved fibre BOM-PD presented here overcomes this limitation by making two key changes to the set-up which resulted in an AM-PM suppression of 60 dB (this work was
4.1
Working principle and noise analysis
The main purpose of the BOM-PD is overcome the limitations caused by excess phase noise due to AM-PM conversion and a low SNR in the photode- tection of an optical pulse train. In order to achieve this, the timing error between the optical pulses and the extracted harmonic of fr is transferred
into an intensity imbalance in the optical domain, before the photodetection. In the following, this process will be described in detail, and the shot noise limit will be derived.
As described in [99], an intensity imbalance which depends on the timing information in the optical domain can be generated by splitting an optical pulse train into two paths and amplitude modulating the pulse trains with a signal from a voltage-controlled oscillator (VCO) that has a 180◦ phase
difference between the two paths. Photodetection of this intensity imbalance with a balanced detector yields an error signal which can phase-lock the VCO to a harmonic of fr. When the PLL is closed, the VCO signal is adjusted
so that its zero crossings overlap with the optical pulses, and the two arms have balanced intensities.
Such a scheme could be realised with a Mach-Zehnder intensity mod- ulator, i.e. a Mach-Zender interferometer with a phase modulator in one arm. However, Kim points out that such a system would not be suitable for long term stable operation due to non-common noise in the two paths of the interferometer caused by temperature and air fluctuations as well as mirror vibrations. This issue can be alleviated by using a Sagnac interfer- ometer with counterpropagating pulse trains so that both beams experience the same fluctuations. As will be discussed in the following, this approach is adopted in the BOM-PD. In the case of the Mach-Zehnder interferometer, the timing error between the VCO and the optical pulse train is trans- ferred into an intensity imbalance between the two interferometer outputs by phase modulating only one of the two beams. When a Sagnac interferom- eter is used, both beams pass through the phase modulator and in order to modulate only one of the counterpropagating beams, a unidirectional phase
4.1 Working principle and noise analysis
Figure 4.1: Schematic of the BOM-PD set-up. EDFA: erbium-doped fibre amplifier; FR: Faraday rotator; QWP: quarter waveplate; VCO: voltage controlled oscillator. All fibres after the polariser are polarisation maintaining.
modulator is employed. Unidirectional phase modulation can be achieved by using a travelling-wave phase modulator in which a microwave driving signal generates an electromagnetic travelling wave that propagates along the electrodes of the modulator. Only the optical beam travelling in the same direction as the microwave signal experiences phase modulation.
In the following, the operation of the BOM-PD will be explained and the shot noise limit and the phase detection sensitivity will be derived (after [99]). A schematic of the BOM-PD set-up can be seen in figure 4.1. The two outputs from the Sagnac interferometer of the BOM-PD can be written as: P1 = Pavgsin2(∆2φ) and P2 = Pavgcos2(∆2φ), where ∆φ is the phase difference between counterpropagating pulses and Pavg is the total average output power of the interferometer. This relation is illustrated in figure 4.2. Taking the pulsed nature of the optical source into consideration, the powers
sin2(∆φ 2) cos2(∆φ 2) ∆φ π 2 Output of the Sagnac in terferometer (a.u.)
Figure 4.2: Dependence of the power of the two outputs of the Sagnac interferometer on the nonreciprocal phase shift between the counterpropagating beams. can also be written as:
P1(t) = Pavg m ∞ X n=0 δ(t−nTr) sin2 ∆φ(t) 2 (4.1) P2(t) = Pavg m ∞ X n=0 δ(t−nTr) cos2 ∆φ(t) 2 (4.2) where Tr is the time between successive pulses, m is the number of pulses
per second and∆φ(t)is the phase difference between the counterpropagating
pulses.
As can be seen from figure 4.2, a phase difference of∆φ= π2 is required
in order to operate the BOM-PD in the balanced condition. To obtain a phase difference between the counterpropagating beams, the symmetry of the Sagnac interferometer has to be broken because the phase modulator only supports unidirectional operation in the microwave region—a DC volt- age supplied to the modulator would result in the same phase shift for both beams—and can therefore not be used for this purpose. In the BOM-PD the symmetry of the Sagnac interferometer is broken by a section consisting of two oppositely oriented 45◦ Faraday rotators and a quarter wave plate.
While everywhere else in the Sagnac interferometer the counterpropagating beams have the same polarisation, between the two Faraday rotators the polarisations are orthogonal. This is illustrated in figure 4.3. By placing
4.1 Working principle and noise analysis
Figure 4.3: Diagram showing how the non-reciprocal phase shift of π
2 is obtained. The red and the blue arrows illustrate the polarisations of the counterpropagating beams. Due to the Faraday rotators, the polarisations are orthogonal at the quarter wave plate. Aligning the fast and slow wave plate axes with the polarisations of the beams therefore yields a non-reciprocal phase shift of π
2.
a quarter wave plate between the Faraday rotators, and aligning the slow and the fast axes of the waveplate with the polarisation axes of the coun- terpropagating beams, the required non-reciprocal phase shift of ∆φ = π2 is
obtained.
This intensity balanced condition of the BOM-PD is key to its work- ing principle because balanced detection of the two outputs of the Sagnac interferometer can be used in order to suppress AM-PM conversion. For the correct operation of the BOM-PD in the balanced condition, the wave- plate has to be correctly aligned with respect to the polarisation axes of the counterpropagating beams. In addition to that, different losses in the non-common paths of the two output arms of the interferometer (P1 enters the balanced detector right after the 50/50 coupler while P2 passes through the circulator before entering the balanced detector) have to be accounted for. Otherwise the BOM-PD is not operated in the balanced condition and the AM-PM suppression will degrade.
In order to phase lock the VCO to a harmonic of the repetition rate, an error signal is derived from the balanced detector by subtracting the average photocurrents. The time dependent phase difference between the counterpropagating beams can be written as
∆φ(t) = Φ0sin (2πfVCOt+θe) +
π
Figure 4.4: Illustration of the phase relation between the optical pulse train and the VCO microwave signal. θeis the phase error between the zero crossing of the
microwave signal and the optical pulse train. Φ0is the amplitude of the phase modulation signal from the VCO.
where fVCO is the frequency of the VCO, Φ0 is the amplitude of the phase modulation signal from the VCO andθe is the phase error between the VCO
signal and the optical pulse train as illustrated in figure 4.4. Substituting for ∆φ(t) in equation 4.1 and 4.2 yields
P1(t) = Pavg m ∞ X n=0 δ(t−nTr) sin2 1 2 Φ0sin (2πfVCOt+θe) + π 2 , (4.4) P2(t) = Pavg m ∞ X n=0 δ(t−nTr) cos2 1 2 Φ0sin (2πfVCOt+θe) + π 2 . (4.5)
When the PLL is closed, the VCO frequency is given by a harmonic N of
the repetition rate, i.e. fVCO =N fr, and the average photocurrents can be
written as hI1i=RPavgsin2 1 2(Φ0sinθe+ π 2) , (4.6) hI2i=RPavgcos2 1 2(Φ0sinθe+ π 2) , (4.7)
where R is the responsivity of the photodetector. The output current hIdi
from the balanced photodetector is
hIdi=hI1i − hI2i=−RPavgcos [Φ0sinθe+
π
4.1 Working principle and noise analysis
For θe <<1this can be approximated as
hIdi ≈RPavgΦ0θe, (4.9)
The error signal derived from the average photocurrents is therefore pro- portional to the phase error in the optical domain θe. The phase detection
sensitivity Kd of the BOM-PD is given by
Kd=
hIdi
θe
=RPavgΦ0. (4.10)
Any amplitude noise on the error signal will be converted into phase