1. Introduction

One common use of ultrastable laser sources is the optical generation and distribution of high fidelity and stable microwave signals. Many methods have been developed to generate a stable RF or microwave signal from optical laser based sources. A few of these techniques include broadband optical frequency comb laser stabilization [1–5], laser stabilization to external cavities by Pound-Drever-Hall methods [6], and locking to atomic or molecular transitions [7]. The goal of these methods is to stabilize the output frequency of the laser, and optically photomix pairs of highly stabilized lasers to generate a stable radio frequency (RF) beat note.

Applications for these stabilized optical RF tone generating systems include molecular spectroscopy [7], atomic spectroscopy [8], and optical clock distribution [9–13]. Furthermore the synchronization of clock and frequency standards across large scale laboratory facilities [14], communication networks, and/or radio-astronomy systems [15] are a natural extension for these stabilized optical sources. For example at the Atacama Large Millimeter Array (ALMA), ultra stable signal distribution is required across the 18 km baseline between the movable antenna array. The optically distributed reference clock’s timing stability needs to be better than 5 × 10⁻¹² (Allan deviation) over 300 seconds or the scientific accuracy of the radio-astronomy data collected will be degraded [15].

Stabilized optical frequency generation and distribution can also be applied to more military-oriented microwave photonic systems [16], fiber optic acoustic sensors [17], and radar systems [18]. For high performance Doppler radar systems, short term local oscillator (LO) frequency instabilities mask the presence of small target return signals. The RF oscillator linewidth is a direct limit on system sensitivity. Narrower, more stable RF oscillator linewidths make finer velocity resolution possible, enabling direct detection of slower moving targets at higher modulation rates and at longer ranges [19]. The minimum detectable velocity of the target is the primary determinate of the stability requirement of the radar system’s LO. Slow moving targets (walking objects) produce small shifts in Doppler frequency (∼ 1 Hz). Therefore phase noise reduction – close to the carrier (≪ 1 Hz) – is required for the detection of the slowest moving targets [20]. By combining the above technologies the optical signal distribution of precision RF signals between spatially diverse bistatic and multistatic radars systems, could enhance system resolution and reduce the need for multiple precision frequency references at each installation.

An alternative method of optically generating a stable RF or microwave signal is to offset frequency lock or offset phase lock two stabilized lasers to an external RF reference standard. Here the frequency of one laser is photomixed with the second laser. Depending on whether a frequency-to-voltage converter (frequency locking) or a phase-to-voltage converter (phase locking) is used, the resultant detected error signal between the laser system beat note and the RF source is used to stabilize the frequency or phase difference between the two lasers. If the pair of lasers are stable enough to be brought into offset phase lock, the deviation in the laser system difference RF frequency (compared to the RF source) is very small and there is little to no frequency slippage or excess linewidth. Using a stabilized RF source (such as a crystal oscillator) or an RF source phase locked to a crystal or atomic frequency standard, stabilizes the optically generated RF beat note even further.

In the paper we first describe linewidth measurements of two highly stable phase locked heterodyned laser signals which are used to generate a stable microwave RF source. For offset frequencies from 65 kHz to 31.7 GHz, and when all the reference oscillators share a common time base, the excess instantaneous linewidth is measured to be < 22.8 μHz. Next we allow the 5 MHz RF reference source to free-run with respect to the instrumentation and the second RF source used for downconverting. When the laser system is phase locked to a free-running RF source, we measure a linewidth of ∼36 μHz over the 651 minute measurement cycle. Then we describe the short to medium term frequency stability measurements of the Allan deviation of fractional frequency fluctuations of this optically generated RF source. At 300 seconds, the Allan deviation of our heterodyned laser system is ∼3 × 10⁻¹⁴, well below the stability requirements of the ALMA [15]. The wavelength of this laser system (at 1.319 μm) is within the telecommunications band and is capable of transporting the stabilized RF tone via length stabilized optical fiber.

2. Linewidth measurements

Depending on the observation time, the frequency stability of an oscillator can be characterized in either the time or frequency domains [21–23]. For short observation times, measurements of the oscillator linewidth, phase noise, and power- or phase-spectral density are used to characterize the oscillator. These measures are indicators of the spectrally driven perturbations of the fundamental frequency. For Doppler radar systems, the RF spectrum is a good qualitative measure of the local oscillator, so we concentrate on linewidth measurements in this section [20, 24]. For longer time intervals, Allan deviation of the fractional frequency fluctuation analysis is typically performed, as it is a useful measure of the longer term time interval stability of a clock or oscillator. Allan deviation measurements are presented in section III.

Since the invention of the laser, there have been many techniques to stabilize the optical frequency of a laser source. Also during this time period, researchers have been optically mixing (heterodyning) laser systems together to generate RF beat frequencies [6]. It has been found that by using two extremely narrow linewidth, ultra-stable lasers, optical generation of very high accuracy RF frequencies results. Extremely wide RF frequency tuning bandwidths from ∼100 Hz up to 100 GHz are readily demonstrated [25].

However, to reach below sub-mHz laser generated RF beat frequency linewidths, three approaches are generally used: these include (but are not limited to) stabilizing the laser cavity to a ultralow expansion (ULE) glass Fabry–Perot optical cavity; locking to an atomic or molecular transition; or offset frequency or phase locking two lasers to a microwave source [26]. All of these techniques effectively use either an external cavity (length) or an external RF frequency standard (time) to stabilize the laser frequency via the speed of light (c = l/t) [27]. We concentrate here on heterodyned laser phase locking because the RF frequency offset can be controlled by a tunable external RF source and a phase discriminator. The tunability of the generated optical beat note is limited only by the microwave components available and the maximum frequency separation of the laser sources. The frequency stability of this system is limited by the stability of the laser pair, the bandwidth and gain of the phase locking electronics, and the stability of the microwave source.

A review of the current literature has found the following reports of sub-mHz laser generated RF beat frequency linewidths for reference. We divide them by locking technique: external optical cavity locking and phase locking to an RF frequency standard. Utilizing a Fabry-Perot optical cavity (FPOC) and using the Pound Drever Hall (PDH) technique [27] to stabilize the locking has repeatedly been shown to be an effective technique for generating an RF beat frequency. The two reports of the narrowest beat frequency linewidths are in [28], where W. C. Swann et al phase locked two CW fiber lasers to a common ULE glass optical cavity. Then a femtosecond Er⁺ doped fiber laser comb was phased locked to the stabilized reference lasers, which produced an 11 MHz beat note with a 1.2 mHz linewidth. In [29], M. Hyodo et al used a Nd:YVO₄ microchip laser actively PDH locked to a FPOC. They measured an upper limit on the excess spectral intensity noise of $16.5\hspace{0.17em}\text{mHz}/\sqrt{\text{Hz}}$. From this value they calculate an estimated 860 μHz beat note linewidth centered at 1 kHz.

Since the offset phase locking of a pair of semiconductor lasers directly can present significant challenges, in [30] Fan et al used an external frequency discriminator to assist in phase locking two grating tuned external cavity 830 nm GaAs diode lasers. At offset locking frequencies of 711 MHz and 1.31 GHz, they mixed down their optically generated RF signal with a second signal generator to 50 kHz and used an HP 3562A dynamic signal analyzer to measure a linewidth of < 1 mHz. In [31], Jun Ye and J. L. Hall phase locked two 1.064 μm Nd:YAG lasers 5 GHz apart, using both a PZT transducer for small frequency control and an external acousto-optic modulator to control fast frequency fluctuations. With all of their RF oscillators sharing a common time base, and using a fast Fourier transform analyzer they report a minimum instantaneous linewidth of their beat note of order 0.95 mHz. In [8], Cheng et al phase locked two external cavity 884 nm diode lasers generating a 85 MHz beat note. With all three RF synthesizer time bases locked together, a fast Fourier transform analyzer was used to measured a beat note instantaneous linewidth of ≤ 0.5 mHz. Finally in [32], Williams et al phase locked a pair of 1.319 μm Nd:YAG nonplanar ring oscillator (NPRO) lasers to a RF synthesizer and demonstrated tuning of the phase locked beat note from 6 to 34 GHz. With both RF synthesizer time bases locked together, they used an HP 3562A dynamic signal analyzer to measure an instantaneous linewidth of ≤ 0.5 mHz at 30 GHz.

One common feature of the above reports is that in order to make sub-mHz linewidth measurements, the stability of the laser locking mechanism - either phase lock or PDH locking - needs to be robust and able to maintain the laser system in question locked together for an extended measurement time. The Nyquist sampling theorem requires that for sub-mHz frequency resolution, kilosecond or longer sampling times of the Fourier transform time record are required. For 1 μHz frequency resolution, megasecond (11.5 day) sampling times of the Fourier transform time record would be required. At extremely long time record lengths, the timing stability of the master clock of the FFT analyzer can become an issue.

In a previous publications where we characterized a pair of 1.319 μm Nd:YAG NPRO lasers [33, 34] we measured the single side band phase noise in excess of the RF source of the optical heterodyned laser signal. We found that for Fourier frequencies offset 5 kHz to 1 MHz from the carrier, unsuppressed laser RIN dominates the phase noise spectrum. For frequencies beyond 20 MHz offset, the RF source and measurement system noise floor were dominant. For close in offset Fourier frequencies within 10 Hz of the carrier, the absolute phase noise of the low phase noise HP 8662A RF source dominated the phased locked optically generated RF signal. Seeking to further understand the phase locked characteristics of this laser system, for close in Fourier components (≪ 1 Hz) of the optically generated beat note, here we investigate the phased locked linewidth as a measure of spectral purity of the heterodyned laser system. We report on the instantaneous beat note linewidth measurements in section 2.3 and then in section 2.4 we unlock the 5 MHz oscillator time bases. The free-running 5 MHz oscillator comparison enables the measurement of an absolute excess RF beat note linewidth, in the configuration of an absolute microwave phase noise measurement system.

2.1. Free-running laser system

Shown in Fig. 1 is a typical measurement of the free-running (not phase-locked) beat frequency signal created by heterodyning the outputs of two Nd:YAG 1.319 μm NPRO lasers. The optical outputs of each laser are combined by a polarization–maintaining fiber optic coupler and detected by an RF photodiode. The measured RF spectrum of this heterodyned laser system is the cross-correlation of each laser’s spectral line shape with the other. For two lasers, with Lorentzian line shapes, the convolution of the two Lorentzian line shapes results in a Lorentzian line shape with a 3 dB down bandwidth or full width half maximum (FWHM) of Δν_L = Δν_L1 + Δν_L2. For two lasers of equal line width, the measured cross-correlation linewidth is Δν_measured = 2 Δν_L [35]. The measured 3 dB cross correlation bandwidth of the free-running laser system, shown in Fig. 1, is ∼8.0± 0.5 kHz which corresponds to a 4–5 kHz linewidth for each individual laser. This linewidth measurement matches the manufacturer’s specification as well as previous reports on this laser configuration [36,37].

 

Fig. 1 Typical free-running beat note between two NPRO 1.319 μm Nd:YAG lasers at a frequency separation of 48 MHz. The full width half maximum is ∼8.0± 0.5 kHz as measured by an Agilent E4888A Series RF spectrum analyzer.

Download Full Size | PDF

2.2. PLL details

One can improve on the free-running linewidth of this laser system by using a phase locked loop (PLL). The PLL provides a feedback signal to maintain a constant phase coherence between the two heterodyned laser sources at a fixed frequency difference. The result is tight phase coherence between the two oscillating laser sources.

A schematic diagram illustrating the experimental heterodyned laser PLL configuration is shown in Fig. 2. This is a implementation of a classic phase locked loop system and is comprised of three optical/RF/electrical component sections: 1) the phase detector which compares the phase of the input signal to that of the output of the voltage controlled oscillator (VCO); 2) the voltage controlled oscillators input voltage changes the frequency of the oscillators output so as to minimize the phase difference between the input signal and the output signal; and 3) the loop filter section which filters the phase detectors output and its signal is injected into the VCO. The optical input to the phase detector section of the PLL is converted to a RF signal by the PLL RF photodiode. This diodes output is fed into the RF port of a double–balanced microwave mixer. The LO port is powered by the electrical output of an RF microwave source, which sets the offset frequency to which the laser system will phase lock. The difference frequency and phase between the LO and RF port exits the intermediate frequency (IF) port of the mixer. The output of the IF port also includes the sum frequency of the LO and RF signals. The voltage controlled oscillator is made up of the heterodyned optical outputs of the two lasers. The lasers are combined by a polarization maintaining fiber optic coupler with one optical output sent to the frequency or stability measuring system. The second optical output of the laser system is again split by a fiber optic coupler with one output attenuated and detected by an RF photodiode which downconverts the optical amplitude modulation tone to an RF electrical signal. This output feeds the phase detector input. The second output is also attenuated and detected by a second photodetector and this RF signal is used to monitor the state of the phased locked RF tone. The PLL filter section input is filtered by a low pass filter set to only pass the difference frequency (LO - RF) and attenuate the sum frequency (LO + RF) outputs of the RF mixer. The phase difference signal is then filtered by a high gain active integrator filter. The output of the PLL filter circuit is injected into the frequency tune port of laser #2. This implements the VCO function, by adjusting the laser system difference frequency and phase. The integration function of the VCO together with the active integrator in the PLL filter together form a two pole second-order phase locking circuit. When the frequency and phase difference between the RF source and the laser difference frequency RF tone is zero, the two lasers are phase locked together. The PLL maintains a constant RF frequency offset and phase coherence between the two laser outputs. When the laser output powers are balanced by each laser’s variable optical attenuator (VOA), the optical output is sinusoidal (at 100% modulation depth), free of harmonics, and frequency spurs.

 

Fig. 2 Schematic diagram of the optical offset phase locked loop heterodyned laser system.

Download Full Size | PDF

The general optical system design [of Fig. 2] and the phase lock loop filter circuitry are unchanged throughout the entire phase locked frequency range. For these measurements the PLL filter circuit has the following parameters: unity gain at 36.5 kHz, a PLL natural frequency of ω_n/2π ∼ 110 kHz, and the PLL dampening coefficient is τ ∼ 1.6. The PLL circuit parameters for these measurements were chosen to insure an optimized balance between PLL phase noise suppression and nonlinear laser voltage controlled oscillator (VCO) response. To insure long term phase lock stability an Analog Devices OPA27E operational amplifier is used because it can drive its high open circuit gain below 10 Hz. The input RF power on both the RF and LO ports of the PLL mixer [in Fig. 2] were carefully balanced to suppress sideband noise power at frequencies away from the central RF peak. The optical fiber and optical couplers used are wavelength selected for 1.319 μm operation. RF sources, RF mixers, and photodetectors with sufficient bandwidth are changed as needed to accommodate the frequency bandwidth of the desired RF beat note signal. Further details of the optical system can be found in [33].

Using an RF spectrum analyzer the detected optical beat note signal is shown in Fig. 3(a) at 65 kHz and Fig. 3(b) at 31.7 GHz. Also shown is the RF spectrum of the RF sources [Fig. 3(a) HP8662A and Fig. 3(b) Agilent 8564E] used to phase lock the laser system. In each case, the optically derived RF beat note is nearly identical to that of the RF source. The major difference across frequencies is that the spectrum analyzer noise floor increases (and dynamic range decreases) 20 dB for the high frequency range measurements.

 

Fig. 3 Typical RF spectrum of the optically generated RF beat note of the heterodyned laser system at (a) 65 kHz frequency offset. The beat note from the optical phase locked loop (OPLL) is compared to the RF output of an HP8662A RF source at 65 kHz which is also used to phase lock the laser system. (b) The laser system beat note at 31.7 GHz frequency offset. The beat note from the OPLL is compared to the RF output of an Agilent E8257D RF source at 31.7 GHz, which is also used to phase lock the laser system. The measurements are made with an Agilent 8564E millimeter-wave spectrum analyzer. In both Figs. 3(a) and 3(b), the measured RF beat note line width is limited by the minimum 1 Hz resolution bandwidth of the spectrum analyzer (sweep time 3 s) with 100× averages. In each case, the photocurrent (1mA) and RF oscillator power are set to yield −22 dBm of RF output power. Note: every other (HP8662A)/ every third (OPLL) data point in these Figs. is plotted for clarity.

Download Full Size | PDF

Despite their utility, direct spectral analysis of the optically generated RF tone by an RF spectrum analyzer is of limited utility due to the limited dynamic range (100 to 130 dB) and the insufficiently narrow bandwidth of the analysis IF filter. The minimum resolution bandwidth of a typical modern spectrum analyzer’s digital IF filter is 1 Hz. This IF bandwidth is too wide to give a detailed view of the generated RF beat note close in to the carrier frequency [38].

2.3. FFT analyzer and downconversion

To get a better understanding of the optically generated RF beat note linewidth a narrower IF filter bandwidth is required, which is available from a fast Fourier transform (FFT) or dynamic signal analyzer (DSA). The FFT/dynamic signal analyzer digitizes the incoming sinusoidal signal at a high sampling rate. The resulting time record is transformed into a frequency spectrum by a FFT algorithm. The resulting frequency spectrum shows the frequency components of the input signal. For our purposes, the longer the sampled time record, the narrower the calculated frequency bin resolution bandwidth (RBW). The limitation of the FFT/DSA technique is the speed and memory depth of the analog to digital converters (ADC). The sampling rate of the ADC typically sets the upper frequency range measured to less than ≤105 kHz. We used an HP 3562A Dynamic Signal Analyzer with an upper frequency limit of 100 kHz and a minimum zoomed span of 20.48 mHz. At the minimum analyzer span of 20.48 mHz there are 800 resolution bins, which yields a minimum BW per bin of 25.6 μHz. Previous measurements of the RF beat note, for a similar laser system, indicated that the 3 dB linewidth was < 0.5 mHz [32].

To be within the frequency measurement range of the DSA, the laser offset frequency was tuned to 65 kHz and phase locked to an HP8662A low phase noise RF source. Schematic diagrams of the measurement are shown in Figs. 4(a) and 4(c) for the direct 65 kHz RF and direct 65 kHz optical PLL (OPLL) linewidth measurements. The external 10 MHz (here a Rb stabilized 10 MHz quartz oscillator) frequency reference is required to maintain a single time base for these measurements. By utilizing a common external 10 MHz reference, the timebases of the RF source and the analyzer are phase locked. This has the effect of allowing the instantaneous linewidth to be measured. Figure 5 shows the center 2 mHz (80 bins) of a 20.48 mHz span (800 bins) of the power spectrum from the FFT dynamic signal analyzer. Due to the sinusoidal periodicity of the OPLL signal, a uniform (or rectangular) windowing function is used for these FFT analyzer measurements. Power spectra of both the optical PLL beat note and of the HP8662A microwave source are shown for comparison. The two 65 kHz sources are difficult to distinguish, demonstrating the high fidelity of the optical PLL beat note and that the OPLL laser system adds little to no excess bandwidth to the detected linewidth. It is noteworthy that the first resolvable bins on either side of the central peak are −45 dB down from the central peak.

 

Fig. 4 Schematic diagrams of the variations of the measurement systems designed to utilize the maximum measurable frequency (100 kHz) of the HP3562A dynamic signal analyzer. (a) Direct RF source measurement diagram. (b) RF source mixed down to 65 kHz, via a second RF source, measurement diagram. (c) Direct optical phase locked loop laser beat note measurement diagram. (d) Optical phase locked loop laser beat note mixed down to 65 kHz, via a second RF source, measurement diagram.

Download Full Size | PDF

 

Fig. 5 Comparison of the 65 kHz beat note instantaneous linewidth of the optical phase locked loop (OPLL) laser system to that of an HP8662A RF source at 65 kHz. Measured with an HP3562A dynamic signal analyzer. 651 min. per sweep with 4x sweep averaging.

Download Full Size | PDF

To measure the instantaneous linewidth of laser beat note frequency at 31.70 GHz, the laser offset frequency was tuned to 31.70 GHz and phase locked to an Agilent E8257D low phase noise RF source. An RF mixer and a second Agilent E8257D RF source set to 31.700065 GHz were used to downconvert the 31.70 GHz signal to a difference frequency of 65 kHz. Again, the external 10 MHz (the Rb stabilized 10 MHz quartz oscillator) frequency reference is necessary to maintain time base synchronization between the two RF sources and the DSA for these measurements. Schematic diagrams of the measurement are shown in Fig. 4(b) for the downconverted 31.7 GHz RF and Fig. 4(d) downconverted 31.7 GHz optical PLL linewidth measurements. The measured downconverted optical signal and the RF comparison signals are both shown in Fig. 6. Again, the RF source and optically generated 31.7 GHz tones are difficult to distinguish, demonstrating the high fidelity of the optical PLL beat note. At this upper frequency, the first resolvable bins on either side of the central peak are −21 dB down from the central peak.

 

Fig. 6 Comparison of the 31.7 GHz beat note (mixed down to 65kHz) instantaneous linewidth of the optical phase locked loop (OPLL) laser system to that of a Agilent E8257D RF source at 31.7 GHz + 65 kHz. The 65 kHz difference frequency is measured with an HP3562A dynamic signal analyzer. 651 min. per sweep with 4x sweep averaging.

Download Full Size | PDF

There are several interesting features to observe about the instantaneous linewidth measurements shown in Figs. 5 and 6 at the two different RF frequencies (65 kHz and 31.7 GHz). At both frequencies, the measured instantaneous linewidth of the optically generated tone is difficult to distinguish from that of the RF source intantaneous linewidth, as the curves overlap. At 31.7 GHz the noise pedestal is measured to be 15 dB higher than for 65 kHz. This is due to the limited dynamic range (80 dB) of the dynamic signal analyzer not being able to correctly measure the 65 kHz noise floor. The noise pedestal for 65 kHz in Fig. 3(a) is −100 dB below the carrier peak, and at 31.7 GHz −75 dB below the carrier peak in Fig. 3(b), again the RF spectrum analyzer has a limited dynamic range (the on screen dynamic range for the Agilent 8564E is limited to ≤ 100 dB). Noise power away from the carrier peak is expected to increase with increasing frequency due to phase noise multiplication at higher frequencies [39].

Lastly, the instantaneous linewidth of the optical beat note is still unresolved at the minimum resolution bandwidth of the dynamic signal analyzer, 25.6 μHz. The bin width resolution is limited by the finite sampling time record of the FFT analyzer. At this minimum resolution bandwidth, the FFT analyzer has a 651 minute long record length. Four time averages are displayed in Figs. 5 and 6, which indicates that the heterodyned laser system was continuously phase locked to the RF source in excess of 43.5 hours of sampling time (∼60 hours total elapsed time), after which the system was manually unlocked. FFT theory [40] tells us that due to the finite number of samples in the FFT process, a window weighting function is typically required to minimize spectral leakage (i.e. reassignment of spectral content into the incorrect frequency bin due to sampling discontinuity at the periodic boundary of the time record). Due to the high purity, stability, and tunability of our generated and detected waveforms, the sinusoidal signal was tuned to the middle of the time record window, allowing us to use a uniform (rectangular) sampling window.

The advantage of the uniform windowing function is that it has the narrowest theoretical bin resolution for a windowing function. The disadvantage of the uniform window is that it can suffers from high side lobe levels, the first of which is 13 dB down from the main peak, for time record content that is not centered in the time record window [40]. As there are no periodic sidelobes in Figs. 5 and 6, this indicates that the detected spectral content is centered in the time record and all of the signal is contained within the central bin. FFT theory also calculates to expect spectral leakage of 6.0 dB down from the central peak at 1.21 bins. Again, as the first bins on either side of the central peak are −45 dB down from the central peak in Fig. 5, and −21 dB down from the central peak in Fig. 6, this again indicates that the detected spectral content is contained within the central bin. For a uniform window, the theoretical 3 dB bandwidth is 0.89 bins wide. Therefore, the heterodyned laser system and the RF sources both have a 3 dB (or full width half maximum (FWHM)) linewidths of < 22.8 μHz (0.89 × 25.6 μHz minimum bin width) [40,41].

Comparing the instantaneous linewidths of Figs. 5 and 6 to the instantaneous linewidth in Fig. 2(b) of [32], shows a marked reduction in the measured 3 dB bandwidth of the laser difference frequency beat note, from 0.5 mHz to ≤ 22.8 μHz. We attribute this 20x reduction in instantaneous beat note linewidth to three likely causes. First, the new active phase lock loop filter circuit increases the low frequency voltage gain up to 20 times at 1 Hz, while maintaining the other phase locked loop operating parameters. This improved loop filter also increases the stability of the heterodyned laser system and increases the tightness of the PLL. The second improvement relates to the fact that the OPLL system closely follows the RF source linewidth and close in phase noise. Known low phase noise RF sources (HP8662A and Agilent E8257D) were used for this report. Williams’s previous report on this laser system used a harmonic frequency mixer to multiply the RF source (an HP 8340B) [42] a factor of eight times (8 × 3.751 GHz = 30.008 GHz) up to 30 GHz. Frequency multiplying n times increases the noise side band power level by n² because there is more power in the noise pedestal relative to the carrier frequency. One result of this multiplication is to increase oscillator linewidth [39]. This linewidth increase was verified by directly measuring, with the dynamic signal analyzer, the multiplied up linewidth of an HP8340B to 31.698 GHz and (downconverting with a Agilent E8257D tuned 65 kHz away) to get an instantaneous linewidth measurement of ∼ 153.6 μHz for the multiplied up RF source. Finally, the time base reference used for these measurements, a Rb stabilized quartz frequency standard, has improved medium term frequency stability over that of the quartz crystal in a standard RF source, that is beyond 10 ks (see section 3.2). This means that the time base clock signal of the dynamic signal analyzer is more accurately timed, and the resulting FFT of the data yields better frequency registry/overlap during the time record collection.

2.4. Free-running RF reference source linewidth

Up to this point, we have attempted to determine if the PLL laser system adds additional frequency noise in excess of the linewidth of the original RF tone. Figures 5 and 6 show little additional linewidth added to the RF sources, when all the RF sources and instrumentation are phase locked to a common 10 MHz external reference oscillator. This shared external reference oscillator maintains a fixed phase coherence between the RF sources and the RF spectrum or dynamic signal analyzers. This has the effect of measuring the instantaneous linewidth of the RF sources at a specific phase, set by the external reference. By removing the external 10 MHz reference, the laser system (phase-locked to an RF source) is allowed to free-run. This is the typical configuration used in a high accuracy optically distributed precision time transfer system.

Figure 7 shows the schematic diagrams used to measure the free-running linewidth of the heterodyned tone between two 5 MHz oven controlled crystal oscillators in Fig 7(a), and with the OPLL inserted in Fig. 7(b). This measurement is used to determine the excess linewidth due to the optical heterodyne laser system. The two 5 MHz reference oscillators are a Frequency and Time Systems, Inc. model 1050A and a Frequency Electronics, Inc. model 1150A. The system clock used to stabilize the DSA for this measurement is the 10 MHz output of a Stanford Research Systems model FS725 Rubidium frequency standard. Specifications and stability measurements of these frequency references are presented in detail in sections 3.2 and 3.3. In practice the two reference oscillators are offset in frequency by 1.83 Hz and are downconverted by an RF mixer. At a 1.83 Hz difference frequency the heterodyned signal is less than the upper 100 kHz measurement frequency limit of the HP 3562A dynamic signal analyzer. The upper side band (at the two oscillators sum frequency) is filtered out by a 1.9 MHz 8-pole RC filter and the difference frequency signal (at ∼1.83 Hz) is passed through to a 10 Hz low pass filter, which is at ∼5× the beat frequency.

 

Fig. 7 Schematic diagram of the free-running heterodyned two oscillator linewidth measurement system. a) with two separate reference oscillators. b) With the optical phase locked loop laser system inserted between one of the reference oscillators and the heterodyning RF mixer. c) Comparison of the linewidth of the 5 MHz beat note (mixed down to 1.83 Hz) of the optical phase locked loop (OPLL) laser system phase locked to a FTS 1050A OCXO 5 MHz source, and a FEI 1150A OCXO 5 MHz source. The downconverted linewidth of the 5 MHz FTS 1050A and the FEI1150A are shown in the background for reference. In all cases a 10 MHz output of a SRS Rb frequency reference is the 10 MHz reference for the HP3562A dynamic signal analyzer. The 1.83 Hz difference frequency is measured with an HP3562A dynamic signal analyzer. One 651 min. sweep shown.

Download Full Size | PDF

Figure 7(c) shows the center 2 mHz (80 bins) of a 20.48 mHz span (800 bins) of the power spectrum from the FFT dynamic signal analyzer. Due to the sinusoidal periodicity of the OPLL signal, a uniform (or rectangular) windowing function is used for these FFT analyzer measurements. The measured downconverted power spectra of both the optical PLL beat note (locked to the FTS 1050A oscillator) and the two free-running reference oscillators are shown for comparison, for one 651 minute sweep. Both signals have their peak scaled to 0 dB for ease of comparison. Within the central ∼800 μHz of the beat note’s main peak, the two 5 MHz oscillators are difficult to distinguish from the optical PLL signal. This again demonstrates the high fidelity of the optical PLL beat note. Outside the central ∼800 μHz, the noise floor of the DSA raises the noise pedestal of both the RF sources and the OPLL signals. The noise pedestal of the optical signal is 5 dB higher that that of the RF sources because of the lower RF power delivered to the mixer by the photodetector. That is the noise floor gets scaled up 5 dB. The free-running 6 dB down linewidth is ∼4 bins wide or ∼102 μHz. The 3 dB linewidth of both the OPLL heterodyned laser system and that of the two 5 MHz oscillators is ∼2 bins wide or ∼51 μHz.

The spectral line shape of Fig. 7(c) fits a Gaussian line shape function. The cross-correlation of two Gaussian line shapes is a Gaussian with a FWHM linewidth of $\mathrm{\Delta }{\nu }_{\mathit{measured}}=\sqrt{2}\mathrm{\Delta }{\nu }_L$ [35]. Therefore the free-running linewidth of the OPLL or (the reference oscillators) is ∼36 μHz over the 651 minute measurement interval of the dynamic signal analyzer. During the course of the linewidth measurement cycle, it is not surprising that there is some phase drift (dϕ/dt) amongst the three free-running microwave standards. This phase drift causes the free-running linewidth to increase over what is measured for the instantaneous linewidth. The measured free-running linewidth of the OPLL laser system closely follows that of the RF source to which the laser system is phase locked.

3. Stability: Allan deviation measurements

The measurement of the beat note linewidth is one measure of the heterodyned laser systems performance, but it is equally important to estimate the frequency stability of the optically generated beat note. Frequency stability measurements are indicators of how well the laser system replicates the reference oscillator frequency over a given time interval. Environmental stressors can cause time dependent drift in either or both the RF reference frequency or the laser system. Stability estimates can be made in either the frequency or time domains. Short term estimates of stability are for fluctuations over intervals of less than a few seconds to tens of seconds, to medium term hours to days and long term days to months to years. Here we will be looking at short to medium term measurements of the fractional frequency fluctuations.

Due to the typical non-stationary nature of oscillator frequency jitter, classical statistics do not often yield meaningful results. Instead, for time interval calculations the “fully overlapping two sample Allan deviation of fractional frequency fluctuations” is calculated from the frequency data. See [43] NIST special publication 1065 and/or [44] NIST technical note 1337 for further information on these calculations.

3.1. Measurement technique

To enhance the minimum time resolution of the electronic frequency and time interval (FTI) analyzer (HP 5371A), the “frequency heterodyne technique” is employed where two stable RF oscillators (reference and signal) are mixed down and only the beat frequency f_osc1 − f_osc2 = Δf_osc1,2 between the two oscillators is detected. With this frequency downconversion the minimum distinguishable resolution of the FTI analyzer is reduced by the downconversion factor, f_osc/Δf_osc1,2 [45]. The heterodyne technique only measures a frequency difference between two reference oscillators, so the accumulated relative phase between the beat signals is calculated first and then the fully overlapped Allan deviation (σ_y (τ)) is computed for each measurement time interval τ.

Figure 8 shows the schematic diagrams used to measure the fractional frequency stability of the heterodyned tone between the RF oscillators in Fig 8(a), and with the OPLL inserted in Fig. 8(b). This measurement set up is used to determine any additional instability due to the optical heterodyne laser system. In practice the two reference oscillators are slightly offset in frequency by 1 to 500 Hz and are downconverted by an RF mixer. The upper side band (the oscillator sum frequency) is filtered out by a 1.9 MHz 8-pole RC filter and the difference frequency signal (at Δf_osc1,2) passed through a tunable 10 to 1000 Hz low pass filter, set at 2 to 5x the beat frequency. A shaper circuit converts the ∼1–200 V/s sine wave into a fast rise-time, stable square-wave with a slew rate of ≥15 V/μs. The signal is detected at fixed time intervals by the frequency and time interval analyzer. The shaping circuit reduces the triggering errors at zero voltage crossing, resulting in reduced time interval errors in the counting hardware [46,47]. With this downconversion method, the stability measured is the combination of the reference oscillator and that of the test oscillator (or of the OPLL driven by the test oscillator). The Allan deviation measured is ${\sigma }_y^2{\left(\tau \right)}_{\text{measured}}={\sigma }_y^2{\left(\tau \right)}_{\text{reference}}+{\sigma }_y^2{\left(\tau \right)}_{\text{test}}$. If the reference oscillators have equal stability, the measured Allan deviation is increased by $\sqrt{2}$, that is ${\sigma }_y{\left(\tau \right)}_{\text{measured}}=\sqrt{2}{\sigma }_y{\left(\tau \right)}_{\text{oscillator}}$. If the reference oscillator is significantly more stable than the test oscillator, σ_y (τ)_measured = σ_y (τ)_{test osc}.

 

Fig. 8 Schematic diagram of the heterodyned two oscillator time interval measurement system. a) with two separate reference oscillators. b) With the optical phase locked loop laser system inserted between one of the reference oscillators and the heterodyning RF mixer.

Download Full Size | PDF

The fractional frequency stability of the OPLL was measured against one low phase noise RF synthesizer (an HP8663A) and three commercially available high stability ovenized 5 MHz quartz oscillators. The first quartz oscillator is the 5 MHz output of a Rubidium stabilized oven controlled crystal oscillator (OCXO) from Stanford Research Systems (SRS model FS725 Rubidium frequency standard) with a minimum specified σ_y (1s) = 2 × 10⁻¹¹, σ_y (10s) = 1 × 10⁻¹¹, and σ_y (100s) = 2 × 10⁻¹². The second quartz oscillator is a Frequency Electronics, Inc. (FEI) precision time and frequency standard model 1150A SC-cut OCXO 5 MHz oscillator (σ_y (τ) < 1 × 10⁻¹¹/day). The third quartz oscillator, a Frequency and Time Systems, Inc. (FTS) precision time and frequency transfer standard model FTS 1050A (now available from Microsemi Corp.) is a high-stability SC-cut OCXO with minimum specified σ_y (τ) ≤ 1 × 10⁻¹² for τ = 1, 10, and 100 s. The FTS oscillator (whose 10 MHz output was also used as the system clock) was voltage tuned +1.83 Hz above 5 MHz to give a nominal downconversion enhancement factor of f/Δf = 2.7 × 10⁶.

3.2. Free-running vs PLL with downconversion

The overlapping Allan deviation of the free-running (not phase locked) heterodyned laser system was measured and is shown in Fig. 9(a) for nominal averaging times (τ₀) of 0.1, 1, 10, 100, and 1000 ms. As a comparison, shown in Fig. 9(b) is the improvement in frequency stability of both a low phase noise RF synthesizer (an HP8663A stabilized by the FS725 Rubidium frequency standard) and the laser system phase locked to said RF synthesizer, using a nominal sampling interval (τ₀) of 1 s. For a 1s averaging time, the frequency stability Δf/f of the free-running heterodyned laser system at a 5 MHz beat frequency difference is σ_y (1s) ∼ 4 × 10⁻³. Upon phase locking to the stabilized RF synthesizer, the frequency stability improves to σ_y (1s) ∼ 2 × 10⁻¹¹. This is a 2 × 10⁸ increase in frequency stability, when phase locked to the stabilized RF synthesizer. Note that there is little to no discernible difference in the Allan deviation measurements between the OPLL system and that of the stabilized RF synthesizer. For short time scales, both the free-running [Fig. 9(a)] and the phased locked [Fig. 9(b)] heterodyned laser system both show a 1/τ rolloff in these stability measurements. This slope is indicative of either white phase modulation or flicker phase modulation noise process (at least at short time scales), which is a common noise term for many oscillator types [44]. After ∼1 s, thermal fluctuations in the free-running laser system causes the laser wavelength difference (frequency difference) to drift. When phase locked to the RF synthesizer, the laser system is far more stable.

 

Fig. 9 a) Overlapping Allan deviation measurements of the heterodyned laser system in the free-running state, at a 5 MHz frequency offset. b) Overlapping Allan deviation of the heterodyned laser system phase locked to the 5MHz output of an HP8663A RF source (phase locked to a SRS Rb 10 MHz frequency standard). The overlapping Allan deviation of the HP8663A source vs FTS1050A source at 5 MHz is also shown in background. In both cases a FTS 1050A OCXO is the 10 MHz reference for the time and frequency interval counter.

Download Full Size | PDF

3.3. 5 MHz quartz and Rb stabilized quartz oscillator comparison

The overlapping Allan deviation σ_y (τ) for three cases is shown in Fig. 10(a). Case 1: the OPLL is driven by the 5 MHz output of the Rb stabilized frequency source and measured w.r.t. the FTS 1050A standard. Case 2: the OPLL driven by the 5 MHz output the FTS 1050A standard and measured w.r.t. the Rb stabilized frequency source. Case 3: the 5 MHz output of Rb frequency source is directly measured w.r.t. to the FTS 1050A standard. At τ = 1s and for all three measurement conditions, σ_y (1s) = 1 × 10⁻¹¹, with a 1/τ rolloff in stability for all three cases. Since the manufacturer of the Rb source specified the Allan deviation to be better than σ_y (1s) = 2 × 10⁻¹¹ and the specified Allan deviation of the FTS1050 source is σ_y (1s) ≤ 1 × 10⁻¹², we can conclude that the Rb stabilized frequency source is the limiting stable oscillator for these measurement conditions. The OPLL adds little to no excess frequency instability to the Rb standard frequency reference.

 

Fig. 10 Overlapping Allan deviation of the heterodyned laser system phase locked to the 5 MHz output of a SRS Rb frequency reference, a FTS 1050A OCXO 5 MHz source, and a FEI 1150A OCXO 5 MHz source. The overlapping Allan deviation of the SRS Rb vs FTS 1050A and the FEI1150A vs the FTS 1050A are shown in the background for reference. In all cases a FTS 1050A OCXO is the 10 MHz reference for the time and frequency interval counter.

Download Full Size | PDF

In Fig. 10(b), three measurements of the overlapping Allan deviation (σ_y (τ)) from two oscillators and the OPLL are shown. Case 1: the OPLL driven by the 5 MHz output of the FEI 1150A oscillator and measured w.r.t. the FTS 1050A standard. Case 2: the OPLL driven by the 5 MHz output the FTS 1050A standard and measured w.r.t. the FEI 1150A oscillator. Case 3: the 5 MHz output of the FEI 1150A oscillator is measured w.r.t. to the FTS 1050A standard. At τ = 1s the heterodyned laser system has higher instability is σ_y (1s) = 8.4 ×10⁻¹² than either of the stabilized quartz frequency transfer standards. The measured combined Allen deviation of the FTS 1050A standard and the FEI 1150A oscillator, at τ = 1s, is σ_y (1s) = 1.3×10⁻¹². Either source is $\sqrt{2}$ smaller or σ_y (1s) < 1 × 10⁻¹², which agrees with the manufacturer’s specification. Beyond 1 ks, the FTS1050A versus FEI1150A σ(τ) curve begins to flatten (σ_y (τ) ∝ constant)) and oscillates. The phase locked laser system does not show this behavior. Instead, the the rolloff in stability begins as 1/τ and continues until 10 ks where it becomes comparable to the quartz oscillators. These curves suggest that, at the 10⁻¹² level, the OPLL laser system and the quartz oscillators have different fundamental noise sources. The increase in the Allan deviation of the OPLL system (w.r.t the quartz oscillators) is likely due to laser wavelength sensitivity to laboratory temperature fluctuations or noise in the electrical power supplies for the heterodyned laser system.

3.4. 10.24 GHz sapphire loaded cavity oscillator comparison

Continuous frequency tunability (from 5 kHz to 51+ GHz) is one feature of the heterodyned laser system, so the laser difference frequency was tuned to match a pair of 10.24 GHz frequency references. These references are two free-running 10.24 GHz sapphire loaded cavity oscillators (SLCO) from Poseidon Scientific Instruments (model SLCO-10.240-BCS). At room temperature these SLCO have low phase noise due to their whispering gallery mode design [48]. This type of whispering gallery mode oscillator (in cryogenic form) is capable of exhibiting a fraction frequency stability of $~5\times {10}^{-6}/\sqrt{\tau }$ for τ = 1s averaging times [49,50]. The Allan deviation of this pair of oscillators was unspecified by the manufacturer. Due to their room temperature operation and the fact that they are not further stabilized by an external Rb standard or a quartz oscillator, the Allan deviation is expected to be higher than their state-of-the-art stabilized cryogenic analogs.

Shown in Fig. 11 are the results of the fully overlapping Allan deviation of the heterodyned laser system phase locked to one of the PSI SCLO oscillators, and the Allan deviation of the two PSI SCLO oscillators versus each other, frequency mixed down to 150 to 500 Hz. The FTS 1050A OCXO at 10 MHz is used as the clock reference for this measurement. At 10.24 GHz, the laser system follows the output of a sapphire loaded cavity oscillator to a stability level of σ_y (1s) = 2.2 × 10⁻¹¹ with a 1/τ^0.7 stability rolloff out to τ =200 seconds where σ_y (200 s) = 6 × 10⁻¹². There is minimal difference in σ_y (τ) with the OPLL system added in or just that of the sapphire oscillators at sampling intervals below 300s. As the two SLCOs are considered identical, σ_y (τ) for each individual oscillator (and the PLL laser system following an individual SLCO oscillator) is $\sqrt{2}$ smaller, or σ_y (1s) = 1.6 × 10⁻¹¹ and σ_y (200 s) = 4.2 × 10⁻¹².

 

Fig. 11 Overlapping Allan deviation of the heterodyned laser system phase locked to one of two PSI sapphire loaded cavity oscillator frequency references at 10.24 GHz. Also shown is the overlapping Allan deviation between PSI sapphire oscillator (1) vs PSI sapphire oscillator (2) at 10.24 GHz, for reference. In both cases a FTS 1050A OCXO is the 10 MHz reference for the measurement system.

Download Full Size | PDF

The oscillations in the overlapping Allan deviation, observed in both the heterodyned laser system and just the RF sources, indicates that there are oscillations in the time interval data at one half the total gate time [43]. In Fig. 10(b) the first oscillation null occurs around 1000 s (15 to 20 minutes) and seems to correspond to the cycling of the environmental control system in the laboratory. For the sapphire oscillators, as shown in Fig. 11, there is a short term (5 to 10 s) and a medium term oscillation null occurring near ∼ 200 s and a longer term oscillation at 2000 s. These oscillations appear directly in the frequency data. The origin of the longer term oscillation appears to be cycling of the environmental control system in the laboratory. The shorter term oscillations originates internal to the housing for the sapphire oscillators. In all cases, the heterodyned laser system follows the frequency deviations of the driving oscillators.

4. Conclusion

We have successfully demonstrated that two 1.319 μm Nd:YAG NPRO lasers can be offset frequency phase locked and produce an RF beat note which has a measured instantaneous linewidth less than 22.8 μHz. The measured instantaneous linewidth follows that of the RF source to which the laser system is phase locked. This measurement is limited by the minimum resolution bandwidth of the dynamic signal analyzer being used for the measurement. When the RF oscillators are in the free-running state (no common 10 MHz reference), the 651 minute 3 dB linewidth of both the 5 MHz sources and the OPLL heterodyned laser system (locked to a 5 MHz source) is ∼36 μHz. In addition, Allan deviation stability measurements of the optically generated RF beat note again indicate that the OPLL system follows the RF source to which the laser system is phase locked. Measurements show that at 5 MHz, the laser system can follow a stabilized quartz reference oscillator to stability levels of σ_y (1s) = 8 × 10⁻¹² and follows a 1/τ rolloff out to τ =10 ks where σ_y (10 ks) = 1 × 10⁻¹⁵. At 10.24 GHz, the laser system follows the output of a sapphire loaded cavity oscillator to stability levels of σ_y (1s) = 1.6 × 10⁻¹¹ with a 1/τ^0.7 stability rolloff out to τ =200 seconds where σ_y (200 s) = 4.2 × 10⁻¹². In both cases, the optically generated RF beat note from the OPLL system closely follows the stability of the driving RF frequency references.