Three-tone modulation for laser stabilization and spectroscopy

Abstract

A method for modulating a laser beam includes electronically synthesizing a three-tone modulation signal and optically modulating the laser beam with the three-tone modulation signal to generate a modulated laser beam. The modulation signal has a center tone with frequency fc, a lower-sideband tone whose frequency fL is less than fc by a modulation frequency fM, and an upper-sideband tone whose frequency fU is greater than fc by the modulation frequency fM. The modulated laser beam includes an optical carrier with frequency fc and phase offset fc, an upper-sideband optical component with frequency f+=f0+fM and phase offset Φ +, and a lower-sideband optical component with frequency f-=f0−fM and phase offset Φ -. The lower-sideband and upper-sideband optical components have the same amplitude. The sum Φ ++Φ −−2Φ 0 equal 180º. The optical carrier, lower-sideband optical component, and upper-sideband optical component have the same optical spatial mode.

Description

PATENT Client Ref. CU6385B-02 Attorney Docket No. UOCO.P2090WO / 00621708 THREE-TONE MODULATION FOR LASER STABILIZATION AND SPECTROSCOPY RELATED APPLICATIONS

[0001] This application claims priority to U.S. Provisional Patent Application No. 63 / 548,182, filed November 11, 2023, which is incorporated herein by reference in its entirety. STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

[0002] This invention was made with government support under grant number PHY- 1734006, awarded by the National Science Foundation, grant number FA8649-21-P-0954, awarded by the U.S. Army Research Office, grant number D18AC00037, awarded by the DOD / DARPA, and grant number 70NANB15H226, awarded by the National Institutes of Health. The government has certain rights in the invention. BACKGROUND

[0003] Pound-Drever-Hall (PDH) cavity-locking has found prevalent uses in precision optical interferometry and laser frequency stabilization. A form of frequency-modulation (FM) spectroscopy, PDH offers superior signal-to-noise recovery, large acquisition dynamic range, wide servo bandwidth, and robust rejection of spurious effects, as compared to other cavity- locking and laser-frequency-stabilization schemes. SUMMARY

[0004] Precision optical interferometry builds on transferring the stability of a length reference to an optical frequency. Generally, an optical oscillator is stabilized to the length reference using the Pound-Drever-Hall (PDH) technique [1–4]. In the PDH technique, modulation sidebands around a carrier act as a phase reference to quantify frequency deviations of the carrier relative to the center of the reference’s spectral resonance. The versatility of the PDH technique has allowed its powerful application in a wide range of optical interferometers, from tabletop experiments seeking to produce low-noise oscillators [5–9] to kilometer-scale gravitational wave interferometers [10–14]. While alternative approaches have been developed for homodyne detection of the reference’s error signal [15–17], the PDH technique is by far the most commonly used and achieves the most robust performance as the carrier and phase- reference sidebands propagate along the same optical path and share the same wavefront.PATENT Attorney Docket No. UOCO.P2090WO / 00621708

[0005] In the PDH technique, the heterodyne beatnote between the cavity leakage reference and the incident field is formed with two out-of-phase modulation sidebands. In the presence of the resonance, this frequency discrimination converts a frequency-modulated (FM) spectrum into an amplitude-modulated field, offering detection sensitivity limited only by the photon shot noise. Away from the cavity resonance, the PDH technique ideally produces a null baseline signal. However, residual amplitude modulation (RAM), which may arise from any of several optical components in the beam path, can produce a fluctuating baseline signal. Reduction of spurious RAM has therefore been a goal in FM spectroscopy, including the development of ultrastable optical interferometers and oscillators [18–21].

[0006] In a typical implementation of the PDH technique, an electro-optic modulator (EOM) is driven with a radio-frequency (RF) signal Ω to produce a series of phase-modulation sidebands around the incident optical carrier,An expansion of the phase-modulation term expresses the incident field as a sum of sidebands, each with an amplitude characterized by a Bessel functionand a modulation depth While the EOM has the convenient feature of placing the first-order sidebands at ±Ω, 180º out- of-phase with each other, it has the deleterious effect that deep modulation generates higher-order sidebands at frequencies ±^^^^Ω with ^^^^ > 1 (see EOM modulation spectrum 204 in FIG. 2).These higher-order sidebands reduce the frequency discrimination slope, introduce RAM, and limit the dynamic range of the carrier used to probe the desired resonance.

[0007] The present embodiments include systems and method for three-tone modulation of a laser beam. These embodiments implement a new approach to the PDH technique in which only the three tones used for frequency discrimination (i.e., the carrier and two first-order sidebands) are generated. For clarity herein, these three tones are referred to as a “PDH triplet,” “modulation triplet,” or “synthetic triplet.” This triplet may then be translated into the optical domain with an optical modulator, such as an acousto-optic modulator (AOM). To ensure that the resulting three optical frequency components have the same spatial mode, the modulated light may be spatially filtered (e.g., with single-mode optical fibers). In some embodiments, the AOM is a fiber-coupled AOM.

[0008] Advantageously, the amplitude and phase of each of the PDH triplet’s three tones may be controlled without introducing unrelated terms from higher-order sidebands,PATENT Attorney Docket No. UOCO.P2090WO / 00621708 thereby circumventing the problems described above for prior-art EOM-based modulation. This feature also provides a higher signal-to-noise ratio (SNR), limited by shot noise. Furthermore, the present embodiments may be implemented without an EOM, thereby avoiding the problem that EOMs generate significant RAM due to their natural crystal birefringence. Prior-art PDH implementations often use active cancellation to prevent RAM from contaminating the amplitude modulation generated by the cavity or atomic and molecular resonances [22–26]. Active cancellation, which leverages the birefringence of the crystal, addresses only the phase quadrature, not the amplitude imbalance between sidebands. Several research efforts have additionally investigated techniques to reduce an EOM’s intrinsic RAM [27–35].

[0009] The present embodiments not only feature less RAM than prior-art EOM-based modulation, but they also provide a natural basis for feedback corrections in the RF domain. These electrical feedback actions are direct and orthogonal, leading to an ideal cavity interrogation beam, limited only by the shot noise in the guide error signal from the RAM photodetector (e.g., see photodetector 160 in FIG. 1). Another advantage of the present embodiments is that all the components of frequency synthesis, demodulation, and control may be integrated into a compact electronic unit, leading to significant reductions of the cost and complexity of performing FM spectroscopy.

[0010] General realizations of PDH locking rarely venture into the regime of ^^^^ > 1, butrecent work with state-of-the-art optical oscillators has identified cases where a suppressedcarrier and high modulation tones (generated with ^^^^ > 1) can lead to lower frequency drift

[0036] in dielectric coatings or reduced light-induced effects in crystalline coatings [37, 38]. The use of low optical carrier power has achieved the best long-term stability of optical oscillators. This is desirable in demonstrating the role of cryogenic crystalline cavities for all-optical timescales

[0039] and searches for novel fundamental physics

[0040] . In the prior-art EOM-basedFM approach, it is more challenging to reduce and stabilize RAM when ^^^^ ≫ 1. To furtherimprove long-term stability, independent and arbitrary amplitude control of the carrier and modulation sidebands is therefore a highly desirable feature. The present embodiments advantageously achieve this goal using control techniques that are much more versatile than what can be achieved with the prior-art EOM-based FM approach. As evidence of these advantages, described below is an experimental demonstration of RAM being actively reduced to the fractional 10-7level relative to the cavity linewidth, even with a small optical carrier. BRIEF DESCRIPTION OF THE FIGURES

[0011] FIG.1 is a schematic diagram of a system that performs three-tone modulationPATENT Attorney Docket No. UOCO.P2090WO / 00621708 of an input laser beam, in embodiments.

[0012] FIG.2 illustrates an experimental setup that was used to compare the three-tone modulation technique of the present embodiments with a prior-art modulation technique.

[0013] FIG. 3A shows plots of cavity transmission for the prior-art modulation technique (top) and the three-tone modulation technique of the present embodiments (bottom). FIG. 3B is a plot comparing the shot-noise-limited signal-to-noise ratio of the prior-art modulation technique and the three-tone modulation technique of the present embodiments.

[0014] FIGS. 4A, 4B, 4C, and 4D illustrate measurements of free-running residual amplitude modulation (RAM).

[0015] FIGS. 5A and 5C are plots of the power spectral density of RAM-induced frequency noise. FIGS.5B and 5D are plots of RAM-induced fractional frequency instability for the respective plots in FIGS.5A and 5C.

[0016] FIG.6 is a flowchart for a method for three-tone modulation of a laser beam, in embodiments.

[0017] FIG.7 is a flowchart of a method 700 for three-tone modulation of a laser beam, in embodiments.

[0018] FIG. 8 is a flowchart of a sub-method for controlling the amplitude and phase offset of a sideband, in embodiments.

[0019] FIG.9 is a flowchart of a sub-method for IQ demodulation, in embodiments. DETAILED DESCRIPTION

[0020] FIG. 1 is a schematic diagram of a system 100 that performs three-tone modulation of an input laser beam 110, in accordance with some of the present embodiments. The system 100 includes a synthesizer 104 that electronically synthesizes, or generates, a center tone 114 that can be represented mathematically as ^^^^^^^^(^^^^) = ^^^^^^^^ cos(2^^^^^^^^^^^^^^^^ + ^^^^^^^^), (2)where ^^^^^^^^, ^^^^^^^^, and ^^^^^^^^denote the amplitude, frequency, and phase offset, respectively, of the center tone 114. As shown in Eqn. 2, the term “tone” refers to a single-frequency sinusoidal waveform. The amplitude ^^^^^^^^and phase offset ^^^^^^^^may vary in time ^^^^ (e.g., due to drift).

[0021] The system 100 also includes a synthesizer 106 that electronically synthesizes a lower-sideband tone 116 that can be represented mathematically as ^^^^^^^^(^^^^) = ^^^^^^^^ cos(2^^^^^^^^^^^^^^^^ + ^^^^^^^^), (3)where ^^^^^^^^, ^^^^^^^^, and ^^^^^^^^denote the amplitude, frequency, and phase offset, respectively, of thePATENT Attorney Docket No. UOCO.P2090WO / 00621708 lower-sideband tone 116. The frequency ^^^^^^^^is less than the frequency ^^^^^^^^by a modulationfrequency ^^^^^^^^, i.e., ^^^^^^^^ = ^^^^^^^^ − ^^^^^^^^. The amplitude ^^^^^^^^ and phase offset ^^^^^^^^ may vary in time ^^^^.

[0022] The system 100 also includes a synthesizer 102 that electronically synthesizes an upper-sideband tone 112 that can be represented mathematically as ^^^^^^^^(^^^^) = ^^^^^^^^ cos(2^^^^^^^^^^^^^^^^ + ^^^^^^^^), (4)where ^^^^^^^^, ^^^^^^^^, and ^^^^^^^^denote the amplitude, frequency, and phase offset, respectively, of the upper-sideband tone 112. The frequency ^^^^^^^^is greater than the frequency ^^^^^^^^by the modulationfrequency ^^^^^^^^, i.e., ^^^^^^^^ = ^^^^^^^^ + ^^^^^^^^. The amplitude ^^^^^^^^ and phase offset ^^^^^^^^ may vary in time ^^^^.

[0023] In the example of FIG.1, the system 100 also includes an adder circuit 120 that sums the tones 112, 114, and 116 to generate a three-tone modulation signal 122 that can be represented mathematically as ^^^^(^^^^) = ^^^^^^^^(^^^^) + ^^^^^^^^(^^^^) + ^^^^^^^^(^^^^)= ^^^^ ( ) ( ) ( )^^^^ cos 2^^^^^^^^^^^^^^^^ + ^^^^^^^^ + ^^^^^^^^ cos 2^^^^^^^^^^^^^^^^ + ^^^^^^^^ + ^^^^^^^^ cos 2^^^^^^^^^^^^^^^^ + ^^^^^^^^The three-tone modulation signal 122 electrically drives an optical modulator 124. The output of the optical modulator 124 is a modulated laser beam 190 having an optical carrier, an upper- sideband optical component, and a lower-sideband optical component. Specifically, the optical carrier, upper-sideband optical component, and lower-sideband optical component are generated by modulating the input laser beam 110 with the center tone 114, upper-sideband tone 112, and lower-sideband tone 116, respectively.

[0024] The optical carrier can be represented mathematically as ^^^^0(^^^^) = ^^^^0 cos(2^^^^^^^0^ ^^^^ + ^^^^0), (6)where ^^^^0, ^^^0^ , and ^^^^0are the amplitude, frequency, and optical phase offset, respectively, of the optical carrier. The frequency ^^^0^ is also referred to as the “carrier” frequency. The upper- sideband optical component can be represented mathematically as ^^^^+(^^^^) = ^^^^+ cos(2^^^^^^^+^ ^^^^ + ^^^^+), (7)where ^^^^+, ^^^+^ , and ^^^^+are the amplitude, frequency, and optical phase offset, respectively, of the upper-sideband optical component. The lower-sideband optical component can be represented mathematically as ^^^^−(^^^^) = ^^^^− cos(2^^^^^^^−^ ^^^^ + ^^^^−) (8)where ^^^^−, ^^^−^ , and ^^^^−are the amplitude, frequency, and optical phase offset, respectively, ofPATENT Attorney Docket No. UOCO.P2090WO / 00621708 the lower-sideband optical component. Like the tones 112 and 114, the frequency ^^^+^ is greater than the carrier frequency ^^^0^ by the modulation frequency ^^^^^^^^. Like the tones 116 and 114, the lower-sideband optical frequency ^^^−^ is less than the carrier frequency ^^^0^ by the modulationfrequency ^^^^^^^^. Mathematically, ^^^±^ = ^^^0^ ± ^^^^^^^^.

[0025] The amplitudes ^^^^^^^^and ^^^^^^^^are similar to each other, although not necessarilyexactly the same (i.e., ^^^^^^^^ ≈ ^^^^^^^^). Furthermore, the lower-sideband tone 116 and upper-sidebandtone 112 are synthesized such that their phase offsets sum to approximately, although notnecessarily exactly, 180° (i.e., ^^^^^^^^ + ^^^^^^^^ ≈ 180°). When the amplitudes ^^^^^^^^ and ^^^^^^^^ are greaterthan ^^^^^^^^, the system 100 operates with a large modulation index. Conversely, when the amplitudes ^^^^^^^^and ^^^^^^^^are less than ^^^^^^^^, the system 100 operates with a small modulation index.

[0026] The system 100 may use feedback to establish two conditions simultaneously. For the first condition, one or both of the amplitudes ^^^^^^^^and ^^^^^^^^are controlled such that theamplitudes ^^^^ and ^^^^ are identical (i.e., ^^^^ = ^^^^ ). The relative p ′+ − + − hase offset ^^^^+ of the upper-sideband optical component is defined as ^^^^′+ = ^^^^+ − ^^^^0. Similarly, the relative phase offset^^^^′ of the lower-sideband optical component is defined as ′− ^^^^− = ^^^^− − ^^^^0. For the secondcondition, one or both of the phase offsets ^^^^^^^^and ^^^^^^^^are controlled such that the relative phaseoffsets ^^^^′ and ^^^^′ sum to 180° (i.e., ^^^^′ + ^^^^′+ − + − = 180°). With these two conditions met, the inputlaser beam 110 is frequency modulated without residual amplitude modulation (RAM). Specifically, any RAM that may be present on the modulated laser beam 190 is too small to detect (e.g., the RAM is less than the optical shot-noise limit of the laser beam 190).

[0027] In the present embodiments, the optical carrier, upper-sideband optical component, and lower-sideband optical component all have the same optical spatial mode, thereby ensuring that both (i) the equality of the amplitudes ^^^^+and ^^^^−and (ii) the 180° sum of the relative phase offsets ^^^^+′and ^^^^−′is the same at all transverse positions of the modulated laser beam 190. FIG.1 shows how the optical carrier, upper-sideband optical component, and lower-sideband optical component can all have the same spatial profile via mode filtering in an optical fiber. Specifically, an input optical fiber 126 couples the input laser beam 110 into the optical modulator 124 while an output optical fiber 128 couples the modulated laser beam 190 out of the optical modulator 124. In embodiments, the optical modulator 124 is a fiber-coupled acousto-optic modulator. However, another type of optical modulator (e.g., a free-space acousto-optic modulator) may be used without departing from the scope hereof. Each of the optical fibers 126 and 128 may be a single-mode, polarization-maintaining fiber.

[0028] To implement the feedback, a free-space fiber coupler 130 couples thePATENT Attorney Docket No. UOCO.P2090WO / 00621708 modulated laser beam 190 into a free-space modulated laser beam 132. A beamsplitter 134 splits off a first portion 138 of the laser beam 132. A photodetector 160 detects the first portion 138, generating a photodetector signal 178 in response. A second portion 136 of the laser beam 132 that propagates through the beamsplitter 134 may then be used for the application at hand (e.g., Pound-Drever-Hall locking to an optical cavity, locking a laser to a Mach-Zehnder interferometer, frequency-modulation laser spectroscopy, modulation transfer spectroscopy, etc.). Advantageously, by splitting the modulated laser beam 132 after the optical modulator 124, any imbalance in the amplitudes ^^^^+and ^^^^−introduced by the optical modulator 124 and output optical fiber 128 can be detected and corrected. Similarly, any deviation from 180° of the sum of the relative phases offsets ^^^^+′and ^^^^−′can be detected and corrected.

[0029] To better understand how the feedback works, consider the RF component ^^^^RF(^^^^) of the photodetector signal 178 at the modulation frequency ^^^^^^^^. The RF component ^^^^RF(^^^^) has two terms: (i) a lower-sideband term ^^^^−(^^^^) arising from interference of the lower- sideband optical component with the optical carrier and (ii) an upper sideband term ^^^^+(^^^^)arising from interference of the upper-sideband optical component with the optical carrier. Mathematically, the RF component ^^^^RF(^^^^) is proportional to the sum of ^^^^−(^^^^) and ^^^^+(^^^^): ^^^^RF(^^^^) ∝ ^^^^+(^^^^) + ^^^^−(^^^^)= ^^^^+^^^^0 cos(2^^^^^^^+^ ^^^^ + ^^^^+) cos(2^^^^^^^0^ ^^^^ + ^^^^0)(9) +^^^^−^^^^0 cos(2^^^^^^^−^ ^^^^ + ^^^^−) cos(2^^^^^^^0^ ^^^^ + ^^^^0).By expanding the products of cosines in Eqn.9 and ignoring the resulting high-frequency termsat frequencies ^^^0^ + ^^^+^ and ^^^0^ + ^^^−^ , Eqn. 9 simplifies toThe right side of Eqn. 10 has two components, one oscillating at the positive frequency +^^^^^^^^and the other oscillating at the negative frequency −^^^^^^^^. There is no residual amplitude modulation when these two components cancel each other. This cancellation will occur whenthe two components have the same amplitude (i.e., ^^^^− = ^^^^+) and are 180° out-of-phase witheach other (i.e., ^^^^′+ − (−^^^^′− ) = ^^^^′ ′+ + ^^^^− = 180°).

[0030] To see how to perform this cancellation via feedback, Eqn.10 simplifies to ^^^^^^^^^^^^(^^^^) ∝ ^^^^0 cos(2^^^^^^^^^^^^^^^^) (^^^^ ′ ′− cos^^^^− + ^^^^+ cos^^^^+ )+ ^^^^ ′0sin(2^^^^^^^^^^^^^^^^) (^^^^− sin^^^^ ′− − ^^^^+ sin^^^^+ ).It is assumed that the relative phase offset ^^^^−′is nominally 0°. Deviations of ^^^^−′from 0° arePATENT Attorney Docket No. UOCO.P2090WO / 00621708denoted ^^^^^^^^′ = ^^^^′ − 0°. To satisf ′ ′− − y ^^^^− + ^^^^+ = 180°, it is also assumed that the relative phaseoffset ^^^^′ is nominally 1 ′ ′ ′+ 80°. Deviations of ^^^^+ from 180° are denoted ^^^^^^^^+ = ^^^^+ − 180°.Substituting these deviations into Eqn.11 and simplifying yields ^^^^^^^^^^^^(^^^^) ∝ ^^^^0 cos(2^^^^^^^^^^^^^^^^) (^^^^− cos ^^^^^^^^′ ′− − ^^^^+ cos ^^^^^^^^+ )(12) +^^^^0 sin(2^^^^^^^^^^^^^^^^) (^^^^− sin ^^^^^^^^′ ′− + ^^^^+ sin ^^^^^^^^+ ).In the limit of small deviations (i.e., ^^^^^^^^′ ≪ 1 and ′ ′− ^^^^^^^^+ ≪ 1), the cosine terms cos ^^^^^^^^− andcos ^^^^^^^^′ can each be Taylor-expanded ′ ′+ to first order, i.e., cos ^^^^^^^^− ≈ 1 and cos ^^^^^^^^+ ≈ 1. Alsoin this limit, sin ^^^^^^^^′ ′ ′ ′− ≈ ^^^^^^^^− and sin ^^^^^^^^+ ≈ ^^^^^^^^+ . Equation 12 then simplifies to^^^^^^^^^^^^(^^^^) ∝ ^^^^0 cos(2^^^^^^^^^^^^^^^^) (^^^^− − ^^^^+)+ ^^^^0 sin 2^^^^^^^^^^^^^^^^ (^^^^−^^^^^^^^′ ′(13) () − + ^^^^+^^^^^^^^+ ).The RF component ^^^^RF(^^^^) may be demodulated to obtain a demodulated in-phase signal ^^^^^^^^(^^^^) ∝(^^^^− − ^^^^+) and a demodulated quadrature signal ^^^^^^^^(^^^^) ∝ (^^^^ ′ ′−^^^^^^^^− + ^^^^+^^^^^^^^+ ). The in-phasesignal ^^^^^^^^(^^^^) serves as an amplitude discriminator that crosses zero where ^^^^− = ^^^^+.Accordingly, the in-phase signal ^^^^^^^^(^^^^) may be used to control, via feedback, one or both of theamplitudes ^^^^^^^^ and ^^^^^^^^ such that ^^^^^^^^(^^^^) = 0. Since ^^^^− and ^^^^+ are both positive, the condition^^^^^^^^(^^^^) = 0 only occurs when ^^^^− = ^^^^+. While this condition is met, the quadrature signalsimplifies to ^^^^^^^^(^^^^) ∝ ^^^^+(^^^^^^^^′− + ^^^^^^^^′+ ). The quadrature signal ^^^^^^^^(^^^^) then serves as a phase-offset discrimator that crosses zero when ^^^^^^^^′ ′ ′ ′− = −^^^^^^^^+ (note that ^^^^^^^^− and ^^^^^^^^+ can each bepositive or negative). Accordingly, ^^^^^^^^(^^^^)may be used to control, via feedback, one or both ofthe phase offsets ^^^^^^^^ and ^^^^^^^^ such that ^^^^^^^^(^^^^) = 0. This condition occurs when ^^^^^^^^′− = −^^^^^^^^′+ ,which is equivalent to ^^^^+ + ^^^^− − 2^^^^0 = 180°. Note that if ^^^^0 = 0, this simplifies to ^^^^+ +^^^^−= 180°. If ^^^^−is equal to 0°, then ^^^^+= 180° (i.e., the phase offsets ^^^^−and ^^^^+differ from each other by 180°).

[0031] In the above derivation, it was assumed that the relative phase offset ^^^^−′was nominally 0° while the relative phase offset ^^^^+′was nominally 180°. Similar results hold for the case where ^^^^+′is nominally 0° and ^^^^−′is nominally 180°. More generally, it is not necessary that ^^^^−or ^^^^+exactly equal 0°. To see this, consider the relative phase offset ^^^^−′to be nominally ^^^^ while the relative phase offset ^^^^+′is nominally 180°−^^^^. In this case, deviations of ^^^^−′from^^^^ are denoted ^^^^^^^^′ = ^^ ′ ′ ′ ′− ^^− − ^^^^ while deviations of ^^^^+ from 180°− ^^^^ are denoted ^^^^^^^^+ = ^^^^+ −(180° − ^^^^). Substituting these deviations into Eqn. 10 and simplifying yields^^^^^^^^^^^^(^^^^) ∝ (14)PATENT Attorney Docket No. UOCO.P2090WO / 00621708 ^^^^0 cos(2^^^^^^^^^^^^^^^^) [cos^^^^ (^^^^ ′ ′− cos ^^^^^^^^− − ^^^^+ cos ^^^^^^^^+ )− sin^^^^ (^^^^− sin ^^^^^^^^′− + ^^^^+ sin ^^^^^^^^′+ )]+^^^^0 sin(2^^^^^^^^^^^^^^^^) [cos^^^^ (^^^^− sin ^^^^^^^^′− + ^^^^+ sin ^^^^^^^^′+ )+ sin^^^^ (^^^^− cos ^^^^^^^^′− − ^^^^ ′+ cos ^^^^^^^^+ )].Note that Eqn. 14 simplifies to Eqn. 12 for the case of ^^^^ = 0°. For ^^^^ = 90°, Eqn. 14 does notproduce a demodulated signal that is proportional to ^^^^− − ^^^^+, and that therefore can be usedfor relative amplitude discrimination and feedback. In between 0° and 90°, Eqn.14 may be usedto find a cutoff value ^^^^^^^^ such that all values of ^^^^ satisfying |^^^^| < ^^^^^^^^ produce a usabledemodulated signal that is proportional to ^^^^− − ^^^^+ and therefore can be used for relativeamplitude discrimination and feedback-based stabilization.

[0032] In some embodiments, the system 100 includes an IQ demodulator 140 that demodulates the photodetector signal 178 with a reference signal 118. The reference signal 118 may be generated, for example, by a reference synthesizer 108. The reference signal 118 is periodic, having the modulation frequency ^^^^^^^^. The reference signal 118 may be a pure tone (i.e., a single-frequency sinusoid) or a signal with multiple tones (e.g., a square wave, triangle wave, pulse train, etc.). The IQ demodulator 140 outputs a demodulated in-phase signal 168 (e.g., see the in-phase signal ^^^^^^^^(^^^^) above) and a demodulated quadrature signal 158 (e.g., see the quadrature signal ^^^^^^^^(^^^^) above). An in-phase servo amplifier 180 transforms (e.g., amplifies and filters) the demodulated in-phase signal 168 into an in-phase servo signal 182. Similarly, a quadrature servo amplifier 170 transforms the demodulated quadrature signal 158 into a quadrature servo signal 172. Each of the servo amplifiers 170 and 180 may have any number and combination of proportional (P), integral (I), and derivative (D) terms or stages (e.g., PI, PID, PI2D, etc.).

[0033] To close the loop for amplitude stabilization, the in-phase servo signal 182 drives an amplitude transducer 184 to vary the amplitude ^^^^^^^^. Alternatively, the amplitude transducer 184 could be used to vary the amplitude ^^^^^^^^(with an extra minus sign introduced into the transfer function). For even higher gain in the feedback loop, the in-phase servo signal 182 can be used to vary both of the amplitudes ^^^^^^^^and ^^^^^^^^simultaneously by driving two of the amplitude transducer 184. The amplitude transducer 184 may be a voltage-controlled attenuator, a voltage-controlled variable-gain amplifier, another type of circuit element that can be electrically controlled to vary gain or attenuation, or a combination of such circuit elements.

[0034] In one embodiment, the amplitude transducer 184 includes a resonant circuit (e.g., an LC tank circuit) with a resonance whose resonant frequency is offset from thePATENT Attorney Docket No. UOCO.P2090WO / 00621708 frequency ^^^^^^^^such that the frequency ^^^^^^^^coincides with a side of the resonance. The amplitude transducer 184 may also include a tuning circuit that, when driven by the in-phase servo signal 182, changes the resonant frequency of the resonant circuit to vary the amplitude ^^^^^^^^of the tone 112 (i.e., the attenuation) as it is transmitted through the resonant circuit. This method of varying the amplitude ^^^^^^^^is accompanied by a second-order variation of the phase ^^^^^^^^.

[0035] To close the loop for stabilizing the phase offsets ^^^^+and ^^^^−, the quadrature servo signal 172 drives a phase transducer 174 to vary the phase offset ^^^^^^^^. Alternatively, the phase transducer 174 could be used to vary the phase offset ^^^^^^^^(with an extra minus sign introduced into the transfer function). For even higher gain in the feedback loop, both of the phase offsets ^^^^^^^^and ^^^^^^^^can be controlled using two of the phase transducer 174. The phase transducer 174 may be a voltage-controlled phase shifter or trimmer, another type of circuit element that can be electrically controlled to vary phase, or a combination thereof.

[0036] In one embodiment, the phase transducer 174 includes a resonant circuit with a resonance whose resonant frequency coincides with the frequency ^^^^^^^^. The phase transducer 174 may also include a tuning circuit that, when driven by the quadrature servo signal 172, changes the resonant frequency of the resonant circuit to vary the phase offset ^^^^^^^^of the upper- sideband tone 112 as it is transmitted through the resonant circuit. This method of varying the phase offset ^^^^^^^^is accompanied by a second-order variation in the amplitude ^^^^^^^^.

[0037] In FIG.1, the IQ demodulator 140 includes a splitter 142 that splits the reference signal 118 into a split reference tone 144 and an in-phase reference tone 154. The IQ demodulator 140 also includes a phase shifter 146 that phase-shifts the split reference tone 144 by 90° to generate a quadrature reference tone 148. The IQ demodulator 140 also includes a splitter 162 that splits the photodetector signal 178 into a first split signal 164 and a second split signal 166. The IQ demodulator 140 also includes an in-phase mixer 156 that demodulates the second split signal 166 with the in-phase reference tone 154 to generate the demodulated in- phase signal 168. The IQ demodulator 140 also includes a quadrature mixer 150 that demodulates the first split signal 164 with the quadrature reference tone 148 to generate the demodulated quadrature signal 158.

[0038] There are many variations of the IQ demodulator 140 that may be implemented without changing the functionality of the system 100. For example, the splitter 142 and phase shifter 146 may be replaced with a 90° hybrid coupler. Alternatively, the 90° phase shifter 146 may be used to phase-shift either the first split signal 164 or the second split signal 166. In this case, the 90° phase shifter and splitter 162 may be replaced with a 90° hybrid coupler. Each ofPATENT Attorney Docket No. UOCO.P2090WO / 00621708 the mixers 150 and 156 may be a passive component (e.g., a double-balanced diode-based mixer) or an active analog component (e.g., based on actively controlled switches). In some embodiments, the IQ demodulator 140 is a two-channel lock-in amplifier. Each of the splitters 142 and 162 may be a hybrid coupler or a different type of power divider, directional coupler, or power splitter.

[0039] In other embodiments, the IQ demodulator 140 is implemented digitally (e.g., via a digital signal processor, microprocessor, field-programmable gate array, or another type of digital circuit or processor). In some of these embodiments, the servo amplifiers 170 and 180 are also implemented digitally. In this case, the system 100 may further include one or more analog-to-digital converters (e.g., to digitize the photodetector signal 178). The system 100 may also include one or more digital-to-analog converters (e.g., for converting the digital outputs of the servo amplifiers 170 and 180 into the analog servo signals 172 and 182, respectively).

[0040] In some embodiments, the outputs of the servo amplifiers 170 and 180 are digital signals that are not converted into analog servo signals. For example, the amplitude transducer 184 may be a digitally controlled programmable attenuator, in which case the in-phase servo signal 182 may be a digital signal. Similarly, the phase transducer may be a digitally controlled programmable phase shifter, in which case the quadrature servo signal 172 may be a digital signal. In some of these embodiments, the digital servo signals 172 and 182 are used to digitally program and update the synthesizer 102 while it is synthesizing the upper-sideband tone 112, thereby varying the amplitude ^^^^^^^^and phase offset ^^^^^^^^.

[0041] In FIG.1, each of the synthesizers 102, 104, 106, and 108 is shown with its own internal time base and therefore may operate as a stand-alone instrument. Each of the synthesizers 102, 104, 106, and 108 may be a function generator, frequency synthesizer, arbitrary waveform generator, or similar type of signal generator known in the art. To ensure synchronicity between the synthesizers 102, 104, 106, and 108, reference timing signals 192 may be used to phase-lock the internal time bases together. In the example of FIG.1, one of the internal time bases serves as a timing reference for all of the synthesizers 102, 104, 106, and 108. This one time base outputs a reference timing signal 192 (e.g., a 10 MHz signal) to the other synthesizers. The internal time bases of these other synthesizers are then phase-locked to this one time base. As an alternative to the example of FIG.1, all of the synthesizers 102, 104, 106, and 108 may be synchronized to the same external time base (not shown in FIG.1).

[0042] In other embodiments, the synthesizers 102, 104, 106, and 108 share one time base. In this case, the synthesizers 102, 104, 106, and 108 may be considered as one piece of equipment having several outputs. Each of the outputs may be generated via direct digitalPATENT Attorney Docket No. UOCO.P2090WO / 00621708 synthesis referenced to the shared time base. As an alternative to the reference synthesizer 108, a portion of the upper-sideband tone 112 (or the lower-sideband tone 116) and a portion of the center tone 114 may be mixed together and filtered to generate the reference signal 118. As an alternative to direct digital synthesis, analog phase-locking techniques may be used to generate the upper-sideband tone 112, the center tone 114, and the lower-sideband tone 116. Experimental Demonstration

[0043] Experimental Techniques

[0044] FIG. 2 illustrates an experimental setup 200 that was used to compare two techniques. The first technique is the three-tone modulation scheme of the present embodiments, implemented with an AOM. The second technique is conventional EOM-based modulation for PDH cavity locking. For clarity in this section, these two techniques are also referred to as the “AOM technique” and the “EOM technique.”

[0045] In the AOM technique, a carrier ^^^0^ and equally spaced modulation sidebands^^^0^ ± Ω (with ^^^0^ = 80 MHz and Ω = 5 MHz throughout this text) are generated via a directdigital synthesizer (DDS). Importantly, the output phase of one of the DDS channels is adjusted such that one of the modulation sidebands is 180º out-of-phase with the carrier, mimicking the electric field output of a phase-modulated EOM. All three tones are referenced to a common 1- GHz clock and a fourth channel on the DDS synthesizes the RF local oscillator (LO) that is used in the demodulation process. The three RF tones drive a fiber-coupled AOM so that the first diffracted orders (at different optical frequencies) are coupled together and propagate in the same fiber. Upon outcoupling through a collimator, all three optical frequency components share an identical spatial path.

[0046] The light out of the AOM is split into two paths. In the first path, light is incident on a photodetector (PD) used for measurement and active stabilization of RAM. In the second path, light is incident on a photodetector for PDH locking via cavity reflection. Each photodetector has resonant gain at the modulation frequency of 5 MHz, further suppressing pickup of any RF signals. In this demonstration, the PDH photodetector is instead used to characterize the out-of-loop (OoL) RAM of the entire system when the laser frequency is tuned sufficiently far away from the cavity resonance and the corresponding baseline fluctuations are measured as the OoL RAM. The cavity used in the experiment is 6-cm long and has a finesse of 497,000, corresponding to a cavity linewidth of 5 kHz. This AOM technique is compared to the EOM technique by replacing the AOM with a lithium niobate (LiNbO3) fiber-coupled waveguide EOM. All other optical elements remain the same.PATENT Attorney Docket No. UOCO.P2090WO / 00621708

[0047] One intrinsic advantage of synthesizing the PDH triplet via three separate tones is the absence of higher-order sidebands generated when phase modulating an EOM (see FIG.3A). Consequently, there is a higher signal-to-noise ratio (SNR) in the former case since there are no additional shot-noise contributions from higher modulation harmonics. This effectis illustrated in FIG.3B, where a high modulation index ^^^^ ≈ 2.3 (corresponding to a small ratioof carrier power ^^^^^^^^to total power ^^^^^^^^^^^^^^^^) is used for the operation of cryogenic silicon cavities with very small long-term frequency drift [36, 38, 39]. In the regime ^^^^^^^^ / ^^^^^^^^^^^^^^^^≪ 1, the shot- noise-limited SNR for PDH detection via an EOM is only 75% of the SNR that can be achieved with the optimal AOM technique.

[0048] FIGS. 4A–4D compare the passive stability of RAM between the EOM and AOM methods. In the prior-art EOM-based PDH technique, a misalignment of the input field polarization to the principal axes of the modulator leads to a polarization-dependent modulation due to differing refractive indices of the two crystal axes. Thermal variations in the crystal birefringence can then be a significant source of RAM and are often the dominant contribution. The modulation RF field additionally has a spatially inhomogeneous distribution inside the EOM crystal, resulting in another source for RAM. Conversely, PDH-triplet synthesis with an AOM is not subject to the same dispersive effects between the carrier and sidebands. Both techniques are sensitive to intra-crystal etalons, which can be mitigated by an angle cut of the crystal faces, but notably the fiber-AOM is found to be relatively insensitive to drifts in the input polarization of the incident light field. The primary source of RAM in the AOM technique is generated during the PDH-triplet synthesis itself. Differential amplitude and phase fluctuations of the synthesizer outputs will appear as noise in the I and Q quadratures at the modulation frequency Ω but are suppressed through the operation of the two RAM servo loops acting directly on the amplitude and phase quadratures of the two sidebands. Replacing optical issues by a stable electronic system in this manner is an advantageous exchange as the free- running RAM can be reduced by careful attention to the RF synthesis, which exhibits a much more robust RAM suppression over a long time scale, as depicted in FIGS.4A–4D.

[0049] State-of-the-art optical oscillators exhibit less than 1×10-6(i.e., sub-ppm) fractional stabilization to the cavity linewidth, which requires active stabilization of the RAM. In the EOM technique, active stabilization can be achieved by temperature controlling the electro-optic crystal and biasing it with a DC voltage that affects the light polarization [30, 31, 33]. Two issues are therefore present with this modulation technique: the modulation device itself generates unacceptable RAM and the actuators for RAM stabilization are not aligned to the in-phase (I) and quadrature (Q) components of the electric field.PATENT Attorney Docket No. UOCO.P2090WO / 00621708

[0050] Even with the superior passive RAM stability demonstrated by the present embodiments, the performance can be easily improved by phase-locking three low-phase-noise crystal oscillators to each other with integer frequency synthesis based on a common clock oscillator, frequency division, and single-sideband-modulation techniques. Presented herein is a unique RAM stabilization scheme to demonstrate that this is not necessary. Specifically, dual quadrature I / Q demodulation is implemented with the signal detected by the RAM photodetector. Rather than tune crystal parameters to affect the output field, feedback is applied directly to the RF outputs of the DDS. An attenuator (implemented by a voltage-controlledchannel conductance of a junction field effect transistor) on the ^^^0^ + Ω channel output receivesa control voltage addressing the error signal of the mixer’s I output. Phase noise at Ω is actively removed by tuning the resonance of an inductor-capacitor (implemented by two varactor diodesconnected in an anti-series configuration) circuit placed at the output of the ^^^0^ + Ω channel.Orthogonality between the mixer’s I and Q outputs is maintained by means of a wire-and-core 3dB quadrature hybrid coupler. The amplitude and phase noise feedback loops are both closed with bandwidths of 120 kHz. The in-loop performance of these stabilization loops is well below the detection shot noise, demonstrating that high suppression of the RAM signal is easily achievable with this orthogonalized and simple control scheme. A comparison of active RAM stabilization between the AOM and EOM techniques is shown in FIGS.5A–5D.

[0051] Results

[0052] Using the experimental setup 200, the free-running and out-of-loop (OoL) RAM performances of the two techniques were compared. This figure of merit ultimately determines the contribution of RAM to the cavity-stabilized laser frequency instability. Two techniqueswere compared first in the regime of conventional PDH cavity locking, where ^^^^ ≈ 1 and thePDH error signal is maximized by setting the sideband power to nearly half of the carrier power. The carrier power ^^^^^^^^is set to be equivalent in both setups (in this case, ^^^^^^^^= 3.3 μW and ^^^^^^^^^^^^^^^^=5.5 μW, or ^^^^ = 1.1) such that the corresponding frequency discrimination slopes for bothapproaches are nearly identical.

[0053] Notably, a significant improvement in free-running RAM control was observed for the AOM technique, as compared to the EOM technique (see FIGS. 4A–4D). Simply adopting the AOM technique without feedback enables control of the optical oscillator to the high 10-17level, a significant standard when compared to state-of-the-art results. High-ppm- level stabilization to the cavity linewidth is obtainable using synthetic FM from the 0.1–100 s timescale relevant for many metrology experiments. The lower RAM of the AOM technique (with feedback disabled) is even more pronounced when reducing the carrier power relative toPATENT Attorney Docket No. UOCO.P2090WO / 00621708 the sideband power (^^^^^^^^ / ^^^^^^^^^^^^^^^^= 0.005, see FIG.4B). With feedback enabled, the resulting OoL RAM signals, as measured via the PDH photodetector, are compared in FIGS.5A and 5D. The fractional frequency noise at bandwidth above 0.1 Hz is photon shot-noise limited to the same value in both techniques. Even in this conventional PDH regime, the AOM technique achieves a lower instability at long integration times, indicating slightly improved performance at lower Fourier frequencies. Electronic noise contributions to RAM stabilization (e.g., detection noise, finite servo gain) are well below the measured OoL traces and the measured instability is attributed to noise on the optical field. The Allan deviation data from the two techniquesconverge to a similar level of noise at averaging times ^^^^ > 4000 s, as shown in FIG. 5B,suggesting the presence of path-length fluctuations that are common to both setups between thein-loop and out-of-loop photodetectors. Basically, for ^^^^ ≈ 1, the two techniques perform at asimilar level. The AOM technique offers only a slight advantage in performance, but a significant advantage in experimental construction and implementation.

[0054] For cryogenic silicon cavities operating at temperatures of 4 K and 16 K [36, 38, 39], long-term cavity instability is observed to improve with significantly reduced optical carrier power coupled into the cavity. Thus, the regime of ^^^^^^^^ / ^^^^^^^^^^^^^^^^≪ 1 was also explored experimentally. The carrier power is reduced to ^^^^^^^^= 100 nW, and again the total optical power ^^^^^^^^^^^^^^^^= 20 μW is identical for both setups (see FIGS.5C and 5D). Note that the SNR obtained from the AOM technique is now 22% higher than that of the EOM technique, and the detected RF signal can be arbitrarily increased with the independent control of sideband amplitude. At high Fourier frequencies, where the fractional frequency noise power spectral density is consistent with the photon shot-noise limit, both techniques show similar instability that is now 10 dB higher owing to the reduced carrier power. However, performance at low Fourier frequencies shows a dramatic advantage of the AOM technique, where the OoL RAM limitation is still well below the measured cavity noise. This extreme limit of low ^^^^^^^^is preferable for obtaining the unparalleled short and long-term stabilities of cryogenic silicon cavities

[0041] .

[0055] The AOM technique, with its direct orthogonalized phase and amplitude control of the PDH triplet, offers an order-of-magnitude improved performance over the EOM technique, where the high modulation index generates large amplitude modulation tones at frequencies ±2Ω and ±3Ω. The significant improvement of the AOM technique indicates that there is a fundamental difference between the ways that the two techniques respond to any out- of-loop noise process. A prominent example is the presence of a parasitic etalon that could be located in the out-of-loop path probed by the PDH detector. Such an effect will generate anPATENT Attorney Docket No. UOCO.P2090WO / 00621708 amplitude and phase imbalance between the modulation sidebands. Noise originating from the etalon instability will be accentuated in the presence of higher order modulation harmonics. Even with the weak carrier power, RAM is still suppressed to the 0.3 ppm level in the AOM technique, and such performance demonstrates the potential for long-term laser-frequency instability well below 1 × 10-16, especially when a higher cavity finesse is employed.

[0056] Conclusion

[0057] In the present embodiments, the PDH triplet is digitally synthesized and frequency-shifted into an optical field via an optical modulator working with the emitted field of a stable single-mode laser. This configuration eliminates higher order sidebands generated via conventional EOM-based phase modulation and consequently offers a higher shot-noise- limited SNR. Direct and independent feedback to amplitude and phase of the modulated electric field are performed in the RF domain rather than the optical. Due to the versatility and robustness of the present embodiments, the carrier power may be independently controlled relative to the sideband amplitude, thereby allowing an optimal optical power to be coupled into the optical cavity.

[0058] The three-tone modulation technique of the present embodiments may be extended to experiments requiring a dynamic sweep of the FM tone. Varying the modulation frequency in an EOM results in an altered modulation depth, but no such effect occurs in the present embodiments. Additionally, retroreflecting the free-space signal before the cavity and allowing it to repass the fiber AOM will heterodyne beatnote at a frequency of 2^^^0^ , which can be used for path-length stabilization. A similar approach enacted in a PDH setup employing an EOM would require a separate AOM for this purpose. It is therefore possible to effectively combine two stabilization loops with a single actuator. Dedicated research efforts have sought to improve the innate RAM associated with EOMs, but here a significant improvement is demonstrated when using the PDH triplet. With active RAM cancellation, the stability exceeds that obtained with conventional EOM-based PDH cavity locking, especially in the regime of low carrier power. This is a direct path forward to achieving a lower frequency drift in cryogenic crystalline cavities employing dielectric coatings and reduction of light-induced noise in those with crystalline coatings. The compact and robust nature of this approach also makes PDH cavity locking and other forms of FM spectroscopy more accessible in a wide range of experiments.

[0059] Figure Captions

[0060] FIG. 2: Optical and electronic experimental setup for the AOM and EOM techniques. ^^^^+: voltage-controlled phase shifter; ^^^^+: voltage-controlled attenuator; Σ: RFPATENT Attorney Docket No. UOCO.P2090WO / 00621708 power combiner; ^^^^DC: Bias DC voltage applied to EOM for active RAM cancellation; PD(RAM): photodetector for detecting RAM; PD(PDH): photodetector for detecting PDH signal; NPBS: non-polarizing beam splitter; PBS: polarizing beam splitter; +S1: positive first- order sideband; -S1: negative first-order sideband; +S2: positive second-order sideband; -S2: negative second-order sideband; +S3: positive third-order sideband; -S3: negative third-order sideband; C: carrier; Amp: amplitude; ^^^^: frequency. Either the AOM or EOM configuration is used for PDH-based laser frequency stabilization.

[0061] FIG.3A: The cavity transmission of EOM (top) and AOM (bottom) techniqueswhen ^^^^^^^^ / ^^^^^^^^^^^^^^^^ = 0.5% for an EOM modulation index ^^^^ ≈ 2.3. Each inset is a plot showing themeasured carrier transmission as the modulated laser is swept in frequency (in MHz). FIG.3B: Comparison of the shot-noise-limited SNR between the EOM and AOM techniques. The totalincident power ^^^^^^^^^^^^^^^^ is fixed and the fraction of carrier power ^^^^^^^^⁄ ^^^^^^^^^^^^^^^^ is varied. A modulationindex ^^^^ = 1 roughly corresponds to ^^^^^^^^⁄ ^^^^^^^^^^^^^^^^ ≈ 0.5. Dots indicate measured results. Linesindicate calculations.

[0062] FIGS.4A–4D: The measurement of the free-running RAM. FIGS.4A and 4C:The power spectral density of RAM-induced frequency noise for high (top, ^^^^^^^^ / ^^^^^^^^^^^^^^^^ = 0.6) andlow (bottom, ^^^^^^^^ / ^^^^^^^^^^^^^^^^= 0.005) carrier power ratio. Dashed lines denote the shot-noise-limited frequency noise while the dotted line denotes Si6 noise as measured in

[0038] . FIGS.4B and 4D: RAM-induced fractional frequency instability for the respective plots in FIGS. 4A and 4C. Free-running RAM for the AOM technique exhibits more than an order-of-magnitude improved performance over the EOM technique at longer integration times in both situations. The right axis plots the data as fractional noise relative to the cavity linewidth.

[0063] FIGS.5A and 5C: The power spectral density of RAM-induced frequency noise for high (top, ^^^^^^^^ / ^^^^^^^^^^^^^^^^= 0.6) and low (bottom, ^^^^^^^^ / ^^^^^^^^^^^^^^^^= 0.005) carrier power ratio. All measurements actively stabilize the RAM to noise detected at the RAM photodetector. The measured signals are OoL RAM contributions detected by the PDH photodetector. The dotted line presents the current level of noise measured on the 6-cm silicon cavity (see

[0038] for details). Dashed lines denote the shot-noise-limited frequency noise. FIGS.5B and 5D: RAM-induced fractional frequency instability for the respective plots in FIGS.5A and 5C. Method Embodiments

[0064] FIG.6 is a flowchart for a method 600 for three-tone modulation of a laser beam, in accordance with some of the present embodiments. In the step 602 of the method 600, a three-tone modulation signal is electronically synthesized. The three-tone modulation signalPATENT Attorney Docket No. UOCO.P2090WO / 00621708 includes (i) a center tone having a center frequency, (ii) a lower-sideband tone having a lower- sideband frequency that is less than the center frequency by a modulation frequency, and (iii) an upper-sideband tone having an upper-sideband frequency that is greater than the center frequency by the modulation frequency.

[0065] In the step 602 of the method 600, an unmodulated laser beam is optically modulated with the three-tone modulation signal to generate a modulated laser beam. The modulated laser beam includes (i) an optical carrier having a carrier frequency, (ii) an upper- sideband optical component whose frequency is greater than the carrier frequency by the modulation frequency, and (iii) a lower-sideband optical component whose frequency that is less than the carrier frequency by the modulation frequency. The optical carrier has a carrier phase offset ^^^^0, the upper-sideband optical component has an upper-sideband optical phase offset ^^^^+, and the lower-sideband optical component has a lower-sideband optical phase offset ^^^^−. The lower-sideband and upper-sideband optical components have the same amplitude andthe sum ^^^^+ + ^^^^− − 2^^^^0 is equal to 180º. The optical carrier, lower-sideband opticalcomponent, and upper-sideband optical component have the same optical spatial mode.

[0066] The system 100 of FIG.1 is one example of a system that performs the method 600 of FIG.6. In this example, the step 602 of the method 600 is performed by the synthesizers 104, 106, and 102 and the adder circuit 120. Specifically, the synthesizer 104 synthesizes the center tone 114 (at frequency ^^^^^^^^), the synthesizer 106 synthesizes the lower-sideband tone 116(at frequency ^^^^^^^^ = ^^^^^^^^ − ^^^^^^^^, where ^^^^^^^^ is the modulation frequency), and the synthesizer 102synthesizes the upper-sideband tone 112 (at frequency ^^^^^^^^ = ^^^^^^^^ + ^^^^^^^^). The adder circuit 120then sums the tones 112, 116, and 114 to create the three-tone modulation signal 122. Also in this example, the step 604 is performed by the optical modulator 124. Specifically, the optical modulator 124 is driven with the three-tone modulation signal 122 to modulate the input laser beam 110, thereby creating the modulated laser beam 190.

[0067] In some embodiments, the method 600 includes an additional step in which the modulated laser beam is used for an application. For example, FIG.6 shows the method 600 with the step 606, in which a portion of the modulated laser beam is coupled into a Fabry-Perot cavity. The optical carrier of modulated laser beam may then be frequency-stabilized to a resonance of the Fabry-Perot cavity. In one example of the step 606, FIG.2 shows a modulated laser beam coupled into a Fabry-Perot cavity for PDH-based laser-frequency stabilization. In another example of how the modulate laser beam may be used for an application, a portion of the modulated laser beam is coupled into a spectrometer for performing spectroscopy (e.g., frequency-modulation laser spectroscopy, modulation transfer spectroscopy, etc.). MorePATENT Attorney Docket No. UOCO.P2090WO / 00621708 generally, the modulated laser beam may be used for several applications simultaneously by splitting the modulated laser beam into several portions, where each of these portions is used for a different one of the several applications.

[0068] FIG.7 is a flowchart of a method 700 for three-tone modulation of a laser beam, in accordance with some of the present embodiments. The method 700 represents certain embodiments of the method 600 of FIG. 6. Thus, the method 700 includes the steps 602 and 604 of the method 600. In some embodiments, the method 700 further includes the step 606.

[0069] In the method 700, the step 602 includes steps 702, 704, 706, and 708. In the step 702, the center tone is electronically synthesized. In the step 704, the lower-sideband tone is electronically synthesized. In the step 706, the upper-sideband tone is electronically synthesized. In the step 708, the center tone, lower-sideband tone, and upper-sideband tone are summed to generate the three-tone modulation signal. In one example of the steps 702, 704, 706, and 708, the synthesizer 104 of FIG.1 synthesizes the center tone 114, the synthesizer 106 synthesizes the lower-sideband tone 116, the synthesizer 102 synthesizes the upper-sideband tone 112, and the adder circuit 120 sums the tones 114, 116, and 112. In this example, each of the tones 114, 116, and 112 is synthesized by its own dedicated synthesizer (e.g., a DDS chip or core), which advantageously allows the amplitude and phase of the upper-sideband tone 112 to be varied independently of the tones 114 and 116. In another example, the tones 114, 116, and 112 are generated by an arbitrary waveform generator. In this example, the amplitude and phase of the upper-sideband tone 112 may be varied independently of the tones 114 and 16 via digital control or programming of the arbitrary waveform generator.

[0070] In the step 710 of the method 700, a portion of the modulated laser beam is detected with a photodetector to generate a photodetector signal. In one example of the step 710, the photodetector 160 of FIG.1 detects the first portion 138 of the laser beam 132, thereby generating the photodetector signal 178.

[0071] In the step 712 of the method 700, an amplitude of the lower-sideband tone or the upper-sideband tone is controlled, based on the photodetector signal, such that the lower- sideband and the upper-sideband optical components have the sample amplitude. In one example of the step 712, the in-phase servo signal 182 of FIG.1 drives the amplitude transducer 184 to vary the amplitude ^^^^^^^^of the upper-sideband tone 112. Alternatively or additionally, the amplitude transducer 184 may be used to vary the amplitude ^^^^^^^^of the lower-sideband tone 116.

[0072] In the step 714 of the method 700, a phase offset of the lower-sideband tone or the upper-sideband tone is controlled, based on the photodetector signal, such that the sum ^^^^++^^^^− − 2^^^^0 is equal to 180º. In one example of the step 714, the servo signal 172 of FIG. 1 drivesPATENT Attorney Docket No. UOCO.P2090WO / 00621708 the phase transducer 174 to vary the phase offset ^^^^^^^^of the upper-sideband tone 112. Alternatively or additionally, the phase transducer 174 may be used to vary the phase offset ^^^^^^^^of the lower-sideband tone 116.

[0073] FIG. 8 is a flowchart of a sub-method 800 for controlling, based on the photodetector signal generated by the step 710 of the method 700, (i) the amplitude of the lower- sideband tone or upper-sideband tone and (ii) the phase offset of the lower-sideband tone or upper-sideband tone. In some of the present embodiments, the method 700 of FIG. 7 is performed using the sub-method 800 for the steps 712 and 714. In the step 802 of the sub- method 800, the photodetector signal is demodulated to generate a demodulated in-phase signal and a demodulated quadrature signal. In the step 804 of the sub-method 800, the demodulated in-phase signal is transformed into an in-phase servo signal. In the step 806 of the sub-method 800, the modulated quadrature signal is transformed into a quadrature servo signal. In the step 808 of the sub-method 800, an amplitude transducer is driven with the in-phase servo signal to vary the amplitude of the lower-sideband tone or the upper-sideband tone. In the step 810 of the sub-method 800, a phase transducer is driven with the quadrature servo signal to vary the phase offset of the lower-sideband tone or the upper-sideband tone.

[0074] In one example of the sub-method 800, the IQ demodulator 140 of FIG. 1 demodulates the photodetector signal 178 to generate the demodulated in-phase signal 168 and the demodulated quadrature signal 158. The in-phase servo amplifier 180 then transforms the demodulated in-phase signal 168 into the in-phase servo signal 182 while the quadrature servo amplifier 170 transforms the demodulated quadrature signal 158 into the quadrature servo signal 172. The in-phase servo signal 182 then drives the amplitude transducer 184 to vary the amplitude ^^^^^^^^of the upper-sideband tone 112. The quadrature servo signal 172 then drives the phase transducer 174 to vary the phase offset ^^^^^^^^of the upper-sideband tone 112. While this example shows both of the transducer 184 and 174 varying the upper-sideband tone 112, one or both of the transducers 184 and 174 may alternatively vary the lower-sideband tone 116.

[0075] FIG. 9 is a flowchart of a sub-method 900 for IQ demodulation of the photodetector signal generated by the step 710 of the method 700. In some of the present embodiments, the sub-method 800 of FIG. 8 is performed using the sub-method 900 for the step 802. In the step 902 of the sub-method 900, in-phase and quadrature reference tones are generated at the modulation frequency. The quadrature reference tone is phase-shifted from the in-phase reference tone by 90°. In the step 904 of the sub-method 900, the photodetector signal is split into a first split signal and a second split signal. In the step 906 of the sub-method 900, the first split signal is demodulated with the in-phase reference tone to generate the demodulatedPATENT Attorney Docket No. UOCO.P2090WO / 00621708 in-phase signal. In the step 908 of the sub-method 900, the second split signal is demodulated with the quadrature reference tone to generate the demodulated quadrature signal.

[0076] In one example of the sub-method 900, the reference synthesizer 108 of FIG.1 synthesizes the reference signal 118. The splitter 142 splits the reference signal 118 into the split reference tone 144 and the in-phase reference tone 154. The phase shifter 146 phase-shifts the split reference tone 144 by 90° to generate the quadrature reference tone 148. The splitter 162 that splits the photodetector signal 178 into the first split signal 164 and the second split signal 166. The quadrature mixer 150 demodulates the first split signal 164 with the quadrature reference tone 148 to generate the demodulated quadrature signal 158. Similarly, the in-phase mixer 156 demodulates the second split signal 166 with the in-phase reference tone 154 to generate the demodulated in-phase signal 168. Combinations of Features

[0077] Features described above as well as those claimed below may be combined in various ways without departing from the scope hereof. The following examples illustrate possible, non-limiting combinations of features and embodiments described above. It should be clear that other changes and modifications may be made to the present embodiments without departing from the spirit and scope of this invention:

[0078] (A1) A method includes electronically synthesizing a three-tone modulation signal and optically modulating an unmodulated laser beam with the three-tone modulation signal to generate a modulated laser beam. The three-tone modulation signal includes a center tone having a center frequency, a lower-sideband tone having a lower-sideband frequency that is less than the center frequency by a modulation frequency, and an upper-sideband tone having an upper-sideband frequency that is greater than the center frequency by the modulation frequency. The modulated laser beam includes an optical carrier having a carrier frequency and a carrier phase offset ^^^^0, an upper-sideband optical component whose frequency is greater than the carrier frequency by the modulation frequency, and a lower-sideband optical component whose frequency is less than the carrier frequency by the modulation frequency. The upper- sideband optical component has an upper-sideband optical phase offset ^^^^+and the lower- sideband optical component has a lower-sideband optical phase offset ^^^^−. The lower-sidebandand upper-sideband optical components have the same amplitude and the sum ^^^^+ + ^^^^− − 2^^^^0is equal to 180º. The optical carrier, lower-sideband optical component, and upper-sideband optical component have the same optical spatial mode.

[0079] (A2) In the method denoted (A1), the method further includes detecting a portionPATENT Attorney Docket No. UOCO.P2090WO / 00621708 of the modulated laser beam with a photodetector to generate a photodetector signal. The method further includes controlling, based on the photodetector signal, an amplitude of the lower-sideband tone or the upper-sideband tone such that the lower-sideband and upper- sideband optical components have the same amplitude. The method further includes controlling, based on the photodetector signal, a phase offset of the lower-sideband tone or theupper-sideband tone such that the sum ^^^^+ + ^^^^− − 2^^^^0 is equal to 180º.

[0080] (A3) In the method denoted (A2), said controlling the amplitude and said controlling the phase include demodulating the photodetector signal to generate a demodulated in-phase signal and a demodulated quadrature signal, transforming the demodulated in-phase signal into an in-phase servo signal, transforming the demodulated quadrature signal into a quadrature servo signal, driving an amplitude transducer with the in-phase servo signal to vary the amplitude of the lower-sideband tone or the upper-sideband tone, and driving a phase transducer with the quadrature servo signal to vary the phase offset of the lower-sideband tone or the upper-sideband tone.

[0081] (A4) In the method denoted (A3), the amplitude transducer varies the amplitude of one of the lower-sideband and upper-sideband tones. Furthermore, the phase transducer varies the phase offset of the one of the lower-sideband and upper-sideband tones.

[0082] (A5) In the method denoted (A3), the amplitude transducer varies the amplitude of one of the lower-sideband and upper-sideband tones. Furthermore, the phase transducer varies the phase offset of the other of the lower-sideband and upper-sideband tones.

[0083] (A6) In any of the methods denoted (A3) to (A5), said demodulating includes generating an in-phase reference tone at the modulation frequency and generating a quadrature reference tone at the modulation frequency. The quadrature reference tone is phase-shifted from the in-phase reference tone by 90°. The method further includes splitting the photodetector signal into a first split signal and a second split signal, demodulating the first split signal with the in-phase reference tone to generate the demodulated in-phase signal, and demodulating the second split signal with the quadrature reference tone to generate the demodulated quadrature signal.

[0084] (A7) In any of the methods denoted (A1) to (A6), said optically modulating includes modulating the laser beam with a fiber-optic-coupled acousto-optic modulator.

[0085] (A8) In any of the methods denoted (A1) to (A7), the method further includes coupling a portion of the modulated laser beam into a Fabry-Perot cavity.

[0086] (A9) In the method denoted (A8), the method further includes frequency- stabilizing the optical carrier to a resonance of the Fabry-Perot cavity.PATENT Attorney Docket No. UOCO.P2090WO / 00621708

[0087] (A10) In any of the methods denoted (A1) to (A9), said electronically synthesizing includes electronically synthesizing the center tone, electronically synthesizing the lower-sideband tone, and electronically synthesizing the upper-sideband tone. Said electronically synthesizing further includes summing the center tone, lower-sideband tone, and upper-sideband tone to generate the three-tone modulation signal.

[0088] (B1) A system includes a synthesizer configured to electronically synthesize a three-tone modulation signal and an optical modulator configured to optically modulate, based on the three-tone modulation signal, an unmodulated laser beam to generate a modulated laser beam. The three-tone modulation signal includes a center tone having a center frequency, a lower-sideband tone having a lower-sideband frequency that is less than the center frequency by a modulation frequency, and an upper-sideband tone having an upper-sideband frequency that is greater than the center frequency by the modulation frequency. The modulated laser beam includes an optical carrier having a carrier frequency and a carrier phase offset ^^^^0, an upper-sideband optical component whose frequency is greater than the carrier frequency by the modulation frequency, and a lower-sideband optical component whose frequency is less than the carrier frequency by the modulation frequency. The upper-sideband optical component has an upper-sideband optical phase offset ^^^^+and the lower-sideband optical component has a lower-sideband optical phase offset ^^^^−. The lower-sideband and upper-sideband opticalcomponents have the same amplitude and the sum ^^^^+ + ^^^^− − 2^^^^0 is equal to 180º. The opticalcarrier, lower-sideband optical component, and upper-sideband optical component have the same optical spatial mode.

[0089] (B2) In the system denoted (B1), the system further includes a photodetector configured to generate a photodetector signal in response to detecting a portion of the modulated laser beam. The system further includes a feedback circuit configured to control, based on the photodetector signal, an amplitude of the lower-sideband tone or the upper-sideband tone such that the lower-sideband and upper-sideband optical components have the same amplitude. The feedback circuit is further configured to control, based on the photodetector signal, a phaseoffset of the lower-sideband tone or the upper-sideband tone such that the sum ^^^^+ + ^^^^− − 2^^^^0is equal to 180º.

[0090] (B3) In the system denoted (B2), the feedback circuit includes a demodulator configured to demodulate the photodetector signal to generate a demodulated in-phase signal and a demodulated quadrature signal; an in-phase servo amplifier configured to transform the demodulated in-phase signal into an in-phase servo signal; a quadrature servo amplifier configured to transform the demodulated quadrature signal into a quadrature servo signal; anPATENT Attorney Docket No. UOCO.P2090WO / 00621708 amplitude transducer configured to vary, based on the in-phase servo signal, the amplitude of the lower-sideband tone or the upper-sideband tone; and a phase transducer configured to vary, based on the quadrature servo signal, the phase offset of the upper-sideband tone or the upper- sideband tone.

[0091] (B4) In the system denoted (B3), the amplitude transducer is configured to vary the amplitude of one of the lower-sideband and upper-sideband tones. The phase transducer is configured to vary the phase offset of the one of the lower-sideband and upper-sideband tones.

[0092] (B5) In the system denoted (B3), the amplitude transducer is configured to vary the amplitude of one of the lower-sideband and upper-sideband tones. Furthermore, the phase transducer is configured to vary the phase offset of the other of the lower-sideband and upper- sideband tones.

[0093] (B6) In any of the systems denoted (B3) to (B5), the amplitude transducer includes a voltage-controlled attenuator. Furthermore, the phase transducer includes a tuning circuit configured to electronically tune a resonance of a resonant circuit.

[0094] (B7) In any of the systems denoted (B3) to (B6), the demodulator includes an IQ demodulator.

[0095] (B8) In any of the systems denoted (B3) to (B7), the demodulator includes a reference synthesizer configured to generate an in-phase reference tone having the modulation frequency and a quadrature reference tone having the modulation frequency. The quadrature reference tone is phase-shifted from the in-phase reference tone by 90°. The demodulator also includes a splitter configured to split the photodetector signal into a first split signal and a second split signal, a first mixer configured to demodulate the first split signal with the in-phase reference tone to generate the demodulated in-phase signal, and a second mixer configured to demodulate the second split signal with the quadrature reference tone to generate the demodulated quadrature signal.

[0096] (B9) In any of the systems denoted (B1) to (B8), the synthesizer includes a direct digital synthesizer.

[0097] (B10) In any of the systems denoted (B1) to (B9), the optical modulator includes a fiber-optic-coupled acousto-optic modulator.

[0098] Changes may be made in the above methods and systems without departing from the scope hereof. It should thus be noted that the matter contained in the above description or shown in the accompanying drawings should be interpreted as illustrative and not in a limiting sense. The following claims are intended to cover all generic and specific features described herein, as well as all statements of the scope of the present method and system, which, as aPATENT Attorney Docket No. UOCO.P2090WO / 00621708 matter of language, might be said to fall therebetween. References [1] R. W. P. Drever et al., “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983). [2] R. V. Pound, “Electronic Frequency Stabilization of Microwave Oscillators,” Rev. Sci. Instrum.17, 490–505 (1946). [3] D. Hils and J. L. Hall, “Ultra-Stable Cavity-Stabilized Lasers with Subhertz Linewidth,” in Frequency Standards and Metrology, A. De Marchi, ed. (Springer Berlin Heidelberg, Berlin, Heidelberg, 1989), pp.162–173. [4] E. D. Black, “An introduction to Pound–Drever–Hall laser frequency stabilization,” Am. J. Phys.69, 79–87 (2001). [5] C. Salomon, D. Hils, and J. L. Hall, “Laser stabilization at the millihertz level,” J. Opt. Soc. Am. B 5, 1576–1587 (1988). [6] B. Julsgaard, A. Walther, S. Kröll, and L. Rippe, “Understanding laser stabilization using spectral hole burning,” Opt. Express 15, 11444–11465 (2007). [7] T. Kessler et al. , “A sub-40-mhz- linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6, 687–692 (2012). [8] D. G. Matei et al., “1.5 μm lasers with sub-10 mHz linewidth,” Phys. Rev. Lett. 118, 263202 (2017). [9] W. Zhang et al., “Ultrastable Silicon Cavity in a Continuously Operating Closed-Cycle Cryostat at 4 K,” Phys. Rev. Lett.119, 243601 (2017).

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Claims

PATENT Attorney Docket No. UOCO.P2090WO / 00621708 CLAIMS What is claimed is:

1. A method, comprising: electronically synthesizing a three-tone modulation signal, the three-tone modulation signal comprising: a center tone having a center frequency; a lower-sideband tone having a lower-sideband frequency that is less than the center frequency by a modulation frequency; and an upper-sideband tone having an upper-sideband frequency that is greater than the center frequency by the modulation frequency; and optically modulating an unmodulated laser beam with the three-tone modulation signal to generate a modulated laser beam, the modulated laser beam comprising: an optical carrier having a carrier frequency and a carrier phase offset ^^^^0; an upper-sideband optical component whose frequency is greater than the carrier frequency by the modulation frequency, the upper-sideband optical component having an upper-sideband optical phase offset ^^^^+; and a lower-sideband optical component whose frequency is less than the carrier frequency by the modulation frequency, the lower-sideband optical component having a lower-sideband optical phase offset ^^^^−; wherein: the lower-sideband and upper-sideband optical components have the same amplitude; the sum ^^^^+ + ^^^^− − 2^^^^0 is equal to 180º; andthe optical carrier, lower-sideband optical component, and upper-sideband optical component have the same optical spatial mode.

2. The method of claim 1, further comprising: detecting a portion of the modulated laser beam with a photodetector to generate aPATENT Attorney Docket No. UOCO.P2090WO / 00621708 photodetector signal; controlling, based on the photodetector signal, an amplitude of the lower-sideband tone or the upper-sideband tone such that the lower-sideband and upper- sideband optical components have the same amplitude; and controlling, based on the photodetector signal, a phase offset of the lower-sideband tone or the upper-sideband tone such that the sum ^^^^+ + ^^^^− − 2^^^^0 is equal to180º.

3. The method of claim 2, wherein said controlling the amplitude and said controlling the phase comprise: demodulating the photodetector signal to generate a demodulated in-phase signal and a demodulated quadrature signal; transforming the demodulated in-phase signal into an in-phase servo signal; transforming the demodulated quadrature signal into a quadrature servo signal; driving an amplitude transducer with the in-phase servo signal to vary the amplitude of the lower-sideband tone or the upper-sideband tone; and driving a phase transducer with the quadrature servo signal to vary the phase offset of the lower-sideband tone or the upper-sideband tone.

4. The method of claim 3, wherein: the amplitude transducer varies the amplitude of one of the lower-sideband and upper- sideband tones; and the phase transducer varies the phase offset of the one of the lower-sideband and upper-sideband tones.

5. The method of claim 3, wherein: the amplitude transducer varies the amplitude of one of the lower-sideband and upper- sideband tones; and the phase transducer varies the phase offset of the other of the lower-sideband and upper-sideband tones.

6. The method of claim 3, wherein said demodulating comprises:PATENT Attorney Docket No. UOCO.P2090WO / 00621708 generating an in-phase reference tone at the modulation frequency; generating a quadrature reference tone at the modulation frequency, the quadrature reference tone being phase-shifted from the in-phase reference tone by 90°; splitting the photodetector signal into a first split signal and a second split signal; demodulating the first split signal with the in-phase reference tone to generate the demodulated in-phase signal; and demodulating the second split signal with the quadrature reference tone to generate the demodulated quadrature signal.

7. The method of claim 1, wherein said optically modulating comprises modulating the laser beam with a fiber-optic-coupled acousto-optic modulator.

8. The method of claim 1, further comprising coupling a portion of the modulated laser beam into a Fabry-Perot cavity.

9. The method of claim 8, further comprising frequency-stabilizing the optical carrier to a resonance of the Fabry-Perot cavity.

10. The method of claim 1, wherein said electronically synthesizing comprises: electronically synthesizing the center tone; electronically synthesizing the lower-sideband tone; electronically synthesizing the upper-sideband tone; and summing the center tone, lower-sideband tone, and upper-sideband tone to generate the three-tone modulation signal.

11. A system, comprising: a synthesizer configured to electronically synthesize a three-tone modulation signal, the three-tone modulation signal comprising: a center tone having a center frequency; a lower-sideband tone having a lower-sideband frequency that is less than the center frequency by a modulation frequency; and an upper-sideband tone having an upper-sideband frequency that is greaterPATENT Attorney Docket No. UOCO.P2090WO / 00621708 than the center frequency by the modulation frequency; and an optical modulator configured to optically modulate, based on the three-tone modulation signal, an unmodulated laser beam to generate a modulated laser beam, the modulated laser beam comprising: an optical carrier having a carrier frequency and a carrier phase offset ^^^^0; an upper-sideband optical component whose frequency is greater than the carrier frequency by the modulation frequency, the upper-sideband optical component having an upper-sideband optical phase offset ^^^^+; and a lower-sideband optical component whose frequency is less than the carrier frequency by the modulation frequency, the lower-sideband optical component having a lower-sideband optical phase offset ^^^^−; wherein: the lower-sideband and upper-sideband optical components have the same amplitude; the sum ^^^^+ + ^^^^− − 2^^^^0 is equal to 180º; andthe optical carrier, lower-sideband optical component, and upper-sideband optical component have the same optical spatial mode.

12. The system of claim 11, further comprising; a photodetector configured to generate a photodetector signal in response to detecting a portion of the modulated laser beam; and a feedback circuit configured to: control, based on the photodetector signal, an amplitude of the lower-sideband tone or the upper-sideband tone such that the lower-sideband and upper-sideband optical components have the same amplitude; and control, based on the photodetector signal, a phase offset of the lower-sideband tone or the upper-sideband tone such that the sum ^^^^+ + ^^^^− − 2^^^^0 isequal to 180º.

13. The system of claim 12, the feedback circuit comprising:PATENT Attorney Docket No. UOCO.P2090WO / 00621708 a demodulator configured to demodulate the photodetector signal to generate a demodulated in-phase signal and a demodulated quadrature signal; an in-phase servo amplifier configured to transform the demodulated in-phase signal into an in-phase servo signal; a quadrature servo amplifier configured to transform the demodulated quadrature signal into a quadrature servo signal; an amplitude transducer configured to vary, based on the in-phase servo signal, the amplitude of the lower-sideband tone or the upper-sideband tone; and a phase transducer configured to vary, based on the quadrature servo signal, the phase offset of the upper-sideband tone or the upper-sideband tone.

14. The system of claim 13, wherein: the amplitude transducer is configured to vary the amplitude of one of the lower- sideband and upper-sideband tones; and the phase transducer is configured to vary the phase offset of the one of the lower- sideband and upper-sideband tones.

15. The system of claim 13, wherein: the amplitude transducer is configured to vary the amplitude of one of the lower- sideband and upper-sideband tones; and the phase transducer is configured to vary the phase offset of the other of the lower- sideband and upper-sideband tones.

16. The system of claim 13, wherein: the amplitude transducer comprises a voltage-controlled attenuator; and the phase transducer comprises a tuning circuit configured to electronically tune a resonance of a resonant circuit.

17. The system of claim 13, the demodulator comprising an IQ demodulator.

18. The system of claim 13, the demodulator comprising: a reference synthesizer configured to generate an in-phase reference tone having the modulation frequency and a quadrature reference tone having the modulationPATENT Attorney Docket No. UOCO.P2090WO / 00621708 frequency, the quadrature reference tone being phase-shifted from the in-phase reference tone by 90°; a splitter configured to split the photodetector signal into a first split signal and a second split signal; a first mixer configured to demodulate the first split signal with the in-phase reference tone to generate the demodulated in-phase signal; and a second mixer configured to demodulate the second split signal with the quadrature reference tone to generate the demodulated quadrature signal.

19. The system of claim 11, the synthesizer comprising a direct digital synthesizer.

20. The system of claim 11, the optical modulator comprising a fiber-optic-coupled acousto-optic modulator.

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