Patent Publication Number: US-9905999-B2

Title: Optical frequency divider based on an electro-optical-modulator frequency comb

Description:
BENEFIT CLAIMS TO RELATED APPLICATIONS 
     This application claims benefit of U.S. provisional App. No. 62/121,257 entitled “Optical Frequency Divider Based on Electro-Optical-Modulator Comb” filed Feb. 26, 2015 in the names of Jiang Li and Kerry Vahala, said provisional application being hereby incorporated by reference as if fully set forth herein. 
    
    
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH 
     This invention was made with government support under Grant No. FA9550-10-1-0284 awarded by the Air Force and under Grant No. W31P4Q-14-1-0001 awarded by the Army. The government has certain rights in the invention. 
    
    
     FIELD OF THE INVENTION 
     The field of the present invention relates to generating microwave-frequency electrical signals and microwave-frequency sources utilizing a dual optical-frequency source and optical frequency division. 
     BACKGROUND 
     Subject matter disclosed or claimed herein may be related to subject matter disclosed in:
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Newbury, “Microwave generation with low residual phase noise from a femtosecond fiber laser with an intracavity electro-optic modulator,” Opt. Express 19, 24387-24395 (2011);   Papp, S. B., Beha, K., DelHaye, P., Quinlan, F., Lee, H., Vahala, K. J., Diddams, S. A., “A microresonator frequency comb optical clock,” arXiv:1309.3525 (2013), Optica Vol. 1, Issue 1, pp. 10-14 (2014);   C. B. Huang, S. G. Park, D. E. Leaird, and A. M. Weiner, “Nonlinearly broadened phase-modulated continuous-wave laser frequency combs characterized using DPSK decoding,” Opt. Express 16, 2520-2527 (2008);   I. Morohashi, T. Sakamoto, H. Sotobayashi, T. Kawanishi, and I. Hosako, “Broadband wavelength-tunable ultrashort pulse source using a Mach-Zehnder modulator and dispersion-flattened dispersion-decreasing fiber,” Opt. Lett. 34, 2297-2299 (2009);   A. Ishizawa, T. Nishikawa, A. Mizutori, H. Takara, A. Takada, T. Sogawa, and M. Koga, “Phase-noise characteristics of a 25-GHz-spaced optical frequency comb based on a phase- and intensity-modulated laser,” Opt. Express 21, 29186-29194 (2013);   S. Suzuki, K. Kashiwagi, Y. Tanaka, Y. Okuyama, T. Kotani, J. Nishikawa, H. Suto, M. Tamura, and T. Kurokawa, “12.5 GHz Near-IR Frequency Comb Generation Using Optical Pulse Synthesizer for Extra-Solar Planet Finder,” in Nonlinear Optics, OSA Technical Digest: Nonlinear Optics Conference (Optical Society of America, 2013), paper NM3A.3;   Young, B., Cruz, F., Itano, W., and Bergquist, J., “Visible Lasers with Subhertz Linewidths,” Phys. Rev. Lett. 82, 3799-3802 (1999);   T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. J. Martin, L. Chen, and J. Ye., “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photon. 6, 687-692 (2012);   Lee, H., Chen, T., Li, J., Yang, K. Y., Jeon, S., Painter, O., and Vahala, K. J., “Chemically etched ultrahigh-Q wedge resonator on a silicon chip,” Nat. Photon. 6, 369-373 (2012);   Li, J., Lee, H., Chen, T., and Vahala, K. J., “Characterization of a high coherence, brillouin microcavity laser on silicon,” Opt. Express 20, 20170-20180 (2012);   J. Li, H. Lee, K. Y. Yang, and K. J. Vahala, “Sideband spectroscopy and dispersion measurement in microcavities,” Opt. Exp. 20, 26337-26344 (2012);   Drever, R., Hall, J. L., Kowalski, F., Hough, J., Ford, G., Munley, A., and Ward, H., “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97-105 (1983);   Gross, M. C., Callahan, P. T., Clark, T. R., Novak, D., Waterhouse, R. B., and Dennis, M. L., “Tunable millimeter-wave frequency synthesis up to 100 GHz by dual-wavelength Brillouin fiber laser,” Opt. Express 18, 13321-13330 (2010);   Callahan, P. T., Gross, M. C., and Dennis, M. L., “Frequency-independent phase noise in a dual-wavelength Brillouin fiber laser,” IEEE J. Quantum Electron. 47, 1142-1150 (2011);   T. Sakamoto, T. Kawanishi, and M. Izutsu, “Asymptotic formalism for ultraflat optical frequency comb generation using a Mach-Zehnder modulator,” Opt. Lett. 32, 1515-1517 (2007);   Dudley, J. M., Genty, G., Coen, Stephane, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135-1184 (2006);   Li, J., Yi, X., Lee, H., Diddams, S., and Vahala, K., “Electro-optical frequency division and stable microwave synthesis,” Science 345, 309-313;   Geng, J., Staines, S., and Jiang, S., “Dual-frequency Brillouin fiber laser for optical generation of tunable low-noise radio frequency/microwave frequency,” Opt. Lett. 33, 16-18 (2008);   Pan, S., and Yao, J., “A wavelength-switchable single-longitudinal-mode dual-wavelength erbium-doped fiber laser for switchable microwave generation,” Opt. Express 17, 5414-5419 (2009);   Taylor, J., Datta, S., Hati, A., Nelson, C., Quinlan, F., Joshi, A., and Diddams, S., “Characterization of Power-to-Phase Conversion in High-Speed P-I-N Photodiodes,” IEEE Photonics Journal 3, 140 (2011);   A. J. Seeds, K. J. Williams, J., Lightwave Technol. 24, 4628-4641 (2006).   J. Yao, J. Lightwave Technol. 27, 314-335 (2009);   G. Carpintero et al., Opt. Lett. 37, 3657-3659 (2012);   U. L. Rohde, Microwave and Wireless Synthesizers: Theory and Design (Wiley, New York, 1997);   E. N. Ivanov, S. A. Diddams, L. Hollberg, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 52, 1068-1074 (2005);   J. Li, H. Lee, K. J. Vahala, Opt. Lett. 39, 287-290 (2014);   U.S. Pat. Pub. No. 2015/0236784 entitled “Stabilized microwave-frequency source” published Aug. 20, 2015 in the names of Vahala, Diddams, Li, Yi, and Lee;   U.S. Pat. Pub. No. 2015/0236789 entitled “Dual-frequency optical source” published Aug. 20, 2015 in the names of Vahala and Li; and   U.S. Pat. Pub. No. 2015/0311662 entitled “Stable microwave-frequency source based on cascaded brillouin lasers” published Oct. 29, 2015 in the names of Vahala and Li.       

     Each of the references listed above is incorporated by reference as if fully set forth herein. 
     SUMMARY 
     A microwave-frequency source for generating an output electrical signal at an output frequency f M  comprises: a dual optical-frequency reference source, an electro-optic sideband generator, one or both of first and second optical bandpass filters, one or both of first and second optical detectors, a reference oscillator, an electrical circuit, and a voltage-controlled electrical oscillator. The dual optical-frequency reference source is arranged so as to generate (i) a first optical reference signal at a first optical reference frequency v 1  and (ii) a second optical reference signal at a second optical reference frequency v 2 &gt;v 1 . The electro-optic sideband generator is arranged so as to (i) receive a third optical reference signal at a third optical reference frequency v 0  and a sideband generator input electrical signal at the frequency f M  and (ii) generate therefrom multiple sideband optical signals at respective sideband optical frequencies of the form v 0 ±nf M , wherein n is an integer. The first optical bandpass filter is arranged so as to transmit the first optical reference signal and a subset of the multiple sideband optical signals including the sideband optical signal at a frequency v 0 −Nf M , wherein N 1  is an integer; the second optical bandpass filter is arranged so as to transmit the second optical reference signal and a subset of the multiple sideband optical signals including the sideband optical signal at a frequency v 0 +N 2 f M , wherein N 2  is an integer. The first optical detector is arranged so as to receive the optical signals transmitted by the first optical bandpass filter and to generate therefrom a first optical detector electrical signal at a first beat frequency f BEAT1 =|v 1 −(v 0 −N 1 f M )|; the second optical detector is arranged so as to receive the optical signals transmitted by the second optical bandpass filter and to generate therefrom a second optical detector electrical signal at a second beat frequency f BEAT2 =|v 2 −(v 0 +N 2 f M )|. 
     The reference oscillator is arranged so as to generate a reference oscillator electrical signal at a reference oscillator frequency f R . The electrical circuit is arranged so as to receive the reference oscillator electrical signal and only one of (A) the first optical detector electrical signal, (B) the second optical detector electrical signal, or (C) a difference electrical signal generated from the first and second optical detector electrical signals by an electrical mixer at an electrical difference frequency f DIFF =f BEAT2 −f BEAT1 =|v 2 −v 1 −(N 1 +N 2 )·f M |. The electrical circuit is further arranged so as to generate from the received electrical signals, using a comparator portion of the electrical circuit, an electrical error signal dependent on relative phase of the electrical signals received by the electrical circuit, and to process the electrical error signal using a loop-filter portion of the electrical circuit. The voltage-controlled electrical oscillator is arranged so as to (i) receive the loop-filtered electrical error signal as a VCO input electrical signal and (ii) generate a VCO output electrical signal at the frequency f M , wherein a first portion of the VCO output electrical signal is received by the electro-optic sideband generator as the sideband generator input electrical signal and a second portion of the VCO output electrical signal forms the output electrical signal of the microwave-frequency source. Reception of the first portion of the VCO output electrical signal by the electro-optic sideband generator as the sideband generator input electrical signal results in the electrical circuit and the voltage-controlled oscillator being coupled in a negative feedback arrangement so as to function as a phase-locked loop. 
     Using the microwave-frequency source, a method for generating a microwave-frequency output electrical signal at the output frequency f M  comprises: (a) generating the first and second optical reference signals; (b) generating the multiple sideband optical signals; (c) transmitting through one or both of the first or second optical bandpass filters one or both of the first or second optical reference signals and one or both of the first or second subsets of the multiple sideband optical signals; (d) generating one or both of the first or second optical detector electrical signals; (e) generating the reference oscillator electrical signal at a reference oscillator frequency f R ; (f) generating and processing the electrical error signal; and (g) receiving the loop-filtered electrical error signal and generating the VCO output electrical signal at the frequency f M . 
     In some examples, the third optical reference frequency v 0  differs from the first optical reference frequency v 1  and from the second optical reference frequency v 2 . The first optical bandpass filter transmits the first optical reference signal and the sideband optical signal at a frequency v 0 −N 1 f M , and the first optical detector generates from those transmitted optical signals the first optical detector electrical signal at the first beat frequency f BEAT1 =|v 1 −(v 0 −N 1 f M )|. The second optical bandpass filter transmits the second optical reference signal and the sideband optical signal at a frequency v 0 +N 2 f M , and the second optical detector generates from those transmitted optical signals the second optical detector electrical signal at the second beat frequency f BEAT2 =|v 2 −(v 0 +N 2 f M )|. The electrical circuit generates the electrical error signal from the reference oscillator electrical signal and the difference electrical signal at the electrical difference frequency f DIFF =f BEAT2 −f BEAT1 =|v 2 −v 1 −(N 1 +N 2 )·f M |. 
     In other examples, a portion of the first optical reference signal serves as the third optical reference signal so that v 0 =v 1 ; the multiple sideband optical signals are generated at respective sideband optical frequencies of the form v 1 ±nf M , wherein n is an integer. The second optical bandpass filter transmits the second optical reference signal and the sideband optical signal at a frequency v 1 +N 2 f M , and the second optical detector generates from those transmitted optical signals the second optical detector electrical signal at the second beat frequency f BEAT2 =|v 2 −(v 1 +N 2 f M )|. The electrical circuit generates the electrical error signal from the reference oscillator electrical signal and the second optical detector electrical signal. 
     In still other examples, a portion of the second optical reference signal serves as the third optical reference signal so that v 0 =v 2 ; the multiple sideband optical signals are generated at respective sideband optical frequencies of the form v 2 ±nf M , wherein n is an integer. The first optical bandpass filter transmits the first optical reference signal and the sideband optical signal at a frequency v 2 −N 1 f M , and the first optical detector generates from those transmitted optical signals the first optical detector electrical signal at the first beat frequency f BEAT1 =|v 1 −(v 2 −N 1 f M )|. The electrical circuit generates the electrical error signal from the reference oscillator electrical signal and the first optical detector electrical signal. 
     Objects and advantages pertaining to optical frequency dividers and optical frequency combs may become apparent upon referring to the example embodiments illustrated in the drawings and disclosed in the following written description or appended claims. 
     This Summary is provided to introduce a selection of concepts in a simplified form that are further described below in the Detailed Description. This Summary is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to be used as an aid in determining the scope of the claimed subject matter. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  illustrates schematically an example of optical frequency division according to the present disclosure. 
         FIG. 2A  illustrates schematically an example of a dual-optical-frequency reference source according to the present disclosure. 
         FIG. 2B  illustrates schematically an example of an optical frequency divider according to the present disclosure. 
         FIG. 3  is a plot illustrating reduction of phase noise resulting from use of the disclosed optical frequency divider. 
     
    
    
     The embodiments depicted are shown only schematically: all features may not be shown in full detail or in proper proportion, certain features or structures may be exaggerated relative to others for clarity, and the drawings should not be regarded as being to scale. The embodiments shown are only examples: they should not be construed as limiting the scope of the present disclosure or appended claims. 
     DETAILED DESCRIPTION OF EMBODIMENTS 
     Microwave- and radio-frequency oscillators are widely used in communications, remote sensing, navigation, radar, scientific measurements, computers, clocks, time standards, and in other areas. They provide a single electrical frequency that can often be controllably tuned. Their performance is characterized using various metrics including power output, frequency tuning range, and frequency stability. The last of these metrics, frequency stability, is crucial to most applications, and the cost of an oscillator is directly related to the level of frequency stability that it provides. The very highest performance oscillators have typically been based on electrical oscillators that employ high-Q dielectric resonators to create a highly stable frequency. Optical detection of two highly coherent laser signals is another well-known approach to generate a stable radio-frequency or microwave-frequency signal; two optical signals coincident on an optical detector (i.e., a photodetector) and having an optical frequency difference within the detection frequency bandwidth of the optical detector produce an electrical output signal from the optical detector at the optical difference frequency of the optical signals, also referred to as a beat note or a beat frequency. A newer approach developed over the last few years employs a revolutionary, all-optical approach to microwave synthesis using a device known as a frequency-comb optical divider (or simply “optical divider”). Optical dividers accept as an input a highly coherent optical signal that oscillates at 100 s of THz. This is normally a laser signal that has been stabilized by an optical reference cavity. This very high input frequency is divided down to a rate in the radio-frequency or microwave-frequency range (100 s of MHz or 100 s of GHz) using the optical divider. This division process is accompanied by a dramatic reduction of the phase noise in the final signal relative to the initial optical signal thereby endowing the radio/microwave signal with remarkable (and record) stability. The frequency-comb optical divider employs a special mode locked laser to achieve this reduction. 
     Disclosed herein are novel, inventive apparatus and methods for achieving optical frequency division and stable operation of a microwave-frequency signal provided by an electrical oscillator having a frequency control input (sometimes called a voltage-controlled oscillator, or VCO). This novel approach is described herein along with preliminary measurements that demonstrate the feasibility of the method. In the inventive approach two optical reference signals, provided by lasers, are frequency-stabilized so that the relative frequency of the lasers (i.e., their optical difference frequency) is as stable as possible (or practicable to achieve performance necessary for a given use or application, i.e., relatively stabilized within an operationally acceptable reference bandwidth). A third optical reference signal is phase modulated using a cascade of phase modulators that are driven by the VCO. In the optical spectrum, this creates a spectrum of sidebands on the optical reference frequency spaced by the modulation frequency. The phase-modulated optical signals can be spectrally broadened to further increase number of sidebands in the side-band frequency spectrum, using an intensity modulator, a dispersion compensator, an optical amplifier (if needed or desired), and a nonlinear optical medium. It is desirable for the frequency separation of the lasers be as large as practicable (for a given use or application) to provide the maximum practicable stabilization of the VCO. However, the frequency separation cannot exceed the range of sidebands generated by the phase modulation cascade (and nonlinear optical broadening, if employed) around the third optical reference frequency. 
     In some examples, one sideband must be close enough in frequency to the first optical reference signal so that a beat frequency between them can be detected using a first photodiode, while another sideband must be close enough in frequency to the second optical reference signal so that a beat frequency between them can be detected using a second photodiode. As described further below, a difference electrical signal generated by an electrical mixer from the two photodiode signals contains phase information arising from the VCO and is used to stabilize the VCO. The inventive method has already enabled generation of highly stable microwave-frequency signals having a phase noise level well below a high performance electrical oscillator at offset frequencies of 10 kHz and 100 kHz. The performance of the inventive method can be substantially improved by further increasing the frequency separation of the two optical reference signals. In examples disclosed herein these optical reference signals are provided by dual pumping of a single high-Q disk resonator to produce stimulated Brillouin oscillation at two distinct wavelengths. The frequency separation is limited by the ability to pump separate lines efficiently within the same cavity, as is discussed further below. Other dual optical-frequency reference sources can be employed, e.g., a reference source comprising two lasers frequency-locked to separate modes of a single reference resonator cavity. 
     Compared to a conventional frequency-comb optical divider, the inventive approach does not require a mode-locked frequency comb generator, which is a sophisticated and costly device. Instead, the inventive method employs relatively simpler and less costly optical components, most of which are available commercially. Also, in many examples the reference optical frequency (i.e., the difference frequency of two optical sources) depends on the relative stability of two resonances within a single resonator. This is, in principle, a more robust reference as technical noise in the system (i.e., arising from instabilities in equipment, as opposed to quantum noise inherent in the physics of the system) is common to both resonances and therefore largely cancelled-out in the optical difference frequency. In contrast, the conventional divider approach relies upon an absolute reference frequency, which is more strongly impacted by technical noise. The principle of operation of the inventive optical frequency divider disclosed herein is also different in that the repetition frequency (i.e., sideband spacing) is set by an electrical VCO as opposed to an optical cavity. As a result it is possible to tune the microwave frequency of the output signal, which is not readily achieved with a conventional frequency-comb optical divider. Also, the conventional frequency-comb divider approach relies upon optical detection of a train of high peak power pulses of light with high bandwidth. The linearity of the photodetection process has been shown to be crucial to attaining frequency stability using this approach and greatly restricts the types of optical detectors that may be employed. In contrast the present invention can employ lower-bandwidth optical detectors or detectors with relaxed linearity requirements. 
     The inventive apparatus and methods disclosed herein for achieving optical frequency division for high-performance microwave-frequency signal generation employ cascaded phase modulation comprising direct phase modulation and also self-phase modulation (if needed or desired to achieve larger division ratios than can be achieve using direct phase modulation alone). In contrast to a comb of spectral lines produced by a mode-locked laser, cascaded phase modulators do not have an intrinsic repetition frequency since there is no optical cavity. While this can endow the cascade-generated comb of sidebands with an arbitrarily chosen line spacing frequency, it also means that optical frequency division must be accomplished in a fashion different from that used with conventional frequency combs. 
       FIGS. 1 and 2A / 2 B illustrate schematically the inventive approach in which two laser lines having sufficiently good (i.e., operationally acceptable) relative frequency stability provide two optical reference signals  120   a / 120   b , at respective optical frequencies v 1  and v 2 , for the microwave-frequency source.  FIG. 1  illustrates the optical spectra of the first and second optical reference signals  120   a / 120   b  and the sidebands  124  generated from a third optical reference signal  122 , and the optical bandpass filters  410   a / 410   b , optical detectors  420   a / 420   b , electrical components  505 / 510 / 520 , and voltage-controlled oscillator (VCO)  600  employed to produce the microwave-frequency signal.  FIG. 2A  illustrates an example of a dual optical-frequency reference source  100 ;  FIG. 2B  illustrates an example of an optical sideband generator. In some examples (described below), the two laser lines  120   a / 120   b  are produced by concomitant Brillouin oscillation in a single high-Q microcavity  110  or fiber-loop cavity (FLC) pumped by two corresponding, independent pump lasers  130   a / 130   b  locked to corresponding modes of the microcavity  110 . Alternatively, the two laser lines  120   a / 120   b  could also be produced by a dual-mode laser, by frequency-locking two lasers to distinct optical modes of a single, common reference cavity. The third optical reference signal  122  (e.g., a third laser line, for example, one of the pump laser lines  130   a  or  130   b ) enters an electro-optic sideband generator where it is phase modulated by a pair of modulators  220 / 230  at a frequency f M  set by a voltage-controlled electrical oscillator  600  (VCO). The sideband spectrum created by the phase modulators  220 / 230  can be further broadened by pulse-forming and self-phase phase modulation in an optical fiber  350 . The multiple sideband optical signals  124  thus formed extend from the third laser line  122  and result in multiple sideband optical signals  124 . 
     The frequency f M  can lie anywhere within the so-called microwave portion of the electromagnetic spectrum, e.g., between about 0.3 GHz and about 300 GHz. In some examples the output frequency f M  is between about 1 GHz and about 100 GHz. 
     One sideband signal  124   a  that is spectrally near the first optical reference signal  120   a  is transmitted along with the signal  120   a  through the first optical bandpass filter  410   a ; those transmitted signals are detected by the first photodetector  420   a  which generates a first optical detector electrical signal  440   a  at a beat frequency f BEAT1 =|v 1 −(v 0 −N 1 f M )|. Another sideband signal  124   b  that is spectrally near the second optical reference signal  120   b  is transmitted along with the signal  120   b  through the second optical bandpass filter  410   b ; those transmitted signals are detected by the second photodetector  420   b  which generates a second optical detector electrical signal  440   b  at a beat frequency f BEAT2 =|v 2 −(v 0 +N 2 f M )|. The two optical detector electrical signals  440   a / 440   b  are difference-frequency mixed in an electrical mixer  505  to produce a difference electrical signal  450  at frequency f DIFF =f BEAT2 −f BEAT1 =|v 2 −v 1 −(N 1 +N 2 )·f M |. As discussed below, that difference electrical signal  450  contains the phase noise of the VCO  600  magnified by the optical division factor (N 1 +N 2 ), the number of sidebands needed to span the frequencies of the first and second optical reference signals  120   a / 120   b . The difference electrical signal  450  therefore provides a suitable error signal for phase-lock-loop control of the VCO  600 . The difference electrical signal  450  is compared to a reference signal produced by a reference oscillator  510 . By adjusting the phase of the VCO  600  to nullify the phase difference, Δφ, the resulting VCO fluctuations are reduced to the following value: φ 2   M ≈(φ 1 −φ 2 ) 2 /(N 1 +N 2 ) 2 ; correlations in the phase noise of the two laser sources (arising, for example, from common-mode technical noise) are cancelled in this approach. The ultimate lower limit of the relative phase noise of the VCO  600  is given by the optical phase noise of the laser lines reduced by the division factor squared (i.e., reduced by a factor of (N 1 +N 2 ) 2 ). Clearly, to reduce phase noise of the electrical VCO  600 , the laser frequency separation should be made as large as possible (or practicable for a given use or application, i.e., operationally acceptable). In a preferred embodiment of the inventive system, this magnitude of the optical difference frequency is determined by the span of the dual-pumped Brillouin lasers. 
     In the example of  FIG. 2A , the first and second optical reference laser signals  120   a / 120   b  are provided by Brillouin laser lines co-lasing within a single silica-on-silicon high-Q disk resonator  110 . The coherence properties of the individual Brillouin laser lines is excellent, and the relative frequency stability of the Brillouin laser lines is enhanced by co-lasing within a common resonator. In the inventive microwave-frequency source the Brillouin laser lines are separated sufficiently that a dual pump configuration is needed. In this example the silica disk resonator  110  is designed and fabricated with a free-spectral-range (FSR) of about 10.890 GHz that substantially matches the Brillouin shift frequency in silica at a pump wavelength of 1550 nm. Other needed, desired, or suitable materials can be employed that exhibit different Brillouin shift frequencies. Each pump laser  130   a / 130   b  (emitting at corresponding pump wavelengths λ 1  and λ 2 ) is frequency-locked to a corresponding distinct resonant optical mode of the disk resonator  110  using the Pound-Drever-Hall (PDH) technique, is coupled into the disk resonator  110  through a circulator  114 , and excites its own corresponding Brillouin laser in the backward-propagating direction at respective optical reference frequencies v 1  and v 2 . The PDH technique is implemented for each pump laser  130   a / 130   b  by employing respective optical bandpass filters  132   a / 132   b , photodetectors  134   a / 134   b , and feedback/servo mechanisms  136   a / 136   b ; control of each pump wavelength can be via direct laser control (as with pump laser  130   b ) or via frequency shifting of the laser output (as with acousto-optic modulator  131   a  shifting the output of pump laser  130   a ). The frequency separation (i.e., the optical difference frequency v 2 −v 1 ) between the two SBS lasers can be readily tuned by tuning the pump lasers  130   a / 130   b  to pump at resonator modes with different azimuthal mode orders. Instead of a disk resonator, a fiber-loop optical resonator (i.e., a fiber-loop cavity or FLC) can be employed. The first and second optical reference signals  120   a / 120   b  are coupled to the optical divider section using the circulator  114 ; other suitable arrangements can be employed, e.g., a beamsplitter or a fiber coupler. 
     To generate multiple sideband optical signals  124  in the example of  FIG. 2B , two cascaded phase modulators  220 / 230  are employed that have relatively low V π  (e.g., ˜3.9V at 12 GHz) and are phase synchronized with an RF phase shifter  234 . The drive signal to the phase modulators  220 / 230  (i.e., the sideband generator input electrical signal) is a portion  620  of the output electrical signal of the VCO  600  at a frequency f M . In one example, up to about 30 electro-optic-modulated (EOM) sidebands (i.e., N 1 +N 2  up to about 30) can be generated using only a first portion  200  of the sideband generator (i.e., using only phase modulators  220 / 230  for generating the sideband signals  124 ), resulting in an optical difference frequency v 2 −v 1  of up to about 327 GHz and optical division by a factor of up to about 30. With frequency division by 30, the phase noise of the microwave-frequency signal  610  will be reduced by about a factor of about 900, relative to phase noise of the optical difference frequency of the optical reference signals  120   a / 120   b , by feedback stabilization of the frequency f M  (described further below). Two phase modulators are employed in the example shown in  FIG. 2B ; however, a single phase modulator can be employed if it provides sufficiently large modulation to produce sufficiently many sidebands. 
     To further enhance the sideband spectral width, additional phase modulators can be used, or phase modulators providing larger phase modulation amplitude can be used. Alternatively, or in addition, so-called continuum generation (e.g., as described in the publications of Huang et al (2008), Morohashi et al (2009), Ishizawa et al (2013, and Suzuki et al (2013), incorporated above) can be employed (as in  FIG. 2B ). An intensity modulator  320  is employed that is driven at frequency f M  by a portion of the VCO signal  620  and synchronized using a phase shifter  324 ; a dispersion compensator  330 , an optical amplifier  340 , and a nonlinear optical medium  350  are also employed. In a typical example, the intensity modulator  320  can comprise an electro-optic Mach-Zehnder interferometer, the dispersion compensator  330  (DC) can comprise a suitable length of suitably dispersive optical fiber (e.g., dispersion-shifted optical fiber or an optical fiber including a dispersion-shifting grating), the optical amplifier  340  can comprise an erbium-doped optical fiber amplifier (EDFA), and the nonlinear optical medium  350  can comprise a suitable length of highly nonlinear optical fiber (HNLF). Other functionally equivalent components can be employed, e.g., an electroabsorption modulator. In this example, continuum generation, cascaded with the phase modulators  220 / 230 , can generate sufficiently many sidebands to enable the two optical reference signals  120   a / 120   b  to be spaced farther apart spectrally than is typically possible using phase modulators alone; in this example, an optical difference frequency v 2 −v 1  can be generated up to 148 FSR or more of the disk resonator  110  (i.e., up to about 1.6 THz apart or more in this example; N 1 +N 2  up to 148 or more; typically limited by the gain bandwidth of the EDFA). With frequency division by 148, the phase noise of the microwave-frequency signal  610  will be reduced by about a factor of over 20,000 relative to phase noise of the optical reference signals  120   a / 120   b  by feedback stabilization of the frequency f M  (described further below). Even greater reduction of phase noise can be achieved by using even larger optical difference frequencies. 
     The first optical bandpass filter  410   a  is arranged so as to transmit the first optical reference signal  120   a  and a subset of the multiple sideband optical signals including the sideband optical signal  124   a  at a frequency v 0 −N 1 f M  (N 1  is an integer). The signals  120   a / 124   a  transmitted by the first optical bandpass filter  410   a  are received by the first optical detector  420   a , which generates therefrom (via linear superposition on the detector) a first optical detector electrical signal  440   a  at a first beat frequency f BEAT1 =|v 1 −(v 0 −N 1 f M )|. The second optical bandpass filter  410   b  is arranged so as to transmit the second optical reference signal  120   b  and a subset of the multiple sideband optical signals including the sideband optical signal  124   b  at a frequency v 0 +N 2 f M  (N 2  is an integer). The signals  120   b / 124   b  transmitted by the first optical bandpass filter  410   b  are received by the second optical detector  420   b , which generates therefrom (via linear superposition on the detector) a second optical detector electrical signal  440   b  at a second beat frequency f BEAT2 =|v 2 −(v 0 +N 2 f M )|. The optical detector electrical signals  440   a / 440   b  are mixed by an electrical mixer  505  to generate a difference electrical signal  450  at an electrical difference frequency f DIFF =f BEAT2 −f BEAT1 =|v 2 −v 1 −(N 1 +N 2 )·f M |. The frequency of the difference electrical signal  450  does not depend on the optical frequency v 0  of the third optical reference signal  122 , nor does the phase noise of the difference electrical signal  450  depend on the phase noise of the optical signal  122 . 
     A reference oscillator  510  generates a reference oscillator electrical signal at frequency f R  (a quartz oscillator operating at about 10 MHz in this example; any suitable reference oscillator can be employed; frequencies f R  from about 1 MHz to about 1 GHz, i.e., &lt;&lt;f M , have been employed). The phase noise contributed by the reference oscillator  510  typically is negligible relative to the phase noise of the optical reference signals  120   a / 120   b , and therefore typically does not present any limit on the phase noise reduction achievable by the optical frequency division process. A comparator portion of an electrical circuit  520  compares the phases of the difference electrical signal  450  and the reference electrical signal and generates an error signal (in any needed, desired, or suitable way), which is then loop-filtered by the circuit  520  (in any needed, desired, or suitable way). The loop-filtered error signal serves as a VCO electrical input signal  530  used for phase-locking the VCO  600  to a subharmonic (i.e., integer submultiple) of the frequency v 2 −v 1 . In the example the VCO  600  comprises a microwave generator operated using the external FM modulation input mode; any suitable type or implementation of a VCO  600  can be employed. Reception of the first portion  620  of the VCO output electrical signal by the electro-optic sideband generator as the sideband generator input electrical signal results in the electrical circuit  520  and the voltage-controlled oscillator  600  being coupled in a negative feedback arrangement so as to function as a phase-locked loop (PLL). Comparator and loop-filter portions of the electrical circuit  520  can be implemented in any needed, desired, or suitable way. 
     In one experimental implementation, a silica microdisk  110  is designed and fabricated so as to exhibit a free spectral range (FSR) about equal to the Brillouin shift frequency in silica (about 10.89 GHz for pumping near 1550 nm). The two pump lasers  130   a / 130   b  are locked to two optical modes of the disk resonator  110  of the same transverse cavity mode family but with different azimuthal mode numbers. The laser  130   a  operates at λ 1  of about 1550.0 nm and the laser  130   b  operates at λ 2  of about 1537.2 nm. Brillouin laser output of the microdisk  110  pumped by the lasers  130   a / 130   b  produces the first and second optical references signals  120   a / 120   b ; the resulting optical reference frequency difference v 2 −v 1  is about 1.61 THz (148 FSRs apart in the microdisk  110 ). In this example the output of the first laser  130   a  at about 1550.0 nm serves as the third optical reference signal  122 . The multiple sideband optical signals  124  in this example span about 50 nm (near 1550 nm), and so are amply sufficient to span the optical frequency difference v 2 −v 1  (about 12.8 nm in this example). The frequency f M  is set to be about equal to the FSR of the microdisk cavity  110 . In that way, the optical frequencies v 1  and v 2  will each lie sufficiently close to a corresponding one of the sidebands to generate a correspondingly low beat frequency, e.g., within the detection bandwidth of the photodetectors  420   a / 420   b.    
     Results from the experimental implementation are shown on  FIG. 3 . The upper curve represents the phase noise of the optical difference signal at frequency v 2 −v 1 . with the lasers locked to resonances of the microdisk  110  but without operation of the optical frequency divider or phase-locked loop circuitry. That phase noise is substantially independent of the frequency separation. The middle curve represents the noise of the microwave-frequency output of the free-running VCO  600 , and the lower curve represents the phase noise of the microwave-frequency output of the VCO  600  with the optical frequency divider operating and providing the loop-filtered error signal as an input to the VCO  600 . The phase noise is reduced by about 40 dB or more over a range of 10 Hz to 10 kHz, compared to the predicted noise reduction of about 42 dB (arising from a division factor of N 1 +N 2 =148; 1/148 2  is equivalent to a reduction of about 42 dB). The bandwidth of the phase-licked loop in this example is about 300 kHz, resulting in the peak in the lower curve at about 300 kHz in  FIG. 3 . When dividing the 1.61 THz Brillouin optical difference frequency by a factor of 148, the achieved phase noise level for the 10.89 GHz carrier is about −103 dBc/Hz at 1 kHz offset frequency and about −118 dBc/Hz at 10 kHz offset frequency. 
     It is advantageous to increase the division ratio to reduce phase noise of the generated microwave-frequency electrical signal. One way to achieve a higher division ratio is to increase the reference optical difference frequency v 2 −v 1 . In some examples, the reference optical difference frequency v 2 −v 1  is greater than about 100 GHz. In some of those examples, the reference optical difference frequency v 2 −v 1  is greater than about 300 GHz, greater than about 1 THz, greater than 10 THz, or greater than 100 THz. Still larger reference optical difference frequencies can be employed as needed, desired, or suitable, or as suitable optical sources are developed or become available. 
     Depending on the reference optical difference frequency v 2 −v 1  and the desired output frequency f M , any suitable division ratio N 1 +N 2  can be employed. In some examples the division ratio N 1 +N 2  is greater than or equal to 10, greater than or equal to 50, greater than or equal to 100, or greater than or equal to 1000. As noted above, a greater division ratio provides greater reduction of phase noise. 
     The optical reference signals  120   a / 120   b  can be provided at any needed, desired, or suitable optical frequencies. In some examples the, first and second optical reference frequencies v 1  and v 2  are each between about 75 THz and about 750 THz (i.e., wavelengths between about 400 nm and about 4 μm), between about 120 THz and about 430 THz (i.e., wavelengths between about 700 nm and about 2.5 μm), or between about 150 THz and about 300 THz (i.e., between about 1 μm and about 2 μm). The latter two ranges can be convenient due to ready availability of fiber-optic and/or solid state sources in those wavelength regions. Other wavelengths can be employed. 
     In some examples, the reference oscillator frequency can be between about 1 MHz and about 1 GHz, between about 5 MHz and about 100 MHz, or between about 10 MHz and about 50 MHz. In some examples, the reference oscillator  510  comprises a crystal oscillator, e.g., a quartz oscillator. In some other examples, the reference oscillator  510  comprises an electrical oscillator, e.g., a frequency-synthesized oscillator. Any suitably stable reference oscillator can be employed that provides sufficiently stable (i.e., operationally acceptable) performance of the microwave-frequency source in a given use or application. 
     In some examples, the dual optical-frequency reference source  100  is stabilized so as to maintain fluctuations of a reference optical difference frequency v 2 −v 1  (i.e., relative frequency stability of the lasers) within an operationally acceptable optical reference bandwidth. In some examples, the operationally acceptable reference bandwidth (given as a bandwidth characterized over a given time interval) is less than about 100 Hz over about a 1 second timescale, or less than about 1 Hz over about a 1 second timescale. In some examples, the operationally acceptable reference bandwidth (given as optical phase noise at a given offset frequency relative to the optical carrier frequency) is about −40 dBc/Hz at 100 Hz offset frequency and about −80 dBc/Hz at 10 kHz offset frequency, or about −80 dBc/Hz at 100 Hz offset frequency and about −125 dBc/Hz at 10 kHz offset frequency. Generally, improved levels of relative stability of the laser sources will translate directly into improvements in the overall frequency stability of the microwave-frequency output signal. Still better stabilized references can be employed as needed, desired, or suitable, or as suitable optical sources are developed or become available. 
     As noted above, in some examples the dual optical-frequency reference source  100  comprises first and second pump laser sources  130   a / 130   b  and an optical resonator (e.g., a disk resonator  110  as in  FIG. 2A , a fiber-loop or Fabry-Perot fiber resonator, or other suitable ring or linear resonator). The free spectral range (FSR) of the optical resonator  110  is substantially equal to a Brillouin shift frequency of the optical resonator (or an integer submultiple thereof). In some examples, the optical resonator comprises silica and the Brillouin shift frequency of the optical resonator is about 10.9 GHz; other needed, desired, or suitable materials can be employed that exhibit different Brillouin shift frequencies. Each one of the first and second pump laser sources  120   a / 120   b  is frequency-locked to a corresponding resonant optical mode of the optical resonator  110 . The first and second optical reference signals  120   a / 120   b  comprise stimulated Brillouin laser (SBL) output generated by optical pumping of the optical resonator simultaneously by the first and second pump laser sources  130   a / 130   b , respectively. In some examples, the free spectral range of the optical resonator  110  is substantially equal to the Brillouin shift frequency of the optical resonator. In some examples, the optical resonator comprises a ring optical resonator such as a disk optical resonator  110  (as in  FIG. 2A ). In other examples, the optical resonator comprises a fiber optical resonator, such as a fiber-loop optical resonator or a linear, Fabry-Perot-type fiber optical resonator (which can include, e.g., fiber Bragg gratings at the pump frequency or the SBL frequency). In some examples, each one of the first and second pump laser sources is frequency-locked to the corresponding resonant optical mode of the resonator  110  by a Pound-Drever-Hall mechanism; any suitable mechanism can be employed, e.g., a Hansch-Couillaud mechanism. 
     In some examples, including those described above, the third optical reference frequency v 0  lies between the first and second optical reference frequencies v 1  and v 2  (i.e., v 1 &lt;v 0 &lt;v 2 ), however, this need not be the case. In other examples the third optical reference frequency v 0  can be less than the first optical reference frequency v 1  (i.e., v 0 &lt;v 1 &lt;v 2 ); in still other examples the third optical reference frequency v 0  can be greater than the second optical reference frequency v 2  (i.e., v 1 &lt;v 2 &lt;v 0 ). In any of those cases, the multiple sideband optical signals  124  must include frequencies that span the range from v 1  to v 2 . To ensure that both first and second optical reference frequencies are sufficiently close to corresponding sideband frequencies, the third optical reference signal  122  can be coupled to the same resonator cavity  110  as the first and second optical reference signals  120   a / 120   b . In the examples described, a portion of the output of one of the pump lasers  130   a / 130   b  can be employed as the third optical reference signal  122  along with Brillouin laser output signals  120   a / 120   b . In other examples, a third laser can be locked to a resonance of the resonator  110 , and output of that third laser, or Brillouin laser output from resonator  110  pumped by the third laser, can be employed as the third optical reference signal  122 . In yet another example, three lasers can each be locked to different resonances of a common optical reference cavity. The frequency f M  in such examples should be an integer submultiple of the FSR of the reference cavity. Any other suitable stabilized optical-frequency reference source, including those described in U.S. Pat. Pub. Nos. 2015/0236784 and 2015/0236789, can be employed that provides needed, desired, or suitable (i.e., operationally acceptable) reference frequencies and optical difference frequency stability of the first, second, and third optical reference signals  120   a ,  120   b , and  122 . 
     In the examples described thus far, the third optical reference frequency v 0  differs from both the first and second optical reference frequencies v 1  and v 2 . In other examples, a portion of the first optical reference signal can serve as the third optical reference signal so that v 0 =v 1 . In such examples the electro-optic sideband generator generates the multiple sideband optical signals at respective sideband optical frequencies of the form v 1 ±nf M  (n is an integer). The second optical bandpass filter  410   b  transmits the second optical reference signal  120   b  and the sideband optical signal  124   b  at a frequency v 1 +N 2 f M . The second optical detector  420   b  generates the second optical detector electrical signal  440   b  at the second beat frequency f BEAT2 =|v 2 −(v 1 +N 2 f M )|. The electrical difference frequency mixer  505  is not needed; the electrical circuit  520  generates the error signal from the reference oscillator electrical signal and the second optical detector electrical signal  440   b . Phase noise of the output electrical signal of the microwave-frequency source at the frequency f M  is reduced by a factor of about (N 2 ) 2  relative to phase noise of the reference difference frequency signal at the reference optical difference frequency v 2 −v 1  of the dual optical-frequency reference source  100 . 
     In still other examples, a portion of the second optical reference signal can serve as the third optical reference signal so that v 0 =v 2 . In such examples the electro-optic sideband generator generates the multiple sideband optical signals at respective sideband optical frequencies of the form v 2 ±nf M  (n is an integer). The first optical bandpass filter  410   a  transmits the first optical reference signal  120   a  and the sideband optical signal  124   a  at a frequency v 2 −N 1 f M . The first optical detector  420   a  generates the first optical detector electrical signal  440   a  at the first beat frequency f BEAT1 =|v 1 −(v 2 −N 1 f M )|. The electrical difference frequency mixer  505  is not needed; the electrical circuit  520  generates the error signal from the reference oscillator electrical signal and the first optical detector electrical signal  440   a . Phase noise of the output electrical signal of the microwave-frequency source at the frequency f M  is reduced by a factor of about (N 1 ) 2  relative to phase noise of the reference difference frequency signal at the reference optical difference frequency v 2 −v 1  of the dual optical-frequency reference source  100 . 
     In addition to the preceding, the following examples fall within the scope of the present disclosure or appended claims: 
     Example 1 
     A microwave-frequency source for generating an output electrical signal at an output frequency f M , the microwave-frequency source comprising: (a) a dual optical-frequency reference source arranged so as to generate (i) a first optical reference signal at a first optical reference frequency v 1  and (ii) a second optical reference signal at a second optical reference frequency v 2 &gt;v 1 ; (b) an electro-optic sideband generator arranged so as to (i) receive a third optical reference signal at a third optical reference frequency v 0  and a sideband generator input electrical signal at the frequency f M  and (ii) generate therefrom multiple sideband optical signals at respective sideband optical frequencies of the form v 0 ±nf M , wherein n is an integer; (c) one or both of (i) a first optical bandpass filter arranged so as to transmit the first optical reference signal and a subset of the multiple sideband optical signals including the sideband optical signal at a frequency v 0 −N 1 f M , wherein N 1  is an integer, or (ii) a second optical bandpass filter arranged so as to transmit the second optical reference signal and a subset of the multiple sideband optical signals including the sideband optical signal at a frequency v 0 +N 2 f M , wherein N 2  is an integer; (d) one or both of (i) a first optical detector arranged so as to receive the optical signals transmitted by the first optical bandpass filter and to generate therefrom a first optical detector electrical signal at a first beat frequency f BEAT1 =|v−(v 0 −N 1 f M )|, or (ii) a second optical detector arranged so as to receive the optical signals transmitted by the second optical bandpass filter and to generate therefrom a second optical detector electrical signal at a second beat frequency f BEAT2 =|v 2 −(v 0 +N 2 f M )|; (e) a reference oscillator arranged so as to generate a reference oscillator electrical signal at a reference oscillator frequency f R ; (f) an electrical circuit arranged so as to (i) receive the reference oscillator electrical signal and only one of (A) the first optical detector electrical signal, (B) the second optical detector electrical signal, or (C) a difference electrical signal generated from the first and second optical detector electrical signals by an electrical frequency mixer at an electrical difference frequency f DIFF =f BEAT2 −f BEAT1 =|v 2 −v 1 −(N 1 +N 2 )·f M |, (ii) generate therefrom, using a comparator portion of the electrical circuit, an electrical error signal dependent on relative phase of the electrical signals received by the electrical circuit, and (iii) process the electrical error signal using a loop-filter portion of the electrical circuit; and (g) a voltage-controlled electrical oscillator arranged so as to (i) receive the loop-filtered electrical error signal as a VCO input electrical signal and (ii) generate a VCO output electrical signal at the frequency f M , wherein a first portion of the VCO output electrical signal is received by the electro-optic sideband generator as the sideband generator input electrical signal and a second portion of the VCO output electrical signal forms the output electrical signal of the microwave-frequency source, (h) wherein reception of the first portion of the VCO output electrical signal by the electro-optic sideband generator as the sideband generator input electrical signal results in the electrical circuit and the voltage-controlled oscillator being coupled in a negative feedback arrangement so as to function as a phase-locked loop. 
     Example 2 
     The microwave-frequency source of Example 1 wherein: (b′) the third optical reference frequency v 0  differs from the first optical reference frequency v 1  and from the second optical reference frequency v 2 ; (c′) the first optical bandpass filter is arranged so as to transmit the first optical reference signal and a subset of the multiple sideband optical signals including the sideband optical signal at the frequency v 0 −N 1 f M , and the second optical bandpass filter is arranged so as to transmit the second optical reference signal and a subset of the multiple sideband optical signals including the sideband optical signal at the frequency v 0 +N 2 f M ; (d′) the first optical detector arranged so as to receive the optical signals transmitted by the first optical bandpass filter and to generate therefrom the first optical detector electrical signal at the first beat frequency f BEAT1 =|v 1 −(v 0 −N 1 f M )|, and the second optical detector arranged so as to receive the optical signals transmitted by the second optical bandpass sideband optical signals and to generate therefrom the second optical detector electrical signal at the second beat frequency f BEAT2 =|v 2 −(v 0 +N 2 f M )|; and (f′) the electrical circuit is arranged so as to (i) receive the reference oscillator electrical signal and the difference electrical signal at the electrical difference frequency f DIFF =f BEAT2 −f BEAT1 =|v 2 −v 1 −(N 1 +N 2 )·f M |, and (ii) generate therefrom, using the comparator portion of the electrical circuit, the electrical error signal dependent on the relative phase of the reference oscillator electrical signal and the difference electrical signal received by the electrical circuit, and (iii) process the electrical error signal using the loop-filter portion of the electrical circuit. 
     Example 3 
     The microwave-frequency source of Example 2 wherein phase noise of the output electrical signal of the microwave-frequency source is reduced by a factor of about (N 1 +N 2 ) 2  relative to phase noise of a reference difference frequency signal at a reference optical difference frequency v 2 −v 1  of the dual optical-frequency reference source. 
     Example 4 
     The microwave-frequency source of Example 1 wherein: (b′) the electro-optic sideband generator is arranged so as to (i) receive a portion of the first optical reference signal as the third optical reference signal so that v 0 =v 1  and (ii) generate therefrom multiple sideband optical signals at respective sideband optical frequencies of the form v 1 ±nf M , wherein n is an integer; (c′) the second optical bandpass filter is arranged so as to transmit the second optical reference signal and a subset of the multiple sideband optical signals including the sideband optical signal at a frequency v 1 +N 2 f M ; (d′) the second optical detector is arranged so as to receive the optical signals transmitted by the second optical bandpass filter and to generate therefrom the second optical detector electrical signal at the second beat frequency f BEAT2 =|v 2 −(v 1 +N 2 f M )|; and (f′) the electrical circuit is arranged so as to (i) receive the reference oscillator electrical signal and the second optical detector electrical signal, and (ii) generate therefrom, using the comparator portion of the electrical circuit, the electrical error signal dependent on the relative phase of the reference oscillator electrical signal and the second optical detector electrical signal received by the electrical circuit, and (iii) process the electrical error signal using the loop-filter portion of the electrical circuit. 
     Example 5 
     The microwave-frequency source of Example 4 wherein phase noise of the output electrical signal of the microwave-frequency source is reduced by a factor of about (N 2 ) 2  relative to phase noise of a reference difference frequency signal at a reference optical difference frequency v 2 −v 1  of the dual optical-frequency reference source. 
     Example 6 
     The microwave-frequency source of Example 1 wherein: (b′) the electro-optic sideband generator is arranged so as to (i) receive a portion of the second optical reference signal as the third optical reference signal so that v 0 =v 2  and (ii) generate therefrom multiple sideband optical signals at respective sideband optical frequencies of the form v 2 ±nf M , wherein n is an integer; (c′) the first optical bandpass filter is arranged so as to transmit the first optical reference signal and a subset of the multiple sideband optical signals including the sideband optical signal at a frequency v 2 −N 1 f M ; (d′) the first optical detector is arranged so as to receive the optical signals transmitted by the first optical bandpass filter and to generate therefrom the first optical detector electrical signal at the first beat frequency f BEAT1 =|v 1 −(v 2 −N 1 f M )|; and (f′) the electrical circuit is arranged so as to (i) receive the reference oscillator electrical signal and the first optical detector electrical signal, and (ii) generate therefrom, using the comparator portion of the electrical circuit, the electrical error signal dependent on the relative phase of the reference oscillator electrical signal and the first optical detector electrical signal received by the electrical circuit, and (iii) process the electrical error signal using the loop-filter portion of the electrical circuit. 
     Example 7 
     The microwave-frequency source of Example 6 wherein phase noise of the output electrical signal of the microwave-frequency source is reduced by a factor of about (N 1 ) 2  relative to phase noise of a reference difference frequency signal at a reference optical difference frequency v 2 −v 1  of the dual optical-frequency reference source. 
     Example 8 
     The microwave-frequency source of any one of Examples 1 through 7 wherein the output frequency f M  is between about 0.3 GHz and about 300 GHz. 
     Example 9 
     The microwave-frequency source of any one of Examples 1 through 7 wherein the output frequency f M  is between about 1 GHz and about 100 GHz. 
     Example 10 
     The microwave-frequency source of any one of Examples 1 through 9 wherein the reference oscillator frequency is between about 1 MHz and about 1 GHz. 
     Example 11 
     The microwave-frequency source of any one of Examples 1 through 9 wherein the reference oscillator frequency is between about 10 MHz and about 100 MHz. 
     Example 12 
     The microwave-frequency source of any one of Examples 1 through 11 wherein the reference oscillator comprises a crystal oscillator. 
     Example 13 
     The microwave-frequency source of any one of Examples 1 through 11 wherein the reference oscillator comprises an electrical oscillator. 
     Example 14 
     The microwave-frequency source of any one of Examples 1 through 13 wherein a reference optical difference frequency v 2 −v 1  is greater than about 100 GHz. 
     Example 15 
     The microwave-frequency source of Example 14 wherein a reference optical difference frequency v 2 −v 1  is greater than about 1 THz. 
     Example 16 
     The microwave-frequency source of Example 14 wherein a reference optical difference frequency v 2 −v 1  is greater than about 10 THz. 
     Example 17 
     The microwave-frequency source of Example 14 wherein a reference optical difference frequency v 2 −v 1  is greater than about 100 THz. 
     Example 18 
     The microwave-frequency source of any one of Examples 1 through 17 wherein N 1 +N 2  is greater than or equal to 10. 
     Example 19 
     The microwave-frequency source of Example 18 wherein N 1 +N 2  is greater than or equal to 50. 
     Example 20 
     The microwave-frequency source of Example 18 wherein N 1 +N 2  is greater than or equal to 100. 
     Example 21 
     The microwave-frequency source of Example 18 wherein N 1 +N 2  is greater than or equal to 1000. 
     Example 22 
     The microwave-frequency source of any one of Examples 1 through 21 wherein the first and second optical reference frequencies v 1  and v 2  are each between about 75 THz and about 750 THz. 
     Example 23 
     The microwave-frequency source of Example 22 wherein the first and second optical reference frequencies v 1  and v 2  are each between about 120 THz and about 430 THz. 
     Example 24 
     The microwave-frequency source of Example 22 wherein the first and second optical reference frequencies v 1  and v 2  are each between about 150 THz and about 300 THz. 
     Example 25 
     The microwave-frequency source of any one of Examples 1 through 24 wherein the dual optical-frequency reference source is stabilized so as to (i) maintain fluctuations of a reference optical difference frequency v 2 −v 1  within an operationally acceptable optical reference bandwidth or (ii) maintain phase noise of a reference optical difference frequency signal within an operationally acceptable reference phase noise level. 
     Example 26 
     The microwave-frequency source of Example 25 wherein the operationally acceptable reference bandwidth is less than about 100 Hz over about a 1 second timescale. 
     Example 27 
     The microwave-frequency source of Example 25 wherein the operationally acceptable reference bandwidth is less than about 1 Hz over about a 1 second timescale. 
     Example 28 
     The microwave-frequency source of any one of Examples 25 through 27 wherein the operationally acceptable reference phase noise level is about −40 dBc/Hz at 100 Hz offset frequency and about −80 dBc/Hz at 10 kHz offset frequency. 
     Example 29 
     The microwave-frequency source of any one of Examples 25 through 27 wherein the operationally acceptable reference phase noise level is about −80 dBc/Hz at 100 Hz offset frequency and about −125 dBc/Hz at 10 kHz offset frequency. 
     Example 30 
     The microwave-frequency source of any one of Examples 1 through 29 wherein (i) the dual optical-frequency reference source comprises first and second pump laser sources and an optical resonator, (ii) a free spectral range of the optical resonator is substantially equal to a Brillouin shift frequency of the optical resonator or an integer submultiple thereof, (iii) each one of the first and second pump laser sources is frequency-locked to a corresponding resonant optical mode of the optical resonator, and (iv) the first and second optical reference signals comprise stimulated Brillouin laser output generated by optical pumping of the optical resonator simultaneously by the first and second pump laser sources, respectively. 
     Example 31 
     The microwave-frequency source of Example 30 wherein the third optical reference signal is comprises a portion of the optical output of one of the first or second pump laser sources. 
     Example 32 
     The microwave-frequency source of any one of Examples 30 or 31 wherein the frequency f M  is about equal to the free spectral range of the optical resonator or an integer submultiple thereof. 
     Example 33 
     The microwave-frequency source of any one of Examples 30 through 32 wherein the free spectral range of the optical resonator is substantially equal to the Brillouin shift frequency of the optical resonator. 
     Example 34 
     The microwave-frequency source of any one of Examples 30 through 33 wherein the optical resonator comprises silica and the Brillouin shift frequency of the optical resonator is about 10.9 GHz. 
     Example 35 
     The microwave-frequency source of any one of Examples 30 through 34 wherein the optical resonator comprises a ring optical resonator. 
     Example 36 
     The microwave-frequency source of Example 35 wherein the ring optical resonator comprises a disk optical resonator. 
     Example 37 
     The microwave-frequency source of any one of Examples 30 through 34 wherein the optical resonator comprises a fiber optical resonator. 
     Example 38 
     The microwave-frequency source of Example 37 wherein the optical resonator comprises a fiber Fabry-Perot optical resonator. 
     Example 39 
     The microwave-frequency source of Example 37 wherein the fiber optical resonator comprises a fiber-loop optical resonator. 
     Example 40 
     The microwave-frequency source of any one of Examples 30 through 39 wherein each one of the first and second pump laser sources is frequency-locked to the corresponding resonant optical mode of the optical resonator by a Pound-Drever-Hall mechanism. 
     Example 41 
     The microwave-frequency source of any one of Examples 1 through 29 wherein the dual optical-frequency reference source comprises a dual-mode laser source. 
     Example 42 
     The microwave-frequency source of Example 41 wherein the dual optical-frequency reference source comprises first and second reference laser sources, wherein the first and second laser sources are each frequency-locked to a corresponding distinct resonant optical mode of a common optical reference cavity. 
     Example 43 
     The microwave-frequency source of Example 42 wherein the third optical reference signal is provided by a third laser source, and the third laser source is frequency-locked to a corresponding resonant optical mode, distinct from the optical modes to which are frequency-locked the first and second laser sources, of the common optical reference cavity. 
     Example 44 
     The microwave-frequency source of any one of Examples 42 or 43 wherein the frequency f M  is about equal to a free spectral range of the common optical reference cavity or an integer submultiple thereof. 
     Example 45 
     The microwave-frequency source of any one of Examples 1 through 44 wherein (i) the electro-optic sideband generator comprises one or more electro-optic phase modulators each driven by a corresponding portion of the sideband generator input electrical signal at the frequency f M , and (ii) the one or more phase modulators are arranged so as to transmit the third optical reference signal so as to generate the multiple optical sideband signals. 
     Example 46 
     The microwave-frequency source of any one of Examples 1 through 45 wherein (i) the electro-optic sideband generator comprises two or more electro-optic phase modulators each driven by a corresponding portion of the sideband generator input electrical signal at the frequency f M , and (ii) the two or more phase modulators are arranged in series so as to sequentially transmit the third optical reference signal so as to generate the multiple optical sideband signals. 
     Example 47 
     The microwave-frequency source of Example 46 wherein pairs of sideband optical signals are generated with n ranging from 2 up to at least 30. 
     Example 48 
     The microwave-frequency source of Example 46 wherein pairs of sideband optical signals are generated with n ranging from 2 up to at least 100. 
     Example 49 
     The microwave-frequency source of any one of Examples 45 through 48 wherein (i) the electro-optic sideband generator further comprises one or more electro-optic phase modulators each driven by a corresponding portion of the sideband generator input electrical signal at the frequency f M , an intensity modulator driven by a corresponding portion of the sideband generator input electrical signal at the frequency f M , a dispersion compensator, an optical amplifier, and a nonlinear optical medium, and (ii) the one or more phase modulators, the intensity modulator, the dispersion compensator, the optical amplifier, and the nonlinear optical medium are arranged in series so as to sequentially, in order, transmit the third optical reference signal so as to generate the multiple optical sideband signals. 
     Example 50 
     The microwave-frequency source of Example 49 wherein pairs of sideband optical signals are generated with n ranging from 2 up to at least 100. 
     Example 51 
     The microwave-frequency source of Example 49 wherein pairs of sideband optical signals are generated with n ranging from 2 up to at least 1000. 
     Example 52 
     The microwave-frequency source of any one of Examples 49 through 51 wherein (i) the intensity modulator comprises an electro-optic Mach-Zehnder modulator, (ii) the dispersion compensator comprises a suitably dispersive optical fiber, (iii) the optical amplifier comprises a doped optical fiber amplifier and (iv) the nonlinear optical medium comprises a nonlinear optical fiber. 
     Example 53 
     A method for generating a microwave-frequency output electrical signal at an output frequency f M , using the apparatus of any one of Examples 1 through 52, the method comprising: (a) using the dual optical-frequency reference source, generating (i) the first optical reference signal at the first optical reference frequency v 1  and (ii) the second optical reference signal at the second optical reference frequency v 2 &gt;v 1 ; (b) using the electro-optic sideband generator, (i) receiving the third optical reference signal at the third optical reference frequency v 0  and the sideband generator input electrical signal at the frequency f M  and (ii) generating therefrom the multiple sideband optical signals at respective sideband optical frequencies of the form v 0 ±nf M , wherein n is an integer; (c) using one or both of the first optical bandpass filter or the second optical bandpass filter, (i) transmitting through the first optical bandpass filter the first optical reference signal and the subset of the multiple sideband optical signals including the sideband optical signal at the frequency v 0 −N 1 f M , wherein N 1  is an integer, or (ii) transmitting through the second optical bandpass filter the second optical reference signal and the subset of the multiple sideband optical signals including the sideband optical signal at the frequency v 0 +N 2 f M , wherein N 2  is an integer; (d) using one or both of a first optical detector or a second optical detector, (i) receiving at the first optical detector the optical signals transmitted by the first optical bandpass filter and generating therefrom the first optical detector electrical signal at the first beat frequency f BEAT1 =|v 1 −(v 0 −N 1 f M )|, or (ii) receiving at the second optical detector the optical signals transmitted by the second optical bandpass filter and generating therefrom the second optical detector electrical signal at the second beat frequency f BEAT2 =|v 2 −(v 0 +N 2 f M )|; (e) using a reference oscillator, generating a reference oscillator electrical signal at a reference oscillator frequency f R ; (f) using an electrical circuit, (i) receiving the reference oscillator electrical signal and only one of (A) the first optical detector electrical signal, (B) the second optical detector electrical signal, or (C) the difference electrical signal generated from the first and second optical detector signals by the electrical frequency mixer at the electrical difference frequency f DIFF =f BEAT2 −f BEAT1 =|v 2 −v 1 −(N 1 +N 2 )·f M |, (ii) generating therefrom, using the comparator portion of the electrical circuit, the electrical error signal dependent on the relative phase of the electrical signals received by the electrical circuit, and (iii) processing the electrical error signal using the loop-filter portion of the electrical circuit; and (g) using the voltage-controlled electrical oscillator, (i) receiving the loop-filtered electrical error signal as the VCO input electrical signal and (ii) generating the VCO output electrical signal at the frequency f M , wherein the first portion of the VCO output electrical signal is received by the electro-optic sideband generator as the sideband generator input electrical signal and the second portion of the VCO output electrical signal forms the output electrical signal of the microwave-frequency source, (h) wherein reception of the first portion of the VCO output electrical signal by the electro-optic sideband generator as the sideband generator input electrical signal results in the electrical circuit and the voltage-controlled oscillator being coupled in a negative feedback arrangement so as to function as a phase-locked loop. 
     It is intended that equivalents of the disclosed example embodiments and methods shall fall within the scope of the present disclosure or appended claims. It is intended that the disclosed example embodiments and methods, and equivalents thereof, may be modified while remaining within the scope of the present disclosure or appended claims. 
     In the foregoing Detailed Description, various features may be grouped together in several example embodiments for the purpose of streamlining the disclosure. This method of disclosure is not to be interpreted as reflecting an intention that any claimed embodiment requires more features than are expressly recited in the corresponding claim. Rather, as the appended claims reflect, inventive subject matter may lie in less than all features of a single disclosed example embodiment. Thus, the appended claims are hereby incorporated into the Detailed Description, with each claim standing on its own as a separate disclosed embodiment. However, the present disclosure shall also be construed as implicitly disclosing any embodiment having any suitable set of one or more disclosed or claimed features (i.e., a set of features that are neither incompatible nor mutually exclusive) that appear in the present disclosure or the appended claims, including those sets that may not be explicitly disclosed herein. In addition, for purposes of disclosure, each of the appended dependent claims shall be construed as if written in multiple dependent form and dependent upon all preceding claims with which it is not inconsistent. It should be further noted that the scope of the appended claims does not necessarily encompass the whole of the subject matter disclosed herein. 
     For purposes of the present disclosure and appended claims, the conjunction “or” is to be construed inclusively (e.g., “a dog or a cat” would be interpreted as “a dog, or a cat, or both”; e.g., “a dog, a cat, or a mouse” would be interpreted as “a dog, or a cat, or a mouse, or any two, or all three”), unless: (i) it is explicitly stated otherwise, e.g., by use of “either . . . or,” “only one of,” or similar language; or (ii) two or more of the listed alternatives are mutually exclusive within the particular context, in which case “or” would encompass only those combinations involving non-mutually-exclusive alternatives. For purposes of the present disclosure and appended claims, the words “comprising,” “including,” “having,” and variants thereof, wherever they appear, shall be construed as open ended terminology, with the same meaning as if the phrase “at least” were appended after each instance thereof, unless explicitly stated otherwise. For purposes of the present disclosure or appended claims, when terms are employed such as “about equal to,” “substantially equal to,” “greater than about,” “less than about,” and so forth, in relation to a numerical quantity, standard conventions pertaining to measurement precision and significant digits shall apply, unless a differing interpretation is explicitly set forth. For null quantities described by phrases such as “substantially prevented,” “substantially absent,” “substantially eliminated,” “about equal to zero,” “negligible,” and so forth, each such phrase shall denote the case wherein the quantity in question has been reduced or diminished to such an extent that, for practical purposes in the context of the intended operation or use of the disclosed or claimed apparatus or method, the overall behavior or performance of the apparatus or method does not differ from that which would have occurred had the null quantity in fact been completely removed, exactly equal to zero, or otherwise exactly nulled. 
     In the appended claims, any labelling of elements, steps, limitations, or other portions of a claim (e.g., (a), (b), (c), etc., or (i), (ii), (iii), etc.) is only for purposes of clarity, and shall not be construed as implying any sort of ordering or precedence of the claim portions so labelled. If any such ordering or precedence is intended, it will be explicitly recited in the claim or, in some instances, it will be implicit or inherent based on the specific content of the claim. In the appended claims, if the provisions of 35 USC § 112(f) are desired to be invoked in an apparatus claim, then the word “means” will appear in that apparatus claim. If those provisions are desired to be invoked in a method claim, the words “a step for” will appear in that method claim. Conversely, if the words “means” or “a step for” do not appear in a claim, then the provisions of 35 USC § 112(f) are not intended to be invoked for that claim. 
     If any one or more disclosures are incorporated herein by reference and such incorporated disclosures conflict in part or whole with, or differ in scope from, the present disclosure, then to the extent of conflict, broader disclosure, or broader definition of terms, the present disclosure controls. If such incorporated disclosures conflict in part or whole with one another, then to the extent of conflict, the later-dated disclosure controls. 
     The Abstract is provided as required as an aid to those searching for specific subject matter within the patent literature. However, the Abstract is not intended to imply that any elements, features, or limitations recited therein are necessarily encompassed by any particular claim. The scope of subject matter encompassed by each claim shall be determined by the recitation of only that claim.