Determining the Angle-of-Arrival of an Radio-Frequency Source with a Rydberg Atom-Based Sensor
Amy K. Robinson, Nikunjkumar Prajapati, Damir Senic, Matthew T. Simons, Joshua A. Gordon, Christopher L. Holloway
DDetermining the Angle-of-Arrival of an Radio-Frequency Source with aRydberg Atom-Based Sensor a) Amy K. Robinson, Nikunjkumar Prajapati, Damir Senic, Matthew T. Simons, and Christopher L. Holloway b) Depart. of Electr. Engin., University of Colorado, Boulder, CO 80305, USA ANSYS, Inc., Boulder, CO, USA National Institute of Standards and Technology, Boulder, CO 80305, USA (Dated: 29 January 2021)
In this work, we demonstrate the use of a Rydberg atom-based sensor for determining the angle-of-arrival of anincident radio-frequency (RF) wave or signal. The technique uses electromagnetically induced transparencyin Rydberg atomic vapor in conjunction with a heterodyne Rydberg atom-based mixer. The Rydberg atommixer measures the phase of the incident RF wave at two different locations inside an atomic vapor cell.The phase difference at these two locations is related to the direction of arrival of the incident RF wave.To demonstrate this approach, we measure phase differences of an incident 19.18 GHz wave at two locationsinside a vapor cell filled with cesium atoms for various incident angles. Comparisons of these measurements toboth full-wave simulation and to a plane-wave theoretical model show that these atom-based sub-wavelengthphase measurements can be used to determine the angle-of-arrival of an RF field.The ability to measure angle-of-arrival (AoA) is ofgreat importance to radar and advanced communicationsapplications. Here we present a method of determin-ing AoA based on Rydberg-atom sensors. Atom-basedsensors have garnered a lot of attention in the past sev-eral years because of their many possible advantages overother conventional technologies. Measurement standardshave evolved towards atom-based measurements over thelast couple decades; most notably length (m), frequency(Hz), and time (s) standards. Recently there has beena great interest in extending this to magnetic and elec-tric (E) field sensors. In particular, since the initiationand completion of DARPA’s QuASAR program, NISTand other groups have made great progress in the devel-opment of Rydberg atom-based radio-frequency (RF) E-field sensors . The Rydberg atom-based sensors nowhave the capability of measuring amplitude, polariza-tion, and phase of the RF field. As such, various ap-plications are beginning to emerge. These include SI-traceable E-field probes , power-sensors , receivers forcommunication signals (AM/FM modulated and digitalphase modulation signals) , and even recording mu-sical instruments . In this paper, we investigate the ca-pability of a Rydberg atom-based sensor for determiningAoA of an incident RF field.The majority of the work on Rydberg atom-basedE-field sensors uses on-resonant electromagnetically in-duced transparency (EIT) and Autler-Townes (AT) split-ting techniques . The concept uses a vapor of alkaliatoms placed in a glass cell (referred to as a “vapor cell”)as a means of detecting and receiving the RF E-field orsignal. The EIT technique involves using two lasers. Onelaser (called a “probe” laser) is used to monitor the opti-cal response of the medium in the vapor cell and a second a) Publication of the U.S. government, not subject to U.S. copy-right. b) Electronic mail: [email protected] laser (called a “coupling” laser) is used to establish a co-herence in the atomic system. When the RF E-field isapplied, it alters the susceptibility of the atomic vaporseen by the probe laser. By detecting the power in theprobe laser propagating through the cell, the RF E-fieldstrength can be determined. This approach has shown tobe very successful for determining the magnitude of anRF E-field. However, an alternative approach is requiredto measure phase, which is necessary to determine AoA.Recently, we developed a heterodyne technique using aRydberg atom-based mixer . In this approach, a ref-erence RF field is applied to the atoms. This referenceRF field is on-resonance with the Rydberg-atom transi-tion, and acts as a local oscillator (LO). The LO fieldcauses the EIT/AT effect in the Rydberg atoms whichis used to down-convert a second, co-polarized RF field(referred to as SIG and is the field for which the phase isdesired). The SIG field is detuned (by a few kHz) fromthe LO field. The frequency difference between the LOand the SIG is an intermediate frequency (IF) and theIF is detected by optically probing the Rydberg atoms.This IF is essentially the beat-note between the LO andSIG frequencies. The phase of the IF signal correspondsdirectly to the relative phase between the LO and SIGsignals. In effect, the atoms down-convert the SIG to theIF, and the phase of SIG is obtained by the probe laserpropagating through the atomic vapor.In order to determine the AoA, the phase ( φ ) of SIGis needed at two different locations, see Fig. 1. Once thephase of SIG is determined at the two different locations,the relationship between AoA (defined as θ in Fig. 1) andthe phase difference at the two locations (location 1 and2 in the figure) can be calculated. Assuming SIG is aplane wave, the relationship between θ and φ is:∆ φ , = φ − φ ≈ k d sin( θ ) : θ ≈ sin − (cid:18) ∆ φ , k d (cid:19) (1)where d is the separation between the two locations, φ , are the phases of SIG at the two locations, k = 2 π/λ , a r X i v : . [ phy s i c s . a t o m - ph ] J a n FIG. 1. Incident plane wave (SIG) onto three locations sepa-rated by d and offset by t . and λ is the wavelength of SIG. This expression assumesthat the line formed by locations 1 and 2 is perpendicularto the line for which the angle θ is measured. If the twolocations (say locations 1 and 3 in Fig. 1) form a line thatis not perpendicular to the line that determine θ , thenthe phase difference between locations 1 and 3 are givenby ∆ φ , = φ − φ ≈ k (cid:112) d + t sin (cid:2) θ + tan − ( t/d ) (cid:3) (2) θ ≈ sin − (cid:18) ∆ φ , k √ d + t (cid:19) − tan − ( t/d ) , (3)where t is defined in Fig. 1. Eqs. (1)-(3) relate AoA tothe measured phase of the SIG and LO signals at twolocations in the cell, assuming that the AoA is definedin a plane orthogonal to the probe laser propagation.Future work will include the measurement AoA in twodimensions, see discussion below.To measure the phase at any two different locations,we generate EIT in two locations inside a vapor cellfilled with Cs, see Fig. 2(a). The probe laser issplit with a beam cube and passed through the vaporcell at two locations. The full power of the couplinglaser is passed through each of the two locations, seeFig. 2(a). The beam directions are chosen to ensurethat at both locations in the cell, the probe and cou-pling lasers are counter-propagating. To generate EIT atthe two locations in the cell, we tune the probe laser tothe D transition for Cs (6 S / -6 P / or wavelength of λ p = 852 .
35 nm) focused to a full-width at half maximum(FWHM) of 390 µ m, with a power of 96 µ W. To pro-duce an EIT signal, we couple to the
Cs 6 P / -58 S / states by applying a counter-propagating coupling laserat λ c = 509 .
26 nm with a power of 60 mW, focused to aFWHM of 450 µ m.The LO and SIG are applied to the vapor cell as shownin Fig. 2(b), where the LO is at a fixed position and theSIG is rotated to different incident directions ( θ ). Weuse a signal generator (SG) to apply a continuous wave(CW) LO field at 19.18 GHz to couple states 58 S / and59 P / . While we use 19.18 GHz in these experiments,this approach can work at carriers from 100 MHz to1 THz (because of the broadband nature of the EIT/AT (a) Laser field schematic(b) antenna arrangement FIG. 2. (a) Schematic of the orientation of the optical fields.The probe beam is split in two by a beam cube, and one cou-pling field is re-circulated using a dichroic mirror to counter-propagate along each probe beam. (b) The LO antenna is sus-pended above the cell, such that the LO field is incident nearlyperpendicular to the line between the two optical beams. TheSIG is held by an adjustable arm to vary the angle of in-cidence, which is measured using an electronic compass at-tached to the horn mount. approach ). A second SG is used to generate a CWSIG field at 19.18 GHz+ f IF (where the f IF =50 kHz)).The output from the two SG are connected to two stan-dard gain horn antennas via RF cables. The LO horn ismounted directly above the vapor cell and is stationary,whereas the SIG horn sits on a rotating arm which setsthe incident angle ( θ ).Two different photodetectors are used to monitor thetwo probe beams that travel through the vapor cell. Theoutput of the photodetectors are sent to an oscilloscopeand a lock-in amplifier. Fig 3(a) shows the beam positionat the two locations inside the vapor cell. These beampositions correspond to locations 1 and 3 as defined inFig. 1, and the phase relationship is given in eq. (3). Inour experiments, d = 2 . t = 0 . (a) (b) FIG. 3. (a)The x - y location of lasers inside vapor cell, wherethe origin is the center of the cell, and (b) Beat-note for twolocations inside the vapor cell. like a mixer and low pass filter in a classic RF hetero-dyne setup. The LO and SIG create a beat-note and theatoms respond directly to this beat-note, which is de-tected by the probe laser transmission measured on thephotodetectors. At each location inside the vapor cell,the total electric field ( E atoms ) is the sum of the LO andSIG fields ( E LO and E SIG ). The atoms demodulate thehigh-frequency ω LO field and the probe transmission asa function of time at locations i and j (1 and 3 as definedin Fig. 1) is given by T ( i,j ) ∝ | E atoms | ≈ E LO + E SIG cos (∆ ω t + φ i,j ) , (4)where φ i,j corresponds to the phase of SIG at locationsi and j, and ∆ ω = ω LO − ω SIG . Once φ i and φ j aredetermined from the probe laser transmissions measuredon the two different photodetectors, the phase difference(∆ φ ) between the two locations is given by∆ φ = φ j − φ i . (5)To be more exact, φ i,j is actually the phase difference(at each location) between the LO and SIG . In theseexperiments, LO is at a fixed location such that a mea-surement of ∆ φ is a measurement of the phase change ofSIG between the two locations.For a given incident angle θ , the beat-notes as mea-sured from the two photodetectors are shown in Fig. 3(b).From the figure, we see the “cosine” behavior as predictedby eq. (4) with a period of 20 µ s (or the IF frequency of50 kHz used in the experiments). In this figure we seethat the two beat-notes are shifted in phase. This is thephase difference for the given incident angles that is de-fined in eq. (3).Using the setup shown in Fig. 2(b), the SIG antennais scanned from θ = ± o . The phase difference (∆ φ )at each θ position was determined and the measured ∆ φ for each incident angle is shown in Fig. 4(a). The errorbars correspond to the standard deviation of 5 data sets.The uncertainties of Rydberg atom based measurementsin general are discussed in Ref. and it is shown in Ref. that the heterodyne Rydberg atom-based mixer approachcan measure the phase to within 1 o . Also shown in this figure are the theoretical results given in eq. (3). Uponcomparing the experimental results to the theoretical re-sults, we see that while the standard deviation for thephase measurement for each incident angle is small (i.e.,small error bars), the measurements do not lie exactlyon the theoretical results. The reason why the data doesnot exactly follow the theoretical model is twofold. First,from Fig. 2(b) we see there are several objects in the ap-paratus used to rotate the SIG antenna. These objectscause scattering which are not accounted for in the the-oretical results. The second reason is due to the vaporcell itself. Because the vapor cell is a dielectric, the RFfields can exhibit multi-reflections inside the cell and RFstanding waves (or resonances) in the field strength candevelop in the cell . Thus, for a given location in-side the cell, the RF field can be larger or smaller thanthe incident field and the phase of the field at a givenlocation will be perturbed as well. Hence, the stand-ing wave can generates differences in the measured AoAwhen compared to the expected sinusoidal relationshipas given in eqs. (1) and (3). Numerical models can beused to investigate this effect. While modeling the entirestructure used to support the SIG antenna is difficult, wecan use full-wave numerical tools to simulate the vaporcell effects.We use ANSYS HFSS (High Frequency StructureSimulator) to simulate only the SIG antenna and thevapor cell (including the plastic vapor-cell holder), seeFig. 5. HFSS convergence criteria was based on the en-ergy of a plane wave to 0.01 W, and the mesh aroundthe cell was seeded using curvilinear approximation, andinside the cell using length-restriction to 1 mm, withfirst order polynomial solving. With this model, we de-termine the phase at location 1 and 3 (as defined inFig. 1) and the ∆ φ , obtained from HFSS are shownin Fig. 4(a). To ensure that the phases are being calcu-lated correctly with the HFSS simulation, we first deter-mine ∆ φ , with no vapor cell present. These results areshown in Fig. 4(a) and match the theoretical calculationclosely, as expected. Now that we have confirmed thatthe HFSS is implemented correctly, the result from theHFSS for the case when the vapor cell is included areshown in Fig. 4(a). We see that the HFSS results (in-cluding the vapor cell) correspond well to the measureddata for angles > -25 o . As with the experimental results,the HFSS results indicate that the vapor cell does per-turb the phase measurement and causes deviation fromthe theoretical results. We see that the HFSS results donot correspond exactly to the measured data over all theangles, but do show the same trends. The deviations be-tween the measured data and HFSS are twofold. First,the exact permittivity ( (cid:15) r ) of the glass is not known, (cid:15) r ranges from 3 to 6 (in this numerical model we as-sume (cid:15) r = 5). Secondly, upon comparing the photo ofthe experimental setup in Fig. 2(b) and the HFSS modelin Fig. 5, we see that not all the objects used to rotatethe SIG antenna are included in the HFSS model. Withthat said, the measured and HFSS model compare well (a) (b) FIG. 4. (a) Experimental and HFSS data for ∆ φ . The errorbars correspond to the standard deviation of 5 data sets, and(b) AoA from the experimental data.FIG. 5. HFSS model for the cell and horn antenna. Thehorn is rotated by an angle Θ around the x − axis, such thatit points towards the cell. The model also shows the vaporcell holder. and follow the same trends, especially for angles > -25 o .There are asymmetries in the apparatus use in the ex-perimenters. For angles < -25 o degrees, the apparatusused to support the SIG antenna and vapor cell in theexperiment begin to influence the phase of the measureddata. The additional scattering caused by this apparatusis not included in the HFSS numerical model. While thevapor cell does perturb the measurement of ∆ φ , , theresults in Fig. 4(a) show that the Rydberg-atom basedsensor can detect the relative phase difference betweentwo locations inside the vapor cell and work as a AoAdetector.With the measured ∆ φ , the AoA can be determinedfrom eq. (3). Fig. 4(b) shows the AoA from the measured φ , for incident angles ranging from θ = ± o . Thesolid line in this figure represents the one-to-one corre-spondence of the incident angle and AoA. The measuredAoA should lay on the line. While the measured AoAfollows this line, it does not exactly lay on the line. Also,in this figure we show the AoA obtained by using theHFSS results for ∆ φ in eq. (3). Here again, we see thatthe HFSS results deviate from the solid line. As with the measured ∆ φ , the deviation in the measured AoA (andthe HFSS results for AoA) is due to the vapor cell per-turbation and due to the supporting apparatus used toexperimental equipment (SIG antenna and vapor cell).This demonstrates that the Rydberg atom-based sensorcan be used to determine AoA of an incident RF signal.While the cell does perturb the AoA measurement,two approaches can be pursued to mitigate this effect.One approach is to design a vapor cell that can minimizeand even eliminate the vapor cell perturbations. Variousgroups are investigating different approaches to modifythe vapor cell used for these Rydberg atom-based sensors.Two examples include the use of vapor cells with honey-comb sides or the use of metamaterials on the sides ofthe vapor cells . A second approach is to use the HFSSresults to calibrate the vapor cell to reduce the perturba-tion effects. This is done by defining a calibration factoras C = AoA HF SS − AoA theory (6)and subtracting this from the measured AoAAoA cal = AoA meas − C (7)where AoA
HF SS , AoA theory , and AoA meas are the AoAobtained from the HFSS results, theory, and experimen-tal results, respectively. Fig. 4(b) shows AoA cal . Whilethere is not a perfect correlation to the solid line withthe calibration based on the HFSS results, we do seethat the calibration did improve the AoA measurement,especially for angles > -25 o . Once again, the deviationsfrom the theory and HFSS simulation for angles < -25 o is due to the asymmetries in associated with the appara-tus to support the experiments. The larger discrepanciescould be handled with a more accurate models of theexperimental setup.This paper demonstrates that it is possible to de-termine the angle of arrival of an RF signal using anatom-based sub-wavelength phase measurement method.While the vapor cell perturbs this measurement, we seethat this effect can be mostly accounted for or at leastexplained, and these Rydberg atom-based sensors havethe capability of measuring the AoA of a incident RFsignal. Future iterations of this experiment will explorethe perimittivity of the glass for more precise modelingof the standing wave in the glass cell. We are also inves-tigating different types of vapor cell designs and beamorientations in order to minimize or eliminate the vaporcell effect on the AoA measurements.Now that we have demonstrated that it is possible todetermine the AoA with a Rydberg atom-based sensor,one can envision (1) developing arrays of these Rydbergatom sensors (2) or sampling the phase at numerous lo-cations inside one vapor cell in order to detect the AoAof more general incidence angles, or for the purpose ofsimultaneously detecting the AoA of several sources atonce. We are currently developing these two types ofsensors and these will be the topic of a future publica-tion. Gordon, J.A., et al., “Quantum-Based SI Traceable Electric-Field Probe,” Proc of , July 25-30, 321-324, 2010. Sedlacek, J.A., et al.,
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