We study the effect of relative motion on a frequency-entangled-based ranging scheme. Two major puzzles arise, i.e., Doppler shift and ambiguity velocity. During condition of rapid relative motion, Doppler shift invalidates the measurement result of this scheme; while during condition of slow relative motion, the ambiguity velocity turns into a major limitation. If relative speed between targets and measurement platform exceeds the ambiguity velocity, an accumulated profile obtained by the coincidence measurement will be distorted, which causes a lower ranging accuracy. Theoretical analysis shows a time-varying delay can be introduced to solve the two major puzzles.
© 2015 Optical Society of America
Distance ranging is widely used in the fields of satellite formation flying and large-scale manufacturing. In distance measuring systems, there are three critical parameters: precision, ambiguity range, and update rare (i.e., the time taken to acquire the data for a given precision). In terms of traditional techniques which are based on Maxwell’s equations, the first two parameters are generally contradictory. For instance, using shorter-wavelength signals, laser interferometer tends to obtain a better precision, however, at the cost of an insufficient ambiguity range .
Multi-wavelength interferometry combines measurements at several optical wavelengths, and can increase the ambiguity range [2, 3], but this technique is still insufficient for measuring a large distance. Recently, femtosecond optical frequency combs have been developed, which can offer a maximum ambiguity range of tens of kilometers [4–6]; however, a complex system and the susceptibility to systematic errors (i.e., spurious reflections and environment noise) are common defects of this scheme .
Employing some ‘quantum trick’ such as frequency entanglement, quantum-enhanced measurement can beat the conventional bounds to the accuracy of measurements [8–10]. Giovannetti et al. demonstrated that using M frequency entangled pulses could theoretically increase the ranging accuracy by a factor of M1/2 compared to using M classical pulses with the same bandwidth. Thus, a frequency-entangled-based ranging (FEB) scheme has been proposed [11, 12], which is generally based on the second-order coherence (SOC) of frequency entangled photons. In fact, when M = 2 in Ref [9, 10], the joint probability for the two detectors to detect the photons pair is just proportional to the SOC function, which corresponds to the principle of the FEB scheme. Besides an ultrahigh ranging precision, the FEB scheme has advantages of high immunity to systematic errors and no ambiguity range. Additionally, it is possible to implement a quantum cryptographic scheme that prevents an eavesdropper from obtaining the information of distance . Considering all of the factors above, this novel scheme may have a promising future in the applications of distance measurement.
Traditional ranging techniques (for example, pulsed radar systems) directly measure the time-of-flight of signals, while the FEB scheme measures the SOC of the photons pair and requires an accumulated time interval [t0, t0 + Ta] spent by the coincidence measurement, (t0 is the time when the detectors initially record the TOAs of the photons pair), which results in a slower update rate; it means that only when t = t0 + Ta, can we obtain the information of distance at time t = t0. The previous works [9–12] paid attention to a model of motionless targets for the FEB scheme. However, most practical situations involve the relative motion between targets and the measuring platform, and the distance we want to measure can vary during the accumulated time interval [t0, t0 + Ta]; because the SOC of the photons pair contains all of the information of distance during [t0, t0 + Ta], the distance information we obtained is no longer the distance at time t = t0. Additionally, Doppler shift due to the relative motion may have an influence on the performance of the FEB scheme. Thus, in this work, we theoretically analyze the major puzzles due to the effect of relative motion, and propose a technique to overcome them.
2. Scheme description
As shown in Fig. 1, Alice located at the measuring platform wants to measure the distance (i.e., d0) from Bob using the FEB scheme. Here Alice and Bob are no longer the consenting parties in a quantum cryptography experiment, Bob is passive and he merely reflects the photons to provide a delay from which the range can be determined. The procedure is as follows: a bunch of continuous-wave (CW) laser is focused on a BBO crystal through a Glan prism, and a pair of frequency entangled photons (i.e., idler photon and signal photon) is then generated, which is known as the spontaneous parametric down-conversion (SPDC). Once generated, the idler photons propagate directly towards detector D1 while signal photons travel through a round trip between Alice and Bob, and are then collected by detector D2. The times of arrival (TOAs) of the signal and idler photons are recorded by the two detectors, and a coincidence measurement is then performed on the time difference of arrival (TDOA) τ of the two photons with a coincidence counts rate (i.e., Pc) proportional to the SOC of the signal-idler photon pair (i.e., G(τ)). After an accumulated time interval Ta, we gains the accumulated profile n(τ) (a distribution histogram of τ), which corresponds to G(τ) .
Since G(τ) is a single-peak function, one can find the maximum coincidence count at a single value (i.e., τmax). Assuming the time taken for the idler and signal photons to be travelled are t1 and t2, respectively, then we can obtain that:
Then, the difference in signal and idler photon paths (i.e., ldif) can be written as follows:
The precision of the FEB scheme is determined by the full-width at half-maximum (FWHM) of G(τ), which depends on the resolution of the coincidence measurement. Using two-photon absorption (TPA) in a GaAs photon-counting module , the FWHM of G(τ) can reach 50 fs. Therefore, the precision of ranging is on the order of 7.5µm.
In this section, the target Bob in Fig. 1 is motionless; therefore, the accumulated profile n(τ) is stable, and τmax can be utilized to measure the distance d. When Bob starts to move, another situation will happen, and details are presented in Section 3.
3 Major puzzles caused by the relative motion
3.1 Effect of Doppler shift
In a scenario of relative motion, for simplicity, we assume Bob moves at a constant speed of v with respect to Alice. Specifying a certain time t during the accumulated time interval [t0, t0 + Ta], the distance d = d0 – v (t - t0), and the difference in the light paths is varying during the time interval.
The SPDC satisfies the condition of degenerate phase matching so that the center frequency of the signal-idler photon pair equals a half of the pump frequency (i.e., ωp); then, the state of signal-idler photon pair can be written as follows:
The signal-idler photon pair then travels a different light paths in a non-dispersive medium and is detected by the two detectors. Doppler shift unquestionably exists in the scenario of relative motion, therefore, the energy of the reflected signal photon is red or blue shifted. As Bob moves towards to Alice, the frequency shift of the reflected signal photon (i.e., Δω2) can be expressed as follows:Eq. (5) can be rewritten as follows:15], namely, with a pass band 1.885 × 1015 – 5.386 × 1015 Hz. The speed of the currently fastest aircraft can reach Mach 20 (i.e., 20 times the speed of sound), and we select a CW pump at a wavelength of 363.8 nm which is the same to that used in the simulation, then the frequency shift can be calculated as 2.3489 × 1011 Hz; thus, it is concluded that even with the fastest aircraft, the frequency of reflected photon cannot detune beyond the spectral response of the detector, and we should not be concerned about it.
Additionally, due to the effect of Doppler shift, the annihilation operator of the reflected signal photon can be affected . As the coincidence counts rate is related to the annihilation operators, this may result in an influence on the distance ranging. Based on Ref , the annihilation operators (i.e., a1 (ω1) and a2 (ω2)) at the detectors 1 and 2, respectively, can be written as follows:Eq. (5).
Then, the electromagnetic field operators at detectors 1 and 2 can be written as follows:17], the coincidence rate is written as follows:Equation (7) can be further represented by the following form [11, 18]:
Equation (13) shows the coincidence rate depends on t2 - χt1 instead of TDOA (i.e., τ = t2 - t1); thus, Doppler shift can invalidate the result of the coincidence measurement of the FEB scheme during condition of rapid relative motion.
3.2 Ambiguity velocity
When relative motion is rapid, the FEB scheme is useless to measure the distance. In this section, we evaluate the performance of the FEB scheme under condition of slow relative motion where the effect of Doppler shift can be ignored; then, a2 (ω2) in Eq. (7) is changed into the following form:
Then, G(t1,t2) in Eq. (11) is rewritten to be:Eq. (12) can be rewritten as follows:
Equation (17) shows the current coincidence counts rate (i.e., Pc) depends on t2 - t1, thus the FEB scheme is valid during condition of slow relative motion; however, a new puzzle arises: Pc is time-variant during [t0, t0 + Ta]. As the value of τ (i.e., τmax), which makes Pc equal its maximum value, is related to the difference in the signal and idler light paths (i.e., ldif (t)); then τmax is shifted along the axis of TDOA during [t0, t0 + Ta], and the accumulated profile n(τ) in Section 2 is distorted after a time interval Ta, which causes a lower ranging accuracy.
Thus, we try to find a situation where the accumulated profile is ‘slightly distorted’. The ‘slightly distorted’ means that the distance d is able to be measured with an accuracy reaching the same order as that achieved by a non-distorted profile. Note that the essential principle of the FEB scheme is measuring the difference in the light path (i.e., ldif); thus, if the distance variation of ldif during the time interval Ta was less than the ranging accuracy, the shift of τmax along the axis of TDOA would be very slight and the ‘slightly distorted’ profile could be obtained:Eq. (3); v is the relative velocity that Bob moves towards Alice.
Since the ranging precision of FEB scheme depends on the half of the FWHM of the accumulated profile, we can obtain:Eq. (2); is the value of τ which makes G(τ) obtain a half of the maximum value.
3.3 Effect of background noise on the FEB scheme
As the idler light path is not exposed to background light, we merely consider the effect of jamming noise that has the same bandwidth on the signal light path; based on Refs [7, 12], the coincidence count rate can be written as follows:7, 12]; then, Eq. (24) can be written further as follows:19], a reasonable definition of signal to noise ratio (SNR) is dividing the expected total coincidence count (i.e., Rs) by the standard deviation of the total counts of two coincidence channels. Ref . shows that both the coincidence photon counts and the background noise counts (i.e., b) obey the Poissonian distribution, then, the standard deviation of the total counts of two coincidence channels can be expressed as follow:12]. The measurement result still has a good performance and the FWHM of the accumulated profile is unchanged, even in an environment that has a noise level 12.84 dB above the signal photons. Additionally, another method can also be used: the extra term 〈Ei〉 〈En〉 can be estimated through measuring the average power of the noise, then it can be eliminated through careful data calibration ; thus, the environment noise cannot affect the coincidence measurement. By the way, we derived the expression of the accumulated time Ta which determines the update rate through Eq. (27):
3.4 Simulation and analysis
3.4.1 Distortion of the profile in a scenario of relative motion
Valencia et al.  and Xiao et al.  have already performed a proof-of-principle experiment on the FEB scheme in a motionless scenario. The signal-idler photon pair were produced from two 0.8-mm-thick type-II BBO crystals pumped by an Ar laser at a wave-length of 363.8 nm. The TDOAs of photons were measured by a coincidence measurement system with a resolution of 4 ps; the coincidence count rate which also contains the dark counts of the two detectors was :
In this simulation, the distance between Alice and Bob at t0 (i.e., d0) was set to 502.5 m, and the reference photon path on-board the platform (i.e., lreference) was set to 5 m. We used the same entangled photon source and coincidence measurement system as above  (i.e., n0 = 5820, ω = 1.8510−3, the accumulated time interval Ta = 10 s, and the FWHM of G(τ) equals 1504.38 ps); thus, the coincidence count rate could be derived using Eq. (29):
The probability of having a joint photo-detection event at two detectors is proportional to the coincidence count rate . Keep this idea in mind, then, we used the normalized pc(τ) as the probability function of the joint-detection of the signal-idler photon pair; after an accumulated time (Ta = 10 s), a histogram of the TDOA of the signal-idler photon pair was obtained and was plotted in Fig. 2, which is the accumulated profile in a motionless scenario. The natural value of the width of this profile is generally in the order of hundreds of femtoseconds (i.e., 10−13 second), which is determined by the bandwidth of the down-converted photons ; while in this work, the measured value of this width is in the order of nanosecond, which is restricted by the detector resolving time.
On the other hand, combining Eqs. (17) and (30), the coincidence count rate in the scenario of relative motion (i.e., pmotion(τ,t)) could be derived as follows:
Equation (31) shows that the probability function of the joint-detection of the signal-idler photon pair during condition of relative motion varies with time. Then, we divided the time interval [0, 10 s] into a great amount of small time bins, in which the corresponding probability functions are different. Selecting three values of the velocity of Bob (i.e., 0 m/s, 0.1 m/s and 1 m/s) and using the same accumulated time (Ta = 10 s), we obtained three accumulated profiles (histograms of the TDOA) in the Fig. 3.
Through the observations shown in Fig. 3, we can conclude: 1) compared with the black curve in the motionless scenario, both of the red and blue curves (correspond to values of 0.1 m/s and 1 m/s, respectively) are distorted; 2) a more rapidly motion of Bob yields a greater broadening of the FWHM of the accumulated profile, and it is thus difficult for us to measure the distance, which coincides with the analysis in Section 3.2; 3) although Bob moves towards Alice, the total number of signal-idler photon pair remains unchanged, which indicates the areas of three curves should be equal; therefore, the peaks of the distorted curve have dropped with the broadening of the FWHM of the distorted profiles, see Fig. 3.
3.4.2 Ambiguity velocity
The scale factor ξ in this simulation was set to 1, which means we loosen the degree of “slightly distortion” to the greatest extent; then, the ambiguity velocity was calculated to be 1.128 × 10−2 m/s, using Eq. (23) with the same parameters (i.e., FWHM = 1504.38 ps, Ta = 10 s) used in Section 3.4.1.
As shown in Fig. 4, the red curve represents the accumulated profile when Bob moves at the ambiguity velocity, and the shape of this profile remains almost unchanged; however, the shape of the blue curve is distorted seriously when Bob moves at a speed approximately ten times faster than the ambiguity velocity.
Additionally, the TDOA shift of the peak of the distorted profile with respect to that of the non-distorted profile inspires us to analyze Eq. (18) at another point of view. Then, we can change Eq. (18) into another form (ξ = 1 in this section):Eq. (2); τ|max[G(τ)] and τ|max[G(τ)]/2 represent the abscissas of the peak (i.e., Opeak in Fig. 4) and the half-maximum point (i.e., Ohalf in Fig. 4) of the non-distorted profile, respectively; τreference is the abscissa of a reference point (i.e., A and B in Fig. 4) of the distorted profile. Then, the two equations are easily derived from Eq. (33):
The reference point is used to judge whether the distorted profile is ‘slightly distorted’: if the displacement of τreference with respect to τ|max[G(τ)] is less than a half of the FWHM of the non-distorted profile (i.e., τreference lies on the right of τ|max[G(τ)]/2), the accumulated profile is ‘slightly distorted’.
While Bob moves at the ambiguity velocity, the reference point A on the red distorted profile in Fig. 4 has the same abscissa as the half-maximum point Ohalf, thus, it can be described as ‘slightly distorted’. However, when Bob moves at a speed exceeding the ambiguity velocity, the reference point B lies on the left of Ohalf. Thus, the blue profile is distorted seriously. The results of the simulation coincide with Eqs. (34) and (35); as the two equations are derived from Eq. (18), on the contrary, the results simultaneously verify the correctness of Eq. (18).
4. Doppler shift and ambiguity velocity cancellation
If the relative speed between Alice and Bob is slower than the ambiguity velocity (i.e., vambiguity), the FEB scheme can be used to measure the distance d. Though we loosen the degree of “slightly distortion” to the greatest extent in Section 3.4.2, the value of the ambiguity velocity is still so small that it severely restricts the applications of the FEB scheme. Thus it is urgent for us to remove the effect of the ambiguity velocity and Doppler shift.
As mentioned in Section 3, Bob moves towards Alice at a speed of v; based on Ref , the motion of Bob performs the same role as a time-varying delay δlB(t), see Fig. 5. Consider the case of linear time dependence, namely:
The motion of Bob simply exists in the light path of the signal photon, thus, we could attempt to introduce another time-varying delay δlA(t) in the signal photon path to cancel the effect of δlB(t). See Fig. 5, the time-varying delay δlA(t) is placed in front of the detector 2, which can be represented as:
Then, let us analyze the evolution of the annihilation operator for the signal photon. Once produced by the photon source, this operator is denoted by as(ω2); one find that as(ω2) can be evolved into aBob(ω2) at Bob’s position (at a distance d1 from Alice) and then into a’2(ω2) at the detector 2’s position (at a distance d2 from Bob). As Bob is far from Alice, for simplicity, we assume δlA(t) and detector 2 are at the same position, i.e., d1 = d2 = d0 at the initial time t0. Therefore, we can obtain:
To eliminate the effect of relative motion, the annihilation operator a’2(ω2) should be independent of the time t, namely:Equation (40) shows the time-varying rate of δlA(t) (i.e., vadd) should equal χv, and χadd can be represented as follows:Eq. (41):
The subsequent procedure of deriving the coincidence rate Pc corresponds to that of Eqs. (7-12); thus, the details are omitted here, and we can obtain:Eq. (13) that is a first-order term related to β. The speed of the currently fastest aircraft can reach 20 times the speed of sound, then the maximum of 1-χ can reach an order of 10−5 while that of 1-γ reaches an order of 10−10. Although Pc in Eq. (44) is still dependent on t2 - γt1 at first sight, due to the extremely small value of 1-γ, Pc can in practice be written as a function of TDOA (i.e., τ):
Finally, the coincidence rate Pc is independent of the time t, and a stable accumulated profile n(τ) of the coincidence measurement can be obtained after a time interval Ta. Thus, we can determine the distance d0 between Alice and Bob at the time t0:
Equation (46) coincides with Eq. (3) in Section 2, which indicates that the effect of relative motion on the FEB scheme has been removed. In summary, a time-varying delay δlA(t) with the time-varying rate of χv can be introduced to solve the two major puzzles.
In this study, we have analyzed the effect of relative motion on the FEB scheme. Two major puzzles have arisen, namely, Doppler shift and ambiguity velocity. When the effect of the Doppler shift is obvious, the SOC of signal-idler photon pair would not depend on the DTOA of photon pairs; thus, the FEB scheme is invalid to measure the distance. When the relative motion is relatively slow, the effect of Doppler shift can be ignored; however, if the relative speed between Alice and Bob exceeds the ambiguity velocity, the accumulated profile will be distorted, which causes a lower ranging accuracy. The ambiguity velocity was calculated as 1.128 cm/s in Section 3.4.2, and this small value would severely restrict the applications of the FEB scheme.
Through the theoretical analysis, introducing a time-varying delay δlA(t), which is placed in front of the detector 2, is able to solve the two major puzzles; the time-varying rate of δlA(t) is directly related to the relative velocity v, which can be determined by the use of time-tagged photons in time correlated singe photon counting: if the absolute time t0 is recorded for the detection of the coincident pairs, then the order in which the counts are entered into the histograms in Figs. 3 and 4 is known; this allows the relative velocity to be determined from an initial guess, since the TDOA can be corrected to account for the relative velocity; then, the correct relative velocity is the one which gives the highest and shortest (i.e. sharpest) peak. Our work has extended the application of the FEB scheme from the static scenario to the motion one, and hopefully, will give some guidance on the practical designs of quantum ranging sensor for measuring the moving target.
This work was supported by the National Natural Science Foundation of China (Grant No. 61172138, No. 61401340), the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2013JQ8040), the Open Research Fund of The Academy of Satellite Application (Grant No. 2014_CXJJ-DH_12), the Shandong Aerospace Innovation Fund (2013JJ04), the Open Fund of Key Laboratory of Precision Navigation and Timing Technology, National Time Service Center, CAS(NO. 2014PNTT07)
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