Abstract
Barreto Lemos et al. [Nature 512, 409–412 (2014)] reported an experiment in which a nondegenerate parametric downconverter and a nondegenerate optical parametric amplifier—used as a wavelengthconverting phase conjugator—were employed to image object transparencies in a manner akin to ghost imaging. Their experiment, however, relied on singlephoton detection, rather than the photoncoincidence measurements employed in ghost imaging with a parametric downconverter source. More importantly, their system formed images despite the photons that passed through the object never being detected. Barreto Lemos et al. interpreted their experiment as a quantum imager, as assuredly it is, owing to its downconverter’s emitting entangled signal and idler beams. We show, however, that virtually all the features of their setup can be realized in a quantummimetic fashion using classicalstate light, specifically a pair of bright pseudothermal beams possessing a phasesensitive cross correlation. Owing to its much higher signaltonoise ratio, our brightsource classical imager could greatly reduce imageacquisition time compared to that of Barreto Lemos et al.‘s quantum system, while retaining the latter’s ability to image with undetected photons.
Introduction
Light is intrinsically quantum mechanical and photodetection is a quantum measurement. Consequently, all imaging is really quantum mechanical. It has long been known, however, that the semiclassical theory of photodetection—in which light is a classical field and the discreteness of the electron charge results in photodetection shot noise—predicts measurement statistics identical to those obtained from quantum theory when the illumination is in a classical state, namely a Glauber coherent state or a classicallyrandom mixture of such states. (See Ref. 1 for a review of quantum versus semiclassical photodetection.) Thus, because experiments whose quantitative behavior is correctly predicted by two disparate theories cannot distinguish between those two theories, it is entirely appropriate that the term quantum imaging be reserved for imagers whose quantitative understanding requires the use of quantum theory. (See Refs. 2, 3, 4 for how a debate on this point has been settled with regards to pseudothermal ghost imaging.)
Threewave mixing in a secondorder nonlinear material is the workhorse of nonclassical lightbeam generation, with spontaneous parametric downconverters producing entangled signal and idler beams^{5}, optical parametric amplifiers producing squeezedvacuum states^{6} and optical parametric oscillators producing photontwin beams^{7}. It follows that imagers using any such sources will be quantum imagers, according to the criterion described in the preceding paragraph. Two such examples are the initial ghostimaging experiment^{8} and quantum optical coherence tomography (QOCT)^{9,10}. Both used continuouswave (cw) spontaneous parametric downconversion (SPDC) sources, whose signal and idler outputs were taken to comprise streams of biphotons detectable by photoncoincidence counting. The joint state obtained from cw SPDC, however, is really a zeromean Gaussian state that is completely characterized by the signal and idler’s nonzero correlations, viz., their phaseinsensitive autocorrelations and their phasesensitive cross correlation^{11}. Signalidler entanglement is then manifest by the phasesensitive cross correlation’s exceeding the classical limit set by the autocorrelations. The biphoton limit ensues for the experiments reported in Refs. 8,10 because cw SPDC emits lowbrightness outputs—average signal and idler photons/secHz much less than 1—whose signalidler photon pairs are easily resolved in time by highspeed photodetectors^{11}.
Phasesensitive cross correlations cannot be sensed in secondorder interference. Hence the aforementioned experiments’ reliance on photoncoincidence counting, which is a fourthorder interference measurement able to sense phasesensitive cross correlations^{12}. More important, for our purpose, is the fact that classical light beams can have phasesensitive cross correlations. Indeed, Ref. 13 showed theoretically and Ref. 14 verified experimentally, that signal and idler beams in a zeromean jointly Gaussian classical state—determined by their nonzero phaseinsensitive autocorrelations and phasesensitive crosscorrelation—yielded ghost images with almost all of the properties of the quantum case.
Threewave mixing can also be used to phaseconjugate a light beam that had only a phasesensitive cross correlation with a companion beam. The resulting conjugate then has a phaseinsensitive cross correlation with that companion, which can be sensed via secondorder interference. This possibility was exploited, theoretically in Ref. 15 and experimentally in Ref. 16, to realize phaseconjugate optical coherence tomography, in which classicalstate signal and idler beams—of the type mentioned in the preceding paragraph—yielded the axial resolution and dispersion immunity afforded by QOCT without the need for nonclassical light.
The upshot of our long introduction is simply this. Quantum imagers employing the entangled signal and idler beams obtained from SPDC often have quantummimetic relatives that use classicalstate signal and idler beams to realize the essential performance characteristics of the original quantum systems. We show that the experiment of Barreto Lemos et al.^{17} is another such example. In particular, by replacing Barreto Lemos et al.‘s entangledstate source with a classicalstate source whose signal and idler beams have a phasesensitive cross correlation, we see that a wavelengthconverting phase conjugator again allows imaging with undetected photons. Moreover, our system can employ highbrightness light, thus leading to a much higher signaltonoise ratio than its quantum counterpart and hence the possibility of greatly reduced imageacquisition time. Before proceeding to establish these results, two explanatory notes concerning “classical imaging with undetected photons” are warranted.
Whereas photons are energy quanta of electromagnetic waves, classical electromagnetic waves are not comprised of photons. So how can we do classical imaging with photons, much less with undetected photons? As explained in our opening paragraph, all imaging is quantum, but we have reserved the term quantum imaging for systems whose statistical descriptions require quantum photodetection theory. Imagers whose statistics can be correctly derived from semiclassical photodetection theory are classicalstate (proper function) imagers, so a more precise description of our work would be to say we will analyze classicalstate imaging with undetected photons. That clarification, however, still leaves us with the question of whether Barreto Lemos et al.‘s quantum imager or our classicalstate imager form their images with undetected photons. Here the quantum explanation—applicable to both the quantum imager (with its nonclassical light source) and the classicalstate imager (the former’s quantummimetic counterpart)—is as follows. Spontaneous parametric downconversion in the wavelengthconverting phase conjugator fissions a small fraction of the pump photons into signalidler photon pairs. But the presence of incoming idler photons causes that threewave mixer to emit additional signalidler photon pairs. Thus, for both imagers considered below, the photons that interacted with the object being imaged are indeed not the photons that are detected.
Results
The setup we shall consider, which mimics that of Ref. 17, is shown in Fig. 1. Here, a signalidler source produces a wavelength λ_{s} signal beam (green beam in Fig. 1) and wavelength λ_{i} idler beam (red beam in Fig. 1) that are separated by a dichroic mirror. The idler beam then propagates through an object transparency and a wavelengthconverting phase conjugator whose output at wavelength λ_{s} is mixed with the signal on a 50–50 beam splitter. The beam splitter’s outputs are detected by cameras whose outputs, we will show, contain positive and negative images of the object transparency, just as reported by Barreto Lemos et al.^{17} for their quantum imager. Lenses, not shown in Fig. 1, image the source’s idlerbeam output onto the object transparency and the idler light transmitted through that transparency onto the phase conjugator. Other lenses, also omitted from Fig. 1, image the source’s signalbeam output and the signalwavelength output from the phase conjugator onto the cameras, completing an equal pathlength interferometer.
For a unified treatment of the quantum and quantummimetic versions of the Fig. 1 setup, we shall employ quantum analysis for both. Quantitatively identical results are obtained for the quantummimetic case when: (1) we use classical wave propagation for the objecttransparency and beamsplitter interactions; (2) we use the classicalnoise model for the action of the wavelengthconverting phase conjugator; and (3) we use the semiclassical (shotnoise plus illumination excessnoise) theory of photodetection. For simplicity, in what follows, we shall discretize in both space and time, i.e., we will take the source’s signal and idler outputs to be a collection of modes with photonannihilation operators for a P × P array of pixels, indexed by (j,k) and a sequence of pulses, indexed by . These modes will be assumed to be in a zeromean jointlyGaussian state whose nonzero correlation functions are
and
where δ_{ab} is the Kronecker delta function. Equations (1, 2, 3) imply pixeltopixel and pulsetopulse statistical independence, with the quantum imager having the maximum quantummechanical phasesensitive cross correlation given its outputs’ average of photons per mode and the classical imager having its maximum classical phasesensitive cross correlation under this same constraint. Note, however, that the SPDC source used to produce the quantum imager’s signal and idler beams operates at low brightness, . On the other hand, the classical imager’s signal and idler beams—which can be obtained by using spatial light modulators to apply phaseconjugate pseudorandom modulations to wavelength λ_{s} and λ_{i} laser beams—can have high brightness, . Thus, to disambiguate these two cases in all that follows, we shall use and to denote the average permode photon numbers for the quantum and classical imagers, respectively.
Regardless of whether the source is quantum or classical, the annihilation operators for the wavelengthλ_{i} modes emerging from the object transparency are
where T_{jk} is the object’s complexvalued field transmission at pixel (j,k) and the are the annihilation operators for a collection of vacuumstate auxiliary modes associated with transmission loss through the object. The annihilation operators for the wavelengthλ_{s} modes emerging from the phase conjugator are then
where G>1 is the conjugator’s gain, assumed to be real valued and the are the annihilation operators for the collection of vacuumstate wavelengthλ_{s} modes at the conjugator’s input. Here, because we have assumed that the gain value is the same for all pixels, it is crucial that the conjugator be a spatiallybroadband device. As such it is very likely to have , e.g., it could be an SPDC system into which the modes are injected. Indeed, if the conjugator uses the same crystal and pump power as the quantum imager’s signalidler source, then will prevail, even for the classical imager. In what follows we will make that assumption. Note that the classical model for the conjugator, when the source emits classicalstate light, is then
where classical complex amplitudes and have replaced the annihilation and creation operators and and is a set of independent, identicallydistributed, zeromean, circulocomplex Gaussian random variables characterized by .
The cameras record
via photon counting (for the lowbrightness quantum system) or shotnoiselimited direct detection (for the highbrightness classical system), where
To compare the images that the quantum and classical systems provide, we first evaluate their ensembleaverage behaviors. It is easy to show, for the states we have assumed, that
Specializing equation (10) to the quantum (subscript q) and classical (subscript c) cases gives
and
where we have used in both cases and for the classical case. Equation (11) agrees with the theory from Ref. 17: cameras C^{+} and C^{−} produce positive and negative images, respectively, of Re(T_{jk}) with 100% fringe visibilities when for a nonzero integer. In comparison with Barreto Lemos et al.‘s quantum imager, equation (12) shows that our classical imager produces positive and negative images of Re(T_{jk}) with much weaker, , fringe visibilities for .
We define the positive and negative images’ signaltonoise ratios (SNRs), for ν=q,c, by
where the subscript in the numerator is used to indicate that we only include the objectrelated portion of the average images, not the objectindependent background terms that appear in equations (11) and (12), in assessing their strengths. Now, because the are in thermal states for both the quantum and classical cases, we immediately find that
and
for and . Thus the positive and negative quantum images have different SNRs, whereas the positive and negative classical images have identical SNRs. More importantly,
For T_{jk}=±1, this SNR ratio approaches , making it seem that the classical imager is vastly inferior to the quantum imager. For , however, this SNR ratio lies between 1/4 and 1/2, indicating that the quantum imager is slightly worse than the classical imager for low transmissivity objects. Such, however, is not the case, as we now demonstrate by examining the behavior of the difference images, , for the quantum and classical systems.
The differenceimage mean values, which follow immediately from equations (11) and (12), are both proportional to Re(T_{jk}), so we define their SNRs by
These SNRs are easily found using
and , . The results,
and
clearly indicate the SNR advantage afforded by the classical imager’s use of a highbrightness source. That advantage can translate into a much shorter imageacquisition time than that of the quantum imager, with its lowbrightness SPDC source. In particular, for all pixels with Re(T_{jk})≠0, we have that increases monotonically with increasing . For this SNR advantage equals and it saturates at as .
The huge improvement in the classical imager’s SNR that accrues from subtracting its negative image from its positive image is easily explained from semiclassical theory. In particular, because and , the noise in the classical system’s positive and negative images is overwhelmingly variance excess noise from the signal light illuminating the two detectors. For the classical system’s difference image, however, the complete correlation between the signalbeam excess noises impinging on the two detectors greatly reduces their impact on the difference image. Indeed, when the signalwavelength output from the phase conjugator is blocked, the difference image’s variance is reduced to , the sum of the two detectors’ shot noises. However, when both beams interfere at the 50–50 beam splitter, there is additional noise in the difference image. Its origin is easily identified if we interpret the difference image as arising from a balancedhomodyne measurement, in which the signal beam in the classical imager acts as a strong local oscillator (LO) that beats with the weak signalwavelength output from the phase conjugator. The extra noise just alluded to is then the LO fluctuations’ random modulation of the beat between the two fields^{18}. Thus, in equation (20), the first term in the denominator comes from the shot noise and the second term in the denominator, which sets the upper bound on the classical imager’s SNR advantage, comes from the modulation noise.
Discussion
SPDC sources can be used to produce a wide variety of entangled and even hyperentangled states. When such a source’s polarizationentangled outputs are used to perform qubit teleportation^{19}, or its quadratureentangled outputs are used to perform continuousvariable teleportation^{20}, the results are fundamentally quantum mechanical, in that classical resources cannot approach the fidelities achievable with entanglementbased teleportation. A somewhat different situation transpires when SPDC sources are used in imagers. Here, previous work on ghost imaging^{13,14} and optical coherence tomography^{15,16} has shown that almost all of the features of SPDCbased systems can be realized with quantummimetic counterparts that use classicalstate light, namely pseudothermal signal and idler beams with a phasesensitive cross correlation. We have demonstrated the same to be true for Barreto Lemos et al.‘s quantum imaging with undetected photons^{17}. In particular, we have analyzed a classicalstate system capable of imaging an object transparency at wavelength λ_{i} by sensing only wavelengthλ_{s} photons, a wavelength for which the object may be opaque. Moreover, our classicalstate system can use a bright source that results in its having a much higher SNR than that of its quantum counterpart, owing to the latter’s use of a lowbrightness source. Thus classical imaging with undetected photons can have a much shorter imageacquisition time than quantum imaging with undetected photons.
To illustrate the behavior of our quantummimetic system, we have performed computer simulations of the positive, negative and difference images, , and . The object transparency, whose T_{jk}^{2} is shown in Fig. 2(a), was a 32×32 MIT logo in which the “M” had field transmission T_{jk}=1/2, the “I” had field transmission T_{jk}=e^{iπ/2}/2, the “T” had field transmission T_{jk}=−1/2 and all other pixels had T_{jk}=0. Figure 2(b–d) are the resulting positive, negative and difference images, respectively, when , and M=100. These images show the characteristics expected from our analysis: (1) low contrast, low SNR in the positive and negative images, but high contrast and high SNR in the difference image; and (2) difference image with positive values for its “M” pixels, nearzero values for its “I” pixels and negative values for its “T” pixels. Note that a quantum difference image of this transparency—taken with the same and M = 100 values—would have an SNR 40 dB lower than that of the classical difference image.
At this juncture some final overarching comments are in order. The work of Barreto Lemos et al.^{17} is, in essence, a spatiallybroadband version of the quantum interference experiment of Zou, Wang and Mandel^{21}. Given that we have shown the former has a quantummimetic (classicalstate) counterpart which retains that system’s essential imaging characteristics, it behooves us to comment on whether a similar situation prevails with respect to the latter. Indeed it does. The Zou, Wang and Mandel experiment is assuredly quantum: it employs nonclassical light and hence requires quantum photodetection theory for its analysis, which those authors accomplish via the biphoton approximation for the postselected outputs from an SPDC source. That said, however, our Gaussianstate treatment of SPDC is more rigorous; for example, it includes the multiplepair emissions that account for the accidental coincidences seen in HongOuMandel interferometry even when detector darkcounts are negligible^{1,12}. Using Gaussianstate analysis, the quantum versus quantummimetic behavior for Ref. 21‘s experiment is explained by the singlepixel, Mpulse, object transmissivity T=e^{iφ} version of our theory, where φ is a controllable phase shift. In particular, from equations (11) and (12) we see that both the Zou, Wang and Mandel experiment and our classicalstate version thereof exhibit sinusoidal fringes as φ is varied, but the nonclassical source provides 100% fringe visibility, while the classicalstate source yields very low fringe visibility.
An additional point, which emerges from our Gaussianstate analyses of^{17,21}, is that both rely on stimulated, rather than spontaneous, emissions from the wavelengthconverting phase conjugator in Fig. 1. This conclusion follows from equation (5) and , which readily yield
for the average photon number of the conjugator’s mth signalpulse output at pixel (j,k). Here, the first term on the right represents signal photons whose emissions were stimulated by the presence of idler photons at the conjugator’s input, while the second term on the right represents signal photons whose emissions occurred spontaneously. That stimulated emissions are responsible for the quantum and classicalstate images we found earlier is then obvious from the resulting nonzero phaseinsensitive cross correlation between the two signalwavelength beams arriving at the beam splitter in Fig. 1, i.e.,
This cross correlation, which must be nonzero for the cameras in Fig. 1 to record an image, vanishes in the absence of stimulated signal emissions from the conjugator.
Equation (22) brings us to our paper’s final point. Just how essential is a wavelengthconverting phase conjugator to quantum and classical imaging with undetected photons? For the quantum case the answer is clear. The phasesensitive cross correlation between the signal and idler beams produced by cw SPDC cannot be sensed in secondorder interference, but the phaseconjugating nature of Fig. 1’s wavelength converter transforms phasesensitive cross correlation into phaseinsensitive cross correlation, which can be sensed in secondorder interference. The same necessity for phase conjugation arises if we replace the SPDC source with a classicalstate source possessing phasesensitive (but not phaseinsensitive) cross correlation between its signal and idler outputs. But why must our source only produce a phasesensitive cross correlation between its two output beams in order to image with undetected photons? The answer is that it need not. Suppose that the source in Fig. 1 produces a wavelength λ_{s} signal beam and a wavelength λ_{i} idler beam whose modal annihilation operators, , are in a zeromean, jointlyGaussian state with nonzero correlation functions given by equations (1), (2) and
Despite this phaseinsensitive crosscorrelation function’s being at the limit set by quantum physics, the joint state of the signal and idler is classical^{22}. Now assume that λ_{s} < λ_{i} and that the wavelengthconverting phase conjugator in Fig. 1 is replaced with a spatiallybroadband upconverter whose output modes at wavelength λ_{s} are given by
where 0 < κ < 1 is the conversion efficiency and the are in their vacuum states. It is easily shown, by paralleling the derivation that we presented earlier for a phasesensitive cross correlation, that cameras C^{+} and C^{−} will record backgroundembedded positive and negative images, respectively, of Re(T_{jk}), even when the object is opaque at the signal wavelength. In fact, having introduced the use of a spatiallybroadband upconverter, we can do classical imaging with undetected photons in an even simpler manner: use laser light at wavelength λ_{i} to uniformly illuminate an object transparency that is opaque at wavelength λ_{s}<λ_{i}. Then, use a spatiallybroadband upconverter—described by equation (24)—to obtain an image of T_{jk}^{2} by casting the wavelength λ_{s} field emerging from this threewave mixer on a camera.
Additional Information
How to cite this article: Shapiro, J. H. et al. Classical Imaging with Undetected Photons. Sci. Rep. 5, 10329; doi: 10.1038/srep10329 (2015).
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Acknowledgements
This work was supported by the US National Science Foundation under Grant No. 1161413.
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J.H.S. and F.N.C.W. did the theory. D.V. performed the computer simulations. All authors contributed to discussions and writing the paper.
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Shapiro, J., Venkatraman, D. & Wong, F. Classical Imaging with Undetected Photons. Sci Rep 5, 10329 (2015). https://doi.org/10.1038/srep10329
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