2. Experimental Setup
To investigate space-time ghost imaging we needed to adjust previous experiments to examine the effect of the scattering phenomena involved in pseudo-thermal light on ghost imaging by adding measurements that resolve the two-photon coherence time of the system. The setup to perform the experiments is shown in
Figure 2. It is similar to previous ghost imaging setups [
1,
2,
15–
19]. The setup differs in that sometimes a faster or slower charged coupled device (CCD) framing rate was implemented and sometimes a faster or slower ground glass rotation rate was used. A variety of space-time ghost imaging experiments were performed with combinations of rotating ground glass rates and CCD timing parameters. The rotation rate of the ground glass ranged from less than 1° per second to greater than 1500 rpm. The CCD timing parameters included frame rates of less than 1 frames per second (fps) to more than 1000 fps and the integration times ranged from micro-seconds to nanoseconds with some timing controls operating in picoseconds. For example at the low end of the experiments CCD integration times of 1 ms, frame rates of approximately 13 fps, and ground glass rotation speeds as low as 1° per second were implemented. We performed these experiments for successive imaging frames so as to track the laser illumination rotating ground glass disturbances over the reference and target fields. The space-time ghost imaging experiments consisted of using the reference field images at each time
tref =
ti and a bucket field measured at a separate time
tb =
ti+Δi where
ti is the time of the
ith measurement. This was performed for each time separation of Δ
i = −
N to +
N, where in some of our experiments
N = 20. We also evaluated the time correlations in the CCD frames. The experimental results showed that it was possible to resolve a ghost image with time separations up to twenty frames, that is Δ
i = ±20. The ghost image of the “ARL” moved to the left or right of the Δ
i = 0 time separation ghost image depending on whether the reference measurement was correlated with an earlier (
tref −
tb < 0) or later (
tref −
tb > 0) “bucket” measurement as depicted at the top of
Figure 2.
These experiments were performed with and without turbulence and used a typical thermal light lensless ghost imaging setup [
1]. The CCD records the secondary image of the primary ghost image [
15,
17] at the reference-image arm which helps make ghost imaging practical for applications. In this experiment, atmospheric turbulence is introduced to the optical paths by the adding heating elements underneath the optical paths operating at a temperature of 550 °C with the refractive index structure parameter in the range
to 10
−9. Note that the illumination and imaging paths traversed differing turbulence realizations each with these high levels of turbulence. In
Figure 2, we illustrate the most serious situation in which turbulence occurs in all optical paths of the setup. Heating of the air causes temporal and spatial fluctuations on its index of refraction that makes the classical image of the object jitter about randomly on the image plane causing a “blurred” picture. Similar to our early experiment [
1], the light source is a typical chaotic pseudo-thermal source, which contains a laser beam and a rotating ground glass diffuser. The thermalized chaotically scattered laser beam, which has a fairly large size on the ground glass (11 mm diameter) in transverse dimension, is split into two by a 50%–50% beamsplitter (BS). One of the beams illuminates an object located at
z1, such as the letters “ARL” as shown in
Figure 2. The scattered and reflected photons from the object are collected and counted by a “bucket” detector, which is simulated by the right-half of the CCD in
Figure 2. The other beam propagates to the ghost image plane of
z2 =
z1 ≃ 1.4 m and the distance from the target to the detectors is ∼1.7 m. Placing a CCD array on the ghost image plane, the CCD array will capture the ghost image of the object if its exposure is gated by the bucket detector [
1]. In this experiment the CCD array will image the target and reference planes located on a sheet of paper. The CCD is moved to a distance to view the ghost image on the glossy white paper. The scattered and reflected light from the glossy white half of the paper, which contains the spatial information for the ghost image, is then captured by the left-half of the high resolution CCD camera operating in the photon counting regime. The CCD camera is focused onto the glossy white paper at the ghost image plane and is gated by the bucket detector for the observation of the secondary ghost image. The hardware circuit and the software program are designed to monitor the outputs of the left-half and the right-half of the CCD individually, as two independent classical cameras, and simultaneously to monitor the gated output of the left-half of the CCD as a ghost camera. In the measurement, the classical image and the secondary ghost image of the object were captured and monitored simultaneously. In addition, measurements were made of the
correlations between intensity deviations (fluctuations) measured at each pixel by detectors 1 and 2 and at each time which can be described by
where the
,
are the measured photon counts at frame
i, and Δ
i is the separation from frame
i. This is essentially the space-time correlation of intensity deviations (fluctuations). Also, the normalized correlation
as a function of Δ
i is given by
where
and
are the standard deviations of
and
.
Equation (2) above is normalized by the standard deviations at each space-time coordinate and represents the normalized space-time correlation of intensity deviations (fluctuations).
4. Theory for Space-Time Ghost Imaging
A theory for space-time ghost imaging is developed below as an extension of conventional spatial ghost imaging. Space-time ghost imaging achieves ghost images with non-coincident measurements in addition to coincident measurements. Conventional spatial ghost imaging only uses coincident measurements at two different photo-sensors, a reference beam sensor and a photon bucket sensor. In thermal Ghost Imaging the beam splitter acts as a quantum operator on the photon quantum wavefunction. In quantum theory, a unitary operator relates the field operators at the input ports of the beamsplitter with the field operators at its output ports as a unitary transformation. The electromagnetic fields propagate from the source to the beamsplitter and then to the photo-detectors, following certain physical rules. With coincidence measurements we expect to be able to extract a ghost image. Ghost Imaging is the result of the turbulence-free point-to-point image-forming correlation, which is caused by two-photon interference: that is, superposition between paired two-photon amplitudes, corresponding to two different yet indistinguishable alternative ways of triggering a joint-detection event by two randomly distributed independent photons. An analysis illustrating the basic concepts follows. The analysis will be divided into three steps. First, we show that the point-to-point correlation between the object plane and the image plane is the result of two-photon interference. Second, we show that this correlation is turbulence-free. Third, we summarize the theory that effectively explains the results.
A joint-detection of two independent point photo-detectors measures the probability of observing a joint-detection event of two photons at space-time points (
r1,
t1) and (
r2,
t2), and is given by the Glauber’s theory of photo-detection [
20,
21],
Here
is the density operator and the quantized thermal field,
E(−)(
rj,
tj) and
E(+)(
rj,
tj) are the negative and positive field operators at space-time coordinate (
rj,
tj). As is well known the coincidence counting rate of two photon counting detectors is proportional to
G(2)(
r1,
t1;
r2,
t2). To calculate the point-to-point correlation between the object plane and the image plane, we need (1) to estimate the state, or the density matrix, of the thermal radiation; and (2) to propagate the field operators from the radiation source to the object and the image planes. We will first estimate the state of thermal radiation at the single-photon level for photon counting measurements to explore the physics behind ghost imaging as two-photon interference. It is important to realize that in addition to timescales representing turbulence, sensor measurements, ground glass rotation and scattering, the lifetime of a radiative process must be considered and it is dependent on the Einstein A coefficient [
22] which is fundamentally quantum in nature. Multiphoton ghost imaging experiments with different Einstein A coefficients will produce interference characteristic of those coefficients and therefore must be considered quantum. Thus, ghost imaging is fundamentally a quantum process and speckle theory does not readily account for such effects. A complete model will have all these space and time effects including the Einstein A coefficient.
A large transverse sized chaotic thermal source can be modeled as a large number of independent and randomly distributed point sub-sources [
1,
2,
15,
16,
23]. Each point sub-source may in turn consist of a large number of independent atoms that are in their ground state, but some can be excited to a higher energy level
E2 and later return back to its ground state
E1. It is reasonable to assume that each atomic transition generates a field in the following single-photon state
Here, |
ϵ| ≪ 1 is the probability amplitude for the atomic transition. Since higher order terms representing multi-photon emissions have terms of
ϵ2,
ϵ3, … and that the magnitude of
ϵ is small these terms fall off rapidly and can be safely neglected. The term
f(
k,
s) = ⟨Ψ
k,s | Ψ⟩ is the probability amplitude for the radiation field to be in the single-photon state of wave number
k and polarization
s:
. We assume a continuous distribution for
k and one polarization for this simplified two-level system. The chaotic nature of the sub-radiations leads to a density operator that can be approximated as:
where
is the transverse wavevector. To simplify the calculation, we will focus on the transverse spatial correlation by assuming single-frequency transitions with monochromatic light as usual. Basically we are modeling the light source as an incoherent statistical mixture of single-photon states and two-photon states with equal probability of having any transverse momentum. The spatial part of the second-order coherence function is thus calculated as:
where
is the transverse coordinate of the
jth photo-detector,
j = 1, 2. The transverse part of the electric field operator can be written as:
where
is the Green’s function, which propagates the field from the source to the photo-detector [
24,
25]. Substituting the field operators into
Equation (9) we have
It is a key point that
Equation (11) indicates interference between two quantum amplitudes, corresponding to two alternatives, different yet indistinguishable, which leads to a joint photo-detection event. This interference involves both arms of the optical setup as well as two distant photo-detection events at
and
, respectively.
Figure 17 schematically illustrates the two alternatives for a pair of mode
and
to produce a joint photo-detection event. It is interesting to see that this superposition plays the same role as the symmetrized wavefunction of identical particles. The superposition of each pair of these amplitudes produces an individual interference in the joint-detection space of
. A large number of these sub-interferences simply add together resulting in a point-to-point
function. It is easy to see that each pair of the two-photon amplitudes illustrated in
Figure 17 will superpose constructively whenever
D1 and
D2 are placed in the positions satisfying
and
z1 ≃
z2; and consequently,
achieves its maximum value as the result of the sum of these individual constructive interferences. In other coordinates, however, the superposition of each individual pair of the two-photon amplitudes may yield different values between constructive maximum and destructive minimum due to unequal optical path propagation, resulting in an averaged background constant sum.
It is straightforward to verify the above interference picture mathematically. The Fresnel near field Green’s function of free-propagation can be approximated as:
where
and
are transverse vectors in the source plane, and the field has propagated from the source to the
plane and
plane in arms 1 and 2, respectively. Substituting the Green’s functions into
Equation (11), the interference term (cross term of
Equation (11)) turns to out be
when choosing
z1 =
z2. Where
ω is the frequency of the electromagnetic radiation and 2
R is the diameter of the imaging lens,
d is the distance from the source to the detector, and
is one half the angular size of illumination source as viewed from the detectors. The normalized second-order correlation function
between the object plane and the image plane is therefore given by
Note that
is not the Green’s function which is also typically written as
g. It is to be realized that the argument of the
somb function is nondimensional. The
δ functions below are understood to be nondimensional where the arguments have been rescaled to allow them to be nondimensional and the normalized vector is
. From the Einstein, Podolsky and Rosen (EPR) [
26] type nonlocal perspective, two photons that created a joint detection event of two distant photo-detectors have equal probabilities to be found at any coordinates on the object plane and the image plane, however, if one of them is observed at a certain position on the object (image) plane, the other one has twice chance to be observed in a unique position on the image (object) plane.
It is now necessary to introduce the ground glass effect directly into the space-time ghost imaging theory. Martienssen and Spiller [
27] demonstrated that quickly rotating ground glass can convert a coherent laser beam into a chaotic radiation source suitable for optical experiments. Martienssen and Spiller characterized the time varying effect of ground glass on coherent radiation. In fact, ground glass is currently being used in fundamental experiments on studying the quantum superposition of coherent and pseudo-thermal light [
11]. In the following we introduce a new model for ground glass that incorporates both time and space phase disturbance effects into the ghost imaging process. Starting from the G
(2) equation we introduce phase disturbances that represent the effects of ground glass at each subsource position as measured at the detector points (see
Figure 18). A relatively wide laser beam intersects a portion of the ground glass and light scatters off many disturbance elements on the ground glass. The scattering radiation contributions summed over the subsource elements and scattering elements interfere in the ghost imaging process. We expect that when the phase disturbances
,
,
and
cancel in the probability amplitude formulation due to alternate and indistinguishable superpositions [
28] at certain measurement space-time points, then the ghost image of the object can readily be seen in the imaging plane. A clear image cannot be formed at those space-time points where the phase disturbances do not cancel,
The cross term is
Also,
It is possible to integrate over
and
first to produce Dirac delta functions.
But the phase disturbances
and
will influence the positions
and
due to the scattering disturbance at shifted time and locations due to the movement of the ground glass at velocity
. Actually, the effect in the measurement plane is proportional to the velocity of the ground glass movement. On the ground glass at
a phase disturbance is related to the position
in the source plane so that
. Consequently, we substitute the position of the phase disturbance motions into the following and obtain
with an equivalent integral over the transverse vectors in the ground glass frame of reference.
Only at the translated positions can the phases
φs and
φs′ end up with a null result and interference creating a space-time ghost image occur. That is
and
The interference term (cross term) turns to out be
As an approximation
Recall that the argument of the
somb function is nondimensional. Similarly, the
δ functions are understood to be nondimensional and the variables have been rescaled to allow them to be nondimensional,
For a time period, the space-time ghost image is turbulence-free. Over long enough time delay as the time separation increases, we expect some spread of the distribution away from a Dirac delta function due to extension beyond the setup coherence time and the introduction of experimental error. We note that the two-photon model is the first to predict this effect of the ground glass with a turbulence-free characteristic. The non-classical character of the equation becomes evident when the
somb function from two-photon second order interference approaches a non-factorizeable
function where the integration is over all possible subsources,
Figure 19a, and the resolution exceeds classical first order resolution. A 4f speckle correlation model,
Figure 19b tends to produce factorizeable (classical)
, where
is the transverse position of the source of the speckle. One has to integrate over classical measurements of
and
so the result tends to introduce classical speckle resolution. The resolution of these correlations is limited by the speckle size since speckle methods involve imaging the speckles. This is due to the speckle models using two classical imaging systems to image the source onto the image and object planes. Furthermore, we have experimentally generated ghost images with sensor photon counts of less than 10 above the background and there is no discernible speckle. Photon counts vary even in two slit interference experiments where photons traverse the slits one or more at a time. Physics does not tend to characterize these as speckle. So the use of speckle models to describe interference phenomena is problematic, whereas the use of multiphoton measurements has a sound characterization based on the fundamental photon measurement work of Glauber.
The two-photon interference model was used because it correctly predicts turbulence-free ghost imaging while speckle models do not. Furthermore, our model incorporates phase information, produced in this case by a rotating ground glass, from two separate times. It is through the two-photon interference of these phases that leads to the generation of a clear image of the translated target.
The same two-photon interference model can be used for nonclassical light such as entangled photons. In fact it can be easily extended to other particles such as electrons and neutrons. The speckle model is not readily adaptable to such cases. While there are phase disturbances from turbulence and from ground glass the model predicts that the adverse effects of the phase disturbances from the turbulence is cancelled whereas the phase disturbances created by the ground glass actually enable the viewing of the object at certain time differences at certain positions where they tend to cancel. Of course, the phase disturbances can be characterized in terms of probability space-time moments and deviations. Thus, our space-time quantum imaging analysis allows quantitative insight into the movement of a phase object, such as ground glass, in the setup. Our approach uses fundamental physics based on the Glauber theory of photo-detection [
20]. Glauber has shown that his model holds not only for the effects of individual photons but also for the effects of time averaged light and particularly he shows that quantum interference is observable from time averaged joint intensity measurements. Furthermore our two-photon interference model of two alternative and indistinguishable ways to generate a joint detection satisfies the Feynmann path integral formulation [
28].
Why use a quantum model for space-time ghost imaging through turbulence rather than a classical speckle based model? First we recognize that all photons are quantum all of the time. When classical approximations are introduced inconsistencies may appear and care must be taken to not destroy or modify the information we are looking for. Using the quantum two-photon and multi-photon interference theory we successfully realized the first ghost image of a remote object and extending the quantum theory guided us to achieve turbulence-free ghost imaging beyond the capabilities of classical approaches. Thus we are in favor of this approach for future explorations as it is consistent with more experiments and introduces fewer errors into the physics formulation. For example, the multi-photon interference approach in ghost imaging allows for diversity in sensor and illumination location, timing, entanglement, size and sensitivity that is difficult to consider with classical approaches. As we all know, any theory claiming to comprehensively model the quantum nature of photons as a classical theory has thus far been shown to be inconsistent. Godel’s second incompleteness theorem is often interpreted to mean that “an inconsistent model can prove everything including its own consistency.” Our turbulence-free ghost imaging experiment cannot be correctly interpreted by a classical model or theory. Turbulence-free ghost imaging is a nonlocal phenomenon, which is the result of a nonlocal interference involving the superposition of two-photon amplitudes representing different alternative and indistinguishable ways for a pair of photons to produce a joint photodetection event at a distance. The insightful quantum theorist Feynmann recognized the importance of having a theory verified by experiments, “It doesn’t matter how beautiful your theory is, it doesn’t matter how smart you are. If it doesn’t agree with experiment, it’s wrong”[
29].