1. Introduction

The quantum state of a single photon, (i.e. the field mode containing a single-photon excitation) can be characterized by the longitudinal state, given by a complex frequency spectrum, and the transverse state, given by a complex spatial field-distribution function in a paraxial regime. For quantum information processing, which uses single-photon sources, complete information about single-photon states can be important [1,2]. A single-photon state can be created from an entangled two-photon state where a measurement of one photon can herald the existence of the other one [3–5]. Quantum state measurement (quantum tomography [6,7]) at the single-photon level has been previously studied by several authors. The single photon nature of an optical wave-packet in a known space-time mode produced by spontaneous parametric down-conversion (SPDC) has been verified [8], and the transverse spatial state of an attenuated laser field at the single-photon level was measured experimentally [9]. Temporal quantum state tomography on single-photon wave packets has been studied [10]. Also examined was the spatial mode matching between a local oscillator beam and a single-photon state obtained by narrow spectral and spatial filtering [11].

The most common method to create a two-photon entangled state is SPDC, where the frequency spectrum of the signal photon and idler photon can be well controlled [3–5]. In this paper, we show that an idler photon can be prepared in a nearly pure state by heralding a signal photon from a well controlled SPDC event, and that the conditional transverse quantum state of the idler photon can be changed by placing a scattering object in the beam path of the signal photon. We can use this property of a well controlled SPDC experiment to obtain complete information about the scattering object. We will explain this in detail later.

Our experiment is closely related to two-photon imaging, also known as ghost imaging [12] and ghost interference [13]. In these experiments, a correlated pair of signal and idler light beam is created from a nonlinear crystal. The signal beam is sent to an object such as a mask (or a double slit). The photons scatted from the object (or going through the slit) are collected by a detector. Unlike a usual imaging experiment, where one might obtain the image of the object from the signal photons that interacted with the object, the image of the object (or an interference pattern resulting from the double slit) can be found from coincidence measurements by putting a second detector in the idler beam path and scanning through the idler beam’s cross section.

The quantum nature of these measurements and the necessity of entanglement for such effects to occur have been debated [14–20] in the literature. Two-photon imaging with thermal light was studied recently [21–23] with discussions about the difference between the quantum light and classical light cases. Even though ghost imaging can be done with thermal light, the best signal-to-noise ratio can be obtained through use of an entangled photon source [24]. Furthermore, the entangled photon source gives us a single-photon state once one photon is detected, and this single-photon state can be used for precision measurements [25]. One goal of the present paper is to unify the understanding of ghost imaging and ghost interference by using the fractional Fourier transform (FrFT). Another is to point out the usefulness of FrFT in complete state reconstruction in the context of two-photon coincidence measurement.

It is now well known that light propagation and its spatial distribution function can be modeled by the FrFT [26–28] and that the FrFT is related to spatial Wigner distribution function (WDF) rotation in phase space [27–30]. Demonstrations of WDF reconstruction for a partially coherent laser beam have been done by using the inverse Radon transform [31–33]. One can also measure the WDF directly at the single-photon level by using a Sagnac interferometer [9]. There it was pointed out that measurements of transverse intensity profiles at many distances from the source are sufficient to reconstruct the phase and amplitude of a coherent field. For a partially coherent field, the method reconstructs the two-point mutual coherence function. For a single-photon field, the mutual coherence function equals the density matrix (or function) and its determination constitutes complete quantum state reconstruction. Reconstruction of transverse states of single photons using the FrFT method has not been reported. Here we take a step toward such a reconstruction, but the time required to acquire sufficient amount of data prevented us from performing a complete reconstruction.

We present a simple theory based on the FrFT and the Radon transform of the WDF. We will show that both ghost interference and ghost imaging are special cases of the Radon transform of the same WDF, and that there exists a more general FrFT relation between the signal and the ilder fields in terms of two-photon detection amplitude. We will also show that the FrFT is a simple and powerful technique for mathematically verifying the advanced-wave picture [34,35]. In the advanced-wave picture, one can picture that the light wave starts from the signal detector and propagates backward to a nonlinear crystal where the two beams were created, and then propagates forward as if reflected from the crystal to the idler detector. We obtain a mathematical expression that explains the advanced-wave picture. In a typical SPDC experiment, the transverse spatial WDF of a single-photon state is determined by the pump transverse profile and the phase-matching condition, but it is modified by the presence of a scattering object in the path of the beam. Because a nearly pure entangled state (in the sense of post selection of detection events) can be generated by SPDC, the single-photon state can also be generated purely if proper spatial filtering is used before detecting the heralding photon [8]. Given such a pure state, from the measurement of the WDF, one can obtain complete information on the amplitude and phase structure of a complex transmission function for the scattering object.

2. Theory

It is well known that beam propagation in free space can be represented by the Fresnel diffraction integral [36]. The Fresnel diffraction integral can be approximated by the FrFT in a paraxial regime [27]. The one dimensional Fresnel integral for optical field propagation from a plane 1 to plane 2 is

$p_1(x\text{'})$and $p_2(x)$are the complex-field distributions of positive-frequency part of the field at plane 1 and plane 2, respectively, and d is the distance between the two planes. x and $x\text{'}$ are the transverse coordinates at plane 1 and plane 2, respectively. We display only one transverse dimension, but all results can be generalized to two dimensions [28,29].

We follow the method of Ozaktas [26] to find the relation between the Fresnel integral and the FrFT. Consider two spherical surfaces with radii$R_1$ and$R_2$, which are tangential to two flat surfaces, as shown below in Fig. 1 .

 

Fig. 1 Relation between complex field distributions $p\text{'}\text{s and }q\text{'}\text{s.}$

Download Full Size | PDF

Let the electric field distributions on the spherical surfaces be$q_1(x\text{'})$and$q_2(x)$, and the electric field distributions on the planes to be $p_1(x\text{'})$ and $p_2(x)$. In paraxial approximation,

Putting these into Eq. (1), we obtain

Now we define geometrical constants $g_1,g_2$ as

If we substitute

The phase-space rotation angle$\phi \equiv a\pi /2$ is determined by the light beam propagation distance and $\widehat{\phi }=\mathrm{sgn}\left(\sin \phi \right)$. When$a=1\left(\phi =\pi /2\right)$, the transform becomes the usual Fourier transform. When $a=2\left(\phi =\pi \right)$, it is a simple inversion transform ; $q\left(\rho \right)->q\left(-\rho \right)$ _. The FrFT kernel, defined as

If a FrFT relation holds, $R_1$, $R_2$ and d must be specified such that the condition $0\le g_1{g}_2\le 1$ holds. Notice that the intensity of the field is, using $\frac{s_1{s}_2}{\lambda d}=\csc \phi$,

Therefore the intensity is also rescaled in this representation. It is worth mentioning that, from Eqs. (2)a), (2b),

Therefore we can represent the beam propagation by the FrFT, with proper scale factors related to the characteristics of linear optical elements. The scaling factors can be calculated when the optical system is specified with certain parameters. In many applications $s_1$, $R_1$ and dare given and ϕ, $s_2$, $R_2$are calculated (see appendix). For instance, free propagation of a laser beam can be represented by a FrFT, from $\phi =0$ (at $d=0$) to $\phi =\pi /2$ (at $d=\infty$). Beam propagation through a single lens can be also represented by a FrFT, where $\phi =\pi /2$at the focal plane of the lens and$\phi =\pi$ at the image plane [26]. At the focal plane, the FrFT becomes the usual Fourier transform, which is the same as going from the near field to the far field in optics. At the imaging plane, the FrFT becomes an inversion, which is the same as going from an object plane to an image plane with a scale factor (magnification) in optics. Intermediate values of φ correspond to other distances, as we demonstrate in the appendix.

An important fact is that, the FrFT has one-to-one correspondence to the Radon transform of the WDF in phase space [27–30,37]. We define the transverse spatial Wigner distribution function for the field distribution $p_1(x)$ at plane 1 as

The Radon transform of an arbitrary function $f(x,y)$ is defined as

So, the Radon transform of the WDF is

The Radon transformation is a rotation followed by integration over an axis. This is called a projection integral, and the result is a marginal density distribution. It is known that [28–30,33] the Radon transform is related to the FrFT as

Therefore, in principle, we can reconstruct the WDF of the field by using the inverse-Radon transformation on several measurements of the intensity distribution$g_{p_1}(x\text{'},\phi =a\pi /2)$at different angleφ. However, it may take measurements at a large number of angles (i.e., propagation distances) to reconstruct a WDF precisely. The number required depends on the complexity of its structure [28,38].

We are interested in the measurement of the transverse WDF of a single photon. An entangled two-photon state can be created from a SPDC source and the detection of one (signal) photon then heralds the existence of the other (idler) photon. The two-photon state that is generated from SPDC using a large crystal with a well collimated pump beam, so that momentum is conserved among the light waves, is close to the EPR state [39].

When one is interested only in two-photon detection, the post-selected quasi-monochromatic two-photon state generated from SPDC using a very thin crystal can be represented by the following two-photon wave function (See, for example [40–42], ).

For the two-photon amplitude, by using Eqs. (17-20) and the commutation relation $\left[\widehat{a}(\kappa ),{\widehat{a}}^{\text{†}}(\kappa \text{'})\right]=\delta (\kappa -\kappa \text{'})$, we obtain

where

This is a kind of EPR state, perfectly correlated in position and loosely anti-correlated in transverse momentum. The spatial entanglement of the two images concerns both the amplitude and the phase [43]. Notice that, unlike the ideal EPR state, the momentum anti-correlation is given by $f(\kappa ),$ which is a Fourier transform of the function $A(x)$ _. $A(x)$is determined by the pump beam angular spectrum and the phase-matching condition in usual SPDC experiments. However, the $A(x)$ can be also given or modified by a scattering object in the beam path, as we will see.

If we consider a classical electric field propagating from a crystal surface and apply FrFT theory, with scale factors $s_0$and$s_1$ so that ${\rho }_0=x_0/s_0$and${\rho }_1=x_1/s_1$,

For the signal and idler waves propagating from the crystal, we write

We can see that the two-photon detection amplitude is determined by the two-photon amplitude at the crystal and the two FrFT kernels. This is the general expression for a two-mode photon state. However, for the case of the SPDC two-photon state, if we have a plane-wave pump beam and a large crystal, then

We have used Eq. (10) at the last step. Therefore the two-photon detection amplitude is now determined by the single FrFT kernel only. When we have a scattering object in the path of the signal beam, we can represent it with a transmission function $A(\rho )$such that

So, from Eq. (30) and Eq. (32), we obtain

This is the relation that holds for the previous ghost imaging and ghost interference experiments. That is, when ${\phi }_1+{\phi }_2=\pi$, $B_{a_1+a_2}({\rho }_1,{\rho }_2)=\delta ({\rho }_1+{\rho }_2)$, so one can get the ghost imaging. And, when${\phi }_1+{\phi }_2=\pi /2$, the FrFT becomes a Fourier transform, so one can get the ghost interference. Furthermore, Eq. (33) shows that the two-photon detection amplitude for any arbitrary two locations – plane 1 (at ${\phi }_1$) and plane 2 (at ${\phi }_2$)– are determined by the single FrFT with the sum of the angle ${\phi }_1+{\phi }_2$, not by the difference of the angles, as if the light field has travelled from plane 1, back to the SPDC crystal, and then forward to plane 2. This provides another mathematical expression, based on wave propagation, for the advanced-wave picture, which was often explained by geometrical ray optics [34,35]. We can use this property to reconstruct the WDF of the idler photon state, as follows.

In practical experimental situations, we have to integrate over the photo-detector area, and it is important to understand its role. Let’s put a collection lens after the scattering object and assume that we put a point detector in the focal plane of the collection lens. Let $x_D$ be the transverse coordinate at the focal plane and${\rho }_D$be a scaled coordinate. Then the field at the focal plane is given by the Fourier transform of the field at the lens,

So, the two-photon amplitude becomes,

Therefore,

One can see that, in general, the detector position ${\rho }_D$ affects the two-photon amplitude.

However, in the case of${\phi }_1+{\phi }_2=\pi$, it becomes

On the other hand, if${\rho }_D=0$ in Eq. (36),

Then we can see that ghost interference appears when ${\phi }_1+{\phi }_2=\pi /2$, where FrFT becomes the usual Fourier transform. In the early ghost imaging experiments with a bucket (i.e., large area) detector [12], what one measured was an integral of the absolute square of the two-photon amplitude at every detector points. Equation (38) shows that one can use the bucket detector because the phase factor $\exp \left[-ib{\rho }_1{\rho }_D\right]$ in Eq. (36) does not contribute to the two-photon coincidence counts. However, for the ghost interference case, the phase factor in Eq. (36) matters. Depending on the location of the detector${\rho }_D$, the phase factor changes and the interference pattern shifts. So the interference will be averaged out if one uses a too large detector, because we integrate over the detector area.

It is important to notice that the detection of a photon at the signal beam detector imposes a specific condition for the two-photon detection amplitude. In quantum terminology, it makes the wave function collapse to a specific condition represented by the phase term associated with the detector location. The point detector at the focal plane of the collection lens works as a mode selector, as shown in Fig. 2 . Because the lens Fourier transforms the beam, each κcomponent of the beam will be focused at a different position $x_D$ at the detector plane as in Fig. 2(a). When we start with an entangled state with a given pump profile, as in Eq. (17), the idler photon after the measurement of the signal photon becomes,

 

Fig. 2 (a) The signal field generated from the pumped BBO crystal with specific κ-value ${\kappa }_0$will be selected by a pinhole at the focal plane followed by a signal detector. The idler field will be collapsed to a single-photon state obtained from the original two-photon state function, as in Eq. (40). (b) The advanced wave picture, as in Eq. (33).

Download Full Size | PDF

So the idler is in a single photon wave packet state, whose state function is given by the original two-photon state function $f({\kappa }_0+{\kappa }_i)$ and the value of the transverse momentum given by ${\kappa }_0$. When there is a scattering object as shown in Fig. 2(b), the same principle applies to the signal plane and idler plane as we found in Eq. (39).

This is the feature of the SPDC two-photon state that we can use to find the characteristics of the scattering object.

3. Experiment

Figure 3 shows the experimental set-up. The 810 nm wavelength pulsed light from a sub-ps mode-locked Ti:Sapphire laser is sent to a type-I BBO crystal for second harmonic generation. The resulting 405 nm light becomes a pump for collinear SPDC at the type II BBO crystal. After splitting the signal and the idler beams by the polarizing beam splitter PBS, the signal beam is sent through a double slit and then collected by 300 mm focal length lens. APD-2 is located in the focal plane for the collected signal light. The idler beam is sent to a 200 mm focal length imaging lens. APD-1 has a small detection area and it is scanned in the transverse direction at several locations. The coincidence time window was 7.5ns. We placed 10 nm bandwidth interference filters in front of the APD’s. Our APD’s have active areas of diameter 175µm. The double slit has slit width 150µm and slit distance 300µm.

 

Fig. 3 Experimental set-up. The type-I BBO is for the SHG and the type-II BBO is for SPDC. The signal and the idler beams are separated by the polarizing beam splitter (PBS). The lens with f = 300 mm is the collection lens and the lens with f = 200 mm is the imaging lens. APD-1 is moved to several longitudinal distances from the lens and then scanned along the transverse direction.

Download Full Size | PDF

APD-2 works as a point-like detector, as shown in Fig. 2(b), with non-zero diameter. We used the collection lens to increase the count rate, without degrading the degree of coherence. For a simple estimate, we consider the SPDC beams as quasi-monochromatic and spatially incoherent light. The Van Cittert-Zernike theorem [36] tells us that the degree of mutual coherence for the two points$\left(X_1,Y_1\right)$, $\left(X_2,Y_2\right)$ a distance R apart in the case of quasi-monochromatic incoherent light source is

$\overline{\lambda }$is the mean wavelength. By adapting the advanced-wave picture, we model APD-2 as an incoherent light source. Considering the detector surface as an effectively Gaussian light source with a diameter of $175\text{ μm}$, placed at the focal point of the collection lens. We may assume that the beam after the lens is well collimated. Because the distance from APD-2 to the lens is 300 mm and the slit is placed close to the lens, we set $\sqrt{{\left(X_1-X_2\right)}^2+{\left(Y_1-Y_2\right)}^2}$ = $300\text{ μm}$ (slit distance) and calculate the degree of coherence as a function ofR _{. We find that} $\left|{\mu }_{12}\right|\ge 0.94$ for $R\ge 300\text{ mm}$, which is enough for the experiment.

In order to compare the experimental data with the theory, we needed to assume a field distribution $p_1(x)$ at the slit. From the intensity measurements of the signal beam passing through one slit when the other one was blocked, we found that the field distribution in the double slit was unbalanced by a 10:8 ratio. The slit width is only 150µm, making it difficult to measure an accurate intensity distribution in the near field, so we used a 10:8 unbalanced, simplified field distribution as shown in Fig. 4 .

 

Fig. 4 The assumed field distribution at the double slit, based on direct measurement of signal photons count rates at the detector APD-2 when one of the two slits was blocked alternatively.

Download Full Size | PDF

The WDF we obtained numerically using MATLAB from this field distribution is shown in Fig. 5 .

 

Fig. 5 Predicted Wigner distribution function calculated from the assumed field distribution in Fig. 4.

Download Full Size | PDF

The Radon transform $g_{p_1}(x\text{'},\phi )$ of this WDF is shown in Fig. 6 . The lines (a), (b),… (f) in Fig. 6 correspond to Radon transform at several specific phase-space angles. Marginal distributions along these lines can be calculated and they are related to the FrFT as shown in Eq. (16). They are shown in Fig. 8 as solid line curves. In order to compare the theory with the experimental situation, we needed to find the proper scaling factor. In the experiment, the angle φ is related to the location of detector APD-1. For a single lens imaging system, the relationship is (see appendix),

 

Fig. 6 The Radon transform of the WDF in Fig. 5. Marginal distributions are calculated along the lines (a), (b),…,(f).

Download Full Size | PDF

 

Fig. 8 Experimental data and the theoretical curves for the marginal distributions at several longitudinal distances. The solid lines are theoretical curves and the filled dots are experimental data.

Download Full Size | PDF

The total angle is $\phi ={\phi }_o+\phi \text{'}$. $d_o$is the distance between the lens and the object (double slit), fis the focal length, $d\text{'}$ is the distance to the measurement location, and φ is the angle in FrFT. We set the scaled parameter $s_o=\sqrt{\lambda f}$. For a 200 mm imaging lens, with wavelength 810 nm, $s_o=402.5\mu m$. Figure 7 shows the relation between the angle and the position.

 

Fig. 7 The FrFT angle (WDF rotation angle) as a function of the detector position $d\text{'}-f$.

Download Full Size | PDF

After finding the detector positions corresponding to the angles shown in Fig. 6, we put APD-1 at positions corresponding to the angles (a), (b), …, (f) shown in Fig. 6 in the idler beam path and measured the coincidence rates while scanning through the cross section of the idler beam. Finally, to include the effect of detector size, we integrated the theoretical curve over a region corresponding to APD-1’s active detector area.

The theoretical curves and experimental data for the marginal distributions are shown in Fig. 8. Solid lines are theoretical curves, and the filled circles are the experimental data. The vertical error bars shown in Fig. 8 are statistical errors and the horizontal error bars represent detector size.

Notice that Fig. 8(a) ($\varphi {\text{=90}}^0$) is the case for the usual ghost interference experiment and the Fig. 8(f) ($\varphi ={180}^0$) is the case for the usual ghost imaging case. These are two special cases of the FrFT and the corresponding Radon transforms are indicated in Fig. 6. Other cases, (b) - (e), are also predicted by Eq. (39). The result shows that the experimental results agree with our theory that the two-photon amplitude propagates according to the FrFT.

If we want to reconstruct the Wigner distribution function completely from measurements, then we would need more data points than we could practically acquire with the realized photon-counting rates. Numerical simulation shows that we need at least 32 scans × 64 data points per each scan to reconstruct the WDF shown in Fig. 5. The complexity of the WDF depends on the shape of the scattering object, so one may need even more data points for more complicated objects.

4. Discussion

We introduced a simple unified theory to show that ghost imaging and ghost interference correspond to two special cases of the FrFT, or the Radon transformation of the WDF, and that the two-photon amplitude is determined by the FrFT at any distance from the SPDC crystal. We can, in principle, reconstruct the WDF of the single-photon state in the idler beam by measuring the Radon transform of the WDF at many longitudinal distances, while the signal photon is detected by a collection lens and a fixed point-like detector. This is possible because the idler photon is entangled with the signal photon as given in Eq. (17) for a pure state. The detection of a signal photon collapses the two-photon state and forces the idler photon into a specific transverse spatial state as described in Eq. (40). The experimental result seems to suggest that the idler photon state is close to a pure state in our experiment. This means that, by using a well characterized scattering object, we can remotely prepare a single-photon wave-packet state in a known form. Conversely, we have also shown that known single-photon state can, in principle, be used to measure the amplitude and the phase of complex scattering object placed in the path of the signal photon.

Appendix

As a good example of the FrFT, consider the free propagation of a laser beam. The WDF of a Gaussian laser beam was successfully measured by Eppich and Peng through use of the relationship between the FrFT and the WDF [33]. Assuming that the phase plane is flat at the position of beam waist, we can set $R_1=\infty$ so that$g_1=1$, and choose$s_1=\sqrt{\pi }w$ to compare with Gaussian beam optics where w is the beam waist. Now, from $g_1\frac{s_1^2}{\lambda d}=\cot \phi$,

And, using $g_2\frac{s_2^2}{\lambda d}=\cot \phi$, we find that

For a lens in the beam path, the effect of the lens is to change the phase of light by $\exp (-i\pi x^2/\lambda f)$, where fis the focal length of the lens. Therefore the field distribution before and after the lens is changed as (Fig. 9 ) [28,29],

 

Fig. 9 The thin lens is represented by a two directional arrow.

Download Full Size | PDF

Let $R_1=\infty$ ($g_1=1$) at the object plane, and then

Also,

The result is shown in Eq. (44) in the text, i.e.,