Abstract

We propose and provide experimental evidence in support of a theory for the remote preparation of a complex spatial state of a single photon. An entangled two-photon source was obtained by spontaneous parametric down-conversion, and a double slit was placed in the path of the signal photon as a scattering object. The signal photon was detected after proper spatial filtering so that the idler photon was prepared in the corresponding single-photon state. By using a two-photon coincidence measurement, we obtained the Radon transform, at several longitudinal distances, of the single-photon Wigner distribution function modified by the double slit. The experimental results are consistent with the idler photon being in a pure state. An inverse Radon transformation can, in principle, be applied to the measured data to reconstruct the modified single-photon Wigner function, which is a complete representation of the amplitude and phase structure of the scattering object.

©2010 Optical Society of America

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 [35]. 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 [35]. 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 [1420] in the literature. Two-photon imaging with thermal light was studied recently [2123] 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 [2628] and that the FrFT is related to spatial Wigner distribution function (WDF) rotation in phase space [2730]. Demonstrations of WDF reconstruction for a partially coherent laser beam have been done by using the inverse Radon transform [3133]. 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

p2(x)=ei2πd/λ(iλd)1/2exp[iπ(xx')2/λd]p1(x')dx'=ei2πd/λ(iλd)1/2exp[i(π/λd)(x22xx'+x'2)]p1(x')dx'.

p1(x')and p2(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' 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 radiiR1 andR2, which are tangential to two flat surfaces, as shown below in Fig. 1 .

 figure: Fig. 1

Fig. 1 Relation between complex field distributions p's and q's.

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Let the electric field distributions on the spherical surfaces beq1(x')andq2(x), and the electric field distributions on the planes to be p1(x') and p2(x). In paraxial approximation,

p1(x')=q1(x')exp(iπx'2/λR1)       (R1<0)
p2(x)=q2(x)exp(iπx2/λR2)        (R2>0).

Putting these into Eq. (1), we obtain

q2(x)=ei2πd/λ(iλd)1/2exp[iπλd(x2(1dR2)2xx'+x'2(1+dR1))]q1(x')dx'.

Now we define geometrical constants g1,g2 as

g11+d/R1,   g21d/R2,
and introduce scale factors s1,s2 such that ρ'=x'/s1, ρ=x/s2, then

q2(ρ)=ei2πd/λs1(iλd)1/2exp[iπλd(g2s22ρ22s1s2ρρ'+g1s12ρ'2)]q1(ρ')dρ'  .

If we substitute

g1s12λd=cotφ,   g2s22λd=cotφ,  and   s1s2λd=cscφ  ,
then Eq. (5) becomes
q2(ρ)=ei2πd/λs1(iλd)1/2exp[iπ(ρ2cot2φ2ρρ'csc2φ+ρ'2cot2φ)]q1(ρ')dρ'={ei2πd/λs1e(πφ^/4φ/2)|sinφ|1/2iλd}×Fa[q1(ρ')]  ,
where the FrFT is defined as

Fa[q(ρ)]ei(πφ^/4φ/2)|sinφ|1/2exp[iπ(ρ2cotφ2ρρ'cscφ+ρ'2cotφ)]q(ρ')dρ'=Ba(ρ,ρ')q(ρ')dρ'.

The phase-space rotation angleφaπ/2 is determined by the light beam propagation distance and φ^=sgn(sinφ). Whena=1(φ=π/2), the transform becomes the usual Fourier transform. When a=2(φ=π), it is a simple inversion transform ; q(ρ)>q(ρ) . The FrFT kernel, defined as

Ba(ρ,ρ')=exp[i(πφ^/4φ/2)]|sinφ|1/2exp[iπ(ρ2cotφ2ρρ'cscφ+ρ'2cotφ)]  
has the following properties:

Ba(ρ,ρ')=Ba(ρ',ρ)  (symmetric)Fa2{Fa1[q(ρ')]}=Fa2+a1[q(ρ')]  (additive)or dρBa1(ρ',ρ)Ba2(ρ,ρ'')=Ba1+a2(ρ',ρ'')  .

If a FrFT relation holds, R1, R2 and d must be specified such that the condition 0g1g21 holds. Notice that the intensity of the field is, using s1s2λd=cscφ,

|q2(ρ)|2={s12|sinφ|λd}×|Fa[q1(ρ')]|2=|s1s2||Fa[q1(ρ')]|2   .

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

|p1(x')|2=|q1(x')|2,|p2(x)|2=|q2(x)|2.

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 s1, R1 and dare given and ϕ, s2, R2are calculated (see appendix). For instance, free propagation of a laser beam can be represented by a FrFT, from φ=0 (at d=0) to φ=π/2 (at d=). Beam propagation through a single lens can be also represented by a FrFT, where φ=π/2at the focal plane of the lens andφ=π 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 [2730,37]. We define the transverse spatial Wigner distribution function for the field distribution p1(x) at plane 1 as

Wp1(x,k)=p1(x+s/2)p1*(xs/2)e2πiskds.

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

g(x',φ)=[f(x,y)]=f(x'cosφx''sinφ,x'sinφ+x''cosφ)dx'',
where x'=xcosφ+ysinφ,x''=xsinφ+ycosφ.

So, the Radon transform of the WDF is

gp1(x',φ)=[Wp1(x,k)]=Wp1(x'cosφk'sinφ,x'sinφ+k'cosφ)dk',
where x'=xcosφ+ksinφ,k'=xsinφ+kcosφ.

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 [2830,33] the Radon transform is related to the FrFT as

|Fa[p1(x)]|2=gp1(x',φ=aπ/2).

Therefore, in principle, we can reconstruct the WDF of the field by using the inverse-Radon transformation on several measurements of the intensity distributiongp1(x',φ=aπ/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 [4042], ).

|Ψ=dκsdκif(κs+κi)a^(κs)a^(κi)|0,
where the functionf(κs+κi) is determined by the pump beam angular spectrum and phase matching condition, and κsandκi are the transverse momentum vectors for the signal and idler fields, respectively. We assume quasi-monochromatic light, separate the time dependence e±iωt, and define the electric field operators simply as
E^(x)=E^(+)(x)+E^()(x),
E^(+)(x)=dκCE(ωκ)a^(κ)exp[iκx]
E^()(x)=dκCE*(ωκ)a^(κ)exp[iκx],
where CE(ωκ)ωκ. If we use a narrow band-pass filter in the detection then we can approximate CE(ωκ)CE, so

E(+)(x)=CEdκ a^(κ)exp[iκx].

For the two-photon amplitude, by using Eqs. (17-20) and the commutation relation [a^(κ),a^(κ')]=δ(κκ'), we obtain

0|E^s(+)(xs)E^i(+)(xi)|Ψ=2(2π)2CE2A(xs)δ(xsxi)
0|E^s(+)(κs)E^i(+)(κi)|Ψ=2CE2f(κs+κi),

where

E^(+)(κ)=CEa^(κ)andf(κ)=dxA(x)exp[iκx].

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(κ), 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 s0ands1 so that ρ0=x0/s0andρ1=x1/s1,

E(+)(ρ1)=Fa[E(+)(ρ0)]=dρ0Ba(ρ1,ρ0)E(+)(ρ0),
where E(+)(ρ0)is the positive frequency part of the electric field at the crystal surface and E(+)(ρ1)is the positive frequency part of the electric field at plane 1 some distance away from the crystal. We may represent the positive frequency part of an electric field operator in plane 1 by using the FrFT relation as
E^(+)(ρ1)=dρ0B(ρ0,ρ1)E^(+)(ρ0),
so that

E^(+)(ρ1)=dρ0B(ρ0,ρ1)E^(+)(ρ0).

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

E^s(+)(ρ1)=dρsBa1(ρs,ρ1)E^s(+)(ρs),
for the signal field at plane 1, and
E^i(+)(ρ2)=dρiBa2(ρi,ρ2)E^i(+)(ρi),
for the idler field at plane 2. ρsandρi are the scaled coordinates for the signal and idler in the crystal. The two-photon detection amplitude for the signal field in plane 1 at arbitrary distance from the nonlinear crystal and the idler field in plane 2 at another arbitrary distance is

0|E^s(+)(ρ1)E^i(+)(ρ2)|Ψ=0|(dρsBa1(ρs,ρ1)E^s(+)(ρs))(dρiBa2(ρi,ρ2)E^i(+)(ρi))|Ψ=dρidρsBa1(ρs,ρ1)Ba2(ρi,ρ2)0|E^s(+)(ρs)E^i(+)(ρi)|Ψ.

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

0|E^s(+)(ρs)E^i(+)(ρi)|Ψ=A'δ(ρsρi),A'=2(2π)2CE2×const.,
from Eq. (21)a). So we have

0|E^s(+)(ρ1)E^i(+)(ρ2)|Ψ=A'dρidρsBa1(ρs,ρ1)Ba2(ρi,ρ2)δ(ρsρi)=A'dρiBa1(ρi,ρ1)Ba2(ρi,ρ2)=A'dρiBa1(ρ1,ρi)Ba2(ρi,ρ2)=A'Ba1+a2(ρ1,ρ2).

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(ρ)such that

E^s'(ρ1)=A(ρ1)E^s(ρ1),
and the two photon amplitude changes as

0|E^s'(+)(ρ1)E^i(+)(ρ2)|Ψ=A(ρ1)0|E^s(+)(ρ1)E^i(+)(ρ2)|Ψ.

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

0|E^s'(+)(ρ1)E^i(+)(ρ2)|Ψ=A'A(ρ1)Ba1+a2(ρ1,ρ2).

This is the relation that holds for the previous ghost imaging and ghost interference experiments. That is, when φ1+φ2=π, Ba1+a2(ρ1,ρ2)=δ(ρ1+ρ2), so one can get the ghost imaging. And, whenφ1+φ2=π/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 φ1) and plane 2 (at φ2)– are determined by the single FrFT with the sum of the angle φ1+φ2, 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 xD be the transverse coordinate at the focal plane andρDbe a scaled coordinate. Then the field at the focal plane is given by the Fourier transform of the field at the lens,

E^s,D(+)(ρD)=dρ1E^s(+)(ρ1)exp[ibρ1ρD],b1λf.

So, the two-photon amplitude becomes,

0|E^s,D(+)(ρD)E^i(+)(ρ2)|Ψ=0|(dρ1E^s(+)(ρ1)exp[ibρ1ρD])E^i(+)(ρ2)|Ψ=dρ10|E^s(+)(ρ1)E^i(+)(ρ2)|Ψexp[ibρ1ρD].

Therefore,

0|E^s,D(+)(ρD)E^i(+)(ρ2)|Ψ=dρ1CE2A(ρ1)Ba1+a2(ρ1,ρ2)exp[ibρ1ρD].

One can see that, in general, the detector position ρD affects the two-photon amplitude.

However, in the case ofφ1+φ2=π, it becomes

0|E^s,D(+)(ρD)E^i(+)(ρ2)|Ψ=CE2dρ1A(ρ1)δ(ρ1+ρ2)exp[ibρ1ρD]=CE2A(ρ2)exp[ibρ2ρD].
ρD goes into the phase factor only, and the two-photon detection probability is
|0|E^s,D(+)(ρD)E^i(+)(ρ2)|Ψ|2=|CE|4|A(ρ2)|2,
which is proportional to the absolute square of the amplitude of the transmission functionA(ρ2) only.

On the other hand, ifρD=0 in Eq. (36),

0|E^s,D(+)(0)E^i(+)(ρ2)|Ψ=dρ1CE2A(ρ1)Ba1+a2(ρ1,ρ2)=CE2Fa1+a2[A(ρ1)].

Then we can see that ghost interference appears when φ1+φ2=π/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[ibρ1ρD] 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 detectorρD, 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 xD 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,

 figure: Fig. 2

Fig. 2 (a) The signal field generated from the pumped BBO crystal with specific κ-value κ0will 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).

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|Ψcollapsed|κ0κ0|dκsdκif(κs+κi)a^(κs)a^(κi)|0=|κ0dκif(κ0+κi)a^(κi)|0.

So the idler is in a single photon wave packet state, whose state function is given by the original two-photon state function f(κ0+κi) and the value of the transverse momentum given by κ0. 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.

 figure: Fig. 3

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.

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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(X1,Y1), (X2,Y2) a distance R apart in the case of quasi-monochromatic incoherent light source is

μ12=(2J1(v)v)eiψ.
J1(v)is the Bessel function of the first kind and first order, and

v=2πλ¯ρR(X1X2)2+(Y1Y2)2,ψ=2πλ¯[(X12+Y12)(X22+Y22)2R].

λ¯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 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 (X1X2)2+(Y1Y2)2 = 300 μm (slit distance) and calculate the degree of coherence as a function ofR . We find that |μ12|0.94 for R300 mm, which is enough for the experiment.

In order to compare the experimental data with the theory, we needed to assume a field distribution p1(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 .

 figure: 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.

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

 figure: Fig. 5

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

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The Radon transform gp1(x',φ) 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),

 figure: Fig. 6

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

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 figure: Fig. 8

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.

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cotφo=so2λdo
cotφ'=λdo2so2[1do+1d'1f]+so2λ[1d'1f].

The total angle is φ=φo+φ'. dois the distance between the lens and the object (double slit), fis the focal length, d' is the distance to the measurement location, and φ is the angle in FrFT. We set the scaled parameter so=λf. For a 200 mm imaging lens, with wavelength 810 nm, so=402.5μm. Figure 7 shows the relation between the angle and the position.

 figure: Fig. 7

Fig. 7 The FrFT angle (WDF rotation angle) as a function of the detector position d'f.

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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) (ϕ=900) is the case for the usual ghost interference experiment and the Fig. 8(f) (ϕ=1800) 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 R1= so thatg1=1, and chooses1=πw to compare with Gaussian beam optics where w is the beam waist. Now, from g1s12λd=cotφ,

φ=tan1(λds12)=tan1[λ(zz0)πw2]=tan1[(zz0)zR],
where zis the beam propagation distance, z0is the beam waist location, and zR=πw2/λ. The scale factor s2(z)at distance zz0is, from the definition s1s2λd=cscφand 1+cot2φ=csc2φ,
s22=(λd)2s12+s12
s22s12=1+d2s14/λ2=1+(zz0)2zR2,
so that

s2=s11+(zz0)2zR2=s1cosφ,

And, using g2s22λd=cotφ, we find that

R2=(zz0)[1+zR2(zz0)2],
which is exactly the same as the radius of curvature found in Gaussian beam optics.

For a lens in the beam path, the effect of the lens is to change the phase of light by exp(iπx2/λf), where fis the focal length of the lens. Therefore the field distribution before and after the lens is changed as (Fig. 9 ) [28,29],

q+(x)=q(x)exp(iπx2/λf),
and its effect is to change the radius of curvature
p(x)=q(x)exp(iπx2/λR),
p+(x)=p(x)exp(iπx2/λf)=q(x)exp[iπx2λ(1R1f)],
so that

 figure: Fig. 9

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

Download Full Size | PPT Slide | PDF

1R+=1R1f.

Let R1= (g1=1) at the object plane, and then

φ0=tan1(λd0s02).

Also,

s=(λd0)2s02+s02,
R=d0(1+s04λ2d02),
which are the same as in free Gaussian beam propagation. The scale factor s=s+ for the thin lens because the field has to be continuous across the lens. The angle φ' at a distance d' from the lens is dertermined by the relations
g+=1+d'R+
and

cotφ'=g+s+2λd'.

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

cotφ'=λdo2so2[1do+1d'1f]+so2λ[1d'1f].

Acknowledgement

This work was supported by the Korea Research Foundation (Contract No. KRF-2006-312-C00551) and by the National Science Foundation (grant PHY-0554842).

References and links

1. E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409(6816), 46–52 (2001). [CrossRef]   [PubMed]  

2. P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79(1), 135–174 (2007). [CrossRef]  

3. T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Heralding single photons from pulsed parametric down-conversion,” Opt. Commun. 246(4-6), 545–550 (2005). [CrossRef]  

4. P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum States,” Phys. Rev. Lett. 100(13), 133601 (2008). [CrossRef]   [PubMed]  

5. P. J. Mosley, J. S. Lundeen, B. J. Smith, and I. A. Walmsley, “Conditional preparation of single photons using parametric down-conversion: A recipe for purity,” N. J. Phys. 10(9), 093011 (2008). [CrossRef]  

6. M. G. Raymer, “Measuring the quantum mechanical wave function,” Contemp. Phys. 38(5), 343–355 (1997). [CrossRef]  

7. A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81(1), 299–332 (2009). [CrossRef]  

8. A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87(5), 050402 (2001). [CrossRef]   [PubMed]  

9. B. J. Smith, B. Killett, M. G. Raymer, I. A. Walmsley, and K. Banaszek, “Measurement of the transverse spatial quantum state of light at the single-photon level,” Opt. Lett. 30(24), 3365–3367 (2005). [CrossRef]  

10. W. Wasilewski, P. Kolenderski, and R. Frankowski, “Spectral density matrix of a single photon measured,” Phys. Rev. Lett. 99(12), 123601 (2007). [CrossRef]   [PubMed]  

11. T. Aichele, A. I. Lvovsky, and S. Schiller, “Optical mode characterization of single photons prepared by means of conditional measurements on a biphoton state,” Eur. Phys. J. D 18(2), 237–245 (2002). [CrossRef]  

12. T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52(5), R3429–R3432 (1995). [CrossRef]   [PubMed]  

13. D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon “ghost” interference and diffraction,” Phys. Rev. Lett. 74(18), 3600–3603 (1995). [CrossRef]   [PubMed]  

14. R. S. Bennink, S. J. Bentley, and R. W. Boyd, ““Two-photon” Coincidence Imaging with a Classical Source,” Phys. Rev. Lett. 89(11), 113601 (2002). [CrossRef]   [PubMed]  

15. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87(12), 123602 (2001). [CrossRef]   [PubMed]  

16. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Entangled-photon Fourier optics,” J. Opt. Soc. Am. B 19(5), 1174–1184 (2002). [CrossRef]  

17. A. Gatti, E. Brambilla, and L. A. Lugiato, “Entangled imaging and wave-particle duality: from the microscopic to the macroscopic realm,” Phys. Rev. Lett. 90(13), 133603 (2003). [CrossRef]   [PubMed]  

18. A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93(9), 093602 (2004). [CrossRef]   [PubMed]  

19. M. D’Angelo, Y. H. Kim, S. P. Kulik, and Y. Shih, “Identifying entanglement using quantum ghost interference and imaging,” Phys. Rev. Lett. 92(23), 233601 (2004). [CrossRef]   [PubMed]  

20. J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78(6), 061802 (2008). [CrossRef]  

21. A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94(6), 063601 (2005). [CrossRef]   [PubMed]  

22. Y. Cai and S. Zhu, “Coincidence fractional Fourier transform implemented with partially coherent light radiation,” J. Opt. Soc. Am. A 22(9), 1798–1804 (2005). [CrossRef]  

23. F. Wang, Y. Cai, and S. He, “Experimental observation of coincidence fractional Fourier transform with a partially coherent beam,” Opt. Express 14(16), 6999–7004 (2006). [CrossRef]   [PubMed]  

24. B. Erkmen and J. Shapiro, “Signal-to-noise ratio of Gaussian-state ghost imaging,” Phys. Rev. A 79(2), 023833 (2009). [CrossRef]  

25. S. V. Polyakov and A. L. Migdall, “Quantum radiometry,” J. Mod. Opt. 56(9), 1045–1052 (2009). [CrossRef]  

26. H. M. Ozaktas and D. Mendlovic, “Fractional Fourier Optics,” J. Opt. Soc. Am. A 12(4), 743–751 (1995). [CrossRef]  

27. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10(10), 2181–2186 (1993). [CrossRef]  

28. M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72(8), 1137–1140 (1994). [CrossRef]   [PubMed]  

29. M. G. Raymer, M. Beck, and D. F. McAlister, “Spatial and temporal optical field reconstruction using phase-space tomography,” in Quantum Optics VI, D. F. Walls and J. D. Harvey, eds. (Springer, 1994).

30. T. Alieva, and M. J. Bastiaans, “Wigner distribution and fractional Fourier transform,” Signal Processing and its Applications, Sixth International Symposium on 2001, 1, 168–169 (2001).

31. D. F. McAlister, M. Beck, L. Clarke, A. Mayer, and M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20(10), 1181–1183 (1995). [CrossRef]   [PubMed]  

32. T. Anhut, B. Karamata, T. Lasser, M. G. Raymer, and L. Wenke, “Measurement of scattered light Wigner functions by phase space tomography and implications for parallel OCT,” in Coherence Domain Optical Methods and Optical Coherence Tomography in Biomedicine VII. Edited by Tuchin, Valery V.; Izatt, Joseph A.; Fujimoto, James G. Proceedings of the SPIE, Volume 4956, 120–128 (2003).

33. B. Eppich and N. Reng, “Measurement of the Wigner distribution function based on the inverse Radon transformation,” Proc. SPIE 2375, 261–268 (1995). [CrossRef]  

34. T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Two-photon geometric optics,” Phys. Rev. A 53(4), 2804–2815 (1996). [CrossRef]   [PubMed]  

35. D. N. Klyshko, Photons and Nonlinear Optics (Gordon and Breach, 1988).

36. M. Born, and E. Wolf, Principles of Optics, 6th Ed. (Cambridge University Press, 1993).

37. Z. Zalevsky and D. Mendlovi, “Fractional Radon transform: definition,” Appl. Opt. 35(23), 4628–4631 (1996). [CrossRef]   [PubMed]  

38. U. Leonhardt and M. Munroe, “Number of phases required to determine a quantum state in optical homodyne tomography,” Phys. Rev. A 54(4), 3682–3684 (1996). [CrossRef]   [PubMed]  

39. A. Einstein, B. Podolsky, and N. Rosen, “Can Quantum-Mechanical Description of Physical Reality be considered complete?” Phys. Rev. 47(10), 777–780 (1935). [CrossRef]  

40. C. Monken, P. Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57(4), 3123–3126 (1998). [CrossRef]  

41. D. S. Tasca, S. P. Walborn, P. H. Souto Ribeiro, and F. Toscano, “Detection of transverse entanglement in phase space,” Phys. Rev. A 78(1), 010304 (2008). [CrossRef]  

42. D. Tasca, S. Walborn, P. Souto Ribeiro, F. Toscano, and P. Pellat-Finet, “Propagation of transverse intensity correlations of a two-photon state,” Phys. Rev. A 79(3), 033801 (2009). [CrossRef]  

43. P. Navez and E. Brambilla, “Spatial entanglement of twin quantum images,” Phys. Rev. A 65, 013813 (2001). [CrossRef]  

References

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  • |

  1. E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409(6816), 46–52 (2001).
    [Crossref] [PubMed]
  2. P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79(1), 135–174 (2007).
    [Crossref]
  3. T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Heralding single photons from pulsed parametric down-conversion,” Opt. Commun. 246(4-6), 545–550 (2005).
    [Crossref]
  4. P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum States,” Phys. Rev. Lett. 100(13), 133601 (2008).
    [Crossref] [PubMed]
  5. P. J. Mosley, J. S. Lundeen, B. J. Smith, and I. A. Walmsley, “Conditional preparation of single photons using parametric down-conversion: A recipe for purity,” N. J. Phys. 10(9), 093011 (2008).
    [Crossref]
  6. M. G. Raymer, “Measuring the quantum mechanical wave function,” Contemp. Phys. 38(5), 343–355 (1997).
    [Crossref]
  7. A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81(1), 299–332 (2009).
    [Crossref]
  8. A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87(5), 050402 (2001).
    [Crossref] [PubMed]
  9. B. J. Smith, B. Killett, M. G. Raymer, I. A. Walmsley, and K. Banaszek, “Measurement of the transverse spatial quantum state of light at the single-photon level,” Opt. Lett. 30(24), 3365–3367 (2005).
    [Crossref]
  10. W. Wasilewski, P. Kolenderski, and R. Frankowski, “Spectral density matrix of a single photon measured,” Phys. Rev. Lett. 99(12), 123601 (2007).
    [Crossref] [PubMed]
  11. T. Aichele, A. I. Lvovsky, and S. Schiller, “Optical mode characterization of single photons prepared by means of conditional measurements on a biphoton state,” Eur. Phys. J. D 18(2), 237–245 (2002).
    [Crossref]
  12. T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52(5), R3429–R3432 (1995).
    [Crossref] [PubMed]
  13. D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon “ghost” interference and diffraction,” Phys. Rev. Lett. 74(18), 3600–3603 (1995).
    [Crossref] [PubMed]
  14. R. S. Bennink, S. J. Bentley, and R. W. Boyd, ““Two-photon” Coincidence Imaging with a Classical Source,” Phys. Rev. Lett. 89(11), 113601 (2002).
    [Crossref] [PubMed]
  15. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87(12), 123602 (2001).
    [Crossref] [PubMed]
  16. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Entangled-photon Fourier optics,” J. Opt. Soc. Am. B 19(5), 1174–1184 (2002).
    [Crossref]
  17. A. Gatti, E. Brambilla, and L. A. Lugiato, “Entangled imaging and wave-particle duality: from the microscopic to the macroscopic realm,” Phys. Rev. Lett. 90(13), 133603 (2003).
    [Crossref] [PubMed]
  18. A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93(9), 093602 (2004).
    [Crossref] [PubMed]
  19. M. D’Angelo, Y. H. Kim, S. P. Kulik, and Y. Shih, “Identifying entanglement using quantum ghost interference and imaging,” Phys. Rev. Lett. 92(23), 233601 (2004).
    [Crossref] [PubMed]
  20. J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78(6), 061802 (2008).
    [Crossref]
  21. A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94(6), 063601 (2005).
    [Crossref] [PubMed]
  22. Y. Cai and S. Zhu, “Coincidence fractional Fourier transform implemented with partially coherent light radiation,” J. Opt. Soc. Am. A 22(9), 1798–1804 (2005).
    [Crossref]
  23. F. Wang, Y. Cai, and S. He, “Experimental observation of coincidence fractional Fourier transform with a partially coherent beam,” Opt. Express 14(16), 6999–7004 (2006).
    [Crossref] [PubMed]
  24. B. Erkmen and J. Shapiro, “Signal-to-noise ratio of Gaussian-state ghost imaging,” Phys. Rev. A 79(2), 023833 (2009).
    [Crossref]
  25. S. V. Polyakov and A. L. Migdall, “Quantum radiometry,” J. Mod. Opt. 56(9), 1045–1052 (2009).
    [Crossref]
  26. H. M. Ozaktas and D. Mendlovic, “Fractional Fourier Optics,” J. Opt. Soc. Am. A 12(4), 743–751 (1995).
    [Crossref]
  27. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10(10), 2181–2186 (1993).
    [Crossref]
  28. M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72(8), 1137–1140 (1994).
    [Crossref] [PubMed]
  29. M. G. Raymer, M. Beck, and D. F. McAlister, “Spatial and temporal optical field reconstruction using phase-space tomography,” in Quantum Optics VI, D. F. Walls and J. D. Harvey, eds. (Springer, 1994).
  30. T. Alieva, and M. J. Bastiaans, “Wigner distribution and fractional Fourier transform,” Signal Processing and its Applications, Sixth International Symposium on 2001, 1, 168–169 (2001).
  31. D. F. McAlister, M. Beck, L. Clarke, A. Mayer, and M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20(10), 1181–1183 (1995).
    [Crossref] [PubMed]
  32. T. Anhut, B. Karamata, T. Lasser, M. G. Raymer, and L. Wenke, “Measurement of scattered light Wigner functions by phase space tomography and implications for parallel OCT,” in Coherence Domain Optical Methods and Optical Coherence Tomography in Biomedicine VII. Edited by Tuchin, Valery V.; Izatt, Joseph A.; Fujimoto, James G. Proceedings of the SPIE, Volume 4956, 120–128 (2003).
  33. B. Eppich and N. Reng, “Measurement of the Wigner distribution function based on the inverse Radon transformation,” Proc. SPIE 2375, 261–268 (1995).
    [Crossref]
  34. T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Two-photon geometric optics,” Phys. Rev. A 53(4), 2804–2815 (1996).
    [Crossref] [PubMed]
  35. D. N. Klyshko, Photons and Nonlinear Optics (Gordon and Breach, 1988).
  36. M. Born, and E. Wolf, Principles of Optics, 6th Ed. (Cambridge University Press, 1993).
  37. Z. Zalevsky and D. Mendlovi, “Fractional Radon transform: definition,” Appl. Opt. 35(23), 4628–4631 (1996).
    [Crossref] [PubMed]
  38. U. Leonhardt and M. Munroe, “Number of phases required to determine a quantum state in optical homodyne tomography,” Phys. Rev. A 54(4), 3682–3684 (1996).
    [Crossref] [PubMed]
  39. A. Einstein, B. Podolsky, and N. Rosen, “Can Quantum-Mechanical Description of Physical Reality be considered complete?” Phys. Rev. 47(10), 777–780 (1935).
    [Crossref]
  40. C. Monken, P. Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57(4), 3123–3126 (1998).
    [Crossref]
  41. D. S. Tasca, S. P. Walborn, P. H. Souto Ribeiro, and F. Toscano, “Detection of transverse entanglement in phase space,” Phys. Rev. A 78(1), 010304 (2008).
    [Crossref]
  42. D. Tasca, S. Walborn, P. Souto Ribeiro, F. Toscano, and P. Pellat-Finet, “Propagation of transverse intensity correlations of a two-photon state,” Phys. Rev. A 79(3), 033801 (2009).
    [Crossref]
  43. P. Navez and E. Brambilla, “Spatial entanglement of twin quantum images,” Phys. Rev. A 65, 013813 (2001).
    [Crossref]

2009 (4)

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81(1), 299–332 (2009).
[Crossref]

B. Erkmen and J. Shapiro, “Signal-to-noise ratio of Gaussian-state ghost imaging,” Phys. Rev. A 79(2), 023833 (2009).
[Crossref]

S. V. Polyakov and A. L. Migdall, “Quantum radiometry,” J. Mod. Opt. 56(9), 1045–1052 (2009).
[Crossref]

D. Tasca, S. Walborn, P. Souto Ribeiro, F. Toscano, and P. Pellat-Finet, “Propagation of transverse intensity correlations of a two-photon state,” Phys. Rev. A 79(3), 033801 (2009).
[Crossref]

2008 (4)

D. S. Tasca, S. P. Walborn, P. H. Souto Ribeiro, and F. Toscano, “Detection of transverse entanglement in phase space,” Phys. Rev. A 78(1), 010304 (2008).
[Crossref]

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78(6), 061802 (2008).
[Crossref]

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum States,” Phys. Rev. Lett. 100(13), 133601 (2008).
[Crossref] [PubMed]

P. J. Mosley, J. S. Lundeen, B. J. Smith, and I. A. Walmsley, “Conditional preparation of single photons using parametric down-conversion: A recipe for purity,” N. J. Phys. 10(9), 093011 (2008).
[Crossref]

2007 (2)

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79(1), 135–174 (2007).
[Crossref]

W. Wasilewski, P. Kolenderski, and R. Frankowski, “Spectral density matrix of a single photon measured,” Phys. Rev. Lett. 99(12), 123601 (2007).
[Crossref] [PubMed]

2006 (1)

2005 (4)

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94(6), 063601 (2005).
[Crossref] [PubMed]

Y. Cai and S. Zhu, “Coincidence fractional Fourier transform implemented with partially coherent light radiation,” J. Opt. Soc. Am. A 22(9), 1798–1804 (2005).
[Crossref]

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Heralding single photons from pulsed parametric down-conversion,” Opt. Commun. 246(4-6), 545–550 (2005).
[Crossref]

B. J. Smith, B. Killett, M. G. Raymer, I. A. Walmsley, and K. Banaszek, “Measurement of the transverse spatial quantum state of light at the single-photon level,” Opt. Lett. 30(24), 3365–3367 (2005).
[Crossref]

2004 (2)

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93(9), 093602 (2004).
[Crossref] [PubMed]

M. D’Angelo, Y. H. Kim, S. P. Kulik, and Y. Shih, “Identifying entanglement using quantum ghost interference and imaging,” Phys. Rev. Lett. 92(23), 233601 (2004).
[Crossref] [PubMed]

2003 (1)

A. Gatti, E. Brambilla, and L. A. Lugiato, “Entangled imaging and wave-particle duality: from the microscopic to the macroscopic realm,” Phys. Rev. Lett. 90(13), 133603 (2003).
[Crossref] [PubMed]

2002 (3)

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Entangled-photon Fourier optics,” J. Opt. Soc. Am. B 19(5), 1174–1184 (2002).
[Crossref]

R. S. Bennink, S. J. Bentley, and R. W. Boyd, ““Two-photon” Coincidence Imaging with a Classical Source,” Phys. Rev. Lett. 89(11), 113601 (2002).
[Crossref] [PubMed]

T. Aichele, A. I. Lvovsky, and S. Schiller, “Optical mode characterization of single photons prepared by means of conditional measurements on a biphoton state,” Eur. Phys. J. D 18(2), 237–245 (2002).
[Crossref]

2001 (4)

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87(12), 123602 (2001).
[Crossref] [PubMed]

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409(6816), 46–52 (2001).
[Crossref] [PubMed]

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87(5), 050402 (2001).
[Crossref] [PubMed]

P. Navez and E. Brambilla, “Spatial entanglement of twin quantum images,” Phys. Rev. A 65, 013813 (2001).
[Crossref]

1998 (1)

C. Monken, P. Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57(4), 3123–3126 (1998).
[Crossref]

1997 (1)

M. G. Raymer, “Measuring the quantum mechanical wave function,” Contemp. Phys. 38(5), 343–355 (1997).
[Crossref]

1996 (3)

T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Two-photon geometric optics,” Phys. Rev. A 53(4), 2804–2815 (1996).
[Crossref] [PubMed]

Z. Zalevsky and D. Mendlovi, “Fractional Radon transform: definition,” Appl. Opt. 35(23), 4628–4631 (1996).
[Crossref] [PubMed]

U. Leonhardt and M. Munroe, “Number of phases required to determine a quantum state in optical homodyne tomography,” Phys. Rev. A 54(4), 3682–3684 (1996).
[Crossref] [PubMed]

1995 (5)

D. F. McAlister, M. Beck, L. Clarke, A. Mayer, and M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20(10), 1181–1183 (1995).
[Crossref] [PubMed]

B. Eppich and N. Reng, “Measurement of the Wigner distribution function based on the inverse Radon transformation,” Proc. SPIE 2375, 261–268 (1995).
[Crossref]

H. M. Ozaktas and D. Mendlovic, “Fractional Fourier Optics,” J. Opt. Soc. Am. A 12(4), 743–751 (1995).
[Crossref]

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52(5), R3429–R3432 (1995).
[Crossref] [PubMed]

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon “ghost” interference and diffraction,” Phys. Rev. Lett. 74(18), 3600–3603 (1995).
[Crossref] [PubMed]

1994 (1)

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72(8), 1137–1140 (1994).
[Crossref] [PubMed]

1993 (1)

1935 (1)

A. Einstein, B. Podolsky, and N. Rosen, “Can Quantum-Mechanical Description of Physical Reality be considered complete?” Phys. Rev. 47(10), 777–780 (1935).
[Crossref]

Abouraddy, A. F.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Entangled-photon Fourier optics,” J. Opt. Soc. Am. B 19(5), 1174–1184 (2002).
[Crossref]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87(12), 123602 (2001).
[Crossref] [PubMed]

Aichele, T.

T. Aichele, A. I. Lvovsky, and S. Schiller, “Optical mode characterization of single photons prepared by means of conditional measurements on a biphoton state,” Eur. Phys. J. D 18(2), 237–245 (2002).
[Crossref]

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87(5), 050402 (2001).
[Crossref] [PubMed]

Bache, M.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93(9), 093602 (2004).
[Crossref] [PubMed]

Banaszek, K.

Beck, M.

D. F. McAlister, M. Beck, L. Clarke, A. Mayer, and M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20(10), 1181–1183 (1995).
[Crossref] [PubMed]

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72(8), 1137–1140 (1994).
[Crossref] [PubMed]

Bennink, R. S.

R. S. Bennink, S. J. Bentley, and R. W. Boyd, ““Two-photon” Coincidence Imaging with a Classical Source,” Phys. Rev. Lett. 89(11), 113601 (2002).
[Crossref] [PubMed]

Benson, O.

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87(5), 050402 (2001).
[Crossref] [PubMed]

Bentley, S. J.

R. S. Bennink, S. J. Bentley, and R. W. Boyd, ““Two-photon” Coincidence Imaging with a Classical Source,” Phys. Rev. Lett. 89(11), 113601 (2002).
[Crossref] [PubMed]

Boyd, R. W.

R. S. Bennink, S. J. Bentley, and R. W. Boyd, ““Two-photon” Coincidence Imaging with a Classical Source,” Phys. Rev. Lett. 89(11), 113601 (2002).
[Crossref] [PubMed]

Brambilla, E.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93(9), 093602 (2004).
[Crossref] [PubMed]

A. Gatti, E. Brambilla, and L. A. Lugiato, “Entangled imaging and wave-particle duality: from the microscopic to the macroscopic realm,” Phys. Rev. Lett. 90(13), 133603 (2003).
[Crossref] [PubMed]

P. Navez and E. Brambilla, “Spatial entanglement of twin quantum images,” Phys. Rev. A 65, 013813 (2001).
[Crossref]

Cai, Y.

Clarke, L.

D’Angelo, M.

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94(6), 063601 (2005).
[Crossref] [PubMed]

M. D’Angelo, Y. H. Kim, S. P. Kulik, and Y. Shih, “Identifying entanglement using quantum ghost interference and imaging,” Phys. Rev. Lett. 92(23), 233601 (2004).
[Crossref] [PubMed]

Dowling, J. P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79(1), 135–174 (2007).
[Crossref]

Einstein, A.

A. Einstein, B. Podolsky, and N. Rosen, “Can Quantum-Mechanical Description of Physical Reality be considered complete?” Phys. Rev. 47(10), 777–780 (1935).
[Crossref]

Eppich, B.

B. Eppich and N. Reng, “Measurement of the Wigner distribution function based on the inverse Radon transformation,” Proc. SPIE 2375, 261–268 (1995).
[Crossref]

Erkmen, B.

B. Erkmen and J. Shapiro, “Signal-to-noise ratio of Gaussian-state ghost imaging,” Phys. Rev. A 79(2), 023833 (2009).
[Crossref]

Frankowski, R.

W. Wasilewski, P. Kolenderski, and R. Frankowski, “Spectral density matrix of a single photon measured,” Phys. Rev. Lett. 99(12), 123601 (2007).
[Crossref] [PubMed]

Franson, J. D.

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Heralding single photons from pulsed parametric down-conversion,” Opt. Commun. 246(4-6), 545–550 (2005).
[Crossref]

Gatti, A.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93(9), 093602 (2004).
[Crossref] [PubMed]

A. Gatti, E. Brambilla, and L. A. Lugiato, “Entangled imaging and wave-particle duality: from the microscopic to the macroscopic realm,” Phys. Rev. Lett. 90(13), 133603 (2003).
[Crossref] [PubMed]

Hansen, H.

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87(5), 050402 (2001).
[Crossref] [PubMed]

He, S.

Jacobs, B. C.

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Heralding single photons from pulsed parametric down-conversion,” Opt. Commun. 246(4-6), 545–550 (2005).
[Crossref]

Killett, B.

Kim, Y. H.

M. D’Angelo, Y. H. Kim, S. P. Kulik, and Y. Shih, “Identifying entanglement using quantum ghost interference and imaging,” Phys. Rev. Lett. 92(23), 233601 (2004).
[Crossref] [PubMed]

Klyshko, D. N.

T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Two-photon geometric optics,” Phys. Rev. A 53(4), 2804–2815 (1996).
[Crossref] [PubMed]

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon “ghost” interference and diffraction,” Phys. Rev. Lett. 74(18), 3600–3603 (1995).
[Crossref] [PubMed]

Knill, E.

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409(6816), 46–52 (2001).
[Crossref] [PubMed]

Kok, P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79(1), 135–174 (2007).
[Crossref]

Kolenderski, P.

W. Wasilewski, P. Kolenderski, and R. Frankowski, “Spectral density matrix of a single photon measured,” Phys. Rev. Lett. 99(12), 123601 (2007).
[Crossref] [PubMed]

Kulik, S. P.

M. D’Angelo, Y. H. Kim, S. P. Kulik, and Y. Shih, “Identifying entanglement using quantum ghost interference and imaging,” Phys. Rev. Lett. 92(23), 233601 (2004).
[Crossref] [PubMed]

Laflamme, R.

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409(6816), 46–52 (2001).
[Crossref] [PubMed]

Leonhardt, U.

U. Leonhardt and M. Munroe, “Number of phases required to determine a quantum state in optical homodyne tomography,” Phys. Rev. A 54(4), 3682–3684 (1996).
[Crossref] [PubMed]

Lohmann, A. W.

Lugiato, L. A.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93(9), 093602 (2004).
[Crossref] [PubMed]

A. Gatti, E. Brambilla, and L. A. Lugiato, “Entangled imaging and wave-particle duality: from the microscopic to the macroscopic realm,” Phys. Rev. Lett. 90(13), 133603 (2003).
[Crossref] [PubMed]

Lundeen, J. S.

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum States,” Phys. Rev. Lett. 100(13), 133601 (2008).
[Crossref] [PubMed]

P. J. Mosley, J. S. Lundeen, B. J. Smith, and I. A. Walmsley, “Conditional preparation of single photons using parametric down-conversion: A recipe for purity,” N. J. Phys. 10(9), 093011 (2008).
[Crossref]

Lvovsky, A. I.

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81(1), 299–332 (2009).
[Crossref]

T. Aichele, A. I. Lvovsky, and S. Schiller, “Optical mode characterization of single photons prepared by means of conditional measurements on a biphoton state,” Eur. Phys. J. D 18(2), 237–245 (2002).
[Crossref]

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87(5), 050402 (2001).
[Crossref] [PubMed]

Mayer, A.

McAlister, D. F.

D. F. McAlister, M. Beck, L. Clarke, A. Mayer, and M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20(10), 1181–1183 (1995).
[Crossref] [PubMed]

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72(8), 1137–1140 (1994).
[Crossref] [PubMed]

Mendlovi, D.

Mendlovic, D.

Migdall, A. L.

S. V. Polyakov and A. L. Migdall, “Quantum radiometry,” J. Mod. Opt. 56(9), 1045–1052 (2009).
[Crossref]

Milburn, G. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79(1), 135–174 (2007).
[Crossref]

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409(6816), 46–52 (2001).
[Crossref] [PubMed]

Mlynek, J.

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87(5), 050402 (2001).
[Crossref] [PubMed]

Monken, C.

C. Monken, P. Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57(4), 3123–3126 (1998).
[Crossref]

Mosley, P. J.

P. J. Mosley, J. S. Lundeen, B. J. Smith, and I. A. Walmsley, “Conditional preparation of single photons using parametric down-conversion: A recipe for purity,” N. J. Phys. 10(9), 093011 (2008).
[Crossref]

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum States,” Phys. Rev. Lett. 100(13), 133601 (2008).
[Crossref] [PubMed]

Munro, W. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79(1), 135–174 (2007).
[Crossref]

Munroe, M.

U. Leonhardt and M. Munroe, “Number of phases required to determine a quantum state in optical homodyne tomography,” Phys. Rev. A 54(4), 3682–3684 (1996).
[Crossref] [PubMed]

Navez, P.

P. Navez and E. Brambilla, “Spatial entanglement of twin quantum images,” Phys. Rev. A 65, 013813 (2001).
[Crossref]

Nemoto, K.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79(1), 135–174 (2007).
[Crossref]

Ozaktas, H. M.

Pádua, S.

C. Monken, P. Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57(4), 3123–3126 (1998).
[Crossref]

Pellat-Finet, P.

D. Tasca, S. Walborn, P. Souto Ribeiro, F. Toscano, and P. Pellat-Finet, “Propagation of transverse intensity correlations of a two-photon state,” Phys. Rev. A 79(3), 033801 (2009).
[Crossref]

Pittman, T. B.

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Heralding single photons from pulsed parametric down-conversion,” Opt. Commun. 246(4-6), 545–550 (2005).
[Crossref]

T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Two-photon geometric optics,” Phys. Rev. A 53(4), 2804–2815 (1996).
[Crossref] [PubMed]

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52(5), R3429–R3432 (1995).
[Crossref] [PubMed]

Podolsky, B.

A. Einstein, B. Podolsky, and N. Rosen, “Can Quantum-Mechanical Description of Physical Reality be considered complete?” Phys. Rev. 47(10), 777–780 (1935).
[Crossref]

Polyakov, S. V.

S. V. Polyakov and A. L. Migdall, “Quantum radiometry,” J. Mod. Opt. 56(9), 1045–1052 (2009).
[Crossref]

Ralph, T. C.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79(1), 135–174 (2007).
[Crossref]

Raymer, M. G.

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81(1), 299–332 (2009).
[Crossref]

B. J. Smith, B. Killett, M. G. Raymer, I. A. Walmsley, and K. Banaszek, “Measurement of the transverse spatial quantum state of light at the single-photon level,” Opt. Lett. 30(24), 3365–3367 (2005).
[Crossref]

M. G. Raymer, “Measuring the quantum mechanical wave function,” Contemp. Phys. 38(5), 343–355 (1997).
[Crossref]

D. F. McAlister, M. Beck, L. Clarke, A. Mayer, and M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20(10), 1181–1183 (1995).
[Crossref] [PubMed]

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72(8), 1137–1140 (1994).
[Crossref] [PubMed]

Reng, N.

B. Eppich and N. Reng, “Measurement of the Wigner distribution function based on the inverse Radon transformation,” Proc. SPIE 2375, 261–268 (1995).
[Crossref]

Ribeiro, P.

C. Monken, P. Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57(4), 3123–3126 (1998).
[Crossref]

Rosen, N.

A. Einstein, B. Podolsky, and N. Rosen, “Can Quantum-Mechanical Description of Physical Reality be considered complete?” Phys. Rev. 47(10), 777–780 (1935).
[Crossref]

Rubin, M. H.

T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Two-photon geometric optics,” Phys. Rev. A 53(4), 2804–2815 (1996).
[Crossref] [PubMed]

Saleh, B. E. A.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Entangled-photon Fourier optics,” J. Opt. Soc. Am. B 19(5), 1174–1184 (2002).
[Crossref]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87(12), 123602 (2001).
[Crossref] [PubMed]

Scarcelli, G.

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94(6), 063601 (2005).
[Crossref] [PubMed]

Schiller, S.

T. Aichele, A. I. Lvovsky, and S. Schiller, “Optical mode characterization of single photons prepared by means of conditional measurements on a biphoton state,” Eur. Phys. J. D 18(2), 237–245 (2002).
[Crossref]

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87(5), 050402 (2001).
[Crossref] [PubMed]

Sergienko, A. V.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Entangled-photon Fourier optics,” J. Opt. Soc. Am. B 19(5), 1174–1184 (2002).
[Crossref]

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. 87(12), 123602 (2001).
[Crossref] [PubMed]

T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Two-photon geometric optics,” Phys. Rev. A 53(4), 2804–2815 (1996).
[Crossref] [PubMed]

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52(5), R3429–R3432 (1995).
[Crossref] [PubMed]

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon “ghost” interference and diffraction,” Phys. Rev. Lett. 74(18), 3600–3603 (1995).
[Crossref] [PubMed]

Shapiro, J.

B. Erkmen and J. Shapiro, “Signal-to-noise ratio of Gaussian-state ghost imaging,” Phys. Rev. A 79(2), 023833 (2009).
[Crossref]

Shapiro, J. H.

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78(6), 061802 (2008).
[Crossref]

Shih, Y.

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94(6), 063601 (2005).
[Crossref] [PubMed]

M. D’Angelo, Y. H. Kim, S. P. Kulik, and Y. Shih, “Identifying entanglement using quantum ghost interference and imaging,” Phys. Rev. Lett. 92(23), 233601 (2004).
[Crossref] [PubMed]

Shih, Y. H.

T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Two-photon geometric optics,” Phys. Rev. A 53(4), 2804–2815 (1996).
[Crossref] [PubMed]

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52(5), R3429–R3432 (1995).
[Crossref] [PubMed]

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon “ghost” interference and diffraction,” Phys. Rev. Lett. 74(18), 3600–3603 (1995).
[Crossref] [PubMed]

Silberhorn, C.

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum States,” Phys. Rev. Lett. 100(13), 133601 (2008).
[Crossref] [PubMed]

Smith, B. J.

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum States,” Phys. Rev. Lett. 100(13), 133601 (2008).
[Crossref] [PubMed]

P. J. Mosley, J. S. Lundeen, B. J. Smith, and I. A. Walmsley, “Conditional preparation of single photons using parametric down-conversion: A recipe for purity,” N. J. Phys. 10(9), 093011 (2008).
[Crossref]

B. J. Smith, B. Killett, M. G. Raymer, I. A. Walmsley, and K. Banaszek, “Measurement of the transverse spatial quantum state of light at the single-photon level,” Opt. Lett. 30(24), 3365–3367 (2005).
[Crossref]

Souto Ribeiro, P.

D. Tasca, S. Walborn, P. Souto Ribeiro, F. Toscano, and P. Pellat-Finet, “Propagation of transverse intensity correlations of a two-photon state,” Phys. Rev. A 79(3), 033801 (2009).
[Crossref]

Souto Ribeiro, P. H.

D. S. Tasca, S. P. Walborn, P. H. Souto Ribeiro, and F. Toscano, “Detection of transverse entanglement in phase space,” Phys. Rev. A 78(1), 010304 (2008).
[Crossref]

Strekalov, D. V.

T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Two-photon geometric optics,” Phys. Rev. A 53(4), 2804–2815 (1996).
[Crossref] [PubMed]

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52(5), R3429–R3432 (1995).
[Crossref] [PubMed]

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon “ghost” interference and diffraction,” Phys. Rev. Lett. 74(18), 3600–3603 (1995).
[Crossref] [PubMed]

Tasca, D.

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Figures (9)

Fig. 1
Fig. 1 Relation between complex field distributions p ' s and  q ' s.
Fig. 2
Fig. 2 (a) The signal field generated from the pumped BBO crystal with specific κ-value κ 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).
Fig. 3
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.
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.
Fig. 5
Fig. 5 Predicted Wigner distribution function calculated from the assumed field distribution in Fig. 4.
Fig. 6
Fig. 6 The Radon transform of the WDF in Fig. 5. Marginal distributions are calculated along the lines (a), (b),…,(f).
Fig. 8
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.
Fig. 7
Fig. 7 The FrFT angle (WDF rotation angle) as a function of the detector position d ' f .
Fig. 9
Fig. 9 The thin lens is represented by a two directional arrow.

Equations (62)

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p 2 ( x ) = e i 2 π d / λ ( i λ d ) 1 / 2 exp [ i π ( x x ' ) 2 / λ d ] p 1 ( x ' ) d x ' = e i 2 π d / λ ( i λ d ) 1 / 2 exp [ i ( π / λ d ) ( x 2 2 x x ' + x ' 2 ) ] p 1 ( x ' ) d x ' .
p 1 ( x ' ) = q 1 ( x ' ) exp ( i π x ' 2 / λ R 1 )         ( R 1 < 0 )
p 2 ( x ) = q 2 ( x ) exp ( i π x 2 / λ R 2 )         ( R 2 > 0 ) .
q 2 ( x ) = e i 2 π d / λ ( i λ d ) 1 / 2 exp [ i π λ d ( x 2 ( 1 d R 2 ) 2 x x ' + x ' 2 ( 1 + d R 1 ) ) ] q 1 ( x ' ) d x ' .
g 1 1 + d / R 1 ,     g 2 1 d / R 2
q 2 ( ρ ) = e i 2 π d / λ s 1 ( i λ d ) 1 / 2 exp [ i π λ d ( g 2 s 2 2 ρ 2 2 s 1 s 2 ρ ρ ' + g 1 s 1 2 ρ ' 2 ) ] q 1 ( ρ ' ) d ρ '    .
g 1 s 1 2 λ d = cot φ ,     g 2 s 2 2 λ d = cot φ ,   and    s 1 s 2 λ d = csc φ    ,
q 2 ( ρ ) = e i 2 π d / λ s 1 ( i λ d ) 1 / 2 exp [ i π ( ρ 2 cot 2 φ 2 ρ ρ ' csc 2 φ + ρ ' 2 cot 2 φ ) ] q 1 ( ρ ' ) d ρ ' = { e i 2 π d / λ s 1 e ( π φ ^ / 4 φ / 2 ) | sin φ | 1 / 2 i λ d } × F a [ q 1 ( ρ ' ) ]    ,
F a [ q ( ρ ) ] e i ( π φ ^ / 4 φ / 2 ) | sin φ | 1 / 2 exp [ i π ( ρ 2 cot φ 2 ρ ρ ' csc φ + ρ ' 2 cot φ ) ] q ( ρ ' ) d ρ ' = B a ( ρ , ρ ' ) q ( ρ ' ) d ρ ' .
B a ( ρ , ρ ' ) = exp [ i ( π φ ^ / 4 φ / 2 ) ] | sin φ | 1 / 2 exp [ i π ( ρ 2 cot φ 2 ρ ρ ' csc φ + ρ ' 2 cot φ ) ]   
B a ( ρ , ρ ' ) = B a ( ρ ' , ρ )   (symmetric) F a 2 { F a 1 [ q ( ρ ' ) ] } = F a 2 + a 1 [ q ( ρ ' ) ]   (additive) or  d ρ B a 1 ( ρ ' , ρ ) B a 2 ( ρ , ρ ' ' ) = B a 1 + a 2 ( ρ ' , ρ ' ' )    .
| q 2 ( ρ ) | 2 = { s 1 2 | sin φ | λ d } × | F a [ q 1 ( ρ ' ) ] | 2 = | s 1 s 2 | | F a [ q 1 ( ρ ' ) ] | 2     .
| p 1 ( x ' ) | 2 = | q 1 ( x ' ) | 2 , | p 2 ( x ) | 2 = | q 2 ( x ) | 2
W p 1 ( x , k ) = p 1 ( x + s / 2 ) p 1 * ( x s / 2 ) e 2 π i s k d s .
g ( x ' , φ ) = [ f ( x , y ) ] = f ( x ' cos φ x ' ' sin φ , x ' sin φ + x ' ' cos φ ) d x ' ' ,
g p 1 ( x ' , φ ) = [ W p 1 ( x , k ) ] = W p 1 ( x ' cos φ k ' sin φ , x ' sin φ + k ' cos φ ) d k ' ,
| F a [ p 1 ( x ) ] | 2 = g p 1 ( x ' , φ = a π / 2 ) .
| Ψ = d κ s d κ i f ( κ s + κ i ) a ^ ( κ s ) a ^ ( κ i ) | 0 ,
E ^ ( x ) = E ^ ( + ) ( x ) + E ^ ( ) ( x ) ,
E ^ ( + ) ( x ) = d κ C E ( ω κ ) a ^ ( κ ) exp [ i κ x ]
E ^ ( ) ( x ) = d κ C E * ( ω κ ) a ^ ( κ ) exp [ i κ x ] ,
E ( + ) ( x ) = C E d κ   a ^ ( κ ) exp [ i κ x ] .
0 | E ^ s ( + ) ( x s ) E ^ i ( + ) ( x i ) | Ψ = 2 ( 2 π ) 2 C E 2 A ( x s ) δ ( x s x i )
0 | E ^ s ( + ) ( κ s ) E ^ i ( + ) ( κ i ) | Ψ = 2 C E 2 f ( κ s + κ i ) ,
E ^ ( + ) ( κ ) = C E a ^ ( κ ) and f ( κ ) = d x A ( x ) exp [ i κ x ]
E ( + ) ( ρ 1 ) = F a [ E ( + ) ( ρ 0 ) ] = d ρ 0 B a ( ρ 1 , ρ 0 ) E ( + ) ( ρ 0 ) ,
E ^ ( + ) ( ρ 1 ) = d ρ 0 B ( ρ 0 , ρ 1 ) E ^ ( + ) ( ρ 0 ) ,
E ^ ( + ) ( ρ 1 ) = d ρ 0 B ( ρ 0 , ρ 1 ) E ^ ( + ) ( ρ 0 ) .
E ^ s ( + ) ( ρ 1 ) = d ρ s B a 1 ( ρ s , ρ 1 ) E ^ s ( + ) ( ρ s ) ,
E ^ i ( + ) ( ρ 2 ) = d ρ i B a 2 ( ρ i , ρ 2 ) E ^ i ( + ) ( ρ i ) ,
0 | E ^ s ( + ) ( ρ 1 ) E ^ i ( + ) ( ρ 2 ) | Ψ = 0 | ( d ρ s B a 1 ( ρ s , ρ 1 ) E ^ s ( + ) ( ρ s ) ) ( d ρ i B a 2 ( ρ i , ρ 2 ) E ^ i ( + ) ( ρ i ) ) | Ψ = d ρ i d ρ s B a 1 ( ρ s , ρ 1 ) B a 2 ( ρ i , ρ 2 ) 0 | E ^ s ( + ) ( ρ s ) E ^ i ( + ) ( ρ i ) | Ψ .
0 | E ^ s ( + ) ( ρ s ) E ^ i ( + ) ( ρ i ) | Ψ = A ' δ ( ρ s ρ i ) , A ' = 2 ( 2 π ) 2 C E 2 × c o n s t . ,
0 | E ^ s ( + ) ( ρ 1 ) E ^ i ( + ) ( ρ 2 ) | Ψ = A ' d ρ i d ρ s B a 1 ( ρ s , ρ 1 ) B a 2 ( ρ i , ρ 2 ) δ ( ρ s ρ i ) = A ' d ρ i B a 1 ( ρ i , ρ 1 ) B a 2 ( ρ i , ρ 2 ) = A ' d ρ i B a 1 ( ρ 1 , ρ i ) B a 2 ( ρ i , ρ 2 ) = A ' B a 1 + a 2 ( ρ 1 , ρ 2 ) .
E ^ s ' ( ρ 1 ) = A ( ρ 1 ) E ^ s ( ρ 1 )
0 | E ^ s ' ( + ) ( ρ 1 ) E ^ i ( + ) ( ρ 2 ) | Ψ = A ( ρ 1 ) 0 | E ^ s ( + ) ( ρ 1 ) E ^ i ( + ) ( ρ 2 ) | Ψ .
0 | E ^ s ' ( + ) ( ρ 1 ) E ^ i ( + ) ( ρ 2 ) | Ψ = A ' A ( ρ 1 ) B a 1 + a 2 ( ρ 1 , ρ 2 ) .
E ^ s , D ( + ) ( ρ D ) = d ρ 1 E ^ s ( + ) ( ρ 1 ) exp [ i b ρ 1 ρ D ] , b 1 λ f .
0 | E ^ s , D ( + ) ( ρ D ) E ^ i ( + ) ( ρ 2 ) | Ψ = 0 | ( d ρ 1 E ^ s ( + ) ( ρ 1 ) exp [ i b ρ 1 ρ D ] ) E ^ i ( + ) ( ρ 2 ) | Ψ = d ρ 1 0 | E ^ s ( + ) ( ρ 1 ) E ^ i ( + ) ( ρ 2 ) | Ψ exp [ i b ρ 1 ρ D ] .
0 | E ^ s , D ( + ) ( ρ D ) E ^ i ( + ) ( ρ 2 ) | Ψ = d ρ 1 C E 2 A ( ρ 1 ) B a 1 + a 2 ( ρ 1 , ρ 2 ) exp [ i b ρ 1 ρ D ] .
0 | E ^ s , D ( + ) ( ρ D ) E ^ i ( + ) ( ρ 2 ) | Ψ = C E 2 d ρ 1 A ( ρ 1 ) δ ( ρ 1 + ρ 2 ) exp [ i b ρ 1 ρ D ] = C E 2 A ( ρ 2 ) exp [ i b ρ 2 ρ D ] .
| 0 | E ^ s , D ( + ) ( ρ D ) E ^ i ( + ) ( ρ 2 ) | Ψ | 2 = | C E | 4 | A ( ρ 2 ) | 2 ,
0 | E ^ s , D ( + ) ( 0 ) E ^ i ( + ) ( ρ 2 ) | Ψ = d ρ 1 C E 2 A ( ρ 1 ) B a 1 + a 2 ( ρ 1 , ρ 2 ) = C E 2 F a 1 + a 2 [ A ( ρ 1 ) ] .
| Ψ c o l l a p s e d | κ 0 κ 0 | d κ s d κ i f ( κ s + κ i ) a ^ ( κ s ) a ^ ( κ i ) | 0 = | κ 0 d κ i f ( κ 0 + κ i ) a ^ ( κ i ) | 0 .
μ 12 = ( 2 J 1 ( v ) v ) e i ψ .
v = 2 π λ ¯ ρ R ( X 1 X 2 ) 2 + ( Y 1 Y 2 ) 2 , ψ = 2 π λ ¯ [ ( X 1 2 + Y 1 2 ) ( X 2 2 + Y 2 2 ) 2 R ] .
cot φ o = s o 2 λ d o
cot φ ' = λ d o 2 s o 2 [ 1 d o + 1 d ' 1 f ] + s o 2 λ [ 1 d ' 1 f ] .
φ = tan 1 ( λ d s 1 2 ) = tan 1 [ λ ( z z 0 ) π w 2 ] = tan 1 [ ( z z 0 ) z R ] ,
s 2 2 = ( λ d ) 2 s 1 2 + s 1 2
s 2 2 s 1 2 = 1 + d 2 s 1 4 / λ 2 = 1 + ( z z 0 ) 2 z R 2 ,
s 2 = s 1 1 + ( z z 0 ) 2 z R 2 = s 1 cos φ ,
R 2 = ( z z 0 ) [ 1 + z R 2 ( z z 0 ) 2 ] ,
q + ( x ) = q ( x ) exp ( i π x 2 / λ f ) ,
p ( x ) = q ( x ) exp ( i π x 2 / λ R ) ,
p + ( x ) = p ( x ) exp ( i π x 2 / λ f ) = q ( x ) exp [ i π x 2 λ ( 1 R 1 f ) ] ,
1 R + = 1 R 1 f .
φ 0 = tan 1 ( λ d 0 s 0 2 ) .
s = ( λ d 0 ) 2 s 0 2 + s 0 2 ,
R = d 0 ( 1 + s 0 4 λ 2 d 0 2 ) ,
g + = 1 + d ' R +
cot φ ' = g + s + 2 λ d ' .
cot φ ' = λ d o 2 s o 2 [ 1 d o + 1 d ' 1 f ] + s o 2 λ [ 1 d ' 1 f ] .

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