1. INTRODUCTION

Quantum states of N photons of wavelength λ can appear as if they had a de Broglie wavelength [1] of ${\lambda }_B=\lambda ∕N$. Examples where light, constituent of N photon states, behaves as if it had a wavelength divided by the photon number have been investigated, e.g., for the diffraction at ordinary gratings [2, 3, 4], in quantum lithography [5], or in related interference schemes [6, 7, 8, 9], and for nonlocal aspects of the de Broglie wavelength of a four-photon state [10]. Such a reduced wavelength can be interesting for an improved spatial resolution in different concepts of advanced imaging methods in the area of quantum imaging [11, 12]. Nonclassical spatial correlations between the photons in general can play a key role [13] for these advanced imaging methods. In this paper we report on the characterization of these spatial correlations of biphotons generated from parametric downconversion by Fraunhofer far-field distributions behind a blazed grating.

The terminology of the photonic de Broglie wavelength is convenient for addressing certain features of the biphoton correlations, but at least in case of the biphoton diffraction at a grating it is not strictly necessary to explain these features. In this paper we use a semi-classical diffraction model containing a measure for the degree of spatial correlation of the photons within one biphoton [14]. Ordinary coherent light without any spatial correlations would interfere to yield only one strong order of diffraction behind a blazed grating. Given a certain degree of spatial correlation between the photons, an additional order of diffraction can be found in the two-photon distribution of the Fraunhofer far field. This order of diffraction relates to the photonic de Broglie wavelength of half the original wavelength of the photons. The visibility of this order of diffraction can be used to characterize the degree of spatial correlation of the biphotons. This characterization can be carried out more easily and efficiently, meaning with higher count-rates, compared with an ordinary transmission grating, since an ideal blazed grating produces only one order of diffraction (see, e.g., [15])—the first order. Thus, the interference of a biphoton beam at a blazed grating creates a diffraction distribution with a hard binary distinction between two different directions (diffraction angles) depending on the degree of correlation of the two photons of a biphoton pair.

Using the blazed grating we demonstrate experimentally the connection between the spatial degree of correlation of the biphotons and the spatial multimode character of the light. The higher the spatial mode order for the propagation of the two photons in the setup, the higher is the degree of correlation between them. It is known that, for a two-photon state, a Schmidt number that is equivalent to more than one Schmidt mode is necessary [16] to describe an entangled state. In our experimental investigation we can simply restrict the number of Schmidt modes used for the illumination of the blazed grating by apertures and investigate the consequences on the biphoton diffraction at the blazed grating.

The paper is organized as follows: In Section 2 the theoretically expected single-photon and two-photon distributions in the Fraunhofer far field of the blazed grating are calculated and discussed. Section 3 in brief describes the realization of the biphoton beam. In Section 4 the experimental results of the diffraction experiments with varying degree of correlation are reported. In Appendix A different measures of correlations, like the degree of correlation, a correlation factor used in [14] and the Schmidt number are compared and related to the generation process of our biphotons, parametric downconversion.

2. CALCULATING THE SINGLE- AND TWO-PHOTON RATE OF DIFFRACTION BEHIND THE BLAZED GRATING

Figure 1 shows the diffraction efficiency of the first order of diffraction (OD) of our blazed (reflection) grating. It has a grating period of $d=25\mu \mathrm{m}$ with a depth of the grooves of $250\mathrm{nm}$ and is manufactured for a blaze wavelength of $500\mathrm{nm}$.

For the calculation of the diffraction at the blazed grating we utilize a semiclassical model as it was applied before by Shimizu et al. [14] to the diffraction of biphotons at ordinary transmission gratings. It allows for a simple phenomenological description of the degree of spatial correlation between the photons and is precise enough to explain and describe our experimental results, as will be seen in Section 4. In Appendix A we discuss the relation of the expected form of the two-photon amplitude from parametric downconversion to the phenomenological description used here.

The two-photon state at the near-field plane of the grating with transverse coordinates $x_1$ and $x_2$ can be expressed as

$C\left(x_1\text{-}{x}_2\right)$ is a function to describe the correlation between the positions of the photons $x_1$ and $x_2$. The two-photon count rate in the near field of the diffracting object can then be expressed by

We first consider the ramifications for the grating in the near-field or an imaged near-field plane. In the case without any correlation, the two-photon amplitude $\tilde{F}(k_{x1},k_{x2})$ becomes separable in functions of $k_{x1}$ and $k_{x2}$. So for equal momenta in the far field, $k_{x1}=k_{x2}=k_x$, the two-photon count rate and single-photon count rate are simply expressed by different powers of the Fourier transform $\mathbf{F}\left[A\right]$ of the single-photon amplitude A, as can be seen from Eqs. (4, 5):

The relationship between partial coherence and partial entanglement has been studied and discussed earlier. There is some analogy of the biphoton wave amplitude with the second-order correlation function for the electrical field (which would be the first-order correlation for the intensity) (see, e.g., [22]). Nonseparability in the biphoton wave amplitude is associated with entanglement. Nonseparability of the second-order correlation function, on the other hand, is connected with loss of coherence and hence loss of interference visibility. Thus, in the limiting case of spatially entangled photons, the single photons totally lose their spatial coherence. However, van Exter et al. [23] point out that, in the near-field imaging case that we have discussed here, the duality discussion is more elaborate. If the coherence function ${\mathrm{G}}^{\left(1\right)}$ is calculated within this semi-classical model as carried out by Simizu et al. [14], it turns out that ${\mathrm{G}}^{\left(1\right)}$ is nonseparable in the perfectly correlated case, and the photons are expected to be completely incoherent in this case.

To illustrate the effect of biphoton diffraction at a blazed grating, we calculate three diffraction cases for different degrees of spatial correlation of the biphotons: a weakly correlated case with $r=3$, a medium correlated case with $r=0.85$, and a strongly correlated case with $r=0.01$. The calculations are carried out exactly for the grating used in our experiments. The distributions are shown in dependence of diffraction angle $q=k_x∕k$ for the direct comparison with our measurements. The first examples shown in Fig. 2 are calculated for an illumination of several grating periods with a spot diameter of $100\mu \mathrm{m}$.

The three diagrams in one row show to the left the two-photon rate as contour plot with the position of two (single photon) detectors at positions $q_1=k_{x1}∕k$ and $q_2=k_{x2}∕k$. The diagram in the middle shows the evaluation of a cut through the contour plot for the two-photon rate with for $k_{x1}=k_{x2}$. Whereas the diagram to the right shows the single-photon rate $R^{\left(1\right)}$ in dependence of $k_x$ The top row of Fig. 2 shows the results in the case of almost uncorrelated photons $(r=3)$. As expected from the explanations given above for the uncorrelated case two- and single-photon rates differ by just the square. There is one strong 1st OD expected for a wavelength of $780\mathrm{nm}$ (red) and a remaining part because of the limited diffraction efficiency in the 0th OD.

For strongly correlated photons (Fig. 2, bottom) one strong OD at the position expected for a 1st OD of light of a wavelength of $390\mathrm{nm}$ (blue) is found, whereas the remaining 1st OD expected for red light $\left(780\mathrm{nm}\right)$ is comparably faint. As expected, the interference contrast in the single-photon rate is almost zero. The ratio of heights of the red and blue OD changes continuously with the degree of the spatial correlation between the photons of a biphoton. The middle row of Fig. 2 shows a calculation for medium degree of correlation $r=0.85$. The interference contrast of the single-photon rate changes in anti-correlation with the relative height of the blue OD in the two-photon rate distribution. Because the diffraction efficiency of the grating does not reach 100%, some photons appear in the 0th-order and the 3rd OD in the blue. The second order of the blue OD falls together with the 1st OD in the red so that they cannot be distinguished.

If the biphoton field in a certain plane is imaged from one plane to another and magnified by a certain factor β the correlation factor r is also expected to scale with β. For example, if the spatial mode diameter is demagnified in an image plane by a factor of two, r is expected to decrease by a factor of two in the image plane, too. Thus, in our experiment, when imaging the nonlinear crystal fact where the biphotons are generated (see Section 3), on the grating we choose smaller spot sizes to obtain a small correlation factor r compared to the grating period d. On the other hand, if fewer grating periods are illuminated, the OD become wider (see Fig. 3 , top). In further anticipation of our experiment we also calculate the single- and two-photon rate for the case of limited angular resolution of the detectors in the Fraunhofer far field of the grating of $10\mathrm{mrad}$. As expected, this causes a decreased contrast in the distributions (see Fig. 3, bottom). The limited angular resolution was incorporated in the calculations by a convolution of the two-photon detection rate $R^{\left(2\right)}(k_x,k_x)$. with a top hat function with width corresponding to our angular resolution.

In far-field illumination of the grating, the two photons of a strongly correlated biphoton do not “pass through” one common period of the grating. Consequently, they do not produce any interference contrast in the two-photon rate for the profile cut of $q_1=q_2$ (see Fig. 4 , bottom). The smaller the degree of spatial correlation, the more the distributions of the count rates of the Fraunhofer diffraction behind the blazed grating in near and far-field illumination assimilate each other as expected (see Fig. 4 top).

3. SETUP TO BUILD A BIPHOTON BEAM

We generate correlated photons from noncollinear type II parametric downconversion [24] in a BBO crystal. The schematic of the setup is shown in Fig. 5 . The BBO crystal is pumped by a frequency doubled, mode-locked picosecond TiSa-laser with a repetition rate of $76\mathrm{MHz}$ and $20\mathrm{mW}$ average power at a wavelength of $390\mathrm{nm}$. The pump spot diameter is $160\mu \mathrm{m}$. The run time differences and walk-off due to the birefringence of the BBO are compensated to first order by a half-wave plate and compensation BBO-crystals [24]. By adjusting the birefringent compensation crystals we can adjust for the phase in the two-photon Bell-state. The phase is adjusted so that the two photons reunite via Hong–Ou–Mandel (HOM) interference [25] to a biphoton beam with 50% probability at the lower and upper port of a beam splitter (see Fig. 5).

We only make use of the biphotons leaving the lower port of the beam splitter. The pairs are detected with a coincidence rate of 10,000 counts/s. We have chosen the noncollinear type II generation process to allow for independent manipulation of the photons before they interfere to produce the biphoton beam. (The full possibility of this independent manipulation is not exploited in the work reported here.) There are no interference filters in the setup. But due to the single-mode acceptance of our fiber-coupled photon detectors behind the grating on one hand, and the angular dependence of the spectral bandwidth of the photons emitted by parametric downconversion on the other hand [26], the bandwidth acceptance of our detection is smaller than $10\mathrm{nm}$.

The interference at the beam splitter is adjusted by an optimization of the visibility of the HOM dip. The visibility of the HOM interference is better than 95%. The photons are detected by single-mode, fiber-coupled single-photon detectors (SPCM-AQRH 15 by PerkinElmer). The crystal plane is imaged onto the grating to allow for a near-field investigation of the generated photons (compare Section 4). This way the strong correlations in the positions of the photons, due to its common origin, should enable a certain fraction of the photons to pass as a pair through one period of the grating (see Section 2). The spot diameter on the grating is $29\mu \mathrm{m}$ resulting in a Rayleigh length of around $1\mathrm{mm}$. Behind the grating the single-photon $R^{\left(1\right)}$ and two-photon coincidence rates $R^{\left(2\right)}$ are observed in a distance of $10\mathrm{cm}$ behind the grating. Since this distance is more than one order of magnitude longer compared to the Rayleigh length of the spot, this is an observation in the Fraunhofer far field to very good approximation.

The angular resolution of the detection is adjusted by a slit aperture in front of the lens of the fiber coupler. Photons of an angular spread of $10\mathrm{mrad}$ are coupled into the single-mode fiber coupler. At the end of the two ports of the fiber coupler the single- and two-photon coincidence count rate is measured by avalanche photo-detectors.

Since the grating is illuminated under an angle θ (see Fig. 5) the exponential terms in Eq. (5) $\exp \left(i{k}_{x}x\right)$ actually would have to be modified for the calculations of the experimental data to $\exp \left(ik[\sin (q+\theta )-\sin \left(\theta \right)]x\right)$. But for small angles this expression transforms back to $\exp \left(ikqx\right)$, which equals $\exp \left(i{k}_{x}x\right)$ because of $q=k_x∕k$ as introduced in Eq. (5). Because we are working at small angles and the facet inclination is small due to the rather large grating period of $25\mu \mathrm{m}$, the diffraction efficiency of the blazed grating is equal within a percent for both polarizations of the incident electrical field, as has been verified experimentally.

4. EXPERIMENTAL OBSERVATION

The far-field distributions behind the grating in the single- and two-photon rate are measured with photon detectors at a fixed position, while the grating was turned around an axis perpendicular to the plane of incidence. Since we scanned only a small angular width of around $100\mathrm{mrad}$, this is equivalent to first order to scanning the detectors in the far field with a fixed grating orientation.

A typical result of such a measurement for the coherence properties of our current biphoton source is shown in Fig. 6 . In the measured two-photon count rate two pronounced and one weaker OD can be observed. The OD with the highest count rate appears at the angular position of the first OD for the original red wavelength $\left(780\mathrm{nm}\right)$ of the photons. In addition, there is a distinct OD at the angular position expected for blue light $\left(390\mathrm{nm}\right)$. The smaller OD observed to the right is the 0th OD. The single-photon count rate shows almost no interference contrast since the spatial correlation of the photons is already too strong (see Section 2). The measured data is plotted together with the calculated count rates for our experimental illumination conditions with the spot size of $29\mu \mathrm{m}$ diameter and $30\mathrm{mrad}$ full-angle incident onto the grating. The best agreement of calculation and experimental data was found for a correlation factor of $\mathrm{r}=0.85$ in this case. This and all following calculations are carried out for the angular resolution of our setup of $10\mathrm{mrad}$. Apart from the correlation width there is no further fit parameter involved. Since the calculation only gives detection probabilities, those were scaled to match the measured peak count rate in the experiment.

To increase the usable degree of spatial correlation, the biphoton source and the conditions of propagation in between the source and the grating have to be improved as was addressed in general in [27], and in specific for our setup in [28]. The degree of spatial correlation in the plane of the grating can be decreased by two different methods. First, if the grating is moved away from the plane of the imaged crystal near field, the degree of correlation factor is expected to become smaller and the correlation factor should become larger. This can easily be achieved by changing the distance z between the grating and the lens in front of the grating (see Fig. 5). In the experiment, the position of the grating has been kept fixed whereas the lens was moved backwards. The count rates in the Fraunhofer far field measured with the first method of correlation variation are shown in Fig. 7 . The correlation factors can be taken from the fit of the measured data. At the lowest level of correlation the blue OD is not visible anymore. This basically means that the photon pairs are not experienced as spatially correlated biphotons by the grating anymore.

For the second method, the degree of spatial correlation of the photons in the near field was weakened by restricting the number of higher-order transverse spatial modes that the correlated photons populate. The degree of entanglement or entanglement entropy in an entangled state of continuous variables is known to grow in a nonlinear fashion with the number of Schmidt modes that can be populated by the photons [27, 29, 30]. In [27] van Exter et al. give a “heuristic” way to derive the Schmidt number of a spatial multimode system in bipartite entanglement (that we motivated recently [28] starting from the entanglement boundary for continuous variable systems given by Mancini et al. [31]). One has to distinguish between the number of Schmidt modes that are actually generated by the parametric downconversion source and the number of modes that can actually be detected and take part in the detected entanglement because of limited detector size and apertures in the setup. According to this, the Schmidt number $K_x$ of detectable or usable modes of a setup is roughly equal to the etendue of the setup or the better known $M^2$ number of modes propagating through the setup to detect the photons. Both measures give the approximate number of spatial modes that are supported by the experimental setup.

The $M^2$ number is defined by the waist radius $w_0$ and far-field divergence θ of a partially coherent beam of wavelength λ by $M^2=\pi w_0\theta ∕\lambda$. This means if we decrease the $M^2$ number of the propagating modes of the two photons, we would expect to reduce the Schmidt number and thereby, as pointed out in Appendix A, the degree of correlation of the photons in the grating plane. In our second series of measurements this restriction of higher-order spatial modes along each propagation path of the two photons is achieved by apertures positioned in the far-field plane behind the first lens after the BBO crystal (see Fig. 5). The diameter of the aperture was varied between $1.5\mathrm{mm}$ and $4\mathrm{mm}$.

The measured count rates in the Fraunhofer far field using the second method are shown in Fig. 8 . Again the calculated distributions for given correlation factors are shown in addition to the measured data. The largest aperture gives the highest contrast for the blue OD in the two-photon rate and the smallest interference contrast in the single-photon rate, whereas the smallest aperture leads to no visibility at all for the blue OD in the two-photon rate and increased interference contrast in the single-photon-rate distribution.

For the different diameters of the apertures, equivalent acceptance $M^2$ numbers for the modes that can transmit through this aperture without too severe losses can be given. The propagation of the modes start at the BBO crystal with a beam waist equal to the pump beam waist (see Section 3). The maximum divergence of the propagating photons generated in the BBO-crystal can be defined by the aperture in the far field of a lens (see Fig. 5). If we define this acceptance $M^2$ number by modes that have a beam radius at the aperture equal to the aperture radius at the aperture’s position, we find the acceptance $M^2$ numbers of $M^2{}_{\mathrm{acc}}=0.6$, 1, 1.6 for the aperture diameters of $1.5\mathrm{mm}$, $2.5\mathrm{mm}$, and the $4\mathrm{mm}$ diameter aperture, respectively. The acceptance $M^2{}_{\mathrm{acc}}=0.6$ simply means that there are already significant losses to be expected for the transverse fundamental mode $\left({\mathrm{TEM}}_{00}\right)$ and an even stronger suppression of the next higher-order mode. Thus, a Schmidt number around one and a vanishing degree of correlation is expected, only. The degree of correlation that can be calculated from Eq. (A16) in Appendix A with the measured correlation factor of $r=3$, is ${\rho }_x=0.05$. Consequently, the OD in the blue in the two-photon rate distribution totally disappeared (see Fig. 8 bottom). For $M^2=1$ the transmission of the ${\mathrm{TEM}}_{00}$ through the aperture should be roughly 86.5%, and there should already be some transmission of the next higher-order mode. In this case ${\rho }_x=0.27$ results. In the measured two-photon-rate distribution, a smaller additional OD becomes visible. For the open aperture with $M^2=1.6$ there should be a distinct transmission of a second spatial mode. The determined correlation factor of $r=0.85$ translates to ${\rho }_x=0.40$, which is a reasonable degree of correlation. As can be seen from Fig. 8 top) there is a distinctive blue OD that is still somewhat smaller than the OD in the red.

Altogether this is an experimental demonstration of the reduction of the dimensionality of entanglement by a restriction of the detected spatial modes explicitly, as theoretically predicted by van Exter et al. [27] and recapitulated in Appendix A for our specific circumstances.

The observed degree of the spatial correlation of the biphotons was mainly limited by the aperture of crystals used for the birefringence compensation that have a free aperture of about $4\mathrm{mm}$ $30\mathrm{cm}$ away from the nonlinear crystal. By using compensation crystals with larger diameter at a position closer to the generation crystal, our correlation could be improved distinctly, and hence the diffraction in the blue would be further increased.

5. CONCLUSION

We demonstrated the diffraction of a biphoton beam at a blazed grating. As known from classical optics, a blazed grating generates a diffraction pattern that shows only one strong order of diffraction (OD). The diffraction angle of the OD is dependent on the wavelength of the photons, which is $780\mathrm{nm}$ (red) in our investigations. In case of spatially correlated biphotons a second strong OD in the far-field distribution of the two-photon coincidence rate arises at the position expected only for light of half of the original wavelength corresponding to $390\mathrm{nm}$ (blue) in our investigations. The higher the degree of spatial correlation of the biphotons the more pronounced becomes the two-photon rate OD at $390\mathrm{nm}$. Thus, the ratio of the heights of the red and the blue OD indicates the degree of correlation of the photons. If the degree of correlation becomes high enough, the two photons of a biphoton will pass through identical grating periods as one quantum object but not separately through different periods.

Since there is only one strong OD caused by the correlated photons, application of the blazed grating might enable the design of a purification filter for photons of a certain spatial correlation width. The edge of such a filter would be set by the grating period.

Furthermore, our results give experimental evidence to the prediction expressed explicitly in [27] that a higher number of spatial modes have to be within the spatial detection bandwidth to enable the observation of a high dimension of spatial entanglement. The blue OD expected for a high degree of spatial correlation width that can be observed in spatial multimode illumination of the grating disappears in case of spatial single-mode illumination of the same parametric downconversion source. Because of the high diffraction efficiency of the blazed grating into only one OD, it is a good tool for the characterization of the spatial correlations of biphotons and possibly also higher-order photon structures such as tri-photons and four entangled photons.

APPENDIX A

In this appendix we will discuss the relation of our phenomenological two-photon function F with the one known from parametric downconversion, $F_{\mathrm{PDC}}$, and the interrelation of our correlation factor with the degree of correlation and the Schmidt number of usable entangled modes.

Following Peters et al. [23], who refers to the original work of Monken et al. [20], we start to express the two-photon probability amplitude as

(A1)$${\tilde{F}}_{\mathit{PDC}}(k_{1\perp },k_{2\perp })\propto {\tilde{E}}_p(k_{1\perp }+k_{2\perp })\tilde{\xi }(k_{1\perp },k_{2\perp }).$$

Again the tilde symbol indicates the spatial Fourier transform of a function. ${\tilde{E}}_p\left(k_{\perp }\right)$ is the momentum representation of the pump field in the BBO crystal in dependence of the transverse wave vectors of signal $\left(k_{1\perp }\right)$ and idler $\left(k_{2\perp }\right)$ photons ${\tilde{E}}_p(k_{1\perp }+k_{2\perp })\propto \exp [-{|k_{1\perp }+k_{2\perp }|}^2∕2{\sigma }_{kp}]$. ${\sigma }_{\mathrm{kp}}$ is standard deviation of the transverse pump wave vector, which relates to the divergence of the pump beam ${\theta }_p$ in paraxial approximation to ${\sigma }_{kp}=\left|{\mathbf{k}}_{\mathbf{p}}\right|{\theta }_p∕2$. $\xi \left(k_{\perp }\right)$ is the phase-matching function in the nonlinear crystal with

(A2)$$\tilde{\xi }(k_{1\perp },k_{2\perp })=\mathrm{sinc}\left[\frac{1}{2}L\Delta k_z(k_{1\perp },k_{2\perp })\right].$$

$\Delta k_z$ is the longitudinal wave vector mismatch and L is the length of the nonlinear crystal. This expression can be expanded for different parametric downconversion geometries [23]. In our case of type II noncollinear phase matching the wave vector mismatch can be expressed up to second order as

(A3)$$\Delta k_z(k_{1\perp },k_{2\perp })\approx {\varphi }_0+\frac{{|k_{1\perp }-k_{2\perp }|}^2}{4n\omega ∕c}+\frac{1}{2}{k}_{2y}\delta .$$

${\varphi }_0$ is the phase mismatch in propagation direction, δ is the walk-off angle for extraordinary polarized light, ω the frequency of the biphotons in the degenerate case, and $n=2n_1{n}_2∕(n_1+n_2)$ [23] is the effective refractive index for the signal and idler photons with refractive index $n_1$ and $n_2$. Since we are only interested in the x-direction perpendicular to the grating period, we can write the two-photon amplitude (as we could have done directly without restriction to the x-components for the noncritical type I and type II cases):

(A4)$${\tilde{F}}_{\mathit{PDC}}(k_{1x},k_{2x})\propto {\tilde{E}}_p(k_{1\perp }+k_{2\perp })\tilde{\xi }(k_{1\perp }-k_{2\perp }).$$

To create some more analytical insight we roughly approximate the $\mathrm{sinc}$-function by a Gaussian function $\mathrm{sinc}\left[\frac{1}{2}L\Delta k_z(k_{1\perp },k_{2\perp })\right]\approx \exp [-\frac{1}{2}L\Delta k_z(k_{1\perp },k_{2\perp })]$, which leads to

(A5)$$\tilde{\xi }(k_{1\perp }-k_{2\perp })\propto \exp [-\frac{{|k_{1\perp }-k_{2\perp }|}^2}{8n\omega ∕\left(Lc\right)}].$$

The inverse Fourier transform gives us the relation of the two-photon field in the near field of the nonlinear crystal:

(A6)$$F_{\mathit{PDC}}(x_1,x_2)\propto \int \int E_p(k_{1\perp }+k_{2\perp })\tilde{\xi }(k_{1\perp }-k_{2\perp })e^{i{k}_{1\perp }{x}_1}{e}^{i{k}_{2\perp }{x}_2}d{k}_{1\perp }d{k}_{2\perp }.$$

This integral can be rewritten to

(A7)$$F_{\mathit{PDC}}(x_1,x_2)\propto \int \int E_p\left(K_1\right)\tilde{\xi }\left(K_2\right)e^{i{K}_1{X}_1}{e}^{i{K}_2{X}_2}d{K}_{1}d{K}_2,$$

with new variables

(A8)$$K_1=\frac{k_{1\perp }+k_{2\perp }}{\sqrt{2}},K_2=\frac{k_{1\perp }-k_{2\perp }}{\sqrt{2}},X_1=\frac{x_1+x_2}{\sqrt{2}},$$

$$X_2=\frac{x_1-x_2}{\sqrt{2}}.$$

This yields

(A9)$$F_{\mathit{PDC}}(x_1,x_2)\propto E_p(x_1+x_2)\xi (x_1-x_2),$$

with

(A10)$$E_p(x_1+x_2)\propto \exp [-\frac{1}{2}\frac{{(x_1+x_2)}^2}{2∕{\sigma }_{kp}^2}],$$

$$\xi (x_1-x_2)\propto \exp [-\frac{{(x_1-x_2)}^2}{Lc∕\left(n\omega \right)}].$$

This shows that in the near field the spatial correlation of the photons is primarily defined via the phase-matching function ξ. In our Gaussian approximation both functions factorize in the two transverse dimension, where we are just interested in the x-dimension. The standard deviations of the two Gaussian distributions can be abbreviated by ${\sigma }_{+}$ and ${\sigma }_{-}$ and then the expected two-photon rate reads

(A11)$$F_{\mathit{PDC}}(x_1,x_2)\propto \exp [-\frac{1}{2}\frac{{(x_1+x_2)}^2}{{\sigma }_{+}^2}]\cdot \exp [-\frac{1}{2}\frac{{(x_1-x_2)}^2}{{\sigma }_{-}^2}].$$

From such a form we can directly give the degree of correlation ${\rho }_x$ (compare [28]):

(A12)$${\rho }_x=\frac{{\sigma }_{-}^2-{\sigma }_{+}^2}{{\sigma }_{-}^2+{\sigma }_{+}^2}.$$

If Eq. (A11) expresses the two-photon function in a plane where the detection takes place, we can deduce the number of detectable or usable Schmidt modes from ${\rho }_x$. Otherwise the two-photon amplitude has to be propagated to the detection plane (see, e.g., Eq. (10) in [23]) and again expressed in a form of Eq. (A11).

If we follow the Schmidt decomposition with Gaussian–Hermite modes like van Exter in [27] then the one-dimensional Schmidt parameter $K_x$ can be determined by $K_x=1∕\sqrt{1-{\rho }_x^2}$ (the authors in [27] write about a separability parameter β, which can be shown to be equal to the degree of correlation ${\rho }_x$. For a measurement method of the number of Schmidt modes in the spectral domain of parametric downconversion, see also [32]).

Our function $A\left(x\right)$ [see Eq. (2)] in the two-photon amplitude contains the transverse field profile $E_{1,2}$ and the complex transmission function $T_{\mathit{bl}}$ of the blazed grating:

(A13)$$A\left(x_1\right)=E_1\left(x_1\right)\cdot T_{\mathit{bl}}\left(x_1\right),$$

so that we can write down our phenomenological two-photon function just before the grating (without $T_{\mathit{bl}}$):

(A14)$$F(x_1,x_2)\propto E_1\left(x_1\right)E_2\left(x_2\right)C(x_1-x_2).$$

With a common field function as expressed in the crystal near field in Eq. (A10) and a different definition of the correlation factor r this two-photon function can be written in the equivalent form of (A11)):

(A15)$$F(x_1,x_2)\propto E_{1,2}(x_1+x_2)C^{\prime }(x_1-x_2).$$

However, for reasons of comparison to the work of Shimizu et al. [14] and to keep the description of the diffraction simple we stick to the two-photon amplitude in the shape of Eq. (A14). But from this form of Eq. (A15) the degree of correlation can be related to the correlation factor r in the function C(x) for a given beam radius of $w_{\mathrm{gr}}=2{\sigma }_{\mathrm{gr}}$ and grating period d by

(A16)$${\rho }_x=\frac{{\sigma }_{gr}^2}{{\sigma }_{gr}^2+r^2{d}^2∕\left(8\ln 2\right)}.$$

To detect spatial entanglement of photons or, more precisely speaking, position-momentum entanglement, correlations not only in the near field but also in the far field have to be observed and surpass a certain degree. This was expressed in general by Mancini et al. [31] and Duan et al. [33] for the conjugated variables of a continuous variable systems. In [34] we discuss a quantitative criterion for the detection of spatial entanglement of photons by considering the degree of correlation in the near and the far field.

ACKNOWLEDGMENTS

We acknowledge fruitful discussions with Carsten Henkel about spatial correlations of photons and very helpful support from Rainer Hultzsch from Carl Zeiss AG regarding the delivery and discussion of the properties of the blazed grating.

 

Fig. 1 Diffraction efficiency for the first order of diffraction in dependence of the wavelengths for the blazed grating used in the experiments (Blaze wavelength $500\mathrm{nm}$, grating period $25\mu \mathrm{m}$).

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Fig. 2 Calculated photon rates for Fraunhofer diffraction behind the blazed grating in near field illumination with a spot diameter of $100\mu \mathrm{m}$ on the grating for a spatial correlation factor of the biphotons of top: $r=3$, middle: $r=0.85$, bottom: $r=0.01$.

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Fig. 3 Calculated photon rates for Fraunhofer diffraction behind the blazed grating in near field illumination with a spot diameter of $29\mu \mathrm{m}$ for a correlation factor of the biphotons of $r=0.85$. Top: angular resolution $0.5\mathrm{mrad}$, bottom: angular resolution $10\mathrm{mrad}$.

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Fig. 4 Calculated photon rates for Fraunhofer diffraction behind the blazed grating in far field illumination for different spatial correlation factors of the biphoton of top; $r=3$; middle, $r=0.85$; bottom, $r=0.01$.

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Fig. 5 Setup to generate a biphoton beam by Hong–Ou–Mandel interference of photons produced from type II noncollinear parametric downconversion (left) and the setup of our grating experiment (right).

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Fig. 6 Single- and two-photon rate measured in the Fraunhofer far field behind the blazed grating (circles) and calculated theoretical expectation (straight line) in case of a correlation factor of $r=0.85$.

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Fig. 7 Two-photon count rate measured in the Fraunhofer far field behind blazed grating (circles) and calculated theoretical expectation (straight line) for increasing correlation factor achieved by moving the image plane of crystal near field away from the grating plane.

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Fig. 8 Two photon and single-photon count rate measured in the Fraunhofer far field behind blazed grating (circles) for decreasing number of illuminating higher-order spatial modes restricted by an aperture of varying diameter and calculated theoretical expectation (straight line) for decreasing correlation width as given. Top; aperture $d=4.0\mathrm{mm}$; middle; $d=2.5\mathrm{mm}$; bottom; $d=1.5\mathrm{mm}$.

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