1. Introduction

Quantum correlations between optical fields produced in spontaneous parametric down conversion (SPDC) were first observed in 1970 by Burnham and Weinberg [1,2]. Since the first demonstration of correlation imaging and interference with an entangled source [3], spatial correlations have been a hot topic in the area of imaging [4–9,13]. More recently, an experiment done in our group allowed the spatial information stored in a phase hologram to be retrieved [10] by measuring quantum spatial correlations between two images formed by twin photons. In the last years, more studies dealt with the propagation of entangled two photons states through scattering medium. This phenomenon has been both theoretically [11,12,14–16] and experimentally [17–21] investigated. From the theoretical point of view, in 1998, Beenakker [11] introduced a convenient formalism of the input-output relations between the incoming and outgoing quantum states. Experimentally, in 2009, Smolka et al. [17] reported the spatial observation of correlations between the scattered modes in a scattering medium and this study showed that the strength of the quantum correlation is related to the number of incident photons. Moreover, in 2010, Peeters et al. [18] showed that, after propagation of entangled photon pairs through a thin scattering medium, the spatial repartition of the coincidence counts rate exhibits a speckle structure which cannot be obtained with the single count rate. The thin scattering medium used in [18] and in our experiments is called in the following a "complex medium", since it can be described by its statistical properties but does not vary randomly with time. More recently, Defienne et al. [21] reported the generation of entangled photon pairs with a tunable degree of spatial entanglement by controlling the spatial coherence of the pump beam with complex diffusers. While in [18] two punctual detectors were scanned over different positions, we propose in this paper to directly image the two detection planes using two electron-multiplying charge coupled device (EMCCD) cameras able to detect single photons [22]. The use of EMCCD cameras in our experiment gives the possibility to obtain a real two-dimensional speckle pattern which would demand a prohibitive time with scanning detectors. Here two configurations are investigated. First, we put a thin diffuser in the crystal near-field and correlations are observed in the far-field of the crystal. Second, a thin diffuser is placed in the crystal far-field and near-field correlations are observed. Finally, we performed stochastic simulations [23] to confirm the experimental results and the spatial structure expected in the theoretical expression of the correlation function, both in the far-field and near-field correlations.

2. Optical speckle pattern with entangled photon pairs

To describe the propagation of light through a complex medium, we consider the two configurations illustrated in Fig. 1. The thin nonlinear crystal lies in the plane ($P$), with a transverse vector $\textbf {r}$. The thin complex medium is assumed being a pure phase-object of transmission $t\left (\textbf {r'}\right ) =e^{i\Phi \left ( \textbf {r'}\right )}$, where $\Phi \left (\textbf {r'}\right )$ is the phase modulation in the transverse plane. It lies in a plane ($P'$) with a transverse vector $\textbf {r'}$. The detectors arrays $EMCCD1$ and $EMCCD2$ lie in the planes ($P_1$) and ($P_2$) with transverse vectors $\textbf {r}_{1,2}$. Let us consider a SPDC light emitted from a planar thin nonlinear crystal illuminated by a monochromatic beam of frequency $\omega _p$ and of amplitude $E_{p}\left ( \textbf {r}\right )$. For a sufficiently thin crystal, we assume that the two photons of the pair are created at the same random place in the nonlinear crystal [10]. Let us name $t_s\left (\textbf {r'}\right )$ and $t_i\left (\textbf {r'}\right )$ the transmission of the thin complex medium for the signal and idler beams. We can write the impulse-response functions of the imaging systems between the crystal and the detection planes for the signal and the idler as [24]:

 

Fig. 1. Two photon speckle generation with entangled photon pairs. The two configurations (a) and (b) are related to the far-field and near-field correlations, respectively. The $\chi ^{(2)}$ nonlinear crystal is pumped by laser pulses at the angular frequency $\omega _p$. The entangled photons (signal and idler) are generated at $\omega _s$ and $\omega _i=\omega _p-\omega _s$. Two separated optical systems with the transfer functions $h_1$ and $h_2$ are used to focus the signal and idler beams onto a thin complex medium that transmits the two beams with the transmissions $t_s$ and $t_i$ for the signal and idler beams respectively. Then, another two separated optical systems with the transfer functions $h_{1}'$ and $h_{2}'$ are used to image the transmitted beams on the $EMCCD1$ and $EMCCD2$ cameras.

Download Full Size | PDF

where $h_{1,2}\left ( \textbf {r}',\textbf {r}\right )$ are the impulse-response functions describing the propagation from the crystal plane up to the plane of the complex medium. Similarly, $h'_{1}\left ( \textbf {r}',\textbf {r}_{1}\right ) )$ and $h'_{2}\left ( \textbf {r}',\textbf {r}_{2}\right )$ describe the propagation from the complex medium to the cameras. If we assume that SPDC is detected at the degeneracy ($\omega _{i}=\omega _{s}=\frac {\omega _{p}}{2}$) with a narrow band interferential filter, we can neglect chromatic dispersion of the whole optical systems. Then, the joint probability of detection of photons at both detectors arrays $EMCCD1$ and $EMCCD2$ is given by [25]:

2.1 Scattering medium in the near-field, far-field correlations

In this imaging configuration where momentum correlations are measured, a $4-f$ optical system is used in both the signal and idler branches to image the signal and idler beams at the same place of the complex medium. Assuming that the phase matching angular bandwidth allows all the photons from the signal and idler beams to be collected by the $4-f$ optical system, we can neglect the effect of the $4-f$ system. In this case, we can consider that the transmission for both beams verifies: $t_s\left ( \textbf {r}'_{s}\right ) =t_{i}\left ( \textbf {r}'_{i}\right ) =t\left ( \textbf {r}'\right )$. Then, two identical lenses are used to record the signal and idler beams onto the detectors arrays $EMCCD1$ and $EMCCD2$. Thereby, $h_{1}'$ and $h_{2}'$ correspond to a two identical $2-f$ impulse-response functions which take the form [26]:

2.2 Scattering medium in the far-field, near-field correlations

In this configuration, the signal and idler beams are transmitted through two sequential and identical $2-f$ optical systems with a thin complex medium in the intermediate Fourier plane between them. The thin complex medium includes two non-identical transmission $t_{s}\left ( \textbf {r}'_{s}\right )$ and $t_{i}\left ( \textbf {r}'_{i}\right )$ since the far-field of the signal and idler beams is transmitted by two different areas of the thin complex medium ($\textbf {r}'_{s}\neq \textbf {r}'_{i}$). Therefore, we can consider that the signal and idler branches comprise a $4-f$ optical with apertures $t_{s}\left ( \textbf {r}'_{s}\right )$ and $t_{i}\left ( \textbf {r}'_{i}\right )$, respectively. Under these considerations, the impulse-response functions in the two branches can be expressed as [25,26]:

3. experimental setup and results

3.1 Far-field correlation: experimental setup

The experimental setup in this subsection is identical to that described in [10], where the hologram is replaced by a thin diffuser.

Figure 2a shows the experimental setup. The collimated pulsed laser at 355 $nm$ (Q-switched Nd:YAG laser, 330 ps pulse duration, 27 $mW$ mean power, $1kHz$ repetition rate and 1.6 $mm$ FWHM beam diameter) illuminates a 0.8 $mm$ long $\beta$-barium borate (BBO) crystal. Entangled photon pairs are generated by noncolinear type-II SPDC interaction where the cross polarized twin beams propagate, outside the crystal, in separate directions because of noncolinear phase matching. The lenses $L_1$ ($f_1$=75 $mm$) and $L_2$ ($f_2$=75 $mm$) ($4-f$) imaging system are used to image the crystal onto the diffuser. The entangled photon pairs transmitted by the thin diffuser are separated because of the noncolinear phase matching. Because the medium used in our experiment is thin, both polarizations experience the same phase shift at a given place. Far-field detection is performed by two EMCCD cameras lying in the focal plane of the lenses $L_3$ ($f_3$=150 $mm$) and $L_4$ ($f_4$=150 $mm$). The exposure time of the EMCCD cameras is equal to 100 ms (i.e. 100 laser shots). Before detection, the SPDC beams are filtered around degeneracy with narrow-band interferential filters $F_1$ and $F_2$ ($@710\,nm, \Delta \lambda =4\,nm$). Figure 2b shows a 3D-profile of an area of $2\times 2\,mm^2$ of the diffusers. The thin complex medium is made of a microscopic slide with one side attacked with fluorydic acid. This process allows the production of diffusers of deep roughness ($\sim 3\,\mu m$) and large waviness profiles (few 100 $\mu m$). The deep roughness profile ensures a spatial phase modulation of the transmitted beams with a large amplitude $(>4\pi )$. The angular aperture of the scattered light is inversly proportional to the typical size of a phase modulation of the diffuser. From Fig. 2c, we can estimate the angular aperture of the scattering medium to 8 $mrad$. This value is smaller than both the numerical aperture of lenses $L_3$ and $L_4$ (170 $mrad$) and the phase matching angular bandwidth (47 $mrad$). This latter value is obtained by multiplying the FWHM ($66\,mm^{-1}$) of the SPDC which is obtained in Fig. 3a by the signal or idler wavelength (710 $nm$). Consequently, we assume that most of the scattered twin photons are collected by the imaging system. To ensure that the diffuser is in the image plane of the crystal, we used a He-Ne laser (no SPDC in this regime) to check the alignment of all optical components and to minimize defocusing. Figure 2c shows the experimental image recorded in the far-field of the diffuser using the He-Ne laser which provides a coherent beam. In Fig. 2c, the speckle pattern carries information both on the coherence properties of the laser light and microscopic details of the diffuser [18]. We determined the size of the speckle pattern and the typical size of the scatterers along a transverse direction. Figure 2c shows the speckle pattern obtained with a FWHM of 8 $mm^{-1}$ in the far-field that corresponds to an average size of the diffuser waviness of 125 $\mu \,m$, in agreement with the waviness of the surface of the diffuser (see Fig. 2b).

 

Fig. 2. (a) Experimental setup: Entangled photon pairs at 710 $nm$ are generated via SPDC in a type-II BBO using a 330 $ps$ pump pulse at 355 $nm$. The crystal is imaged onto the thin diffuser with a $4-f$ optical system. The entangled photon pairs (signal and idler) transmitted by the diffuser are detected and resolved spatially in the far-field on two EMCCD cameras. (b): 3D-profile of the diffuser and (c): Far-field intensity of the coherent scattered light (in a.u). $\rvert V\rangle$ and $\rvert H\rangle$: the vertical and horizontal polarizations. $(P_1)$ and $(P_2)$: the Fourier plane and $(P')$ the image plane. D: dichroic mirror, $F_3$ and $F_4$: the interferential filters.

Download Full Size | PDF

 

Fig. 3. Without diffuser,(a) average photon number in single far-field images (signal or idler) of SPDC and (b) measured correlation function in dB between 100 twin images. With diffuser, (c) average photon number in single far-field images (signal or idler) of SPDC and (d) measured correlation function in dB over 40 000 twin images. With stochastic simulations, (e) average photon number in single far-field images (signal or idler) of SPDC and (f) correlation function issued from 10 000 stochastic simulations with diffuser.

Download Full Size | PDF

Now, we discuss the case where SPDC is used. First, we measure the correlation function with a set of 100 twin images without the diffuser in order to characterize the bi-photon state. The method to calculate the correlation function between twin photons in images has been detailed in [8,27]. Typically, the correlation function $G^{(2)}\left ( \textbf {r}_{1},\textbf {r}_{2}\right )$ is obtained by averaging the normalized intercorrelation functions calculated between each pair of twin images. Figure 3a shows a typical single far-field image of SPDC simply conditioned by phase matching. The momentum spatial correlation function is calculated by summing the cross-correlations of the signal image of a pair with the 180 ^∘ rotated idler image of the same pair. Figure 3b shows a narrow correlation peak in the correlation function without the diffusing medium. This correlation peak gives access to the degree of correlation, i.e the ratio between the number of photons detected in pairs and the total number of photons [10]. In this experiment, the value of the degree of correlation without diffuser is equal to 0.22. This value is close to the effective quantum efficiency $0.26$ of the entire detection system and the integral of the correlation peak $0.23$ obtained in [28]. The slight difference can be attributed to the additional losses induced by the two supplementary lenses necessary to image the diffusing medium. By approximating the correlation peak to a two dimensional Gaussian profile, we can calculate the conditional variances, which correspond to the widths of the normalized cross correlations peak along the $x$ and $y$ axis. For more details about the calculation of the conditional variances, see [29]. From the normalized cross correlation in momentum calculated with a set of 100 twin images (see Fig. 3b), we obtained $\sigma _{\nu _x}^2=5,00.10^{-5}\,\mu m^{-2}\hbar ^2$ along $x$ axis and $\sigma _{\nu _y}^2=1,05.10^{-5}\,\mu m^{-2}\hbar ^2$ along $y$ axis. Secondly, we considered the diffuser in the image plane of the crystal as shown in the experimental setup (see Fig. 2a). Similarly to Fig. 3a without diffuser, Fig. 3c and Fig. 3e show typical single (signal or idler only) far-field images of SPDC simply conditioned by phase matching. Moreover, Fig. 3a; Fig. 3c and Fig. 3e show that the profile of the SPDC beam scattered by the diffuser is not significantly widened. For the correlation function, Fig. 3d and Fig. 3f show a speckle pattern. The presence of the speckle pattern is in agreement with the predictions of Eq. 6 and also consistent with the works done by Peeters et al. [18] where, by scanning the positions of both detectors independently in the horizontal line, a far-field correlation peak appears around a null value of the sum coordinates. As we did with the laser beam in Fig. 2c, we determined the speckle pattern size in Fig. 3d and we get a FWHM of 8,4 $mm^{-1}$ that corresponds to an average size of the diffuser waviness of 119 $\mu m$, in agreement with the waviness of the surface the diffuser. We emphasize that the correlation speckle pattern has the same size as the speckle pattern obtained with coherent light. The degree of correlation with diffuser estimated in Fig. 3d is 0.17. To explain the difference between the degree of correlation without diffuser and with diffuser, we have considered that a part of correlations is lost because of the reflection or absorption effects induced by the diffuser and also of the fact that few scattered photons are not collected by the imaging system. Compared to the case of classical light in Fig. 2c and to simulations in Fig. 3f, the speckle pattern measured in Fig. 5d does not present a good contrast. The low contrast is due to some geometrical aberations and some defocusing induced by the optical components, and also to the bandwidth of the interferential filters. As these effects contribute to reduce the degree of spatial entanglement, the contrast of the correlation speckle is also related to the dimensionality of the bi-photon state. Moreover, because of the noncolinear type-II SPDC interaction, the two entangled photons have not the same incidence angle in the plane where the thin scattering medium is inserted. All these elements montioned above could contribute to reduce the speckle contrast. The absence of a speckle pattern in Fig. 3c and Fig. 3e in the single images of one beam of the SPDC light results from the incoherent character of one beam of the SPDC light. In fact, it is well known and it has been demontrated in [30] that the photon number distribution of incoherent light obeys a thermal statistics and it is almost impossible to experimentally distinguish this statistics from a poissonian one when the brighness is small. In this case, spatial correlations vanish. Moreover, experimental images of SPDC beams are the result of the incoherent sum of a very large number of temporal modes. In our case, the number of time integrated temporal modes is about 400.000. This large number is related to the exposure time of EMCCD cameras to the bandwith of interferential filters and to the pump pulse duration [31,32]. In this case, the thermal statistic of a single temporal mode is no more measurable.

3.2 Near-field correlation: experimental setup

In this second part, we report the near-field correlations between entangled photon of pairs transmitted through the diffuser. As mentioned in section 2.2, the far-field of the crystal is imaged onto the diffuser and the spatial correlations are measured between near-field images of the crystal.

The experimental setup, depicted in Fig. 4, is similar to that described in Fig. 2a, when the $4-f$ imaging system is replaced by a $2-f$ imaging system ($L_1$:$f_1$=150 $mm$). In this case, the position of the diffuser lies in the far-field of the crystal and the EMCCD cameras image the near-field. The transverse coordinates $x'$ and $y'$ in the near-field plane are related to the transverse coordinates in spatial frequency units $\nu _x$ and $\nu _y$ in the far-field plane by $x'=\lambda f_{1}\nu _x$ and $y'=\lambda f_{1}\nu _y$ , where $\lambda =710\,nm$ is the signal or idler wavelength and $f_1$ the focal length of the far-field imaging lens. As we did before, first we removed the diffuser in the experimental setup (see Fig. 4) and we measured the correlation function with a set of 100 twin images.

 

Fig. 4. Experimental setup:Entangled photon pairs at 710 $nm$ are generated via SPDC in a type-II BBO using 330 $ps$ pump pulse at 355 $nm$. Lens $L_1$ ($f_1$=150 $mm$) is used to image the far field of the crystal onto the diffuser placed in the Fourier plane. The entangled photon pairs (signal and idler) transmitted by the diffuser are detected in the near field on the EMCCD1 and EMCCD2 cameras using lenses $L_2$ ($f_2$=150 $mm$) and $L_3$ ($f_3$=150 $mm$) respectively. $\rvert V\rangle$ and $\rvert H\rangle$: the vertical and horizontal polarizations. $(P_1)$ and $(P_2)$: the image plane and $(P')$ the Fourier plane. D: a dichroic mirror, $F_1$ and $F_2$: interferential filters.

Download Full Size | PDF

Figure 5a shows a single near-field image of the SPDC conditioned by both phase matching and the pump beam size. As written in section 3.1 and in agreement with [18], the absence of any speckle pattern is consistent with the incoherent character of the light formed by a single beam of the entangled light. Similarly to the far-field correlations (see Fig. 3b), the near-field correlations without diffuser (see Fig. 5b) show a narrow peak. The degree of correlation deduced from this peak is equal to 0.23, slightly above that obtained in Fig. 3b. Similarly to the far-field correlations, we calculated the conditional variances in near-field using Fig. 5b. We obtained $\sigma _{x}^2=171\,\mu m^2$ along $x$ axis and $\sigma _{y}^2=72\,\mu m^2$ along $y$ axis. The asymmetry observed in both position and momentum correlation measurements results from the effect of the width of the interferential filters. This effect leads to an anisotropy, i.e an enlargement of the correlation peaks along $x$ direction in both spaces (Fig. 3b and 5b). Using the conditional variances both in near-field and far-field, we can estimate the Schmidt number of the biphoton state in both transverse dimensions as [33]:

 

Fig. 5. Without diffuser, (a) average photon number in single near-field images(signal or idler) of SPDC and (b) measured correlation function in dB between 100 twin images. With diffuser, (c) average photon number in single near-field images (signal or idler)of SPDC and (d) measured correlation function in dB over 70 000 twin images. With stochastic simulations, (e) average photon number in single near-field images (signal or idler) of SPDC and (f) correlation issued from 10 000 stochastic simulations with diffuser.

Download Full Size | PDF

4. Conclusion

We have experimentally studied the spatial correlations, imaged by two EMCCD cameras, of a biphoton state transmitted by a thin complex medium lying either in the far-field or in the near-field of a source of entangled photons. As expected, our studies have shown the absence of one photon speckle in the quantum light transmitted through a scattering medium. For the entangled photon pairs, near-field and far-field spatial correlations have been evidenced. In both configurations, the correlation function of the entangled photon pairs exhibits a speckle pattern. Our results are in perfect agreement with [18], where the correlations were obtained by scanning two single detectors. The estimation of the degree of correlation without diffuser and with diffuser has shown that a small part of the correlations is lost in presence of a scattering medium. The degrees of correlation measured with the thin complex medium in the two configurations (i.e 0,17 in far-field and 0,16 in near-field) are close to those measured without complex medium (i.e 0,22 in far-field and 0,23 in near-field). From this fact, we conclude that the total number of photons detected in pairs is only slightly lowered after propagation through the thin complex medium. A development of the present work could be the control of the coincidence speckle pattern by using spatial light modulator (SLM) [20,34].