In this paper we show how after the generation of parametric down-conversion radiation (PDC) in the very high gain pulsed regime, we are able to reconstruct the pump via up-conversion of the twin beams originated from that PDC process. The peculiarity of the experiment is the ultra-broad spectral and angular bandwidth sent into the process of sum frequency mixing thanks to an achromatic imaging technique from the exit face of the PDC crystal using off-axis parabolic mirrors. The recorded spectra presented illustrate the high visibility recombination of the intense phase-conjugated signal and idler beams and pave the way for the investigation of both the spatial and temporal properties of the near field biphoton amplitude.
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The parametric down-conversion (PDC) process obtained by pumping a χ 2 non linear crystal permits to generate from a low divergence narrowband visible pump beam, angularly and spectrally broadband signal and idler infrared (IR) radiation with well defined space-time properties  and satisfying the so-called phase-matching relations. The reverse process or up-conversion of such a broadband field in an identical crystal is generally phase-mismatched. In fact, large spatial walk-off and temporal group-velocity mismatch (GVM) limits the back conversion of its ultra fine spatial and temporal structure into the visible range . However, when the up-conversion involves two perfectly phase conjugated signal and idler components, their spatio-temporal amplitude-and-phase fluctuations mutually cancel, because of their anticorrelation, permitting the initial low-divergence and narrow-band pump to be re-generated. Notably, the absence of any fine structure in the resulting large beam and long pulse render the walk-off and GVM almost ineffective, and so the process more efficient than the up-conversion of each individual component. For this reason, the pump reconstruction capability provides a meaningful indicator of phase conjugation, of its amount, and its characteristics (e.g. the robustness vs signal-idler delay). It is also worth noting that diffraction and dispersion exactly compensate for phase-matched signal and idler photons. As shown in [3, 4], thanks to the compensation of diffraction and dispersion, the spatio-temporal correlation of twin photons at the output face of the PDC crystal is featured by a localized X-shaped structure, with a temporal localization in the femtosecond range.
Up to now this non linear interaction technique (up-conversion) has been used in the low gain regime, for the study of the temporal coherence properties of the parametric radiation , and of the ultrafast non classical temporal correlation between the entangled photon pairs [6–11], with integrated measurements based on single-photon counting techniques. From the theoretical side, studies on time resolved pair-wise up-conversion have been reported in Refs. [12,13]. The PDC bandwidth collected in these experiments is generally limited by the angular acceptance, the dispersion, or the losses of the optical components used in the experiments. In addition note that in the above context the spatial properties of the up-converted field have never been addressed.
In this paper, we illustrate an experiment where for the first time, to the best of our knowledge, an ultra broad spectral and angular bandwidth of the PDC radiation generated in the very high gain regime (G ≥ 106) is reconverted through SFG into a non linear crystal, giving rise to the reconstruction of the low divergence narrowband pump, or more precisely to the spatial gain profile of the parametric down conversion process. The twin beam radiation is recombined into the SFG crystal thanks to an achromatic imaging technique using off-axis parabolic (OAP) mirrors, permitting thus to recreate the PDC field in amplitude and phase at the entrance of the crystal. The exact 4f -diffraction free- imaging system, preserves the cancellation of diffraction and dispersion for phase-matched photons that will recombine through the non linear process of sum frequency generation. Moreover the CCD based spatio-temporal diagnostics used in this work allows us to enhance the signal to noise contrast. The high visibility recombination of the intense phase-conjugated signal and idler beams, and the high spatio-temporal resolution accessed by the ultra broad PDC bandwidth used, are essential requirements and constitute the first step for the planned experimental investigation of the twin beams spatio-temporal correlation at the output of the PDC crystal, theoretically analyzed in .
2. Experimental set-up
The experimental layout is shown in Fig. 1. A 4mm long β-Barium-Borate (BBO) type I crystal is pumped for PDC by a 1 ps, 527.5 nm second harmonic pulse from a 10 Hz repetition rate, Nd:Glass laser (Twinkle, Light Conversion Ltd.), initially collimated down to 700 μm, with an energy of about 500 μJ. A 2mm glass filter placed just after the crystal is used to completely remove the pump (transmittivity T > 90% in the 750–1300 nm wavelength region). The PDC crystal, cut for perfect collinear phase-matching (PM) at the degenerate wavelength, 1055 nm (θcut = 22.93°), is tuned for collinear emission of the degenerate signal and idler wavelength.
The PDC radiation is sent into the imaging system, composed by two identical 90 degrees, gold OAP mirrors, with equivalent focal length of f = 17.8cm. The exit face of the crystal is imaged (f-2f-f system) onto the entrance face of a second BBO identical to the first, and tuned at the same phase-matching conditions so that the twin beam up conversion process can efficiently take place. Although necessiting a very careful alignment, with a proper tip to tip orientation, the parabolic mirrors used in the experiment guarantee an achromatic imaging without any distortions for objects size up to a few mm, as carefully studied by Malone and coworkers in . Moreover the reflectivity of these mirrors is greater than 85% for λ > 700nm and reaches 98% over the whole infrared spectral region starting at 1μm. Because of these characteristics and thanks to the big angular acceptance of the OAP mirrors (outer diameter 5 cm), we can estimate that of the very large spatiotemporal bandwidth of the radiation emitted by the first BBO and collected by the imaging system, at least 600 nm (in the range 800nm – 1400nm) participate -thanks to energy conservation- to the generation through SFG of a new field of same frequency of the original pump.
In order to verify that a large bandwidth of the PDC radiation is injected into the second crystal for reconversion through SFG, we have recorded its far-field (angle, wavelength) power spectrum just before the second non linear crystal. For this purpose we have used an imaging spectrometer (Oriel Instruments, MS260i) whose input slit was placed in the focal plane of a 20 cm lens that collects the radiation to be injected into the SFG crystal. Single shot far-field spectra have been recorded by means of a 14 bit CCD camera (Sony ICX205AL, WincamD) in the near infrared region, and by a thermo-electrically cooled InGaAs detector (Xenics) in the infrared region above 900 nm. The results shown in Fig. 2a and Fig. 2b respectively, present the typical spectrum of the down-converted radiation with the characteristic speckles reflecting the fact that the parametric down conversion process is initiated by vacuum. Note that the spectra presented are constituted by different images, which have been recorded over different spectral regions (dispersion was given by a 600 l/mm grating) and have then been put together in a single figure. Moreover it is worth mentioning that the portions of the spectra at wavelengths below 850 nm and above 1150 nm, are characterized by lower intensity (also due to the lower quantum efficiency of the detectors in those spectral regions), and have thus been recorded by removing the neutral filters present in front of the detectors. The saturated speckles in those regions are just a mean to visualize the PDC radiation emitted in the related bandwidth portions. Finally due to the transmission drop below 750 nm and above 1400 nm of the filter placed after the PDC crystal, no radiation is transmitted at those wavelengths.
3. Sum frequency mixing of twin beams
In the down converted beam with circular symmetry (typical of type I phase-matching, when the pump is weakly focused), transverse momentum conservation from phase-matching conditions implies that a given photon (e.g. signal) will have its pair member (idler) on the opposite side of the beam center. Therefore the up-conversion process corresponds to the synchronous recombination of the phase-conjugated twin beams (twin photons being generated with symmetric transverse wave-vectors).
The diagnostics of the up converted field has been made both in the spatial far field (angular) domain and in the spatiotemporal domain. Figure 3 shows the angular distribution of the up-converted radiation recorded in single shot with a 10 nm interference filter (IF) centered at 530 nm, in the far-field of a 20 cm focal length lens, by the CCD camera used before, and for three different tuning angles of the second BBO crystal. The green pump beam reconstructed as expected in the forward propagation direction (k⃗ = 0⃗) through frequency mixing, was evident by eye. It is clear in Fig. 3b where the peak intensity is five times higher than the background, corresponding to a visibility (Imax – Imin)/(Imax + Imin) of about 66 % (second BBO tuned identically to the first). Figure 3a and Fig. 3c show the evident loss of visibility when that second crystal is tuned in different phase-matching conditions. An extensive theoretical analysis of the spatio-temporal properties of the up-converted radiation will be given in .
The incoherent background in Fig. 3 propagates at many different k⃗ ≠ 0⃗, and originates from the up-conversion of un-paired PDC photons that are not phase-conjugated (i.e. they are produced in different elementary down-conversion events along the crystal), or from the possible frequency doubling of part of the PDC spectrum. The spatially coherent central peak is fixed and reproducible and is generated thanks to the sum frequency mixing of all the IR phase-conjugated twin contributions propagating in all those directions satisfying the momentum conservation. When the SFG crystal tuning is modified the recombination of the PDC radiation is evidently less efficient, at the expenses of a higher incoherent background, which is characterized by a speckled structure changing from shot to shot (following the typical speckle structure of far field PDC light).
The pump field reconstruction experiment through sum frequency mixing has also been repeated for a 1mm diameter (FWHM) collimated input pump (energy ∼500 μJ) sent into the first non linear crystal for parametric down conversion. The spatial spectrum of the reconstructed green field after the SFG crystal is reported in Fig. 4a with a comparison with the original pump spectrum (shown in the white inset box). Figure 4b illustrates the corresponding transverse far-field profile, evidencing in this case a signal to noise ratio of about 23, corresponding to a visibility of 92%. The images have been recorded by means of the same WincamD camera, but without interference filter in this case. As expected the reconstructed far field pump is slightly narrower than in Fig. 3b. Moreover it is interesting to note that the spatial far field profile of the original pump (injected into the first crystal) is much narrower than the reconstructed one. This can be understood when considering that the reconstructed coherent profile at the very output of the SFG non linear crystal (near field) reproduces in fact the parametric gain spatial profile of the vacuum field amplification process in the first crystal. Indeed the signal and idler beam size does not map the pump-beam profile but the actual parametric amplification gain profile G(r) ∼ cosh 2[σA(r)L] [15, 16] (L being the crystal length, A the pump field amplitude and σ a parameter proportional to the non linear susceptibility), as long as filtering due to the phase-matching does not take place. The narrowing of the near field gain spatial profile with very high gain explains the enlargement of the corresponding far field spatial profile of the reconstructed pump. We find an enlargement factor of 3 in accordance with numerical simulations (data not shown). The higher visibility of the reconstructed pump spatial profile in the case of Fig. 4b with respect to the case illustrated in Fig. 3b, follows from the fact that the input pump beam on the PDC crystal is in the first case larger (1mm in the case of Fig. 4, 700 μm for Fig. 3). When all the other parameters (in particular the crystal length) are fixed, a larger spatial pump profile implies a larger number K of independent spatial modes contributing to the PDC emission. Qualitatively, the coherent component of the intensity of the up-converted light grows quadratically with the number of modes Icoh ≈ K 2, because independent modes sum up coherently, while the incoherent component grows only linearly Iincoh ≈ K, leading thereby to an enhancement in visibility when the number of modes grows.
From energy measurements performed in the far field of the second crystal (about 30 cm away from the output face), for a 700 μm FWHM input pump beam and for two different pump energy conditions (notably 500 μJ and 615 μJ), we estimate that the reconstructed pump (selected by means of a pinhole) contains at most about 40% of the energy of the total up converted radiation. For instance in the higher energy case we measured about 155 nJ of energy by selecting the coherent peak with a ⋍1 mm diameter pinhole and 400 nJ when the pinhole was open; while in the lower energy case, we measured about 85 nJ and 237 nJ respectively.
Quantitative spectral measurements have confirmed that the reconstructed pump is indeed centered at 527.5nm, as shown for instance by the curve of Fig. 5b recorded by a fibre spectrometer (Ocean Optics) placed in the k⃗ = 0⃗ direction at about 30 cm from the SFG crystal output (case of 700 μm FWHM input pump). As expected when twin signal and idler photons are frequency mixed, their mutual fluctuations cancel each other because of their anticorrelations giving rise to a sharp spectrum, as already observed in . Note that the blue-shifted background observable in Fig. 5 is the incoherent contribution collected by the spectrometer.
An even more interesting result is obtained by measuring the single-shot spatio-temporal spectrum in the angular and wavelength (ϑ, λ) domain, recorded by means of the imaging spectrometer (MS 260i, Lot-Oriel) combined with the CCD. This is presented in Fig. 5a, recorded when the two BBO crystals are identically tuned in a slightly non collinear PM configuration, illustrating the possibility of selecting both angularly and spectrally, the coherent reconstructed pump with respect to the incoherent contribution. As shown in detail in  the incoherent spatio-temporal spectrum has a maximum along a plane in the (k⃗, ω) space (where k⃗ is the radiation wave vector and ω the frequency), whose location can be modified by tuning the angle of the second (SFG) crystal with respect to the PDC crystal tuning angle. As a consequence, it is possible to find tuning angles of the second crystal such that the incoherent background temporal spectrum is shifted away from the pump frequency. Also notice that the vertical entrance slit of the imaging spectrometer is orthogonal to the walk off plane, and that we collect only a limited portion of the incoherent spectrum. The frequency limited incoherent background selected in this way by the slit does not lie, for the chosen phase-matching, exactly underneath the pump frequency, permitting thus to enhance for any input pump conditions the experimental visibility, by isolating the 527.7 nm peak with respect to the other frequency contributions. In the case of Fig. 5 we can estimate a visibility of about 90% to be compared with the visibility of 66% of Fig. 3b. Table 1 summarizes the results for the different measurements settings used.
Note that a measurement of the total incoherent background spectrum performed by collecting the output radiation by means of a microscope objective and then focusing it on the fiber interferometer has shown that this spectrum extends over almost 100 nm (in collinear PM conditions from 500 till 600 nm wavelength, with a 50 nm wide FWHM bandwidth). This is in accordance with the results obtained from 3D numerical simulations of the experiment , in particular using the same ultrabroad PDC bandwidth as the experimental one shown in Fig. 2. On the basis of a theoretical modeling of the SFG process that takes into account the phase-matching conditions inside the SFG crystal, it can be shown that this 100nm bandwidth of the incoherent spectrum of up conversion is limited by the interplay of spatial walk-off and GVM that affects the injected PDC radiation and the generated SFG field. In contrast, the reconstruction of the coherent pump beam generated from twin photon pairs is not influenced by those linear propagation effects, also investigated by Jedrkiewicz et al. in . For this reason, the portion of PDC bandwidth that contributes to the building of the coherent component of the SFG spectrum is in principle unlimited. The brightness of the sharp coherent component at the pump wavelength (compared with that of the incoherent background) in Fig. 5b suggests that the involved PDC bandwidth is indeed very large, typically several hundreds of nanometers. Note that the observations presented here above are well supported and clarified by a theoretical model .
Finally we note that by covering part of the output PDC beam (and thus by cutting some of the paired phase-conjugated components) the visibility of the pump decreases, and goes to zero when only half portion of the cones are let through the imaging system. On the other hand, putting a stopper at the center of the far field PDC conical radiation, does not prevent from the reconstruction of the pump, even if the visibility is lower.
In conclusion, this work illustrates the efficient recombination via sum frequency mixing of the high intensity synchronous phase-conjugated broadband parametric down conversion twin beams, by achromatic imaging with off axis parabolic mirrors. The exact 4f-imaging system, permitting to recreate both in amplitude and phase the PDC field at the entrance of the SFG crystal, preserves the cancelation of diffraction and dispersion of phase-matched photons that recombine for the reconstruction of the pump beam with high visibility. Measurements of the spatial spectral profile of the reconstructed pump after the second non linear crystal have been performed by means of a CCD camera in different phase-matching and input pump beam conditions. Moreover the CCD based spectral diagnostics with an imaging spectrometer shows the possibility to obtain a high visibility reconstructed pump for any input pump beam conditions. In addition to be a tool for the test of preservation of phase-conjugation of two fields over a huge bandwidth, the SFG process as implemented here opens new perspectives for the experimental investigation of the spatiotemporal features of the near field biphoton amplitude, currently under progress.
We acknowledge financial support from the Future and Emerging Technologies (FET) program within the Seventh Framework Program for Research of the European Commission, under the FET-Open grant agreement HIDEAS (Grant No. FP7-ICT-221906).
References and links
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