1. Introduction

Spontaneous parametric down-conversion (SPDC) is the process by which photons in a laser pump beam incident on a nonlinear optical crystal (BBO) spontaneously decay into pairs of correlated photons that can be simultaneously entangled in energy, momentum, and polarization (for type-II SPDC) [1]. Entangled states of light such as this have been utilized successfully in a wide variety of experiments ranging from fundamental investigations of the foundations of quantum mechanics [2–4], implementations for quantum information manipulation and applications in communication technology [5–9], absolute calibration of single-photon detectors [10–14], to characterization of optical materials [15]. The convenience of SPDC has made it the common method for generation of entangled pairs for use in these experiments.

The general theoretical aspects of the quantum mechanical correlations between the signal and idler photons that constitute a down-converted pair are well known and thoroughly studied [16–18]. The properties of SPDC photons with respect to various parameters affecting their generation and the interaction that takes place inside the crystal have also been explored. Effects resulting specifically from the spectral properties of the pump beam, on the spatial coherence of the down-converted beams, and the consequences of broadband pulsed pumping have been investigated in [19–21]. In addition, the role of the spatial properties of the pump, particularly that of focused pumping [22, 23], have been demonstrated to be of crucial importance for applications that require coupling of signal-idler modes into a pair of transmission channels, such as single-mode fibers [24, 25]. Furthermore, the spatial characteristics and evolution of the single-photon SPDC image generated under focused pumping have recently been studied [26, 27]. It has also been shown that parameters like the pump beam waist and the pump beam wave front affect the “entanglement area” and the “entanglement time” of SPDC photons [28], and consequently have an impact on the interaction of entangled light with matter, where interesting non-classical effects had been predicted theoretically [29], and recently demonstrated experimentally [30].

In general, these studies on the effects of spatial properties of the pump beam on SPDC output state, mentioned above, have concentrated on the consequent changes in the correlations between signal-idler pairs, and particularly, the role of pump divergence has been the main focus of investigation. However, the effect of focused pump intensity, especially under constant pump divergence, on overall SPDC photon yield (the efficiency of the down conversion process) has not been investigated. This is mainly because of the highly scattering nature of the SPDC process, whose output is governed by the phase-matching conditions, which are a manifestation of momentum and energy conservation; k _p=k _s+k _i, ω _p=ω _s+ω _i. These conditions stipulate that down-converted photons are emitted with significant probability only when Δk _j L _j=(k _p-k _s-k _i)_j L _j is close to zero, where j=x, y, z. Here, L _j is the length of the crystal in dimension j, k denotes wave-vector, and subscripts p, s, i refer to the pump field, signal photon, and idler photon, respectively. It follows that the diversification introduced into the incoming pump wave-vectors due to nonzero pump divergence can affect the SPDC output state, but position of the pump focus with respect to nonlinear crystal surface, cannot. In other words, translating the focal plane of a pump focusing lens via a linear stage, and hence changing the pump intensity, is not expected to alter SPDC output yield because pump divergence is constant throughout this process [27].

While all applications that utilize SPDC as a source of entangled photon pairs can benefit from higher brightness, or increased entangled photon yield of the SPDC process, maximizing this entangled photon flux becomes especially important in practical aspects of experiments that investigate the interaction of entangled photons with matter, mentioned earlier, such as entangled multi-photon absorption.

In this contribution, we present the results of an experimental investigation, in which, the SPDC photon yield is measured under focused pumping conditions where the pump beam divergence is invariant and down-conversion efficiency is observed to exhibit a strong dependence on pump beam intensity. We also develop a theoretical model incorporating the geometrical aspects of the pump field for comparison purposes and show by numerical simulations that we have found to agree with our experimental findings. This result is important because the efficiency of the SPDC process is not expected to depend on position of pump beam waist [31, 32] or show enhancement through focusing [33]. The following section describes our experimental conditions. The theory section will summarize our theoretical considerations, which will be followed by the results, discussions and finally concluding remarks.

2. Experimental

Our experimental setup, illustrated in Fig. 1, utilizes the second-harmonic of a femtosecond Ti:sapphire laser system as a pump source for the generation of down-converted photons in free space. The laser system (Spectra-Physics MAITAI) delivers 100-fs pulses with a repetition rate of 82 MHz at λ=800 nm. A neutral-density filter wheel just after the laser system is used to control power admitted into the rest of the setup. The laser output is frequency-doubled by focusing onto a 1-mm thick β-Barium Borate (BBO) nonlinear crystal, which is subsequently collimated into a spot size of ~2-mm diameter. The 400-nm light is then separated from the redundant fundamental by means of a dichroic mirror (DM), followed by a Brewster-angled prism.

 

Fig. 1. Schematic representation of our experimental set-up. PD are photodetectors used to monitor fundamental and second harmonic pump power, DM denote dichroic mirrors utilized for spectral selection, PBS is a Glan-laser polarization beam splitter, and IF is a narrow-band interference filter.

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A blue filter (BF) is utilized to minimize background intensity before pumping the second BBO crystal for generation of SPDC photons. The 400-nm pump is focused onto the input surface of this second BBO crystal via an 8-cm focal length lens which is mounted on a translating stage that allows us to vary the distance between the lens and the crystal continuously ranging from 1 cm to 9.5 cm. In our experiments, we have utilized BBO crystals of two different thicknesses for the generation of SPDC photons: one 0.5-mm thick (BBO₁), and the other 2-mm thick (BBO₂), both of which are cut for type-II SPDC at φ=42° with respect to the center pump direction (z). The crystals are tilted slightly to achieve collinear phase-matching. The output surface of the BBO crystal lies on the focal plane of a successive 6-cm focal length lens that serves to collimate the SPDC photons which are characteristically emitted into two cones. The remaining pump beam is then removed by means of another dichroic mirror, and the SPDC photons are further selected by a spectral filter (IF) with 25-nm bandwidth centered at 800 nm, which also constitutes a ~25-mm spatial aperture that is matched or surpassed by all subsequent optic elements. The reason for the use of such a wide aperture is to guarantee the collection of the whole SPDC pattern.

In order to test for good collection of the whole SPDC profile, we have measured transmitted photon flux through a 1-mm pinhole that was scanned across an image plane perpendicular to the SPDC photons’ propagation path after the collimating lens. The measurements were then combined into composite images, given in Fig. 2 below. These scans were repeated for two different separations between the pump focusing lens and BBO down-conversion crystal of d=10mm and d=f=80mm, representing both extremes of focusing sharpness for our experimental setup; loose, and sharpest possible focusing, respectively. As is seen from these images, the overall size of the profile is ~13 mm vertically across both rings, with an increase of up to ~3 mm as the focusing lens-BBO separation decreases, which is well within the confines of the aperture, and both focusing schemes depict the whole of the SPDC pattern. Note that the SPDC profile exhibits asymmetric broadening of the signal and idler cones due to non-zero pump divergence. This effect is expected and has been studied in detail in [26] and [27].

Immediately after the spectral selection filter, the signal (e-ray plane of BBO) and idler (o-ray plane of BBO) modes of SPDC light are separated with a Glan-laser polarization beam-splitter (PBS) before being focused onto free-space single-photon-sensitive avalanche photodiodes (PerkinElmer Optoelectronics SPCM-AQR-13) by 6-cm focal length collection lenses. The single-photon counting modules have a detection efficiency of ~56% at 800 nm. The output pulses from the photodetectors can then be counted either individually (single-count rate), or fed through a fast coincidence circuit (Ortec NIM7400) with a 10-nsec coincidence window, for coincidence-count rate measurements, through a data acquisition circuit. Since subsequent optic elements after the spectral filter (IF) all match the 25-mm aperture size, and images of the down-converted profile have been verified with the aforementioned pinhole-scans, both photodetectors are able to “see” the whole of their respective signal or idler SPDC patterns under all focusing schemes utilized in our experiments.

 

Fig. 2. Image of the SPDC spatial profile in our experimental configuration under (a) sharpest focusing of pump beam (d=f=80mm), and (b) loose focusing (d=10mm) condition, generated by measuring transmitted flux through a 1-mm pinhole that is scanned across an image plane normal to the beam path after the collimating lens. Both axes denote position of the pinhole in millimeters. The poor resolution of the image is due to the relatively wide pinhole aperture that was used.

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The objective of the experiment is to record single and coincidence count rates as a function of pump power for various separations (d) between the pump focusing lens and BBO crystal. Due to unavoidable imperfection in the experimental alignment, varying the distance from down-conversion crystal to the focusing lens, via motion of the stage the lens is mounted on, causes the location of the signal and idler collection spots to shift on the focal planes of their respective collection lenses prior to detection. This, combined with the very small active areas (ø175 µm) of the single-photon counting modules, requires that both photodetectors be repositioned to reacquire maximum counts after each adjustment of the focusing lens-BBO distance. To this end, the single-photon counting modules are mounted on free-space, 3-axis motion stages that are controlled electronically, and the collection system is re-optimized after each iteration of the distance d.

3. Theoretical background

In this section we present a theoretical analysis of the down-conversion efficiency for photon pairs generated via type-II SPDC under focused pumping conditions. The aim is to express transverse spatial parameters in terms of the separation between down-conversion crystal and pump focusing lens and derive the subsequent SPDC two-photon probability amplitude [34] which will govern the rate of SPDC photons detected in coincidence as a function of input pump intensity.

Pittman et al. [22] have analyzed the spatial correlations of the generated ordinary (idler) and extraordinary (signal) photons under focused CW pumping by describing the transverse-plane pump field profile as a Gaussian intensity distribution and expressing the SPDC two-photon state as having a spherical-like contribution in the momentum phase-matching condition. Following their methodology, we carry out a similar analysis incorporating also the angular frequency envelope for a broadband pulsed pump beam, expressing the classical pump field as

where ê is the pump extraordinary polarization direction, A(ω _p) determines pump spectral width Δω _p, and E _p(r⊥) is the pump amplitude distribution in the transverse plane, approximated by [35]

with σ ² _P≈c/ω _p(d-f-i(λ ⁰ _p/πw ² ₀)f ²). Here, f is the focal length of the lens, w ₀ is the pump beam waist before focusing, λ ⁰ _p is the pump center wavelength, and d is the variable distance between the lens and input face of the crystal.

Adopting the pump beam center direction as the z-axis in subsequent calculations, it is further required that the dependence on transverse components of pump wave vector be removed from the z-component of the pump wave vector, k _pz, by introducing a new pump field quantity, K _p≡(ω _p/c)n _e(ω _p), which is equal to the magnitude of the pump k vector if it were exactly parallel to the z direction, where n _e,_o(ω) respectively are the extraordinary and ordinary refractive indices of the crystal. As a result, the pump field amplitude inside the crystal can now be written as;

Here, it is assumed that the pump diameter is relatively small compared to the focal length of the lens such that the paraxial approximation [36] can be applied. Further, a thin crystal approximation [37] must also be applied for the above description to be valid, without which the simplification of the pump field into computable terms is extremely complicated. Completing the square in the momentum-space Gaussian integration in Eq. (3) yields a spherical-like description for the pump field, rather than the usual plane-wave model considered in most treatments of SPDC [22]. It should also be observed that, due to the focal-length dependent term in the denominator of the exponential in Eq. (2), the plane-wave model is recovered in the very long focal length (f→∞) limit, as expected.

The two-photon output state is then calculated to first order in perturbation theory with an interaction Hamiltonian derived from the standard form; [38]

where V is the volume of interaction for the classical pump field described by Eq. (3), H.C. denotes the Hermitian conjugate, â†_kj with j=_{s, i} are the creation operators for the signal and idler modes, respectively, and electric susceptibility χ of the nonlinear crystal, along with all unimportant constants have been absorbed into factor A ₁. When the volume integration is broken up into the multiplication of an area integral and integration over the crystal length L, the transverse integration over the r⊥-dependent terms can be approximated as a Gaussian over an infinite range, provided the crystal cross sectional area is large compared to the pump beam waist. It is then possible to use the technique of first order perturbation theory, and the two-photon state is calculated as [1];

where the new constant factor A ₂ has absorbed the constants generated in the transverse integration of (4) and A ₁.

In the detection scheme utilized for our experiment, the average coincidence-count rate, defined by the Glauber formulation [39], will depend directly on the square of the two-photon probability amplitude;

where the analytical fields E ⁽⁺⁾ _1,2 describe the free-space field operators at detector APD ₁ for the ordinary-polarized idler beam, and at detector APD ₂ for the extraordinary-polarized signal beam. As such, E ₁ ⁽⁺⁾ incorporates the annihilation operator â _ki of the idler mode, and E ₂ ⁽⁺⁾ contains the annihilation operator â _ks for the signal mode. Since the detection scheme utilized in our experiment selects near-degenerate wavelengths before the signal and idler pairs are separated, both fields have identical angular frequency integrations to describe the transmission interval of the interference filter employed. Thus, combining (5) and (6), we are able to establish a description of the coincidence counting rate of SPDC photons generated in our setup as dependent on the geometry of the focused pump beam, i.e. the separation between pump focusing lens and nonlinear crystal, d.

Note that under conditions where focal-length of the pump-focusing lens is invariant such that pump-beam divergence does not change and average pump power is kept constant, translation of focusing lens to vary d leads only to variation of pump spot size on the input face of the down-conversion crystal and changes the pump intensity distribution. As a result, Eq. (5) and (6) express, effectively, the coincidence-count rate as a function of input pump intensity.

4. Results and discussion

In Fig. 2 we show, for down-conversion crystals of two different thicknesses, the measured SPDC coincidence-count rates and numerical simulations based on the theoretical model developed in the previous section [Eq. (5) and (6)] for various focusing lens-BBO separations. The experimental data has been normalized for comparison purposes. Maximum count rates obtained in coincidence under optimized conditions were ~3.60×10⁴ cps for BBO₁, and ~1.24×10⁵ cps for BBO₂. Due to the complexity of equations (4) and (5), the theoretical modeling was carried out numerically instead of analytically for near-degenerate wavelengths of the signal and idler photons, detection restricted to the bandwidth of the interference filter (800±12.5 nm), with pump wave vectors defined in terms of measured pump spectral width (<13 nm), at normal incidence to the BBO surface. Note that these simulations are carried out for varying distances between down-conversion crystal and focusing lens, and not for varying focal lengths (pump divergences), where, in the long focal length limit, the theoretical model would represent a plane-wave pump field and all dependencies on pump geometry in SPDC output state would vanish. In this regime, the output yield of the SPDC process would correspond to the dashed plots given in Fig. 3, irrespective of the value of d. Note that while each trace exhibits linear behavior as a function of pump power, the different focusing schemes result in different slopes, or efficiency, of the SPDC process. This is in contrast to previous theoretical predictions [31–33], where SPDC output has been reported to be independent of the position of pump beam waist. The inclusion of the spherical-like geometrical factor in our theoretical model allows for us to calculate the SPDC yield as a function of the separation, d.

The theoretical simulations and experimental results for both cases show good qualitative agreement in their linear dependence on input pump power. The linear character of this dependence is interpreted as evidence that the parametric down-conversion in our experimental configuration is within the spontaneous regime. The quantitative difference between computed theoretical yield and measured SPDC coincidence counting rate for thinner nonlinear crystal BBO1 [Fig. 3(a)] is attributed to optical losses not related to aperture size, and alignment imperfections. However, in the case of the thicker crystal BBO₂, the observed coincidence-count rate is slightly higher than the theoretical yield predicted [Fig. 3(b)]. We suggest that this is due to the larger thickness of nonlinear crystal, which is roughly an order of magnitude thicker than the metric required so that the thin-crystal approximation remains applicable; in this regime, the pump field description given by Eq. (3) is no longer sufficient to describe the interaction inside the crystal completely and additive terms that had been dropped for ease of numerical calculation have a more significant effect compared to the case of the thinner (0.5-mm) crystal.

 

Fig. 3. Normalized coincidence-count rate data from (a) 0.5-mm thick, and (b) 2.0-mm thick BBO crystals measured as a function of percent peak pump power for three focusing conditions of the pump: d=80mm (squares), d=38mm (dots), and d=15mm (triangles). Solid lines represent numerical calculations of each case, and dashed lines show the theoretical prediction for the plane-wave (f→∞) limit.

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It should be noted here that the presence of “extra near-infrared fluorescence” originating from the BBO crystal has previously been reported under focused CW pumping conditions for pump divergence exceeding 32 mrad in [26]. Though the origin and nature of this excess light has not been investigated quantitatively, it is presumed not to be a result of the SPDC process. The pump divergence obtained from the optical setup employed in our experiment (25±1 mrad) is smaller than this threshold value, excluding the possibility of detecting uncorrelated photons and generating false coincidences. Furthermore, this same communication had reported a decrease in SPDC single-count rate under conditions where entangled pairs were collected through a “bucket” detection scheme which utilizes narrow apertures in front of single-photon counting modules. This decrease, however, is attributed to the inhomogeneous broadening of the SPDC spatial profile, which, combined with the application of narrow apertures for collection, results in diminished collection efficiency of signal-idler pairs. As described previously in the experimental section, our setup utilizes a wide aperture (ø 2.54 cm) in order to circumvent collection drawbacks of this kind.

The phase matching argument, as well as previous theoretical studies [31–33], suggests that change in the pump beam waist at the nonlinear crystal will not have an impact on the SPDC output yield. To test this argument, we have also investigated the dependence of single-count rates of the signal and idler photons on position of pump beam waist under fixed pump power. In Fig. 4 we show single-count rates as a function of focusing lens-BBO separation, acquired at peak pump power. As expected, the counting rates for both the signal and idler modes of the SPDC process exhibit an increase toward corresponding maxima as the pump focusing scheme approaches the sharpest configuration, and a decrease after. Note that the data also suggests a symmetry with respect to the d=f=80 mm (sharp) focusing condition, however, collection of counting-rate data for separations d>95 mm was not possible with our current experimental setup due to physical space constraints. Because Eq. (5) and (6) describe specifically the coincidence-count rate and hence are not directly applicable to single-count data, we have tested adherence of these plots to a form similar to that of the spherical-like geometrical factor in Eq. (3). Solid lines in Fig. 4 represent analytical fits of the experimental data to such functions. The experimental data is found to show good agreement with this form and suggests that the SPDC output photon yield shows a non-linear dependence on pump beam intensity for both thin and thick BBO crystals.

 

Fig. 4. Intensity dependence of signal and idler single-count rates as a function of focusing lens-BBO separation for (a) 0.5-mm thick, and (b) 2.0-mm thick nonlinear crystals. Note that the scale for the thick crystal is ca. 3 times larger. Solid lines for each trace are fits to a parametrized exponential of a form matching that of the spherical-like envelope of the pump field. The mismatch in signals from the two detectors is due to the presence of extra optical loss along APD2 path which is not present in APD1. The insets illustrate progression from small separations between focusing lens and BBO to where the separation is equal to one focal-length of the pump focusing lens. The distance between the BBO and collimating lens is fixed at L=60mm. Because pump-beam divergence is constant throughout this translation, the angular spread of the SPDC spatial profile does not change. However, due to the wider cross-sectional area of interaction at small distances, the size of the profile at a constant distance from the BBO output surface gets narrower as one approaches d=f=80mm limit. A wide aperture is utilized to compensate for this change and ensure collection of the whole down-conversion profile at all d values.

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5. Conclusions

Our experimental findings show unambiguously that a pump beam focused into higher intensities enhances the SPDC output yield, despite the constant pump divergence. As opposed to previous publications that had reported no change or even reduced yield under such focused pumping conditions, our experimental setup is designed to collect the entire SPDC pattern, and thus enhancement of output yield is seen to originate from better down-conversion efficiency at higher pump intensity. Accordingly, almost one order of magnitude of enhancement can be achieved under optimized conditions. Furthermore, this enhanced flux of photons is ascribed solely to the SPDC process, because the pump divergence utilized in our experiments is small compared to the divergences at which generation of excess, non-SPDC photons had been reported previously [26].

The comparison of theoretical simulations with our experimental results have shown that adopting a spherical-like description for the pump field, as originally proposed in [22], serves to minimize the discrepancy between theoretical predictions and experimental observations as opposed to a plane-wave model.