Using the approach of coupled wave equations, we consider spontaneous parametric down conversion (SPDC) in the narrow-band regime and its relationship to classical nonlinear processes such as sum-frequency generation. We find simple expressions in terms of mode-overlap integrals for the absolute pair-production rate into single-spatial modes and simple relationships between the efficiencies of the classical and quantum processes. The results, obtained with Green function techniques, are not specific to any geometry or nonlinear crystal. The theory is applied to both degenerate and nondegenerate SPDC. We also find a time-domain expression for the correlation function between filtered signal and idler fields.
Using the approach of coupled wave equations, we consider spontaneous parametric down conversion (SPDC) in the narrow-band regime and its relationship to classical nonlinear processes such as sum-frequency generation. We find simple expressions in terms of mode-overlap integrals for the absolute pair-production rate into single-spatial modes and simple relationships between the efficiencies of the classical and quantum processes. The results, obtained with Green function techniques, are not specific to any geometry or nonlinear crystal. The theory is applied to both degenerate and nondegenerate SPDC. We also find a time-domain expression for the correlation function between filtered signal and idler fields.
The process of spontaneous parametric down-conversion (SPDC) inX(2)-materials, in which a pump (P) photon of wavelength λpcan split into a pair of entangled signal (S) and idler (I) photons of lower energy, is an instrumental tool in optical quantum technologies. Recently, there has been a growing interest in pair generation in sub-wavelength-thin nonlinear crystals [1], [2], especially because they are not restricted by the phase-matching condition and exhibit SPDC in a very wide spectral and angular range. This also motivates their use in quantum imaging applications, for pushing the resolution of such schemes to the limit [3], [4]. Despite its seemingly simple appearance, there has not yet been a comprehensive theoretical analysis of photon-pair generation in a thin unstructured nonlinear slab. Specifically, a proper theory should embed the Fabry-Pérot type of interferences that the signal and idler photons experience in such a nonlinear etalon during SPDC. In our work, we develop the theory for description of SPDC in thin nonlinear etalons and calculate far-field properties of the generated pairs. We use the Green's function (GF) quantization approach for description of pairgeneration [5], combined with analytical expressions for the GF of multilayer systems [6], to develop our formalism. Because of the generality of the GF method, this formalism can describe any thickness of the slab, ultra-thin or thick, and far-field radiation in the non-paraxial regime. Fig. 1 (a) schematic of spdc in a nonlinear etalon.ϵ1,ϵ2andϵ3are linear permittivities of the mediums. Medium 2 with thicknesshhas a second-orderχ(2)nonlinearity. (b-e) Joint detection probabilities of different detection configurations and etalon thicknesses.
Optimizing the brightness of a spontaneous parametric down-conversion (SPDC) source is an important task for many quantum information applications. We investigate the optimal focusing conditions to maximize the number of photons produced in an SPDC process and coupled with single-mode fibers. We provide a general expression for the two-photon wave function, generalizing previous known results, by considering collinear and noncollinear emission. We present analytical expressions for our results in the thin-crystal limit and clarify the relation between different focusing conditions already existing in the literature. Differently from what was previously reported, we show that the optimal ratio between the pump waist and the generated photons waist depends on the emission angle: It is 1/2 for collinear degenerate emission and approaches 1/2 for larger collection angles. The role of spectral filters is also analyzed. We support and enrich our discussion with numerical simulations, performed for type-I SPDC in a $\ensuremath{\beta}$ barium borate crystal. For this type of emission, we also investigate the role of the transverse walk-off outside the thin-crystal regime.
A theoretical investigation of the performance of single-mode coupled spontaneous parametric downconversion sources is proposed, which only requires very few assumptions of practical interest: quasi-degenerate collinear generation and narrow bandwidth obtained through spectral filtering. Other assumptions, such as pump-beam spatial and temporal envelopes, target single-mode profile and size, and nonlinear susceptibility distribution, are only taken into account in the final step of the computation, thus making the theory general and flexible. Figures of merit for performance include absolute coupled brightness and conditional coupling efficiency. Their optimization is investigated using functions that only depend on dimensionless parameters, so that the results provide the best experimental configuration for a whole range of design choices (e.g., crystal length, pump power, etc.). A particular application of the theory is validated by an experimental optimization obtained under compatible assumptions. A comparison with other works and proposals for numerically implementing the theory under less stringent assumptions is also provided.
Abstract As a tutorial, we examine the absolute brightness and number statistics of photon pairs generated in spontaneous parametric down-conversion (SPDC) from first principles. In doing so, we demonstrate how the diverse implementations of SPDC can be understood through a single common framework, and use this to derive straightforward formulas for the biphoton generation rate (pairs per second) in a variety of different circumstances. In particular, we consider the common cases of both collimated and focused Gaussian pump beams in a bulk nonlinear crystal, as well as in nonlinear waveguides and micro-ring resonators. Furthermore, we examine the number statistics of down-converted light using a non-perturbative approximation (the multi-mode squeezed vacuum), to provide quantitative formulas for the relative likelihood of multi-pair production events, and explore how the quantum state of the pump affects the subsequent statistics of the down-converted light. Following this, we consider the limits of the undepleted pump approximation, and conclude by performing experiments to test the effectiveness of our theoretical predictions for the biphoton generation rate in a variety of different sources.
A concept for an ultra-broad-band optical parametric amplifier or oscillator has emerged as a by-product of a theoretical study in fundamental quantum optics. The study was originally intended to address the question of whether the two-photon temporal correlation function of light [in particular, light produced by spontaneous parametric down conversion (SPDC)] can be considerably narrower than the inverse of the spectral width (bandwidth) of the light. The answer to the question was found to be negative. More specifically, on the basis of the universal integral relations between the quantum two-photon temporal correlation and the classical spectrum of light, it was found that the lower limit of two-photon correlation time is set approximately by the inverse of the bandwidth. The mathematical solution for the minimum two-photon correlation time also provides the minimum relative frequency dispersion of the down-converted light components; in turn, the minimum relative frequency dispersion translates to the maximum bandwidth, which is important for the design of an ultra-broad-band optical parametric oscillator or amplifier. In the study, results of an analysis of the general integral relations were applied in the case of an optically nonlinear, frequency-dispersive crystal in which SPDC produces collinear photons. Equations were found for the crystal orientation and pump wavelength, specific for each parametric-down-converting crystal, that eliminate the relative frequency dispersion of collinear degenerate (equal-frequency) signal and idler components up to the fourth order in the frequency-detuning parameter
Spontaneous parametric downconversion (SPDC) using periodically poled nonlinear optical crystals under the quasiphase- matching condition has found wide use in quantum optics. High efficiencies and good coupling to single-mode fibers resulted from using channel waveguides in crystals. It is often desirable to have a very narrow bandwidth for the signal and idler photons, but under the typical operating conditions, phase matching dictates the bandwidth of the SPDC to be of the order of <1 nm. This occurs because the co-propagating signal and idler photons are entangled, and an increase of the signal wave-vector is compensated by a decrease of the idler wave-vector. One way to reduce the bandwidth is by forming either external or internal cavities. Additionally, bandwidth reduction is possible without cavities when the signal and idler are counter-propagating, and the changes in the wave-vector with frequency are additive. To accomplish this a domain inversion on the wavelength scale is required. In this work, we experimentally demonstrate SPDC in one-dimensional KTP-based waveguides with sub-micron poling for forward and backward interactions. Some of the spectral features of the generated light are accounted for by mode coupling theory in periodically poled waveguides but other features are as yet not explained.