We present a theoretical and experimental analysis of the joint effects of the transverse electric field distribution and of the nonlinear crystal characteristics on the properties of photon pairs generated by spontaneous parametric downconversion (SPDC). While it is known that for a sufficiently short crystal the pump electric field distribution fully determines the joint signal–idler properties, for longer crystals the nonlinear crystal properties also play an important role. In this paper we present experimental measurements of the angular spectrum (AS) and of the conditional angular spectrum (CAS) of photon pairs produced by SPDC, carried out through spatially resolved photon counting. In our experiment we control whether or not the source operates in the short-crystal regime through the degree of pump focusing, and explicitly show how the AS and CAS measurements differ in these two regimes. Our theory provides an understanding of the boundary between these two regimes and also predicts the corresponding differing behaviors.
We present a theoretical and experimental analysis of the joint effects of the transverse electric field distribution and of the nonlinear crystal characteristics on the properties of photon pairs generated by spontaneous parametric downconversion (SPDC). While it is known that for a sufficiently short crystal the pump electric field distribution fully determines the joint signal–idler properties, for longer crystals the nonlinear crystal properties also play an important role. In this paper we present experimental measurements of the angular spectrum (AS) and of the conditional angular spectrum (CAS) of photon pairs produced by SPDC, carried out through spatially resolved photon counting. In our experiment we control whether or not the source operates in the short-crystal regime through the degree of pump focusing, and explicitly show how the AS and CAS measurements differ in these two regimes. Our theory provides an understanding of the boundary between these two regimes and also predicts the corresponding differing behaviors.
We present an experimental and theoretical study of type I, frequency-degenerate spontaneous parametric down-conversion (SPDC) with a Bessel-Gauss pump in which we include both paraxial and nonparaxial pump beam configurations. We present measurements of the SPDC angular spectrum (AS), of the conditional angular spectrum (CAS) of signal-mode single photons as heralded by the detection of an idler photon, and of the transverse wave-vector signal-idler correlations (TWC). We show that as the pump is made increasingly nonparaxial, the AS acquires a nonconcentric double-cone structure, with the CAS shape depending on the azimuthal location of the heralding detector, while the signal-idler wave-vector correlation region splits into characteristic doublet stripes, representing as yet unexplored nontrivial, nonlocal quantum correlations between the signal and idler photons. Our work provides further understanding of SPDC with a particular class of structured pump beams, and we believe that the controlled presence of double wave-vector correlations represents an interesting resource for photon-pair quantum-state engineering.
This paper presents a study of the correlations in wave vector space of photon pairs generated by type I spontaneous parametric down conversion using a Gaussian pump beam. The analysis covers both moderate focused and highly focused regimes, paying special attention to the angular spectrum and the conditional angular spectrum. Simple analytic expressions are derived that allow a detailed study of the dependence of these spectra on the waist of the source and the length of the nonlinear crystal. These expressions are in good agreement with numerical expectations and reported experimental results. They are used to make a systematic search of optimization parameters that improve the feasibility of using highly focused Gaussian beams to generate idler and signal photons with predetermined mean values and spread of their transverse wave vectors.
Abstract Spontaneous parametric down-conversion (SPDC) is the most widely used process for generating photon pairs entangled in various degrees of freedom such as polarization, time-energy, position-transverse momentum, and angle-orbital angular momentum (OAM). In SPDC, a pump photon interacts with a non-linear optical crystal and splits into two entangled photons called the signal and the idler photons. The SPDC process has been studied extensively in the last few decades for various pump and crystal configurations, and the entangled photon pairs produced by SPDC have been used in numerous experimental studies on quantum entanglement and entanglement-based real-world quantum-information applications. In this tutorial article, we present a thorough study of phase matching in BBO crystals for spontaneous parametric down-conversion and thereby also investigate the generation of entangled photons in such crystals. First, we present a theoretical derivation of two-photon wavefunction produced by SPDC in the frequency and transverse momentum bases. We then discuss in detail the effects due to various crystal and pump parameters including the length of the crystal, the angle between the optic axis and the pump propagation direction, the pump incidence angle on the crystal surface, the refraction at the crystal surfaces, and the pump propagation direction inside the crystal. These effects are extremely relevant in experimental situations. We then present our numerical and experimental results in order to illustrate how various experimental parameters affect the phase matching and thus the generation of entangled photons. Finally, using the two-photon wavefunction in the transverse wave-vector basis, we show how to derive the two-photon wavefunction in the OAM basis and thereby calculate the two-photon angular Schmidt spectrum. We expect this article to be useful for researchers working in various capacities with entangled photons generated by SPDC in BBO crystals.
Spontaneous parametric down-conversion (SPDC) is one of the most important sources of entangled photons in quantum optics and underpins a wide range of applications in quantum communication, quantum imaging, quantum sensing, and photonic quantum information processing. In this nonlinear optical process, a pump photon is converted into a pair of lower-energy photons that exhibit strong correlations in multiple degrees of freedom. While the generation of photon pairs is governed by energy and momentum conservation, the resulting biphoton state possesses a rich spatial structure whose properties are determined by the interplay among the pump field, the nonlinear medium, and subsequent propagation. This thesis investigates the spatial properties of entangled photon pairs generated through SPDC, focusing on the evolution of their correlations in the near field and on the behavior of orbital-angular-momentum (OAM) entangled states under tight focusing conditions. A common theme throughout this work is the examination of physical effects that emerge when the full biphoton wavefunction is retained rather than reduced through the simplifying approximations often employed in theoretical treatments. The principal contribution of this thesis concerns the near-field propagation of SPDC photon pairs. The spatial correlations of SPDC-generated photons are commonly analyzed within the thin-crystal approximation, where the biphoton wavefunction is assumed to factorize into independent functions of the sum and difference coordinates. We demonstrate that birefringence-induced transverse walk-off breaks this factorization and introduces an intrinsic coupling between these coordinates. More importantly, this coupling survives free-space propagation and remains observable even for crystals that nominally satisfy the thin-crystal condition. As a result, the joint spatial correlations develop a characteristic tapering near the crystal image plane that cannot be captured by conventional factorized models. Numerical simulations and experimental measurements are found to be in excellent agreement, revealing a previously overlooked feature of SPDC and providing a more complete description of photon-pair generation and propagation in birefringent nonlinear media. The second part of the thesis explores the propagation of OAM-entangled photon pairs beyond the paraxial regime. While OAM correlations are typically studied under weak-focusing conditions, many emerging quantum technologies require structured quantum states to be manipulated in highly confined optical systems. We show that tight focusing modifies the OAM correlation spectrum, leading to a broadening of the accessible mode distribution. This broadening exhibits distinct transverse and longitudinal characteristics and arises from the emergence of nonparaxial field components absent in conventional scalar descriptions. These results provide new insight into the behavior of structured quantum light in strongly focused optical fields and establish a framework for understanding OAM entanglement beyond the paraxial approximation. Together, the results presented in this thesis reveal previously unexplored aspects of the structure and propagation of entangled photon pairs. By moving beyond widely used approximations, this work uncovers physical effects that influence the generation, evolution, and measurement of spatially entangled states, with implications for quantum imaging, high-dimensional quantum communication, and the broader development of structured-light-based quantum technologies.
In this work we study, for the spontaneous parametric downconversion process, how the transverse amplitude of the pump may be transferred to one of the emitted photons in a given pair when heralded by the detection of the remaining photon. We present a theoretical description of this process along with a discussion of the short-crystal regime within which faithful transfer of the transverse amplitude occurs. We show experimental results for Gaussian and Bessel–Gauss pump beams. In both cases, we verify that the transverse amplitude mechanism occurs and for the former also shows the effects of a departure from the short crystal regime. For the latter we show that our heralded single photons exhibit a non-diffractive behavior, and for orders l = 1 and l = 2 we show that these heralded single photons are in fact vortices with orbital angular momentum transferred from the pump.
We experimentally study the spatial properties of the field generated by spontaneous parametric down-conversion (SPDC) when the pump laser beam is strongly focused in the nonlinear crystal. Special attention is paid to classical intensity measurements with a CCD camera. We introduce the concept of a classical equivalent source that replicates all the coherence properties of SPDC light and explains all our experimental results. We show that, in contrast with experiments with a well-collimated pump, here both the phase-matching conditions and the position of the focusing plane determine the measured intensity profiles in the image plane of the crystal. The transition from the near-field regime to the far field is investigated. Measurements of two-photon correlations under strong focusing are also presented and the special features thereof are discussed.