A detailed analysis of four-wave mixing (FWM) and its required phase-matching conditions in graded-index multimode optical fibers is presented. The analysis is relevant mainly for spontaneous frequency conversion and photon pair generation via FWM, when the fiber length is of the order of a meter or shorter, and the signal and idler are generated at large frequency separations from the pump. An analytic expression for calculating the phase mismatch occurring as a result of the waveguide dispersion among the pump, signal, and idler beams is derived, and it agrees well with independent numerical analysis. This expression simplifies calculations considerably, as it allows the prediction of the wavelengths of the signal and idler by using only the mode group number of the spatial modes involved in the FWM process and the refractive index profile of the fiber. The scaling of the signal–pump–idler frequency separations with the mode group numbers is explored. The abundance of the configuration degeneracies of the FWM process in graded-index multimode fibers indicates that nearly identical signal and idler frequencies can be obtained for many different pump, signal, and idler spatial mode profile combinations.
A detailed analysis of four-wave mixing (FWM) and its required phase-matching conditions in graded-index multimode optical fibers is presented. The analysis is relevant mainly for spontaneous frequency conversion and photon pair generation via FWM, when the fiber length is of the order of a meter or shorter, and the signal and idler are generated at large frequency separations from the pump. An analytic expression for calculating the phase mismatch occurring as a result of the waveguide dispersion among the pump, signal, and idler beams is derived, and it agrees well with independent numerical analysis. This expression simplifies calculations considerably, as it allows the prediction of the wavelengths of the signal and idler by using only the mode group number of the spatial modes involved in the FWM process and the refractive index profile of the fiber. The scaling of the signal–pump–idler frequency separations with the mode group numbers is explored. The abundance of the configuration degeneracies of the FWM process in graded-index multimode fibers indicates that nearly identical signal and idler frequencies can be obtained for many different pump, signal, and idler spatial mode profile combinations.
In this work, we report the modeling and the experimental demonstration of intermodal spontaneous as well as stimulated four-wave mixing (FWM) in silicon waveguides.In intermodal FWM, the phase-matching condition is achieved by exploiting the different dispersion profiles of the optical modes in a multimode waveguide.Since both the energy and the wave vectors have to be conserved in the FWM process, this leads to a wide tunability of the generated photon wavelength, allowing us to achieve a large spectral conversion.We measured several waveguides that differ by their widths and demonstrate large signal generation spanning from the pump wavelength (1550 nm) down to 1202 nm.A suited setup evidences that the different waves propagated indeed on different order modes, which supports the modeling.Despite observing a reduced efficiency with respect to intramodal FWM due to the decreased modal overlap, we were able to show a maximum spectral distance between the signal and idler of 979.6 nm with a 1550 nm pump.Our measurements suggest the intermodal FWM is a viable means for large wavelength conversion and heralded photon sources.
The possibility to perform broad wavelength generation and conversion is of great interest in classical and quantum optics. With the introduction of integrated nonlinear silicon photonics, these tasks have been made even more viable thanks to the combination of the high nonlinear refractive index of silicon with the high confinement provided by the photonic waveguides. Exploiting this platform, several solutions for wavelength conversion have been investigated, for both optical processing [1] and applications in the Mid Infrared (MIR) [2]. The possibility to perform broad and controllable wavelength generation can be exploited also for creating integrated sources of quantum states of light based on single photons [3] or correlated photon pairs [4]. Both the generation and conversion can be accomplished by means of Four Wave Mixing (FWM). FWM is a nonlinear optical process in which two pump photons at frequency are converted into signal and idler photons at frequencies and ω/ respectively (with ωI/ωS). In the particular case of wavelength conversion, FWM is stimulated by a seed wave at the same wavelength of the signal (idler) that will be converted to the wavelength of the idler (signal). FWM has to satisfy the energy conservation, according to which the relation ωP+ ωP= ωS+ ωIholds, and its efficiency is ruled by the phase matching condition. This latter condition is related to the phase mismatch among the waves involved in the FWM process, that is evaluated as Δk = kp+ kp−kS−kI, where kP, kS, kIare the wavevectors for the pump, the signal and the idler, respectively. When Δk = 0, the phase matching condition is fulfilled and the efficiency of FWM is maximized. What is important for wavelength generation and conversion is the spectral position of the phase matching condition, which determines the wavelength at which the maximum efficiency of the conversion or generation process is achieved. Because of this, techniques for the control of the phase matching condition have been developed, focusing on the possibility to tailor the Group Velocity Dispersion (GVD) through a proper engineering of the waveguide geometry [5]. While these techniques consider only the first order waveguide mode, here we propose to exploit the higher order modes propagating inside a multimode waveguide. The dispersion of the higher order modes can be used to tune the spectral position of the phase matching condition for the FWM process, achieving controllable and large spectral translation in both Stimulated Four Wave Mixing (sFWM) and Spontaneous Four Wave Mixing (SFWM).
We present a study of spontaneously arisen spatially entangled photon pairs via intermodal four-wave mixing in a graded-index multimode optical fiber. Unique dispersive features of the fiber allow spectral indistinguishability of two different phase-matched processes, producing entangled pairs of spatial qubits. The bases are realized as superpositions of orthogonal transverse fiber modes having opposite parities. In particular, we take into consideration the spectral properties of the processes by examining their joint spectral amplitudes. It is shown that illuminating graded-index optical fiber with different pump wavelengths has an impact upon efficiency parameters accordingly the degree of spatial entanglement and gives rise to photon pairs with various spectral purities. Photons with higher spectral purity enable desired single-photon based interactions to take place, whereas photons with lower spectral purity exhibit hybrid entanglement in frequency and transverse mode. We also discuss Wigner function formalism and parity-displacement-based realization to characterize spatial properties of the states, as well as to verify quantum entanglement through a violation of Clauser-Horne-Shimony-Holt inequality.
We present a study of the spectral properties of photon pairs generated through the process of spontaneous four-wave mixing (SFWM) in single mode fiber. Our analysis assumes narrowband pumps, which are allowed to be frequency degenerate or nondegenerate. Based on this analysis, we derive conditions on the pump frequencies and on the fiber dispersion parameters which guarantee the generation of ultrabroadband photon pairs. Such photon pairs are characterized by (i) a very large degree of entanglement, and (ii) a very high degree of temporal synchronization between the signal and idler photons. Through a numerical exercise, we find that the use of photonic crystal fiber (PCF) facilitates the fulfilment of the conditions for ultrabroadband photon pair generation; in particular, the spectral region in which emission occurs can be adjusted to particular needs through an appropriate choice of the PCF parameters. In addition, we present a quantum interference effect, resulting from indistinguishable pathways to the same outcome, which can occur when pumping a SFWM source with multiple spectral lines.
Summary form only given. Single-photon sources are key components in applications of photonic quantum technologies such as quantum key distribution (QKD) [1]. One way of realizing single-photon sources is generation of photon pairs (PP) using spontaneous four-wave mixing (FWM): two photons from a pump p annihilate and create two side-band photons at frequencies determined partly by the energy conservation 2ωρ= ω1+ ω2, where ωp,ω1,ω2are the frequencies of the pump and the two side-bands, respectively, and partly by the phase-matching condition. PP generated spontaneously arrive at indeterministic times but even so, they are useful for QKD because one of the photons can be heralded by detecting the other. The heralded photons are then used for transmitting the quantum key.We propose a higher order mode scheme for generating PP in silica fibers; the side-band photons are generated at center wavelength λ1 = 810 nm in the linearly polarized LP02-mode and at λ2 = 1550 nm in the LP01mode, respectively, by pumping in the LP11-mode at λ7, = 1064 nm, where laser sources are readily available. The 810 nm photon is detectable with relatively cheap and efficient silicon-based single-photon detectors, and the heralded 1550 nm photon is mode-matched with transmission fibers in the low-loss spectral window. We design a silica fiber that phase matches these modes and wavelengths while it also reduces the impacts of core radius fluctuations (CRF) caused by fabrication imperfections; the fiber design is inspired by B. P.-P. Kuo et al. [3] and a sketch of the refractive index profile is shown in Fig. 1(a). The scalar wave equation is solved using appropriate Sellmeier data for Germanium-doped silica and the calculated effective indexes for the three modes versus wavelength are shown in Fig. 1(b) together with the core and cladding refractive indexes; the inset shows the wavelength configuration of the pump and two side-bands; phase matching is achieved when the effective indexes of the three modes lie on a straight line [2]. The nonlinear overlap of the waves in this configuration is approximately 18% of the corresponding overlap where all waves are in the LP01-mode.
Based on the dispersion property of a given photonic crystal fiber (PCF), we study how to directly generate photon pairs with minimized frequency correlation via pulse-pumped spontaneous four-wave mixing. After illustrating why the intensity correlation function g (2) of individual signal (idler) photons can be used to reliably characterize the frequency correlation of photon pairs, we numerically investigate the dependence of g (2) under various kinds of experimental conditions. The results show that to minimize the frequency correlation, the experimental parameters should be properly optimized by balancing the influences of the high-order dispersion and the intrinsic sinc oscillation of phase matching function, apart from the satisfaction of the specified phase matching condition and the usage of transform-limited pump pulses. To verify the calculated results, we conduct two series of experiments by regulating the pump to respectively satisfy the asymmetric and symmetric group velocity matching conditions in our 0.6 m-long PCF. In both cases, the measured values of g (2) are less than the calculated results due to the inhomogeneity of the PCF; however, the experimental results qualitatively agree with the numerical simulations. Our investigation is very useful in fiber-based quantum state engineering.