We analyze the effect of atmospheric Kolmogorov turbulence on entangled orbital angular momentum states generated by parametric down-conversion. We calculate joint and signal photon detection probabilities and obtain numerically their dependence on the mode-width-to-Fried-parameter ratio. We demonstrate that entangled photons are less robust to the effects of Kolmogorov turbulence compared to single photons. In contrast, signal photons are more robust than single photons in the lowest-order mode.We also obtain numerically a scaling relation between the value of the mode-width-to-Fried-parameter ratio for which the joint detection probabilities is a maximum and the momentum mismatch between signal and idler photons after propagation through the medium. PACS codes: 42.50.Dv; 42.68.Bz IntroductionQuantum communication utilizing the phenomenon of entanglement has many applications such as quantum teleportation [1], quantum cryptography [2, 3] and superdense coding [4]. Multidimensional entangled orbital angular momentum (OAM) states can be utilized to generate arbitrary base-N quantum digits thereby allowing the implementation of higher capacity optical communication systems. This feature has generated considerable interest with respect to encoding quantum and classical information [2,5]. In addition it has been shown that the (OAM) of photons can be used to encode data onto a laser beam for transmitting information in free-space optical systems [6]. The performance of free-space optical communication systems is however severely degraded by decoherence caused by atmospheric turbulence in the optical channel, resulting in fluctuations in both the intensity and phase of the received signal. Recently, C. Paterson [7] estimated the theoretical performance limits for communication links based on the OAM of single photons with no entanglement. It was demonstrated that for Kolmogorov atmospheric statistics this system is significantly affected, even in the weak turbulence regime, with narrower and lower-order modes being more robust. Spontaneous parametric downconversion generates pairs of down-converted photons entangled in OAM [5]. It has been demonstrated that Laguerre-Gaussian (LG) modes possess well defined orbital angular momemta [8]. In the paraxial approximation, these modes are the eigenstates of the orbital angular momentum operator with eigenvalue equal to h l per photon. Xi-Fen Ren et al [9] calculated the superposition coefficients of down converted correlated LG modes and showed that this probability amplitude decreases almost exponentially with increasing OAM. In this paper we investigate the effects of Kolmogorov turbulence on entangled OAM photon states by calculating the joint detection probability of entangled LG modes. We compare the effects of Kolmogorov turbulence on entangled photon pairs and single photons.
We present our results on optical waveguides formed by thermal diffusion of ions in glass. It was found that the peak of the ion-exchanged region can be shifted into the substrate interior by limiting the diffusion process. We also found that low loss films (<0.1 dB/cm) can be fabricated using this process and that the modal losses in these films do not agree with those losses predicted by existing theories. Also, the ion-exchange process has proved to be a simple means for fabricating tapered-edge couplers.
PHYSICAL REVIEW LETTERS 6 DECEMBER 1976 sponding nd amplitudes and coherent addition of the pure Coulomb amplitude. 11 This amounts to neglecting the Coulomb modifications of the strong nd amplitudes which, in the present formalism, means neglect of all shorter-range Coulomb effects in V ,(M) and G 0 (/J) . However, we are now in the position to check the reliability of such a procedure. For this purpose we have included in Fig. 2 the cross sections obtained in such a way. It appears that at low energies this approximate treatment is rather unsatisfactory, whereas it may become more reliable at higher energies.In conclusion we emphasize once more that our approach to the Coulomb corrections in pd scattering is not only mathematically correct but also well suited for practical applications with no need for drastic approximations. (This is in contrast to the use of the formalism of Ref. 2 made in Ref. 4. Indeed, the results obtained there bear only a faint resemblance to ours or to the experimental data.) Apart from employing separable nuclear potentials, only one approximation has been made in the present calculations. Namely, the effective potentials and Green's functions which determine, via Eq. (7), the Coulomb-modified strong amplitude Tsc (ll) 9 are evaluated to lowest order in e 2 only. However, the neglect of higher-order terms can and will be checked. Furthermore, the difficult question which is bound to plague most other approaches of how many partial waves in the pp subsystem should be taken into account never does arise in our method where, characteristically, Frequency upconversion using third-harmonic generation and four-wave mixing has received attention in recent years for the generation of coherent radiation in the vacuum ultraviolet (VUV) region of the spectrum. 1 " 3 Such processes have been used to produce coherent radiation at wavelengths as short as 57.0 nm. 4 The generation of coherent light in the extreme ultraviolet region by third-order processes becomes increasingly difficult because of the scarcity of intense coherent sources at the required pumping wavelengths. the full three-dimensional Coulomb potential is built in.The development of frequency conversion techniques utilizing higher-order nonlinearities offers an attractive alternative to this approach, since it would allow larger steps along the frequency scale to be made in a single conversion process.Several of these processes have been suggested in the literature. 5 * 6 Although reasonable conversion efficiencies have been predicted for some of these interactions, the only published experimental evidence of such processes has been the fifth-The generation of coherent radiation of 53.2 nm by fifth-harmonic conversion of laser pulses at 266.1 nm in both Ne and He is reported.1540
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