Abstract
Stimulated parametric downconversion is a nonlinear optical process that can be used for phase conjugation and frequency conversion of an optical field. A precise description of the outgoing stimulated field has been developed for the case where the input pump and seed fields are coherent. However, partially coherent beams can have interesting and important characteristics that are absent in coherent beams. One example is the twist phase, a novel optical phase that can appear in partially coherent Gaussian beams and gives rise to a nonzero orbital angular momentum. Here, we consider stimulated downconversion for partially coherent input fields. As a case study, we use twisted Gaussian SchellModel beams as the seed and pump beams in stimulated parametric downconversion. It is shown both theoretically and experimentally that the stimulated idler beam can be written as a twisted Gaussian SchellModel beam, where the beam parameters are determined entirely by the seed and pump. When the pump beam is coherent, the twist phase of the idler is the conjugate of that of the seed. These results could be useful for the correction of wavefront distortion such as in atmospheric turbulence in optical communication channels, and synthesis of partially coherent beams.
1 Introduction
Parametric nonlinear processes play a central role in classical and quantum optics, as they can give rise to amplification, modulation and frequency conversion of optical fields. Typically, the input fields are assumed to be spatially and temporally coherent. However, many modern light sources display only partial transverse coherence. Moreover, partially coherent beams can have interesting and useful characteristics. One example is the Twisted Gaussian Schellmodel (TGSM) beam [1, 2], which carry a socalled twist phase, and could be useful in communication through turbulence [3, 4] and imaging [5].
The role of partial transverse coherence in nonlinear optics has been studied for the case of spontaneous parametric downconversion [6–17]. A related process, known as stimulated parametric downconversion (StimPDC), occurs when a pump and a seed beam interact in a medium with a nonzero secondorder susceptibility [18]. They produce a third field, known as the stimulated or idler field, which can be the phase conjugate of the seed [19–22]. Phase conjugation can be used to correct wavefront distortion [23], with applications in laser construction [24, 25], highresolution imaging [26], and communications [27–29]. StimPDC has also been used to produce quantum states of light [30–33]. Using an appropriate source, StimPDC has been used for phase conjugation and frequency conversion of vector beams [22, 34], [35], [36], and as a tool for probing the spontaneous regime with an increased signal [37].
Even though StimPDC has been recently addressed in several different contexts, including studies using structured light, the role of partial transverse coherence in the process has not yet been addressed. Here, we consider partially coherent StimPDC, with the aim of elucidating the relationship between the transverse degree of coherence as well as the twist phase of the pump, signal and idler beams. This knowledge will help to enhance the applications of TGSM beams. Section 2 develops a general description of the output idler field in terms of monochromatic but partially coherent pump and seed beams. We then apply this to the case in which both input fields are described by the TGSM [1, 2]. In Section 3 we present results of a StimPDC experiment using a TGSM seed beam with a coherent pump beam, observing that the idler beam is the phase conjugate of the seed.
2 Partially coherent StimPDC
Let us first recall StimPDC using spatially coherent beams in a single, thin nonlinear crystal. For a sufficiently strong seed beam, and considering that the crystal length ℓ _{ z } is much smaller than the Rayleigh length of the pump beam z _{R}, it was shown in Refs. [19, 35] that the output idler field is described by
where
where P and S are the field profiles of the pump and seed beams, respectively. The above expressions assume beams with perfect transverse coherence.
A simple and intuitive approach to StimPDC with partially coherent beams is to consider that every partially coherent beam can be decomposed as a convex sum of coherent beams. Thus, for partially coherent pump and seed beams, this results in a statistical mixture of coherent StimPDC processes, each with coherent pump beam and seed profiles given by P
_{
j
}(r) and
where the weighting coefficients λ _{ j } and β _{ k } are nonnegative and each sum to unity. One can recognize
as a coherent mode decomposition [38] of the CSD function of a partially coherent pump beam. A similar expression can be written for the seed beam. Moreover, we can recognize the CSD function of the idler field (3) as the product of the CSD function of the pump beam with the complex conjugate of the CSD function of the seed beam:
We note that this expression is quite general, considering only that the pump and the seed be monochromatic paraxial fields, incident on a thin nonlinear crystal. In the next section, we apply this result to the special case of TGSM beams.
2.1 StimPDC with twisted Schellmodel beams
There are several types of partially coherent beams. Perhaps the most wellknown is the Gaussian Schellmodel (GSM) [38], whose intensity and degreeofcoherence profile follow Gaussian distributions. A more general class are the TGSM beams [1, 2, 39, 40], which include a novel twistphase μ. The TGSM beams reduce to GSM beams when μ = 0. These are Gaussian beams, which at the beam waist are described by the CSD function
where w is the beam waist, δ is the transverse coherence length, and R is the radius of curvature. In the nearfield, the intensity distribution has variance
Considering both the seed and pump as TGSM beams, and using Eq. (6) in Eq. (5), the idler field can be written as
suggesting that the output idler field can be described by a TGSM beam with beam waist
transverse coherence length
phase curvature
and twist phase
where i, s, p stand for idler, signal (seed) and pump, respectively. Taking the absolute value of Eq. (11) and using (9), we note that the positivity constraint on the twist phase is satisfied, namely:
showing that the output idler field can indeed be understood as a bona fide TGSM beam. The above equations show that the parameters of the TGSM seed and pump beams can be used to engineer TGSM idler beams.
Equation (9) gives the coherence length of the idler field as a function of the coherence lengths of the seed and pump beams, from which we see that the transverse coherence length of the idler field is always less than or equal to that of the least coherent beam δ
_{
i
} ≤ min(δ
_{
s
}, δ
_{
p
}). For example, if δ
_{
s
} = δ
_{
p
}, then the coherence length of the idler is
When ω
_{
p
} = ω
_{
s
} and δ
_{
p
} = δ
_{
s
}, the ratio of coherence length and beam waist of the idler field remain the same as that of the pump or signal (seed) photon (δ
_{
i
}/w
_{
i
} = δ
_{
s
}/w
_{
s
} = δ
_{
p
}/w
_{
p
}). However, the beam waist and coherence length of the idler field are each related to those of the pump/seed by a factor of
When the pump beam has zero twist phase (μ
_{
p
} = 0), then from Eq. (11) the idler twist phase μ
_{
i
} = −(k
_{
s
}/k
_{
i
})μ
_{
s
}, so that the twist phase of signal and idler are equal in amplitude when the fields are degenerate. The idler twist phase is thus the opposite of the seed, but the two beams are not necessarily complex conjugates of one another, due to the other beam parameters. Specifying further that the pump beam described by a coherent plane wave (
showing that the outgoing idler beam is the phase conjugate of the incoming seed beam. In the next section, we confirm these theoretical results in a StimPDC experiment.
The term phase conjugation is very often related to the idea of a light beam reflected by a phase conjugation mirror, which produces a copy of the input beam propagating backwards in time. In the context of StimPDC, it is also possible to produce an idler beam that is a copy of the seed beam evolving backwards in time. However, it is not a reflection, and this occurs when the pump can be approximated by a plane wave. A more general way of understanding the phase conjugation that occurs in StimPDC is by envisioning the idler beam as the field resulting from the scattering of the seed beam in a modulator described by the pump beam, which propagates backwards in time. The effective modulation provided by the pump can change the wavelength, amplitude and phase of the idler as compared to the input seed beam.
3 Experiment
Figure 1 shows a sketch of the experimental setup. It is a typical StimPDC experiment [34, 36], where a 2 mm long BBO (BetaBariumBorate) nonlinear crystal is pumped by a diode laser oscillating at 405 nm. The phase matching conditions for the crystal are type I, so that the vertically polarized pump gives rise to horizontally polarized signal and idler at 780 and 840 nm, respectively. In order to achieve stimulated emission in the idler mode, a seed laser beam (780 nm) is input into the seed mode.
The key difference from previous StimPDC experiments is the use of a partially coherent seed beam, which is prepared as a TGSM beam. It is produced by sending the initially coherent seed laser onto a spatial light modulator (SLM) that is broadcasting a sequence of phaserandomized images, following the methods demonstrated in Refs. [41–43]. To achieve partial coherence, we use a video composed of 300 images played at 15 frames per second, giving a total sampling time of 20 s. Each image is given by a blazed grating pattern to maximize transmission into the first diffraction order. Upon the blazed grating, we imprint a phase profile corresponding to
with
and weight function
and the parameter
In what follows, we classify TGSM beams according to the normalized twist phase τ = μkδ
^{2}, such that −1 ≤ τ ≤ 1. We prepare the seed beam with normalized twist phases τ
_{
s
} = 0, ±1. For each twist phase, we perform measurements for seed and idler beams while varying the transverse coherence length δ
_{
s
} of the seed. In a first set of measurements, we measure the farfield variance
4 Results
4.1 Coherence transfer
The first set of measurements was carried out with the goal of studying the transverse coherence properties of the stimulated idler beam, as a function of the properties of the seed beam. With the coherent and collimated pump beam, and the seed beam prepared as a TGSM beam, from Section 2.1 we expect δ
_{
i
} ≈ δ
_{
s
} and μ
_{
i
} = −k
_{
s
}
μ
_{
s
}/k
_{
i
} ≈ −μ
_{
s
} since the fields are nearly degenerate (λ
_{
i
}/λ
_{
s
} ∼ 1.08). A common method for studying the coherence properties of TGSM beams is through the divergence of the beam [44]. Here, we use the CCD camera to measure the variance of the intensity distribution in the farfield
showing that the beam diverges faster for nonzero twist phase. Figure 2 shows plots of the farfield variance
As an additional test of the coherence properties, in a second set of measurements we have analysed the visibility of the interference fringes in a doubleslit experiment as a function of the transverse coherence length. The doubleslit aperture was imprinted over the TGSM images on the SLM, and this profile was imaged onto the crystal plane. The CCD camera was again placed in the farfield of the doubleslit, as shown in Figure 1. The resulting interference fringes were accumulated over 20 s (all 300 TGSM frames). The visibility
4.2 Transfer and conjugation of twist phase
The results presented in Figure 2 confirm the presence of twist phase in the seed and idler beams given in Eq. (14). What remains is to observe the phase conjugation relation between the TGSM seed and idler beams. To do so, we exploit the fact that the 2D doubleslit interference pattern is rotated by an amount proportional to the twist phase, with farfield intensity pattern give by:
where f = 400 mm is the focal length of the farfield optical system (lenses L2 and L3), and 2d is the distance between the centers of the slits. Thus, the central interference maximum lies along y = (fμ)x. We can thus use the interference patterns to estimate not only the amplitude but also the sign of the twist phase.
Figure 4 shows interference patterns for both seed and idler fields for normalized twist phase of the seed beam τ _{ s } = 0, ±1, and two values of the transverse coherence. For a qualitative demonstration we choose two coherence length values that, combined between visibility and rotation angle, best illustrate the rotation. For τ _{ s } = 1, we obtained the results presented in Figure 4a and d for the seed and idler beams, respectively. In these cases, as in Figure 4c and f with τ _{ s } = −1, for δ _{ s } = 0.56 mm, we can observe the seed and idler beams rotate in opposite directions. For τ _{ s } = 0, as in Figure 4b and e, we observe very little rotation of the images, as expected. We include the results for δ _{ s } = 5.0 mm, which show only very small tilt, to further demonstrate that observed rotation is a result of the twist phase (μ ∝ 1/δ ^{2}), and not other aspects of the beam.
To quantify the rotations, we use a Radon transform of the images to identify the rotation angle as the one which maximizes the marginal visibility of the interference pattern. We obtained data given in Table 1, where we calculate μ = θ _{ μ }/f after converting θ _{ μ } to radians, for both seed and idler. Also given is the theoretical value of the twist phase for the seed beam. We can see that the numerical values obtained are consistent with the TGSM idler beam being the phase conjugate of the TGSM seed beam.
Theory (×10^{3} mm^{−1})  μ _{ s } = 0.40  μ _{ s } = 0  μ _{ s } = −0.40 

μ _{ s } (×10^{3} mm^{−1})  0.26 ± 0.04  −0.02 ± 0.04  −0.13 ± 0.04 
μ _{ i } (×10^{3} mm^{−1})  −0.15 ± 0.04  −0.04 ± 0.04  0.22 ± 0.04 
Theory (×10^{3} mm^{−1})  μ _{ s } = 0.005  μ _{ s } = 0  μ _{ s } = −0.005 
μ _{ s } (×10^{3} mm^{−1})  0.04 ± 0.04  0.00 ± 0.04  0.04 ± 0.04 
μ _{ i } (×10^{3} mm^{−1})  0.02 ± 0.04  0.02 ± 0.04  −0.02 ± 0.04 

The top (bottom) three rows correspond to coherence length δ _{ s } = 0.56 mm (δ _{ s } = 5.0 mm). The expected twist phase for δ _{ s } = 5.0 mm is smaller than our experimental precision. The idler beam takes on a twist phase that is consistent with the conjugation of the seed twist phase.
5 Conclusions
We have studied StimPDC in the case where the pump and seed beams have partial transverse coherence. In particular, when the pump and seed beams are described by the TGSM, we have shown analytically that the stimulated idler beam is also a TGSM beam, with properties determined by those of the pump and seed. These results were investigated experimentally using a pump beam with large transverse coherence. In this case, we experimentally confirm our theoretical results, showing that the coherence properties of the idler are determined by the seed beam alone. We have also demonstrated that the TGSM idler beam is the phase conjugate of the TGSM seed beam. These results open up the possibility of using stimulated downconversion in schemes for correction of distortions in the wavefront of partially coherent beams. Our results motivate the use of the GSM and TGSM fields in communication protocols or imaging scenarios where light beams propagate through disturbing media. An interesting avenue for future research concerns polarizationinduced spatial incoherence, and/or partial polarization, which can be addressed using a twocrystal source as in Refs. [34, 36]. We note that the control of the degree of polarization might allow for the generation of TGSMlike beams without the use of moviemasks in an SLM.
Funding source: Fondo de Fomento al Desarrollo Cientifico y Tecnológico
Award Identifier / Grant number: 1190901
Award Identifier / Grant number: 1190933
Award Identifier / Grant number: 1200266
Funding source: National Research Foundation http://dx.doi.org/10.13039/501100001321
Funding source: Comisión Nacional de Investigación Cientifica y Tecnológica
Award Identifier / Grant number: ICN17_012
Funding source: Fundação de Amparo à Pesquisa e Inovação do Estado de Santa Catarina
Funding source: Conselho Nacional de Desenvolvimento Cientifico e Tecnológico
Funding source: Coordenação de Aperfeiçoamento de Pessoal de Nível Superior http://dx.doi.org/10.13039/501100002322

Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

Research funding: This work was funded by the Chilean agencies Fondo Nacional de Desarrollo Científico y Tecnológico (FONDECYT – DOI 501100002850) (1190901, 1190933, 1200266); National Agency of Research and Development (ANID) Millennium Science Initiative Program—ICN17012; the Brazilian agencies Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES – DOI 501100002322), Fundação de Amparo à Pesquisa do Estado de Santa Catarina (FAPESC – DOI 501100005667), Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq – DOI 501100003593), Instituto Nacional de Ciência e Tecnologia de Informação Quântica (INCTIQ 465469/20140); the National Research Foundations of South Africa.

Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
References
[1] R. Simon and N. Mukunda, “Twisted Gaussian Schellmodel beams,” J. Opt. Soc. Am. A, vol. 10, p. 95, 1993. https://doi.org/10.1364/josaa.10.000095.Search in Google Scholar
[2] R. Simon and N. Mukunda, “Twist phase in Gaussianbeam optics,” J. Opt. Soc. Am. A, vol. 15, p. 2373, 1998. https://doi.org/10.1364/josaa.15.002373.Search in Google Scholar
[3] F. Wang and Y. Cai, “Secondorder statistics of a twisted Gaussian Schellmodel beam in turbulent atmosphere,” Opt. Express, vol. 18, p. 24661, 2010. https://doi.org/10.1364/oe.18.024661.Search in Google Scholar
[4] M. Zhou, W. Fan, and G. Wu, “Evolution properties of the orbital angular momentum spectrum of twisted Gaussian Schellmodel beams in turbulent atmosphere,” J. Opt. Soc. Am. A, vol. 37, p. 142, 2020. https://doi.org/10.1364/josaa.37.000142.Search in Google Scholar
[5] Y. Liu, X. Liu, L. Liu, F. Wang, Y. Zhang, and Y. Cai, “Ghost imaging with a partially coherent beam carrying twist phase in a turbulent ocean: a numerical approach,” Appl. Sci., vol. 9, 2019, Art no. 3023. https://doi.org/10.3390/app9153023.Search in Google Scholar
[6] G. Lima, F. TorresRuiz, L. Neves, A. Delgado, C. Saavedra, and S. Pádua, “Generating mixtures of spatial qubits,” Opt. Commun., vol. 281, p. 5058, 2008. https://doi.org/10.1016/j.optcom.2008.06.050.Search in Google Scholar
[7] A. K. Jha and R. W. Boyd, “Spatial twophoton coherence of the entangled field produced by downconversion using a partially spatially coherent pump beam,” Phys. Rev. A, vol. 81, p. 013828, 2010. https://doi.org/10.1103/physreva.81.053832.Search in Google Scholar
[8] M. E. Olvera and S. FrankeArnold, arXiv: Quantum Physics, 2015.Search in Google Scholar
[9] Y. Ismail, S. Joshi, and F. Petruccione, “Polarizationentangled photon generation using partial spatially coherent pump beam,” Sci. Rep., vol. 7, p. 12091, 2017. https://doi.org/10.1038/s41598017123766.Search in Google Scholar PubMed PubMed Central
[10] E. Giese, R. Fickler, W. Zhang, L. Chen, and R. W. Boyd, “Influence of pump coherence on the quantum properties of spontaneous parametric downconversion,” Phys. Scripta, vol. 93, p. 084001, 2018. https://doi.org/10.1088/14024896/aace12.Search in Google Scholar
[11] H. Defienne and S. Gigan, “Spatially entangled photonpair generation using a partial spatially coherent pump beam,” Phys. Rev. A, vol. 99, p. 053831, 2019. https://doi.org/10.1103/physreva.99.053831.Search in Google Scholar
[12] W. Zhang, R. Fickler, E. Giese, L. Chen, and R. W. Boyd, “Influence of pump coherence on the generation of positionmomentum entanglement in optical parametric downconversion,” Opt. Express, vol. 27, p. 20745, 2019. https://doi.org/10.1364/oe.27.020745.Search in Google Scholar
[13] S. Joshi and B. Kanseri, “Spatial coherence properties of down converted biphoton field generated using partially coherent pump beam,” Optik, vol. 217, p. 164941, 2020.10.1016/j.ijleo.2020.164941Search in Google Scholar
[14] B. Kanseri and P. Sharma, “Effect of partially coherent pump on the spatial and spectral profiles of downconverted photons,” J. Opt. Soc. Am. B, vol. 37, p. 505, 2020. https://doi.org/10.1364/josab.376565.Search in Google Scholar
[15] S. P. Phehlukwayo, M. L. Umuhire, Y. Ismail, S. Joshi, and F. Petruccione, “Influence of coincidence detection of a biphoton state through freespace atmospheric turbulence using a partially spatially coherent pump,” Phys. Rev. A, vol. 102, p. 033732, 2020. https://doi.org/10.1103/physreva.102.033732.Search in Google Scholar
[16] L. Hutter, G. Lima, and S. P. Walborn, “Boosting entanglement generation in downconversion with incoherent illumination,” Phys. Rev. Lett., vol. 125, p. 193602, 2020. https://doi.org/10.1103/physrevlett.125.193602.Search in Google Scholar PubMed
[17] L. Hutter, E. S. Gómez, G. Lima, and S. P. Walborn, “Partially coherent spontaneous parametric downconversion: twisted Gaussian biphotons,” AVS Quantum Sci., vol. 3, p. 031401, 2021. https://doi.org/10.1116/5.0058681.Search in Google Scholar
[18] L. Wang, X. Zou, and L. Mandel, “Observation of induced coherence in twophoton downconversion,” J. Opt. Soc. Am. B, vol. 8, p. 978, 1991. https://doi.org/10.1364/josab.8.000978.Search in Google Scholar
[19] P. H. S. Souto Ribeiro, S. Pádua, and C. H. Monken, “Image and coherence transfer in the stimulated downconversion process,” Phys. Rev. A, vol. 60, p. 5074, 1999. https://doi.org/10.1103/physreva.60.5074.Search in Google Scholar
[20] P. H. Souto Ribeiro, D. P. Caetano, M. P. Almeida, J. A. Huguenin, B. Coutinho dos Santos, and A. Z. Khoury, “Observation of image transfer and phase conjugation in stimulated downconversion,” Phys. Rev. Lett., vol. 87, p. 133602, 2001. https://doi.org/10.1103/physrevlett.87.133602.Search in Google Scholar
[21] D. P. Caetano, M. P. Almeida, P. H. Souto Ribeiro, J. A. O. Huguenin, B. Coutinho dos Santos, and A. Z. Khoury, “Conservation of orbital angular momentum in stimulated downconversion,” Phys. Rev. A, vol. 66, p. 041801, 2002. https://doi.org/10.1103/physreva.66.041801.Search in Google Scholar
[22] A. G. de Oliveira, M. F. Arruda, W. C. Soares, et al.., “Phase conjugation and mode conversion in stimulated parametric downconversion with orbital angular momentum: a geometrical interpretation,” Braz. J. Phys., vol. 49, p. 10, 2019. https://doi.org/10.1007/s1353801806144.Search in Google Scholar
[23] B. Y. Zel’Dovich, V. Popovichev, V. Ragul’Skii, and F. Faizullov, Landmark Papers on Photorefractive Nonlinear Optics, Singapore, World Scientific, 1995, pp. 303–306.10.1142/9789812832047_0033Search in Google Scholar
[24] R. McFarlane and D. Steel, “Laser oscillator using resonator with selfpumped phaseconjugate mirror,” Opt. Lett., vol. 8, p. 208, 1983. https://doi.org/10.1364/ol.8.000208.Search in Google Scholar PubMed
[25] M. Gower, “KrF laser amplifier with phaseconjugate Brillouin retroreflectors,” Opt. Lett., vol. 7, p. 423, 1982. https://doi.org/10.1364/ol.7.000423.Search in Google Scholar PubMed
[26] M. D. Levenson, “Highresolution imaging by wavefront conjugation,” Opt. Lett., vol. 5, p. 182, 1980. https://doi.org/10.1364/ol.5.000182.Search in Google Scholar PubMed
[27] K. R. MacDonald, W. R. Tompkin, and R. W. Boyd, “Passive oneway aberration correction using fourwave mixing,” Opt. Lett., vol. 13, p. 485, 1988. https://doi.org/10.1364/ol.13.000485.Search in Google Scholar PubMed
[28] R. W. Boyd, K. R. MacDonald, and M. S. Malcuit, Laser Wavefront Control, vol. 1000, International Society for Optics and Photonics, 1989, pp. 69–81.10.1117/12.960250Search in Google Scholar
[29] S. Set, S. Yamashita, M. Ibsen, et al.., “Ultrahigh bit rate optical phase conjugation/wavelength conversion in DSF and SOA with novel configuration incorporating inline fibre DFB lasers,” Electron. Lett., vol. 34, p. 1681, 1998. https://doi.org/10.1049/el:19981168.10.1049/el:19981168Search in Google Scholar
[30] A. Zavatta, S. Viciani, and M. Bellini, “Quantumtoclassical transition with singlephotonadded coherent states of light,” Science, vol. 306, p. 660, 2004. https://doi.org/10.1126/science.1103190.Search in Google Scholar PubMed
[31] A. Kolkiran and G. S. Agarwal, “Quantum interferometry using coherent beam stimulated parametric downconversion,” Opt. Express, vol. 16, p. 6479, 2008. https://doi.org/10.1364/oe.16.006479.Search in Google Scholar PubMed
[32] M. Barbieri, N. Spagnolo, M. G. Genoni, et al.., “NonGaussianity of quantum states: an experimental test on singlephotonadded coherent states,” Phys. Rev. A, vol. 82, p. 063833, 2010. https://doi.org/10.1103/physreva.82.063833.Search in Google Scholar
[33] T. Kiesel, W. Vogel, M. Bellini, and A. Zavatta, “Nonclassicality quasiprobability of singlephotonadded thermal states,” Phys. Rev. A, vol. 83, p. 032116, 2011. https://doi.org/10.1103/physreva.83.032116.Search in Google Scholar
[34] A. G. de Oliveira, M. F. Z. Arruda, W. C. Soares, et al.., “Realtime phase conjugation of vector vortex beams,” ACS Photonics, vol. 7, p. 249, 2020. https://doi.org/10.1021/acsphotonics.9b01524.Search in Google Scholar
[35] A. de Oliveira, N. Rubiano da Silva, R. Medeiros de Araújo, P. Souto Ribeiro, and S. Walborn, “Quantum optical description of phase conjugation of vector vortex beams in stimulated parametric downconversion,” Phys. Rev. Appl., vol. 14, p. 024048, 2020. https://doi.org/10.1103/physrevapplied.14.024048.Search in Google Scholar
[36] N. Rubiano da Silva, A. de Oliveira, M. Arruda, et al.., “Stimulated parametric downconversion with vector vortex beams,” Phys. Rev. Appl., vol. 15, p. 024039, 2021. https://doi.org/10.1103/physrevapplied.15.024039.Search in Google Scholar
[37] L. A. Rozema, C. Wang, D. H. Mahler, et al.., “Characterizing an entangledphoton source with classical detectors and measurements,” Optica, vol. 2, p. 430, 2015. https://doi.org/10.1364/optica.2.000430.Search in Google Scholar
[38] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, New York, Cambridge University Press, 1995.10.1017/CBO9781139644105Search in Google Scholar
[39] R. Simon, K. Sundar, and N. Mukunda, “Twisted Gaussian Schellmodel beams I Symmetry structure and normalmode spectrum,” J. Opt. Soc. Am. A, vol. 10, p. 2008, 1993. https://doi.org/10.1364/josaa.10.002008.Search in Google Scholar
[40] K. Sundar, R. Simon, and N. Mukunda, “Twisted Gaussian Schellmodel beams II Spectrum analysis and propagation characteristics,” J. Opt. Soc. Am. A, vol. 10, p. 2017, 1993. https://doi.org/10.1364/josaa.10.002017.Search in Google Scholar
[41] R. Wang, S. Zhu, Y. Chen, H. Huang, Z. Li, and Y. Cai, “Experimental synthesis of partially coherent sources,” Opt. Lett., vol. 45, p. 1874, 2020. https://doi.org/10.1364/ol.388307.Search in Google Scholar PubMed
[42] C. Tian, S. Zhu, H. Huang, Y. Cai, and Z. Li, “Customizing twisted Schellmodel beams,” Opt. Lett., vol. 45, p. 5880, 2020. https://doi.org/10.1364/ol.405149.Search in Google Scholar
[43] H. Wang, X. Peng, H. Zhang, et al.., “Experimental synthesis of partially coherent beam with controllable twist phase and measuring its orbital angular momentum,” Nanophotonics, 2021. https://doi.org/10.1515/nanoph20210432.Search in Google Scholar
[44] A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schellmodel beams,” J. Opt. Soc. Am. A, vol. 11, p. 1818, 1994. https://doi.org/10.1364/josaa.11.001818.Search in Google Scholar
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