Pu and Korotkova (Optics Communications 283, 1693, 2010) derived propagation laws for the spectral density and polarization of stochastic electromagnetic beams — modeled as electromagnetic Gaussian Schell-model (EMGSM) sources — traveling through linear, deterministic media whose complex wave numbers Kx(ω)K_x(\omega), Ky(ω)K_y(\omega) for the two polarization components are constant along the propagation direction, using paraxial ABCD (ray-matrix) propagation. Their analysis rests on explicit simplifying assumptions that delineate five extension directions: (1) longitudinally varying, stratified, or otherwise inhomogeneous media; (2) statistically random (turbulent) media; (3) nonparaxial or full-vector Maxwell propagation; (4) nonlinear medium response such as gain saturation or an intensity-dependent refractive index; and (5) experimental validation and source classes beyond EMGSM. Directions (1), (2), (5), and the free-space part of (3) have direct treatments in the same cross-spectral-density-matrix framework [4–8, 10, 11, 16–18, 21]; no direct treatment is known for stochastic electromagnetic beams (fluctuating polarization, 2x2 cross-spectral-density matrix) in media with nonlinear response, or for a full-vector nonparaxial treatment in linear gain or absorbing media. The problem is therefore the surviving core: derive and validate propagation laws for the cross-spectral-density matrix of stochastic electromagnetic beams in media with nonlinear response — gain-saturated active media or Kerr-type media with an intensity-dependent refractive index — and under full-vector nonparaxial propagation in linear gain or absorbing media, such that the published linear paraxial laws are recovered in the limits specified in the Problem Statement; the nonlinear and nonparaxial regimes, and the two nonlinearity classes, are independent targets, each sufficient on its own.

Background

A stochastic (random) electromagnetic beam is a beam whose electric field components fluctuate statistically in space and time; its second-order properties are described, at a single frequency ω\omega, by the 2x2 cross-spectral density (CSD) matrix with entries Wij(r1,r2,ω)W_{ij}(\mathbf{r}_1,\mathbf{r}_2,\omega), where the indices ii and jj label two mutually orthogonal transverse polarization components. The spectral density (the spectrum at a point) is S(r,ω)=Wxx+WyyS(\mathbf{r},\omega)=W_{xx}+W_{yy}; the spectral degree of coherence between two points is the normalized trace of the CSD matrix; and the degree of polarization at a point is P(r,ω)=1−4det⁡W/(Tr W)2P(\mathbf{r},\omega)=\sqrt{1-4\det W/\left(\mathrm{Tr}\,W\right)^2}, ranging from 0 (unpolarized) to 1 (fully polarized). Wolf's unified theory of coherence and polarization (2003) established that coherence and polarization are two aspects of this one matrix, and that all three quantities generally change on propagation, even in free space, through correlation-induced effects.

A workhorse source model is the electromagnetic Gaussian Schell-model (EMGSM) beam, for which the spectral density of each component is Gaussian and the degrees of coherence between all component pairs are Gaussian functions of the separation of the two points. Its propagation through deterministic first-order (ABCD) optical systems — systems, such as free space, thin lenses, and quadratic media, in which the transverse ray coordinates transform linearly under paraxial (small-angle) propagation — can be computed by the generalized Huygens–Fresnel integral.

Pu and Korotkova (2010) combined these tools for a medium with gain or absorption, modeled by complex wave numbers Kx(ω)K_x(\omega), Ky(ω)K_y(\omega) whose real parts are refractive indices and whose imaginary parts describe gain or absorption; the two polarization components may be amplified or attenuated differently (dichroism). They derived how the spectral density, the degree of coherence, and the degree of polarization of EMGSM beams evolve in such media, showing for example that the degree of polarization can be controlled by the dispersion of the complex wave numbers. The analysis rests on explicit simplifying assumptions: the medium is linear, deterministic, and uniform along the propagation direction (KxK_x, KyK_y independent of zz), and propagation is paraxial within the ABCD formalism, with free-space propagation used in the numerical examples. These assumptions are the scope limits of the 2010 theory; the problem asks for the theory to be extended beyond them.

Substantial adjacent work exists. Turbulent (statistically random) media were treated within the same CSD framework by Salem, Korotkova, Dogariu, and Wolf (2004) for atmospheric turbulence, by Korotkova, Salem, and Wolf (2004), Korotkova, Salem, Dogariu, and Wolf (2005), and Korotkova and Gbur (2007), and by Pu and Korotkova themselves (2009) for the degree of cross-polarization in a turbulent atmosphere. Korotkova and Farwell (2010, 2011) extended this to oceanic turbulence, whose power spectrum mixes temperature and salinity fluctuations; Yao, Toselli, and Korotkova (2014) and Korotkova (2015) treated anisotropic turbulence. Related spectral analyses for deterministic media with general refractive properties, including negative-phase (negative-index) materials, were given by Tong and Korotkova (2010). Lahiri and Wolf (2013) developed the propagation theory for beams of any state of coherence and polarization through multilayered stratified media. Nonparaxial stochastic electromagnetic beams in free space were analyzed by Zhu and Zhao (2009) and Ding and coauthors (2011). Source classes beyond EMGSM include electromagnetic multi-Gaussian, cosine-Gaussian, vortex, and rectangular multi-Gaussian Schell-model beams (Mei, Korotkova, and Shchepakina 2012; Mei and Korotkova 2013; Chen and Pu 2008; Chen and Zhao 2016), and Avramov-Zamurovic and coauthors (2016) reported laboratory measurements of electromagnetic Bessel-Gaussian Schell-model beams in a turbulent channel. For the nonlinear regime, the adjacent literature treats mutually incoherent vector soliton pairs in saturable or self-focusing media (Litchinitser and coauthors, 1999; Yang and Pelinovsky, 2003; Ponomarenko and Agrawal, 2004; Motzek and coauthors, 2004) and deterministic filamentation instability of electromagnetic beams (Sodha and Faisal, 2017), but does not deliver propagation laws for the CSD matrix, spectral density, or polarization in the unified coherence–polarization framework; follow-up work on partially coherent beams in gain or absorbing media (Yang and coauthors, 2021) keeps the medium response linear.

Problem Statement

Pu and Korotkova (2010) obtained the evolution laws for the 2x2 cross-spectral-density (CSD) matrix and the derived spectral density, spectral degree of coherence, and degree of polarization for linear, deterministic, z-uniform gain/absorbing media under paraxial ABCD propagation. Extending this propagation theory to more general propagation environments is the problem posed here: (i) media whose complex wave numbers vary along the propagation direction or are spatially non-uniform, such as stratified or inhomogeneous media; (ii) statistically random (turbulent) media; (iii) nonparaxial propagation or situations requiring a full-vector Maxwell treatment; (iv) media with nonlinear response, such as gain saturation or an intensity-dependent refractive index; and (v) experimental validation and source classes beyond the EMGSM family. Because (i), (ii), (v), and the free-space portion of (iii) already have direct treatments in the literature [4–8, 10, 11, 16–18, 21], the surviving open core is the following. For a stochastic electromagnetic beam with a general admissible 2x2 CSD matrix (fluctuating polarization; the EMGSM family is the case underlying the published laws), derive and validate propagation laws for the CSD matrix in media with nonlinear response — either a gain-saturated active medium or a Kerr-type medium with an intensity-dependent refractive index, each independently sufficient — and, independently, for full-vector nonparaxial propagation in linear gain or absorbing media with complex wave numbers Kx(ω)K_x(\omega), Ky(ω)K_y(\omega). Each of these two regimes is a separately sufficient target, and within the nonlinear regime either nonlinearity class likewise suffices on its own; no target is conditional on another. For the regime treated, the law must give the full CSD matrix WijW_{ij} and the derived spectral density SS, spectral degree of coherence μ\mu, and degree of polarization PP, and must reduce to the known linear, deterministic, z-uniform, paraxial results of Pu and Korotkova in the limit of vanishing nonlinear response (zero intensity-dependent index; linear gain/absorption) for the nonlinear regime, and in the paraxial (small-angle) limit for the nonparaxial regime.

References

  1. Jixiong Pu, Olga Korotkova (2010). Spectral and polarization properties of stochastic electromagnetic beams propagating in gain or absorbing media. Optics Communications 283, 1693-1706. DOI: 10.1016/j.optcom.2009.12.068, https://doi.org/10.1016/j.optcom.2009.12.068
  2. Emil Wolf (2003). Unified theory of coherence and polarization of random electromagnetic beams. Physics Letters A 312, 263-267. DOI: 10.1016/S0375-9601(03)00684-4, https://doi.org/10.1016/S0375-9601(03)00684-4
  3. Emil Wolf (2003). Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation. Optics Letters 28, 1078-1080. DOI: 10.1364/OL.28.001078, https://doi.org/10.1364/OL.28.001078
  4. Mohamed Salem, Olga Korotkova, Aristide Dogariu, Emil Wolf (2004). Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere. Waves in Random Media 14, 513-523. DOI: 10.1088/0959-7174/14/4/003, https://doi.org/10.1088/0959-7174/14/4/003
  5. Olga Korotkova, Mohamed Salem, Emil Wolf (2004). The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence. Optics Communications 233, 225-230. DOI: 10.1016/j.optcom.2004.01.005, https://doi.org/10.1016/j.optcom.2004.01.005
  6. O. Korotkova, M. Salem, A. Dogariu, E. Wolf (2005). Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere. Waves in Random and Complex Media 15, 353-364. DOI: 10.1080/17455030500184511, https://doi.org/10.1080/17455030500184511
  7. Olga Korotkova, Greg Gbur (2007). Angular spectrum representation for propagation of random electromagnetic beams in a turbulent atmosphere. Journal of the Optical Society of America A 24, 2728-2736. DOI: 10.1364/JOSAA.24.002728, https://doi.org/10.1364/JOSAA.24.002728
  8. Jixiong Pu, Olga Korotkova (2009). Propagation of the degree of cross-polarization of a stochastic electromagnetic beam through the turbulent atmosphere. Optics Communications 282, 1691-1698. DOI: 10.1016/j.optcom.2009.01.029, https://doi.org/10.1016/j.optcom.2009.01.029
  9. Ziyang Chen, Jixiong Pu (2008). Stochastic electromagnetic vortex beam and its propagation. Physics Letters A 372, 2734-2740. DOI: 10.1016/j.physleta.2007.12.038, https://doi.org/10.1016/j.physleta.2007.12.038
  10. Yingbin Zhu, Daomu Zhao (2009). Stokes parameters and degree of polarization of nonparaxial stochastic electromagnetic beams. Physics Letters A 373, 1595-1598. DOI: 10.1016/j.physleta.2009.02.060, https://doi.org/10.1016/j.physleta.2009.02.060
  11. Chaoliang Ding, Liuzhan Pan, Zhao Zhiguo, Zhang Yongtao, Wang Haixia (2011). Changes in the spectral degree of polarization of nonparaxial stochastic electromagnetic pulsed beams. Optics Communications 284, 5072-5077. DOI: 10.1016/j.optcom.2011.07.035, https://doi.org/10.1016/j.optcom.2011.07.035
  12. Zhisong Tong, Olga Korotkova (2010). Stochastic electromagnetic beams in positive- and negative-phase materials. Optics Letters 35, 175-177. DOI: 10.1364/OL.35.000175, https://doi.org/10.1364/OL.35.000175
  13. Zhisong Tong, Olga Korotkova (2010). Far-field analysis of spectral shifts in Gaussian Schell-model beams propagating through media with arbitrary refractive properties. Journal of Optics 12, 095708. DOI: 10.1088/2040-8978/12/9/095708, https://doi.org/10.1088/2040-8978/12/9/095708
  14. Olga Korotkova, Nathan Farwell (2010). Polarization changes in stochastic electromagnetic beams propagating in the oceanic turbulence. Proceedings of SPIE 7588, 75880S. DOI: 10.1117/12.846740, https://doi.org/10.1117/12.846740
  15. Olga Korotkova, Nathan Farwell (2011). Effect of oceanic turbulence on polarization of stochastic beams. Optics Communications 284, 1740-1746. DOI: 10.1016/j.optcom.2010.12.024, https://doi.org/10.1016/j.optcom.2010.12.024
  16. Mayukh Lahiri, Emil Wolf (2013). Propagation of electromagnetic beams of any state of spatial coherence and polarization through multilayered stratified media. Journal of the Optical Society of America A 30, 2547-2554. DOI: 10.1364/JOSAA.30.002547, https://doi.org/10.1364/JOSAA.30.002547
  17. Zhangrong Mei, Olga Korotkova, Elena Shchepakina (2012). Electromagnetic multi-Gaussian Schell-model beams. Journal of Optics 15, 025705. DOI: 10.1088/2040-8978/15/2/025705, https://doi.org/10.1088/2040-8978/15/2/025705
  18. Zhangrong Mei, Olga Korotkova (2013). Electromagnetic cosine-Gaussian Schell-model beams in free space and atmospheric turbulence. Optics Express 21, 27246-27257. DOI: 10.1364/OE.21.027246, https://doi.org/10.1364/OE.21.027246
  19. Min Yao, Italo Toselli, Olga Korotkova (2014). Propagation of electromagnetic stochastic beams in anisotropic turbulence. Optics Express 22, 31608-31616. DOI: 10.1364/OE.22.031608, https://doi.org/10.1364/OE.22.031608
  20. Olga Korotkova (2015). Polarization changes in light beams trespassing anisotropic turbulence. Optics Letters 40, 3077-3080. DOI: 10.1364/OL.40.003077, https://doi.org/10.1364/OL.40.003077
  21. S. Avramov-Zamurovic, C. Nelson, S. Guth, O. Korotkova, R. Malek-Madani (2016). Experimental study of electromagnetic Bessel-Gaussian Schell Model beams propagating in a turbulent channel. Optics Communications 359, 207-215. DOI: 10.1016/j.optcom.2015.09.078, https://doi.org/10.1016/j.optcom.2015.09.078
  22. Xudong Chen, Daomu Zhao (2016). Propagation properties of electromagnetic rectangular multi-Gaussian Schell-model beams in oceanic turbulence. Optics Communications 372, 137-143. DOI: 10.1016/j.optcom.2016.03.082, https://doi.org/10.1016/j.optcom.2016.03.082
  23. Biling Zhang, Yonggen Xu, Youquan Dan, Xueru Deng, Zhengquan Zhao (2017). Beam spreading and M2-factor of electromagnetic Gaussian Schell-model beam propagating in inhomogeneous atmospheric turbulence. Optik 149, 398-408. DOI: 10.1016/j.ijleo.2017.09.061, https://doi.org/10.1016/j.ijleo.2017.09.061
  24. Dajun Liu, Yaochuan Wang (2018). Properties of a random electromagnetic multi-Gaussian Schell-model vortex beam in oceanic turbulence. Applied Physics B 124. DOI: 10.1007/s00340-018-7048-0, https://doi.org/10.1007/s00340-018-7048-0
  25. Chen Yang, Xin Guo, Miaojun Guo, Jing Wang, Meiling Duan, Jinhong Li (2021). Influence of gain or absorption media on transmission of partially coherent vortex beams. Journal of the Optical Society of America A 38, 675-684. DOI: 10.1364/JOSAA.420800, https://doi.org/10.1364/JOSAA.420800
  26. Fei Zhuang, Xinyue Du, Daomu Zhao (2011). Polarization modulation for a stochastic electromagnetic beam passing through a chiral medium. Optics Letters 36, 2683-2685. DOI: 10.1364/OL.36.002683, https://doi.org/10.1364/OL.36.002683
  27. Natalia M. Litchinitser, Wiesław Królikowski, Nail N. Akhmediev, Govind P. Agrawal (1999). Asymmetric partially coherent solitons in saturable nonlinear media. Physical Review E 60, 2377-2380. DOI: 10.1103/PhysRevE.60.2377, https://doi.org/10.1103/PhysRevE.60.2377
  28. Jianke Yang, Dmitry E. Pelinovsky (2003). Stable vortex and dipole vector solitons in a saturable nonlinear medium. Physical Review E 67, 016608. DOI: 10.1103/PhysRevE.67.016608, https://doi.org/10.1103/PhysRevE.67.016608
  29. Ningjing Xiang (2026). The Spreading and Wander of a Gaussian Schell-Model Beam Through Oceanic Turbulence. Photonics 13, 478. DOI: 10.3390/photonics13050478, https://doi.org/10.3390/photonics13050478
  30. Sergey A. Ponomarenko, Govind P. Agrawal (2004). Asymmetric incoherent vector solitons. Physical Review E 69, 036604. DOI: 10.1103/PhysRevE.69.036604, https://doi.org/10.1103/PhysRevE.69.036604
  31. Kristian Motzek, Friedemann Kaiser, José R. Salgueiro, Yuri Kivshar, Cornelia Denz (2004). Incoherent vector vortex-mode solitons in self-focusing nonlinear media. Optics Letters 29, 2285. DOI: 10.1364/OL.29.002285, https://doi.org/10.1364/OL.29.002285
  32. M. S. Sodha, M. Faisal (2017). Filamentation instability of electromagnetic beams in nonlinear media: a tutorial review. Proceedings of the Indian National Science Academy 94. DOI: 10.16943/ptinsa/2017/49220, https://doi.org/10.16943/ptinsa/2017/49220
  33. Naiyue Zhang, Yu Deng, Donglin Wang, Xiaoling Ji (2025). Characteristics of partially coherent radially polarized beams in strongly nonlocal nonlinear media. SSRN preprint. DOI: 10.2139/ssrn.5343875, https://doi.org/10.2139/ssrn.5343875