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Abstract
We propose a near-infrared image recovery method based on modulation instability in the photorefractive semiconductor CdZnTe:V. The formation mechanism of modulation instability in CdZnTe:V is discussed, and the theoretical gain model is derived. Theoretical results of optical image recovery at 1 µm and 1.5 µm wavelengths demonstrate that the maximum cross-correlation gain is 2.6 with a signal to noise intensity ratio of 0.1. These results suggest that our method could be one of potential aids for near-infrared imaging.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In nonlinear media, spatial modulation instability (MI) occurs if small spatial perturbations are modulated by self-focusing [1–3]. Due to MI, small perturbations are amplified and filaments or spots patterns form as the light propagation. Therefore, the pattern of MI can be treated as a series of spatial solitons [4]. Incoherent MI appears if the nonlinearity exceeds a threshold that depends on the incoherence of light [1,2]. The key of incoherent MI is non-instantaneous media. In such media, the response time (>0.1 s) is much longer than the average phase fluctuation time (∼ms) of incoherent light, and therefore the refractive index change is determined by the time averaged light intensity. Typically, when a set of perturbations are seeded artificially onto a uniform beam, visibility of the perturbations would be enhanced by induced MI in the competition between different modes of the perturbations [5].
Image recovery is of particular interest in the domain of imaging [6–14]. Incoherent MI has shown the ability in visible image recovery by introducing a photorefractive SBN (SrBaNb2O6) crystal into the imaging system [11–14]. Within the crystal, modes of the image signal induce MI for incoherent noise by tiny refractive index modulation. The method is categorized as a kind of stochastic resonance because it can be treated as an energy transfer of noise to signals [14–18]. Compared with other image recovery technologies, using MI to recover noisy images is more effective against same-frequency noise and low signal to noise ratio conditions [14].
Near-infrared light is widely used in communication, metrology, and detection technologies [19–24]. However, the photorefractive sensitivity to near-infrared in SBN crystals is ∼106 times lower than that to visible light. As a result, high light intensity (up to 102 W/cm2) is needed if near-infrared MI is to be observed [25,26]. Semiconductors CdZnTe:V and InP:Fe exhibit photorefractivity at 0.9 to 1.5 µm wavelengths with much lower light intensity (∼mW/cm2) [27–41]. On account of small electro-optic coefficients, photorefractivity in semiconductors needs to be enhanced by other properties. Considering that both holes and electrons contribute to the transport in InP:Fe and CdZnTe:V crystals (two-carriers photorefractive media), an electron-hole resonance effect which leads to extraordinary space charge field modulation and dramatically enhances the photorefractive effect has been confirmed [31–36]. The two-carriers band transport model has successfully described the enhanced two-wave mixing effect and temporal dynamic of space charge transport during the formation of spatial solitons [33,40]. In InP:Fe, temperature determines the resonance condition, which in CdZnTe:V is much easier to control by background illumination [31,32]. Here we develop a theoretical method to study incoherent spatial MI in CdZnTe:V, and we then discuss that how MI gain in CdZnTe:V influences spatial modes within the noisy images. In our simulation, the image is hidden by additive noise, and 1 µm and 1.5 µm light are both considered.
2. Theory
The photorefractive effect in CdZnTe:V (with deep levels V2+/V3+) can be described by the following one-dimensional two-carriers band transport model [33]
E is the space charge field, and the space variable x is parallel to the field direction. ND and NA are the shallow level densities of donors and acceptors respectively. n, jn, µn, Dn, and cn are the concentration, the current density, the charge mobility, the diffusion constant, and the capture coefficient of electrons, respectively, while p, jp, µp, Dp, and cp are those of holes. nT and pT are the densities of captured electrons and holes within the deep levels V2+ and V3+ respectively, and NT is the total vanadium density. e is the elementary charge and ɛ is the dielectric permittivity. Emission rate coefficients of electrons and holes en and ep can be defined as [33]
I0 and Ib are the intensities of the signal beam and the uniform background illumination perpendicular to the signal beam respectively. σn and σp are the photoionization cross sections of the signal beam, while ${\sigma ^{\prime}_n}$ and ${\sigma ^{\prime}_p}$ are those of the background beam. The sketch of the energy band for a clear illustration of the parameters is shown in Fig. 1.
Dark irradiance Id in CdZnTe:V less than 300 µW/cm2 suggests that the thermal emission rate eth is negligible compared to the optical emission if light intensity is large enough [29]. Carrier diffusion can be neglected in Eq. (1b) and (1c) when a large applied electric field is needed. Substituting Eq. (1d) and (1e) into (1f), we get a form at the steady state
which indicates that the total current density is constant. integrating with respect to x, it becomes here the 0 subscript identifies the boundary values, and E0 is the applied electric field. Hence, $n_{0}=e_{n 0} n_{T 0} / c_{n} p_{T 0}$ and $p_{0}=e_{p 0} p_{T 0} / c_{p} n_{T 0}$, where $e_{n 0}=\sigma_{n}^{\prime} I_{b}$, $e_{p 0}=\sigma_{p}^{\prime} I_{b}$, nT0 = ND − NA, and pT0 = NT − nT0.Assuming the spatial variations of the space charge field, space charge densities, and carrier densities are small enough, Eq. (1d) and (1e) yield
Substituting the expressions of n, p, n0, and p0 into Eq. (4), we find the saturable space charge field depending on I0 and Ib
andIf the electron-hole resonance condition is nearly met (i.e., eppT ≈ ennT) [33], Eq. (1f) reduces to
at the steady state. The combination of Eq. (8) and Eq. (4) yieldsTo simplify the system, spatial variations of nT, pT, and I0 are not considered in our derivation. Nonetheless, they are crucial if the distribution of the space charge field E(x) is concerned [40]. It can be confirmed that Esat establishes a refractive index distribution supporting self-focusing, while the gradient of the refractive index is lifted by Er. The simplified illustration is shown in Fig. 2. The refractive index distribution is given by n(x) = n0 − Δn(x) and $\Delta n(x) = (1/2){{n ^{3}_0}}\gamma_{41}E(x)$, when the signal beam is polarized along the [$1\bar{1}0$] direction of the crystal and propagates along [110], and the electric field is applied along [001]. Here n0 is the initial refractive index and γ41 is the electro-optic coefficient.
Fig. 2. Simplified illustration of refractive index distributions. The black dot line shows the intensity with a Gaussian distribution (or it could be replaced by any other shape that supports self-focusing). The green dash dot line represents the index distribution influenced by saturable nonlinearity under I(x) (such as in SBN crystals). The pink solid line represents the index distribution in CdZnTe:V: on the basis of saturable nonlinearity, the resonant space charge field creates a sharp index decay, which induces an increased index gradient for self-focusing.
MI gain of incoherent light can be derived from the first-order paraxial wave equation [2]
With reasonable parameters of CdZnTe:V crystals in Table 1 [27,30], MI gain as a function of transverse spatial frequencies for different external conditions are shown in Fig. 3. The signal light with a larger wavelength experiences stronger diffraction and weaker refractive index modulation within crystals, which causes lower gain (Fig. 3(a)). The strength of photorefractive nonlinearity is determined directly by an applied electric field (Fig. 3(b)), whereas incoherence represents the statistical dephasing and restricts MI (Fig. 3(c)). In ferroelectric photorefractive crystals, a background illumination is used to tune the saturation degree [1]. Specifically, MI gain in CdZnTe:V is dominated by I0 / Ir. Fig. 3(d) shows rapid growth of gain when I0 is close to Ir. This agrees with the coherent MI experiments in CdZnTe:V of Ref. [41], where gain is nearly quintupled as the signal intensity is increased from I0 = 0.16 mW/cm2 to 0.32 mW/cm2.
Fig. 3. MI gain coefficient vs transverse spatial frequency for different values of (a) the signal beam wavelength, (b) the applied electric field, (c) the spatial coherence length, and (d) the ratio of the signal intensity to the resonance intensity. Other parameters are fixed at (a) E0 = 6 kV/cm, I0 / Ir = 0.75, and lc = 100 µm; (b) λ0 = 1 µm, I0 / Ir = 0.75, and lc = 100 µm; (c) λ0 = 1 µm, E0 = 6 kV/cm, and I0 / Ir = 0.75; (d) λ0 = 1 µm, E0 = 6 kV/cm, and lc = 100 µm.
Table 1. Related parameters of CdZnTe:V
3. Results
Schematic diagram of optical image recovery based on MI is shown in Fig. 4. A dc applied electric field across the [001] direction of the CdZnTe:V crystal provides the crystal with photorefractive nonlinearity. A linearly polarized (polarization along [$1\bar{1}0$]) cw laser beam at 1064 nm (or 1550 nm) is split into two beams using a non-polarizing beam splitter. One of the beams is sent through a rotating diffuser to create incoherent noise light. The coherence of noise light can be controlled by changing the beam diameter onto the diffuser [1]. The other beam across a resolution chart acts as an original image signal. After adjusting the signal to noise intensity ratio by an attenuator, two beams are recombined and launched onto the (110) face of the crystal. The other cw laser beam at 1550 nm (or 1064 nm) propagating along [$1\bar{1}0$] is expanded and plays a role of uniform background illumination. The background beam is polarized along [001], which guarantees that the background beam cannot induce any index change. Photorefractive response time τ ∼ E / I should be controlled by background intensity to support incoherent MI. The output image is captured by a charge-coupled device (CCD) camera.
Fig. 4. Schematic diagram. An optical image is hidden by incoherent noise light. An applied voltage controls the nonlinear strength of the CdZnTe:V crystal, and the background illumination controls the resonance intensity. The output face of the crystal is imaged into a CCD camera.
As Fig. 3 shows, spatial MI causes different levels of amplification for different spatial frequencies k⊥. For an original image contains a set of spatial frequencies, the optimal performance occurs if the main spatial frequencies of the image fall into the range with the maximum MI gain. First, numerical results for λ0 = 1064 nm are compared in Fig. 5, and Fig. 6 reveals the matches between the spatial frequencies of resolution bars and the gain curves. The intensity ratio of the signal to noise is 0.1 on the input face; the periods of the bars are 100 µm (k⊥ = 0.063 µm-1) and 65 µm (k⊥ = 0.097 µm-1) respectively; other parameters are fixed at I0 / Ir = 0.75, lc = 100 µm, and the propagating distance of the signal beam in the crystal d = 7 mm. Since there is signal-noise coupling during the nonlinear process, the signal to noise intensity ratio is no longer a proper indicator for output quality. Therefore, we use the cross-correlation coefficient (0∼1) for image quality estimation
Fig. 5. Simulation results of the output images at 1064 nm for different values of the applied electric field. The signal to noise intensity ratio on the input face 0.1. The periods of the bars (b)–(e) 100 µm and (g)–(j) 65 µm. The area within the yellow box is used for cross-correlation calculations. Cross-correlation coefficients (f) 0.117, (b) 0.121, (c) 0.162, (d) 0.186, (e) 0.177, (g) 0.124, (h) 0.140, (i) 0.268, and (j) 0.292. Cross-correlation gains (b) 1.035, (c) 1.383, (d) 1.591, (e) 1.517, (g) 1.062, (h) 1.199, (i) 2.294, and (j) 2.501.
Fig. 6. MI gain coefficient vs transverse spatial frequency for different values of the applied electric field. The signal wavelength λ0 = 1064 nm. The dot line k⊥ = 0.063 µm−1 and the dash dot line k⊥ = 0.097 µm−1 correspond to the bars with a 100 µm period and a 65 µm period respectively.
Fig. 7. Cross-correlation coefficient vs applied electric field for different periods of the bars. The signal wavelength λ0 = 1064 nm and the signal to noise intensity ratio on the input face 0.1.
Figure 8 shows the dependence of image recovery performances on the signal to noise intensity ratio on the input face. According to the theory, growth of the signal comes from the pump of background noise. Therefore, effective gain could be limited if the energy of noise is not high enough, which reflects on the cross-correlation gain (Fig. 8(j)). On the contrary, gain is not reliant on the noise intensity if the signal intensity is weak enough, and the recovered images decline in quality with the decreasing signal intensity (Figs. 8(f)–(i)). However, better performances under a lower signal to noise intensity ratio would achieve provided certain approaches such as the temporal modulation of background illumination are used to improve gain [41]. The results at 1550 nm show the same trend as 1064 nm while the applied field is raised from E0 = 0 kV/cm to 15 kV/cm (Fig. 9). 1550 nm light is obviously more insensitive to the changes of field strength. Nevertheless, output images similar to 1064 nm are obtained with ∼1.8 times higher applied fields.
Fig. 8. Simulation results of the output images at 1064 nm for different signal to noise intensity ratios on the input face. The frequency of the bars k⊥ = 0.097 µm−1, (a)–(e) E0 = 0 kV/cm, and (f)–(j) E0 = 6.5 kV/cm. Cross-correlation coefficients (a) 0.047, (b) 0.082, (c) 0.117, (d) 0.228, (e) 0.331, (f) 0.112, (g) 0.193, (h) 0.268, (i) 0.466, and (j) 0.531. Cross-correlation gains (f) 2.371, (g) 2.359, (h) 2.294, (i) 2.039, and (j) 1.604.
Fig. 9. Simulation results of the output images at 1550 nm for different values of the applied electric field. The frequency of the bars k⊥ = 0.084 µm−1 and the signal to noise intensity ratio on the input face 0.1. Cross-correlation coefficients (a) 0.117, (b) 0.125, (c) 0.186, (d) 0.278, and (e) 0.308. Cross-correlation gains (b) 1.073, (c) 1.594, (d) 2.380, and (e) 2.636.
4. Discussion and conclusion
The ratio I0 / Ib at which electric-hole resonance occurs is related to nT and pT, which depend on ND and NA according to Eq. (1a). In particular, electric-hole resonance can barely happen if $n_T >> p_T$ or $p_T >> n_T$. Therefore, a varying doping of shallow and deep impurities could be the cause of the distinction between samples. However, doping with shallow impurities As (acceptors) or Cl (donors) could be a convenient way to control these parameters in CdZnTe:V [30]. For effective image recovery, parameters such as the applied field have to be adjusted with respect to the spatial frequencies of the original image. This causes the method difficult to work on a complex image with a wide spatial frequency band. Since the use of MI is a pure physical process, building a specific algorithm would be promising to address this limitation.
In conclusion, we have studied the effects of electrons and holes transport on the formation of spatial MI in semiconductor CdZnTe:V crystals, which is better for low power level applications at near-infrared than SBN crystals. Electron-hole resonance controlled by light intensity promotes MI by inducing an increased index gradient. After that, some examples of image recovery at 1 µm and 1.5 µm wavelengths in CdZnTe:V have been presented. On account of the mode competition, image quality depends on the matches between modes of the original image and MI gain as a function of the spatial frequency. In particular, a noisy image containing an original image on a smaller scale is able to achieve higher image quality under larger nonlinearity. The proposed method has potential applications for near-infrared imaging technologies such as LiDAR, night vision, and biomedical imaging.
Funding
National Natural Science Foundation of China (61775234, 61975232).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. D. Kip, M. Soljacic, M. Segev, E. Eugenieva, and D. N. Christodoulides, “Modulation Instability and Pattern Formation in Spatially Incoherent Light Beams,” Science 290(5491), 495–498 (2000). [CrossRef]
2. M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, “Modulation Instability of Incoherent Beams in Noninstantaneous Nonlinear Media,” Phys. Rev. Lett. 84(3), 467–470 (2000). [CrossRef]
3. M.-F. Shih, C.-C. Jeng, F.-W. Sheu, and C.-Y. Lin, “Spatiotemporal Optical Modulation Instability of Coherent Light in Noninstantaneous Nonlinear Media,” Phys. Rev. Lett. 88(13), 133902 (2002). [CrossRef]
4. M. Segev, M. Shih, and G. C. Valley, “Photorefractive screening solitons of high and low intensity,” J. Opt. Soc. Am. B 13(4), 706–718 (1996). [CrossRef]
5. J. Klinger, H. Martin, and Z. Chen, “Experiments on induced modulational instability of an incoherent optical beam,” Opt. Lett. 26(5), 271–273 (2001). [CrossRef]
6. X. Li, J. A. Greenberg, and M. E. Gehm, “Single-shot multispectral imaging through a thin scatterer,” Optica 6(7), 864–871 (2019). [CrossRef]
7. G. Barbastathis, A. Ozcan, and G. Situ, “On the use of deep learning for computational imaging,” Optica 6(8), 921–943 (2019). [CrossRef]
8. O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8(10), 784–790 (2014). [CrossRef]
9. X. Feng, H. Liu, N. Huang, Z. Wang, and Y. Zhang, “Reconstruction of noisy images via stochastic resonance in nematic liquid crystals,” Sci. Rep. 9(1), 3976 (2019). [CrossRef]
10. Y. Zhang, Z. Wang, N. Huang, and H. Liu, “Magneto-optically reorientation-induced image reconstruction in bulk nematic liquid crystals,” Opt. Express 29(11), 17581–17590 (2021). [CrossRef]
11. D. V. Dylov, L. Waller, and J. W. Fleischer, “Instability-driven recovery of diffused images,” Opt. Lett. 36(18), 3711–3713 (2011). [CrossRef]
12. Z. Wang, H. Liu, N. Huang, Y. Zhang, and J. Chi, “Nonlinear reconstruction of weak optical diffused images under turbid water,” Opt. Lett. 44(14), 3502–3505 (2019). [CrossRef]
13. Y. Zhang, H. Liu, N. Huang, and Z. Wang, “White-Light Image Reconstruction via Seeded Modulation Instability,” Phys. Rev. Appl. 12(5), 054005 (2019). [CrossRef]
14. D. V. Dylov and J. W. Fleischer, “Nonlinear self-filtering of noisy images via dynamical stochastic resonance,” Nat. Photonics 4(5), 323–328 (2010). [CrossRef]
15. S. Fauve and F. Heslot, “Stochastic resonance in a bistable system,” Phys. Lett. A 97(1-2), 5–7 (1983). [CrossRef]
16. F. Chapeau-Blondeau, “Input-output gains for signal in noise in stochastic resonance,” Phys. Lett. A 232(1-2), 41–48 (1997). [CrossRef]
17. H. Chen and P. K. Varshney, “Theory of the Stochastic Resonance Effect in Signal Detection—Part II: Variable Detectors,” IEEE Trans. Signal Process. 56(10), 5031–5041 (2008). [CrossRef]
18. Z. Shao, Z. Yin, H. Song, W. Liu, X. Li, J. Zhu, K. Biermann, L. L. Bonilla, H. T. Grahn, and Y. Zhang, “Fast Detection of a Weak Signal by a Stochastic Resonance Induced by a Coherence Resonance in an Excitable GaAs/Al_0.45Ga_0.55As Superlattice,” Phys. Rev. Lett. 121(8), 086806 (2018). [CrossRef]
19. C. Yan, Y. Zhang, and Z. Guo, “Recent progress on molecularly near-infrared fluorescent probes for chemotherapy and phototherapy,” Coord. Chem. Rev. 427, 213556 (2021). [CrossRef]
20. Y. Gu, Z. Guo, W. Yuan, M. Kong, Y. Liu, Y. Liu, Y. Gao, W. Feng, F. Wang, J. Zhou, D. Jin, and F. Li, “High-sensitivity imaging of time-domain near-infrared light transducer,” Nat. Photonics 13(8), 525–531 (2019). [CrossRef]
21. Z. Guo, M. Wang, A. A. Agyekum, J. Wu, Q. Chen, M. Zuo, H. R. El-Seedi, F. Tao, J. Shi, Q. Ouyang, and X. Zou, “Quantitative detection of apple watercore and soluble solids content by near infrared transmittance spectroscopy,” J. Food Eng. 279, 109955 (2020). [CrossRef]
22. K. Tuong Ly, R.-W. Chen-Cheng, H.-W. Lin, Y.-J. Shiau, S.-H. Liu, P.-T. Chou, C.-S. Tsao, Y.-C. Huang, and Y. Chi, “Near-infrared organic light-emitting diodes with very high external quantum efficiency and radiance,” Nat. Photonics 11(1), 63–68 (2017). [CrossRef]
23. A. Zampetti, A. Minotto, and F. Cacialli, “Near-Infrared (NIR) Organic Light-Emitting Diodes (OLEDs): Challenges and Opportunities,” Adv. Funct. Mater. 29(21), 1807623 (2019). [CrossRef]
24. M. T. Hasan, B. H. Lee, C.-W. Lin, A. McDonald-Boyer, R. Gonzalez-Rodriguez, S. Vasireddy, U. Tsedev, J. Coffer, A. M. Belcher, and A. v Naumov, “Near-infrared emitting graphene quantum dots synthesized from reduced graphene oxide for in vitro/in vivo/ex vivo bioimaging applications,” 2D Mater. 8(3), 035013 (2021). [CrossRef]
25. M. Wesner, C. Herden, D. Kip, E. Kratzig, and P. Moretti, “Photorefractive steady state solitons up to telecommunication wavelengths in planar SBN waveguides,” Opt. Commun. 188(1-4), 69–76 (2001). [CrossRef]
26. D. Wolfersberger and D. Tranca, “2D infrared self-focusing in bulk photorefractive SBN,” Opt. Mater. Express 1(7), 1178–1184 (2011). [CrossRef]
27. A. Zerrai, G. Marrakchi, and G. Bremond, “Electrical and optical characteristics of deep levels in vanadium-doped Cd0.96Zn0.04Te materials by photoinduced current, capacitance, and photocapacitance transient spectroscopies,” J. Appl. Phys. 87(9), 4293–4302 (2000). [CrossRef]
28. M. Tapiero, “Effect of zinc in codoped photorefractive CdTeZn:V,” in Photorefractive Fiber and Crystal Devices: Materials, Optical Properties, and Applications VIII, F. T. S. Yu and R. Guo, eds. (SPIE, 2002), 4803, pp. 293–300.
29. J.-Y. Moisan, N. Wolffer, O. Moine, P. Gravey, G. Martel, A. Aoudia, E. Repka, Y. Marfaing, and R. Triboulet, “Characterization of photorefractive CdTe:V: high two-wave mixing gain with an optimum low-frequency periodic external electric field,” J. Opt. Soc. Am. B 11(9), 1655–1667 (1994). [CrossRef]
30. L. A. de Montmorillon, P. Delaye, G. Roosen, H. B. Rjeily, F. Ramaz, B. Briat, J. G. Gies, J. P. Zielinger, M. Tapiero, H. J. von Bardeleben, T. Arnoux, and J. C. Launay, “Correlations between microscopic properties and the photorefractive response for vanadium-doped CdTe,” J. Opt. Soc. Am. B 13(10), 2341–2351 (1996). [CrossRef]
31. P. Pogany, H. J. Eichler, and M. H. Ali, “Two-wave mixing gain enhancement in photorefractive CdZnTe:V by optically stimulated electron–hole resonance,” J. Opt. Soc. Am. B 15(11), 2716–2720 (1998). [CrossRef]
32. T. Schwartz, Y. Ganor, T. Carmon, R. Uzdin, S. Shwartz, M. Segev, and U. El-Hanany, “Photorefractive solitons and light-induced resonance control in semiconductor CdZnTe,” Opt. Lett. 27(14), 1229–1231 (2002). [CrossRef]
33. G. Picoli, P. Gravey, C. Ozkul, and V. Vieux, “Theory of two-wave mixing gain enhancement in photorefractive InP:Fe: A new mechanism of resonance,” J. Appl. Phys. 66(8), 3798–3813 (1989). [CrossRef]
34. G. Picoli, P. Gravey, and C. Ozkul, “Model for resonant intensity dependence of photorefractive two-wave mixing in InP:Fe,” Opt. Lett. 14(24), 1362–1364 (1989). [CrossRef]
35. M. Chauvet, S. A. Hawkins, G. J. Salamo, M. Segev, D. F. Bliss, and G. Bryant, “Self-trapping of planar optical beams by use of the photorefractive effect in InP:Fe,” Opt. Lett. 21(17), 1333–1335 (1996). [CrossRef]
36. M. Chauvet, S. A. Hawkins, G. J. Salamo, M. Segev, D. F. Bliss, and G. Bryant, “Self-trapping of two-dimensional optical beams and light-induced waveguiding in photorefractive InP at telecommunication wavelengths,” Appl. Phys. Lett. 70(19), 2499–2501 (1997). [CrossRef]
37. R. Uzdin, M. Segev, and G. J. Salamo, “Theory of self-focusing in photorefractive InP,” Opt. Lett. 26(20), 1547–1549 (2001). [CrossRef]
38. N. Fressengeas, N. Khelfaoui, C. Dan, D. Wolfersberger, G. Montemezzani, H. Leblond, and M. Chauvet, “Roles of resonance and dark irradiance for infrared photorefractive self-focusing and solitons in bipolar InP:Fe,” Phys. Rev. A 75(6), 063834 (2007). [CrossRef]
39. D. Wolfersberger, N. Khelfaoui, C. Dan, N. Fressengeas, and H. Leblond, “Fast photorefractive self-focusing in InP:Fe semiconductor at infrared wavelengths,” Appl. Phys. Lett. 92(2), 021106 (2008). [CrossRef]
40. F. Devaux and M. Chauvet, “Three-dimensional numerical model of the dynamics of photorefractive beam self-focusing in InP:Fe,” Phys. Rev. A 79(3), 033823 (2009). [CrossRef]
41. S. Shwartz, M. Segev, E. Zolotoyabko, and U. El-Hanany, “Spatial modulation instability driven by light-enhanced nonlinearities in semiconductor CdZnTe:V crystals,” Appl. Phys. Lett. 93(10), 101116 (2008). [CrossRef]
