Research Article  Open Access
Qiang Xu, Renxian Li, Yuanyuan Zhang, Yiping Han, Zhensen Wu, "Vector Rayleigh Diffraction of HighPower Laser Diode Beam in Optical Communication", International Journal of Optics, vol. 2020, Article ID 7059638, 7 pages, 2020. https://doi.org/10.1155/2020/7059638
Vector Rayleigh Diffraction of HighPower Laser Diode Beam in Optical Communication
Abstract
Laser diodes (LDs) are widely used in optical wireless communication (OWC) and optical networks, and proper theoretical models are needed to precisely describe the complicated beam field of LDs. A novel mathematical model is proposed to describe the vectorial field of nonparaxial LD beams. Laser beam propagation is studied using the vector Rayleigh diffraction integrals, and the stationary phase method is used to find the asymptotic expansion of diffraction integral. The farfield distribution of the LD beam in the plane parallel and perpendicular to the junction is considered in detail, and the computed intensity distributions of the theory are compared with the corresponding measurements. This model is precise for single transverse model beam of LDs and can be applied to describe the LD beams in OWC and optical networks.
1. Introduction
Considering that laser diodes (LDs) are the efficient light source and easy to integrate, LDenabled optical wireless communication (OWC) is an emerging technology for realizing highconfidentiality and highspeed pointtopoint (PtP), vehicletovehicle, and whitelighting data access links in freespace communication [1–6], indoor communication [7, 8], underwater communication [9–11], and optical networks [12–16].
However, the output beam quality of LDs is relatively poor, such as astigmatism, high beam asymmetry, and large beam divergence [17–19], in many applications, and proper theoretical models are needed to precisely describe the optical field distribution of LDs.
The problem of laser propagation is mainly dealt through paraxial approximation. However, the output facet of LDs is extremely small, and their beams are divergent and asymmetrical. The rigid optical field distributions cannot be calculated from the paraxial approximation, and the longitudinal component in beam propagation direction should be considered. Thus, the vector theory for nonparaxial beams should be used to precisely describe beam fields of LDs. Several models, such as exponential Gaussian function [20–22], Hermite–Gaussian model [23], nonparaxial diffraction of vectorial Gaussian wave [24, 25], plane waves with a small aperture [26], propagation of LD beams in the optical system [27–29], and polarization of LD beams [30], are used to describe the beam field of LDs. However, no theoretical model is used for all cases because of the complicated beam field of LDs. Thus, a novel model should be developed to precisely describe the output field of LDs, which is the aim of this paper.
2. Vectorial Electric Field of LD Beam
Considering that transverse electric modes are usually excited in LD, is identified with the component of the electric field vector, and a source beam is linearly polarized at the plane z = 0:where p and q are related to the waveguide structure of LDs, and , in which is the beam wavelength in the active layer of LDs, d_{x} is the waveguide width in the x direction, and d_{y} is the waveguide width in the y direction, and is a constant.
Beam propagation is governed by the vector Rayleigh diffraction integrals that provide the field expression in the entire halfspace . When the boundary condition at the plane z = 0 is given, the field takes the following form [24, 26]:where ( is the vector in beam output plane), ( is the beam propagation vector), are the unit vectors in the x, y, and zdirections, respectively, andin which k is the wavenumber related to wavelength by . Substituting equation (4) into equation (3) yields [24, 26]
We expand into a series, keeping the first, second, and thirdorder series expansions [31]:where , replace in equation (5) by equation (6), and replace in equation (5) by :
For large k (), rapidly oscillates, and such rapid oscillations over the range of integration indicate that the integrand averages to approximately zero, except near the stationary phase. Thus, the stationary phase method is used to find the asymptotic expansion of the diffraction integral.
The corresponding diffraction integral is approximated by [32]
for , and for .wherewhere and are the stationary phase points, and we have
Lettingwe find the stationary phase pointsand , , and .
Thus,and ,
Substituting equations (13)–(16) into equation (7) yields
Equation (17) represents the expression of vector theory for nonparaxial LD beam.
The intensity profiles can be given byand the total intensity can be expressed as
The intensity of LD beams can be investigated in two vertical planes. In the plane perpendicular to the junction (i.e., y = 0), as shown in Figure 1, the substitution of y = 0 into equation (19) yields
In the plane parallel to the junction (i.e., x = 0), as shown in Figure 1, the total intensity can be expressed as follows:
3. Experimental Procedure
The experiments were performed to examine the theoretical results using three highpower LDs (USHIO HL63391DC, TOSHIBA TOLD9441MC, and USHIO HL63290HD). The parameters are shown in Table 1.

As shown in Figure 2, the intensity profiles of laser beam were measured through a pinhole scan (radius is 100 μm) and a photodiode (LSGSPDUL0.25, 0.25 mm visible light PIN photodiode, wavelength 500–880 nm, and 0.25 mm active diameter) behind the hole. The photodiode moved along the straight lines parallel to the output facet of the LDs’ chip in the x–z and y–z planes, where z = 50 mm. The uncertainty of measurements is less than 1%.
Figure 3 shows the measurements and theoretical beam profiles of HL63391DC, and the intensity curve of the theory agrees with the experimental data in most portions. Figure 4 shows the light intensity profiles of TOLD9441MC. The calculated profiles agree well with the experimental data in most portions, except for the discrepancies in the lowintensity value regions in the x–z plane. The theoretical curve agrees well with the experimental data in the y–z plane. Figure 5 shows the light intensity profiles of HL63290HD, and the discrepancies of theory and measurement of this LD are greater than those of HL63391DC and TOLD9441MC because only single transverse mode exists in HL63391DC and TOLD9441MC, whereas multitransverse modes exist in HL63290HD. Thus, the output light field of the latter is more complicated, the shape intensity of two lobes appears in the y–z plane, and the theoretical curve does not fit the measurement.
(a)
(b)
(a)
(b)
(a)
(b)
Compared with the previous models of LD beam, including Hermite–Gaussian model [23], Gaussian model [25, 27, 28], elliptical Gaussian model [24, 29], and negative exponential Gaussian model [22, 30], the novel output model in this article is more precise for single transverse model beam. For the calculation of the vector Rayleigh diffraction integrals, we expand into a series by keeping the first, second, and third series expansions. The calculations make the diffraction integral of beam distribution with large divergence more reliable compared with the first two expansions in the article [24, 26].
4. Conclusion
A novel theoretical model for the nonparaxial vectorial field of highpower LDs was proposed, and the beam parameters were related to the structure of LDs’ waveguide. Highorder approximations of the diffraction integral were calculated on the basis of the vector Rayleigh diffraction integrals, the fields parallel and perpendicular to beam propagation direction were considered, and the beam intensities of three highpower LDs beam were measured. The mathematical model provided a good fit to the experimental data of single transverse model beam of LDs. This mathematical model can be used to describe the beam propagation and shape of LDs in OWC and optical networks.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (61975158) and Aeronautical Science Foundation of China (20180181).
References
 C.T. Tsai, C.H. Cheng, H.C. Kuo, and G.R. Lin, “Toward highspeed visible laser lighting based optical wireless communications,” Progress in Quantum Electronics, vol. 67, Article ID 100225, 2019. View at: Publisher Site  Google Scholar
 C. Shen, J. A. HolguinLerma, A. A. Alatawi et al., “GroupIIInitride superluminescent diodes for solidstate lighting and highspeed visible light communications,” IEEE Journal of Selected Topics in Quantum Electronics, vol. 25, no. 6, pp. 1–10, 2019. View at: Publisher Site  Google Scholar
 J. Kosman, O. Almer, T. A. Abbas et al., “29.7 A 500 Mb/s46.1 dBm CMOS SPAD receiver for laser diode visiblelight communications,” in Proceedings of the IEEE International SolidState Circuits Conference, pp. 468–470, San Francisco, CA, USA, 2019 February. View at: Publisher Site  Google Scholar
 M. H. M. Shamim, M. A. Shemis, C. Shen et al., “Analysis of optical injection on red and blue laser diodes for high bitrate visible light communication,” Optics Communications, vol. 449, pp. 79–85, 2019. View at: Publisher Site  Google Scholar
 S. Gwyn, S. Watson, S. Viola et al., “GaNbased distributed feedback laser diodes for optical communications,” in Proceedings of SPIEThe International Society for Optical Engineering, Lisbon, Portugal, 2019. View at: Google Scholar
 H.Y. Wang, Y.C. Chi, and G.R. Lin, “Dualmode laser diode carrier with orthogonal polarization and singlemode modulation for remotenode heterodyne MMWRoF,” Optics Letters, vol. 41, no. 20, pp. 4676–4679, 2016. View at: Publisher Site  Google Scholar
 C. Li, X. Zhang, E. Tangdiongga et al., “Costefficient halfduplex 10 Gbit/s alloptical indoor optical wireless communication enabled by a lowcost Fabry–Perot laser/photodetector,” Optics Letters, vol. 44, no. 5, pp. 1158–1161, 2019. View at: Publisher Site  Google Scholar
 J. Fakidis, S. Videv, S. Kucera, H. Claussen, and H. Haas, “Indoor optical wireless power transfer to small cells at nighttime,” Journal of Lightwave Technology, vol. 34, no. 13, pp. 3236–3258, 2016. View at: Publisher Site  Google Scholar
 Y. Chen, M. Kong, T. Ali et al., “26 m/5.5 Gbps airwater optical wireless communication based on an OFDMmodulated 520nm laser diode,” Optics Express, vol. 25, no. 13, pp. 14760–14765, 2017. View at: Publisher Site  Google Scholar
 J. Wang, C. Lu, S. Li, and Z. Xu, “100 m/500 Mbps underwater optical wireless communication using an NRZOOK modulated 520 nm laser diode,” Opt. Express, vol. 27, no. 9, pp. 12171–12181, 2019. View at: Publisher Site  Google Scholar
 X. Liu, S. Yi, X. Zhou et al., “345 m underwater optical wireless communication with 270 Gbps data rate based on a green laser diode with NRZOOK modulation,” Optics Express, vol. 25, no. 22, p. 27937, 2017. View at: Publisher Site  Google Scholar
 W.S. Tsai, H.H. Lu, H.W. Wu et al., “500 Gb/s PAM4 FSOUWOC convergent system with a R/G/B fivewavelength polarizationmultiplexing scheme,” IEEE Access, vol. 8, pp. 16913–16921, 2020. View at: Publisher Site  Google Scholar
 A. Majumdar, C. Mandal, and S. Gangopadhyay, “Laser diode to singlemode circular core parabolic index fiber coupling via upsidedown tapered hyperbolic microlens on the tip of the fiber: prediction of coupling optics by abcd matrix formalism,” Journal of Optical Communications, vol. 40, no. 3, pp. 171–180, 2019. View at: Publisher Site  Google Scholar
 Fiber optic active components and devices, Performance standards, Part 3, Modulatorintegrated laser diode transmitters for 40Gbit/s fiber optic transmission systems, 1–16, 2018.
 S. Watson, S. Gwyn, S. Viola et al., “InGaN/GaN laser diodes and their applications,” in Proceedings of the 2018 20th International Conference on Transparent Optical Networks, Bucharest, Romania, 2018 July. View at: Google Scholar
 N. Kim, M. Park, S. An, T.S. Kim, W. S. Han, and O.K. Kwon, “25 Gbps electroabsorption modulated dfb laser diodes for digital fronthaul network,” in Proceedings of the 2018 23rd OptoElectronics and Communications Conference, Jeju Island, Republic of Korea, July 2018. View at: Publisher Site  Google Scholar
 K. Lei, X. Qin, H. Liu, and M. Ni, “Analysis and modeling of melt pool morphology for high power diode laser cladding with a rectangle beam spot,” Optics & Lasers in Engineering, vol. 110, pp. 89–99, 2018. View at: Publisher Site  Google Scholar
 M. U. Hoque, “Enhancement of optical fiber coupling based on collimated and focused laser diode beam with micro aspherical planoconvex lens fabricated by excimer laser,” Fiber & Integrated Optics, vol. 37, no. 5, pp. 264–276, 2018. View at: Publisher Site  Google Scholar
 H. Tan, H. Meng, X. Ruan, W. Du, and Z. Wang, “Highpower direct diode laser output by spectral beam combining,” Laser Physics, vol. 28, no. 3, Article ID 035802, 2018. View at: Publisher Site  Google Scholar
 X. Zeng and A. Naqwi, “Farfield distribution of doubleheterostructure diode laser beams,” Applied Optics, vol. 32, no. 24, pp. 4491–4500, 1993. View at: Publisher Site  Google Scholar
 S. Nemoto, “Experimental evaluation of a new expression for the far field of a diode laser beam,” Applied Optics, vol. 33, no. 27, pp. 6387–6390, 1994. View at: Publisher Site  Google Scholar
 H. Dong, S. Shi, and S. Chen, “Analysis of error yielded by scalar approximation to the properties of laser diode beams,” Applied Optics, vol. 45, no. 21, pp. 5160–5163, 2006. View at: Publisher Site  Google Scholar
 Z. Zhao, K. Duan, and B. Lü, “Nonequiphase Hermite–Gaussian model of diode laser beams,” Optik, vol. 119, no. 4, pp. 167–170, 2008. View at: Publisher Site  Google Scholar
 K. Duan and B. Lü, “Propagation properties of vectorial elliptical Gaussian beams beyond the paraxial approximation,” Optics & Laser Technology, vol. 36, no. 6, pp. 489–496, 2004. View at: Publisher Site  Google Scholar
 X. Zeng, C. Liang, and Y. An, “Farfield propagation of an offaxis Gaussian wave,” Applied Optics, vol. 38, no. 30, pp. 6253–6256, 1999. View at: Publisher Site  Google Scholar
 K. Duan and B. Lü, “Nonparaxial diffraction of vectorial plane waves at a small aperture,” Optics & Laser Technology, vol. 37, no. 3, pp. 193–197, 2005. View at: Publisher Site  Google Scholar
 M. R. H. Rad, F. D. Kashani, M. M. Eftekhari, and M. R. Mahzoun, “Characterizing the divergence properties of the laser diode beams propagation through collimator and aperture ABCD optical system,” Optics & Laser Technology, vol. 42, no. 8, pp. 1269–1275, 2010. View at: Publisher Site  Google Scholar
 F. Kashani, M. R. H. Rad, Z. Firozzadeh, and M. R. Mahzoun, “Beam propagation analysis of a multilaser diode FSO system through free space,” Journal of Optics, vol. 13, no. 10, Article ID 105709, 2011. View at: Publisher Site  Google Scholar
 H. Sun, “Simple mathematical model for designing laser diode focusing optics with a large numerical aperture,” Optical Engineering, vol. 53, no. 10, pp. 105105.1–105105.7, 2014. View at: Publisher Site  Google Scholar
 Q. Xu, J. Wang, Y. Han, and Z. Wu, “Vectorial analytical description of the polarized light of a highpower laser diode,” Applied Optics, vol. 52, no. 8, pp. 1711–1715, 2013. View at: Publisher Site  Google Scholar
 A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Optics Communications, vol. 202, no. 1–3, pp. 17–20, 2002. View at: Publisher Site  Google Scholar
 J. Stamnes, “Waves, rays, and the method of stationary phase,” Optics Express, vol. 10, no. 16, pp. 740–751, 2002. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2020 Qiang Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.