#### 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 far-field 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, LD-enabled optical wireless communication (OWC) is an emerging technology for realizing high-confidentiality and high-speed point-to-point (PtP), vehicle-to-vehicle, and white-lighting data access links in free-space 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 half-space . 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 *z*-directions, 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 third-order 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 high-power 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 *μ*m) and a photodiode (LSGSPD-UL0.25, 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 low-intensity 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.

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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 high-power LDs was proposed, and the beam parameters were related to the structure of LDs’ waveguide. High-order 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 high-power 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).