Vector Rayleigh Diffraction of High-Power Laser Diode Beam in Optical Communication

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. 'e 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. 'is model is precise for single transverse model beam of LDs and can be applied to describe the LD beams in OWC and optical networks.

However, the output beam quality of LDs is relatively poor, such as astigmatism, high beam asymmetry, and large beam divergence [17][18][19], in many applications, and proper theoretical models are needed to precisely describe the optical field distribution of LDs. e 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. e rigid optical field distributions cannot be calculated from the paraxial approximation, and the longitudinal component in beam propagation direction should be considered. us, the vector theory for nonparaxial beams should be used to precisely describe beam fields of LDs. Several models, such as exponential Gaussian function [20][21][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][28][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. us, a novel model should be developed to precisely describe the output field of LDs, which is the aim of this paper.

Vectorial Electric Field of LD Beam
Considering that transverse electric modes are usually excited in LD, E(x, y) is identified with the component E y (x, y) 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, (1/p) � 1.22(λ n /d x ) × 10 −3 and (1/q) � 1.22(λ n /d y ) × 10 −3 , in which λ n 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 E 0 is a constant.
Beam propagation is governed by the vector Rayleigh diffraction integrals that provide the field expression in the entire half-space z > 0. When the boundary condition at the plane z � 0 is given, the field takes the following form [24,26]: where r 0 � x 0 i + y 0 j (r 0 is the vector in beam output plane), r � xi + yj + zκ (r is the beam propagation vector), i, j, κ are the unit vectors in the x-, y-, and z-directions, respectively, and in which k is the wavenumber related to wavelength λ by k � 2π/λ. Substituting equation (4) into equation (3) yields [24,26] We expand exp(ik|r − r 0 |) into a series, keeping the first-, second-, and third-order series expansions [31]: where r � ���������� x 2 + y 2 + z 2 , replace exp(ik|r − r 0 |) in equation (5) by equation (6), and replace |r − r 0 | in equation (5) by r: For large k (10 4 mm − 1 ), exp[−ik((xx 0 + yy 0 )/r) + ik(((y 2 + x 2 )x 2 0 + (x 2 + z 2 )y 2 0 − 2xyx 0 y 0 )/2r 3 )] rapidly oscillates, and such rapid oscillations over the range of integration indicate that the integrand averages to approximately zero, except near the stationary phase. us, the stationary phase method is used to find the asymptotic expansion of the diffraction integral. e corresponding diffraction integral is approximated by [32] 2 International Journal of Optics x s ,y s < 0 , where x s and y s are the stationary phase points, and we have Letting we find the stationary phase points and (z 2 g/zx 2 0 ) � ((y 2 + z 2 )/r 3 ), (z 2 g/zy 2 0 ) � ((x 2 + z 2 )/r 3 ), and (z 2 g/zx 0 zy 0 ) � (xy/r 3 ). us, and Substituting equations (13)-(16) into equation (7) yields Equation (17) represents the expression of vector theory for nonparaxial LD beam. e intensity profiles can be given by International Journal of Optics 3 and the total intensity can be expressed as I(x, y, z) � I x (x, y, z) + I y (x, y, z) + I z (x, y, z) e 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:

Experimental Procedure
e experiments were performed to examine the theoretical results using three high-power LDs (USHIO HL63391DC, TOSHIBA TOLD9441MC, and USHIO HL63290HD). e 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 (LSGSPD-UL0.25, 0.25 mm visible light PIN photodiode, wavelength 500-880 nm, and 0.25 mm active diameter) behind the hole. e 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. e 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.    e 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. e 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. us, 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.
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 E y (x 0 , y 0 ) � E 0 exp(−p|x 0 | (3/2) − qy 2 0 ) in this article is more precise for single transverse model beam. For the calculation of the vector Rayleigh diffraction integrals, we expand exp(ik|r − r 0 |) into a series by keeping the first, second, and third series expansions. e calculations make the diffraction integral of beam distribution with large divergence more reliable compared with the first two expansions in the article [24,26]. International Journal of Optics

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. e mathematical model provided a good fit to the experimental data of single transverse model beam of LDs. is mathematical model can be used to describe the beam propagation and shape of LDs in OWC and optical networks.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.  International Journal of Optics