Abstract
Based on the vectorial Rayleigh-Sommerfeld diffraction integral formulae, analytical expressions for a vectorial elliptical Gaussian beam’s nonparaxial propagating in free space are derived and used to investigate target beam’s propagation properties. As a special case of nonparaxial propagation, the target beam’s paraxial propagation has also been examined. The relationship of vectorial elliptical Gaussian beam’s intensity distribution and nonparaxial effect with elliptic coefficient and waist width related parameter has been analyzed. Results show that no matter what value of elliptic coefficient is, when parameter is large, nonparaxial conclusions of elliptical Gaussian beam should be adopted; while parameter is small, the paraxial approximation of elliptical Gaussian beam is effective. In addition, the peak intensity value of elliptical Gaussian beam decreases with increasing the propagation distance whether parameter is large or small, and the larger the elliptic coefficient is, the faster the peak intensity value decreases. These characteristics of vectorial elliptical Gaussian beam might find applications in modern optics.
1. Introduction
With the development of laser technology, research on semiconductor lasers [1], microoptical technologies [2–4], and highly focusing field [5–7] has become deeper. In practical application, the problem that would be confronted is of a beam with large divergence angle or small spot size that is of the order of light wavelength. In this case, the theory of optical propagation and transformation based on paraxial approximation is no longer valid [8], and it needs strict electromagnetic field theory to solve the problem of beam’s nonparaxial propagation. In recent decades, several research methods about solving beam’s nonparaxial propagation have been developed, such as vectorial Rayleigh-Sommerfeld diffraction integral method [9], perturbation power series method [10], transition operators [11], angular spectrum representation [12], and virtual source point technique [13]. And vectorial Rayleigh-Sommerfeld diffraction method has been used to treat various beam’s nonparaxial propagation problems [14–17].
An elliptical Gaussian beam can be radiated and realized by semiconductor diode laser [18]. In the past few years, some nonparaxial propagation properties of vectorial elliptical Gaussian beams have been reported, such as the far-field beam divergence angle [19], diffracted at a circular and a rectangular aperture [20, 21]. Since the semiconductor laser beam has a large divergence angle, it would become necessary to consider the target beam’s nonparaxial propagation. In this work, we use the vectorial Rayleigh-Sommerfeld diffraction integral formulae to solve the nonparaxial propagation of a vectorial elliptical Gaussian beam. Target beam’s nonparaxial propagation analytical expressions are derived and used to investigate its propagation properties, including the evolution of intensity and shape of elliptical Gaussian beam with different elliptic coefficient and different waist width related parameter , and the relationships of elliptical Gaussian beam’s nonparaxial effect and its intensity distributions with elliptic coefficient as well as parameter are analyzed.
2. Nonparaxial Propagation of Vectorial Elliptical Gaussian Beams in Free Space
Let us consider the incident field of elliptical Gaussian beam, which is polarized in the direction and can be defined bywhere and and are the unit vectors in and directions, respectively. is a constant, is the waist width, and is elliptic coefficient, which denotes the ratio of elliptical Gaussian beam’s waist width in and directions.
According to the vectorial Rayleigh-Sommerfeld diffraction integral formulae, the nonparaxial propagation of light beam in the half-space turns out to be [9]where and denotes the unit vector in direction. are components of the vector along , , and directions in an arbitrary plane , respectively,where is the wave number and is the incident wavelength. When , can be approximately expanded into [19]So (3) can be expressed aswhere
Substituting (5) into (2a)–(2c), we obtainSubstituting (1) into (6a), we can obtainBy utilizing the following integral formula [20] (7) can be expressed as follows:with and being given bySimilarly, substituting (1) into (6b) and (6c), and recalling integral formula (8), we can obtain other elements of the elliptical Gaussian beam:The intensity distribution of nonparaxial propagation of the elliptical Gaussian beam at the point can be expressed as follows:where , , and are the intensity distributions of the , , and components of the field, respectively.
The paraxial propagation of elliptical Gaussian beam can be dealt with as a special case by using the paraxial expansionAccordingly, (7) can be reduced towhereEquations (9) and (12) are the main analytical results for elliptical Gaussian beam’s nonparaxial propagating in free space, and (15) is the paraxial analytical formula for elliptical Gaussian beam’s paraxial propagating in free space.
3. Numerical Simulations and Analysis
In order to confirm the relationship of elliptical Gaussian beam’s intensity distribution and nonparaxial effects with elliptic coefficient as well as parameter , according to the analytical expressions obtained above, we have carried out the numerical simulations of intensity distributions of vectorial elliptical Gaussian beam’s nonparaxial propagating in free space. For the convenience of comparison, the light peak intensity in the input plane is set to 1. The propagation distance is normalized to , where is the Rayleigh distance, and the incident wavelength is 632.8 nm.
The evolution behavior of intensity distributions of nonparaxial elliptical Gaussian beams with several elliptic coefficients in different observation planes is depicted in Figures 1 and 2, which correspond to two different waist width related parameters , respectively. From Figure 1, for small value —that is, elliptical Gaussian beam’s waist width is large—one can see that all the normalized intensity distributions of nonparaxial elliptical Gaussian beams would preserve Gaussian type when the propagation distance ranges from to , while for large value —that is, elliptical Gaussian beam’s waist width is small (see Figure 2)—we can find that, with the increase of propagation distance , the transverse intensity profiles turn into Gaussian-like shape quickly. Besides, numerical results also show that the peak intensity value decreases when the propagation distance increases, and the larger the value of elliptic coefficient is, the faster the peak intensity value decreases, no matter whether is large or small.
Figure 3 gives the intensity distributions of elliptical Gaussian beam in the plane for different parameter . The elliptic coefficient is fixed to 0.8, 1, and 1.5 from the first row to the third row, respectively. The corresponding longitudinal component of nonparaxial elliptical Gaussian beam and paraxial result of elliptical Gaussian beam are also depicted together for comparison. From Figures 3(a1)–3(a3), one can see that no matter what value of the elliptic coefficient is, for small value of , is very small and can be neglected; hence, the curves of total intensity distribution and corresponding paraxial result are almost coincident. While the parameters are increased to 0.3 (see Figures 3(b1)–3(b3)), becomes strong, so and began to show slight difference. When is further increased to 0.5, becomes more strong, and the difference between and increased obviously (see Figures 3(c1)–3(c3)). As a result, no matter what the value of is, the nonparaxial conclusions of the elliptical Gaussian beam should be considered when is large. Conversely, the paraxial approximation of elliptical Gaussian beam is valid when is small. Furthermore, the light peak intensity value will decrease with increasing the elliptic coefficient in the same observation plane, no matter what value of is. However, the larger the value of parameters is, the smaller the spot size of beam is, no matter what value of elliptic coefficient is.
Figure 4 shows the contour graphs of intensity distributions , , and of nonparaxial elliptical Gaussian beams for elliptic coefficient in the plane , and the corresponding paraxial result is also given in Figure 4. The parameter is chosen as 0.1, 0.3, and 0.5 from the first row to the third row, respectively. As shown in Figure 3, when is small, the longitudinal component is very small and can be neglected; hence the beam profiles of total intensity distribution and corresponding paraxial result are visibly similar. Figures 4(a1)–4(a4) also show that can be neglected, and the paraxial approximation is valid when is small. However, when is chosen as 0.3 (see Figures 4(b1)–4(b4)), becomes strong, and the spots of and show a little distinction. As is further increased to 0.5, and show obvious difference (see Figures 4(c1)–4(c4)). In other words, the contribution of the longitudinal component would become significant, and the nonparaxial conclusions of elliptical Gaussian beam should be adopted when is large.
4. Conclusions
In this paper, based on the vectorial Rayleigh-Sommerfeld diffraction integral formulae, we have derived the analytical expressions for a vectorial elliptical Gaussian beam’s nonparaxial propagating in free space, and the paraxial approximation expression has also been examined as a special case. The evolution of the beam’s intensity and shape with different elliptic coefficient and different waist width related parameter is illustrated by numerical examples. Results show that, with increasing propagation distance , all contours of the transverse cross sections of nonparaxial propagation of the elliptical Gaussian beams preserve Gaussian type when is small, while all contours of the transverse cross sections of nonparaxial propagation of the elliptical Gaussian beams would change to Gaussian-like type when is large. Meanwhile, whether parameter is large or small, the peak intensity value decreased with increasing the propagation distance, and the larger the elliptic coefficient is, the faster the peak intensity value decreases. In addition, numerical results also show that no matter what value of elliptic coefficient is, when parameter is small, the paraxial approximation of elliptical Gaussian beam is effective; when parameter is large, the nonparaxial conclusions of the elliptical Gaussian beam should be adopted. These characteristics of vectorial elliptical Gaussian beam might find applications in modern optics.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the National Science Foundation of China (11547002 and 11447235), Jiangxi Provincial Natural Science Foundation of China (20142BAB212003), and China Scholarship Council 201508360027.