Nonparaxial Propagation of Vectorial Elliptical Gaussian Beams

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 α and waist width related parameter f ω has been analyzed. Results show that no matter what value of elliptic coefficient α is, when parameter f ω is large, nonparaxial conclusions of elliptical Gaussian beam should be adopted; while parameter f ω 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 f ω is large or small, and the larger the elliptic coefficient α is, the faster the peak intensity value decreases. These characteristics of vectorial elliptical Gaussian beam might find applications in modern optics.


Introduction
With the development of laser technology, research on semiconductor lasers [1], microoptical technologies [2][3][4], and highly focusing field [5][6][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light wavelength.In this case, the theory of optical propagation and transformation based on paraxial approximation is no longer valid [8], and it needs strict electromagnetic field theory to solve the problem of beam's nonparaxial propagation.In recent decades, several research methods about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][15][16][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.

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 by where r 0 =  0 i +  0 j and i and j are the unit vectors in  and  directions, respectively. 0 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  > 0 turns out to be [9] where r = i + j + k and k denotes the unit vector in  direction. ,, (, , ) are components of the  vector along , , and  directions in an arbitrary plane , respectively, where  = 2/ is the wave number and  is the incident wavelength.When |r − r 0 | ≫ , |r − r 0 | can be approximately expanded into [19]     r − r 0     ≈  + So (3) can be expressed as where  = ( 2 +  2 +  2 ) 1/2 .Substituting (5) into (2a)-(2c), we obtain Substituting ( 1) into (6a), we can obtain By utilizing the following integral formula [20] (7) can be expressed as follows: with  and  being given by Similarly, 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 expansion Accordingly, (7) can be reduced to where Equations ( 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.
International Journal of Optics

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  = 0 is set to 1.The propagation distance is normalized to /  , where   =  2 / 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   = 0.01-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  = 0 to  = 20  , while for large value   = 0.5-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  = 10  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   = 0.1,   is very small and can be neglected; hence, the curves of total intensity 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  = 1.5 in the plane  = 10  , 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 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.

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.
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.