# Nonparaxial Propagation of Vectorial Elliptical Gaussian Beams

Authors: Wang Xun; Huang Kelin; Liu Zhirong; Zhao Kangyi
Journal: International Journal of Optics
 (2016)

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## 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 coefficientα 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.


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## Body

## 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 In this case, the theory of optical propagation and transformation based on paraxial approximation is no longer valid [8], and it needs strict electromagnetic field theory to solve the problem of beam’s nonparaxial propagation.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 α and different waist width related parameter f
ω, and the relationships of elliptical Gaussian beam’s nonparaxial effect and its intensity distributions with elliptic coefficient α as well as parameter f
ω 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 thex direction and can be defined by(1)
E
x
r
0
,
0
E
y
r
0
,
0
=
E
x
x
0
,
y
0
,
0
0
=
E
0
exp
⁡
-
x
0
2
+
α
y
0
2
ω
2
0
,where r
0
=
x
0
i
+
y
0
j and i and j are the unit vectors in x and y directions, respectively. E
0 is a constant, ω is the waist width, and α is elliptic coefficient, which denotes the ratio of elliptical Gaussian beam’s waist width in x and y directions.According to the vectorial Rayleigh-Sommerfeld diffraction integral formulae, the nonparaxial propagation of light beam in the half-spacez
>
0 turns out to be [9](2a)
E
x
x
,
y
,
z
=
-
1
2
π
∬
-
∞
∞
E
x
x
0
,
y
0
,
0
∂
R
r
,
r
0
∂
z
d
x
0
d
y
0
,
(2b)
E
y
x
,
y
,
z
=
-
1
2
π
∬
-
∞
∞
E
y
x
0
,
y
0
,
0
∂
R
r
,
r
0
∂
z
d
x
0
d
y
0
,
(2c)
E
z
x
,
y
,
z
=
1
2
π
∬
-
∞
∞
E
x
x
0
,
y
0
,
0
∂
R
r
,
r
0
∂
x
+
E
y
x
0
,
y
0
,
0
d
x
0
d
y
0
∂
R
r
,
r
0
∂
y
,where r
=
x
i
+
y
j
+
z
k and k denotes the unit vector in z direction. E
x
,
y
,
z
(
x
,
y
,
z
) are components of the E vector along x, y, and z directions in an arbitrary plane z, respectively,(3)
R
r
,
r
0
=
exp
⁡
i
k
r
-
r
0
r
-
r
0
,where k
=
2
π
/
λ is the wave number and λ is the incident wavelength. When r
-
r
0
≫
λ, r
-
r
0 can be approximately expanded into [19](4)
r
-
r
0
≈
r
+
x
0
2
+
y
0
2
-
2
x
x
0
-
2
y
y
0
2
r
.So (3) can be expressed as(5)
R
r
,
r
0
=
1
r
exp
⁡
i
k
r
+
x
0
2
+
y
0
2
-
2
x
x
0
-
2
y
y
0
2
r
,where r
=
(
x
2
+
y
2
+
z
2
)
1
/
2
.Substituting (5) into (2a)–(2c), we obtain(6a)
E
x
x
,
y
,
z
=
-
i
z
λ
r
2
∬
-
∞
∞
E
x
x
0
,
y
0
,
0
exp
⁡
i
k
r
+
r
0
2
-
2
x
x
0
-
2
y
0
y
2
r
d
x
0
d
y
0
,
(6b)
E
y
x
,
y
,
z
=
-
i
z
λ
r
2
∬
-
∞
∞
E
y
x
0
,
y
0
,
0
exp
⁡
i
k
r
+
r
0
2
-
2
x
x
0
-
2
y
0
y
2
r
d
x
0
d
y
0
,
(6c)
E
z
x
,
y
,
z
=
i
λ
r
2
∬
-
∞
∞
x
-
x
0
E
x
x
0
,
y
0
,
0
+
y
-
y
0
E
y
x
0
,
y
0
,
0
exp
⁡
i
k
r
+
r
0
2
-
2
x
x
0
-
2
y
0
y
2
r
d
x
0
d
y
0
.Substituting (1) into (6a), we can obtain(7)
E
x
x
,
y
,
z
=
-
i
z
E
0
λ
r
2
exp
⁡
i
k
r
∫
-
∞
∞
exp
⁡
-
x
0
2
ω
0
2
+
i
k
x
0
2
2
r
-
i
k
x
x
0
r
d
x
0
∫
-
∞
∞
exp
⁡
-
α
2
y
0
2
ω
0
2
+
i
k
y
0
2
2
r
-
i
k
y
y
0
r
d
y
0
.By utilizing the following integral formula [20](8)
∫
-
∞
∞
x
n
exp
⁡
-
μ
x
2
+
2
υ
x
d
x
=
n
!
π
μ
υ
μ
n
exp
⁡
υ
2
μ
∑
s
=
0
n
/
2
1
n
-
2
s
!
s
!
μ
4
υ
2
s
,
R
e
μ
>
0
, (7) can be expressed as follows:(9)
E
x
x
,
y
,
z
=
i
E
0
π
z
λ
r
2
p
q
exp
⁡
i
k
r
exp
⁡
k
2
4
r
2
x
2
p
+
y
2
q
,with p and q being given by(10a)
p
=
k
2
f
ω
2
-
i
k
2
r
,
(10b)
q
=
α
2
k
2
f
ω
2
-
i
k
2
r
,
(10c)
f
ω
=
1
k
ω
.Similarly, substituting (1) into (6b) and (6c), and recalling integral formula (8), we can obtain other elements of the elliptical Gaussian beam:(11)
E
y
x
,
y
,
z
=
0
,
(12)
E
z
x
,
y
,
z
=
i
E
0
π
x
λ
r
2
p
q
·
1
+
i
k
2
p
r
exp
⁡
i
k
r
exp
⁡
-
k
2
4
r
2
x
2
p
+
y
2
q
.The intensity distribution of nonparaxial propagation of the elliptical Gaussian beam at the point (
x
,
y
,
z
) can be expressed as follows:(13)
I
x
,
y
,
z
=
I
x
x
,
y
,
z
+
I
y
x
,
y
,
z
+
I
z
x
,
y
,
z
=
E
x
x
,
y
,
z
2
+
E
y
x
,
y
,
z
2
+
E
z
x
,
y
,
z
2
,where I
x
(
x
,
y
,
z
), I
y
(
x
,
y
,
z
), and I
z
(
x
,
y
,
z
) are the intensity distributions of the x, y, and z 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(14)
r
≈
z
+
x
2
+
y
2
2
z
.Accordingly, (7) can be reduced to(15)
E
p
x
,
y
,
z
=
i
E
0
π
λ
z
p
′
q
′
exp
⁡
i
k
z
exp
⁡
i
k
x
2
+
y
2
2
z
exp
⁡
k
2
4
z
2
x
2
p
′
+
y
2
q
′
,where(16a)
p
′
=
k
2
f
ω
2
-
i
k
2
z
,
(16b)
q
′
=
α
2
k
2
f
ω
2
-
i
k
2
z
.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.


## 3. Numerical Simulations and Analysis

In order to confirm the relationship of elliptical Gaussian beam’s intensity distribution and nonparaxial effects with elliptic coefficientα as well as parameter f
ω, 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 z
=
0 is set to 1. The propagation distance is normalized to z
/
z
r, where z
r
=
π
ω
2
/
λ 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 coefficientsα in different observation planes is depicted in Figures 1 and 2, which correspond to two different waist width related parameters f
ω, respectively. From Figure 1, for small value f
ω
=
0.01—that is, elliptical Gaussian beam’s waist width ω 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 z
=
0 to z
=
20
z
r, while for large value f
ω
=
0.5—that is, elliptical Gaussian beam’s waist width ω is small (see Figure 2)—we can find that, with the increase of propagation distance z, 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 α is, the faster the peak intensity value decreases, no matter whether f
ω is large or small.Figure 1
Intensity distribution of elliptical Gaussian beam forf
ω
=
0.01 as a function of x in free space in different planes: (a) z
=
0; (b) z
=
z
r; (c) z
=
5
z
r; (d) z
=
10
z
r; (e) z
=
20
z
r. Several curves correspond to different elliptic coefficient α: α
=
0.8 (dashed curve); α
=
1 (solid curve); α
=
1.5 (dotted curve).Figure 2
Intensity distribution of elliptical Gaussian beam forf
ω
=
0.5 as a function of x in free space in different planes: (a) z
=
0; (b) z
=
z
r; (c) z
=
5
z
r; (d) z
=
10
z
r; (e) z
=
20
z
r. Several curves correspond to different elliptic coefficient α: α
=
0.8 (dashed curve); α
=
1 (solid curve); α
=
1.5 (dotted curve).Figure3 gives the intensity distributions of elliptical Gaussian beam in the plane z
=
10
z
r for different parameter f
ω. The elliptic coefficient α is fixed to 0.8, 1, and 1.5 from the first row to the third row, respectively. The corresponding longitudinal component I
z of nonparaxial elliptical Gaussian beam and paraxial result I
p 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 α is, for small value of f
ω
=
0.1, I
z is very small and can be neglected; hence, the curves of total intensity distribution I and corresponding paraxial result I
p are almost coincident. While the parameters f
ω are increased to 0.3 (see Figures 3(b1)–3(b3)), I
z becomes strong, so I and I
p began to show slight difference. When f
ω is further increased to 0.5, I
z becomes more strong, and the difference between I and I
p increased obviously (see Figures 3(c1)–3(c3)). As a result, no matter what the value of α is, the nonparaxial conclusions of the elliptical Gaussian beam should be considered when f
ω is large. Conversely, the paraxial approximation of elliptical Gaussian beam is valid when f
ω is small. Furthermore, the light peak intensity value will decrease with increasing the elliptic coefficient α in the same observation plane, no matter what value of f
ω is. However, the larger the value of parameters f
ω is, the smaller the spot size of beam is, no matter what value of elliptic coefficient α is.Figure 3
Intensity distributionsI
(
x
,
y
,
z
), I
z
(
x
,
y
,
z
), and I
p
(
x
,
y
,
z
) of elliptical Gaussian beam in the plane z
=
10
z
r for different parameters f
ω and elliptic coefficient α: (a1) f
ω
=
0.1, α
=
0.8; (b1) f
ω
=
0.3, α
=
0.8; (c1) f
ω
=
0.5, α
=
0.8; (a2) f
ω
=
0.1, α
=
1; (b2) f
ω
=
0.3, α
=
1; (c2) f
ω
=
0.5, α
=
1; (a3) f
ω
=
0.1, α
=
1.5; (b3) f
ω
=
0.3, α
=
1.5; (c3) f
ω
=
0.5, α
=
1.5.Figure4 shows the contour graphs of intensity distributions I
x, I
z, and I of nonparaxial elliptical Gaussian beams for elliptic coefficient α
=
1.5 in the plane z
=
10
z
r, and the corresponding paraxial result I
p is also given in Figure 4. The parameter f
ω 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 f
ω is small, the longitudinal component I
z is very small and can be neglected; hence the beam profiles of total intensity distribution I and corresponding paraxial result I
p are visibly similar. Figures 4(a1)–4(a4) also show that I
z can be neglected, and the paraxial approximation is valid when f
ω is small. However, when f
ω is chosen as 0.3 (see Figures 4(b1)–4(b4)), I
z becomes strong, and the spots of I and I
p show a little distinction. As f
ω is further increased to 0.5, I and I
p show obvious difference (see Figures 4(c1)–4(c4)). In other words, the contribution of the longitudinal component I
z would become significant, and the nonparaxial conclusions of elliptical Gaussian beam should be adopted when f
ω is large.Figure 4
Contour graphs of intensity distributionsI
x
x
,
y
,
z
,
I
z
(
x
,
y
,
z
), I
(
x
,
y
,
z
), and I
p
(
x
,
y
,
z
) of elliptical Gaussian beam with elliptic coefficient α
=
1.5 in the plane z
=
10
z
r for different parameters f
ω: (a1)–(a4) f
ω
=
0.1; (b1)–(b4) f
ω
=
0.3; (c1)–(c4) f
ω
=
0
.
5.


## 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 coefficientα and different waist width related parameter f
ω is illustrated by numerical examples. Results show that, with increasing propagation distance z, all contours of the transverse cross sections of nonparaxial propagation of the elliptical Gaussian beams preserve Gaussian type when f
ω 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 f
ω is large. Meanwhile, whether parameter f
ω is large or small, the peak intensity value decreased with increasing the propagation distance, and the larger the elliptic coefficient α is, the faster the peak intensity value decreases. In addition, numerical results also show that no matter what value of elliptic coefficient α is, when parameter f
ω is small, the paraxial approximation of elliptical Gaussian beam is effective; when parameter f
ω 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.



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*Source: 6427141-2016-02-18.xml*