Hollow sinh-Gaussian beams and their paraxial properties

Qiongge Sun; Keya Zhou; Guangyu Fang; Guoqiang Zhang; Zhengjun Liu; Shutian Liu

doi:10.1364/OE.20.009682

1. Introduction

Beams with dark-hollow characteristic have always received intensive attention and extensive investigation [1,2]. The high order Bessel beams are famous examples, and various efficient methods to generate and control the Bessel beams have been proposed [3–11]. As an alternative model of dark-hollow beams, the mathematical expression of hollow Gaussian beams was proposed and studied [12–14]. Recently, the Hypergeometric-Gaussian (HyGG) beams which present single ringed intensity distributions have been experimentally generated, which are the eigenstates of the photo orbital angular momentum [15–20]. Because of their inherent characteristic, these dark-hollow beams have a large range of potential applications in atom guiding, trapping and focusing [21–28].

As a special case of Hermite-sinusoidal-Gaussian beams [29–32], the propagation properties of conventional sinh-Gaussian (CsG) beams and pulses in Kerr medium were reported recently [33,34]. In this article, we introduce new sinh-Gaussian beams called hollow sinh-Gaussian beams which are different from CsG beams. This new beams can be used as mathematical model to describe dark-hollow beams, and are characterized by single ringed transverse intensity distributions. Their intensity pattern presents a central dark hollow surrounded by one bright ring whose size can be controlled by the order of beams. Based on the Collins integral, analytical propagation equation of HsG beams through an ABCD optical system have been derived. In special case, the beams can be regarded as combination of a series of special HyGG beams, LG modes and MLG modes. By using different values of beam width in $x$ and $y$ directions, the elliptical HsG beams have also been introduced. Numerical simulations of HsG beams and elliptical HsG beams in free space propagation have been performed by using the Fresnel transform. Their propagation characteristics have also been illustrated graphically.

2. Hollow sinh-Gaussian beams

We define the electric field of the HsG beams at the original plane of $z = 0$ as follows:

E_{n} (r, 0) = \sinh^{n} (\frac{r}{ω}) \exp (- \frac{r^{2}}{ω^{2}}),

where

n

(

n = 0, 1, 2, \dots

) stands for the order of the HsG beams. Obviously, the hyperbolic sine function in HsG beams is associated with the variable

r

instead of

x

or

y

in CsG beams, which leads to the key evolution of hollow property. If

n = 0

, the above equation is reduced to the fundamental Gaussian beam with beam waist of

ω

. If

n = 1

, the above equation is a new kind of sinh-Gaussian beams, the intensity distribution of this new beams is different from that of the CsG beams, as we know that the latter has spots intensity pattern, but the intensity distribution of new HsG beams is a single bright ring. By using the Fresnel transform of Eq. (1), this characteristic has been presented. In Fig. 1(a) , the first order of HsG beams has a dark-hollow intensity distribution at the original plane.

Fig. 1 Normalized intensity distributions of HsG beams in free space propagation at different distances. (a)-(d) z = 0, (e)-(h) z = z_R. The parameters are λ = 632.8nm, ω = 1mm and z = 4.965m.

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The electric field of the HsG beams at the original plane can be rewritten in the following form

E_{n} (r, 0) = \sum_{m = 0}^{n} a_{m} b_{m} \exp [\frac{- {(r + c_{m})}^{2}}{ω^{2}}],

with the coefficients given by

a_{m} = {(- 1)}^{m} 2^{- n} (\begin{matrix} n \\ m \end{matrix}), b_{m} = \exp [{(m - \frac{n}{2})}^{2}], c_{m} = ω (m - \frac{n}{2}),

which implies the

n

-th order HsG beam can be produced simply by superposition of several decentered Gaussian beams with the same waist width

ω

, whose centers are located at the positions

(- c_{m}, 0)

.

In the framework of the paraxial approximation, the propagation of the laser beams through the paraxial ABCD optical system can be treated by the Collins integral formula. In a cylindrical coordinate system, the electric field in the output plane can be described as

U (r, z) = \frac{i}{λ B} \exp (- i k z) \int_{0}^{2 π} \int_{0}^{\infty} \exp {- \frac{i k}{2 B} [A r'^{2} - 2 r r' \cos (θ - θ') + D r^{2}]} U_{0} (r', 0) r' d r' d θ',

where

k = 2 π / λ

is the wave number and

λ

is the wavelength,

(r, θ)

and

(r', θ')

stand for the radial and azimuthal angle coordinates in input and output planes, respectively.

Substituting Eq. (2) into Eq. (4) and applying the integral formula of the Bessel function

J_{0} (t) = \frac{1}{2 π} \int_{0}^{2 π} \exp (i t \cos φ) d φ,

we have

\begin{matrix} E_{n} (r, z) = \frac{i}{λ B} \exp (- i k z) \exp (- \frac{i k D r^{2}}{2 B}) \sum_{m = 0}^{n} a_{m} b_{m} \int_{0}^{\infty} \exp (- \frac{i k A r'^{2}}{2 B}) \\ \times J_{0} (\frac{k r r'}{B}) \exp [\frac{- {(r' + c_{m})}^{2}}{ω^{2}}] r' d r' . \end{matrix}

We expand the exponential part into Taylor series, and use the integral formula of the hypergeometric Kummer function to evaluate the beams. When $Re (μ + ν) > 0$ and $Re (a^{2}) > 0$ , the integral formula will be of the following form [35]

\begin{array}{l} \int_{0}^{\infty} t^{μ} \exp (- a^{2} x^{2}) J_{ν} (p t) d t \\ = Γ (\frac{μ + v + 1}{2}) \frac{p^{ν}}{2^{v + 1} a^{μ + v + 1} Γ (ν + 1)} {}_{1}F_{1} (\frac{μ + ν + 1}{2}, ν + 1; - \frac{p^{2}}{4 a^{2}}), \end{array}

where

{}_{1}F_{1} (a, b; x)

denotes the confluent hypergeometric function and

Γ (x)

is the gamma function, respectively. After recalling the recurrent relations for Kummer function and employing the straightforward integration, we obtain

\begin{matrix} E_{n} (r, z) = \frac{i ω^{2}}{2 λ B} \exp (- i k z) \exp (\frac{- i k r^{2}}{2 q}) \sum_{m= 0}^{n} \sum_{s = 0}^{\infty} a_{m} \frac{{(n - 2 m)}^{s}}{s!} Γ (1 + s / 2) \\ \times {(\frac{B / q_{0}}{A + B / q_{0}})}^{1 + \frac{s}{2}} {}_{1}F_{1} (- \frac{s}{2}, 1; \frac{- i k r^{2}}{2 B (A + B / q_{0})}), \end{matrix}

where

q_{0} = i k ω^{2} / 2

is beam

q

parameter at original plane

z = 0

. On the output plane at the distance

z

,

q

becomes

q = \frac{A + B / q_{0}}{C + D / q_{0}},

χ

is beam size parameter at output plane taking the following form

χ^{2} = 2 B (A + B / q_{0}) .

Equation (8) is the general transformation and propagation formula for HsG beams through the paraxial ABCD optical system, which will provide a convenient way to treat the transformation and propagation of HsG beams.

We continue to study the intensity distribution properties of HsG beams in the free-space propagation. The transfer matrix for free space of distance $z$ writes as

[\begin{matrix} A & B \\ C & D \end{matrix}] = [\begin{matrix} 1 & z \\ 0 & 1 \end{matrix}] .

Substituting the above relation into Eq. (8), we obtain the field distribution of the HsG beams in free space at different propagation distance as follows:

E_{n} (r, z) = \frac{1}{2 π} \exp (- i k z) \sum_{s = 0}^{\infty} {| HyGG 〉}_{s β (β = 0)}^{n},

where

{| HyGG 〉}_{s β (β = 0)}^{n} = \sum_{m = 0}^{n} a_{m} \frac{{(n - 2 m)}^{s}}{s!} {| HyGG 〉}_{s β (β = 0)} .

Equation (12) stands for the combination of a series of hypergeometric-Gaussian beams, which can be described as [16]

\begin{matrix} {| HyGG 〉}_{s β (β = 0)} = Γ (1 + s / 2) {(1 + z / q_{0})}^{- 1 - s / 2} {(z / q_{0})}^{s / 2} \\ \times \exp [\frac{- r^{2}}{ω^{2} (1 + z / q_{0})}] {}_{1}F_{1} (- \frac{s}{2}, 1; \frac{- r^{2}}{ω^{2} (1 + z / q_{0}) z / q_{0}}) . \end{matrix}

Obviously, ${| HyGG 〉}_{s β (β = 0)}$ presents a series of hypergeometric-Gaussian beam with the parameters $β = 0$ and $s \geq 0$ . We can conclude that the HsG beams can be expressed as the linear superposition of a series of HyGG beams with $β = 0$ under the condition of $r \neq 0$ . At the beam center $r = 0$ where the Kummer function ${}_{1}F_{1} (a, b; 0) = 1$ , we can still expect small intensity distribution on axis, by considering the coefficients inside the summation sign in Eq. (8). The coefficient $a_{m}$ presents symmetry values with opposite sign, so does the $n - 2 m$ to the given power of $s$ . When $n$ is a big number, which means more modes are involved, the infinite summation sign of $s$ will take effect and lead to weaken the intensity on axis. In order to demonstrate this characteristic, the Fresnel transform of Eq. (1) has been implemented graphically in Fig. 2 . At the original plane, the HsG beams have no intensity on optical axis. In Fig. 2(b), as the beams propagate apart from the plane of $z = 0$ and arrive at $z = 2 z_{R}$ , the beam with an order of $n = 3$ has on-axis intensity, but the higher order of beams still hold their single ringed intensity distribution without change. This is because higher order of HsG beams means the superposition of more different modes, which can enhance the single ringed characteristic and weaken the on-axis intensity in initial propagation.

Fig. 2 Normalized radial intensity distributions of HsG beams with different orders in free space propagation at two different distances. (a) z = 0, (b) z = 2z_R. The parameters are λ = 632.8nm, ω = 1mm and z_R = 4.965m.

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When $s$ is a nonnegative integer, the HyGG beam can be expressed in the complete basis of LG modes [16], that is ${| HyGG 〉}_{s β (β = 0)} = \sum_{p = 0}^{\infty} A_{s p} {| LG 〉}_{p β (β = 0)}$ , where $A_{s p}$ is superposition coefficients. Obviously, the HsG beams can also be expanded as combination of a series of ${| LG 〉}_{p β (β = 0)}$ modes. For special cases, when $s = 0$ , as HyGG beam reduces to ${TEM}_{00}$ Gassian mode [16], the HsG beams can be described as superposition of ${TEM}_{00}$ Gaussian modes and therefore have the on-axis intensity. Moreover, when $s$ is a nonnegative even integer, because the HyGG beam can further reduce to the modified LG (MLG) mode whose intensity pattern at original plane presents a single ringed characteristic [16], the HsG beams can also be expressed as superposition of MLG modes. Actually, the HsG beams can be treated as the fractional order of hollow Gaussian beams and realized by spatial filtering in the fractional Fourier transform domain.

3. Numerical results and analyses of the HsG beams

The numerical simulation of HsG beams has been made by Fresnel transform with the same parameters, $λ = 632.8 nm$ and $ω = 1 mm$ . $z_{R} = 4.965 m$ is the Rayleigh distance. The normalized intensity distributions of the HsG beams at $z = 0$ and $z = z_{R}$ planes have been respectively presented in Fig. 1. We can conclude from Figs. 1(a)-1(d) that this new beams are characterized by a single ringed intensity pattern, and the radius of the ring increases with the order of beams. We can see from Figs. 1(e)-1(h) that the seventh and eighth order HsG beams preserve their dark-hollow intensity pattern, but the first and the fourth order HsG beams lose their single ringed profiles because of the Fresnel diffraction.

The same characteristic has also been demonstrated in Fig. 2, which presents the normalized two-dimensional intensity distribution of the HsG beams in free space propagation at $z = 0$ and $z = 2 z_{R}$ planes. For example, when $n = 6$ , the intensity pattern of the beams is one bright ring surrounding a central bright spot, but when $n = 9$ , it maintains its single ringed intensity pattern. This means the higher order beams are more likely to remain their dark-hollow pattern in the free space propagation with Fresnel diffraction. The lower order beams always lose their original intensity pattern after the free space propagation, and then become the bright ring surrounding the central bright spot. From the Fig. 2(a), we can also conclude that the radius of the bright ring increases in a nearly uniform way, so does the radius of the hollow area. Moreover, the width of the bright ring will become wider in the propagation as long as the higher order beams preserve its intensity pattern.

In order to compare the propagation properties of beams with different orders, in Fig. 3 , three-dimensional simulations of the HsG beams have been made. We adopt $n = 5$ as the lower order and $n = 10$ as the higher order for comparison. In the Figs. 3(a)-3(c) where the order is relatively lower, the beam cannot preserve its original intensity pattern in long distance propagation as discussed above. In Figs. 3(d)-3(f), when the order is higher, the beam can maintain its single ringed intensity distribution in a long propagation distance beyond ten meters, and then the intensity distribution of this beam presents several bright rings surrounding the central bright spot. This characteristic of higher order HsG beams also means that with the increase of the order, the propagation distance preserving the single ringed intensity pattern will increase.

Fig. 3 Comparison of the beam propagation properties with different orders in free space. (a)-(c) n = 5, (d)-(f) n = 10. The parameters are λ = 632.8nm, ω = 1mm and z_R = 4.965m.

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In Fig. 4 , the dynamic simulation of HsG beams in free space propagation has been presented. For both the lower order and comparatively higher order beams, their intensity patterns gradually become the central maximum after propagation in free space, which is to say, the single ringed intensity pattern disappears, and the on-axis intensity becomes maximal that presents a central bright spot. This is because the HsG beam is not a pure mode but can be treated as the combination of a series of the HyGG beams (or LG modes). These different modes evolve differently with respect to distance, and they overlap and interfere in comparatively long distance propagation, thus causing this central bright spot intensity pattern. Higher order of HsG beam can preserve single ring intensity distribution in long propagation distance without focusing, while the relatively low order of HsG beams is more likely to lose this intensity distribution in near field.

Fig. 4 Dynamic simulations of HsG beams in free space propagation. (a) n = 3 (Media 1), (b) n = 10 (Media 2). The parameters are λ = 632.8nm, ω = 1mm, z_R = 4.965m.

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4. Numerical results and analyses of HsG beams with elliptical symmetries

We define the electric field of the elliptical HsG beams at the original plane of $z = 0$ in Cartesian coordinate system as follow:

E_{n} (x, y, 0) = \sinh^{n} [{(\frac{x^{2}}{ω_{x}^{2}} + \frac{y^{2}}{ω_{y}^{2}})}^{1 / 2}] \exp (- \frac{x^{2}}{ω_{x}^{2}} - \frac{y^{2}}{ω_{y}^{2}}),

where

ω_{j}

with

j = x

or

y

denotes the beam width at

x

or

y

direction. The above equation represents the elliptical HsG beams at the original plane, and clearly reveals the elliptical HsG beams have elliptical dark-hollow intensity patterns at the original plane. Comparing the Eq. (1) and Eq. (15), the previously defined radial coordinate system in Eq. (1) is converted to a Cartesian one. By using the relation

r^{2} = x^{2} / ω_{x}^{2} + y^{2} / ω_{y}^{2}

with the required scale factor

p = ω_{y} / ω_{x}

, the Eq. (15) will be obtained. Then we can conclude the elliptical HsG beams can be realized by spatial filtering of HsG beams with coordinate conversion and scaling elements.

By using the same method and parameters as the circular HsG beams, we study the propagation properties of elliptical HsG beams in free space. The two-dimensional intensity distributions of elliptical HsG beams at the distance of $z = 0$ and $z = z_{R}$ are shown in Fig. 5 , beam width pairs are $ω_{x} = 1 mm$ , $ω_{y} = 1.5 mm$ and $ω_{x} = 1.5 mm$ , $ω_{y} = 1 mm$ , respectively. As analyzed above, in Figs. 5(a)-5(d) the elliptical HsG beams with different orders obviously present the dark-hollow intensity pattern with elliptical symmetries at the original plane. In Figs. 5(e)-5(h) they propagate to the plane $z = z_{R}$ in free space and still possess the elliptical dark-hollow intensity pattern.

Fig. 5 Comparison of the elliptical beam propagation properties for different values of $n$ and $ω$ at two different distances. (a)-(d) z = 0, (e)-(h) z = z_R. The parameters are λ = 632.8nm and z_R = 4.965m.

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In Fig. 6 , we present the three-dimensional propagation of elliptical HsG beams with order of $n = 4$ and $n = 10$ in free space. Some properties are similar to those of the circular HsG beams. First, at the original plane, the beams still present elliptical dark hollow intensity profiles. Second, after the beams propagate a long distance in free space, their intensity distributions become anomalous, and eventually change to the central maximum intensity profiles. Moreover, the higher order beams seem to have more disturbances in their intensity distribution. This is also because the HsG beam is not the pure mode, and higher order of the beam leads to more HyGG beams (or LG modes) which are the combination of the beam, these different modes overlap and interfere in beam propagation, thus causing complex intensity pattern. Furthermore, the dynamic simulations of the free space propagation of elliptical HsG beams ( $n = 2$ and $n = 7$ ) have been given in Fig. 7 to demonstrate the above analyses.

Fig. 6 Normalized three-dimensional intensity distributions of the elliptical HsG beams in free space propagation at different distances. (a)-(c) n = 4, ω_x = 1.2mm, ω_y = 1.2mm. (d)-(f) n = 10, ω_x = 1.2mm, ω_y = 1mm. The parameters are λ = 632.8nm and z_R = 4.965m.

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Fig. 7 Dynamic simulations of elliptical HsG beams in free space propagation. (a) n = 2, ω_x = 1mm, ω_y = 1.5mm (Media 3), (b) n = 7, ω_x = 1.5mm, ω_y = 1mm (Media 4). The parameters are λ = 632.8nm and z_R = 4.965m.

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Conclusion

In conclusion, we have introduced a new model of dark-hollow beams named hollow sinh-Gaussian beams which are characterized by bright single ringed intensity distribution along propagation. The radius of the bright ring can be controlled by the order of HsG beams, the size of the ring increases as $n$ increases in a nearly uniform way at the original plane, which means the area of dark region in intensity distribution of beams increases with order $n$ . The shape of the bright ring can be modulated by using different values of $ω_{x}$ and $ω_{y}$ , which confers HsG beams to elliptical HsG beams.

Based on the Collins integral, the analytical propagation equation of HsG beams through the paraxial ABCD optical system has been derived. The HsG beam is not pure mode but can be regarded as the combination of a series of HyGG beams with the same $β$ and any integer $s \geq 0$ . Moreover, when $s$ is a nonnegative integer, the HsG beams can also be treated as combination of a series of LG modes. For special cases, when $s = 0$ , the HsG beams can be described as the superposition of ${TEM}_{00}$ Gassian modes. When $s$ is a nonnegative even integer, the HsG beams will be expressed as superposition of a series of MLG modes.

Numerical stimulations have been made to represent the propagation characteristics of HsG beams and elliptical HsG beams in free space by means of Fresnel transform method. Because the beams are not pure mode, but the combination of different modes, these different modes evolve differently, and bring special characteristics to HsG beams. In propagation, higher order of beams can preserve single ringed intensity pattern in longer propagation distance; while the relatively low order of HsG beams is more likely to lose this intensity pattern, and their on-axis intensity becomes maximum. That is to say, the propagation distance of the beams that preserves single ringed intensity distribution increases with the order of beams. Higher order of beams means more modes involved, and these modes emphasize the single ringed intensity distribution of HsG beams. Moreover, after HsG beams propagate a long distance and lose their single ringed intensity pattern, higher order of beams seems to have more disturbances when the on-axis intensity becomes maximum, this is because more modes are involved, they evolve differently after propagating in a long distance, and eventually cause more disturbances. Due to their interesting properties, especially for their inherent central hollow region, the HsG beams can be served as an idea model of dark-hollow beams and have potential usage in atom guiding and trapping.

Acknowledgments

This work was supported by the National Basic Research Program of China (Grant No. 2011CB301801), the National Natural Science Foundation of China (Grant Nos. 10974039, 11047153, 10904027, 61008039, and 11104049), the Doctoral Program of Higher Education of China (Grant No. 20102302120009), and the Fundamental Research Funds for the Central Universities of China (Grant No.2009038).

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Hollow sinh-Gaussian beams and their paraxial properties

Abstract

1. Introduction

2. Hollow sinh-Gaussian beams

3. Numerical results and analyses of the HsG beams

4. Numerical results and analyses of HsG beams with elliptical symmetries

Conclusion

Acknowledgments

References and links

Supplementary Material (4)

Cited By

Figures (7)

Equations (15)

Optics Express