Far-field properties of electromagnetic elliptical Gaussian vortex beams

Optics Communications 283 (2010) 3578–3584

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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m

Far-field properties of electromagnetic elliptical Gaussian vortex beams Yamei Luo a,b,⁎, Baida Lü b a b

Department of Biomedical Engineering, Luzhou Medical College, Luzhou 646000, China Institute of Laser Physics and Chemistry, Sichuan University, Chengdu 610064, China

a r t i c l e

i n f o

Article history: Received 14 January 2010 Received in revised form 1 March 2010 Accepted 25 May 2010 Keywords: Electromagnetic elliptical Gaussian vortex beam Edge dislocation Energy flux distribution

a b s t r a c t The explicit far-field expressions for the TE and TM terms and energy flux distributions of electromagnetic elliptical Gaussian vortex beams are derived and their far-field properties including the phase singularities and energy flux distributions of the TE and TM terms and whole beam are studied in detail. It is shown that there exist two edge dislocations. The number and position of edge dislocations and energy flux distributions are dependent on the amplitude ratio and waist width ratio of electromagnetic elliptical Gaussian vortex beams. The analytical results are illustrated numerically. © 2010 Elsevier B.V. All rights reserved.

1. Introduction It is known that the general solution of Maxwell's equations can be expressed as a sum of the TE and TM terms by virtue of the vectorial angular spectrum representation, where the electric field of the TE term and the magnetic field of the TM term are transverse to the propagation axis, respectively [1–4]. In the far field the TE and TM terms are orthogonal to each other and separable. The isolated TE term may find applications in improving the density of optical storage [5]. The vectorial structure of a variety of beams, such as Gaussian beams, Laguerre–Gaussian beams, Hermite–Laguerre– Gaussian beams, hollow Gaussian beams and radially polarized beams etc. in the far field was studied analytically and numerically [5–12]. On the other hand, optical beams carrying phase singularities have attracted much interest from both fundamental theoretical and potential applicative aspects [13–24]. There exist the screw dislocation (vortex) with spiral phase, the edge dislocation with π-phase shift along a transverse plane, and the hybrid dislocation. The freespace propagation of an array of optical vortices embedded in a Gaussian beam was dealt with by Indebetouw who found that the relative position of vortices with same topological charges is invariant, whereas pairs of vortices with opposite charges attract and annihilate each other [14]. Maleev and Swartzlander described the motion of composite optical vortices in the superimposed field of two noncollinear Gaussian vortex beams and showed that the position of the resulting composite vortex can be controlled by

⁎ Corresponding author. Department of Biomedical Engineering, Luzhou Medical College, Luzhou 646000, China. Tel.: + 86 28 85412819; fax: + 86 28 85412322. E-mail addresses: [email protected] (Y. Luo), [email protected] (B. Lü). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.05.053

varying the relative phase or distance between the composing beams [21]. Petrov studied the interaction of a vortex and an edge dislocation in free-space propagation and observed the edge dislocation bending and break up into vortices [18,19]. Very recently, we showed that for the case of the vortex-edge dislocation interaction in the presence of an astigmatic lens the astigmatism additionally affects the vortex evolution [24]. This paper is devoted to the study of the far-field properties of electromagnetic elliptical Gaussian vortex beams. In Section 2, based on the vectorial angular spectrum representation and stationary phase method, the analytical far-field expressions for the TE and TM terms of electromagnetic elliptical Gaussian vortex beams are derived. The phase singularities of the electric x-component field of the TE term are analyzed in Section 3. Section 4 presents the energy flux distributions of the TE and TM terms and whole beam. The results are illustrated numerically and interpreted physically. In Section 5 the main results obtained in this paper are summarized, and a comparison with the previous work is made. 2. Theoretical model The electric field of an electromagnetic elliptical Gaussian vortex beam at the plane z = 0 reads as "

2

2

ðx−bÞ + y Ex ðx; y; 0Þ = E0x ½ðx−bÞ + iy exp − w20x

# ;

ð1aÞ

# ðx−bÞ2 + y2 ; Ey ðx; y; 0Þ = E0y ½ðx−bÞ + iy exp − w20y

ð1bÞ

"

Y. Luo, B. Lü / Optics Communications 283 (2010) 3578–3584

3579

Fig. 1. Contour lines of phase of the x-component ETEx (a) η = 0, (b) η = 0.8, (c) η = 1.6, (d) η = − 0.8, and (e) η = − 1.6. The calculation parameters are seen in the text.

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where w0x, w0y are waist widths and E0x, E0y are amplitude constants in the x and y directions, respectively, b denotes the off-axis distance in the x direction. Performing the Fourier transform, the vectorial angular spectrum of the field at z = 0 is expressed as 1 ∬ Ex ðx; y; 0Þ exp½−ikðpx + qyÞdxdy; λ2 ∞

ð2aÞ

1 Ay ðp; qÞ = 2 ∬ Ey ðx; y; 0Þ exp½−ikðpx + qyÞdxdy; λ ∞

ð2bÞ

Ax ðp; qÞ =

Ay ðp; qÞ =

  i E0x kπw 40x ðq−ipÞ kh 2 2 2 i4bp + kw ; ð3aÞ exp − p + q 0x 4 2λ2 E0y kπw 40y ðq−ipÞ 2λ2

ETM ðrÞ = ∬ ∞

1 2 2 ½pAx ðp; qÞ + qAy ðp; qÞ½pγi + qγj−ðp + q Þk γðp2 + q2 Þ

× exp½ikðpx + qy + γzÞdpdq;

HTM ðrÞ = −

where λ denotes the wave length related to the wave number by λ = 2π / k. The substitution from Eqs. (1a), (1b) into Eqs. (2a), (2b) yields Ax ðp; qÞ =

and

rffiffiffi ε 1 ½pAx ðp; qÞ + qAy ðx; yÞ½qi−pj ∬ μ ∞ γðp2 + q2 Þ

ð6aÞ

ð6bÞ

× exp½ikðpx + qy + γzÞdpdq;

r = xi + yj + zk with i, j, k being unit vectors in the x, y, z directions, respectively, γ = (1 − p2 − q2)1/2, H is the magnetic field vector, ε and μ denote the electric permittivity and the magnetic permeability of the medium, respectively.

  i kh 2 2 2 : ð3bÞ exp − i4bp + kw0y p + q 4

According to the vectorial structure of electromagnetic beams, an arbitrary electromagnetic field is composed of TE and TM terms, i.e., [10] EðrÞ = ETE ðrÞ + ETM ðrÞ;

ð4aÞ

HðrÞ = HTE ðrÞ + HTM ðrÞ;

ð4bÞ

where ETE ðrÞ = ∬ ∞

1 ½qAx ðp; qÞ−pAy ðp; qÞðqi−pjÞ exp½ikðpx + qy + γzÞdpdq; p 2 + q2

ð5aÞ HTE ðrÞ =

rffiffiffi ε 1 2 2 ½qAx ðp; qÞ−pAy ðp; qÞ½pγi + qγj−ðp + q Þk ∬ μ ∞ p2 + q2 × exp½ikðpx + qy + γzÞdpdq;

Fig. 2. Contour lines of phase of the x-component ETEx for b = 50w0x.

ð5bÞ

Fig. 3. Contour lines of phase of the x-component ETEx (a) a = 2 and (b) a = 0.3.

Y. Luo, B. Lü / Optics Communications 283 (2010) 3578–3584

Fig. 4. Energy flux distributions of (a), (d), (g) bSzN

TE,

(b), (e), (h) bSzNTM, and (c), (f), (i) bSzNwhole, (a)–(f) a = 1, and (g)–(i) a = 0.25.

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In the far field the condition kr = k(x2 + y2 + z2)1/2 → ∞ is fulfilled. Using the stationary phase method [25], from Eqs. (5a), (5b) and (6a), (6b) we obtain ETE ðrÞ =

HTE ðrÞ =

" #  

2 2 2 2 k w0y ðx + y Þ kπzðx + iyÞ bx 4 E w x exp − exp ikr 1− 0y 0y 4r 2 2λr 3 ðx2 + y2 Þ r2 " # 2 2 2 2 k w 0x ðx + y Þ ð7aÞ 4 −E0x w0x y exp − ðyi−xjÞ; 4r 2

f g

qffiffiffiffiffiffiffi kπzðx + iyÞ ε=μ

" #  

k2 w20y ðx2 + y2 Þ bx 4 E0y w0y x exp − exp ikr 1− 2 2 4r r 2λr ðx + y Þ " # 2 2 2 2 k w0x ðx + y Þ 4 2 2 ð7bÞ ½xzi + yzj−ðx + y Þk; −E0x w0x y exp − 4r 2 4

2

2

f g

and " #  

2 2 2 2 k w0y ðx + y Þ kπðx + iyÞ bx 4 E w y exp − exp ikr 1− 0y 0y 2λr3 ðx2 + y2 Þ r2 4r2 " # 2 2 2 2 k w0x ðx + y Þ 4 2 2 ð8aÞ ½xzi + yzj−ðx + y Þk; + E0x w0x x exp − 4r2

f g

ETM ðrÞ = −

HTM ðrÞ =

qffiffiffiffiffiffiffi kπðx + iyÞ ε=μ

" #  

2 2 2 2 k w 0y ðx + y Þ bx 4 E w y exp − exp ikr 1− 0y 0y r2 4r 2 2λr 2 ðx2 + y2 Þ " # 2 2 2 2 k w 0x ðx + y Þ 4 ð8bÞ ½yi−xj: + E0x w0x x exp − 4r2

f g

Eqs. (7a), (7b) and (8a), (8b) are analytical vectorial expressions for the TE and TM terms in the far field, which depend on the waist widths w0x, w0y, amplitude constants E0x, E0y and off-axis distance b. In Eqs. (7a), (7b) and (8a), (8b) the variables x and y are not separable, which represents the general feature of nonparaxial beams [26].

}

3. Singular behavior of phase The TE and TM terms in Eqs. (7a), (7b) and (8a), (8b) are orthogonal in the far field and separable. The phase distributions of the x, y and z components of the TE and TM terms can be obtained from Eqs. (7a), (7b) and (8a), (8b). As an illustrative example, we study the phase singularities of the x component of ETE. The contour lines of phase are determined by ϕETE x = arctan

Fig. 4 (continued).

  Im½ETE x ðx; y; zÞ = const: Re½ETE x ðx; y; zÞ

ð9Þ

where Re and Im are the real and imaginary parts of ETEx, respectively. In the following numerical calculations we focus our attention on the dependence of phase singularities on the amplitude ratio η = E0x/E0y, waist width ratio a = w0x/w0y and off-axis distance b, and take z = 1000λ and ε / μ = 1 (in free space). The contour lines of phase of the x-component ETEx for different values of the amplitude ratio η are depicted in Fig. 1(a) η = 0, (b) η = 0.8, (c) η = 1.6, (d) η = − 0.8, (e) η = −1.6, where the other calculation parameters are w0x = w0y = 0.2λ and b = 10w0x. For η = 0 in (a) the initial beam is linearly polarized in the y direction. For such a case there exist two edge dislocations x = 0 (y axis) and y = 0 (x axis), across which a π-phase shift occurs. With increasing |η|, the edge dislocation on the x axis remains, but the y axis is no longer an edge dislocation. For η N 0 the other edge dislocation clockwise rotates about the origin (0, 0) with an increase of η from η = 0.8 in (b) to η = 1.6 in (c). For η b 0 the other edge dislocation anti-clockwise rotates about the origin (0, 0) as |η| is increased from |η| = 0.8 in (d) to |η| = 1.6 in (e). These edge dislocations for η and − η are symmetrical about the y axis and approach the x axis with a further increase of |η|.

Y. Luo, B. Lü / Optics Communications 283 (2010) 3578–3584

Fig. 2 gives contour lines of phase of ETEx for b = 50w0x, and the other calculation parameters are the same as in Fig. 1(b). A comparison of Fig. 2 and 1(b) shows that the off-axis distance affects the phase distribution of ETEx, but does not affect the number and position of edge dislocations. The contour lines of phase of ETEx for different values of the waist width ratio a is plotted in Figs. 3(a) a = 2 and (b) a = 0.3, where w0x = 0.2λ and the other calculation parameters are the same as in Fig. 1(b). From Figs. 3(a), (b) and Fig. 1(b) it follows that there are two edge dislocations, one is located on the x axis, and the other anticlockwise rotates about the origin (0, 0) as the waist width ratio a is decreased from a = 2, 1 to a = 0.3 and approaches the y axis; it can be seen that for η b 0 the other edge dislocation clockwise rotates and approaches the y axis with decreasing a (not shown). The results can be explained as follows. The position of phase singularities obeys [18] Re½ETE x ðx; y; zÞ = 0;

ð10aÞ

Im½ETE x ðx; y; zÞ = 0:

ð10bÞ

" # " 2 2 2 #) k w 0y ðx + y2 Þ k2 w 20x ðx2 + y2 Þ 4 4 xy w 0y x exp −ηw 0x y exp = 0; 4r 2 4r 2 ð11aÞ 2

y

"

# " 2 2 2 #) k w 0y ðx + y2 Þ k2 w 20x ðx2 + y2 Þ 4 4 −ηw0x y exp = 0: w 0y x exp 4r 2 4r2 ð11bÞ

Eq. (11a), (11b) does not contain the parameter b. Therefore, the offaxis distance b does not affect the number and position of phase singularities. In addition, y = 0 satisfies Eqs. (11a), (11b) i.e., the x axis is a dislocation. From the phase analysis we see that the y axis is an edge dislocation irrespective of the values of η and a. Furthermore, for η = 0, x = 0 also satisfy Eqs. (11a), (11b) thus y axis is an edge dislocation. For a = 1 in Fig. 1(a)–(e), y = x / η is a solution of Eqs. (11a), (11b). It means that the straight line with slope 1 / η is an edge dislocation, which approaches the y axis with an decrease of |η|, and the sign of η determines the rotation direction of the edge dislocation as η varies. For 2 2

k w0x ð1−a−2 Þðx2 + y2 Þ x the general case the equation y = exp 4 2 4r ηa determines the position of the other edge dislocation, which approaches the y axis with decreasing a from a = 2, 1 in Fig. 3(a) and Fig. 1(b), respectively, to a = 0.3 in Fig. 3(b). The phase distributions and phase singularities for the y component of ETE and for HTE, ETM and HTM can be analyzed in a similar way, and are not shown in this paper. 4. Energy flux distributions The energy flux distributions of the TE and TM terms at the z plane are expressed in terms of the z component of their time-average Poynting vector as 〈Sz 〉TE =

1  Re½ETE ðrÞ  HTEðrÞz ; 2

ð12aÞ

〈Sz 〉TM =

1  Re½ETM ðrÞ  HTM ðrÞz ; 2

ð12bÞ

with the asterisk being the complex conjugate, and the energy flux distribution of the whole beam is 〈Sz 〉whole = 〈Sz 〉TE + 〈Sz 〉TM :

The substitution from Eqs. (7a), (7b) and (8a), (8b) into Eqs. (12a), (12b), (12c) yields " # k2 ðx2 + y2 Þðw20x + w20y Þ k2 π2 z3 exp − 8λ2 r 7 2r 2 ( " # " 2 2 2 #) 2 2 2 k w0y ðx + y2 Þ k w0x ðx + y2 Þ 4 4 −E ; w y exp × E0y w0y x exp 0x 0x 4r 2 4r 2

〈Sz 〉TE =

ð13aÞ "

2

2

2

2

# 2 + w0y Þ

2 2 k ðx + y Þðw0x k π z exp − 8λ2 r 5 2r2 ( " # " 2 2 2 #) 2 2 2 2 2 k w0y ðx + y Þ k w0x ðx + y Þ 4 4 w x exp + E ; × E0y w0y y exp 0x 0x 4r 2 4r2

〈Sz 〉TM =

ð13bÞ 〈Sz 〉whole =

" # 2 2 2 2 2 2 2 k ðx + y Þðw0x + w0y Þ k π z 4 4 2 2 exp − 2E0x E0y w0x w0y xyðr −z Þ 2 7 2 2r 8λ r " 2 2 # " # 2 2 2 k ðx + y Þðw 0x + w0y Þ k2 w20x ðx2 + y2 Þ 2 8 2 2 2 2 × exp + E0y w0y ðr y + x z Þ exp 2 2 4r 2r " 2 2 2 # 2 k w0y ðx + y Þ 2 8 2 2 2 2 ð13cÞ : + E0x w0x ðr x + y z Þ exp 2r 2

f

g

On substituting Eq. (7a) into Eqs. (10a), (10b) we obtain (

(

3583

ð12cÞ

Fig. 4(a)–(i) represents the energy flux distributions of bSzNTE in (a), (d), bSzNTM in (b), (e), and bSzNwhole in (c), (f), where the calculation parameters in (a)–(c) are the same as those in Fig. 1(b), those in (d)–(f) are the same as those in Fig. 1(b) except for η = 1.6, those in (g)–(i) are the same as those in Fig. 1(b) except for a = 0.25. As can be seen, there exist two axes, about which the energy flux distribution bSzNTE or bSzNTM is symmetrically located, respectively. Two symmetrical axes are determined by Eqs. (13a) and (13b), and are expressed as y = x / η for bSzNTE and y = −ηx for bSzNTM when a = 1, thus the two symmetrical axes are orthogonal to each other. On the axis y = x / η, bSzNTE = 0 and on the axis y = −ηx, bSzNTM = 0. With increasing η from η = 0.8 in (a), (b) to η = 1.6 in (d), (e)bSzNTE, bSzNTM and their symmetrical axes clockwise rotate about the origin (0, 0), whereas bSzNTE, bSzNTM and their symmetrical axes anti-clockwise rotate about the origin (0, 0) with increasing |η| for η b 0 (not shown). The energy flux distribution of the whole beam bSzNwhole shows an elliptical profile with a central dark core in (c), (f). As the waist width ratio is decreased from a = 1 in (a)–(c) to a = 0.25 in (g)–(i), the symmetrical axes of bSzNTE and bSzNTM approach the y and x axes respectively, and bSzNwhole becomes nearly a circular distribution with a small central dark core, because from Eq. (13c) it turns out that bSzNwhole|x = y = 0 = 0. In addition, Eqs. (13a), (13b), (13c) indicate that bSzNTE, bSzNTM and bSzNwhole are independent of the off-axis distance b. 5. Conclusions In this paper, based on the vectorial angular spectrum representation and stationary phase method, the analytical far-field expressions for the TE and TM terms and energy flux distributions of electromagnetic elliptical Gaussian vortex beams have been derived and their far-field properties have been studied in detail. In the far field the optical vortex nested in initial electromagnetic elliptical Gaussian beams disappears and two edge dislocations are present, one is located on the x axis and the position of the other edge dislocation depends on the amplitude ratio η and waist width ratio a. With decreasing |η| the other edge dislocation approaches the y axis and in the limiting case of η = 0 coincides with the y axis. The other edge dislocation approaches the y axis as the waist width ratio a decreases. There exist two orthogonal symmetrical axes y = x / η and y = −ηx for the energy flux distributions bS zNTE and bSzNTM, respectively. With increasing waist width ratio a, the two symmetrical axes approach the y and x axes, respectively, and the energy flux

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distribution of the whole beam bSzNwhole varies from elliptical profile to nearly circular one. The off-axis distance affects the phase distribution of ETEx, but the number and position of edge dislocations and energy flux distributions bSzNTE, bSzNTM and bSzNwhole are independent of b. In comparison with Refs. [5–12], where the vectorial structure of different beams in the far field was analyzed, in our paper the far-field properties of electromagnetic elliptical Gaussian beams carrying an optical vortex, in particular their phase singular behavior have been studied, and the far-field energy flux distributions due to the effect of the vortex differ from those in Ref. [5], but for the case of electromagnetic elliptical vortex-free Gaussian beams, our results reduce to those in Ref. [5]. Interesting is that, as compared with Refs. [18,19], where the interaction of a vortex and an edge dislocation was investigated, and the creation of vortices is accompanied by the disappearance of the edge dislocation, however, in our paper two edge dislocations are present in the far field, while the vortex nested in initial beams disappears. The results obtained in this paper would be useful for a deep understanding of the propagation dynamics of screw and edge dislocations and far-field properties of electromagnetic vortex beams and for their potential applications. Acknowledgement This work was supported by the National Natural Science Foundation of China (NSFC) under grant No. 10874125 and the Natural Science Foundation of Luzhou Medical College.

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