Phase singularities of nonparaxial cosh-Gaussian vortex beams diffracted by a rectangular aperture

Optics & Laser Technology 43 (2011) 1264–1269

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Phase singularities of nonparaxial cosh-Gaussian vortex beams diffracted by a rectangular aperture Xiaoxu Lian, Baida Lu¨ n Institute of Laser Physics and Chemistry, Sichuan University, Chengdu 610064, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 30 December 2010 Received in revised form 13 March 2011 Accepted 14 March 2011 Available online 31 March 2011

Analytical expression for the propagation of nonparaxial cosh-Gaussian (ChG) beams diffracted by a rectangular aperture is derived based on the vector Rayleigh–Sommerfeld diffraction integrals and expansion of the aperture window function into a finite sum of complex Gaussian functions, and used to study the phase singularities of nonparaxial diffracted ChG vortex beams. The pair creation, annihilation, motion of phase singularities in the diffracted field and the dependence of position and number of phase singularities on the aperture and beam parameters, as well as on the beam nonparaxiality are illustrated by numerical examples. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Phase singularity Aperture diffraction

1. Introduction Recently, optical beams carrying phase singularities have attracted much interest because of their theoretical importance in modern optics and potential applications in optical manipulation, atom trapping, high-resolution metrology optical communications, etc. [1,2]. The propagation of optical vortex beams in diffracted field was extensively studied [3–8]. The stability and propagation of Gaussian vortex beams diffracted by a half-plane screen were analyzed in Refs. [3–5], and the reconstruction of an optical vortex embedded in a Laguerre–Gauss beam is observed experimentally in the diffracted field under certain conditions [6]. The spatial correlation properties and evolution of correlation vortices of partially coherent vortex beams diffracted by an aperture were investigated in [7], and the result showed that the correlation vortices may appear in the diffracted field even when the vortex core is stopped by the aperture. The diffraction of Gaussian vortex beams passing through an aperture-iris diaphragm and comparison with the experiment were dealt with in [8]. The above work was limited to the paraxial approximation, which is no longer valid when the beam width is comparable to or less than the wavelength [9]. More recently, the polarization singularities of nonparaxial Gaussian vortex beams diffracted by a half-plane screen were analyzed in [10]. The motivation of the present paper is to study the propagation dynamics of phase singularities of nonparaxial vortex beams in the diffracted field. In Section 2, taking the nonparaxial cosh-Gaussian (ChG) beams as an illustrative example, the analytical expression for the

n

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propagation of nonparaxial ChG vortex beams diffracted by a rectangular aperture is derived, and the propagation expression of paraxial diffracted ChG vortex beams is treated as a special case. The evolution of phase singularities embedded in diffracted ChG vortex beams is illustrated in Section 3. Section 4 deals with the dependence of phase singularities on the aperture truncation and beam parameters such as the off-axis displacement, waist width and decentered parameter. The propagation of phase singularities in free space is analyzed in Section 5. Finally, Section 6 summarizes the main results obtained in this paper.

2. Diffraction of nonparaxial ChG vortex beams by a rectangular aperture Consider a linearly polarized ChG vortex beam in the source place z¼0 whose electric field E ¼Ex(x0,y0,0)i þEy(x0,y0,0)j reads as [11,12] ! x2 þy2 Ex ðx0 ,y0 ,0Þ ¼ exp  0 2 0 cos hðO0 x0 Þcos hðO0 y0 Þ½ðx0 aÞ w0 þisgnðmÞðy0 bÞ9m9 , Ey ðx0 ,y0 ,0Þ ¼ 0,

ð1aÞ ð1bÞ

where w0 and O0 denote the waist width in the Gaussian part and parameter in the cosh part, respectively a, b are off-axial displacements, i, j are unit vectors in the x and y directions, respectively, m is the topological charge and takes þ1 in this paper. Experimentally, a ChG beam can be generated in the ring resonator with a ChG apodized aperture [13], or by superimposing two decentered Gaussian beams [14].Then, the ChG vortex

X. Lian, B. L¨ u / Optics & Laser Technology 43 (2011) 1264–1269

beam can be obtained using computer-generated holograms [15], etc. Assume that a rectangular aperture with half-width d is located in the plane z¼0. The window function T(x0,y0) of the aperture can be expanded into a finite sum of complex Gaussian function [16] x  y  0 0 Tðxo ,yo Þ ¼ rect rect 2d 2d   10   10 X Gn x2 X Gn x 2 Fn1 exp  12 Fn2 exp  22 , ð2Þ ¼ d d n ¼1 n ¼1 1

2

where Fn1, Fn2, Gn1 and Gn2 are seen in [16]. The field just behind the aperture is written as

E0y ðx0 ,y0 ,0Þ ¼ 0,

ð3bÞ

According to the vector Rayleigh–Sommerfeld diffraction integrals [17], the field in the half plane z 40 is expressed as Z 1Z 1 1 @Gðr,r0 Þ dx0 dy0 , Ex ðx,y,zÞ ¼  E0 ðx0 ,y0 ,0Þ ð4aÞ 2p 1 1 x @z Ey ðx,y,zÞ ¼ 

Ez ðx,y,zÞ ¼

1 2p

1 2p

Z

1 1

1

Z

1 1

E0y ðx0 ,y0 ,0Þ

@Gðr,r0 Þ dx0 dy0 , @z

ð4bÞ

1

@Gðr,r0 Þ ½E0x ðx0 ,y0 ,0Þ @x 1 1  @Gðr,r0 Þ 0 dx0 dy0 , þ Ey ðx0 ,y0 ,0Þ @x

ð4cÞ

where r0 ¼x0iþy0j, r ¼xiþyj þzk, k denotes the unit vector in the z direction and [17,18] Gðr,r0 Þ ¼

expðik9rr0 9Þ , 9rr0 9

9rr 0 9  r þ

Fn1 Fn2

n1 ¼ 1 n2 ¼ 1

izpw20 expðikrÞ 8lr 2 ðp1 p2 Þ3=2

exp

( " #) 1 ðq1 aÞ2 ðq2 aÞ2 þ p1 p2 4

½2p1 p2 P1 P2 ða þ ibÞ þ w0 ðp2 P2 Q1 þ ip1 P1 Q2 Þ,

Ey ðx,y,zÞ ¼ 0 , Ez ðx,y,zÞ ¼

ð7aÞ ð7bÞ

10 10 X X

Fn1 Fn2

ipw40 expðikrÞ 5=2 3=2

( " #) 1 ðq1 aÞ2 ðq2 aÞ2 þ p1 p2 4

exp

16lr 2 p1 p2 ( 2p1 p2 P2 Q1 ðaþ ibÞ 2p1 x   þ ip1 Q1 Q2 þ 2p1 p2 P1 P2 ðaþ ibÞ w0 w20 n1 ¼ 1 n2 ¼ 1

n þ w0 ðp2 P2 Q1 þ ip1 P1 Q2 Þ þp2 P2 ðq1 aÞ2 þ 2p1 )   i q1 a h ðq1 þ aÞ2 þ2p1 þ exp , p1

ð7cÞ

where a ¼ O0w0 is the decentered parameter, and pj ¼ 1 þ

Gnj

 2

ikw20 , 2r

ð9Þ

Eq. (7a) reduces to 10 10 X X

Fn1 Fn2

ipw20 expðikzÞ

exp

2  iðx þ y2 Þ lz

8lzðp01 p02 Þ3=2 ( " #) 1 ðq01 aÞ2 ðq02 aÞ2  0 0 0 0  exp þ 2p1 p2 P1 P2 ða þ ibÞ 4 p01 p02 þw0 ðp02 P20 Q10 þip01 P10 Q20 Þ,

ð10Þ

where p0j ¼ 1 þ

ikw20 ,  2z ðd=w0 Þ Gnj

2

0  q a Pj0 ¼ 1 þ exp p10 ,

ikw0 y q02 ¼ ,  0 z q a Qj0 ¼ q0j a þ exp 10 ðq0j þ aÞ p1 q01 ¼

1

ikw0 x , z

ðj ¼ 1,2Þ

ð11Þ

It is readily shown that Eq. (10) is consistent using the Fresnel diffraction integral.

3. Evolution of phase singularities of nonparaxial ChG vortex beams in the diffracted field The position of phase singularities is given by [19]

x20 þy20 2xx0 2yy0

10 10 X X

x2 þ y2 , 2z

ð5Þ

, ð6Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r 0 ¼ x20 þy20 , r ¼ x2 þ y2 þz2 and k denote the wave number related to the wavelength by k¼ 2p/l. On substituting Eqs. (1)–(3) into Eq. (4), tedious but straightforward integral calculations deliver Ex ðx,y,zÞ ¼

r  zþ

n1 ¼ 1 n2 ¼ 1

ð3aÞ

Z

Eqs. (7a)–(7c) with Eq. (8) indicate that in the nonparaxial diffracted field the transverse electric-field component Ey(x,y,z) is still zero, but the longitudinal electric-field component Ez(x,y,z) appears. Ex(x,y,z) and Ez(x,y,z) in the z plane depend on the aperture half-width d, and beam parameters w0, a and a, b in general. Under the paraxial approximation

Epx ðx,y,zÞ ¼

E0x ðx0 ,y0 ,0Þ ¼ Ex ðx0 ,y0 ,0ÞTðx0 ,y0 Þ,

Z

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Pj ¼ 1 þexp

ðd=w0 Þ ikw0 x ikw0 y , q2 ¼ , q1 ¼ r  r qj a Qj ¼ qj a þ exp ðqj þ aÞ pj

  qj a pj

ðj ¼ 1,2Þ

ð8Þ

Re½Ex ðx,y,zÞ ¼ Im½Ex ðx,y,zÞ ¼ 0,

ð12aÞ

Re½Ez ðx,y,zÞ ¼ Im½Ez ðx,y,zÞ ¼ 0:

ð12bÞ

where Re and Im denote the real and imaginary parts of Ex(x,y,z) and Ez(x,y,z), respectively. In the following we consider only the phase singularities (optical vortices) in the x component Ex(x,y,z). Fig. 1 gives the phase singularities versus the relative propagation distance z/ZR (ZR ¼ kw20 =2—Rayleigh range) in the first quadrant, where the topological charge m of phase singularities is determined by analyzing vorticity of phase contours around singularities (not shown) [15] and m¼ þ 1( 1) is labeled by open (black) dot, respectively, and the calculation parameters are a ¼b¼0, d/ pffiffiffi w0 ¼3, w0 ¼1.2l and a ¼ 2. From Eq. (7a) it turns that Ex(  y,x)¼ Ex(  x,  y)¼Ex(y,  x)¼0 when a¼b ¼0 and Ex(x,y)¼ 0, so that the phase singularities in the rest three quadrants can be obtained in consideration of the symmetry. Fig. 1 indicates that in the region z/ZRA[4, 4.69] there are five phase singularities with m¼ þ1, þ1, þ1,  1 and  1, respectively, and the phase singularity located at the origin (0,0) remains unchanged upon propagation. The result is true because the substitution from a ¼b¼0 and x¼y¼ 0 into Eq. (7a) yields Ex(0,0,z) ¼0. Additionally, two pairs of oppositely charged phase singularities (m¼ 71) appear and move from (0.81w0, 16.25w0), (18.87w0, 18.02w0), (8.37w0, 15.86w0) and (6.96w0, 14.45w0) in the plane z/ZR ¼ 4 to (0.75w0, 18.67w0), (21.56w0, 20.31w0), (9.1w0, 17.66w0) and (8.65w0, 17.17w0) in the plane z/ZR ¼4.6, respectively. In the region z/ZRA(4.69,4.93] two phase singularities with opposite charge m¼ 71 approach and annihilate each other in the region z/ZRA(4.93,6] a pair of oppositely charged phase singularities occurs, and there are five phase singularities with m¼ þ1, þ1, þ1,  1 and  1. Therefore,

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Fig. 2. Distance D between a pair of phase singularities versus z/ZR.

Fig. 1. Evolution of phase singularities in the nonparaxial diffracted field. The calculation parameters are seen in the text. ‘‘J’’m¼ þ1, ‘‘K’’m¼  1.

the total topological charge is equal to that in the source plane z¼0, and is conserved in the evolution process. The variation of the distance D/l between a pair of oppositely charged phase singularities positioned at (8.37w0, 15.86w0) and (6.96w0, 14.45w0) in the plane z/ZR ¼4 versus z/ZR is plotted in Fig. 2. As can be seen, D/l decreases with increasing z/ZR, and becomes very small, for example, D/l ¼0.31 in z/ZR ¼4.67, resulting in the subwavelength structure, and finally the pair annihilation takes place in z/ZR ¼4.695.

4. Dependence of phase singularities on the aperture truncation and beam parameters The dependence of phase singularities on the relative truncation parameter d/w0 is depicted in Fig. 3, where the calculation pffiffiffi parameters are z ¼4ZR, a ¼b¼0, d/w0 ¼ 3, w0 ¼1.2l and a ¼ 2. It is seen that there are nine phase singularities at d/w0 ¼2, and a pair of oppositely charged phase singularities annihilates each other both at d/w0 ¼2.1 and d/w0 ¼2.2, and a pair of oppositely charged phase singularities appears at d/w0 ¼ 2.2. The pair

Fig. 3. Position and number of phase singularities versus d/w0.

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Fig. 4. Position and number of phase singularities versus b/w0. Fig. 5. Position and number of phase singularities versus w0/l in the nonparaxial case.

annihilation, creation and motion of phase singularities subsequently take place with increasing d/w0, but the total topological charge remains unchanged. The position and number of phase singularities versus the relative off-axis displacement b/w0pisffiffiffi given in Fig. 4, where z¼5ZR, a ¼0, d/w0 ¼3, w0 ¼1.2l and a ¼ 2. It is noted that for such a case the symmetry is violated because ba0, but the similar variation behavior of phase singularities like pair production, annihilation and motion of phase singularities is observed with varying b/w0. For example, there are seventeen phase singularities at b¼0, and two pairs of oppositely charged phase singularities appear and a pair of oppositely charged phase singularities annihilates at b/w0 ¼0.1, and so on. In addition, the phase singularity in the source plane z¼0 moves from (0, 0) to (  1.44w0, 0.35w0) as b/w0 is increased from 0 to 0.6. In the process the topological charge is conserved. The position and number of phase singularities versus the relative width w0/l is plotted in Fig. 5 using Eqs. (7a) and (12), where the pffiffifficalculation parameters are z ¼100l, a¼b¼ 0, d ¼20l and a ¼ 2. Fig. 6 gives the corresponding paraxial result using Eqs. (10) and (12), and the calculation parameters are the same as those in Fig. 5. From Fig. 5 we can see the pair production,

annihilation and motion of phase singularities in the region w0/lA[2.2, 3.42], but the phase singularity located at (0,0) remains unchanged because b¼ 0. The number of phase singularities is decreased from seven to three with an increase of w0/l from 2.2 to 3.42, then there is only one phase singularity located at (0, 0), in the region w0/lA(3.42,4.0]. Compared with Fig. 6, it delivers a different result about the position and number in the region w0/lA[2.2, 3.42], and the same result in the region w0/lA(3.42,4.0]. It means that the paraxial approximation holds true as w0/l 43.42, and does not deliver the correct result as w0/l r3.42. Similar results can be obtained by varying the decentered parameter a, and are omitted.

5. Propagation of phase singularities in free space The evolution of phase singularities of a nonparaxaial ChG vortex beam in the free-space propagation is illustrated in Fig. 7 using Eqs. (7a) and (12), where the pffiffiffi calculation parameters are a¼b ¼0, d-N, w0 ¼1.2l and a ¼ 2. As can be seen, the phase singularity with m ¼ þ1 at the origin (0,0) remains unchanged

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Fig. 6. Position and number of phase singularities versus w0/l in the paraxial case. Fig. 7. Evolution of phase singularities in the nonparaxial free-space propagation.

upon propagation, and a pair of oppositely charged phase singularities occurs in the region z/ZRA[7.5, 11], which moves from (4.84w0, 10.72w0) and (5.91w0, 10.36w0) to (3.91w0, 16.65w0) and (11.45w0, 14.31w0) as z/ZR increase from 7.5 to 11. For comparison, the position and number of phase singularities versus z/ZR in the paraxial case using Eqs. (10a) and (12) are depicted in Fig. 8, where the calculation parameters are the same as those in Fig. 7 except for w0 ¼4l. It is seen that the phase singularity with m¼ þ1 at the origin (0, 0) still remains in the paraxial propagation, and the pair production is present in the region z/ZRA[8.1, 11], which moves from (4.98w0, 10.67w0) and (5.79w0, 10.34w0) to (3.97w0, 15.31w0) and (10.18w0, 13.2w0) with an increase of z/ZR from 8.1 to 11. In comparison with Figs. 7 and 8, there are three or one phase singularity at z/ZR ¼7.5 and the phase singularities are located at (4.84w0, 10.72w0), (5.91w0, 10.36w0), (0, 0) or (0, 0), depending on the nonparaxial or paraxial propagation. Therefore, the position and number of phase singularities may be different in the nonparaxial and paraxial cases. Additionally, a comparison of Figs. 1 and 7 shows that the aperture diffraction also affects the position and number of phase singularities. For example, in the region z/ZRA[4,6] the pair production and annihilation are present in the diffracted

field, whereas there exists only one phase singularity located at (0,0) in the free space propagation.

6. Concluding remarks In this paper, the phase singularities of nonparaxial ChG vortex beams diffracted by a rectangular aperture have been studied in detail. It has been shown that the pair production, annihilation and motion of phase singularities take place in the diffracted field. The position and number of phase singularities depend on the aperture truncation and beam parameters, such as the off-axis displacement, waist width and decentered parameter, and additionally depend on the beam nonparaxiality. There are richer dynamic evolution behaviors in the diffracted field than that in free space. The topological charge is conserved in the evolution process and in the variation of the aperture truncation and vortex beam parameters. To ensure the validity of the results, direct integral calculations of Eq. (4) were performed, showing the consistency with those using Eq. (7a). The results obtained in this paper would be useful for a deep understanding of the dynamical behavior of phase singularities in the nonparaxial

X. Lian, B. L¨ u / Optics & Laser Technology 43 (2011) 1264–1269

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applications, the nonparaxial trapping would have advantages over paraxial trapping [21].

Acknowledgment This work was supported by the National Nature Science Foundation of China (NSFC) under Grant no. 10874125.

References

Fig. 8. Evolution of phase singularities in the paraxial free-space propagation.

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