Propagation dynamics of abruptly autofocusing Airy beams with optical vortices

doi:10.1364/OE.20.018579

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Propagation dynamics of abruptly autofocusing Airy beams with optical vortices

Yunfeng Jiang, Kaikai Huang, and Xuanhui Lu »View Author Affiliations

Optics Express, Vol. 20, Issue 17, pp. 18579-18584 (2012)
http://dx.doi.org/10.1364/OE.20.018579

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▸ Article Outline
– Abstract
1. Introduction
2. Abruptly autofocusing characteristics of CAB with an OV
3. CAB with two OVs
4. Conclusion
– References and links

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Abstract

We investigated the propagation dynamics of the Circular Airy Beams (CAB) with optical vortices (OVs) by numerical calculation. Comparing to the common CAB, the maximum intensity of CAB with vortices can be increased greatly at the focal plane and its focal intensity profile is doughnut-shaped when an on-axis vortex is imposed. The case for an off-axis OV and multiple OVs have been investigated as well. We demonstrate that two opposite OVs will annihilate exactly at the focal plane, with the focal intensity is highly increased.

1. Introduction

The Circular Airy Beam (CAB) is of great interest due to its unique abruptly autofocusing characteristics [1

N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045–4047 (2010). [CrossRef] [PubMed]

–5

D. Chremmos, Z. Chen, D. N. Christodoulides, and N. K. Efremidis, “Abruptly autofocusing and autodefocusing optical beams with arbitrary caustics,” Phys. Rev. A 85(2), 023828 (2012). [CrossRef]

]. This beam abruptly focuses its energy right before the focal point while maintaining a low intensity profile until that very point. The abruptly autofocusing feature of the CAB has great potential applications in biomedical treatment or optical micromanipulation [1

N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045–4047 (2010). [CrossRef] [PubMed]

]. Recently, CAB has been used to trap and guide micro particles [6

P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing Airy beams,” Opt. Lett. 36(15), 2883–2885 (2011). [CrossRef] [PubMed]

]. Comparing to the common setup of optical tweezers, an objective lens with weaker magnification is used due to its own abruptly autofocusing property. But because the intensity profile at the focal point is not hollow, CAB is not suitable for trapping the micro particle whose refractive index is smaller than the surrounding medium. And this beam could not “ignite” physical process related to the orbital angular momentum of the light at the intended area. Therefore, the abruptly autofocusing Airy beam with its intensity profile at focal spot being hollow or carried with orbital angular momentum can greatly extend its application in various fields. Based on our recent work, this task can be possibly realized by introducing optical vortices (OVs) into CAB.

The OV is a screw dislocation in the phase fronts and its amplitude is zero in the core. It also carries orbital angular momentum, which is determined by the topological charge. Because of its interesting characteristics and useful applications in optical tweezers [7

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995). [CrossRef] [PubMed]

, 8

K. T. Gahagan and G. A. Swartzlander Jr., “Optical vortex trapping of particles,” Opt. Lett. 21(11), 827–829 (1996). [CrossRef] [PubMed]

], optical communications [9

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001). [CrossRef] [PubMed]

, 10

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004). [CrossRef] [PubMed]

] and other fields, the OV has been extensively investigated in recent years. The intensity gradient and phase gradient are regarded as two main factors affecting the propagation of the OV [11

D. Rozas, C. T. Law, and G. A. Swartzlander Jr., “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B 14(11), 3054–3065 (1997). [CrossRef]

]. OVs would translate, rotate or annihilate in the background beam. When nested in different kinds of host beam, or the same kind of host beam with different parameters, their propagation dynamics will be observably different as the consequence of different profile of the intensity and the phase. OVs will also influence the property of the background beam. The fundamental Gaussian beam is the most common background beam, and the propagation dynamics of OVs nested is determined by the location of the OV relative to the host beam and the waist size of the Gaussian beam [12

G. Indebetouw, “Optical Vortices and Their Propagation,” J. Mod. Opt. 40(1), 73–87 (1993). [CrossRef]

]. Recently, the propagation of an OV superimposed on Airy beams has been investigated [13

H. T. Dai, Y. J. Liu, D. Luo, and X. W. Sun, “Propagation dynamics of an optical vortex imposed on an Airy beam,” Opt. Lett. 35(23), 4075–4077 (2010). [CrossRef] [PubMed]

]. It is found that the OV would shift in transverse direction with twice the velocity of the Airy beam. The propagation dynamics of OV carried by Laguerre-Gaussian beams, partially coherent beams have also been investigated [14

F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre–Gaussian beams,” Opt. Commun. 250(4-6), 218–230 (2005). [CrossRef]

–16

S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A 18(1), 150–156 (2001). [CrossRef] [PubMed]

]. However, the propagation of the OV superimposed onto CAB has never been investigated before.

As mentioned above, it is worthwhile to investigate the propagation characteristics of CAB with OVs. In this work the propagation dynamics of CAB with an on-axis OV has been analyzed first, followed by the case for the off-axis OV. Finally, CAB carrying two OVs has been studied through numerical calculation, where the overlap effect and annihilating effect of the OVs are observed.

2. Abruptly autofocusing characteristics of CAB with an OV

The electric fields of the initial CAB superimposed by a spiral phase in cylindrical coordinate can be expressed as [1

N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045–4047 (2010). [CrossRef] [PubMed]

, 12

G. Indebetouw, “Optical Vortices and Their Propagation,” J. Mod. Opt. 40(1), 73–87 (1993). [CrossRef]

]

u (r, φ, z = 0) = C \cdot A i (\frac{r_{0} - r}{w}) \exp (a \frac{r_{0} - r}{w}) \cdot {(r e^{i φ} - r_{k} e^{i φ_{k}})}^{l},

(1)

where r₀ is related to the radial position of the main Airy ring, w is a radial scale; a is the decaying parameter; (r_k,

φ_{k}

) denotes the location of the OV, l represents the topological charge of the OV; C is a constant related to the beam power. The propagation of the beam with the initial condition of

u (r, φ, z = 0) = g (r) e^{i l φ}

can be computed in terms of the followinglth-order Hankel transform pair:

u (r, φ, z) = 2 π e^{i l φ} \int_{0}^{\infty} \tilde{g} (k) J_{l} (2 π k r) e^{2 i π z \sqrt{λ^{- 2} - k^{2}}} k d k,

(2)

\tilde{g} (k) = 2 π \int_{0}^{\infty} g (r) J_{l} (2 π k r) r d r .

(3)

The analytic expression for

u (r, φ, z)

does not exist. Fortunately, we can use the quasi-discrete method proposed by Yu [17

L. Yu, M. Huang, M. Chen, W. Chen, W. Huang, and Z. Zhu, “Quasi-discrete Hankel transform,” Opt. Lett. 23(6), 409–411 (1998). [CrossRef] [PubMed]

] and Guizar-SIcairos [18

M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields,” J. Opt. Soc. Am. A 21(1), 53–58 (2004). [CrossRef] [PubMed]

] to numerically calculate the Hankel transforms in Eq. (2) and Eq. (3). This method is demonstrated to be much more accurate and efficient than other existing methods. We assume r₀ = 1mm, w = 0.08mm, a = 0.1 throughout this paper, unless it is stated otherwise. The power for different beams is 1W. We also assume the wavelength is 1064nm.

Figure 1 shows the propagation dynamics of CAB with an on-axis OV, the topological charges are l = 1 and l = 2 respectively. The results are compared with common CAB (Fig. 1(c)). We can see that all these beams are focused at certain point, which is called the focal point. The hollow region disappears at the focal point for the common CAB; whereas for CAB with OV, the hollow region in the center would persist even at the focal point or after the focal ppoint. In our case, the focal point is located at about z = 0.275m. The location of focal pointseems not related to the topological charge. After the focal plane, the intensity decreases with oscillation, which is caused by the subsequent Airy rings.

Fig. 1 Propagation dynamics of CAB with an on-axis OV in r-z plane: (a) l = 2; (b) l = 1. (c) Propagation dynamics of common CAB.

Figures 2(a) and 2(b) show the intensity profile at the initial plane. We can find that the number of Airy rings increases with l. The doughnut profiles at the focal plane are shown in Figs. 2(c) and 2(d). At the focal plane, the intensity profile of CAB with an on-axis OV is similar to the Laguerre Gaussian beam, which indicates its potential applications in optical micromanipulation, especially in optical spanners. The intensity distributions at the initial plane and the focal plane are showed in Figs. 2(e) and 2(f). For simplicity, we assume the maximum intensity at the initial plane is I₀, the maximum intensity at an arbitrary transverse plane is I_m. We can see that I₀ of the beam without OV is larger than that of the beam with OV either at the initial plane or at the focal plane, when the beams possess the same total energy. For the beam with an OV, I₀ increases with l, and I_m at the focal plane are nearly equivalent.

Fig. 2 Intensity profile of CAB with an on-axis OV at different transverse planes: (a) l = 2, at the initial plane; (b) l = 1, at the initial plane; (c) l = 2, at the focal plane; (d) l = 1, at the focal plane. (e) The intensity distribution at the initial plane. (f) The intensity distribution at the focal plane.

The abruptly autofocusing property of CAB with an OV can be clearly seen in Fig. 3 , where the changes of the intensity and the radius of the beam during propagation are analyzed. The increase of the maximum intensity during propagation can be expressed by the ratio of I_m to I₀. From Fig. 3(a), we can see that the ratio I_m/I₀ remains around to be unity at first for each of the three beams, but it rapidly increase shortly before focal point (z = 0.25m). For instance, the peak intensity of CAB with an OV of l = 2 can increase by 96 times in our case, which is much greater than CAB without an OV. The beam with a greater topological charge will be much more increased at the focal plane, mainly because I₀ for the beam with a greater l is smaller as we can see in Fig. 2(e).

Fig. 3 Abruptly autofocusing properties of CAB with an on-axis OV: (a) I_m/I₀ as a function of z; (b) radius of maximum light intensity as a function of z; (c) the maximum value of I_m/I₀ that the beams can reach during propagation for different values of r₀.

From Fig. 3(b), we can see that the radii of main Airy ring of the three beams gradually decrease following the same parabolic trajectory until z = 0.25m. But the radius suddenly decreases at certain position in consequence of the abrupt autofocusing property. For the common CAB, this radius rapidly decreases to zero near z = 0.25m; for the beam with OV, this radius decreases to about 1/10 of the initial radius. We can also find that the radius of the beam at the focal plane is greater with a larger l.

We denote the maximum value of I_m/I₀ that the beam could reach during propagation by (I_m/I₀)_max. The value of (I_m/I₀)_max of the CAB with OV can be modulated by varying initial radius r₀ or the decaying parameter a. Figure 3(c) shows the value of (I_m/I₀)_max that the beamreaches during propagation with different r₀. For all the three beams, as r₀ increases, (I_m/I₀)_max also increases and takes an extreme value for a certain r₀; when r₀ continues to grow, it starts to decrease. For example, (I_m/I₀)_max for the beam with l = 2 take its maximum value 131.6 at r₀ = 0.62mm. From Fig. 3(c), we can also know that the maximum intensity of the beam with a larger l can be much more increased with an appropriate value of r₀. However, the maximum value of (I_m/I₀)_max that the beam with l = 1 could reach is a little smaller than the common CAB could reach.

An off-axis OV will be forced to move to near the center because of the autofocusing property of CAB, as we can see in Figs. 4(a) and 4(b), where we assume r_k = 0.5mm,

φ_{k} = 0.

The distance between the OV and the beam center is 0.5mm initially, but it becomes about 0.01mm at the focal plane. However, the OV could not be at the beam center exactly, and the beam profile would remain asymmetric throughout.

Fig. 4 Intensity profile of CAB with an off-axis OV with l = 1: (a) at the initial plane; (b) at the focal plane.

3. CAB with two OVs

Since the off-axis OV superimposed on CAB would be forced to move to near the center, so we considered if multiple OVs superimposed on CAB would overlap or collide during propagation. Let’s take the case of two OVs for example. We assume the topological charges of the two OVs are ( + 1, + 1) or ( + 1,-1), and they are distributed symmetrical about the origin in x-axis. So the fields in the initial plane can be expressed as [11

D. Rozas, C. T. Law, and G. A. Swartzlander Jr., “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B 14(11), 3054–3065 (1997). [CrossRef]

u (r, φ, z = 0) = C \cdot A i (\frac{r_{0} - r}{w}) \exp (a \frac{r_{0} - r}{w}) \cdot (r e^{i φ} + r_{k}) (r e^{\pm i φ} - r_{k}) .

(5)

Substituting Eq. (5) into Eqs. (1) and (2), we can obtain the fields of CAB with two OVs at different positions. We also assume r_k = 0.5mm in our numerical calculation.

From Figs. 5(a) and 5(b), we can see that when two same OVs are imposed, both of them have the tendency of moving toward the centre. The OVs overlap each other at the focal plane, but they do not coincide exactly, which also could be seen from the phase pattern. From Figs. 5(c) and 5(d), we can see that distance between the OVs is changed from 1mm to about 0.03mm, and they also rotate around the axis about 90 degrees at the focal plane.

Fig. 5 Intensity profiles and phase patterns of CAB with two OVs of ( + 1, + 1) at different positions: (a) intensity profile at the initial plane; (b) intensity profile at the focal plane; (c) phase pattern at the initial plane; (d) phase pattern at the focal plane.

But when two opposite OVs are imposed, they will collide and annihilate at the focal plane as we can see in Figs. 6(a) and 6(b). The phase patterns are showed in Figs. 6(c) and 6(d). At the focal plane, the intensity profile becomes a round light spot as the common CAB, the OVs vanish and the light intensity is increased. The value of I_m/I₀ could reach 300 (Fig. 7 ). We notice that this value is far larger than that exhibited in Fig. 3(a). This is because the size of the beam is decreased dramatically when the two OVs vanish. The great enhancement of the peak intensity may have potential applications in optical micromanipulation or other fields. However, we should note that the two OVs discussed here are imposed inside the main Airy ring of the CAB. The cases that OVs are imposed at arbitrary transverse positions of the CAB are more complicated, which will be discussed in the future.

Fig. 6 Intensity profile and phase pattern of the CAB with two OVs of ( + 1,-1) at different positions: (a) intensity profile at the initial plane; (b) intensity profile at the focal plane; (c) phase pattern at the initial plane; (d) phase pattern at the focal plane.

Fig. 7 Changes of I_m/I₀ of the CAB with two OVs of ( + 1,-1) during propagation.

4. Conclusion

In conclusion, we have investigated the propagation dynamics of CAB with OVs. When an on-axis OV is imposed, a doughnut profile will appear at the focal point, and the light intensity increases. It is found that the beam with a greater topological charge will be much more increased at the focal plane. The abruptly increase of the intensity and the abruptly decrease of the radius show the abruptly autofocusing property of the CAB with OV. Besides the advantage of enhanced autofocusing contrast and abruptness exhibited by common CAB, the CAB with OV also has its own unique applications. For example, because its focal intensity profile remains hollow, it can be used to manipulate the particles whose refractive indices are lower than the surrounding medium. Moreover, it can be used in optical spanners or other fields because it carries orbital angular momentum.

The off-axis OVs nested in CAB will move to near the center because of the autofocusing property of CAB. The two same OVs could not overlap on each other perfectly at the focal plane. The two opposite OVs will collide and annihilate at the focal plane. Accompanied with the disappearance of the OVs, the maximum light intensity is greatly increased, which may have potential applications in many fields.

Acknowledgment

This work is supported by the National Basic Research Program of China (Grant No.2012CB921602) and National Natural Science Foundation of China (Grant No.10974177) and the program of International S&T Cooperation of China (2010DFA04690).

References and links

1.	N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045–4047 (2010). [CrossRef] [PubMed]
2.	I. Chremmos, N. K. Efremidis, and D. N. Christodoulides, “Pre-engineered abruptly autofocusing beams,” Opt. Lett. 36(10), 1890–1892 (2011). [CrossRef] [PubMed]
3.	D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. 36(10), 1842–1844 (2011). [CrossRef] [PubMed]
4.	I. Chremmos, P. Zhang, J. Prakash, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Fourier-space generation of abruptly autofocusing beams and optical bottle beams,” Opt. Lett. 36(18), 3675–3677 (2011). [CrossRef] [PubMed]
5.	D. Chremmos, Z. Chen, D. N. Christodoulides, and N. K. Efremidis, “Abruptly autofocusing and autodefocusing optical beams with arbitrary caustics,” Phys. Rev. A 85(2), 023828 (2012). [CrossRef]
6.	P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing Airy beams,” Opt. Lett. 36(15), 2883–2885 (2011). [CrossRef] [PubMed]
7.	H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995). [CrossRef] [PubMed]
8.	K. T. Gahagan and G. A. Swartzlander Jr., “Optical vortex trapping of particles,” Opt. Lett. 21(11), 827–829 (1996). [CrossRef] [PubMed]
9.	G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001). [CrossRef] [PubMed]
10.	G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004). [CrossRef] [PubMed]
11.	D. Rozas, C. T. Law, and G. A. Swartzlander Jr., “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B 14(11), 3054–3065 (1997). [CrossRef]
12.	G. Indebetouw, “Optical Vortices and Their Propagation,” J. Mod. Opt. 40(1), 73–87 (1993). [CrossRef]
13.	H. T. Dai, Y. J. Liu, D. Luo, and X. W. Sun, “Propagation dynamics of an optical vortex imposed on an Airy beam,” Opt. Lett. 35(23), 4075–4077 (2010). [CrossRef] [PubMed]
14.	F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre–Gaussian beams,” Opt. Commun. 250(4-6), 218–230 (2005). [CrossRef]
15.	S. M. Baumann, D. M. Kalb, L. H. MacMillan, and E. J. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express 17(12), 9818–9827 (2009). [CrossRef] [PubMed]
16.	S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A 18(1), 150–156 (2001). [CrossRef] [PubMed]
17.	L. Yu, M. Huang, M. Chen, W. Chen, W. Huang, and Z. Zhu, “Quasi-discrete Hankel transform,” Opt. Lett. 23(6), 409–411 (1998). [CrossRef] [PubMed]
18.	M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields,” J. Opt. Soc. Am. A 21(1), 53–58 (2004). [CrossRef] [PubMed]

OCIS Codes
(050.1940) Diffraction and gratings : Diffraction
(260.0260) Physical optics : Physical optics
(050.4865) Diffraction and gratings : Optical vortices

ToC Category:
Physical Optics

History
Original Manuscript: June 11, 2012
Revised Manuscript: July 20, 2012
Manuscript Accepted: July 25, 2012
Published: July 30, 2012

Citation
Yunfeng Jiang, Kaikai Huang, and Xuanhui Lu, "Propagation dynamics of abruptly autofocusing Airy beams with optical vortices," Opt. Express 20, 18579-18584 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-17-18579

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References

N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett.35(23), 4045–4047 (2010). [CrossRef] [PubMed]
I. Chremmos, N. K. Efremidis, and D. N. Christodoulides, “Pre-engineered abruptly autofocusing beams,” Opt. Lett.36(10), 1890–1892 (2011). [CrossRef] [PubMed]
D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett.36(10), 1842–1844 (2011). [CrossRef] [PubMed]
I. Chremmos, P. Zhang, J. Prakash, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Fourier-space generation of abruptly autofocusing beams and optical bottle beams,” Opt. Lett.36(18), 3675–3677 (2011). [CrossRef] [PubMed]
D. Chremmos, Z. Chen, D. N. Christodoulides, and N. K. Efremidis, “Abruptly autofocusing and autodefocusing optical beams with arbitrary caustics,” Phys. Rev. A85(2), 023828 (2012). [CrossRef]
P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing Airy beams,” Opt. Lett.36(15), 2883–2885 (2011). [CrossRef] [PubMed]
H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett.75(5), 826–829 (1995). [CrossRef] [PubMed]
K. T. Gahagan and G. A. Swartzlander., “Optical vortex trapping of particles,” Opt. Lett.21(11), 827–829 (1996). [CrossRef] [PubMed]
G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett.88(1), 013601 (2001). [CrossRef] [PubMed]
G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express12(22), 5448–5456 (2004). [CrossRef] [PubMed]
D. Rozas, C. T. Law, and G. A. Swartzlander., “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B14(11), 3054–3065 (1997). [CrossRef]
G. Indebetouw, “Optical Vortices and Their Propagation,” J. Mod. Opt.40(1), 73–87 (1993). [CrossRef]
H. T. Dai, Y. J. Liu, D. Luo, and X. W. Sun, “Propagation dynamics of an optical vortex imposed on an Airy beam,” Opt. Lett.35(23), 4075–4077 (2010). [CrossRef] [PubMed]
F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre–Gaussian beams,” Opt. Commun.250(4-6), 218–230 (2005). [CrossRef]
S. M. Baumann, D. M. Kalb, L. H. MacMillan, and E. J. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express17(12), 9818–9827 (2009). [CrossRef] [PubMed]
S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A18(1), 150–156 (2001). [CrossRef] [PubMed]
L. Yu, M. Huang, M. Chen, W. Chen, W. Huang, and Z. Zhu, “Quasi-discrete Hankel transform,” Opt. Lett.23(6), 409–411 (1998). [CrossRef] [PubMed]
M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields,” J. Opt. Soc. Am. A21(1), 53–58 (2004). [CrossRef] [PubMed]

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