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Topological charge determination of linearly polarized Lorentz-Gaussian vortex beam
Accepted Manuscript Title: Topological charge determination of linearly polarized Lorentz-Gaussian vortex beam Author: Qiufang Zhan Rongfu Zhang Yu Miao Guanxue Wang Xinmiao Lu Xiumin Gao PII: DOI: Reference:
S0030-4026(16)31168-8 http://dx.doi.org/doi:10.1016/j.ijleo.2016.10.013 IJLEO 58281
To appear in: Received date: Accepted date:
3-8-2016 3-10-2016
Please cite this article as: Qiufang Zhan, Rongfu Zhang, Yu Miao, Guanxue Wang, Xinmiao Lu, Xiumin Gao, Topological charge determination of linearly polarized Lorentz-Gaussian vortex beam, Optik - International Journal for Light and Electron Optics http://dx.doi.org/10.1016/j.ijleo.2016.10.013 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Title page Topological charge determination of linearly polarized Lorentz-Gaussian vortex beam Qiufang Zhana, Rongfu Zhanga, Yu Miaoa, Guanxue Wanga, Xinmiao Lub, Xiumin Gaoa,b* * Corresponding author: Xiumin Gao Telephone: 86-86919029 Email: [email protected] a
Shanghai Key Lab of Modern Optical System, and School of Optical-Electrical and
Computer Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China. b
Hangzhou Dianzi University, Hangzhou 310018, China
Abstract Focusing properties of linearly polarized Lorentz-Gaussian vortex beams was investigated by vector diffraction theory, from which topological charge determination method is proposed. Results show that the focal pattern can be altered considerably by topological charge, and for certain condition, the number of dark spots along one coordinate direction equals to topological charge, and the number of intensity lines crossing focal pattern along the other coordinate direction minus topological charge is one. The intensity lines across focal pattern get weaker for higher topological charge, which leads to the boundaries among dark spots become blurring. Focal shift value of the most out dark peak increases on increasing topological charge, and the dependence curve is monotonous. Topological charge can be obtained according to dependence curve even for fractional topological charge.
1
Keywords: Lorentz-Gaussian beam; Focusing properties; Optical vortex; Vector beam
Text 1 Introduction Lorentz-Gaussian beams have attracted much attention due to their propagation properties and beam characteristics [1-4]. Lorentz-Gaussian beams can be used to describe the output beams from diode lasers, in which the far field distribution in the direction normal to the junction plane approaches a Lorentzian function, and in the direction parallel to the junction it may be approximated by a Gaussian function [5-7]. Nemoto measured the far field of diode lasers and show that the field distribution in the directions normal and parallel to the junction plane agree well with Lorentzian and Gaussian functions, respectively [8]. Fractional Fourier transform, focal shift, beam propagation factors of a Lorentz-Gauss beam were investigated in free space [9-11]. And propagation of Lorentz-Gauss beams in crystal and fractional Fourier transform optical systems has also been explored [12, 13]. Recently, elegant super Lorentz-Gauss beams were represented [14], and focusing properties of Lorentz-Gaussian beam with trigonometric function modulation was investigated by vector diffraction theory [15]. Optical vortex has grown rapidly because they have many interesting properties and promising applications [16-20]. And optical vortex was also introduced in Lorentz-Gaussian beams. Focusing properties of linearly polarized Lorentz-Gaussian beam with one on-axis optical vortex was investigated [21]. Zhou explored propagation property of a Lorentz-Gauss vortex beam in a strongly nonlocal nonlinear media [22]. Wigner distribution function of a
2
Lorentz-Gauss vortex beam was also exploited [23]. Topological charge of optical vortices is important parameter for optical vortex beam, and measurement of the topological charge is necessary in many optical systems. Leach and co-workers proposed an interferometric method for measuring the orbital angular momentum of single photons [24]. Fractional optical vortex can also be measured by a ring-type multi-pinhole interferometer [25]. Han and Zhao analyzed the spatial spectrum of the diffraction intensity pattern of an ideal Bessel beam and found an implicit rule that the number of the bright rings in the spatial spectrum is equal to the topological charge of the Bessel beam [26]. Measuring the topological charge of integer and fraction vortices using multipoint plates was also proposed [27]. Recently, Measurement methods of the topological charge include tilted convex lens[28], diffraction gratings [29], and cylindrical lens [30]. Though the Lorentz-Gaussian beams with optical vortex were investigated, much attention focuses on the propagating properties. To the best of our knowledge, the relation between the focusing properties of Lorentz-Gauss vortex beams with topological charge has not been studied systematically, which in fact is one kind of measurement method for topological charge, so topological charge determination of linearly polarized Lorentz-Gaussian vortex beam was investigated. The focusing principle is given in section 2. And the section 3 shows the results and discussions. The conclusions are summarized in section 4. 2 Determination principle of topological charge Corresponding to Lorentz-Gauss beam shown in figure 1 of reference 21, the amplitude distributions of the electric field in the directions parallel and normal to the junction are the Loreatzian and Gaussian functions, respectively [1-9, 21]
3
x2 E0 x0 , y0 E0 x0 E0 y0 exp 02 0
2 2 0 2 0 y0
(1)
where parameters 0 and 0 are the 1 e -width of the Gaussian distribution and the half width of the Lorentzian distribution, respectively. When the Lorentz-Gauss beam contains one on-axis optical vortex, the focusing properties has been investigated, which shows the focal pattern evolution. In order to calculate the focusing properties easily and clearly, the equation (1) can be rewritten in the form as [21],
cos 2 sin 2 1 exp exp im E0 , 2 2 2 2 NA w sin sin x 1 NA 2 y2
(2)
where r0 is the radial coordinate, and is the azimuthal angle. m is the charge number of the optical vortex, and wx 0 rp is called relative beam waist in y coordinate direction and also called as relative Gauss parameter. r p is the outer radius of optical aperture in focusing system, f is focal length of the focusing system. NA is the numerical aperture of the focusing system, defined as the multiple of refractive index of surrounding medium and sine value of aperture angle. y 0 rp is called relative beam waist in x coordinate direction, and can be called relative Lorentz parameter. It is assumed that the incident Lorentz-Gauss beam is linearly polarized along the x axis. According to vector diffraction theory, the electric field in focal region can be written as [31, 32, 21]
1 E , , z cos sin 2 1 cos x
cos sin cos 1 y cos sin z
4
cos 2 sin 2 1 exp exp im 2 2 2 2 NA wx 1 sin sin NA 2 y2 exp ik sin cos exp ikz cos sin dd (3) where 0, 2 , 0, arcsin( NA) . Vectors x, y, and z are the unit vectors in the x, y, and z directions, respectively. It is clear that the incident beams is depolarized and has three components (Ei, Ej and Ek) in x, y, and z directions, respectively. The variables , , and z are the cylindrical coordinates of an observation point in focal region. The optical intensity distributions in focal region can be calculated from Eq. (3). It can be seen from In fact that focal patterns can be used to determinate the topological charge, in this article, we show the relation between focusing properties and topological charge by analyzing focal pattern in details. 3 Results and discussion It should be noted that the distance unit in all figures is wavelength of incident beam in vacuum without loss of validity and generality. Figure 1 gives the intensity distributions of Lorentz-Gauss vortex beam under condition of NA 0.9 , x 0.3 , y 0.3 , and different m .
It can be seen from the figure that the focal pattern evolves considerably on
increasing topological charge of Lorentz-Gauss vortex beam. There is one annular shape for case of m 1 , and when topological charge increases up to 2, focal pattern changes into one annular shape with one lines cross it along x coordinate direction, which leads to two dark spots along y coordinate direction, as shown in figure 1(b). Then the topological charge increases continuously, there are two lines across focal pattern and two dark spots comes into
5
being under condition of m 3 . For the similar focal pattern evolution process on increasing topological charge, we can see that the number of dark spots along y coordinate direction equals to topological charge, in the other words, the number of intensity lines crossing focal pattern along x coordinate direction minus topological charge is one. In addition, the intensity lines across focal pattern get weaker for higher topological charge, which leads to the boundaries among dark spots become blurring. Therefore, dependence of focal pattern evolution on topological charge is disciplinary, which can be used to determined topological charge from focal pattern of Lorentz-Gauss vortex beams. It can be seen from above focal pattern evolution process, the relation between topological charge and focal pattern may reduce to the relation between topological charge and intensity curve along y coordinate axis. In practice, topological charge determination of linearly polarized Lorentz-Gaussian vortex beam may be carried out by analyzing intensity curve along y coordinate axis. On the other hand, though the number of intensity lines crossing focal pattern along x coordinate direction is related to the topological charge, it may be more convenient to find intensity minimum than intensity peaks because the intensity difference among dark spots is relative smaller than that among intensity peaks, as illustrated in figure 1. In order to get deeper insight into the topological charge determination of linearly polarized Lorentz-Gaussian vortex beams, the intensity curves along y coordinate of Lorentz-Gauss vortex beam under condition of NA 0.9 , x 0.3 , y 0.3 , and different integral topological charge are also calculated and shown in figure 2. Figure 2(a) gives intensity curves along y coordinate of Lorentz-Gauss vortex beam under condition of m 1 and
6
m 2 , from which it can be seen that the minimum value of dark peaks is smaller than unified maximum value. Therefore, dark peaks are very remarkable. For other intensity curves along y coordinate axis, we can see that all dark peaks are remarkable and their number equals to topological charge. From above intensity distributions, all topological charge of linearly polarized Lorentz-Gaussian vortex beams is integral. Now, we focused on the fractional topological charge. Figure 3 illustrates intensity curves along y coordinate axis of Lorentz-Gauss vortex beam under condition of NA 0.9 , x 0.3 , y 0.3 , and topological charge with step 0.2 from 1 to 3. It can be seen that the dark peak shifts along y coordinate axis towards negative y coordinate direction when topological charge increases from 1 to 2, as shown in figure3(a).
And when topological charge increases continuously from 2 to 3, the
moving shifts continuously along y coordinate axis, as illustrated in figure 3(b). From the focal shift on increasing topological charge, it can be employed to determine fractional topological charge by analyzing value of focal shift, which in fact is one kind of determine method for fractional topological charge. Now, we summarized the dependence curve of topological charge on increasing focal shift under condition of NA 0.9 , x 0.3 , y 0.3 , as illustrated in figure 4. It should be noted that the focal shift refer to the deviation distance of dark peak from original coordinate point, so the value of focal shift is positive, and the dark peak refers to the most out one. It can be seen from this figure that the focal shift value increases on increasing topological charge, and the dependence curve is monotonous. Therefore, for certain condition, when we measure the focal shift, we can get value of topological charge according to dependence curve,
7
and the measurement resolution is related to experimental signal measurement resolution of focal shift, and to the dependence curve resolution. In fact, focal patterns can be changed by choosing different numerical aperture and beam parameters. Figure 5 gives intensity distributions of Lorentz-Gauss vortex beam under condition of NA 0.4 , x 0.3 , y 0.3 , and (a) m 1 , (b) m 6 , respectively. It can be seen from this figure that intensity lines crossing annular focal region are still clear, which shows the topological charge measurement is easy to carry out. Figure 6 gives intensity distributions under condition of NA 0.9 , x 0.5 , y 0.5 , and different topological charge, from which we can see unremarkable intensity lines crossing focal region. Therefore, numerical aperture and beam parameters may affect measurement resolution, which should be considered in measurement process. 4 Conclusions The focusing properties of linearly polarized Lorentz-Gaussian vortex beams with one on-axis optical vortex was investigated by vector diffraction theory, and according to focal pattern evolution, one kind of topological charge determination method is proposed. Results show that the number of dark spots along one coordinate direction equals to topological charge for certain condition. The number of intensity lines crossing focal pattern along the other coordinate direction minus topological charge is one. Focal shift value of the most out dark peak increases on increasing topological charge, and the dependence curve is monotonous. Therefore, we can get value of topological charge according to dependence curve. The focal patterns for other value of numerical aperture and beam parameters are also investigated, which shows that focal patterns can be changed considerably. Therefore, in order to analyzing
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topological charge practically, effect of numerical aperture and beam parameters are also considered in certain degree, which will be explored in detail. Acknowledgement This work was partly supported by National Basic Research Program of China (973 Program) (Grant No. 2015CB352001), National Nature Science Foundation of China (61378035, 61405053), and 151Talent Project of Zhejiang Province (12-2-008). References [1] G. Zhou, Analytical vectorial structure of a Lorentz-Gauss beam in the far field, Appl. Phys. B. 93 (2008) 891-899. [2] C. Zhao, and Y. Cai, Paraxial propagation of Lorentz and Lorentz-Gauss beams in uniaxial crystals orthogonal to the optical axis, J. Mod. Opt. 57 (2010) 375-384. [3] Y. Jiang, K. Huang, and X. Lu, Radiation force of highly focused Lorentz-Gauss beams on a Rayleigh particle, Opt. Express. 19 (2011) 9708-9713. [4] X. Wang, Z. Liu, and D. Zhao, nonparaxial propagation of Lorentz-Gauss beams in uniaixal crystal orthogonal to the optical axis, J. Opt. Soc. Am. A. 31 (2014) 872-878. [5] W. P. Dumke, The angular beam divergence in double-heterojunction lasers with very thin active regions, IEEE J. Quantum Electron. 7 (1975) 400-402. [6] H. C. Casey, Jr. and M. B. Pannish, Heterostructure lasers (Academic, New York, 1978). [7] A. Naqwi, and F. Durst, Focusing of diode laser beams: a simple mathematical model, Appl. Opt. 29 (1990) 1780-1785. [8] S. Nemoto, Experimental evaluation of a new expression for the far field of a diode laser beam, Appl. Opt. 33 (1994) 6387-6392.
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[9] G. Zhou, Fractional Fourier transform of Lorentz-Gauss beams, J. Opt. Soc. Am. A. 26 (2009) 350-355. [10] G. Zhou, Focal shift of focused truncated Lorentz-Gauss beam, J. Opt. Soc. Am. A. 25 (2008) 2594-2599. [11] G. Zhou, Beam propagation factors of a Lorentz-Gauss beam, Appl. Phys. B. 96 (2009) 149-153. [12] X. Wang, Z. Liu, and D. Zhao, nonparaxial propagation of Lorentz-Gauss beams in uniaixal crystal orthogonal to the optical axis, J. Opt. Soc. Am. A. 31 (2014) 872-878. [13] W. Du, C. Zhao, and Y. Cai, Propagation of Lorentz and Lorentz-Gauss beams through an apertured fractional Fourier transform optical system, Optics and Lasers in Engineering, 49 (2011) 25-31. [14] Q. Sun, G. Zhang, C. Li, Y. Yang, L. Gao, Z. Liu, and S. Liu, Elegant super Lorentz-Gauss beams, Optik. 126 (2015) 5339-5342. [15] H. Wang, X. Gao, X. Lu, B. Li, and G. Qian, Focusing properties of Lorentz-Gaussian beam with trigonometric function modulation, Optik. 127 (2016) 9436-9443. [16] D. Rozas, Z. S. Sacks, and G. A. Swartzlander, Jr., Experimental Observation of Fluidlike Motion of Optical Vortices, Phys. Rev. Lett. 79 (1997) 3399-3402. [17] E. Brasselet, Optical Vortices from Liquid Crystal Droplets, Phys. Rev. Lett. 103 (2009) 103903. [18] X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, S. Yu, Integrated Compact Optical Vortex Beam Emitters, Science. 338(2012) 363-366.
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[19] M. R. Dennis, R. P. King, B. Jack, K. O’Holleran, and M. J. Padgett, Isolated optical vortex knots, Nature Physics. 6 (2010) 118-121. [20] D. Garoli, P. Zilio, Y. Gorodetski, F. Tantussi, and F. D. Angelis, Optical vortex beam generator at nanoscale level, Scientific Reports. 6 (2016) 29547. [21] F. Rui, D. Zhang, M. Ting, X. Gao, and S. Zhuang, Focusing of linearly polarized Lorentz-Gauss beam with one optical vortex, Optik. 124 (2013) 2969-2973. [22] G. Zhou, Propagation property of a Lorentz-Gauss vortex beam in a strongly nonlocal nonlinear media, Optics Communication. 330 (2014) 106-112. [23] A. Torre, Wigner distribution function of a Lorentz-Gauss vortex beam: alternative approach. Appl. Phys. B. 122 (2016) 55. [24] J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, J. Courtial, Measuring the orbital angular momentum of a single photon, Phys. Rev. Lett. 88 ( 2002) 257901. [25] G. X. Wei, Y. Y. Liu, S. G. Fu, and P. Wang, Measurement of Fractional Optical Vortex by a Ring-Type Multi-Pinhole Interferometer, Advanced Materials Research. 433(2012) 6339-6344. [26] Y. Han, and G. Zhao, Measuring the topological charge of optical vortices with an axicon, Opt. Lett. 36 (2011) 2017-2019. [27] Y. Li, H. Liu, Z. Chen, J. Pu, B. Yao, Measuring the topological charge of integerand fraction vortices using multipoint plates, Optical Review. 18(2011) 7-12. [28] P. Vaity, J. Banerji, and R. P. Singh, Measuring the topological charge of an optical vortex by using a tilted convex lens, Phys. Lett. A. 377 (2013) 1154-1156. [29] M. Liu, Measuring the fractional and integral topological charges of the optical vortex
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beams using a diffraction grating, Optik. 126 (2015) 5263-5268. [30] Y. Peng, X. Gan, P. Ju, Y. Wang, and J. Zhao, Measuring topological charges of optical vortices with multi-singularity using a cylindrical lens, Chin. Phys. Lett. 32 (2015) 56-59. [31] M. Gu, Advanced Optical Imaging Theory (Springer, Heidelberg, 2000). [32] D. Ganic, X. Gan, and M. Gu, Focusing of doughnut laser beams by a high numerical-aperture objective in free space, Opt. Express. 11 (2003) 2747-2752.
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Captions of figures Figure 1 Intensity distributions of Lorentz-Gauss vortex beam under condition of NA 0.9 ,
x 0.3 , y 0.3 , and (a) m 1 , (b) m 2 , (c) m 3 , (d) m 4 , (e) m 5 , and (f) m 6 , respectively. Figure 2 Intensity curves along y coordinate of Lorentz-Gauss vortex beam under condition of NA 0.9 , x 0.3 , y 0.3 , and integral topological charge (a) m 1 , m 2 , (b) m 3 , m 4 , (c) m 5 , m 6 , Figure 3 Intensity curves along y coordinate of Lorentz-Gauss vortex beam under condition of NA 0.9 , x 0.3 , y 0.3 , and topological charge with step 0.2 from (a) 1 to 2, (b) 2 to 3, respectively. Figure 4 Dependence curve of topological charge on increasing focal shift under condition of
NA 0.9 , x 0.3 , y 0.3 . Figure 5 Intensity distributions of Lorentz-Gauss vortex beam under condition of NA 0.4 ,
x 0.3 , y 0.3 , and (a) m 1 , (b) m 6 , respectively. Figure 6 Intensity distributions of Lorentz-Gauss vortex beam under condition of NA 0.9 ,
x 0.5 , y 0.5 , and (a) m 1 , (b) m 2 , (c) m 3 , and (d) m 4 , respectively.
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Illustrations Figure 1
(a)
(b)
(c)
(d)
(e)
(f)
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Figure 2
(a)
(b)
(c) 15
Figure 3
(a)
(b) Figure 4
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Figure 5
(a)
(b)
(a)
(b)
(c)
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Figure 6
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S0030-4026(16)31168-8 http://dx.doi.org/doi:10.1016/j.ijleo.2016.10.013 IJLEO 58281
To appear in: Received date: Accepted date:
3-8-2016 3-10-2016
Please cite this article as: Qiufang Zhan, Rongfu Zhang, Yu Miao, Guanxue Wang, Xinmiao Lu, Xiumin Gao, Topological charge determination of linearly polarized Lorentz-Gaussian vortex beam, Optik - International Journal for Light and Electron Optics http://dx.doi.org/10.1016/j.ijleo.2016.10.013 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Title page Topological charge determination of linearly polarized Lorentz-Gaussian vortex beam Qiufang Zhana, Rongfu Zhanga, Yu Miaoa, Guanxue Wanga, Xinmiao Lub, Xiumin Gaoa,b* * Corresponding author: Xiumin Gao Telephone: 86-86919029 Email: [email protected] a
Shanghai Key Lab of Modern Optical System, and School of Optical-Electrical and
Computer Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China. b
Hangzhou Dianzi University, Hangzhou 310018, China
Abstract Focusing properties of linearly polarized Lorentz-Gaussian vortex beams was investigated by vector diffraction theory, from which topological charge determination method is proposed. Results show that the focal pattern can be altered considerably by topological charge, and for certain condition, the number of dark spots along one coordinate direction equals to topological charge, and the number of intensity lines crossing focal pattern along the other coordinate direction minus topological charge is one. The intensity lines across focal pattern get weaker for higher topological charge, which leads to the boundaries among dark spots become blurring. Focal shift value of the most out dark peak increases on increasing topological charge, and the dependence curve is monotonous. Topological charge can be obtained according to dependence curve even for fractional topological charge.
1
Keywords: Lorentz-Gaussian beam; Focusing properties; Optical vortex; Vector beam
Text 1 Introduction Lorentz-Gaussian beams have attracted much attention due to their propagation properties and beam characteristics [1-4]. Lorentz-Gaussian beams can be used to describe the output beams from diode lasers, in which the far field distribution in the direction normal to the junction plane approaches a Lorentzian function, and in the direction parallel to the junction it may be approximated by a Gaussian function [5-7]. Nemoto measured the far field of diode lasers and show that the field distribution in the directions normal and parallel to the junction plane agree well with Lorentzian and Gaussian functions, respectively [8]. Fractional Fourier transform, focal shift, beam propagation factors of a Lorentz-Gauss beam were investigated in free space [9-11]. And propagation of Lorentz-Gauss beams in crystal and fractional Fourier transform optical systems has also been explored [12, 13]. Recently, elegant super Lorentz-Gauss beams were represented [14], and focusing properties of Lorentz-Gaussian beam with trigonometric function modulation was investigated by vector diffraction theory [15]. Optical vortex has grown rapidly because they have many interesting properties and promising applications [16-20]. And optical vortex was also introduced in Lorentz-Gaussian beams. Focusing properties of linearly polarized Lorentz-Gaussian beam with one on-axis optical vortex was investigated [21]. Zhou explored propagation property of a Lorentz-Gauss vortex beam in a strongly nonlocal nonlinear media [22]. Wigner distribution function of a
2
Lorentz-Gauss vortex beam was also exploited [23]. Topological charge of optical vortices is important parameter for optical vortex beam, and measurement of the topological charge is necessary in many optical systems. Leach and co-workers proposed an interferometric method for measuring the orbital angular momentum of single photons [24]. Fractional optical vortex can also be measured by a ring-type multi-pinhole interferometer [25]. Han and Zhao analyzed the spatial spectrum of the diffraction intensity pattern of an ideal Bessel beam and found an implicit rule that the number of the bright rings in the spatial spectrum is equal to the topological charge of the Bessel beam [26]. Measuring the topological charge of integer and fraction vortices using multipoint plates was also proposed [27]. Recently, Measurement methods of the topological charge include tilted convex lens[28], diffraction gratings [29], and cylindrical lens [30]. Though the Lorentz-Gaussian beams with optical vortex were investigated, much attention focuses on the propagating properties. To the best of our knowledge, the relation between the focusing properties of Lorentz-Gauss vortex beams with topological charge has not been studied systematically, which in fact is one kind of measurement method for topological charge, so topological charge determination of linearly polarized Lorentz-Gaussian vortex beam was investigated. The focusing principle is given in section 2. And the section 3 shows the results and discussions. The conclusions are summarized in section 4. 2 Determination principle of topological charge Corresponding to Lorentz-Gauss beam shown in figure 1 of reference 21, the amplitude distributions of the electric field in the directions parallel and normal to the junction are the Loreatzian and Gaussian functions, respectively [1-9, 21]
3
x2 E0 x0 , y0 E0 x0 E0 y0 exp 02 0
2 2 0 2 0 y0
(1)
where parameters 0 and 0 are the 1 e -width of the Gaussian distribution and the half width of the Lorentzian distribution, respectively. When the Lorentz-Gauss beam contains one on-axis optical vortex, the focusing properties has been investigated, which shows the focal pattern evolution. In order to calculate the focusing properties easily and clearly, the equation (1) can be rewritten in the form as [21],
cos 2 sin 2 1 exp exp im E0 , 2 2 2 2 NA w sin sin x 1 NA 2 y2
(2)
where r0 is the radial coordinate, and is the azimuthal angle. m is the charge number of the optical vortex, and wx 0 rp is called relative beam waist in y coordinate direction and also called as relative Gauss parameter. r p is the outer radius of optical aperture in focusing system, f is focal length of the focusing system. NA is the numerical aperture of the focusing system, defined as the multiple of refractive index of surrounding medium and sine value of aperture angle. y 0 rp is called relative beam waist in x coordinate direction, and can be called relative Lorentz parameter. It is assumed that the incident Lorentz-Gauss beam is linearly polarized along the x axis. According to vector diffraction theory, the electric field in focal region can be written as [31, 32, 21]
1 E , , z cos sin 2 1 cos x
cos sin cos 1 y cos sin z
4
cos 2 sin 2 1 exp exp im 2 2 2 2 NA wx 1 sin sin NA 2 y2 exp ik sin cos exp ikz cos sin dd (3) where 0, 2 , 0, arcsin( NA) . Vectors x, y, and z are the unit vectors in the x, y, and z directions, respectively. It is clear that the incident beams is depolarized and has three components (Ei, Ej and Ek) in x, y, and z directions, respectively. The variables , , and z are the cylindrical coordinates of an observation point in focal region. The optical intensity distributions in focal region can be calculated from Eq. (3). It can be seen from In fact that focal patterns can be used to determinate the topological charge, in this article, we show the relation between focusing properties and topological charge by analyzing focal pattern in details. 3 Results and discussion It should be noted that the distance unit in all figures is wavelength of incident beam in vacuum without loss of validity and generality. Figure 1 gives the intensity distributions of Lorentz-Gauss vortex beam under condition of NA 0.9 , x 0.3 , y 0.3 , and different m .
It can be seen from the figure that the focal pattern evolves considerably on
increasing topological charge of Lorentz-Gauss vortex beam. There is one annular shape for case of m 1 , and when topological charge increases up to 2, focal pattern changes into one annular shape with one lines cross it along x coordinate direction, which leads to two dark spots along y coordinate direction, as shown in figure 1(b). Then the topological charge increases continuously, there are two lines across focal pattern and two dark spots comes into
5
being under condition of m 3 . For the similar focal pattern evolution process on increasing topological charge, we can see that the number of dark spots along y coordinate direction equals to topological charge, in the other words, the number of intensity lines crossing focal pattern along x coordinate direction minus topological charge is one. In addition, the intensity lines across focal pattern get weaker for higher topological charge, which leads to the boundaries among dark spots become blurring. Therefore, dependence of focal pattern evolution on topological charge is disciplinary, which can be used to determined topological charge from focal pattern of Lorentz-Gauss vortex beams. It can be seen from above focal pattern evolution process, the relation between topological charge and focal pattern may reduce to the relation between topological charge and intensity curve along y coordinate axis. In practice, topological charge determination of linearly polarized Lorentz-Gaussian vortex beam may be carried out by analyzing intensity curve along y coordinate axis. On the other hand, though the number of intensity lines crossing focal pattern along x coordinate direction is related to the topological charge, it may be more convenient to find intensity minimum than intensity peaks because the intensity difference among dark spots is relative smaller than that among intensity peaks, as illustrated in figure 1. In order to get deeper insight into the topological charge determination of linearly polarized Lorentz-Gaussian vortex beams, the intensity curves along y coordinate of Lorentz-Gauss vortex beam under condition of NA 0.9 , x 0.3 , y 0.3 , and different integral topological charge are also calculated and shown in figure 2. Figure 2(a) gives intensity curves along y coordinate of Lorentz-Gauss vortex beam under condition of m 1 and
6
m 2 , from which it can be seen that the minimum value of dark peaks is smaller than unified maximum value. Therefore, dark peaks are very remarkable. For other intensity curves along y coordinate axis, we can see that all dark peaks are remarkable and their number equals to topological charge. From above intensity distributions, all topological charge of linearly polarized Lorentz-Gaussian vortex beams is integral. Now, we focused on the fractional topological charge. Figure 3 illustrates intensity curves along y coordinate axis of Lorentz-Gauss vortex beam under condition of NA 0.9 , x 0.3 , y 0.3 , and topological charge with step 0.2 from 1 to 3. It can be seen that the dark peak shifts along y coordinate axis towards negative y coordinate direction when topological charge increases from 1 to 2, as shown in figure3(a).
And when topological charge increases continuously from 2 to 3, the
moving shifts continuously along y coordinate axis, as illustrated in figure 3(b). From the focal shift on increasing topological charge, it can be employed to determine fractional topological charge by analyzing value of focal shift, which in fact is one kind of determine method for fractional topological charge. Now, we summarized the dependence curve of topological charge on increasing focal shift under condition of NA 0.9 , x 0.3 , y 0.3 , as illustrated in figure 4. It should be noted that the focal shift refer to the deviation distance of dark peak from original coordinate point, so the value of focal shift is positive, and the dark peak refers to the most out one. It can be seen from this figure that the focal shift value increases on increasing topological charge, and the dependence curve is monotonous. Therefore, for certain condition, when we measure the focal shift, we can get value of topological charge according to dependence curve,
7
and the measurement resolution is related to experimental signal measurement resolution of focal shift, and to the dependence curve resolution. In fact, focal patterns can be changed by choosing different numerical aperture and beam parameters. Figure 5 gives intensity distributions of Lorentz-Gauss vortex beam under condition of NA 0.4 , x 0.3 , y 0.3 , and (a) m 1 , (b) m 6 , respectively. It can be seen from this figure that intensity lines crossing annular focal region are still clear, which shows the topological charge measurement is easy to carry out. Figure 6 gives intensity distributions under condition of NA 0.9 , x 0.5 , y 0.5 , and different topological charge, from which we can see unremarkable intensity lines crossing focal region. Therefore, numerical aperture and beam parameters may affect measurement resolution, which should be considered in measurement process. 4 Conclusions The focusing properties of linearly polarized Lorentz-Gaussian vortex beams with one on-axis optical vortex was investigated by vector diffraction theory, and according to focal pattern evolution, one kind of topological charge determination method is proposed. Results show that the number of dark spots along one coordinate direction equals to topological charge for certain condition. The number of intensity lines crossing focal pattern along the other coordinate direction minus topological charge is one. Focal shift value of the most out dark peak increases on increasing topological charge, and the dependence curve is monotonous. Therefore, we can get value of topological charge according to dependence curve. The focal patterns for other value of numerical aperture and beam parameters are also investigated, which shows that focal patterns can be changed considerably. Therefore, in order to analyzing
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topological charge practically, effect of numerical aperture and beam parameters are also considered in certain degree, which will be explored in detail. Acknowledgement This work was partly supported by National Basic Research Program of China (973 Program) (Grant No. 2015CB352001), National Nature Science Foundation of China (61378035, 61405053), and 151Talent Project of Zhejiang Province (12-2-008). References [1] G. Zhou, Analytical vectorial structure of a Lorentz-Gauss beam in the far field, Appl. Phys. B. 93 (2008) 891-899. [2] C. Zhao, and Y. Cai, Paraxial propagation of Lorentz and Lorentz-Gauss beams in uniaxial crystals orthogonal to the optical axis, J. Mod. Opt. 57 (2010) 375-384. [3] Y. Jiang, K. Huang, and X. Lu, Radiation force of highly focused Lorentz-Gauss beams on a Rayleigh particle, Opt. Express. 19 (2011) 9708-9713. [4] X. Wang, Z. Liu, and D. Zhao, nonparaxial propagation of Lorentz-Gauss beams in uniaixal crystal orthogonal to the optical axis, J. Opt. Soc. Am. A. 31 (2014) 872-878. [5] W. P. Dumke, The angular beam divergence in double-heterojunction lasers with very thin active regions, IEEE J. Quantum Electron. 7 (1975) 400-402. [6] H. C. Casey, Jr. and M. B. Pannish, Heterostructure lasers (Academic, New York, 1978). [7] A. Naqwi, and F. Durst, Focusing of diode laser beams: a simple mathematical model, Appl. Opt. 29 (1990) 1780-1785. [8] S. Nemoto, Experimental evaluation of a new expression for the far field of a diode laser beam, Appl. Opt. 33 (1994) 6387-6392.
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[9] G. Zhou, Fractional Fourier transform of Lorentz-Gauss beams, J. Opt. Soc. Am. A. 26 (2009) 350-355. [10] G. Zhou, Focal shift of focused truncated Lorentz-Gauss beam, J. Opt. Soc. Am. A. 25 (2008) 2594-2599. [11] G. Zhou, Beam propagation factors of a Lorentz-Gauss beam, Appl. Phys. B. 96 (2009) 149-153. [12] X. Wang, Z. Liu, and D. Zhao, nonparaxial propagation of Lorentz-Gauss beams in uniaixal crystal orthogonal to the optical axis, J. Opt. Soc. Am. A. 31 (2014) 872-878. [13] W. Du, C. Zhao, and Y. Cai, Propagation of Lorentz and Lorentz-Gauss beams through an apertured fractional Fourier transform optical system, Optics and Lasers in Engineering, 49 (2011) 25-31. [14] Q. Sun, G. Zhang, C. Li, Y. Yang, L. Gao, Z. Liu, and S. Liu, Elegant super Lorentz-Gauss beams, Optik. 126 (2015) 5339-5342. [15] H. Wang, X. Gao, X. Lu, B. Li, and G. Qian, Focusing properties of Lorentz-Gaussian beam with trigonometric function modulation, Optik. 127 (2016) 9436-9443. [16] D. Rozas, Z. S. Sacks, and G. A. Swartzlander, Jr., Experimental Observation of Fluidlike Motion of Optical Vortices, Phys. Rev. Lett. 79 (1997) 3399-3402. [17] E. Brasselet, Optical Vortices from Liquid Crystal Droplets, Phys. Rev. Lett. 103 (2009) 103903. [18] X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, S. Yu, Integrated Compact Optical Vortex Beam Emitters, Science. 338(2012) 363-366.
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Captions of figures Figure 1 Intensity distributions of Lorentz-Gauss vortex beam under condition of NA 0.9 ,
x 0.3 , y 0.3 , and (a) m 1 , (b) m 2 , (c) m 3 , (d) m 4 , (e) m 5 , and (f) m 6 , respectively. Figure 2 Intensity curves along y coordinate of Lorentz-Gauss vortex beam under condition of NA 0.9 , x 0.3 , y 0.3 , and integral topological charge (a) m 1 , m 2 , (b) m 3 , m 4 , (c) m 5 , m 6 , Figure 3 Intensity curves along y coordinate of Lorentz-Gauss vortex beam under condition of NA 0.9 , x 0.3 , y 0.3 , and topological charge with step 0.2 from (a) 1 to 2, (b) 2 to 3, respectively. Figure 4 Dependence curve of topological charge on increasing focal shift under condition of
NA 0.9 , x 0.3 , y 0.3 . Figure 5 Intensity distributions of Lorentz-Gauss vortex beam under condition of NA 0.4 ,
x 0.3 , y 0.3 , and (a) m 1 , (b) m 6 , respectively. Figure 6 Intensity distributions of Lorentz-Gauss vortex beam under condition of NA 0.9 ,
x 0.5 , y 0.5 , and (a) m 1 , (b) m 2 , (c) m 3 , and (d) m 4 , respectively.
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Illustrations Figure 1
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Figure 2
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Figure 5
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