Measuring the fractional and integral topological charges of the optical vortex beams using a diffraction grating

Measuring the fractional and integral topological charges of the optical vortex beams using a diffraction grating

Accepted Manuscript Title: Measuring the fractional and integral topological charges of the optical vortex beams using a diffraction grating Author: M...

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Accepted Manuscript Title: Measuring the fractional and integral topological charges of the optical vortex beams using a diffraction grating Author: Man Liu PII: DOI: Reference:

S0030-4026(15)01060-8 http://dx.doi.org/doi:10.1016/j.ijleo.2015.09.004 IJLEO 56209

To appear in: Received date: Accepted date:

24-9-2014 6-9-2015

Please cite this article as: M. Liu, Measuring the fractional and integral topological charges of the optical vortex beams using a diffraction grating, Optik - International Journal for Light and Electron Optics (2015), http://dx.doi.org/10.1016/j.ijleo.2015.09.004 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.

Measuring the fractional and integral topological charges of the optical vortex beams using a

1) (School of science, Qilu university of technology Jinan 250353, China)

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Man Liu1) 2)

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diffraction grating

2) (College of Physics and Electronics, Shandong Normal University, Jinan 250014, People’s Republic Corresponding author: [email protected], [email protected]

Abstract

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of China)

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The diffraction properties of intensity and phase of the optical vortex beam

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through the diffraction gratings are studied. It is found that the profiles of the zero-order light spot in the center is similar to the +1 and -1 diffraction orders, and the

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distributions of light spots is related to the topological charges of vortex beam. For the

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fractional topological changes, the profiles of light spots is become long in the y

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direction, which parallel to the direction of slits of grating, and the patterns for positive and negative of topological charges are mirrored in the x axis. For the integral

topological changes there is no difference between the patterns for positive and negative of topological charges, in this case, we can easily determined the topological charges from the patterns of phase. The results may be used to detect the fractional and integral topological charges of vortex beams. Keywords: vortex beams, diffraction grating, intensity distribution, phase distribution PACS: 42.25.Fx, 42.30.Ms 1. Introduction As we all know, all beams with an azimuthal phase term, exp(ilθ ) of which 1

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Laguerre-Gaussian beams are an example, have an orbital angular momentum of l per photon, where θ is the azimuthal angle and l (positive or negative) is the

topological charge [1]. If the phase around the center of vortex beam increases in

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clockwise direction, the sign of topological charge is negative; if the phase around the

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center of vortex beam is increases in the counterclockwise direction, the sign of

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topological charge is positive. The optical vortex beam carrying orbital angular momentum not only can be transferred to the micro particle, to drive the particles

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rotation, but also can realize the capture of micron, sub micron particles[2], quantum information processing[3-8], atomic operation[9-12], micro operation[13-15] and

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biological sciences[16,17]. In recent years, it has attracted more attention in astronomical applications [18,19]. The topological charges of vortex beams can be

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applied to the optical information coding and transmission, especially the fractional topological charges, this new coding method has the advantages of high capacity, high

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security and so on, so the measurement of the topological charge of vortex beam is very important. Detection the topological charge of vortex beam is hot research topic in recent years [19-21].

In this paper, the characteristics of the intensity and phase distributions of

Laguerre-Gaussian (LG) beam passing through a diffraction grating in the Fraunhofer diffraction region are simulated. It is found that the light spot of zero order in each center is similar to the +1 and -1 orders. For fractional topological charges, the light spots are all parallel to the direction of slit of grating and the intensity distribution patterns are symmetric in the x axis with the fractional topological charges of vortex

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beams are opposite; for integral topological charges, the intensity distribution patterns are also related to the topological charges of incident optical vortex beams, while in this case there is no difference between the intensity distribution patterns for positive

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and negative topological charges, making it impossible to differentiate between

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positive and negative topological charges, in this case, it can be distinguished from

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the phase distribution patterns. So this method can easily to probe the topological charges of optical vortex beams directly by observing the bright spots in intensity

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distribution patterns or the number of phase vortices in the central region in phase distribution patterns.

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2. Theoretical study the intensity and phase distributions in the Fraunhofer diffraction region

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First, we study the distribution characteristic of intensity and phase in the far field

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produced by optical vortex beams through a diffraction grating. Laguerre-Gauss beam

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is the most typical optical vortex beam, in the z  0 incident plane, which have the complex amplitude can be written as

Ei ( x0 , y0 )  (

x02  y02 l x2  y2 ) exp( 0 2 0 ) exp(ilθ ) ω ω

(1)

where ω is the radius of the beam waist, l is the topological charge of vortex beam, and ( x0 , y0 ) is coordinate. The complex amplitude of LG beams passed through a grating at a point in the observation plane can be expressed as E0 ( x0 , y0 )  Ei ( x0 , y0 )t ( x0 , y0 )

(2)

where Ei ( x0 , y0 ) is the light field before the diffraction grating, t ( x0 , y0 ) is amplitude transmittance function, which can be given by

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x0 x d x  2d x  ( N  1)d )  rect( 0 )rect( 0 )   rect( 0 ) a a a a N 1 x  md   rect( 0 ) a m=0

t ( x0 , y0 )  rect(

(3)

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where a is the slit width , d is the grating constant, and N is the number of identical slits, Let the incident LG beam be centered at the origin. Then the far field complex

( x 2  y02 ) xx  yy0 1 exp(ikz ) exp[ik 0 ] E0 ( x0 , y0 ) exp(ik 0 )dx0 dy0  iλz 2z z

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E ( x, y , z ) 

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amplitude of the LG beam diffracted by the diffraction grating can be expressed as

(4)

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where z is the distance from the diffraction plane to the observation plane, k  2π / λ is the wave vector. The intensity and phase at point ( x, y ) are given

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by I ( x, y, z )  E ( x, y, z ) E  ( x, y, z ) and θ  arg( x  iy ) , respectively. Using the above theory, the intensity and phase distributions of LG beams

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diffracted by the diffraction grating in the observation plane can be obtained. In the calculations, the wavelength of the illuminating light is set at 632.8nm corresponding

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to that of the He-Ne laser. The waist size of the beam ω is 20.0μm . The slit width a is 2.0μm , the grating constant d is 4.0μm , and the number of slits is 25 .The distance z is set at 20cm . The range of the observation plane in which the intensity is

calculated is 1.2  1.2cm 2 with 501 501 sampling points.

Figure1. The left patter is the diffraction grating; the right pattern is the intensity distribution of LG beams with l=+3.0 on plane z=0.

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The range of the diffraction grating is 100 100μm2 with 501 501 sampling points. The grating constant d  4.0μm and light intensity profile of LG beam with l =+3.0 are shown in Figure 1. The gray-scale values take arbitrary units and the number of

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gray-scale level is 32.

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Next, we analyze the distribution characteristics of intensity and phase of

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Laguerre-Gaussian beam with integral and fractional topological charges passing through a diffraction grating in the Fraunhofer observation plane.

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3. Analysis the simulation results

3.1. The characteristics of diffraction intensity distribution in far field

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In order to analyze the relationship between the topological charge of LG beam and intensity distribution in far field, first, we simulated the intensity distributions of

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LG beams with the opposite topological charges l= 1, 2, 3, 4 and  5 passing through a diffraction grating in the Fraunhofer diffraction region and shown them in

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Figures 2, respectively. The gray-scale values take arbitrary units, but those values are same in each pattern, and the number of gray-scale level is also 32. In left column, from top to bottom corresponding to the topological charges l= 1, 2, 3, 4 and  5 ,

respectively; in right column, from top to bottom corresponding to the topological charges l= 1, 2,3, 4 and 5 , respectively. From Figure 2, one can seen that the number of the light spots is equal to 3 in each pattern, around those light spots there are small bright spots which similar to the speckle grains, the light spot in each center corresponding to the zero order light spot of diffraction of grating in far field, and the two sides light spots corresponding to the

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+1 and -1 orders, and the profile of light spots are similar to each other. Except the topological charge is equal to 1 , at the each

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center of light spots,

where

the

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cores,

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there are small dark

intensity value is zero,

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which corresponding to the center of vortex

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beams. It is seen that

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three light spots in each

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the bright spots around

pattern are becoming

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smaller gradually and

the increased number with the increase of integral

topological

charge, around the light

spots there are many small

bright

spots

arranged in circular arc and

circular

profile,

Figure 2. The intensity distribution patterns in far field behind the grating illuminated by the LG beams with l=-1, l=+1, l=-2, l=+2, l=-3, l=+3, l=-4, l=+4.0, l=-5 and l=+5.0, respectively.

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those small bright spots are similar to the spots in speckle field, and the phenomena of cluster appear with the increase of topological charge. The intensity distribution patterns are all symmetric with respect to the origin. In this case there is no difference

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between the pattern for positive and negative topological charges, making it

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impossible to differentiate between negative and positive topological charges of

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vortex beam. In Figure 2, for the charges

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topological

l= 3,  5 , it can be seen

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that the small bright

distribution from

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light spots show radial

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spots around the central

the

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center to the outward, the small bright spots

form the bright fringes, and

the

number

of

bright fringes is four

times of the topological charges

of

optical

vortex beam. For the topological

Figure3. The intensity distribution patterns in far field behind the grating illuminated by the LG beams with l=-1.5, l=+1.5, l=-2.5, l=+2.5, l=-3.5, l=+3.5, l=-4.5 and l=+4.5, respectively. 7

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charges l  2,  4 , the number of small bright spots around the central light spots is just eight times of the topological charge of optical vortex beam. In order to observe the relationship between the positive and negative fractional

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topological charges, whose intensity distributions are shown in Fig.3, and have the

from

top

to

bottom

corresponding

to

the

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same parameters as shown in Fig.2, except for the topological charges. In left column, topological

charges

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l= 1.5, 2.5, 3.5 and  4.5 , respectively; in right column, from top to bottom

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corresponding to the topological charges l= 1.5, 2.5, 3.5 and  4.5 , respectively.

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Comparing the Figure 2 with Figure 3, we find that the small dark cores in

Figure 4. The intensity distribution patterns behind a diffraction grating illuminated by the LG beams with the fractional topological charges l=2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8 and 2.9, respectively. 8

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Figure 2 disappeared in Figure 3, the light spots are becoming longer gradually with the increase of fractional topological charge, which are all parallel to the direction of slit of grating and the intensity distribution patterns are mirrored in the x axis with the

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fractional topological charges of vortex beam are opposite, and the number of the

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light spots is also equal to 3 in each pattern, the small bright spots around those light

with the increase of fractional topological charge.

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spots in each pattern are also becoming smaller gradually and the increased number

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Next, we discuss the evolution of the far field intensity of LG beam with the fractional topological charges. The intensity distribution patterns behind a diffraction

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grating illuminated by the LG beams with the fractional topological charges l=2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8 and 2.9 are presented in Figure 4, respectively.

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In Figure 4, one can seen that light spots in y direction are becoming longer gradually when the fractional charges are increased from 2.1 to 2.5 and then shorter

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with the changes of the fractional charges from 2.6 to 2.9. At the same time, the number of small bright spots around three bright spots is increased. In addition, arrangement of the small bright spots around the central light spots show circle profile. However, the brightness of the light spots in each center of each pattern is nearly unchanged.

3.2. The characteristics of diffraction phase distribution in far field The phase distribution patterns behind a diffraction grating illuminated by the LG beams with the topological charges l=4 and 5 are shown in Figure 5, respectively, corresponding to the intensity patterns in Figure 2. Phase patterns represented in grey

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scale, and the grayscale level number is 8, so the phase interval is π / 4 . The phase changes from minimum to maximum presenting from dark to white. In order to clearly observe the phase distribution in central region, we enlarged the white squared

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region in left column and show them in right column.

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In the left column of Figure 5, there are three phase vortices corresponding to the

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three black cores in Figure 2, around which the phase distribution is uniform, from

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each center there are many chains, on which the positive and negative phase vortices

Figure 5. The left column are the phase distribution patterns in far field behind the grating illuminated by the LG beams with l=4 and l=5, respectively, and the right column are the enlarged phase distribution corresponding to the squared region in left column.

are alternating appear. At the edge of each phase pattern, the phase distribution is similar to those in speckle field, whose structure becoming more and more complex with the increase of the topological charge. In right column, at each central region, the 10

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an

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cr

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phase vortices split into 4 or 5 single-charge vortices, respectively. From the top-right

Figure 6. The phase distribution patterns in far field behind the grating illuminated by the LG beams with the half-integer values of topological charge. The insert in each pattern show the phase distribution of the central region.

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pattern of Fig.5, we can see that the phase around a central region changes by four multiple of 2 , at the central region of the insert there are four phase vortices, they have the same sign and separated from each other. Around the center, there are 32

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phase vortices have alternating signs connected with those four phase vortices in

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central region. In bottom-right pattern of Fig.5, there is one positive phase vortices in

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center, adjacent to it there are four positive phase vortices. Around those vortices, there are 40 phase vortices have alternating signs connected with the five phase

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vortices in center region. Around them, the phase changes by five times of 2 . Figure 6 shows the phase distribution patterns with the topological charges are

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equal to half-integer. The left column corresponding to the topological charge l=-1.5,-2.5, -3.5 and -4.5, and the right column corresponding to the topological

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charge l=1.5, 2.5, 3.5 and 4.5, respectively. The insert in each pattern show the phase distributions of the central region of zero-order diffraction patterns. The positive

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vortices are depicted by white circles, and a black circle represents a negative vortex. From Figure 6, we can seen, there are three strip region parallel with the y axis in

each pattern, which center is a fractional phase vortex. The patterns for left columns and the right column are mirrored in the x axis. For the topological charge l  1.5 ,

we can clearly see that the each center has two separated negative (positive) phase vortices, they are surrounded by three positive (negative) phase vortices; for the topological charge l  2.5 , the each center has three separated negative (positive) phase vortices, they are surrounded by six positive (negative) phase vortices; for the topological charge l  3.5 , the each center has four separated negative (positive)

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phase vortices, they are surrounded by nine positive (negative) phase vortices; for the topological charge l  4.5 , at each center has five separated negative (positive) phase vortices, they are surrounded by twelve positive (negative) phase vortices. The

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number of the phase vortices in each central region is exactly equal to the integer part

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of the fractional topological charge of incident beam plus one, and around the center

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the number of phase vortices is 3 times of the integer part of the fractional topological

Figure 7. The phase distribution patterns for various topological charge values from l=2 to l=3, the topological charge interval is 0.1

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charge. The phase distribution patterns are all symmetric with respect to the y axis. Finally, we discuss the formed process of the phase vortices in the center. The phase distribution for various topological charge values from l=2 to l=3, the

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difference is 0.1. The size of the patterns are all 2  2mm 2 with 501by 501 sampling

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points.

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In Figure7, for l=2, one can see that there are two phase vortices in center region, whose phase increment in counterclockwise direction, the phase in the black dashed

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circle is also incremented counterclockwise. When the topological charge is non-integer, the two phase vortices at the center are slightly separation, and deviated

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from the center, and at the same time moving upward in y direction. The increment direction of phase near the phase vortices has not changes, which is moving slowly

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upward, until the topological charge is 3. Around the phase vortices in the center, there are many curved chains, on which the negative and positive phase vortices are

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alternating appear. 4. Conclusions

In conclusion, we have analyzed theoretically the far field intensity and the phase

distributions of the Laguerre-Gaussian beams with integral and fractional topological charges through a diffraction grating in detail. The profiles of the distributions of light spots and the phase are related to the topological charges of the optical vortex beam, while the diffraction pattern of an optical vortex beam by a multipoint interferometer is no difference between the positive and negative topological charges for an even number of points. These results may provide a theoretical basis for experiments

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probing the topological charges of an optical vortex beam, and may have reference

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value for analysis the propagation characteristics of optical vortex beam in far field.

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