Measuring the fractional topological charge of LG beams by using interference intensity analysis

Optics Communications 334 (2015) 235–239

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Optics Communications journal homepage: www.elsevier.com/locate/optcom

Measuring the fractional topological charge of LG beams by using interference intensity analysis Xinzhong Li a,b,n, Yuping Tai c, Fangjie Lv a, Zhaogang Nie d a

School of Physics and Engineering, Henan University of Science and Technology, Luoyang 471003, China State Key Laboratory of Transient Optics and Photonics, Chinese Academy of Sciences, Xi’an 710119, China c School of Chemical Engineering and Pharmaceutics, Henan University of Science and Technology, Luoyang 471003, China d School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore b

art ic l e i nf o

a b s t r a c t

Article history: Received 16 June 2014 Received in revised form 3 August 2014 Accepted 8 August 2014 Available online 2 September 2014

We demonstrate a method to measure the fractional topological charge (TC) of Laguerre–Gaussian (LG) beams by analyzing the interference intensity patterns between the vortex beam and its conjugate beam. By this method, the magnitude of integer and fractional TCs is quantitatively measured by using a simple unified formula. The proposed method can measure TCs up to 60. & 2014 Elsevier B.V. All rights reserved.

Keywords: Singular optics Optical vortices Fractional topological charge Measurement Interference intensity analysis

1. Introduction Optical vortices in scalar optical fields, in which the intensities are zero and phases are uncertain, have recently attracted attention in quantum information communication [1,2], micro-particle manipulation [3,4], and optical wrenches [5], among others [6–11]. An optical vortex carries an orbital angular momentum (OAM) of mћ [where m is the topological charge (TC)]. Recently, considerable attention has been paid to study the OAM properties. Therefore, measuring the TC of m is an important aspect in related studies. Two kinds of methods, namely, interference [12–15] and diffraction [16–20], can be used to measure TC. A commonly interferometric method is to interfere the measured vortex beams using a Mach– Zehnder interferometer [13–15], in which the interferometric patterns reveal the TC values of the optical vortices. Among these studies, Leach et al. measured the TC of a single photon using a Mach–Zehnder interferometer with a Dove prism placed in each arm [14]. Further, this method is able to measure the total angular momentum and simultaneously measure the spin and orbital angular momentum of single photons [15]. However, the interferometric methods are often involved with a complicated interferometric setup and unsatisfactory interference patterns.

n Corresponding author at: School of Physics and Engineering, Henan University of Science and Technology, Luoyang 471003, China E-mail address: [email protected] (X. Li).

http://dx.doi.org/10.1016/j.optcom.2014.08.020 0030-4018/& 2014 Elsevier B.V. All rights reserved.

To solve this problem, diffraction methods are always employed. Recently, an outstanding work is the triangular aperture diffraction method [16], which measures TC by simply counting the number of diffraction spots along the triangular lattice. Another method, annular aperture diffraction [17], quantitatively measures TCs by counting the number of bright rings. Several studies have used multi-pinhole plate [18], axicon [19], and spherical bi-convex lens [20] to determine TC. However, these methods including interference and diffraction only realize integer and half integer TC measurement [21,22]. To the best of our knowledge, only a few studies have reported on the effective measurement of fractional TCs. In this letter, we report a new method to measure the fractional TCs of Laguerre–Gaussian (LG) vortex beams by using a modified Mach–Zehnder interferometer with a Dove prism. Analyzing the interference intensity patterns between the vortex beam and its conjugate beam enables measurement of the integer and fractional TCs with the use of a simple formula. Moreover, this new method can be used to measure TC up to 60.

2. Theoretical background and experimental Considering an LG vortex beam, the simplified expression in cylindrical coordinates can be written as [23] !  jmj

 r r2 E1 r; φ ¼  exp  2  exp imφ ð1Þ w0 w0

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where w0 is the waist width of the vortex beam, and m is the TC. We rewrite m ¼m0 þ ε, where m0 and ε are the integer and fractional parts of the TC, respectively. The conjugate wave of this vortex beam can be expressed as !  jmj

 r r2 ð2Þ  exp  2  exp imφ E2 r; φ ¼ w0 w0 The interference intensity of these two vortex beams at the imaging plane yields
 

2 I r; φ ¼ E1 r; φ þ E2 r; φ  !  2jmj 
 r 2r 2 ð3Þ ¼2  exp  2  1 þ cos 2mφ w0 w0 The experimental setup is shown in Fig. 1. A He–Ne laser with 632.8 nm wavelength is used as the light source. After the application of the beam expander (BE, 10  ), the Gaussian beam becomes a plane wave that illuminates a computer hologram with controllable pixels written in a spatial light modulator (SLM, Holoeye LC-2012). This modulator contains the forked grating with the LG vortex beam determined by Eq. (1). A dove prism (Edmund Optics Inc., ♯49-427) is inserted into one optical path of the Mach–Zehnder interferometer and thus changes the sign of the TC and produces the conjugate wave of the LG beam. A CCD camera (Basler acA1600–20 gm/gc, pixel size: 4.4 mm  4.4 mm) is used to record and save the interference patterns into a computer.

3. Results and discussions By numerically using Eq. (3), we obtain the interference patterns between the LG vortex beam and its conjugate beam. We demonstrate the interference intensity patterns for integer TCs of m ¼ 2 and m¼ 5 in Fig. 2. In this case, the coordinate origin is located at the center point of each image. The bright fringes of n (where n is the number of the bright fringes) in Fig. 2 are arranged in a flower-like structure with four and 10 petals, twice that of the order of TCs. This rule is suitable for other integer TCs [24,25]. Therefore, we can measure the TCs by counting the number of bright fringes, m ¼n/2. The doughnut structure of the LG beam causes the bright fringes to be symmetrically arranged on a circle and plotted using white curves. The radius r0 of the circle increases with the order of the TCs, which is determined by the first factor in Eq. (3) [26]. The intensity change on two lines in the interference pattern is then analyzed. A horizontal line passes through the origin and the left vertical tangent line of the doughnut circle of the LG beam, are respectively represented by red and green curves, similarly hereinafter. The intensity on the horizontal line shows two peaks. The intensity of each peak obeys the Gaussian function distribution determined by the second factor in Eq. (3). Moreover, the values of

these two peaks are equal because of the circular symmetry of the fringe pattern. For integer TC measurement, this method can be used to measure higher integer TC values than previous methods [20,27]. In our experiment, the order of integer TC is measured up to m ¼60, and the result is presented in Fig. 3. The experimental result is consistent with the numerical simulation result. The number of the bright fringes is 120, and the order of TC is 60. By carefully arranging the experimental elements, larger values of TCs 4 60 are obtained, if necessary. For fractional TC measurement, the interference patterns are shown in Fig. 4. The patterns show that the TCs increase from 2.1 to 3 by increments of 0.1. When m is within 2 om r2.5, the bright fringe on the left vertical tangent lines of the doughnut circle of the LG beam gradually splits into two petals, which results in the formation of one new fringe. Meanwhile, the petal distribution gradually becomes uniform and exhibits rotational symmetry when m reaches 2.5. By contrast, when m is within 2.5 o mo 3, one new bright fringe is gradually generated at the same position. The intensity change on the left tangent lines explicitly reflects the progress of these two new fringes. Media 1 Fig. 4 shows only one symmetric axis for each pattern, which is plotted by a red horizontal dash line passes through the origin. The intensity change on the symmetric axes for m ¼ 2.1-3 shown by the red solid curves in Fig. 4 is then analyzed. Obviously, two peaks are observed on each red curve except for the curve of m ¼2.5. The intensity ratio of the two peaks on each red curve, V, is defined as the smaller peak value divided by the larger peak value. In this case, the domain of V is within [0 1]. The normalized intensity of the red solid curves in the interferometric patterns are re-plotted in Fig. 5. When m ¼2.5, only one peak is observed. The other peak is gradually formed as m decreases from 2.5 to 2.0 or increases from 2.5 to 3.0, which indicates that one petal annihilates or appears, respectively. The calculated peak intensity ratio V has the same value for the ε pairs 0.120.9, 0.220.8, 0.320.7, and 0.420.6, corresponding to V ¼0.9045, 0.6545, 0.3455, and 0.0955, respectively. If the bright fringe number is odd, then the value of ε is less than 0.5; otherwise, the value of ε is greater than 0.5. A comprehensive analysis of Eq. (3) reveals that the two fringes on the symmetrical axis correspond to the coordinates φ ¼ 0 and φ ¼ π, respectively. By substituting these values into Eq. (3), the value of V is determined by V¼[cos(εn2π) þ1]/2. V is equal to zero if ε is 0.5, in which case one peak disappears. Through simple arithmetic, the fractional part ε can be calculated by ε ¼ acos (2V 1)/(2π) when the bright fringe number is odd. Conversely, the value of ε is calculated by ε ¼  acos(2V  1)/(2π) when the bright fringe number is even. Further numerical results show that the peak intensity ratio V takes the same value for equal fractional parts but different values for integer parts of two TCs. This rule can be observed in Media 1,

Fig. 1. Schematic of the experimental setup employed to measure the TCs of the optical vortices. BE: Beam Expander; SLM: Spatial Light Modulator; BS1, BS2: Beam splitters; M1, M2: Mirrors.

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Fig. 2. Interference intensity patterns for integer TCs of m¼ 2 and m¼ 5.

Fig. 3. Interference patterns of the TC of m¼ 60: (a) Numerical simulation and (b) experimental results.

Fig. 4. Numerical simulations of interference patterns obtained for m¼ 2.1-3 by steps of m¼0.1. Change in interference patterns when TCs increase from 0 to 5 by a step of 0.1 (Media 1). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

in which the interference patterns change when TCs increase from 0 to 5 in steps of 0.1. These results indicate that the fractional TCs, including integer and half integer TCs, can be obtained in three steps. The first step is determining the integer part TCs by using the formula m0 ¼(n  1)/2 or m0 ¼n/2, where n is the number of bright fringes for odd and even values of n, respectively. The second step is obtaining the fractional part of the TCs by using ε ¼acos(2V  1)/(2π) or ε ¼  acos(2V 1)/(2π) for odd and even

values of n, respectively. The third step is measuring the fractional TCs by using the following expression, ( m ¼ m0 þ ε ¼

ðn  1Þ=2 þ arccosð2V 1Þ=ð2π Þ; n=2  arccosð2V  1Þ=ð2π Þ;

n is odd n is even

ð4Þ

To verify the effectiveness of this method, we experimentally measure the fractional TCs that change from 2.1 to 3.0 by

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Video 1. Change in interference patterns when TCs increase from 0 to 5 by a step of 0.1. A video clip is available online. Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.optcom.2014.08.020.

Fig. 5. Intensity curves on the symmetric axes of the interference patterns in Fig. 4. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

increments of 0.1. The results are shown in Figs. 6 and 7. Fig. 6 shows the interference intensity patterns, in which the red dashed lines denote the symmetry axis of each pattern. The symmetry axes rotate at an angle because of dove prism insertion. The wavefront phase of the conjugate wave is delayed by approximately 2π/λn34.104 mm compared with the vortex wave. In this case, the manner by which to confirm the symmetry axis of the fringe pattern is important. However, the fringes of m ¼2.5 are not uniformly distributed obviously for the experimental results of Fig. 6. This phenomenon is attributed to the two beams mismatches as a result of misalignment in the experimental elements. Moreover, the aberrations and imperfections of the optical elements is another reason to disturb the interference patterns. When this method is used to measure the order of TCs, carefully arranging the experimental elements is therefore needed. The V values of the two peaks on the symmetry axes are obtained by using image-processing techniques (Fig. 7). After counting the number of bright fringes n, the fractional TCs, m, can be easily calculated using Eq. (4). In this context, the order of the TCs is positive, that is, m is the magnitude of the TC. Given that this method uses the vortex beam and its conjugate wave interference, the sign of the TC is not determined but can be easily measured by other methods, such as triangle diffraction.

Fig. 7. Ratio V between the values of the two peaks (dashed lines in Fig. 6) versus the TCs.

Fig. 6. Experimental interference patterns for the fractional TCs m ¼2.1-3 in steps of m ¼0.1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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The proposed method can be used at a resolution of 0.1. Higher resolution can be achieved if necessary, but adjusting the interference elements becomes more difficult. Moreover, the proposed method for fractional TC measurement is based on the analysis of the LG vortex beam. Further studies are needed to measure fractional TCs of other optical vortex beams by using the proposed method. 4. Conclusions In summary, a new fractional interference intensity-based TC measurement method is presented. This method measures the integer TC up to 60. This technique can be used in various applications, such as optical tweezers. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant no. 61205086), Open Research Fund of State Key Laboratory of Transient Optics and Photonics, Chinese Academy of Sciences (SKLST201203), and Key Research Project of Education Minister of Henan Province (no. 12B140006). References [1] G. Molina-Terriza, J.P. Torres, L. Torner, Nat. Phys. 3 (2007) 305. [2] G. Molina-Terriza, A. Vaziri, J. Řeháček, Z. Hradil, A. Zeilinger, Phys. Rev. Lett. 92 (2004) 167903.

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