Local topological charge analysis of electromagnetic vortex beam based on empirical mode decomposition

Xiaonnan Hui; Shilie Zheng; Weite Zhang; Xiaofeng Jin; Hao Chi; Xianmin Zhang

doi:10.1364/OE.24.005423

1. Introduction

Since Allen et al. recognized in 1992 that the light beams with the transverse azimuthal dependence of exp(−jmφ) carry orbital angular momentum (OAM) [1], this fundamental physical property of electromagnetic waves is widely investigated [2–5 ], and applied in many fields, such as micro-machines driving, atoms trapping, stimulated emission depletion microscopy, rotational Doppler shift, and high transmission capacity communication [6–11 ]. It is well known that OAM-carrying beams have the features of helicoidal wave-front and an amplitude null in the center of the beam. Due to the OAM beam divergence, the ‘dark zone’ around the amplitude null will become larger with beam traveling. The divergence will also be more serious with the increase of the topological charge m. The divergence and imperfection of the OAM beams make it significant to investigate the local topological charges. In the last decade, various means to sort and detect OAM mode or superposition modes have been proposed with whole angular aperture receiving scheme [11–14 ] or partial angular aperture receiving scheme [15]. However, the Fourier transform based de-multiplexing is basically restrained by the uncertainty principle [16], and cannot achieve high angular resolution and mode resolution simultaneously. The empirical mode decomposition (EMD) is one process of Hilbert-Huang transform (HHT). Unlike the Fourier transform which is a priori-defined basis transformation, HHT is a posteriori-defined bases method, which is not limited by the uncertainty principle [17] and has been widely applied in many research fields [17–19 ].

In this paper, we demonstrate an analysis method for local topological charges of vortex beams with superposition OAM modes based on EMD. The bases of the electromagnetic wave can be constructed by the intrinsic mode functions (IMFs) from the EMD, and thus each local topological charge can be respectively defined. Both the azimuth angle and local topological charge can be obtained with high resolution, and the OAM spectrum can be defined. The radiation of the multi-ring resonator OAM antenna with multiple OAM modes is numerically simulated and experimentally measured, and the local topological charge spectra are analyzed. The results confirm the validity of the EMD based method.

2. Theoretical principle

For an n-OAM modes superposed beam, the co-polarization electric field along the azimuth angle can be presented as

E (φ) = \sum_{i = 1}^{n} A_{i} \exp (- j m_{i} φ)

where A_i is the amplitude of OAM mode i, and m_i is the topological charge. Under the Fourier transform based OAM spectrum definition, the spectrum can be presented as

S (m_{k}) = | \int_{- π}^{π} E (φ) \exp [- j (- m_{k}) φ] d φ |

where m_k∈(−∞, ∞). Obviously, the information of azimuth angle will be lost under this definition. In order to obtain the azimuth angle information, the common method is to insert an angular aperture or employ the windowed Fourier transform. When only one OAM mode is considered, that is to say n = 1 in the Eq. (1), the local OAM mode spectrum can be presented as

S (m_{k}) = | \int_{φ_{0} - Δ φ / 2}^{φ_{0} + Δ φ / 2} E (φ) \exp [- j (- m_{k}) φ] d φ | = A_{1} Δ φ \sin c[(m_{1} - m_{k}) Δ φ / 2]

The envelope of the spectrum is the sinc function, as shown in Fig. 1 . The red bar chart is the OAM mode spectrum of m ₁ = 2 with the partial angular receiving aperture Δφ = 10°. It is clear that the spectrum expands greatly, so that the topological charge can hardly be identified. The blue bar chart is the spectrum of the superposition OAM field of m_i = 2, −6, and 17, with the partial angular receiving aperture Δφ = 10°. It does not show any information of the topological charges. So, the Fourier transform based OAM spectrum definition is restricted by the uncertainty principle and hardly to achieve high OAM mode resolution and angular resolution simultaneously.

Fig. 1 The topological charge spectra with partial angular receiving aperture Δφ = 10°. Red bar: the spectrum of m = 2, and blue bar: the spectrum of superposed filed of m = 2, −6, and 17.

Download Full Size | PDF

Here, we demonstrate a new definition of local topological charges for vortex beams based on EMD. For n = 1, i.e. only one OAM mode is considered, the local topological charge m(φ) and the corresponding field amplitude a(φ) can be easily defined as

m (φ) = \frac{d θ (φ)}{d (φ)},

where

θ (φ) = arc \tan [\frac{E_{i m} (φ)}{E_{r e} (φ)}]

and

a (φ) = \sqrt{E_{r e} {(φ)}^{2} + E_{i m} {(φ)}^{2}}

where E_re(φ) and E_im(φ) are the real part and the imaginary part of E(φ), respectively. When n>1, i.e. there are multiple OAM modes superposition, the EMD process are introduced to decompose the superposed fields of E_re(φ) and E_im(φ). With the IMFs from EMD, both the local topological charge m(φ) and the amplitude a(φ) can be obtained.

Take E_re(φ) as the example, the EMD process can be operated as follows. Supposing the real part of a superposition OAM field is shown as the blue solid curve in Fig. 2 , firstly, the local maximum values of E_re(φ) are identified, and connected with a cubic spline, as the upper envelop (red dash line) in Fig. 2. Then, repeat this operation to the local minimum values. The lower envelope line (green dot line) can be obtained. The mean value (black dash dot line) of the upper and lower envelopes is designated as m ₁₁. The difference between E_re(φ) and m ₁₁ is described as h _11,

Fig. 2 An example of the real part of a superposition OAM field (blue solid line) to be analyzed. Upper envelop (UE, red dash line), lower envelop (LE, green dot line), and the mean value curve (black dash dot line).

Download Full Size | PDF

h_{11} = E_{r e} (φ) - m_{11}

For the EMD, the IMF definition should meet two conditions: (a) the sum of maxima and minima in the data must either equal with the zero-crossings or differ at most by one; (b) the mean value of the two envelopes defined by the maxima and the minima is zero.

In practice, h ₁₁ cannot meet the above IMF definition, hence the sifting processes should be operated. h ₁₁ is usually treated as the proto-data for the first sifting process. By repeating the above EMD procedures, h ₁₂ can be obtained,

h_{12} = h_{11} - m_{12}

where m ₁₂ is the mean value obtained from h ₁₁.

After k times’ iterating, h ₁ _k can meet the above two IMF definition, thus,

h_{1 k} = h_{1 (k - 1)} - m_{1 k}

where m ₁₂ is the mean value obtained from h _1(k-1). h ₁ _k is the first IMF of E_re(φ) designated as c_re ₁.

As IMF₁ is obtained, the residue r_re ₁ can be obtained by subtracting the IMF₁ from the original data.

r_{r e 1} = E_{r e} (φ) - c_{r e 1}

Since r_re ₁ may also consist in some longer period oscillations, the same decomposition continues, and the following IMFs can be obtained. Repeat the sifting processing described by Eq. (6) to Eq. (9), until the final residue r_ren is either the mean trend or a constant.

Till now, all IMFs (c_rei) are obtained with this method, and the E_re(φ) can be presented as

E_{r e} (φ) = \sum_{i = 1}^{n} c_{r e i} + r_{r e n}

Similarily, the imaginary part E_im(φ) can be presented as

E_{i m} (φ) = \sum_{i = 1}^{n} c_{i m i} + r_{i m n}

Combined with Eqs. (4) and (5) , for each pair of c_rei and c_imi, the local topological charge and amplitude can be obtained as

m_{i} (φ) = \frac{d θ_{i} (φ)}{d φ}, w h e r e θ_{i} (φ) = arc \tan (\frac{c_{i m i}}{c_{r e i}})

and

a_{i} (φ) = \sqrt{c_{r e i}^{2} + c_{i m i}^{2}}

The field can be expressed as

E (φ) = \sum_{i = 1}^{n} a_{i} (φ) \exp [- j \int m_{i} (φ) d φ]

here, the residuals are not considered. After the EMD process, the local topological charges of the superposed OAM beam can be defined and calculated by Eqs. (12)-(14) .

3. Simulation and experiment

To verify the definition method, supposing a superposition field, which contains the OAM modes of −6, 2, and 17 with the amplitude ratio of 2.5:4.5:3. The real part of the field is shown as the blue solid line in Fig. 2. The field is discretized by 360 samples with a step of 1°. The real part and the imaginary part are processed by the EMD respectively, and the results of IMFs are shown in Fig. 3 . It can be seen that the data are mainly decomposed into 3 IMFs, namely 3 topological oscillations. The components with topological charges m = 17, −6 and 2 are decomposed into IMF₁, IMF₂, and IMF₃, respectively.

Fig. 3 The IMFs of the field. (a) IMFs of the real part, and (b) IMFs of the imaginary part.

Download Full Size | PDF

With the definition of local topological charge, the analytic expression of the field can be written as Eq. (14), so the local topological charge spectrum can be calculated, as shown in Fig. 4(a) . The local topological charge spectrum demonstrates the relationship of azimuth angle (horizon axis), calculated topological charge number (vertical axis), and the amplitude (color). It can be seen from Fig. 4(a), there are 3 horizontal lines representing the components of m = −6, 2, and 17 along the azimuth angle. Although there is some small oscillating variation of the local topological charges, which is caused by the error of the EMD decomposition, its exactness is tolerable. The amplitudes denoted by the color show good agreement with the set ratio of 2.5:4.5:3. Therefore, the local topological charge information can be identified from the spectrum easily. The azimuth angle resolution depends on the sampling resolution (1° in this case). The local topological charges can also be integrated along azimuth angle. Figure 4(b) shows the integrated value within 10° from 171° to 180°. Compared with the result based on Fourier transform shown in Fig. 1, the local topological charge spectrum presents a significant meaning for OAM mode analysis. It should be noted that the intensity peaks in Fig. 4(b) do not agree with the amplitudes’ ratio of 2.5:4.5:3. The reason lies that the fluctuations of the calculated mode spectra [the yellow, red and cyan spectra in Fig. 4(a)] widen the mode peaks in different way, which makes the intensity of the mode peak different with the original amplitude ratio. However the integral values for each mode peak are 2.75, 4.90, and 3.29, respectively, which agree with the set ratio of 2.5:4.5:3.

Fig. 4 The theoretical calculation of (a) the local topological charge spectrum, and (b) the 10° integrated local topological charge spectrum.

Download Full Size | PDF

To verify the calculation results, the simulation and experiment are performed. A multi-ring resonator OAM antenna with center frequency of 10 GHz is designed based on our previous works [20, 21 ], as shown in Fig. 5(a) . There are three resonators, namely resonators A, B, and C, respectively, which radiate the microwave beams carrying OAM modes m = 2, −6, and 17, respectively. It is known that the azimuthal phase distributions of radial, azimuthal and axial components of the electric field radiated by the circular traveling-wave antenna are the same in the cylindrical coordinate system [20]. Hence we can only use the radial polarized field to calculate the local topological charge. Figure 5(b) is the simulation results by CST Microwave Studio. It shows the real part of electric field in radial polarization. The observation window is 500mm × 500mm lies in the plane 300 mm away from the antenna. The data are extracted along a circle with radius of 120 mm, shown as the black circle in Fig. 5(b). The chamfer edge around the ring slots are carefully designed to make sure that the divergence of the three OAM waves are very similar, hence the effective intensity of all the OAM waves can be obtained. The amplitude ratio for the three OAM modes m = −6, 2, and 17 is also designed to be 2.5:4.5:3, which can be controlled by feeding power. Using the EMD method, the local topological charge spectrum is obtained and shown in Fig. 6(a) . The spectrum exhibits the distribution of topological charge along the azimuth angle clearly, which fits well with the theoretically calculated spectrum shown in Fig. 4(a) except that the oscillation variation of the local topological charge along the azimuth angle is a bit severe and the intensity of high-order OAM mode is not so uniform. This result is reasonable and further verifies the validity of this EMD based method because the generated OAM fields from the simulated antenna is not so perfect as the calculated fields as Fig. 3 shows, especially for the high-order OAM mode. Figure 6(b) shows the 10° integrated local topological charge spectrum, the integral values for each mode peak are 2.58, 4.88, and 3.48, respectively, which is also consistent with the theoretical values shown in Fig. 4(b).

Fig. 5 The simulation model and the simulation field results. (a) The multi-resonator OAM antenna model, and (b) the simulation results of the real part of electric field in radial polarization.

Download Full Size | PDF

Fig. 6 The simulation results. (a) The local topological charge spectrum, and (b) the 10° integrated local topological charge spectrum.

Download Full Size | PDF

To do the further verification, an experiment is carried out. A two-resonator basecd OAM antenna is designed and fabricated with copper, as shown in Fig. 7(a) . A beam carrying OAM modes m = −2 and 3 can be generated simultaneously. This OAM antenna is settled on a rotation platform and thus the near-field electric field along a circle can be measured. The measurement plane is 400 mm away from the OAM antenna, and the circle radius is 200 mm. The radial polarized field is detected by an open-end waveguide, and the real part and the imaginary part of the fields are recorded by a vector network analyzer (R&S ZVA-67). The field is sampled with an angular step of 2°. The measured data are then processed with the above described method. Figure 7(b) shows the local topological charge spectrum and Fig. 7(c) shows the integrated spectrum between 171° to 180°. It is seen that the topological charges are mainly distributed at the modes m = −2 and 3. Since the field is sampled every 2°, the azimuth angular resolution achieves about 2°. The resolution of the calculated topological charge depends on the calculation setting, the phase accuracy is set to 0.2°, which means 0.2 is the mode resolution in the spectral diagram.

Fig. 7 The experimental results. (a) Two-resonator OAM antenna, (b) the local topological charge spectrum, and (c) the 10° integrated local topological charge spectrum.

Download Full Size | PDF

It can be seen that this method can be easily applied in the radio frequency, as the real part and the imaginary part can be obtained by a vector network analyzer. However, it is rather challenging to apply this method in optical frequencies. To get the real part of the field in optical regime, one common solution is to employ an interferometer, and then acquire the real part with a camera or a PD array. Afterwards, EMD can be performed to get the IMFs. The imaginary part of each IMFs can be obtained by Hilbert transform.

4. Conclusion

We demonstrate a method to define and calculate the local topological charges of the superposed multi-mode OAM beam. The EMD method decomposes the field into the IMFs which represent the variation of the field along the azimuth angle. Based on the EMD method, the local topological charge spectrum can be calculated. The angular resolution achieves as high as 1° in the simulation and 2° in the experiment, and the topological charge number resolution is set as 0.2. Compared with the Fourier transform, the EMD based method can achieve high angular and mode resolution simultaneously, especially for the fields in non-stationary or those superpositions of OAM modes that present discontinuities and irregularities. It offers a powerful analysis tool for the applications of electromagnetic orbital angular momentum.

Acknowledgments

This work was supported by the National Basic Research Program of China (973 program) under Grant 2014CB340005, and the Natural Science Foundation of China under Grant 61371030 and 61571391.

References and links

1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef] [PubMed]

2. S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev. 2(4), 299–313 (2008). [CrossRef]

3. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011). [CrossRef]

4. M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A, Pure Appl. Opt. 6(2), 259–268 (2004). [CrossRef]

5. G. Parisi, E. Mari, F. Spinello, F. Romanato, and F. Tamburini, “Manipulating intensity and phase distribution of composite Laguerre-Gaussian beams,” Opt. Express 22(14), 17135–17146 (2014). [CrossRef] [PubMed]

6. G. Knöner, S. Parkin, T. A. Nieminen, V. L. Y. Loke, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Integrated optomechanical microelements,” Opt. Express 15(9), 5521–5530 (2007). [CrossRef] [PubMed]

7. T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997). [CrossRef]

8. T. J. Gould, S. T. Hess, and J. Bewersdorf, “Optical nanoscopy: from acquisition to analysis,” Annu. Rev. Biomed. Eng. 14(1), 231–254 (2012). [CrossRef] [PubMed]

9. M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013). [CrossRef] [PubMed]

10. J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012). [CrossRef]

11. F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14(3), 033001 (2012). [CrossRef]

12. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002). [CrossRef] [PubMed]

13. C. Schulze, A. Dudley, D. Flamm, M. Duparre, and A. Forbes, “Measurement of the orbital angular momentum density of light by modal decomposition,” New J. Phys. 15(7), 073025 (2013). [CrossRef]

14. M. P. J. Lavery, D. J. Robertson, A. Sponselli, J. Courtial, N. K. Steinhoff, G. A. Tyler, A. E. Willner, and M. J. Padgett, “Efficient measurement of an optical orbital angular momentum spectrum comprising more than 50 states,” New J. Phys. 15(1), 013024 (2013). [CrossRef]

15. S. Zheng, X. Hui, J. Zhu, H. Chi, X. Jin, S. Yu, and X. Zhang, “Orbital angular momentum mode-demultiplexing scheme with partial angular receiving aperture,” Opt. Express 23(9), 12251–12257 (2015). [CrossRef] [PubMed]

16. S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, “Uncertainty principle for angular position and angular momentum,” New J. Phys. 6, 103 (2004). [CrossRef]

17. N. E. Huang and S. S. P. Shen, The Hilbert-Huang Transform and Its Application (World Scientific Publishing, 2005).

18. N. E. Huang and Z. Wu, “A review on Hilbert-Huang transform: method and its applications to geophysical studies,” Rev. Geophys. 46, RG000228 (2008).

19. X. Hui, S. Zheng, J. Zhou, H. Chi, X. Jin, and X. Zhang, “Hilbert-Huang transform time-frequency analysis in ϕ-OTDR distributed sensor,” IEEE Photon. Technol. Lett. 26, 2403–2406 (2014). [CrossRef]

20. S. Zheng, X. Hui, X. Jin, H. Chi, and X. Zhang, “Transmission characteristics of a twisted radio wave based on circular traveling-wave antenna,” IEEE Trans. Antenn. Propag. 63(4), 1530–1536 (2015). [CrossRef]

21. X. Hui, S. Zheng, Y. Chen, Y. Hu, X. Jin, H. Chi, and X. Zhang, “Multiplexed millimeter wave communication with dual orbital angular momentum (OAM) mode antennas,” Sci. Rep. 5, 10148 (2015). [CrossRef] [PubMed]

Local topological charge analysis of electromagnetic vortex beam based on empirical mode decomposition

Abstract

1. Introduction

2. Theoretical principle

3. Simulation and experiment

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (7)

Equations (15)

Optics Express