Determining topological charge based on an improved Fizeau interferometer

Shengwei Cui; Bin Xu; Saiyu Luo; Huiying Xu; Zhiping Cai; Zhengqian Luo; Jixiong Pu; Sabino Chávez-Cerda

doi:10.1364/OE.27.012774

1. Introduction

Vortex beams are widely studied as a typical high-order Laguerre-Gaussian mode, which possess particular helical phase structures exp(il $ϕ$ ) and central singularities [1,2]. The topological charge l refers to the number of 2π phase cycle around the singularity, meanwhile, every photon of the vortex beam carries an orbital angular momentum (OAM) of l $ℏ$ . Because of the specific helical characteristics, vortex beams are widely used in quantum-information coding [3–5], multiplexing OAM optical communication [6–8], detecting a spinning object [9] and optical tweezers and spanners [10–13].

Therefore, determining topological charge becomes a research hot topic in recent years, and varies techniques are proposed [14–21]. The typical and classic determining method is based on Mach-Zehnder interferometer, which can easily to distinguish both the sign and magnitude of topological charge [14,22,23]. As we all know, Mach-Zehnder interferometer is a very sensitive configuration and needs a complex equipment, thus the topological charge determining methods of triangular aperture, double slits interference, wedged optical flat, and cylindrical lens mode converters were proposed and widely used as a convenient way. However, these convenient methods have their own disadvantages, especially for determining the handedness of vortex beams.

As an eminently suitable method, Fizeau interferometer has been extensively applied to measure optical flatness, which places two reflecting surfaces facing each other to form interference fringes [24,25]. Due to its unique compact structure, Fizeau interferometer is much easier to be realized with a great accuracy, as well as less sensitive to air turbulence or mechanical vibrations than other interferometers, such as Mach-Zehnder and Michelson configurations.

In this paper, we propose an improved Fizeau interferometer to determine both the sign and magnitude of topological charge by simply using a suitable coated flat-concave mirror. We verified the phase of vortex, Hermite-Gaussian and elliptical vortex beams both in theory and experiment. Excitingly, the inference fringes are shown as distinct as Mach-Zehnder interferometer. Using this determining method, we can easily get the information of both sign and magnitude of topological charge.

2. Improved Fizeau interferometer

The schematic diagram is shown in Fig. 1(a). A vortex beam obliquely incidences to a suitable coated flat-concave mirror, which has a low reflectance on flat surface and high reflectance on curved surface. The vortex beam remains its shape after the front flat surface reflection, but carries a spherical phase after reflected by the rear curved surface. Due to the obliquely incident angle, the centers of the two reflection beams will separate completely after a short distance transmission. Then using a charge-coupled device (CCD) to detect the interference fringes.

Fig. 1 Schematic diagram and geometry diagram for Fizeau interferometer determining the topological charge.

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The field amplitude E of incident vortex beam can be expressed as [20,26]

E (r, ϕ, z) = ε \exp [- i k r^{2} / 2 R (z)] \exp (- i k z) \exp (- i l ϕ),

where z is the distance, r is the radius, ϕ is the azimuthal angle, l is the topological charge and the wavefront curvature R(z) is the radius of the curvature of the wavefront. After the vortex beam reflected by the flat and curved surface, the fields respectively are

E (r, ϕ, z_{1}) = \sqrt{η_{1}} ε \exp [- i k r^{2} / 2 R (z_{1})] \exp (- i k_{1} z_{1}) \exp (- i l ϕ),

E (r, ϕ, z_{2}) = (\frac{R_{0}}{R_{0} + z_{2}}) \sqrt{η_{2}} ε \exp [- \frac{i k r^{2}}{2 [R (z_{2}) + R_{0}]}] \exp (- i k_{2} z_{2}) \exp (- i l ϕ),

where η₁, η₂ are the reflectances on flat and curved surfaces, respectively. R₀ is the radius of curved surface. The optical path difference D between two reflect beams can be expressed as

D = n l_{1} + n l_{2} + D_{2} - D_{1},

in which, n is the refractive index of the mirror, as shown in Fig. 1(b), we have

l_{1} = \frac{d_{1}}{\cos {\sin^{- 1} [(\sin α) / n]}},

l_{2} = \frac{d_{1}}{\cos {\sin^{- 1} [(\sin α) / n] + 2 β}},

d_{1} = d_{0} + R_{0} (1 - \cos β),

D_{2} = \sqrt{D_{0}^{2} + D_{1}^{2} \cos^{2} α},

D_{0} = D_{1} \sin α - l_{1} - l_{2},

The two reflect beams can interfere with each other when the coherent length is longer than the optical path difference. In normal condition, the thickness of the flat-concave mirror is no more than 6 mm as well as the optical path difference D is less than 20 mm. That means, illuminating with a laser source can easily get the interference fringes by overlapping the two reflect beams. Due to the thickness of the flat-concave mirror, there always be an optical path difference between two reflect beams. If we want to determine the topological charge of a short coherent length vortex beam, we need the mirror as thin as possible.

According to Eq. (3), the vortex light field after the rear curved surface carries a spherical phase, as shown in Fig. 2 (a). The interference area with the reflect beam by flat surface is marked by a green square. Figure 2(b) shows the phase with the center of the interference area, which can be regarded as a standard spherical phase. Actually, in experiment, the distance between interference area with the vortex singularity is much further than shown in Fig. 2. Moreover, due to the spherical light intensity $I_{c u r v e d} \propto {(\frac{R_{0}}{R_{0} + z_{2}})}^{2}$ , which means the intensity is inversely proportional to the square of the distance. Through adjusting the distance between the mirror and CCD we always can get the two reflect intensities comparable $(I_{f l a t} \approx I_{c u r v e d})$ as well as distinct inference fringes.

Fig. 2 Vortex spherical phases on-axis and off-axis, the interference area is marked by the green square.

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The simulated interference fringes of different vortex beams with spherical beam are shown in Fig. 3. Figures 3(a1-d1) are phase profiles and Figs. 3(a2-d2) are corresponding interference fringes. Moreover, Figs. 3(a) (b) are on-axis topological charge of 1 and −4, Figs. 3(c) (d) are off-axis topological charge of 1 and −1, respectively. We can determine the on-axis topological charge by the number and direction of spiral fringes from the central singularity, as well as the off-axis topological charge by off-center fork fringes.

Fig. 3 Simulated interference fringes of different vortex beams with spherical beam. (a) (b) on-axis l = 1, −4, and (c) (d) off-axis l = 1, −1 respectively.

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3. Experimental demonstration

The experimental setup is shown in Fig. 4, in which a He-Ne laser is used as a coherent light source. After a collimating system and a linear polarizer, the laser beam incidences on a phase-only spatial light modulator (SLM, PLUTO | Holoeye Photonics AG) to get a vortex phase. Then we use a flat-concave mirror to generate vortex beam and spherical beam interference fringes, which can be recorded by a conventional CCD. The diameter of the flat-concave mirror is 1 inch, thickness is about 3 mm and the radius of concave side is 100 mm. The reflectances of the mirror on flat and curved surfaces are 5% and 90%, respectively. The distance between flat-concave mirror and CCD is about 400 mm.

Fig. 4 Experimental setup for determining topological charge by using an improved Fizeau interferometer.

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As shown in Fig. 5, we measured different orders of vortex beams to verify our proposed method. From the interference fringes, we can accurately determine the topological charges of vortex beams Figs. 5(a, b, c) are ± 1, ± 4, and ± 10. The corresponding intensities are shown in Figs. 5(a1, b1, c1), which show the size of central singularity will be larger with increasing the topological charge. These interference fringes are as distinct as Mach-Zehnder inference fringes [14,22,23], which can determine both sign and magnitude of topological charge simply.

Fig. 5 Intensities, phase profiles and corresponding interference fringes of different vortex beams. (a1-a3) intensities of l = ± 1, ± 4, and ± 10, (a2) (a3) l = 1, (a4) (a5) l = −1, (b2) (b3) l = 4, (b4) (b5) l = −4, (c2) (c3) l = 10, (c4) (c5) l = −10.

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The highest vortex beam we determined in this experiment is l = 30, as shown in Fig. 6. Due to the central phase of l = 30 is too tight to be distinguished by the SLM, the center of the vortex beam is not totally dark any more, as shown in Fig. 6(a). However, the interference fringes show a very good result, in which there are exactly 30 clockwise spiral fringes from the singularity, as shown in Fig. 6(c).

Fig. 6 Intensity, phase profile and interference fringes of l = 30 vortex beam.

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Additionally, we determined the Hermite and elliptical vortex beams by using the same method. Hermite-Gaussian mode is the eigen solution of the paraxial wave equation deduced from Maxwell’s equations by means of the paraxial approximation in rectangular coordinates [27]. Elliptical vortex beam is a superposition of a number of vortices, which has OAM on the off-axis [28]. However, using most existing convenient methods is hard to measure off-axis topological charge. The phase profiles and their corresponding interference fringes are shown in Fig. 7. Figure 7(a) is HG_2,0 mode, 7(b) has two off-axis l = 1 vortices, 7(c) has one on-axis l = 1 and two off-axis l = −1 vortices, 7(d) has four off-axis l = −1 vortices. From theoretical simulations Figs. 7(a2-d2) and the experiment results Figs. 7(a3-d3), the phase information of Hermite-Gaussian and elliptical vortex beams can be determined similarly with vortex beams. The sign and magnitude of off-axis topological charge also can be determined by off-axis fork fringes.

Fig. 7 Phase profiles and corresponding interference fringes of Hermite-Gaussian and elliptical vortex beams. (a) HG_2,0 mode, (b-d) elliptical vortex beams.

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4. Conclusion

In conclusion, based on an improved Fizeau interferometer, we realized determining topological charges of vortex beams by using a suitable coated flat-concave mirror. Because of the two reflected surfaces are relatively static, the interference is very stable and not sensitive to external condition. Vortex, Hermite-Gaussian, and elliptical vortex beams are commendably verified by this method. Moreover, the topological charge of l = 30 is also determined in experiment. Note that other kinds of spherical interference fringes can be obtained by this approach. However, this method only can be used in single-OAM detecting. For applying to multiplexing OAM communication, this method will require further improvement.

Funding

National Natural Science Foundation of China (NSFC) (11674269, 91750115); Natural Science Foundation of Fujian Province of China (2018J01108); Principal Fund of Xiamen University (20720180082); International Collaborative Laboratory of 2D Materials for Optoelectronics Science and Technology, Shenzhen University (2DMOST2018026); Scientific Research Foundation for Returned Scholars, Ministry of Education of China.

References

1. L. Allen, M. W. Beijersbergen, R. J. 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. M. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. 121(1-3), 36–40 (1995). [CrossRef]

3. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001). [CrossRef] [PubMed]

4. J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4(4), 282–286 (2008). [CrossRef]

5. A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8(3), 234–238 (2014). [CrossRef]

6. A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015). [CrossRef]

7. Y. Ren, Z. Wang, P. Liao, L. Li, G. Xie, H. Huang, Z. Zhao, Y. Yan, N. Ahmed, A. Willner, M. P. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, I. B. Djordjevic, M. A. Neifeld, and A. E. Willner, “Experimental characterization of a 400 Gbit/s orbital angular momentum multiplexed free-space optical link over 120 m,” Opt. Lett. 41(3), 622–625 (2016). [CrossRef] [PubMed]

8. Z. S. Eznaveh, J. C. A. Zacarias, J. E. A. Lopez, K. Shi, G. Milione, Y. Jung, B. C. Thomsen, D. J. Richardson, N. Fontaine, S. G. J. O. e. Leon-Saval, and R. A. Correa, “Photonic lantern broadband orbital angular momentum mode multiplexer,” Opt. Express 26(23), 30042–30051 (2018). [CrossRef] [PubMed]

9. M. P. 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. N. Simpson, L. Allen, and M. Padgett, “Optical tweezers and optical spanners with Laguerre–Gaussian modes,” J. Mod. Opt. 43(12), 2485–2491 (1996). [CrossRef]

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

12. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011). [CrossRef]

13. M. J. Padgett, “Orbital angular momentum 25 years on [Invited],” Opt. Express 25(10), 11265–11274 (2017). [CrossRef] [PubMed]

14. N. Heckenberg, R. McDuff, C. Smith, H. Rubinsztein-Dunlop, and M. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24(9), S951–S962 (1992). [CrossRef]

15. 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]

16. H. I. Sztul and R. R. Alfano, “Double-slit interference with Laguerre-Gaussian beams,” Opt. Lett. 31(7), 999–1001 (2006). [CrossRef] [PubMed]

17. J. M. Hickmann, E. J. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105(5), 053904 (2010). [CrossRef] [PubMed]

18. Q. S. Ferreira, A. J. Jesus-Silva, E. J. Fonseca, and J. M. Hickmann, “Fraunhofer diffraction of light with orbital angular momentum by a slit,” Opt. Lett. 36(16), 3106–3108 (2011). [CrossRef] [PubMed]

19. S. N. Alperin, R. D. Niederriter, J. T. Gopinath, and M. E. Siemens, “Quantitative measurement of the orbital angular momentum of light with a single, stationary lens,” Opt. Lett. 41(21), 5019–5022 (2016). [CrossRef] [PubMed]

20. B. Khajavi and E. J. Galvez, “Determining topological charge of an optical beam using a wedged optical flat,” Opt. Lett. 42(8), 1516–1519 (2017). [CrossRef] [PubMed]

21. L. A. Melo, A. J. Jesus-Silva, S. Chávez-Cerda, P. H. S. Ribeiro, and W. C. J. S. R. Soares, “Direct measurement of the topological charge in elliptical beams using diffraction by a triangular aperture,” Sci. Rep. 8(1), 6370 (2018). [CrossRef] [PubMed]

22. Z. F. Zhiqiang Fang, Y. Y. Yao Yao, K. X. Kegui Xia, and J. L. Jianlang Li, “Simple Nd:YAG laser generates vector and vortex beam,” Chin. Opt. Lett. 13(3), 031405 (2015). [CrossRef]

23. D. J. Kim and J. W. Kim, “Direct generation of an optical vortex beam in a single-frequency Nd:YVO₄ laser,” Opt. Lett. 40(3), 399–402 (2015). [CrossRef] [PubMed]

24. R. Bünnagel, H. A. Oehring, and K. Steiner, “Fizeau interferometer for measuring the flatness of optical surfaces,” Appl. Opt. 7(2), 331–335 (1968). [CrossRef] [PubMed]

25. K. Hibino, B. F. Oreb, P. S. Fairman, and J. Burke, “Simultaneous measurement of surface shape and variation in optical thickness of a transparent parallel plate in wavelength-scanning Fizeau interferometer,” Appl. Opt. 43(6), 1241–1249 (2004). [CrossRef] [PubMed]

26. M. E. Riley and M. A. Gusinow, “Laser beam divergence utilizing a lateral shearing interferometer,” Appl. Opt. 16(10), 2753–2756 (1977). [CrossRef] [PubMed]

27. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5(10), 1550–1567 (1966). [CrossRef] [PubMed]

28. S. Chavez-Cerda, J. C. Gutierrez-Vega, and GHC. New, “Elliptic vortices of electromagnetic wave fields,” Opt. Lett. 26(22), 1803–1805 (2001). [CrossRef] [PubMed]

Determining topological charge based on an improved Fizeau interferometer

Abstract

1. Introduction

2. Improved Fizeau interferometer

3. Experimental demonstration

4. Conclusion

Funding

References

Cited By

Figures (7)

Equations (9)

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