We propose a new method to determine topological charge by using an improved Fizeau interferometer. This interferometer is very easy to realize, as well as interference fringes are very distinct. Phases of vortex, Hermite-Gaussian, and elliptical vortex beams are experimentally verified using this method. It provides a convenient way to determine the sign and magnitude of topological charge. This method may have some potential applications in space optical communication.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Vortex beams are widely studied as a typical high-order Laguerre-Gaussian mode, which possess particular helical phase structures exp(il) 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. 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  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.
The field amplitude E of incident vortex beam can be expressed as [20,26]Fig. 1(b), we have
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 , 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 as well as distinct inference fringes.
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.
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.
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.
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).
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 . Elliptical vortex beam is a superposition of a number of vortices, which has OAM on the off-axis . 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 HG2,0 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.
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.
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.
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