# Winding light beams along elliptical helical trajectories

###### Abstract

Conventional caustic methods in real or Fourier space produced accelerating optical beams only with convex trajectories. We develop a superposition caustic method capable of winding light beams along non-convex trajectories. We ascertain this method by constructing a one-dimensional (1D) accelerating beam moving along a sinusoidal trajectory, and subsequently extending to two-dimensional (2D) accelerating beams along arbitrarily elliptical helical trajectories. We experimentally implement the method with a compact and robust integrated optics approach by fabricating micro-optical structures on quartz glass plates to perform the spatial phase and amplitude modulation to the incident light, generating beam trajectories highly consistent with prediction. The theoretical and implementation methods can in principle be extended to the construction of accelerating beams with a wide variety of non-convex trajectories, thereby opening up a new route of manipulating light beams for fundamental research and practical applications.

The seemingly counter-intuitive discovery that light fields with appropriate initial field distributions such as Airy distribution Siviloglou et al. (2007); Siviloglou and Christodoulides (2007) capable of winding light beams as propagating along curved trajectories known as accelerating beams in free space has led to extensive research interests, as such optical phenomena may enable potential applications including optical micro-manipulation Baumgartl et al. (2008), laser plasma filamentation Polynkin et al. (2009), ’light bullets’ Siviloglou and Christodoulides (2007); Chong et al. (2010); Abdollahpour et al. (2010), imaging Jia et al. (2014), and plasmonics Salandrino and Christodoulides (2010); Liu et al. (2011); Minovich et al. (2011); Li et al. (2011); Lin et al. (2012). Nevertheless, Airy beams only move along parabolic trajectories, which limit the flexibility in practical applications, so a number of researches have been carried out to extend the variety of beam trajectories. Direct phase engineering in real space under the paraxial approximation successfully constructed non-broadening beams moving along arbitrary convex trajectories at the cost of giving up propagation invariance Greenfield et al. (2011). Going beyond the paraxial condition in real space enabled large-angle bending of the trajectoryFroehly et al. (2011). These works can be categorized as the caustic method in real space. In addition to phase engineering in real space, similar works have also been carried out in Fourier space Hu et al. (2013, 2013); Bongiovanni et al. (2015). However, only convex trajectories are realized based on these caustic methods because the first derivative of the trajectory is required to be single-valued. Limited types of non-convex trajectories are obtained by other approaches, such as by superposing some specific beams (e.g., circular beams) to construct periodic accelerating beams over an overall convex trajectory Hu et al. (2013); Kaminer et al. (2012); GREENFIELD et al. (2012); Mathis et al. (2013) or by imposing a spiral-shape phase or amplitude modulation to incident beams Jarutis et al. (2009); Morris et al. (2010); Matijošius et al. (2010); Liu et al. (2014).

On the other hand, most of the experimental realization of wingding light beams has been based on phase-only spatial light modulators (SLM). In order to achieve high quality beams with low side-lobes, both amplitude and phase modulation would be required. Some schemes have been proposed to implement both phase and amplitude modulations in the generation of accelerating beams. One scheme is to encode both phase and amplitude modulation into a phase-only SLM Mathis et al. (2013), which is more suitable for a binary amplitude modulation, otherwise it will introduce unwanted phase change and noise from other diffraction orders. Another scheme is to position a printed amplitude mask Hu et al. (2013) or a linear filter Schley et al. (2014) close to the phase-only SLM to realize additional amplitude modulation, but accurate alignment between the amplitude and phase modulation elements is a major problem in such a configuration. Techniques of generating structured light beams have experienced a general evolution from bulk optical elements toward compact and robust integrated optical devices, exemplified in the case of vortex beam in Refs. Cai et al. (2012); Sun et al. (2015). Applying such integrated approaches to the generation of accelerating beams should also enable the engineering of the optical phase, amplitude and even polarization in a more accurate, compact and robust fashion Lin et al. (2013).

In this work, we demonstrate the winding of light beams along arbitrarily elliptical helical trajectories in free space [Fig. 1(a)] by developing a superposition caustic method and by means of an integrated optics approach.

A helical beam’s 2D accelerating trajectory can be separated into two orthogonal 1D sinusoidal trajectories under the paraxial approximation. It is noted that the meaning of 1D and 2D here refers to the transverse profile of a light field and should not be confused with surface plasmonic waves propagating in a 2D plane Salandrino and Christodoulides (2010); Liu et al. (2011); Minovich et al. (2011); Li et al. (2011); Lin et al. (2012). Hence a 1D-accelerating beam with a sinusoidal trajectory is to be constructed first. Our method is based on the caustic method in Fourier space, where an accelerating beam moving along a curved trajectory is constructed by the interference of plane waves. Figures 1(b) and 1(c) present this interference process in a simplified light-ray picture with each red ray tangent to the predesigned trajectory representing the plane wave in need sup (). The relation between a predesigned trajectory and its necessary plane waves in previous caustic method Hu et al. (2013) can be summarized as (see supplmental Material for a detailed derivation sup ())

(1) |

where and are the spatial frequency and wave number in free space, and are the amplitude and phase of the initial spatial spectrum , and is the predesigned trajectory. Only if is single-valued can we obtain the necessary initial spatial spectrum, which limits the trajectory to be convex as mentioned before [Fig. 1(c)].

To construct non-convex beams, we break this limitation by dividing the entire trajectory into several segments so that each segment is convex and thus corresponds to an initial spatial spectrum according to Eq. 1. If, over the space of one particular segment, the intensity along the main lobe of this segment is much higher than the side lobes in the same space resulting from the other segments, the interference between these segments along the main lobe is negligible, and the entire non-convex trajectory can be obtained by simple linear superposition of all the initial spatial spectra corresponding to all segments, which can be expressed as

(2) |

where the index indicates the ordinal of the segments along the propagation () direction. In this manner, several positions on the trajectory are mapped onto the same spatial frequency, which is different from the one-to-one correspondence for convex trajectories constructed previously Greenfield et al. (2011); Hu et al. (2013).

Using this superposition caustic method, we are able to connect the beam trajectory with a specific initial phase and amplitude distribution. To perform the necessary phase and amplitude modulation to the incident light beam in experiment, integrated optics approach is exploited to fabricate a compact and robust optical element. The integrated optics element is fabricated on a quartz glass plate with a size of 20 mm 20 mm 0.5 mm and the effective area with patterns is 9.606 mm 9.606 mm. This is divided into 601 601 pixels with each pixel to be 16 m 16 m, representing sufficiently high sampling rates in the Fourier space. The phase modulation is implemented by aligned 16-level etching into the fused quartz plate. In each step, the structure is defined using photolithography in a positive photoresist (Shipley S1805), followed by a reactive ion etching process to transfer the pattern into the quartz plate, with each etch depth to be around 87 nm deep. The process is repeated for 15 times, overlaid using pre-fabricated alignment marks, so that the phase shift caused by each pixel is defined by the number of times it is etched. The amplitude modulation is realized by controlling the duty ratio of the partial metal coverage over the pixel. More specifically, each pixel is sub-divided into an grid with each grid to be 2 m 2 m, and the number of grids with metal coverage is controlled to realize specific transmittance for the whole pixel. Here nine different patterns are used to achieve a 9-level quasi-grey-scale [Fig. 2(a)]. These patterns are designed with the appropriate symmetry in mind in order to minimize additional diffraction effects. The structure is patterned in the S1805 resist using photolithography. After development, an 80-nm Chromium layer was coated using electron beam evaporation and then followed by the standard lift-off process [Fig. 2(b)]. The thickness of the whole micro-optical structures is less than 1.5 m for the wavelength of 632.8 nm, with the alignment of the phase and amplitude modulation patterns to sub-micron precision, resulting in a highly compact and robust device for accelerating beam generation. Three fabricated elements for winding light beams are shown in Fig. 2(c).

After completion of the fabrication, an experimental setup, as shown in Fig. 2(d), is used to generate the pre-designed accelerating beams as well as investigating their propagation dynamics in free space. A He-Ne laser operating at 632.8 nm emits a linearly polarized Gaussian beam that is expanded and collimated by an objective lens and lens 1. An aperture is used to obtain a quasi-plane wave before illuminating the fabricated quartz plate. After transmitting through the plate, the beam passes through lens 2 with a focal length = 500 mm, placed at a distance behind the quartz plate to perform the optical Fourier transform. The pre-designed accelerating beam is formed around a distance f behind lens 2, where a CMOS camera is placed to record the propagation dynamics of this beam by moving along the linear translating rail.

For the 1D sinusoidal beam, the designed initial phase and amplitude modulations varying along only one direction are shown in Fig. 3(a). The beam propagation resulting from this modulation is numerically simulated with a Beam Propagation Method (OptiBPM) [Fig. 3(b)]. A plot of the transverse position of intensity maxima along the z-direction shows that the trajectory fits very well with the sinusoidal curve, apart from a linear term representing a tilt of the beam axis [Fig. 3(c)]. The experimentally measured beam pattern and trajectory are plotted [Fig. 3(b) and 3(c)] and the trajectory coincides very well with the simulated trajectory, only the linear tilt is slightly different resulting from a slight tilt of the incident beam in the experiment. Moreover, the intensity variance along the main lobe shown in Supplementary sup () Fig. S3 also has good agreement.

Due to the variables of and being separable under the paraxial condition, it is straightforward to construct 2D accelerating beams simply by multiplying two 1D field distributions along and direction Siviloglou and Christodoulides (2007); Bongiovanni et al. (2015). Therefore, 2D accelerating beams here with arbitrarily elliptical helical trajectories can be constructed based on two 1D sinusoidal trajectories in and planes respectively. If the two 1D sinusoids are in phase, the helical beam degenerates into a 2D sinusoidal beam, whose projection on the plane is a line along the angle. The experimental result is shown in Fig. 4(a) and its projections to the and planes are similar to the 1D sine beam. Although its projection onto the transverse plane along the angle broadens slightly to an ellipsoid indicating a small phase difference of which may result from a slight tilt of the spherical lenses in the experimental setup leading to unequal focal lengths, the experimental results are still highly consistent with the simulation (Supplementary sup () Fig. S4).

Furthermore, if a phase shift is introduced between the trajectories in the and planes, winding light beams along arbitrarily elliptical helical trajectories can be realized with the eccentricity ranging from to . For demonstration, we choose to be and thus the eccentricity becomes zero, corresponding to a circular helical beam. The necessary phase and amplitude modulation as well as the experimental result are shown in Fig. 4. In this case, except one obvious point deviating from the fitted curve resulting from the imperfect fabrication of the quartz glass plate (detailed analysis in Supplementary sup () Fig. S5), the trajectory also coincides well with the simulation results (Supplementary sup () Fig. S6).

In conclusion, by way of a superposition caustic method, we break the one-to-one correspondence between the real space position on the beam trajectory and the spatial frequency in the Fourier space caustic method, which enables the construction of accelerating beams whose main lobe moves along non-convex sinusoidal trajectories. We experimentally realize 1D sinusoidal accelerating beams using this method. We further construct 2D accelerating beams with arbitrarily elliptical helical trajectories by superimposing two 1D sinusoidal trajectories in the and planes and controlling their phase shift. In principle, such a method could be applied to the construction of accelerating beams along a wide variety of other non-convex trajectories.

We employ an integrated optics approach to the experimental implementation of winding light beams by fabricating micro-optical structures on quartz glass plates to simultaneously provide the designed phase and amplitude modulation with high precision. The method is compact and robust, as confirmed by the high consistency between the experiment and the simulation results, and paves the way for the reliable generation of various accelerating beams by means of photonic integration for practical applications in the future.

This work is supported by the National Basic Research Program of China (973 Program) (2014CB340000 and 2012CB315702), the Natural Science Foundations of China (61323001, 61490715, 51403244, and 11304401), the Natural Science Foundation of Guangdong Province (2014A030313104), and the Fundamental Research Funds for the Central Universities of China (Sun Yat-sen University: 13lpgy65, 15lgpy04, 15lgzs095, 15lgjc25, and 16lgjc16). Y. Chen would also like to thank the Specialized Research Fund for the Doctoral Program of Higher Education of China (20130171120012). Y. Wen thanks the Undergraduate Training Programs for Innovation and Entrepreneurship (201302071) and SPE’s Undergraduate Scientific Research Program (2013011). The authors thank Lin Liu, Chunchuan Yang and Zengkai Shao for technical assistance in micro-fabrication, as well as Pengfei Xu, Jiangbo Zhu, Yi Wang, Guoxuan Zhu, and Johannes Herrnsdorf (University of Strathclyde, UK) for helpful discussions.

## References

- Siviloglou et al. (2007) G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett., 99, 213901 (2007).
- Siviloglou and Christodoulides (2007) G. A. Siviloglou and D. N. Christodoulides, Opt. Lett., 32, 979 (2007).
- Baumgartl et al. (2008) J. Baumgartl, M. Mazilu, and K. Dholakia, Nat. Photon., 2, 675 (2008).
- Polynkin et al. (2009) P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science, 324, 229 (2009).
- Chong et al. (2010) A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photon., 4, 103 (2010).
- Abdollahpour et al. (2010) D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett., 105, 253901 (2010).
- Jia et al. (2014) S. Jia, J. C. Vaughan, and X. Zhuang, Nat.Photon, 8, 302 (2014).
- Salandrino and Christodoulides (2010) A. Salandrino and D. N. Christodoulides, Opt. Lett., 35, 2082 (2010).
- Liu et al. (2011) W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, Opt. Lett., 36, 1164 (2011).
- Minovich et al. (2011) A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, Phys. Rev. Lett., 107, 116802 (2011).
- Li et al. (2011) L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, Phys. Rev. Lett., 107, 126804 (2011).
- Lin et al. (2012) J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, Phys. Rev. Lett., 109, 093904 (2012).
- Greenfield et al. (2011) E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys. Rev. Lett., 106, 213902 (2011).
- Froehly et al. (2011) L. Froehly, F. Courvoisier, A. Mathis, M. Jacquot, L. Furfaro, R. Giust, P. A. Lacourt, and J. M. Dudley, Opt. Express, 19, 16455 (2011).
- Hu et al. (2013) Y. Hu, D. Bongiovanni, Z. Chen, and R. Morandotti, Phys. Rev. A, 88, 043809 (2013a).
- Hu et al. (2013) Y. Hu, D. Bongiovanni, Z. Chen, and R. Morandotti, Opt. Lett., 38, 3387 (2013b).
- Bongiovanni et al. (2015) D. Bongiovanni, Y. Hu, B. Wetzel, R. A. Robles, G. G. Mendoza, E. A. Martipanameå¸½o, Z. Chen, and R. Morandotti, Sci. Rep., 5, 13197 (2015).
- Kaminer et al. (2012) I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, Phys. Rev. Lett., 108, 163901 (2012).
- GREENFIELD et al. (2012) E. Greenfield, I. Kaminer, and M. Segev, in Frontiers in Optics 2012/Laser Science XXVIII (Optical Society of America, 2012) p. FW1A.2.
- Mathis et al. (2013) A. Mathis, F. Courvoisier, R. Giust, L. Furfaro, M. Jacquot, L. Froehly, and J. M. Dudley, Opt. Lett., 38, 2218 (2013).
- Jarutis et al. (2009) V. Jarutis, A. Matijošius, P. D. Trapani, and A. Piskarskas, Opt. Lett., 34, 2129 (2009).
- Morris et al. (2010) J. E. Morris, T. Čižmár, H. I. C. Dalgarno, R. F. Marchington, F. J. Gunn-Moore, and K. Dholakia, J. Opt., 12, 124002 (2010).
- Matijošius et al. (2010) A. Matijošius, V. Jarutis, and A. Piskarskas, Opt. Express, 18, 8767 (2010).
- Liu et al. (2014) H. Liu, M. Q. Mehmood, K. Huang, L. Ke, H. Ye, P. Genevet, M. Zhang, A. Danner, S. P. Yeo, C.-W. Qiu, and J. Teng, Adv. Opt. Mater., 2, 1193 (2014).
- Schley et al. (2014) R. Schley, I. Kaminer, E. Greenfield, R. Bekenstein, Y. Lumer, and M. Segev, Nat. Commun., 5, 5189 (2014).
- Cai et al. (2012) X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, Science, 338, 363 (2012).
- Sun et al. (2015) J. Sun, A. Yaacobi, M. Moresco, D. D. Coolbaugh, and M. R. Watts, in CLEO: 2015 Postdeadline Paper Digest (Optical Society of America, 2015) p. JTh5A.5.
- Lin et al. (2013) J. Lin, P. Genevet, M. A. Kats, N. Antoniou, and F. Capasso, Nano Lett., 13, 4269 (2013).
- (29) See Supplemental Material for Further Theoretical Analysis and Simulation Results in Correspondence.

Supplemental_Materials.pdf,1

Supplemental_Materials.pdf,2

Supplemental_Materials.pdf,3