Investigation of propagation dynamics of truncated vector vortex beams

P. SRINIVAS,1,2 P. CHITHRABHANU,2,3 NIJIL LAL,2,4 R.P. SINGH,2 B. SRINIVASAN1* 1Department of Electrical Engineering, IIT Madras, Chennai 600036, Tamil Nadu, India 2Physical Research Laboratory, Ahmedabad, Gujarat 380009, India 3Centre for Quantum Technologies, National University of Singapore, Block S15, 3 Science Drive 2, 117543, Singapore 4Indian Institute of Technology Gandhinagar, Palaj, Gandhinagar 382355, India *Corresponding author: balajis@ee.iitm.ac.in

Received XX Month XXXX; revised XX Month, XXXX; accepted XX Month XXXX; posted XX Month XXXX (Doc. ID XXXXX); published XX Month XXXX

In this paper, we experimentally investigate the propagation dynamics of truncated vector vortex beams generated using a Sagnac interferometer. Upon focusing, the truncated vector vortex beam is found to regain its original intensity structure within the Rayleigh range. In order to explain such behavior, the propagation dynamics of truncated vector vortex beam is simulated by decomposing it into sum of integral charge beams with associated complex weights. We also show that the polarization of the truncated composite vector vortex beam is preserved all along the propagation axis. Experimental observations are consistent with theoretical predictions based on previous literature and are also in good agreement with our simulation results. The results hold importance as vector vortex modes are eigenmodes of the optical fiber. © 2017 Optical Society of America

OCIS codes: (080.4865) Optical Vortices; (260.6042) Singular optics

http://dx.doi.org/10.1364/OL.99.099999

The ability to individually manipulate spin angular momentum (SAM) and orbital angular momentum (OAM) of optical beams, leading to the so-called vector vortex beams [1] are of much interest for several applications [2–5]. Such vector vortex beams, which are also known as composite vector vortex beams are the superposition of the spin-orbit aligned and anti-aligned beams and have unique focusing properties [6]. Hence these beams are potentially useful in applications such as high resolution imaging [7], study of molecular orientations [8] and 3D beam shaping [9]. Of particular interest is the fact that such spin-orbit aligned or anti-aligned beams are eigenmodes of a standard optical fiber [10] and stable propagation of these modes over few kilometers [11] have been demonstrated. In this study, we experimentally demonstrate the intensity as well as polarization structure evolution of the truncated vector vortex beams not only at the focus but beyond the focal plane as well. We generate vector vortex beams using Sagnac interferometer and truncate it with the help of an intensity mask placed right after a converging lens. We

observe that the truncated vector vortex beam regains its original intensity structure within the Rayleigh range and the truncated part reappears in the far field with an orientation diametrically opposite to its initial location. We explain this behavior by considering the rotation dynamics of an untruncated vortex beam. We validate the experimental results with simulations wherein the truncated vector vortex beam is expressed as a superposition of multiple untruncated vortex beams using the modal decomposition method [12]. Recently, a similar approach has been demonstrated to carry out a spatial spectrum analysis using scalar vortex beams enabling a robust extraction of physical object information [13]. Different techniques have been proposed to generate vector vortex beams [14–16]. In our work, we chose the modified polarizing Sagnac interferometer due to its inherent immunity to surrounding vibrations owing to its common-path configuration [17]. The schematic of the experimental setup used is shown in Fig. 1, where the output of polarizing beam splitter (PBS) is a superposition of two linearly polarized vortex beams.

Fig. 1. (a) Sagnac configuration to generate vector vortex beams P: Polarizer, SPP: Spiral phase plate, HWP: Half wave plate,

QWP: Quarter wave plate, PBS: Polarizing beam splitter. Polarization map (red markers) for charge 5 spin-orbit anti-

aligned mode overlapped on the corresponding intensity profile. (b) from simulation (c) from experiment. The yellow

circles and arrows are highlighting the sink and source nodes respectively.

The beam is then passed through a quarter wave plate (QWP) to get the superposition of spin-orbit aligned beams or spin-orbit anti-aligned beams. The relative phase shift between the two orthogonal components obtained because of PBS (δ1) and QWP (δ2) is compensated using a half wave plate (HWP). The vector vortex beams mentioned above have been experimentally demonstrated as shown in Fig. 1 (c) by using a spiral phase plate (SPP) with charge 𝑙 = 5 and QWP at an angle of 135°. As mentioned briefly in the Introduction, it is important to note that the obtained polarization structure of the spin-orbit anti-aligned beam, exp (𝑖5∅)(𝑥 − 𝑖�̂�) + exp (−𝑖5∅)(�̂� + 𝑖�̂�) is consistent with that of the 𝐸𝐻4,1

𝑒 hybrid mode of an optical fiber. The polarization map presented in Fig. 1(c) as red markers is experimentally obtained through Stokes polarimetry [18]. Before studying the propagation dynamics of truncated vector vortex beams, we investigated the propagation dynamics of vector vortex beams without any truncation. For this investigation, we chose a charge 5 spin-orbit anti-aligned (Fig. 1(c)) as the propagation dynamics may be captured in our camera with better resolution for a higher charge vortex beam. The charge 5 vector vortex beam shown in Fig. 1(c) is then focused using a converging lens of focal length 20 cm and its intensity profile at different locations are captured to study its propagation without any truncation. As shown in Fig. 2(a), the vortex intensity distribution of the beam is maintained through the focusing point and up to a length corresponding to 7 times the Rayleigh range. We then analyzed the propagation dynamics of vector vortex beam with a perturbation. A knife edge is placed immediately after the lens to block a part of the beam as shown in Fig. 2 and the beam profile is captured along the propagation path. Fig. 2(b) shows the intensity profiles captured along the propagation when quarter part of the vector vortex beam is blocked using the knife edge. It is clearly evident that the truncated beam regained its original intensity structure in the Rayleigh range [19]. Furthermore, it is interesting to note that upon further propagation through a length corresponding to a few times the Rayleigh range, the quarter blocked part (defect) has re-appeared at a location diametrically opposite to the initial defect location while retaining the original polarization structure. The same experiment has been repeated by blocking half of the beam using the knife edge and similar results are obtained as shown in Fig. 2(c). In order to verify any dependency of propagation dynamics on the vortex charge, we repeated the experiment by blocking quarter part of charge 2 spin-orbit anti-aligned vector vortex beam and observed the exact same behavior. The above observations raise a few interesting questions. Foremost among them are: how does the truncated vector vortex beam regains its original structure in the Rayleigh region? Why does it go back to its truncated structure in the far field with an orientation diagonally opposite to the original truncated beam? How does the polarization evolve

Fig. 2. Intensity profiles of charge 5 vector vortex beam propagated through a lens of focal length 20 cm. (a) un-

truncated beam, (b) with quarter part of the beam blocked knife edge (KE), (c) with half part of the beam blocked.

along the propagation axis? An earlier study [20] on such vector vortex beams is limited only to the focal plane and does not explain the propagation characteristics. To address the above questions on the propagation characteristics of vector vortex beams, we experimentally observed the propagation dynamics of the two orthogonal components of charge 5 vector vortex beam individually by separating them using a QWP and a PBS. Fig. 3 shows the propagation of the LCP and RCP components which are associated to vortex charge of -5 and +5 respectively through snapshots of the intensity pattern at several locations. It is clearly observed that the +5 and -5 charge vortex beams rotate in opposite directions, but at the same rate. At the Rayleigh range, the two vortex beams appear at diametrically opposite orientation. Furthermore, their orientations are aligned in the far field (7zr in Fig. 3).

To understand the above observation, we need to consider the rotation of the vortex beam on propagation. As explained in [21], the rotation (𝜗) is due to the rate of change of azimuthal component of the Poynting vector, typically attributed to the Gouy phase and is given by

𝜗 =|𝑙|

𝑙𝑎𝑟𝑐𝑡𝑎𝑛 (

𝑧

𝑧𝑟) (1)

where 𝑙 is the OAM charge, 𝑧 is the beam propagation direction and 𝑧𝑟 is the beam Rayleigh range. As such, the vortex beam with charge +𝑙 rotates clockwise by 𝜋 ⁄ 4 radians travelling till −𝑧𝑟 from −∞ and then rotates clockwise by 𝜋 ⁄ 2 radians within the Rayleigh range (2𝑧𝑟) and again a 𝜋 ⁄ 4 radians clockwise rotation on travelling from +𝑧𝑟 to +∞. As a whole the beam rotates in clockwise direction by 𝜋 radians on travelling from −∞ 𝑡𝑜 + ∞.

Similarly vortex beam with charge – 𝑙 rotates in anti-clockwise direction upon propagation and rotates by 𝜋 radians on travelling from −∞ 𝑡𝑜 + ∞. Hence on propagation the vector vortex beam which consists of both the charge components appears to be reconstructed at the Rayleigh range and also the defect re-appears diametrically opposite to the initial location. It is also evident from the Equ. 1 that the rotation of vortex beams is independent of the magnitude of vortex charge.

Fig. 3. Propagation of individual beams of charge 5 vector vortex beam by blocking a quarter part of the beam with KE;

vortex beam of charge - 5 rotates in anti-clockwise and vortex beam of 5 rotates in clockwise direction.

However, it is to be noted that the above discussion pertains only to an un-truncated single charge vortex beam. As such, any theoretical description of the observation illustrated in Fig. 3 for the truncated vortex beam needs to be connected to the existing theory of rotational dynamics of a single un-truncated vortex beam. This is possible only through a decomposition of the truncated vortex charge into the constituent un-truncated single charge vortex beams through a modal decomposition technique. In this technique, the truncated vortex beam given in Equ. 2 is expressed as a sum of integral charge vortex beams with their corresponding complex weights as shown in Equ. 3.

𝛹𝑙,𝑝𝑇 (𝜌, ∅) = 𝑇(∅) 𝛹𝑙,𝑝(𝜌, ∅) (2)

𝛹𝑙,𝑝

𝑇 (𝜌, ∅) = ∑ ∑ 𝑊𝑙′,𝑝′𝑢𝑙′,𝑝′(𝜌, ∅)∞𝑙′=−∞

∞𝑝′=0 (3)

Fig. 4. Relative power distribution representing the decomposition across neighboring charges for the quarter truncated and half truncated charge 5 vector vortex beam

where 𝛹𝑙,𝑝𝑇 and 𝛹𝑙,𝑝 are the electric fields corresponding to

truncated and un-truncated vortex beams respectively, 𝑇(∅) refers to the truncation function, 𝑢𝑙′,𝑝′ is the electric

field corresponding to the integral charge vortex beam, and 𝑊𝑙′,𝑝′ is the complex weight of the integral charge vortex

beam which may be quantified as

𝑊𝑙′,𝑝′ = ∫ ∫ 𝑢𝑙′,𝑝′∗ (𝜌, ∅)𝛹𝑙,𝑝

𝑇 (𝜌, ∅)𝜌𝑑∅ 𝑑𝜌2𝜋

0

∞

0(4)

It should be noted that the truncated vortex beams are normalized before calculating the modal weights. Complex modal weights of +5 charge quarter truncated vortex beam decomposed at a relatively long distance (10zr, corresponding to location of knife edge) before the focus are shown in Table 1. As expected, charge 5 possesses a much higher weight compared to the other charges. The relative weight distribution is also symmetric around charge 5, with a phase shift. We also find that the weight distribution and relative phase is quite consistent before and after focus, as expected based on their intensity profiles. The relative power distribution based on the above weight distribution is shown in Fig. 4 for the quarter and half truncated vortex beams. As expected, the quarter truncated vortex beam has a narrower distribution compared to the half truncated beam and both distributions are symmetric around charge 5. Complex modal weights of -5 charge quarter part truncated vortex beam has similar modal amplitudes but with different phase values (not shown). However, the relative power distribution of +5 and -5 charge quarter part truncated beam remains the same.

Based on the above vortex charge distribution, we are now able to propagate the individual charges collinearly and superpose them at different locations to observe the resultant intensity profile. Fig. 5 illustrates such a position-dependent intensity profile for the quarter-truncated +5, and -5 charge vortex beams. As predicted by Equ. 1, we observe a rotation of the individual charges after the beams propagate through the focal point. A superposition of the two opposite charge vortex beams also show how the composite beam is able to regain the original untruncated structure over the Rayleigh region. The above simulations are found to precisely match the experimental observations illustrated in Figs. 2b and 3. As such, we can conclude that the rotation dynamics of the individual charge vortex beam is the key to explaining the “healing” of the composite vortex beam in the Rayleigh range as well as the rotation of the truncated beam in the far field after focusing.

2 3 4 5 6 7 80

20

40

60

80

Vortex charge

Re

lative

po

we

r in

%

Quarter part truncated

Half part truncated

Table 1. Modal weights obtained by decomposing quarter part truncated charge +5 vector vortex beam

Vortex charge (l’) 2 3 4 5 6 7 8

Complex weight

before focus

0.07∠ − 2.68 0.16∠1.38 0.26∠ − 0.90 0.87 0.26∠0.90 0.17∠ − 1.40 0.08∠2.73

after focus

0.07∠ − 2.03 0.16∠1.76 0.26∠ − 0.67 0.87 0.26∠0.67 0.17∠ − 1.76 0.08∠1.98

Fig. 5. Numerically simulated propagation dynamics at various locations of quarter part truncated (a) +5 charge vortex beam,

(b)-5 charge vortex beam and (c) composite charge 5 vector vortex beam.

The discussion above has focused only on the intensity evolution of the vector vortex beam. For such beams, it is also important to analyze the polarization evolution along the propagation axis. To this end, we have experimentally obtained the polarization structure through the use of a Stokes polarimetry [18] at different transverse planes along the propagation direction as shown in Fig. 6. We have also compared it with simulation results obtained by modal decomposition technique for a charge 5 vector vortex beam.

Fig. 6. Polarization structure evolution of quarter-truncated charge 5 vector vortex beam on propagation; experimental

results (top row), numerical simulations (bottom row). As mentioned before, the vector vortex beam which is the superposition of RCP (l= +5) and LCP (l= -5) vortex beams has linear polarization structure. As seen from the highlighted sections corresponding to the overlap of the two constituent charges in Fig. 6, the polarization structure of the beam remains linearly polarized along through the propagation (-7zr to +7zr). It is interesting to note that in the region adjoining the highlighted sections in the azimuthal plane i.e., the regions where we observe only one of the constituent charges, we observe circular polarization corresponding to the individual charges. The above observations are quite consistent with our simulations, corroborating the observation that the polarization structure of the composite beam is preserved all along the propagation direction.

In conclusion, we have investigated the propagation dynamics of truncated vector vortex beams which are generated using a Sagnac interferometer. We observe that the truncated vector vortex beam regains its original intensity structure within the Rayleigh range and the

truncated part reappears in the far field with an orientation diametrically opposite to its initial location. Such an experimental observation has been explained by decomposing the truncated vector vortex beam into a sum of multiple untruncated single charge vortex beams whose rotation dynamics are well known. Such a methodology is potentially applicable to explain the propagation of vortex beams with other types of perturbations as well. More importantly, the vector vortex modes are consistent with the eigenmodes of an optical fiber and the results reported may offer deeper insights in modeling their propagation.

Funding. Asian Office of Aerospace Research and

Development (AOARD) (FA2386-16-1-4077).

Acknowledgment. Authors acknowledge Pidishetty

Shankar, Harishankar Ramachandran from IIT Madras for technical discussions and Raghu Dharmavarapu from IIT Madras, Pravin Vaity from Physical Research Laboratory for help in setting up initial experiments.

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