Open Access
Add to BibSonomy
Abstract
We introduce a novel and simple modulation technique to tailor optical beams with a customized amount of orbital angular momentum (OAM). The technique is based on the modulation of the angular spectrum of a seed beam, which allows us to specify in an independent manner the value of OAM and the shape of the resulting beam transverse intensity. We experimentally demonstrate our method by arbitrarily shaping the radial and angular intensity distributions of Bessel and Laguerre-Gauss beams, while their OAM value remains constant. Our experimental results agree with the numerical and theoretical predictions.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Beam shaping studies the techniques to modify the amplitude [1], phase [2] and polarization [3] of a light beam. Particular attention has been devoted to manipulate the amount of orbital angular momentum (OAM). The OAM of light beams was presented by Allen et al. [4] in his seminal paper about Laguerre-Gauss beams. It is worth noticing that OAM existence is not restricted to LG beams and it can be found in other beam modes, such as Mathieu, Ince-Gaussian and Bessel beams [5–7]. The total OAM depends on the transverse amplitude and phase distribution of the light beam. In the particular case where the light beam contains an azimuthal phase distribution of the form $\exp (i\ell \varphi )$, it turns out that the total OAM is directly associated to $\ell$, which is known as the topological charge. The light OAM has found applications in optical trapping and particle manipulation, where the OAM is used to control the particles without steering the laser beam [8,9]. OAM states can be used as carriers of information and hence they provide potential applications in free-space and fiber optic communications [10–12].
There are several methods to generate light beams with OAM. We can use spiral phase plates (SPP), superposition of Hermite-Gauss modes, Pancharatnam-Berry phase optical elements (e.g. q-plates), and spatial light modulators (SLM) [13]. These methods usually generate light beams containing integer values of OAM, however they can be used to generate custom values of OAM, e.g. fractional values [14,15]. These can also be realized in the optical spectrum using plasmonic vortex elements [16], whereas in the extreme ultraviolet spectrum through high harmonic generation [17]. Fractional OAM have a high potential to increase the information density in both classical and quantum communications [12]. An application of fractional OAM is shown by Alexeyev et al. [18], where they study the propagation of a fractional vortex beam through an optical fiber. Furthermore, fractional OAM generated by SPP can be used to study two-photon entanglement [19].
In this work, we introduce a versatile method to tailor the transverse shape of a light beam, while maintaining its OAM fixed at a given custom value. Our experimental method finds its theoretical basis on the work by Martinez-Castellanos et al. [20]. The method consists in the proper modification of a seed beam in the Fourier domain. The seed beam does not carry OAM, but the resulting beam contains the desired amount of OAM and inherit the propagation properties of the seed beam. Therefore, our method provides the additional benefit of controlling the propagation behaviour. For example, we can generate non-diffracting beams with fractional OAM which are stables on propagation. In contrast, paraxial beams with fractional OAM generated with a non-integer SPP show an unstable structure on propagation [14].
2. Description of the beam shaping approach
It has been demonstrated that the $z$ component of the average OAM per unit power, per unit length, and per unit angle of a paraxial beam $U(\mathbf {r},z)$ is given by [20]
Equation (1) can be further simplified for functions of the separable form $\mathcal {A}(\rho ,\phi )=R(\rho )\exp [i \Omega (\phi )]$, where the radial part $R(\rho )$ is a square-integrable function in the transverse plane. This condition leads to [20]
We also consider that $\Omega$ has the form where $\Theta (\phi )$ is a particular phase modulation and $m\in \mathbb {R}$. Expressing $\Omega (\phi )$ in this form yields The previous equation is one of the first important results of our method. It provides a simple expression to calculate the average OAM value. Notice that the first term in Eq. (4) is a constant parameter. Therefore, if we can find a phase modulation $\Theta$ such that the second term $\int \partial _{\phi } \Theta \,\mathrm {d}\phi =0$, we would be able to modify the beam shape without changing its OAM content. With that objective in mind, we set the following phase modulation: where $a,b$, and $c$ are positive real numbers. With this choice for $\Theta$, it turns out thatThere is an intuitive interpretation of $U=F^{-1}[\mathcal {A}\tilde {U}_{0}]$ in combination with $\Omega =m\phi + \Theta$ in Eq. (3). The resulting beam $U$ can be seen as the result of $\Omega$ diffracted by the angular spectrum of the seed beam. The value of OAM is given by the spiral phase $m\phi$, whereas the shape of $|U|^{2}$ is determined by the symmetries of $\Theta$. Notice that $m\in \mathbb {R}$ is not necessarily an integer number. This interpretation is illustrated in Fig. 1.
Fig. 1. Intuitive interpretation of the modulation process. The phase modulation $\Omega$ in Eq. (3) is equivalent to the superposition of a spiral phase with a wavefront $\Theta$. The spiral phase determines the value of $J_{z}$ whereas $\Theta$ shapes the intensity of the resulting beam $U$, which results from the diffraction of $\Omega$ with the angular spectrum of the seed beam.
3. Experimental arrangement and results
Figure 2 shows a schematic of the experimental setup. We apply the technique developed by Arrizon et al. [21], which uses an amplitude-only liquid crystal spatial light modulator (SLM) to generate arbitrary complex fields. Given the algebraic function $\mathcal {A}$, the beam amplitude $U$ is calculated with the inverse Fourier transform of $\mathcal {A}\tilde {U}_{0}$, i.e. $U=F^{-1}[\mathcal {A}\tilde {U}_{0}]$ (cf. [20]). We use a collimated and linearly polarized He-Ne laser ($632.8$ nm) and a transmissive SLM (HOLOEYE model LC2002). The spatial filter system (SF) before the SLM cleans the beam transverse intensity and creates a uniform Gaussian beam. The beam polarization is oriented parallel to the SLM director axis using a half-wave plate (HWP). A linear polarizer (LP) is placed after the SLM to eliminate the unmodulated polarization component. The beam $U$ is encoded in a computer generated hologram (CGH) which is displayed in the SLM. A combination of a 4f system with an aperture is implemented to decode $U$ from the first diffraction order, as shown in Fig. 2(a). The intensity pattern $|U|^{2}$ is recorded using a CCD camera. The phase information, i.e. $\arg (U)$, is obtained from the interference pattern between the first and zeroth diffraction orders, as shown in Fig. 2(b). We implemented a phase retrieval algorithm following the work by Takeda et al. [22].
Fig. 2. Scheme of the experimental setup to generate the structured beam. (a) detection scheme and (b) phase retrieval scheme. Laser: He-Ne laser with $\lambda =632.8$ nm, HWP: half-wave plate, SF: spatial filter, LP: linear polarizer, SLM: spatial light modulator, lenses with focal lengths $f_{1}=15$ cm and $f_{2}=5$ cm, CCD: camera.
In the examples that follow, we use three types of seed beams: a Gaussian beam (GB) $\mathrm {GB}= A \exp [-(2r/w_{0})^{2}]$, where $A$ is a normalization constant such that $\iint |\mathrm {GB}|^{2} \mathrm {d}x\mathrm {d}y=1$, a normalized Laguerre-Gauss beam (LGB)
Figure 3 shows the results for the case with $m=3.5$ and $\Theta = 0$ in Eq. (3), which gives $J_{z}=3.5$. Thus, the transverse modulation is defined by the seed beams. First, second and third rows show the resulting beam profiles for the GB, LGB, and BB seeds, respectively. For the LGB case we set $n=5$. The figure shows the simulated beam profile using $U=F^{-1}[\mathcal {A}\tilde {U}_{0}]$ and the experimental measurements. In both cases we calculate the value of $J_{z}$ using the formal definition [25]
Fig. 3. Intensity and phase for a structured beam with $J_{z}=3.5$ given by Eq. (4) with $\Theta =0$. GB: Gaussian-type beam due to a Gaussian seed with a radial variation $R(\rho )=\rho ^{2}$. LGB: Laguerre-Gauss type beam generated with an LG seed [Eq. (7)]. BB: Nondiffracting type beam with a zeroth-order Bessel seed. (a) Simulation generated with $U=F^{-1}[\mathcal {A}\tilde {U}_{0}]$, and (b) experimental results. The numerical value is calculated with Eq. (8) for both the simulation and the experimental measurements. It is equal in both cases.
Figure 4 shows our second example. We set again the value of $J_{z}=3.5$ but we choose $\Theta =2\sin (5\phi )$. The simulations are shown in Fig. 4(a) and the experimental measurements in (b). Notice that the phase modulation $\Theta$ is revealed in the pentagonal symmetry of the intensity pattern. In general, the shape is determined by the oscillation frequency of the sinusoidal function. Therefore, we can infer that the parameter $b$ in Eq. (5) determines the transverse symmetry of the resulting beam, whereas the seed function determines the radial shape and the propagation characteristics (which is either paraxial or non-diffracting). The calculated value of $J_{z}$ for the GB and BB seeds shows excellent agreement between the simulation and the experimental measurements. However, the LGB seed presents a disagreement of about 0.38 between the simulation and the experiment.
Fig. 4. Intensity and phase for a structured beam with $J_{z}=3.5$ using Eq. (4) with $\Theta =2\sin (5\phi )$. GB: Gaussian seed. LGB: Laguerre-Gauss seed. BB: Bessel seed. (a) Simulation generated with $U=F^{-1}[\mathcal {A}\tilde {U}_{0}]$, and (b) experimental results. The numerical values indicate the calculated $J_{z}$ according to Eq. (8).
Our last example shows a beam profile with a triangular symmetry. It is generated by choosing $\Theta =8 \sin ^{3}(\phi )$ (thus $a=8$, $b=1$ and $c=3$ in Eq. (5)). Once more, we set the value of $J_{z}$ to $3.5$. In this case, we notice that the triangular symmetry is related to the parameter $c$. By comparing with the previous results, we can infer that the transverse shape is determined by the maximum oscillation frequency of the sinusoidal modulation, which is given by the product of $b$ with $c$ (we notice this by expanding the sinusoidal function in its Fourier components and we observe that the maximum frequency in the expansion is given by $bc$). The simulation and experimental measurements are shown in Figs. 5(a) and 5(b), respectively. Similarly, there is a good agreement between the simulations and experimental values of $J_{z}$.
Fig. 5. Intensity and phase for a structured beam with $J_{z}=3.5$ using Eq. (4) with $\Theta =8\sin ^{3}(\phi )$. GB: Gaussian seed. LGB: Laguerre-Gauss seed. BB: Bessel seed. (a) Simulation generated with $U=F^{-1}[\mathcal {A}\tilde {U}_{0}]$, and (b) experimental results.
An important characteristic of our method is that the resulting beam inherits the propagation properties of the seed beam. Therefore, it is expected that non-diffracting-type beams preserve their transverse shape on propagation compare to paraxial-type beams. To demonstrate this property, we have measured the transverse intensity of a Bessel-type beam (see Fig. 5(a)) and a Gaussian type beam (see Fig. 4) at different propagation distances, which are measured with respect to the image plane of the lens f2 (see Fig. 2). The results are shown in Fig. 6. From the results, it is clear that the non-diffracting-type beam is more stable on propagation with respect to the paraxial-type beam.
Fig. 6. Experimental measurements of the transverse intensity for a Bessel-type beam and a Gaussian-type beam at different propagation distances. Notice that the non-diffracting properties of the Bessel seed are inherited in the resulting beam.
4. Conclusions
We have introduced a method that allows us to control in an independent manner the symmetry of the beam transverse intensity and its amount of OAM. We found a simple expression to modulate the angular spectrum of a seed beam without affecting the OAM. The modulation is given by Eq. (5), which provides the shaping parameters $a,b$ and $c$. From Figs. 4 and 5, we can see that the shaping parameters $b,c$ determines the polygonal shape of the beam. In general, the shape is determined by the product of $b$ with $c$. The parameter $a$ resembles a focusing factor. For example, in the Fourier domain the operation of a cylindrical lens is given by $\exp (i \sin ^{2}(\phi )/f)$, where $f$ is the focal length. Thus, by comparison, $a\equiv 1/f$ behaves as a focusing parameter. We emphasize that the value of $J_{z}$ is ultimately determined by the seed beam and the algebraic function $\mathcal {A}$, as it is prescribed by Eq. (8). However, Eq. (4) gives a simple expression where it is straightforward to see the relevant parameters that can be modified without affecting the OAM. Furthermore, we show that the resulting beam inherits the propagation properties of the seed beam. Our method can find applications in optical information and optical tweezers. In the former case, the amount of OAM can be used to encode information whereas the shape can be used for detection [12]. In optical tweezers can be used to control the particle trajectories without steering the laser beam [9,26].
Funding
Consejo Nacional de Ciencia y Tecnología (257517, 280181, 293471, 295239, APN2016-3140); Ministerstwo Nauki i Szkolnictwa Wyższego (DIA 2016 0079 45); Narodowe Centrum Nauki (UMO-2018/28/T/ST2/00125).
Acknowledgments
JMH acknowledges partial support from CONACyT, México. JMH and DLM thank Antonio Morales-Hernández and Alejandra Padilla-Camargo for their help with the SLM.
References
1. F. M. Dickey, Laser Beam Shaping: Theory and Techniques (CRC, 2014).
2. C. Rosales-Guzmán and A. Forbes, How to shape light with spatial light modulators (SPIE, 2017).
3. B. Perez-Garcia, C. López-Mariscal, R. I. Hernandez-Aranda, and J. C. Gutiérrez-Vega, “On-demand tailored vector beams,” Appl. Opt. 56(24), 6967 (2017). [CrossRef]
4. 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]
5. I. A. Litvin, A. Dudley, and A. Forbes, “Poynting vector and orbital angular momentum density of superpositions of bessel beams,” Opt. Express 19(18), 16760–16771 (2011). [CrossRef]
6. C. López-Mariscal, J. C. Gutiérrez-Vega, G. Milne, and K. Dholakia, “Orbital angular momentum transfer in helical mathieu beams,” Opt. Express 14(9), 4182–4187 (2006). [CrossRef]
7. M. A. Bandres and J. C. Gutiérrez-Vega, “Ince–gaussian beams,” Opt. Lett. 29(2), 144–146 (2004). [CrossRef]
8. J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104(10), 103601 (2010). [CrossRef]
9. Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100(1), 013602 (2008). [CrossRef]
10. J. P. Torres and L. Torner, Twisted Photons: Applications of Light with Orbital Angular Momentum (Wiley-VCH, 2011).
11. H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19(1), 013001 (2017). [CrossRef]
12. 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 (2015). [CrossRef]
13. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011). [CrossRef]
14. J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16(2), 993–1006 (2008). [CrossRef]
15. 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]
16. Y. Wang, P. Zhao, X. Feng, Y. Xu, F. Liu, K. Cui, W. Zhang, and Y. Huang, “Dynamically sculpturing plasmonic vortices: from integer to fractional orbital angular momentum,” Sci. Rep. 6(1), 36269 (2016). [CrossRef]
17. A. Turpin, L. Rego, A. Picón, J. S. Román, and C. Hernández-Garcá, “Extreme ultraviolet fractional orbital angular momentum beams from high harmonic generation,” Sci. Rep. 7(1), 43888 (2017). [CrossRef]
18. C. N. Alexeyev, A. O. Kovalyova, A. F. Rubass, A. V. Volyar, and M. A. Yavorsky, “Transmission of fractional topological charges via circular arrays of anisotropic fibers,” Opt. Lett. 42(4), 783–786 (2017). [CrossRef]
19. S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “How to observe high-dimensional two-photon entanglement with only two detectors,” Phys. Rev. Lett. 92(21), 217901 (2004). [CrossRef]
20. I. Martinez-Castellanos and J. C. Gutiérrez-Vega, “Shaping optical beams with non-integer orbital-angular momentum: a generalized differential operator approach,” Opt. Lett. 40(8), 1764–1767 (2015). [CrossRef]
21. V. Arrizón, G. Méndez, and D. S. de La-Llave, “Accurate encoding of arbitrary complex fields with amplitude-only liquid crystal spatial light modulators,” Opt. Express 13(20), 7913–7927 (2005). [CrossRef]
22. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982). [CrossRef]
23. J. Mendoza-Hernández, M. L. Arroyo-Carrasco, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Laguerre-gauss beams versus bessel beams showdown: peer comparison,” Opt. Lett. 40(16), 3739–3742 (2015). [CrossRef]
24. J. Mendoza-Hernández, M. A. Carrasco, M. M. Otero, S. Chávez-Cerda, and M. I. Castillo, “New asymmetric propagation invariant beams obtained by amplitude and phase modulation in frequency space,” J. Mod. Opt. 61(sup1), S46–S56 (2014). [CrossRef]
25. L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Engineering Technology, Physical Sciences, 2003).
26. A. Serrano-Trujillo and V. R. Cortés, “Engineering the intensity trajectory and phase gradient of light beams,” Proc. SPIE 10744, 1074404 (2018). [CrossRef]
