Characterization of topological charge and orbital angular momentum of shaped optical vortices Anderson M. Amaral,1,2 Edilson L. Falcão-Filho,1 and Cid B. de Araújo1,∗ 1 Departamento
de Física, Universidade Federal de Pernambuco, 50670-901 Recife, PE, Brazil 2 [email protected] ∗ [email protected]
Abstract: Optical vortices (OV) are usually associated to cylindrically symmetric light beams. However, they can have more general geometries that extends their applicability. Since the typical experimental characterization methods are not appropriate for OV with arbitrary shapes, we discuss in this work how the definitions of the classical orbital angular momentum and the topological charge can be used to retrieve these informations in the general case. The concepts discussed are experimentally demonstrated and may be specially useful in areas such as optical tweezers and plasmonics. © 2014 Optical Society of America OCIS codes: (260.6042) Singular optics; (050.4865) Optical vortices.
References 1. 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, 8185–8189 (1992). 2. J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nature Photon. 6, 488–496 (2012). 3. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001). 4. W. N. Plick, M. Krenn, R. Fickler, S. Ramelow, and A. Zeilinger, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87, 033806 (2013). 5. A. Jesacher, C. Maurer, S. Fuerhapter, A. Schwaighofer, S. Bernet, and M. Ritsch-Marte, “Optical tweezers of programmable shape with transverse scattering forces,” Opt. Commun. 281, 2207–2212 (2008). 6. M. Chen, M. Mazilu, Y. Arita, E. Wright, and K. Dholakia, “Dynamics of microparticles trapped in a perfect vortex beam,” Opt. Lett. 38, 4919–4922 (2013). 7. H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10, 529–536 (2010). 8. A. Rury, “Coherent control of plasmonic spectra using the orbital angular momentum of light,” Phys. Rev. B 88, 205132 (2013). 9. K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, and T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12, 3645–3649 (2012). 10. Y. Gorodetski, A. Drezet, C. Genet, and T. W. Ebbesen, “Generating far-field orbital angular momenta from near-field optical chirality,” Phys. Rev. Lett. 110, 203906 (2013). 11. E. Brasselet, G. Gervinskas, G. Seniutinas, and S. Juodkazis, “Topological shaping of light by closed-path nanoslits,” Phys. Rev. Lett. 111, 193901 (2013). 12. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011). 13. S. Chávez-Cerda, M. Padgett, I. Allison, G. New, J. Gutiérrez-Vega, A. O’Neil, I. MacVicar, and J. Courtial, “Holographic generation and orbital angular momentum of high-order Mathieu beams,” J. Opt. B: Quantum and Semiclass. Opt. 4, S52–S57 (2002).
#222475 - $15.00 USD Received 5 Sep 2014; revised 13 Nov 2014; accepted 16 Nov 2014; published 26 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030315 | OPTICS EXPRESS 30315
14. M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6, 259–268 (2004). 15. J. Leach, E. Yao, and M. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004). 16. F. S. Roux, “Distribution of angular momentum and vortex morphology in optical beams,” Opt. Commun. 242, 45–55 (2004). 17. A. Y. Bekshaev, M. V. Vasnetsov, and M. S. Soskin, “Description of the morphology of optical vortices using the orbital angular momentum and its components,” Opt. Spectrosc. 100, 910–915 (2006). 18. Q. Zhan, “Properties of circularly polarized vortex beams,” Opt. Lett. 31, 867–869 (2006). 19. E. Abramochkin and V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102, 336–350 (1993). 20. E. Abramochkin and V. Volostnikov, “Spiral-type beams: optical and quantum aspects,” Opt. Commun. 125, 302–323 (1996). 21. A. M. Amaral, E. L. Falcão-Filho, and C. B. de Araújo, “Shaping optical beams with topological charge,” Opt. Lett. 38, 1579–1581 (2013). 22. G. Berkhout and M. Beijersbergen, “Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects,” Phys. Rev. Lett. 101, 100801 (2008). 23. J. M. Hickmann, E. J. S. 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, 053904 (2010). 24. M. Soskin, V. Gorshkov, M. Vasnetsov, J. Malos, and N. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997). 25. T. Ando, N. Matsumoto, Y. Ohtake, Y. Takiguchi, and T. Inoue, “Structure of optical singularities in coaxial superpositions of Laguerre-Gaussian modes,” J. Opt. Soc. Am. A 27, 2602–2612 (2010). 26. C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, “Measurement of the orbital angular momentum density of light by modal decomposition,” New J. Phys. 15, 073025 (2013). 27. M. Nakahara, Geometry, Topology and Physics (Taylor & Francis, 2003). 28. E. G. Abramochkin and V. G. Volostnikov, “Spiral light beams,” Physics-Uspekhi 47, 1177–1203 (2004). 29. M. V. Berry, “Paraxial beams of spinning light,” Proc. SPIE 3487, 6 (1998). 30. M. V. Berry, “Optical currents,” J. Opt. A: Pure Appl. Opt. 11, 094001 (2009). 31. J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt. Lett. 28, 872–874 (2003). 32. R. Pugatch, M. Shuker, O. Firstenberg, A. Ron, and N. Davidson, “Topological stability of stored optical vortices,” Phys. Rev. Lett. 98, 203601 (2007). 33. 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, 156–160 (1982). 34. K. Ahnert and M. Abel, “Numerical differentiation of experimental data: local versus global methods,” Comput. Phys. Commun. 177, 764–774 (2007). 35. M. Dennis, “Rows of optical vortices from elliptically perturbing a high-order beam,” Opt. Lett. 31, 1325–1327 (2006). 36. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
1.
Introduction
Optical vortices (OV) have been extensively studied since the seminal work by Allen et al. [1] and are applied in topics as diverse as classical and quantum communications [2–4], optical tweezers [5,6] and plasmonics [7–11]. OV are mainly characterized by their topological charge (TC) q, which is related to the beam phase profile, and their orbital angular momentum (OAM) l, which also involves the intensity profile [12]. Most of previously referred applications use cylindrical OV (c-OV), which exhibiths azimuthal symmetry and are characterized by well defined values of l and q, with l = q, as Bessel or Laguerre-Gauss beams. However, by analyzing OAM and TC for shaped OV (s-OV) it can be seen that their values might differ, as in Mathieu beams [4, 13], fractional OV [14, 15] or in noncanonical OV [16, 17]. It should be stressed that the inequality l 6= q can be achieved by considering only the wavefront. This is physically distinct from the difference between the total angular momentum and TC that can be observed in azimuthally polarized light beams [18]. A proper understanding of s-OV is of great interest because s-OV may extend the current applications of c-OV, as initially shown in [19, 20] and more recently discussed in [21]. For example, it is possible to control transverse forces in optical tweezers [5], or selectively match specific surface plasmon modes [7, 11].
#222475 - $15.00 USD Received 5 Sep 2014; revised 13 Nov 2014; accepted 16 Nov 2014; published 26 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030315 | OPTICS EXPRESS 30316
The most simple techniques to characterize the TC in c-OV consists in analyzing the diffraction pattern due to a designed screen [22, 23]. However, since the impinging light profile must match the screen profile, by changing the OV profile these techniques tend to measure only part of the total TC. For example, in [21] it is discussed an approach to design s-OV and when the triangular slit technique [23] was used to characterize the TC, it was observed that it was usually inadequate for non-cylindrical beams. Another usual procedure to measure the TC consists of counting the number of spirals when interfering an OV with a reference wave [24]. This allows in principle an analysis of the total TC of an s-OV, but without any information about its spatial distribution. A more complete method to characterize TC requires the simultaneous measurement of the beam amplitude and phase, as in [15, 25]. Even though phase retrieval increase the complexity of the experimental system, this approach allows a deeper analysis of the beam properties. To measure the classical OAM of a beam, one may perform a modal decomposition [26] or a simultaneous retrieval of the beam phase and amplitude. In this work we describe how the OAM and the TC can be measured in beams with arbitrarily shaped OV cores, and the paper is structured as follows. In sec. 2 we discuss how the OAM and the TC can be evaluated in OV with generic geometries. The expression for the classical OAM is given, and the local OAM (`-OAM) concept is introduced. In sec. 3 we discuss an experimental setup in which the OV amplitude and phase are measured for various beams. Also it is introduced in sec. 3 a procedure to obtain the `-OAM and the OAM from the experimentally measured amplitude and phase profiles. In sec. 4 we demonstrate this approach for s-OV with cores shaped as a line, a corner and a triangle. Section 5 contains a summary of the work. 2. `-OAM and OAM in optical vortices In cylindrical coordinates r = r (ρ, φ , z) a paraxial monochromatic field propagating in vacuum along zˆ may be represented by the following vector potential A (r,t) = εˆ
2µ0 P0 ωk
1 2
A (r) exp i [χ (r) + kz − ωt] ,
(1)
where P0 is the optical power, µ0 is the vacuum permeability, ω and k are the light angular frequency and wave number, respectively. The remaining terms are the beam transverse phase profile χ (r) and A (r) is the vector potential amplitude envelope, normalized such that ´ 2 z=z0 ρdρdφ |A (r)| = 1 for any z0 . Considering a linearly polarized OV, the propagation of A (r,t) is then properly accounted by the scalar terms in the slowly varying envelope approximation, ∂ 2 A (r) eiχ(r) = 0, (2) ∇⊥ − 2ik ∂z where ∇2⊥ operates only over the transverse coordinates ρ, φ . The total TC, QT , contained inside a contour C of radius c, at the plane z = z0 , i.e. on the plane ρ, φ perpendicular to the propagation direction, may be written as [14, 27] ˛ 1 QT = dx · ~∇⊥ χ (r) , (3) 2π C where dx is the infinitesimal displacement along C. QT gives the number of times that the beam phase pass through the interval [0, 2π] following the curve C. If the contour C does not pass over a singularity, QT is an integer even when χ (r) has discontinuities [14]. For the purposes of this work, a single TC is positioned at any point r in the plane z0 such that Re {A (r,t)} = Im {A (r,t)} = 0 and the integral in
#222475 - $15.00 USD Received 5 Sep 2014; revised 13 Nov 2014; accepted 16 Nov 2014; published 26 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030315 | OPTICS EXPRESS 30317
Eq. (3) is nonzero when C is a circle of infinitesimal radius encircling r. For example, supposing that the OV of interest has a single dark core, C may be chosen as a circle of radius ρ = c. Mathematically, the previous definition of TC can be stated more precisely in terms of the Weierstrass canonical product. An arbitrary OV can be represented by the product between an envelope with a Gaussian decay and an entire analytical function of a single complex variable f (u) [19, 20, 28]. In terms of the Weierstrass canonical product, it is possible to represent f (u) as a product of its zeros, f (u) = ∏Ni=1 (u − ui )qi , where ui and qi are, respectively the position of the zero and its associated TC. Infinitesimally close to ui , and considering u = ui + δ , f (δ ) = δ qi C (δ ) ≈ δ qi C (0), since C (δ ) must vary very weakly with δ for |δ | 1. Then, it is simple to verify through Eq. (3) that the TC associated with ui is precisely qi . From the definition in Eq. (3), QT is independent of the shape of C. This choice has some advantages, as the ability to observe a spatial profile relative to the TC distribution, and it is directly related to the beam `-OAM. Notice that in this case Eq. (3) simplifies to QT =
1 2π
ˆ
2π
w (ρ = c, φ , z0 ) dφ ,
(4)
0
where w (r) is the `-OAM [13], w (r) =
∂χ (r) . ∂φ
(5)
The evaluation of w (r) is important because it associates a designed OV phase structure to the experimentally obtained profile, both qualitatively (spatial profile) and quantitatively (through Eq. 4). w (r) characterizes the OV morphology [16, 17], and will be used here to associate a spatial profile to the beam topological properties. It should be remarked that the `-OAM can assume any value in principle. In the general case, the connection between the `-OAM and the TC is through Eq. (4) . The TC is the average value of the `-OAM through the contour C. Therefore if w (r) > QT at some portion of the curve C, there must be another portion in which w (r) < QT . Another quantity of general interest in OV is the classical n OAM density of light, Lzo , along the propagation direction zˆ. It can be shown that Lz = ε02ω Re A (r,t) −i ∂∂φ A∗ (r,t) [29], and by direct substitution of Eq. (1), we obtain P0 ∂χ ∗ Lz = A (r) A (r) . (6) ωc ∂φ Since P0 = N h¯ ω, where N is the number of photons carrying energy h¯ ω impinging on the plane ρφ per second, the `-OAM value per photon at a given position is h¯ w = h¯ ∂ χ/∂ φ [13]. Considering that the product A ∗ (r) A (r) gives the probability of finding a photon at a given point, the semi-classical average OAM per photon may be determined from Eq. (6) as ˆ ∂χ h¯ hli = h¯ ρdρdφ A ∗ (r) A (r) . (7) ∂φ A comparison between Eqs. (4) and (7) shows QT 6= hli in general. We therefore remark that usually there is no direct relation between OAM and total TC [30]. Even for beams with azimuthal symmetry, it is possible to obtain exotic configurations in which these quantities are distinct. For example, by embedding a Laguerre-Gauss beam characterized by a TC minner > 0 inside the core of another Laguerre-Gauss beam characterized by a TC mouter > minner > 0, it can be observed that the superposition of these beams leads to QT = mouter , while hli < mouter . Also, although the intensity profile of a beam is related to its TC distribution [14,21,31], it does
#222475 - $15.00 USD Received 5 Sep 2014; revised 13 Nov 2014; accepted 16 Nov 2014; published 26 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030315 | OPTICS EXPRESS 30318
not carry information about the topological or OAM properties of a beam [32]. Therefore in a general OV, by measuring QT one does not necessarily have information about hli and viceversa. However, both quantities are related to the `-OAM which can be obtained from χ (r). Thus we emphasize that it is important to determine the `-OAM for the characterization of the classical OAM and TC in arbitrary OV beams. 3.
Experimental determination of OAM and TC of linearly polarized light
The initial step to obtain the experimental properties of the TC distribution and the OAM profile of a general OV consists in measuring the light amplitude and phase profiles. Using the experimental setup shown in Fig. 1(a) it is possible to produce OV using the spatial light modulator (SLM) and characterize their amplitude and phase using the CCD camera. The OV are generated in the SLM by applying a hologram χSLM which contains the OV phase profile χOV superimposed with a blazed grating χOV . The blazed grating directs the OV of interest to the SLM first diffracted order. By using a spatial filter, it is possible to select only the SLM first diffracted order and block unwanted light from the SLM. If the Arm-2 mirror is off the beam line, only the intensity profile IOV (r) of the produced OV will reach the CCD camera. In this p case, the amplitude profile is given by A (r) = IOV (r). The CCD was positioned at the SLM image plane, which we defined as z = 0 cm.
(a)
Arm 1 SLM
Arm 2 off
Arm 2 on
Intensity measurement
Interference measurements
Laser 805 nm
Iris
Retrieved phase profile
CCD
f=30 cm
0
z
f=100 cm
0
Intensity (a.u.)
1
0
Arm 2 off
Arm 2 on
f=25 cm
(b)
Phase
2π
Fig. 1. (a) Experimental setup (not to scale). The output of a fiber coupled laser diode emitting at 805 nm is collimated with a lens with long focal distance ( f = 25 cm), producing a nearly plane wave. The collimated light goes to a Michelson interferometer (MI) in which the arm 1 contains a SLM (Hamamatsu - LCOS X10468-02). The MI arm 2 provides the plane wave reference when the mirror is on the beam line, or allows intensity measurements when the mirror is off the beam line. The reference (red line) and modulated (blue line) beams have a small relative angle and are spatially filtered and then imaged on a CCD camera (Thorlabs - DCC1240M) positioned at the SLM image plane (z = 0 cm). (b) Fluxogram of the experimental procedure to determine the beam amplitude and phase profiles, as described in the text.
To measure the phase profile, χ (r), we followed an approach similar to that described in [15, 25]. We add to χSLM a spatial DC phase offset φOffset ∈ [0, 2π] and place the MI Arm-2 mirror on the beam line. The intensity pattern at the CCD plane for a perfectly coherent light consists of p (8) ICCD (φOffset ) = IOV + IRef − 2 IOV IRef cos (χRel + φOffset ) , where IRef is the intensity profile of the reference beam, and χRel is the relative phase between the reference beam and the OV. The position r is implicitly considered in each term for improved clarity. Since φOffset can be adjusted a priori on the phase mask, one may retrieve the spatial phase profile by virtue of Fourier orthogonality. Multiplying the measured intensity profile #222475 - $15.00 USD Received 5 Sep 2014; revised 13 Nov 2014; accepted 16 Nov 2014; published 26 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030315 | OPTICS EXPRESS 30319
ICCD (φOffset ) by exp (−iφOffset ), and numerically integrating over φOffset one has 1 2π
ˆ
2π
p dφOffset ICCD (φOffset ) e−iφOffset = − IOV IRef eiχRel ,
(9)
0
and the argument of the resulting complex number is χRel (r) + π. The above approach allows the retrieval of χRel at each CCD pixel, and it is a simple and effective way to retrieve spatial phase profiles. To obtain a better signal/noise ratio, we used 10 values of φOffset [15]. Two relevant characteristics of this method are worth noticing. When the reference wave is an ideal plane wave propagating √ parallel to the modulated beam,√χRel (r) = χOV (r) + constant, and since IOV → 0 near a TC, IOV is amplified by a factor of IRef . Therefore, although this experimental phase-retrieval method requires more data and acquisition time than those based on the Fourier transform method [25, 33], it shows an improved sensitivity, the output is automatically determined in complex notation and it does not require filtering in the momentum space. Usually the reference beam is not an ideal plane wave propagating parallel to the modulated beam. However, this poses no problem to the approach described above. Since any optical phase retrieval method will always measure a relative phase, it is possible to design a reference n (r) = wave at the SLM. For an OV labeled by the superscript n, one has in general that χRel n (r) + χ 0 (r). If a flat phase profile is applied to the SLM (no OV), this returns the overall χOV Rel 0 (r). By subtracting χ 0 (r) from χ n (r) it is possible to obtain only the OV phase phase χRel Rel Rel profile. After measuring the spatial profiles of amplitude and phase, the `-OAM, w (r), OAM, hli, and TC, QT , may be calculated. To compute the azimuthal derivative in the `-OAM , and to avoid the unphysical discontinuities when the phase χ goes from 0 to 2π, w (r) was determined using the identity ∂ ∂ ∂χ = Re e−iχ −i x − y eiχ , (10) w (r) = ∂φ ∂y ∂x where the derivatives were calculated via a Fourier spectral method with a smoothing Gaussian filter [34]. Notice that the derivatives in Eq. (10) act over the well behaved function exp (iχ), which is insensitive to the modulus 2π phase jumps that may be present in the originally retrieved phase profile χ. This fact must be remarked because the unphysical discontinuities are significant when ∂ χ/∂ φ is calculated directly. To ensure the numerical stability when evaluating QT from the phase profile via Eq. (4), it is necessary´ to weaken the condition ρ = c. This can be achieved by generalizing ∞ w (ρ = c, φ , z0 ) = 0 ρdρ p (ρ) w (ρ, φ , z0 ), where p (ρ) is a radial probability distribution strongly concentrated near ρ = c. For simplicity we assume that for a sufficiently small thickness ε, p (ρ) = 0, if |ρ − c| > ε, and p (ρ) = (2ερ)−1 , if |ρ − c| ≤ ε. This approach allows the line integral in Eq. (4) to be approximated by a surface integral over the plane z = z0 ˆ 1 QT = ρdρdφ p (ρ) w (ρ, φ , z0 ) . (11) 2π To gain insight on the meaning of the quantities discussed above, in Fig. 2 it is shown how the quantities discussed above behave in an experimental measurement of a c-OV for a beam obtained by applying a TC of 5 in the SLM. Notice that the amplitude profile is very uniform in Fig. 2(a) and the measured phase profile in Fig. 2(b) is consistent to that expected for usual c-OV. The semi-classical OAM density is given by the integrand of Eq. (7) and can be obtained by multiplying the result of Eq. (10) with the measured intensity profile. The normalized ex#222475 - $15.00 USD Received 5 Sep 2014; revised 13 Nov 2014; accepted 16 Nov 2014; published 26 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030315 | OPTICS EXPRESS 30320
(e) 1
10
OAM density
2π 1
Phase
(d)
(c)
Amplitude
(a)
5
0
0
0
0
(b)
Fig. 2. Application of the concepts developed in sec. 3 to a c-OV for χOV = 5φ . Experimentally measured amplitude (a) and phase (b) beam profiles. OAM density profile (c), from which it was determined that hli = 4.9. Experimental (d) and theoretically expected (e) `-OAM profiles for this beam. For this beam it was measured QT = 5.0. 2π 1 OAM density
(c) 1
12
Phase
(b)
Amplitude
(a)
0
0
0
0
6
Fig. 3. Typical experimental s-OV profiles of amplitude and phase, and the corresponding `-OAM and OAM density. The data represent TC distributed over a line (a), a corner (b) and a triangle (c). The values of hli and QT were calculated, respectively, by applying Eqs. (7) and (11) to the experimental data.
perimental OAM density, Lz , for the c-OV shown in Fig. 2(c) is flat, up to experimental error. Another quantity of interest is the average semi-classical OAM per photon hli, which is obtained by performing the integral in Eq. (7). For the c-OV from Fig. 2, it was measured that hli = 4.9. Figs. 2(d)-(e) show the experimental and theoretically expected `-OAM profiles, respectively, for this c-OV. It can be noticed that there is a good agreement between both in the color scale. The experimental determination of the beam TC, QT , was performed using Eq. (11) with c = 0.85rmax , ε = 0.1rmax , where rmax is the external radius of the amplitude profile. As a result, it was measured that the produced c-OV has QT = 5.0. 4.
Experimental results
Without loss of generality to the concepts discussed in the previous sections, we experimentally characterized the `-OAM and the OAM density in the s-OV described in [21]. The s-OV dark core profiles are shaped as a line, a corner and a triangle due to a distribution of TC with the same geometry on the SLM phase mask. These s-OV are prototypes for more complex OV geometries, and have adjustable total TC. Some typical amplitude and phase profiles of these s-OV at z = 0 cm are shown in Fig. 3. The results for each TC distribution are discussed in the subsections below. From the experimental amplitude and phase profiles, hli was calculated by applying Eq. (7), while for QT it was used Eq. (11) with c = 0.85rmax , ε = 0.1rmax , where rmax is the external radius of the amplitude profile.
#222475 - $15.00 USD Received 5 Sep 2014; revised 13 Nov 2014; accepted 16 Nov 2014; published 26 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030315 | OPTICS EXPRESS 30321
2π 1 OAM density
(c) 1
12
Phase
(b)
Amplitude
(a)
0
0
0
0
6
Fig. 4. Experimental data for TC lines with different line lengths and a fixed total applied TC= 10 at z = 0 cm. The line length increases from (a) to (c), and the respective OV are in the regimes of high TC density, elongated OV core and small TC density.
4.1.
Linear distributions of TC
A linear distribution of TC forms a prototype for more complex s-OV core profiles [21], and therefore it is discussed first. Although this TC distribution discussed here is designed on the SLM, its worth noticing that they can appear naturally from c-OV perturbed by the optical system [35]. The OV core of a TC line of infinitesimal length has a circular shape. By increasing the TC line length and keeping the total TC fixed, the core becomes elongated until a point in which the multiple OV cores of unit charge become discernible. For a fixed TC of 10 these three regimes can be visualized in Figs. 4(a)-(c) as, high TC density, elongated OV core and small TC density, respectively. An important remark is that the measured value of QT obtained from Eq. (11) agrees with the applied value for all the geometries shown in Fig. 4 and also in all the other cases discussed below. Another observation is that the `-OAM magnitude, |w (r)|, is reduced near the center of the TC line. This can be understood by noticing that between adjacent equally-charged OV the azimuthal phase variation is smaller than if both OV were at the same point. Far from the TC distribution the phase profile must depend only on the value of QT and therefore the `-OAM, w (r), should have similar magnitudes in all regimes [21]. This can be observed in Figs. 4(a)-(c). The OAM densities follows the beam intensity profile but are reduced near the OV core. The reduced OAM density, Lz , follows the `-OAM, w (r), reduction, and as a result we observed that hli ≤ QT . Also, it was investigated how the linear distributions of TC propagate, and a typical example is shown in Fig. 5. Notice that the OV core rotates and expands under propagation, but remains line-shaped. The rotation is due to the Guoy phase shift [28, 36], and can be seen as a signature of the structural stability of the OV profiles under propagation [28]. Another important point is that both QT and hli are conserved under propagation. This is important because establishes that the approach discussed in this work does not depend on the CCD position. 4.2.
Corner- and triangle-shaped distributions of TC
After discussing the behavior of a TC line, the next step in complexity are two TC lines. The data for a corner-shaped distribution of TC is shown in Fig. 6. By fixing the geometry of the TC distribution, the OV core also can be characterized by the three regimes discussed for TC lines as illustrated in Fig. 6(a). A high TC density was observed when QT ≥ 12, since the OV core profile becomes more rounded and tends to a circular shape. At smaller TC densities, QT ≤ 8, it can be observed that multiple OV cores are distinguishable. When QT = 10, the OV core is corner-shaped. From the `-OAM profiles, it can be seen that the `-OAM is very small inside the triangle determined by the corner extremities. This fact can also be inferred from the relative uniformity of the phase profiles, and implies that the OAM density core profile is more rounded
#222475 - $15.00 USD Received 5 Sep 2014; revised 13 Nov 2014; accepted 16 Nov 2014; published 26 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030315 | OPTICS EXPRESS 30322
(c) 1
OAM density
2π 1 12
Phase
(b)
Amplitude
(a)
0
0
0
0
6
Fig. 5. Measurements corresponding to a TC line at distinct z planes. z = 0 cm, 5 cm, 10 cm respectively in (a-c). Notice that both QT and hli are conserved under propagation.
6
8
10
12
2π 1
0 0 14 Applied TC
(b)
12
0
0
Measured values
14
OAM density
Phase
Amplitude
(a) 1
12 10
6
8 6 6
8
10
12
14
Applied TC
Fig. 6. (a) Profiles of amplitude, phase, OAM density and LOAM for a corner-shaped TC distribution with a fixed geometry and varying total applied TC. (b) Relation between the measured QT and hli in terms of the applied TC at the SLM. All measurements were taken at z = 0 cm.
6
9
12
15
0 0 Applied TC
12
0
0
20
Measured values
OAM density
3
(b)
2π 1
Phase
Amplitude
(a) 1
15
6
10 5 5
10
15
Applied TC
20
Fig. 7. (a) Profiles of amplitude, phase, OAM density and `-OAM for a triangle-shaped TC distribution with a fixed geometry and varying total applied TC. (b) Relation between the measured QT and hli in terms of the applied TC at the SLM. All measurements were taken at z = 0 cm.
#222475 - $15.00 USD Received 5 Sep 2014; revised 13 Nov 2014; accepted 16 Nov 2014; published 26 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030315 | OPTICS EXPRESS 30323
than the OV core at the amplitude profile. This conclusion is important for optical tweezers applications, in which someone wants to transport a dielectric particle along a specified route. On the other hand, it can be seen in Fig. 6(b) that the measured values of QT corresponds to the applied TC values and for these beams the reduction in hli with respect to a c-OV is small. The experimental results for a triangle-shaped TC distribution are shown in Fig. 7. In Fig. 7(a) it can be seen that the OV core profiles are triangle-shaped and the intensity profile regimes are high TC density at QT ≥ 12, elongated OV core at QT = 9 and small TC density at QT ≤ 6. The `-OAM and the OAM density profiles follows the TC distribution geometry, and therefore are suitable for applications in optical tweezers. Also it can be seen that, similarly to the cornershaped TC distribution, QT corresponds to the applied value of TC and for these beams the reduction in hli with respect to a c-OV is smaller than the previous case. 5.
Summary
In this paper it was described how orbital angular momentum (OAM) and topological charge (TC) are distinct physical quantities even for linearly polarized light, that usually have degenerate values for beams with azimuthal symmetry, i.e., cylindrical optical vortices (c-OV). A general approach for determining the OAM and the TC of a beam was developed for shaped optical vortices (s-OV) in terms of the light beam amplitude and phase profiles and the concept of local OAM (`-OAM) was used as a quantity to characterize the OV morphology. This is important because contrarily to c-OV, in s-OV the beam TC is not concentrated at a single point, and therefore the OAM and TC do not have degenerate values; the OAM is not necessarily homogeneous over the beam profile. Then, it followed a discussion on the experimental setup and how to retrieve the light spatial phase profile. Some fundamental aspects to determine the `-OAM and the TC from the experimental data were discussed. The developed concepts were applied without loss of generality to the set of s-OV discussed in [21]. It can be seen that the characterization technique proposed in this work is applicable in s-OV with general geometries by retrieving data of both OAM and TC. It should be remarked that the TC measurement approach described here does not depend on the matching of the beam spatial profile with a specific geometry, as in [22–24], and is therefore more suited for non-cylindrical OV. The introduced concept of `-OAM, w (r), demonstrates its importance in obtaining the TC, QT , and clarified the behavior of the OAM density profile in corner-shaped s-OV. For the line- and triangle-shaped TC distributions it was observed that both amplitude and OAM density profiles followed the designed geometry. The s-OV characterization procedure described above can be of special interest in optical tweezers and plasmonics. Since s-OV can be used to control the transverse forces in optical tweezers [5], the designed particle trajectory can have any shape, and a general characterization procedure as the one presented here is necessary. Also, the matching of an OV to a surface plasmon may requires non-cylindrical OV [11]. Therefore, our approach is relevant to characterize the OV produced by plasmonic structures. Finally, since the herein discussed procedure is general, it can be applied also to c-OV as Bessel or Laguerre-Gauss beams. Acknowledgments We acknowledge the financial support from the Brazilian agencies Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Fundação de Amparo à Ciência e Tecnologia do Estado de Pernambuco (FACEPE). The work was performed in the framework of the National Institute of Photonics (INCT de Fotônica) project and PRONEX/CNPq/FACEPE. We also acknowledge helpful discussions with Dr. L. Pruvost. A. M. A. also thanks Dr. W. Löffler for inciting an extension of the previous work on s-OV.
#222475 - $15.00 USD Received 5 Sep 2014; revised 13 Nov 2014; accepted 16 Nov 2014; published 26 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030315 | OPTICS EXPRESS 30324
de Física, Universidade Federal de Pernambuco, 50670-901 Recife, PE, Brazil 2 [email protected] ∗ [email protected]
Abstract: Optical vortices (OV) are usually associated to cylindrically symmetric light beams. However, they can have more general geometries that extends their applicability. Since the typical experimental characterization methods are not appropriate for OV with arbitrary shapes, we discuss in this work how the definitions of the classical orbital angular momentum and the topological charge can be used to retrieve these informations in the general case. The concepts discussed are experimentally demonstrated and may be specially useful in areas such as optical tweezers and plasmonics. © 2014 Optical Society of America OCIS codes: (260.6042) Singular optics; (050.4865) Optical vortices.
References 1. 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, 8185–8189 (1992). 2. J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nature Photon. 6, 488–496 (2012). 3. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001). 4. W. N. Plick, M. Krenn, R. Fickler, S. Ramelow, and A. Zeilinger, “Quantum orbital angular momentum of elliptically symmetric light,” Phys. Rev. A 87, 033806 (2013). 5. A. Jesacher, C. Maurer, S. Fuerhapter, A. Schwaighofer, S. Bernet, and M. Ritsch-Marte, “Optical tweezers of programmable shape with transverse scattering forces,” Opt. Commun. 281, 2207–2212 (2008). 6. M. Chen, M. Mazilu, Y. Arita, E. Wright, and K. Dholakia, “Dynamics of microparticles trapped in a perfect vortex beam,” Opt. Lett. 38, 4919–4922 (2013). 7. H. Kim, J. Park, S.-W. Cho, S.-Y. Lee, M. Kang, and B. Lee, “Synthesis and dynamic switching of surface plasmon vortices with plasmonic vortex lens,” Nano Lett. 10, 529–536 (2010). 8. A. Rury, “Coherent control of plasmonic spectra using the orbital angular momentum of light,” Phys. Rev. B 88, 205132 (2013). 9. K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, and T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12, 3645–3649 (2012). 10. Y. Gorodetski, A. Drezet, C. Genet, and T. W. Ebbesen, “Generating far-field orbital angular momenta from near-field optical chirality,” Phys. Rev. Lett. 110, 203906 (2013). 11. E. Brasselet, G. Gervinskas, G. Seniutinas, and S. Juodkazis, “Topological shaping of light by closed-path nanoslits,” Phys. Rev. Lett. 111, 193901 (2013). 12. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011). 13. S. Chávez-Cerda, M. Padgett, I. Allison, G. New, J. Gutiérrez-Vega, A. O’Neil, I. MacVicar, and J. Courtial, “Holographic generation and orbital angular momentum of high-order Mathieu beams,” J. Opt. B: Quantum and Semiclass. Opt. 4, S52–S57 (2002).
#222475 - $15.00 USD Received 5 Sep 2014; revised 13 Nov 2014; accepted 16 Nov 2014; published 26 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030315 | OPTICS EXPRESS 30315
14. M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6, 259–268 (2004). 15. J. Leach, E. Yao, and M. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004). 16. F. S. Roux, “Distribution of angular momentum and vortex morphology in optical beams,” Opt. Commun. 242, 45–55 (2004). 17. A. Y. Bekshaev, M. V. Vasnetsov, and M. S. Soskin, “Description of the morphology of optical vortices using the orbital angular momentum and its components,” Opt. Spectrosc. 100, 910–915 (2006). 18. Q. Zhan, “Properties of circularly polarized vortex beams,” Opt. Lett. 31, 867–869 (2006). 19. E. Abramochkin and V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102, 336–350 (1993). 20. E. Abramochkin and V. Volostnikov, “Spiral-type beams: optical and quantum aspects,” Opt. Commun. 125, 302–323 (1996). 21. A. M. Amaral, E. L. Falcão-Filho, and C. B. de Araújo, “Shaping optical beams with topological charge,” Opt. Lett. 38, 1579–1581 (2013). 22. G. Berkhout and M. Beijersbergen, “Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects,” Phys. Rev. Lett. 101, 100801 (2008). 23. J. M. Hickmann, E. J. S. 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, 053904 (2010). 24. M. Soskin, V. Gorshkov, M. Vasnetsov, J. Malos, and N. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997). 25. T. Ando, N. Matsumoto, Y. Ohtake, Y. Takiguchi, and T. Inoue, “Structure of optical singularities in coaxial superpositions of Laguerre-Gaussian modes,” J. Opt. Soc. Am. A 27, 2602–2612 (2010). 26. C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, “Measurement of the orbital angular momentum density of light by modal decomposition,” New J. Phys. 15, 073025 (2013). 27. M. Nakahara, Geometry, Topology and Physics (Taylor & Francis, 2003). 28. E. G. Abramochkin and V. G. Volostnikov, “Spiral light beams,” Physics-Uspekhi 47, 1177–1203 (2004). 29. M. V. Berry, “Paraxial beams of spinning light,” Proc. SPIE 3487, 6 (1998). 30. M. V. Berry, “Optical currents,” J. Opt. A: Pure Appl. Opt. 11, 094001 (2009). 31. J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt. Lett. 28, 872–874 (2003). 32. R. Pugatch, M. Shuker, O. Firstenberg, A. Ron, and N. Davidson, “Topological stability of stored optical vortices,” Phys. Rev. Lett. 98, 203601 (2007). 33. 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, 156–160 (1982). 34. K. Ahnert and M. Abel, “Numerical differentiation of experimental data: local versus global methods,” Comput. Phys. Commun. 177, 764–774 (2007). 35. M. Dennis, “Rows of optical vortices from elliptically perturbing a high-order beam,” Opt. Lett. 31, 1325–1327 (2006). 36. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
1.
Introduction
Optical vortices (OV) have been extensively studied since the seminal work by Allen et al. [1] and are applied in topics as diverse as classical and quantum communications [2–4], optical tweezers [5,6] and plasmonics [7–11]. OV are mainly characterized by their topological charge (TC) q, which is related to the beam phase profile, and their orbital angular momentum (OAM) l, which also involves the intensity profile [12]. Most of previously referred applications use cylindrical OV (c-OV), which exhibiths azimuthal symmetry and are characterized by well defined values of l and q, with l = q, as Bessel or Laguerre-Gauss beams. However, by analyzing OAM and TC for shaped OV (s-OV) it can be seen that their values might differ, as in Mathieu beams [4, 13], fractional OV [14, 15] or in noncanonical OV [16, 17]. It should be stressed that the inequality l 6= q can be achieved by considering only the wavefront. This is physically distinct from the difference between the total angular momentum and TC that can be observed in azimuthally polarized light beams [18]. A proper understanding of s-OV is of great interest because s-OV may extend the current applications of c-OV, as initially shown in [19, 20] and more recently discussed in [21]. For example, it is possible to control transverse forces in optical tweezers [5], or selectively match specific surface plasmon modes [7, 11].
#222475 - $15.00 USD Received 5 Sep 2014; revised 13 Nov 2014; accepted 16 Nov 2014; published 26 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030315 | OPTICS EXPRESS 30316
The most simple techniques to characterize the TC in c-OV consists in analyzing the diffraction pattern due to a designed screen [22, 23]. However, since the impinging light profile must match the screen profile, by changing the OV profile these techniques tend to measure only part of the total TC. For example, in [21] it is discussed an approach to design s-OV and when the triangular slit technique [23] was used to characterize the TC, it was observed that it was usually inadequate for non-cylindrical beams. Another usual procedure to measure the TC consists of counting the number of spirals when interfering an OV with a reference wave [24]. This allows in principle an analysis of the total TC of an s-OV, but without any information about its spatial distribution. A more complete method to characterize TC requires the simultaneous measurement of the beam amplitude and phase, as in [15, 25]. Even though phase retrieval increase the complexity of the experimental system, this approach allows a deeper analysis of the beam properties. To measure the classical OAM of a beam, one may perform a modal decomposition [26] or a simultaneous retrieval of the beam phase and amplitude. In this work we describe how the OAM and the TC can be measured in beams with arbitrarily shaped OV cores, and the paper is structured as follows. In sec. 2 we discuss how the OAM and the TC can be evaluated in OV with generic geometries. The expression for the classical OAM is given, and the local OAM (`-OAM) concept is introduced. In sec. 3 we discuss an experimental setup in which the OV amplitude and phase are measured for various beams. Also it is introduced in sec. 3 a procedure to obtain the `-OAM and the OAM from the experimentally measured amplitude and phase profiles. In sec. 4 we demonstrate this approach for s-OV with cores shaped as a line, a corner and a triangle. Section 5 contains a summary of the work. 2. `-OAM and OAM in optical vortices In cylindrical coordinates r = r (ρ, φ , z) a paraxial monochromatic field propagating in vacuum along zˆ may be represented by the following vector potential A (r,t) = εˆ
2µ0 P0 ωk
1 2
A (r) exp i [χ (r) + kz − ωt] ,
(1)
where P0 is the optical power, µ0 is the vacuum permeability, ω and k are the light angular frequency and wave number, respectively. The remaining terms are the beam transverse phase profile χ (r) and A (r) is the vector potential amplitude envelope, normalized such that ´ 2 z=z0 ρdρdφ |A (r)| = 1 for any z0 . Considering a linearly polarized OV, the propagation of A (r,t) is then properly accounted by the scalar terms in the slowly varying envelope approximation, ∂ 2 A (r) eiχ(r) = 0, (2) ∇⊥ − 2ik ∂z where ∇2⊥ operates only over the transverse coordinates ρ, φ . The total TC, QT , contained inside a contour C of radius c, at the plane z = z0 , i.e. on the plane ρ, φ perpendicular to the propagation direction, may be written as [14, 27] ˛ 1 QT = dx · ~∇⊥ χ (r) , (3) 2π C where dx is the infinitesimal displacement along C. QT gives the number of times that the beam phase pass through the interval [0, 2π] following the curve C. If the contour C does not pass over a singularity, QT is an integer even when χ (r) has discontinuities [14]. For the purposes of this work, a single TC is positioned at any point r in the plane z0 such that Re {A (r,t)} = Im {A (r,t)} = 0 and the integral in
#222475 - $15.00 USD Received 5 Sep 2014; revised 13 Nov 2014; accepted 16 Nov 2014; published 26 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030315 | OPTICS EXPRESS 30317
Eq. (3) is nonzero when C is a circle of infinitesimal radius encircling r. For example, supposing that the OV of interest has a single dark core, C may be chosen as a circle of radius ρ = c. Mathematically, the previous definition of TC can be stated more precisely in terms of the Weierstrass canonical product. An arbitrary OV can be represented by the product between an envelope with a Gaussian decay and an entire analytical function of a single complex variable f (u) [19, 20, 28]. In terms of the Weierstrass canonical product, it is possible to represent f (u) as a product of its zeros, f (u) = ∏Ni=1 (u − ui )qi , where ui and qi are, respectively the position of the zero and its associated TC. Infinitesimally close to ui , and considering u = ui + δ , f (δ ) = δ qi C (δ ) ≈ δ qi C (0), since C (δ ) must vary very weakly with δ for |δ | 1. Then, it is simple to verify through Eq. (3) that the TC associated with ui is precisely qi . From the definition in Eq. (3), QT is independent of the shape of C. This choice has some advantages, as the ability to observe a spatial profile relative to the TC distribution, and it is directly related to the beam `-OAM. Notice that in this case Eq. (3) simplifies to QT =
1 2π
ˆ
2π
w (ρ = c, φ , z0 ) dφ ,
(4)
0
where w (r) is the `-OAM [13], w (r) =
∂χ (r) . ∂φ
(5)
The evaluation of w (r) is important because it associates a designed OV phase structure to the experimentally obtained profile, both qualitatively (spatial profile) and quantitatively (through Eq. 4). w (r) characterizes the OV morphology [16, 17], and will be used here to associate a spatial profile to the beam topological properties. It should be remarked that the `-OAM can assume any value in principle. In the general case, the connection between the `-OAM and the TC is through Eq. (4) . The TC is the average value of the `-OAM through the contour C. Therefore if w (r) > QT at some portion of the curve C, there must be another portion in which w (r) < QT . Another quantity of general interest in OV is the classical n OAM density of light, Lzo , along the propagation direction zˆ. It can be shown that Lz = ε02ω Re A (r,t) −i ∂∂φ A∗ (r,t) [29], and by direct substitution of Eq. (1), we obtain P0 ∂χ ∗ Lz = A (r) A (r) . (6) ωc ∂φ Since P0 = N h¯ ω, where N is the number of photons carrying energy h¯ ω impinging on the plane ρφ per second, the `-OAM value per photon at a given position is h¯ w = h¯ ∂ χ/∂ φ [13]. Considering that the product A ∗ (r) A (r) gives the probability of finding a photon at a given point, the semi-classical average OAM per photon may be determined from Eq. (6) as ˆ ∂χ h¯ hli = h¯ ρdρdφ A ∗ (r) A (r) . (7) ∂φ A comparison between Eqs. (4) and (7) shows QT 6= hli in general. We therefore remark that usually there is no direct relation between OAM and total TC [30]. Even for beams with azimuthal symmetry, it is possible to obtain exotic configurations in which these quantities are distinct. For example, by embedding a Laguerre-Gauss beam characterized by a TC minner > 0 inside the core of another Laguerre-Gauss beam characterized by a TC mouter > minner > 0, it can be observed that the superposition of these beams leads to QT = mouter , while hli < mouter . Also, although the intensity profile of a beam is related to its TC distribution [14,21,31], it does
#222475 - $15.00 USD Received 5 Sep 2014; revised 13 Nov 2014; accepted 16 Nov 2014; published 26 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030315 | OPTICS EXPRESS 30318
not carry information about the topological or OAM properties of a beam [32]. Therefore in a general OV, by measuring QT one does not necessarily have information about hli and viceversa. However, both quantities are related to the `-OAM which can be obtained from χ (r). Thus we emphasize that it is important to determine the `-OAM for the characterization of the classical OAM and TC in arbitrary OV beams. 3.
Experimental determination of OAM and TC of linearly polarized light
The initial step to obtain the experimental properties of the TC distribution and the OAM profile of a general OV consists in measuring the light amplitude and phase profiles. Using the experimental setup shown in Fig. 1(a) it is possible to produce OV using the spatial light modulator (SLM) and characterize their amplitude and phase using the CCD camera. The OV are generated in the SLM by applying a hologram χSLM which contains the OV phase profile χOV superimposed with a blazed grating χOV . The blazed grating directs the OV of interest to the SLM first diffracted order. By using a spatial filter, it is possible to select only the SLM first diffracted order and block unwanted light from the SLM. If the Arm-2 mirror is off the beam line, only the intensity profile IOV (r) of the produced OV will reach the CCD camera. In this p case, the amplitude profile is given by A (r) = IOV (r). The CCD was positioned at the SLM image plane, which we defined as z = 0 cm.
(a)
Arm 1 SLM
Arm 2 off
Arm 2 on
Intensity measurement
Interference measurements
Laser 805 nm
Iris
Retrieved phase profile
CCD
f=30 cm
0
z
f=100 cm
0
Intensity (a.u.)
1
0
Arm 2 off
Arm 2 on
f=25 cm
(b)
Phase
2π
Fig. 1. (a) Experimental setup (not to scale). The output of a fiber coupled laser diode emitting at 805 nm is collimated with a lens with long focal distance ( f = 25 cm), producing a nearly plane wave. The collimated light goes to a Michelson interferometer (MI) in which the arm 1 contains a SLM (Hamamatsu - LCOS X10468-02). The MI arm 2 provides the plane wave reference when the mirror is on the beam line, or allows intensity measurements when the mirror is off the beam line. The reference (red line) and modulated (blue line) beams have a small relative angle and are spatially filtered and then imaged on a CCD camera (Thorlabs - DCC1240M) positioned at the SLM image plane (z = 0 cm). (b) Fluxogram of the experimental procedure to determine the beam amplitude and phase profiles, as described in the text.
To measure the phase profile, χ (r), we followed an approach similar to that described in [15, 25]. We add to χSLM a spatial DC phase offset φOffset ∈ [0, 2π] and place the MI Arm-2 mirror on the beam line. The intensity pattern at the CCD plane for a perfectly coherent light consists of p (8) ICCD (φOffset ) = IOV + IRef − 2 IOV IRef cos (χRel + φOffset ) , where IRef is the intensity profile of the reference beam, and χRel is the relative phase between the reference beam and the OV. The position r is implicitly considered in each term for improved clarity. Since φOffset can be adjusted a priori on the phase mask, one may retrieve the spatial phase profile by virtue of Fourier orthogonality. Multiplying the measured intensity profile #222475 - $15.00 USD Received 5 Sep 2014; revised 13 Nov 2014; accepted 16 Nov 2014; published 26 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030315 | OPTICS EXPRESS 30319
ICCD (φOffset ) by exp (−iφOffset ), and numerically integrating over φOffset one has 1 2π
ˆ
2π
p dφOffset ICCD (φOffset ) e−iφOffset = − IOV IRef eiχRel ,
(9)
0
and the argument of the resulting complex number is χRel (r) + π. The above approach allows the retrieval of χRel at each CCD pixel, and it is a simple and effective way to retrieve spatial phase profiles. To obtain a better signal/noise ratio, we used 10 values of φOffset [15]. Two relevant characteristics of this method are worth noticing. When the reference wave is an ideal plane wave propagating √ parallel to the modulated beam,√χRel (r) = χOV (r) + constant, and since IOV → 0 near a TC, IOV is amplified by a factor of IRef . Therefore, although this experimental phase-retrieval method requires more data and acquisition time than those based on the Fourier transform method [25, 33], it shows an improved sensitivity, the output is automatically determined in complex notation and it does not require filtering in the momentum space. Usually the reference beam is not an ideal plane wave propagating parallel to the modulated beam. However, this poses no problem to the approach described above. Since any optical phase retrieval method will always measure a relative phase, it is possible to design a reference n (r) = wave at the SLM. For an OV labeled by the superscript n, one has in general that χRel n (r) + χ 0 (r). If a flat phase profile is applied to the SLM (no OV), this returns the overall χOV Rel 0 (r). By subtracting χ 0 (r) from χ n (r) it is possible to obtain only the OV phase phase χRel Rel Rel profile. After measuring the spatial profiles of amplitude and phase, the `-OAM, w (r), OAM, hli, and TC, QT , may be calculated. To compute the azimuthal derivative in the `-OAM , and to avoid the unphysical discontinuities when the phase χ goes from 0 to 2π, w (r) was determined using the identity ∂ ∂ ∂χ = Re e−iχ −i x − y eiχ , (10) w (r) = ∂φ ∂y ∂x where the derivatives were calculated via a Fourier spectral method with a smoothing Gaussian filter [34]. Notice that the derivatives in Eq. (10) act over the well behaved function exp (iχ), which is insensitive to the modulus 2π phase jumps that may be present in the originally retrieved phase profile χ. This fact must be remarked because the unphysical discontinuities are significant when ∂ χ/∂ φ is calculated directly. To ensure the numerical stability when evaluating QT from the phase profile via Eq. (4), it is necessary´ to weaken the condition ρ = c. This can be achieved by generalizing ∞ w (ρ = c, φ , z0 ) = 0 ρdρ p (ρ) w (ρ, φ , z0 ), where p (ρ) is a radial probability distribution strongly concentrated near ρ = c. For simplicity we assume that for a sufficiently small thickness ε, p (ρ) = 0, if |ρ − c| > ε, and p (ρ) = (2ερ)−1 , if |ρ − c| ≤ ε. This approach allows the line integral in Eq. (4) to be approximated by a surface integral over the plane z = z0 ˆ 1 QT = ρdρdφ p (ρ) w (ρ, φ , z0 ) . (11) 2π To gain insight on the meaning of the quantities discussed above, in Fig. 2 it is shown how the quantities discussed above behave in an experimental measurement of a c-OV for a beam obtained by applying a TC of 5 in the SLM. Notice that the amplitude profile is very uniform in Fig. 2(a) and the measured phase profile in Fig. 2(b) is consistent to that expected for usual c-OV. The semi-classical OAM density is given by the integrand of Eq. (7) and can be obtained by multiplying the result of Eq. (10) with the measured intensity profile. The normalized ex#222475 - $15.00 USD Received 5 Sep 2014; revised 13 Nov 2014; accepted 16 Nov 2014; published 26 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030315 | OPTICS EXPRESS 30320
(e) 1
10
OAM density
2π 1
Phase
(d)
(c)
Amplitude
(a)
5
0
0
0
0
(b)
Fig. 2. Application of the concepts developed in sec. 3 to a c-OV for χOV = 5φ . Experimentally measured amplitude (a) and phase (b) beam profiles. OAM density profile (c), from which it was determined that hli = 4.9. Experimental (d) and theoretically expected (e) `-OAM profiles for this beam. For this beam it was measured QT = 5.0. 2π 1 OAM density
(c) 1
12
Phase
(b)
Amplitude
(a)
0
0
0
0
6
Fig. 3. Typical experimental s-OV profiles of amplitude and phase, and the corresponding `-OAM and OAM density. The data represent TC distributed over a line (a), a corner (b) and a triangle (c). The values of hli and QT were calculated, respectively, by applying Eqs. (7) and (11) to the experimental data.
perimental OAM density, Lz , for the c-OV shown in Fig. 2(c) is flat, up to experimental error. Another quantity of interest is the average semi-classical OAM per photon hli, which is obtained by performing the integral in Eq. (7). For the c-OV from Fig. 2, it was measured that hli = 4.9. Figs. 2(d)-(e) show the experimental and theoretically expected `-OAM profiles, respectively, for this c-OV. It can be noticed that there is a good agreement between both in the color scale. The experimental determination of the beam TC, QT , was performed using Eq. (11) with c = 0.85rmax , ε = 0.1rmax , where rmax is the external radius of the amplitude profile. As a result, it was measured that the produced c-OV has QT = 5.0. 4.
Experimental results
Without loss of generality to the concepts discussed in the previous sections, we experimentally characterized the `-OAM and the OAM density in the s-OV described in [21]. The s-OV dark core profiles are shaped as a line, a corner and a triangle due to a distribution of TC with the same geometry on the SLM phase mask. These s-OV are prototypes for more complex OV geometries, and have adjustable total TC. Some typical amplitude and phase profiles of these s-OV at z = 0 cm are shown in Fig. 3. The results for each TC distribution are discussed in the subsections below. From the experimental amplitude and phase profiles, hli was calculated by applying Eq. (7), while for QT it was used Eq. (11) with c = 0.85rmax , ε = 0.1rmax , where rmax is the external radius of the amplitude profile.
#222475 - $15.00 USD Received 5 Sep 2014; revised 13 Nov 2014; accepted 16 Nov 2014; published 26 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030315 | OPTICS EXPRESS 30321
2π 1 OAM density
(c) 1
12
Phase
(b)
Amplitude
(a)
0
0
0
0
6
Fig. 4. Experimental data for TC lines with different line lengths and a fixed total applied TC= 10 at z = 0 cm. The line length increases from (a) to (c), and the respective OV are in the regimes of high TC density, elongated OV core and small TC density.
4.1.
Linear distributions of TC
A linear distribution of TC forms a prototype for more complex s-OV core profiles [21], and therefore it is discussed first. Although this TC distribution discussed here is designed on the SLM, its worth noticing that they can appear naturally from c-OV perturbed by the optical system [35]. The OV core of a TC line of infinitesimal length has a circular shape. By increasing the TC line length and keeping the total TC fixed, the core becomes elongated until a point in which the multiple OV cores of unit charge become discernible. For a fixed TC of 10 these three regimes can be visualized in Figs. 4(a)-(c) as, high TC density, elongated OV core and small TC density, respectively. An important remark is that the measured value of QT obtained from Eq. (11) agrees with the applied value for all the geometries shown in Fig. 4 and also in all the other cases discussed below. Another observation is that the `-OAM magnitude, |w (r)|, is reduced near the center of the TC line. This can be understood by noticing that between adjacent equally-charged OV the azimuthal phase variation is smaller than if both OV were at the same point. Far from the TC distribution the phase profile must depend only on the value of QT and therefore the `-OAM, w (r), should have similar magnitudes in all regimes [21]. This can be observed in Figs. 4(a)-(c). The OAM densities follows the beam intensity profile but are reduced near the OV core. The reduced OAM density, Lz , follows the `-OAM, w (r), reduction, and as a result we observed that hli ≤ QT . Also, it was investigated how the linear distributions of TC propagate, and a typical example is shown in Fig. 5. Notice that the OV core rotates and expands under propagation, but remains line-shaped. The rotation is due to the Guoy phase shift [28, 36], and can be seen as a signature of the structural stability of the OV profiles under propagation [28]. Another important point is that both QT and hli are conserved under propagation. This is important because establishes that the approach discussed in this work does not depend on the CCD position. 4.2.
Corner- and triangle-shaped distributions of TC
After discussing the behavior of a TC line, the next step in complexity are two TC lines. The data for a corner-shaped distribution of TC is shown in Fig. 6. By fixing the geometry of the TC distribution, the OV core also can be characterized by the three regimes discussed for TC lines as illustrated in Fig. 6(a). A high TC density was observed when QT ≥ 12, since the OV core profile becomes more rounded and tends to a circular shape. At smaller TC densities, QT ≤ 8, it can be observed that multiple OV cores are distinguishable. When QT = 10, the OV core is corner-shaped. From the `-OAM profiles, it can be seen that the `-OAM is very small inside the triangle determined by the corner extremities. This fact can also be inferred from the relative uniformity of the phase profiles, and implies that the OAM density core profile is more rounded
#222475 - $15.00 USD Received 5 Sep 2014; revised 13 Nov 2014; accepted 16 Nov 2014; published 26 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030315 | OPTICS EXPRESS 30322
(c) 1
OAM density
2π 1 12
Phase
(b)
Amplitude
(a)
0
0
0
0
6
Fig. 5. Measurements corresponding to a TC line at distinct z planes. z = 0 cm, 5 cm, 10 cm respectively in (a-c). Notice that both QT and hli are conserved under propagation.
6
8
10
12
2π 1
0 0 14 Applied TC
(b)
12
0
0
Measured values
14
OAM density
Phase
Amplitude
(a) 1
12 10
6
8 6 6
8
10
12
14
Applied TC
Fig. 6. (a) Profiles of amplitude, phase, OAM density and LOAM for a corner-shaped TC distribution with a fixed geometry and varying total applied TC. (b) Relation between the measured QT and hli in terms of the applied TC at the SLM. All measurements were taken at z = 0 cm.
6
9
12
15
0 0 Applied TC
12
0
0
20
Measured values
OAM density
3
(b)
2π 1
Phase
Amplitude
(a) 1
15
6
10 5 5
10
15
Applied TC
20
Fig. 7. (a) Profiles of amplitude, phase, OAM density and `-OAM for a triangle-shaped TC distribution with a fixed geometry and varying total applied TC. (b) Relation between the measured QT and hli in terms of the applied TC at the SLM. All measurements were taken at z = 0 cm.
#222475 - $15.00 USD Received 5 Sep 2014; revised 13 Nov 2014; accepted 16 Nov 2014; published 26 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030315 | OPTICS EXPRESS 30323
than the OV core at the amplitude profile. This conclusion is important for optical tweezers applications, in which someone wants to transport a dielectric particle along a specified route. On the other hand, it can be seen in Fig. 6(b) that the measured values of QT corresponds to the applied TC values and for these beams the reduction in hli with respect to a c-OV is small. The experimental results for a triangle-shaped TC distribution are shown in Fig. 7. In Fig. 7(a) it can be seen that the OV core profiles are triangle-shaped and the intensity profile regimes are high TC density at QT ≥ 12, elongated OV core at QT = 9 and small TC density at QT ≤ 6. The `-OAM and the OAM density profiles follows the TC distribution geometry, and therefore are suitable for applications in optical tweezers. Also it can be seen that, similarly to the cornershaped TC distribution, QT corresponds to the applied value of TC and for these beams the reduction in hli with respect to a c-OV is smaller than the previous case. 5.
Summary
In this paper it was described how orbital angular momentum (OAM) and topological charge (TC) are distinct physical quantities even for linearly polarized light, that usually have degenerate values for beams with azimuthal symmetry, i.e., cylindrical optical vortices (c-OV). A general approach for determining the OAM and the TC of a beam was developed for shaped optical vortices (s-OV) in terms of the light beam amplitude and phase profiles and the concept of local OAM (`-OAM) was used as a quantity to characterize the OV morphology. This is important because contrarily to c-OV, in s-OV the beam TC is not concentrated at a single point, and therefore the OAM and TC do not have degenerate values; the OAM is not necessarily homogeneous over the beam profile. Then, it followed a discussion on the experimental setup and how to retrieve the light spatial phase profile. Some fundamental aspects to determine the `-OAM and the TC from the experimental data were discussed. The developed concepts were applied without loss of generality to the set of s-OV discussed in [21]. It can be seen that the characterization technique proposed in this work is applicable in s-OV with general geometries by retrieving data of both OAM and TC. It should be remarked that the TC measurement approach described here does not depend on the matching of the beam spatial profile with a specific geometry, as in [22–24], and is therefore more suited for non-cylindrical OV. The introduced concept of `-OAM, w (r), demonstrates its importance in obtaining the TC, QT , and clarified the behavior of the OAM density profile in corner-shaped s-OV. For the line- and triangle-shaped TC distributions it was observed that both amplitude and OAM density profiles followed the designed geometry. The s-OV characterization procedure described above can be of special interest in optical tweezers and plasmonics. Since s-OV can be used to control the transverse forces in optical tweezers [5], the designed particle trajectory can have any shape, and a general characterization procedure as the one presented here is necessary. Also, the matching of an OV to a surface plasmon may requires non-cylindrical OV [11]. Therefore, our approach is relevant to characterize the OV produced by plasmonic structures. Finally, since the herein discussed procedure is general, it can be applied also to c-OV as Bessel or Laguerre-Gauss beams. Acknowledgments We acknowledge the financial support from the Brazilian agencies Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Fundação de Amparo à Ciência e Tecnologia do Estado de Pernambuco (FACEPE). The work was performed in the framework of the National Institute of Photonics (INCT de Fotônica) project and PRONEX/CNPq/FACEPE. We also acknowledge helpful discussions with Dr. L. Pruvost. A. M. A. also thanks Dr. W. Löffler for inciting an extension of the previous work on s-OV.
#222475 - $15.00 USD Received 5 Sep 2014; revised 13 Nov 2014; accepted 16 Nov 2014; published 26 Nov 2014 (C) 2014 OSA 1 December 2014 | Vol. 22, No. 24 | DOI:10.1364/OE.22.030315 | OPTICS EXPRESS 30324
