Manipulating intensity and phase distribution of composite Laguerre-Gaussian beams G. Parisi,1,* E. Mari,1 F. Spinello,1,3 F. Romanato,1,2and F. Tamburini1,2 1 Twistoff s.r.l., via della Croce Rossa 112, I-35129 Padova, Italy Department of Physics and Astronomy, University of Padova, via Marzolo 8, I-35100 Padova, Italy 3 Department of Information Engineering, University of Padova, via Gradenigo 5B I-35131 Padova, Italy * [email protected] 2
Abstract: We propose a method to manipulate the intensity and phase distributions of a beam with non-zero orbital angular momentum (OAM). We investigate the superposition of coherent consecutive OAM modes, with concordant topological charges values, showing that it is possible to predict and control the phase and the radial and angular dimension of the resulting beam by acting on the number of superposed modes (N) and on their minimum value of the OAM ( mmin ). A general analysis from the Wigner function formalism is adopted for the geometric characterization of the beam. ©2014 Optical Society of America OCIS codes: (140.0140) Lasers and laser optics; (080.2468) First-order optics.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
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). D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92(14), 143905 (2004). T. Ongarello, G. Parisi, D. Garoli, E. Mari, P. Zilio, and F. Romanato, “Focusing dynamics on circular distributed tapered metallic waveguides by means of plasmonic vortex lenses,” Opt. Lett. 37(21), 4516–4518 (2012). A. Vaziri, G. Weihs, and A. Zeilinger, “Superpositions of the orbital angular momentum for applications in quantum experiments,” J. Opt. B Quantum Semiclassical Opt. 4(2), S47–S51 (2002). E. Serabyn, D. Mawet, and R. Burruss, “An image of an exoplanet separated by two diffraction beamwidths from a star,” Nature 464(7291), 1018–1020 (2010). E. Mari, F. Tamburini, G. A. Swartzlander, Jr., A. Bianchini, C. Barbieri, F. Romanato, and B. Thidé, “SubRayleigh optical vortex coronagraphy,” Opt. Express 20(3), 2445–2451 (2012). S. M. Baumann, D. M. Kalb, L. H. MacMillan, and E. J. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express 17(12), 9818–9827 (2009). A. M. C. Aguirre and T. Alieva, “Orbital angular momentum Density of beam given as superposition of Hermite-Laguerre-Guass function,” in PIERS Proceedings, (Marrakesh, Marocco, 2011), pp. 250–254. T. Ando, N. Matsumoto, Y. Ohtake, Y. Takiguchi, and T. Inoue, “Structure of optic singularities in coaxial superposition of Laguerre-Gaussian modes,” J. Opt. Soc. Am. A 27(12), 2602–2612 (2010). I. D. Maleev and G. A. Swartzlander, Jr., “Composite optical vortices,” J. Opt. Soc. Am. B 20(6), 1169–1176 (2003). I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, 2007). W. H. Carter, “Spot size and divergence for Hermite Gaussian beams of any order,” Appl. Opt. 19(7), 1027– 1029 (1980). R. L. Phillips and L. C. Andrews, “Spot size and divergence for Laguerre Gaussian beams of any order,” Appl. Opt. 22(5), 643–644 (1983). T. Alieva, C. Alejandro, and M. J. Bastiaans, “Mathematical formalism in wave optics,” in Mathematical Optics, V. Lakshminarayanan, M. L. Calvo, T. Alieva ed. (CRC Press, 2013). D. Dragoman, “The Wigner distribution function in optic and optoelectronics,” Prog. Optics 37, 1–56 (1997). R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 (1993). K. Sundar, N. Mukunda, and R. Simon, “Coherent-mode decomposition of general anisotropic Gaussian Schellmodel beams,” J. Opt. Soc. Am. A 12(3), 560–569 (1995). R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian-Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36(8), 3868–3880 (1987).
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19. M. A. Alonso, “Wigner functions in optics: describing beams as ray bundles and pulses as particle ensembles,” Adv. Opt. Photon. 3(4), 272–365 (2011). 20. Y. A. Anan’ev and A. Y. Bekshaev, “Theory of intensity moments for arbitrary light beams,” Opt. Spectrosc. 76, 558–568 (1994). 21. A. Y. Bekshaev, “Intensity moments of a laser beam formed by superposition of Hermite-Gaussian modes,” Fotoelektronika 8, 2–13 (1999). 22. R. Simon and G. S. Agarwal, “Wigner representation of Laguerre--Gaussian beams,” Opt. Lett. 25(18), 1313– 1315 (2000). 23. A. Camara and T. Alieva, in PIERS Proceedings, (Cambrige, MA, 2010), pp. 505–508. 24. J. Alda, “ Laser and Gaussian Beam Propagation and Transformation,” in Encyclopedia of Optical Engineering, R. G. Driggers ed. (CRC Press, 2003). 25. D. Judge, “On the uncertainty relation for Lz φ , ” Phys. Lett. 5(3), 189 (1963).
1. Introduction Beams carrying orbital angular momentum (OAM) are characterized by the presence of optical vortices that are topological defects characterized by phase singularities in the wavefront propagation [1]. In correspondence of these phase singularities the field is null and so the intensity distribution presents the typical doughnut-shaped pattern. In paraxial approximation, these beams can be mathematically described by a superposition of LaguerreGaussian (LG) beams. A generic LG mode is characterized by two parameters, m and p, the first is the azimuthal index that gives the number of twists of the helical wave-front within a wavelength, and the second is the number of radial nodes present in the LG mode, as reported in Eq. (1): | m|
2
ikr 1 r 2 |m| 2r 2 − r 2 / w( z )2 − 2 R ( z ) i ( 2 p + m +1)ζ ( z ) − imφ u p , m (r , φ , z ) = A e e e e , (1) L p 2 w( z ) w( z ) w( z )
the variables r, φ , and z are the cylindrical coordinates, A is the normalization factor, w( z ) = w0 1 + ( z / zr ) 2 is the beam radius, w0 is the beam-waist, L p
m
the correspondent
generalized Laguerre polynomial, k = 2π / λ the wave number, zr = π w / λ is the Rayleigh 2 0
distance. The quantities R ( z ) = z[1 + ( zr / z ) 2 ] and r ≠ 0 are the radius of curvature of the wave-front and the Gouy phase respectively. The phase of the mode varies linearly with the azimuthal angle as expressed by the term e − imφ . OAM beams and their phase singularities have found practical applications in many fields such as radar [2], nanotechnology [3], quantum experiments [4] and astronomy [5,6]. The possibility of manipulation the intensity distribution and the embedded phase in OAM beams can offer useful applications when one needs to control the beam shaping and its phase content. In this paper we present a method to manipulate the intensity distribution and the phase of a beam with non-zero orbital angular momentum (OAM). In particular, we show that it is possible to concentrate the intensity distribution in a restricted angular domain and to embed a certain phase variation within the concentrated beam through a smart combination of LG modes. Some examples of analysis on the superposition of LG modes have been analytically and experimentally investigated in deep [7–10]. Here we present the study, both in amplitude and phase, of the scalar field generated by a superposition of a given number N of coherent OAM modes with integer, concordant and consecutive value of azimuthal index. 2. Normalization factor
In this section, we focus our attention to the role played by the normalization factor of a superposed beam. A generic superposition of coherent and coaxial OAM modes, each of them with integer and non-zero value of topological charge [1], can be written as
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utot (r , φ , z ) = m u p , m (r , φ , z )eiδ m ,
(2)
where um (r , φ ) is the m-th beam and δ m is the relative phase with respect to the reference beam. Here we deal with LG modes having the same beam radius and the same relative phase (i.e., δ m = 0 ); we set p = 0 and all the value of m are concordant and consecutive. With these prescriptions, we can write the coherent superposition of N concordant and consecutive modes in Eq. (1) as follows: ikr 2
1 − r 2 / w ( z )2 − 2 R ( z ) utot (r , φ , z ) = um = e e w( z ) mmin mmax
mmax
Amα m e (
i 2 m +1)ζ ( z ) − imφ
e
,
(3)
m = mmin
where, the maximum term in the sum is mmax = mmin + N − 1 , the beam radius w( z ) is fixed and equal for all the superposed modes, the parameter α ( z ) = 2r / w( z ) is related to the spatial beam distribution and mmin gives the minimum OAM value of the set of the superposed OAM modes. By way of example, we analyze two distinct cases of coherent superposition of OAM modes at the beam waist plane (z = 0) that do not and do keep in consideration the normalization of each beam to the unitary power, respectively • Am is a pure constant ( = 1 for convenience). As a consequence, it is possible to expand the series in Eq. (3) into a new formulation with help of [11], utot (r , φ , 0) =
1 − r 2 / w02 α (0) 2 N − 2α (0) N cos( Nφ ) + 1 α (0) mmin e −imminφ e . α (0) 2 − 2α (0) cos φ + 1 w0
(4)
• Am = 2 / π m ! instead, depends on the OAM value, m, of each single mode and it is given by the direct superposition of the terms in Eq. (3), giving 1 − r 2 / w02 ut o t (r , φ , 0) = e w0
Here, the presence of a factor, 1/
mmax
m = mmin ≠ 0
| m|
2 r 2 − imφ e . π m ! w0
(5)
m ! , in the term Am normalizes each single beam to the
unitary power, giving a different intensity and phase distribution with respect to those having constant amplitude factor Am , as reported in Fig. 1.
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Fig. 1. Plots of the superposition of concordant, consecutive and coherent OAM modes. In the left column are reported the plots with
Am = 2 / (π m !)
Normalized intensity of scalar field for N = 16, d).
mmin = 1
. Right column:
a), b) and for N = 7,
Am = 1 .
mmin = 1 c),
As shown in Fig. 1, the resulting intensity distributions are no longer symmetric with respect to the propagation axis and, those weighted by the factorial term, 1/
m ! , result
mainly concentrated in a very restricted angular domain. On the other hand, with a constant Am term, it is not possible to concentrate the power of the beam within a restricted angular region. For the purpose of this paper we will investigate only the mathematical properties of those OAM beams superposition for which the intensity is concentrated in a restricted angular domain, namely when Am = 2 / (π m !) .
3. Divergence of superposed beams with Wigner formalism
In this section, we adopt a formulation that allows to predict the angular, ϑ far , and radial position, s(z), at fixed z-axis plane, of the resulting beam with respect to the optical axis (see Fig. 2).
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Fig. 2. Schematic representation of angular, ϑ far and radial, s(z), position of the resulting beam.
We utilize the Wigner function formalism to express the radial and angular divergence for a coherent superposition of integer, concordant and consecutive OAM beams. Let us recall that the circular spot size of a Gaussian beam, λ z 2 w( z ) = w0 1 + 2 π w0
1/ 2
(6)
has a simple physical meaning only for the fundamental mode when the light beam has a Gaussian intensity profile [12,13] and can be thus related to the divergence of the beam itself. It is evident that Eq. (6) is independent of order m of the LG modes, so for Gaussian beams with higher order (m ≥ 1) , this expression gives an inaccurate estimate of the beam geometric properties in the propagation. To overcome this approximation and obtain a more detailed description of the beam superposition, here we adopted an analytic approach based on the Wigner distribution of intensity and moments of the scalar field. The Wigner distribution contains information regarding both the spatial irradiance distribution and its angular spectrum. The Wigner distribution is a tool widely used in optics and applied for the analysis of coherent and partially coherent beams. A more detailed description of the Wigner function can be found in [14–19]. The first application of the theory of intensity moments to arbitrary light beams, in particular applied to an arbitrary superposition of Hermite-Gaussian (HG) modes, was discussed in [20,21], whereas the Wigner representation of LG beams can be found in [22]. This latter representation is based on the remarkable unitary relationship between HG and LG modes [17]. Finally, the intensity moment matrix for an arbitrary superposition of hybrid Hermite-Laguerre-Gaussian (HLG) modes can be found in [23]. Following the approach in [22,23], we express the Wigner distribution for the scalar field expanded in terms of a series of rotational-symmetric intensity higher LG modes of the type as in Eq. (3) at the beam waist plane, with consecutive and concordant angular index as TOT WLG (r , p;0) =
mmax
mmax
W(
m = mmin n ≥ m
m,n) LG
(r , p;0),
(7)
where the summing term is given by
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m−n m−n m π w0 2 2 ( −1) 4 m ! x − iy m,n ( ) WL(G ) ( r , p; 0) = sign m n p ip + − − ( ) y x n ! w0 λ λ2 x +y wπ + 2 2 2 λ w0 2
m−n
Lm
2
2 0
2
(p
2 x
2
)
2π x +y wπ ( xp y − ypx ) exp −2 −2 2 2 w0 λ λ 2
2
+ py -
2 0
(8) 2
(p
2 x
)
+ py , 2
( m,n) being the quantity WLG the Wigner function (WF) of the mixing term between two arbitrary coherent LG modes with angular indexes m and n, respectively. In particular, the case n = m corresponds to that of a single mode and, r ≡ [ x, y ] and p ≡ px , p y are the transversal
variable in coordinate and momentum space, respectively. Besides Eq. (8) is written only for z = 0, the free propagation (z>0) can be accounted for by means of a simple change of arguments i.e., W (r , p, z ) = W (r − λ zp, p, 0) . This is a general property of the Wigner formalism [14]. Through Eq. (7), we obtain the principal results of the current work. It is worth to be noted that in virtue of the above relation, quantities written after propagation could be calculated also neglecting the Gouy and the radius of curvature term since they have a null contribution at z = 0. Getting into the details, we first calculate the total intensity according to the normalization factor in Eq. (5) that is invariant under free space propagation, I =
mmax
4 m = mmin
( m,m ) WLG (r , p;0 )drdp = N .
(9)
Finally, we write the analytical expression for the radial and angular position of the resulting beam spot by integrating on momentum and space coordinates respectively and then by averaging the resulting WF over the polar coordinates. We can write an analytical expression for the spot position of the emitted beam at fixed z-axis s( z ) =
2 r2 I
1
mmax 1∞ π ∞ ∞ 2 N + 2mmin + 1 ( m,m) (r , p; z )dr 2 dφr dp = w( z ) 2 = r 2 WLG , (10) 2 I 0 −π −∞ −∞ m = mmin
and for the far field angle of divergence, that is expected to be invariant to axial position z, as the Eq. (9)
ϑ far =
2 pr2 I
1
mmax 1∞ π ∞ ∞ 2 λ 2 N + 2mmin + 1 (m,m) (r , p;0)dp 2 dφ p dr = = p 2 WLG . (11) π 2 w02 2 I 0 −π −∞ −∞ m = mmin
( m,n) Equations (9)-(11) have non-zero values for WLG with n = m. In fact the mixing terms integrate to zero because of the orthogonality condition proper of the OAM modes [1]. Moreover ϑ far = lim z →∞ s ( z ) / z [16]. The factor 2, which is present in the first line of both
Eqs. (10) and (11), has been first introduced in [13] for the characterization of the illuminated area of a single LG mode. This area has been defined as a circle having as radial dimension the standard deviation of 2r / I on the plane of constant z. A similar definition can be found in [24] minding that, for n = m, one obtains that r 2 = 2 x 2 = 2 y 2 holds. We calculate the average value of the second order quantities with respect to the origin of coordinates by adopting the assumption used in [13], i.e., by setting the averaged quantities x , y , r to zero, considering the standard deviation with respect to the origin of the coordinates instead of the physical “center of mass”. In this way, the illuminated area of
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the total beam in any plane of constant z is contained within a circle having circumference of 2π s . This area linearly increases with the distance z when it is far from the Rayleigh distance, which is the typical behavior of Gaussian beams [21]. For N = 1 and mmi n = 0 that describes the fundamental Gaussian mode, Eq. (10) reduces to the circular spot size described by Eq. (6). In Fig. 3, we show some examples of the evolution of s(z) at different distances z for different number of superimposed OAM modes, having fixed the beam waist and mmin = 1 .
Fig. 3. Examples of s(z) at different distances for different number of superimposed OAM modes. All the OAM modes have the same beam waist and their m-values are concordant and consecutive.
We notice that the bigger is N the larger are s(z) and ϑ far . We also focus our attention to the dependence of the terms s(z) and ϑ far on the smallest value of the superposed OAM modes, mmin .The larger is the value mmin , the farther becomes the illuminated spot from the optical axis.
4. Angular aperture
In this section we find an analytical expression for the angular (azimuthal) aperture of the superposed beams that shows the dependence on the number of the superposed modes N and on the minimum value of OAM, mmin . To quantify how much the spot is angularly concentrated we have found a quantity, Θ , analogous to s(z), that describe the azimuthal angular coordinate φ . The total azimuthal angle of the intensity pattern Θ is given by 1
Θ=2
π2
=2
3
φ
+
2
1 N
mmax mmax 1∞ π ∞ ∞ 2 ( m,n) = 2 φ 2 WLG (r , p;0 )rdrdφ dp I 0 −π −∞ −∞ n = mmin m = mmin
mmax
mmax
m = mmin n > m
n+m 1 Γ + 2 2 , 2 ( n − m ) m !n !
4 ( −1)
n−m
(12)
where Γ is the Gamma function. Differently from Eqs. (9)-(11) the mixing terms in Eq. (12) do not integrate to zero because of the φ 2 term as part of the integrand function. The intensity 2
distribution, ut ot , normalized as Eq. (5) (see also Fig. 1) on a plane perpendicular to the
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propagation direction z presents an azimuthal symmetry with respect to the axis defined by φ = 0 at z = 0 . This stems from the fact that the product of the mixing terms of the intensity are ∝ f (r ) cos[φ (n − m)] with −π < φ < π , and so the mean value of φ is null, namely, φ = 0 . The effect of the Gouy term of Eq. (1) is a 0 ≤ ζ ( z ) < π / 2 rotation of the total intensity pattern, without affecting the field distribution, and so the intensity distribution at fixed z keeps the azimuthal symmetry but with respect to the set of axis defined by φ ' = φ + ζ ( z ) [7]. If one wants to calculate the integral of Eq. (12) for z>0, it is mathematically convenient to define the origin of the angular coordinate as the physical symmetry axis φ ' = φ + ζ ( z ) due to the Gouy effect of rotation. In this way the product of the mixing terms of the total intensity results ∝ f (r ) cos[(n − m)φ '] with −π < φ ' < π and the mean value φ ' is null. In this way one can remove the dependence on φ ' , with the same result of Eq. (12), by calculating
2
φ '2 − φ ' , which is a clear measure of the standard
deviation of the azimuthal spread angle. In other words a rigid rotation of the intensity pattern, as the only effect of Gouy phase when (z>0), does not modify the angular aperture Θ , as expected. From these argumentations we calculated Eq. (12) at z = 0, for convenience. Analogously to the definition of the variable s(z), the parameter Θ describes the angular standard deviation within which the illuminated area is angularly confined. In fact, the angular aperture is preserved during the beam propagation. It is also inversely depends on the number of superposed OAM modes N. The larger is N, the smaller is the angular aperture of the illuminated spot results. In Fig. 4 we show the relationship between N and the angular aperture of the spot.
Fig. 4. Graphic representation of the relationship between the angular aperture and the number of superimposed OAM modes at several values of
mmin . Inset: exemplified plot for Θ and
s(z).
We notice that the parameter Θ decreases as N increases and it is quite unaffected by the lowest order beam mmin . This implies that the second term in Eq. (12) has a remarkable dependence on N while the factor (−1) n − m is the only quantity responsible for the interference pattern between the mixing terms creating the angular confinement. In other words, it is possible to control the angular aperture of a superposition of coherent, concordant and consecutive LG beams by acting only (in a first approximation) on the number of modes N. #209600 - $15.00 USD (C) 2014 OSA
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It is also worth to be notice that the first term in Eq. (12) ( π 2 / 3 ) corresponding to the angular deviation of free propagating single modes have a quantum analogy with the statistical uncertainty in angular position for pure states as pointed out in [25]. 5. Concentration of the phase variation
In this section we show how to control the phase embedded within the angular region described by Θ , analyzing the effect on the phase of the superposition of concordant, consecutive and coherent OAM modes. In Fig. 5 we report examples of phase map, plotted on a plane with z = 0, for this kind of superposition.
Fig. 5. Phase (a) and intensity (b) plot for N = 5 and mmin for N = 5 and mmin
= 1 . Phase (c) and Intensity (d) plot
= 3.
The value of the total topological charge observed in the plots is equal to the maximum number of OAM modes, mmax , whereas the number of singularities is equal to N. The location of the phase vortices in this field is obtained by finding the set of points (rs , φs ) where the following condition is satisfied [7,9]:
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ℑ( Etot (r , φ )) a tan = 0, ℜ( Etot (r , φ ))
(13)
where Etot (r , φ ) is given by Eq. (5) and the term s gives the s-th singularity. One of the vortex is always located at the center (r = 0) and this singularity is of the order mmin as can be deduced from Eq. (3); for r ≠ 0 the other singularities, hereafter called secondary singularities [9], are present in a number of mmax − mmin . As a consequence, in the central zone, the phase is distributed as the phase of a pure OAM of value mmin whereas in the zone far from the center (starting from a certain value of r ≡ r that corresponds to maximum value of the set rs i.e., the radial position of the secondary singularity, which depends on the particular choice of the parameters N and mmin ) the whole phase is distributed as the phase of a pure OAM of value mmax . Given the definition of r , we have numerically verified that the maximum of the illuminated area of the total beam lies on r > r , so the corresponding phase falls in the zone where mmax is dominant. We now define the angular parameter β as the angle within a variation of 360° of the phase profile is included. We can also define the effective phase variation within the restricted angular domain of angular aperture Θ as Δφeff = Θ / β . For r >> r the angular parameter β is given by 360° / mmax . So, it is possible, with good accuracy, to predict the total phase variation within a fixed angle when the intensity distribution is far from the optical axis. The above condition r >> r is fulfilled for large values of mmin (see Eqs. (10) or (11)). In particular, we have numerically verified that it is fully satisfied just when mmin > 2 . On the contrary in the case of mmin = 1, 2 the expression 360° / mmax is no longer a good approximation for β . In fact, for these values of mmin , the illuminated spot is concentrated in a zone where r r . In this zone the phase variation is not equally distributed and so the angle embedding the 360° phase variation, β , is not constant. We analyze some examples to show the effective phase variation embedded in the region with angular aperture Θ . In particular, we fixed N to 4 and 8 that in good approximation, as stated above, is equivalent to fix the angular aperture of the spot in Θ and we varied mmin . Table 1. Resuming table
N
4 8
mmin
mmax
Θ (deg)
2 9 15 2 9 15
5 12 18 9 16 22
99° 96° 96° 72° 69° 68°
β = 360° / mmax (deg) 72° 30° 20° 40° 22.5° 16.4°
βm (deg) 91° 32° 20° 52° 25° 17°
Δφeff = Θ / β m (deg) 1.3 3.2 4 1.8 2.9 4.2
We report in Table 1 the cases for mmin = 2, 9 and 15. Numerical data confirm that the angular spot of light in Θ is quite the same for different values of mmin having fixed N. On the contrary, the most striking effect obtained by varying mmin , and so mmax , is to concentrate the phase variation in the angular region described by the parameter Θ as can be seen by considering the parameter β . In addition to β we report another parameter, β m , which represents the angle embedding the 360° phase variation at r = s ( z ) obtained from numerical
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calculations. The quantity Δφeff = Θ / β m gives the effective number of the phase variations of 360° inside the angular region delimited by Θ . From the comparison between β and β m it is confirmed that 360° / mmax is not a good approximation for low mmin ( = 2 in the reported examples) whereas for larger mmin it is a meaningful quantity which can be used for the prediction of the number of phase variation within a fixed angle Θ of interest. In Fig. 6 we show the phase embedded in Θ , measured at r = s ( z ) for the cases resumed in Table 1. Each line represents, for different mmin , how many times the phase changes between 0 and 360° within the angular region Θ . The value of Θ sets the angular limits reported in abscissa.
Fig. 6. Simulation of the phase variation embedded in the angular region Θ measured at distance s from the center for three different values of
mmin .(a) N = 4 and (b) N = 8.
6. Conclusions
In this work we have showed that it is possible to control the dimension of the intensity distribution of a beam with non-zero OAM and the phase embedded in the spot area through the superposition of coherent, consecutive and concordant OAM modes. This smart combination of LG modes, weighted by a factorial normalization factor, allows to concentrate the beam in a restricted angular region. We have investigated the geometric characteristics of the total superposed beam, adopting the Wigner function formalism. We have derived the analytical expressions for the radial and azimuthal extension on the transversal plane of the intensity field and its far field divergence. We have also found an analytical expression for the angular aperture of the superposed beams. All these parameters depend on the number N of the superposed OAM modes and on the lowest value of topological charge in the set of modes. Moreover, once fixed the number of states to be superposed, it is possible to control the number of 360° phase variation embedded within the field angular aperture by acting on the lowest topological charge of the set. We also verified that when the lowest OAM value in the set of modes is greater than two it is possible to calculate the number of 360° phase
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variation within the restricted angular region. This can open new possibilities in controlling the beam shaping and its phase content for EM application. Acknowledgments
We thank C. G. Someda, G. Colombo and B. Thidè for the interesting discussion during this work. G. Parisi is grateful to A. Ya. Bekshaev for the suggested and interesting readings from which the paper has taken benefit. We also acknowledge financial support and collaboration of SIAE Microelectronics.
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Abstract: We propose a method to manipulate the intensity and phase distributions of a beam with non-zero orbital angular momentum (OAM). We investigate the superposition of coherent consecutive OAM modes, with concordant topological charges values, showing that it is possible to predict and control the phase and the radial and angular dimension of the resulting beam by acting on the number of superposed modes (N) and on their minimum value of the OAM ( mmin ). A general analysis from the Wigner function formalism is adopted for the geometric characterization of the beam. ©2014 Optical Society of America OCIS codes: (140.0140) Lasers and laser optics; (080.2468) First-order optics.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
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). D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92(14), 143905 (2004). T. Ongarello, G. Parisi, D. Garoli, E. Mari, P. Zilio, and F. Romanato, “Focusing dynamics on circular distributed tapered metallic waveguides by means of plasmonic vortex lenses,” Opt. Lett. 37(21), 4516–4518 (2012). A. Vaziri, G. Weihs, and A. Zeilinger, “Superpositions of the orbital angular momentum for applications in quantum experiments,” J. Opt. B Quantum Semiclassical Opt. 4(2), S47–S51 (2002). E. Serabyn, D. Mawet, and R. Burruss, “An image of an exoplanet separated by two diffraction beamwidths from a star,” Nature 464(7291), 1018–1020 (2010). E. Mari, F. Tamburini, G. A. Swartzlander, Jr., A. Bianchini, C. Barbieri, F. Romanato, and B. Thidé, “SubRayleigh optical vortex coronagraphy,” Opt. Express 20(3), 2445–2451 (2012). S. M. Baumann, D. M. Kalb, L. H. MacMillan, and E. J. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express 17(12), 9818–9827 (2009). A. M. C. Aguirre and T. Alieva, “Orbital angular momentum Density of beam given as superposition of Hermite-Laguerre-Guass function,” in PIERS Proceedings, (Marrakesh, Marocco, 2011), pp. 250–254. T. Ando, N. Matsumoto, Y. Ohtake, Y. Takiguchi, and T. Inoue, “Structure of optic singularities in coaxial superposition of Laguerre-Gaussian modes,” J. Opt. Soc. Am. A 27(12), 2602–2612 (2010). I. D. Maleev and G. A. Swartzlander, Jr., “Composite optical vortices,” J. Opt. Soc. Am. B 20(6), 1169–1176 (2003). I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, 2007). W. H. Carter, “Spot size and divergence for Hermite Gaussian beams of any order,” Appl. Opt. 19(7), 1027– 1029 (1980). R. L. Phillips and L. C. Andrews, “Spot size and divergence for Laguerre Gaussian beams of any order,” Appl. Opt. 22(5), 643–644 (1983). T. Alieva, C. Alejandro, and M. J. Bastiaans, “Mathematical formalism in wave optics,” in Mathematical Optics, V. Lakshminarayanan, M. L. Calvo, T. Alieva ed. (CRC Press, 2013). D. Dragoman, “The Wigner distribution function in optic and optoelectronics,” Prog. Optics 37, 1–56 (1997). R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 (1993). K. Sundar, N. Mukunda, and R. Simon, “Coherent-mode decomposition of general anisotropic Gaussian Schellmodel beams,” J. Opt. Soc. Am. A 12(3), 560–569 (1995). R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian-Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36(8), 3868–3880 (1987).
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19. M. A. Alonso, “Wigner functions in optics: describing beams as ray bundles and pulses as particle ensembles,” Adv. Opt. Photon. 3(4), 272–365 (2011). 20. Y. A. Anan’ev and A. Y. Bekshaev, “Theory of intensity moments for arbitrary light beams,” Opt. Spectrosc. 76, 558–568 (1994). 21. A. Y. Bekshaev, “Intensity moments of a laser beam formed by superposition of Hermite-Gaussian modes,” Fotoelektronika 8, 2–13 (1999). 22. R. Simon and G. S. Agarwal, “Wigner representation of Laguerre--Gaussian beams,” Opt. Lett. 25(18), 1313– 1315 (2000). 23. A. Camara and T. Alieva, in PIERS Proceedings, (Cambrige, MA, 2010), pp. 505–508. 24. J. Alda, “ Laser and Gaussian Beam Propagation and Transformation,” in Encyclopedia of Optical Engineering, R. G. Driggers ed. (CRC Press, 2003). 25. D. Judge, “On the uncertainty relation for Lz φ , ” Phys. Lett. 5(3), 189 (1963).
1. Introduction Beams carrying orbital angular momentum (OAM) are characterized by the presence of optical vortices that are topological defects characterized by phase singularities in the wavefront propagation [1]. In correspondence of these phase singularities the field is null and so the intensity distribution presents the typical doughnut-shaped pattern. In paraxial approximation, these beams can be mathematically described by a superposition of LaguerreGaussian (LG) beams. A generic LG mode is characterized by two parameters, m and p, the first is the azimuthal index that gives the number of twists of the helical wave-front within a wavelength, and the second is the number of radial nodes present in the LG mode, as reported in Eq. (1): | m|
2
ikr 1 r 2 |m| 2r 2 − r 2 / w( z )2 − 2 R ( z ) i ( 2 p + m +1)ζ ( z ) − imφ u p , m (r , φ , z ) = A e e e e , (1) L p 2 w( z ) w( z ) w( z )
the variables r, φ , and z are the cylindrical coordinates, A is the normalization factor, w( z ) = w0 1 + ( z / zr ) 2 is the beam radius, w0 is the beam-waist, L p
m
the correspondent
generalized Laguerre polynomial, k = 2π / λ the wave number, zr = π w / λ is the Rayleigh 2 0
distance. The quantities R ( z ) = z[1 + ( zr / z ) 2 ] and r ≠ 0 are the radius of curvature of the wave-front and the Gouy phase respectively. The phase of the mode varies linearly with the azimuthal angle as expressed by the term e − imφ . OAM beams and their phase singularities have found practical applications in many fields such as radar [2], nanotechnology [3], quantum experiments [4] and astronomy [5,6]. The possibility of manipulation the intensity distribution and the embedded phase in OAM beams can offer useful applications when one needs to control the beam shaping and its phase content. In this paper we present a method to manipulate the intensity distribution and the phase of a beam with non-zero orbital angular momentum (OAM). In particular, we show that it is possible to concentrate the intensity distribution in a restricted angular domain and to embed a certain phase variation within the concentrated beam through a smart combination of LG modes. Some examples of analysis on the superposition of LG modes have been analytically and experimentally investigated in deep [7–10]. Here we present the study, both in amplitude and phase, of the scalar field generated by a superposition of a given number N of coherent OAM modes with integer, concordant and consecutive value of azimuthal index. 2. Normalization factor
In this section, we focus our attention to the role played by the normalization factor of a superposed beam. A generic superposition of coherent and coaxial OAM modes, each of them with integer and non-zero value of topological charge [1], can be written as
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utot (r , φ , z ) = m u p , m (r , φ , z )eiδ m ,
(2)
where um (r , φ ) is the m-th beam and δ m is the relative phase with respect to the reference beam. Here we deal with LG modes having the same beam radius and the same relative phase (i.e., δ m = 0 ); we set p = 0 and all the value of m are concordant and consecutive. With these prescriptions, we can write the coherent superposition of N concordant and consecutive modes in Eq. (1) as follows: ikr 2
1 − r 2 / w ( z )2 − 2 R ( z ) utot (r , φ , z ) = um = e e w( z ) mmin mmax
mmax
Amα m e (
i 2 m +1)ζ ( z ) − imφ
e
,
(3)
m = mmin
where, the maximum term in the sum is mmax = mmin + N − 1 , the beam radius w( z ) is fixed and equal for all the superposed modes, the parameter α ( z ) = 2r / w( z ) is related to the spatial beam distribution and mmin gives the minimum OAM value of the set of the superposed OAM modes. By way of example, we analyze two distinct cases of coherent superposition of OAM modes at the beam waist plane (z = 0) that do not and do keep in consideration the normalization of each beam to the unitary power, respectively • Am is a pure constant ( = 1 for convenience). As a consequence, it is possible to expand the series in Eq. (3) into a new formulation with help of [11], utot (r , φ , 0) =
1 − r 2 / w02 α (0) 2 N − 2α (0) N cos( Nφ ) + 1 α (0) mmin e −imminφ e . α (0) 2 − 2α (0) cos φ + 1 w0
(4)
• Am = 2 / π m ! instead, depends on the OAM value, m, of each single mode and it is given by the direct superposition of the terms in Eq. (3), giving 1 − r 2 / w02 ut o t (r , φ , 0) = e w0
Here, the presence of a factor, 1/
mmax
m = mmin ≠ 0
| m|
2 r 2 − imφ e . π m ! w0
(5)
m ! , in the term Am normalizes each single beam to the
unitary power, giving a different intensity and phase distribution with respect to those having constant amplitude factor Am , as reported in Fig. 1.
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Received 4 Apr 2014; revised 18 Jun 2014; accepted 21 Jun 2014; published 7 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.017135 | OPTICS EXPRESS 17137
Fig. 1. Plots of the superposition of concordant, consecutive and coherent OAM modes. In the left column are reported the plots with
Am = 2 / (π m !)
Normalized intensity of scalar field for N = 16, d).
mmin = 1
. Right column:
a), b) and for N = 7,
Am = 1 .
mmin = 1 c),
As shown in Fig. 1, the resulting intensity distributions are no longer symmetric with respect to the propagation axis and, those weighted by the factorial term, 1/
m ! , result
mainly concentrated in a very restricted angular domain. On the other hand, with a constant Am term, it is not possible to concentrate the power of the beam within a restricted angular region. For the purpose of this paper we will investigate only the mathematical properties of those OAM beams superposition for which the intensity is concentrated in a restricted angular domain, namely when Am = 2 / (π m !) .
3. Divergence of superposed beams with Wigner formalism
In this section, we adopt a formulation that allows to predict the angular, ϑ far , and radial position, s(z), at fixed z-axis plane, of the resulting beam with respect to the optical axis (see Fig. 2).
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Fig. 2. Schematic representation of angular, ϑ far and radial, s(z), position of the resulting beam.
We utilize the Wigner function formalism to express the radial and angular divergence for a coherent superposition of integer, concordant and consecutive OAM beams. Let us recall that the circular spot size of a Gaussian beam, λ z 2 w( z ) = w0 1 + 2 π w0
1/ 2
(6)
has a simple physical meaning only for the fundamental mode when the light beam has a Gaussian intensity profile [12,13] and can be thus related to the divergence of the beam itself. It is evident that Eq. (6) is independent of order m of the LG modes, so for Gaussian beams with higher order (m ≥ 1) , this expression gives an inaccurate estimate of the beam geometric properties in the propagation. To overcome this approximation and obtain a more detailed description of the beam superposition, here we adopted an analytic approach based on the Wigner distribution of intensity and moments of the scalar field. The Wigner distribution contains information regarding both the spatial irradiance distribution and its angular spectrum. The Wigner distribution is a tool widely used in optics and applied for the analysis of coherent and partially coherent beams. A more detailed description of the Wigner function can be found in [14–19]. The first application of the theory of intensity moments to arbitrary light beams, in particular applied to an arbitrary superposition of Hermite-Gaussian (HG) modes, was discussed in [20,21], whereas the Wigner representation of LG beams can be found in [22]. This latter representation is based on the remarkable unitary relationship between HG and LG modes [17]. Finally, the intensity moment matrix for an arbitrary superposition of hybrid Hermite-Laguerre-Gaussian (HLG) modes can be found in [23]. Following the approach in [22,23], we express the Wigner distribution for the scalar field expanded in terms of a series of rotational-symmetric intensity higher LG modes of the type as in Eq. (3) at the beam waist plane, with consecutive and concordant angular index as TOT WLG (r , p;0) =
mmax
mmax
W(
m = mmin n ≥ m
m,n) LG
(r , p;0),
(7)
where the summing term is given by
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m−n m−n m π w0 2 2 ( −1) 4 m ! x − iy m,n ( ) WL(G ) ( r , p; 0) = sign m n p ip + − − ( ) y x n ! w0 λ λ2 x +y wπ + 2 2 2 λ w0 2
m−n
Lm
2
2 0
2
(p
2 x
2
)
2π x +y wπ ( xp y − ypx ) exp −2 −2 2 2 w0 λ λ 2
2
+ py -
2 0
(8) 2
(p
2 x
)
+ py , 2
( m,n) being the quantity WLG the Wigner function (WF) of the mixing term between two arbitrary coherent LG modes with angular indexes m and n, respectively. In particular, the case n = m corresponds to that of a single mode and, r ≡ [ x, y ] and p ≡ px , p y are the transversal
variable in coordinate and momentum space, respectively. Besides Eq. (8) is written only for z = 0, the free propagation (z>0) can be accounted for by means of a simple change of arguments i.e., W (r , p, z ) = W (r − λ zp, p, 0) . This is a general property of the Wigner formalism [14]. Through Eq. (7), we obtain the principal results of the current work. It is worth to be noted that in virtue of the above relation, quantities written after propagation could be calculated also neglecting the Gouy and the radius of curvature term since they have a null contribution at z = 0. Getting into the details, we first calculate the total intensity according to the normalization factor in Eq. (5) that is invariant under free space propagation, I =
mmax
4 m = mmin
( m,m ) WLG (r , p;0 )drdp = N .
(9)
Finally, we write the analytical expression for the radial and angular position of the resulting beam spot by integrating on momentum and space coordinates respectively and then by averaging the resulting WF over the polar coordinates. We can write an analytical expression for the spot position of the emitted beam at fixed z-axis s( z ) =
2 r2 I
1
mmax 1∞ π ∞ ∞ 2 N + 2mmin + 1 ( m,m) (r , p; z )dr 2 dφr dp = w( z ) 2 = r 2 WLG , (10) 2 I 0 −π −∞ −∞ m = mmin
and for the far field angle of divergence, that is expected to be invariant to axial position z, as the Eq. (9)
ϑ far =
2 pr2 I
1
mmax 1∞ π ∞ ∞ 2 λ 2 N + 2mmin + 1 (m,m) (r , p;0)dp 2 dφ p dr = = p 2 WLG . (11) π 2 w02 2 I 0 −π −∞ −∞ m = mmin
( m,n) Equations (9)-(11) have non-zero values for WLG with n = m. In fact the mixing terms integrate to zero because of the orthogonality condition proper of the OAM modes [1]. Moreover ϑ far = lim z →∞ s ( z ) / z [16]. The factor 2, which is present in the first line of both
Eqs. (10) and (11), has been first introduced in [13] for the characterization of the illuminated area of a single LG mode. This area has been defined as a circle having as radial dimension the standard deviation of 2r / I on the plane of constant z. A similar definition can be found in [24] minding that, for n = m, one obtains that r 2 = 2 x 2 = 2 y 2 holds. We calculate the average value of the second order quantities with respect to the origin of coordinates by adopting the assumption used in [13], i.e., by setting the averaged quantities x , y , r to zero, considering the standard deviation with respect to the origin of the coordinates instead of the physical “center of mass”. In this way, the illuminated area of
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the total beam in any plane of constant z is contained within a circle having circumference of 2π s . This area linearly increases with the distance z when it is far from the Rayleigh distance, which is the typical behavior of Gaussian beams [21]. For N = 1 and mmi n = 0 that describes the fundamental Gaussian mode, Eq. (10) reduces to the circular spot size described by Eq. (6). In Fig. 3, we show some examples of the evolution of s(z) at different distances z for different number of superimposed OAM modes, having fixed the beam waist and mmin = 1 .
Fig. 3. Examples of s(z) at different distances for different number of superimposed OAM modes. All the OAM modes have the same beam waist and their m-values are concordant and consecutive.
We notice that the bigger is N the larger are s(z) and ϑ far . We also focus our attention to the dependence of the terms s(z) and ϑ far on the smallest value of the superposed OAM modes, mmin .The larger is the value mmin , the farther becomes the illuminated spot from the optical axis.
4. Angular aperture
In this section we find an analytical expression for the angular (azimuthal) aperture of the superposed beams that shows the dependence on the number of the superposed modes N and on the minimum value of OAM, mmin . To quantify how much the spot is angularly concentrated we have found a quantity, Θ , analogous to s(z), that describe the azimuthal angular coordinate φ . The total azimuthal angle of the intensity pattern Θ is given by 1
Θ=2
π2
=2
3
φ
+
2
1 N
mmax mmax 1∞ π ∞ ∞ 2 ( m,n) = 2 φ 2 WLG (r , p;0 )rdrdφ dp I 0 −π −∞ −∞ n = mmin m = mmin
mmax
mmax
m = mmin n > m
n+m 1 Γ + 2 2 , 2 ( n − m ) m !n !
4 ( −1)
n−m
(12)
where Γ is the Gamma function. Differently from Eqs. (9)-(11) the mixing terms in Eq. (12) do not integrate to zero because of the φ 2 term as part of the integrand function. The intensity 2
distribution, ut ot , normalized as Eq. (5) (see also Fig. 1) on a plane perpendicular to the
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propagation direction z presents an azimuthal symmetry with respect to the axis defined by φ = 0 at z = 0 . This stems from the fact that the product of the mixing terms of the intensity are ∝ f (r ) cos[φ (n − m)] with −π < φ < π , and so the mean value of φ is null, namely, φ = 0 . The effect of the Gouy term of Eq. (1) is a 0 ≤ ζ ( z ) < π / 2 rotation of the total intensity pattern, without affecting the field distribution, and so the intensity distribution at fixed z keeps the azimuthal symmetry but with respect to the set of axis defined by φ ' = φ + ζ ( z ) [7]. If one wants to calculate the integral of Eq. (12) for z>0, it is mathematically convenient to define the origin of the angular coordinate as the physical symmetry axis φ ' = φ + ζ ( z ) due to the Gouy effect of rotation. In this way the product of the mixing terms of the total intensity results ∝ f (r ) cos[(n − m)φ '] with −π < φ ' < π and the mean value φ ' is null. In this way one can remove the dependence on φ ' , with the same result of Eq. (12), by calculating
2
φ '2 − φ ' , which is a clear measure of the standard
deviation of the azimuthal spread angle. In other words a rigid rotation of the intensity pattern, as the only effect of Gouy phase when (z>0), does not modify the angular aperture Θ , as expected. From these argumentations we calculated Eq. (12) at z = 0, for convenience. Analogously to the definition of the variable s(z), the parameter Θ describes the angular standard deviation within which the illuminated area is angularly confined. In fact, the angular aperture is preserved during the beam propagation. It is also inversely depends on the number of superposed OAM modes N. The larger is N, the smaller is the angular aperture of the illuminated spot results. In Fig. 4 we show the relationship between N and the angular aperture of the spot.
Fig. 4. Graphic representation of the relationship between the angular aperture and the number of superimposed OAM modes at several values of
mmin . Inset: exemplified plot for Θ and
s(z).
We notice that the parameter Θ decreases as N increases and it is quite unaffected by the lowest order beam mmin . This implies that the second term in Eq. (12) has a remarkable dependence on N while the factor (−1) n − m is the only quantity responsible for the interference pattern between the mixing terms creating the angular confinement. In other words, it is possible to control the angular aperture of a superposition of coherent, concordant and consecutive LG beams by acting only (in a first approximation) on the number of modes N. #209600 - $15.00 USD (C) 2014 OSA
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It is also worth to be notice that the first term in Eq. (12) ( π 2 / 3 ) corresponding to the angular deviation of free propagating single modes have a quantum analogy with the statistical uncertainty in angular position for pure states as pointed out in [25]. 5. Concentration of the phase variation
In this section we show how to control the phase embedded within the angular region described by Θ , analyzing the effect on the phase of the superposition of concordant, consecutive and coherent OAM modes. In Fig. 5 we report examples of phase map, plotted on a plane with z = 0, for this kind of superposition.
Fig. 5. Phase (a) and intensity (b) plot for N = 5 and mmin for N = 5 and mmin
= 1 . Phase (c) and Intensity (d) plot
= 3.
The value of the total topological charge observed in the plots is equal to the maximum number of OAM modes, mmax , whereas the number of singularities is equal to N. The location of the phase vortices in this field is obtained by finding the set of points (rs , φs ) where the following condition is satisfied [7,9]:
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ℑ( Etot (r , φ )) a tan = 0, ℜ( Etot (r , φ ))
(13)
where Etot (r , φ ) is given by Eq. (5) and the term s gives the s-th singularity. One of the vortex is always located at the center (r = 0) and this singularity is of the order mmin as can be deduced from Eq. (3); for r ≠ 0 the other singularities, hereafter called secondary singularities [9], are present in a number of mmax − mmin . As a consequence, in the central zone, the phase is distributed as the phase of a pure OAM of value mmin whereas in the zone far from the center (starting from a certain value of r ≡ r that corresponds to maximum value of the set rs i.e., the radial position of the secondary singularity, which depends on the particular choice of the parameters N and mmin ) the whole phase is distributed as the phase of a pure OAM of value mmax . Given the definition of r , we have numerically verified that the maximum of the illuminated area of the total beam lies on r > r , so the corresponding phase falls in the zone where mmax is dominant. We now define the angular parameter β as the angle within a variation of 360° of the phase profile is included. We can also define the effective phase variation within the restricted angular domain of angular aperture Θ as Δφeff = Θ / β . For r >> r the angular parameter β is given by 360° / mmax . So, it is possible, with good accuracy, to predict the total phase variation within a fixed angle when the intensity distribution is far from the optical axis. The above condition r >> r is fulfilled for large values of mmin (see Eqs. (10) or (11)). In particular, we have numerically verified that it is fully satisfied just when mmin > 2 . On the contrary in the case of mmin = 1, 2 the expression 360° / mmax is no longer a good approximation for β . In fact, for these values of mmin , the illuminated spot is concentrated in a zone where r r . In this zone the phase variation is not equally distributed and so the angle embedding the 360° phase variation, β , is not constant. We analyze some examples to show the effective phase variation embedded in the region with angular aperture Θ . In particular, we fixed N to 4 and 8 that in good approximation, as stated above, is equivalent to fix the angular aperture of the spot in Θ and we varied mmin . Table 1. Resuming table
N
4 8
mmin
mmax
Θ (deg)
2 9 15 2 9 15
5 12 18 9 16 22
99° 96° 96° 72° 69° 68°
β = 360° / mmax (deg) 72° 30° 20° 40° 22.5° 16.4°
βm (deg) 91° 32° 20° 52° 25° 17°
Δφeff = Θ / β m (deg) 1.3 3.2 4 1.8 2.9 4.2
We report in Table 1 the cases for mmin = 2, 9 and 15. Numerical data confirm that the angular spot of light in Θ is quite the same for different values of mmin having fixed N. On the contrary, the most striking effect obtained by varying mmin , and so mmax , is to concentrate the phase variation in the angular region described by the parameter Θ as can be seen by considering the parameter β . In addition to β we report another parameter, β m , which represents the angle embedding the 360° phase variation at r = s ( z ) obtained from numerical
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calculations. The quantity Δφeff = Θ / β m gives the effective number of the phase variations of 360° inside the angular region delimited by Θ . From the comparison between β and β m it is confirmed that 360° / mmax is not a good approximation for low mmin ( = 2 in the reported examples) whereas for larger mmin it is a meaningful quantity which can be used for the prediction of the number of phase variation within a fixed angle Θ of interest. In Fig. 6 we show the phase embedded in Θ , measured at r = s ( z ) for the cases resumed in Table 1. Each line represents, for different mmin , how many times the phase changes between 0 and 360° within the angular region Θ . The value of Θ sets the angular limits reported in abscissa.
Fig. 6. Simulation of the phase variation embedded in the angular region Θ measured at distance s from the center for three different values of
mmin .(a) N = 4 and (b) N = 8.
6. Conclusions
In this work we have showed that it is possible to control the dimension of the intensity distribution of a beam with non-zero OAM and the phase embedded in the spot area through the superposition of coherent, consecutive and concordant OAM modes. This smart combination of LG modes, weighted by a factorial normalization factor, allows to concentrate the beam in a restricted angular region. We have investigated the geometric characteristics of the total superposed beam, adopting the Wigner function formalism. We have derived the analytical expressions for the radial and azimuthal extension on the transversal plane of the intensity field and its far field divergence. We have also found an analytical expression for the angular aperture of the superposed beams. All these parameters depend on the number N of the superposed OAM modes and on the lowest value of topological charge in the set of modes. Moreover, once fixed the number of states to be superposed, it is possible to control the number of 360° phase variation embedded within the field angular aperture by acting on the lowest topological charge of the set. We also verified that when the lowest OAM value in the set of modes is greater than two it is possible to calculate the number of 360° phase
#209600 - $15.00 USD (C) 2014 OSA
Received 4 Apr 2014; revised 18 Jun 2014; accepted 21 Jun 2014; published 7 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.017135 | OPTICS EXPRESS 17145
variation within the restricted angular region. This can open new possibilities in controlling the beam shaping and its phase content for EM application. Acknowledgments
We thank C. G. Someda, G. Colombo and B. Thidè for the interesting discussion during this work. G. Parisi is grateful to A. Ya. Bekshaev for the suggested and interesting readings from which the paper has taken benefit. We also acknowledge financial support and collaboration of SIAE Microelectronics.
#209600 - $15.00 USD (C) 2014 OSA
Received 4 Apr 2014; revised 18 Jun 2014; accepted 21 Jun 2014; published 7 Jul 2014 14 July 2014 | Vol. 22, No. 14 | DOI:10.1364/OE.22.017135 | OPTICS EXPRESS 17146
