Abstract

Many approaches for producing optical vortices have been developed both for fundamental interests of science and for engineering applications. In particular, the approach with direct excitation of several emitters has a potential to control the topological charges with a control of the source conditions without any modifications of structures of the system. In this paper, we investigate the propagation properties of the optical vortices emitted from a collectively polarized electric dipole array as a simple model of the several emitters. Using an analytical approach based on the Jacobi-Anger expansion, we derive a relationship between the topological charge of the optical vortices and the source conditions of the emitter, and clarify and report our new finding; there exists an intrinsic split of the singular points in the electric field due to the spin-orbit interaction of the dipole fields.

© 2015 Optical Society of America

1. Introduction

Optical vortices, which carry orbital angular momentum of light, are characterized by a spatially distributed phase profile of eilϕ where ϕ is an azimuthal angle on the plane orthogonal to the propagation direction of light, and l is the topological charge of optical vortices [1–3]. Optical vortices has been actively studied over a wide range of fields such as optical manipulation [4], optical sensing [5], laser fabrication [6] and classical and quantum information processing [7,8]. In order to generate and control the optical vortices, the mode conversion of electromagnetic field has been mainly proposed in various structural devices. The spatial light modulators were used to generate a Laguerre-Gaussian beam with an arbitrary topological charge by holographic patterns [9, 10]. The pinhole plates with various arrangements were designed to create arrays of optical vortices by illuminating with a plane wave [11, 12]. The spin-to-orbital angular momentum conversion were performed using artificial optical devices such as Q-plate [13] and a plasmonic metasurface [14]. The compact optical vortex emitters were fabricated to convert the optical modes from whispering-gallery-modes to optical vortices [15, 16]. On the other hand, the direct excitation of several emitters for producing optical vortices were recently studied in a ring molecules [17] and an antenna array [18]. This method has a potential to produce optical vortices by controlling the source conditions of the emitters without any modification of the system structure.

In this paper, we analytically investigate the propagation properties of the optical vortices generated by a collectively polarized electric dipole array, which becomes a simple model for the direct excitation of several emitters. We derive an analytical solution of the electromagnetic field decomposed by vortex fields via several approximations, and clarify the following properties: (1) there is a certain relationship between the topological charge of the optical vortices and the source conditions of the emitters, and (2) There exists an intrinsic split of the singular points in the electric field due to the spin-orbit interaction of light between the collective electric dipole moments and the intrinsic radiation patterns of each emitter without any split of the singular point in the magnetic field.

2. Theory

2.1. Far-field electromagnetic field decomposed by Jacobi-Anger expansion

In our system, N infinitesimal electric dipoles are placed on the x–y plane with the radius of a such that the center of the circle is located at the origin of the coordinate. The coordinate of the j-th dipole is given by (a cosφj, a sinφj, 0) where the azimuthal angle φj=2π(j1)N. The electric dipole moment of all of the electric dipoles is represented by a collective polarization vector p = p(sxex + syey)eiqφj where the real value p shows the amplitude of the electric dipole moment, the complex parameters of sx, sy represent the polarization of each dipole with the normalized amplitude sx2+sy2=1 and eiqφj is the phase factor of each electric dipole with an integer q which decides the topological property of the initial phase distribution of the system. The source condition of the system is characterized by the three parameters; the number of dipoles in the array N, the polarization of each dipole (sx, sy) and the initial topological factor q. Figure 1(a) depicts an illustration of our system with the source condition of (sx, sy) = (1, 0), N = 3 and q = 1. An analytical solution of the electromagnetic field generated by the electric dipole array is given by the superposition of each electric dipole field [19],

E(r)=j=1Neik|ξj|4πε0{k2(ξj|ξj|×p)×ξj|ξj|2+[3ξj|ξj|(ξj|ξj|p)p](1|ξj|3ik|ξj|2)}
H(r)=j=1Nck2eik|ξj|4π(ξj|ξj|×p)1|ξj|(11ik|ξj|)
where ξjrRj, Rj is the coordinate vector of the jth electric dipoles, k is the magnitude of wavevector, ε0 is the dielectric constant in the vacuum and the terms of the harmonic time evolution eiωt is omitted here. Because the profile of the electromagnetic field emitted from the dipole array is symmetrical about the xy plane including the origin, we focus on the optical vortices propagating to the positive direction of the z axis in the following discussion.

 

Fig. 1 Optical vortices of x component of electric field generated by an electric dipole array which is collectively polarized to x direction at z = 100λ. (a) An illustration of our system.(b),(d) and (f) Phase distribution of electric field in the case of (N.q) = (3, 1), (5, 2) and (7, 3) respectively. (c), (e) and (f) Phase distribution of magnetic field in the case of (N.q) = (3, 1), (5, 2) and (7, 3) respectively.

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In order to verify the possibility for the generation of the optical vortices in our system, we calculate the phase distribution of the electromagnetic field in the case where the electric dipoles are collectively polarized to x direction (sx = 1, sy = 0). Figures 1(b)–1(g) shows the phase distribution of the x component of the electric and the y component of the magnetic field with the three different couple of the parameters (N, q) = (3, 1), (5, 2) and (7, 3) at z = 100λ. The horizontal and the vertical axes are scaled by the wavelength of the electromagnetic field emitted from each dipole λ. The magnetic field has a singular point at the center of the xy plane with a topological charge q. On the other hand, the singular points of the electric field appears with the split in the case of the higher order of q. The distance between the singular points is as long as the wavelength of the electromagnetic field. Although the exact solutions of Eqs. (1) and (2) give us a certain distribution of electromagnetic filed, however, it is not clear for the relationship between the propagation properties of the optical vortices and the source conditions.

In order to discuss the singular points in the electric fields split in the far-field with the distance as long as the wavelength of λ, we perform two approximations: the far-field approximation in which the higher terms of λ/za/z are neglected and the near-axis approximation in which the higher terms of λ/zx/zy/z are neglected. These two approximations are suitable for analyzing the internal phase structure of the optical vortices in the far-field regime. At first, we perform the approximation to the function of the phase factor in a cylindrical coordinate (ρ, ϕ, z) by omitting the higher orders than the second order of x/z, y/z, a/z, λ/z. Moreover, we perform the Jacobi-Anger expansion to expand the electromagnetic field by the superposition of the vortex field which propagates along the z axis.

eik|ξj|exp[ikz(1+ρ2+a22z2)]exp[ikρazcos(ϕϕjπ)]=eikψ(z)m(i)mJm(kρaz)eim(ϕϕj)
where ψ(z)z(1+ρ2+a22z2) is the phase factor which is independent on the j and the azimuthal angle of ϕ. Secondly, we approximately evaluate the first vector terms of electric and magnetic field. In this approximation, we also omit the terms which are higher than the second order of x/z, y/z, a/z, λ/z.
(ξj|ξj|×p)×ξj|ξj|2peiqφjz3(sxz232sxξx212sxξy2syξxξysyz232syξy212syξx2sxξxξyz(sxξx+syξy))Λjzeiqφj
(ξj|ξj|×p)1|ξj|peiqφjz3(syz2+sy(ξx2+ξy2)sxz2sx(ξx2+ξy2)sxξyz+syξxz)Πjzeiqφj
The terms of Λj and Πj imply a presence of the spin-orbit interaction of light due to the non-paraxial emission of the electromagnetic fields. By calculating the sum of Λjei(qm)φj and Πjei(qm)φj (the detail calculation is shown in the Appendix A), the far-field and the near-axis solution of the electromagnetic field is obtained as the following;
E(r)k24πε0zeikψ(z)Q(sxFQ0(r)+FQSOI(r)syFQ0(r)iFQSOI(r)GQSOI,+(r))
H(r)ck4πzeikψ(z)Q(syFQ0(r)sxFQ0(r)iGQSOI,(r))
where
FQ0(r)=[(1ρ2+a2z2)fQ+ρaz2(fQ+1+fQ1))]eiQϕ,
FQSOI(r)=s[ρa2z2fQ+1ρ24z2fQa24z2fQ+2]ei(Q+2)ϕ+s+[ρa2z2fQ1ρ24z2fQa24z2fQ2]ei(Q2)ϕ,
GQSOI,±(r)=s(ρ2zfQa2zfQ+1)ei(Q+1)ϕ±s+(ρ2zfQa2zfQ1)ei(Q1)ϕ,
QqnN is an integer defined by the system parameters of N, q and an arbitrary integer of n, the summation of Q is taken from n = −∞ to n = ∞, fQfQ(ρ,z)=(i)QJQ(kρa2z) and s± = sx ± isy. The term of ”SOI” stands for the spin-orbit interaction because the FQSOI(r) and GQSOI,±(r) include the multiplication terms including the opposite sign of the spin term s± and the orbit term ei(Q∓1)ϕ or ei(Q∓2)ϕ. Using the polynomial expansion of the Bessel function, the lowest order of kρa/z is determined by the order of the absolute value of Q.
JQ(kρaz)=(1)sgn(Q)12|Q|t=0(1)tt!(t+|Q|)!(kρa2z)2t+|Q|(1)sgn(Q)12|Q|1|Q|!(kρa2z)|Q|
where sgn(x) is the sign function. Here, we define an integer Qmin such that |Qmin| becomes the minimum integer of Q(n). If the minimum absolute value of Q does not uniquely defined such that Q(n0) = −Qmin and Q(n1) = Qmin, there are no optical vortices due to the superposition of the clockwise and the anti-clockwise vortices. For example, this situation appears in the case of Qmin = 0. On the other hand, in the case where the minimum absolute value is uniquely defined by n, each term of the electromagnetic field are simply expressed as a form including optical vortices with several terms for each Qmin.
QFQ0(r)(i)|Qmin||Qmin|!(1)sgn(Qmin1)2|Qmin|(kρa2z)QmineiQminϕ
QFQSOI(r){0(Qmin=±1)s+(i)Qmin2(Qmin2)!(kρa2z)Qmin2a24z2ei(Qmin2)ϕ(Qmin2)(1)|Qmin+2|si|Qmin+2||Qmin+2|!(kρa2z)|Qmin+2|a24z2ei(Qmin+2)ϕ(Qmin2)
QGQSOI,±(r){s+(i)Qmin1(Qmin1)!(kρa2z)Qmin1a2zei(Qmin1)ϕ(Qmin1)(1)|Qmin+1|si|Qmin+1||Qmin+1|!(kρa2z)|Qmin+1|a2zei(Qmin+1)ϕ(Qmin1)
The analytical solution of the electromagnetic field expressed by the vortex fields in the Eqs. (12)(14) is one of our main results. In the far-field and the near-axis regime, we obtain a certain relationship between the source conditions and the topological charge of the optical vortices. Figure 2 show the value of Qmin with respect to the system parameters (N, q) [Fig. 2(a)], and several phase profiles calculated from the exact solutions in Eqs. (1) and (2) fixing the collective polarization to (sx = 1, sy = 0) [Figs. 2(b)–2(d)]. Qmin obviously corresponds to the topological charge calculated from the exact solution, and becomes a good index for characterizing the dipole array as an optical vortex emitter corresponds to the topological charge calculated from the exact solution.

 

Fig. 2 The topological charge of the optical vortex with respect to the source conditions with the collective polarization of (sx = 1, sy = 0). (a) The relationship between Qmin and (N, q). The phase profiles of the electric and the magnetic fields are shown in (b) ∼ (d) with the source conditions of (N, q) = (9, 5), (N, q) = (9, 4) and (N, q) = (9, 1) respectively.

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2.2. Intrinsic split of singular points in electric field with higher order topological charge

In the case where |Qmin| ≥ 2, the transverse components of electric fields are represented by the sum of the two terms. This implies that the singular points of the transverse components are not located at the center of xy plane. For simplicity, we consider only the case of Qmin ≥ 2 in the following discussion. The transverse components of the electric field are given by

Ex(r)k24πε0z(ikρ)Qmin2(Qmin2)!(a2z)Qminei(Qmin2)ϕ[sxQmin(Qmin1)(kρ)2ei2ϕ+s+]
Ey(r)k24πε0z(ikρ)Qmin2(Qmin2)!(a2z)Qminei(Qmin2)ϕ[syQmin(Qmin1)(kρ)2ei2ϕ+is+].
The singular points with the topological charge Qmin − 2 are located at the center of xy plane. In addition, there are non-trivial singular points hidden in the sum of the final terms. In order to expand a precise calculation for the coordinate of all singular points, we express the compolex polarization paramter as a state vector on the Poincare sphere, sx=cosβ2, sy=sinβ2eiγ where each angle β and γ are bounded as 0 ≤ βπ and 0 ≤ γ ≤ 2π respectively. There are two non-trivial singular points with the topological charge one in the coordinate of ζi=(ζxi,ζyi) where i ∈ {1, 2} shows the label of the two singular points (the detail calculation is shown in the Appendix B).
ζ1(β,γ)={(ζ+(β,γ),ζ(β,γ))(0γπ2,3π2γ2π)(ζ+(β,γ),ζ(β,γ))(π2γ2π)
ζ2(β,γ)={(ζ+(β,γ),ζ(β,γ))(0γπ2,3π2γ2π)(ζ+(β,γ),ζ(β,γ))(π2γ2π)
ζ±(β,γ)±λQmin(Qmin1)2π21sinβsinγcosβ2±(tanβ2sinγ1)
Because the coordinates of the singular point are independent on z coordinate, all singular points propagate in parallel to the z axis in the far-field regime. Equations (17)(19) are completely characterized the intrinsic split of the singular points in the electric field of Ex with respect to the source condition of the emitter. If the electric dipoles are horizontally polarized (β = 0 and γ = 0), the singular points in the electric field of Ex are vertically split with the distance of the singular points simply expressed by
D=λQmin(Qmin1)2π.
On the other hand, if the electric dipoles are left (right) circularly polarized in the case of the positive (negative) Qmin, the singlar points in the electric field are not split and are located at the center of the xy plane with the topological charge Qmin. We can make a single optical vortex with the topological charge Qmin propagating on the z axis. Several correspondence between the transverse coordinate of the singular points and the collective polarization of the electric dipoles on the Poincare sphere are shown in Fig. 3.

 

Fig. 3 The correspondence between the coordinates of the singular points (right) and the collective polarizations on the Poincare sphere (left). The dots of same color in both of the left and the right fiure shows the same angle of β and γ. The colered lines in the right figure show the trajectory in the same Qmin. The characters in the left figure indicate the collective polarization states. H: horizontal polarization, V: vertical polarization, D: diagonal polarization, A: anti-diagonal polarization, L: left circular polarization and R: right circular polarization.

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The split of the higher-order optical vortex has been discussed in the holograhic method [20, 21]. In general, the higher-order vortex is easily split into several vortices with the topological charge one under the practical fluctuation and incompleteness. We emphasize that if the perfect system removing any practical problems is assumed for generation of the optical vortices, nevertheless, there exists intrinsic split of the singular points in the electric field in the dipole array due to the spin-orbit interaction caused by the radiation patterns of each electric dipole.

3. Numerical simulation

To verify the consistency of our analytical approach with several approximations, we numerically calculate a topological charge around one of the singular points ζ1(β, γ) from the exact solution of Eqs. (1) and (2). The topological charge that we numerically calculated is defined as

QN12πcdlArg[Ex(r)]=12πxπdϕ1Arg[Ex(RNcosϕ1,RNsinϕ1,z)]
where RN is a radius of closed path for integration and ϕ1 is the azimuthal angle around ζ1(β, γ). The topological charge depends on the z coordinate when the state of the polarization is fixed to β = 0 and γ = 0 [Fig. 4(a)]. The topological charges for each Qmin reach to one when the z coordinate is over several ten times of the wavelength. This implies that there is a split vortex with a topological charge one in the coordinate of the singular points given by Eqs. (17)(19) around the far-field regime, and the singular points propagate in parallel to the z axis. Figure 4(b) shows the numerical topological charges under Qmin = 2. At all of the points without the case where β = γ = π/2, the singular point with the topological charge one appears. This implies that the singular points split into two, and one of them propagates along z axis with the coordinate of ζ1(β, γ). On the other hand, there is an optical vortex with the topological charge two which has no splitting at the center of the xy plane in the case of β = γ = π/2 because the coordinate of singular points degenerate into ζ1(β, γ) = (0, 0). All numerical result calculated from the exact solutions is supported by our analytical approach with the far-field and the near-axis approximations.

 

Fig. 4 The results of the numerical calculations (a) the topological charges around the coordinate of ζ1(0, 0) for each z. (b) the topological charges around the coordinate of ζ1(β, γ) at z = 100λ for each β and γ.

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4. Discussion

We mainly focused on the phase distribution of the electromagnetic field to characterize the structures of the optical vortices generated by the electric dipole array. On the other hand, the energy distribution is also important to discuss the function of the system. Figure 5 shows the energy density in the zx plane calculated from the exact solutions for each source condition Qmin with the polarization condition of (sx, sy) = (1, 0). The larger is the area where the energy becomes weak around the z axis, the larger is the topological charge of the optical vortex. It stems from the terms of amplitude including the Bessel function with the order of its topological charge.

 

Fig. 5 Amplitude profiles of the optical vortex in each topological charge.

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An electric dipole is the most fundamental element for emitting electromagnetic field to the far-field with a wide range of wavelengths. For the implementation of our optical vortex generator, it is necessary to fix the position of each dipole on a circle and to control each phase. It is possible to easily implement our system in radiowave or microwave regions because the phase of the electric dipoles can be directly controlled with conventional systems. In addition, the intrinsic split of the singular points in the electric field can be experimentally detectable due to the long wavelengths. Our system has a possibility to switch the topological charges only by controlling the phase distributions of the electric dipoles q without the number of the electric dipoles N.

5. Conclusion

In this paper, we discuss the propagation properties of the optical vortices emitted from a collectively polarized dipole array. Under the far-field and the near-axis approximations, we clarify the following two properties: (1) there is a certain relationship between the source conditions and the topological charge of the optical vortices, (2) There exists an intrinsic split of the singular points in the electric field due to the spin-orbit interaction of light between the collective electric dipole moments and the intrinsic radiation patterns of each emitter without any split of the singular point in the magnetic field. These analytical results give us a certain perspective for controlling the topological charge of optical vortices.

Appendix A: Derivation of the far-field solution

In order to derive the far-field solution by decomposing the optical vortices which propagates on the z axis, we start from the electric and the magnetic field indicated by the superposition of the field emitted from each dipole

E(r)k24πε0zm=(i)mJm(kρaz)eimϕj=1NΛjei(qm)φj
H(r)ck4πzm=(i)mJm(kρaz)eimϕj=1NΠjei(qm)φj,
where Λj and Πj are the spin-orbit interaction terms defined in Eqs. (4) and (5) respectively. We obtain the selection rules for the topological charges given by the Dirac’s delta via the direct calculation of the sum of Λj and Πj.
[Λj]x=sx(13ρ22z2cos2ϕ3a22z2cos2φj+3ρaz2cosϕcosφjρ22z2sin2ϕa22z2sin2φj+ρaz2sinϕsinφj)sy{ρ2z2cosϕsinϕ+a2z2cosφjsinφjρaz2(cosφsinφj+cosφjsinφ)}=sx(1ρ2+a2z2ρ22z2cos2ϕa22z2cos2φjs+3ρaz2cosϕcosφj+ρaz2sinϕsinφj)sy(ρ22z2sin2ϕ+a22z2sin2φjρa2z2sin(ϕ+φj))
j=1N[Λj]xei(qm)φj=n=sx[(1ρ2+a2z2)δqm,nN+ρ24z2(ei2ϕ+ei2ϕ)δqm,nNa24z2(δqm+2,nN+δqm2,nN)+3ρa4z2(eiϕ+eiϕ)(δqm+1,nN+δqm1,nN)ρa4z2(eiϕeiϕ)(δqm+1,nNδqm1,nN)]n=sy[ρ2i4z2(ei2ϕei2ϕ)δqm,nN+a2i4z2(δqm+2,nNδqm2,nN)ρai2z2(eiϕδqm+1,nNeiϕδqm1,nN)]
j=1N[Λj]yei(qm)φj=n=sy[(1ρ2+a2z2)δqm,nN+ρ24z2(ei2ϕ+ei2ϕ)δqm,nN+a24z2(δqm+2,nN+δqm2,nN)3ρa4z2(eiϕeiϕ)(δqm+1,nNδqm1,nN)+ρa4z2(eiϕ+eiϕ)(δqm+1,nN+δqm1,nN)]n=sx[ρ2i4z2(ei2ϕei2ϕ)δqm,N+a2i4z2(δqm+2,nNδqm2,nN)ρai2z2(eiϕδqm+1,nNeiϕδqm1,nN)]
j=1N[Λj]zei(qm)φj=n=[sx{ρ2z(eiϕ+eiϕ)δqm,nNa2z(δqm+1,nN+δqm1,nN)}+sy{ρi2z(eiϕeiϕ)δqm,nNai2z(δqm+1,nNδqm1,nN)}]
Finally, all component of the electric field is calculated as
Ex(r)k24πε0zQsx(fQρ2+a2z2fQ+ρaz2((fQ+1+fQ1)))eiQϕ+Qs[ρa2z2fQ+1ρ24z2fQa24z2fQ+2]ei(Q+2)ϕ+Qs+[ρa2z2fQ1ρ24z2fQa24z2fQ2]ei(Q2)ϕ,
Ey(r)k24πε0zQsy(fQρ2+a2z2fQ+ρaz2((fQ+1+fQ1)))eiQϕiQs[ρa2z2fQ+1ρ24z2fQa24z2fQ+2]ei(Q+2)ϕiQs+[ρa2z2fQ1ρ24z2fQa24z2fQ2]ei(Q2)ϕ,
Ez(r)k24πε0zQ[s(ρ2zfQa2zfQ+1)ei(Q+1)ϕ+s+(ρ2zfQa2zfQ1)ei(Q1)ϕ],
where fnfn(x) ≡ (−i)nJn(x). As a same manner, the magnetic field is calculated as
Πj=1z2(syz2+sy(ξx2+ξy2)sxz2sx(ξx2+ξy2)sxξyz+syξxz),
j=1N[Πj]xei(qm)φj=syn[(1ρ2+a2z2)δqm,nN+ρaz2(eiϕδqm1,nN+eiϕδqm+1,nN)],
j=1N[Πj]yei(qm)φj=sxn[(1ρ2+a2z2)δqm,nN+ρaz2(eiϕδqm1,nN+eiϕδqm+1,nN)],
j=1N[Πj]zei(qm)φj=isn(ρ2zeiϕδqm,nNa2zδqm+1,nN)is+n(ρ2zeiϕδqm,nNa2zδqm1,nN),
Hx(r)=ck4πzsyeiQϕn[(1ρ2+a2z2)fQ+ρaz2(fQ1+fQ+1)],
Hy(r)=ck4πzsxeiQϕn[(1ρ2+a2z2)fQ+ρaz2(fQ1+fQ+1)],
Hz(r)=ick4πz[sei(Q+1)ϕn(ρ2zfQfQ+1)s+ei(Q1)ϕn(ρ2zfQa2zfQ1)].

Appendix B: Derivation of the singular points in Ex

In the case Qmin ≥ 2, the Ex given in Eq. (15) includes the non-trivial vortices. In order to find the coordinate of the singular points of the non-trivial vortices, we derive the product form of e1e2 from the final terms of Eq. (15) where ϕ1 and ϕ2 are defined as new azimuthal angles in cylindrical coordinates achieved by translating the z axis into the radial directions respectively. At first, we define A=k2Qmin(Qmin1), and expand to the Cartesian coordinate to find the transformation factor ζx and ζy.

Asxρ2ei2ϕ+s+=[Acosβ2(x2y2+i2xy)+cosβ2+isinβ2cosγsinβ2sinγ]=[Acosβ2(x2y2)+cosβ2sinβ2sinγ+i(2Axycosβ2+sinβ2sinγ)]
The phase factor ϕ′ is calculated as
ϕ=tan1[Im[G]Re[G]]=tan1[2Axycosβ2+sinβ2cosγAcosβ2(x2y2)+cosβ2sinβ2sinγ]=tan1[2xy+1Atanβ2cosγ(x2y2)+1A1Atanβ2sinγ].
According to a property of the inverse trigonometric function, the phase factor is separated as a form of sum.
tan1[yζyxζx+y+ζyx+ζx1yζyxζxy+ζyx+ζx]=tan1y+ζyx+ζx+tan1yζyxζx
Finally, we can find the translation factor ζx,ζy from the following simultaneous equations:
ζxζy=12Atanβ2cosγ,
ζy2ζx2=1A1Atanβ2sinγ.
As a result, we obtain the coordinate of the singular points.
ζ1(β,γ)={(ζ+(β,γ),ζ(β,γ))(0γπ2,3π2γ2π)(ζ+(β,γ),ζ(β,γ))(π2γ2π)
ζ2(β,γ)={(ζ+(β,γ),ζ(β,γ))(0<γπ2,3π2γ2π)(ζ+(β,γ),ζ(β,γ))(π2γ2π)
ζ±(β,γ)±λQmin(Qmin1)2π21sinβsinγcosβ2±(tanβ2sinγ1)

Acknowledgments

This work was supported by Program for Leading Graduate Schools: ”Interactive Materials Science Cadet Program”.

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10. N. Matsumoto, T. Ando, T. Inoue, Y. Ohtake, N. Fukuchi, and T. Hara, “Generation of high-quality higher-order laguerre-gaussian beams using liquid-crystal-on-silicon spatial light modulators,” J. Opt. Soc. Am. A 25, 1642–1651 (2008). [CrossRef]  

11. K. Nicholls and J. Nye, “Three-beam model for studying dislocations in wave pulses,” J. Phys. A: Math. Gen. 20, 4673 (1987). [CrossRef]  

12. Z. Li, M. Zhang, G. Liang, X. Li, X. Chen, and C. Cheng, “Generation of high-order optical vortices with asymmetrical pinhole plates under plane wave illumination,” Opt. Express 21, 15755–15764 (2013). [CrossRef]   [PubMed]  

13. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006). [CrossRef]   [PubMed]  

14. E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3, e167 (2014). [CrossRef]  

15. A. A. Savchenkov, A. B. Matsko, I. Grudinin, E. A. Savchenkova, D. Strekalov, and L. Maleki, “Optical vortices with large orbital momentum: generation and interference,” Opt. Express 14, 2888–2897 (2006). [CrossRef]   [PubMed]  

16. X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated compact optical vortex beam emitters,” Science 338, 363–366 (2012). [CrossRef]   [PubMed]  

17. M. D. Williams, M. M. Coles, D. S. Bradshaw, and D. L. Andrews, “Direct generation of optical vortices,” Phys. Rev. A 89, 033837 (2014). [CrossRef]  

18. X. Ma, M. Pu, X. Li, C. Huang, W. Pan, B. Zhao, J. Cui, and X. Luo, “Optical phased array radiating optical vortex with manipulated topological charges,” Opt. Express 23, 4873–4879 (2015). [CrossRef]   [PubMed]  

19. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).

20. I. Basistiy, V. Y. Bazhenov, M. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993). [CrossRef]  

21. F. Ricci, W. Löffler, and M. van Exter, “Instability of higher-order optical vortices analyzed with a multi-pinhole interferometer,” Opt. Express 20, 22961–22975 (2012). [CrossRef]   [PubMed]  

References

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  1. L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
    [Crossref] [PubMed]
  2. G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
    [Crossref]
  3. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011).
    [Crossref]
  4. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
    [Crossref] [PubMed]
  5. M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using lightfs orbital angular momentum,” Science 341, 537–540 (2013).
    [Crossref] [PubMed]
  6. T. Omatsu, K. Chujo, K. Miyamoto, M. Okida, K. Nakamura, N. Aoki, and R. Morita, “Metal microneedle fabrication using twisted light with spin,” Opt. Express 18, 17967–17973 (2010).
    [Crossref] [PubMed]
  7. J. Wang, J.-Y. 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,” Nat. Photonics 6, 488–496 (2012).
    [Crossref]
  8. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
    [Crossref] [PubMed]
  9. N. Heckenberg, R. McDuff, C. Smith, and A. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
    [Crossref] [PubMed]
  10. N. Matsumoto, T. Ando, T. Inoue, Y. Ohtake, N. Fukuchi, and T. Hara, “Generation of high-quality higher-order laguerre-gaussian beams using liquid-crystal-on-silicon spatial light modulators,” J. Opt. Soc. Am. A 25, 1642–1651 (2008).
    [Crossref]
  11. K. Nicholls and J. Nye, “Three-beam model for studying dislocations in wave pulses,” J. Phys. A: Math. Gen. 20, 4673 (1987).
    [Crossref]
  12. Z. Li, M. Zhang, G. Liang, X. Li, X. Chen, and C. Cheng, “Generation of high-order optical vortices with asymmetrical pinhole plates under plane wave illumination,” Opt. Express 21, 15755–15764 (2013).
    [Crossref] [PubMed]
  13. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
    [Crossref] [PubMed]
  14. E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3, e167 (2014).
    [Crossref]
  15. A. A. Savchenkov, A. B. Matsko, I. Grudinin, E. A. Savchenkova, D. Strekalov, and L. Maleki, “Optical vortices with large orbital momentum: generation and interference,” Opt. Express 14, 2888–2897 (2006).
    [Crossref] [PubMed]
  16. X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated compact optical vortex beam emitters,” Science 338, 363–366 (2012).
    [Crossref] [PubMed]
  17. M. D. Williams, M. M. Coles, D. S. Bradshaw, and D. L. Andrews, “Direct generation of optical vortices,” Phys. Rev. A 89, 033837 (2014).
    [Crossref]
  18. X. Ma, M. Pu, X. Li, C. Huang, W. Pan, B. Zhao, J. Cui, and X. Luo, “Optical phased array radiating optical vortex with manipulated topological charges,” Opt. Express 23, 4873–4879 (2015).
    [Crossref] [PubMed]
  19. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).
  20. I. Basistiy, V. Y. Bazhenov, M. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
    [Crossref]
  21. F. Ricci, W. Löffler, and M. van Exter, “Instability of higher-order optical vortices analyzed with a multi-pinhole interferometer,” Opt. Express 20, 22961–22975 (2012).
    [Crossref] [PubMed]

2015 (1)

2014 (2)

E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3, e167 (2014).
[Crossref]

M. D. Williams, M. M. Coles, D. S. Bradshaw, and D. L. Andrews, “Direct generation of optical vortices,” Phys. Rev. A 89, 033837 (2014).
[Crossref]

2013 (2)

Z. Li, M. Zhang, G. Liang, X. Li, X. Chen, and C. Cheng, “Generation of high-order optical vortices with asymmetrical pinhole plates under plane wave illumination,” Opt. Express 21, 15755–15764 (2013).
[Crossref] [PubMed]

M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using lightfs orbital angular momentum,” Science 341, 537–540 (2013).
[Crossref] [PubMed]

2012 (3)

J. Wang, J.-Y. 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,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated compact optical vortex beam emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

F. Ricci, W. Löffler, and M. van Exter, “Instability of higher-order optical vortices analyzed with a multi-pinhole interferometer,” Opt. Express 20, 22961–22975 (2012).
[Crossref] [PubMed]

2011 (1)

2010 (1)

2008 (1)

2007 (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[Crossref]

2006 (2)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref] [PubMed]

A. A. Savchenkov, A. B. Matsko, I. Grudinin, E. A. Savchenkova, D. Strekalov, and L. Maleki, “Optical vortices with large orbital momentum: generation and interference,” Opt. Express 14, 2888–2897 (2006).
[Crossref] [PubMed]

2003 (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref] [PubMed]

2001 (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

1993 (1)

I. Basistiy, V. Y. Bazhenov, M. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[Crossref]

1992 (2)

N. Heckenberg, R. McDuff, C. Smith, and A. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[Crossref] [PubMed]

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
[Crossref] [PubMed]

1987 (1)

K. Nicholls and J. Nye, “Three-beam model for studying dislocations in wave pulses,” J. Phys. A: Math. Gen. 20, 4673 (1987).
[Crossref]

Ahmed, N.

J. Wang, J.-Y. 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,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Allen, L.

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
[Crossref] [PubMed]

Ando, T.

Andrews, D. L.

M. D. Williams, M. M. Coles, D. S. Bradshaw, and D. L. Andrews, “Direct generation of optical vortices,” Phys. Rev. A 89, 033837 (2014).
[Crossref]

Aoki, N.

Barnett, S. M.

M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using lightfs orbital angular momentum,” Science 341, 537–540 (2013).
[Crossref] [PubMed]

Basistiy, I.

I. Basistiy, V. Y. Bazhenov, M. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[Crossref]

Bazhenov, V. Y.

I. Basistiy, V. Y. Bazhenov, M. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[Crossref]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
[Crossref] [PubMed]

Boyd, R. W.

E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3, e167 (2014).
[Crossref]

Bradshaw, D. S.

M. D. Williams, M. M. Coles, D. S. Bradshaw, and D. L. Andrews, “Direct generation of optical vortices,” Phys. Rev. A 89, 033837 (2014).
[Crossref]

Cai, X.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated compact optical vortex beam emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

Chen, X.

Cheng, C.

Chujo, K.

Coles, M. M.

M. D. Williams, M. M. Coles, D. S. Bradshaw, and D. L. Andrews, “Direct generation of optical vortices,” Phys. Rev. A 89, 033837 (2014).
[Crossref]

Cui, J.

De Leon, I.

E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3, e167 (2014).
[Crossref]

Dolinar, S.

J. Wang, J.-Y. 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,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Fazal, I. M.

J. Wang, J.-Y. 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,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Fukuchi, N.

Grier, D. G.

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref] [PubMed]

Grudinin, I.

Hara, T.

Heckenberg, N.

Huang, C.

Huang, H.

J. Wang, J.-Y. 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,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Inoue, T.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).

Johnson-Morris, B.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated compact optical vortex beam emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

Karimi, E.

E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3, e167 (2014).
[Crossref]

Lavery, M. P.

M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using lightfs orbital angular momentum,” Science 341, 537–540 (2013).
[Crossref] [PubMed]

Li, X.

Li, Z.

Liang, G.

Löffler, W.

Luo, X.

Ma, X.

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

Maleki, L.

Manzo, C.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref] [PubMed]

Marrucci, L.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref] [PubMed]

Matsko, A. B.

Matsumoto, N.

McDuff, R.

Miyamoto, K.

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[Crossref]

Morita, R.

Nakamura, K.

Nicholls, K.

K. Nicholls and J. Nye, “Three-beam model for studying dislocations in wave pulses,” J. Phys. A: Math. Gen. 20, 4673 (1987).
[Crossref]

Nye, J.

K. Nicholls and J. Nye, “Three-beam model for studying dislocations in wave pulses,” J. Phys. A: Math. Gen. 20, 4673 (1987).
[Crossref]

O’Brien, J. L.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated compact optical vortex beam emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

Ohtake, Y.

Okida, M.

Omatsu, T.

Padgett, M. J.

M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using lightfs orbital angular momentum,” Science 341, 537–540 (2013).
[Crossref] [PubMed]

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011).
[Crossref]

Pan, W.

Paparo, D.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref] [PubMed]

Pu, M.

Qassim, H.

E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3, e167 (2014).
[Crossref]

Ren, Y.

J. Wang, J.-Y. 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,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Ricci, F.

Savchenkov, A. A.

Savchenkova, E. A.

Schulz, S. A.

E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3, e167 (2014).
[Crossref]

Smith, C.

Sorel, M.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated compact optical vortex beam emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

Soskin, M.

I. Basistiy, V. Y. Bazhenov, M. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[Crossref]

Speirits, F. C.

M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using lightfs orbital angular momentum,” Science 341, 537–540 (2013).
[Crossref] [PubMed]

Spreeuw, R.

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
[Crossref] [PubMed]

Strain, M. J.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated compact optical vortex beam emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

Strekalov, D.

Thompson, M. G.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated compact optical vortex beam emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[Crossref]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[Crossref]

Tur, M.

J. Wang, J.-Y. 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,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Upham, J.

E. Karimi, S. A. Schulz, I. De Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light Sci. Appl. 3, e167 (2014).
[Crossref]

van Exter, M.

Vasnetsov, M. V.

I. Basistiy, V. Y. Bazhenov, M. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[Crossref]

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

Wang, J.

J. Wang, J.-Y. 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,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated compact optical vortex beam emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

White, A.

Williams, M. D.

M. D. Williams, M. M. Coles, D. S. Bradshaw, and D. L. Andrews, “Direct generation of optical vortices,” Phys. Rev. A 89, 033837 (2014).
[Crossref]

Willner, A. E.

J. Wang, J.-Y. 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,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Woerdman, J.

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
[Crossref] [PubMed]

Yan, Y.

J. Wang, J.-Y. 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,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Yang, J.-Y.

J. Wang, J.-Y. 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,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Yao, A. M.

Yu, S.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated compact optical vortex beam emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

Yue, Y.

J. Wang, J.-Y. 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,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Zeilinger, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

Zhang, M.

Zhao, B.

Zhu, J.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated compact optical vortex beam emitters,” Science 338, 363–366 (2012).
[Crossref] [PubMed]

Adv. Opt. Photon. (1)

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Figures (5)

Fig. 1
Fig. 1 Optical vortices of x component of electric field generated by an electric dipole array which is collectively polarized to x direction at z = 100λ. (a) An illustration of our system.(b),(d) and (f) Phase distribution of electric field in the case of (N.q) = (3, 1), (5, 2) and (7, 3) respectively. (c), (e) and (f) Phase distribution of magnetic field in the case of (N.q) = (3, 1), (5, 2) and (7, 3) respectively.
Fig. 2
Fig. 2 The topological charge of the optical vortex with respect to the source conditions with the collective polarization of (sx = 1, sy = 0). (a) The relationship between Qmin and (N, q). The phase profiles of the electric and the magnetic fields are shown in (b) ∼ (d) with the source conditions of (N, q) = (9, 5), (N, q) = (9, 4) and (N, q) = (9, 1) respectively.
Fig. 3
Fig. 3 The correspondence between the coordinates of the singular points (right) and the collective polarizations on the Poincare sphere (left). The dots of same color in both of the left and the right fiure shows the same angle of β and γ. The colered lines in the right figure show the trajectory in the same Qmin. The characters in the left figure indicate the collective polarization states. H: horizontal polarization, V: vertical polarization, D: diagonal polarization, A: anti-diagonal polarization, L: left circular polarization and R: right circular polarization.
Fig. 4
Fig. 4 The results of the numerical calculations (a) the topological charges around the coordinate of ζ1(0, 0) for each z. (b) the topological charges around the coordinate of ζ1(β, γ) at z = 100λ for each β and γ.
Fig. 5
Fig. 5 Amplitude profiles of the optical vortex in each topological charge.

Equations (45)

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E ( r ) = j = 1 N e i k | ξ j | 4 π ε 0 { k 2 ( ξ j | ξ j | × p ) × ξ j | ξ j | 2 + [ 3 ξ j | ξ j | ( ξ j | ξ j | p ) p ] ( 1 | ξ j | 3 i k | ξ j | 2 ) }
H ( r ) = j = 1 N c k 2 e i k | ξ j | 4 π ( ξ j | ξ j | × p ) 1 | ξ j | ( 1 1 i k | ξ j | )
e i k | ξ j | exp [ i k z ( 1 + ρ 2 + a 2 2 z 2 ) ] exp [ i k ρ a z cos ( ϕ ϕ j π ) ] = e i k ψ ( z ) m ( i ) m J m ( k ρ a z ) e i m ( ϕ ϕ j )
( ξ j | ξ j | × p ) × ξ j | ξ j | 2 p e i q φ j z 3 ( s x z 2 3 2 s x ξ x 2 1 2 s x ξ y 2 s y ξ x ξ y s y z 2 3 2 s y ξ y 2 1 2 s y ξ x 2 s x ξ x ξ y z ( s x ξ x + s y ξ y ) ) Λ j z e i q φ j
( ξ j | ξ j | × p ) 1 | ξ j | p e i q φ j z 3 ( s y z 2 + s y ( ξ x 2 + ξ y 2 ) s x z 2 s x ( ξ x 2 + ξ y 2 ) s x ξ y z + s y ξ x z ) Π j z e i q φ j
E ( r ) k 2 4 π ε 0 z e i k ψ ( z ) Q ( s x F Q 0 ( r ) + F Q SOI ( r ) s y F Q 0 ( r ) i F Q SOI ( r ) G Q SOI , + ( r ) )
H ( r ) c k 4 π z e i k ψ ( z ) Q ( s y F Q 0 ( r ) s x F Q 0 ( r ) i G Q SOI , ( r ) )
F Q 0 ( r ) = [ ( 1 ρ 2 + a 2 z 2 ) f Q + ρ a z 2 ( f Q + 1 + f Q 1 ) ) ] e i Q ϕ ,
F Q SOI ( r ) = s [ ρ a 2 z 2 f Q + 1 ρ 2 4 z 2 f Q a 2 4 z 2 f Q + 2 ] e i ( Q + 2 ) ϕ + s + [ ρ a 2 z 2 f Q 1 ρ 2 4 z 2 f Q a 2 4 z 2 f Q 2 ] e i ( Q 2 ) ϕ ,
G Q SOI , ± ( r ) = s ( ρ 2 z f Q a 2 z f Q + 1 ) e i ( Q + 1 ) ϕ ± s + ( ρ 2 z f Q a 2 z f Q 1 ) e i ( Q 1 ) ϕ ,
J Q ( k ρ a z ) = ( 1 ) sgn ( Q ) 1 2 | Q | t = 0 ( 1 ) t t ! ( t + | Q | ) ! ( k ρ a 2 z ) 2 t + | Q | ( 1 ) sgn ( Q ) 1 2 | Q | 1 | Q | ! ( k ρ a 2 z ) | Q |
Q F Q 0 ( r ) ( i ) | Q min | | Q min | ! ( 1 ) sgn ( Q min 1 ) 2 | Q min | ( k ρ a 2 z ) Q min e i Q min ϕ
Q F Q SOI ( r ) { 0 ( Q min = ± 1 ) s + ( i ) Q min 2 ( Q min 2 ) ! ( k ρ a 2 z ) Q min 2 a 2 4 z 2 e i ( Q min 2 ) ϕ ( Q min 2 ) ( 1 ) | Q min + 2 | s i | Q min + 2 | | Q min + 2 | ! ( k ρ a 2 z ) | Q min + 2 | a 2 4 z 2 e i ( Q min + 2 ) ϕ ( Q min 2 )
Q G Q SOI , ± ( r ) { s + ( i ) Q min 1 ( Q min 1 ) ! ( k ρ a 2 z ) Q min 1 a 2 z e i ( Q min 1 ) ϕ ( Q min 1 ) ( 1 ) | Q min + 1 | s i | Q min + 1 | | Q min + 1 | ! ( k ρ a 2 z ) | Q min + 1 | a 2 z e i ( Q min + 1 ) ϕ ( Q min 1 )
E x ( r ) k 2 4 π ε 0 z ( i k ρ ) Q min 2 ( Q min 2 ) ! ( a 2 z ) Q min e i ( Q min 2 ) ϕ [ s x Q min ( Q min 1 ) ( k ρ ) 2 e i 2 ϕ + s + ]
E y ( r ) k 2 4 π ε 0 z ( i k ρ ) Q min 2 ( Q min 2 ) ! ( a 2 z ) Q min e i ( Q min 2 ) ϕ [ s y Q min ( Q min 1 ) ( k ρ ) 2 e i 2 ϕ + i s + ] .
ζ 1 ( β , γ ) = { ( ζ + ( β , γ ) , ζ ( β , γ ) ) ( 0 γ π 2 , 3 π 2 γ 2 π ) ( ζ + ( β , γ ) , ζ ( β , γ ) ) ( π 2 γ 2 π )
ζ 2 ( β , γ ) = { ( ζ + ( β , γ ) , ζ ( β , γ ) ) ( 0 γ π 2 , 3 π 2 γ 2 π ) ( ζ + ( β , γ ) , ζ ( β , γ ) ) ( π 2 γ 2 π )
ζ ± ( β , γ ) ± λ Q min ( Q min 1 ) 2 π 2 1 sin β sin γ cos β 2 ± ( tan β 2 sin γ 1 )
D = λ Q min ( Q min 1 ) 2 π .
Q N 1 2 π c d l Arg [ E x ( r ) ] = 1 2 π x π d ϕ 1 Arg [ E x ( R N cos ϕ 1 , R N sin ϕ 1 , z ) ]
E ( r ) k 2 4 π ε 0 z m = ( i ) m J m ( k ρ a z ) e i m ϕ j = 1 N Λ j e i ( q m ) φ j
H ( r ) c k 4 π z m = ( i ) m J m ( k ρ a z ) e i m ϕ j = 1 N Π j e i ( q m ) φ j ,
[ Λ j ] x = s x ( 1 3 ρ 2 2 z 2 cos 2 ϕ 3 a 2 2 z 2 cos 2 φ j + 3 ρ a z 2 cos ϕ cos φ j ρ 2 2 z 2 sin 2 ϕ a 2 2 z 2 sin 2 φ j + ρ a z 2 sin ϕ sin φ j ) s y { ρ 2 z 2 cos ϕ sin ϕ + a 2 z 2 cos φ j sin φ j ρ a z 2 ( cos φ sin φ j + cos φ j sin φ ) } = s x ( 1 ρ 2 + a 2 z 2 ρ 2 2 z 2 cos 2 ϕ a 2 2 z 2 cos 2 φ j s + 3 ρ a z 2 cos ϕ cos φ j + ρ a z 2 sin ϕ sin φ j ) s y ( ρ 2 2 z 2 sin 2 ϕ + a 2 2 z 2 sin 2 φ j ρ a 2 z 2 sin ( ϕ + φ j ) )
j = 1 N [ Λ j ] x e i ( q m ) φ j = n = s x [ ( 1 ρ 2 + a 2 z 2 ) δ q m , n N + ρ 2 4 z 2 ( e i 2 ϕ + e i 2 ϕ ) δ q m , n N a 2 4 z 2 ( δ q m + 2 , n N + δ q m 2 , n N ) + 3 ρ a 4 z 2 ( e i ϕ + e i ϕ ) ( δ q m + 1 , n N + δ q m 1 , n N ) ρ a 4 z 2 ( e i ϕ e i ϕ ) ( δ q m + 1 , n N δ q m 1 , n N ) ] n = s y [ ρ 2 i 4 z 2 ( e i 2 ϕ e i 2 ϕ ) δ q m , n N + a 2 i 4 z 2 ( δ q m + 2 , n N δ q m 2 , n N ) ρ a i 2 z 2 ( e i ϕ δ q m + 1 , n N e i ϕ δ q m 1 , n N ) ]
j = 1 N [ Λ j ] y e i ( q m ) φ j = n = s y [ ( 1 ρ 2 + a 2 z 2 ) δ q m , n N + ρ 2 4 z 2 ( e i 2 ϕ + e i 2 ϕ ) δ q m , n N + a 2 4 z 2 ( δ q m + 2 , n N + δ q m 2 , n N ) 3 ρ a 4 z 2 ( e i ϕ e i ϕ ) ( δ q m + 1 , n N δ q m 1 , n N ) + ρ a 4 z 2 ( e i ϕ + e i ϕ ) ( δ q m + 1 , n N + δ q m 1 , n N ) ] n = s x [ ρ 2 i 4 z 2 ( e i 2 ϕ e i 2 ϕ ) δ q m , N + a 2 i 4 z 2 ( δ q m + 2 , n N δ q m 2 , n N ) ρ a i 2 z 2 ( e i ϕ δ q m + 1 , n N e i ϕ δ q m 1 , n N ) ]
j = 1 N [ Λ j ] z e i ( q m ) φ j = n = [ s x { ρ 2 z ( e i ϕ + e i ϕ ) δ q m , n N a 2 z ( δ q m + 1 , n N + δ q m 1 , n N ) } + s y { ρ i 2 z ( e i ϕ e i ϕ ) δ q m , n N a i 2 z ( δ q m + 1 , n N δ q m 1 , n N ) } ]
E x ( r ) k 2 4 π ε 0 z Q s x ( f Q ρ 2 + a 2 z 2 f Q + ρ a z 2 ( ( f Q + 1 + f Q 1 ) ) ) e i Q ϕ + Q s [ ρ a 2 z 2 f Q + 1 ρ 2 4 z 2 f Q a 2 4 z 2 f Q + 2 ] e i ( Q + 2 ) ϕ + Q s + [ ρ a 2 z 2 f Q 1 ρ 2 4 z 2 f Q a 2 4 z 2 f Q 2 ] e i ( Q 2 ) ϕ ,
E y ( r ) k 2 4 π ε 0 z Q s y ( f Q ρ 2 + a 2 z 2 f Q + ρ a z 2 ( ( f Q + 1 + f Q 1 ) ) ) e i Q ϕ i Q s [ ρ a 2 z 2 f Q + 1 ρ 2 4 z 2 f Q a 2 4 z 2 f Q + 2 ] e i ( Q + 2 ) ϕ i Q s + [ ρ a 2 z 2 f Q 1 ρ 2 4 z 2 f Q a 2 4 z 2 f Q 2 ] e i ( Q 2 ) ϕ ,
E z ( r ) k 2 4 π ε 0 z Q [ s ( ρ 2 z f Q a 2 z f Q + 1 ) e i ( Q + 1 ) ϕ + s + ( ρ 2 z f Q a 2 z f Q 1 ) e i ( Q 1 ) ϕ ] ,
Π j = 1 z 2 ( s y z 2 + s y ( ξ x 2 + ξ y 2 ) s x z 2 s x ( ξ x 2 + ξ y 2 ) s x ξ y z + s y ξ x z ) ,
j = 1 N [ Π j ] x e i ( q m ) φ j = s y n [ ( 1 ρ 2 + a 2 z 2 ) δ q m , n N + ρ a z 2 ( e i ϕ δ q m 1 , n N + e i ϕ δ q m + 1 , n N ) ] ,
j = 1 N [ Π j ] y e i ( q m ) φ j = s x n [ ( 1 ρ 2 + a 2 z 2 ) δ q m , n N + ρ a z 2 ( e i ϕ δ q m 1 , n N + e i ϕ δ q m + 1 , n N ) ] ,
j = 1 N [ Π j ] z e i ( q m ) φ j = i s n ( ρ 2 z e i ϕ δ q m , n N a 2 z δ q m + 1 , n N ) i s + n ( ρ 2 z e i ϕ δ q m , n N a 2 z δ q m 1 , n N ) ,
H x ( r ) = c k 4 π z s y e i Q ϕ n [ ( 1 ρ 2 + a 2 z 2 ) f Q + ρ a z 2 ( f Q 1 + f Q + 1 ) ] ,
H y ( r ) = c k 4 π z s x e i Q ϕ n [ ( 1 ρ 2 + a 2 z 2 ) f Q + ρ a z 2 ( f Q 1 + f Q + 1 ) ] ,
H z ( r ) = i c k 4 π z [ s e i ( Q + 1 ) ϕ n ( ρ 2 z f Q f Q + 1 ) s + e i ( Q 1 ) ϕ n ( ρ 2 z f Q a 2 z f Q 1 ) ] .
A s x ρ 2 e i 2 ϕ + s + = [ A cos β 2 ( x 2 y 2 + i 2 x y ) + cos β 2 + i sin β 2 cos γ sin β 2 sin γ ] = [ A cos β 2 ( x 2 y 2 ) + cos β 2 sin β 2 sin γ + i ( 2 A x y cos β 2 + sin β 2 sin γ ) ]
ϕ = tan 1 [ Im [ G ] Re [ G ] ] = tan 1 [ 2 A x y cos β 2 + sin β 2 cos γ A cos β 2 ( x 2 y 2 ) + cos β 2 sin β 2 sin γ ] = tan 1 [ 2 x y + 1 A tan β 2 cos γ ( x 2 y 2 ) + 1 A 1 A tan β 2 sin γ ] .
tan 1 [ y ζ y x ζ x + y + ζ y x + ζ x 1 y ζ y x ζ x y + ζ y x + ζ x ] = tan 1 y + ζ y x + ζ x + tan 1 y ζ y x ζ x
ζ x ζ y = 1 2 A tan β 2 cos γ ,
ζ y 2 ζ x 2 = 1 A 1 A tan β 2 sin γ .
ζ 1 ( β , γ ) = { ( ζ + ( β , γ ) , ζ ( β , γ ) ) ( 0 γ π 2 , 3 π 2 γ 2 π ) ( ζ + ( β , γ ) , ζ ( β , γ ) ) ( π 2 γ 2 π )
ζ 2 ( β , γ ) = { ( ζ + ( β , γ ) , ζ ( β , γ ) ) ( 0 < γ π 2 , 3 π 2 γ 2 π ) ( ζ + ( β , γ ) , ζ ( β , γ ) ) ( π 2 γ 2 π )
ζ ± ( β , γ ) ± λ Q min ( Q min 1 ) 2 π 2 1 sin β sin γ cos β 2 ± ( tan β 2 sin γ 1 )

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