Abstract

We investigate optical singularities in coaxial superpositions of two Laguerre–Gaussian (LG) modes with a common beam waist from the viewpoints of a general formulation of phase structure, experimental generation of various superposition beams, and evaluation of the generated beams’ fidelity. By applying a holographic phase-amplitude modulation scheme using a phase-modulation-type spatial light modulator, output fidelity beyond 0.960 was observed under several typical conditions. Additionally, an elliptic-type folded singularity, which provides a different class of phase structures from familiar helical singularities, was predicted and observed in a superposition involving two LG modes of both radially and azimuthally higher orders.

© 2010 Optical Society of America

1. INTRODUCTION

Phase singularities associated with Laguerre–Gaussian (LG) modes [1] have been popular subjects of topological structures in the optics field. Above all, the behavior of the phase singularity according to light propagation [2], sometimes referred to as an optical vortex (OV), is a central issue of singular optics. More recently, knot structures in OVs have been reported in both theoretical [3, 4] and experimental [5] works to advance the field of singular optics. In addition, the alignment, splitting, and merging of OVs in coaxial superpositions of radially lowest-order LG modes have been analyzed in the context of propagation properties of the LG-mode superposition [6, 7, 8, 9, 10, 11]. For the development of singular optics, it is significant to establish quantitative methods for generating and controlling the superposition of multiple LG modes.

So far, holographic methods using spatial light modulators (SLMs) have been widely applied to generate pure LG beams of superior quality [12, 13]. However, holographic methods modulating only phase structures of lights frequently cause deviation of beam patterns from ideal ones, especially for multi-mode superpositions. In fact, superpositions of LG modes were generated by optically adding component beams with a beam splitter [11] or by introducing another trick into the holographic method to enable amplitude modulation [5, 14, 15] in the studies of OVs. Nevertheless, there have been no reports on a universal and quantitative method for generating the multi-mode superpositions as well as the evaluation of output beam fidelity.

In this paper, we report a comprehensive study on the generation and evaluation of LG-mode superpositions involving formulation and classification of optical singularities in two-mode superpositions of general LG modes. Extending the formulations in previous studies [6, 7, 8, 9, 10, 11], we established general formulas for the position and phase structure of singularities in the superpositions involving radially higher-order LG modes. Various LG-mode superpositions were experimentally generated with the help of a simultaneous phase-amplitude modulation scheme [16] using a phase-modulation-type SLM. The output fidelity was also evaluated as a squared inner product between the observed and theoretical phase-amplitude distributions of an output light through observation of the output beam pattern and phase profile. As a result, we achieved the holographic generation of LG-mode superpositions with fidelity of more than 0.960 in some typical cases. We also demonstrate the experimental generation of elliptic-type folded singularities [17, 18], which can appear only in superpositions involving both radially and azimuthally higher-order LG modes.

2. FORMULATION OF SINGULARITIES IN TWO-LG-MODE SUPERPOSITIONS

Optical singularities are defined as zero points of light amplitudes. In this study, we investigate the structure of optical singularities in scalar complex amplitudes, which are obtained as solutions of the Helmholtz equation with scalar-wave and envelope-function approximations.

LG modes compose a complete set of orthogonal bases for paraxial light propagation in the cylindrical coordinates (r,ϕ,z), with z as a propagation direction. The complex amplitude of the radially pth- and azimuthally lth-order LG mode is defined as

upl(r,ϕ,z)=(1)pwz[2πp!(p+|l|)!]1/2(2r2wz2)|l|/2exp(r2wz2)Lp|l|(2r2wz2)eilϕ  exp[i2zkw02r2wz2]exp[i(2p+|l|+1)arctan(2zkw02)],
where wz denotes the beam radius at z and is defined as
wz=w0[1+(2zkw02)2]1/2,
with w0 as a beam radius parameter at the beam waist position (z=0).

In the following, we mainly study phase singularities in the cross-sectional beam profile at the beam waist position and omit the z coordinate in expressions. Moreover, r is rescaled as 2r/w0r for notational simplicity.

2A. Two-Mode Superposition of Azimuthally Same-Order LG Modes

The simplest examples of two-mode superposition are superpositions of two LG modes having identical azimuthal mode indices. Although this situation appears to be trivial, there are some suitable examples for noting the variety of beam structures in mode superpositions.

For example, we consider the following superposition of two LGpl=0 modes:

u(r,ϕ)=up=1l=0(r,ϕ)+up=0l=0(r,ϕ)=π1/2w0er2/2r2.
This complex amplitude presents the same beam pattern as the LGp=0l=2 mode at the beam waist; in particular, r=0 becomes an amplitude zero point.

It is obvious that the phase keeps a constant value around r=0 in Eq. (2). Phase singularities of lights are usually defined by the amplitude zero points because the phase becomes indefinite at the zero points. However, the present example gives a phase profile that is continuous and differentiable almost everywhere except the origin, which composes a measure zero set in the two-dimensional space. In this context, the zero point of Eq. (2) gives a nonsingular zero point. To determine whether an amplitude zero point provides the phase singularity in practice, direct investigation of phase structure around the zero point is required.

The following example shows another type of nonsingular zero point:

u(r,ϕ)=up=2l(r,ϕ)+[|l|+22(|l|+1)]1/2up=0l(r,ϕ)=[π(|l|+2)!]1/2w0er2/2r|l|eilϕ×(r|l|+2)2(r+|l|+2)2.
It should be noted that the ϕ-dependence of u(r,ϕ) is determined only by eilϕ, suggesting that the beam pattern is always circularly symmetric. Hence the zero points of u(r,ϕ) construct concentric circles, which resemble the radial discontinuity in pure radially higher-order LG beams. However, the zero line of Eq. (3), i.e., r=|l|+2, is accompanied with a different phase structure from those of ordinary radial phase discontinuities: u(r,ϕ) in Eq. (3) does not change its signature according to whether r exceeds the radial zero point, while the signature of light amplitude changes around the radial discontinuities in the pure radially higher-order LG modes. As a result, the phase structure of Eq. (3) is the same as that of LGp=0l modes, but the amplitude distribution is completely different.

From the experimental viewpoint, mode superpositions like Eqs. (2, 3) cannot be generated via holographic methods that attach only the phase distributions of desired beams to an incident light. Therefore, light modulation of both amplitude and phase is necessary for experimental generation of superposition beams.

2B. General Formula for Two-LG-Mode Superpositions

For a general coaxial superposition of two LG modes, a scalar complex amplitude is written as follows:

u(r,ϕ)=cp,lupl(r,ϕ)+cp,lupl(r,ϕ)=(2π)1/2er2/2w0[Cp,lr|l|Lp|l|(r2)eilϕ+Cp,lr|l|Lp|l|(r2)eilϕ].
Here, cp,l is a complex linear-combination coefficient, while Cp,l is a complex number satisfying Cp,l=(1)p[p!/(p+|l|)!]1/2cp,l. To analyze singularities in the cross section of a propagating light, we assume 0|l||l| and deform the above equation to obtain
u(r,ϕ)=(2π)1/2er2/2w0r|l|eilϕCp,leiδF(r,ϕ),
where we define a polynomial
F(r,ϕ)=r|l||l|Lp|l|(r2)ei[(ll)ϕ+δ]+ALp|l|(r2),
with a real constant A=|Cp,l|/|Cp,l| and a relative phase factor δ=arg(Cp,l/Cp,l). Here, we note that the phase factor δ works only to rotate the total beam profile, since the ϕ-dependence of the beam profile is determined by the exponential factor in Eq. (6) and δ only gives an offset to ϕ. Hence we can omit δ without loss of generality in the case of two-mode superposition.

Positions of phase singularities are determined from Eq. (5). Obviously, r=0 gives a phase singularity, which arises from the r|l|eilϕ term. The total phase change around the zero point is calculated by introducing polar coordinates (ε,ϕ) around the origin [Fig. 1a ] and substituting them into Eq. (5). Here, ε is assumed to be small to remove the influence of other zero points. The phase value Ψ at position (ε,ϕ) is given as

Ψ(ϕ)=arctan(Im  u(ε,ϕ)Re  u(ε,ϕ))=arctan(ε|l|[c0+O(ε)]sin(lϕ)ε|l|[c0+O(ε)]cos(lϕ))lϕ   (ε0),
where Re  u and Im  u indicate the real and imaginary parts of u, respectively, and c0 denotes the constant term in F(r,ϕ) (c0 always exists from the general property of Laguerre polynomials). In the above derivation, we choose suitable branches of arctan(x) considering its π arbitrariness and discontinuity at x=π/2,3π/2. From Eq. (7), the phase Ψ varies from zero to 2lπ according to the change of ϕ from zero to 2π around the origin, meaning that the central zero is attached with a helical phase singularity of order l [note that the complex amplitude of LGpl is defined by Eq. (1): the term eilϕ exhibits a counterclockwise helix of phasefront for positive l]. Such central phase singularities commonly appear in the coaxial superpositions of more than two LG modes having radial mode indices larger than unity.

In addition to the center singularity, zero points of F(r,ϕ) [Eq. (6)] determine other amplitude zero points. In this sense, F(r,ϕ) can be referred to as a zero-determining polynomial for the off-centered zero points. If (ρ,α) (ρ>0) is a zero point of F(r,ϕ), the positions of other zero points are given as follows in the polar coordinates:

(ρ,α+2πjll)   (j=0,1,,|ll|1),
which is derived from the symmetry of the exponential term in Eq. (6). These zero points are placed on a circle of radius ρ to divide the circle into |ll| equal parts. Generally, there can exist more than two different ρ’s depending on the definite form of the polynomial F(r,ϕ), but the angular positions of zero points always satisfy α=0 or π/(ll) (mod  2π/|ll|) as shown in Appendix A. In any case, the total beam profile of a two-mode superposition maintains an |ll|th-order circular symmetry.

Supposing that (ρ,α) (ρ>0) is one of the zero points of F(r,ϕ), the phase structure around the zero point is expressed in terms of the polar coordinates (ε,φ) around (ρ,α) while assuming that ε is an infinitesimal variable [Fig. 1b]. For notational simplicity, we choose the origin of angular variables ϕ and φ as the direction of the zero point (i.e., α). With these preparations, the following relations between (ρ,α) and (ε,φ) are satisfied:

r2=ρ2+ε2+2ρε  cos  φ,
cos  ϕ=12rρ(r2+ρ2ε2)=112ε2ρ2sin2φ+O(ε4/ρ4),
both of which are derived from the cosine rule. Above all, Eq. (9) leads to the relationship
sin  ϕ=ερsin  φ+O(ε3/ρ3)ϕ=ερsin  φ+O(ε3/ρ3),
which describes infinitesimal behavior of the angular variable around the zero point.

Substituting Eqs. (8, 10) into Eq. (5) and taking the lowest-order terms of Taylor series expansion with respect to ε/ρ, we obtain

u(ε,φ)=(2π)1/2eρ2/2w0ρ|l|Cp,l[ρF(ρ,α)cos  φi(ll)ρ|l||l|Lp|l|(ρ2)sin  φ]ερ+O(ε2/ρ2),
where
F(ρ,α)|rF(r,ϕ)|r=ρ,ϕ=α.
Regarding the ∓ signature in Eq. (11), − and + signatures express the cases corresponding to α=0 and π/(ll), respectively. In the above derivation, we made use of the fact
F(ρ,α)=0,
er2/2r|l|eilϕ=eρ2/2ρ|l|+O(ε/ρ).
Finally, the phase structure around an off-centered singularity in a superposition of two LG modes is generally expressed as
Ψ(φ)=arctan((ll)ρ|l||l|Lp|l|(ρ2)ρF(ρ,α)tan  φ)+O(ε/ρ),
which is obtained in a similar way to the derivation of Eq. (7).

2C. Two-Mode Superposition of Radially Lowest-Order LG Modes

As a direct example of the results in Subsection 2B, the singularities in superpositions of two p=0 LG beams are discussed in this subsection. Although parts of the discussion are commonly cited in the previous reports [6, 7, 8, 9, 10, 11], it is important to confirm the consistency between the previous and present studies. Since the structure of central singularity has already been elucidated in the previous subsection, we focus our attention on the analysis of off-centered singularities. Noting that Lp=0|l|(x)=1, an equation determining zero points becomes

F(r,ϕ)=r|l||l|ei(ll)ϕ+A=0.
Since r>0, it should be obvious that all solitary zero points appear at ϕ=α satisfying
cos(ll)α=sgn(A).
In the present case, |ll| zero points appear at (|A|1/(|l||l|),α+2πn/|ll|) (n=0,1,,|ll|), and there are no other zero points; this result meets the requirement imposed on the position of the zero point in Appendix A. Moreover, the above discussion on the position of singularity gives consistent expressions with those reported in the previous reports [6, 7, 8, 9, 10, 11].

To continue the analysis, we observe the phase change along a circular path around an off-centered zero point by calculating Ψ(ϕ) according to Eq. (15). Assuming A<0 for notational simplicity, we obtain α=0 and

F(ρ,α)=(|l||l|)ρ|l||l|   (ρ|l||l|=|A|),
which leads to
Ψ(φ)=arctan(ll|l||l|tan  φ)+O(ε/ρ).
To obtain the phase change along an infinitesimal loop around one of the off-centered zeros, it is necessary to take into account of the discontinuities of Ψ(φ) at φ=π/2 and 3π/2, i.e.,
Ψ(φ)φ=0φ=2π=Ψ(φ)0π/2+Ψ(φ)π/23π/2+Ψ(φ)3π/22πsgn(ll|l||l|)2π   (ε0).
This result clarifies that the order of singularity is ±1 for all off-centered singularities.

Equation (18) suggests another property regarding the phase structure of off-centered singularities. If |l| and |l| have the same signature, Eq. (18) is further simplified to give

Ψ(φ)=sgn(l)φ+O(ε/ρ),
meaning that the phase varies like a helix with a uniform pitch around the off-centered singularity for ll>0 [Fig. 2a ]. Contrarily, the pitch of the helical phase profile is nonuniform for ll<0 [Fig. 2b]. This property also affects the beam pattern around the off-centered singularity.

2D. Folded Singularity in Two-Mode Superposition of General LG Modes

Although we derived Eq. (15) for phase structure around zero points in superpositions of two LG modes, the formula fails in the case of F(ρ,α)=0, which can occur for superpositions involving radially higher-order LG modes. This failure can be repaired by taking further higher-order terms of the Taylor series expansion Eq. (11), although the phase behaves differently from popular helical phase singularities (sometimes referred to as “regular singularities”) to provide higher-order or folded singularities [17, 18]. However, the term “higher order” may cause confusion with the order of helical phase singularity, and thus we adopt the term “folded singularity” in the following discussion. It is possible to design the folded singularities as superpositions of three or more Hermite–Gaussian (HG) modes [6] (see Appendix B), but the realization of folded singularity in two-mode superposition is available only in the following example.

The simplest folded singularity is observed in a series of two-mode superpositions defined as

u(r,ϕ)=u1l(r,ϕ)+[|l|!(|l|+1)!]1/2(|l|+1)24u0l(r,ϕ),
when two integers l and l satisfy |l||l|=2. Following the discussion in Subsection 2B, Eq. (21) provides the following zero-determining polynomial:
F(r,ϕ)=r2(r2|l|1)ei(ll)ϕ+(|l|+1)24,
from which we notice that the zero points of F(r,ϕ) appear at
(ρ,α)=((|l|+1)/2,2πj/(ll))   (j=0,1,,|ll|1).
In the above example, F(r,ϕ) and F(r,ϕ) vanish simultaneously at the above zero points. Hence, the expression for the phase distribution, Eq. (13), which arises from the first-order term of Taylor series expansion, fails; and higher-order terms are required to extract the proper information of phase distribution.

To further investigate the phase structure of Eq. (21), we again perform infinitesimal analysis around the zero points. Substituting Eqs. (8, 10) into Eq. (21) and taking the lowest-order real and imaginary terms with respect to ε, we finally obtain the following expression in the Cartesian coordinates (x=ε  cos  ϕ and y=ε  sin  ϕ):

u(x,y)2(|l|+1)2x2(|l|+1)[(ll)2+1]y2+i(ll)(|l|+12)3/2y,
which is classified as the structure of elliptic singularity [18]. Figure 3 exhibits a schematic contour plot and bird’s-eye view of the phase distribution around the elliptic-type folded singularity.

The phase distribution of Eq. (24) takes the form of

Ψ(φ)=arctan((ll)2εa  sin  φ4a2cos2φ+b2sin2φ),
where we define positive real constants a=[(|l|+1)/2]1/2 and b=[(ll)2+1]1/2. Noting that the argument of inverse tangent in Eq. (25) is kept finite so far as ε0, Ψ(φ) remains in the same branch during the change of ϕ. This means that Ψ(φ) is a continuous single-valued function except at the zero point (pictured in Fig. 3), and that the phase change vanishes along arbitrary finite closed paths around the zero point. The existence of such folded singularities has been theoretically predicted [17, 18], but their definite construction in the paraxial regime provides a significant material for experimental verification.

Before proceeding to the next section, we analyze the additional phase singularities in the present example. The complex amplitude of Eq. (21) has a different type of amplitude zero points at (ρ±,α), where α=π(2j+1)/(ll) (j=0,1,,|ll|1) and

ρ±=|l|+1(12±12)1/2.
These zero points satisfy F(ρ±,α)=0 and F(ρ±,α)0 to be classified as regular phase singularities; however, the point (ρ,α) violates the requirement of ρ>0 for physically existing zero points. Consequently, there are |ll| zero points on a circle of radius r=ρ+, each of which zero points is a regular phase singularity of the first order.

3. EXPERIMENTAL RESULTS AND DISCUSSION

3A. Experimental Setup

As mentioned above, beam patterns of multi-mode superpositions often deviate from ideal ones when generated by holographic methods modulating only phase structures of lights. This problem becomes critical in the study of OVs, where the alignment of phase singularities is determined from intensity zero points in the cross-sectional beam profiles. To overcome the problem, holographic phase patterns were prepared following the simultaneous phase-amplitude modulation scheme reported in previous studies [16, 19]. The basic concept of the phase-amplitude modulation scheme is to superimpose a desired phase profile on a blazed phase pattern that is locally amplitude-modulated according to a desired amplitude profile. Following this scheme, we obtain arbitrary superpositions of fundamental modes as the first-order diffraction outputs. Moreover, the blazed phase pattern has a pitch of 4 pixels on the SLM surface, and the local modulation of the blazed phase pattern was performed based on the Kirk–Jones approach [19].

We performed experiments with a Mach–Zehnder interferometer as shown in Fig. 4 . A light from a diode-pumped green laser (wavelength: 532 nm) is modified to a uniform flat-wavefront beam through a spatial filter, a collimator lens, and an aperture. The uniform flat-wavefront beam, referred to as a “top-hat” beam in the following, is divided into two paths with a beam splitter (BS1), one of which is used as a reference light for the Mach–Zehnder interferometer. The transmitted top-hat beam is projected on the SLM, and the holographic output light is separated by BS2. The output light is transferred to an observation regime through two convex lenses (L1 and L2) having focal lengths of f1 and f2. L1 and L2 are aligned to construct a 4f system, where the output light is observed at the beam waist position using an image sensor. A pinhole is placed at the focal plane of the 4f system to pick up only the first-order diffraction output.

To generate a fringe-interference pattern, the output light is combined by BS3 with the reference light. Once the fringe-interference pattern is observed, a phase distribution of the output light can be obtained via the Fourier-transform method [20, 21, 22, 23]. We can also observe an output beam profile, i.e., distribution of squared light amplitude, by blocking the reference path with a shutter, meaning that both amplitude and phase information is available in the experiment. On the other hand, the theoretical complex amplitude can be determined from the observed beam profile via a fitting calculation while choosing a total energy scaling factor, a beam center position, and a beam waist size as fitting parameters [12, 13]. Finally, we can evaluate the fidelity of the observed beam as a squared absolute value of the inner product between the fitted theoretical complex amplitude and the observed complex amplitude.

3B. Two-Mode Superposition of Azimuthally Same-Order LG Beams

In considering pure modes, each element of LG (or HG) modes is attached with a different phase profile. However, we can construct different light beams with a common phase profile by introducing superpositions of azimuthally same-order LG modes, meaning that traditional phase-only modulation schemes are inadequate. Hence, such superpositions provide suitable examples for demonstrating the necessity of simultaneous phase-amplitude modulation in the holographic generation of general superposition beams.

The simplest example is an equal superposition of LGp=0l=0 and LGp=1l=0 modes in Eq. (2). Figure 5 presents the experimental and fitted theoretical profiles for this example. The beam pattern is similar to that of the LGp=0l=2 mode, while the phase profile shows a flat one, meaning that this amplitude zero point is accompanied with no phase singularity. The output fidelity can be evaluated from the inner product of experimental and theoretical light amplitudes to give 0.979. This result also suggests that amplitude-only modulation is available in practice with a phase-modulation SLM with the present simultaneous phase-amplitude modulation scheme.

We note that a multi-mode superposition does not always keep its beam pattern during propagation. In fact, z-directional propagation of the superposition of LGp=0l=0 and LGp=1l=0 can be described as follows with the help of Eq. (1):

|u(r,ϕ,z)|2=π21wz2e2rz2{rz4+2(1rz2)[1(1+16z2kw02)1/2]}{π21w02e2r02r04(z0)π21wz2e2rz2(rz21)2(z),}
where we define rz=r/wz. Equation (27) denotes the difference between beam profiles at the beam waist and Fraunhofer domain.

Another typical example is 3LGp=0l=1+2LGp=2l=1, which is obtained by setting l=1 in Eq. (3). Figure 6 shows the experimental and fitted profiles, between which we observe agreement regarding both squared amplitude and phase. Moreover, fidelity evaluated as a squared inner product of experimental and theoretical amplitudes is 0.963. Phase ambiguity appears as a circle in the phase profile, which is considered to arise from the intensity-zero ring, where the Fourier method cannot give a definite phase value.

Analytical description for the superposition of azimuthally same-order LG modes is extendable to the superposition of more than three LG modes. In the case of the N-mode superposition of radially lth-order LG beams, a general form of complex amplitude becomes

u(r,φ)=(2π)1/2er2/2w0r|l|eilϕF(r,ϕ),
with
F(r,ϕ)=Cp1,leiδ1Lp1|l|(r2)+Cp2,leiδ2Lp2|l|(r2)++CpN,leiδNLpN|l|(r2)
as a [2  max(p1,p2,,pN)]th-order polynomial. Here, F(r,ϕ) is independent of ϕ, meaning that u(r,ϕ)=0 is satisfied for arbitrary ϕ at r={ρRρ>0,F(ρ,ϕ)=0}, i.e., all zero points are distributed continuously on concentric circles.

Since F(r,ϕ) is a complex-coefficient polynomial, the fundamental theorem of algebra allows the following expression:

F(r,ϕ)(rρ1)n1(rρ2)n2(rρj)nj,
where ni (i=1,2,,j) is a positive integer and ρi is generally a complex number. If there exist positive real values (here we denote one of them as ρm) in the set of {ρi}, r=ρm draws an intensity-zero circle, around which the phase structure is different according to whether nm is odd or even. For odd nm, u(r,ϕ) suffers from π phase change when r passes through ρm. On the other hand, u(r,ϕ) keeps its phase around r=ρm for even nm. The latter case introduces a continuous phase distribution almost everywhere around r=ρm, suggesting the existence of nonsingular amplitude zero points.

3C. Two-Mode Superposition of Radially Lowest-Order LG Beams

In previous reports [15], two-mode superpositions of radially lowest-order LG beams were experimentally generated via a holographic method that modulates phase distributions while taking into account of amplitude distributions of output beams. To demonstrate the fidelity achieved by the present phase-amplitude modulation scheme for superpositions of two radially lowest-order LG modes, we demonstrate two examples proposed by Franke-Arnold et al. [15] for an optical atom trap in quantum simulation experiments.

Figure 7 shows beam patterns and phase profiles of an equal superposition of LGp=0l=11 and LGp=0l=3 modes, while Fig. 8 shows that of LGp=0l=5 and LGp=0l=3 modes. We observe a proper correspondence between the experimental and fitted profiles for both beam profiles and phase patterns, although the phase profiles appear to contain constant offsets. The calculated fidelity gives 0.967 and 0.971 for Figs. 7, 8, respectively, which indicates the superior ability of the present phase-amplitude modulation scheme for generation of superposition beams.

Additionally, the observed phase profiles in both Figs. 7, 8 present the splittings of central third-order regular phase singularities into three first-order regular phase singularities. The splittings of higher-order regular phase singularities are generally seen in azimuthally higher-order LG beams [5, 21, 23], and they appear to be reduced when aberration is properly removed. However, it is nearly impossible to completely reduce the splittings of higher-order regular phase singularities.

3D. Folded Singularity in Two-Mode Superposition of General LG Modes

Folded phase singularities appear specifically in the superpositions of two LG modes involving both radially and azimuthally higher-order ones. The simplest implementation of the folded phase singularity is given by Eq. (21) with l=0, i.e., the superposition is defined by LGp=1l=2+27/32LGp=0l=0.

Figure 9 shows the beam patterns and phase profiles of the superposition defined by LGp=1l=2+27/32LGp=0l=0. In Fig. 9, the phase structure of the two folded phase singularities, which are aligned horizontally in the observed plane, is properly reproduced. On the contrary, the phase structure of regular phase singularities, which are aligned vertically, reproduces the structure of azimuthally first-order regular phase singularity but exhibits deviation from the ideal one. Nevertheless, the fidelity evaluated from the inner product of experimental and theoretical amplitudes is 0.968, indicating agreement between the experimental result and the theoretical expression.

Practically speaking, the deviation of a phase profile in the regime of small light amplitude produces only a minor contribution to the experimentally derived output fidelity. It is necessary to enhance both precision of the phase-profile measurements and the dynamic range of the beam-pattern measurements to improve the quantitativity of the experimentally derived fidelity.

3E. Superposition of Three or More Radially Lowest-Order LG Beams

When the azimuthal mode index l of the component modes has the same signature, we can extend the previous discussion (Subsection 2C) to superpositions of more than three LGp=0l modes. Here we consider the N-mode superposition of LGp=0l modes with real combination coefficients {C0,li} and phase factors {δi} (i=0,1,,n). This case appears to be very limited but provides a universal strategy for aligning regular singularities on demand. In this case, a complex amplitude is expressed as

u(r,ϕ)=(2π)1/2er2/2w0[C0,l1r|l1|ei(l1ϕ+δ1)++C0,lNr|lN|ei(lNϕ+δN)],
where the polynomial between a pair of square brackets [referred to as F(r,ϕ) following the previous notation] determines the zero points of this light amplitude. For the present, we assume li>0 for all i=1,2,,N. Introducing a complex variable z=reiϕ and complex combination coefficients C̃0,li=C0,lieiδi, F(r,ϕ) can be rewritten as a complex-coefficient polynomial of the complex variable z:
F̃(z)=C̃0,l1zl1+C̃0,l2zl2++C̃0,lNzlN(zγ1)n1(zγ2)n2(zγj)nj,
where the final expression is again given from the fundamental theorem of algebra. Here, we note that all γi’s correspond to physically existing zero points at (|γi|,arg(γi)) (i=1,2,,j) in polar coordinates.

Recalling the discussion on the central phase singularity in the two-mode superposition of LG0l modes (Subsection 2B), the phase distribution is determined in a similar way: obviously, the zero point at z=γi is attached with a helical phase singularity of order ni. For example,

u(r,ϕ)=up=0l=4(r,ϕ)13up=0l=2(r,ϕ)+126up=0l=0(r,ϕ)=123πer2/2w0(reiϕ+1)2(reiϕ1)2
describes a complex amplitude providing two second-order helical phase singularities of at (r,ϕ)=(1,0) and (1,π).

Figure 10 presents the beam patterns and phase profiles of the superposition defined by Eq. (33). Splitting of the second-order phase singularity cannot be removed completely but is almost fully suppressed. In practice, light amplitudes are close to zero around the singularity, and thus the deviation of the phase around the singularity produces little effect on the total fidelity. The fidelity evaluated from the inner product of experimental and theoretical amplitudes is 0.968.

This example suggests versatile placement of arbitrary-order regular singularities as long as the order of singularities has the same signature. First, we establish the factorized form of light amplitude as in Eq. (32) according to the position (γi) and topological charge (ni) of the ith singularity. Then, we expand the factorized amplitude expression to obtain a polynomial in the form of Eq. (31). Finally, we obtain appropriate linear combination coefficients of component LG modes for the designed array of singularities. In the case of li<0, the above discussion can be applied similarly by defining z=reiϕ. The only difference is that the order of helical phase singularity becomes negative, i.e., the zero point corresponding to z=γi is attached with a helical phase singularity of the nith order in Eq. (32).

Before closing this section, we mention the numerical precision of fidelity. In this study, the output fidelity was described up to the third digit because statistical error appeared at the fourth digit. However, fidelity did not always become zero but showed a value of a few percent when calculating the inner product between an observed output-light amplitude and a theoretical amplitude orthogonal to the corresponding fitted amplitude. This is considered to be due to the precision and stability of measurements, and further study on the quantitativity of fidelity is required.

4. SUMMARY AND CONCLUSION

In this study, we established a general formulation for phase singularities in coaxial superpositions of two LG modes to reveal that phase structures different from regular phase singularities in pure LG modes can appear around amplitude zero points, e.g., nonuniform helical phase singularity, folded phase singularity, and nonsingular continuous phase distribution. Experimental generation of superposition beams was also reported using a holographic generation scheme enabling both amplitude and phase modulation. With the help of Fourier conversion of fringe interference patterns, the phase profiles of the generated superposition beams were also observed as well as the beam patterns. The mode purities of the experimentally generated superposition beams were derived as squared absolute values of inner products between the experimental amplitude-phase distributions and the theoretically fitted ones. Finally, mode output fidelity higher than 0.960 was obtained for all examples in the present study, suggesting the advantage of the present beam-generation scheme and measurement system.

APPENDIX A: REQUIREMENTS FOR THE POSITIONS OF OFF-CENTERED ZERO POINTS IN TWO-MODE SUPERPOSITION

In this appendix, we derive general properties of the positions of solitary zero points appearing at off-centered positions in the superposition of two LG modes. The off-centered zero points are defined as solutions of F(r,ϕ)=0 [Eq. (6)]. We start from the assumption that ei[(ll)ϕ+δ] is not a real number, i.e., ei[(ll)ϕ+δ]=eiη with η0,π. In this case, the complex-valued equation F(r,ϕ)=0 becomes two real-valued equations:

r|l||l|Lp|l|(r2)sin  η=0,
r|l||l|Lp|l|(r2)cos  η+ALp|l|(r2)=0.
Since sin  η0 is assumed, it follows that Lp|l|(r2)=0 at zero points from the first equation in Eq. (A1) (here the zero point at r=0 is out of our interest). Straightforwardly, we also obtain Lp|l|(r2)=0. If there exists r=ρ satisfying these two equations, F(ρ,ϕ)=0 is always true for arbitrary ϕ. This means that such zero points compose a circular line (radial singularity) rather than a set of solitary zero points. Conversely, solitary zero points can appear only at ϕ=δ/(ll) or (πδ)/(ll).

APPENDIX B: FOLDED SINGULARITIES IN SUPERPOSITIONS OF HERMITE–GAUSSIAN MODES

The folded singularities [17] are classified by asymptotic forms of light amplitudes around the singularities. In the discussion of Subsection 2D, a complex amplitude is asymptotically expressed as a second-order polynomial of x and y around an elliptic-type folded singularity. It is obvious that the asymptotic form of complex amplitude must be more than second order to satisfy the condition of folded singularity in Subsection 2D [F(r,ϕ)=F(r,ϕ)=0 at the singularity]. In this sense, the folded singularity discussed in the text is one of the “lowest-order” folded singularities. According to the normal form of a second-order polynomial in two-dimensional space, folded singularities can be classified into the following three types:

  1. Elliptic-type folded singularity:
    u(x,y)x2+y2+iy,
  2. Parabolic-type folded singularity:
    u(x,y)x2+iy,
  3. Hyperbolic-type folded singularity:
    u(x,y)x2y2+iy.

Following the construction of singularities in [6], the folded singularities are generally described as superpositions of HG modes, i.e., elements of a complete set of orthogonal basis for solutions of the paraxial light-propagation equation in the Cartesian coordinates (x,y,z). Here, we denote the (n,m)th-order HG mode as HGnm, where n and m indicate the mode indices regarding x and y directions, respectively. The three types of folded singularity are expressed as the following superpositions of HG modes (2xx,2yy):

  1. Elliptic-type folded singularity:
    HG20+HG02+2HG00+iHG01,
  2. Parabolic-type folded singularity:
    HG20+12HG00+iHG01,
  3. Hyperbolic-type folded singularity:
    HG20HG02+iHG01.

The above expressions suggest experimental implementation of folded singularities. On the other hand, it is reported that only the hyperbolic-type folded singularity is allowed for a solution of the Helmholtz equation [18].

To clarify whether the folded singularities are available in practice, we performed experimental generation of the folded singularities. Figures 11, 12, 13 show the beam patterns and phase profiles of elliptic-, parabolic-, and hyperbolic-type folded phase singularities, respectively. Although the experimentally observed phase profile of the hyperbolic singularity appears to contain a constant offset, we can observe the correspondence of beam patterns and phase profiles between experimental results and theoretical models. The output fidelity reaches the values of more than 0.975 for all folded singularities, suggesting the proper generation of all three types of folded singularities.

Various folded singularities are possible, e.g., singularities around which complex amplitudes are asymptotically written as polynomials of more than third order and singularities at which F(r,ϕ)=F(r,ϕ)==F(n)(r,ϕ)=0 (n3) is satisfied. Such “higher-order” folded singularities can appear in superpositions of more than three LG modes to present various specific properties that are out of the scope of this study. However, discussion in Subsection 2B should be extended straightforwardly to the analysis of higher-order folded singularities.

ACKNOWLEDGMENTS

The authors thank T. Hiruma, A. Hiruma, Y. Suzuki, T. Hara, and S. Ohsuka of Hamamatsu Photonics K.K. for their encouragement throughout this work, as well as H. Toyoda and T. Ohtsu for helpful discussions.

 

Fig. 1 Polar coordinates around zero points on the cross section of light propagation: (a) for the zero point at the beam center, i.e., the origin O, and (b) for the zero point placed at off-centered position P apart from O by distance ρ.

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Fig. 2 Schematic contour plots of the phase profiles described by Eq. (18) for (a) (l=2,l=1) and (b) (l=2,l=1).

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Fig. 3 (a) Schematic contour plot of the phase distribution for complex amplitude Eq. (21) around the neighborhood of a zero point indicated by Eq. (22) (l=2 and l=0). (b) Bird’s-eye view of (a).

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Fig. 4 Schematic diagram of experimental setup: SF, spatial filter; CL, collimator lens; BS13, beam splitters; L1,2, convex lenses with focal lengths of f1 and f2, respectively. Precise phase modulation is achieved by a liquid-crystal-on-silicon (LCOS)-type SLM. Optical path distance from the SLM surface to L1 is adjusted as f1 to construct the 4f system.

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Fig. 5 Beam patterns and phase profiles of equal superposition of LGp=0l=0 and LGp=1l=0 modes. Fidelity evaluated from the inner product of experimental and theoretical amplitudes is 0.979.

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Fig. 6 Beam patterns and phase profiles of superposition of 3LGp=0l=1+2LGp=2l=1 modes. Fidelity evaluated from the inner product of experimental and theoretical amplitudes is 0.963.

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Fig. 7 Beam patterns and phase profiles of equal superposition of LGp=0l=11 and LGp=0l=3 modes. Fidelity evaluated from the inner product of experimental and fitted theoretical amplitudes is 0.967.

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Fig. 8 Beam patterns and phase profiles of equal superposition of LGp=0l=5 and LGp=0l=3 modes. Fidelity evaluated from the inner product of experimental and fitted theoretical amplitudes is 0.971.

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Fig. 9 Beam patterns and phase profiles of superposition defined by LGp=1l=2+27/32LGp=0l=0. Mode purity evaluated from the inner product of experimental and theoretical amplitudes is 0.968.

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Fig. 10 Beam patterns and phase profiles of superposition of Eq. (33). Fidelity evaluated from the inner product of experimental and theoretical amplitudes is 0.968.

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Fig. 11 Beam patterns and phase profiles of elliptic-type phase singularity. Mode purity evaluated from the inner product of experimental and theoretical amplitudes is 0.979.

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Fig. 12 Beam patterns and phase profiles of parabolic-type phase singularity. Mode purity evaluated from the inner product of experimental and theoretical amplitudes is 0.976.

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Fig. 13 Beam patterns and phase profiles of hyperbolic-type phase singularity. Mode purity evaluated from the inner product of experimental and theoretical amplitudes is 0.975.

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1. L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003). [CrossRef]  

2. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974). [CrossRef]  

3. M. V. Berry and M. R. Dennis, “Knotting and unknotting of phase singularities: Helmholtz waves, paraxial waves and waves in 2+1 spacetime,” J. Phys. A 34, 8877–8888 (2001). [CrossRef]  

4. M. V. Berry and M. R. Dennis, “Knotted and linked phase singularities in monochromatic waves,” Proc. R. Soc. London, Ser. A 457, 2251–2263 (2001). [CrossRef]  

5. J. Leach, M. R. Dennis, J. Coutial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005). [CrossRef]  

6. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993). [CrossRef]  

7. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997). [CrossRef]  

8. F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre–Gaussian beams,” Opt. Commun. 250, 218–230 (2005). [CrossRef]  

9. J. Hamazaki, Y. Mineta, K. Oka, and R. Morita, “Direct observation of Gouy phase shift in a propagating optical vortex,” Opt. Express 14, 8382–8392 (2006). [CrossRef]   [PubMed]  

10. V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007). [CrossRef]  

11. 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, 9818–9827 (2009). [CrossRef]   [PubMed]  

12. Y. Ohtake, T. Ando, N. Fukuchi, N. Matsumoto, H. Ito, and T. Hara, “Universal generation of higher-order multiringed Laguerre–Gaussian beams by using a spatial light modulator,” Opt. Lett. 32, 1411–1413 (2007). [CrossRef]   [PubMed]  

13. 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]  

14. J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam,” Opt. Lett. 25, 191–193 (2000). [CrossRef]  

15. S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Öhberg, and A. S. Arnold, “Optical Ferris wheel for ultracold atoms,” Opt. Express 15, 8619–8625 (2007). [CrossRef]   [PubMed]  

16. T. Ando, Y. Ohtake, N. Matsumoto, T. Inoue, and N. Fukuchi, “Mode purities of Laguerre–Gaussian beams generated via complex-amplitude modulation using phase-only spatial light modulators,” Opt. Lett. 34, 34–36 (2009). [CrossRef]  

17. J. F. Nye, “Unfolding of higher-order wave dislocations,” J. Opt. Soc. Am. A 15, 1132–1138 (1998). [CrossRef]  

18. J. Adachi and G. Ishikawa, “Classification of phase singularities for complex scalar waves and their bifurcations,” Nonlinearity 20, 1907–1925 (2007). [CrossRef]  

19. J. P. Kirk and A. L. Jones, “Phase-only complex-valued spatial filter,” J. Opt. Soc. Am. 61, 1023–1028 (1971). [CrossRef]  

20. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982). [CrossRef]  

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

22. J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004). [CrossRef]  

23. C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, “High-resolution measurement of phase singularities produced by computer-generated holograms,” Opt. Commun. 242, 163–169 (2004). [CrossRef]  

References

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  1. L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).
    [CrossRef]
  2. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
    [CrossRef]
  3. M. V. Berry and M. R. Dennis, “Knotting and unknotting of phase singularities: Helmholtz waves, paraxial waves and waves in 2+1 spacetime,” J. Phys. A 34, 8877–8888 (2001).
    [CrossRef]
  4. M. V. Berry and M. R. Dennis, “Knotted and linked phase singularities in monochromatic waves,” Proc. R. Soc. London, Ser. A 457, 2251–2263 (2001).
    [CrossRef]
  5. J. Leach, M. R. Dennis, J. Coutial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
    [CrossRef]
  6. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
    [CrossRef]
  7. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
    [CrossRef]
  8. F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre–Gaussian beams,” Opt. Commun. 250, 218–230 (2005).
    [CrossRef]
  9. J. Hamazaki, Y. Mineta, K. Oka, and R. Morita, “Direct observation of Gouy phase shift in a propagating optical vortex,” Opt. Express 14, 8382–8392 (2006).
    [CrossRef] [PubMed]
  10. V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007).
    [CrossRef]
  11. 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, 9818–9827 (2009).
    [CrossRef] [PubMed]
  12. Y. Ohtake, T. Ando, N. Fukuchi, N. Matsumoto, H. Ito, and T. Hara, “Universal generation of higher-order multiringed Laguerre–Gaussian beams by using a spatial light modulator,” Opt. Lett. 32, 1411–1413 (2007).
    [CrossRef] [PubMed]
  13. 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]
  14. J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam,” Opt. Lett. 25, 191–193 (2000).
    [CrossRef]
  15. S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Öhberg, and A. S. Arnold, “Optical Ferris wheel for ultracold atoms,” Opt. Express 15, 8619–8625 (2007).
    [CrossRef] [PubMed]
  16. T. Ando, Y. Ohtake, N. Matsumoto, T. Inoue, and N. Fukuchi, “Mode purities of Laguerre–Gaussian beams generated via complex-amplitude modulation using phase-only spatial light modulators,” Opt. Lett. 34, 34–36 (2009).
    [CrossRef]
  17. J. F. Nye, “Unfolding of higher-order wave dislocations,” J. Opt. Soc. Am. A 15, 1132–1138 (1998).
    [CrossRef]
  18. J. Adachi and G. Ishikawa, “Classification of phase singularities for complex scalar waves and their bifurcations,” Nonlinearity 20, 1907–1925 (2007).
    [CrossRef]
  19. J. P. Kirk and A. L. Jones, “Phase-only complex-valued spatial filter,” J. Opt. Soc. Am. 61, 1023–1028 (1971).
    [CrossRef]
  20. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  21. I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
    [CrossRef]
  22. J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
    [CrossRef]
  23. C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, “High-resolution measurement of phase singularities produced by computer-generated holograms,” Opt. Commun. 242, 163–169 (2004).
    [CrossRef]

2009

2008

2007

Y. Ohtake, T. Ando, N. Fukuchi, N. Matsumoto, H. Ito, and T. Hara, “Universal generation of higher-order multiringed Laguerre–Gaussian beams by using a spatial light modulator,” Opt. Lett. 32, 1411–1413 (2007).
[CrossRef] [PubMed]

S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Öhberg, and A. S. Arnold, “Optical Ferris wheel for ultracold atoms,” Opt. Express 15, 8619–8625 (2007).
[CrossRef] [PubMed]

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007).
[CrossRef]

J. Adachi and G. Ishikawa, “Classification of phase singularities for complex scalar waves and their bifurcations,” Nonlinearity 20, 1907–1925 (2007).
[CrossRef]

2006

2005

J. Leach, M. R. Dennis, J. Coutial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre–Gaussian beams,” Opt. Commun. 250, 218–230 (2005).
[CrossRef]

2004

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[CrossRef]

C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, “High-resolution measurement of phase singularities produced by computer-generated holograms,” Opt. Commun. 242, 163–169 (2004).
[CrossRef]

2003

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).
[CrossRef]

2001

M. V. Berry and M. R. Dennis, “Knotting and unknotting of phase singularities: Helmholtz waves, paraxial waves and waves in 2+1 spacetime,” J. Phys. A 34, 8877–8888 (2001).
[CrossRef]

M. V. Berry and M. R. Dennis, “Knotted and linked phase singularities in monochromatic waves,” Proc. R. Soc. London, Ser. A 457, 2251–2263 (2001).
[CrossRef]

2000

1998

1997

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

1993

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

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

1982

1974

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

1971

Adachi, J.

J. Adachi and G. Ishikawa, “Classification of phase singularities for complex scalar waves and their bifurcations,” Nonlinearity 20, 1907–1925 (2007).
[CrossRef]

Allen, L.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).
[CrossRef]

Ando, T.

Arlt, J.

Arnold, A. S.

Barnett, S. M.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).
[CrossRef]

Basistiy, I. V.

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

Baumann, S. M.

Bazhenov, V. Yu.

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

Berry, M. V.

M. V. Berry and M. R. Dennis, “Knotting and unknotting of phase singularities: Helmholtz waves, paraxial waves and waves in 2+1 spacetime,” J. Phys. A 34, 8877–8888 (2001).
[CrossRef]

M. V. Berry and M. R. Dennis, “Knotted and linked phase singularities in monochromatic waves,” Proc. R. Soc. London, Ser. A 457, 2251–2263 (2001).
[CrossRef]

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

Coutial, J.

J. Leach, M. R. Dennis, J. Coutial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

Dändliker, R.

C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, “High-resolution measurement of phase singularities produced by computer-generated holograms,” Opt. Commun. 242, 163–169 (2004).
[CrossRef]

Dennis, M. R.

J. Leach, M. R. Dennis, J. Coutial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

M. V. Berry and M. R. Dennis, “Knotted and linked phase singularities in monochromatic waves,” Proc. R. Soc. London, Ser. A 457, 2251–2263 (2001).
[CrossRef]

M. V. Berry and M. R. Dennis, “Knotting and unknotting of phase singularities: Helmholtz waves, paraxial waves and waves in 2+1 spacetime,” J. Phys. A 34, 8877–8888 (2001).
[CrossRef]

Ellinas, D.

Flossmann, F.

F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre–Gaussian beams,” Opt. Commun. 250, 218–230 (2005).
[CrossRef]

Franke-Arnold, S.

Fukuchi, N.

Galvez, E. J.

Girkin, J. M.

Gorshkov, V. N.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

Hamazaki, J.

Hara, T.

Heckenberg, N. R.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

Herzig, H. P.

C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, “High-resolution measurement of phase singularities produced by computer-generated holograms,” Opt. Commun. 242, 163–169 (2004).
[CrossRef]

Ina, H.

Indebetouw, G.

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

Inoue, T.

Ishikawa, G.

J. Adachi and G. Ishikawa, “Classification of phase singularities for complex scalar waves and their bifurcations,” Nonlinearity 20, 1907–1925 (2007).
[CrossRef]

Ito, H.

Ivanovskyy, A. A.

C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, “High-resolution measurement of phase singularities produced by computer-generated holograms,” Opt. Commun. 242, 163–169 (2004).
[CrossRef]

Jones, A. L.

Kalb, D. M.

Khonina, S. N.

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007).
[CrossRef]

Kirk, J. P.

Kobayashi, S.

Kotlyar, V. V.

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007).
[CrossRef]

Leach, J.

S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Öhberg, and A. S. Arnold, “Optical Ferris wheel for ultracold atoms,” Opt. Express 15, 8619–8625 (2007).
[CrossRef] [PubMed]

J. Leach, M. R. Dennis, J. Coutial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[CrossRef]

Lembessis, V. E.

MacMillan, L. H.

Maier, M.

F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre–Gaussian beams,” Opt. Commun. 250, 218–230 (2005).
[CrossRef]

Malos, J. T.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

Matsumoto, N.

Mineta, Y.

Morita, R.

Nye, J. F.

J. F. Nye, “Unfolding of higher-order wave dislocations,” J. Opt. Soc. Am. A 15, 1132–1138 (1998).
[CrossRef]

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

Öhberg, P.

Ohtake, Y.

Oka, K.

Padgett, M. J.

S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Öhberg, and A. S. Arnold, “Optical Ferris wheel for ultracold atoms,” Opt. Express 15, 8619–8625 (2007).
[CrossRef] [PubMed]

J. Leach, M. R. Dennis, J. Coutial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[CrossRef]

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).
[CrossRef]

J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam,” Opt. Lett. 25, 191–193 (2000).
[CrossRef]

Rockstuhl, C.

C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, “High-resolution measurement of phase singularities produced by computer-generated holograms,” Opt. Commun. 242, 163–169 (2004).
[CrossRef]

Salt, M. G.

C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, “High-resolution measurement of phase singularities produced by computer-generated holograms,” Opt. Commun. 242, 163–169 (2004).
[CrossRef]

Schwarz, U. T.

F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre–Gaussian beams,” Opt. Commun. 250, 218–230 (2005).
[CrossRef]

Skidanov, R. V.

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007).
[CrossRef]

Soifer, V. A.

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007).
[CrossRef]

Soskin, M. S.

C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, “High-resolution measurement of phase singularities produced by computer-generated holograms,” Opt. Commun. 242, 163–169 (2004).
[CrossRef]

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

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

Takeda, M.

Vasnetsov, M. V.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

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

Wright, A. J.

Yao, E.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[CrossRef]

J. Mod. Opt.

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. A

M. V. Berry and M. R. Dennis, “Knotting and unknotting of phase singularities: Helmholtz waves, paraxial waves and waves in 2+1 spacetime,” J. Phys. A 34, 8877–8888 (2001).
[CrossRef]

New J. Phys.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[CrossRef]

J. Leach, M. R. Dennis, J. Coutial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

Nonlinearity

J. Adachi and G. Ishikawa, “Classification of phase singularities for complex scalar waves and their bifurcations,” Nonlinearity 20, 1907–1925 (2007).
[CrossRef]

Opt. Commun.

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

C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, “High-resolution measurement of phase singularities produced by computer-generated holograms,” Opt. Commun. 242, 163–169 (2004).
[CrossRef]

F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre–Gaussian beams,” Opt. Commun. 250, 218–230 (2005).
[CrossRef]

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

Proc. R. Soc. London, Ser. A

M. V. Berry and M. R. Dennis, “Knotted and linked phase singularities in monochromatic waves,” Proc. R. Soc. London, Ser. A 457, 2251–2263 (2001).
[CrossRef]

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

Other

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics, 2003).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Polar coordinates around zero points on the cross section of light propagation: (a) for the zero point at the beam center, i.e., the origin O, and (b) for the zero point placed at off-centered position P apart from O by distance ρ.

Fig. 2
Fig. 2

Schematic contour plots of the phase profiles described by Eq. (18) for (a) ( l = 2 , l = 1 ) and (b) ( l = 2 , l = 1 ) .

Fig. 3
Fig. 3

(a) Schematic contour plot of the phase distribution for complex amplitude Eq. (21) around the neighborhood of a zero point indicated by Eq. (22) ( l = 2 and l = 0 ). (b) Bird’s-eye view of (a).

Fig. 4
Fig. 4

Schematic diagram of experimental setup: SF, spatial filter; CL, collimator lens; BS 1 3 , beam splitters; L 1 , 2 , convex lenses with focal lengths of f 1 and f 2 , respectively. Precise phase modulation is achieved by a liquid-crystal-on-silicon (LCOS)-type SLM. Optical path distance from the SLM surface to L 1 is adjusted as f 1 to construct the 4 f system.

Fig. 5
Fig. 5

Beam patterns and phase profiles of equal superposition of LG p = 0 l = 0 and LG p = 1 l = 0 modes. Fidelity evaluated from the inner product of experimental and theoretical amplitudes is 0.979.

Fig. 6
Fig. 6

Beam patterns and phase profiles of superposition of 3 LG p = 0 l = 1 + 2 LG p = 2 l = 1 modes. Fidelity evaluated from the inner product of experimental and theoretical amplitudes is 0.963.

Fig. 7
Fig. 7

Beam patterns and phase profiles of equal superposition of LG p = 0 l = 11 and LG p = 0 l = 3 modes. Fidelity evaluated from the inner product of experimental and fitted theoretical amplitudes is 0.967.

Fig. 8
Fig. 8

Beam patterns and phase profiles of equal superposition of LG p = 0 l = 5 and LG p = 0 l = 3 modes. Fidelity evaluated from the inner product of experimental and fitted theoretical amplitudes is 0.971.

Fig. 9
Fig. 9

Beam patterns and phase profiles of superposition defined by LG p = 1 l = 2 + 27 / 32 LG p = 0 l = 0 . Mode purity evaluated from the inner product of experimental and theoretical amplitudes is 0.968.

Fig. 10
Fig. 10

Beam patterns and phase profiles of superposition of Eq. (33). Fidelity evaluated from the inner product of experimental and theoretical amplitudes is 0.968.

Fig. 11
Fig. 11

Beam patterns and phase profiles of elliptic-type phase singularity. Mode purity evaluated from the inner product of experimental and theoretical amplitudes is 0.979.

Fig. 12
Fig. 12

Beam patterns and phase profiles of parabolic-type phase singularity. Mode purity evaluated from the inner product of experimental and theoretical amplitudes is 0.976.

Fig. 13
Fig. 13

Beam patterns and phase profiles of hyperbolic-type phase singularity. Mode purity evaluated from the inner product of experimental and theoretical amplitudes is 0.975.

Equations (44)

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u p l ( r , ϕ , z ) = ( 1 ) p w z [ 2 π p ! ( p + | l | ) ! ] 1 / 2 ( 2 r 2 w z 2 ) | l | / 2 exp ( r 2 w z 2 ) L p | l | ( 2 r 2 w z 2 ) e i l ϕ   exp [ i 2 z k w 0 2 r 2 w z 2 ] exp [ i ( 2 p + | l | + 1 ) arctan ( 2 z k w 0 2 ) ] ,
w z = w 0 [ 1 + ( 2 z k w 0 2 ) 2 ] 1 / 2 ,
u ( r , ϕ ) = u p = 1 l = 0 ( r , ϕ ) + u p = 0 l = 0 ( r , ϕ ) = π 1 / 2 w 0 e r 2 / 2 r 2 .
u ( r , ϕ ) = u p = 2 l ( r , ϕ ) + [ | l | + 2 2 ( | l | + 1 ) ] 1 / 2 u p = 0 l ( r , ϕ ) = [ π ( | l | + 2 ) ! ] 1 / 2 w 0 e r 2 / 2 r | l | e i l ϕ × ( r | l | + 2 ) 2 ( r + | l | + 2 ) 2 .
u ( r , ϕ ) = c p , l u p l ( r , ϕ ) + c p , l u p l ( r , ϕ ) = ( 2 π ) 1 / 2 e r 2 / 2 w 0 [ C p , l r | l | L p | l | ( r 2 ) e i l ϕ + C p , l r | l | L p | l | ( r 2 ) e i l ϕ ] .
u ( r , ϕ ) = ( 2 π ) 1 / 2 e r 2 / 2 w 0 r | l | e i l ϕ C p , l e i δ F ( r , ϕ ) ,
F ( r , ϕ ) = r | l | | l | L p | l | ( r 2 ) e i [ ( l l ) ϕ + δ ] + A L p | l | ( r 2 ) ,
Ψ ( ϕ ) = arctan ( Im   u ( ε , ϕ ) Re   u ( ε , ϕ ) ) = arctan ( ε | l | [ c 0 + O ( ε ) ] sin ( l ϕ ) ε | l | [ c 0 + O ( ε ) ] cos ( l ϕ ) ) l ϕ     ( ε 0 ) ,
( ρ , α + 2 π j l l )     ( j = 0 , 1 , , | l l | 1 ) ,
r 2 = ρ 2 + ε 2 + 2 ρ ε   cos   φ ,
cos   ϕ = 1 2 r ρ ( r 2 + ρ 2 ε 2 ) = 1 1 2 ε 2 ρ 2 sin 2 φ + O ( ε 4 / ρ 4 ) ,
sin   ϕ = ε ρ sin   φ + O ( ε 3 / ρ 3 ) ϕ = ε ρ sin   φ + O ( ε 3 / ρ 3 ) ,
u ( ε , φ ) = ( 2 π ) 1 / 2 e ρ 2 / 2 w 0 ρ | l | C p , l [ ρ F ( ρ , α ) cos   φ i ( l l ) ρ | l | | l | L p | l | ( ρ 2 ) sin   φ ] ε ρ + O ( ε 2 / ρ 2 ) ,
F ( ρ , α ) | r F ( r , ϕ ) | r = ρ , ϕ = α .
F ( ρ , α ) = 0 ,
e r 2 / 2 r | l | e i l ϕ = e ρ 2 / 2 ρ | l | + O ( ε / ρ ) .
Ψ ( φ ) = arctan ( ( l l ) ρ | l | | l | L p | l | ( ρ 2 ) ρ F ( ρ , α ) tan   φ ) + O ( ε / ρ ) ,
F ( r , ϕ ) = r | l | | l | e i ( l l ) ϕ + A = 0.
cos ( l l ) α = sgn ( A ) .
F ( ρ , α ) = ( | l | | l | ) ρ | l | | l |     ( ρ | l | | l | = | A | ) ,
Ψ ( φ ) = arctan ( l l | l | | l | tan   φ ) + O ( ε / ρ ) .
Ψ ( φ ) φ = 0 φ = 2 π = Ψ ( φ ) 0 π / 2 + Ψ ( φ ) π / 2 3 π / 2 + Ψ ( φ ) 3 π / 2 2 π sgn ( l l | l | | l | ) 2 π     ( ε 0 ) .
Ψ ( φ ) = sgn ( l ) φ + O ( ε / ρ ) ,
u ( r , ϕ ) = u 1 l ( r , ϕ ) + [ | l | ! ( | l | + 1 ) ! ] 1 / 2 ( | l | + 1 ) 2 4 u 0 l ( r , ϕ ) ,
F ( r , ϕ ) = r 2 ( r 2 | l | 1 ) e i ( l l ) ϕ + ( | l | + 1 ) 2 4 ,
( ρ , α ) = ( ( | l | + 1 ) / 2 , 2 π j / ( l l ) )     ( j = 0 , 1 , , | l l | 1 ) .
u ( x , y ) 2 ( | l | + 1 ) 2 x 2 ( | l | + 1 ) [ ( l l ) 2 + 1 ] y 2 + i ( l l ) ( | l | + 1 2 ) 3 / 2 y ,
Ψ ( φ ) = arctan ( ( l l ) 2 ε a   sin   φ 4 a 2 cos 2 φ + b 2 sin 2 φ ) ,
ρ ± = | l | + 1 ( 1 2 ± 1 2 ) 1 / 2 .
| u ( r , ϕ , z ) | 2 = π 2 1 w z 2 e 2 r z 2 { r z 4 + 2 ( 1 r z 2 ) [ 1 ( 1 + 16 z 2 k w 0 2 ) 1 / 2 ] } { π 2 1 w 0 2 e 2 r 0 2 r 0 4 ( z 0 ) π 2 1 w z 2 e 2 r z 2 ( r z 2 1 ) 2 ( z ) , }
u ( r , φ ) = ( 2 π ) 1 / 2 e r 2 / 2 w 0 r | l | e i l ϕ F ( r , ϕ ) ,
F ( r , ϕ ) = C p 1 , l e i δ 1 L p 1 | l | ( r 2 ) + C p 2 , l e i δ 2 L p 2 | l | ( r 2 ) + + C p N , l e i δ N L p N | l | ( r 2 )
F ( r , ϕ ) ( r ρ 1 ) n 1 ( r ρ 2 ) n 2 ( r ρ j ) n j ,
u ( r , ϕ ) = ( 2 π ) 1 / 2 e r 2 / 2 w 0 [ C 0 , l 1 r | l 1 | e i ( l 1 ϕ + δ 1 ) + + C 0 , l N r | l N | e i ( l N ϕ + δ N ) ] ,
F ̃ ( z ) = C ̃ 0 , l 1 z l 1 + C ̃ 0 , l 2 z l 2 + + C ̃ 0 , l N z l N ( z γ 1 ) n 1 ( z γ 2 ) n 2 ( z γ j ) n j ,
u ( r , ϕ ) = u p = 0 l = 4 ( r , ϕ ) 1 3 u p = 0 l = 2 ( r , ϕ ) + 1 2 6 u p = 0 l = 0 ( r , ϕ ) = 1 2 3 π e r 2 / 2 w 0 ( r e i ϕ + 1 ) 2 ( r e i ϕ 1 ) 2
r | l | | l | L p | l | ( r 2 ) sin   η = 0 ,
r | l | | l | L p | l | ( r 2 ) cos   η + A L p | l | ( r 2 ) = 0.
u ( x , y ) x 2 + y 2 + i y ,
u ( x , y ) x 2 + i y ,
u ( x , y ) x 2 y 2 + i y .
HG 20 + HG 02 + 2 HG 00 + i HG 01 ,
HG 20 + 1 2 HG 00 + i HG 01 ,
HG 20 HG 02 + i HG 01 .

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