Generalized hyperboloid structures of polarization singularities
in Laguerre-Gaussian vector fields
T. H. Lu, Y. F. Chen,
*
and K. F. HuangDepartment of Electrophysics, National Chiao Tung University, Hsinchu, Taiwan 30050
共Received 10 August 2007; published 12 December 2007兲
We present the propagation-dependent polarization vector fields by use of an isotropic microchip laser with the longitudinal-transverse coupling and the entanglement of the polarization states. With the coherent super-position of orthogonal circularly polarized vortex modes which are made up of two Laguerre-Gaussian modes with different order, the experimental three-dimensional vector fields can be reconstructed analytically. From the theoretical analyses, the generalized structures of singularities such as V points, C lines, and L surfaces can be clearly demonstrated. Importantly, the projections of C lines on the transverse plane are found to form the intriguing petal structures.
DOI:10.1103/PhysRevA.76.063809 PACS number共s兲: 42.25.Ja, 02.40.Xx, 03.65.Vf, 42.60.Jf
I. INTRODUCTION
Singular optics which includes phase and polarization sin-gularities has become an important topic in modern physics to understand the physics of light关1,2兴. Recently, a consid-erable number of studies have been focused on experimental and theoretical results of phase singularities in scalar fields, known as wave front dislocations, such as optical vortices 关3兴, vortex lattices in superconductors 关4兴, quantum and mi-crowave billiards 关5兴, quantum Hall effects 关6兴, and linear and nonlinear optics关7–9兴. In addition to phase singularities in scalar fields, there are two types of polarization singulari-ties in vector fields of paraxial optical beams, known as wave front disclinations, to be discussed: Vector singularities and Stokes singularities关10兴. Vector singularities 共V points兲 are stationary points at which the orientation of the electric vec-tor of a linearly polarized vecvec-tor field becomes undefined. The importance of the vector singularities has been explored in the optical coherent waves with the representation of spa-tial structures and polarization states 关11–14兴. Recently, the complicated V point structure has been studied from the low-order关15兴 and high-order 关16兴 space-dependent linearly po-larized fields in transversely isotropic laser systems. How-ever, the mapping of vector field singularities onto the scalar field vortices leads to many new consequences关10兴.
The more general state of optical field with two orthogo-nal components is elliptically polarized state which leads to two special conditions of Stokes singularities: C lines, on which the field is circularly polarized and the orientations of the major and minor axes of the ellipse are undefined, and L surfaces, on which the field is linearly polarized and the handedness of the ellipse are undefined关17兴. In paraxial op-tics, C lines present as isolated points in the observation plane and L surfaces present as continuous lines, L lines, which separate regions of right-handed and left-handed po-larization关18–20兴. With the experimental results of micro-waves 关10兴 and optical waves 关21–24兴, the importance of polarization singularities of elliptically polarized fields has been revealed.
Recently, a diode-pumped microchip laser has been em-ployed to generate the propagation-dependent polarization vector fields with the longitudinal-transverse coupling and the entanglement of the polarization states 关25兴. However, the characteristics of polarization singularities are revealed with the theoretical wave representation only in the condition of single-ring wave pattern. In this work, we demonstrate the general expression of the multiple structures of polarization singularities embedded in the multiring vector wave patterns. With the coherent superposition of orthogonal circularly po-larized vortex modes composed of two Laguerre-Gaussian 共LG兲 modes with different order, the general structures of the polarization singularities are systematically analyzed. The theoretical analyses reveal that the projection of the C lines on the transverse plane displays the intriguing petal struc-tures. From the analytical results of the singularities, the po-larization states of the experimental LG vector fields under propagation can be clearly demonstrated.
II. EXPERIMENTAL SETUP AND RESULTS In this experiment, the laser system was a diode-pumped Nd: YVO4microchip laser and the resonator was formed by a spherical mirror and a gain medium. The spherical mirror was a 10 mm radius-of-curvature concave mirror with anti-reflection coating at the pumping wavelength on the entrance face 共R⬍0.2%兲, high-reflection coating at lasing wave-length 共R⬎99.8%兲, and high-transmission coating at the pumping wavelength on the other surface 共T⬎95%兲. The gain medium was a 2.0 at. % Nd: YVO4 crystal with the length of 2 mm. The laser crystal was precisely cut along the
c axis for high-level transverse isotropy 关26兴. One planar
surface of the laser crystal was coated for antireflection at the pumping and lasing wavelengths; the other surface was coated to be an output coupler with the reflectivity of 99%. The pump source was a 1 W 808 nm fiber-coupled laser di-ode with a core diameter of 100m and a numerical aper-ture of 0.2. A focusing lens was used to reimage the pump beam into the laser crystal. The pump spot radius was con-trolled to be in the range 50– 200m. The effective cavity length was set in the range 9.6– 9.9 mm to form a nearly *FAX:共886-35兲 725230; [email protected]
hemispherical resonator, in which the fundamental cavity mode size was approximately 20m. Since the pump-to-mode size ratio was significantly greater than unity, a variety of high-order transverse modes could be generated.
The pump power was controlled to be near lasing thresh-old to maintain the single mode in the cavity to explore the characteristic of polarization. To measure the far-field pat-tern, the output beam was directly projected on a paper screen at a distance of⬃50 cm from the rear cavity mirror and the scattered light was captured by a digital camera. Figures 1共a兲–1共c兲 show three experimental far-field trans-verse patterns which are represented as flower modes with different transverse radial index. Not only the single-ring but also the multiring is the general transverse mode formed by the propagation-dependent polarization states to prevail in the laser cavity. The fundamental mode is not excited be-cause the pump-to-mode size ratio is significantly greater than unity and then the lasing threshold of fundamental mode is higher than that of high-order transverse modes. A micro-scope objective lens mounted on a translation stage was used to reimage the tomographic transverse patterns at different propagation position onto a CCD camera. Figure2 displays the polarization-resolved transverse patterns at three different propagation positions: z = 0, z = zR, and zzR, where the ZR is the Rayleigh range and ZR= 1.28 mm. It can be found that the polarization-resolved patterns represent as an azimuthally polarized flower mode at the beam waist共z=0兲, whereas it turns out to be like a radially polarized flower mode at the far field共zzR兲. Moreover, the polarization state at z=zR was
confirmed to behave as a circularly polarized flower mode by use of a quarter-wave plate. The polarization-resolved trans-verse modes formed by the three-dimensional共3D兲 coherent vector field provide an important aspect to explore the phys-ics of polarization singularities. It is worthwhile to mention that the lasing modes are propagation-dependent polarization vector fields which are generated from the nearly hemi-spherical cavity. The following analysis will substantiate that the longitudinal-transverse coupling with the entanglement of the polarization states leads to the formation of 3D coher-ent vector fields in the isotropic laser cavity. Therefore, the generalized structures of polarization singularities in coher-ent vector fields with longitudinal-transverse coupling can be clearly revealed with the theoretical analysis.
III. ANALYTICAL WAVE FUNCTIONS FOR EXPERIMENTAL PATTERNS AND POLARIZATION SINGULARITIES
According to the lasing modes represented as flower modes in the transverse patterns, we start from the LG mode to be the basis of the experimental results. The wave function of LG mode with longitudinal index s, transverse radial in-dex p, and transverse azimuthal inin-dex l in cylindrical coor-dinates 共,, z兲 is given by ⌿p,l,s共,, z兲=eil⌽p,l,s共, z兲, where ⌽p,l,s共,z兲 =
冑
2p! 共p + 兩l兩兲! 1 w共z兲冉
冑
2 w共z兲冊
兩l兩 Lp兩l兩冉
22 w共z兲2冊
⫻exp冋
− 2 w共z兲2册
exp再
− ikp,l,sz冋
1 + 2 2共z2+ z R 2兲册
冎
⫻exp关i共2p + 兩l兩 + 1兲G共z兲兴 共1兲 where w共z兲=w0冑
1 +共z/zR兲2, w0 is the beam radius at the waist, zR=w02/ is the Rayleigh range, L p
l共·兲 are the associ-ated Laguerre polynomials, kp,l,s is the wave number, and G共z兲=tan−1共z/zR兲 is the Gouy phase. In the resonator with the effective length L, the wave number kp,l,s is given by
kp,l,sL =关s+共2p+兩l兩兲共⌬fT/⌬fL兲兴, where ⌬fL= c/2L is the
(a) (b) (c)
FIG. 1.共Color online兲 Experimental far-field transverse patterns with different radial index p and azimuthal index l: 共a兲 共p,l兲 =共1,66兲; 共b兲 共p,l兲=共2,41兲; 共c兲 共p,l兲=共7,100兲. 0 z R z z R zz
FIG. 2. 共Color online兲 Polarization-resolved transverse patterns for the experimental re-sult with the index 共p,l兲=共1,39兲 at three different propagation po-sitions: z = 0, z = zR, and zzR, where zR is the Rayleigh range.
The arrows indicate the transmis-sion axis of the polarizer.
longitudinal mode spacing and ⌬fT is the transverse mode spacing. It has been verified 关27兴 that the longitudinal-transverse coupling and mode-locking effect can lead to the frequency locking among different transverse modes with the help of different longitudinal orders when the ratio⌬fT/⌬fL is close to a simple fractional. As a result, the configuration of the nearly hemispherical cavity refers to be ⌬fT/⌬fL ⬇1/2, and the group of LG modes ⌿p,l+2k,s−k共,, z兲, with
k = 0 , 1 , 2 , 3 . . . , forms an important family of frequency
de-generate states. With LG modes as the basis, the experimen-tal vector fields can be decomposed into a coherent superpo-sition of orthogonal circularly polarized helical modes E៝ = ER共,, z兲aˆR+ EL共,, z兲aˆL, where
ER共,,z兲 = 关⌿p,−共l+1兲,s−1共,,z兲 − ⌿p,l−1,s共,,z兲兴/
冑
2, 共2兲EL共,,z兲 = 关⌿p,l+1,s−1共,,z兲 − ⌿p,−共l−1兲,s共,,z兲兴/
冑
2, 共3兲 and aˆR=共aˆx− iaˆy兲/冑
2 and aˆL=共aˆx+ iaˆy兲/冑
2 are the helical ba-sis unit vectors for the right- and left-handed circulation po-larizations, respectively. Figure 3 displays the numerically reconstructed patterns for the experimental results shown in Fig.2. There is a good agreement between the reconstructed and experimental patterns. From this point of view, the cir-cularly polarized vortex modes indeed play an important role to form the propagation-dependent polarization vector fields. Equations共2兲 and 共3兲 indicate that each circularly polarized component of the vector fields is composed of two LG modes with different order. It is worthwhile to mention that the frequency locking of two LG modes with different azi-muthal orders arises from the longitudinal-transverse cou-pling in a nearly hemispherical cavity.After some algebra, Eqs.共2兲 and 共3兲 for the general con-dition can be simplified as
ER共,,z兲 = 关˜l2e−i2lei2G共z兲− 1兴e i共l−1兲⌽ p,l−1,s共,z兲/
冑
2, 共4兲 EL共,,z兲 = 关˜l 2ei2lei2G共z兲− 1兴e−i共l−1兲⌽
p,l−1,s共,z兲/
冑
2, 共5兲 where ˜l2=关冑
2/w共z兲兴2关1/冑
共l + p兲共l + p + 1兲兴 ⫻冋
Ll+1p冉
22 w共z兲2冊
冒
Lp l−1冉
22 w共z兲2冊
册
. 共6兲In the basis of circular polarizations, the condition for left-handed and right-left-handed C point loci can be given by
ER共,, z兲=0 and EL共,, z兲=0, respectively. For the paraxial 3D vector fields, the trajectories of C singularities can be expressed as the parametric curves with z as a vari-able. In addition to the central singularity at the origin, the expression in the bracket of Eq.共4兲 indicates the left-handed
C point trajectories are determined by the following two
con-ditions: 共1兲 ˜l2= 1 and e−i2lei2G共z兲= 1; 共2兲 ˜
l
2= −1 and
e−i2lei2G共z兲= −1. In general, there are 2p + 1 solutions of the
exact radius which the C points are symmetrically embedded in. Note that for p = 0 there are 2l peripheral left-handed C points symmetrically arrayed on a circle of radius 0 =
冑
冑
l共l+1兲w共z兲/冑
2 at angles m=关G共z兲+m兴/l with m = 0 , 1 , 2 , . . . , 2l − 1 and 2l peripheral right-handed C points on the same circle of radius at angles m=关−G共z兲+m兴/l with m = 0 , 1 , 2 , . . . , 2l − 1. The brief case of p = 0 has been verified to be in good agreement with experimental results 关25兴. Besides p=0, the theoretical solution of radius with radial index p can be solved analytically for the cases p = 1 – 3. Further, we analyzed the case of pⱖ1. For p=1, the three solutions of radius can be expressed analytically:FIG. 3. Numerically recon-structed patterns for the experi-mental results shown in Fig.2.
1= 1 2
冑
2 + l +冑
共l + 1兲共l + 2兲 −冑
6 + 7l + 2l 2+ 4冑
共l + 1兲共l + 2兲 − 2l冑
共l + 1兲共l + 2兲w共z兲, 共7兲 2= 1 2冑
2 + l +冑
共l + 1兲共l + 2兲 +冑
6 + 7l + 2l 2+ 4冑
共l + 1兲共l + 2兲 − 2l冑
共l + 1兲共l + 2兲w共z兲, 共8兲 3= 1 2冑
2 + l −冑
共l + 1兲共l + 2兲 +冑
6 + 7l + 2l 2− 4冑
共l + 1兲共l + 2兲 + 2l冑
共l + 1兲共l + 2兲w共z兲. 共9兲On the one hand, there are 2l peripheral left-handed and 2l peripheral right-handed C points symmetrically arrayed at anglesm=关G共z兲+m兴/l and m=关−G共z兲+m兴/l, respec-tively, with m = 0 , 1 , 2 , . . . , 2l − 1 according to the circle ra-dius in the situation of˜l2= 1, and, on the other hand, there are 2l peripheral left-handed and 2l peripheral right-handed
C points symmetrically arrayed at anglesm=关2G共z兲+共2m + 1兲兴/2l and m=关−2G共z兲+共2m+1兲兴/2l, respectively, with m = 0 , 1 , 2 , . . . , 2l − 1 according to the circle radius in the situation of ˜l2= −1. As aresult, there are 2l共2p+1兲 left-handed C points and 2l共2p+1兲 right-left-handed C points embed-ded in the polarization-dependent vector field. Therefore, C lines singularities embedded in the propagation-dependent polarization vector field with p = 0 form the hyperboloid structure. The theoretical results of the view from the propa-gation direction to the beam waist of the general structures of the C lines singularities with p = 1 – 3 and l = 1 – 6 are repre-sented in Figs.4–6. The different color of C line singularities represents the different allowable circle of radius according to the radial index p of the transverse modes. Therefore, the different radial position of the C line singularity with the same color implies the different propagation position of the propagation-dependent polarization vector field. The mini-mum of the radial position represents the beam waist and the maximum of the radial position represents the far field.
Another important and interesting feature is that the ex-perimental 3D polarization vector fields at the beam waist and far field which are made up of two linearly polarized modes with different spatial structures. For the general con-dition, the experimental vector field can be given by E៝ = Ex共,, p兲xˆ+Ey共,, p兲yˆ, where
Ex共,,z兲 = ⌽p,l−1,s共,0兲兵˜l 2 ei2G共z兲cos关共l + 1兲兴 − cos关共l − 1兲兴其
冑
2 共10兲 and Ey共,,z兲 = ⌽p,l−1,s共,0兲兵˜l 2 ei2G共z兲sin关共l + 1兲兴 + sin关共l − 1兲兴其/冑
2. 共11兲 The transverse vector field at beam waist and far field can be verified to possess the V point singularities that are generally described in terms of the field of the angle function⌰共x,y兲 = arctan共Ey/Ex兲 关10,28兴, where Exand Eyare the scalar com-ponents of the vector field along the x and y axes. The vor-tices of⌰共x,y兲 are the vector singularities at which the ori-entation of the electric vector is undefined. Figures7–9show the angle pattern⌰共x,y兲 of the numerical vector field at the far field. Consistently, the V point singularities are right at(a) (b) (c)
(d) (e) (f)
FIG. 4.共Color online兲 Structure of the C line singularities of the theoretical vector field from the view of propagation direction to the beam waist with the same radial index p = 0 and different azimuthal index l: 共a兲 共p,l兲=共0,1兲; 共b兲 共p,l兲=共0,2兲; 共c兲 共p,l兲=共0,3兲; 共d兲 共p,l兲=共0,4兲; 共e兲 共p,l兲=共0,5兲; 共f兲 共p,l兲=共0,6兲.
(a) (b) (c)
(d) (e) (f)
FIG. 5.共Color online兲 Structure of the C line singularities of the theoretical vector field from the view of propagation direction to the beam waist with the same radial index p = 1 and different azimuthal index l: 共a兲 共p,l兲=共1,1兲; 共b兲 共p,l兲=共1,2兲; 共c兲 共p,l兲=共1,3兲; 共d兲 共p,l兲=共1,4兲; 共e兲 共p,l兲=共1,5兲; 共f兲 共p,l兲=共1,6兲.
the intersections of the right-handed and left-handed C lines shown in Fig.4–6. With Eqs. 共10兲 and 共11兲 and some alge-bra, there are 2l peripheral V points symmetrically arrayed at angles m= m/l on a circle of radius of the condition
˜l2= 1 and 2l peripheral V points symmetrically arrayed at anglesm=共2m+1兲/2l on a circle of radius of the con-dition˜l2= −1 with m = 0 , 1 , 2 , . . . , 2l − 1 at the beam waist in addition to the central singularity at the origin. The Gouy phase plays a vital role to transform the singularities between
V points and C points under propagation of the 3D vector
field. Consequently, there are 2l peripheral V points sym-metrically arrayed at anglesm=共2m+1兲/2l on a circle of radius of the condition˜l2= 1 and 2l peripheral V points symmetrically arrayed at angles m= m/l on a circle of radiusof the condition˜l2= −1 with m = 0 , 1 , 2 , . . . , 2l − 1 at the far field in addition to the central singularity at the origin. Intriguingly, each peripheral V point with the winding num-ber of 1 is transformed to two different handed C points with
the winding number of 1/2. Apparently, the winding num-bers are conserved during the singularity transformation and under the vector field propagation关29兴. Figure10depicts the characteristics of the C line and V point singularities of an experimental result. It can be found that the structure of C lines shown in Fig. 10共b兲 forms the hyperboloid with multilayer in the radial direction. The theoretical pattern of the view from the propagation direction to the beam waist of the structures of the C lines singularities forms a kind of fascinating petal pattern corresponding to the experimental transverse pattern shown in Fig.10共a兲.
Besides C line and V point singularities, there is L surface singularity embedded in the propagation-dependent polariza-tion vector fields with the longitudinal-transverse coupling and the entanglement of the polarization states. The L singu-larities can be determined by the conditions 兩ER兩2=兩EL兩2. With Eqs. 共4兲 and 共5兲, it can be found that there are 4l L surfaces on the -z plane with the azimuthal angles at n = n/共2l兲, where n=0,1,2, ... ,4l−1. Figure11displays the
(a) (b) (c)
(d) (e) (f)
FIG. 6.共Color online兲 Structure of the C line singularities of the theoretical vector field from the view of propagation direction to the beam waist with the same radial index p = 2 and different azimuthal index l: 共a兲 共p,l兲=共2,1兲; 共b兲 共p,l兲=共2,2兲; 共c兲 共p,l兲=共2,3兲; 共d兲 共p,l兲=共2,4兲; 共e兲 共p,l兲=共2,5兲; 共f兲 共p,l兲=共2,6兲.
(a) (b) (c)
(d) (e) (f)
FIG. 7.共Color online兲 Numerical patterns of the angle function at the far field of the same radial index p = 0 and different azimuthal index l: 共a兲 共p,l兲=共0,1兲; 共b兲 共p,l兲=共0,2兲; 共c兲 共p,l兲=共0,3兲; 共d兲 共p,l兲=共0,4兲; 共e兲 共p,l兲=共0,5兲; 共f兲 共p,l兲=共0,6兲.
(a) (b) (c)
(d) (e) (f)
FIG. 8.共Color online兲 Numerical patterns of the angle function at the far field of the same radial index p = 1 and different azimuthal index l: 共a兲 共p,l兲=共1,1兲; 共b兲 共p,l兲=共1,2兲; 共c兲 共p,l兲=共1,3兲; 共d兲 共p,l兲=共1,4兲; 共e兲 共p,l兲=共1,5兲; 共f兲 共p,l兲=共1,6兲.
(a) (b) (c)
(d) (e) (f)
FIG. 9.共Color online兲 Numerical patterns of the angle function at the far field of the same radial index p = 2 and different azimuthal index l: 共a兲 共p,l兲=共2,1兲; 共b兲 共p,l兲=共2,2兲; 共c兲 共p,l兲=共2,3兲; 共d兲 共p,l兲=共2,4兲; 共e兲 共p,l兲=共2,5兲; 共f兲 共p,l兲=共2,6兲.
vector and polarization singularities with the analytical rep-resentation of the transverse pattern with the radial and azi-muthal index共p,l兲 to be 共0,4兲 from the view of the propa-gation direction to the beam waist. The different radial position of the figure implies the different propagation posi-tion of the 3D polarizaposi-tion vector field. The minimum of the radial position represents the beam waist and the maximum of the radial position represents the far field. From the ana-lytical structures of the singularities, the polarization state of the experimental 3D vector field under propagation can be clearly revealed. From the loci of C lines, it can be confirmed that L surfaces separate regions of right-handed and left-handed polarization and V points locate on the intersection of right-handed and left-handed polarization.
It is worthwhile to give a more detailed comparison be-tween theory and experimental results. The present hyperbo-loid structures of polarization singularities are directly de-rived from Eqs.共2兲 and 共3兲 in which the two different LG modes are superposed with equal amplitude. For general cases of experimental results, however, the amplitude of the two LG modes can be somewhat different. Nevertheless, with the same theoretical analysis, the distributions of the polarization singularities can be certainly found to be topo-logically invariant. In other words, the hyperboloid structure of polarization singularities represents a characteristic feature of resonant laser modes emitted from degenerate cavities. On the other hand, more complicated phase singularities, such as link and knot structures, can be produced by using a Gauss-ian laser beam illuminating a hologram or a phase modulator 关30,31兴. However, these complex structures are not at all related to the fundamental aspects of laser resonators.
The present polarization singularities are explored based on the paraxial approximation in which the longitudinal elec-tric field is neglected. For a rigorous point of view, it is more appropriate to analyze the experimental polarization singu-larities with the full 3D electric field. Recently, Berry 关18兴 has confirmed that the separations between two singularities obtained with the paraxial approximation and the full 3D fields are generally much smaller than the wavelength. Therefore, the present findings are almost not affected by neglecting the longitudinal field.
IV. CONCLUSION
In conclusion, we have used an isotropic microchip laser to generate the propagation-dependent polarization vector fields with the longitudinal-transverse coupling and the en-tanglement of the polarization states. It is found that the ex-perimental 3D coherent vector fields can be reconstructed by the orthogonal circularly polarized vortex mode which is made up of two Laguerre-Gaussian共LG兲 modes with differ-ent order. With the analytical represdiffer-entation, the general structures for the singularities of the C lines, V points, and L surfaces can be systematically analyzed. In general, there are 2p + 1 solutions of the radius which the C lines and V points are symmetrically embedded in and the theoretical solutions of the radius can be represented analytically for the cases p = 0 – 3. Importantly, the theoretical analyses reveal that the trajectories of the C lines projected on the transverse plane displays the intriguing petal structures. Furthermore, the po-larization states of the experimental LG vector fields under propagation can be clearly demonstrated. The generalized structures of the polarization singularities in coherent vector fields may provide some useful insights into the nature of the waves.
FIG. 11. 共Color online兲 Diagram of the representation of the polarization state under propagation corresponding to the singulari-ties of C lines共blue line兲, V points 共white points at far field and pink points at beam waist兲, and L surfaces 共yellow dashed lines兲.
(a) (b) (c) x y z (d)
FIG. 10. 共Color online兲 共a兲 Experimental far-field pattern with radial and azimuthal index共p,l兲=共1,12兲. 共b兲 Structure of C line singularities of the correspondent 3D vector field.共c兲 Structure of the C line singularities from the view of propagation direction to the beam waist.共d兲 Numerical pattern of the angle function at the far field.
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