Hollow vortex Gaussian beam expressed in terms of cylindrical wave

The electrical field of hollow vortex Gaussian beam (HVGB) with an arbitrary combination of the beam order and topological charge is formulated by using the cylindrical wave spectrum representation (CWSR), which satisfies Maxwell’s equations rigorously and allows to study analytically and numerically the evolution of the beam along its propagation in the source region and the near/far- field regions. It is found that in the source region there exists a sharp and very narrow peak, due to the contribution of evanescent waves when the beam order and the topological charge satisfy specific relations. The effect of the spiral phase plate (SPP) in generating a non-vortex Gaussian beam to a vortex one is explained. Besides, it is found that the divergence of the beam is mainly determined by the topological charge and the hollow structure is mainly determined by the beam order.


Introduction
Optical tweezers have been applied for various fields in physics, chemistry, and biology [1,2].The dark hollow beams (DHBs) [3], such as the doughnut-shaped beams, Laguerre-Gaussian beams (LGBs), Bessel-Gauss beams (BGBs), have received great interest [4][5][6][7] because the null intensity in the near-axis region reduces the scattering force and the beams may be used to trap the particle whose refractive index is less than that of the surrounding material [8,9].During propagation the dark region of the DHB may gradually decrease and finally disappear, which hinders the application of the beam [10].To prevent from the shortcoming, the hollow vortex beams (HVBs) which carry optical vortexes or orbital angular momentum (OAM) are proposed.As an example of this, figure 1 illustrates the evolution of a non-vortex hollow Gaussian beam along propagation.The beam profile in the transverse plane which is very close to the beam waist exhibits a ring-shaped structure.Namely, the intensity of the beam on the axis is null and which gradually increases along the z-axis, reaches the peak and then turns to decrease.The hollow structure disappears when the propagation distance is sufficiently large.As a comparison, the vortex Gaussian beam carrying a spiral phase is shown in figure 2. The hollow structure does not disappear upon propagation, owing to the existence of the optical vortex or the OAM.
Experimentally the OAM can be produced by simply imposing a phase structure onto the DHB with the help of spatial light modulator (SLM) [11].In fact, the arbitrary spatial phase and amplitude modulation patterns can be obtained with the transform of traditional beam to the desired optical modes with the help of SLM [12][13][14].
Other tools may also be employed to generate the HVBs, such as computer-generated-holograms [3,15,16], q-plates [17], novel material structure [18,19] and spiral phase plates (SPP) [20,21].The SPP is an optical element whose transmittance may be described by the spiral phase ( ) where f is the azimuthal angle and p is the topological charge number associating with the orbital angular momentum of the beam.For example, the SPP may be produced with a transparent plastic disk with a helical surface, as sketched in figure 1 of [22].The readers who are interested in the generation and detection of OAM may refer to references, e.g.[23].
The hollow vortex Gaussian beam (HVGB) is one of the most prominent HVB models [24][25][26][27], having a single ring-shaped structure in its cross section, as shown in figure 2, and thus being much simpler than those of the LGB and the BGB.As it can be seen in appendix, either the LGB or the BGB may be taken as the linear combination of the HVGBs.The near-axis dark region of the HVGB exists in both the near and far fields along propagation [24].The HVGBs possess both spin angular momentum and orbital angular momentum and thus find applications ranging from optical trapping, optical micro-manipulation [24,[28][29][30], to quantum information [31], etc The beam profile of the HVGB in its waist plane may be characterized by Here, r and f are the radial and azimuthal coordinates in the cylindrical coordinate system respectively; w 0 is the waist radius of the beam, the integer l denotes the beam order and p is  the topological charge number.Usually, the beam order l of the HVGB is assumed to be even while the topological charge p may be even and odd, e.g.[24,[32][33][34][35][36][37][38].However, as a special case of the HVGB with = l p, the ⁎ TEM l 0 doughnut hollow beam may have odd beam orders [30,[39][40][41].More generally in [42,43], the beam profile of the HVGB in its waist plane is expressed in terms of ( wherein l 1 and l 2 both are arbitrary integers.In this case, the topological charge is less than or equal to the beam order.Simulation of the vector diffraction of linear polarized light beam by the SPP was reported in [44].Researches on the vortex Gaussian beams (VGBs) generated by the SPP, using Fraunhofer diffraction or Fresnel diffraction, were given in [45][46][47].Based on the Rayleigh-Sommerfeld integral, the nonparaxial diffraction of a VGB with initial radial polarization and an arbitrary integer topological charge p was reported in [48].The intensity on the optical axis is null when the topological charge is bigger than 1.However, the beam with =  p 1, while focusing with a lens, produces a circular subwavelength spot.The propagation of * TEM l 0 doughnut hollow beam was studied in the paraxial regime [49].The phase and polarization singularities of nonparaxial VGBs were explored in [50].Using the angular spectrum method, the VGB was decomposed into the transverse electric and transverse magnetic terms in the source region strictly in [51], and the propagation dynamics of a vector vortex field with azimuthally inhomogeneous states of polarization was studied.The interaction between the phase singularity and the polarization singularity leads to the creation or annihilation of the optical field in the central region [52].The intensity distribution in helical modes of light, varying with the topological charge, was discussed in [45].
The evolution of the HVGB during its propagation depend closely on both the beam order and the topological charge, which will be investigated in this work.Distinct from previous work, the electric field of the HVGB with an arbitrary combination of the beam order and topological charge is expressed in terms of cylindrical wave spectrum (CWS) which rigorously satisfies Maxwell's equations.The application of the CWS method allows to study the beam field in the source-region (i.e. the region very close to the waist plane in subwavelength scale), which includes the contribution from both the propagating Bessel modes and the evanescent waves.Special attention will be paid to the dependence of the electric field on the beam order and the topological charge.Based on the CWSR, numerical calculations are performed and the characteristics of the beam field are studied.Besides, the mechanism of the SPP in transforming a non-vortex Gaussian beam (NVGB) to a VGB is discussed.
The rest parts of this paper are organized as follows.In section 2, the expression of the electrical field of the VGB is derived using the CWSR.Section 3 shows the numerical results, together with analytical discussion on the CWSR of the electrical field.Concluding remarks are given in section 4.

The electromagnetic field of the VGB
For simplicity, we assume that the VGB propagates along z-axis and is centered at the origin of the coordinate system.A pair of vector potentials A and ¢ A are introduced to describe the beam having axisymmetric structure

A r e e r
A r e e r 1 x x y y x y y x where ( ) y r is the scalar potential function which describes the beam profile, p x and p y are the parameters describing the state of polarization of the beam.Using the negative time dependence where w is the angular frequency, the relation between the electric field E and the vector potentials A and ¢ A under the Lorenz gauge may be established as [36,52] where p l = k 2 is the wavenumber in which l is the wavelength.The magnetic field H can be obtained from Faraday's law, i.e., ( ) in which m is the magnetic permeability.When the scalar potential function ( ) y r satisfies the Helmholtz equation y y the harmonic electric and magnetic fields ( ) E H , and the vector potentials ( ) ¢ A, A satisfy the vector Helmholtz equation in which C equal E, H, A and ¢ A respectively.Besides, we have  ⋅ = E 0 and  ⋅ = H 0. According to [24,32,36], the scalar potential function of the VGB in the plane = z 0 may be generally given as where l and p are non-negative integers denoting the beam order and topological charge number respectively, ( ) = G e l 2 l l 0 2 is the normalization constant; w 0 is the beam waist radius.The cylindrical coordinates ( ) r f , satisfy the relations r = + x y 1 The CWSR of the scalar potential function ( ) y r f z , , , following [54], reads as In equation (4), k t and k z are respectively the transverse and longitudinal components of the wave vector k satisfying the relation where a is the angle between the wave vector k and z-axis, and both the components are real.The integration in this subinterval corresponds to the contribution of propagating Bessel modes (or the propagating waves, PWs).In the second subinterval, we have is an imaginary number.In this case, the Bessel mode may be expressed as showing the exponential attenuation upon propagation.This indicates that the integration in this interval represents the contribution of evanescent Bessel modes (or evanescent waves, EWs) [55,56].The contribution of the evanescent Bessel mode to the beam field is very small when the propagation distance z is much larger than the wavelength l and thus may be omitted.The CWS ( ) f k m t is the Fourier-Bessel transform of the scalar potential function ( ) y r f , , 0 that has been given in equation (3), and it can be obtained as ; ; Substituting equation (7) into equation (4), the summation over m can be cancelled, leading to Inserting equation (8) into equation (1) and then into equation (2), after a straightforward deduction (for more details please refer to [41]), we have the expression of the electric field of the VGB where ( ) = s kw 0 1 is the confinement parameter of the beam and the integration variable is replaced with c = k k. t The integrand reads as  and ˜( ) The spherical coordinates ( ) q f r, , satisfy ( ) q = -z r cos 1  and 1 The integrand ( ) c T in equation (10) 2 and the power function c + , p 1 characterize the profile of the VGB.Especially, the last two terms reflect the effects from the beam order l and the topological charge p.For example, when = p l, the CHF equals 1.Thus the VGB is in fact the * TEM l 0 doughnut hollow beam.More specially, when = = p l 0, it is then the fundamental Gaussian beam.The CWSR provides a method for numerically studying the beam field of the VGB with an arbitrary combination of the beam order and the topological charge number within the source-region, i.e. the propagation distance z is less than the wavelength.Also, it may be used to calculate the beam field in the near-and far-regions.
Similarly to equation (8), the integration interval in equation ( 9) may be divided into two parts: the first is the subinterval c   0 1 for the PWs and the other is c > 1 for the EWs.It should be noted that the integrand in the subinterval c   0 1 has bounded values.However, when c > >1, the variation of the integrand along with the increase of c shows a close dependence on the beam parameters, especially on the beam order l and the topological charge p.When c > >1, the components of ˆ( ) c v p E , , may be approximated to Since the magnitudes of the terms in all the square brackets of equation ( 12) have bounded values, the magnitude of ˆ( is approximately proportional to c , 2 if we limit our discussion on the beam axis (i.e.v = 0) and the topological charge  p 2. According to formula 13.2.2 in [58], the CHF in equation ( 9) may be expressed in terms of an infinite series, where for convenience we define ( ) =q l p 2, and the Pochhammer symbol ( ) When q is a non-negative integer, the summation in equation ( 13) is truncated at = j q.Therefore when c > >1, the value of the CHF is dominantly determined by ( ) Correspondingly, the CWS 2 in value.This suggests that the magnitude of the CWS, i.e., ( ) | ( )| c c = T T , tends to decrease when c > >1 and thus has bounded value in the whole integration interval.Therefore, the integration of equation ( 9) may be numerically carried out and as a result the electric field is finite in value.On the contrary, when ( ) = -¹ q l p 2 0, 1, 2, ,  the CHF asymptotically tends to (formula 13.2.23 in [58]) Therefore, when c > >1, the magnitude of the CWS ( ) In the region > z 0, ( ) c T tends to decrease while c > >1 and thus has bounded value in the whole integration interval.However, in the waist plane = z 0, ( ) c T is characterized by c -l 1 only.It increases linearly along with the increase of c when = l 0 but keeps almost unchanged when = l 1.In such two cases, the integral in equation ( 9) is not convergent and hence the electric field is infinitely large.
It is noteworthy that in the discussion above, by letting v = 0 (corresponding to the electric field on the beam axis r = 0), the dependence of the Bessel function on the variable c [see equation (12)] is not included.In case v ¹ 0 (or equivalently r ¹ 0), the Bessel functions are enveloped by a factor c -1 2 when c > >1.Correspondingly, the CWS ( ) c T of the VGB, when 1 2 in value.Therefore, the integral in equation ( 9) is convergent when = l 1, meaning that the electric field is finite in magnitude.
To summarize, the electric field ( ) E r of the VGB has been formulated in terms of the CWSR, as shown in equation (9).The integrand ( ) c T , known as the CWS, has bounded values when > z 0 and hence the integration may be carried out numerically, resulting into finite values of the electric field.However, in the plane = z 0, the situation is somehow different.The magnitude of the CWS ( ) c T , when ( ) = -¹ q l p 2 0, 1, 2, and = l 0, 1, may become very large or keep unchanged in the range c > >1.Besides, it should be noted that the VGB with ¹ p 0 can be generated from the NVGB (i.e., = p 0) with the help of a SPP carrying the spiral phase ( ) . In this sense, the effect of the SPP may be investigated by comparing the CWS of the VGB and the NVGB having the same beam order and the same beam waist radius.Namely, the VGB distinguishes from the NVGB in its CWS characterized by the product of the CHF ( ) and the power function c + , p 1 in which the topological charge p may equal zero or not.Numerical examples will be given and discussed in the next section.

Numerical results and discussions
The electric field of the VGB, expressed in the CWSR, contains the contribution from both the propagating and evanescent waves.It has been discussed in section 2 that the CWS of the beam is closely dependent on the beam parameters, especially the beam order and the topological charge.Besides, it is worth noting that the evanescent waves attenuate exponentially while propagating so that their contribution may be omitted in the far-field.Nevertheless, the contribution of evanescent waves is very strong in the near-field, leading to some interesting phenomena in the region very close to the beam waist (the source region).Therefore, the numerical calculations below will first focus on the field in the source region in which both the propagating and evanescent waves are included.In section 3.1, special attention is paid to the CWS on the beam axis which is very close to the beam waist.Correspondingly, the electric field on the beam axis is discussed in section 3.2, and afterwards in section 3.3 we extend our calculations to the region very close to the beam waist.Finally, in section 3.4, the propagation of the beam is studied in the far-field, i.e. the region where the propagation distance z is much larger than the wavelength l of the beams.

CWS of the VGB and NVGB
In this subsection, the CWS of the VGB and the NVGB will be studied numerically.Having said that the VGB may be generated with the help of SPP from the NVGB which has the same beam order, a comparison of the CWS between the VGB and the NVGB may explain the role the SPP plays in generating the VGB.
In the first part, we will focus on the electric field on the beam axis, i.e., r = 0 or v = 0. Since ˜( ) for  p 3, see equation (11).This indicates that the VGB whose topological charge is equal to or larger than 3 has null intensity on the beam axis.However, the situation for < p 3 is somehow different.When = p 0 or 2, we have ˆ= E 0 z so that the field on the beam axis contains only the contributions from Êx and Ê , y implying that the electrical field on the beam axis is transversely polarized.When = p 1, we have ˆ= = E E 0 x y and thus the beam field on the beam axis contains Êz only.In the following numerical calculations and discussion, we assume that beam waist radius m = w 3 m 0 (or equivalently the confinement parameter » s 0.0336), the wavelength l m = 0.6328 m and polarization parameters ( ) ( ) = p p , 1,0, i.e., the beam is linearly polarized.
Case 1.The NVGB with = p 0 When topological charge = p 0, we have ˆ( z Therefore, the electric field on the beam axis may be written as where the CWS, according to equation (10), is Numerical results of the logarithm of the CWS, i.e.
| ( )| c T log , 10 in the range [ ] c Î 0, 1.8 are shown in figures 3, 4 and 5 for different values of the beam order l.Meanwhile, the topological charge is fixed at = p 0, i.e. the beam is non-vortex beam.At first, we assume = z 0. For the beam orders being even, the CWS oscillates in the range c <  0 0.2 and then decreases very fast when c increases, as shown in figure 3.However, the situation for the odd beam orders is quite different, see figure 4. The CWS first oscillates in the range c <  0 0.3 and then decreases monotonically at a rate much slower than that in figure 3(a).This indicates that the CWS of the NVGBs with the even beam orders distributes narrower than that of the beams with the odd beam orders.Attention should be paid to the exception which occurs when = l 1.The magnitude of the CWS keeps almost unchanged state in the range c > 1, see the black curve with square dots in figure 4(a).In such a case, the contribution of the EWs to the beam field is very large.Relevant discussion has been made in the text below equation (14).Now let us consider the influence of the propagation distance z on the CWS ( ) c T .In the range [ ] c Î 0, 1 (corresponding to the PWs), the propagation of the beam along z-axis only introduces a phase shift to the CWS.
Therefore, the electric field on the beam axis may be obtained as where the CWS reads as Comparing between the curve with open square dots in figure 6 and the curve with closed square dots in figure 3, which corresponds to the VGB with ( ) ( ) = p l , 1,0 and the NVGB with ( ) ( ) = p l , 0,0 respectively, we may find that the CWS of the VGB is much wider than that of the NVGB.The distribution of the CWS of the VGB is significantly compressed in the range of c < 0.1 and meanwhile the CWS is enhanced in the range c > 0.1.Since the VGB may be generated from the NVGB by using a SPP, the effect of the SPP may be understood by the variation from the CWS of NVGB to the CWS of VGB.Namely, it transforms part of the PWs in the range of small c to the EWs, and thus produces the hollow structure of the beam.
The CWS of the VGBs with = p 1 and  l pare calculated too.It is found that when the beam order l is odd (i.e.= l 1, 3, 5), the CWS varies in the way very similar to those shown in figure 3; and for even l (i.e.= l 2, 4, 6), the CWS varies similarly to those plotted in figure 4.This can be understood by looking back at equation (10) in which the CWS is characterized by the CHF whose first argument is ( ) p l 2. It suggests that the distribution of the CWS, especially in the range c > 1, is dependent on the combination of the beam order and the topological charge (namely their difference), but not one of them.Numerical results reveal that the magnitude of the CWS when < p lis much smaller than when  p l.This implies that the SPP when  p lcan modulate the beam profile very significantly.On the contrary, when < p l, the effect of the SPP on the beam is somehow weak.
z Therefore, the electric field on the beam axis is obtained as and the integrand is Numerical results show that the CWS of even beam orders l are quite similar to those in figure 3 and the CWS of odd beam orders l are quite similar to those in figure 4, when =  l p 2. However, the situations of = l 0 and = l 1 (i.e., < = l p 2) are different, as shown in figure 7. When = z 0 (corresponding to the curves with open square dots), the CWS of the VGB with = l 0 increases linearly along with the increasing c, but for = l 1 it increases first, turns to decrease and then keeps almost unchanged in the range c > 0.5.The curves with closed circular dots exhibit the variation of the CWS for m = z 5 m,in which the CWS behave in the same way as those for = z 0 in the range c  1.However, both the CWS attenuate exponentially when c > 1.The dependence of the CWS on the beam order when = p 2 is similar to the situation when = p 1. On the whole, it can be found that the CWS of the beam satisfying -= l p 0, 2, 4,distributes narrowly in the range c <  0 0.2.The CWS in the range c > 1 is so small that can be neglected.The CWS for -= l p 1, 3, 5,is somehow wider and mainly distributes in the range c <  0 0.3.On the contrary, when > p lthe CWS distributes very widely, the contribution of the EWS to the beam field cannot be omitted.Besides, it should be noted that the propagator 2 in the CWS works as a phase term in the range c  1 but it becomes an exponentially decaying function for c > 1.Therefore, the contribution of the EWs fades away when z is much larger than the wavelength l.
The property of the CWS may be understood mathematically by looking at the discussion at the end of section 2. Namely, the CWS is mainly determined by the CHF 2 which is governed by the first argument.Since the VGB can be taken as a product of the NVGB by imposing a SPP carrying the spiral phase ( ) f ip exp , the effect of the SPP can be explained from the variation of the CWS.It is well known that the property of the vortex beams is usually described, using the paraxial wave approximation, and the propagation dynamics is explained in the theory of scalar wave diffraction, for instance in [46,49,[59][60][61][62].In this paper, the VGB is studied, using the vector description of electric field (i.e., the CWSR) which satisfies rigorously Maxwell's equations and allows to study the electric field in the source-region.The comparison between the CWS of the NVGB and the VGB having the same beam order offers an alternative way to understand the effect of SPP.Namely, the SPP transforms the PWs of the beam to the EWS and thus maintains the stability of the hollow structure while the beam propagates.
The numerical results above are limited to the beam axis, i.e. r = 0 or v = 0, and hence the relation between the Bessel function ( ) v J q and the variable c is not included.The Bessel function ( ) v J , u when v rc = k is sufficiently large, may be asymptotically written as ( ) Therefore, according to equation (12), we have ~( where the subscript = j x y z , , .Accordingly, in the range of large c, the Cartesian components of ( ) c T in equation ( 10) may be approximately rewritten as

When
( ) =q l p 2 is a nonnegative integer, the CHF is a sum of finite series truncated at the order q [see equation (13)] so that the dependence of the CWS on the variable c may be characterized by the term 2 when c > >1.In such a case, the CWS distributes narrowly, i.e., the EWs contribute little to the electric field, and hence the integral can be numerically calculated.
On the contrary, when > p lor -= l p 1, 3, 5, ,  the CHF is a sum of infinite series, whose asymptotic expression is given in equation (14).By substituting equation (14) into equation (21), we find that the dependence of the CWS on the variable c is characterized by Therefore, in the plane = z 0, when = l 0, the CWS is approximately proportional to c 1 2 for c > >1, growing up when χ increases.But when  l 1 it is proportional to c -, .The above discussion suggests that, for the beams with = l 0 and  p 1, the integral for calculating the electric field in the plane = z 0 results into infinitely large values, due to the contribution of EWs.Similarly, for the beams with >  p l 1 (or simultaneously satisfying the relations  p 0 and -= l p 1, 3,), the electric field has finite values in the plane = z 0 except for the positions very close to the origin of the coordinates (i.e.= r 0).It is noteworthy that the above discussion is valid for  p 3 too, though the electric field on the beam axis is zero.

Electric field on the beam axis
In this subsection, the electric field on the beam axis is calculated and discussed.The electric field of the NVGB ( i.e. = p 0) can be numerically calculated by using equations (15) and (16).Numerical calculation is very fast in the cases shown in figures 3 and 5(a), but is comparatively slower for those in figures 4 and 5(b).Especially, catastrophe in integration (the integration does not converge to a limited value) occurs when = l 1 and = z 0. Owing to the large distribution of the CWS in the range from c = 1 to +¥, the integration does not converge, which implies that | ( )| ¥ E 0, 0, 0 .Numerical results of the electric field on the beam axis are shown in figure 9 1 m, 0, 0.1 m .Different from the CWS shows in the figures above, the CWS plotted here correspond to the field out of the beam axis.As we have indicated in the first paragraph of section 3.1 that when  p 3 the electric field of the VGB is always zero on the beam axis.However, when  p 2, the VGB has non-zero electric field on the beam axis, listed as follows: (1) For the beams with ( ) ( ) = p l , 0,1,( ) 1, 0 , ( ) 2, 0 and ( ) 2, 1 , the electric field on the beam axis is infinitely large at = z 0 and it attenuates exponentially along propagation, due to the significant contribution of the EWs to the beam field in the range l < z .
(2) For the beams satisfying = = l p 0, 1, 2, the contribution of EWs can be neglected and the electric field on the beam axis is peaked at = z 0. The amplitude of the electric field on the beam axis decreases when the topological charge increases.(4) when -= l p 2, 4, (i.e.positive even integers), the contribution of the EWs may be neglected.The electric field on the beam axis undergoes a procedure of first increasing and then decreasing during its propagation.For example, ( ) ( Besides, the VGB may be generated by simply imposing a phase structure onto the NVGB with the help of SLM such as a SPP characterized by ( ) f ip exp [11,[15][16][17][18][19][20][21], see also the end of section 2. The phase structure modulates a NVGB into a VGB by changing the CWS of the beam.For example, the CWS of the NVGB with ( ) ( ) = p l , 0,0 [see the black curve with closed square dots in figure 3] distributes mainly in the range [ ] c Î 0, 0.2 and the contribution of EWs is so weak that can be neglected.The CWS of the VGBs with  x 0 (on the beam axis, please see also figure 10), which is produced by the EWs.The full width at half maximum (FWHM) of the peak in this plane is about m 0.023 m (about 0.036 times to the wavelength) and the maximum of the electric field is about 25.Since the peak is contributed by the EWs, it attenuates quickly along the beam's propagates.The strong peak in the vicinity of the beam waist exhibits the characteristics, such as sub-wavelength in width, strong enhancement and short attenuation distance, may find its applications in high resolution imaging [63] and high density storage [64].
When the propagation distance z is sufficiently large, only the PWs contribute to the electric field, which results into a hollow structure of the beam.The calculation is repeated for beams with = l 0 and = p 2, 3.The results have the same characteristics.Besides, it is found that, as the topological charge increases, the magnitude of the narrow peak increases and meanwhile the hollow structure increases.The dependence of the electric field of the VGB on the topological charge may be easily understood.The VGB with = l 0 and  p 1 can be generated from the NVGB with ( ) ( ) = p l , 0,0 by using a SPP ( ) , transforming part of the PWs into EWs and thus produces the narrow peak and the hollow structure of the beam.
The electric fields for the beams with ( ) ( ) = p l , 0,1 and ( ) ( ) = p l , 3,1 are shown in figure 13, for m = z 0.001 m, m 0.01 m and m 0.1 m respectively.It can be seen, the electric fields exhibit peaks at the places very near to the beam axis in both the cases, which are contributed from the EWs and attenuate quickly along propagation.Compared with the beam with ( ) ( ) = p l , 1,0 showed in figure 12, the peaks in figure 13 are much weaker and fade away much earlier in propagation, due to the beam order is higher (i.e., = l 1).Besides, it may be found that the peak in figure 13   field is so small (typically less than - 10 93 ) that can be neglected.Each beam exhibits a near-axis dark region, but the electric field on the beam axis (i.e., = x 0) is slightly bigger than 0 [see also figure 9(b)].Additional calculations reveal that the increase of the beam order or the topological charge (or both of them) may enlarge the near-axis dark region and meanwhile reduce the electric field on the beam axis.
All in all, the electric field of the VGB in the range m  z 1 m depends closely on both the beam order and the topological charge.Three cases have been studied, as summarized below: (1) For the beams satisfying > = p l 0, the EWs contribute to the electric field greatly, leading to an extremely narrow and strong peak on the beam axis.The peak attenuates quickly when z increases.It is found that the increase of the topological charge enhances the narrow peak and meanwhile widens the hollow structure of the beam.
(2) When the beam order and the topological charge satisfy the relation >  p l 1 (or satisfy the relations  l 1 and -= l p 1, 3, simultaneously), the EWs contribute to the electric field, which leads to a very narrow peak on the beam axis and a hollow beam structure.
(3) For the beams satisfying the relation -= l p 0, 2, 4, ,  the EWs contribute to the electric field is very weak so that it does not produce the peak on the beam axis.However, the hollow structure of the beam becomes wider when either the beam order or the topological charge increases (or both of them increase).Since the VGB may be generated by imposing a SPP in the waist plane of the NVGB, the dependence of the electric field of the beam on the topological charge may be understood by comparing between the CWS of the NVGB and the VGB having the same beam order.The SPP transforms part of the PWs of the NVGB to EWs of the VGB, depending on the topological charge number.
In case (1), the NVGB with = l 0 has maximal electric field on the beam axis, therefore the effect of the SPP is very strong.However, in cases (2) and (3) where the beam order is  l 1, the effect of the SPP is much weaker than in case (1), revealing the effect of the beam order on the beam profile.
It is interesting to point out that, in the theory of scale wave diffraction, the SPP (also called diffractive optical element) makes the core of the vortex remain dark as the beam propagates, which may be understood from the point of view of destructive interference between rays diffracted into the core [59].However, by using the vector CWSR, we have found that the electric field of the VGB exhibits a strong and narrow peak in the vicinity of the beam center (sub-wavelength region) when the beam order and the topological charge satisfy a specific relation , as shown in figures 12 and 13.This phenomenon, to the best of our knowledge, has not been published.The beam order l changes from 1 to 4. It can be seen the axisymmetric dark region exists only in the near field when z is small.The width of the dark region increases when the beam order increases.During propagation, the dark region of the beam disappears gradually, showing a dependence on the beam order.Namely, the dark region disappears earlier in propagation when the beam order is lower.On the axis and in the region nearby, the electric field increases along z-axis and then turns to decrease.The on-axis peak occurs earlier when the beam order is lower (also see figure 9 and the relevant discussion in section 3.2).The numerical results here confirm that the NVGB cannot keep its hollow structure in the far field, as discussed in [10].
When the propagation distance is much larger than the wavelength, the contribution of the EWs can be omitted.Therefore, the electric field in the range m  z 1 m contains the PWs only.In the view of CWS, the field is the interference between the CWS, which is governed by the propagator ( ) c ikz exp 1 . 2This explains the evolution of the electric field upon propagation.Figure 16 Figure 17 shows the evolution of the VGBs along propagation.propagating, the beam profile varies in a way depending on the beam order and the topological charge.In figure 17(a), the topological charge is set at = p 1 and the beam order increases from = l 2 to 5. It can be found that in the near-field the dark region is much wider when the beam order is large, which is also displayed in figure 15 for the NVGBs.Along propagation, the width of the dark region decreases gradually and then turns to increase slightly.The dark region does not disappear in propagation, nevertheless its width is very small in the far-field because the topological charge number is 1 here.In all the cases in figure 17(a), the beam divergence is very small and keeps almost unchanged while the beam order increases.This suggests that the beam order does not affect the divergence of the beam effectively.To have an impression on the beam profile in the transverse plane, figure 18 .However, the increase of the topological charge produces side lobes outside the main ring structure in the near field, as shown in figure 18(b).Besides, it is found that the beam diverges more largely and the electric field decays faster when the topological charge increases.
Figure 17(c) illustrates the evolution of the electric fields for the * TEM l 0 doughnut hollow beams, in which the beam order is identical to the topological charge, increasing from 1 to 4. The electric field keeps the axisymmetric

Conclusions
In this paper, the CWSR of electric field of the HVGB is formulated by introducing the potential functions in the waist plane.The CWSR describes the VGB in the spectrum space in which the beam field is taken as the spectrum of the propagating and evanescent Bessel modes.Such a representation describes the field satisfying Maxwell's equations rigorously and thus allows to study the beam field in the source region, in which the contribution of evanescent waves is included.The beam field is characterized by the beam waist radius, the beam order and the topological charge.The dependence of the beam field on the beam order and the topological charge is analytically discussed and numerically studied.It is found that, in the near field region (especially in the source region) behind the waist plane, the contribution of EWs to the electric field is significant, when the beam order and the topological charge satisfy specific conditions, such as ( ) ( ) = p l , 0,1, > p land so on.The contribution of EWs produces a very narrow and sharp peak in the near-axis region, which decays quickly along propagation.The effect of the SPP in generating a VGB from a NVGB is discussed by comparing the CWS of the beams.It is found that, when > p l, part of the PWs of the NVGB is transformed into the EWs of the VGB.The phenomenon in the near region, to our knowledge, has not been published elsewhere.
The beam profile depends on the beam order and the topological charge closely.It is found that when = p 0 and > l 0 the hollow structure exists in the near field only, indicating that the so-called hollow non-vortex Gaussian beam is not hollow in the far field.The divergence of the beam in its propagation is mainly determined by the topological charge but the size of the hollow structure (in the near field) is mainly decided by the beam order.Namely, the beam diverges largely when the topological charge increases.The hollow structure is wider when the beam order is bigger.It is interesting that, when > = l p 1, as shown in figure 17(a), the hollow structure in the near field is wider and it becomes narrower along propagation.The beam profile exhibits some side lobes during propagation when ¹ l p.When < l pthe side lobes exist outside the ring structure in the near field.On the contrary, when < p lthe side lobes exist inside the ring structure and along propagation the side lobes become strong meanwhile the ring structure disappears, which makes the beam size narrower.As to the * TEM l 0 doughnut hollow beam (i.e.= l p), the beam profile keeps the ring structure, whose size increases along propagation.p l increasing from 1 to 4. In all the cases, the hollow structure does not disappear upon propagation.Attention should be paid to the characteristics of the beam that the width of the hollow structure in the near-field depends mainly on the beam order but the beam divergence is mainly dependent on the topological charge.which indicates that the LGB can be expressed as a linear superposition of the VGBs.The above analytical deduction demonstrates that the VGB is the fundamental mode of the LGB and the BGB.

Figure 1 .
Figure 1.Evolution of a non-vortex hollow Gaussian beam along propagation.The graph on the upside is in xz-plane and the panels below it are in the transverse planes, the left of which is very close to the beam waist, showing the hollow structure, and the right one is in the far-field region where the hollow structure disappears.

Figure 2 .
Figure 2. Evolution of a vortex hollow Gaussian beam upon propagation.The graph on the upside is in xz-plane and the two panels below it are in the transverse planes.The hollow structure does not disappear while propagating.

Figure 3 .
Figure 3.The CWS for the NVGBs (i.e. the beam with = p 0) in the waist plane (i.e. the plane with m = z 0 m), calculated with equation (16).The beam orders l equal 0, 2 and 4, respectively.(a) Logarithm of the CWS; (b) the CWS in the range c =  0 0.20.It indicates that the contribution of EWs to the beam field is very small so that it can be omitted.In all the cases, the CWS distributes very narrowly in the forward directions less than a < 0.2 rad.

Figure 4 . 2
Figure 4.The CWS for the NVGBs (i.e. the beam with = p 0) in the waist plane m = z 0 m,also calculated with equation (16).The beam orders here are 1, 3 and 5 respectively.(a) Logarithm of the CWS; (b) the CWS in the range c =  0 0.3.Attention should be paid to the difference between the CWS here and those in figure 3, due to the different beam orders.Especially, the black curve with square dots in subfigure figure 4(a) shows that the CWS keeps unchanged when c > 1, which indicates the strong contribution form EWs.
Numerical results of the CWS of the VGB with ( ) ( ) = p l , 1,0,calculated with equation (18) for both = z 0 and m = z 5 m,are shown in figure 6.In the range c > 1, the CWS increases linearly along with the increase of c, see the curve with open square dots.The variation of the CWS of the VGB with ( ) ( ) = p l , 1,0 in this range is evidently different from that of the NVGB with ( ) ( ) = p l , 0,1,as shown in figure 4(a).However, when m = z 5 m the CWS exponentially decays in the range c > 1, see the curve with closed circular dots in figure 6.In the range c  1, the CWS for both = z 0 and m = z 5 m should coincide with each other.The difference on the plot is because the curve with open square dots is plotted in linear scale corresponding to the ordinate on the left side and the curve with closed circular dots are drawn in logarithmic scale labeled on the right side.

Figure 5 .
Figure 5.The CWS for the NVGBs at the transverse plane m = z 5 m.The beam orders in (a) are 0, 2 and 4, but in (b) are 1, 3 and 5.The contribution of the EWs is smaller than that in figures 3(a) and 4(a).The CWS with even beam orders distribute more narrowly than those of odd beam orders.

Figure 6 .
Figure 6.The CWS calculated with equation (18) for the VGB with ( ) ( ) = p l , 1 ,0 at = z 0 and m = z 5 m.The curve with open square dots corresponds to the ordinate on the left side and the one with closed circular dots corresponds to the ordinate on the right side.

l 1 2 1 .
decreasing when χ increases.Numerical examples are shown in figure 8(a) for the beams with ( ) ( ) Nevertheless, when > z 0, the CWS decrease quickly in the range c > >1 due to the exponential function ( ) c -kz exp , as shown figure 8(b) (a), where the beam parameters are same as those in figures 3, 4 and 5.More details of the beam field in the vicinity of the beam waist ranging from = z 0 to m 0.2 m may be observed in figure 9(b).All the curves with closed dots denote the electric field containing the contributions from both the PWs and the EWs, while those with open dots in figure 9(b) contain

Figure 7 . 1 . 2
Figure 7.The CWS calculated with equation (20) for the VGBs with (a) ( ) ( ) = p l , 2 ,0;(b) ( ) ( ) = p l , 2 ,1.The curves with open square dots are plotted in linear scale, corresponding to the ordinate on the left side.The curves with closed circular dots are drawn in logarithmic scale labeled on the right side.

Figure 9 .
Figure 9. Numerical results of the electrical field for the NVGBs on the beam axis, where = p 0. The curves with closed dots contain both PWs and EWs, and those with open dots contain the PWs only: (a) the electric field on the beam axis; (b) a comparison in absence and presence of the EWs in the vicinity of the waist plane.Attention should be paid to the black curve with close circular dots which exhibits a very strong field in the vicinity of the beam waist.This will be further discussed in figure 13(a) in the next subsection.

Figure 11
illustrates the numerical results of the electric field of the VGBs with = p 2 and different beam orders.It can be seen from figure 11(a) that the on-axis electric fields decrease monotonically along z-axis when  p l.The EWs contribute much in the range l < z for = l 0, 1, 3 (i.e., -¹ l p 0, 2, 4,).Especially, when  p l[see the red and black curves with closed dots in figure 11(b)], the contribution for EWs is very large and attenuates exponentially.Besides, the comparison between figure 9(a), figure 10(a) and figure 11(a) suggests that the electric field on the beam axis is lower and lower when the topological charge p increases from 0 to 2.

Figure 10 . 1 .
Figure 10.Numerical results of the electrical field of the VGB on the beam axis, where = p 1. The curves with closed dots contain both the PWs and EWs, and those with open dots contain the PWs only: (a) the electric field on the beam axis; (b) a comparison in absence and presence of the EWs in the vicinity of the waist plane.The large difference between the curves with open/closed square dots reveals the contribution of the EWs, these will be further discussed in figure 12 in different transverse planes which are very close to the beam waist.

Figure 11 .
Figure 11.Numerical results of electrical field of the VGB on the beam axis, where = p 2. The curves with closed dots contain both PWs and EWs, and those with open dots contain the PWs only; (a) the electric field on the beam axis; (b) a comparison in absence and presence of the EWs in the vicinity of the waist plane.Attention should be paid to the strong electric field at the beam waist when > p l.

( 3 ) 3
When -= l p 1, 3, (i.e.positive odd integers), the EWs contribute visibly to the electric field in the range l < z .Examples in this case are ( in figure 11(b).

f i exp and ( ) f i exp 2 ,
respectively shown in figure 6 and figure 7(a) (see the black curves with open square dots), distribute very widely and the contributions from EWs are significant.Since the VGBs with ( can be generated from the NVGB by using the SPP carrying the spiral phases ( ) the effect of the SPP is reflected in the widening of the CWS of the beam.Namely, part of the PWs of the NVGB is transformed into the EWs of the VGB, leading to a very strong electric field behind the SPP which attenuates exponentially during propagation (see the red curves with closed square dots in figures 10 and 11).

3. 3 .
Electric field in the range close to the waist plane In this subsection, the electric fields of the VGB in different transverse planes which are very close to the waist plane are discussed.Numerical results of the electric field on the x-axis are shown in figure 12 for the beam with ( and m 1 m.The electric fields in figures 12(a) and (b) contain respectively the contributions from PWs and EWs.It can be seen, along the beam's propagation, the electric field contributed by PWs does not change much but the field contributed from EWs attenuates quickly.The electric fields of the beam including the contribution of both the PWs and the EWs are plotted in figure 12(c).The electric field in the plane m = z 0.01 m exhibits a very strong and narrow peak at =

p l , 3 , 1 3
(b) for the beam with ( ) ( ) = is higher than the one in figure13(a) for meanwhile the dark region in the former is slightly wider than the latter, which explains that the VGB with ( ) ( ) = p l , 3,1 may be generated from the NVGB with ( ) ( ) = p l , 0,1 by using the SPP ( ) f i exp 3 which transforms part of the PWs of the NVGB into EWs of the VGB.Finally, the peak in figure13(b) has null electric field on the beam axis because the topological charge is = p 3 in this case.It has been discussed above that, for the beams satisfying -= l p 0, 2, 4, ,  the CWS distributes narrowly in the range [ ] c Î 0, 0.2 and the contribution of the EWs to the electric field is extremely small.As an example of this, numerical results for the beams with ( are shown in figure14, where m = z 0.001 m, m 0.01 m and m 0.1 m respectively.In all these cases, the contribution of the EWs to the electric

Figure 12 .
Figure 12.The electric field of the VGB with ( ) ( ) = p l , 1 ,0 in different transverse planes very close to the beam waist: (a) contribution of the PWs; (b) contribution of the EWs; (b) the field containing both the PWs and EWs.The sharp peak is very strong, the width of the peak is sub-wavelength and attenuates very quickly along propagation.

Figure 13 . 1 .
Figure 13.(A) comparison of the electric fields in different transverse planes close to the waist plane.The beam order and the topological charge are: (a) ( ) ( ) = p l , 0 ,1;(b) ( ) ( ) = p l , 3 ,1.The evolution of the on-axis peak for the VGB with ( ) ( ) = p l , 0 ,1 along propagation is shown in figure 9.

3. 4 .m 8 to mm 8 .w m 3 . 0 Figure 15
Figure15shows the electrical fields | | E of the NVGBs (i.e.= p 0) in xz-plane.The beam order l changes from 1 to 4. It can be seen the axisymmetric dark region exists only in the near field when z is small.The width of the dark region increases when the beam order increases.During propagation, the dark region of the beam disappears gradually, showing a dependence on the beam order.Namely, the dark region disappears earlier in propagation when the beam order is lower.On the axis and in the region nearby, the electric field increases along
(a) shows the distribution of the electric field of the NVGB with ( ) ( ) = p l , 0,3 on x-axis where the propagation distance is m = z 15 m and figure 16(b) displays the field distribution at m = z 80 m.The ring structured field of the non-vortex hollow Gaussian beam in the source-region evolves to a bell-shaped distribution when the propagation distance is sufficiently large.
(a) displays the distribution of the electric field along x-axis for the VGB with ( ) The dark region is narrow and it should be noted the electric field at m = x 0 m (i.e. on the beam axis) is non-zero.For the electric field on the beam axis, please look back at figure 10.In figure 17(b), the beam order is fixed at = l 1 but the topological charge varies from = p 2 to 5. In all the cases, the beam keeps the axisymmetric hollow structure while propagating.The variation of topological charge does not visibly affect the transverse size of the beam in the near field, i.e. m  z m 1

Figure 15 .z m 30 .2
Figure 15.The evolution of the electric field of the NVGB along propagation.The plots are in xz-plane.It can be seen that the dark region exists in the near-field and the width increases when the beam order increases.The hollow structure gradually disappears during propagation, forming the bell-shaped distribution in the transverse plane.

Figure 16 .
Figure 16.The distribution of the electric field of the NVGB with ( ) ( ) = p l , 0 ,3 along x-axis. (a) m = z 15 m; (b) m = z 80 m.Attention should be paid to the evolution of the beam profile along propagation.

Figure 17 .
Figure 17.The evolution of the electric field of the VGBs along propagation: (a) the topological charge is fixed at = p 1 but the beam order increases from 2 to 5; (b) the beam order is fixed at = l 1 and the topological charge varies from 2 to 5; (c) the * TEM l 0 doughnut hollow beam, with = p l increasing from 1 to 4. In all the cases, the hollow structure does not disappear upon propagation.Attention should be paid to the characteristics of the beam that the width of the hollow structure in the near-field depends mainly on the beam order but the beam divergence is mainly dependent on the topological charge.
given in equation(3).We can see from equation (A2) that the BGB is expressed as a linear superposition of the VGBs.
The integral interval may be divided into two parts: the first one covers the range from = k 0In the first subinterval, the transverse component k t and longitudinal component k z of the wave vector k may be written as t to ¥ and a summation over different mode number m.t