Abstract

The general analytical formula for the propagation of the power-exponent-phase vortex (PEPV) beam through a paraxial ABCD optical system is derived. On that basis the evolution of the intensity distribution of such a beam in free space and the focusing system is investigated. In addition, some experiments are carried out, which verify the theoretical predictions. Both of the theoretical and experimental results show that the beam’s profile can be modulated by the topological charge and the power order of the PEPV beam.

© 2016 Optical Society of America

1. Introduction

Nowadays, considerable interest has been exhibited in the propagation and interaction of optical beams through random media, and characterization of the propagation properties offers a good reference point for evaluating the effect of a beam in practical environment. In some circumstances, it is desirable to have a laser beam with a different irradiance distribution from the one originally coming out of the laser cavity, such as laser welding, laser processing, and laser medical applications.

Since the pioneering work of Nye and Berry on phase singularities in optical fields revealed the existence of such interesting structures as phase dislocations and optical vortices, the research on optical vortices has been investigated extensively both in theory and experiment for its wide practical applications, such as astronomical, micromanipulation of particles and atom guiding [1–4]. Optical vortices are a peculiar type of beam which has one or more singularities on the transverse beam profile [5–7] and an azimuthal phase dependence exp(ilθ), where l is an integer number and refers to the topological charge (azimuthal index) of the field [8], which is related to the orbital angular momentum (OAM) L of photos by the relationship L=l. Over the past decades, apart from the canonical vortices, which carry a spiral phase that varies uniformly with azimuthal angle, several kinds of noncanonical vortices, such as nonsymmetric vortices [9, 10], Mathieu vortices [11] and fractional vortices [12–14], have also been investigated in order to explore the properties and applications of OAM. It has been reported that the OAM could be used for quantum computing [15] and moving particles [16].

Power-exponent-phase vortex (PEPV) beam is a kind of noncanonical vortex that characterized with power-exponent-phase [17, 18]. It has been illustrated that the PEPV beam carries OAM, which is determined by the topological charge [17]. The simplest way to generate the PEPV beam is to employ a spatial light modulator (SLM) [18]. This method is more interesting and popular because it has the advantage of providing dynamic and programmable modulations. In this paper, we experimentally generate the PEPV beam by means of phase modulation through a SLM, that is, we directly impose a power-exponent phase into a Gaussian beam in such way. And the intensity properties of PEPV beam in the cases of free space and focusing system are theoretically and experimentally studied.

2. Theory

It is assumed that the electric field of the PEPV beam in the source plane has a form of

E(0)(r,φ)=A0exp(r2w2)exp[i2mπ(φ2π)n],
where A0 and w refer, respectively, to the characteristic amplitude and waist width of the incident beam, n denotes the power order of the beam, φ denotes the azimuth angle with m being the topological charge of the PEPV beam. In the case of n=1, the PEPV beam reduces to the traditional canonical vortex beam. For the sake of convenience, A0 equals to 1.

In the situation of the paraxial approximation, the electric field in the transverse plane z=const>0 through an ABCD paraxial optical system can be studied with the help of the generalized Collins formula [19, 20]

E(ρ,θ,z)=ik2πB002πE(0)(r,φ)exp{ik2B[Ar22rρcos(θφ)+Dρ2]}rdrdφ,
where A, B, C and D denote the elements of the transfer matrix of the optical system, and k is the wave number. In addition, the phase term exp(ikL0) has been omitted in Eq. (2).

On substituting from Eq. (1) into Eq. (2), it follows that

E(ρ,θ,z)=ik2πBexp(ikDρ22B)002πexp(r2w2)exp(ikAr22B)×exp[ikρrBcos(θφ)]exp[i2mπ(φ2π)n]rdrdφ.

And using the following formulas [21]

exp[ikrρBcos(φθ)]=h=ihJh(krρB)exp[ih(φθ)],Jl(x)=(1)lJl(x),Jl(x)=p=0(1)p1p!Γ(l+p+1)(x2)l+2p,Γ(x)=0exp(t)tx1dt,exp(sxn)=j=0sjxnjj!,
where Jl(.) represents the lth integer order Bessel function of the first kind, and Γ(x) is the Gamma function.

On substituting from Eq. (4) into Eq. (3) and taking the integrations, one obtains the final analytical expression of the electric field in the output plane as

E(ρ,θ,z)=(i2λBR)exp(ikD2Bρ2){exp(k2ρ24B2R)M0+l=1[p=0Γ(p+l/2+1)p!Γ(p+l+1)(k2ρ24B2R)p+l/2]×[exp(ilθ)Ml+exp(ilθ)Ml]},
where R=ikA2B+1w2, and

Ml=02πexp{i[2πm(φ2π)n+lφ]}dφ={j=0h=0ij+hmjlh(2π)j+h+1j!h!(nj+h+1),l0,j=0ijmj(2π)j+1j!(nj+1),l=0.

The expression of the intensity of the PEPV beam is given by

I(ρ,θ,z)=E*(ρ,θ,z)E(ρ,θ,z).

3. Experimental generation and propagation properties of the PEPV beam through free space and a focusing system

Since transforming a laser beam into an arbitrary complex field could be done by means of phase modulation of SLM, which is a versatile and convenient way for modulating optical field, we use this method for generating a typical PEPV beam and carry out experimental study of its intensity properties to verify the theoretical results. The experimental setup for generating the PEPV beam and measuring its optical intensity under the situations of propagating through free space and a focused system is shown in Fig. 1. A straightforward way for producing the PEPV beam is to impose a power-exponent phase into a Gaussian beam, in such a way as to modulate the complex field and optical intensity distribution. In our scheme, the reflective phase SLM is controlled by a personal computer (PC1), which is used to input the holograph into the SLM, and is illuminated by a linearly polarized He-Ne laser beam with λ=633nm. In addition, the beam is conditioned by a beam expander, and the waist width of the beam is set to be 1mm. The beam reflected by the SLM is regarded as the PEPV beam source. Then the intensity of the beam is detected by a charge-coupled device (CCD) after propagating through the distance z. In order to study its focusing properties, a thin lens with focal length f=300mm is seated before the CCD, which is located at the focal plane. A personal computer (PC2) connects the CCD, which is used to measure the corresponding intensity. The separation between the SLM and the thin lens is s. To avoid the reflection of the SLM, we select the first-order diffraction of the beam from the SLM. The patterns of the holograph for generating the PEPV beam of different power orders and topological charges are displayed in Fig. 2.

 

Fig. 1 Experimental setup for generating a PEPV beam and measuring its intensity properties in free space and focusing system. BE, beam expander; P, polarizer; SLM, spatial light modulator; L, thin lens; PC1, PC2, personal computers.

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Fig. 2 The holographs for the generation of the PEPV beam with different power orders n and topological charges m. (a) n=2,m=1; (b) n=2,m=2; (c) n=2,m=6; (d) n=4,m=2.

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In the following, we will investigate the intensity properties of PEPV beam propagating through free space and a focused system. We first consider the case of free space, in which the transfer matrix of distance z reads as [20]

(ABCD)=(1z01).

On substituting from Eq. (8) into Eq. (5), it reduces to

E(ρ,θ,z)=(iE02λzR)exp(ik2zρ2){exp(k2ρ24z2R)M0+l=1[p=0Γ(p+l/2+1)p!Γ(p+l+1)(k2ρ24z2R)p+l/2]×[exp(ilθ)Ml+exp(ilθ)Ml]},
where R=ik2z+1w2.

The distributions of the intensity illuminated by a PEPV beam at propagation distance z in free space are theoretically and experimentally shown in Fig. 3, where the dependencies on the power order n are illustrated. Figures 3(e)-3(h) represent the experimental results, which are well consistent with the theoretical calculation results shown in Figs. 3(a)-3(d). From these figures, we find that the intensity of the PEPV beam is closely determined by n. Moreover, being different from the intensity distribution of the traditional canonical vortex beam, which has a “doughnut-like” profile and contains a circular dark core with zero amplitude along the optical axis, the intensity pattern of the PEPV beam seem like the letter “C”, as Figs. 3(a) and 3(e) depicted. In Figs. 3(b) and 3(f), the “C-like” pattern evolves into a thwart-wise “U-like” pattern, and the PEPV beam has an oval dark core. When the power order n is gradually increased to 10, the intensity pattern would gradually evolve into a skew “L-like” pattern, or saying the “boomerang-like” pattern. Furthermore, the dark cores of these PEPV beam shifts away from the optical axis with the increase of the power order.

 

Fig. 3 Theoretical (a-d) and experimental (e-h) results of the intensity distributions of the PEPV beam with m=1 at the propagation distance z=0.7m for different values of power order n. (a, e)n=2; (b, f) n=4; (c, g) n=6; (d, h) n=10.

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In order to examine the influence of the power order n more explicitly, the intensity distributions of the PEPV beam with large power orders (n>>1) are theoretically studied and displayed in Fig. 4. Comparing Fig. 4(a) with Fig. 3(d), we can find that for the larger power order, the upper part of the “L-like” pattern would be brighter and wider, while the lower part of the pattern becomes narrower and shorter. When n is large enough [see Figs. 4(b, c)], the upper part of the pattern would dominate the whole pattern and the lower part would vanish. In addition, it should be noticed that the center of the beam returns back to the optical axis. Particularly, the profile of the PEPV beam with n=200 takes approximately a Gaussian form. This phenomenon could be well explained by the property of the phase function appearing in Eq. (1): when the power order n takes an extreme large value, the term of (φ/2π)n would approach to zero except the case that φ is very close to 2π, thus the phase function in Eq. (1) would be almost a constant. According to Eq. (2), when the phase function is a constant, the beam's profile would take a Gaussian form [22]. Hence, the PEPV beam would evolve into a Gaussian beam when n is great enough.

 

Fig. 4 Theoretical results of the optical intensity distribution of the PEPV beam with large power order. (a) n=20; (b) n=50; (c) n=100; (d) n=200. The other parameters are the same as in Fig. 3.

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The theoretical and experimental results for the intensity distributions and the corresponding phase contour of the PEPV beam with the power order n=2 versus the topological charge m are shown in Fig. 5. In Figs. 5(a) and 5(e), the patterns of optical intensity distribution are a “C-like” optical arc with an intensity spot at its upper end. As the topological charge m increases, the pattern of the optical intensity enlarges, and the energy of the beam starts to concentrate on the intensity spot. Meanwhile, the intensity of the lower part is weaken. Therefore, the topological charge could modulate the size of the optical pattern. In Figs. 5(i-l), it is obvious that the phase singularities are surrounded by the intensity arc, and the phase singularities are discrete on propagation, showing a different picture from the pattern of phase of the beam source and the canonical vortex.

 

Fig. 5 Theoretical (a-d) and experimental (e-f) results for the optical intensity distribution and the phase contour (i-l) of the PEPV beam with different topological charges when the power order n=2 and the propagation distance z=0.7m. (a, e, i) m=2; (b, f, j) m=4; (c, g, k) m=6; (d, h, l) m=10. The locations of singularities are labeled by white circles.

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As another particular example, we will now consider the focusing properties of the PEPV beam. The corresponding transfer matrix between the source plane and the observed plane in the focusing system is [20]

(ABCD)=(1f01)(101/f1)(1s01)=(0f1/f1s/f).

On substituting from Eq. (10) into Eq. (5), we obtain the following expression for the electric field of the PEPV beam in the focused plane

E(ρ,θ,z)=(iE0w22λf)exp[ik(fs)2f2ρ2]{exp[(kρw2f)2]M0+l=1[p=0Γ(p+l/2+1)p!Γ(p+l+1)(ikρw2f)2p+l]×[exp(ilθ)Ml+exp(ilθ)Ml]}.

The focused intensity distributions of the PEPV beam are shown in Fig. 6. It can be seen from Fig. 6 that the main part of the focused intensity likes a kidney in shape, and a bright optical intensity spot is formed inside the “kidney-like” pattern. In addition, such “kidney-like” patterns are accompanied by some dark cores on the left side, and some dim optical spots and lines might appear besides the “kidney-like” pattern.

 

Fig. 6 The theoretically (a-c) and experimentally (d-f) focused intensity distributions of the PEPV beam. (a, d) m=2, n=3; (b, e) m=3, n=4; (c, f) m=4, n=3.

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4. Error analysis

In order to evaluate the error between the results of numerical simulation and experiment, we can consider the error of the following equation, which is a part of the Eq. (5):

l=1[p=0Γ(p+l/2+1)p!Γ(p+l+1)(k2ρ24B2R)p+l/2]×[exp(ilθ)Ml+exp(ilθ)Ml].

Since the computer could not sum up infinite terms, it is necessary to set the upper summation limits for the indices p, l, h and j.

4.1 Selecting the suitable upper summation limit for p

According to the properties of Gamma function and Stirling’s formula, when p is great enough, we have

Γ(p+l/2+1)Γ(p+1)~Γ(p+l+1)Γ(p+l/2+1),Γ(x)~2πx(xe)x,
and we define the upper limit as pmax.

First let us discuss the part containing p in Eq. (12), which can be rewritten as

p=0Sp=p=0pmaxSp+p=pmax+1Sp,
where Sp=Γ(p+l/2+1)p!Γ(p+l+1)(k2ρ24B2R)p+l/2. Since the first term of the right-hand side in above equation has been used for simulation, only the error of the second term need to be estimated.

On substituting from Eq. (13) into the second term of the right-hand side of Eq. (14), the error Δp is:

Δp=p=pmax+1Γ(p+l/2+1)p!Γ(p+l+1)(k2ρ24B2R)p+l/2~p=pmax+11Γ(p+l/2+1)(k2ρ24B2R)p+l/2~p=pmax+112π(p+l/2+1)[k2ρ2e4B2R(p+l/2+1)]p+l/2.

Theoretically, the inequation |k2ρ2e4B2Rpmax|<1 need to be satisfied to make sure the summation is convergent. According to the parameters in the experiments, we can find out that |k2ρ2e4B2R|<85, thus we could set the maximum summation index pmaxas 100.

The error of this part is

Δp~p=pmax+112π(pmax)(k2ρ2e4B2Rpmax)p~1010.

4.2 Selecting a suitable upper summation limit lmax

We could find lmax through estimating the results of the Eq. (6) before the simulation. As an example, we would estimate Eq. (6) under the circumstance of m=2,n=2.

Approximately, we have

Ml=02πexp{i[2πm(φ2π)n+lφ]}dφq=0sexp{2πi[m(qs)n+lqs]}2πs,
where s is set as 400.

Since |Ml| would be close to zero when |l| is great enough, and we only choose the terms that dominate the simulation results, we would care about the ratio between |Ml| and |Ml|max, the maximum value of |Ml|. Thus, ln|Ml| would be helpful for selecting a suitable lmax (see Fig. 7).

 

Fig. 7 The relationship between estimated value of ln|Ml| and l. Because of the error of Eq. (17), the curve becomes nearly flat when |l|>30.

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According to the results, we can find that when |l|>15, the ratio between |Ml| and |Ml|max is less than e(3.5)1.40.008. Thus, we set lmax as 15 in the simulation, and it would be accurate enough.

4.3 Choosing maximum values for indices h and j in Eq. (6)

Defining Ml,j,h=ij+hmjlh(2π)j+h+1j!h!(nj+h+1), and Eq. (6) becomes

j=0h=0ij+hmjlh(2π)j+h+1j!h!(nj+h+1)=j=0h=0Ml,j,h,
dividing the summation into four parts:
j=0h=0Ml,j,h=j=0jmaxh=0hmaxMl,j,h+j=jmax+1h=0hmaxMl,j,h+j=0jmaxh=hmax+1Ml,j,h+j=jmax+1h=hmax+1Ml,j,h,
we can see that the error of this part is

ΔM=j=jmax+1h=0hmaxMl,j,h+j=0jmaxh=hmax+1Ml,j,h+j=jmax+1h=hmax+1Ml,j,h.

When both j and h are great enough, according to the Stirling’s formula, we have

Ml,j,h=ij+hmjlh(2π)j+h+1j!h!(nj+h+1)~e2(i2mπej+1)j(i2lπeh+1)h(nj+h+1)(j+1)(h+1)(j+1)(h+1).

Similarly, in order to make the summations are convergent, both terms 2mπej+1 and 2lπeh+1 should be less than 1. As a result, it is carried out that jmax>2πme69 and hmax>2πle=256. Thus, we set jmax as 80 and hmax as 300.

We can estimate the error produced by the last summation term with an approximation:

Ml,j,h=2π(2imπ)j(2ilπ)hj!h!(njmax+hmax+1),
thus, we have

j=jmax+1h=hmax+12π(2imπ)j(2ilπ)hj!h!(njmax+hmax+1)~2π(2×10-128×10-12i)(4×10158×1015)=5×10251025i.

Another approximation for estimating the first term of the error is:

Ml,j,h~2π(2imπ)jj!(2ilπ)h(njmax+1)h!,
thus

j=jmax+1h=0hmaxMl,j,h~h=0hmax2π(2ilπ)h(njmax+1)h![exp(2imπ)j=0jmax(2imπ)jj!]~2π(njmax+1)×(2×10-128×10-12i)=8×10-143×10-13i.

The approximation for estimating the second term is:

Ml,j,h~2π(2imπ)j(2ilπ)h(hmax+1)j!h!,
thus

j=0jmaxh=hmax+1Ml,j,h~j=0jmaxh=hmax+12π(2imπ)j(2ilπ)h(hmax+1)j!h!=8×10172×1016i.

We sum up the three terms and find that the total error |ΔMl| is approximately equal to 4×1013, which is negligible and all the Ml we take into consideration in our simulation are much greater than |ΔMl|.

4.4 Estimating the error caused by limitation of l and the relative error of Eq. (5)

The error caused by the range of l is (neglect the error caused by limitation of p, which is quite small):

Δl~l=lmax+1[p=0Γ(p+l/2+1)p!Γ(p+l+1)(k2ρ24B2R)p+l/2]×[exp(ilθ)Ml+exp(ilθ)Ml]~l=lmax+1(k2ρ24B2R)l/2[p=01p!(k2ρ24B2R)p]×|Mlmax|~exp(k2ρ24B2R)|Mlmax|(k2ρ24B2R)(lmax+1)/21(k2ρ24B2R)1/2~2×103.

As a result, since the magnitude of the results of Eq. (5) is approximately 101~101, we could see that the relative error of the simulation is approximately 2%.

5. Discussions

We will discuss the propagation dynamics of PEP vortices now. In order to discuss the evolution of intensity distribution and phase contour, we simulate the PEPV beam with different propagation distances z under the case of topological charge m is 4 and the power exponent number n is 3.

Theoretical distributions of intensity and phase of a PEPV beam with topological charge m=4 and power order n=3 at different propagation distances z in free space are shown in Fig. 8. In Fig. 8, we can see that the intensity distribution of z=1mis a “C-like” pattern. As the propagation distance increases, the lower part of the pattern would become dimmer, and the pattern would gradually evolve into a “bean-like” intensity spot. In addition, the phase singularities are getting farther and farther away from the axis.

 

Fig. 8 Theoretical (a-d) results for the optical intensity distribution and the phase contour (e-h) of the PEPV beam at different propagation distances z with m=4, n=3 . (a, e) z=1m; (b, f) z=2m; (c, g) z=4m; (d, h) z=8m. The locations of singularities are labeled by white circles.

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In some other researches on vortex beams, vortices would experience a rotation due to Guoy phase [23, 24]. Since the Collins formalism does not govern Guoy phase, it may be not so suitable for illustrating the phenomenon of some kinds of vortex beams that could be influenced by Guoy phase significantly. However, in our research, the evolution of PEP vortices is more likely to be determined by the phase factor exp[i2πm(φ2π)n]. Since all the PEP vortices are on the optical axis when they are in the source plane, and Eq. (5) shows that when z>0 and ρ=0, the electric field is not zero and the phase singularities are not on the axis, which means that the vortices would not on the optical axis once they leave the source plane. That is due to the fact that the integral of azimuthal angle φ is not zero when z>0 and ρ=0. In other words, the initial motion of the phase singularities is determined by the phase factor. In addition, in Fig. 5, we can see that as the topological charge m increases, the average distance between the axis and phase singularities is likely to be greater, however the Guoy phase of the Gaussian beam in the experiment are the same. Thus, the phase factor should be the major contributor of PEP vortices’ evolution.

6. Conclusions

In conclusion, we have derived an analytical propagation formula for the PEPV beam passing through a paraxial ABCD optical system. On this basis, the properties of the intensity of such a beam on propagation in free space and the focused system are investigated respectively. In addition, we also carried out the generation of the PEPV beam experimentally and measured its optical intensity distribution in these two cases. Experimental results are well consistent with the theoretical predictions, and it is shown that the power order of the beam can change the shape of the PEPV beam and the topological charge governs the size of the beam’s pattern. In the far field (or in the focal plane), the intensity distribution would present a “kidney-like” shape accompanied with several dark cores. Furthermore, the phase singularities are moving and discrete on propagation, and we discuss the reason of this phenomenon.

Due to this kind of unique propagation properties, the PEPV beam could be used in optical tweezers, optical manipulation and medical applications and have some advantages. One of the advantages of PEPV beam is that it could trap a group of tiny particles and then release them just through adjusting the parameters, such as the power n and the topological charge m. For example, since PEPV beam’s profile is a ring-like pattern when n is 1, tiny particles could be trapped by PEPV beam with power n=1. When these particles are going to be released, a gap could be produced on the optical ring through increasing PEPV beam’s power n, so that these particles could be pushed out of the optical trap by the non-uniform gradient force of the PEPV beam. In addition, since the location of the gap could be determined, the direction of these particles’ escaping motion could be estimated or also determined.

Funding

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11274273 and 11474253) and the Fundamental Research Funds for the Central Universities (2016FZA3004).

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References

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  1. P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73(5), 403–408 (1989).
    [Crossref]
  2. K. T. Gahagan and G. A. Swartzlander., “Optical vortex trapping of particles,” Opt. Lett. 21(11), 827–829 (1996).
    [Crossref] [PubMed]
  3. J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71(4), 549–554 (2000).
    [Crossref]
  4. C. Barbieri, F. Tamburini, G. Anzolin, A. Bianchini, E. Mari, A. Sponselli, G. Umbriaco, M. Prasciolu, F. Romanato, and P. Villoresi, “Light’s orbital angular momentum and optical vortices for astronomical coronagraphy from ground and space telescopes,” Earth Moon Planets 105(2–4), 283–288 (2009).
    [Crossref]
  5. A. Y. Bekshaev and A. I. Karamoch, “Spatial characteristics of vortex light beams produced by diffraction gratings with embedded phase singularity,” Opt. Commun. 281(6), 1366–1374 (2008).
    [Crossref]
  6. Z. S. Sacks, D. Rozas, and G. A. Swartzlander., “Holographic formation of optical-vortex filaments,” J. Opt. Soc. Am. B 15(8), 2226–2234 (1998).
    [Crossref]
  7. Y. S. Rumala and A. E. Leanhardt, “Multiple-beam interference in a spiral phase plate,” J. Opt. Soc. Am. B 30(3), 615–621 (2013).
    [Crossref]
  8. L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
    [Crossref] [PubMed]
  9. T. J. Alexander, A. A. Sukhorukov, and Y. S. Kivshar, “Asymmetric vortex solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 93(6), 063901 (2004).
    [Crossref] [PubMed]
  10. G. Molina-Terriza, E. M. Wright, and L. Torner, “Propagation and control of noncanonical optical vortices,” Opt. Lett. 26(3), 163–165 (2001).
    [Crossref] [PubMed]
  11. H. Li and J. Yin, “Generation of a vectorial Mathieu-like hollow beam with a periodically rotated polarization property,” Opt. Lett. 36(10), 1755–1757 (2011).
    [Crossref] [PubMed]
  12. I. V. Basistiy, V. A. Pas ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A, Pure Appl. Opt. 6(5), S166–S169 (2004).
    [Crossref]
  13. S. Tao, X. C. Yuan, J. Lin, X. Peng, and H. Niu, “Fractional optical vortex beam induced rotation of particles,” Opt. Express 13(20), 7726–7731 (2005).
    [Crossref] [PubMed]
  14. J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16(2), 993–1006 (2008).
    [Crossref] [PubMed]
  15. K. T. Kapale and J. P. Dowling, “Vortex phase qubit: Generating arbitrary, counterrotating, coherent superpositions in Bose-Einstein condensates via optical angular momentum beams,” Phys. Rev. Lett. 95(17), 173601 (2005).
    [Crossref] [PubMed]
  16. J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt. Lett. 28(11), 872–874 (2003).
    [Crossref] [PubMed]
  17. P. Li, S. Liu, T. Peng, G. Xie, X. Gan, and J. Zhao, “Spiral autofocusing Airy beams carrying power-exponent-phase vortices,” Opt. Express 22(7), 7598–7606 (2014).
    [Crossref] [PubMed]
  18. W. Luo, S. Cheng, and Z. Yuan, “Power-exponent-phase vortices for manipulating particles,” Acta Opt. Sin. 34(11), 1109001 (2014).
    [Crossref]
  19. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60(9), 1168–1177 (1970).
    [Crossref]
  20. S. Wang and D. Zhao, Matrix Optics (CHEP-Springer, Beijing, 2000).
  21. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, 1980).
  22. J. Alda, “Laser and Gaussian beam propagation and transformation,” in Encyclopedia of Optical Engineering (Dekker, 2003), 999–1013.
  23. S. M. Baumann, D. M. Kalb, L. H. MacMillan, and E. J. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express 17(12), 9818–9827 (2009).
    [Crossref] [PubMed]
  24. Y. S. Rumala, “Propagation of structured light beams after multiple reflections in a spiral phase plate,” Opt. Eng. 54(11), 111306 (2015).
    [Crossref]

2015 (1)

Y. S. Rumala, “Propagation of structured light beams after multiple reflections in a spiral phase plate,” Opt. Eng. 54(11), 111306 (2015).
[Crossref]

2014 (2)

P. Li, S. Liu, T. Peng, G. Xie, X. Gan, and J. Zhao, “Spiral autofocusing Airy beams carrying power-exponent-phase vortices,” Opt. Express 22(7), 7598–7606 (2014).
[Crossref] [PubMed]

W. Luo, S. Cheng, and Z. Yuan, “Power-exponent-phase vortices for manipulating particles,” Acta Opt. Sin. 34(11), 1109001 (2014).
[Crossref]

2013 (1)

2011 (1)

2009 (2)

C. Barbieri, F. Tamburini, G. Anzolin, A. Bianchini, E. Mari, A. Sponselli, G. Umbriaco, M. Prasciolu, F. Romanato, and P. Villoresi, “Light’s orbital angular momentum and optical vortices for astronomical coronagraphy from ground and space telescopes,” Earth Moon Planets 105(2–4), 283–288 (2009).
[Crossref]

S. M. Baumann, D. M. Kalb, L. H. MacMillan, and E. J. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express 17(12), 9818–9827 (2009).
[Crossref] [PubMed]

2008 (2)

A. Y. Bekshaev and A. I. Karamoch, “Spatial characteristics of vortex light beams produced by diffraction gratings with embedded phase singularity,” Opt. Commun. 281(6), 1366–1374 (2008).
[Crossref]

J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16(2), 993–1006 (2008).
[Crossref] [PubMed]

2005 (2)

K. T. Kapale and J. P. Dowling, “Vortex phase qubit: Generating arbitrary, counterrotating, coherent superpositions in Bose-Einstein condensates via optical angular momentum beams,” Phys. Rev. Lett. 95(17), 173601 (2005).
[Crossref] [PubMed]

S. Tao, X. C. Yuan, J. Lin, X. Peng, and H. Niu, “Fractional optical vortex beam induced rotation of particles,” Opt. Express 13(20), 7726–7731 (2005).
[Crossref] [PubMed]

2004 (2)

I. V. Basistiy, V. A. Pas ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A, Pure Appl. Opt. 6(5), S166–S169 (2004).
[Crossref]

T. J. Alexander, A. A. Sukhorukov, and Y. S. Kivshar, “Asymmetric vortex solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 93(6), 063901 (2004).
[Crossref] [PubMed]

2003 (1)

2001 (1)

2000 (1)

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71(4), 549–554 (2000).
[Crossref]

1998 (1)

1996 (1)

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

1989 (1)

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73(5), 403–408 (1989).
[Crossref]

1970 (1)

Alexander, T. J.

T. J. Alexander, A. A. Sukhorukov, and Y. S. Kivshar, “Asymmetric vortex solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 93(6), 063901 (2004).
[Crossref] [PubMed]

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Anzolin, G.

C. Barbieri, F. Tamburini, G. Anzolin, A. Bianchini, E. Mari, A. Sponselli, G. Umbriaco, M. Prasciolu, F. Romanato, and P. Villoresi, “Light’s orbital angular momentum and optical vortices for astronomical coronagraphy from ground and space telescopes,” Earth Moon Planets 105(2–4), 283–288 (2009).
[Crossref]

Arlt, J.

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71(4), 549–554 (2000).
[Crossref]

Barbieri, C.

C. Barbieri, F. Tamburini, G. Anzolin, A. Bianchini, E. Mari, A. Sponselli, G. Umbriaco, M. Prasciolu, F. Romanato, and P. Villoresi, “Light’s orbital angular momentum and optical vortices for astronomical coronagraphy from ground and space telescopes,” Earth Moon Planets 105(2–4), 283–288 (2009).
[Crossref]

Barnett, S. M.

Basistiy, I. V.

I. V. Basistiy, V. A. Pas ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A, Pure Appl. Opt. 6(5), S166–S169 (2004).
[Crossref]

Baumann, S. M.

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Bekshaev, A. Y.

A. Y. Bekshaev and A. I. Karamoch, “Spatial characteristics of vortex light beams produced by diffraction gratings with embedded phase singularity,” Opt. Commun. 281(6), 1366–1374 (2008).
[Crossref]

Bianchini, A.

C. Barbieri, F. Tamburini, G. Anzolin, A. Bianchini, E. Mari, A. Sponselli, G. Umbriaco, M. Prasciolu, F. Romanato, and P. Villoresi, “Light’s orbital angular momentum and optical vortices for astronomical coronagraphy from ground and space telescopes,” Earth Moon Planets 105(2–4), 283–288 (2009).
[Crossref]

Cheng, S.

W. Luo, S. Cheng, and Z. Yuan, “Power-exponent-phase vortices for manipulating particles,” Acta Opt. Sin. 34(11), 1109001 (2014).
[Crossref]

Collins, S. A.

Coullet, P.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73(5), 403–408 (1989).
[Crossref]

Curtis, J. E.

Dholakia, K.

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71(4), 549–554 (2000).
[Crossref]

Dowling, J. P.

K. T. Kapale and J. P. Dowling, “Vortex phase qubit: Generating arbitrary, counterrotating, coherent superpositions in Bose-Einstein condensates via optical angular momentum beams,” Phys. Rev. Lett. 95(17), 173601 (2005).
[Crossref] [PubMed]

Flossmann, F.

Franke-Arnold, S.

Gahagan, K. T.

Galvez, E. J.

Gan, X.

Gil, L.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73(5), 403–408 (1989).
[Crossref]

Götte, J. B.

Grier, D. G.

Hitomi, T.

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71(4), 549–554 (2000).
[Crossref]

Kalb, D. M.

Kapale, K. T.

K. T. Kapale and J. P. Dowling, “Vortex phase qubit: Generating arbitrary, counterrotating, coherent superpositions in Bose-Einstein condensates via optical angular momentum beams,” Phys. Rev. Lett. 95(17), 173601 (2005).
[Crossref] [PubMed]

Karamoch, A. I.

A. Y. Bekshaev and A. I. Karamoch, “Spatial characteristics of vortex light beams produced by diffraction gratings with embedded phase singularity,” Opt. Commun. 281(6), 1366–1374 (2008).
[Crossref]

Kivshar, Y. S.

T. J. Alexander, A. A. Sukhorukov, and Y. S. Kivshar, “Asymmetric vortex solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 93(6), 063901 (2004).
[Crossref] [PubMed]

Leanhardt, A. E.

Li, H.

Li, P.

Lin, J.

Liu, S.

Luo, W.

W. Luo, S. Cheng, and Z. Yuan, “Power-exponent-phase vortices for manipulating particles,” Acta Opt. Sin. 34(11), 1109001 (2014).
[Crossref]

MacMillan, L. H.

Mari, E.

C. Barbieri, F. Tamburini, G. Anzolin, A. Bianchini, E. Mari, A. Sponselli, G. Umbriaco, M. Prasciolu, F. Romanato, and P. Villoresi, “Light’s orbital angular momentum and optical vortices for astronomical coronagraphy from ground and space telescopes,” Earth Moon Planets 105(2–4), 283–288 (2009).
[Crossref]

Molina-Terriza, G.

Niu, H.

O’Holleran, K.

Padgett, M. J.

Pas ko, V. A.

I. V. Basistiy, V. A. Pas ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A, Pure Appl. Opt. 6(5), S166–S169 (2004).
[Crossref]

Peng, T.

Peng, X.

Prasciolu, M.

C. Barbieri, F. Tamburini, G. Anzolin, A. Bianchini, E. Mari, A. Sponselli, G. Umbriaco, M. Prasciolu, F. Romanato, and P. Villoresi, “Light’s orbital angular momentum and optical vortices for astronomical coronagraphy from ground and space telescopes,” Earth Moon Planets 105(2–4), 283–288 (2009).
[Crossref]

Preece, D.

Rocca, F.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73(5), 403–408 (1989).
[Crossref]

Romanato, F.

C. Barbieri, F. Tamburini, G. Anzolin, A. Bianchini, E. Mari, A. Sponselli, G. Umbriaco, M. Prasciolu, F. Romanato, and P. Villoresi, “Light’s orbital angular momentum and optical vortices for astronomical coronagraphy from ground and space telescopes,” Earth Moon Planets 105(2–4), 283–288 (2009).
[Crossref]

Rozas, D.

Rumala, Y. S.

Y. S. Rumala, “Propagation of structured light beams after multiple reflections in a spiral phase plate,” Opt. Eng. 54(11), 111306 (2015).
[Crossref]

Y. S. Rumala and A. E. Leanhardt, “Multiple-beam interference in a spiral phase plate,” J. Opt. Soc. Am. B 30(3), 615–621 (2013).
[Crossref]

Sacks, Z. S.

Slyusar, V. V.

I. V. Basistiy, V. A. Pas ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A, Pure Appl. Opt. 6(5), S166–S169 (2004).
[Crossref]

Soskin, M. S.

I. V. Basistiy, V. A. Pas ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A, Pure Appl. Opt. 6(5), S166–S169 (2004).
[Crossref]

Sponselli, A.

C. Barbieri, F. Tamburini, G. Anzolin, A. Bianchini, E. Mari, A. Sponselli, G. Umbriaco, M. Prasciolu, F. Romanato, and P. Villoresi, “Light’s orbital angular momentum and optical vortices for astronomical coronagraphy from ground and space telescopes,” Earth Moon Planets 105(2–4), 283–288 (2009).
[Crossref]

Spreeuw, R. J.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Sukhorukov, A. A.

T. J. Alexander, A. A. Sukhorukov, and Y. S. Kivshar, “Asymmetric vortex solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 93(6), 063901 (2004).
[Crossref] [PubMed]

Swartzlander, G. A.

Tamburini, F.

C. Barbieri, F. Tamburini, G. Anzolin, A. Bianchini, E. Mari, A. Sponselli, G. Umbriaco, M. Prasciolu, F. Romanato, and P. Villoresi, “Light’s orbital angular momentum and optical vortices for astronomical coronagraphy from ground and space telescopes,” Earth Moon Planets 105(2–4), 283–288 (2009).
[Crossref]

Tao, S.

Torner, L.

Umbriaco, G.

C. Barbieri, F. Tamburini, G. Anzolin, A. Bianchini, E. Mari, A. Sponselli, G. Umbriaco, M. Prasciolu, F. Romanato, and P. Villoresi, “Light’s orbital angular momentum and optical vortices for astronomical coronagraphy from ground and space telescopes,” Earth Moon Planets 105(2–4), 283–288 (2009).
[Crossref]

Vasnetsov, M. V.

I. V. Basistiy, V. A. Pas ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A, Pure Appl. Opt. 6(5), S166–S169 (2004).
[Crossref]

Villoresi, P.

C. Barbieri, F. Tamburini, G. Anzolin, A. Bianchini, E. Mari, A. Sponselli, G. Umbriaco, M. Prasciolu, F. Romanato, and P. Villoresi, “Light’s orbital angular momentum and optical vortices for astronomical coronagraphy from ground and space telescopes,” Earth Moon Planets 105(2–4), 283–288 (2009).
[Crossref]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Wright, E. M.

Xie, G.

Yin, J.

Yuan, X. C.

Yuan, Z.

W. Luo, S. Cheng, and Z. Yuan, “Power-exponent-phase vortices for manipulating particles,” Acta Opt. Sin. 34(11), 1109001 (2014).
[Crossref]

Zhao, J.

Acta Opt. Sin. (1)

W. Luo, S. Cheng, and Z. Yuan, “Power-exponent-phase vortices for manipulating particles,” Acta Opt. Sin. 34(11), 1109001 (2014).
[Crossref]

Appl. Phys. B (1)

J. Arlt, T. Hitomi, and K. Dholakia, “Atom guiding along Laguerre-Gaussian and Bessel light beams,” Appl. Phys. B 71(4), 549–554 (2000).
[Crossref]

Earth Moon Planets (1)

C. Barbieri, F. Tamburini, G. Anzolin, A. Bianchini, E. Mari, A. Sponselli, G. Umbriaco, M. Prasciolu, F. Romanato, and P. Villoresi, “Light’s orbital angular momentum and optical vortices for astronomical coronagraphy from ground and space telescopes,” Earth Moon Planets 105(2–4), 283–288 (2009).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

I. V. Basistiy, V. A. Pas ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A, Pure Appl. Opt. 6(5), S166–S169 (2004).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (2)

Opt. Commun. (2)

A. Y. Bekshaev and A. I. Karamoch, “Spatial characteristics of vortex light beams produced by diffraction gratings with embedded phase singularity,” Opt. Commun. 281(6), 1366–1374 (2008).
[Crossref]

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73(5), 403–408 (1989).
[Crossref]

Opt. Eng. (1)

Y. S. Rumala, “Propagation of structured light beams after multiple reflections in a spiral phase plate,” Opt. Eng. 54(11), 111306 (2015).
[Crossref]

Opt. Express (4)

Opt. Lett. (4)

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Phys. Rev. Lett. (2)

T. J. Alexander, A. A. Sukhorukov, and Y. S. Kivshar, “Asymmetric vortex solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 93(6), 063901 (2004).
[Crossref] [PubMed]

K. T. Kapale and J. P. Dowling, “Vortex phase qubit: Generating arbitrary, counterrotating, coherent superpositions in Bose-Einstein condensates via optical angular momentum beams,” Phys. Rev. Lett. 95(17), 173601 (2005).
[Crossref] [PubMed]

Other (3)

S. Wang and D. Zhao, Matrix Optics (CHEP-Springer, Beijing, 2000).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, 1980).

J. Alda, “Laser and Gaussian beam propagation and transformation,” in Encyclopedia of Optical Engineering (Dekker, 2003), 999–1013.

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Figures (8)

Fig. 1
Fig. 1 Experimental setup for generating a PEPV beam and measuring its intensity properties in free space and focusing system. BE, beam expander; P, polarizer; SLM, spatial light modulator; L, thin lens; PC1, PC2, personal computers.
Fig. 2
Fig. 2 The holographs for the generation of the PEPV beam with different power orders n and topological charges m . (a) n = 2 , m = 1 ; (b) n = 2 , m = 2 ; (c) n = 2 , m = 6 ; (d) n = 4 , m = 2 .
Fig. 3
Fig. 3 Theoretical (a-d) and experimental (e-h) results of the intensity distributions of the PEPV beam with m = 1 at the propagation distance z = 0.7 m for different values of power order n . (a, e) n = 2 ; (b, f) n = 4 ; (c, g) n = 6 ; (d, h) n = 10 .
Fig. 4
Fig. 4 Theoretical results of the optical intensity distribution of the PEPV beam with large power order. (a) n = 20 ; (b) n = 50 ; (c) n = 100 ; (d) n = 200 . The other parameters are the same as in Fig. 3.
Fig. 5
Fig. 5 Theoretical (a-d) and experimental (e-f) results for the optical intensity distribution and the phase contour (i-l) of the PEPV beam with different topological charges when the power order n = 2 and the propagation distance z = 0.7 m . (a, e, i) m = 2 ; (b, f, j) m = 4 ; (c, g, k) m = 6 ; (d, h, l) m = 10 . The locations of singularities are labeled by white circles.
Fig. 6
Fig. 6 The theoretically (a-c) and experimentally (d-f) focused intensity distributions of the PEPV beam. (a, d) m = 2 , n = 3 ; (b, e) m = 3 , n = 4 ; (c, f) m = 4 , n = 3 .
Fig. 7
Fig. 7 The relationship between estimated value of ln | M l | and l . Because of the error of Eq. (17), the curve becomes nearly flat when | l | > 30 .
Fig. 8
Fig. 8 Theoretical (a-d) results for the optical intensity distribution and the phase contour (e-h) of the PEPV beam at different propagation distances z with m = 4 , n = 3 . (a, e) z = 1 m ; (b, f) z = 2 m ; (c, g) z = 4 m ; (d, h) z = 8 m . The locations of singularities are labeled by white circles.

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

E ( 0 ) ( r , φ ) = A 0 exp ( r 2 w 2 ) exp [ i 2 m π ( φ 2 π ) n ] ,
E ( ρ , θ , z ) = i k 2 π B 0 0 2 π E ( 0 ) ( r , φ ) exp { i k 2 B [ A r 2 2 r ρ cos ( θ φ ) + D ρ 2 ] } r d r d φ ,
E ( ρ , θ , z ) = i k 2 π B exp ( i k D ρ 2 2 B ) 0 0 2 π exp ( r 2 w 2 ) exp ( i k A r 2 2 B ) × exp [ i k ρ r B cos ( θ φ ) ] exp [ i 2 m π ( φ 2 π ) n ] r d r d φ .
exp [ i k r ρ B cos ( φ θ ) ] = h = i h J h ( k r ρ B ) exp [ i h ( φ θ ) ] , J l ( x ) = ( 1 ) l J l ( x ) , J l ( x ) = p = 0 ( 1 ) p 1 p ! Γ ( l + p + 1 ) ( x 2 ) l + 2 p , Γ ( x ) = 0 exp ( t ) t x 1 d t , exp ( s x n ) = j = 0 s j x n j j ! ,
E ( ρ , θ , z ) = ( i 2 λ B R ) exp ( i k D 2 B ρ 2 ) { exp ( k 2 ρ 2 4 B 2 R ) M 0 + l = 1 [ p = 0 Γ ( p + l / 2 + 1 ) p ! Γ ( p + l + 1 ) ( k 2 ρ 2 4 B 2 R ) p + l / 2 ] × [ exp ( i l θ ) M l + exp ( i l θ ) M l ] } ,
M l = 0 2 π exp { i [ 2 π m ( φ 2 π ) n + l φ ] } d φ = { j = 0 h = 0 i j + h m j l h ( 2 π ) j + h + 1 j ! h ! ( n j + h + 1 ) , l 0 , j = 0 i j m j ( 2 π ) j + 1 j ! ( n j + 1 ) , l = 0.
I ( ρ , θ , z ) = E * ( ρ , θ , z ) E ( ρ , θ , z ) .
( A B C D ) = ( 1 z 0 1 ) .
E ( ρ , θ , z ) = ( i E 0 2 λ z R ) exp ( i k 2 z ρ 2 ) { exp ( k 2 ρ 2 4 z 2 R ) M 0 + l = 1 [ p = 0 Γ ( p + l / 2 + 1 ) p ! Γ ( p + l + 1 ) ( k 2 ρ 2 4 z 2 R ) p + l / 2 ] × [ exp ( i l θ ) M l + exp ( i l θ ) M l ] } ,
( A B C D ) = ( 1 f 0 1 ) ( 1 0 1 / f 1 ) ( 1 s 0 1 ) = ( 0 f 1 / f 1 s / f ) .
E ( ρ , θ , z ) = ( i E 0 w 2 2 λ f ) exp [ i k ( f s ) 2 f 2 ρ 2 ] { exp [ ( k ρ w 2 f ) 2 ] M 0 + l = 1 [ p = 0 Γ ( p + l / 2 + 1 ) p ! Γ ( p + l + 1 ) ( i k ρ w 2 f ) 2 p + l ] × [ exp ( i l θ ) M l + exp ( i l θ ) M l ] } .
l = 1 [ p = 0 Γ ( p + l / 2 + 1 ) p ! Γ ( p + l + 1 ) ( k 2 ρ 2 4 B 2 R ) p + l / 2 ] × [ exp ( i l θ ) M l + exp ( i l θ ) M l ] .
Γ ( p + l / 2 + 1 ) Γ ( p + 1 ) ~ Γ ( p + l + 1 ) Γ ( p + l / 2 + 1 ) , Γ ( x ) ~ 2 π x ( x e ) x ,
p = 0 S p = p = 0 p max S p + p = p max + 1 S p ,
Δ p = p = p max + 1 Γ ( p + l / 2 + 1 ) p ! Γ ( p + l + 1 ) ( k 2 ρ 2 4 B 2 R ) p + l / 2 ~ p = p max + 1 1 Γ ( p + l / 2 + 1 ) ( k 2 ρ 2 4 B 2 R ) p + l / 2 ~ p = p max + 1 1 2 π ( p + l / 2 + 1 ) [ k 2 ρ 2 e 4 B 2 R ( p + l / 2 + 1 ) ] p + l / 2 .
Δ p ~ p = p max + 1 1 2 π ( p max ) ( k 2 ρ 2 e 4 B 2 R p max ) p ~ 10 10 .
M l = 0 2 π exp { i [ 2 π m ( φ 2 π ) n + l φ ] } d φ q = 0 s exp { 2 π i [ m ( q s ) n + l q s ] } 2 π s ,
j = 0 h = 0 i j + h m j l h ( 2 π ) j + h + 1 j ! h ! ( n j + h + 1 ) = j = 0 h = 0 M l , j , h ,
j = 0 h = 0 M l , j , h = j = 0 j max h = 0 h max M l , j , h + j = j max + 1 h = 0 h max M l , j , h + j = 0 j max h = h max + 1 M l , j , h + j = j max + 1 h = h max + 1 M l , j , h ,
Δ M = j = j max + 1 h = 0 h max M l , j , h + j = 0 j max h = h max + 1 M l , j , h + j = j max + 1 h = h max + 1 M l , j , h .
M l , j , h = i j + h m j l h ( 2 π ) j + h + 1 j ! h ! ( n j + h + 1 ) ~ e 2 ( i 2 m π e j + 1 ) j ( i 2 l π e h + 1 ) h ( n j + h + 1 ) ( j + 1 ) ( h + 1 ) ( j + 1 ) ( h + 1 ) .
M l , j , h = 2 π ( 2 i m π ) j ( 2 i l π ) h j ! h ! ( n j max + h max + 1 ) ,
j = j max + 1 h = h max + 1 2 π ( 2 i m π ) j ( 2 i l π ) h j ! h ! ( n j max + h max + 1 ) ~ 2 π ( 2 × 10 -12 8 × 10 -12 i ) ( 4 × 10 15 8 × 10 15 ) = 5 × 10 25 10 25 i .
M l , j , h ~ 2 π ( 2 i m π ) j j ! ( 2 i l π ) h ( n j max + 1 ) h ! ,
j = j max + 1 h = 0 h max M l , j , h ~ h = 0 h max 2 π ( 2 i l π ) h ( n j max + 1 ) h ! [ exp ( 2 i m π ) j = 0 j max ( 2 i m π ) j j ! ] ~ 2 π ( n j max + 1 ) × ( 2 × 10 -12 8 × 10 -12 i ) = 8 × 10 -14 3 × 10 -13 i .
M l , j , h ~ 2 π ( 2 i m π ) j ( 2 i l π ) h ( h max + 1 ) j ! h ! ,
j = 0 j max h = h max + 1 M l , j , h ~ j = 0 j max h = h max + 1 2 π ( 2 i m π ) j ( 2 i l π ) h ( h max + 1 ) j ! h ! = 8 × 10 17 2 × 10 16 i .
Δ l ~ l = l max + 1 [ p = 0 Γ ( p + l / 2 + 1 ) p ! Γ ( p + l + 1 ) ( k 2 ρ 2 4 B 2 R ) p + l / 2 ] × [ exp ( i l θ ) M l + exp ( i l θ ) M l ] ~ l = l max + 1 ( k 2 ρ 2 4 B 2 R ) l / 2 [ p = 0 1 p ! ( k 2 ρ 2 4 B 2 R ) p ] × | M l max | ~ exp ( k 2 ρ 2 4 B 2 R ) | M l max | ( k 2 ρ 2 4 B 2 R ) ( l max + 1 ) / 2 1 ( k 2 ρ 2 4 B 2 R ) 1 / 2 ~ 2 × 10 3 .

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