Abstract

On the basis of the vectorial Rayleigh diffraction integral, the analytical expressions for the electromagnetic fields of the radially polarized beams diffracted at a circular aperture are derived, which helps us investigate the propagation properties of the apertured radially polarized beams in the nonparaxial and paraxial regimes. The unapertured and paraxial cases can be viewed as the special cases of the general result obtained in this paper. The analyses indicate that the nonparaxiality of the apertured radially polarized beams depends on the ratio of the waist width to the wavelength and the truncation parameter. In addition, the truncation parameter and the beam order have a great impact on the beam diffraction effect and the beam evolution behavior.

© 2010 OSA

1. Introduction

It is well known that radial and azimuthal polarizations can be treated as two particular cases of the novel nonuniformly polarized beams introduced by Gori [1]. The cylindrical symmetry of the electric field vectors for the radially and azimuthally polarized beams make them draw growing attention to some applications such as electron acceleration, particle trapping, scanning optical microscopy, elimination of thermally induced birefringence effects, laser cutting and material processing [27]. Because of these unique advantages, various methods for generating the radially polarized beams have been reported [812]. The radially polarized Laguerre-Gaussian beams (LGBs) can be obtained by use of a radial Brewster window inside a circularly symmetric laser resonator [8].

The standard paraxial theory based on the parabolic equation fails at describing the beam propagation when the beam waist is comparable or even smaller than the wavelength. Therefore, the study of the beam propagation in the nonparaxial regime becomes increasingly important. For example, Deng et al. presented the nonparaxial propagation and the analytical vectorial structure of the radially polarized light beams by the vectorial Rayleigh diffraction integral and the vector angular spectrum method, respectively [1315]. Yan et al. gave the description of the radially polarized Gaussian beam and Laguerre-Gaussian beam beyond the paraxial approximation on the basis of the complex-source-point spherical waves approach and the perturbative Lax series expansion method, respectively [16, 17]. However, previous research has been restricted to the unapertured case. There are usually apertures in the practical optical systems, thus one may be curious to know the nonparaxial properties of the apertured radially polarized beams. In this paper, we represent the propagation properties of the radially polarized LGBs diffracted at a circular aperture in the nonparaxial and paraxial regimes. The nonparaxial behavior and the beam diffraction effect should be both taken into account under the existence of the aperture.

The paper is organized as follow: In Section 2, the analytical expressions for the electromagnetic fields of the radially polarized LGBs diffracted at a circular aperture are derived. In Section 3, numerical results are presented to illustrate the propagation properties in detail. The main conclusion is given in Section 4.

2. Propagation of radially polarized beams diffracted at a circular aperture

In the Cartesian coordinate system, the transverse electric field distribution of a radially polarized LGB at the plane z=0 has the form [8]

En10(x0,y0,0)=En1x0(x0,y0,0)ex+En1y0(x0,y0,0)ey
En1x0(x0,y0,0)=2E0x0ω0Ln1(2ρ02ω02)exp(ρ02ω02)
En1y0(x0,y0,0)=2E0y0ω0Ln1(2ρ02ω02)exp(ρ02ω02)
where ex and ey are the unit vectors in the x and y directions, respectively, ρ0=(x02+y02)1/2, ω0 is the waist width of the fundamental mode, Ln1() is the Laguerre polynomial of order n and index 1, and E0 is an amplitude constant. Equation (1) implies that the radially polarized beams have ring intensity distribution and their electric field vectors at the plane of the beam cross section point to the radial direction. Assume that a circular aperture with radius a is located at the plane z=0. The transverse electric field just behind the aperture reads as
En1x(x0,y0,0)=2E0x0ω0Ln1(2ρ02ω02)exp(ρ02ω02)t(x0,y0)
En1y(x0,y0,0)=2E0y0ω0Ln1(2ρ02ω02)exp(ρ02ω02)t(x0,y0)
t(x0,y0)={1x02+y02a20otherwise
where t(x0,y0) is the window function of the aperture.

According to the vectorial Rayleigh diffraction integral [18], the forward monochromatic electric field propagating in a homogeneous medium of refractive index can be given once its transverse components are known at the plane z=0.

En1x(r)=12π+En1x(x0,y0,0)G(r,r0)zdx0dy0
En1y(r)=12π+En1y(x0,y0,0)G(r,r0)zdx0dy0
En1z(r)=12π+[En1x(x0,y0,0)G(r,r0)x+En1y(x0,y0,0)G(r,r0)y]dx0dy0
where r0=x0ex+y0ey and r=xex+yey+zez, ez is the unit vector in the z direction,
G(r,r0)=exp(ik|rr0|)|rr0|
k is the wave number, and |rr0| can be approximately expanded into [14,19]
|rr0|r+x02+y022xx02yy02r
where r=(x2+y2+z2)1/2. Substituting Eqs. (4), (5), (6), (10), (11) into Eqs. (7)-(9) and transforming the Cartesian coordinate system to the cylindrical one, one obtains
En1x(r)=2E0kzxρexp(ikr)r2ω00aρ02Ln1(2ρ02ω02)exp(gρ02)J1(kρρ0r)dρ0
En1y(r)=2E0kzyρexp(ikr)r2ω00aρ02Ln1(2ρ02ω02)exp(gρ02)J1(kρρ0r)dρ0
En1z(r)=2E0kexp(ikr)r2ω0[ρ0aρ02Ln1(2ρ02ω02)exp(gρ02)J1(kρρ0r)dρ0i0aρ03Ln1(2ρ02ω02)exp(gρ02)J0(kρρ0r)dρ0]
where ρ=(x2+y2)1/2, g=ik/(2r)1/ω02, J0() and J1() are the zeroth-order and the first-order Bessel functions of the first kind, respectively. To the best of our knowledge, previous study dealt with the apertured Gaussian beams with linear polarization [19,20], but it has not be extend to inhomogeneous polarization. In this paper, Eqs. (12)-(14) depict the nonparaxial propagation for the electric field of the apertured radially polarized LGBs in the integral form.

When one expands the Bessel functions and the Laguerre polynomial into series and recalls the incomplete gamma function [21],

Γ(α,x)=xettα1dt
the integrals of Eqs. (12)-(14) turn out to be
En1(r)=2E0kexp(ikr)r2l=0nu=0(n+1)!2l1(l+1)!l!(nl)!ω02l+1{zρ(xex+yey)1u!(u+1)!(kρ2r)2u+1+ez[ρu!(u+1)!(kρ2r)2u+1+iu!u!(kρ2r)2u]}[Γ(l+u+2,ga2)(l+u+1)!]gl+u+2
The analytical expression for the electric field of the apertured radially polarized LGBs is presented by Eq. (16), which indicates that the electric field retains the cylindrical symmetry under the propagation.

By taking the curl of Eq. (16), the analytical expression for the corresponding magnetic field reads as

Hn1(r)=i2εμE0exp(ikr)(yexxey)l=0nu=0(n+1)!2l1(l+1)!l!(nl)!ω02l+1×{(ikr2)r4[ρu!(u+1)!(kρ2r)2u+1iu!u!(kρ2r)2u][Γ(l+u+2,ga2)(l+u+1)!]gl+u+2+[2+(ikr2)z2r2]1ρr2u!(u+1)!(kρ2r)2u+1[Γ(l+u+2,ga2)(l+u+1)!]gl+u+2+k2r5[(i(3ρ2r2ρ)1u!(u+1)!(kρ2r)2u+1+z2u!(u+2)!(kρ2r)2u+2+1u!u!(kρ2r)2u)×[Γ(l+u+3,ga2)(l+u+2)!]gl+u+3+z2u!u!(kρ2r)2u[Γ(l+u+2,ga2)(l+u+1)!]gl+u+2]kz22r5[(iρu!(u+1)!(kρ2r)2u+1+1u!(u+2)!(kρ2r)2u+2)[Γ(l+u+3,ga2)(l+u+2)!]gl+u+3+1u!u!(kρ2r)2u[Γ(l+u+2,ga2)(l+u+1)!]gl+u+2]}
where ε is the electric permittivity, and μ is the magnetic permeability. Equations (16) and (17) are the basic result obtained in this paper, which describe the nonparaxial propagation of the radially polarized LGBs diffracted at a circular aperture under the condition zω0. From Eq. (17), one can find that the magnetic field of the radially polarized beams takes on the azimuthal polarization and its longitudinal component vanishes. For the unapertured case a, Eq. (16) are consistent with Eqs. (10)-(12) in Ref. 13, which described the nonparaxial propagation of a radially polarized beam in free space.

Within the framework of the paraxial approximation, one can expand r into series and keep the first and second terms

rz+x2+y22z
Replacing r of the exponential part in Eqs. (16) and (17) with Eq. (18) and the other terms with z, one obtains
En1p(r)=2E0kexp[ik(z+ρ22z)]l=0nu=0(n+1)!2l1(l+1)!l!(nl)!ω02l+1{1ρz(xex+yey)×1u!(u+1)!(kρ2z)2u+1+ez1z2[ρu!(u+1)!(kρ2z)2u+1+iu!u!(kρ2z)2u]}×[Γ(l+u+2,ta2)(l+u+1)!]tl+u+2
Hn1p(r)=i2εμE0exp[ik(z+ρ22z)](yexxey)l=0nu=0(n+1)!2l1(l+1)!l!(nl)!ω02l+1×{(ikz2)z4[ρu!(u+1)!(kρ2z)2u+1iu!u!(kρ2z)2u][Γ(l+u+2,ta2)(l+u+1)!]tl+u+2+ikρzu!(u+1)!(kρ2z)2u+1[Γ(l+u+2,ta2)(l+u+1)!]tl+u+2+k2z5[(i(3ρ2z2ρ)1u!(u+1)!(kρ2z)2u+1+z2u!(u+2)!(kρ2z)2u+2+1u!u!(kρ2z)2u)×[Γ(l+u+3,ta2)(l+u+2)!]tl+u+3+z2u!u!(kρ2z)2u[Γ(l+u+2,ta2)(l+u+1)!]tl+u+2]k2z3[(iρu!(u+1)!(kρ2z)2u+1+1u!(u+2)!(kρ2z)2u+2)[Γ(l+u+3,ta2)(l+u+2)!]tl+u+3+1u!u!(kρ2z)2u[Γ(l+u+2,ta2)(l+u+1)!]tl+u+2]}
where t=ik/(2z)1/ω02. Equations (19) and (20) are the propagation expressions for the electromagnetic fields of the apertured radially polarized LGBs under the paraxial approximation.

3. Numerical results and discussions

In order to illustrate the nonparaxial behavior and the beam diffraction effect of the radially polarized LGBs through a circular aperture, detailed numerical calculations have been performed using the formulae derived in Section 2. All length measurements are in units of wavelengths, E0=1, k=2π/λ, zR=πω02/λ is the Rayleigh length, and δ=a/ω0 is the truncation parameter. The energy flux distribution at the plane z=const can be described by the z component of the time-average Poynting vector. Sz=12Re[En1(r)×Hn1*(r)]z and Szp=12Re[En1p(r)×Hn1p*(r)]z denote the nonparaxial and paraxial energy flux distributions of the apertured radially polarized LGBs, respectively, where Re is the real part and the asterisk means the complex conjugate. The energy flux distributions of the apertured radially polarized beams at the plane z=10zR are presented for different parameters of n=0, ω0/λ=2, δ=2 in Fig. 1 ; n=0, ω0/λ=1, δ=2 in Fig. 2 ; n=0, ω0/λ=1, δ=0.5 in Fig. 3 ; n=2, ω0/λ=2, δ=2 in Fig. 4 ; n=2, ω0/λ=1, δ=2 in Fig. 5 and n=2, ω0/λ=1, δ=0.5 in Fig. 6 . The panels (a) and (b) in Figs. 1-6 exhibit the nonparaxial and paraxial energy flux distributions versus x/λ and y/λ. Solid and dashed curves in the panel (c) corresponding to the panels (a) and (b) are the cross-section energy flux distributions at y=0.

 

Fig. 1 Nonparaxial (a) and paraxial (b) energy flux distributions of the apertured radially polarized beams versusx/λ and y/λ at the plane z=10zR for n=0, ω0/λ=2, δ=2. Solid and dashed curves in the panel (c) denote the corresponding cross-section energy flux distributions at y=0 for the nonparaxial and paraxial cases, respectively.

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Fig. 2 The same legend as in Fig. 1 except that the parameters n=0, ω0/λ=1, δ=2.

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Fig. 3 The same legend as in Fig. 1 except that the parameters n=0, ω0/λ=1, δ=0.5.

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Fig. 4 The same legend as in Fig. 1 except that the parameters n=2, ω0/λ=2, δ=2.

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Fig. 5 The same legend as in Fig. 1 except that the parameters n=2, ω0/λ=1, δ=2.

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Fig. 6 The same legend as in Fig. 1 except that the parameters n=2, ω0/λ=1, δ=0.5.

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Figures 1-6 show that the energy flux distributions of the apertured radially polarized beams have the cylindrical symmetry and the dark center. It is easy to find that both the ratio of the waist width to the wavelength and the truncation parameter affect the nonparaxial behavior of the apertured radially polarized beams, and the beam diffraction effect increases with the truncation parameter decreasing. In Fig. 1, there is little difference between the nonparaxial and the paraxial energy flux distributions for a large value of ω0/λ, whereas the difference in Fig. 2 is no longer negligible with the parameter ω0/λ decreasing. As the truncation parameter decreases, the difference becomes more obvious in Fig. 3, additionally, the beam diffraction effect caused by a circular aperture should be considered. For the higher-order radially polarized beams (see Fig. 4-6), the beam order n enhances the nonparaxiality and the beam diffraction effect. As illustrated in Figs. 4 and 5, the brightest position of the nonparaxial and paraxial energy flux distributions exists at the outer ring due to the circular aperture effect. A comparison between Figs. 3 and 6 indicates that the beam order n only influences the magnitude of the energy flux, whereas their profiles keep the same when the truncation parameter becomes small enough. This phenomenon explains that the decreasing aperture radius makes the initial fields be confined into smaller area. The circular aperture blocks the outer ring part of the radially polarized beams, but the central part is maintained.

To further weigh the effect of the waist width, the aperture radius and the beam order on the nonparaxial behavior of the apertured radially polarized LGBs, the normalized maximum intensity of the longitudinal component of the electric field Iz,max(x,0,10zR)/Ix,max(x,0,10zR) versus ω0/λ are plotted in Fig. 7(a) for different values of δ=0.5 (solid curve), 1 (dashed curve), 3 (dotted curve), n=0 and in Fig. 7(b) for different values of n=0 (solid curve), 1 (dashed curve), 2 (dotted curve), δ=1, where Iz(x,0,z)=|Ez(x,0,z)|2 and Ix(x,0,z)=|Ex(x,0,z)|2 denote the intensity distributions of the longitudinal z and transverse x components of the electric field, respectively. From Fig. 0.7, one can note that Iz,max(x,0,10zR)/Ix,max(x,0,10zR) decreases with the increasing ratio of the waist width to the wavelength, the increasing truncation parameter and the decreasing beam order. Particularly, when ω0/λ>2.8, Iz,max(x,0,10zR)/Ix,max(x,0,10zR) is less than 0.94% for n=0 and δ=3. It implies that the longitudinal component of the electric field compared with the transverse component can be neglected and the paraxial approximation is applicable.

 

Fig. 7 Normalized maximum intensity of the longitudinal component of the electric field Iz,max(x,0,10zR)/Ix,max(x,0,10zR) versus ω0/λ in the cases of (a) δ=0.5 (solid curve), 1 (dashed curve), 3 (dotted curve), n=0 and (b) n=0 (solid curve), 1 (dashed curve), 2 (dotted curve), δ=1.

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4. Conclusion

In conclusion, the propagation properties of the radially polarized beams through a circular aperture have been studied in the nonparaxial and paraxial regimes. The analytical expressions for the electromagnetic fields have been derived based on the vectorial Rayleigh diffraction integral. For two particular cases, the general result obtained in this paper can reduce to the unapertured and paraxial ones. The numerical results show that both the ratio of the waist width to the wavelength and the truncation parameter play an important role in the nonparaxiality of the apertured radially polarized beams, moreover, the truncation parameter and the beam order influence the beam diffraction effect and the beam evolution behavior. The normalized maximum intensity of the longitudinal component of the electric field was depicted for different parameters, which implies that the longitudinal component of the electric field compared with the transverse component can be increasingly neglected with the increasing ratio of the waist width to the wavelength, the increasing truncation parameter and the decreasing beam order.

Acknowledgements

This work was supported by the National Key Technology Research and Development Program under grant 2007BAF11B01.

References and links

1. F. Gori, “Polarization basis for vortex beams,” J. Opt. Soc. Am. A 18(7), 1612–1617 (2001). [CrossRef]  

2. Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88(9), 095005 (2002). [CrossRef]   [PubMed]  

3. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-15-3377. [CrossRef]   [PubMed]  

4. K. S. Youngworth and T. G. Brown, “Inhomogeneous polarization in scanning optical microscopy,” Proc. SPIE 3919, 75–85 (2000). [CrossRef]  

5. I. Moshe, S. Jackel, and A. Meir, “Production of radially or azimuthally polarized beams in solid-state lasers and the elimination of thermally induced birefringence effects,” Opt. Lett. 28(10), 807–809 (2003). [CrossRef]   [PubMed]  

6. V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32(13), 1455–1461 (1999). [CrossRef]  

7. M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys., A Mater. Sci. Process. 86(3), 329–334 (2007). [CrossRef]  

8. A. A. Tovar, “Production and propagation of cylindrical polarized Laguerre-Gaussian laser beams,” J. Opt. Soc. Am. A 15(10), 2705–2711 (1998). [CrossRef]  

9. M. A. Ahmed, J. Schulz, A. Voss, O. Parriaux, J. C. Pommier, and T. Graf, “Radially polarized 3kW beam from a CO2 laser with an intracavity resonant grating mirror,” Opt. Lett. 32(13), 1824–1826 (2007). [CrossRef]   [PubMed]  

10. K. J. Moh, X. C. Yuan, J. Bu, R. E. Burge, and B. Z. Gao, “Generating radial or azimuthal polarization by axial sampling of circularly polarized vortex beams,” Appl. Opt. 46(30), 7544–7551 (2007). [CrossRef]   [PubMed]  

11. T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80(6), 707–713 (2005). [CrossRef]  

12. G. MacHavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarization beams,” Opt. Commun. 281(4), 732–738 (2008). [CrossRef]  

13. D. Deng, “Nonparaxial propagation of radially polarized light beams,” J. Opt. Soc. Am. B 23(6), 1228–1234 (2006). [CrossRef]  

14. D. Deng, Q. Guo, L. Wu, and X. Yang, “Propagation of radially polarized elegant light beams,” J. Opt. Soc. Am. B 24(3), 636–643 (2007). [CrossRef]  

15. D. Deng and Q. Guo, “Analytical vectorial structure of radially polarized light beams,” Opt. Lett. 32(18), 2711–2713 (2007). [CrossRef]   [PubMed]  

16. S. Yan and B. Yao, “Accurate description of a radially polarized Gaussian beam,” Phys. Rev. A 77(2), 023827 (2008). [CrossRef]  

17. S. Yan and B. Yao, “Description of a radially polarized Laguerre-Gauss beam beyond the paraxial approximation,” Opt. Lett. 32(22), 3367–3369 (2007). [CrossRef]   [PubMed]  

18. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1966).

19. B. Lü and K. Duan, “Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture,” Opt. Lett. 28(24), 2440–2442 (2003). [CrossRef]   [PubMed]  

20. K. Duan and B. Lü, “Nonparaxial analysis of far-field properties of Gaussian beams diffracted at a circular aperture,” Opt. Express 11(13), 1474–1480 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-13-1474. [CrossRef]   [PubMed]  

21. I. S. Gradshteyn, and I. M. Ryzhik, Table of Integrals, Series, and Products, (Academic Press, New York, 1994).

References

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  • |

  1. F. Gori, “Polarization basis for vortex beams,” J. Opt. Soc. Am. A 18(7), 1612–1617 (2001).
    [CrossRef]
  2. Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88(9), 095005 (2002).
    [CrossRef] [PubMed]
  3. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-15-3377 .
    [CrossRef] [PubMed]
  4. K. S. Youngworth and T. G. Brown, “Inhomogeneous polarization in scanning optical microscopy,” Proc. SPIE 3919, 75–85 (2000).
    [CrossRef]
  5. I. Moshe, S. Jackel, and A. Meir, “Production of radially or azimuthally polarized beams in solid-state lasers and the elimination of thermally induced birefringence effects,” Opt. Lett. 28(10), 807–809 (2003).
    [CrossRef] [PubMed]
  6. V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32(13), 1455–1461 (1999).
    [CrossRef]
  7. M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys., A Mater. Sci. Process. 86(3), 329–334 (2007).
    [CrossRef]
  8. A. A. Tovar, “Production and propagation of cylindrical polarized Laguerre-Gaussian laser beams,” J. Opt. Soc. Am. A 15(10), 2705–2711 (1998).
    [CrossRef]
  9. M. A. Ahmed, J. Schulz, A. Voss, O. Parriaux, J. C. Pommier, and T. Graf, “Radially polarized 3kW beam from a CO2 laser with an intracavity resonant grating mirror,” Opt. Lett. 32(13), 1824–1826 (2007).
    [CrossRef] [PubMed]
  10. K. J. Moh, X. C. Yuan, J. Bu, R. E. Burge, and B. Z. Gao, “Generating radial or azimuthal polarization by axial sampling of circularly polarized vortex beams,” Appl. Opt. 46(30), 7544–7551 (2007).
    [CrossRef] [PubMed]
  11. T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80(6), 707–713 (2005).
    [CrossRef]
  12. G. MacHavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarization beams,” Opt. Commun. 281(4), 732–738 (2008).
    [CrossRef]
  13. D. Deng, “Nonparaxial propagation of radially polarized light beams,” J. Opt. Soc. Am. B 23(6), 1228–1234 (2006).
    [CrossRef]
  14. D. Deng, Q. Guo, L. Wu, and X. Yang, “Propagation of radially polarized elegant light beams,” J. Opt. Soc. Am. B 24(3), 636–643 (2007).
    [CrossRef]
  15. D. Deng and Q. Guo, “Analytical vectorial structure of radially polarized light beams,” Opt. Lett. 32(18), 2711–2713 (2007).
    [CrossRef] [PubMed]
  16. S. Yan and B. Yao, “Accurate description of a radially polarized Gaussian beam,” Phys. Rev. A 77(2), 023827 (2008).
    [CrossRef]
  17. S. Yan and B. Yao, “Description of a radially polarized Laguerre-Gauss beam beyond the paraxial approximation,” Opt. Lett. 32(22), 3367–3369 (2007).
    [CrossRef] [PubMed]
  18. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1966).
  19. B. Lü and K. Duan, “Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture,” Opt. Lett. 28(24), 2440–2442 (2003).
    [CrossRef] [PubMed]
  20. K. Duan and B. Lü, “Nonparaxial analysis of far-field properties of Gaussian beams diffracted at a circular aperture,” Opt. Express 11(13), 1474–1480 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-13-1474 .
    [CrossRef] [PubMed]
  21. I. S. Gradshteyn, and I. M. Ryzhik, Table of Integrals, Series, and Products, (Academic Press, New York, 1994).

2008 (2)

G. MacHavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarization beams,” Opt. Commun. 281(4), 732–738 (2008).
[CrossRef]

S. Yan and B. Yao, “Accurate description of a radially polarized Gaussian beam,” Phys. Rev. A 77(2), 023827 (2008).
[CrossRef]

2007 (6)

2006 (1)

2005 (1)

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80(6), 707–713 (2005).
[CrossRef]

2004 (1)

2003 (3)

2002 (1)

Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88(9), 095005 (2002).
[CrossRef] [PubMed]

2001 (1)

2000 (1)

K. S. Youngworth and T. G. Brown, “Inhomogeneous polarization in scanning optical microscopy,” Proc. SPIE 3919, 75–85 (2000).
[CrossRef]

1999 (1)

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32(13), 1455–1461 (1999).
[CrossRef]

1998 (1)

Ahmed, M. A.

M. A. Ahmed, J. Schulz, A. Voss, O. Parriaux, J. C. Pommier, and T. Graf, “Radially polarized 3kW beam from a CO2 laser with an intracavity resonant grating mirror,” Opt. Lett. 32(13), 1824–1826 (2007).
[CrossRef] [PubMed]

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80(6), 707–713 (2005).
[CrossRef]

Brown, T. G.

K. S. Youngworth and T. G. Brown, “Inhomogeneous polarization in scanning optical microscopy,” Proc. SPIE 3919, 75–85 (2000).
[CrossRef]

Bu, J.

Burge, R. E.

Deng, D.

Duan, K.

Feurer, T.

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys., A Mater. Sci. Process. 86(3), 329–334 (2007).
[CrossRef]

Gao, B. Z.

Glur, H.

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80(6), 707–713 (2005).
[CrossRef]

Gori, F.

Graf, T.

M. A. Ahmed, J. Schulz, A. Voss, O. Parriaux, J. C. Pommier, and T. Graf, “Radially polarized 3kW beam from a CO2 laser with an intracavity resonant grating mirror,” Opt. Lett. 32(13), 1824–1826 (2007).
[CrossRef] [PubMed]

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80(6), 707–713 (2005).
[CrossRef]

Guo, Q.

Jackel, S.

G. MacHavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarization beams,” Opt. Commun. 281(4), 732–738 (2008).
[CrossRef]

I. Moshe, S. Jackel, and A. Meir, “Production of radially or azimuthally polarized beams in solid-state lasers and the elimination of thermally induced birefringence effects,” Opt. Lett. 28(10), 807–809 (2003).
[CrossRef] [PubMed]

Keitel, C. H.

Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88(9), 095005 (2002).
[CrossRef] [PubMed]

Lü, B.

Lumer, Y.

G. MacHavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarization beams,” Opt. Commun. 281(4), 732–738 (2008).
[CrossRef]

MacHavariani, G.

G. MacHavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarization beams,” Opt. Commun. 281(4), 732–738 (2008).
[CrossRef]

Meier, M.

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys., A Mater. Sci. Process. 86(3), 329–334 (2007).
[CrossRef]

Meir, A.

G. MacHavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarization beams,” Opt. Commun. 281(4), 732–738 (2008).
[CrossRef]

I. Moshe, S. Jackel, and A. Meir, “Production of radially or azimuthally polarized beams in solid-state lasers and the elimination of thermally induced birefringence effects,” Opt. Lett. 28(10), 807–809 (2003).
[CrossRef] [PubMed]

Moh, K. J.

Moser, T.

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80(6), 707–713 (2005).
[CrossRef]

Moshe, I.

G. MacHavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarization beams,” Opt. Commun. 281(4), 732–738 (2008).
[CrossRef]

I. Moshe, S. Jackel, and A. Meir, “Production of radially or azimuthally polarized beams in solid-state lasers and the elimination of thermally induced birefringence effects,” Opt. Lett. 28(10), 807–809 (2003).
[CrossRef] [PubMed]

Nesterov, A. V.

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32(13), 1455–1461 (1999).
[CrossRef]

Niziev, V. G.

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32(13), 1455–1461 (1999).
[CrossRef]

Parriaux, O.

M. A. Ahmed, J. Schulz, A. Voss, O. Parriaux, J. C. Pommier, and T. Graf, “Radially polarized 3kW beam from a CO2 laser with an intracavity resonant grating mirror,” Opt. Lett. 32(13), 1824–1826 (2007).
[CrossRef] [PubMed]

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80(6), 707–713 (2005).
[CrossRef]

Pigeon, F.

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80(6), 707–713 (2005).
[CrossRef]

Pommier, J. C.

Romano, V.

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys., A Mater. Sci. Process. 86(3), 329–334 (2007).
[CrossRef]

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80(6), 707–713 (2005).
[CrossRef]

Salamin, Y. I.

Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88(9), 095005 (2002).
[CrossRef] [PubMed]

Schulz, J.

Tovar, A. A.

Voss, A.

Wu, L.

Yan, S.

Yang, X.

Yao, B.

Youngworth, K. S.

K. S. Youngworth and T. G. Brown, “Inhomogeneous polarization in scanning optical microscopy,” Proc. SPIE 3919, 75–85 (2000).
[CrossRef]

Yuan, X. C.

Zhan, Q.

Appl. Opt. (1)

Appl. Phys. B (1)

T. Moser, H. Glur, V. Romano, F. Pigeon, O. Parriaux, M. A. Ahmed, and T. Graf, “Polarization-selective grating mirrors used in the generation of radial polarization,” Appl. Phys. B 80(6), 707–713 (2005).
[CrossRef]

Appl. Phys., A Mater. Sci. Process. (1)

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys., A Mater. Sci. Process. 86(3), 329–334 (2007).
[CrossRef]

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

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

J. Phys. D (1)

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32(13), 1455–1461 (1999).
[CrossRef]

Opt. Commun. (1)

G. MacHavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarization beams,” Opt. Commun. 281(4), 732–738 (2008).
[CrossRef]

Opt. Express (2)

Opt. Lett. (5)

Phys. Rev. A (1)

S. Yan and B. Yao, “Accurate description of a radially polarized Gaussian beam,” Phys. Rev. A 77(2), 023827 (2008).
[CrossRef]

Phys. Rev. Lett. (1)

Y. I. Salamin and C. H. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88(9), 095005 (2002).
[CrossRef] [PubMed]

Proc. SPIE (1)

K. S. Youngworth and T. G. Brown, “Inhomogeneous polarization in scanning optical microscopy,” Proc. SPIE 3919, 75–85 (2000).
[CrossRef]

Other (2)

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1966).

I. S. Gradshteyn, and I. M. Ryzhik, Table of Integrals, Series, and Products, (Academic Press, New York, 1994).

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Figures (7)

Fig. 1
Fig. 1

Nonparaxial (a) and paraxial (b) energy flux distributions of the apertured radially polarized beams versus x / λ and y / λ at the plane z = 10 z R for n = 0 , ω 0 / λ = 2 , δ = 2 . Solid and dashed curves in the panel (c) denote the corresponding cross-section energy flux distributions at y = 0 for the nonparaxial and paraxial cases, respectively.

Fig. 2
Fig. 2

The same legend as in Fig. 1 except that the parameters n = 0 , ω 0 / λ = 1 , δ = 2 .

Fig. 3
Fig. 3

The same legend as in Fig. 1 except that the parameters n = 0 , ω 0 / λ = 1 , δ = 0.5 .

Fig. 4
Fig. 4

The same legend as in Fig. 1 except that the parameters n = 2 , ω 0 / λ = 2 , δ = 2 .

Fig. 5
Fig. 5

The same legend as in Fig. 1 except that the parameters n = 2 , ω 0 / λ = 1 , δ = 2 .

Fig. 6
Fig. 6

The same legend as in Fig. 1 except that the parameters n = 2 , ω 0 / λ = 1 , δ = 0.5 .

Fig. 7
Fig. 7

Normalized maximum intensity of the longitudinal component of the electric field I z , max ( x , 0 , 10 z R ) / I x , max ( x , 0 , 10 z R ) versus ω 0 / λ in the cases of (a) δ = 0.5 (solid curve), 1 (dashed curve), 3 (dotted curve), n = 0 and (b) n = 0 (solid curve), 1 (dashed curve), 2 (dotted curve), δ = 1.

Equations (20)

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E n 1 0 ( x 0 , y 0 , 0 ) = E n 1 x 0 ( x 0 , y 0 , 0 ) e x + E n 1 y 0 ( x 0 , y 0 , 0 ) e y
E n 1 x 0 ( x 0 , y 0 , 0 ) = 2 E 0 x 0 ω 0 L n 1 ( 2 ρ 0 2 ω 0 2 ) exp ( ρ 0 2 ω 0 2 )
E n 1 y 0 ( x 0 , y 0 , 0 ) = 2 E 0 y 0 ω 0 L n 1 ( 2 ρ 0 2 ω 0 2 ) exp ( ρ 0 2 ω 0 2 )
E n 1 x ( x 0 , y 0 , 0 ) = 2 E 0 x 0 ω 0 L n 1 ( 2 ρ 0 2 ω 0 2 ) exp ( ρ 0 2 ω 0 2 ) t ( x 0 , y 0 )
E n 1 y ( x 0 , y 0 , 0 ) = 2 E 0 y 0 ω 0 L n 1 ( 2 ρ 0 2 ω 0 2 ) exp ( ρ 0 2 ω 0 2 ) t ( x 0 , y 0 )
t ( x 0 , y 0 ) = { 1 x 0 2 + y 0 2 a 2 0 o t h e r w i s e
E n 1 x ( r ) = 1 2 π + E n 1 x ( x 0 , y 0 , 0 ) G ( r , r 0 ) z d x 0 d y 0
E n 1 y ( r ) = 1 2 π + E n 1 y ( x 0 , y 0 , 0 ) G ( r , r 0 ) z d x 0 d y 0
E n 1 z ( r ) = 1 2 π + [ E n 1 x ( x 0 , y 0 , 0 ) G ( r , r 0 ) x + E n 1 y ( x 0 , y 0 , 0 ) G ( r , r 0 ) y ] d x 0 d y 0
G ( r , r 0 ) = exp ( i k | r r 0 | ) | r r 0 |
| r r 0 | r + x 0 2 + y 0 2 2 x x 0 2 y y 0 2 r
E n 1 x ( r ) = 2 E 0 k z x ρ exp ( i k r ) r 2 ω 0 0 a ρ 0 2 L n 1 ( 2 ρ 0 2 ω 0 2 ) exp ( g ρ 0 2 ) J 1 ( k ρ ρ 0 r ) d ρ 0
E n 1 y ( r ) = 2 E 0 k z y ρ exp ( i k r ) r 2 ω 0 0 a ρ 0 2 L n 1 ( 2 ρ 0 2 ω 0 2 ) exp ( g ρ 0 2 ) J 1 ( k ρ ρ 0 r ) d ρ 0
E n 1 z ( r ) = 2 E 0 k exp ( i k r ) r 2 ω 0 [ ρ 0 a ρ 0 2 L n 1 ( 2 ρ 0 2 ω 0 2 ) exp ( g ρ 0 2 ) J 1 ( k ρ ρ 0 r ) d ρ 0 i 0 a ρ 0 3 L n 1 ( 2 ρ 0 2 ω 0 2 ) exp ( g ρ 0 2 ) J 0 ( k ρ ρ 0 r ) d ρ 0 ]
Γ ( α , x ) = x e t t α 1 d t
E n 1 ( r ) = 2 E 0 k exp ( i k r ) r 2 l = 0 n u = 0 ( n + 1 ) ! 2 l 1 ( l + 1 ) ! l ! ( n l ) ! ω 0 2 l + 1 { z ρ ( x e x + y e y ) 1 u ! ( u + 1 ) ! ( k ρ 2 r ) 2 u + 1 + e z [ ρ u ! ( u + 1 ) ! ( k ρ 2 r ) 2 u + 1 + i u ! u ! ( k ρ 2 r ) 2 u ] } [ Γ ( l + u + 2 , g a 2 ) ( l + u + 1 ) ! ] g l + u + 2
H n 1 ( r ) = i 2 ε μ E 0 exp ( i k r ) ( y e x x e y ) l = 0 n u = 0 ( n + 1 ) ! 2 l 1 ( l + 1 ) ! l ! ( n l ) ! ω 0 2 l + 1 × { ( i k r 2 ) r 4 [ ρ u ! ( u + 1 ) ! ( k ρ 2 r ) 2 u + 1 i u ! u ! ( k ρ 2 r ) 2 u ] [ Γ ( l + u + 2 , g a 2 ) ( l + u + 1 ) ! ] g l + u + 2 + [ 2 + ( i k r 2 ) z 2 r 2 ] 1 ρ r 2 u ! ( u + 1 ) ! ( k ρ 2 r ) 2 u + 1 [ Γ ( l + u + 2 , g a 2 ) ( l + u + 1 ) ! ] g l + u + 2 + k 2 r 5 [ ( i ( 3 ρ 2 r 2 ρ ) 1 u ! ( u + 1 ) ! ( k ρ 2 r ) 2 u + 1 + z 2 u ! ( u + 2 ) ! ( k ρ 2 r ) 2 u + 2 + 1 u ! u ! ( k ρ 2 r ) 2 u ) × [ Γ ( l + u + 3 , g a 2 ) ( l + u + 2 ) ! ] g l + u + 3 + z 2 u ! u ! ( k ρ 2 r ) 2 u [ Γ ( l + u + 2 , g a 2 ) ( l + u + 1 ) ! ] g l + u + 2 ] k z 2 2 r 5 [ ( i ρ u ! ( u + 1 ) ! ( k ρ 2 r ) 2 u + 1 + 1 u ! ( u + 2 ) ! ( k ρ 2 r ) 2 u + 2 ) [ Γ ( l + u + 3 , g a 2 ) ( l + u + 2 ) ! ] g l + u + 3 + 1 u ! u ! ( k ρ 2 r ) 2 u [ Γ ( l + u + 2 , g a 2 ) ( l + u + 1 ) ! ] g l + u + 2 ] }
r z + x 2 + y 2 2 z
E n 1 p ( r ) = 2 E 0 k exp [ i k ( z + ρ 2 2 z ) ] l = 0 n u = 0 ( n + 1 ) ! 2 l 1 ( l + 1 ) ! l ! ( n l ) ! ω 0 2 l + 1 { 1 ρ z ( x e x + y e y ) × 1 u ! ( u + 1 ) ! ( k ρ 2 z ) 2 u + 1 + e z 1 z 2 [ ρ u ! ( u + 1 ) ! ( k ρ 2 z ) 2 u + 1 + i u ! u ! ( k ρ 2 z ) 2 u ] } × [ Γ ( l + u + 2 , t a 2 ) ( l + u + 1 ) ! ] t l + u + 2
H n 1 p ( r ) = i 2 ε μ E 0 exp [ i k ( z + ρ 2 2 z ) ] ( y e x x e y ) l = 0 n u = 0 ( n + 1 ) ! 2 l 1 ( l + 1 ) ! l ! ( n l ) ! ω 0 2 l + 1 × { ( i k z 2 ) z 4 [ ρ u ! ( u + 1 ) ! ( k ρ 2 z ) 2 u + 1 i u ! u ! ( k ρ 2 z ) 2 u ] [ Γ ( l + u + 2 , t a 2 ) ( l + u + 1 ) ! ] t l + u + 2 + i k ρ z u ! ( u + 1 ) ! ( k ρ 2 z ) 2 u + 1 [ Γ ( l + u + 2 , t a 2 ) ( l + u + 1 ) ! ] t l + u + 2 + k 2 z 5 [ ( i ( 3 ρ 2 z 2 ρ ) 1 u ! ( u + 1 ) ! ( k ρ 2 z ) 2 u + 1 + z 2 u ! ( u + 2 ) ! ( k ρ 2 z ) 2 u + 2 + 1 u ! u ! ( k ρ 2 z ) 2 u ) × [ Γ ( l + u + 3 , t a 2 ) ( l + u + 2 ) ! ] t l + u + 3 + z 2 u ! u ! ( k ρ 2 z ) 2 u [ Γ ( l + u + 2 , t a 2 ) ( l + u + 1 ) ! ] t l + u + 2 ] k 2 z 3 [ ( i ρ u ! ( u + 1 ) ! ( k ρ 2 z ) 2 u + 1 + 1 u ! ( u + 2 ) ! ( k ρ 2 z ) 2 u + 2 ) [ Γ ( l + u + 3 , t a 2 ) ( l + u + 2 ) ! ] t l + u + 3 + 1 u ! u ! ( k ρ 2 z ) 2 u [ Γ ( l + u + 2 , t a 2 ) ( l + u + 1 ) ! ] t l + u + 2 ] }

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