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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 17 — Aug. 17, 2009
  • pp: 14865–14871
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Comparative studies of paraxial and nonparaxial vectorial elegant Laguerre-Gaussian beams

Zhangrong Mei and Juguan Gu  »View Author Affiliations


Optics Express, Vol. 17, Issue 17, pp. 14865-14871 (2009)
http://dx.doi.org/10.1364/OE.17.014865


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Abstract

Analytical paraxial and nonparaxial propagation expressions for vectorial elegant Laguerre-Gaussian (eLG) beam together with its even and odd modes are introduced by use of the vectorial Rayleigh-Sommerfeld formulas and the relations between eLG and elegant Hermite-Gaussian (eHG) modes. The propagation features of vectorial eLG beams are studied and analyzed comparatively in the paraxial and nonparaxial regimes with vivid illustration. It is shown that the propagation behavior of nonparaxial vectorial eLG beams is notably different from that of paraxial cases.

© 2009 OSA

1. Introduction

The concept of elegant beams was introduced by Siegman to describe the eHG beams [1

1. A. E. Siegman, “Hermite-Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63(9), 1093–1094 (1973). [CrossRef]

], and then was extended to study the eLG beams by Takenaka et al [2

2. T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximate,” J. Opt. Soc. Am. A 2(6), 826–829 (1985). [CrossRef]

4

4. S. Saghafi and C. J. R. Sheppard, “Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998). [CrossRef]

]. The elegant solutions are also eigenmodes of the paraxial wave equation, but they differ from the standard solutions in that the former contain polynomials with a complex argument, but in the latter the argument is real. Some authors have tried to extend paraxial eLG solutions to the nonparaxial generalization of eLG beams. Takenaka et al obtained the correction terms of eLG beams in scalar theory by using the perturbation method proposed by Lax et al [2

2. T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximate,” J. Opt. Soc. Am. A 2(6), 826–829 (1985). [CrossRef]

]. Seshadri et al gave the first three orders of nonparaxial corrections for the corresponding paraxial eLG beam [5

5. S. R. Seshadri, “Virtual source for a Laguerre-Gauss beam,” Opt. Lett. 27(21), 1872–1874 (2002). [CrossRef]

,6

6. M. A. Bandres and J. C. Gutiérrez-Vega, “Higher-order complex source for elegant Laguerre-Gaussian waves,” Opt. Lett. 29(19), 2213–2215 (2004). [CrossRef] [PubMed]

]. Kim and Lee obtained the power-series solutions of eLG beams by use of Lommel’s lemma in vector theory [7

7. H. C. Kim and Y. H. Lee, “Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation,” Opt. Commun. 169(1-6), 9–16 (1999). [CrossRef]

]. Recently, April gave an explicit closed form for nonparaxial eLG beams, which is an exact solution of the Helmholtz equation and can be expressed as linear combination of a finite number of analytic functions [8

8. A. April, “Nonparaxial elegant Laguerre-Gaussian beams,” Opt. Lett. 33(12), 1392–1394 (2008). [CrossRef] [PubMed]

]. In this paper, by the representations of eLG modes in terms of the Hermite polynomials and the vectorial Rayleigh-Sommerfeld formulas [9

9. R. Borghi and M. Santarsiero, “Nonparaxial propagation of spirally polarized optical beams,” J. Opt. Soc. Am. A 21, 2029–2037 (2004). [CrossRef]

11

11. Z. Mei and D. Zhao, “Nonparaxial analysis of vectorial Laguerre-Bessel-Gaussian beams,” Opt. Express 15(19), 11942–11951 (2007). [CrossRef] [PubMed]

], we obtain the analytical propagation expressions of vectorial eLG beam along with its even and odd modes in the nonparaxial and paraxial regime, which could enlighten the main features of nonparaxial propagation of eLG beams, and then investigate and analyze comparatively their propagation properties with vivid numerical examples.

2. Theoretical analyses

In the following we shall consider the free-space propagation of a quasi-monochromatic electromagnetic beam whose mean propagation axis coincides with the z axis of a cylindrical Cartesian reference frame (ρ;z), where ρ=xe^x+ye^y with e^α(α=x,y,z) as the unit vector along the α axis. The quantities E(ρ,z)=Ex(ρ,z)e^x+Ey(ρ,z)e^y  and Ez(ρ,z) will denote the transverse and longitudinal components, respectively. We consider a beam that, across the plane z = 0, is characterized by the transverse electric field E(ρ0,0) . The total electric field in the half-space z>0 can be derived by the vectorial Rayleigh-Sommerfeld formulas [9

9. R. Borghi and M. Santarsiero, “Nonparaxial propagation of spirally polarized optical beams,” J. Opt. Soc. Am. A 21, 2029–2037 (2004). [CrossRef]

]
E(ρ,z)=12πE(ρ0,0)z[exp(ikR)R]d2ρ0,
(1a)
Ez(ρ,z)=12π{Ex(ρ0,0)x[exp(ikR)R]+Ey(ρ0,0)y[exp(ikR)R]}d2ρ0,
(1b)
where k is the wave number, and R=(|ρρ0|2+z2)1/2 can be approximately expanded into R=r+(x02+y022xx02yy0)/2r, which holds true under the conditions w0<2λ and zw02/2λ, w 0 is the waist width and r=(x2+y2+z2)1/2 [12

12. A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202(1-3), 17–20 (2002). [CrossRef]

].

Consider an eLG beam with polarization parallel to x direction whose transverse electric field E(ρ0,0) at the source plane z = 0 takes the form [2

2. T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximate,” J. Opt. Soc. Am. A 2(6), 826–829 (1985). [CrossRef]

4

4. S. Saghafi and C. J. R. Sheppard, “Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998). [CrossRef]

]
E(ρ0,0)=(ρ0w0)mLnm(ρ02w02)exp(ρ02w02)exp(imθ0)e^x,
(2)
where Lnm denotes the Laguerre polynomial with radial number n and azimuthal parameter m, ρ 0 is the polar radius and θ 0 is the azimuthal angle. When an eLG beam has to integrated over Cartesian coordinates as Eq. (1), the polar coordinates (ρ, θ, z) in Eq. (2) need to be transform into rectangular coordinates (x, y, z). Based on the following relation expressing Laguerre in terms of Hermite polynomials [13

13. I. Kimel and L. R. Elias, “Relations Between Hermite and Laguerre Gaussian Modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993). [CrossRef]

]
eimθρmLnm(ρ2)=(1)n22n+mn!t=0ns=0mis(nt)(ms)HM1(x)HN1(y),
(3)
where M 1 = 2t + m-s, N 1 = 2n-2t + s, Eq. (2) can be expressed as sum of eHG modes:
E(ρ0,0)=(1)n22n+mn!t=0ns=0mis(nt)(ms)HM1(x0w0)HN1(y0w0)exp(x02+y02w02)e^x.
(4)
Substituting from Eq. (4) into Eq. (1), and recalling the following integral formula
exp[(x1x2)22u]Hm(x1)dx1=2πu(12u)m2Hm(x212u),
(5)
after tedious integral calculation, we obtain
E(ρ,z)=i(1)nzC222n+mn!kr2f2exp(ikr)t=0ns=0mis(nt)(ms)PM1+N1HM1(C1Px)                  ×HN1(C1Py)exp[C122C2(x2+y2)]e^x,
(6a)
Ez(ρ,z)=i(1)nC222n+mn!kr2f2exp(ikr)t=0ns=0mis(nt)(ms)PM1+N1[xHM1(C1Px)                 P2kfHM1+1(C1Px)M1kfPHM11(C1Px)]HN1(C1Py)exp[C122C2(x2+y2)],
(6b)
where C1=kf/(1+2ikrf2), C2=ikrf2/(1+2ikrf2), P=12C2 and f=1/kw0. Equation (6) provides a general propagation expression for the field of nonparaxial vectorial eLG beams.

Within the paraxial regime, we often neglect the longitudinal component of the beam field. Expanding r into series and keeping the first and second terms, i.e., rz + (x 2 + y 2)/2z, Eq. (6) reduces to the field distribution of paraxial eLG beams
Ep(ρ,z)=i(1)nC2p22n+mn!kzf2exp(ikz)exp[ik2z(x2+y2)]t=0ns=0mis(nt)(ms)PpM1+N1                   ×HM1(C1pPpx)HN1(C1pPpy)exp[C1p22C2p(x2+y2)]e^x,
(7)
where C1p=kf/(1+2ikzf2), C2p=ikzf2/(1+2ikzf2) and Pp=12C2p.

The eLG modes can alternatively defined with non-negative values of m as the following two types. Superscript “e” in the field distribution E ( e )(ρ,z) stands for even mode with the even function cos() for the azimuthal dependence and superscript “o” in E ( o )(ρ,z) stands for odd mode with the odd function sin(). Based on the following relations [13

13. I. Kimel and L. R. Elias, “Relations Between Hermite and Laguerre Gaussian Modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993). [CrossRef]

]
ρmLnm(ρ2)cos(mθ)=(1)n22n+mn!t=0ns=0[m/2](1)s(nt)(m2s)HM2(x)HN2(y),
(8)
even modes of eLG beams at the source plane z = 0 can be expressed as
Ε(e)(ρ0,0)=(1)n22n+mn!t=0ns=0[m/2](1)s(nt)(m2s)HM2(x0w0)HN2(y0w0)exp(x02+y02w02)e^x,
(9)
where M 2 = 2t + m-2s and N 2 = 2n-2t + 2s. Following the same procedure obtained Eqs. (6), the nonparaxial propagation expression for the even modes is given by
E(e)(ρ,z)=i(1)nzC222n+mn!kr2f2exp(ikr)t=0ns=0[m/2](1)s(nt)(m2s)PM2+N2                  ×HM2(C1Px)HN2(C1Py)exp[C122C2(x2+y2)]e^x,
(10a)
Ez(e)(ρ,z)=i(1)nC222n+mn!kr2f2exp(ikr)t=0ns=0[m/2](1)s(nt)(m2s)PM2+N2[xHM2(C1Px)              P2kfHM2+1(C1Px)M2kfPHM21(C1Px)]HN2(C1Py)exp[C122C2(x2+y2)].
(10b)
In the paraxial approximation, Eq. (10) reduces to
Ep(e)(ρ,z)=i(1)nC2p22n+mn!kzf2exp(ikz)exp[ik2z(x2+y2)]t=0ns=0[m/2](1)s(nt)(m2s)                   ×PpM2+N2HM2(C1pPpx)HN2(C1pPpy)exp[C1p22C2p(x2+y2)]e^x.
(11)
For odd modes of eLG beams, after applying the following formula [13

13. I. Kimel and L. R. Elias, “Relations Between Hermite and Laguerre Gaussian Modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993). [CrossRef]

]
ρmLnm(ρ2)sin(mθ)=(1)n22n+mn!t=0ns=0[(m1)/2](1)s(nt)(m2s+1)HM3(x)HN3(y),
(12)
the field distribution at the source plane z = 0 arrives at
E(o)(ρ0,0)=(1)n22n+mn!t=0ns=0[(m1)/2](1)s(nt)(m2s+1)HM3(x0w0)HN3(y0w0)exp(x02+y02w02)e^x,
(13)
where M 3 = 2t + m-2s-1 and N 3 = 2n-2t + 2s + 1. Similarly, the nonparaxial propagation expression for the odd modes of eLG beams is given by
E(o)(ρ,z)=i(1)nzC222n+mn!kr2f2exp(ikr)t=0ns=0[(m1)/2](1)s(nt)(m2s+1)PM3+N3                   ×HM3(C1Px)HN3(C1Py)exp[C122C2(x2+y2)]e^x,
(14a)
Ez(o)(ρ,z)=i(1)nC222n+mn!kr2f2exp(ikr)t=0ns=0[(m1)/2](1)s(nt)(m2s+1)PM3+N3[xHM3(C1Px)                  P2kfHM3+1(C1Px)M3kfPHM31(C1Px)]HN3(C1Py)exp[C122C2(x2+y2)].
(14b)
Consequently, in the paraxial limit, Eqs. (14) reduces to
Ep(o)(ρ,z)=i(1)nC2p22n+mn!kzf2exp(ikz)exp[ik2z(x2+y2)]t=0ns=0[(m1)/2](1)s(nt)(m2s+1)                  ×PpM3+N3HM3(C1pPpx)HN3(C1pPpy)exp[C1p22C2p(x2+y2)]e^x.
(15)
These are the main analytical results of this paper, which provide the propagtion equations and exhibit the peculiar propagation properties of vectorial eLG beams within the nonparaxial and paraxial regime.

From the above paraxial results, the paraxial fundamental Gaussian mode yields
E(x,y,z)=iw0w(z)exp[ikz(x2+y2)w2(z)+ik2z(x2+y2)],
(16)
where w(z)=w0(1+z2/zR2)1/2, zR=πw02/λ is the Rayleigh length.

3. Numerical calculation results and comparative analyses

It worth making a more farther comparison between the nonparaxial eLG beams and its paraxial approximation, the three-dimensional intensity distributions and corresponding contour graphs of nonparaxial and paraxial eLG beam with n = 2, m = 3 for f = 0.2 at the plane z = 15zR are plotted by using Eqs. (6) and (7), as shown in Fig. 3
Fig. 3 The three-dimensional intensity distribution (a) and corresponding contour graph (b) of a nonparaxial vectorial eLG beam with n = 2, m = 3 for f = 0.2 at the plane z = 15zR, and its corresponding paraxial result (c) and (d).
. Figure 3(a) and (b) express the nonparaxial eLG beams, (c) and (d) are the paraxial cases. We can see from Fig. 3 that the paraxial eLG beam is a circular symmetrical dark hollow beam and the nonparaxial eLG beam is an elliptical symmetrical dark hollow beam. Figure 4
Fig. 4 The three-dimensional intensity distribution (a) and corresponding contour graph (b) of the even mode of a nonparaxial vectorial eLG beam with n = 2, m = 3 for f = 0.2 at the plane z = 15zR, and its corresponding paraxial result (c) and (d).
is plotted by using Eqs. (10) and (11) for even modes and Fig. 5
Fig. 5 As Fig. 4, but for the odd mode.
is plotted by using Eqs. (14) and (15) for odd modes, the calculation parameters are the same as those in Fig. 3. From Figs. 4-5, it can be seen that the intensity patterns of even and odd modes are notable different from those of the paraxial cases. The paraxial intensity patterns of even and odd modes are circular symmetry and approximately similar except for the difference of spatial orientation of beam lobes. While for the nonparaxial case, the intensity distributions of eLG beams lose their circular symmetry and become elliptical symmetry, and that the intensity patterns of even and odd modes also have obvious difference, due to the longitudinal component to the strength of the electric field become large and cannot be neglected when the parameter f is large value.

4. Conclusion

In the present paper the free-space propagation features of vectorial eLG beams have been investigated and analyzed comparatively in the paraxial and in the nonparaxial regimes. Based on the vectorial Rayleigh-Sommerfeld formulas and the relations between eLG and eHG modes, the analytical propagation expressions of a vectorial eLG beam along with its even and odd modes have been derived. The corresponding paraxial approximation expressions have been dealt with as special case of our general results. With the help of numerical calculations, the propagation properties of the nonparaxial vectorial eLG beams have been studied and analyzed comparatively with the paraxial results. The numerical results have shown that the parameter f plays an important role in determining the nonparaxial propagation properties of vectorial eLG beams. Under the condition that f<0.05 the paraxial approximation for eLG beams with n = 2, m = 3 is allowable. Otherwise, one must take into account the vector structure of eLG beams. The condition is different for different order eLG modes, also different from the case of fundamental Gaussian beam. Moreover, the intensity patterns of a nonparaxial vevtorial eLG beam along with its even and odd modes are notable different from those of the conventional transverse-mode patterns in the paraxial regime.

Acknowledgments

This work was supported by the Zhejiang Provincial Natural Science Foundation of China (Y6090247, Y606320), Huzhou Civic Natural Science Fund of Zhejiang Province of China (2009C50064) and National Natural Science Foundation of China (10874150).

References and links

1.

A. E. Siegman, “Hermite-Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63(9), 1093–1094 (1973). [CrossRef]

2.

T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximate,” J. Opt. Soc. Am. A 2(6), 826–829 (1985). [CrossRef]

3.

E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 3(4), 465–469 (1986). [CrossRef]

4.

S. Saghafi and C. J. R. Sheppard, “Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998). [CrossRef]

5.

S. R. Seshadri, “Virtual source for a Laguerre-Gauss beam,” Opt. Lett. 27(21), 1872–1874 (2002). [CrossRef]

6.

M. A. Bandres and J. C. Gutiérrez-Vega, “Higher-order complex source for elegant Laguerre-Gaussian waves,” Opt. Lett. 29(19), 2213–2215 (2004). [CrossRef] [PubMed]

7.

H. C. Kim and Y. H. Lee, “Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation,” Opt. Commun. 169(1-6), 9–16 (1999). [CrossRef]

8.

A. April, “Nonparaxial elegant Laguerre-Gaussian beams,” Opt. Lett. 33(12), 1392–1394 (2008). [CrossRef] [PubMed]

9.

R. Borghi and M. Santarsiero, “Nonparaxial propagation of spirally polarized optical beams,” J. Opt. Soc. Am. A 21, 2029–2037 (2004). [CrossRef]

10.

K. Duan, B. Wang, and B. Lü, “Propagation of Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 22(9), 1976–1980 (2005). [CrossRef]

11.

Z. Mei and D. Zhao, “Nonparaxial analysis of vectorial Laguerre-Bessel-Gaussian beams,” Opt. Express 15(19), 11942–11951 (2007). [CrossRef] [PubMed]

12.

A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202(1-3), 17–20 (2002). [CrossRef]

13.

I. Kimel and L. R. Elias, “Relations Between Hermite and Laguerre Gaussian Modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993). [CrossRef]

OCIS Codes
(050.1970) Diffraction and gratings : Diffractive optics
(140.3300) Lasers and laser optics : Laser beam shaping

ToC Category:
Physical Optics

History
Original Manuscript: June 2, 2009
Revised Manuscript: July 28, 2009
Manuscript Accepted: August 2, 2009
Published: August 6, 2009

Citation
Zhangrong Mei and Juguan Gu, "Comparative studies of paraxial and nonparaxial vectorial elegant Laguerre-Gaussian beams," Opt. Express 17, 14865-14871 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-17-14865


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References

  1. A. E. Siegman, “Hermite-Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63(9), 1093–1094 (1973). [CrossRef]
  2. T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximate,” J. Opt. Soc. Am. A 2(6), 826–829 (1985). [CrossRef]
  3. E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 3(4), 465–469 (1986). [CrossRef]
  4. S. Saghafi and C. J. R. Sheppard, “Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998). [CrossRef]
  5. S. R. Seshadri, “Virtual source for a Laguerre-Gauss beam,” Opt. Lett. 27(21), 1872–1874 (2002). [CrossRef]
  6. M. A. Bandres and J. C. Gutiérrez-Vega, “Higher-order complex source for elegant Laguerre-Gaussian waves,” Opt. Lett. 29(19), 2213–2215 (2004). [CrossRef] [PubMed]
  7. H. C. Kim and Y. H. Lee, “Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation,” Opt. Commun. 169(1-6), 9–16 (1999). [CrossRef]
  8. A. April, “Nonparaxial elegant Laguerre-Gaussian beams,” Opt. Lett. 33(12), 1392–1394 (2008). [CrossRef] [PubMed]
  9. R. Borghi and M. Santarsiero, “Nonparaxial propagation of spirally polarized optical beams,” J. Opt. Soc. Am. A 21, 2029–2037 (2004). [CrossRef]
  10. K. Duan, B. Wang, and B. Lü, “Propagation of Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 22(9), 1976–1980 (2005). [CrossRef]
  11. Z. Mei and D. Zhao, “Nonparaxial analysis of vectorial Laguerre-Bessel-Gaussian beams,” Opt. Express 15(19), 11942–11951 (2007). [CrossRef] [PubMed]
  12. A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202(1-3), 17–20 (2002). [CrossRef]
  13. I. Kimel and L. R. Elias, “Relations Between Hermite and Laguerre Gaussian Modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993). [CrossRef]

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