## Comparative studies of paraxial and nonparaxial vectorial elegant Laguerre-Gaussian beams

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

## 2. Theoretical analyses

*z*axis of a cylindrical Cartesian reference frame

*α*axis. The quantities

*z*= 0, is characterized by the transverse electric field

*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]

*k*is the wave number, and

*w*

_{0}is the waist width and

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]

*x*direction whose transverse electric field

*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. 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]

*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]

*M*

_{1}= 2

*t*+

*m*-

*s*,

*N*

_{1}= 2

*n*-2t +

*s*, Eq. (2) can be expressed as sum of eHG modes:Substituting from Eq. (4) into Eq. (1), and recalling the following integral formulaafter tedious integral calculation, we obtain

*r*into series and keeping the first and second terms, i.e.,

*r*≈

*z*+ (

*x*

^{2}+

*y*

^{2})/2

*z*, Eq. (6) reduces to the field distribution of paraxial eLG beams

*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(

*mθ*) for the azimuthal dependence and superscript “

*o*” in

**E**

^{(}

^{o}^{)}(

**ρ**,

*z*) stands for odd mode with the odd function sin(

*mθ*). 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]

*z*= 0 can be expressed aswhere

*M*

_{2}= 2

*t*+

*m*-2

*s*and

*N*

_{2}= 2

*n*-2t + 2

*s*. Following the same procedure obtained Eqs. (6), the nonparaxial propagation expression for the even modes is given by

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

*z*= 0 arrives at

*M*

_{3}= 2

*t*+

*m*-2

*s*-1 and

*N*

_{3}= 2

*n*-2t + 2

*s*+ 1. Similarly, the nonparaxial propagation expression for the odd modes of eLG beams is given by

## 3. Numerical calculation results and comparative analyses

*n*= 2,

*m*= 3 for different value

*f*, as functions of the normalized transverse coordinate

*x/λ*, evaluated at the transverse plane

*z*= 15

*z*, where

_{R}*I*= |

*E*|

_{x}^{2}+ |

*E*|

_{y}^{2+}|

*E*|

_{z}^{2}. For convenience of comparison, the corresponding paraxial results

*I*(dotted curve) and the contribution

_{p}*z*longitudinal component to the strength of the electric field expressed

*I*(dashdotted curve) are compiled together. One can see from Fig. 1 that when the value of parameter

_{z}*f*is very small, the nonparaxial results by using Eqs. (6) coincide with the paraxial results by using Eqs. (7) quite well, so that for this case the paraxial approximate holds true and the

*z*component is very small and can be negligible. However, with the increasing

*f*, the difference between the paraxial and nonparaxial transverse components becomes obvious, and that the longitudinal component

*I*become large and cannot be neglected. It means that for the large values of parameter

_{z}*f*the vectorial nonparaxial approach instead of the scalar paraxial one should employed, and the vectorial nonparaxial behavior of eLG beams should be taken into consideration. Figure 2 shows the normalized strength distribution of a fundamental Gaussian beam at the transverse plane

*z*= 15

*z*for different

_{R}*f*value. It is worth noting that the fundamental Gaussian beam compared to the higher-order eLG beams with

*n*= 2,

*m*= 3, paraxial approximation provides better accuracy for values of

*f*as high as 0.25, which is consistent with the previous results. However, toward the eLG as shown in Fig. 1, for values of

*f*as low as 0.05, the difference between the nonparaxial results

*I*and paraxial results

*I*have been large, and that the condition is change with the radial number

_{p}*n*and azimuthal parameter

*m*.

*n*= 2,

*m*= 3 for

*f*= 0.2 at the plane

*z*= 15

*z*are plotted by using Eqs. (6) and (7), as shown in Fig. 3 . 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 is plotted by using Eqs. (10) and (11) for even modes and Fig. 5 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

_{R}*f*is large value.

## 4. Conclusion

*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

## References and links

1. | A. E. Siegman, “Hermite-Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. |

2. | T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximate,” J. Opt. Soc. Am. A |

3. | E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am. A |

4. | S. Saghafi and C. J. R. Sheppard, “Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes,” J. Mod. Opt. |

5. | S. R. Seshadri, “Virtual source for a Laguerre-Gauss beam,” Opt. Lett. |

6. | M. A. Bandres and J. C. Gutiérrez-Vega, “Higher-order complex source for elegant Laguerre-Gaussian waves,” Opt. Lett. |

7. | H. C. Kim and Y. H. Lee, “Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation,” Opt. Commun. |

8. | A. April, “Nonparaxial elegant Laguerre-Gaussian beams,” Opt. Lett. |

9. | R. Borghi and M. Santarsiero, “Nonparaxial propagation of spirally polarized optical beams,” J. Opt. Soc. Am. A |

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 |

11. | Z. Mei and D. Zhao, “Nonparaxial analysis of vectorial Laguerre-Bessel-Gaussian beams,” Opt. Express |

12. | A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. |

13. | I. Kimel and L. R. Elias, “Relations Between Hermite and Laguerre Gaussian Modes,” IEEE J. Quantum Electron. |

**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

- A. E. Siegman, “Hermite-Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63(9), 1093–1094 (1973). [CrossRef]
- 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]
- E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 3(4), 465–469 (1986). [CrossRef]
- 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]
- S. R. Seshadri, “Virtual source for a Laguerre-Gauss beam,” Opt. Lett. 27(21), 1872–1874 (2002). [CrossRef]
- 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]
- 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]
- A. April, “Nonparaxial elegant Laguerre-Gaussian beams,” Opt. Lett. 33(12), 1392–1394 (2008). [CrossRef] [PubMed]
- R. Borghi and M. Santarsiero, “Nonparaxial propagation of spirally polarized optical beams,” J. Opt. Soc. Am. A 21, 2029–2037 (2004). [CrossRef]
- 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]
- Z. Mei and D. Zhao, “Nonparaxial analysis of vectorial Laguerre-Bessel-Gaussian beams,” Opt. Express 15(19), 11942–11951 (2007). [CrossRef] [PubMed]
- 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]
- 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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