## Nonparaxial propagation of an elegant Laguerre-Gaussian beam orthogonal to the optical axis of a uniaxial crystal |

Optics Express, Vol. 20, Issue 15, pp. 17160-17173 (2012)

http://dx.doi.org/10.1364/OE.20.017160

Acrobat PDF (5933 KB)

### Abstract

Analytical expressions of the nonparaxial propagation of even and odd elegant Laguerre-Gaussian beams orthogonal to the optical axis of a uniaxial crystal are derived. The intensity distributions of even and odd elegant Laguerre-Gaussian beams and their three components propagating in uniaxial crystals orthogonal to the optical axis are illustrated by numerical examples. Even though one of the two transversal components of even and odd elegant Laguerre-Gaussian beams in the input plane is set to be zero, this transversal and the longitudinal components have nonzero intensities and cannot be neglected upon propagation inside the uniaxial crystal. The evolution laws of even and odd elegant Laguerre-Gaussian beams propagating in uniaxial crystals orthogonal to the optical axis are also demonstrated. The intensity distributions of even and odd elegant Laguerre-Gaussian beams can be modulated by the uniaxial crystal, which is beneficial to the some applications involving in the special beam profile.

© 2012 OSA

## 1. Introduction

1. 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]

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

17. H. Xu, Z. Cui, and J. Qu, “Propagation of elegant Laguerre-Gaussian beam in non-Kolmogorov turbulence,” Opt. Express **19**(22), 21163–21173 (2011). [CrossRef] [PubMed]

18. F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express **17**(25), 22366–22379 (2009). [CrossRef] [PubMed]

20. Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Polarization properties of partially coherent electromagnetic elegant Laguerre-Gaussian beams in turbulent atmosphere,” Appl. Phys. B **102**(4), 937–944 (2011). [CrossRef]

21. M. A. Porras, R. Borghi, and M. Santarsiero, “Relationship between elegant Laguerre-Gauss and Bessel-Gauss beams,” J. Opt. Soc. Am. A **18**(1), 177–184 (2001). [CrossRef] [PubMed]

22. R. Borghi, “Elegant Laguerre-Gauss beams as a new tool for describing axisymmetric flattened Gaussian beams,” J. Opt. Soc. Am. A **18**(7), 1627–1633 (2001). [CrossRef] [PubMed]

23. A. Ciattoni, G. Cincotti, and C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A **19**(4), 792–796 (2002). [CrossRef] [PubMed]

32. G. Zhou, R. Chen, and X. Chu, “Propagation of Airy beams in uniaxial crystals orthogonal to the optical axis,” Opt. Express **20**(3), 2196–2205 (2012). [CrossRef] [PubMed]

## 2. Theoretical derivations

*z*-axis is taken to be the propagation axis. The optical axis of the uniaxial crystal coincides with the

*x*-axis. The input plane is

*z*= 0 and the observation plane is

*z*. The dielectric tensor of the uniaxial crystal is described bywhere

*n*and

_{o}*n*are the ordinary and the extraordinary refractive indices, respectively. The elegant Laguerre-Gaussian beam can be divided into two types: the even and the odd modes. First, we consider the even elegant Laguerre-Gaussian beam. The even elegant Laguerre-Gaussian beam considered here is linearly polarized in the

_{e}*x*-direction and is incident on a uniaxial crystal in the plane

*z*= 0. The even elegant Laguerre-Gaussian beam in the input plane

*z*= 0 takes the form [1

1. 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]

*ρ*_{0}=

*x*

_{0}

*e**+*

_{x}*y*

_{0}

*e**. Vectors*

_{y}

*e**and*

_{x}

*e**are the unit vectors in the*

_{y}*x*- and

*y*-directions, respectively.

*n*and

*m*are the radial and the angular mode numbers, respectively. The symbol

*w*

_{0}is the waist radius of the Gaussian part.

33. 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}=

*m*+ 2

*l*-2

*s*and

*N*

_{1}= 2

*n*-2

*l*+ 2

*s*. The expression [

*m*/2] denotes the integer part of

*m*/2, and

*M*

_{1}. The expressions

34. A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A **20**(11), 2163–2171 (2003). [CrossRef] [PubMed]

**=**

*k**k*

_{x}

*e**+*

_{x}*k*

_{y}

*e**,*

_{y}**=**

*ρ**x*

*e**+*

_{x}*y*

*e**, and*

_{y}*k*

_{0}= 2

*π*/

*λ*is the wave number with

*λ*being the optical wavelength.

*z*= 0 and is given bywhere

*j*=

*x*or

*y*(hereafter).

*k*and

_{ez}*k*are defined byWhen the beam waist radius of the even elegant Laguerre-Gaussian beam is comparable with the wavelength, the paraxial approach is not accurate enough. A nonparaxial laser beam can be produced from a paraxial laser beam by its focusing with a high numerical aperture, and is commonly encountered in holography microscopy and microlithography. When the field in a uniaxial crystal is nonparaxial, Eq. (5) can be expressed as a sum of the dominant paraxial result and a nonparaxial correction term [34

_{oz}34. A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A **20**(11), 2163–2171 (2003). [CrossRef] [PubMed]

*E*(

_{x}**,**

*ρ**z*) is just the paraxial result.

*E*(

_{y}**,**

*ρ**z*) and

*E*(

_{z}**,**

*ρ**z*) are the first nonparaxial correction to the paraxial result. The nonparaxial correction contains all the nonparaxial features of the beam and can be easily evaluated from the knowledge of the paraxial result [34

34. A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A **20**(11), 2163–2171 (2003). [CrossRef] [PubMed]

*µ*is an arbitrary integer. The auxiliary parameters are defined as where

*z*is the Rayleigh length of the Gaussian part. As shown in Eqs. (19)-(21), the polynomials in three components of the even elegant Laguerre-Gaussian beam have complex argument, which can be compared with those of the standard elegant Laguerre-Gaussian beam propagating in the uniaxial crystal orthogonal to the optical axis by using the method suggested in Ref [36

_{r}36. S. Saghafi, C. J. R. Sheppard, and J. A. Piper, “Characterising elegant and standard Hermite-Gaussian beam modes,” Opt. Commun. **191**(3-6), 173–179 (2001). [CrossRef]

*z*= 0 is described byBased on the following expansion [33

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

## 3. Numerical calculations and analyses

*m*= 3,

*n*= 2,

*w*

_{0}=

*λ*, and the ordinary refractive index is fixed to be 2.616. The extraordinary refractive index is variable. Figures 1 –3 represent the contour graphs of the intensity distribution of the three components of an even elegant Laguerre-Gaussian beam propagating in the uniaxial crystal orthogonal to the optical axis at several observation planes.

*e*is set to be 1.5 in Figs. 1–3. The observation planes are

*z*= 0.1

*z*,

_{r}*z*=

*z*,

_{r}*z*= 3

*z*, and

_{r}*z*= 5

*z*, respectively. The intensity is proportional to the scale of 1/

_{r}*λ*

^{2}, which is not marked in the figures. The beam profile of the

*x*-component of an even elegant Laguerre-Gaussian beam has six lobes. Upon propagation orthogonal to the optical axis of the uniaxial crystal, the spreading of the beam profile in the

*y*-direction is far slower than that in the

*x*-direction, which is caused by the anisotropic effect of the crystal. As a result, the beam profile in the uniaxial crystal is elongated in the

*x*-direction. The magnitude of the intensity of the six lobes close to the input plane is equivalent. However, the equality of magnitude of the intensity of the six lobes is broken upon propagation in the uniaxial crystal. At first the magnitude of the intensity of the two middle lobes is smaller than that of the other four lobes. When the observation plane is far enough, the magnitude of the intensity of the four top and bottom lobes is smaller than that of the two middle lobes. Though the

*y*-component of an even elegant Laguerre-Gaussian beam in the input plane is set to be zero, it is no longer equal to zero upon propagation in the uniaxial crystal, which can be interpreted as follow. The

*y*-component is affected by the

*x*-component because of the dependence on

32. G. Zhou, R. Chen, and X. Chu, “Propagation of Airy beams in uniaxial crystals orthogonal to the optical axis,” Opt. Express **20**(3), 2196–2205 (2012). [CrossRef] [PubMed]

*y*-component of an even elegant Laguerre-Gaussian beam apparently varies upon propagation in the uniaxial crystal. Upon propagation in the uniaxial crystal, the two central lobes including the top and bottom side lobes will diminish until disappear. Instead, the four bilateral lobes gradually become the dominant lobes. Compared with the magnitude of the

*x*-component, the magnitude of the

*y*-component is small. When propagating from the observation plane

*z*= 0.1

*z*to the observation plane

_{r}*z*=

*z*, the beam profile of the longitudinal component undergoes a process of the split of the lobes. When propagating from the observation plane

_{r}*z*=

*z*to the observation plane

_{r}*z*= 3

*z*, the two lobes in the

_{r}*y*-axis gradually diminish until disappear, and the two lobes in the

*x*-axis evolve from the accessory lobes into the dominant lobes. When the observation plane is larger than 3

*z*, the beam profile of the longitudinal component keeps stable. The magnitude of the longitudinal component is smaller than that of the

_{r}*x*-component but larger than that of the

*y*-component. The contour graph of the intensity distribution of an even elegant Laguerre-Gaussian beam propagating in the uniaxial crystal orthogonal to the optical axis at several observation planes is shown in Fig. 4 where

*e*= 1.5. As

*x*-component. However, the details in the beam profile between the even elegant Laguerre-Gaussian beam and its

*x*-component are different. Moreover, the magnitude of the intensity of the even elegant Laguerre-Gaussian beam is larger than that of its

*x*-component, which sometimes cannot be distinguished through the comparison of the corresponding figures as the label of the intensity is marked in the rank.

*z =*3

*z*of different uniaxial crystals. The uniaxial crystals can be divided into two kinds. One is the positive uniaxial crystal, which corresponds to

_{r}*e*>1. The other is the negative uniaxial crystal, which corresponds to

*e*<1. As to a uniaxial crystal, the value of

*e*is fixed. For simplicity, here a model which is the combination of two kinds of uniaxial crystals is used to illustrate the effect of the uniaxial crystals.

*e*= 1 corresponds to the isotropic mediums. When

*e*= 1, the beam profile of the even elegant Laguerre-Gaussian beam is symmetric and the magnitude of the intensity of the six lobes is equivalent. When

*e*is not equal to unity, the beam profile becomes an astigmatic beam. When

*e*<1, the beam profile is elongated in the

*y*-direction, and the magnitude of the intensity of the top and bottom lobes is larger than that of the two middle lobes. When

*e*>1, the beam profile is elongated in the

*x*-direction, and the magnitude of the intensity of the top and bottom lobes is smaller than that of the two middle lobes. With increasing the deviation of

*e*from unity, the elongation of the beam spot augments, and the difference of the magnitude of the intensity between the middle lobes and the other lobes also increases.

*e*= 1.5. When

*z*= 0.1

*z*, the beam profile of the

_{r}*x*-component of the odd elegant Laguerre-Gaussian beam is that of the even elegant Laguerre-Gaussian beam rotating 90° from the

*y*-axis. Upon propagation in the uniaxial crystal, the magnitude of the intensity of the lobes in the

*y*-axis quickly decreases and is smaller than that of the four bilateral lobes. When

*z*= 0.1

*z*, the beam profile of the

_{r}*y*-component of the odd elegant Laguerre-Gaussian beam is somewhat similar to that of the even elegant Laguerre-Gaussian beam rotating 90° from the

*y*-axis. However, the magnitude of the intensity slightly decreases. When

*z*=

*z*or

_{r}*z*= 3

*z*, the beam profile of the

_{r}*y*-component of the odd elegant Laguerre-Gaussian beam is completely different from that of the even elegant Laguerre-Gaussian beam. In these cases, the beam profile of the

*y*-component of the odd elegant Laguerre-Gaussian beam takes on an annular shape. When

*z*= 5

*z*, the beam profile of the

_{r}*y*-component of the odd elegant Laguerre-Gaussian beam is somewhat similar to that of the even elegant Laguerre-Gaussian beam. The beam profile of the longitudinal component of the odd elegant Laguerre-Gaussian beam is always completely different from that of the even elegant Laguerre-Gaussian beam. The beam profile of the longitudinal component of the odd elegant Laguerre-Gaussian beam has eight lobes. When

*z*= 3

*z*, each two lobes is mutually connected to form a group. Upon propagation in the uniaxial crystal, the magnitude of the intensity of the four central lobes diminishes and that of the four bilateral lobes increases. Figure 9 represents the contour graph of the intensity distribution of an odd elegant Laguerre-Gaussian beam propagating in the uniaxial crystal orthogonal to the optical axis at several observation planes. Comparing Fig. 9(d) with Fig. 6(d), the magnitude of the intensity denotes that the

_{r}*y*- and

*z*-components cannot be ignored. Figure 10 shows the contour graph of the intensity of an odd elegant Laguerre-Gaussian beam in the observation plane

*z =*3

*z*of different uniaxial crystals. When

_{r}*e*<1, the magnitude of the intensity of the lobes in the

*y*-axis is larger than that of the four bilateral lobes. When

*e*>1, the magnitude of the intensity of the lobes in the

*y*-axis is smaller than that of the four bilateral lobes. When

*e*<1, the beam profile of the odd elegant Laguerre-Gaussian beam and its components is elongated in the

*y*-direction. When

*e*>1, the beam profile of the odd elegant Laguerre-Gaussian beam and its components is elongated in the

*x*-direction. Comparing Figs. 1–5 with Figs. 6–10, one can find that the even and odd elegant Laguerre-Gaussian beams have their respective evolution laws.

## 4. Conclusions

*y*-component of even and odd elegant Laguerre-Gaussian beams in the input plane is set to be zero, the

*y*-component emerges upon propagation in the uniaxial crystal. Moreover, the

*y*- and

*z*-components cannot be ignored. The even and odd elegant Laguerre-Gaussian beams propagating in uniaxial crystal become astigmatic beams. The beam profile of even and odd elegant Laguerre-Gaussian beams in the uniaxial crystal is elongated in the

*x*- or

*y*-direction, which is determined by the ratio of the extraordinary refractive index to the ordinary refractive index. With the increase of the deviation of the ratio of the extraordinary refractive index to the ordinary refractive index from unity, the elongation of the beam profile also increases. The even and odd elegant Laguerre-Gaussian beams propagating in the uniaxial crystal have their respective evolution laws. As the beam profile of even and odd elegant Laguerre-Gaussian beams can be modulated by the uniaxial crystal, this research is beneficial to the some applications involving in the special beam profile.

## Acknowledgments

## References and links

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

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

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

4. | S. Luo and B. Lü, “Propagation of the kurtosis parameter of elegant Hermite-Gaussian and Laguerre-Gaussian beams passing through |

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

6. | Z. Mei and D. Zhao, “Propagation of Laguerre-Gaussian and elegant Laguerre-Gaussian beams in apertured fractional Hankel transform systems,” J. Opt. Soc. Am. A |

7. | D. Zhao, Z. Mei, J. Gu, H. Mao, L. Chen, and S. Wang, “Propagation characteristics of truncated standard and elegant Laguerre-Gaussian beams,” Proc. SPIE |

8. | Z. Mei and D. Zhao, “The generalized beam propagation factor of truncated standard and elegant Laguerre-Gaussian beams,” J. Opt. A, Pure Appl. Opt. |

9. | Y. Pagani and W. Nasalski, “Diagonal relations between elegant Hermite-Gaussian and Laguerre-Gaussian beam fields,” Opto-Electron. Rev. |

10. | J. C. Gutiérrez-Vega, “Fractionalization of optical beams: II. Elegant Laguerre-Gaussian modes,” Opt. Express |

11. | D. Deng, Q. Guo, L. Wu, and X. Yang, “Propagation of radially polarized elegant light beams,” J. Opt. Soc. Am. B |

12. | Z. Mei, “The elliptical elegant Laguerre-Gaussian beam and its propagation through aligned and misaligned paraxial optical systems,” Optik (Stuttg.) |

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

14. | Z. Mei and J. Gu, “Comparative studies of paraxial and nonparaxial vectorial elegant Laguerre-Gaussian beams,” Opt. Express |

15. | W. Nasalski, “Elegant Hermite-Gaussian and Laguerre-Gaussian beams at a dielectric interface,” Opt. Appl. |

16. | J. Qu, Y. Zhong, Z. Cui, and Y. Cai, “Elegant Laguerre-Gaussian beam in a turbulent atmosphere,” Opt. Commun. |

17. | H. Xu, Z. Cui, and J. Qu, “Propagation of elegant Laguerre-Gaussian beam in non-Kolmogorov turbulence,” Opt. Express |

18. | F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express |

19. | F. Wang, Y. Cai, H. T. Eyyuboglu, and Y. K. Baykal, “Average intensity and spreading of partially coherent standard and elegant Laguerre-Gaussian beams in turbulent atmosphere,” PIER |

20. | Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Polarization properties of partially coherent electromagnetic elegant Laguerre-Gaussian beams in turbulent atmosphere,” Appl. Phys. B |

21. | M. A. Porras, R. Borghi, and M. Santarsiero, “Relationship between elegant Laguerre-Gauss and Bessel-Gauss beams,” J. Opt. Soc. Am. A |

22. | R. Borghi, “Elegant Laguerre-Gauss beams as a new tool for describing axisymmetric flattened Gaussian beams,” J. Opt. Soc. Am. A |

23. | A. Ciattoni, G. Cincotti, and C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A |

24. | B. Lü and S. Luo, “Propagation properties of three-dimensional flatted Gaussian beams in uniaxially anisotropic crystals,” Opt. Laser Technol. |

25. | D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. |

26. | D. Liu and Z. Zhou, “Generalized stokes parameters of stochastic electromagnetic beams propagating through uniaxial crystals orthogonal to the optical axis,” J. Opt. A, Pure Appl. Opt. |

27. | B. Tang, “Hermite-cosine-Gaussian beams propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A |

28. | D. Liu and Z. Zhou, “Propagation of partially polarized, partially coherent beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D |

29. | C. Zhao and Y. Cai, “Paraxial propagation of Lorentz and Lorentz-Gauss beams in uniaxial crystals orthogonal to the optical axis,” J. Mod. Opt. |

30. | J. Li, Y. Chen, Y. Xin, and S. Xu, “Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D |

31. | J. Li, Y. Xin, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, and F. Chen, “Diffraction of Gaussian vortex beam in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys. |

32. | G. Zhou, R. Chen, and X. Chu, “Propagation of Airy beams in uniaxial crystals orthogonal to the optical axis,” Opt. Express |

33. | I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. |

34. | A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A |

35. | I. S. Gradshteyn and I. M. Ryzhik, |

36. | S. Saghafi, C. J. R. Sheppard, and J. A. Piper, “Characterising elegant and standard Hermite-Gaussian beam modes,” Opt. Commun. |

**OCIS Codes**

(260.1180) Physical optics : Crystal optics

(260.1960) Physical optics : Diffraction theory

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: May 23, 2012

Revised Manuscript: June 28, 2012

Manuscript Accepted: July 9, 2012

Published: July 12, 2012

**Citation**

Yongzhou Ni and Guoquan Zhou, "Nonparaxial propagation of an elegant Laguerre-Gaussian beam orthogonal to the optical axis of a uniaxial crystal," Opt. Express **20**, 17160-17173 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-15-17160

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### References

- T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximate,” J. Opt. Soc. Am. A2(6), 826–829 (1985). [CrossRef]
- E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am.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(10), 1999–2009 (1998). [CrossRef]
- S. Luo and B. Lü, “Propagation of the kurtosis parameter of elegant Hermite-Gaussian and Laguerre-Gaussian beams passing through ABCD systems,” Optik (Stuttg.)113(5), 227–231 (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]
- Z. Mei and D. Zhao, “Propagation of Laguerre-Gaussian and elegant Laguerre-Gaussian beams in apertured fractional Hankel transform systems,” J. Opt. Soc. Am. A21(12), 2375–2381 (2004). [CrossRef] [PubMed]
- D. Zhao, Z. Mei, J. Gu, H. Mao, L. Chen, and S. Wang, “Propagation characteristics of truncated standard and elegant Laguerre-Gaussian beams,” Proc. SPIE5639, 149–158 (2004). [CrossRef]
- Z. Mei and D. Zhao, “The generalized beam propagation factor of truncated standard and elegant Laguerre-Gaussian beams,” J. Opt. A, Pure Appl. Opt.6(11), 1005–1011 (2004). [CrossRef]
- Y. Pagani and W. Nasalski, “Diagonal relations between elegant Hermite-Gaussian and Laguerre-Gaussian beam fields,” Opto-Electron. Rev.13, 51–60 (2005).
- J. C. Gutiérrez-Vega, “Fractionalization of optical beams: II. Elegant Laguerre-Gaussian modes,” Opt. Express15(10), 6300–6313 (2007). [CrossRef] [PubMed]
- D. Deng, Q. Guo, L. Wu, and X. Yang, “Propagation of radially polarized elegant light beams,” J. Opt. Soc. Am. B24(3), 636–643 (2007). [CrossRef]
- Z. Mei, “The elliptical elegant Laguerre-Gaussian beam and its propagation through aligned and misaligned paraxial optical systems,” Optik (Stuttg.)118(8), 361–366 (2007). [CrossRef]
- A. April, “Nonparaxial elegant Laguerre-Gaussian beams,” Opt. Lett.33(12), 1392–1394 (2008). [CrossRef] [PubMed]
- Z. Mei and J. Gu, “Comparative studies of paraxial and nonparaxial vectorial elegant Laguerre-Gaussian beams,” Opt. Express17(17), 14865–14871 (2009). [CrossRef] [PubMed]
- W. Nasalski, “Elegant Hermite-Gaussian and Laguerre-Gaussian beams at a dielectric interface,” Opt. Appl.40, 615–622 (2010).
- J. Qu, Y. Zhong, Z. Cui, and Y. Cai, “Elegant Laguerre-Gaussian beam in a turbulent atmosphere,” Opt. Commun.283(14), 2772–2781 (2010). [CrossRef]
- H. Xu, Z. Cui, and J. Qu, “Propagation of elegant Laguerre-Gaussian beam in non-Kolmogorov turbulence,” Opt. Express19(22), 21163–21173 (2011). [CrossRef] [PubMed]
- F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express17(25), 22366–22379 (2009). [CrossRef] [PubMed]
- F. Wang, Y. Cai, H. T. Eyyuboglu, and Y. K. Baykal, “Average intensity and spreading of partially coherent standard and elegant Laguerre-Gaussian beams in turbulent atmosphere,” PIER103, 33–56 (2010). [CrossRef]
- Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Polarization properties of partially coherent electromagnetic elegant Laguerre-Gaussian beams in turbulent atmosphere,” Appl. Phys. B102(4), 937–944 (2011). [CrossRef]
- M. A. Porras, R. Borghi, and M. Santarsiero, “Relationship between elegant Laguerre-Gauss and Bessel-Gauss beams,” J. Opt. Soc. Am. A18(1), 177–184 (2001). [CrossRef] [PubMed]
- R. Borghi, “Elegant Laguerre-Gauss beams as a new tool for describing axisymmetric flattened Gaussian beams,” J. Opt. Soc. Am. A18(7), 1627–1633 (2001). [CrossRef] [PubMed]
- A. Ciattoni, G. Cincotti, and C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A19(4), 792–796 (2002). [CrossRef] [PubMed]
- B. Lü and S. Luo, “Propagation properties of three-dimensional flatted Gaussian beams in uniaxially anisotropic crystals,” Opt. Laser Technol.36(1), 51–56 (2004). [CrossRef]
- D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun.281(2), 202–209 (2008). [CrossRef]
- D. Liu and Z. Zhou, “Generalized stokes parameters of stochastic electromagnetic beams propagating through uniaxial crystals orthogonal to the optical axis,” J. Opt. A, Pure Appl. Opt.11(6), 065710 (2009). [CrossRef]
- B. Tang, “Hermite-cosine-Gaussian beams propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A26(12), 2480–2487 (2009). [CrossRef] [PubMed]
- D. Liu and Z. Zhou, “Propagation of partially polarized, partially coherent beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D54(1), 95–101 (2009). [CrossRef]
- C. Zhao and Y. Cai, “Paraxial propagation of Lorentz and Lorentz-Gauss beams in uniaxial crystals orthogonal to the optical axis,” J. Mod. Opt.57(5), 375–384 (2010). [CrossRef]
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