## The analytical vectorial structure of a nonparaxial Gaussian beam close to the source

Optics Express, Vol. 16, Issue 6, pp. 3504-3514 (2008)

http://dx.doi.org/10.1364/OE.16.003504

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### Abstract

The description of a nonparaxial Gaussian beam is made directly staring with the Maxwell’s equations. The vector angular spectrum method is used to resolve the Maxwell’s equations. As the vector angular spectrum can be decomposed into the two terms in the frequency domain, the nonparaxial Gaussian beam is also expressed as a sum of two terms. One term is the electric field transverse to the propagation axis, and the other term is the associated magnetic field transverse to the propagation axis. By means of mathematical techniques, the analytical expressions for the two terms in the source region have been derived without any approximation. The influence of the evanescent plane wave on the vectorial structure is also investigated. The results are analyzed with numerical example. This research is useful to the optical trapping and the optical manipulation.

© 2008 Optical Society of America

## 1. Introduction

1. K. Varjú, A. P. Kovács, K. Osvay, and G. Kurdi, “Angular dispersion of femtosecond pulses in a Gaussian beam,” Opt. Lett. **27**, 2034–2036 (2002). [CrossRef]

2. V. Delaubert, N. Treps, C. C. Harb, P. K. Lam, and H. -A. Bachor, “Quantum measurements of spatial conjugate variables: displacement and tilt of a Gaussian beam,” Opt. Lett. **31**, 1537–1539 (2006). [CrossRef] [PubMed]

3. A. Ashkin, “Optical trapping and manipulation of neutral particles using lasers,” Proc. Natl. Acad. Sci. U.S.A. **94**, 4853–4860 (1997). [CrossRef] [PubMed]

4. R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian-Maxwell beams,” J. Opt. Soc. Am. A **3**, 536–540 (1986). [CrossRef]

8. R. Martínez-Herrero, P. Mejías, S. Bosch, and A. Carnicer, “Vectorial structure of nonparaxial electromagnetic beams,” J. Opt. Soc. Am. A **18**, 1678–1680 (2001). [CrossRef]

11. G. Zhou, “Analytical vectorial structure of Laguerre-Gaussian beam in the far field,” Opt. Lett. **31**, 2616–2618 (2006). [CrossRef] [PubMed]

14. Patrick C. Chaument, “Fully vectorial highly nonparaxial beam clost to the waist,” J. Opt. Soc. Am. A **23**, 3197–3202 (2007). [CrossRef]

## 2. The analytical vectorial structure close to the source

*z*-axis is taken to be the propagation axis. The half space

*z*≥0 is a free space. Note that a nonparaxial Gaussian beam is treated to be linearly polarized in both theoretical and practical applications [15,16], the transverse electric field of a nonparaxial Gaussian beam at the source plane

*z*= 0 takes the form as

*w*

_{0}is the Gaussian half width.

*α*is the linearly polarized angle with respect to the

*x*-axis. The time dependent factor exp(-i

*ωt*) is omitted in the Eq. (1), and

*ω*is the circular frequency. Here, the description of the nonparaxial Gaussian beam is made directly starting with the Maxwell’s equations:

**=**

*r**x*

**x**+

*y*

**y**+

*z*

**z**, and

*k*= 2

*π*/

*λ . λ*is the light wavelength.

**x**,

**y**and

**z**are unit vectors in the

*x*-,

*y*- and

*z*-directions, respectively.

*(*

**E***) and*

**r***(*

**H***) are the propagating electromagnetic fields of the nonparaxial Gaussian beam. Transforming from the space domain into the frequency domain, Eqs. (2)–(4) become*

**r***=*

**L****x**

*ikp*+

**y**

*ikq*+

**z**∂/∂

*z*.

*p*/

*λ*and

*q*/

*λ*are the transverse frequencies.

**(**

*E͂**p*,

*q*,

*z*) and

**(**

*H͂**p*,

*q*,

*z*) denote the spatial Fourier transforms of

**(**

*E***) and**

*r**(*

**H****), respectively. The solutions of Eqs. (5)–(7) are expressed in the form as**

*r**=*

**s***p*

**x**+

*q*

**y**+

*γ*

**z**, and

*γ*= (1-

*p*

^{2}-

*q*

^{2})

^{1/2}. The vector angular spectrum

*A*(

*p*,

*q*) reads as

*A*(

_{x}*p*,

*q*),

*A*(

_{y}*p*,

*q*) and

*A*(

_{z}*p*,

*q*) are the transverse and the longitudinal components of the vector angular spectrum, respectively. The electric field of the nonparaxial Gaussian beam propagating toward half free space

*z*≥0 can be obtained [17

17. C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Heilmann, and M. L. Schattenburg, “Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximations,” J. Opt. Soc. Am. A **19**, 404–412 (2002). [CrossRef]

*f*= 1/

*kw*

_{0}, and

*b*= (

*p*

^{2}+

*q*

^{2})

^{1/2}. The longitudinal component of the vector angular spectrum is given by the orthogonal relation

**·**

*s**(*

**A***p*,

*q*)=0 and turns out to be

*e*_{1}and

*e*_{2}can be defined as follows

**,**

*s*

*e*_{1}and

*e*_{2}form a mutually perpendicular right–handed system

*(*

**A***p*,

*q*) can be decomposed into the two terms [8–9

8. R. Martínez-Herrero, P. Mejías, S. Bosch, and A. Carnicer, “Vectorial structure of nonparaxial electromagnetic beams,” J. Opt. Soc. Am. A **18**, 1678–1680 (2001). [CrossRef]

*E*_{TE}(

**) and**

*r*

*E*_{TM}(

**) given by**

*r**φ*= tan

^{-1}(

*q*/

*p*), Eq. (18) can be rewritten as follow

*θ*= tan

^{-1}(

*y*/

*x*), and

*ρ*= (

*x*

^{2}+

*y*

^{2})

^{1/2}. The values of

*b*<1 correspond to the homogeneous plane waves propagating at angles sin

^{-1}

*b*with respect to the

*z*-axis, whereas values of

*b*>1 the evanescent plane waves. Under the integration process, the following integral formula is satisfied [18]

*J*is the

_{n}*n*-th order Bessel function of the first kind, and

*n*is an arbitrary integer. Therefore, the integral expression of the first term turns out to be

*δ*= 2

*θ*-

*α*, and

*T*(

_{n}*) given by*

**r****) given by**

*r**z*≥0.

*z*is smaller or of the order of the wavelength. Accordingly, the homogeneous and the evanescent plane waves must be both taken into account for the contribution to the electric field. By transforming the integral variable from

*b*to

*γ*, Eqs. (23) and (27) can be rewritten as follows:

*j*is an arbitrary integer.

*F*(·) is the Faddeev function and can be calculated by the procedure suggested by Ref. [21

21. G. P. M. poppe and C. M. J. Wijers, “More efficient computation of the complex error function,” ACM Trans. Math. Softw. **16**, 38–46 (1990). [CrossRef]

*β*=

*x*,

*y*, and

*A*= exp(-1/4

*f*

^{2})/4

*f*

^{2}. The corresponding analytical expression of the second term in the source region turns out to be

*µ*=

*x*,

*y*,

*z*and

**(**

*E***)=0 . Within the paraxial regime, the longitudinal electric field is far smaller comparing with the transverse electric field. Therefore, the longitudinal electric field vanishes and the electric field is treated to be lain in planes orthogonal to the propagation axis. This treatment provides good results within the paraxial framework. In the nonparaxial regime, however, the magnitude of the longitudinal electric field is comparable with that of the transverse electric field. Accordingly, the longitudinal electric field can not be omitted. As the description of a nonparaxial Gaussian beam is made directly starting with the Maxwell’s equations, Eqs. (41)–(48) consequentially satisfy the exact Maxwell’s equations and also the vector Helmholtz wave equation for the electric field. But, the two terms can not be produced alone. Only their sum constitutes a realizable laser beam. The electric field of a nonparaxial Gaussian beam can be decomposed into**

*r**E*,

_{x}*E*and

_{y}*E*in the spatial domain, while the decomposition of the electric field into the two terms is carried out in terms of the frequency domain, which is just an alternative approach of expression.

_{z}14. Patrick C. Chaument, “Fully vectorial highly nonparaxial beam clost to the waist,” J. Opt. Soc. Am. A **23**, 3197–3202 (2007). [CrossRef]

## 3. The numerical results and analyses

*w*

_{0}is set to be

*λ*/2. The plane

*z*=

*λ*/4 is selected as the reference plane.

*α*is considered to be 0°, which denotes that the initial transverse electric field is polarized along the

*x*-axis. The difference between the

*y*components of the two terms is only a minus sign. Thus, the

*y*component of the second term, which is same as Figs. 2(d)–2(f), is omitted in Fig. 3. The magnitude of amplitude of the

*x*component is larger than that of the

*y*component. Moreover, their patterns are completely different. Compared with that of the propagation wave, the contribution of the evanescent wave to the transverse components of the two terms is small enough to be negligible. As to the second term, the magnitude of amplitude of the

*z*component is comparable with that of the

*x*component. Moreover, the contribution of the evanescent wave can not be neglected. There is a minus sign between the evanescent part and the real portion of the propagation part. As a result, the magnitude of amplitude of the

*z*component is smaller than that of the corresponding propagation part. The contribution of evanescent wave to the

*z*component of the second term is considerable in the magnitude. Due to the Bessel function, the amplitude distribution of the evanescent part shows circles.

*T*denotes the ratio in percentage between the intensity of the evanescent part and the intensity of the first term or the latter term, which is shown in Fig. 5. The symmetry axis of

*T*for the first term is just consistent with the direction of the linearly polarized angle, while that for the latter term is along the orientation perpendicular to the direction of the linearly polarized angle. Combined with Figs. 2 and 3, the contribution of evanescent wave must be taken into account for description of the second term. Figure 6 investigates the effect of the propagation distance

*z*on the nonparaxial Gaussian beam and its two terms. All the parameters except for the propagation distance

*z*keep invariable. When the propagation distance

*z*increases, the magnitude of the intensities for the whole beam and its two terms decreases. As expected, the corresponding beam spots augment. The evolution of a nonparaxial Gaussian beam and its two terms upon the propagation is apparently revealed.

## 4. Conclusions

*E*,

_{x}*E*and

_{y}*E*in the spatial domain, while the decomposition of the electric field into the two terms is carried out in terms of the frequency domain, which offers an alternative approach to investigate the vectorial properties of the nonparaxial Gaussian beam. The vectorial structure of the nonparaxial Gaussian beam is emerged in the integral form. By means of mathematical techniques, the analytical expressions for the two terms in the source region have been derived without any approximation. The influence of the evanescent wave on the vectorial structure is analyzed with numerical example. The contribution of evanescent wave to the

_{z}*z*component of the second term is considerable in the magnitude. However, the contribution of the evanescent wave to the transverse components of the two terms is small enough to be negligible. This research reveals the vectorial composition of a nonparaxial Gaussian beam close to the source and is useful to the optical trapping and the optical manipulation.

## Acknowledgments

## References and links

1. | K. Varjú, A. P. Kovács, K. Osvay, and G. Kurdi, “Angular dispersion of femtosecond pulses in a Gaussian beam,” Opt. Lett. |

2. | V. Delaubert, N. Treps, C. C. Harb, P. K. Lam, and H. -A. Bachor, “Quantum measurements of spatial conjugate variables: displacement and tilt of a Gaussian beam,” Opt. Lett. |

3. | A. Ashkin, “Optical trapping and manipulation of neutral particles using lasers,” Proc. Natl. Acad. Sci. U.S.A. |

4. | R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian-Maxwell beams,” J. Opt. Soc. Am. A |

5. | Dennis G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. |

6. | S. R. Seshadri, “Electromagnetic Gaussian beam,” J. Opt. Soc. Am. A |

7. | C. J. R. Sheppard and S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A |

8. | R. Martínez-Herrero, P. Mejías, S. Bosch, and A. Carnicer, “Vectorial structure of nonparaxial electromagnetic beams,” J. Opt. Soc. Am. A |

9. | P. M. Mejías, R. Martínez-Herrero, G. Piquero, and J. M. Movilla, “Parametric characterization of the spatial structure of non-uniformly polarized laser beams,” Prog. Quantum Electron. |

10. | H. Guo, J. Chen, and S. Zhuang, “Vector plane wave spectrum of an arbitrary polarized electromagnetic wave,” Opt. Express |

11. | G. Zhou, “Analytical vectorial structure of Laguerre-Gaussian beam in the far field,” Opt. Lett. |

12. | G. Zhou, L. Chen, and Y. Ni, “Vectorial structure of non-paraxial linearly polarized Gaussian beam in far field,” Chin. Phys. Lett. |

13. | G. Zhou, K. Zhu, and F. Liu, “Analytical structure of the TE and TM terms of paraxial Gaussian beam in the near field,” Opt. Commun. |

14. | Patrick C. Chaument, “Fully vectorial highly nonparaxial beam clost to the waist,” J. Opt. Soc. Am. A |

15. | A. E. Siegman, |

16. | P. W. Milonni and J. H. Eberly, |

17. | C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Heilmann, and M. L. Schattenburg, “Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximations,” J. Opt. Soc. Am. A |

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

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

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

21. | G. P. M. poppe and C. M. J. Wijers, “More efficient computation of the complex error function,” ACM Trans. Math. Softw. |

**OCIS Codes**

(260.2110) Physical optics : Electromagnetic optics

(260.5430) Physical optics : Polarization

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: January 17, 2008

Revised Manuscript: February 10, 2008

Manuscript Accepted: February 29, 2008

Published: March 3, 2008

**Virtual Issues**

Vol. 3, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

Guoquan Zhou, "The analytical vectorial structure of a nonparaxial Gaussian beam close to the source," Opt. Express **16**, 3504-3514 (2008)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-16-6-3504

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### References

- K. Varjú, A. P. Kovács, K. Osvay, and G. Kurdi, "Angular dispersion of femtosecond pulses in a Gaussian beam," Opt. Lett. 27, 2034-2036 (2002). [CrossRef]
- V. Delaubert, N. Treps, C. C. Harb, P. K. Lam, and H. -A. Bachor, "Quantum measurements of spatial conjugate variables: displacement and tilt of a Gaussian beam," Opt. Lett. 31, 1537-1539 (2006). [CrossRef] [PubMed]
- A. Ashkin, "Optical trapping and manipulation of neutral particles using lasers," Proc. Natl. Acad. Sci. U.S.A. 94, 4853-4860 (1997). [CrossRef] [PubMed]
- R. Simon, E. C. G. Sudarshan, and N. Mukunda, "Gaussian-Maxwell beams," J. Opt. Soc. Am. A 3, 536-540 (1986). [CrossRef]
- Dennis G. Hall, "Vector-beam solutions of Maxwell’s wave equation," Opt. Lett. 21, 9-11 (1996). [CrossRef] [PubMed]
- S. R. Seshadri, "Electromagnetic Gaussian beam," J. Opt. Soc. Am. A 15, 2712-2719 (1998). [CrossRef]
- C. J. R. Sheppard and S. Saghafi, "Electromagnetic Gaussian beams beyond the paraxial approximation," J. Opt. Soc. Am. A 16, 1381-1386 (1999). [CrossRef]
- R. Martínez-Herrero, P. Mejías, S. Bosch, and A. Carnicer, "Vectorial structure of nonparaxial electromagnetic beams," J. Opt. Soc. Am. A 18, 1678-1680 (2001). [CrossRef]
- P. M. Mejías, R. Martínez-Herrero, G. Piquero, and J. M. Movilla, "Parametric characterization of the spatial structure of non-uniformly polarized laser beams," Prog. Quantum Electron. 26, 65-130 (2002). [CrossRef]
- H. Guo, J. Chen and S. Zhuang, "Vector plane wave spectrum of an arbitrary polarized electromagnetic wave," Opt. Express 14, 2095-2100 (2006). [CrossRef] [PubMed]
- G. Zhou, "Analytical vectorial structure of Laguerre-Gaussian beam in the far field," Opt. Lett. 31, 2616-2618 (2006). [CrossRef] [PubMed]
- G. Zhou, L. Chen, and Y. Ni, "Vectorial structure of non-paraxial linearly polarized Gaussian beam in far field," Chin. Phys. Lett. 23, 1180-1183 (2006). [CrossRef]
- G. Zhou, K. Zhu, F. Liu, "Analytical structure of the TE and TM terms of paraxial Gaussian beam in the near field," Opt. Commun. 276, 37-43 (2007). [CrossRef]
- P. C. Chaument, "Fully vectorial highly nonparaxial beam clost to the waist," J. Opt. Soc. Am. A 23, 3197-3202 (2007). [CrossRef]
- A. E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986), Chaps. 16-17.
- P. W. Milonni and J. H. Eberly, Lasers (Wiley Interscience, New York, 1988), sec. 14.5.
- C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Heilmann, and M. L. Schattenburg, "Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximations," J. Opt. Soc. Am. A 19, 404-412 (2002). [CrossRef]
- I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, (Academic Press, New York, 1980), p. 952.
- I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, corrected and enlarged edition (Academic press, New York, 1980), p. 951.
- I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, corrected and enlarged edition (Academic press, New York, 1980), pp. 1064-1067.
- G. P. M. Poppe and C. M. J. Wijers, "More efficient computation of the complex error function," ACM Trans. Math. Software 16, 38-46 (1990). [CrossRef]

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