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

As a typical beam model, Gaussian beam receives considerable interest [1, 2]. Within the paraxial framework, a Gaussian beam is described by the approximate solution of Helmholtz equation. For beams with the small beam spot size comparable with the light wavelength and/or the large divergence angle, however, the paraxial approximation fails. The nonparaxial Gaussian beam can be very useful in many areas such as the optical manipulation [3]. The accurate description of a nonparaxial Gaussian beam should be directly derived from the Maxwell’s equations. The Gaussian-beam-type solutions to the Maxwell’s equations have been constructed by using many different methods [4–7], one of which namely the vector angular spectrum method is a suitable one. As the vector angular spectrum can be decomposed into the two terms in the frequency domain, a nonparaxial Gaussian beam is essentially 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 [8–10]. The two terms offer an alternative approach to investigate the vectorial properties of a nonparaxial Gaussian beam. In the present paper, therefore, the two terms are called as the vectorial structure of a nonparaxial Gaussian beam. As the divergence condition of the electric field should be satisfied and the polarized direction of every plane wave component must be perpendicular to its own wave vector, the vectorial structure of a nonparaxial Gaussian beam is unique. From the vector plane wave spectrum representation of the electromagnetic beam, one may distinguish three different propagation regions: the source region, the near field and the far field. As the analytical vectorial structure of nonparaxial and paraxial Gaussian beams has been examined in the near and far fields [11–13], one may be curious to know it close to the source. Moreover, the spatial orientation relationship between the two terms is varied upon propagation. Once the analytical expressions of vectorial structure in three propagation regions are presented, one can master the propagation characteristics of the two terms. In the source region, the homogeneous and the evanescent plane waves should be both taken into account for a complete description of a nonparaxial Gaussian beam [14]. By means of mathematical techniques, therefore, the analytical vectorial structure of a noparaxial Gaussian beam is presented in the source region. The influence of the evanescent plane wave on the vectorial structure of a nonparaxial Gaussian beam is also investigated in the source region.

2. The analytical vectorial structure close to the source

In the Cartesian coordinate system, the 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

(Ex(x0,y0,0)Ey(x0,y0,0))=(cosαsinα)exp(ρ02w02),

where ρ0=(x02+y02)1/2 , w 0 is the Gaussian half width. α is the linearly polarized angle with respect to the x-axis. The time dependent factor exp(-iωt) 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:

×E(r)ikH(r)=0,
×H(r)+ikE(r)=0,
·E(r)=·H(r)=0,

where r=x x+y y+z z, and k = 2π/λ . λ is the light wavelength. x, y and z are unit vectors in the x-, y- and z-directions, respectively. E(r) andH(r) are the propagating electromagnetic fields of the nonparaxial Gaussian beam. Transforming from the space domain into the frequency domain, Eqs. (2)–(4) become

L×E˜(p,q,z)ikH˜(p,q,z)=0,
L×H˜(p,q,z)+ikE˜(p,q,z)=0,
L·E˜(p,q,z)=L·H˜(p,q,z)=0,

where L = x ikp+y ikq+z∂/∂z. p/λ and q/λ are the transverse frequencies. (p,q,z) and (p,q,z) denote the spatial Fourier transforms of E(r) and H(r), respectively. The solutions of Eqs. (5)–(7) are expressed in the form as

E˜(p,q,z)=A(p,q)exp(ikγz),
H˜(p,q,z)=[s×A(p,q)]exp(ikγz),

where 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(p,q)=Ax(p,q)x+Ay(p,q)y+Az(p,q)z,

where Ax(p,q), Ay(p,q) and Az(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]

E(r)=E˜(p,q,z)exp[ik(px+qy)]dpdq=A(p,q)exp(ikr.s)dpdq.

According to the boundary condition of initial transverse electric field, the transverse components of the vector angular spectrum are found to be

(Ax(p,q)Ay(p,q))=14πf2exp(b24f2)(cosαsinα),

where 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

Az(p,q)=pAx(p,q)+qAy(p,q)γ=pcosα+qsinα4πf2γexp(b24f2).

In the frequency domain, two unit vectors e 1 and e 2 can be defined as follows

e1=qbxpby,e2=bx+bybz.

Therefore, the three unit vectors s, e 1 and e 2 form a mutually perpendicular right–handed system

s×e1=e2,e1×e2=s,e2×s=e1.

In this system, the vector angular spectrum A(p,q) can be decomposed into the two terms [8–9]

A(p,q)=[A(p,q).e1]e1+[A(p,q).e2]e2.
 

Fig. 1. Scheme of the definition of unit vectors and the decomposition of the vector angular spectrum.

Download Full Size | PPT Slide | PDF

Figure 1 is the scheme of the definition of unit vectors and the decomposition of the vector angular spectrum. As a result, the propagating electric field of the nonparaxial Gaussian beam is essentially expressed as a sum of two terms:

E(r)=ETE(r)+ETM(r),

with E TE (r) and E TM (r) given by

(ETE(r)ETM(r))=14πf21bexp(b24f2)((qcosαpsinα)e1(pcosα+qsinα)e1)exp(ikr.s)dpdq.

The first term in Eq. (18) is the electric field transverse to the propagation axis, and the magnetic field associated with the latter term can be verified to be also transverse to the propagation axis. By introducing a new parameter φ = tan-1(q/p), Eq. (18) can be rewritten as follow

(ETE(r)ETM(r))=14πf2002πexp(b24f2)(sin(φα)(sinφxcosφy)cos(φα)[γ(cosφx+sinφy)bz])exp(ikγz)
×exp[ikρbcos(φθ)]bdbdφ,

where 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]

Jn(kρb)=12π02πexp[ikρbcos(φθ)+in(φθπ2)]dφ,

where Jn is the 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

ETEx(r)=14f2[cosαT0(r)+cosδT2(r)],
ETEy(r)=14f2[sinαT0(r)+sinδT2(r)],

with δ = 2θ-α, and Tn(r) given by

Tn(r)=0exp(b24f2)exp(ikγz)Jn(kρb)bdb.

Similarly, the integral expression of the second term reads as

ETMx(r)=14f2[cosαT0(r)cosδT2(r)],
ETMy(r)=14f2[sinαT0(r)sinδT2(r)],
ETMz(r)=icos(θα)2f2Ω(r),

with Ω(r) given by

Ω(r)=0exp(b24f2)exp(ikγz)J1(kρb)b2γdb.

The vectorial structure of a nonparaxial Gaussian beam obtained here is applicable to the half space z≥0.

In the source region, 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:

Tn(r)=exp(14f2)(010+i)exp(γ24f2)exp(ikγz)Jn(kρ1γ2)γdγ,
Ω(r)=exp(14f2)(010+i)exp(γ24f2)exp(ikγz)J1(kρ1γ2)1γ2.

As the integration is inconvenient to perform, the following Taylor expansions are valid [19]

Jn(kρ1γ2)=(2)nl=0m=0l+n2(1)mCl(l+n2)!γ2m(l+n)!m!(lm+n2)!,nis an even integer,
J1(kρ1γ2)1γ2=2l=0m=0l+1(1)mClγ2mm!(l+1m)!,

where C l = (-1)l (/2)2l/l!. Eqs. (28) and (29) turn out to be

Tn(r)=(2)nexp(14f2)l=0m=0l+n2(1)mCl(l+n2)!(l+n)!m!(lm+n2)!(010+i)exp(γ24f2)exp(ikγz)γ2m+1,
Ω(r)=2exp(14f2)l=0m=0l+1(1)mClm!(l+1m)!(010+i)exp(γ24f2)exp(ikγz)γ2m.

To obtain the integration, one should first perform the propagation part [20]

Ijpro=01exp(γ24f2)exp(ikγz)γj=2f2[exp(14f2)exp(ikz)ikzIj1pro(j1)Ij2pro],

with I0pro and I1pro given by

I0pro=ifπ[F(izw0)exp(14f2)exp(ikz)F(izw0+kw02)],
I1pro=2f2[exp(14f2)exp(ikz)1ikzI0pro],

where j is an arbitrary integer. F(·) is the Faddeev function and can be calculated by the procedure suggested by Ref. [21]. Secondly, the evanescent part yields [20]

Ijeva=0+iexp(γ24f2)exp(ikγz)γj=(i2f)j+1j!Dj+1(2zw0),

where D j+1(·) is related to the parabolic cylinder function. The recurrence relation of D j+1(·) is found to be

D1(2zw0)=π2F(izw0),
D2(2zw0)=12zw0D1(2zw0),
Dj+1(2zw0)=1j[Dj1(2zw0)2zw0Dj(2zw0)],

Therefore, the analytical expression of the first term in source region reads as

ETEβ(r)=ETEβpro(r)+ETEβeva(r),

with β = x, y, and

(ETExpro(r)ETExeva(r)ETEypro(r)ETEyeva(r))=A(cosα2cosδsinα2sinδ)(l=0m=0l(1)mClI2m+1prom!(lm)!l=0m=0l(1)m+1ClI2m+1evam!(lm)!l=0m=0l+1(1)mClI2m+1pro(l+2)m!(l+1m)!l=0m=0l+1(1)m+1ClI2m+1eva(l+2)m!(l+1m)!)

where A = exp(-1/4f 2)/4f 2. The corresponding analytical expression of the second term in the source region turns out to be

ETMμ(r)=ETMμpro(r)+ETMμeva(r),

with µ = x, y, z and

(ETMxpro(r)ETMxeva(r)ETMypro(r)ETMyeva(r))=A(cosα2cosδsinα2sinδ)(l=0m=0l(1)mClI2m+1prom!(lm)!l=0m=0l(1)m+1ClI2m+1evam!(lm)!l=0m=0l+1(1)mClI2m+1pro(l+2)m!(l+1m)!l=0m=0l+1(1)m+1ClI2m+1eva(l+2)m!(l+1m)!),

as well as

(ETMzpro(r)ETMzeva(r))=ikρcos(θα)(l=0m=0l+1(1)mClI2mprom!(l+1m)!)l=0m=0l+1(1)m+1ClI2mevam!(l+1m)!).

As to the whole beam, it can be given by the sum of the two terms:

Ex(r)=Expro(r)+Exeva(r)=2Acosα(l=0m=0l(1)mClI2m+1prom!(lm)!+l=0m=0l(1)m+1ClI2m+1evam!(lm)!),
Ey(r)=Eypro(r)+Eyeva(r)=2Asinα(l=0m=0l(1)mClI2m+1prom!(lm)!+l=0m=0l(1)m+1ClI2m+1evam!(lm)!),
Ez(r)=ETMz(r).

Here, the longitudinal electric field stems from the divergence theorem of the electric field ∇·;E(r)=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 Ex, Ey and Ez 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.

The analytical expressions for the two terms are obtained without any approximation, which allows one to investigate them in the observation point close to the source. All above presented series are alternating and absolute convergent. Moreover, the series associated with the propagating wave converge more quickly than those associated with the evanescent waves [14].

3. The numerical results and analyses

To intuitively show the contribution of the evanescent and the propagation waves to the vectorial structure, the amplitude distributions of the components for the two terms are depicted in Figs. 2 and 3. The 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.

 

Fig. 2. The amplitude distribution of the first term in the reference plane z = λ/4. w 0 = λ/2, and α = 0°. (a) The propagation part of the x component; (b) The evanescent part of the x component; (c) The x component; (d) The propagation part of the y component; (e) The evanescent part of the y component; (f) The y component.

Download Full Size | PPT Slide | PDF

 

Fig. 3. The amplitude distribution of the second term in the reference plane z = λ/4. w 0 = λ/2, and α = 0°. (a) The propagation part of the x component; (b) The evanescent part of the x component; (c) The x component; (d) The propagation part of the z component; (e) The evanescent part of the z component; (f) The z component.

Download Full Size | PPT Slide | PDF

Figure 4 presents the intensity distributions of the two terms and the whole beam as well as the crossed term. All the parameters keep invariable. The contribution of the evanescent wave is embedded in Fig. 4. The two terms are not orthogonal to each other in the source region. Accordingly, the sum of the intensities of the two terms is not equal to that of the whole beam. The intensity of the crossed term, which results from the unorthogonality of the two terms, is also plotted in Fig. 4(d). The magnitude of the intensity of the crossed term is equal to the half of the whole beam. As the two terms are orthogonal to each other in the far field, it can be concluded that the intensity of the crossed term will gradually decrease upon propagation. To further weigh the contribution of the evanescent wave to the two terms, a new parameter is defined as follow:

T={ETExeva(r)2+ETEyeva(r)2ETE(r)2,forthefirsttermETMxeva(r)2+ETMyeva(r)2+ETMzeva(r)2ETM(r)2,forthelatterterm.

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.

 

Fig. 4. The intensity distribution in the reference plane z = λ/4. w 0 = λ/2, and α = 0°. (a) The first term; (b) The second term; (c) The whole beam; (d) The crossed term.

Download Full Size | PPT Slide | PDF

 

Fig. 5. z = λ/4, w 0 = λ/2, and α = 0°. (a) The percentage of the intensity of the evanescent part for the first term; (b) The percentage of the intensity of the evanescent part for the latter term.

Download Full Size | PPT Slide | PDF

 

Fig. 6. The intensity distribution in the different reference planes. The top row denotes z = λ/2, and the bottom row z = λ. w 0 = λ/2, and α = 0°. (a) and (d) The first term; (b) and (e) The second term; (c) and (f) The whole beam.

Download Full Size | PPT Slide | PDF

4. Conclusions

In this paper, 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. The electric field of a nonparaxial Gaussian beam can be decomposed into Ex, Ey and Ez 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 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

This work was supported by Scientific Research Fund of Zhejiang Provincial Education Department under grant 20060677. The author is indebted to the reviewers for valuable comments.

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. 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]  

5. Dennis G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. 21, 9–11 (1996). [CrossRef]   [PubMed]  

6. S. R. Seshadri, “Electromagnetic Gaussian beam,” J. Opt. Soc. Am. A 15, 2712–2719 (1998). [CrossRef]  

7. C. J. R. Sheppard and S. Saghafi, “Electromagnetic Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 16, 1381–1386 (1999). [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]  

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. 26, 65–130 (2002). [CrossRef]  

10. 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]  

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

12. 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]  

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. 276, 37–43 (2007). [CrossRef]  

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

15. A. E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986), Chaps. 16–17.

16. P. W. Milonni and J. H. Eberly, Lasers (Wiley Interscience, New York, 1988), sec. 14.5.

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]  

18. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, (Academic Press, New York, 1980), p. 952.

19. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, corrected and enlarged edition (Academic press, New York, 1980), p. 951.

20. 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.

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]  

References

  • View by:
  • |
  • |
  • |

  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]
  5. Dennis G. Hall, "Vector-beam solutions of Maxwell’s wave equation," Opt. Lett. 21, 9-11 (1996).
    [CrossRef] [PubMed]
  6. S. R. Seshadri, "Electromagnetic Gaussian beam," J. Opt. Soc. Am. A 15, 2712-2719 (1998).
    [CrossRef]
  7. C. J. R. Sheppard and S. Saghafi, "Electromagnetic Gaussian beams beyond the paraxial approximation," J. Opt. Soc. Am. A 16, 1381-1386 (1999).
    [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]
  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. 26, 65-130 (2002).
    [CrossRef]
  10. 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]
  11. G. Zhou, "Analytical vectorial structure of Laguerre-Gaussian beam in the far field," Opt. Lett. 31, 2616-2618 (2006).
    [CrossRef] [PubMed]
  12. 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]
  13. 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]
  14. P. C. Chaument, "Fully vectorial highly nonparaxial beam clost to the waist," J. Opt. Soc. Am. A 23, 3197-3202 (2007).
    [CrossRef]
  15. A. E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986), Chaps. 16-17.
  16. P. W. Milonni and J. H. Eberly, Lasers (Wiley Interscience, New York, 1988), sec. 14.5.
  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]
  18. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, (Academic Press, New York, 1980), p. 952.
  19. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, corrected and enlarged edition (Academic press, New York, 1980), p. 951.
  20. 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.
  21. 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]

2007 (2)

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]

2006 (4)

2002 (3)

2001 (1)

1999 (1)

1998 (1)

1997 (1)

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

1996 (1)

1990 (1)

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]

1986 (1)

Ashkin, A.

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

Bachor, H. -A.

Bosch, S.

Carnicer, A.

Chaument, P. C.

Chen, C. G.

Chen, J.

Chen, L.

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]

Delaubert, V.

Ferrera, J.

Guo, H.

Hall, Dennis G.

Harb, C. C.

Heilmann, R. K.

Konkola, P. T.

Kovács, A. P.

Kurdi, G.

Lam, P. K.

Liu, F.

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]

Martínez-Herrero, R.

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]

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]

Mejías, P.

Mejías, P. M.

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]

Movilla, J. M.

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]

Mukunda, N.

Ni, Y.

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]

Osvay, K.

Piquero, G.

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]

Poppe, G. P. M.

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]

Saghafi, S.

Schattenburg, M. L.

Seshadri, S. R.

Sheppard, C. J. R.

Simon, R.

Sudarshan, E. C. G.

Treps, N.

Varjú, K.

Wijers, C. M. J.

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]

Zhou, G.

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]

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]

Zhu, K.

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]

Zhuang, S.

ACM Trans. Math. Softw. (1)

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]

Chin. Phys. Lett. (1)

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]

J. Opt. Soc. Am. A (6)

Opt. Commun. (1)

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]

Opt. Express (1)

Opt. Lett. (4)

Proc. Natl. Acad. Sci. U.S.A. (1)

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

Prog. Quantum Electron. (1)

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]

Other (5)

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.

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.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1.

Scheme of the definition of unit vectors and the decomposition of the vector angular spectrum.

Fig. 2.
Fig. 2.

The amplitude distribution of the first term in the reference plane z = λ/4. w 0 = λ/2, and α = 0°. (a) The propagation part of the x component; (b) The evanescent part of the x component; (c) The x component; (d) The propagation part of the y component; (e) The evanescent part of the y component; (f) The y component.

Fig. 3.
Fig. 3.

The amplitude distribution of the second term in the reference plane z = λ/4. w 0 = λ/2, and α = 0°. (a) The propagation part of the x component; (b) The evanescent part of the x component; (c) The x component; (d) The propagation part of the z component; (e) The evanescent part of the z component; (f) The z component.

Fig. 4.
Fig. 4.

The intensity distribution in the reference plane z = λ/4. w 0 = λ/2, and α = 0°. (a) The first term; (b) The second term; (c) The whole beam; (d) The crossed term.

Fig. 5.
Fig. 5.

z = λ/4, w 0 = λ/2, and α = 0°. (a) The percentage of the intensity of the evanescent part for the first term; (b) The percentage of the intensity of the evanescent part for the latter term.

Fig. 6.
Fig. 6.

The intensity distribution in the different reference planes. The top row denotes z = λ/2, and the bottom row z = λ. w 0 = λ/2, and α = 0°. (a) and (d) The first term; (b) and (e) The second term; (c) and (f) The whole beam.

Equations (50)

Equations on this page are rendered with MathJax. Learn more.

( E x ( x 0 , y 0 , 0 ) E y ( x 0 , y 0 , 0 ) ) = ( cos α sin α ) exp ( ρ 0 2 w 0 2 ) ,
× E ( r ) ik H ( r ) = 0 ,
× H ( r ) + ik E ( r ) = 0 ,
· E ( r ) = · H ( r ) = 0 ,
L × E ˜ ( p , q , z ) ik H ˜ ( p , q , z ) = 0 ,
L × H ˜ ( p , q , z ) + ik E ˜ ( p , q , z ) = 0 ,
L · E ˜ ( p , q , z ) = L · H ˜ ( p , q , z ) = 0 ,
E ˜ ( p , q , z ) = A ( p , q ) exp ( ikγz ) ,
H ˜ ( p , q , z ) = [ s × A ( p , q ) ] exp ( ikγz ) ,
A ( p , q ) = A x ( p , q ) x + A y ( p , q ) y + A z ( p , q ) z ,
E ( r ) = E ˜ ( p , q , z ) exp [ ik ( px + qy ) ] dpdq = A ( p , q ) exp ( ik r . s ) dpdq .
( A x ( p , q ) A y ( p , q ) ) = 1 4 π f 2 exp ( b 2 4 f 2 ) ( cos α sin α ) ,
A z ( p , q ) = p A x ( p , q ) + q A y ( p , q ) γ = p cos α + q sin α 4 π f 2 γ exp ( b 2 4 f 2 ) .
e 1 = q b x p b y , e 2 = b x + b y b z .
s × e 1 = e 2 , e 1 × e 2 = s , e 2 × s = e 1 .
A ( p , q ) = [ A ( p , q ) . e 1 ] e 1 + [ A ( p , q ) . e 2 ] e 2 .
E ( r ) = E TE ( r ) + E TM ( r ) ,
( E TE ( r ) E TM ( r ) ) = 1 4 π f 2 1 b exp ( b 2 4 f 2 ) ( ( q cos α p sin α ) e 1 ( p cos α + q sin α ) e 1 ) exp ( ik r . s ) dpdq .
( E TE ( r ) E TM ( r ) ) = 1 4 π f 2 0 0 2 π exp ( b 2 4 f 2 ) ( sin ( φ α ) ( sin φ x cos φ y ) cos ( φ α ) [ γ ( cos φ x + sin φ y ) b z ] ) exp ( ikγz )
× exp [ i k ρ b cos ( φ θ ) ] b d b d φ ,
J n ( k ρb ) = 1 2 π 0 2 π exp [ ik ρb cos ( φ θ ) + in ( φ θ π 2 ) ] d φ ,
E TE x ( r ) = 1 4 f 2 [ cos α T 0 ( r ) + cos δT 2 ( r ) ] ,
E TE y ( r ) = 1 4 f 2 [ sin αT 0 ( r ) + sin δT 2 ( r ) ] ,
T n ( r ) = 0 exp ( b 2 4 f 2 ) exp ( ikγz ) J n ( kρb ) bdb .
E TM x ( r ) = 1 4 f 2 [ cos αT 0 ( r ) cos δT 2 ( r ) ] ,
E TM y ( r ) = 1 4 f 2 [ sin αT 0 ( r ) sin δT 2 ( r ) ] ,
E TM z ( r ) = i cos ( θ α ) 2 f 2 Ω ( r ) ,
Ω ( r ) = 0 exp ( b 2 4 f 2 ) exp ( ikγz ) J 1 ( kρb ) b 2 γ db .
T n ( r ) = exp ( 1 4 f 2 ) ( 0 1 0 + i ) exp ( γ 2 4 f 2 ) exp ( ikγz ) J n ( k ρ 1 γ 2 ) γdγ ,
Ω ( r ) = exp ( 1 4 f 2 ) ( 0 1 0 + i ) exp ( γ 2 4 f 2 ) exp ( ikγz ) J 1 ( k ρ 1 γ 2 ) 1 γ 2 .
J n ( k ρ 1 γ 2 ) = ( 2 ) n l = 0 m = 0 l + n 2 ( 1 ) m C l ( l + n 2 ) ! γ 2 m ( l + n ) ! m ! ( l m + n 2 ) ! , n is an even integer,
J 1 ( k ρ 1 γ 2 ) 1 γ 2 = 2 l = 0 m = 0 l + 1 ( 1 ) m C l γ 2 m m ! ( l + 1 m ) ! ,
T n ( r ) = ( 2 ) n exp ( 1 4 f 2 ) l = 0 m = 0 l + n 2 ( 1 ) m C l ( l + n 2 ) ! ( l + n ) ! m ! ( l m + n 2 ) ! ( 0 1 0 + i ) exp ( γ 2 4 f 2 ) exp ( ikγz ) γ 2 m + 1 ,
Ω ( r ) = 2 exp ( 1 4 f 2 ) l = 0 m = 0 l + 1 ( 1 ) m C l m ! ( l + 1 m ) ! ( 0 1 0 + i ) exp ( γ 2 4 f 2 ) exp ( ikγz ) γ 2 m .
I j pro = 0 1 exp ( γ 2 4 f 2 ) exp ( ikγz ) γ j = 2 f 2 [ exp ( 1 4 f 2 ) exp ( ikz ) ikz I j 1 pro ( j 1 ) I j 2 pro ] ,
I 0 pro = if π [ F ( iz w 0 ) exp ( 1 4 f 2 ) exp ( ikz ) F ( iz w 0 + kw 0 2 ) ] ,
I 1 pro = 2 f 2 [ exp ( 1 4 f 2 ) exp ( ikz ) 1 ikz I 0 pro ] ,
I j eva = 0 + i exp ( γ 2 4 f 2 ) exp ( ikγz ) γ j = ( i 2 f ) j + 1 j ! D j + 1 ( 2 z w 0 ) ,
D 1 ( 2 z w 0 ) = π 2 F ( iz w 0 ) ,
D 2 ( 2 z w 0 ) = 1 2 z w 0 D 1 ( 2 z w 0 ) ,
D j + 1 ( 2 z w 0 ) = 1 j [ D j 1 ( 2 z w 0 ) 2 z w 0 D j ( 2 z w 0 ) ] ,
E TE β ( r ) = E TE β pro ( r ) + E TE β eva ( r ) ,
( E TE x pro ( r ) E TE x eva ( r ) E TE y pro ( r ) E TE y eva ( r ) ) = A ( cos α 2 cos δ sin α 2 sin δ ) ( l = 0 m = 0 l ( 1 ) m C l I 2 m + 1 pro m ! ( l m ) ! l = 0 m = 0 l ( 1 ) m + 1 C l I 2 m + 1 eva m ! ( l m ) ! l = 0 m = 0 l + 1 ( 1 ) m C l I 2 m + 1 pro ( l + 2 ) m ! ( l + 1 m ) ! l = 0 m = 0 l + 1 ( 1 ) m + 1 C l I 2 m + 1 eva ( l + 2 ) m ! ( l + 1 m ) ! )
E TM μ ( r ) = E TM μ pro ( r ) + E TM μ eva ( r ) ,
( E TM x pro ( r ) E TM x eva ( r ) E TM y pro ( r ) E TMy eva ( r ) ) = A ( cos α 2 cos δ sin α 2 sin δ ) ( l = 0 m = 0 l ( 1 ) m C l I 2 m + 1 pro m ! ( l m ) ! l = 0 m = 0 l ( 1 ) m + 1 C l I 2 m + 1 eva m ! ( l m ) ! l = 0 m = 0 l + 1 ( 1 ) m C l I 2 m + 1 pro ( l + 2 ) m ! ( l + 1 m ) ! l = 0 m = 0 l + 1 ( 1 ) m + 1 C l I 2 m + 1 eva ( l + 2 ) m ! ( l + 1 m ) ! ) ,
( E TM z pro ( r ) E TM z eva ( r ) ) = ik ρ cos ( θ α ) ( l = 0 m = 0 l + 1 ( 1 ) m C l I 2 m pro m ! ( l + 1 m ) ! ) l = 0 m = 0 l + 1 ( 1 ) m + 1 C l I 2 m eva m ! ( l + 1 m ) ! ) .
E x ( r ) = E x pro ( r ) + E x eva ( r ) = 2 A cos α ( l = 0 m = 0 l ( 1 ) m C l I 2 m + 1 pro m ! ( l m ) ! + l = 0 m = 0 l ( 1 ) m + 1 C l I 2 m + 1 eva m ! ( l m ) ! ) ,
E y ( r ) = E y pro ( r ) + E y eva ( r ) = 2 A sin α ( l = 0 m = 0 l ( 1 ) m C l I 2 m + 1 pro m ! ( l m ) ! + l = 0 m = 0 l ( 1 ) m + 1 C l I 2 m + 1 eva m ! ( l m ) ! ) ,
E z ( r ) = E TM z ( r ) .
T = { E TE x eva ( r ) 2 + E TE y eva ( r ) 2 E TE ( r ) 2 , for the first term E TM x eva ( r ) 2 + E TM y eva ( r ) 2 + E TM z eva ( r ) 2 E TM ( r ) 2 , for the latter term .

Metrics