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

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    [CrossRef]

2007

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

2002

2001

1999

1998

1997

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

1990

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

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, 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, "Analytical vectorial structure of Laguerre-Gaussian beam in the far field," Opt. Lett. 31, 2616-2618 (2006).
[CrossRef] [PubMed]

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.

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.

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

Opt. Commun.

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

Opt. Lett.

Proc. Natl. Acad. Sci. U.S.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]

Prog. Quantum Electron.

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

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.

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.

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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)

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( 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 .

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