Abstract

A parabolic mirror illuminated with an incident collimated beam whose axis of propagation does not exactly coincide with the axis of revolution of the mirror shows distortion and strong coma. To understand the behavior of such a focused beam, a detailed description of the electric field in the focal region of a parabolic mirror illuminated with a beam having a nonzero angle of incidence is required. We use the Richards–Wolf vector field equation to investigate the electric energy density distribution of a beam focused with a parabolic mirror. The explicit aberration function of this focused field is provided along with numerically calculated electric energy densities in the focal region for different angles of incidence. The location of the peak intensity, the Strehl ratio and the full-width at half-maximum as a function of the angle of incidence are given and discussed. The results confirm that the focal spot of a strongly focused beam is affected by severe coma, even for very small tilting of the mirror. This analysis provides a clearer understanding of the effect of the angle of incidence on the focusing properties of a parabolic mirror as such a focusing device is of growing interest in microscopy.

© 2011 OSA

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References

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    [CrossRef]
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    [CrossRef]
  33. G. Rodríguez-Morales and S. Chávez-Cerda, “Exact nonparaxial beams of the scalar Helmholtz equation,” Opt. Lett. 29(5), 430–432 (2004).
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  34. J. Sheldakova, A. Kudryashov, and T. Y. Cherezova, “Femtosecond laser beam improvement: correction of parabolic mirror aberrations by means of adaptive optics,” Proc. SPIE 6872, 687203 (2008).
    [CrossRef]

2010 (2)

A. April and M. Piché, “4π Focusing of TM(01) beams under nonparaxial conditions,” Opt. Express 18(21), 22128–22140 (2010).
[CrossRef] [PubMed]

M. Avendaño-Alejo, L. Castañeda, and I. Moreno, “Caustics and wavefronts by multiple reflections in a circular surface,” Am. J. Phys. 78(11), 1195–1198 (2010).
[CrossRef]

2009 (1)

2008 (5)

N. Bokor and N. Davidson, “4π focusing with single paraboloid mirror,” Opt. Commun. 281(22), 5499–5503 (2008).
[CrossRef]

J. Stadler, C. Stanciu, C. Stupperich, and A. J. Meixner, “Tighter focusing with a parabolic mirror,” Opt. Lett. 33(7), 681–683 (2008).
[CrossRef] [PubMed]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary coma on the focusing of a Laguerre–Gaussian beam by a high numerical aperture system; vectorial diffraction theory,” J. Opt. A, Pure Appl. Opt. 10(7), 075008 (2008).
[CrossRef]

H. Kawauchi, Y. Kozawa, and S. Sato, “Generation of radially polarized Ti:sapphire laser beam using a c-cut crystal,” Opt. Lett. 33(17), 1984–1986 (2008).
[CrossRef] [PubMed]

J. Sheldakova, A. Kudryashov, and T. Y. Cherezova, “Femtosecond laser beam improvement: correction of parabolic mirror aberrations by means of adaptive optics,” Proc. SPIE 6872, 687203 (2008).
[CrossRef]

2007 (1)

T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272(2), 314–319 (2007).
[CrossRef]

2005 (1)

C. Varin, M. Piché, and M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(2 ), 026603 (2005).
[CrossRef] [PubMed]

2004 (2)

2003 (2)

C. Debus, M. A. Lieb, A. Drechsler, and A. J. Meixner, “Probing highly confined optical fields in the focal region of a high NA parabolic mirror with subwavelength spatial resolution,” J. Microsc. 210(3), 203–208 (2003).
[CrossRef] [PubMed]

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

2001 (4)

2000 (3)

1997 (1)

C. J. R. Sheppard, “Vector diffraction in paraboloidal mirrors with Seidel aberrations: effects of small object displacements,” Opt. Commun. 138(4-6), 262–264 (1997).
[CrossRef]

1996 (1)

R. Kant, “Vector diffraction in paraboloidal mirrors with Seidel aberrations. I: Spherical aberration, curvature of field aberration and distortion,” Opt. Commun. 128(4-6), 292–306 (1996).
[CrossRef]

1995 (1)

R. Kant, “An analytical method of vector diffraction for focusing optical systems with Seidel aberrations II: astigmatism and coma,” J. Mod. Opt. 42(2), 299–320 (1995).
[CrossRef]

1993 (1)

C. J. R. Sheppard and M. Gu, “Imaging by a high aperture optical system,” J. Mod. Opt. 40(8), 1631–1651 (1993).
[CrossRef]

1990 (1)

1987 (1)

1981 (1)

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39(4), 205–210 (1981).
[CrossRef]

1979 (1)

1977 (1)

C. J. R. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” IEE J. Microwaves, Opt. Acoust. 1(4), 129–132 (1977).
[CrossRef]

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[CrossRef]

April, A.

Avendaño-Alejo, M.

M. Avendaño-Alejo, L. Castañeda, and I. Moreno, “Caustics and wavefronts by multiple reflections in a circular surface,” Am. J. Phys. 78(11), 1195–1198 (2010).
[CrossRef]

Barakat, R.

Bokor, N.

Brown, T. G.

Castañeda, L.

M. Avendaño-Alejo, L. Castañeda, and I. Moreno, “Caustics and wavefronts by multiple reflections in a circular surface,” Am. J. Phys. 78(11), 1195–1198 (2010).
[CrossRef]

Chávez-Cerda, S.

Cherezova, T. Y.

J. Sheldakova, A. Kudryashov, and T. Y. Cherezova, “Femtosecond laser beam improvement: correction of parabolic mirror aberrations by means of adaptive optics,” Proc. SPIE 6872, 687203 (2008).
[CrossRef]

Choudhury, A.

C. J. R. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” IEE J. Microwaves, Opt. Acoust. 1(4), 129–132 (1977).
[CrossRef]

Courjon, D.

T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272(2), 314–319 (2007).
[CrossRef]

Davidson, N.

De Koninck, Y.

Debus, C.

C. Debus, M. A. Lieb, A. Drechsler, and A. J. Meixner, “Probing highly confined optical fields in the focal region of a high NA parabolic mirror with subwavelength spatial resolution,” J. Microsc. 210(3), 203–208 (2003).
[CrossRef] [PubMed]

A. Drechsler, M. A. Lieb, C. Debus, A. J. Meixner, and G. Tarrach, “Confocal microscopy with a high numerical aperture parabolic mirror,” Opt. Express 9(12), 637–644 (2001).
[CrossRef] [PubMed]

Dehez, H.

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000).
[CrossRef]

Drechsler, A.

C. Debus, M. A. Lieb, A. Drechsler, and A. J. Meixner, “Probing highly confined optical fields in the focal region of a high NA parabolic mirror with subwavelength spatial resolution,” J. Microsc. 210(3), 203–208 (2003).
[CrossRef] [PubMed]

A. Drechsler, M. A. Lieb, C. Debus, A. J. Meixner, and G. Tarrach, “Confocal microscopy with a high numerical aperture parabolic mirror,” Opt. Express 9(12), 637–644 (2001).
[CrossRef] [PubMed]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000).
[CrossRef]

Ford, D. H.

Gannaway, J.

C. J. R. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” IEE J. Microwaves, Opt. Acoust. 1(4), 129–132 (1977).
[CrossRef]

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000).
[CrossRef]

Grosjean, T.

T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272(2), 314–319 (2007).
[CrossRef]

Gu, M.

C. J. R. Sheppard and M. Gu, “Imaging by a high aperture optical system,” J. Mod. Opt. 40(8), 1631–1651 (1993).
[CrossRef]

Howard, J. E.

Ichikawa, H.

Kant, R.

R. Kant, “Vector diffraction in paraboloidal mirrors with Seidel aberrations. I: Spherical aberration, curvature of field aberration and distortion,” Opt. Commun. 128(4-6), 292–306 (1996).
[CrossRef]

R. Kant, “An analytical method of vector diffraction for focusing optical systems with Seidel aberrations II: astigmatism and coma,” J. Mod. Opt. 42(2), 299–320 (1995).
[CrossRef]

Kawauchi, H.

Kikuta, H.

Kimura, W. D.

Kozawa, Y.

Kudryashov, A.

J. Sheldakova, A. Kudryashov, and T. Y. Cherezova, “Femtosecond laser beam improvement: correction of parabolic mirror aberrations by means of adaptive optics,” Proc. SPIE 6872, 687203 (2008).
[CrossRef]

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000).
[CrossRef]

Li, Y.

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39(4), 205–210 (1981).
[CrossRef]

Lieb, M. A.

Meixner, A. J.

Moreno, I.

M. Avendaño-Alejo, L. Castañeda, and I. Moreno, “Caustics and wavefronts by multiple reflections in a circular surface,” Am. J. Phys. 78(11), 1195–1198 (2010).
[CrossRef]

Piché, M.

Porras, M. A.

C. Varin, M. Piché, and M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(2 ), 026603 (2005).
[CrossRef] [PubMed]

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000).
[CrossRef]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[CrossRef]

Rodríguez-Morales, G.

Sato, S.

Senthilkumaran, P.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary coma on the focusing of a Laguerre–Gaussian beam by a high numerical aperture system; vectorial diffraction theory,” J. Opt. A, Pure Appl. Opt. 10(7), 075008 (2008).
[CrossRef]

Sheldakova, J.

J. Sheldakova, A. Kudryashov, and T. Y. Cherezova, “Femtosecond laser beam improvement: correction of parabolic mirror aberrations by means of adaptive optics,” Proc. SPIE 6872, 687203 (2008).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard, “Vector diffraction in paraboloidal mirrors with Seidel aberrations: effects of small object displacements,” Opt. Commun. 138(4-6), 262–264 (1997).
[CrossRef]

C. J. R. Sheppard and M. Gu, “Imaging by a high aperture optical system,” J. Mod. Opt. 40(8), 1631–1651 (1993).
[CrossRef]

C. J. R. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” IEE J. Microwaves, Opt. Acoust. 1(4), 129–132 (1977).
[CrossRef]

Singh, K.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary coma on the focusing of a Laguerre–Gaussian beam by a high numerical aperture system; vectorial diffraction theory,” J. Opt. A, Pure Appl. Opt. 10(7), 075008 (2008).
[CrossRef]

Singh, R. K.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary coma on the focusing of a Laguerre–Gaussian beam by a high numerical aperture system; vectorial diffraction theory,” J. Opt. A, Pure Appl. Opt. 10(7), 075008 (2008).
[CrossRef]

Stadler, J.

Stanciu, C.

Stupperich, C.

Tarrach, G.

Tidwell, S. C.

Török, P.

Varga, P.

Varin, C.

C. Varin, M. Piché, and M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(2 ), 026603 (2005).
[CrossRef] [PubMed]

Wolf, E.

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39(4), 205–210 (1981).
[CrossRef]

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[CrossRef]

Youngworth, K. S.

Am. J. Phys. (1)

M. Avendaño-Alejo, L. Castañeda, and I. Moreno, “Caustics and wavefronts by multiple reflections in a circular surface,” Am. J. Phys. 78(11), 1195–1198 (2010).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. B (1)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light—theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

IEE J. Microwaves, Opt. Acoust. (1)

C. J. R. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” IEE J. Microwaves, Opt. Acoust. 1(4), 129–132 (1977).
[CrossRef]

J. Microsc. (1)

C. Debus, M. A. Lieb, A. Drechsler, and A. J. Meixner, “Probing highly confined optical fields in the focal region of a high NA parabolic mirror with subwavelength spatial resolution,” J. Microsc. 210(3), 203–208 (2003).
[CrossRef] [PubMed]

J. Mod. Opt. (2)

R. Kant, “An analytical method of vector diffraction for focusing optical systems with Seidel aberrations II: astigmatism and coma,” J. Mod. Opt. 42(2), 299–320 (1995).
[CrossRef]

C. J. R. Sheppard and M. Gu, “Imaging by a high aperture optical system,” J. Mod. Opt. 40(8), 1631–1651 (1993).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

R. K. Singh, P. Senthilkumaran, and K. Singh, “Effect of primary coma on the focusing of a Laguerre–Gaussian beam by a high numerical aperture system; vectorial diffraction theory,” J. Opt. A, Pure Appl. Opt. 10(7), 075008 (2008).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (6)

N. Bokor and N. Davidson, “4π focusing with single paraboloid mirror,” Opt. Commun. 281(22), 5499–5503 (2008).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179(1-6), 1–7 (2000).
[CrossRef]

T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272(2), 314–319 (2007).
[CrossRef]

R. Kant, “Vector diffraction in paraboloidal mirrors with Seidel aberrations. I: Spherical aberration, curvature of field aberration and distortion,” Opt. Commun. 128(4-6), 292–306 (1996).
[CrossRef]

C. J. R. Sheppard, “Vector diffraction in paraboloidal mirrors with Seidel aberrations: effects of small object displacements,” Opt. Commun. 138(4-6), 262–264 (1997).
[CrossRef]

E. Wolf and Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39(4), 205–210 (1981).
[CrossRef]

Opt. Express (4)

Opt. Lett. (5)

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

C. Varin, M. Piché, and M. A. Porras, “Acceleration of electrons from rest to GeV energies by ultrashort transverse magnetic laser pulses in free space,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(2 ), 026603 (2005).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

Proc. R. Soc. Lond. A Math. Phys. Sci. (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[CrossRef]

Proc. SPIE (1)

J. Sheldakova, A. Kudryashov, and T. Y. Cherezova, “Femtosecond laser beam improvement: correction of parabolic mirror aberrations by means of adaptive optics,” Proc. SPIE 6872, 687203 (2008).
[CrossRef]

Other (3)

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006), Chap. 3.

J. J. Stamnes, Waves in Focal Regions (Hilger, 1986), Sec. 16.1.2.

E. W. Weisstein, “Catacaustic.” From MathWorld — A Wolfram Web Resource. http://mathworld.wolfram.com/Catacaustic.html .

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Figures (6)

Fig. 1
Fig. 1

An exact ray tracing shows that, when the incident rays of a collimated beam are not perfectly parallel to the axis of revolution of the mirror, the focused rays form a catacaustic.

Fig. 2
Fig. 2

The height x of a ray of the incident collimated beam is related to the polar angle α and the focal length f of the parabolic mirror.

Fig. 3
Fig. 3

The electric energy density distribution We in the focal plane (z = 0) of a focused TM01 beam by a parabolic mirror for which ka = 1 with kfδ equal to (a) 0, (b) 1.5π, and (c) 3π.

Fig. 4
Fig. 4

Electric energy density profiles along the X-axis (y = 0) in the focal plane (z = 0) for selected values of ka and kfδ. Note the different scale for each plot.

Fig. 5
Fig. 5

Electric energy density distribution along the X-axis (y = 0) in the focal plane (z = 0) of a focused TM01 beam for which for ka = 7, for kfδ = 0 and kfδ = 6π. (a) | E x | 2 , (b) | E z | 2 , and (c) | E x | 2 + | E z | 2 . The profiles are normalized by the maximum intensity of | E x | 2 + | E z | 2 .

Fig. 6
Fig. 6

(a) The Strehl ratio, (b) the full-width at half-maximum (FWHM), and (c) the position x max of the peak intensity of the electric energy density profile in the focal plane (z = 0) as a function of the parameter kfδ, for ka = 1 (red curves), for ka = 4 (green curves), and for ka = 7 (blue curves).

Equations (17)

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E ( r , ϕ , z ) = 1 j 2 π 0 2 π 0 α max E o ( α ) exp [ j k Γ ( α , β ) ] sin α d α d β ,
E o ( α , β ) a ^ ( α , β ) E o q ( α ) l 0 ( α ) ,
Γ ( α , β ) Φ ( α , β ) r sin α cos ( β ϕ ) + z cos α ,
q ( α ) = 2 1 + cos α = sec 2 ( 1 2 α ) ,
g ( α ) = 2 sin α 1 + cos α = 2 tan ( 1 2 α ) .
Φ ( α , β ) = 2 f δ tan ( 1 2 α ) cos β f δ sin α cos β + 1 4 f δ sin 3 α cos β +
Γ ( α , β ) = sin α [ r cos ϕ cos β f δ g ( α ) sin α cos β + r sin ϕ sin β ] + z cos α .
Γ ( α , β ) = r ˜ ( α ) sin α cos [ β ϕ ˜ ( α ) ] + z cos α ,
r ˜ ( α ) [ r 2 + f 2 δ 2 g 2 ( α ) sin 2 α 2 r f δ g ( α ) sin α cos ϕ ] 1 / 2 ,
tan ϕ ˜ ( α ) r sin ϕ r cos ϕ f δ g ( α ) / sin α .
a ^ ( α , β ) = a ^ x cos α cos β + a ^ y cos α sin β + a ^ z sin α ,
{ E x E y E z } = E o j 2 π 0 2 π 0 α max q ( α ) l 0 ( α ) { cos α cos β cos α sin β sin α } × exp { j k r ˜ ( α ) sin α cos [ β ϕ ˜ ( α ) ] } exp ( j k z cos α ) sin α d α d β .
0 2 π exp [ j k r sin α cos ( β ϕ ) ] { cos ( m β ) sin ( m β ) } d β = 2 π j m J m ( k r sin α ) { cos ( m ϕ ) sin ( m ϕ ) } ,
{ E x E y E z } = E o 0 α max q ( α ) l 0 ( α ) sin α { cos α cos ϕ ˜ ( α ) J 1 [ k r ˜ ( α ) sin α ] cos α sin ϕ ˜ ( α ) J 1 [ k r ˜ ( α ) sin α ] j sin α J 0 [ k r ˜ ( α ) sin α ] } exp ( j k z cos α ) d α ,
cos ϕ ˜ ( α ) = r cos ϕ f δ g ( α ) / sin α r ˜ ( α ) ,
sin ϕ ˜ ( α ) = r sin ϕ r ˜ ( α ) .
l 0 ( α ) = r w o exp ( r 2 w o 2 ) = f g ( α ) w o exp [ f 2 g 2 ( α ) w o 2 ] ,

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