Abstract

Vector diffraction theory is applied to the case of focused TEM00 Gaussian beams passing through a spatially limiting aperture in order to investigate the propagation of these clipped focused-Gaussian beams. Beam distributions at different axial distances show that a traditional M2 propagation model cannot be used for the propagation of clipped focus-Gaussian beams. Using Luneberg’s vector diffraction theory and Fresnel approximations, an analytical model for the on-axis transverse and longitudinal electric fields and intensity distributions is presented including predictions of the maximum obtainable intensity. In addition, an analytical expression is provided for the longitudinal component of the electric field of a TEM00 mode unperturbed Gaussian beam. Experimental results are also presented and compared to the model’s predictions.

© 2010 Optical Society of America

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References

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  1. H. Kogelnik and T. Li, "Laser beam resonators," Proc. IEEE 54, 1312-1329 (1966).
    [CrossRef]
  2. A. Yariv, Quantum Electronics, Third Edition, (John Wiley & Sons, New York, NY, 1989.)
  3. W. T. Sifvast, Laser Fundamentals, (Cambridge University Press, New York, NY, 1996.)
  4. A. E. Siegman, Lasers, (University Science Books, Mill Valley, CA, 1986.)
  5. M. W. Sasnett, Physics and Technology of Laser Resonators, (Hilger, New York, NY 1989.)
  6. A. E. Siegman, "New Developments in Laser Resonators," Proc. SPIE 1224, 2-12 (1990).
    [CrossRef]
  7. I. Ghebregziabher and B. C. Walker, "Effect of focal geometry on radiation from atomic ionization in an ultrastrong and ultrafast laser field," Phys. Rev. A 76, 023415 (2007).
    [CrossRef]
  8. J. M. P. Coelho, M. A. Abreu, and F. C. Rodrigues, "Modelling the spot shape influence on high-speed transmission lap welding of thermoplastic films," J. Opt. Lasers Eng. 46, 55-61 (2007).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  12. 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]
  13. B. Lu and K. Duan, "Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture," Opt. Lett. 28, 2440-2442 (2003).
    [CrossRef] [PubMed]
  14. K. Duan and B. Lu, "Vectorial nonparaxial propagation equation of elliptical Gaussian beams in the presence of a rectangular aperture," J. Opt. Soc. Am. A 21, 1613-1620 (2004).
    [CrossRef]
  15. K. Duan and B. Lu, "Polarization properties of vectorial nonparaxial Gaussian beams in the far field," Opt. Lett. 2005, 308-310 (2005).
    [CrossRef]
  16. G. Bekefi, "Diffraction of electromagnetic waves by an aperture in a large screen," J. Appl. Phys. 24, 1123-1130 (1953).
    [CrossRef]
  17. S. Guha and G. D. Gillen, "Description of light propagation through a circular aperture using nonparaxial vector diffraction theory," Opt. Express 13, 1424-1447 (2005).
    [CrossRef] [PubMed]
  18. G. D. Gillen, K. Baughman, and S. Guha, "Application of Hertz vector diffraction theory to the diffraction of focused Gaussian beams and calculations of focal parameters," Opt. Express 17,1478-1492 (2009).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  21. L. Cicchitelli, H. Hora and R. Postle, "Longitudinal field components for laser beams in vacuum," Phys. Rev. A 41, 3727-3732 (1990).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  24. L. Novotny, M. R. Beversluis, K. S. Youngworth and T. G. Brown, "Longitudinal field modes probed by single molecules," Phys. Rev. Lett. 86, 5251-5254 (2001).
    [CrossRef] [PubMed]
  25. S. Takeuchi, R. Sugihara and K. Shimoda, "Electron acceleration by longitudinal electric field of a Gaussian laser beam," J. Phys. Soc. Jpn. 63, 1186-1193 (1994).
    [CrossRef]
  26. J. P. Paquette and J. L. Chaloupka, "Effect of realistic focal conditions on the strong-field ionization of helium," Phys. Rev. A 79, 043410 (2009).
    [CrossRef]
  27. G. D. Gillen and S. Guha, "Modeling and propagation of near-field diffraction patterns: a more complete approach," Am. J. Phys. 72, 1195-1201 (2004).
    [CrossRef]
  28. R. K. Luneberg, Mathematical theory of optics, (University of California Press, Berkeley, California, 1964.)
  29. A. J. Campillo, J. E. Pearson, S. L. Shapiro, and N. J. Terrell, Jr., "Fresnel diffraction effects in the design of high-power laser systems," Appl. Phys. Lett. 23, 85-87 (1973).
    [CrossRef]

2009 (2)

2007 (2)

I. Ghebregziabher and B. C. Walker, "Effect of focal geometry on radiation from atomic ionization in an ultrastrong and ultrafast laser field," Phys. Rev. A 76, 023415 (2007).
[CrossRef]

J. M. P. Coelho, M. A. Abreu, and F. C. Rodrigues, "Modelling the spot shape influence on high-speed transmission lap welding of thermoplastic films," J. Opt. Lasers Eng. 46, 55-61 (2007).
[CrossRef]

2005 (4)

2004 (2)

K. Duan and B. Lu, "Vectorial nonparaxial propagation equation of elliptical Gaussian beams in the presence of a rectangular aperture," J. Opt. Soc. Am. A 21, 1613-1620 (2004).
[CrossRef]

G. D. Gillen and S. Guha, "Modeling and propagation of near-field diffraction patterns: a more complete approach," Am. J. Phys. 72, 1195-1201 (2004).
[CrossRef]

2003 (1)

2002 (1)

2001 (1)

L. Novotny, M. R. Beversluis, K. S. Youngworth and T. G. Brown, "Longitudinal field modes probed by single molecules," Phys. Rev. Lett. 86, 5251-5254 (2001).
[CrossRef] [PubMed]

1996 (1)

J. J. Macklin, J. K. Trautman, T. D. Harris and L. E. Brus, "Imaging and time-resolved spectroscopy of single molecules at an interface," Science 272, 255-258 (1996).
[CrossRef]

1994 (1)

S. Takeuchi, R. Sugihara and K. Shimoda, "Electron acceleration by longitudinal electric field of a Gaussian laser beam," J. Phys. Soc. Jpn. 63, 1186-1193 (1994).
[CrossRef]

1990 (3)

A. E. Siegman, "New Developments in Laser Resonators," Proc. SPIE 1224, 2-12 (1990).
[CrossRef]

L. Cicchitelli, H. Hora and R. Postle, "Longitudinal field components for laser beams in vacuum," Phys. Rev. A 41, 3727-3732 (1990).
[CrossRef] [PubMed]

M. O. Scully, "A simple laser linac," Appl. Phys. B 51, 238-241 (1990).
[CrossRef]

1979 (1)

1975 (1)

M. Lax, W. H. Louisell and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975).
[CrossRef]

1973 (1)

A. J. Campillo, J. E. Pearson, S. L. Shapiro, and N. J. Terrell, Jr., "Fresnel diffraction effects in the design of high-power laser systems," Appl. Phys. Lett. 23, 85-87 (1973).
[CrossRef]

1972 (1)

1966 (1)

H. Kogelnik and T. Li, "Laser beam resonators," Proc. IEEE 54, 1312-1329 (1966).
[CrossRef]

1953 (1)

G. Bekefi, "Diffraction of electromagnetic waves by an aperture in a large screen," J. Appl. Phys. 24, 1123-1130 (1953).
[CrossRef]

Abreu, M. A.

J. M. P. Coelho, M. A. Abreu, and F. C. Rodrigues, "Modelling the spot shape influence on high-speed transmission lap welding of thermoplastic films," J. Opt. Lasers Eng. 46, 55-61 (2007).
[CrossRef]

Agrawal, G. P.

Baughman, K.

Bekefi, G.

G. Bekefi, "Diffraction of electromagnetic waves by an aperture in a large screen," J. Appl. Phys. 24, 1123-1130 (1953).
[CrossRef]

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth and T. G. Brown, "Longitudinal field modes probed by single molecules," Phys. Rev. Lett. 86, 5251-5254 (2001).
[CrossRef] [PubMed]

Brown, T. G.

L. Novotny, M. R. Beversluis, K. S. Youngworth and T. G. Brown, "Longitudinal field modes probed by single molecules," Phys. Rev. Lett. 86, 5251-5254 (2001).
[CrossRef] [PubMed]

Brus, L. E.

J. J. Macklin, J. K. Trautman, T. D. Harris and L. E. Brus, "Imaging and time-resolved spectroscopy of single molecules at an interface," Science 272, 255-258 (1996).
[CrossRef]

Campillo, A. J.

A. J. Campillo, J. E. Pearson, S. L. Shapiro, and N. J. Terrell, Jr., "Fresnel diffraction effects in the design of high-power laser systems," Appl. Phys. Lett. 23, 85-87 (1973).
[CrossRef]

Carter, W. H.

Chaloupka, J. L.

J. P. Paquette and J. L. Chaloupka, "Effect of realistic focal conditions on the strong-field ionization of helium," Phys. Rev. A 79, 043410 (2009).
[CrossRef]

Chen, C. G.

Cicchitelli, L.

L. Cicchitelli, H. Hora and R. Postle, "Longitudinal field components for laser beams in vacuum," Phys. Rev. A 41, 3727-3732 (1990).
[CrossRef] [PubMed]

Coelho, J. M. P.

J. M. P. Coelho, M. A. Abreu, and F. C. Rodrigues, "Modelling the spot shape influence on high-speed transmission lap welding of thermoplastic films," J. Opt. Lasers Eng. 46, 55-61 (2007).
[CrossRef]

Duan, K.

Ferrera, J.

Ghebregziabher, I.

I. Ghebregziabher and B. C. Walker, "Effect of focal geometry on radiation from atomic ionization in an ultrastrong and ultrafast laser field," Phys. Rev. A 76, 023415 (2007).
[CrossRef]

Gillen, G. D.

Guha, S.

Harris, T. D.

J. J. Macklin, J. K. Trautman, T. D. Harris and L. E. Brus, "Imaging and time-resolved spectroscopy of single molecules at an interface," Science 272, 255-258 (1996).
[CrossRef]

Heilmann, R. K.

Hora, H.

L. Cicchitelli, H. Hora and R. Postle, "Longitudinal field components for laser beams in vacuum," Phys. Rev. A 41, 3727-3732 (1990).
[CrossRef] [PubMed]

Kogelnik, H.

H. Kogelnik and T. Li, "Laser beam resonators," Proc. IEEE 54, 1312-1329 (1966).
[CrossRef]

Konkola, P. T.

Lax, M.

M. Lax, W. H. Louisell and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975).
[CrossRef]

Li, T.

H. Kogelnik and T. Li, "Laser beam resonators," Proc. IEEE 54, 1312-1329 (1966).
[CrossRef]

Li, Y.

Louisell, W. H.

M. Lax, W. H. Louisell and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975).
[CrossRef]

Lu, B.

Macklin, J. J.

J. J. Macklin, J. K. Trautman, T. D. Harris and L. E. Brus, "Imaging and time-resolved spectroscopy of single molecules at an interface," Science 272, 255-258 (1996).
[CrossRef]

McKnight, W. B.

M. Lax, W. H. Louisell and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975).
[CrossRef]

Novotny, L.

L. Novotny, M. R. Beversluis, K. S. Youngworth and T. G. Brown, "Longitudinal field modes probed by single molecules," Phys. Rev. Lett. 86, 5251-5254 (2001).
[CrossRef] [PubMed]

Paquette, J. P.

J. P. Paquette and J. L. Chaloupka, "Effect of realistic focal conditions on the strong-field ionization of helium," Phys. Rev. A 79, 043410 (2009).
[CrossRef]

Pattanayak, D. N.

Pearson, J. E.

A. J. Campillo, J. E. Pearson, S. L. Shapiro, and N. J. Terrell, Jr., "Fresnel diffraction effects in the design of high-power laser systems," Appl. Phys. Lett. 23, 85-87 (1973).
[CrossRef]

Postle, R.

L. Cicchitelli, H. Hora and R. Postle, "Longitudinal field components for laser beams in vacuum," Phys. Rev. A 41, 3727-3732 (1990).
[CrossRef] [PubMed]

Rodrigues, F. C.

J. M. P. Coelho, M. A. Abreu, and F. C. Rodrigues, "Modelling the spot shape influence on high-speed transmission lap welding of thermoplastic films," J. Opt. Lasers Eng. 46, 55-61 (2007).
[CrossRef]

Schattenburg, M. L.

Scully, M. O.

M. O. Scully, "A simple laser linac," Appl. Phys. B 51, 238-241 (1990).
[CrossRef]

Shapiro, S. L.

A. J. Campillo, J. E. Pearson, S. L. Shapiro, and N. J. Terrell, Jr., "Fresnel diffraction effects in the design of high-power laser systems," Appl. Phys. Lett. 23, 85-87 (1973).
[CrossRef]

Shimoda, K.

S. Takeuchi, R. Sugihara and K. Shimoda, "Electron acceleration by longitudinal electric field of a Gaussian laser beam," J. Phys. Soc. Jpn. 63, 1186-1193 (1994).
[CrossRef]

Siegman, A. E.

A. E. Siegman, "New Developments in Laser Resonators," Proc. SPIE 1224, 2-12 (1990).
[CrossRef]

Sugihara, R.

S. Takeuchi, R. Sugihara and K. Shimoda, "Electron acceleration by longitudinal electric field of a Gaussian laser beam," J. Phys. Soc. Jpn. 63, 1186-1193 (1994).
[CrossRef]

Takeuchi, S.

S. Takeuchi, R. Sugihara and K. Shimoda, "Electron acceleration by longitudinal electric field of a Gaussian laser beam," J. Phys. Soc. Jpn. 63, 1186-1193 (1994).
[CrossRef]

Terrell, N. J.

A. J. Campillo, J. E. Pearson, S. L. Shapiro, and N. J. Terrell, Jr., "Fresnel diffraction effects in the design of high-power laser systems," Appl. Phys. Lett. 23, 85-87 (1973).
[CrossRef]

Trautman, J. K.

J. J. Macklin, J. K. Trautman, T. D. Harris and L. E. Brus, "Imaging and time-resolved spectroscopy of single molecules at an interface," Science 272, 255-258 (1996).
[CrossRef]

Walker, B. C.

I. Ghebregziabher and B. C. Walker, "Effect of focal geometry on radiation from atomic ionization in an ultrastrong and ultrafast laser field," Phys. Rev. A 76, 023415 (2007).
[CrossRef]

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth and T. G. Brown, "Longitudinal field modes probed by single molecules," Phys. Rev. Lett. 86, 5251-5254 (2001).
[CrossRef] [PubMed]

Am. J. Phys. (1)

G. D. Gillen and S. Guha, "Modeling and propagation of near-field diffraction patterns: a more complete approach," Am. J. Phys. 72, 1195-1201 (2004).
[CrossRef]

Appl. Phys. B (1)

M. O. Scully, "A simple laser linac," Appl. Phys. B 51, 238-241 (1990).
[CrossRef]

Appl. Phys. Lett. (1)

A. J. Campillo, J. E. Pearson, S. L. Shapiro, and N. J. Terrell, Jr., "Fresnel diffraction effects in the design of high-power laser systems," Appl. Phys. Lett. 23, 85-87 (1973).
[CrossRef]

J. Appl. Phys. (1)

G. Bekefi, "Diffraction of electromagnetic waves by an aperture in a large screen," J. Appl. Phys. 24, 1123-1130 (1953).
[CrossRef]

J. Opt. Lasers Eng. (1)

J. M. P. Coelho, M. A. Abreu, and F. C. Rodrigues, "Modelling the spot shape influence on high-speed transmission lap welding of thermoplastic films," J. Opt. Lasers Eng. 46, 55-61 (2007).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

J. Phys. Soc. Jpn. (1)

S. Takeuchi, R. Sugihara and K. Shimoda, "Electron acceleration by longitudinal electric field of a Gaussian laser beam," J. Phys. Soc. Jpn. 63, 1186-1193 (1994).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

K. Duan and B. Lu, "Polarization properties of vectorial nonparaxial Gaussian beams in the far field," Opt. Lett. 2005, 308-310 (2005).
[CrossRef]

B. Lu and K. Duan, "Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture," Opt. Lett. 28, 2440-2442 (2003).
[CrossRef] [PubMed]

Phys. Rev. A (4)

M. Lax, W. H. Louisell and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975).
[CrossRef]

L. Cicchitelli, H. Hora and R. Postle, "Longitudinal field components for laser beams in vacuum," Phys. Rev. A 41, 3727-3732 (1990).
[CrossRef] [PubMed]

I. Ghebregziabher and B. C. Walker, "Effect of focal geometry on radiation from atomic ionization in an ultrastrong and ultrafast laser field," Phys. Rev. A 76, 023415 (2007).
[CrossRef]

J. P. Paquette and J. L. Chaloupka, "Effect of realistic focal conditions on the strong-field ionization of helium," Phys. Rev. A 79, 043410 (2009).
[CrossRef]

Phys. Rev. Lett. (1)

L. Novotny, M. R. Beversluis, K. S. Youngworth and T. G. Brown, "Longitudinal field modes probed by single molecules," Phys. Rev. Lett. 86, 5251-5254 (2001).
[CrossRef] [PubMed]

Proc. IEEE (1)

H. Kogelnik and T. Li, "Laser beam resonators," Proc. IEEE 54, 1312-1329 (1966).
[CrossRef]

Proc. SPIE (1)

A. E. Siegman, "New Developments in Laser Resonators," Proc. SPIE 1224, 2-12 (1990).
[CrossRef]

Science (1)

J. J. Macklin, J. K. Trautman, T. D. Harris and L. E. Brus, "Imaging and time-resolved spectroscopy of single molecules at an interface," Science 272, 255-258 (1996).
[CrossRef]

Other (5)

R. K. Luneberg, Mathematical theory of optics, (University of California Press, Berkeley, California, 1964.)

A. Yariv, Quantum Electronics, Third Edition, (John Wiley & Sons, New York, NY, 1989.)

W. T. Sifvast, Laser Fundamentals, (Cambridge University Press, New York, NY, 1996.)

A. E. Siegman, Lasers, (University Science Books, Mill Valley, CA, 1986.)

M. W. Sasnett, Physics and Technology of Laser Resonators, (Hilger, New York, NY 1989.)

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Figures (9)

Fig. 1.
Fig. 1.

Theoretical setup used in this investigation where ωo is the unperturbed minimum beam half-width, ωa is the unperturbed beam half-width in the aperture plane, rc is the radius of the clipping aperture, and zG is axial the distance from the aperture to the unperturbed beam waist.

Fig. 2.
Fig. 2.

Calculated e -2 beam half-widths of the intensity of a clipped focused-Gaussian beam using GHVDT (diamonds) and fits to the calculations using a traditional M2 model (solid lines) for various clipping ratios, γ, between values of 3 and 1. All figures are for ωo = 5μm, zG = 10 mm, and λ = 780 nm. The values of M2 fit for each of the four cases are (a) 1.00, (b) 1.02,(c) 1.12, and (d) 1.32.

Fig. 3.
Fig. 3.

Beam intensity profiles in the focal plane, z = zG , for (a) |Ex |2 with γ = 3, (b) |EZ |2 with γ = 3, (c) |Ex |2 with γ = 1, and (d) |EZ |2 with γ = 1. The color scalings of (a) and (c) are both normalized to the peak intensity of (a), while the color scalings of (b) and (d) are both normalized to the peak intensity of (b). All figures are for ωo = 5μm, zG = 10 mm and λ = 780 nm.

Fig. 4.
Fig. 4.

(a) On-axis intensity, (b) radial intensity for an on-axis position of the on-axis minimum at z = 9.385 mm, (c) beam profile of |Ex|2 for z = 9.385 mm, and (d) the beam profile of |EZ |2 for z = 9.385 mm. All figures are for γ = 1, ωo = 5μm,zG = 10 mm, and λ = 780 nm.

Fig. 5.
Fig. 5.

Calculated peak intensities versus the clipping ratio using the full analytical expression, Eq. 34, and the simpler approximated expression, Eq. 38

Fig. 6.
Fig. 6.

Calculated on-axis intensities using the analytical model for clipping ratios of 3, 2, 1.5, 1, and 0.5. The dashed vertical line corresponds to the unperturbed focal plane.

Fig. 7.
Fig. 7.

Experimentally measured on-axis intensities for clipping ratios of 3, 2, 1,5, 1, and 0.5. The abrupt cutoff ∼ 6 mm is due to physical limitations between the aperture mount and the face of the beam profiler. All intensities are normalized to the peak of an unperturbed beam.

Fig. 8.
Fig. 8.

Calculated and experimentally measured minimum beam e -2 half-widths of the intensity. Calculations were performed using the general integral solution Eq. 8.

Fig. 9.
Fig. 9.

Calculated and experimentally measured peak intensities. Calculations were performed using the full analytical expression, Eq. 34, and experimentally measured peak intensities are normalized to the peak intensity of an unperturbed beam.

Equations (39)

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

S ( U V V U ) · n ̂ ds = v ( U 2 V V 2 U ) dv ,
E ( r ) = 1 2 π S o E ( r o ) ( G z ) S o d s o
G = e ikρ ρ
ρ 2 = r r o 2 = ( x x o ) 2 + ( y y o ) 2 + ( z z o ) 2 .
G z = ikz ( 1 + 1 ikρ ) e ikρ ρ 2 .
E ( x o , y o , z o ) = E o ( 1 + z o z G q o ) e ik ( x o 2 + y o 2 ) 2 ( q o + z o z G ) i k ( z o z G )
q o = i π ω o 2 n λ
E x ( x , y , z ) = ikz E o e ik z G 2 π ( 1 z G q o ) e ik ( x o 2 + y o 2 ) 2 ( q o z G ) e ikρ ρ 2 ( 1 + 1 ikρ ) d x o d y o
E y ( r ) = 1 2 π S o E y ( r o ) ( G z ) S o d s o ,
E z ( r ) = 1 2 π S o [ E x ( r o ) ( G x ) S o + E y ( r o ) ( G y ) S o ] d s o ,
E y ( x , y , z ) = 0
E z x y z = ik E o e ik z G 2 π ( 1 z G q o ) e ik ( x o 2 + y o 2 ) 2 ( q o z G ) e ikρ ρ 2 ( 1 + 1 ikρ ) ( x x o ) d x o d y o .
E x r θ z = ikz E o e ik z G 2 π ( 1 z G q o ) e ik r o 2 2 ( q o z G ) e ikρ ρ 2 r o dr o o ,
E z r θ z = ik E o e ik z G 2 π ( 1 z G q o ) e ik r o 2 2 ( q o z G ) e ikρ ρ 2 ( r cos θ r o cos θ o ) r o dr o o
E x r θ z = ik E o e ik ( z G z ) e ik r 2 2 z z ( 1 z G q o ) e ik ( 1 2 ( q o z G ) + 1 2 z ) r o 2 J 0 ( kr r o z ) r o d r o .
E z r θ z = ik E o cos θe ik ( z G z ) e ik r 2 2 z z 2 ( 1 z G q o ) e ik ( 1 2 ( q o z G ) + 1 2 z ) r o 2
× [ r J 0 ( kr r o z ) i r o J 1 ( kr r o z ) ] r o d r o .
E x ( r , θ , z ) = A e ik r 2 2 z e a r o 2 J 0 ( β r o ) r o d r o
E z ( r , θ , z ) = A e ik r 2 2 z cos θ z e a r o 2 [ r J 0 ( β r o ) i r o J 1 ( β r o ) ] r o d r o ,
A = ik E o e ik ( z G z ) z ( 1 z G q o ) ,
a = ik ( 1 2 ( q o z G ) + 1 2 z ) , and β = kr z .
E x ( r , θ , z ) = E o ( 1 + z z G q o ) e ik r 2 2 ( q o + z z G ) ik ( z z G )
E z ( r , θ , z ) = E o cos θ q o ( 1 + z z G q o ) 2 e ik r 2 2 ( q o + z z G ) ik ( z z G )
E z ( r , θ , z ) = E x ( r , θ , z ) r cos θ ( q o + z z G ) .
E z ( x , y , z ) E x ( x , y , z ) = x ( q o + z z G ) .
E z E x = ix z R .
z R = π ω o 2 n λ .
( E z E x ) max = π 2 ω o .
E x 2 = E o 2 1 + ( z z G ) 2 Z R 2 e 2 r 2 ω 2 ( z ) ,
E z 2 = E x 2 r 2 cos 2 θ z R 2 ( 1 + ( z z G ) 2 z R 2 ) .
E x ( 0 , θ , z ) = A 0 r c e a r o 2 r o d r o .
E z ( 0 , θ , z ) = 0 .
E x ( z ) = E G ( z ) [ 1 e a r c 2 ] .
E G ( z ) = A 2 a = E o 1 + z z G q o e ik ( z z G )
I ( z ) = I G ( z ) [ 1 + e 2 γ 2 2 e γ 2 cos ( k r c 2 2 [ 1 R a 1 z ] ) ]
I G ( z ) = E o 2 1 + ( z z G ) 2 z R 2
R a = z G ( 1 + z R 2 z G 2 ) .
γ = r c ω a where ω a 2 = ω o 2 ( 1 + z G 2 z R 2 ) .
I max ( γ ) z≈ z G = I G max [ 1 + e 2 γ 2 2 e γ 2 ] .

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