Abstract

The mechanism of the nonparaxial propagation for tightly focused beams is investigated in the view of the influence of the higher-orders of diffraction (HOD). The HOD induce novel propagation characteristics which are crucially different from those predicted by the traditional paraxial theory. Based on the management of HOD, we propose an approach on controlling the intensity pattern of the focus to satisfy the application requirements.

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
  4. N. Bokor and N. Davidson, “Toward a spherical spot distribution with 4pi focusing of radially polarized light,” Opt. Lett. 29, 1968–1970 (2004).
    [CrossRef] [PubMed]
  5. 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]
  6. J. Pang, Y. K. Ho, X. Q. Yuan, N. Cao, Q. Kong, P. X. Wang, and L. Shao, “Subluminous phase velocity of a focused laser beam and vacuum laser acceleration,” Phys. Rev. E 66, 066501 (2002).
    [CrossRef]
  7. P. Sprangle, E. Esarey, A. Ting, and G. Joyce, “Laser wakefield acceleration and relativistic optical guiding,” Appl. Phys. Lett. 53, 2146–2148 (1988).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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  17. K. Reivelt and P. Saari, “Optically realizable localized wave solutions of the homogeneous scalar wave equation,” Phys. Rev. E 65, 046622 (2002).
    [CrossRef]
  18. S. Orlov, A. Piskarskas, and A. Stabinis, “Localized optical subcycle pulses in dispersive media,” Opt. Lett. 27, 2167–2169 (2002)
    [CrossRef]
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    [CrossRef]
  20. M. A. Porras, R. Borghi, and M. Santarsiero, “Suppression of dispersion broadening of light pulses with Bessel–Gauss beams,” Opt. Commun. 206, 235–241 (2003).
    [CrossRef]
  21. M. A. Porras, “Diffraction-free and dispersion-free pulsed beam propagation in dispersive media,” Opt. Lett. 26, 1364–1366 (2001)
    [CrossRef]
  22. M. A. Porras and P. Di Trapani, “Localized and stationary light wave modes in dispersive media,” Phys. Rev. E 69, 066606 (2004).
    [CrossRef]
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    [CrossRef]
  24. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

2010

S. Orlov and U. Peschel, “Complex source beam: a tool to describe highly focused vector beams analytically,” Phys. Rev. A 82, 063820 (2010).
[CrossRef]

2009

2008

2007

2006

2004

N. Bokor and N. Davidson, “Toward a spherical spot distribution with 4pi focusing of radially polarized light,” Opt. Lett. 29, 1968–1970 (2004).
[CrossRef] [PubMed]

S. Orlov and U. Peschel, “Angular dispersion of diffraction-free optical pulses in dispersive medium,” Opt. Commun. 240, 1–8 (2004).
[CrossRef]

M. A. Porras and P. Di Trapani, “Localized and stationary light wave modes in dispersive media,” Phys. Rev. E 69, 066606 (2004).
[CrossRef]

2003

R. Butkus, S. Orlov, A. Piskarskas, V. Smilgevičius, and A. Stabinis, “Localization of optical wave packets by linear parametric amplification,” Opt. Commun. 227, 237–243 (2003).
[CrossRef]

M. A. Porras, R. Borghi, and M. Santarsiero, “Suppression of dispersion broadening of light pulses with Bessel–Gauss beams,” Opt. Commun. 206, 235–241 (2003).
[CrossRef]

R. Borghi and M. Santarsiero, “Summing Lax series for nonparaxial beam propagation,” Opt. Lett. 28, 774–776 (2003).
[CrossRef] [PubMed]

2002

J. Pang, Y. K. Ho, X. Q. Yuan, N. Cao, Q. Kong, P. X. Wang, and L. Shao, “Subluminous phase velocity of a focused laser beam and vacuum laser acceleration,” Phys. Rev. E 66, 066501 (2002).
[CrossRef]

K. Reivelt and P. Saari, “Optically realizable localized wave solutions of the homogeneous scalar wave equation,” Phys. Rev. E 65, 046622 (2002).
[CrossRef]

S. Orlov, A. Piskarskas, and A. Stabinis, “Localized optical subcycle pulses in dispersive media,” Opt. Lett. 27, 2167–2169 (2002)
[CrossRef]

2001

M. A. Porras, “Diffraction-free and dispersion-free pulsed beam propagation in dispersive media,” Opt. Lett. 26, 1364–1366 (2001)
[CrossRef]

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]

2000

A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. 177, 9–13 (2000).
[CrossRef]

1988

P. Sprangle, E. Esarey, A. Ting, and G. Joyce, “Laser wakefield acceleration and relativistic optical guiding,” Appl. Phys. Lett. 53, 2146–2148 (1988).
[CrossRef]

1975

M. Lax, W. H. Loisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics , 3rd ed. (Academic, 2001).

Bandres, M. A.

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]

Bokor, N.

Borghi, R.

R. Borghi and M. Santarsiero, “Summing Lax series for nonparaxial beam propagation,” Opt. Lett. 28, 774–776 (2003).
[CrossRef] [PubMed]

M. A. Porras, R. Borghi, and M. Santarsiero, “Suppression of dispersion broadening of light pulses with Bessel–Gauss beams,” Opt. Commun. 206, 235–241 (2003).
[CrossRef]

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]

Butkus, R.

R. Butkus, S. Orlov, A. Piskarskas, V. Smilgevičius, and A. Stabinis, “Localization of optical wave packets by linear parametric amplification,” Opt. Commun. 227, 237–243 (2003).
[CrossRef]

Cao, N.

J. Pang, Y. K. Ho, X. Q. Yuan, N. Cao, Q. Kong, P. X. Wang, and L. Shao, “Subluminous phase velocity of a focused laser beam and vacuum laser acceleration,” Phys. Rev. E 66, 066501 (2002).
[CrossRef]

Ciattoni, A.

A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. 177, 9–13 (2000).
[CrossRef]

Crosignani, B.

A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. 177, 9–13 (2000).
[CrossRef]

Davidson, N.

Deng, D.

Di Porto, P.

A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. 177, 9–13 (2000).
[CrossRef]

Di Trapani, P.

M. A. Porras and P. Di Trapani, “Localized and stationary light wave modes in dispersive media,” Phys. Rev. E 69, 066606 (2004).
[CrossRef]

Esarey, E.

P. Sprangle, E. Esarey, A. Ting, and G. Joyce, “Laser wakefield acceleration and relativistic optical guiding,” Appl. Phys. Lett. 53, 2146–2148 (1988).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Guo, Q.

Gutiérrez-Vega, J. C.

Ho, Y. K.

J. Pang, Y. K. Ho, X. Q. Yuan, N. Cao, Q. Kong, P. X. Wang, and L. Shao, “Subluminous phase velocity of a focused laser beam and vacuum laser acceleration,” Phys. Rev. E 66, 066501 (2002).
[CrossRef]

Joyce, G.

P. Sprangle, E. Esarey, A. Ting, and G. Joyce, “Laser wakefield acceleration and relativistic optical guiding,” Appl. Phys. Lett. 53, 2146–2148 (1988).
[CrossRef]

Kong, Q.

J. Pang, Y. K. Ho, X. Q. Yuan, N. Cao, Q. Kong, P. X. Wang, and L. Shao, “Subluminous phase velocity of a focused laser beam and vacuum laser acceleration,” Phys. Rev. E 66, 066501 (2002).
[CrossRef]

Lan, S.

Lax, M.

M. Lax, W. H. Loisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Loisell, W. H.

M. Lax, W. H. Loisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

McKnight, W. B.

M. Lax, W. H. Loisell, 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]

Orlov, S.

S. Orlov and U. Peschel, “Complex source beam: a tool to describe highly focused vector beams analytically,” Phys. Rev. A 82, 063820 (2010).
[CrossRef]

S. Orlov and U. Peschel, “Angular dispersion of diffraction-free optical pulses in dispersive medium,” Opt. Commun. 240, 1–8 (2004).
[CrossRef]

R. Butkus, S. Orlov, A. Piskarskas, V. Smilgevičius, and A. Stabinis, “Localization of optical wave packets by linear parametric amplification,” Opt. Commun. 227, 237–243 (2003).
[CrossRef]

S. Orlov, A. Piskarskas, and A. Stabinis, “Localized optical subcycle pulses in dispersive media,” Opt. Lett. 27, 2167–2169 (2002)
[CrossRef]

Pang, J.

J. Pang, Y. K. Ho, X. Q. Yuan, N. Cao, Q. Kong, P. X. Wang, and L. Shao, “Subluminous phase velocity of a focused laser beam and vacuum laser acceleration,” Phys. Rev. E 66, 066501 (2002).
[CrossRef]

Peschel, U.

S. Orlov and U. Peschel, “Complex source beam: a tool to describe highly focused vector beams analytically,” Phys. Rev. A 82, 063820 (2010).
[CrossRef]

S. Orlov and U. Peschel, “Angular dispersion of diffraction-free optical pulses in dispersive medium,” Opt. Commun. 240, 1–8 (2004).
[CrossRef]

Piskarskas, A.

R. Butkus, S. Orlov, A. Piskarskas, V. Smilgevičius, and A. Stabinis, “Localization of optical wave packets by linear parametric amplification,” Opt. Commun. 227, 237–243 (2003).
[CrossRef]

S. Orlov, A. Piskarskas, and A. Stabinis, “Localized optical subcycle pulses in dispersive media,” Opt. Lett. 27, 2167–2169 (2002)
[CrossRef]

Porras, M. A.

M. A. Porras and P. Di Trapani, “Localized and stationary light wave modes in dispersive media,” Phys. Rev. E 69, 066606 (2004).
[CrossRef]

M. A. Porras, R. Borghi, and M. Santarsiero, “Suppression of dispersion broadening of light pulses with Bessel–Gauss beams,” Opt. Commun. 206, 235–241 (2003).
[CrossRef]

M. A. Porras, “Diffraction-free and dispersion-free pulsed beam propagation in dispersive media,” Opt. Lett. 26, 1364–1366 (2001)
[CrossRef]

Reivelt, K.

K. Reivelt and P. Saari, “Optically realizable localized wave solutions of the homogeneous scalar wave equation,” Phys. Rev. E 65, 046622 (2002).
[CrossRef]

Saari, P.

K. Reivelt and P. Saari, “Optically realizable localized wave solutions of the homogeneous scalar wave equation,” Phys. Rev. E 65, 046622 (2002).
[CrossRef]

Salamin, Y. I.

Santarsiero, M.

R. Borghi and M. Santarsiero, “Summing Lax series for nonparaxial beam propagation,” Opt. Lett. 28, 774–776 (2003).
[CrossRef] [PubMed]

M. A. Porras, R. Borghi, and M. Santarsiero, “Suppression of dispersion broadening of light pulses with Bessel–Gauss beams,” Opt. Commun. 206, 235–241 (2003).
[CrossRef]

Sepke, S.

Shao, L.

J. Pang, Y. K. Ho, X. Q. Yuan, N. Cao, Q. Kong, P. X. Wang, and L. Shao, “Subluminous phase velocity of a focused laser beam and vacuum laser acceleration,” Phys. Rev. E 66, 066501 (2002).
[CrossRef]

Smilgevicius, V.

R. Butkus, S. Orlov, A. Piskarskas, V. Smilgevičius, and A. Stabinis, “Localization of optical wave packets by linear parametric amplification,” Opt. Commun. 227, 237–243 (2003).
[CrossRef]

Sprangle, P.

P. Sprangle, E. Esarey, A. Ting, and G. Joyce, “Laser wakefield acceleration and relativistic optical guiding,” Appl. Phys. Lett. 53, 2146–2148 (1988).
[CrossRef]

Stabinis, A.

R. Butkus, S. Orlov, A. Piskarskas, V. Smilgevičius, and A. Stabinis, “Localization of optical wave packets by linear parametric amplification,” Opt. Commun. 227, 237–243 (2003).
[CrossRef]

S. Orlov, A. Piskarskas, and A. Stabinis, “Localized optical subcycle pulses in dispersive media,” Opt. Lett. 27, 2167–2169 (2002)
[CrossRef]

Ting, A.

P. Sprangle, E. Esarey, A. Ting, and G. Joyce, “Laser wakefield acceleration and relativistic optical guiding,” Appl. Phys. Lett. 53, 2146–2148 (1988).
[CrossRef]

Umstadter, D.

Wang, P. X.

J. Pang, Y. K. Ho, X. Q. Yuan, N. Cao, Q. Kong, P. X. Wang, and L. Shao, “Subluminous phase velocity of a focused laser beam and vacuum laser acceleration,” Phys. Rev. E 66, 066501 (2002).
[CrossRef]

Yang, X.

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]

Yuan, X. Q.

J. Pang, Y. K. Ho, X. Q. Yuan, N. Cao, Q. Kong, P. X. Wang, and L. Shao, “Subluminous phase velocity of a focused laser beam and vacuum laser acceleration,” Phys. Rev. E 66, 066501 (2002).
[CrossRef]

Appl. Phys. B

Y. I. Salamin, “Fields of a Gaussian beam beyond the paraxial approximation,” Appl. Phys. B 86, 319–326 (2007).
[CrossRef]

Appl. Phys. Lett.

P. Sprangle, E. Esarey, A. Ting, and G. Joyce, “Laser wakefield acceleration and relativistic optical guiding,” Appl. Phys. Lett. 53, 2146–2148 (1988).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. 177, 9–13 (2000).
[CrossRef]

S. Orlov and U. Peschel, “Angular dispersion of diffraction-free optical pulses in dispersive medium,” Opt. Commun. 240, 1–8 (2004).
[CrossRef]

M. A. Porras, R. Borghi, and M. Santarsiero, “Suppression of dispersion broadening of light pulses with Bessel–Gauss beams,” Opt. Commun. 206, 235–241 (2003).
[CrossRef]

R. Butkus, S. Orlov, A. Piskarskas, V. Smilgevičius, and A. Stabinis, “Localization of optical wave packets by linear parametric amplification,” Opt. Commun. 227, 237–243 (2003).
[CrossRef]

Opt. Lett.

Phys. Rev. A

M. Lax, W. H. Loisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

S. Orlov and U. Peschel, “Complex source beam: a tool to describe highly focused vector beams analytically,” Phys. Rev. A 82, 063820 (2010).
[CrossRef]

Phys. Rev. E

M. A. Porras and P. Di Trapani, “Localized and stationary light wave modes in dispersive media,” Phys. Rev. E 69, 066606 (2004).
[CrossRef]

J. Pang, Y. K. Ho, X. Q. Yuan, N. Cao, Q. Kong, P. X. Wang, and L. Shao, “Subluminous phase velocity of a focused laser beam and vacuum laser acceleration,” Phys. Rev. E 66, 066501 (2002).
[CrossRef]

K. Reivelt and P. Saari, “Optically realizable localized wave solutions of the homogeneous scalar wave equation,” Phys. Rev. E 65, 046622 (2002).
[CrossRef]

Phys. Rev. Lett.

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. SPIE

M. A. Bandres and J. C. Gutiérrez-Vega, “Generalized Ince Gaussian beams,” Proc. SPIE 6290, 62900S (2006).
[CrossRef]

Other

G. P. Agrawal, Nonlinear Fiber Optics , 3rd ed. (Academic, 2001).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

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Figures (4)

Fig. 1
Fig. 1

Rows 1 and 2: the evolution of the intensity distribution of the transverse component of a tightly focused Gaussian beam in spatial domain resulted from the paraxial theory and the nonparaxial theory, respectively. Rows 3 and 4: the evolution of the intensity (dashed line), the phase (dotted line), and the chirp (solid line) in the angular spectrum domain resulted from the paraxial theory and the nonparaxial theory, respectively. The distance between the entrance plane and the pseudo-waist plane is zs = 30zR . The beam width of the pseudo-waist resulted from the paraxial theory is w 0 = λ = 1μm.

Fig. 2
Fig. 2

(a) The comparison between the evolution of the beam width for the transverse component of the Gaussian beam (defined based on the second-order moment) resulted from the nonparaxial theory and from the paraxial theory for various zs . (b) Solid lines: the beam width at the real-waist (Δz = zf ) and the pseudo-waist (Δz = zs ) vs zs . Dashed line: the distance between the real-waist and pseudo-waist vs zs . Parameters are the same as those in Fig. 1 except zs .

Fig. 3
Fig. 3

Dashed line and dash-dotted line: the evolution of the beam width (defined based on the second-order moment) for the transverse component of the focused Gaussian beam without the pre-added nonlinear chirp, resulted from the paraxial and nonparaxial theory, respectively. Solid line: the evolution of the width for the beam with an appropriate pre-added nonlinear chirp, resulted from the nonparaxial theory. Maps around are the corresponding intensity distributions. Parameters are the same as those in Fig. 1 except zs = −60zR .

Fig. 4
Fig. 4

The sketch of an apparatus for managing the HOD.

Equations (38)

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

E = 0 .
E ˜ , | | ( z 2 ) = E ˜ , | | ( z 1 ) exp ( i k r r + i k z Δ z ) ,
E ˜ | | = 1 k z k E ˜ e z .
E ˜ , | | ( z 2 ) = E ˜ , | | ( z 1 ) exp ( i k r r + i k z Δ z ) d k x d k y .
k z = k + 1 2 ! β 2 k r 2 + 1 4 ! β 4 k r 4 + ,
E , | | ( p ) ( z 2 ) = E ˜ , | | ( z 1 ) exp ( i k r r + ik Δ z + i 2 β 2 Δ z k r 2 ) d k x d k y ,
E , | | ( n p ) ( z 2 ) = E ˜ , | | ( z 1 ) exp ( i k r r + ik Δ z + i 2 ! β 2 Δ z k r 2 + i 4 ! β 4 Δ z k r 4 + ) d k x d k y ,
E ( z 1 ) = A 0 exp [ 1 + iC w z 1 2 ( x 2 + y 2 ) + i ϕ z 1 ] e x ,
E ˜ ( z 1 ) = A ( k x , k y ) exp [ i C w z 1 2 4 ( 1 + C 2 ) k r 2 + i ϕ z 1 ] e x ,
E ˜ | | ( z 1 ) = k x k z A ( k x , k y ) exp [ i C w z 1 2 4 ( 1 + C 2 ) k r 2 + i ϕ z 1 ] e z ,
A ( k x , k y ) = A 0 w z 1 2 2 ( 1 + i C ) exp [ w z 1 2 4 ( 1 + C 2 ) k r 2 ] .
E ˜ ( p ) ( z 2 ) = A ( k x , k y ) exp [ i ( C w z 1 2 4 ( 1 + C 2 ) + 1 2 β 2 Δ z ) k r 2 + i ϕ z 1 + ik Δ z ] e x ,
E ˜ | | ( p ) ( z 2 ) = k x k A ( k x , k y ) exp [ i ( C w z 1 2 4 ( 1 + C 2 ) + 1 2 β 2 Δ z ] ) k r 2 + i ϕ z 1 + ik Δ z ] ] e z ,
Ω p ( Δ z ) = C w z 1 2 4 ( 1 + C 2 ) + 1 2 β 2 Δ z
F ^ { Λ ( k r / a 1 ) exp ( i b 1 k r 2 ) } = Λ ( r / a 2 ) exp ( i b 2 r 2 ) ,
E ( p ) ( z 2 ) = w 0 w exp ( r 2 w 2 ) exp ( ik r 2 2 R i ϕ ) e x ,
E | | ( p ) ( z 2 ) = 2 i w 0 x w 3 k ( 1 + i Δ z / z R ) exp ( r 2 w 2 ) exp ( i k r 2 2 R i ϕ ) e z ,
Ω p ( z s ) = 0.
E ( np ) ( z 2 ) = A ( k x , k y ) exp ( i Ω np + i ϕ z 1 + ik Δ z ) e x d k x d k y ,
E | | ( np ) ( z 2 ) = k x k z A ( k x , k y ) exp ( i Ω np + i ϕ z 1 + ik Δ z ) e z d k x d k y .
Ω np ( k r , Δ z ) = [ C w z 1 2 4 ( 1 + C 2 ) + 1 2 ! β 2 Δ z ] k r 2 + 1 4 ! β 4 Δ z k r 4 +
F ^ { Λ ( k r / a 1 ) exp ( i b 1 k r 2 i c 1 k r 4 + ... ) } Λ ( r / a 2 ) exp ( i b 2 r 2 i c 2 r 4 + ... ) .
Ω np ( k r , z s ) = 1 4 ! β 4 z s k r 4 + 1 6 ! β 6 z s k r 6 + .
Δ r ( z s ) = k r Ω np ( k r , z s ) = 1 3 ! β 4 z s k r 3 + 1 5 ! β 6 z s k r 5 + .
Δ r ( Δ z ) = k r Ω np ( k r , Δ z ) = 2 [ C w z 1 2 4 ( 1 + C 2 ) + 1 2 ! β 2 Δ z ] k r + 1 3 ! β 4 Δ z k r 3 +
E ˜ , | | ( z 1 ) = E ˜ , | | ( z 1 ) exp ( Δ β z s ) ,
Δ β = k z k 1 2 ! β 2 k r 2 = 1 4 ! β 4 k r 4 + 1 6 ! β 6 k r 6 +
E ˜ , | | ( z 1 ) = E ˜ , | | ( p ) ( z 1 + z s ) exp ( ik z s i 2 ! β 2 z s k r 2 i 4 ! β 4 z s k r 4 ... ) .
E , | | ( np ) ( z 2 ) = E ˜ , | | ( z 1 ) exp ( i k r r + i k Δ z + i 2 ! β 2 Δ z k r 2 + i 4 ! β 4 Δ z k r 4 + ... ) d k x d k y = E ˜ , | | ( p ) ( z 1 + z s ) exp ( i Ω np ) exp ( i k r r ) d k x d k y ,
Ω np ( k r , Δ z ) = ( ik + i 2 ! β 2 k r 2 + i 4 ! β 4 k r 4 + ... ) ( Δ z z S ) .
E , | | ( np ) ( z 1 + z s ) = E , | | ( p ) ( z 1 + z s ) ,
Δ r ( Δ z ) = k r Ω np ( k r , Δ z ) = ( β 2 Δ z k r + 1 3 ! β 4 Δ z k r 3 + ... ) ( Δ z z s ) .
Γ = β n + 2 ( n + 2 ) ! k r n + 2 β n n ! k r n = n 1 n + 2 k r 2 k 2
E , | | ( x 1 , y 1 , z 1 ) = exp ( ikf ) i λ f E , | | ( x 0 , y 0 , z 0 ) exp { i k 2 f [ ( x 1 x 0 ) 2 + ( y 1 y 0 ) 2 ] } dxdy .
E , | | ( x 1 , y 1 , z 1 ) = t E , | | ( x 1 , y 1 , z 1 ) = exp ( i δ ) exp [ i k 2 f ( x 1 2 + y 1 2 ) ] E , | | ( x 1 , y 1 , z 1 ) ,
E , | | ( x 1 , y 1 , z 1 ) = exp ( ikf + i δ ) i λ f E , | | ( x 0 , y 0 , z 0 ) exp [ i k 2 f ( x 0 2 + y 0 2 ) ] exp [ i k f ( x 0 x 1 + y 0 y 1 ) ] d x 0 d y 0 ,
E ˜ , | | ( k x , k y , z 1 ) = exp ( ikf + i δ ) i λ f exp [ i f 2 k ( k x 2 + k y 2 ) ] E , | | ( k x , k y , z 0 ) ,
k x = k f x 0 , k y = k f y 0 .

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