Abstract

We present a theoretical analysis of the field distribution in the focal plane of a dispersionless, high numerical aperture (NA) aplanatic lens for an x-polarized short pulse. We compare the focused pulse spatial distribution with that of a focused continuous wave (CW) field and its temporal distribution with the profile of the incident pulse. Regardless of the aberration free nature of the focusing aplanatic lens, the temporal width of the focused pulse widens considerably for incident pulses with durations on the order of a few cycles due to the frequency-dependent nature of diffraction phenomena, which imposes a temporal diffraction limit for focused short pulses. The spatial distribution of the focused pulse is also affected by this dependence and is altered with respect to the diffraction limited distribution of the CW incident field. We have analyzed pulses with flat top and Gaussian spatial irradiance profiles and found that the focused pulse temporal widening is less for the Gaussian spatial irradiance pulse, whereas the spatial distribution variation is similar in both cases. We present results of the focused pulsewidth as a function of the NA for the two spatial irradiance distributions, which show that the Gaussian irradiance pulse outperforms the flat top pulse at preserving the incident pulse duration.

© 2014 Optical Society of America

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References

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  1. W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21, 1369–1377 (2003).
    [CrossRef]
  2. S. W. Hell, “Toward fluorescence nanoscopy,” Nat. Biotechnol. 21, 1347–1355 (2003).
    [CrossRef]
  3. M. Ross, S. I. Green, and J. Brand, “Short-pulse optical communications experiments,” Proc. IEEE 58, 1719–1726 (1970).
    [CrossRef]
  4. A. M. Weiner, “Femtosecond optical pulse shaping and processing,” Prog. Quantum Electron. 19, 161–237 (1995).
    [CrossRef]
  5. M. D. Perry, B. C. Stuart, P. S. Banks, M. D. Feit, V. Yanovsky, and A. M. Rubenchik, “Ultrashort-pulse laser machining of dielectric materials,” J. Appl. Phys. 85, 6803–6810 (1999).
    [CrossRef]
  6. R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2, 219–225 (2008).
    [CrossRef]
  7. M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419, 803–807 (2002).
    [CrossRef]
  8. A. Stolow, A. E. Bragg, and D. M. Neumark, “Femtosecond time-resolved photoelectron spectroscopy,” Chem. Rev. 104, 1719–1758 (2004).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2013 (2)

N. C. Bruce, M. Rosete-Aguilar, O. G. Rodríguez-Herrera, J. Garduño-Mejía, and R. Ortega-Martínez, “Spatial chirp in the focusing of few-optical-cycle pulses by a mirror,” J. Mod. Opt. 60, 1037–1044 (2013).
[CrossRef]

S. Anaya-Vera, L. García-Martínez, M. Rosete-Aguilar, N. C. Bruce, and J. Garduño-Mejía, “Temporal spreading generated by diffraction in the focusing of ultrashort light pulses with perfectly conducting spherical mirrors,” J. Opt. Soc. Am. A 30, 1620–1626 (2013).
[CrossRef]

2012 (1)

2010 (1)

2009 (1)

2008 (1)

R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2, 219–225 (2008).
[CrossRef]

2005 (1)

L. E. Helseth, “Strongly focused polarized light pulse,” Phys. Rev. E 72, 047602 (2005).
[CrossRef]

2004 (1)

A. Stolow, A. E. Bragg, and D. M. Neumark, “Femtosecond time-resolved photoelectron spectroscopy,” Chem. Rev. 104, 1719–1758 (2004).
[CrossRef]

2003 (3)

K. M. Romallosa, J. Bantang, and C. Saloma, “Three-dimensional light distribution near the focus of a tightly focused beam of few-cycle pulses,” Phys. Rev. A 68, 033812 (2003).
[CrossRef]

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21, 1369–1377 (2003).
[CrossRef]

S. W. Hell, “Toward fluorescence nanoscopy,” Nat. Biotechnol. 21, 1347–1355 (2003).
[CrossRef]

2002 (1)

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419, 803–807 (2002).
[CrossRef]

1999 (2)

M. D. Perry, B. C. Stuart, P. S. Banks, M. D. Feit, V. Yanovsky, and A. M. Rubenchik, “Ultrashort-pulse laser machining of dielectric materials,” J. Appl. Phys. 85, 6803–6810 (1999).
[CrossRef]

U. Morgner, F. X. Kärtner, S. H. Cho, Y. Chen, H. A. Haus, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, “Sub-two-cycle pulses from a Kerr-lens mode-locked Ti:sapphire laser,” Opt. Lett. 24, 411–413 (1999).
[CrossRef]

1995 (1)

A. M. Weiner, “Femtosecond optical pulse shaping and processing,” Prog. Quantum Electron. 19, 161–237 (1995).
[CrossRef]

1992 (1)

1970 (1)

M. Ross, S. I. Green, and J. Brand, “Short-pulse optical communications experiments,” Proc. IEEE 58, 1719–1726 (1970).
[CrossRef]

1959 (2)

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London A 253, 349–357 (1959).
[CrossRef]

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

Anaya-Vera, S.

Angelow, G.

Banks, P. S.

M. D. Perry, B. C. Stuart, P. S. Banks, M. D. Feit, V. Yanovsky, and A. M. Rubenchik, “Ultrashort-pulse laser machining of dielectric materials,” J. Appl. Phys. 85, 6803–6810 (1999).
[CrossRef]

Bantang, J.

K. M. Romallosa, J. Bantang, and C. Saloma, “Three-dimensional light distribution near the focus of a tightly focused beam of few-cycle pulses,” Phys. Rev. A 68, 033812 (2003).
[CrossRef]

Born, M.

M. Born and E. Wolf, “Geometrical theory of optical imaging,” Principles of Optics (Cambridge University, 1999), pp. 179–180.

Bragg, A. E.

A. Stolow, A. E. Bragg, and D. M. Neumark, “Femtosecond time-resolved photoelectron spectroscopy,” Chem. Rev. 104, 1719–1758 (2004).
[CrossRef]

Brand, J.

M. Ross, S. I. Green, and J. Brand, “Short-pulse optical communications experiments,” Proc. IEEE 58, 1719–1726 (1970).
[CrossRef]

Bruce, N. C.

N. C. Bruce, M. Rosete-Aguilar, O. G. Rodríguez-Herrera, J. Garduño-Mejía, and R. Ortega-Martínez, “Spatial chirp in the focusing of few-optical-cycle pulses by a mirror,” J. Mod. Opt. 60, 1037–1044 (2013).
[CrossRef]

S. Anaya-Vera, L. García-Martínez, M. Rosete-Aguilar, N. C. Bruce, and J. Garduño-Mejía, “Temporal spreading generated by diffraction in the focusing of ultrashort light pulses with perfectly conducting spherical mirrors,” J. Opt. Soc. Am. A 30, 1620–1626 (2013).
[CrossRef]

Chen, Y.

Cho, S. H.

Dainty, C.

Dainty, J. C.

Drescher, M.

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419, 803–807 (2002).
[CrossRef]

Estrada-Silva, F. C.

Feit, M. D.

M. D. Perry, B. C. Stuart, P. S. Banks, M. D. Feit, V. Yanovsky, and A. M. Rubenchik, “Ultrashort-pulse laser machining of dielectric materials,” J. Appl. Phys. 85, 6803–6810 (1999).
[CrossRef]

Fujimoto, J. G.

García-Martínez, L.

Garduño-Mejía, J.

Gattass, R. R.

R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2, 219–225 (2008).
[CrossRef]

Green, S. I.

M. Ross, S. I. Green, and J. Brand, “Short-pulse optical communications experiments,” Proc. IEEE 58, 1719–1726 (1970).
[CrossRef]

Haus, H. A.

Heinzmann, U.

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419, 803–807 (2002).
[CrossRef]

Hell, S. W.

S. W. Hell, “Toward fluorescence nanoscopy,” Nat. Biotechnol. 21, 1347–1355 (2003).
[CrossRef]

Helseth, L. E.

L. E. Helseth, “Strongly focused polarized light pulse,” Phys. Rev. E 72, 047602 (2005).
[CrossRef]

Hentschel, M.

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419, 803–807 (2002).
[CrossRef]

Ippen, E. P.

Kärtner, F. X.

Kempe, M.

Kenny, F.

Kienberger, R.

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419, 803–807 (2002).
[CrossRef]

Kleineberg, U.

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419, 803–807 (2002).
[CrossRef]

Krausz, F.

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419, 803–807 (2002).
[CrossRef]

Lara, D.

Lin, J.

Mazur, E.

R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2, 219–225 (2008).
[CrossRef]

Morgner, U.

Neumark, D. M.

A. Stolow, A. E. Bragg, and D. M. Neumark, “Femtosecond time-resolved photoelectron spectroscopy,” Chem. Rev. 104, 1719–1758 (2004).
[CrossRef]

Ortega-Martínez, R.

N. C. Bruce, M. Rosete-Aguilar, O. G. Rodríguez-Herrera, J. Garduño-Mejía, and R. Ortega-Martínez, “Spatial chirp in the focusing of few-optical-cycle pulses by a mirror,” J. Mod. Opt. 60, 1037–1044 (2013).
[CrossRef]

F. C. Estrada-Silva, J. Garduño-Mejía, M. Rosete-Aguilar, C. J. Román-Moreno, and R. Ortega-Martínez, “Aberration effects on femtosecond pulses generated by nonideal achromatic doublets,” Appl. Opt. 48, 4723–4734 (2009).
[CrossRef]

Perry, M. D.

M. D. Perry, B. C. Stuart, P. S. Banks, M. D. Feit, V. Yanovsky, and A. M. Rubenchik, “Ultrashort-pulse laser machining of dielectric materials,” J. Appl. Phys. 85, 6803–6810 (1999).
[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. London A 253, 358–379 (1959).
[CrossRef]

Robinson, G.

E. T. Whittaker and G. Robinson, The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. (Dover, 1967), pp. 156–158.

Rodríguez-Herrera, O. G.

Romallosa, K. M.

K. M. Romallosa, J. Bantang, and C. Saloma, “Three-dimensional light distribution near the focus of a tightly focused beam of few-cycle pulses,” Phys. Rev. A 68, 033812 (2003).
[CrossRef]

Román-Moreno, C. J.

Rosete-Aguilar, M.

Ross, M.

M. Ross, S. I. Green, and J. Brand, “Short-pulse optical communications experiments,” Proc. IEEE 58, 1719–1726 (1970).
[CrossRef]

Rubenchik, A. M.

M. D. Perry, B. C. Stuart, P. S. Banks, M. D. Feit, V. Yanovsky, and A. M. Rubenchik, “Ultrashort-pulse laser machining of dielectric materials,” J. Appl. Phys. 85, 6803–6810 (1999).
[CrossRef]

Rudolph, W.

Saloma, C.

K. M. Romallosa, J. Bantang, and C. Saloma, “Three-dimensional light distribution near the focus of a tightly focused beam of few-cycle pulses,” Phys. Rev. A 68, 033812 (2003).
[CrossRef]

Scheuer, V.

Scrinzi, A.

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419, 803–807 (2002).
[CrossRef]

Stamm, U.

Stolow, A.

A. Stolow, A. E. Bragg, and D. M. Neumark, “Femtosecond time-resolved photoelectron spectroscopy,” Chem. Rev. 104, 1719–1758 (2004).
[CrossRef]

Stuart, B. C.

M. D. Perry, B. C. Stuart, P. S. Banks, M. D. Feit, V. Yanovsky, and A. M. Rubenchik, “Ultrashort-pulse laser machining of dielectric materials,” J. Appl. Phys. 85, 6803–6810 (1999).
[CrossRef]

Tschudi, T.

Uiberacker, M.

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419, 803–807 (2002).
[CrossRef]

Webb, W. W.

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21, 1369–1377 (2003).
[CrossRef]

Weiner, A. M.

A. M. Weiner, “Femtosecond optical pulse shaping and processing,” Prog. Quantum Electron. 19, 161–237 (1995).
[CrossRef]

Westerwalbesloh, T.

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419, 803–807 (2002).
[CrossRef]

Whittaker, E. T.

E. T. Whittaker and G. Robinson, The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. (Dover, 1967), pp. 156–158.

Wilhelmi, B.

Williams, R. M.

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21, 1369–1377 (2003).
[CrossRef]

Wolf, E.

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

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London A 253, 349–357 (1959).
[CrossRef]

M. Born and E. Wolf, “Geometrical theory of optical imaging,” Principles of Optics (Cambridge University, 1999), pp. 179–180.

Yakovlev, V.

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419, 803–807 (2002).
[CrossRef]

Yanovsky, V.

M. D. Perry, B. C. Stuart, P. S. Banks, M. D. Feit, V. Yanovsky, and A. M. Rubenchik, “Ultrashort-pulse laser machining of dielectric materials,” J. Appl. Phys. 85, 6803–6810 (1999).
[CrossRef]

Zipfel, W. R.

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21, 1369–1377 (2003).
[CrossRef]

Appl. Opt. (1)

Chem. Rev. (1)

A. Stolow, A. E. Bragg, and D. M. Neumark, “Femtosecond time-resolved photoelectron spectroscopy,” Chem. Rev. 104, 1719–1758 (2004).
[CrossRef]

J. Appl. Phys. (1)

M. D. Perry, B. C. Stuart, P. S. Banks, M. D. Feit, V. Yanovsky, and A. M. Rubenchik, “Ultrashort-pulse laser machining of dielectric materials,” J. Appl. Phys. 85, 6803–6810 (1999).
[CrossRef]

J. Mod. Opt. (1)

N. C. Bruce, M. Rosete-Aguilar, O. G. Rodríguez-Herrera, J. Garduño-Mejía, and R. Ortega-Martínez, “Spatial chirp in the focusing of few-optical-cycle pulses by a mirror,” J. Mod. Opt. 60, 1037–1044 (2013).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Nat. Biotechnol. (2)

W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21, 1369–1377 (2003).
[CrossRef]

S. W. Hell, “Toward fluorescence nanoscopy,” Nat. Biotechnol. 21, 1347–1355 (2003).
[CrossRef]

Nat. Photonics (1)

R. R. Gattass and E. Mazur, “Femtosecond laser micromachining in transparent materials,” Nat. Photonics 2, 219–225 (2008).
[CrossRef]

Nature (1)

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419, 803–807 (2002).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. A (1)

K. M. Romallosa, J. Bantang, and C. Saloma, “Three-dimensional light distribution near the focus of a tightly focused beam of few-cycle pulses,” Phys. Rev. A 68, 033812 (2003).
[CrossRef]

Phys. Rev. E (1)

L. E. Helseth, “Strongly focused polarized light pulse,” Phys. Rev. E 72, 047602 (2005).
[CrossRef]

Proc. IEEE (1)

M. Ross, S. I. Green, and J. Brand, “Short-pulse optical communications experiments,” Proc. IEEE 58, 1719–1726 (1970).
[CrossRef]

Proc. R. Soc. London A (2)

E. Wolf, “Electromagnetic diffraction in optical systems I. An integral representation of the image field,” Proc. R. Soc. London A 253, 349–357 (1959).
[CrossRef]

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

Prog. Quantum Electron. (1)

A. M. Weiner, “Femtosecond optical pulse shaping and processing,” Prog. Quantum Electron. 19, 161–237 (1995).
[CrossRef]

Other (2)

M. Born and E. Wolf, “Geometrical theory of optical imaging,” Principles of Optics (Cambridge University, 1999), pp. 179–180.

E. T. Whittaker and G. Robinson, The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. (Dover, 1967), pp. 156–158.

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Figures (7)

Fig. 1.
Fig. 1.

Diagram of the light focused by a high-NA system. The origin of the coordinate system is in the geometrical focus of the lens.

Fig. 2.
Fig. 2.

Comparison between the normalized irradiance temporal profile of the incident pulse, I0(t), and the 10, 4.5, and 2.7 fs focused pulses with a flat top spatial irradiance distribution (Ω, NA=0.95, u=0).

Fig. 3.
Fig. 3.

Comparison between cross sections of the normalized irradiance spatial profile in the (a) x and (b) y directions of the focal plane (u=0) for a flat top spatial irradiance CW laser, I0(v), and the 10, 4.5, and 2.7 fs focused pulses with the same irradiance distribution (Ω, NA=0.95).

Fig. 4.
Fig. 4.

Normalized duration of the flat top spatial irradiance focused pulse as a function of the NA for incident pulses of 10, 4.5, and 2.7 fs. τ and τf are the durations of the incident and focused pulses, respectively, measured at 1/e.

Fig. 5.
Fig. 5.

Comparison between the normalized irradiance temporal profile of the incident pulse, I0(t), and the 10, 4.5, and 2.7 fs focused pulses with a Gaussian spatial irradiance distribution (Ω=1.18mm, NA=0.95, u=0).

Fig. 6.
Fig. 6.

Comparison between cross sections of the normalized irradiance spatial profile in the (a) x and (b) y directions of the focal plane (u=0) for a Gaussian spatial irradiance CW laser, I0(v), and the 10, 4.5, and 2.7 fs focused pulses with the same irradiance distribution (Ω=1.18mm, NA=0.95).

Fig. 7.
Fig. 7.

Normalized duration of the Gaussian spatial irradiance focused pulse as a function of the NA for incident pulses of 10, 4.5, and 2.7 fs. τ and τf are the durations of the incident and focused pulses, respectively, measured at 1/e.

Tables (1)

Tables Icon

Table 1. Focused Pulse Temporal and Spatial Widths for Incident Pulses of 20, 10, 4.5, and 2.7 fs with Flat Top and Gaussian Spatial Irradiance Distributions Focused by an Aplanatic Lens with NA=0.95a

Equations (23)

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

E(P)=ik2πΓa(sx,sy)szexp(ik[Φ(sx,sy)+s^·rp])dsxdsy,
H(P)=ik2πΓb(sx,sy)szexp(ik[Φ(sx,sy)+s^·rp])dsxdsy,
E(x,y,z;Δω)=ik2πΓa(sx,sy)szP(sx,sy)A(Δω)×exp(ik[Φ(sx,sy)+s^·rp])dsxdsy.
A(Δω)=A0exp([T(Δω)2]2),
P(sx,sy)={1,ifsx,syΓ0,otherwise,
a(sx,sy)=a0(sx,sy)A(sx,sy),
A(sx,sy)=exp(x02+y02Ω2),
x0=ρcosϕ,y0=ρsinϕ,
ρ=fsinθ,
A(θ,ϕ)=exp(f2sin2θΩ2).
P(θ,ϕ)={1,ifθα0,otherwise,
a0(θ,ϕ)=12([(cosθ+1)+(cosθ1)cos2ϕ](cosθ1)sin2ϕ2sinθcosϕ).
Ex(v,u,ϕp;Δω)=ik02(1+Δωω0)A0exp([T(Δω)2]2)×0αcosθsinθ(cosθ+1)exp(f2sin2θΩ2)×exp(iucosθsin2α(1+Δωω0))J0(vsinθsinα(1+Δωω0))dθ+ik02(1+Δωω0)A0exp([T(Δω)2]2)cos2ϕp×0αcosθsinθ(cosθ1)exp(f2sin2θΩ2)×exp(iucosθsin2α(1+Δωω0))J2(vsinθsinα(1+Δωω0))dθ,
v=k0ρpsinα,u=k0zpsin2α,
k=k0(1+Δωω0),
Ey(v,u,ϕp;Δω)=ik02(1+Δωω0)A0exp([T(Δω)2]2)sin2ϕp×0αcosθsinθ(cosθ1)exp(f2sin2θΩ2)×exp(iucosθsin2α(1+Δωω0))J2(vsinθsinα(1+Δωω0))dθ,
Ez(v,u,ϕp;Δω)=k0(1+Δωω0)A0exp([T(Δω)2]2)cosϕp×0αcosθsinθ(sinθ)exp(f2sin2θΩ2)×exp(iucosθsin2α(1+Δωω0))J1(vsinθsinα(1+Δωω0))dθ.
Ex(v,u,ϕp;Δω)=iK0[I0(v,u;Δω)I2(v,u;Δω)cos2ϕp],Ey(v,u,ϕp;Δω)=+iK0I2(v,u;Δω)sin2ϕp,Ez(v,u,ϕp;Δω)=2K0I1(v,u;Δω)cosϕp,
K0=k02(1+Δωω0)A0exp([T(Δω)2]2),
I0(v,u;Δω)=0αcosθsinθ(cosθ+1)exp(f2sin2θΩ2)×exp(iucosθsin2α(1+Δωω0))J0(vsinθsinα(1+Δωω0))dθ,I1(v,u;Δω)=0αcosθsinθ(sinθ)exp(f2sin2θΩ2)×exp(iucosθsin2α(1+Δωω0))J1(vsinθsinα(1+Δωω0))dθ,I2(v,u;Δω)=0αcosθsinθ(cosθ1)exp(f2sin2θΩ2)×exp(iucosθsin2α(1+Δωω0))J2(vsinθsinα(1+Δωω0))dθ.
E(v,u,ϕp;t)=d(Δω)exp(i(Δω)t)E(v,u,ϕp;Δω).
I(t)=02π0|E(v,u,ϕp;t)|2vdvdϕp
I(v)=|E(v,u,ϕp;t)|2dt,

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