Abstract

We assess several widely used vector models of a Gaussian laser beam in the context of more accurate vector diffraction integration. For the analysis, we present a streamlined derivation of the vector fields of a uniformly polarized beam reflected from an ideal parabolic mirror, both inside and outside of the resulting focus. This exact solution to Maxwell’s equations, first developed in 1920 by V. S. Ignatovsky, is highly relevant to high-intensity laser experiments since the boundary conditions at a focusing optic dictate the form of the focus in a manner analogous to a physical experiment. In contrast, many models simply assume a field profile near the focus and develop the surrounding vector fields consistent with Maxwell’s equations. In comparing the Ignatovsky result with popular closed-form analytic vector models of a Gaussian beam, we find that the relatively simple model developed by Erikson and Singh in 1994 provides good agreement in the paraxial limit. Models involving a Lax expansion introduce a divergences outside of the focus while providing little if any improvement in the focal region. Extremely tight focusing produces a somewhat complicated structure in the focus, and requires the Ignatovsky model for accurate representation.

© 2017 Optical Society of America

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References

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2016 (3)

2015 (1)

Y. I. Salamin, “Simple analytical derivation of the fields of an ultrashort tightly focused linearly polarized laser pulse,” Phys. Rev. A 92, 063818 (2015).
[Crossref]

2014 (3)

N. Powers, I. Ghebregziabher, G. Golovin, C. Liu, S. Chen, S. Banerjee, J. Zhang, and D. Umstadter, “Quasi-monoenergetic and tunable X-rays from a laser-driven Compton light source,” Nature Photonics 8, 28–31 (2014).
[Crossref]

C.P. Ridgers, J.G. Kirk, R. Duclous, T.G. Blackburn, C.S. Brady, K. Bennette, T.D. Arbere, and A.R. Bell, “Modelling gamma-ray photon emission and pair production in high-intensity laser-matter interactions,” J. Comput. Phys. 260, 273–285 (2014).
[Crossref]

N.V. Zamfir, “Nuclear Physics with 10PW laser beams at extreme light infrastructure–nuclear physics (ELI-NP),” Eur. Phys. J. Special Topics 223, 1221–1227 (2014).
[Crossref]

2013 (1)

I. Ghebregziabher, B. A. Shadwick, and D. Umstadter, “Spectral bandwidth reduction of Thomson scattered light by pulse chirping,” Phys. Rev. Spec. Top. - Accel. Beams 16, 030705 (2013).
[Crossref]

2012 (1)

I. Gonoskov, A. Aiello, S. Heugel, and G. Leuchs, “Dipole pulse theory: Maximizing the field amplitude from 4π focused laser pulses,” Phys. Rev. A 86, 053836 (2012).
[Crossref]

2011 (1)

G. Mourou and T. Tajima, “More intense, shorter pulses,” Science 331, 41–42 (2011).
[Crossref] [PubMed]

2008 (3)

K. I. Popov, V. Yu. Bychenkov, W. Rozmus, and R. D. Sydora, “Electron vacuum acceleration by a tightly focused laser pulse,” Phys. Plasmas 15, 013108 (2008).
[Crossref]

N. Bokor and N. Davidson, “4π Focusing with single paraboloid mirror,” Opt. Comm. 281, 5499–5503 (2008).
[Crossref]

V. Yanovsky, V. Chvykov, G. Kalinchenko, P. Rousseau, T. Planchon, T. Matsuoka, A. Maksimchuk, J. Nees, G. Cheriaux, G. Mourou, and K. Krushelnick, “Ultra-high intensity- 300-TW laser at 0.1 Hz repetition rate,” Opt. Express 16, 2109–2114 (2008).
[Crossref] [PubMed]

2007 (3)

H. Luo, S. Liu, Z. Lin, and C. T. Chan, “Method for accurate description of a radially polarized Gaussian laser beam beyond the paraxial approximation,” Opt. Lett. 32, 1692–1694 (2007).
[Crossref] [PubMed]

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

S. G. Bochkarev and Y. Yu. Bychenkov, “Acceleration of electrons by tightly focused femtosecond laser pulses,” Quantum Electron. 37, 273–284 (2007).
[Crossref]

2006 (1)

2004 (1)

2002 (1)

Y. Salamin and C. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88, 095005 (2002).
[Crossref] [PubMed]

2001 (2)

2000 (1)

1999 (1)

H.-C. Kim and Y. H. Lee, “Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation,” Opt. Comm. 169, 9–16 (1999).
[Crossref]

1998 (3)

P. Varga and P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Comm. 152, 108–118 (1998).
[Crossref]

B. Quesnel and P. Mora, “Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum,” Phys. Rev. E 58, 3719–3732 (1998).
[Crossref]

M. Scully and M. Zubairy, “Simple laser accelerator: Optics and particle dynamics,” Phys. Rev. A 152, 108–118 (1998).

1994 (3)

W. Erikson and S. Singh, “Polarization properties of Maxwell-Gaussian laser beams,” Phys. Rev. E 49, 5778–5786 (1994).
[Crossref]

W. Hsu and R. Barakat, “Stratton-Chu vectorial diffraction of electromagnetic fields by apertures with application to small-Fresnel-number systems,” J. Opt. Soc. Am. A 26, 623–629 (1994).
[Crossref]

C. Sheppard and K. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41, 1495–1505 (1994).
[Crossref]

1990 (1)

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

1989 (1)

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[Crossref]

1987 (1)

1985 (1)

T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. A 2, 826–829 (1985).
[Crossref]

1984 (1)

1979 (1)

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[Crossref]

1977 (1)

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

1975 (1)

M. Lax, W. Louisell, and W. McKnight,“From Maxwell to paraxial wave optics,” Phys.Rev.A 11, 1365–1370 (1975).

1974 (1)

A. Yoshida and T. Asakura, “Electromagnetic Field near the focus of Gaussian Beams,” Optik 41, 281–292 (1974).

1959 (2)

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. 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. A 253, 358–379 (1959).
[Crossref]

1939 (1)

J. A. Stratton and L. J. Chu, “Diffraction Theory of Electromagnetic Waves,” Phys. Rev. 56, 99–107 (1939).
[Crossref]

1920 (3)

V. S. Ignatovsky, “The relationship between geometric and wave optics and diffraction of an azimuthally symmetric beam,” Trans. Opt. Inst. Petrograd 1, 3 (1920).

V. S. Ignatovsky, “Diffraction by a lens with arbitrary aperture,” Trans. Opt. Inst. Petrograd 1, 4 (1920).

V. S. Ignatovsky, “Diffraction by a parabolic mirror having arbitrary opening,” Trans. Opt. Inst. Petrograd 1, 5 (1920).

1909 (1)

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 30, 755–776 (1909).
[Crossref]

Aiello, A.

I. Gonoskov, A. Aiello, S. Heugel, and G. Leuchs, “Dipole pulse theory: Maximizing the field amplitude from 4π focused laser pulses,” Phys. Rev. A 86, 053836 (2012).
[Crossref]

Alexander, D. R.

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[Crossref]

Arbere, T.D.

C.P. Ridgers, J.G. Kirk, R. Duclous, T.G. Blackburn, C.S. Brady, K. Bennette, T.D. Arbere, and A.R. Bell, “Modelling gamma-ray photon emission and pair production in high-intensity laser-matter interactions,” J. Comput. Phys. 260, 273–285 (2014).
[Crossref]

Asakura, T.

A. Yoshida and T. Asakura, “Electromagnetic Field near the focus of Gaussian Beams,” Optik 41, 281–292 (1974).

Bahk, S.

Banerjee, S.

N. Powers, I. Ghebregziabher, G. Golovin, C. Liu, S. Chen, S. Banerjee, J. Zhang, and D. Umstadter, “Quasi-monoenergetic and tunable X-rays from a laser-driven Compton light source,” Nature Photonics 8, 28–31 (2014).
[Crossref]

Barakat, R.

W. Hsu and R. Barakat, “Stratton-Chu vectorial diffraction of electromagnetic fields by apertures with application to small-Fresnel-number systems,” J. Opt. Soc. Am. A 26, 623–629 (1994).
[Crossref]

R. Barakat, “Diffracted electromagnetic fields in the neighborhood of the focus of a paraboloidal mirror having a central obscuration,” Appl. Opt. 26, 3790–3795 (1987).
[Crossref] [PubMed]

Barton, J. P.

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[Crossref]

Bell, A.R.

C.P. Ridgers, J.G. Kirk, R. Duclous, T.G. Blackburn, C.S. Brady, K. Bennette, T.D. Arbere, and A.R. Bell, “Modelling gamma-ray photon emission and pair production in high-intensity laser-matter interactions,” J. Comput. Phys. 260, 273–285 (2014).
[Crossref]

Bennette, K.

C.P. Ridgers, J.G. Kirk, R. Duclous, T.G. Blackburn, C.S. Brady, K. Bennette, T.D. Arbere, and A.R. Bell, “Modelling gamma-ray photon emission and pair production in high-intensity laser-matter interactions,” J. Comput. Phys. 260, 273–285 (2014).
[Crossref]

Blackburn, T.G.

C.P. Ridgers, J.G. Kirk, R. Duclous, T.G. Blackburn, C.S. Brady, K. Bennette, T.D. Arbere, and A.R. Bell, “Modelling gamma-ray photon emission and pair production in high-intensity laser-matter interactions,” J. Comput. Phys. 260, 273–285 (2014).
[Crossref]

Bochkarev, S. G.

S. G. Bochkarev and Y. Yu. Bychenkov, “Acceleration of electrons by tightly focused femtosecond laser pulses,” Quantum Electron. 37, 273–284 (2007).
[Crossref]

Bokor, N.

N. Bokor and N. Davidson, “4π Focusing with single paraboloid mirror,” Opt. Comm. 281, 5499–5503 (2008).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Sec. 8.8.1.
[Crossref]

Borot, A.

G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space-time characterization of ultra-intense femtosecond laser beams,” Nature Photonics 10, 547–553 (2016).
[Crossref]

Brady, C.S.

C.P. Ridgers, J.G. Kirk, R. Duclous, T.G. Blackburn, C.S. Brady, K. Bennette, T.D. Arbere, and A.R. Bell, “Modelling gamma-ray photon emission and pair production in high-intensity laser-matter interactions,” J. Comput. Phys. 260, 273–285 (2014).
[Crossref]

Chan, C. T.

Chen, S.

N. Powers, I. Ghebregziabher, G. Golovin, C. Liu, S. Chen, S. Banerjee, J. Zhang, and D. Umstadter, “Quasi-monoenergetic and tunable X-rays from a laser-driven Compton light source,” Nature Photonics 8, 28–31 (2014).
[Crossref]

Cheriaux, G.

Choudhury, A.

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

Chu, L. J.

J. A. Stratton and L. J. Chu, “Diffraction Theory of Electromagnetic Waves,” Phys. Rev. 56, 99–107 (1939).
[Crossref]

Chvykov, V.

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]

Coburn, C.

Cunningham, E.

Davidson, N.

N. Bokor and N. Davidson, “4π Focusing with single paraboloid mirror,” Opt. Comm. 281, 5499–5503 (2008).
[Crossref]

Davis, L. W.

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[Crossref]

Debye, P.

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 30, 755–776 (1909).
[Crossref]

Duclous, R.

C.P. Ridgers, J.G. Kirk, R. Duclous, T.G. Blackburn, C.S. Brady, K. Bennette, T.D. Arbere, and A.R. Bell, “Modelling gamma-ray photon emission and pair production in high-intensity laser-matter interactions,” J. Comput. Phys. 260, 273–285 (2014).
[Crossref]

Erikson, W.

W. Erikson and S. Singh, “Polarization properties of Maxwell-Gaussian laser beams,” Phys. Rev. E 49, 5778–5786 (1994).
[Crossref]

Fainman, Y.

Fukumitsu, O.

T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. A 2, 826–829 (1985).
[Crossref]

Gallet, V.

G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space-time characterization of ultra-intense femtosecond laser beams,” Nature Photonics 10, 547–553 (2016).
[Crossref]

Gannaway, J.

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

Ghebregziabher, I.

N. Powers, I. Ghebregziabher, G. Golovin, C. Liu, S. Chen, S. Banerjee, J. Zhang, and D. Umstadter, “Quasi-monoenergetic and tunable X-rays from a laser-driven Compton light source,” Nature Photonics 8, 28–31 (2014).
[Crossref]

I. Ghebregziabher, B. A. Shadwick, and D. Umstadter, “Spectral bandwidth reduction of Thomson scattered light by pulse chirping,” Phys. Rev. Spec. Top. - Accel. Beams 16, 030705 (2013).
[Crossref]

Gobert, O.

G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space-time characterization of ultra-intense femtosecond laser beams,” Nature Photonics 10, 547–553 (2016).
[Crossref]

Golovin, G.

N. Powers, I. Ghebregziabher, G. Golovin, C. Liu, S. Chen, S. Banerjee, J. Zhang, and D. Umstadter, “Quasi-monoenergetic and tunable X-rays from a laser-driven Compton light source,” Nature Photonics 8, 28–31 (2014).
[Crossref]

Gonoskov, I.

I. Gonoskov, A. Aiello, S. Heugel, and G. Leuchs, “Dipole pulse theory: Maximizing the field amplitude from 4π focused laser pulses,” Phys. Rev. A 86, 053836 (2012).
[Crossref]

Heugel, S.

I. Gonoskov, A. Aiello, S. Heugel, and G. Leuchs, “Dipole pulse theory: Maximizing the field amplitude from 4π focused laser pulses,” Phys. Rev. A 86, 053836 (2012).
[Crossref]

Holmes, M. H.

M. H. Holmes, Introduction to Perturbation Methods, 2nd Ed. (Springer, 2013).
[Crossref]

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]

Hsu, W.

W. Hsu and R. Barakat, “Stratton-Chu vectorial diffraction of electromagnetic fields by apertures with application to small-Fresnel-number systems,” J. Opt. Soc. Am. A 26, 623–629 (1994).
[Crossref]

Ignatovsky, V. S.

V. S. Ignatovsky, “Diffraction by a lens with arbitrary aperture,” Trans. Opt. Inst. Petrograd 1, 4 (1920).

V. S. Ignatovsky, “Diffraction by a parabolic mirror having arbitrary opening,” Trans. Opt. Inst. Petrograd 1, 5 (1920).

V. S. Ignatovsky, “The relationship between geometric and wave optics and diffraction of an azimuthally symmetric beam,” Trans. Opt. Inst. Petrograd 1, 3 (1920).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998), Eq. (10.87).

Kalinchenko, G.

Kalintchenko, G.

Keitel, C.

Y. Salamin and C. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88, 095005 (2002).
[Crossref] [PubMed]

Kim, H.-C.

H.-C. Kim and Y. H. Lee, “Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation,” Opt. Comm. 169, 9–16 (1999).
[Crossref]

Kirk, J.G.

C.P. Ridgers, J.G. Kirk, R. Duclous, T.G. Blackburn, C.S. Brady, K. Bennette, T.D. Arbere, and A.R. Bell, “Modelling gamma-ray photon emission and pair production in high-intensity laser-matter interactions,” J. Comput. Phys. 260, 273–285 (2014).
[Crossref]

Krushelnick, K.

Larkin, K.

C. Sheppard and K. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41, 1495–1505 (1994).
[Crossref]

Lax, M.

M. Lax, W. Louisell, and W. McKnight,“From Maxwell to paraxial wave optics,” Phys.Rev.A 11, 1365–1370 (1975).

Lee, Y. H.

H.-C. Kim and Y. H. Lee, “Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation,” Opt. Comm. 169, 9–16 (1999).
[Crossref]

Leuchs, G.

I. Gonoskov, A. Aiello, S. Heugel, and G. Leuchs, “Dipole pulse theory: Maximizing the field amplitude from 4π focused laser pulses,” Phys. Rev. A 86, 053836 (2012).
[Crossref]

Levy, U.

Lieb, M. A.

Lin, Z.

Liu, C.

N. Powers, I. Ghebregziabher, G. Golovin, C. Liu, S. Chen, S. Banerjee, J. Zhang, and D. Umstadter, “Quasi-monoenergetic and tunable X-rays from a laser-driven Compton light source,” Nature Photonics 8, 28–31 (2014).
[Crossref]

Liu, S.

Louisell, W.

M. Lax, W. Louisell, and W. McKnight,“From Maxwell to paraxial wave optics,” Phys.Rev.A 11, 1365–1370 (1975).

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics, (University of California Press, 1966), Sec. 46.

Luo, H.

Maksimchuk, A.

Matsuoka, T.

McKnight, W.

M. Lax, W. Louisell, and W. McKnight,“From Maxwell to paraxial wave optics,” Phys.Rev.A 11, 1365–1370 (1975).

Meixner, A. J.

Mora, P.

B. Quesnel and P. Mora, “Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum,” Phys. Rev. E 58, 3719–3732 (1998).
[Crossref]

Mourou, G.

Nees, J.

Pariente, G.

G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space-time characterization of ultra-intense femtosecond laser beams,” Nature Photonics 10, 547–553 (2016).
[Crossref]

Peatross, J.

Planchon, T.

Popov, K. I.

K. I. Popov, V. Yu. Bychenkov, W. Rozmus, and R. D. Sydora, “Electron vacuum acceleration by a tightly focused laser pulse,” Phys. Plasmas 15, 013108 (2008).
[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]

Powers, N.

N. Powers, I. Ghebregziabher, G. Golovin, C. Liu, S. Chen, S. Banerjee, J. Zhang, and D. Umstadter, “Quasi-monoenergetic and tunable X-rays from a laser-driven Compton light source,” Nature Photonics 8, 28–31 (2014).
[Crossref]

Quéré, F.

G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space-time characterization of ultra-intense femtosecond laser beams,” Nature Photonics 10, 547–553 (2016).
[Crossref]

Quesnel, B.

B. Quesnel and P. Mora, “Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum,” Phys. Rev. E 58, 3719–3732 (1998).
[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. A 253, 358–379 (1959).
[Crossref]

Ridgers, C.P.

C.P. Ridgers, J.G. Kirk, R. Duclous, T.G. Blackburn, C.S. Brady, K. Bennette, T.D. Arbere, and A.R. Bell, “Modelling gamma-ray photon emission and pair production in high-intensity laser-matter interactions,” J. Comput. Phys. 260, 273–285 (2014).
[Crossref]

Rousseau, P.

Rozmus, W.

K. I. Popov, V. Yu. Bychenkov, W. Rozmus, and R. D. Sydora, “Electron vacuum acceleration by a tightly focused laser pulse,” Phys. Plasmas 15, 013108 (2008).
[Crossref]

Salamin, Y.

Y. Salamin and C. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88, 095005 (2002).
[Crossref] [PubMed]

Salamin, Y. I.

Y. I. Salamin, “Simple analytical derivation of the fields of an ultrashort tightly focused linearly polarized laser pulse,” Phys. Rev. A 92, 063818 (2015).
[Crossref]

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

Scully, M.

M. Scully and M. Zubairy, “Simple laser accelerator: Optics and particle dynamics,” Phys. Rev. A 152, 108–118 (1998).

Sepke, S. M.

Shadwick, B. A.

I. Ghebregziabher, B. A. Shadwick, and D. Umstadter, “Spectral bandwidth reduction of Thomson scattered light by pulse chirping,” Phys. Rev. Spec. Top. - Accel. Beams 16, 030705 (2013).
[Crossref]

Shamir, J.

Sheppard, C.

C. Sheppard and K. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41, 1495–1505 (1994).
[Crossref]

Sheppard, C. J. R.

C. J. R. Sheppard, “High-aperture beams,” J. Opt. Soc. Am. A 18, 1579–1587 (2001).
[Crossref]

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

Silberberg, Y.

Singh, S.

W. Erikson and S. Singh, “Polarization properties of Maxwell-Gaussian laser beams,” Phys. Rev. E 49, 5778–5786 (1994).
[Crossref]

Stamns, J. J.

J. J. Stamns, Waves in Focal Regions, (IOP Publishing, 1986), Sec. 16.1.

Stratton, J. A.

J. A. Stratton and L. J. Chu, “Diffraction Theory of Electromagnetic Waves,” Phys. Rev. 56, 99–107 (1939).
[Crossref]

Sydora, R. D.

K. I. Popov, V. Yu. Bychenkov, W. Rozmus, and R. D. Sydora, “Electron vacuum acceleration by a tightly focused laser pulse,” Phys. Plasmas 15, 013108 (2008).
[Crossref]

Tajima, T.

G. Mourou and T. Tajima, “More intense, shorter pulses,” Science 331, 41–42 (2011).
[Crossref] [PubMed]

Takenaka, T.

T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. A 2, 826–829 (1985).
[Crossref]

Török, P.

P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. I. Theory,” J. Opt. Soc. Am. A 17, 2081–2089 (2000).
[Crossref]

P. Varga and P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Comm. 152, 108–118 (1998).
[Crossref]

Umstadter, D.

N. Powers, I. Ghebregziabher, G. Golovin, C. Liu, S. Chen, S. Banerjee, J. Zhang, and D. Umstadter, “Quasi-monoenergetic and tunable X-rays from a laser-driven Compton light source,” Nature Photonics 8, 28–31 (2014).
[Crossref]

I. Ghebregziabher, B. A. Shadwick, and D. Umstadter, “Spectral bandwidth reduction of Thomson scattered light by pulse chirping,” Phys. Rev. Spec. Top. - Accel. Beams 16, 030705 (2013).
[Crossref]

Umstadter, D. P.

Varga, P.

P. Varga and P. Török, “Focusing of electromagnetic waves by paraboloid mirrors. I. Theory,” J. Opt. Soc. Am. A 17, 2081–2089 (2000).
[Crossref]

P. Varga and P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Comm. 152, 108–118 (1998).
[Crossref]

Ware, M.

Wolf, E.

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. 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. A 253, 358–379 (1959).
[Crossref]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Sec. 8.8.1.
[Crossref]

Yanovsky, V.

Yokota, M.

T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. A 2, 826–829 (1985).
[Crossref]

Yoshida, A.

A. Yoshida and T. Asakura, “Electromagnetic Field near the focus of Gaussian Beams,” Optik 41, 281–292 (1974).

Yu. Bychenkov, V.

K. I. Popov, V. Yu. Bychenkov, W. Rozmus, and R. D. Sydora, “Electron vacuum acceleration by a tightly focused laser pulse,” Phys. Plasmas 15, 013108 (2008).
[Crossref]

Yu. Bychenkov, Y.

S. G. Bochkarev and Y. Yu. Bychenkov, “Acceleration of electrons by tightly focused femtosecond laser pulses,” Quantum Electron. 37, 273–284 (2007).
[Crossref]

Zamfir, N.V.

N.V. Zamfir, “Nuclear Physics with 10PW laser beams at extreme light infrastructure–nuclear physics (ELI-NP),” Eur. Phys. J. Special Topics 223, 1221–1227 (2014).
[Crossref]

Zhang, J.

N. Powers, I. Ghebregziabher, G. Golovin, C. Liu, S. Chen, S. Banerjee, J. Zhang, and D. Umstadter, “Quasi-monoenergetic and tunable X-rays from a laser-driven Compton light source,” Nature Photonics 8, 28–31 (2014).
[Crossref]

Zubairy, M.

M. Scully and M. Zubairy, “Simple laser accelerator: Optics and particle dynamics,” Phys. Rev. A 152, 108–118 (1998).

Ann. Phys. (1)

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 30, 755–776 (1909).
[Crossref]

Appl. Opt. (2)

Appl. Phys. B (1)

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

Eur. Phys. J. Special Topics (1)

N.V. Zamfir, “Nuclear Physics with 10PW laser beams at extreme light infrastructure–nuclear physics (ELI-NP),” Eur. Phys. J. Special Topics 223, 1221–1227 (2014).
[Crossref]

IEEE J. Microwave Opt. Acoust. (1)

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

J. Appl. Phys. (1)

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[Crossref]

J. Comput. Phys. (1)

C.P. Ridgers, J.G. Kirk, R. Duclous, T.G. Blackburn, C.S. Brady, K. Bennette, T.D. Arbere, and A.R. Bell, “Modelling gamma-ray photon emission and pair production in high-intensity laser-matter interactions,” J. Comput. Phys. 260, 273–285 (2014).
[Crossref]

J. Mod. Opt. (1)

C. Sheppard and K. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41, 1495–1505 (1994).
[Crossref]

J. Opt. Soc. A (1)

T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. A 2, 826–829 (1985).
[Crossref]

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

Nature Photonics (2)

N. Powers, I. Ghebregziabher, G. Golovin, C. Liu, S. Chen, S. Banerjee, J. Zhang, and D. Umstadter, “Quasi-monoenergetic and tunable X-rays from a laser-driven Compton light source,” Nature Photonics 8, 28–31 (2014).
[Crossref]

G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space-time characterization of ultra-intense femtosecond laser beams,” Nature Photonics 10, 547–553 (2016).
[Crossref]

Opt. Comm. (3)

P. Varga and P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Comm. 152, 108–118 (1998).
[Crossref]

H.-C. Kim and Y. H. Lee, “Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation,” Opt. Comm. 169, 9–16 (1999).
[Crossref]

N. Bokor and N. Davidson, “4π Focusing with single paraboloid mirror,” Opt. Comm. 281, 5499–5503 (2008).
[Crossref]

Opt. Express (2)

Opt. Lett. (4)

Optik (1)

A. Yoshida and T. Asakura, “Electromagnetic Field near the focus of Gaussian Beams,” Optik 41, 281–292 (1974).

Phys. Plasmas (1)

K. I. Popov, V. Yu. Bychenkov, W. Rozmus, and R. D. Sydora, “Electron vacuum acceleration by a tightly focused laser pulse,” Phys. Plasmas 15, 013108 (2008).
[Crossref]

Phys. Rev. (1)

J. A. Stratton and L. J. Chu, “Diffraction Theory of Electromagnetic Waves,” Phys. Rev. 56, 99–107 (1939).
[Crossref]

Phys. Rev. A (5)

Y. I. Salamin, “Simple analytical derivation of the fields of an ultrashort tightly focused linearly polarized laser pulse,” Phys. Rev. A 92, 063818 (2015).
[Crossref]

M. Scully and M. Zubairy, “Simple laser accelerator: Optics and particle dynamics,” Phys. Rev. A 152, 108–118 (1998).

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. Gonoskov, A. Aiello, S. Heugel, and G. Leuchs, “Dipole pulse theory: Maximizing the field amplitude from 4π focused laser pulses,” Phys. Rev. A 86, 053836 (2012).
[Crossref]

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[Crossref]

Phys. Rev. E (2)

W. Erikson and S. Singh, “Polarization properties of Maxwell-Gaussian laser beams,” Phys. Rev. E 49, 5778–5786 (1994).
[Crossref]

B. Quesnel and P. Mora, “Theory and simulation of the interaction of ultraintense laser pulses with electrons in vacuum,” Phys. Rev. E 58, 3719–3732 (1998).
[Crossref]

Phys. Rev. Lett. (1)

Y. Salamin and C. Keitel, “Electron acceleration by a tightly focused laser beam,” Phys. Rev. Lett. 88, 095005 (2002).
[Crossref] [PubMed]

Phys. Rev. Spec. Top. - Accel. Beams (1)

I. Ghebregziabher, B. A. Shadwick, and D. Umstadter, “Spectral bandwidth reduction of Thomson scattered light by pulse chirping,” Phys. Rev. Spec. Top. - Accel. Beams 16, 030705 (2013).
[Crossref]

Phys.Rev.A (1)

M. Lax, W. Louisell, and W. McKnight,“From Maxwell to paraxial wave optics,” Phys.Rev.A 11, 1365–1370 (1975).

Proc. R. Soc. A (2)

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. 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. A 253, 358–379 (1959).
[Crossref]

Quantum Electron. (1)

S. G. Bochkarev and Y. Yu. Bychenkov, “Acceleration of electrons by tightly focused femtosecond laser pulses,” Quantum Electron. 37, 273–284 (2007).
[Crossref]

Science (1)

G. Mourou and T. Tajima, “More intense, shorter pulses,” Science 331, 41–42 (2011).
[Crossref] [PubMed]

Trans. Opt. Inst. Petrograd (3)

V. S. Ignatovsky, “The relationship between geometric and wave optics and diffraction of an azimuthally symmetric beam,” Trans. Opt. Inst. Petrograd 1, 3 (1920).

V. S. Ignatovsky, “Diffraction by a lens with arbitrary aperture,” Trans. Opt. Inst. Petrograd 1, 4 (1920).

V. S. Ignatovsky, “Diffraction by a parabolic mirror having arbitrary opening,” Trans. Opt. Inst. Petrograd 1, 5 (1920).

Other (9)

∫02πdϕ′cosnϕ′e±iacos(ϕ′−ϕ)=2π(±i)ncosnϕJn(a), ∫02πdϕ′sinnϕ′e±iacos(ϕ′−ϕ)=2π(±i)nsinnϕJn(a)

J. J. Stamns, Waves in Focal Regions, (IOP Publishing, 1986), Sec. 16.1.

R. K. Luneburg, Mathematical Theory of Optics, (University of California Press, 1966), Sec. 46.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Sec. 8.8.1.
[Crossref]

Note that Jn′(x)=12[Jn−1(x)−Jn+1(x)] and 2nJn (x) = x [Jn−1 (x) + Jn+1 (x)].

M. H. Holmes, Introduction to Perturbation Methods, 2nd Ed. (Springer, 2013).
[Crossref]

ru.wikipedia.org , search V. S. Ignatowsky; en.wikipedia.org , search V. S. Ignatowski.

∫−∞∞du eiau2=(1+i)π2a

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998), Eq. (10.87).

Supplementary Material (6)

NameDescription
» Visualization 1: MP4 (4309 KB)      Animation of the Ignatovsky model
» Visualization 2: MP4 (3751 KB)      Animation of the Singh model for a tight focus
» Visualization 3: MP4 (4524 KB)      Animation of the Davis model
» Visualization 4: MP4 (4788 KB)      Animation of the Barton model
» Visualization 5: MP4 (6338 KB)      Animation of the Quesnel model
» Visualization 6: MP4 (3917 KB)      Animation of the Singh model for a loose focus

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Figures (10)

Fig. 1
Fig. 1

Incident light (x-polarized) reflected by a parabolic mirror. s- and p-polarized components of the field reflect as plane waves about the local surface normal .

Fig. 2
Fig. 2

Field components for a Gaussian-envelope beam with Eenv (x′, y′) = E0eρ′2/w2, immediately after reflection from a parabolic mirror. The fields are computed for f/(2w) = 0.5 (NA = 0.8) on the surface of the mirror as a function of x′ and y′ (horizontal and vertical in the figure). The rapidly-varying phase factors in (6) and (7) are removed for clarity. Each box has dimensions 4f × 4f, where f is the focal length of the mirror.

Fig. 3
Fig. 3

(a) Transfer of fields onto a spherical surface prior to diffraction calculation. (b) Direct use of fields at parabolic surface in diffraction calculation.

Fig. 4
Fig. 4

Field components at z = 0 for the Gaussian beam from Fig. 2 focused with f/(2w) = 0.5 (NA = 0.8). The fields are computed in the plane z = 0. Each box has dimensions 4λ × 4λ, where λ = 2π/k, and E0 denotes the maximum value of Ex.

Fig. 5
Fig. 5

Animation showing field components for Gaussian beam focused with f/(2w) = 1 (NA = 0.47) computed with the Ignatovsky model (see Visualization 1). The dashed lines in the upper frames show the locations of line-outs that are plotted beneath. The ρ dimensions on the lower graphs correspond to the ρ dimensions of the square images above. The phase displayed by the colors in the images is relative to the phase of the x-component of the field: Ex = |Ex|ex. E0 denotes the maximum value of Ex at z = 0.

Fig. 6
Fig. 6

Animation showing the Singh model for the same parameters used in Fig. 5 ( Visualization 2). For comparison, the Ignatovsky result from Fig. 5 is also plotted in the lower frames.

Fig. 7
Fig. 7

Animation similar to Fig. 6 using the Davis model ( Visualization 3).

Fig. 8
Fig. 8

Animation similar to Fig. 6 using the Barton model ( Visualization 4).

Fig. 9
Fig. 9

Animation similar to Fig. 6 using the Quesnel model ( Visualization 5).

Fig. 10
Fig. 10

Animation of the Singh model similar to Fig. 6 using f/(2w) = 2 ( Visualization 6).

Equations (34)

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

z = f + ρ 2 4 f and ρ = x 2 + y 2 .
n ^ ( x , y ) = 1 2 f ( x x ^ + y y ^ ) + z ^ 1 + ρ 2 4 f 2 ,
E i ( x , y ) = x ^ E env ( x , y ) e i ( k ( z + f ) ω t ) .
k ^ ( x , y ) = 1 f ( x x ^ + y y ^ ) + ( 1 ρ 2 4 f 2 ) z ^ 1 + ρ 2 4 f 2 .
p ^ ( E ) ( x , y ) = ( 1 x 2 y 2 4 f 2 ) x ^ x y 2 f 2 y ^ + x f z ^ 1 + ρ 2 4 f 2
E surface ( r ) ( x , y ) = p ^ ( E ) ( x , y ) E env ( x , y ) e i ( k ρ 2 4 f ω t )
B surface ( r ) ( x , y ) = p ^ ( B ) ( x , y ) E env ( x , y ) c e i ( k ρ 2 4 f ω t )
p ^ ( B ) ( x , y ) = x y 2 f 2 x ^ + ( 1 + x 2 y 2 4 f 2 ) y ^ + y f z ^ 1 + ρ 2 4 f 2
p ^ ( E ) × p ^ ( B ) = k ^ , k ^ p ^ ( E ) = 0 , and k ^ p ^ ( B ) = 0 .
E ( x , y , z ) = S d A [ E ( r ) ( n ^ G ) G ( n ^ ) E ( r ) ]
R = | r r | = r 1 2 r r r 2 + r 2 r 2 r r r r ,
E ( x , y , z ) = i k e i ( k f ω t ) 2 π f d x d y 1 + ρ 2 4 f 2 E env ( x , y ) p ^ ( E ) ( x , y ) e i k k ^ ( x , y ) r
B ( x , y , z ) = i k e i ( k f ω t ) 2 π f d x d y 1 + ρ 2 4 f 2 E env ( x , y ) c p ^ ( B ) ( x , y ) e i k k ^ ( x , y ) r
d x d y 0 ρ d ρ 0 2 π d ϕ .
E ( x , y , z ) = i k f e i ( k f ω t ) [ x ^ ( I 0 + x 2 y 2 ρ 2 I 2 ) + y ^ 2 x y ρ 2 I 2 i z ^ x ρ I 1 ]
I 0 = 0 π d θ E e ( ρ ) J 0 ( k ρ sin θ ) sin θ e i k z cos θ I 1 = 2 0 π d θ E e ( ρ ) ξ ( θ ) J 1 ( k ρ sin θ ) sin θ e i k z cos θ I 2 = 0 π d θ E e ( ρ ) ξ ( θ ) J 2 ( k ρ sin θ ) sin θ e i k z cos θ
B ( x , y , z ) = i k f c e i ( k f ω t ) [ x ^ 2 x y ρ 2 I 2 + y ^ ( I 0 x 2 y 2 ρ 2 I 2 ) i z ^ y ρ I 1 ]
ψ ( 0 ) = z 0 exp ( k 2 ρ 2 ) ,
E Singh = E 0 [ x ^ + x y 2 2 y ^ i x z ^ ] ψ ( 0 ) e i ( k z ω t )
B Singh = E 0 c [ x y 2 2 x ^ + y ^ i y z ^ ] ψ ( 0 ) e i ( k z ω t )
( 2 x 2 + 2 y 2 + 2 i k z ) ψ revised = 2 ψ trial z 2
E Davis = E 0 [ ( 1 + x 2 2 k ρ 4 8 3 ) x ^ + x y 2 y ^ i ( x x k 2 + x ρ 2 3 k x ρ 4 8 4 ) z ^ ] ψ ( 0 ) e i ( k z ω t )
B Davis = E 0 c [ ( 1 + ρ 2 2 2 k ρ 4 8 3 ) y ^ i ( y + y k 2 + y ρ 2 2 3 k y ρ 4 8 4 ) z ^ ] ψ ( 0 ) e i ( k z ω t )
E ( x , y ) / c = x ^ B y ( y , x ) y ^ B x ( y , x ) + z ^ B z ( y , x ) c B ( x , y ) = x ^ E y ( y , x ) + y ^ E x ( y , x ) + z ^ E z ( y , x )
E Barton = E 0 [ ( 1 + ρ 2 + 2 x 2 4 2 k ρ 4 8 3 + α x ) x ^ + ( x y 2 2 + α y ) y ^ i ( x + 3 x ρ 2 4 3 k x ρ 4 8 4 + α z ) z ^ ] ψ ( 0 ) e i ( k z ω t )
B Barton = E 0 c [ ( x y 2 2 + β x ) x ^ + ( 1 + ρ 2 + 2 y 2 4 2 k ρ 4 8 3 + β y ) y ^ i ( y + 3 y ρ 2 4 3 k y ρ 4 8 4 + β z ) z ^ ] ψ ( 0 ) e i ( k z ω t )
α x = ρ 4 + 4 x 2 ρ 2 8 4 3 k ρ 6 + 2 k x 2 ρ 4 32 5 + k 2 ρ 8 128 6 α y = x y ρ 2 2 4 k x y ρ 6 16 5 α z = 5 x ρ 4 8 5 5 k x ρ 6 32 6 + k 2 x ρ 8 128 7 β x = α y β y = ρ 4 + 4 y 2 ρ 2 8 4 3 k ρ 6 + 2 k y 2 ρ 4 32 5 + k 2 ρ 8 128 6 β z = 5 y ρ 4 8 5 5 k y ρ 6 32 6 + k 2 y ρ 8 128 7
E Quesnel = E 0 e i ω t [ x ^ ( I ˜ 1 + x 2 y 2 k ρ 3 I ˜ 2 + y 2 ρ 2 I ˜ 3 ) + y ^ x y ρ 2 ( 2 k ρ I ˜ 2 I ˜ 3 ) i z ^ x ρ I ˜ 4 ]
B Quesnel = E 0 c e i ω t [ x ^ x y ρ 2 ( 2 k ρ I ˜ 2 I ˜ 3 ) + y ^ ( I ˜ 1 + x 2 y 2 k ρ 3 I ˜ 2 + x 2 ρ 2 I ˜ 3 ) i z ^ x ρ I ˜ 4 ]
I ˜ 1 = a 0 1 e a b 2 ( 1 + 1 b 2 ) e i k z 1 b 2 J 0 ( k ρ b ) b d b I ˜ 2 = a 0 1 e a b 2 1 1 b 2 e i k z 1 b 2 J 1 ( k ρ b ) b 2 d b I ˜ 3 = a 0 1 e a b 2 1 1 b 2 e i k z 1 b 2 J 0 ( k ρ b ) b 3 d b
I ˜ 4 = a 0 1 e a b 2 ( 1 + 1 1 b 2 ) e i k z 1 b 2 J 1 ( k ρ b ) b 2 d b
E mirror = i k e i ( k ρ 2 4 f ω t ) 2 π f d x d y 1 + ρ 2 4 f 2 E env ( x , y ) p ^ ( E ) ( x , y ) e i k 2 f ( x x ) 2 + ( y y ) 2 1 + ρ 2 4 f 2
E mirror e i ( k ρ 2 4 f ω t ) E env ( x , y ) p ^ ( E ) ( x , y ) [ i k 2 π f ( 1 + ρ 2 4 f 2 ) d x d y e i k 2 f ( x x ) 2 + ( y y ) 2 ( 1 + ρ 2 4 f 2 ) ]
Δ θ ~ λ / f

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