Abstract

The focusing property of a focal spot of a femtosecond laser pulse is presented under tight focusing conditions (below f-number of 1). The spatial and temporal intensity distributions of a focused electric field are calculated by vector diffraction integrals and coherent superposition method. The validity of the calculation method is examined by comparing the intensity distribution obtained under a high f-number condition to that obtained with the fast Fourier transform method that assumes the scalar paraxial approximation. The spatial and temporal modifications under tight focusing conditions are described for a focused femtosecond laser pulse. The calculation results show that a peak intensity of about 2.5x1024 W/cm2 can be achievable by tightly focusing a 12-fs, 10 PW laser pulse with a f/0.5 parabolic optic. The precise information on intensity distributions of a femtosecond focal spot obtained under a tight focusing condition will be crucial in assessing a focused intensity and in describing the motion of charged particles under an extremely strong electric field in ultra-relativistic and/or relativistic laser matter-interaction studies.

© 2015 Optical Society of America

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References

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2014 (1)

T. M. Jeong and J. Lee, “Femtosecond petawatt laser,” Ann. Phys. (Berlin) 526(3–4), 157–172 (2014).
[Crossref]

2013 (3)

N. D. Powers, I. Ghebregziabher, G. Golovin, C. Liu, S. Chen, S. Banerjee, J. Zhang, and D. P. Umstadter, “Quasi-monoenergetic and tunable X-rays from a laser-driven Compton light source,” Nat. Photonics 8(1), 28–31 (2013).
[Crossref]

I. J. Kim, K. H. Pae, C. M. Kim, H. T. Kim, J. H. Sung, S. K. Lee, T. J. Yu, I. W. Choi, C.-L. Lee, K. H. Nam, P. V. Nickles, T. M. Jeong, and J. Lee, “Transition of Proton Energy Scaling Using an Ultrathin Target Irradiated by Linearly Polarized Femtosecond Laser Pulses,” Phys. Rev. Lett. 111(16), 165003 (2013).
[Crossref] [PubMed]

H. T. Kim, K. H. Pae, H. J. Cha, I. J. Kim, T. J. Yu, J. H. Sung, S. K. Lee, T. M. Jeong, and J. Lee, “Enhancement of Electron Energy to the Multi-GeV Regime by a Dual-Stage Laser-Wakefield Accelerator Pumped by Petawatt Laser Pulses,” Phys. Rev. Lett. 111(16), 165002 (2013).
[Crossref] [PubMed]

2012 (4)

D. Margarone, O. Klimo, I. J. Kim, J. Prokůpek, J. Limpouch, T. M. Jeong, T. Mocek, J. Pšikal, H. T. Kim, J. Proška, K. H. Nam, L. Stolcová, I. W. Choi, S. K. Lee, J. H. Sung, T. J. Yu, and G. Korn, “Laser-Driven Proton Acceleration Enhancement by Nanostructured Foils,” Phys. Rev. Lett. 109(23), 234801 (2012).
[Crossref] [PubMed]

K. Ta Phuoc, S. Corde, C. Thaury, V. Malka, A. Tafzi, J. P. Goddet, R. C. Shah, S. Sebban, and A. Rousse, “All-optical Compton gamma-ray source,” Nat. Photonics 6(5), 308–311 (2012).
[Crossref]

J. Liu, J. Tan, T. Wilson, and C. Zhong, “Rigorous theory on elliptical mirror focusing for point scanning microscopy,” Opt. Express 20(6), 6175–6184 (2012).
[Crossref] [PubMed]

T. J. Yu, S. K. Lee, J. H. Sung, J. W. Yoon, T. M. Jeong, and J. Lee, “Generation of high-contrast, 30 fs, 1.5 PW laser pulses from chirped-pulse amplification Ti:sapphire laser,” Opt. Express 20(10), 10807–10815 (2012).
[Crossref] [PubMed]

2011 (1)

2010 (2)

2008 (1)

2006 (1)

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78(2), 309–371 (2006).
[Crossref]

2005 (2)

S.-W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Characterization of focal field formed by a large numerical aperture paraboloidal mirror and generation of ultra-high intensity (1022W/cm2),” Appl. Phys. B 80(7), 823–832 (2005).
[Crossref]

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

2002 (2)

Y. I. Salamin and C. H. Keitel, “Electron Acceleration by a Tightly Focused Laser Beam,” Phys. Rev. Lett. 88(9), 095005 (2002).
[Crossref] [PubMed]

Y. I. Salamin, G. R. Mocken, and C. H. Keitel, “Electron scattering and acceleration by a tightly focused laser beam,” Phys. Rev. 5(10), 101301 (2002).

2000 (2)

1939 (1)

J. Stratton and L. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56(1), 99–107 (1939).
[Crossref]

Audebert, P.

Bahk, S.-W.

S.-W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Characterization of focal field formed by a large numerical aperture paraboloidal mirror and generation of ultra-high intensity (1022W/cm2),” Appl. Phys. B 80(7), 823–832 (2005).
[Crossref]

Banerjee, S.

N. D. Powers, I. Ghebregziabher, G. Golovin, C. Liu, S. Chen, S. Banerjee, J. Zhang, and D. P. Umstadter, “Quasi-monoenergetic and tunable X-rays from a laser-driven Compton light source,” Nat. Photonics 8(1), 28–31 (2013).
[Crossref]

Buffechoux, S.

Bulanov, S. V.

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78(2), 309–371 (2006).
[Crossref]

Cha, H. J.

H. T. Kim, K. H. Pae, H. J. Cha, I. J. Kim, T. J. Yu, J. H. Sung, S. K. Lee, T. M. Jeong, and J. Lee, “Enhancement of Electron Energy to the Multi-GeV Regime by a Dual-Stage Laser-Wakefield Accelerator Pumped by Petawatt Laser Pulses,” Phys. Rev. Lett. 111(16), 165002 (2013).
[Crossref] [PubMed]

Chen, S.

N. D. Powers, I. Ghebregziabher, G. Golovin, C. Liu, S. Chen, S. Banerjee, J. Zhang, and D. P. Umstadter, “Quasi-monoenergetic and tunable X-rays from a laser-driven Compton light source,” Nat. Photonics 8(1), 28–31 (2013).
[Crossref]

Cheriaux, G.

Choi, I. W.

I. J. Kim, K. H. Pae, C. M. Kim, H. T. Kim, J. H. Sung, S. K. Lee, T. J. Yu, I. W. Choi, C.-L. Lee, K. H. Nam, P. V. Nickles, T. M. Jeong, and J. Lee, “Transition of Proton Energy Scaling Using an Ultrathin Target Irradiated by Linearly Polarized Femtosecond Laser Pulses,” Phys. Rev. Lett. 111(16), 165003 (2013).
[Crossref] [PubMed]

D. Margarone, O. Klimo, I. J. Kim, J. Prokůpek, J. Limpouch, T. M. Jeong, T. Mocek, J. Pšikal, H. T. Kim, J. Proška, K. H. Nam, L. Stolcová, I. W. Choi, S. K. Lee, J. H. Sung, T. J. Yu, and G. Korn, “Laser-Driven Proton Acceleration Enhancement by Nanostructured Foils,” Phys. Rev. Lett. 109(23), 234801 (2012).
[Crossref] [PubMed]

Chu, L.

J. Stratton and L. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56(1), 99–107 (1939).
[Crossref]

Chvykov, V.

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(3), 2109–2114 (2008).
[Crossref] [PubMed]

S.-W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Characterization of focal field formed by a large numerical aperture paraboloidal mirror and generation of ultra-high intensity (1022W/cm2),” Appl. Phys. B 80(7), 823–832 (2005).
[Crossref]

Corde, S.

K. Ta Phuoc, S. Corde, C. Thaury, V. Malka, A. Tafzi, J. P. Goddet, R. C. Shah, S. Sebban, and A. Rousse, “All-optical Compton gamma-ray source,” Nat. Photonics 6(5), 308–311 (2012).
[Crossref]

Fuchs, J.

Ghebregziabher, I.

N. D. Powers, I. Ghebregziabher, G. Golovin, C. Liu, S. Chen, S. Banerjee, J. Zhang, and D. P. Umstadter, “Quasi-monoenergetic and tunable X-rays from a laser-driven Compton light source,” Nat. Photonics 8(1), 28–31 (2013).
[Crossref]

Goddet, J. P.

K. Ta Phuoc, S. Corde, C. Thaury, V. Malka, A. Tafzi, J. P. Goddet, R. C. Shah, S. Sebban, and A. Rousse, “All-optical Compton gamma-ray source,” Nat. Photonics 6(5), 308–311 (2012).
[Crossref]

Golovin, G.

N. D. Powers, I. Ghebregziabher, G. Golovin, C. Liu, S. Chen, S. Banerjee, J. Zhang, and D. P. Umstadter, “Quasi-monoenergetic and tunable X-rays from a laser-driven Compton light source,” Nat. Photonics 8(1), 28–31 (2013).
[Crossref]

Jeong, T. M.

T. M. Jeong and J. Lee, “Femtosecond petawatt laser,” Ann. Phys. (Berlin) 526(3–4), 157–172 (2014).
[Crossref]

H. T. Kim, K. H. Pae, H. J. Cha, I. J. Kim, T. J. Yu, J. H. Sung, S. K. Lee, T. M. Jeong, and J. Lee, “Enhancement of Electron Energy to the Multi-GeV Regime by a Dual-Stage Laser-Wakefield Accelerator Pumped by Petawatt Laser Pulses,” Phys. Rev. Lett. 111(16), 165002 (2013).
[Crossref] [PubMed]

I. J. Kim, K. H. Pae, C. M. Kim, H. T. Kim, J. H. Sung, S. K. Lee, T. J. Yu, I. W. Choi, C.-L. Lee, K. H. Nam, P. V. Nickles, T. M. Jeong, and J. Lee, “Transition of Proton Energy Scaling Using an Ultrathin Target Irradiated by Linearly Polarized Femtosecond Laser Pulses,” Phys. Rev. Lett. 111(16), 165003 (2013).
[Crossref] [PubMed]

D. Margarone, O. Klimo, I. J. Kim, J. Prokůpek, J. Limpouch, T. M. Jeong, T. Mocek, J. Pšikal, H. T. Kim, J. Proška, K. H. Nam, L. Stolcová, I. W. Choi, S. K. Lee, J. H. Sung, T. J. Yu, and G. Korn, “Laser-Driven Proton Acceleration Enhancement by Nanostructured Foils,” Phys. Rev. Lett. 109(23), 234801 (2012).
[Crossref] [PubMed]

T. J. Yu, S. K. Lee, J. H. Sung, J. W. Yoon, T. M. Jeong, and J. Lee, “Generation of high-contrast, 30 fs, 1.5 PW laser pulses from chirped-pulse amplification Ti:sapphire laser,” Opt. Express 20(10), 10807–10815 (2012).
[Crossref] [PubMed]

J. H. Sung, S. K. Lee, T. J. Yu, T. M. Jeong, and J. Lee, “0.1 Hz 1.0 PW Ti:sapphire laser,” Opt. Lett. 35(18), 3021–3023 (2010).
[Crossref] [PubMed]

Kalinchenko, G.

Kalintchenko, G.

S.-W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Characterization of focal field formed by a large numerical aperture paraboloidal mirror and generation of ultra-high intensity (1022W/cm2),” Appl. Phys. B 80(7), 823–832 (2005).
[Crossref]

Keitel, C. H.

Y. I. Salamin, G. R. Mocken, and C. H. Keitel, “Electron scattering and acceleration by a tightly focused laser beam,” Phys. Rev. 5(10), 101301 (2002).

Y. I. Salamin and C. H. Keitel, “Electron Acceleration by a Tightly Focused Laser Beam,” Phys. Rev. Lett. 88(9), 095005 (2002).
[Crossref] [PubMed]

Kim, C. M.

I. J. Kim, K. H. Pae, C. M. Kim, H. T. Kim, J. H. Sung, S. K. Lee, T. J. Yu, I. W. Choi, C.-L. Lee, K. H. Nam, P. V. Nickles, T. M. Jeong, and J. Lee, “Transition of Proton Energy Scaling Using an Ultrathin Target Irradiated by Linearly Polarized Femtosecond Laser Pulses,” Phys. Rev. Lett. 111(16), 165003 (2013).
[Crossref] [PubMed]

Kim, H. T.

I. J. Kim, K. H. Pae, C. M. Kim, H. T. Kim, J. H. Sung, S. K. Lee, T. J. Yu, I. W. Choi, C.-L. Lee, K. H. Nam, P. V. Nickles, T. M. Jeong, and J. Lee, “Transition of Proton Energy Scaling Using an Ultrathin Target Irradiated by Linearly Polarized Femtosecond Laser Pulses,” Phys. Rev. Lett. 111(16), 165003 (2013).
[Crossref] [PubMed]

H. T. Kim, K. H. Pae, H. J. Cha, I. J. Kim, T. J. Yu, J. H. Sung, S. K. Lee, T. M. Jeong, and J. Lee, “Enhancement of Electron Energy to the Multi-GeV Regime by a Dual-Stage Laser-Wakefield Accelerator Pumped by Petawatt Laser Pulses,” Phys. Rev. Lett. 111(16), 165002 (2013).
[Crossref] [PubMed]

D. Margarone, O. Klimo, I. J. Kim, J. Prokůpek, J. Limpouch, T. M. Jeong, T. Mocek, J. Pšikal, H. T. Kim, J. Proška, K. H. Nam, L. Stolcová, I. W. Choi, S. K. Lee, J. H. Sung, T. J. Yu, and G. Korn, “Laser-Driven Proton Acceleration Enhancement by Nanostructured Foils,” Phys. Rev. Lett. 109(23), 234801 (2012).
[Crossref] [PubMed]

Kim, I. J.

H. T. Kim, K. H. Pae, H. J. Cha, I. J. Kim, T. J. Yu, J. H. Sung, S. K. Lee, T. M. Jeong, and J. Lee, “Enhancement of Electron Energy to the Multi-GeV Regime by a Dual-Stage Laser-Wakefield Accelerator Pumped by Petawatt Laser Pulses,” Phys. Rev. Lett. 111(16), 165002 (2013).
[Crossref] [PubMed]

I. J. Kim, K. H. Pae, C. M. Kim, H. T. Kim, J. H. Sung, S. K. Lee, T. J. Yu, I. W. Choi, C.-L. Lee, K. H. Nam, P. V. Nickles, T. M. Jeong, and J. Lee, “Transition of Proton Energy Scaling Using an Ultrathin Target Irradiated by Linearly Polarized Femtosecond Laser Pulses,” Phys. Rev. Lett. 111(16), 165003 (2013).
[Crossref] [PubMed]

D. Margarone, O. Klimo, I. J. Kim, J. Prokůpek, J. Limpouch, T. M. Jeong, T. Mocek, J. Pšikal, H. T. Kim, J. Proška, K. H. Nam, L. Stolcová, I. W. Choi, S. K. Lee, J. H. Sung, T. J. Yu, and G. Korn, “Laser-Driven Proton Acceleration Enhancement by Nanostructured Foils,” Phys. Rev. Lett. 109(23), 234801 (2012).
[Crossref] [PubMed]

Klimo, O.

D. Margarone, O. Klimo, I. J. Kim, J. Prokůpek, J. Limpouch, T. M. Jeong, T. Mocek, J. Pšikal, H. T. Kim, J. Proška, K. H. Nam, L. Stolcová, I. W. Choi, S. K. Lee, J. H. Sung, T. J. Yu, and G. Korn, “Laser-Driven Proton Acceleration Enhancement by Nanostructured Foils,” Phys. Rev. Lett. 109(23), 234801 (2012).
[Crossref] [PubMed]

Kodama, R.

Kon, A.

Korn, G.

D. Margarone, O. Klimo, I. J. Kim, J. Prokůpek, J. Limpouch, T. M. Jeong, T. Mocek, J. Pšikal, H. T. Kim, J. Proška, K. H. Nam, L. Stolcová, I. W. Choi, S. K. Lee, J. H. Sung, T. J. Yu, and G. Korn, “Laser-Driven Proton Acceleration Enhancement by Nanostructured Foils,” Phys. Rev. Lett. 109(23), 234801 (2012).
[Crossref] [PubMed]

Krushelnick, K.

Lee, C.-L.

I. J. Kim, K. H. Pae, C. M. Kim, H. T. Kim, J. H. Sung, S. K. Lee, T. J. Yu, I. W. Choi, C.-L. Lee, K. H. Nam, P. V. Nickles, T. M. Jeong, and J. Lee, “Transition of Proton Energy Scaling Using an Ultrathin Target Irradiated by Linearly Polarized Femtosecond Laser Pulses,” Phys. Rev. Lett. 111(16), 165003 (2013).
[Crossref] [PubMed]

Lee, J.

T. M. Jeong and J. Lee, “Femtosecond petawatt laser,” Ann. Phys. (Berlin) 526(3–4), 157–172 (2014).
[Crossref]

I. J. Kim, K. H. Pae, C. M. Kim, H. T. Kim, J. H. Sung, S. K. Lee, T. J. Yu, I. W. Choi, C.-L. Lee, K. H. Nam, P. V. Nickles, T. M. Jeong, and J. Lee, “Transition of Proton Energy Scaling Using an Ultrathin Target Irradiated by Linearly Polarized Femtosecond Laser Pulses,” Phys. Rev. Lett. 111(16), 165003 (2013).
[Crossref] [PubMed]

H. T. Kim, K. H. Pae, H. J. Cha, I. J. Kim, T. J. Yu, J. H. Sung, S. K. Lee, T. M. Jeong, and J. Lee, “Enhancement of Electron Energy to the Multi-GeV Regime by a Dual-Stage Laser-Wakefield Accelerator Pumped by Petawatt Laser Pulses,” Phys. Rev. Lett. 111(16), 165002 (2013).
[Crossref] [PubMed]

T. J. Yu, S. K. Lee, J. H. Sung, J. W. Yoon, T. M. Jeong, and J. Lee, “Generation of high-contrast, 30 fs, 1.5 PW laser pulses from chirped-pulse amplification Ti:sapphire laser,” Opt. Express 20(10), 10807–10815 (2012).
[Crossref] [PubMed]

J. H. Sung, S. K. Lee, T. J. Yu, T. M. Jeong, and J. Lee, “0.1 Hz 1.0 PW Ti:sapphire laser,” Opt. Lett. 35(18), 3021–3023 (2010).
[Crossref] [PubMed]

Lee, S. K.

H. T. Kim, K. H. Pae, H. J. Cha, I. J. Kim, T. J. Yu, J. H. Sung, S. K. Lee, T. M. Jeong, and J. Lee, “Enhancement of Electron Energy to the Multi-GeV Regime by a Dual-Stage Laser-Wakefield Accelerator Pumped by Petawatt Laser Pulses,” Phys. Rev. Lett. 111(16), 165002 (2013).
[Crossref] [PubMed]

I. J. Kim, K. H. Pae, C. M. Kim, H. T. Kim, J. H. Sung, S. K. Lee, T. J. Yu, I. W. Choi, C.-L. Lee, K. H. Nam, P. V. Nickles, T. M. Jeong, and J. Lee, “Transition of Proton Energy Scaling Using an Ultrathin Target Irradiated by Linearly Polarized Femtosecond Laser Pulses,” Phys. Rev. Lett. 111(16), 165003 (2013).
[Crossref] [PubMed]

D. Margarone, O. Klimo, I. J. Kim, J. Prokůpek, J. Limpouch, T. M. Jeong, T. Mocek, J. Pšikal, H. T. Kim, J. Proška, K. H. Nam, L. Stolcová, I. W. Choi, S. K. Lee, J. H. Sung, T. J. Yu, and G. Korn, “Laser-Driven Proton Acceleration Enhancement by Nanostructured Foils,” Phys. Rev. Lett. 109(23), 234801 (2012).
[Crossref] [PubMed]

T. J. Yu, S. K. Lee, J. H. Sung, J. W. Yoon, T. M. Jeong, and J. Lee, “Generation of high-contrast, 30 fs, 1.5 PW laser pulses from chirped-pulse amplification Ti:sapphire laser,” Opt. Express 20(10), 10807–10815 (2012).
[Crossref] [PubMed]

J. H. Sung, S. K. Lee, T. J. Yu, T. M. Jeong, and J. Lee, “0.1 Hz 1.0 PW Ti:sapphire laser,” Opt. Lett. 35(18), 3021–3023 (2010).
[Crossref] [PubMed]

Limpouch, J.

D. Margarone, O. Klimo, I. J. Kim, J. Prokůpek, J. Limpouch, T. M. Jeong, T. Mocek, J. Pšikal, H. T. Kim, J. Proška, K. H. Nam, L. Stolcová, I. W. Choi, S. K. Lee, J. H. Sung, T. J. Yu, and G. Korn, “Laser-Driven Proton Acceleration Enhancement by Nanostructured Foils,” Phys. Rev. Lett. 109(23), 234801 (2012).
[Crossref] [PubMed]

Liu, C.

N. D. Powers, I. Ghebregziabher, G. Golovin, C. Liu, S. Chen, S. Banerjee, J. Zhang, and D. P. Umstadter, “Quasi-monoenergetic and tunable X-rays from a laser-driven Compton light source,” Nat. Photonics 8(1), 28–31 (2013).
[Crossref]

Z. Wang, C. Liu, Z. Shen, Q. Zhang, H. Teng, and Z. Wei, “High-contrast 116 PW Ti:sapphire laser system combined with a doubled chirped-pulse amplification scheme and a femtosecond optical-parametric amplifier,” Opt. Lett. 36(16), 3194–3196 (2011).
[Crossref] [PubMed]

Liu, J.

Maksimchuk, A.

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(3), 2109–2114 (2008).
[Crossref] [PubMed]

S.-W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Characterization of focal field formed by a large numerical aperture paraboloidal mirror and generation of ultra-high intensity (1022W/cm2),” Appl. Phys. B 80(7), 823–832 (2005).
[Crossref]

Malka, V.

K. Ta Phuoc, S. Corde, C. Thaury, V. Malka, A. Tafzi, J. P. Goddet, R. C. Shah, S. Sebban, and A. Rousse, “All-optical Compton gamma-ray source,” Nat. Photonics 6(5), 308–311 (2012).
[Crossref]

Margarone, D.

D. Margarone, O. Klimo, I. J. Kim, J. Prokůpek, J. Limpouch, T. M. Jeong, T. Mocek, J. Pšikal, H. T. Kim, J. Proška, K. H. Nam, L. Stolcová, I. W. Choi, S. K. Lee, J. H. Sung, T. J. Yu, and G. Korn, “Laser-Driven Proton Acceleration Enhancement by Nanostructured Foils,” Phys. Rev. Lett. 109(23), 234801 (2012).
[Crossref] [PubMed]

Matsuoka, T.

Mocek, T.

D. Margarone, O. Klimo, I. J. Kim, J. Prokůpek, J. Limpouch, T. M. Jeong, T. Mocek, J. Pšikal, H. T. Kim, J. Proška, K. H. Nam, L. Stolcová, I. W. Choi, S. K. Lee, J. H. Sung, T. J. Yu, and G. Korn, “Laser-Driven Proton Acceleration Enhancement by Nanostructured Foils,” Phys. Rev. Lett. 109(23), 234801 (2012).
[Crossref] [PubMed]

Mocken, G. R.

Y. I. Salamin, G. R. Mocken, and C. H. Keitel, “Electron scattering and acceleration by a tightly focused laser beam,” Phys. Rev. 5(10), 101301 (2002).

Mourou, G.

Mourou, G. A.

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78(2), 309–371 (2006).
[Crossref]

S.-W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Characterization of focal field formed by a large numerical aperture paraboloidal mirror and generation of ultra-high intensity (1022W/cm2),” Appl. Phys. B 80(7), 823–832 (2005).
[Crossref]

Nakatsutsumi, M.

Nam, K. H.

I. J. Kim, K. H. Pae, C. M. Kim, H. T. Kim, J. H. Sung, S. K. Lee, T. J. Yu, I. W. Choi, C.-L. Lee, K. H. Nam, P. V. Nickles, T. M. Jeong, and J. Lee, “Transition of Proton Energy Scaling Using an Ultrathin Target Irradiated by Linearly Polarized Femtosecond Laser Pulses,” Phys. Rev. Lett. 111(16), 165003 (2013).
[Crossref] [PubMed]

D. Margarone, O. Klimo, I. J. Kim, J. Prokůpek, J. Limpouch, T. M. Jeong, T. Mocek, J. Pšikal, H. T. Kim, J. Proška, K. H. Nam, L. Stolcová, I. W. Choi, S. K. Lee, J. H. Sung, T. J. Yu, and G. Korn, “Laser-Driven Proton Acceleration Enhancement by Nanostructured Foils,” Phys. Rev. Lett. 109(23), 234801 (2012).
[Crossref] [PubMed]

Nees, J.

Nickles, P. V.

I. J. Kim, K. H. Pae, C. M. Kim, H. T. Kim, J. H. Sung, S. K. Lee, T. J. Yu, I. W. Choi, C.-L. Lee, K. H. Nam, P. V. Nickles, T. M. Jeong, and J. Lee, “Transition of Proton Energy Scaling Using an Ultrathin Target Irradiated by Linearly Polarized Femtosecond Laser Pulses,” Phys. Rev. Lett. 111(16), 165003 (2013).
[Crossref] [PubMed]

Pae, K. H.

I. J. Kim, K. H. Pae, C. M. Kim, H. T. Kim, J. H. Sung, S. K. Lee, T. J. Yu, I. W. Choi, C.-L. Lee, K. H. Nam, P. V. Nickles, T. M. Jeong, and J. Lee, “Transition of Proton Energy Scaling Using an Ultrathin Target Irradiated by Linearly Polarized Femtosecond Laser Pulses,” Phys. Rev. Lett. 111(16), 165003 (2013).
[Crossref] [PubMed]

H. T. Kim, K. H. Pae, H. J. Cha, I. J. Kim, T. J. Yu, J. H. Sung, S. K. Lee, T. M. Jeong, and J. Lee, “Enhancement of Electron Energy to the Multi-GeV Regime by a Dual-Stage Laser-Wakefield Accelerator Pumped by Petawatt Laser Pulses,” Phys. Rev. Lett. 111(16), 165002 (2013).
[Crossref] [PubMed]

Piché, M.

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

Planchon, T.

Planchon, T. A.

S.-W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Characterization of focal field formed by a large numerical aperture paraboloidal mirror and generation of ultra-high intensity (1022W/cm2),” Appl. Phys. B 80(7), 823–832 (2005).
[Crossref]

Porras, M. A.

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

Powers, N. D.

N. D. Powers, I. Ghebregziabher, G. Golovin, C. Liu, S. Chen, S. Banerjee, J. Zhang, and D. P. Umstadter, “Quasi-monoenergetic and tunable X-rays from a laser-driven Compton light source,” Nat. Photonics 8(1), 28–31 (2013).
[Crossref]

Prokupek, J.

D. Margarone, O. Klimo, I. J. Kim, J. Prokůpek, J. Limpouch, T. M. Jeong, T. Mocek, J. Pšikal, H. T. Kim, J. Proška, K. H. Nam, L. Stolcová, I. W. Choi, S. K. Lee, J. H. Sung, T. J. Yu, and G. Korn, “Laser-Driven Proton Acceleration Enhancement by Nanostructured Foils,” Phys. Rev. Lett. 109(23), 234801 (2012).
[Crossref] [PubMed]

Proška, J.

D. Margarone, O. Klimo, I. J. Kim, J. Prokůpek, J. Limpouch, T. M. Jeong, T. Mocek, J. Pšikal, H. T. Kim, J. Proška, K. H. Nam, L. Stolcová, I. W. Choi, S. K. Lee, J. H. Sung, T. J. Yu, and G. Korn, “Laser-Driven Proton Acceleration Enhancement by Nanostructured Foils,” Phys. Rev. Lett. 109(23), 234801 (2012).
[Crossref] [PubMed]

Pšikal, J.

D. Margarone, O. Klimo, I. J. Kim, J. Prokůpek, J. Limpouch, T. M. Jeong, T. Mocek, J. Pšikal, H. T. Kim, J. Proška, K. H. Nam, L. Stolcová, I. W. Choi, S. K. Lee, J. H. Sung, T. J. Yu, and G. Korn, “Laser-Driven Proton Acceleration Enhancement by Nanostructured Foils,” Phys. Rev. Lett. 109(23), 234801 (2012).
[Crossref] [PubMed]

Rousse, A.

K. Ta Phuoc, S. Corde, C. Thaury, V. Malka, A. Tafzi, J. P. Goddet, R. C. Shah, S. Sebban, and A. Rousse, “All-optical Compton gamma-ray source,” Nat. Photonics 6(5), 308–311 (2012).
[Crossref]

Rousseau, P.

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(3), 2109–2114 (2008).
[Crossref] [PubMed]

S.-W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Characterization of focal field formed by a large numerical aperture paraboloidal mirror and generation of ultra-high intensity (1022W/cm2),” Appl. Phys. B 80(7), 823–832 (2005).
[Crossref]

Salamin, Y. I.

Y. I. Salamin and C. H. Keitel, “Electron Acceleration by a Tightly Focused Laser Beam,” Phys. Rev. Lett. 88(9), 095005 (2002).
[Crossref] [PubMed]

Y. I. Salamin, G. R. Mocken, and C. H. Keitel, “Electron scattering and acceleration by a tightly focused laser beam,” Phys. Rev. 5(10), 101301 (2002).

Sebban, S.

K. Ta Phuoc, S. Corde, C. Thaury, V. Malka, A. Tafzi, J. P. Goddet, R. C. Shah, S. Sebban, and A. Rousse, “All-optical Compton gamma-ray source,” Nat. Photonics 6(5), 308–311 (2012).
[Crossref]

Shah, R. C.

K. Ta Phuoc, S. Corde, C. Thaury, V. Malka, A. Tafzi, J. P. Goddet, R. C. Shah, S. Sebban, and A. Rousse, “All-optical Compton gamma-ray source,” Nat. Photonics 6(5), 308–311 (2012).
[Crossref]

Shen, Z.

Stolcová, L.

D. Margarone, O. Klimo, I. J. Kim, J. Prokůpek, J. Limpouch, T. M. Jeong, T. Mocek, J. Pšikal, H. T. Kim, J. Proška, K. H. Nam, L. Stolcová, I. W. Choi, S. K. Lee, J. H. Sung, T. J. Yu, and G. Korn, “Laser-Driven Proton Acceleration Enhancement by Nanostructured Foils,” Phys. Rev. Lett. 109(23), 234801 (2012).
[Crossref] [PubMed]

Stratton, J.

J. Stratton and L. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56(1), 99–107 (1939).
[Crossref]

Sung, J. H.

H. T. Kim, K. H. Pae, H. J. Cha, I. J. Kim, T. J. Yu, J. H. Sung, S. K. Lee, T. M. Jeong, and J. Lee, “Enhancement of Electron Energy to the Multi-GeV Regime by a Dual-Stage Laser-Wakefield Accelerator Pumped by Petawatt Laser Pulses,” Phys. Rev. Lett. 111(16), 165002 (2013).
[Crossref] [PubMed]

I. J. Kim, K. H. Pae, C. M. Kim, H. T. Kim, J. H. Sung, S. K. Lee, T. J. Yu, I. W. Choi, C.-L. Lee, K. H. Nam, P. V. Nickles, T. M. Jeong, and J. Lee, “Transition of Proton Energy Scaling Using an Ultrathin Target Irradiated by Linearly Polarized Femtosecond Laser Pulses,” Phys. Rev. Lett. 111(16), 165003 (2013).
[Crossref] [PubMed]

D. Margarone, O. Klimo, I. J. Kim, J. Prokůpek, J. Limpouch, T. M. Jeong, T. Mocek, J. Pšikal, H. T. Kim, J. Proška, K. H. Nam, L. Stolcová, I. W. Choi, S. K. Lee, J. H. Sung, T. J. Yu, and G. Korn, “Laser-Driven Proton Acceleration Enhancement by Nanostructured Foils,” Phys. Rev. Lett. 109(23), 234801 (2012).
[Crossref] [PubMed]

T. J. Yu, S. K. Lee, J. H. Sung, J. W. Yoon, T. M. Jeong, and J. Lee, “Generation of high-contrast, 30 fs, 1.5 PW laser pulses from chirped-pulse amplification Ti:sapphire laser,” Opt. Express 20(10), 10807–10815 (2012).
[Crossref] [PubMed]

J. H. Sung, S. K. Lee, T. J. Yu, T. M. Jeong, and J. Lee, “0.1 Hz 1.0 PW Ti:sapphire laser,” Opt. Lett. 35(18), 3021–3023 (2010).
[Crossref] [PubMed]

Ta Phuoc, K.

K. Ta Phuoc, S. Corde, C. Thaury, V. Malka, A. Tafzi, J. P. Goddet, R. C. Shah, S. Sebban, and A. Rousse, “All-optical Compton gamma-ray source,” Nat. Photonics 6(5), 308–311 (2012).
[Crossref]

Tafzi, A.

K. Ta Phuoc, S. Corde, C. Thaury, V. Malka, A. Tafzi, J. P. Goddet, R. C. Shah, S. Sebban, and A. Rousse, “All-optical Compton gamma-ray source,” Nat. Photonics 6(5), 308–311 (2012).
[Crossref]

Tajima, T.

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78(2), 309–371 (2006).
[Crossref]

Tan, J.

Teng, H.

Thaury, C.

K. Ta Phuoc, S. Corde, C. Thaury, V. Malka, A. Tafzi, J. P. Goddet, R. C. Shah, S. Sebban, and A. Rousse, “All-optical Compton gamma-ray source,” Nat. Photonics 6(5), 308–311 (2012).
[Crossref]

Török, P.

Umstadter, D. P.

N. D. Powers, I. Ghebregziabher, G. Golovin, C. Liu, S. Chen, S. Banerjee, J. Zhang, and D. P. Umstadter, “Quasi-monoenergetic and tunable X-rays from a laser-driven Compton light source,” Nat. Photonics 8(1), 28–31 (2013).
[Crossref]

Varga, P.

Varin, C.

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

Wang, Z.

Wei, Z.

Wilson, T.

Yanovsky, V.

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(3), 2109–2114 (2008).
[Crossref] [PubMed]

S.-W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Characterization of focal field formed by a large numerical aperture paraboloidal mirror and generation of ultra-high intensity (1022W/cm2),” Appl. Phys. B 80(7), 823–832 (2005).
[Crossref]

Yoon, J. W.

Yu, T. J.

H. T. Kim, K. H. Pae, H. J. Cha, I. J. Kim, T. J. Yu, J. H. Sung, S. K. Lee, T. M. Jeong, and J. Lee, “Enhancement of Electron Energy to the Multi-GeV Regime by a Dual-Stage Laser-Wakefield Accelerator Pumped by Petawatt Laser Pulses,” Phys. Rev. Lett. 111(16), 165002 (2013).
[Crossref] [PubMed]

I. J. Kim, K. H. Pae, C. M. Kim, H. T. Kim, J. H. Sung, S. K. Lee, T. J. Yu, I. W. Choi, C.-L. Lee, K. H. Nam, P. V. Nickles, T. M. Jeong, and J. Lee, “Transition of Proton Energy Scaling Using an Ultrathin Target Irradiated by Linearly Polarized Femtosecond Laser Pulses,” Phys. Rev. Lett. 111(16), 165003 (2013).
[Crossref] [PubMed]

D. Margarone, O. Klimo, I. J. Kim, J. Prokůpek, J. Limpouch, T. M. Jeong, T. Mocek, J. Pšikal, H. T. Kim, J. Proška, K. H. Nam, L. Stolcová, I. W. Choi, S. K. Lee, J. H. Sung, T. J. Yu, and G. Korn, “Laser-Driven Proton Acceleration Enhancement by Nanostructured Foils,” Phys. Rev. Lett. 109(23), 234801 (2012).
[Crossref] [PubMed]

T. J. Yu, S. K. Lee, J. H. Sung, J. W. Yoon, T. M. Jeong, and J. Lee, “Generation of high-contrast, 30 fs, 1.5 PW laser pulses from chirped-pulse amplification Ti:sapphire laser,” Opt. Express 20(10), 10807–10815 (2012).
[Crossref] [PubMed]

J. H. Sung, S. K. Lee, T. J. Yu, T. M. Jeong, and J. Lee, “0.1 Hz 1.0 PW Ti:sapphire laser,” Opt. Lett. 35(18), 3021–3023 (2010).
[Crossref] [PubMed]

Zhang, J.

N. D. Powers, I. Ghebregziabher, G. Golovin, C. Liu, S. Chen, S. Banerjee, J. Zhang, and D. P. Umstadter, “Quasi-monoenergetic and tunable X-rays from a laser-driven Compton light source,” Nat. Photonics 8(1), 28–31 (2013).
[Crossref]

Zhang, Q.

Zhong, C.

Ann. Phys. (Berlin) (1)

T. M. Jeong and J. Lee, “Femtosecond petawatt laser,” Ann. Phys. (Berlin) 526(3–4), 157–172 (2014).
[Crossref]

Appl. Phys. B (1)

S.-W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Characterization of focal field formed by a large numerical aperture paraboloidal mirror and generation of ultra-high intensity (1022W/cm2),” Appl. Phys. B 80(7), 823–832 (2005).
[Crossref]

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

Nat. Photonics (2)

N. D. Powers, I. Ghebregziabher, G. Golovin, C. Liu, S. Chen, S. Banerjee, J. Zhang, and D. P. Umstadter, “Quasi-monoenergetic and tunable X-rays from a laser-driven Compton light source,” Nat. Photonics 8(1), 28–31 (2013).
[Crossref]

K. Ta Phuoc, S. Corde, C. Thaury, V. Malka, A. Tafzi, J. P. Goddet, R. C. Shah, S. Sebban, and A. Rousse, “All-optical Compton gamma-ray source,” Nat. Photonics 6(5), 308–311 (2012).
[Crossref]

Opt. Express (3)

Opt. Lett. (3)

Phys. Rev. (2)

Y. I. Salamin, G. R. Mocken, and C. H. Keitel, “Electron scattering and acceleration by a tightly focused laser beam,” Phys. Rev. 5(10), 101301 (2002).

J. Stratton and L. Chu, “Diffraction theory of electromagnetic waves,” Phys. Rev. 56(1), 99–107 (1939).
[Crossref]

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

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

Phys. Rev. Lett. (4)

Y. I. Salamin and C. H. Keitel, “Electron Acceleration by a Tightly Focused Laser Beam,” Phys. Rev. Lett. 88(9), 095005 (2002).
[Crossref] [PubMed]

I. J. Kim, K. H. Pae, C. M. Kim, H. T. Kim, J. H. Sung, S. K. Lee, T. J. Yu, I. W. Choi, C.-L. Lee, K. H. Nam, P. V. Nickles, T. M. Jeong, and J. Lee, “Transition of Proton Energy Scaling Using an Ultrathin Target Irradiated by Linearly Polarized Femtosecond Laser Pulses,” Phys. Rev. Lett. 111(16), 165003 (2013).
[Crossref] [PubMed]

H. T. Kim, K. H. Pae, H. J. Cha, I. J. Kim, T. J. Yu, J. H. Sung, S. K. Lee, T. M. Jeong, and J. Lee, “Enhancement of Electron Energy to the Multi-GeV Regime by a Dual-Stage Laser-Wakefield Accelerator Pumped by Petawatt Laser Pulses,” Phys. Rev. Lett. 111(16), 165002 (2013).
[Crossref] [PubMed]

D. Margarone, O. Klimo, I. J. Kim, J. Prokůpek, J. Limpouch, T. M. Jeong, T. Mocek, J. Pšikal, H. T. Kim, J. Proška, K. H. Nam, L. Stolcová, I. W. Choi, S. K. Lee, J. H. Sung, T. J. Yu, and G. Korn, “Laser-Driven Proton Acceleration Enhancement by Nanostructured Foils,” Phys. Rev. Lett. 109(23), 234801 (2012).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78(2), 309–371 (2006).
[Crossref]

Other (3)

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

L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge university press, 1995) chap. 4, p. 149.

http://www.extreme-light-infrastructure.eu/

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Figures (9)

Fig. 1
Fig. 1 On-axis focusing scheme for an aberrated laser beam with a low f-number parabolic mirror.
Fig. 2
Fig. 2 Expressions for electric and magnetic fields in the tight imaging scheme with ellipsoidal mirror.
Fig. 3
Fig. 3 Line plots of intensity distributions calculated with fast FT (red line) and vectorial diffraction integral (blue line). An x-polarized EM field is used as an input and | E x ( x,y,z ) | 2 are plotted in the positive x-axis.
Fig. 4
Fig. 4 Intensity distributions for an x-polarized laser beam (a) with and (b) without a wavefront aberration under various focusing conditions. In the figure, the horizontal axis is the x-axis and the vertical axis is the y-axis.
Fig. 5
Fig. 5 Intensity distributions with tight imaging scheme using a parabolic and ellipsoidal mirrors. The first row represents intensities obtained by focusing an aberrated EM field with a f/3 parabolic mirror. The second row represents intensities on the ellipsoidal mirror. The third and fourth rows represent intensities imaged with ellipsoidal mirrors having a different length of the minor axis. In the figure, the horizontal axis is the x-axis and the vertical axis is the y-axis.
Fig. 6
Fig. 6 (a) Spectrum of a femtosecond laser pulse, (b) Spectral phase induced by a broadband mirror, (c) Temporal pulse profile calculated through the fast FT with spectrum and spectral phase.
Fig. 7
Fig. 7 (a) Spatial intensity distributions and (b) temporal intensity distributions. In Fig. 7(a), line plots of spatial intensity distributions of focal spots for a monochromatic and a coherent polychromatic (broadband) electric field are shown under f/3 focusing optic. In Fig. 7(b), line plots of temporal intensity distributions of focal spots for a broadband electric field are shown for the cases calculated with fast FT and diffraction integrals under f/3 and f/0.5.
Fig. 8
Fig. 8 (a) Phase difference between electric fields coming from the on-axis and an off-axis position on a parabolic mirror. Intensity distributions for a monochromatic 800-nm electric field focused by (b) f/3 optic and (c) f/0.5. The phase difference shorten The Rangleigh range and affects the pulse duration along z-axis at a moment.
Fig. 9
Fig. 9 Spatio-temporal intensity distributions of a focal spot focused with (a) a f/3 parabolic mirror and (b) a f/0.5 parabolic mirror. The intensity distribution in the XY plane means the spatial intensity distribution of the focal spot and the intensity distribution in the XZ plane means the temporal distribution of the focal spot. +/− 6 μm in the z-axis corresponds to +/− 20 fs in time. Under tight focusing condition, electric fields of a focal spot are compressed in both spatial and temporal domains. The peak intensities are described in figures.

Equations (17)

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E x ( x P , y P , z P ) θm π 0 2π d θ S d ϕ S E inc ( θ S , ϕ S )exp{ ikφ( x P , y P , z P , θ S , ϕ S ) } 2fsin θ S ( 1cos θ S ) , ×{ 1 sin θ S cos ϕ S 1cos θ S ( 1 1cos θ S i2kf ) 2fsin θ S cos ϕ S x P ( 1cos θ S ) 2f }
E y ( x P , y P , z P ) θm π 0 2π d θ S d ϕ S E inc ( θ S , ϕ S )exp{ iφ( x P , y P , z P , θ S , ϕ S ) } 2fsin θ S ( 1cos θ s ) 2 , ×{ sin θ S cos ϕ S ( 1 1cos θ S i2kf ) 2fsin θ S sin ϕ S y P ( 1cos θ S ) 2f }
E z ( x P , y P , z P ) θm π 0 2π d θ S d ϕ S E inc ( θ S , ϕ S )exp{ iφ( x P , y P , z P , θ S , ϕ S ) } 2fsin θ S ( 1cos θ S ) 2 , ×sin θ S cos ϕ S { 1( 1 1cos θ S i2kf ) 2fcos θ S z P ( 1cos θ S ) 2f }
φ( x P , y P , z P , θ S , ϕ S )=k( z P cos θ S + x P sin θ S cos ϕ S + y P sin θ S sin ϕ S ).
E inc ( θ S , ϕ S )= E 0 ( θ S , ϕ S )exp{ ik W inc ( θ n , ϕ S ) }.
θ n = cos θ S 1cos θ S 1cos θ m cos θ m tan( π θ S ) tan( π θ m ) .
2 n ^ ( S n ^ ) S =( x ^ sin θ S cos ϕ S + y ^ sin θ S sin ϕ S z ^ cos θ S ) W S ( θ n , ϕ S )
n x = sin θ S cos ϕ S [ 2( 1cos θ S ) ] 1/2 , n y = sin θ S sin ϕ S [ 2( 1cos θ S ) ] 1/2 , and n z = ( 1cos θ S 2 ) 1/2 .
[ 2 n ^ ( S n ^ ) S ] k ρ = 2π λ W S ( θ n , ϕ S ){ sin 2 θ S cos2 ϕ S + cos 2 θ S }
E x S ( θ ' S ,ϕ ' S )~ Σ d x P d y P { H y ( x P , y P , z 0 )( 1 1 ik R P )( E x ( x P , y P , z 0 )cosθ ' S + E z ( x P , y P , z 0 ) x P R P sinθ ' S cosϕ ' S R P ) } exp[ iφ' ] R P ,
E y S ( θ ' S ,ϕ ' S )~ Σ d x P d y P { H x ( x P , y P , z 0 )( 1 1 ik R P )( E y ( x P , y P , z 0 )cosθ ' S + E z ( x P , y P , z 0 ) y P R P sinθ ' S sinϕ ' S R P ) } exp[ iφ' ] R P ,
E z S ( θ ' S ,ϕ ' S )~ Σ d x S d y S ( 1 1 ik R P )( E x ( x P , y P , z 0 ) x P R P sinθ ' S cosϕ ' S R P + E y ( x P , y P , z 0 ) y P R P sinθ ' S sinϕ ' S R P + E z ( x P , y P , z 0 )cosθ ' S ) exp[ iφ' ] R P ,
φ'( x P , y P , z P , θ S , ϕ S )=k( x P sinθ ' S cosϕ ' S y P sinθ ' S sinϕ ' S ).
E x EM ( x,y,z )~ Σ dθdϕ ρsinθexp( iku ) ( a/ρ ) 2 sin 2 θ { { H z S ( θ,ϕ ) sinθsinϕ 1 e 2 H y S ( θ,ϕ ) ( a/ρ ) 2 sin 2 θ }+ ( 1 1 ikρ ) Δx ρ { E x S ( θ,ϕ ) sinθcosϕ 1 e 2 E y S ( θ,ϕ ) sinθsinϕ 1 e 2 + E z S ( θ,ϕ ) ( a/ρ ) 2 sin 2 θ } } ,
E y EM ( x,y,z )~ Σ dθdϕ ρsinθexp( iku ) ( a/ρ ) 2 sin 2 θ { { H x S ( θ,ϕ ) ( a/ρ ) 2 sin 2 θ + H z S ( θ,ϕ ) sinθcosϕ 1 e 2 }+ ( 1 1 ikρ ) Δy ρ { E x S ( θ,ϕ ) sinθcosϕ 1 e 2 E y S ( θ,ϕ ) sinθsinϕ 1 e 2 + E z S ( θ,ϕ ) ( a/ρ ) 2 sin 2 θ } } ,
E z EM ( x,y,z )~ Σ dθdϕ ρsinθexp( iku ) ( a/ρ ) 2 sin 2 θ { { H y S ( θ,ϕ ) sinθcosϕ 1 e 2 + H x S ( θ,ϕ ) sinθsinϕ 1 e 2 }+ ( 1 1 ikρ ) Δz ρ { E x S ( θ,ϕ ) sinθcosϕ 1 e 2 E y S ( θ,ϕ ) sinθsinϕ 1 e 2 + E z S ( θ,ϕ ) ( a/ρ ) 2 sin 2 θ } } .
E x,y,z ( x P , y P , z P )= W λ1 exp( i α λ1 ) E x,y,z ( λ 1 : x P , y P , z P )+ W λ2 exp( i α λ2 ) E x,y,z ( λ 2 : x P , y P , z P )+. + W λn exp( i α λn ) E x,y,z ( λ n : x P , y P , z P )

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