Abstract

In this paper, we present a systematic investigation of the characterization of tightly focused vector fields formed by an off-axis parabolic mirror. Based on the Stratton-Chu integral of Green’s theorem, the rigorous diffraction integrals that generate vector fields within and outside the focus arising from a collimated beam incident on an idealized parabolic mirror were derived in detail. In addition, explicit analytical expressions for the far-field vector diffraction electric and magnetic fields suitable for an off-axis parabolic mirror were also obtained. It is shown that there are significant differences in the vector diffraction characterizations between on- and off-axis parabolic mirrors. When the off-axis rate is greater than 4, the longitudinal field is predominant, and the maximum peak intensity ratio between the longitudinal field and the transverse field at the focus is approximately 103. This property is valuable for all applications in which a strong longitudinal field component is desirable. The effective focal length increases with increasing off-axis rate, and the depth of focus and the convergence angle of the focused beam are strongly dependent onf/2ω: smaller values of f/2ω lead to shorter focal depths and larger beam convergence (or tighter focusing).

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (1)

2016 (1)

G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space-time characterization of ultra-intense femtosecond laser beams,” Nat. Photonics 10(8), 547–553 (2016).
[Crossref]

2015 (2)

2014 (2)

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

N. V. Zamfir, “Nuclear Physics with 10PW laser beams at extreme light infrastructure–nuclear physics (ELI-NP),” Eur. Phys. J. Spec. Top. 223(6), 1221–1227 (2014).
[Crossref]

2011 (3)

G. Mourou and T. Tajima, “Physics. More intense, shorter pulses,” Science 331(6013), 41–42 (2011).
[Crossref] [PubMed]

A. V. Korzhimanov, A. A. Gonoskov, E. A. Khazanov, and A. M. Sergeev, “Horizons of petawatt laser technology,” Phys.- Usp. 54(1), 9–28 (2011).
[Crossref]

A. April, P. Bilodeau, and M. Piché, “Focusing a TM(01) beam with a slightly tilted parabolic mirror,” Opt. Express 19(10), 9201–9212 (2011).
[Crossref] [PubMed]

2010 (1)

2008 (4)

J. Stadler, C. Stanciu, C. Stupperich, and A. J. Meixner, “Tighter focusing with a parabolic mirror,” Opt. Lett. 33(7), 681–683 (2008).
[Crossref] [PubMed]

N. Bokor and N. Davidson, “4π Focusing with single paraboloid mirror,” Opt. Commun. 281(22), 5499–5503 (2008).
[Crossref]

K. I. Popov, V. Yu. Bychenkov, W. Rozmus, and R. D. Sydora, “Electron vacuum acceleration by a tightly focused laser pulse,” Phys. Plasmas 15(1), 013108 (2008).
[Crossref]

N. A. M. Hafz, T. M. Jeong, W. Choi, S. K. Lee, K. H. Pae, V. V. Kulagin, J. H. Sung, T. J. Yu, K. H. Hong, D. K. Ko, and J. Lee, “Stable generation of GeV-class electron beam from self–guided laser–plasma channels,” Nat. Photonics 2(9), 571–577 (2008).
[Crossref]

2007 (2)

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 64540O (2007).
[Crossref]

F. Merenda, J. Rohner, J. M. Fournier, and R. P. Salathé, “Miniaturized high-NA focusing-mirror multiple optical tweezers,” Opt. Express 15(10), 6075–6086 (2007).
[Crossref] [PubMed]

2006 (3)

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

W. P. Leemans, B. Nagler, A. J. Gonsalves, C. Tyth, K. Nakamura, C. G. R. Geddes, E. Esarey, C. B. Schroeder, and S. M. Hooker, “Gev electron beams from a centimetre-scale accelerator,” Nat. Phys. 2(10), 696–699 (2006).
[Crossref]

U. M. Mescheder, C. Estan, G. Somogyi, and M. Freudenreich, “Distortion optimized focusing mirror device with large aperture,” Sens. Actuators A Phys. 130–131, 20–27 (2006).
[Crossref]

2005 (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 intensities (1022W/cm2),” Appl. Phys. B 80(7), 823–832 (2005).

2004 (2)

2001 (2)

2000 (2)

1987 (1)

1977 (1)

C. J. R. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” IEE J. Microw. Opt. Acoust. 1(4), 129–132 (1977).
[Crossref]

1959 (2)

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959).
[Crossref]

B. Richards and E. Wolf, “Electromagnetic di_raction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

1939 (1)

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

Andreeva, V. A.

April, A.

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 intensities (1022W/cm2),” Appl. Phys. B 80(7), 823–832 (2005).

S. W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Generation and characterization of the highest laser intensities (10(22) W/cm2),” Opt. Lett. 29(24), 2837–2839 (2004).
[PubMed]

Barakat, R.

Berrondo, M.

Bilodeau, P.

Bokor, N.

Borot, A.

G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space-time characterization of ultra-intense femtosecond laser beams,” Nat. Photonics 10(8), 547–553 (2016).
[Crossref]

Brase, J. M.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 64540O (2007).
[Crossref]

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]

Bychenkov, V. Yu.

K. I. Popov, V. Yu. Bychenkov, W. Rozmus, and R. D. Sydora, “Electron vacuum acceleration by a tightly focused laser pulse,” Phys. Plasmas 15(1), 013108 (2008).
[Crossref]

Choi, W.

N. A. M. Hafz, T. M. Jeong, W. Choi, S. K. Lee, K. H. Pae, V. V. Kulagin, J. H. Sung, T. J. Yu, K. H. Hong, D. K. Ko, and J. Lee, “Stable generation of GeV-class electron beam from self–guided laser–plasma channels,” Nat. Photonics 2(9), 571–577 (2008).
[Crossref]

Choudhury, A.

C. J. R. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” IEE J. Microw. Opt. Acoust. 1(4), 129–132 (1977).
[Crossref]

Chu, L. J.

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

Chvykov, V.

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 intensities (1022W/cm2),” Appl. Phys. B 80(7), 823–832 (2005).

S. W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Generation and characterization of the highest laser intensities (10(22) W/cm2),” Opt. Lett. 29(24), 2837–2839 (2004).
[PubMed]

Combs, R. L.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 64540O (2007).
[Crossref]

Couairon, A.

Davidson, N.

Debus, C.

Drechsler, A.

Esarey, E.

W. P. Leemans, B. Nagler, A. J. Gonsalves, C. Tyth, K. Nakamura, C. G. R. Geddes, E. Esarey, C. B. Schroeder, and S. M. Hooker, “Gev electron beams from a centimetre-scale accelerator,” Nat. Phys. 2(10), 696–699 (2006).
[Crossref]

Estan, C.

U. M. Mescheder, C. Estan, G. Somogyi, and M. Freudenreich, “Distortion optimized focusing mirror device with large aperture,” Sens. Actuators A Phys. 130–131, 20–27 (2006).
[Crossref]

Fochs, S. N.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 64540O (2007).
[Crossref]

Fournier, J. M.

Freudenreich, M.

U. M. Mescheder, C. Estan, G. Somogyi, and M. Freudenreich, “Distortion optimized focusing mirror device with large aperture,” Sens. Actuators A Phys. 130–131, 20–27 (2006).
[Crossref]

Gallet, V.

G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space-time characterization of ultra-intense femtosecond laser beams,” Nat. Photonics 10(8), 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,” IEE J. Microw. Opt. Acoust. 1(4), 129–132 (1977).
[Crossref]

Geddes, C. G. R.

W. P. Leemans, B. Nagler, A. J. Gonsalves, C. Tyth, K. Nakamura, C. G. R. Geddes, E. Esarey, C. B. Schroeder, and S. M. Hooker, “Gev electron beams from a centimetre-scale accelerator,” Nat. Phys. 2(10), 696–699 (2006).
[Crossref]

Gobert, O.

G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space-time characterization of ultra-intense femtosecond laser beams,” Nat. Photonics 10(8), 547–553 (2016).
[Crossref]

Gonoskov, A. A.

A. V. Korzhimanov, A. A. Gonoskov, E. A. Khazanov, and A. M. Sergeev, “Horizons of petawatt laser technology,” Phys.- Usp. 54(1), 9–28 (2011).
[Crossref]

Gonsalves, A. J.

W. P. Leemans, B. Nagler, A. J. Gonsalves, C. Tyth, K. Nakamura, C. G. R. Geddes, E. Esarey, C. B. Schroeder, and S. M. Hooker, “Gev electron beams from a centimetre-scale accelerator,” Nat. Phys. 2(10), 696–699 (2006).
[Crossref]

Hafz, N. A. M.

N. A. M. Hafz, T. M. Jeong, W. Choi, S. K. Lee, K. H. Pae, V. V. Kulagin, J. H. Sung, T. J. Yu, K. H. Hong, D. K. Ko, and J. Lee, “Stable generation of GeV-class electron beam from self–guided laser–plasma channels,” Nat. Photonics 2(9), 571–577 (2008).
[Crossref]

Hong, K. H.

N. A. M. Hafz, T. M. Jeong, W. Choi, S. K. Lee, K. H. Pae, V. V. Kulagin, J. H. Sung, T. J. Yu, K. H. Hong, D. K. Ko, and J. Lee, “Stable generation of GeV-class electron beam from self–guided laser–plasma channels,” Nat. Photonics 2(9), 571–577 (2008).
[Crossref]

Hooker, S. M.

W. P. Leemans, B. Nagler, A. J. Gonsalves, C. Tyth, K. Nakamura, C. G. R. Geddes, E. Esarey, C. B. Schroeder, and S. M. Hooker, “Gev electron beams from a centimetre-scale accelerator,” Nat. Phys. 2(10), 696–699 (2006).
[Crossref]

Hurd, R. L.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 64540O (2007).
[Crossref]

Jeong, T. M.

T. M. Jeong, S. Weber, B. Le Garrec, D. Margarone, T. Mocek, and G. Korn, “Spatio-temporal modification of femtosecond focal spot under tight focusing condition,” Opt. Express 23(9), 11641–11656 (2015).
[Crossref] [PubMed]

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

N. A. M. Hafz, T. M. Jeong, W. Choi, S. K. Lee, K. H. Pae, V. V. Kulagin, J. H. Sung, T. J. Yu, K. H. Hong, D. K. Ko, and J. Lee, “Stable generation of GeV-class electron beam from self–guided laser–plasma channels,” Nat. Photonics 2(9), 571–577 (2008).
[Crossref]

Jukna, V.

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 intensities (1022W/cm2),” Appl. Phys. B 80(7), 823–832 (2005).

S. W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Generation and characterization of the highest laser intensities (10(22) W/cm2),” Opt. Lett. 29(24), 2837–2839 (2004).
[PubMed]

Khazanov, E. A.

A. V. Korzhimanov, A. A. Gonoskov, E. A. Khazanov, and A. M. Sergeev, “Horizons of petawatt laser technology,” Phys.- Usp. 54(1), 9–28 (2011).
[Crossref]

Ko, D. K.

N. A. M. Hafz, T. M. Jeong, W. Choi, S. K. Lee, K. H. Pae, V. V. Kulagin, J. H. Sung, T. J. Yu, K. H. Hong, D. K. Ko, and J. Lee, “Stable generation of GeV-class electron beam from self–guided laser–plasma channels,” Nat. Photonics 2(9), 571–577 (2008).
[Crossref]

Korn, G.

Korzhimanov, A. V.

A. V. Korzhimanov, A. A. Gonoskov, E. A. Khazanov, and A. M. Sergeev, “Horizons of petawatt laser technology,” Phys.- Usp. 54(1), 9–28 (2011).
[Crossref]

Kosareva, O. G.

Kulagin, V. V.

N. A. M. Hafz, T. M. Jeong, W. Choi, S. K. Lee, K. H. Pae, V. V. Kulagin, J. H. Sung, T. J. Yu, K. H. Hong, D. K. Ko, and J. Lee, “Stable generation of GeV-class electron beam from self–guided laser–plasma channels,” Nat. Photonics 2(9), 571–577 (2008).
[Crossref]

LaFortune, K. N.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 64540O (2007).
[Crossref]

Le Garrec, B.

Lee, J.

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

N. A. M. Hafz, T. M. Jeong, W. Choi, S. K. Lee, K. H. Pae, V. V. Kulagin, J. H. Sung, T. J. Yu, K. H. Hong, D. K. Ko, and J. Lee, “Stable generation of GeV-class electron beam from self–guided laser–plasma channels,” Nat. Photonics 2(9), 571–577 (2008).
[Crossref]

Lee, S. K.

N. A. M. Hafz, T. M. Jeong, W. Choi, S. K. Lee, K. H. Pae, V. V. Kulagin, J. H. Sung, T. J. Yu, K. H. Hong, D. K. Ko, and J. Lee, “Stable generation of GeV-class electron beam from self–guided laser–plasma channels,” Nat. Photonics 2(9), 571–577 (2008).
[Crossref]

Leemans, W. P.

W. P. Leemans, B. Nagler, A. J. Gonsalves, C. Tyth, K. Nakamura, C. G. R. Geddes, E. Esarey, C. B. Schroeder, and S. M. Hooker, “Gev electron beams from a centimetre-scale accelerator,” Nat. Phys. 2(10), 696–699 (2006).
[Crossref]

Lieb, M.

Maksimchuk, 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 intensities (1022W/cm2),” Appl. Phys. B 80(7), 823–832 (2005).

S. W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Generation and characterization of the highest laser intensities (10(22) W/cm2),” Opt. Lett. 29(24), 2837–2839 (2004).
[PubMed]

Margarone, D.

Meixner, A.

Meixner, A. J.

Merenda, F.

Mescheder, U. M.

U. M. Mescheder, C. Estan, G. Somogyi, and M. Freudenreich, “Distortion optimized focusing mirror device with large aperture,” Sens. Actuators A Phys. 130–131, 20–27 (2006).
[Crossref]

Mocek, T.

Mourou, G.

G. Mourou and T. Tajima, “Physics. More intense, shorter pulses,” Science 331(6013), 41–42 (2011).
[Crossref] [PubMed]

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 intensities (1022W/cm2),” Appl. Phys. B 80(7), 823–832 (2005).

S. W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Generation and characterization of the highest laser intensities (10(22) W/cm2),” Opt. Lett. 29(24), 2837–2839 (2004).
[PubMed]

Nagler, B.

W. P. Leemans, B. Nagler, A. J. Gonsalves, C. Tyth, K. Nakamura, C. G. R. Geddes, E. Esarey, C. B. Schroeder, and S. M. Hooker, “Gev electron beams from a centimetre-scale accelerator,” Nat. Phys. 2(10), 696–699 (2006).
[Crossref]

Nakamura, K.

W. P. Leemans, B. Nagler, A. J. Gonsalves, C. Tyth, K. Nakamura, C. G. R. Geddes, E. Esarey, C. B. Schroeder, and S. M. Hooker, “Gev electron beams from a centimetre-scale accelerator,” Nat. Phys. 2(10), 696–699 (2006).
[Crossref]

Nesa, F.

Olivier, S. S.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 64540O (2007).
[Crossref]

Pae, K. H.

N. A. M. Hafz, T. M. Jeong, W. Choi, S. K. Lee, K. H. Pae, V. V. Kulagin, J. H. Sung, T. J. Yu, K. H. Hong, D. K. Ko, and J. Lee, “Stable generation of GeV-class electron beam from self–guided laser–plasma channels,” Nat. Photonics 2(9), 571–577 (2008).
[Crossref]

Panov, N. A.

Pariente, G.

G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space-time characterization of ultra-intense femtosecond laser beams,” Nat. Photonics 10(8), 547–553 (2016).
[Crossref]

Pax, P. H.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 64540O (2007).
[Crossref]

Peatross, J.

Piché, M.

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 intensities (1022W/cm2),” Appl. Phys. B 80(7), 823–832 (2005).

S. W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Generation and characterization of the highest laser intensities (10(22) W/cm2),” Opt. Lett. 29(24), 2837–2839 (2004).
[PubMed]

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(1), 013108 (2008).
[Crossref]

Quéré, F.

G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space-time characterization of ultra-intense femtosecond laser beams,” Nat. Photonics 10(8), 547–553 (2016).
[Crossref]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic di_raction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

Rohner, J.

Rotter, M. D.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 64540O (2007).
[Crossref]

Rousseau, P.

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 intensities (1022W/cm2),” Appl. Phys. B 80(7), 823–832 (2005).

S. W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Generation and characterization of the highest laser intensities (10(22) W/cm2),” Opt. Lett. 29(24), 2837–2839 (2004).
[PubMed]

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(1), 013108 (2008).
[Crossref]

Salathé, R. P.

Schroeder, C. B.

W. P. Leemans, B. Nagler, A. J. Gonsalves, C. Tyth, K. Nakamura, C. G. R. Geddes, E. Esarey, C. B. Schroeder, and S. M. Hooker, “Gev electron beams from a centimetre-scale accelerator,” Nat. Phys. 2(10), 696–699 (2006).
[Crossref]

Sergeev, A. M.

A. V. Korzhimanov, A. A. Gonoskov, E. A. Khazanov, and A. M. Sergeev, “Horizons of petawatt laser technology,” Phys.- Usp. 54(1), 9–28 (2011).
[Crossref]

Sheppard, C. J. R.

C. J. R. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” IEE J. Microw. Opt. Acoust. 1(4), 129–132 (1977).
[Crossref]

Shipilo, D. E.

Smith, D.

Somogyi, G.

U. M. Mescheder, C. Estan, G. Somogyi, and M. Freudenreich, “Distortion optimized focusing mirror device with large aperture,” Sens. Actuators A Phys. 130–131, 20–27 (2006).
[Crossref]

Stadler, J.

Stanciu, C.

Stratton, J. A.

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

Stupperich, C.

Sung, J. H.

N. A. M. Hafz, T. M. Jeong, W. Choi, S. K. Lee, K. H. Pae, V. V. Kulagin, J. H. Sung, T. J. Yu, K. H. Hong, D. K. Ko, and J. Lee, “Stable generation of GeV-class electron beam from self–guided laser–plasma channels,” Nat. Photonics 2(9), 571–577 (2008).
[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(1), 013108 (2008).
[Crossref]

Tajima, T.

G. Mourou and T. Tajima, “Physics. More intense, shorter pulses,” Science 331(6013), 41–42 (2011).
[Crossref] [PubMed]

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

Tarrach, G.

Török, P.

Tyth, C.

W. P. Leemans, B. Nagler, A. J. Gonsalves, C. Tyth, K. Nakamura, C. G. R. Geddes, E. Esarey, C. B. Schroeder, and S. M. Hooker, “Gev electron beams from a centimetre-scale accelerator,” Nat. Phys. 2(10), 696–699 (2006).
[Crossref]

Varga, P.

Ware, M.

Weber, S.

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic di_raction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959).
[Crossref]

Yamamoto, R. M.

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 64540O (2007).
[Crossref]

Yanovsky, V.

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 intensities (1022W/cm2),” Appl. Phys. B 80(7), 823–832 (2005).

S. W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovsky, “Generation and characterization of the highest laser intensities (10(22) W/cm2),” Opt. Lett. 29(24), 2837–2839 (2004).
[PubMed]

Yu, T. J.

N. A. M. Hafz, T. M. Jeong, W. Choi, S. K. Lee, K. H. Pae, V. V. Kulagin, J. H. Sung, T. J. Yu, K. H. Hong, D. K. Ko, and J. Lee, “Stable generation of GeV-class electron beam from self–guided laser–plasma channels,” Nat. Photonics 2(9), 571–577 (2008).
[Crossref]

Zamfir, N. V.

N. V. Zamfir, “Nuclear Physics with 10PW laser beams at extreme light infrastructure–nuclear physics (ELI-NP),” Eur. Phys. J. Spec. Top. 223(6), 1221–1227 (2014).
[Crossref]

Ann. Phys. (Berlin) (1)

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

Appl. Opt. (1)

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 intensities (1022W/cm2),” Appl. Phys. B 80(7), 823–832 (2005).

Eur. Phys. J. Spec. Top. (1)

N. V. Zamfir, “Nuclear Physics with 10PW laser beams at extreme light infrastructure–nuclear physics (ELI-NP),” Eur. Phys. J. Spec. Top. 223(6), 1221–1227 (2014).
[Crossref]

IEE J. Microw. Opt. Acoust. (1)

C. J. R. Sheppard, A. Choudhury, and J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” IEE J. Microw. Opt. Acoust. 1(4), 129–132 (1977).
[Crossref]

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

Nat. Photonics (2)

N. A. M. Hafz, T. M. Jeong, W. Choi, S. K. Lee, K. H. Pae, V. V. Kulagin, J. H. Sung, T. J. Yu, K. H. Hong, D. K. Ko, and J. Lee, “Stable generation of GeV-class electron beam from self–guided laser–plasma channels,” Nat. Photonics 2(9), 571–577 (2008).
[Crossref]

G. Pariente, V. Gallet, A. Borot, O. Gobert, and F. Quéré, “Space-time characterization of ultra-intense femtosecond laser beams,” Nat. Photonics 10(8), 547–553 (2016).
[Crossref]

Nat. Phys. (1)

W. P. Leemans, B. Nagler, A. J. Gonsalves, C. Tyth, K. Nakamura, C. G. R. Geddes, E. Esarey, C. B. Schroeder, and S. M. Hooker, “Gev electron beams from a centimetre-scale accelerator,” Nat. Phys. 2(10), 696–699 (2006).
[Crossref]

Opt. Commun. (1)

N. Bokor and N. Davidson, “4π Focusing with single paraboloid mirror,” Opt. Commun. 281(22), 5499–5503 (2008).
[Crossref]

Opt. Express (8)

A. April and M. Piché, “4π Focusing of TM(01) beams under nonparaxial conditions,” Opt. Express 18(21), 22128–22140 (2010).
[Crossref] [PubMed]

T. M. Jeong, S. Weber, B. Le Garrec, D. Margarone, T. Mocek, and G. Korn, “Spatio-temporal modification of femtosecond focal spot under tight focusing condition,” Opt. Express 23(9), 11641–11656 (2015).
[Crossref] [PubMed]

F. Merenda, J. Rohner, J. M. Fournier, and R. P. Salathé, “Miniaturized high-NA focusing-mirror multiple optical tweezers,” Opt. Express 15(10), 6075–6086 (2007).
[Crossref] [PubMed]

A. Drechsler, M. Lieb, C. Debus, A. Meixner, and G. Tarrach, “Confocal microscopy with a high numerical aperture parabolic mirror,” Opt. Express 9(12), 637–644 (2001).
[Crossref] [PubMed]

M. Lieb and A. Meixner, “A high numerical aperture parabolic mirror as imaging device for confocal microscopy,” Opt. Express 8(7), 458–474 (2001).
[Crossref] [PubMed]

J. Peatross, M. Berrondo, D. Smith, and M. Ware, “Vector fields in a tight laser focus: comparison of models,” Opt. Express 25(13), 13990–14007 (2017).
[Crossref] [PubMed]

A. Couairon, O. G. Kosareva, N. A. Panov, D. E. Shipilo, V. A. Andreeva, V. Jukna, and F. Nesa, “Propagation equation for tight-focusing by a parabolic mirror,” Opt. Express 23(24), 31240–31252 (2015).
[Crossref] [PubMed]

A. April, P. Bilodeau, and M. Piché, “Focusing a TM(01) beam with a slightly tilted parabolic mirror,” Opt. Express 19(10), 9201–9212 (2011).
[Crossref] [PubMed]

Opt. Lett. (3)

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(1), 013108 (2008).
[Crossref]

Phys. Rev. (1)

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

Phys.- Usp. (1)

A. V. Korzhimanov, A. A. Gonoskov, E. A. Khazanov, and A. M. Sergeev, “Horizons of petawatt laser technology,” Phys.- Usp. 54(1), 9–28 (2011).
[Crossref]

Proc. R. Soc. Lond. A Math. Phys. Sci. (2)

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 349–357 (1959).
[Crossref]

B. Richards and E. Wolf, “Electromagnetic di_raction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

Proc. SPIE (1)

K. N. LaFortune, R. L. Hurd, S. N. Fochs, M. D. Rotter, P. H. Pax, R. L. Combs, S. S. Olivier, J. M. Brase, and R. M. Yamamoto, “Technical challenges for the future of high energy lasers,” Proc. SPIE 6454, 64540O (2007).
[Crossref]

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]

Science (1)

G. Mourou and T. Tajima, “Physics. More intense, shorter pulses,” Science 331(6013), 41–42 (2011).
[Crossref] [PubMed]

Sens. Actuators A Phys. (1)

U. M. Mescheder, C. Estan, G. Somogyi, and M. Freudenreich, “Distortion optimized focusing mirror device with large aperture,” Sens. Actuators A Phys. 130–131, 20–27 (2006).
[Crossref]

Other (2)

R. N. Wilson, Reflecting Telescope Optics (Springer, 2004).

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

Supplementary Material (2)

NameDescription
» Visualization 1       Animation showing the contour plots of the electromagnetic field intensity distribution of a focused square super-Gaussian top-hat beam polarized along the +x direction by an off-axis parabolic mirror for different off-axis rate.
» Visualization 2       Movie showing the contour plots of the electromagnetic field intensity distribution of a focused circular super-Gaussian top-hat beam polarized along the +x direction by an off-axis parabolic mirror for different off-axis rate.

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Figures (8)

Fig. 1
Fig. 1 Schematic illustrating reflection of off-axis parabolic mirror. (a) 3D view of paraboloid reflection and Cartesian coordinate systems. (b) Meridional section of off-axis parabolic mirror. Focus of off-axis parabolic mirror coincides with origin of Cartesian coordinate system.
Fig. 2
Fig. 2 Animation showing contour plots of electromagnetic field intensity distribution of focused square super-Gaussian top-hat beam polarized along + x direction by OAP for different off-axis rates (see Visualization 1). Fields are computed in focal plane (x'-y' plane with z P =0). Dashed lines show transverse locations of contours that are plotted in Fig. 3. Intensity is indicated by the “heat” of the color. Numerical computations based on Eqs. (34), (35), and (4).
Fig. 3
Fig. 3 Contour plots of electric field intensity distribution along dashed transverse locations shown in Fig. 2 as function of off-set h. (a) | E x | 2 along x' axis for 0h3600 mm, (b) | E z | 2 along x' axis for 0h3600 mm, (c) | E y | 2 along y' axis for 0h3600 mm, (d) | E x | 2 along x' axis for 1400h1800 mm(or in the vicinity of the rotation angle φ=π/2 ), and (e) | E z | 2 along x' axis for 0h100 mm (or 0α0.6). Intensity is indicated by the “heat” of the color. Numerical computations based on Eqs. (34), (35), and (4).
Fig. 4
Fig. 4 Peak intensity of electric field components of focused square super-Gaussian top-hat beam by OAP as function of (a) off-axis rate α and (b) rotation angle φ. Dependence of effective focal length (EFL) f of OAP system on rotation angle is also shown in (b). Fields are computed in focal plane (x'-y' plane with z P =0). A logarithmic scale for intensity of field components ( | E x | 2 , | E y | 2 , and | E z | 2 ) is used in (a), but an absolute value scale for intensity of field components is used in (b). Cyan dashed line shows | E z | 2 = | E x | 2 case, magenta dash-dotted line location of the maximum peak intensity of | E z | 2 and | E y | 2 , and olive dash-dot-dotted line location of the maximum dark center of | E x | 2 .
Fig. 5
Fig. 5 3D intensity distribution of total diffraction electric field of focused square super-Gaussian top-hat beam by OAP with α=1.844 (rotation angle φ=0.36 rad). (a) | E | 2 in coordinate system S ( x , y , z ). (b) | E | 2 in coordinate system S(x,y,z). Intensity is indicated by the “heat” of the color. Dashed line in (b) represents propagation direction of focused field.
Fig. 6
Fig. 6 Contour plots of electric field intensity distribution along z' axis (or depth of focus) of focused square super-Gaussian top-hat beam by OAP as function of h. (a) | E x | 2 for 0h3600 mm(or 2.5 f / 2ω 15.2) and x =0, y =0 and (b) | E z | 2 for 0h3600 mm(or 2.5 f / 2ω 15.2) and x =0, y =0. Intensity is indicated by the “heat” of the color.
Fig. 7
Fig. 7 Contour plots of focused electric field intensity distribution in focal plane along x' and z' axes as function of f / 2ω ( f is fixed). (a) | E x | 2 along x' axis for 0.42 f / 2ω 50 and y =0, z =0, (b) | E z | 2 along x' axis for 0.42 f / 2ω 50 and y =0, z =0, (c) | E x | 2 along z' axis for 0.42 f / 2ω 50and x =0, y =0, and (d) | E z | 2 along z' axis for 0.42 f / 2ω 50 and x =0, y =0. Intensity is indicated by the “heat” of the color.
Fig. 8
Fig. 8 Movie showing contour plots of electromagnetic field intensity distribution of focused circular super-Gaussian top-hat beam polarized along + x direction by OAP for different off-axis rates (see Visualization 2). Fields are computed in focal plane (x'-y' plane with z P =0). Intensity is indicated by the “heat” of the color. Numerical computations based on Eqs. (34), (35), and (3).

Equations (35)

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z o = x o 2 + y o 2 4f f,
n ^ ( x o , y o )= 1 2f ( x o x ^ + y o y ^ )+ z ^ 1+ x o 2 + y o 2 4 f 2 ,
Σ: ( x o h ) 2 + y o 2 R 2 ( for a circular offaxis surface ),
Σ:R x o hR and R y o R ( for a square offaxis surface ) ,
E r,n = E i,n = n ^ ( n ^ E i ),
H r,t = H i,t = H i H i,n ,
E r,t = E i,t =( E i E r,n ),
H r,n = H i,n = n ^ ( n ^ H i ).
E r =2 n ^ ( n ^ E i ) E i ,
H r = H i 2 n ^ ( n ^ H i ).
E= E i + E r =2 n ^ ( n ^ E i ).
H= H i + H r =2 H i 2 n ^ ( n ^ H i ).
E( P )= 1 4π S OAP [ iωμ( n ^ ×H )G+( n ^ ×E )×G+( n ^ E )G ] dA,
H( P )= 1 4π S OAP [ iωε( E× n ^ )G+( n ^ ×H )×G+( n ^ H )G ] dA,
r OP =| r P r O |= [ ( x P x o ) 2 + ( y P y o ) 2 + ( z P z o ) 2 ] 1/2 .
G=ik(1 1 ik r OP ) G r OP [ ( x o x P ) x ^ +( y o y P ) y ^ +( z o z P ) z ^ ] = ik(1 1 ik r OP ) G r OP ( Δx x ^ +Δy y ^ +Δz z ^ ) =ik(1 1 ik r OP ) G r OP r OP
dA= [ 1+ ( z o x o ) 2 + ( z o y o ) 2 ] 1/2 d x o d y o = ( 1+ x o 2 + y o 2 4 f 2 ) 1/2 d x o d y o .
E i =( Ψ 0x x ^ + Ψ 0y y ^ )exp( iωtik z i ),
H i = 1 η ( Ψ 0y x ^ Ψ 0x y ^ )exp( iωtik z i ),
E( P )= ikexp( iωt ) 2π S OAP exp( ik z o ) d x o d y o ×{ [ ( 1( 1 1 ik r OP ) x o 2f Δx r OP ) Ψ 0x G( 1 1 ik r OP ) y o 2f Δx r OP Ψ 0y G ] x ^ + [ ( 1( 1 1 ik r OP ) y o 2f Δy r OP ) Ψ 0y G( 1 1 ik r OP ) x o 2f Δy r OP Ψ 0x G ] y ^ +( x o 2f Ψ 0x + y o 2f Ψ 0y ) [ 1( 1 1 ik r OP ) Δz r OP ]G z ^ }
H( P )= ikexp( iωt ) 2π S OAP exp( ik z o ) η ( 1 1 ik r OP ) G r OP d x o d y o ×{ [ ( Δy x o 2f ) Ψ 0x +( x o 2 + y o 2 4f f Δy y o 2f ) Ψ 0y ] x ^ + [ ( x o 2 + y o 2 4f f Δx x o 2f ) Ψ 0x +( Δx y o 2f ) Ψ 0y ] y ^ +( Δy Ψ 0x Δx Ψ 0y ) z ^ }
S= E r × H r = 1 η ( Ψ 0x 2 + Ψ 0y 2 )exp[ 2i( ωt+k z o ) ]( x o r o x ^ y o r o y ^ z o r o z ^ ).
E F ( P )= ikexp( i2kfiωt ) 2π S OAP exp( ik r O · r P / r o ) d x o d y o ×{ [ ( 1 r o x o 2 2f r o 2 ) Ψ 0x x o y o 2f r o 2 Ψ 0y ] x ^ , + [ ( 1 r o y o 2 2f r o 2 ) Ψ 0y x o y o 2f r o 2 Ψ 0x ] y ^ +( x o Ψ 0x + y o Ψ 0y ) 1 r o 2 z ^ }
H F ( P )= ikexp( i2kfiωt ) 2π S OAP exp( ik r O · r P / r o ) η d x o d y o ×{ [ ( x o y o 2f r o 2 ) Ψ 0x +( x o 2 y o 2 4f r o 2 f r o 2 ) Ψ 0y ] x ^ . + [ ( x o 2 y o 2 4f r o 2 + f r o 2 ) Ψ 0x +( x o y o 2f r o 2 ) Ψ 0y ] y ^ +( y o Ψ 0x x o Ψ 0y ) 1 r o 2 z ^ }
ik r O · r P / r o =ik( x o r o x P y o r o y P z o r o z P ),
ik r O · r P / r o =ik( x o cosφ+ z o sinφ r o x P y o r o y P x o sinφ+ z o cosφ r o z P ),
α= ( x o cosφ+ z o sinφ )/ r o ,
β= y o / r o ,
γ= ( x o sinφ+ z o cosφ )/ r o ,
x o = 2f( αcosφ+γsinφ ) ( 1+αsinφ+γcosφ ) ,
y o = 2f( β ) ( 1+αsinφ+γcosφ ) ,
z o = 2f( αsinφγcosφ ) ( 1+αsinφ+γcosφ ) .
J= 4 f 2 γ ( 1+αsinφ+γcosφ ) 2 .
E ' F ( P )= ikexp( i2kfiωt ) 2π S OAP ' Jexp[ ik( αx ' P +β y P +γz ' P ) ]dαdβ ×{ [ ( 1 r o x o 2 2f r o 2 ) Ψ 0x x o y o 2f r o 2 Ψ 0y ] x ^ ' + [ ( 1 r o y o 2 2f r o 2 ) Ψ 0y x o y o 2f r o 2 Ψ 0x ] y ^ ' , +( x o Ψ 0x + y o Ψ 0y ) 1 r o 2 z ^ ' }
H F ( P )= ikexp( i2kfiωt ) 2π S OAP ' Jexp[ ik( α x P +β y P +γ z P ) ] η dαdβ ×{ [ ( x o y o 2f r o 2 ) Ψ 0x +( x o 2 y o 2 4f r o 2 f r o 2 ) Ψ 0y ] x ^ + [ ( x o 2 y o 2 4f r o 2 + f r o 2 ) Ψ 0x +( x o y o 2f r o 2 ) Ψ 0y ] y ^ , +( y o Ψ 0x x o Ψ 0y ) 1 r o 2 z ^ }

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