Abstract

We study the Gouy phase of a scalar wavefield that is focused by a lens suffering from primary spherical aberration. It is found that the Gouy phase has different behaviors at the two sides of the intensity maximum. This results in a systematic increase of the successive wavefront spacings around the diffraction focus. Since all lenses have some amount of spherical aberration, this observation has implications for optical calibration and metrology.

© 2013 Optical Society of America

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  19. W. Zhu, A. Agrawal, and A. Nahata, Opt. Express 15, 9995 (2007).
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  20. H. C. Kandpal, S. Raman, and R. Mehrotra, Opt. Lasers Eng. 45, 249 (2007).
    [CrossRef]
  21. F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 92, 113001 (2004).
    [CrossRef]
  22. L. Robertsson, Metrologia 44, 35 (2007).
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  23. I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, Nat. Photonics 3, 351 (2009).
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  28. C. J. R. Sheppard, Opt. Lett. 25, 1660 (2000).
    [CrossRef]

2013

2012

2011

X. Pang, T. D. Visser, and E. Wolf, Opt. Commun. 284, 5517 (2011).
[CrossRef]

X. Pang, G. Gbur, and T. D. Visser, Opt. Lett. 36, 2492 (2011).
[CrossRef]

B. Andreas, L. Ferroglio, K. Fujii, N. Kuramoto, and G. Mana, Metrologia 48, S104 (2011).
[CrossRef]

2010

2009

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, Nat. Photonics 3, 351 (2009).
[CrossRef]

2007

W. Zhu, A. Agrawal, and A. Nahata, Opt. Express 15, 9995 (2007).
[CrossRef]

H. C. Kandpal, S. Raman, and R. Mehrotra, Opt. Lasers Eng. 45, 249 (2007).
[CrossRef]

L. Robertsson, Metrologia 44, 35 (2007).
[CrossRef]

2006

2005

2004

J. H. Chow, G. de Vine, M. B. Gray, and D. E. McClelland, Opt. Lett. 29, 2339 (2004).
[CrossRef]

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef]

2000

C. J. R. Sheppard, Opt. Lett. 25, 1660 (2000).
[CrossRef]

R. W. McGowan, R. A. Cheville, and D. Grischkowskya, Appl. Phys. Lett. 76, 670 (2000).
[CrossRef]

1999

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, Phys. Rev. Lett. 83, 3410 (1999).
[CrossRef]

1998

1995

1989

1956

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. B 69, 823 (1956).

1891

L. G. Gouy, Annales des Chimie et de Physique 6e séries 24, 145 (1891).

1890

L. G. Gouy, Comptes Rendus hebdomadaires des Séances de l’Académie des Sciences 110, 1251 (1890).

Agrawal, A.

Andreas, B.

B. Andreas, L. Ferroglio, K. Fujii, N. Kuramoto, and G. Mana, Metrologia 48, S104 (2011).
[CrossRef]

Baltuska, A.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge University, 1999).

Cheville, R. A.

R. W. McGowan, R. A. Cheville, and D. Grischkowskya, Appl. Phys. Lett. 76, 670 (2000).
[CrossRef]

Chow, J. H.

Coddington, I.

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, Nat. Photonics 3, 351 (2009).
[CrossRef]

Creath, K.

de Vine, G.

Feng, S.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, Phys. Rev. Lett. 83, 3410 (1999).
[CrossRef]

Ferroglio, L.

B. Andreas, L. Ferroglio, K. Fujii, N. Kuramoto, and G. Mana, Metrologia 48, S104 (2011).
[CrossRef]

Fischer, D. F.

Foley, J. T.

Fujii, K.

B. Andreas, L. Ferroglio, K. Fujii, N. Kuramoto, and G. Mana, Metrologia 48, S104 (2011).
[CrossRef]

Gatto, A.

Gbur, G.

Goulielmakis, E.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef]

Gouy, L. G.

L. G. Gouy, Annales des Chimie et de Physique 6e séries 24, 145 (1891).

L. G. Gouy, Comptes Rendus hebdomadaires des Séances de l’Académie des Sciences 110, 1251 (1890).

Gray, M. B.

Grischkowskya, D.

R. W. McGowan, R. A. Cheville, and D. Grischkowskya, Appl. Phys. Lett. 76, 670 (2000).
[CrossRef]

Hamazaki, J.

Kadlec, C.

Kadlec, F.

Kandpal, H. C.

H. C. Kandpal, S. Raman, and R. Mehrotra, Opt. Lasers Eng. 45, 249 (2007).
[CrossRef]

Krausz, F.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef]

Kuramoto, N.

B. Andreas, L. Ferroglio, K. Fujii, N. Kuramoto, and G. Mana, Metrologia 48, S104 (2011).
[CrossRef]

Kužel, P.

Larkin, K. G.

Lezius, M.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef]

Lindner, F.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef]

Linfoot, E. H.

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. B 69, 823 (1956).

Mana, G.

B. Andreas, L. Ferroglio, K. Fujii, N. Kuramoto, and G. Mana, Metrologia 48, S104 (2011).
[CrossRef]

Martelli, P.

Martinelli, M.

McClelland, D. E.

McGowan, R. W.

R. W. McGowan, R. A. Cheville, and D. Grischkowskya, Appl. Phys. Lett. 76, 670 (2000).
[CrossRef]

Mehrotra, R.

H. C. Kandpal, S. Raman, and R. Mehrotra, Opt. Lasers Eng. 45, 249 (2007).
[CrossRef]

Mineta, Y.

Moneta, G.

Morita, R.

Nahata, A.

Nemec, H.

Nenadovic, L.

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, Nat. Photonics 3, 351 (2009).
[CrossRef]

Neumaier, K. R.

Newbury, N. R.

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, Nat. Photonics 3, 351 (2009).
[CrossRef]

Oka, K.

Pang, X.

Paulus, G. G.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef]

Raman, S.

H. C. Kandpal, S. Raman, and R. Mehrotra, Opt. Lasers Eng. 45, 249 (2007).
[CrossRef]

Robertsson, L.

L. Robertsson, Metrologia 44, 35 (2007).
[CrossRef]

Rudd, J. V.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, Phys. Rev. Lett. 83, 3410 (1999).
[CrossRef]

Ruffin, A. B.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, Phys. Rev. Lett. 83, 3410 (1999).
[CrossRef]

Sackmann, E.

Sheppard, C. J. R.

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Adam Hilger, 1986).

Swann, W. C.

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, Nat. Photonics 3, 351 (2009).
[CrossRef]

Tacca, M.

Visser, T. D.

Walther, H.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef]

Whitaker, J. F.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, Phys. Rev. Lett. 83, 3410 (1999).
[CrossRef]

Wiegand, G.

Winful, H. G.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, Phys. Rev. Lett. 83, 3410 (1999).
[CrossRef]

Wolf, E.

X. Pang, T. D. Visser, and E. Wolf, Opt. Commun. 284, 5517 (2011).
[CrossRef]

T. D. Visser and E. Wolf, Opt. Commun. 283, 3371 (2010).
[CrossRef]

J. T. Foley and E. Wolf, Opt. Lett. 30, 1312 (2005).
[CrossRef]

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. B 69, 823 (1956).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge University, 1999).

Zhu, W.

Annales des Chimie et de Physique 6e séries

L. G. Gouy, Annales des Chimie et de Physique 6e séries 24, 145 (1891).

Appl. Opt.

Appl. Phys. Lett.

R. W. McGowan, R. A. Cheville, and D. Grischkowskya, Appl. Phys. Lett. 76, 670 (2000).
[CrossRef]

Comptes Rendus hebdomadaires des Séances de l’Académie des Sciences

L. G. Gouy, Comptes Rendus hebdomadaires des Séances de l’Académie des Sciences 110, 1251 (1890).

J. Opt. Soc. Am. A

Metrologia

B. Andreas, L. Ferroglio, K. Fujii, N. Kuramoto, and G. Mana, Metrologia 48, S104 (2011).
[CrossRef]

L. Robertsson, Metrologia 44, 35 (2007).
[CrossRef]

Nat. Photonics

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, Nat. Photonics 3, 351 (2009).
[CrossRef]

Opt. Commun.

T. D. Visser and E. Wolf, Opt. Commun. 283, 3371 (2010).
[CrossRef]

X. Pang, T. D. Visser, and E. Wolf, Opt. Commun. 284, 5517 (2011).
[CrossRef]

Opt. Express

Opt. Lasers Eng.

H. C. Kandpal, S. Raman, and R. Mehrotra, Opt. Lasers Eng. 45, 249 (2007).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, Phys. Rev. Lett. 83, 3410 (1999).
[CrossRef]

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef]

Proc. Phys. Soc. B

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. B 69, 823 (1956).

Other

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge University, 1999).

J. J. Stamnes, Waves in Focal Regions (Adam Hilger, 1986).

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Figures (6)

Fig. 1.
Fig. 1.

Illustrating the notation.

Fig. 2.
Fig. 2.

Axial intensity distribution for different values of the spherical aberration parameter, A0=0 (blue curve), A0=λ (red curve), and A0=3.5λ (olive curve). Here, and in all the following examples, a/f is taken to be 1/2.

Fig. 3.
Fig. 3.

Gouy phase of the field along the axis for different values of the aberration parameter A0.

Fig. 4.
Fig. 4.

Gouy phase δ(0,A0) at the geometrical focus (red curve) and the Gouy phase δ(2kA0,A0) at the diffraction focus (blue curve) as functions of the aberration parameter A0.

Fig. 5.
Fig. 5.

Gouy phase and the intensity distribution of the field along the axis for the case A0=λ/4.

Fig. 6.
Fig. 6.

Gouy phase and the intensity distribution of the field along the axis for the case A0=λ/4.

Tables (1)

Tables Icon

Table 1. Wavefront Spacings (in Free-Space Wavelengths λ) near the Diffraction Focus for Different Amounts of Spherical Aberration, for the Case a/f=1/2

Equations (18)

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

U(P)=iλAeikffSeik[Φ+s]sdS,
Φ(ρ)=A0ρ4,
u=kz(af)2,
v=k(x2+y2)1/2af.
U(u,v;A0)=C01J0(ρv)ei(uρ2/2+kA0ρ4)ρdρ,
U*(u,v;A0)=U(u,v;A0),
|U(u,0;A0)|2=|U(u,0;A0)|2,
arg[U(u,v;A0)]+arg[U(u,v;A0)]=π.(mod2π).
arg[U(u,v)]+arg[U(u,v)]=π,
U(u;A0)=C(1)3/4π4kA0eiu2/16kA0{erfi[(1)1/4(4kA0u)4kA0]+erfi[(1)1/4u4kA0]},
δ(u;A0)=arg[U(u;A0)]sign(u)kR,
kR=k|z|=(fa)2|u|,
sign(x)={1ifx<0,1ifx>0.
δ(u;A0)+δ(u;A0)=π(mod2π).
U(0;A0)=ikA(af)2(1)3/4π4kA0erfi[(1)1/4kA0].
U(2kA0;A0)=ikA(af)2(1)3/4π2kA0ei2kA0(f/a)2×eikA0/4erfi(ikA0/2),
U(u;A0)=ikA(af)2eikf01eik(s+A0ρ4)ρdρ,
s=f[1+(ufka2)2+2uka2f2a2ρ2]1/2.

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