Abstract

Second-harmonic generation was studied theoretically for the vector singularities (daisy modes) of linearly polarized light and for the elliptic singularities (C points) of elliptically polarized light. Topological charge doubling for C points and daisy modes, similar to that found for vortices, was found. Unlike for vortices, however, it was found that for both C points and daisy modes the sign of the charge is reversed; for C points the photon spin or handedness (right or left) is also reversed; and for daisy modes the number of intensity petals is not doubled. These findings are all unexpected because neither charge nor spin nor orbital angular momentum is conserved for C points, whereas a daisy mode in which the number of intensity petals does not equal twice the magnitude of the charge is anomalous.

© 2002 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  20. N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965), Chap. 4.

2002 (2)

2001 (3)

2000 (1)

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Y. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
[CrossRef] [PubMed]

1997 (2)

F. Prati, G. Tissoni, M. S. Miguel, and N. B. Abraham, Opt. Commun. 143, 133 (1997).
[CrossRef]

A. K. J. Van Doorn, M. P. Van Exter, A. M. Van Der Lee, and J. P. Woerdman, Phys. Rev. A 55, 1473 (1997).
[CrossRef]

1996 (1)

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, Phys. Rev. A 54, R3742 (1996).
[CrossRef]

1993 (1)

I. V. Basistiy, V. Yu Bazhenov, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 103, 422 (1993).
[CrossRef]

1992 (2)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, Opt. Lett. 17, 221 (1992).
[CrossRef]

1987 (1)

J. V. Hajnal, Proc. R. Soc. London Ser. A 414, 433 (1987).
[CrossRef]

1974 (1)

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
[CrossRef]

1962 (2)

J. A. Giordmaine, Phys. Rev. Lett. 8, 19 (1962).
[CrossRef]

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, Phys. Rev. Lett. 8, 21 (1962).
[CrossRef]

Abraham, N. B.

F. Prati, G. Tissoni, M. S. Miguel, and N. B. Abraham, Opt. Commun. 143, 133 (1997).
[CrossRef]

Agullo-Lopez, F.

Allen, L.

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, Phys. Rev. A 54, R3742 (1996).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Basistiy, I. V.

I. V. Basistiy, V. Yu Bazhenov, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 103, 422 (1993).
[CrossRef]

Bazhenov, V. Yu

I. V. Basistiy, V. Yu Bazhenov, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 103, 422 (1993).
[CrossRef]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Berry, M. V.

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
[CrossRef]

Bloembergen, N.

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965), Chap. 4.

Calvo, G. F.

Denz, C.

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Y. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
[CrossRef] [PubMed]

Dholakia, K.

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, Phys. Rev. A 54, R3742 (1996).
[CrossRef]

Freund, I.

I. Freund, Opt. Commun. 201, 251 (2002).
[CrossRef]

I. Freund, Opt. Lett. 26, 1996 (2001).
[CrossRef]

Geisser, M.

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Y. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
[CrossRef] [PubMed]

Giordmaine, J. A.

J. A. Giordmaine, Phys. Rev. Lett. 8, 19 (1962).
[CrossRef]

Hajnal, J. V.

J. V. Hajnal, Proc. R. Soc. London Ser. A 414, 433 (1987).
[CrossRef]

Heckenberg, N. R.

Kivshar, Y. S.

D. Neshev, G. McCarthy, W. Krolikowski, E. A. Ostrovskaya, Y. S. Kivshar, G. F. Calvo, and F. Agullo-Lopez, Opt. Lett. 15, 1185 (2001).
[CrossRef]

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Y. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
[CrossRef] [PubMed]

Konukhov, A. I.

A. I. Konukhov and L. A. Melnikov, J. Opt. B 3, S139 (2001).
[CrossRef]

Krolikowski, W.

D. Neshev, G. McCarthy, W. Krolikowski, E. A. Ostrovskaya, Y. S. Kivshar, G. F. Calvo, and F. Agullo-Lopez, Opt. Lett. 15, 1185 (2001).
[CrossRef]

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Y. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
[CrossRef] [PubMed]

Luther-Davies, B.

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Y. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
[CrossRef] [PubMed]

Maker, P. D.

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, Phys. Rev. Lett. 8, 21 (1962).
[CrossRef]

McCarthy, G.

D. Neshev, G. McCarthy, W. Krolikowski, E. A. Ostrovskaya, Y. S. Kivshar, G. F. Calvo, and F. Agullo-Lopez, Opt. Lett. 15, 1185 (2001).
[CrossRef]

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Y. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
[CrossRef] [PubMed]

McDuff, R.

Melnikov, L. A.

A. I. Konukhov and L. A. Melnikov, J. Opt. B 3, S139 (2001).
[CrossRef]

Miguel, M. S.

F. Prati, G. Tissoni, M. S. Miguel, and N. B. Abraham, Opt. Commun. 143, 133 (1997).
[CrossRef]

Molina-Terriza, G.

Neshev, D.

Nisenoff, M.

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, Phys. Rev. Lett. 8, 21 (1962).
[CrossRef]

Nye, J. F.

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
[CrossRef]

J. F. Nye, Natural Focusing and the Fine Structure of Light (IOP, Bristol, UK, 1999).

Ostrovskaya, E. A.

D. Neshev, G. McCarthy, W. Krolikowski, E. A. Ostrovskaya, Y. S. Kivshar, G. F. Calvo, and F. Agullo-Lopez, Opt. Lett. 15, 1185 (2001).
[CrossRef]

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Y. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
[CrossRef] [PubMed]

Padgett, M. J.

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, Phys. Rev. A 54, R3742 (1996).
[CrossRef]

Petrov, D. V.

Prati, F.

F. Prati, G. Tissoni, M. S. Miguel, and N. B. Abraham, Opt. Commun. 143, 133 (1997).
[CrossRef]

Recolons, J.

Savage, C. M.

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, Phys. Rev. Lett. 8, 21 (1962).
[CrossRef]

Simpson, N. B.

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, Phys. Rev. A 54, R3742 (1996).
[CrossRef]

Singh, S.

S. Singh, in Handbook of Lasers, R. J. Presley, ed. (Chemical Rubber Company, Cleveland, Ohio, 1971), Chap. 18.

Smith, C. P.

Soskin, M. S.

I. V. Basistiy, V. Yu Bazhenov, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 103, 422 (1993).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2001), Vol. 42, Chap. 4, Sec. 10.

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Terhune, R. W.

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, Phys. Rev. Lett. 8, 21 (1962).
[CrossRef]

Tissoni, G.

F. Prati, G. Tissoni, M. S. Miguel, and N. B. Abraham, Opt. Commun. 143, 133 (1997).
[CrossRef]

Torner, L.

Van Der Lee, A. M.

A. K. J. Van Doorn, M. P. Van Exter, A. M. Van Der Lee, and J. P. Woerdman, Phys. Rev. A 55, 1473 (1997).
[CrossRef]

Van Doorn, A. K. J.

A. K. J. Van Doorn, M. P. Van Exter, A. M. Van Der Lee, and J. P. Woerdman, Phys. Rev. A 55, 1473 (1997).
[CrossRef]

Van Exter, M. P.

A. K. J. Van Doorn, M. P. Van Exter, A. M. Van Der Lee, and J. P. Woerdman, Phys. Rev. A 55, 1473 (1997).
[CrossRef]

Vasnetsov, M. V.

I. V. Basistiy, V. Yu Bazhenov, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 103, 422 (1993).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2001), Vol. 42, Chap. 4, Sec. 10.

Weilnau, C.

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Y. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
[CrossRef] [PubMed]

White, A. G.

Woerdman, J. P.

A. K. J. Van Doorn, M. P. Van Exter, A. M. Van Der Lee, and J. P. Woerdman, Phys. Rev. A 55, 1473 (1997).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

J. Opt. B (1)

A. I. Konukhov and L. A. Melnikov, J. Opt. B 3, S139 (2001).
[CrossRef]

Opt. Commun. (3)

I. V. Basistiy, V. Yu Bazhenov, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 103, 422 (1993).
[CrossRef]

F. Prati, G. Tissoni, M. S. Miguel, and N. B. Abraham, Opt. Commun. 143, 133 (1997).
[CrossRef]

I. Freund, Opt. Commun. 201, 251 (2002).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. A (3)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, Phys. Rev. A 54, R3742 (1996).
[CrossRef]

A. K. J. Van Doorn, M. P. Van Exter, A. M. Van Der Lee, and J. P. Woerdman, Phys. Rev. A 55, 1473 (1997).
[CrossRef]

Phys. Rev. Lett. (3)

W. Krolikowski, E. A. Ostrovskaya, C. Weilnau, M. Geisser, G. McCarthy, Y. S. Kivshar, C. Denz, and B. Luther-Davies, Phys. Rev. Lett. 85, 1424 (2000).
[CrossRef] [PubMed]

J. A. Giordmaine, Phys. Rev. Lett. 8, 19 (1962).
[CrossRef]

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, Phys. Rev. Lett. 8, 21 (1962).
[CrossRef]

Proc. R. Soc. London Ser. A (2)

J. V. Hajnal, Proc. R. Soc. London Ser. A 414, 433 (1987).
[CrossRef]

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
[CrossRef]

Other (4)

M. S. Soskin and M. V. Vasnetsov, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2001), Vol. 42, Chap. 4, Sec. 10.

S. Singh, in Handbook of Lasers, R. J. Presley, ed. (Chemical Rubber Company, Cleveland, Ohio, 1971), Chap. 18.

J. F. Nye, Natural Focusing and the Fine Structure of Light (IOP, Bristol, UK, 1999).

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965), Chap. 4.

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Figures (2)

Fig. 1
Fig. 1

Daisy modes, Eqs. (1) with ax=ay, γ=π/8, b=3. (a), (b), (d), (e) Vector fields. (c), (f) Intensities. The vectors of a positive (negative) daisy mode with charge +q (charge -q) rotate by 2qπ in the same (opposite) direction as a path that encloses the singularity. These rotations may be most easily followed along the outer boundary for each figure. For positive daisy modes the streamlines form 2q-1 vector petals; for negative modes, 2q+1 hyperbolic sectors. The intensity distribution of a daisy mode normally contains 2q maxima (intensity petals) alternating with 2q saddle points. (a)–(c) Daisy modes at input frequency ω. (a) q=+2 mode with two vector petals. (b) q=-2 mode with six hyperbolic sectors. (c) Intensity of the modes in (a) and (b), showing the expected four maxima and four saddle points surrounding zero intensity at the origin. (d)–(f) Output daisy modes at 2ω produced by SHG of the adjacent input daisy modes in (a)–(c). Under SHG, charge q is doubled and reversed in sign. (d) q=-4 daisy mode with 10 hyperbolic sectors. (e) q=+4 daisy mode with six vector petals. (f) Instead of eight maxima and eight saddle points, this anomalous 2ω intensity distribution shows four maxima and four saddle points at exactly the same locations as the ω intensity distribution in (c). This behavior is characteristic of SHG of real vector fields, for which the intensity at 2ω is the square of the intensity at w.

Fig. 2
Fig. 2

C points. (a), (b), (d), (e) Ellipse fields. (c), (f) Intensity distributions. Charge IC of a C point (central filled circles) equals the number of 2π rotations of the surrounding ellipses along a closed path that encircles the point. The surrounding ellipses rotate in the same (opposite) direction as the path for a positive (negative) C point. Right-handed, h=+1 (left handed, h=-1) ellipses are shown by solid (dotted) curves. (a)–(c) C points of input ellipse fields at frequency ω created with (a) eR=G2, eL=V1G2, which produces a C point with IC=+1/2, h=+1 [Eq. (2)], and (b) eR=V1G2, eL=G2, which produces a C point with IC=-1/2, h=-1, where eR and eL are the right and left circular components of the field, respectively, G2=exp-br2, with b=2, is the Gaussian envelope of the host beam, and V1=x+iy. (c) Intensity of (a) and of (b). (d)–(f) C points of output ellipse fields at 2ω produced by SHG of adjacent input C points in (a)–(c). Under SHG the charge IC of a C point is doubled, and its sign is reversed, as is the sign of its handedness h. (d) IC=-1,h=-1. (e) IC=+1,h=+1. (f) Intensity of (d) and (e). Because the input fields are complex, the output intensity in (f) at 2ω is not the square of the input intensity at ω in (c), unlike the case for Figs. 1(c) and 1(f).

Equations (7)

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

ex=axrn cosnθexp-br2, ey=±ayrn sinnθ+γexp-br2,
IC=-hR,LqR,L/2,
Ex=-2Cd22exey,    Ey=Cd22ey2-ex2.
Ex=±r2n sin2nθexp-2br2, Ey=-r2n cos2nθexp-2br2.
ψx=ψR+ψL/21/2,    ψy=iψR-ψL/21/2,
ψR=ψx-iψy/21/2,    ψL=ψx+iψy/21/2.
ER=i21/2eL2,    EL=-i21/2eR2.

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