Abstract

The components of electric field of the third harmonic beam, generated in isotropic medium with cubic nonlinearity by a monochromatic light beam carrying polarization singularity of an arbitrary type, are found analytically. The relation between C-points characteristics in the fundamental and signal beams are determined, as well as the impact of the phase mismatch on the shape of the C-lines.

© 2017 Optical Society of America

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Polarization singularities in superposition of vector beams

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Opt. Express 21(7) 8972-8986 (2013)

References

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  1. P. Bey, J. Giuliani, and H. Rabin, “Linear and circular polarized laser radiation in optical third harmonic generation,” Phys. Lett. A 26, 128–129 (1968).
    [Crossref]
  2. K. S. Grigoriev, N. Y. Kuznetsov, E. B. Cherepetskaya, and V. A. Makarov, “Second harmonic generation in isotropic chiral medium with nonlocality of nonlinear optical response by heterogeneously polarized pulsed beams,” Opt. Express 25, 6253–6262 (2017).
    [Crossref] [PubMed]
  3. I. A. Perezhogin, K. S. Grigoriev, N. N. Potravkin, E. B. Cherepetskaya, and V. A. Makarov, “Transfer efficiency of angular momentum in sum-frequency generation and control of its spin and orbital parts by varying polarization and frequency of fundamental beams,” Laser Phys. Lett. 14, 085401 (2017).
    [Crossref]
  4. B. Hao and J. Leger, “Polarization beam shaping,” Appl. Opt. 46, 8211–8217 (2007).
    [Crossref] [PubMed]
  5. R. Won, “Optical physics: Shaping the topology of light,” Nat. Photon. 8, 8 (2014).
    [Crossref]
  6. F. Cardano, E. Karimi, L. Marrucci, C. de Lisio, and E. Santamato, “Generation and dynamics of optical beams with polarization singularities,” Opt. Express 21, 8815–8820 (2013).
    [Crossref] [PubMed]
  7. K. E. Ballantine, J. F. Donegan, and P. R. Eastham, “There are many ways to spin a photon: Half-quantization of a total optical angular momentum,” Sci. Adv. 2, e1501748 (2016).
    [Crossref] [PubMed]
  8. M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Chapter 5 singular optics: Optical vortices and polarization singularities,” (Elsevier, 2009), pp. 293–363.
  9. J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Second-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes,” Phys. Rev. A 56, 4193–4196 (1997).
    [Crossref]
  10. A. Beržanskis, A. Matijošius, A. Piskarskas, V. Smilgevičius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–380 (1998).
    [Crossref]
  11. W. T. Buono, L. F. C. Moraes, J. A. O. Huguenin, C. E. R. Souza, and A. Z. Khoury, “Arbitrary orbital angular momentum addition in second harmonic generation,” New J. Phys. 16, 093041 (2014).
    [Crossref]
  12. E. J. Galvez, B. L. Rojec, V. Kumar, and N. K. Viswanathan, “Generation of isolated asymmetric umbilics in light’s polarization,” Phys. Rev. A 89, 031801 (2014).
    [Crossref]
  13. R. S. Tasgal and Y. B. Band, “Third-harmonic generation in isotropic media by focused pulses,” Phys. Rev. A 70, 053810 (2004).
    [Crossref]
  14. K. S. Grigoriev, V. A. Makarov, and I. A. Perezhogin, “Polarization singularities in a sum-frequency light beam generated by a bichromatic singular beam in the bulk of an isotropic nonlinear chiral medium,” Phys. Rev. A 92, 023814 (2015).
    [Crossref]

2017 (2)

K. S. Grigoriev, N. Y. Kuznetsov, E. B. Cherepetskaya, and V. A. Makarov, “Second harmonic generation in isotropic chiral medium with nonlocality of nonlinear optical response by heterogeneously polarized pulsed beams,” Opt. Express 25, 6253–6262 (2017).
[Crossref] [PubMed]

I. A. Perezhogin, K. S. Grigoriev, N. N. Potravkin, E. B. Cherepetskaya, and V. A. Makarov, “Transfer efficiency of angular momentum in sum-frequency generation and control of its spin and orbital parts by varying polarization and frequency of fundamental beams,” Laser Phys. Lett. 14, 085401 (2017).
[Crossref]

2016 (1)

K. E. Ballantine, J. F. Donegan, and P. R. Eastham, “There are many ways to spin a photon: Half-quantization of a total optical angular momentum,” Sci. Adv. 2, e1501748 (2016).
[Crossref] [PubMed]

2015 (1)

K. S. Grigoriev, V. A. Makarov, and I. A. Perezhogin, “Polarization singularities in a sum-frequency light beam generated by a bichromatic singular beam in the bulk of an isotropic nonlinear chiral medium,” Phys. Rev. A 92, 023814 (2015).
[Crossref]

2014 (3)

W. T. Buono, L. F. C. Moraes, J. A. O. Huguenin, C. E. R. Souza, and A. Z. Khoury, “Arbitrary orbital angular momentum addition in second harmonic generation,” New J. Phys. 16, 093041 (2014).
[Crossref]

E. J. Galvez, B. L. Rojec, V. Kumar, and N. K. Viswanathan, “Generation of isolated asymmetric umbilics in light’s polarization,” Phys. Rev. A 89, 031801 (2014).
[Crossref]

R. Won, “Optical physics: Shaping the topology of light,” Nat. Photon. 8, 8 (2014).
[Crossref]

2013 (1)

2007 (1)

2004 (1)

R. S. Tasgal and Y. B. Band, “Third-harmonic generation in isotropic media by focused pulses,” Phys. Rev. A 70, 053810 (2004).
[Crossref]

1998 (1)

A. Beržanskis, A. Matijošius, A. Piskarskas, V. Smilgevičius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–380 (1998).
[Crossref]

1997 (1)

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Second-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes,” Phys. Rev. A 56, 4193–4196 (1997).
[Crossref]

1968 (1)

P. Bey, J. Giuliani, and H. Rabin, “Linear and circular polarized laser radiation in optical third harmonic generation,” Phys. Lett. A 26, 128–129 (1968).
[Crossref]

Allen, L.

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Second-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes,” Phys. Rev. A 56, 4193–4196 (1997).
[Crossref]

Ballantine, K. E.

K. E. Ballantine, J. F. Donegan, and P. R. Eastham, “There are many ways to spin a photon: Half-quantization of a total optical angular momentum,” Sci. Adv. 2, e1501748 (2016).
[Crossref] [PubMed]

Band, Y. B.

R. S. Tasgal and Y. B. Band, “Third-harmonic generation in isotropic media by focused pulses,” Phys. Rev. A 70, 053810 (2004).
[Crossref]

Beržanskis, A.

A. Beržanskis, A. Matijošius, A. Piskarskas, V. Smilgevičius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–380 (1998).
[Crossref]

Bey, P.

P. Bey, J. Giuliani, and H. Rabin, “Linear and circular polarized laser radiation in optical third harmonic generation,” Phys. Lett. A 26, 128–129 (1968).
[Crossref]

Buono, W. T.

W. T. Buono, L. F. C. Moraes, J. A. O. Huguenin, C. E. R. Souza, and A. Z. Khoury, “Arbitrary orbital angular momentum addition in second harmonic generation,” New J. Phys. 16, 093041 (2014).
[Crossref]

Cardano, F.

Cherepetskaya, E. B.

K. S. Grigoriev, N. Y. Kuznetsov, E. B. Cherepetskaya, and V. A. Makarov, “Second harmonic generation in isotropic chiral medium with nonlocality of nonlinear optical response by heterogeneously polarized pulsed beams,” Opt. Express 25, 6253–6262 (2017).
[Crossref] [PubMed]

I. A. Perezhogin, K. S. Grigoriev, N. N. Potravkin, E. B. Cherepetskaya, and V. A. Makarov, “Transfer efficiency of angular momentum in sum-frequency generation and control of its spin and orbital parts by varying polarization and frequency of fundamental beams,” Laser Phys. Lett. 14, 085401 (2017).
[Crossref]

Courtial, J.

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Second-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes,” Phys. Rev. A 56, 4193–4196 (1997).
[Crossref]

de Lisio, C.

Dennis, M. R.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Chapter 5 singular optics: Optical vortices and polarization singularities,” (Elsevier, 2009), pp. 293–363.

Dholakia, K.

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Second-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes,” Phys. Rev. A 56, 4193–4196 (1997).
[Crossref]

Donegan, J. F.

K. E. Ballantine, J. F. Donegan, and P. R. Eastham, “There are many ways to spin a photon: Half-quantization of a total optical angular momentum,” Sci. Adv. 2, e1501748 (2016).
[Crossref] [PubMed]

Eastham, P. R.

K. E. Ballantine, J. F. Donegan, and P. R. Eastham, “There are many ways to spin a photon: Half-quantization of a total optical angular momentum,” Sci. Adv. 2, e1501748 (2016).
[Crossref] [PubMed]

Galvez, E. J.

E. J. Galvez, B. L. Rojec, V. Kumar, and N. K. Viswanathan, “Generation of isolated asymmetric umbilics in light’s polarization,” Phys. Rev. A 89, 031801 (2014).
[Crossref]

Giuliani, J.

P. Bey, J. Giuliani, and H. Rabin, “Linear and circular polarized laser radiation in optical third harmonic generation,” Phys. Lett. A 26, 128–129 (1968).
[Crossref]

Grigoriev, K. S.

I. A. Perezhogin, K. S. Grigoriev, N. N. Potravkin, E. B. Cherepetskaya, and V. A. Makarov, “Transfer efficiency of angular momentum in sum-frequency generation and control of its spin and orbital parts by varying polarization and frequency of fundamental beams,” Laser Phys. Lett. 14, 085401 (2017).
[Crossref]

K. S. Grigoriev, N. Y. Kuznetsov, E. B. Cherepetskaya, and V. A. Makarov, “Second harmonic generation in isotropic chiral medium with nonlocality of nonlinear optical response by heterogeneously polarized pulsed beams,” Opt. Express 25, 6253–6262 (2017).
[Crossref] [PubMed]

K. S. Grigoriev, V. A. Makarov, and I. A. Perezhogin, “Polarization singularities in a sum-frequency light beam generated by a bichromatic singular beam in the bulk of an isotropic nonlinear chiral medium,” Phys. Rev. A 92, 023814 (2015).
[Crossref]

Hao, B.

Huguenin, J. A. O.

W. T. Buono, L. F. C. Moraes, J. A. O. Huguenin, C. E. R. Souza, and A. Z. Khoury, “Arbitrary orbital angular momentum addition in second harmonic generation,” New J. Phys. 16, 093041 (2014).
[Crossref]

Karimi, E.

Khoury, A. Z.

W. T. Buono, L. F. C. Moraes, J. A. O. Huguenin, C. E. R. Souza, and A. Z. Khoury, “Arbitrary orbital angular momentum addition in second harmonic generation,” New J. Phys. 16, 093041 (2014).
[Crossref]

Kumar, V.

E. J. Galvez, B. L. Rojec, V. Kumar, and N. K. Viswanathan, “Generation of isolated asymmetric umbilics in light’s polarization,” Phys. Rev. A 89, 031801 (2014).
[Crossref]

Kuznetsov, N. Y.

Leger, J.

Makarov, V. A.

K. S. Grigoriev, N. Y. Kuznetsov, E. B. Cherepetskaya, and V. A. Makarov, “Second harmonic generation in isotropic chiral medium with nonlocality of nonlinear optical response by heterogeneously polarized pulsed beams,” Opt. Express 25, 6253–6262 (2017).
[Crossref] [PubMed]

I. A. Perezhogin, K. S. Grigoriev, N. N. Potravkin, E. B. Cherepetskaya, and V. A. Makarov, “Transfer efficiency of angular momentum in sum-frequency generation and control of its spin and orbital parts by varying polarization and frequency of fundamental beams,” Laser Phys. Lett. 14, 085401 (2017).
[Crossref]

K. S. Grigoriev, V. A. Makarov, and I. A. Perezhogin, “Polarization singularities in a sum-frequency light beam generated by a bichromatic singular beam in the bulk of an isotropic nonlinear chiral medium,” Phys. Rev. A 92, 023814 (2015).
[Crossref]

Marrucci, L.

Matijošius, A.

A. Beržanskis, A. Matijošius, A. Piskarskas, V. Smilgevičius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–380 (1998).
[Crossref]

Moraes, L. F. C.

W. T. Buono, L. F. C. Moraes, J. A. O. Huguenin, C. E. R. Souza, and A. Z. Khoury, “Arbitrary orbital angular momentum addition in second harmonic generation,” New J. Phys. 16, 093041 (2014).
[Crossref]

O’Holleran, K.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Chapter 5 singular optics: Optical vortices and polarization singularities,” (Elsevier, 2009), pp. 293–363.

Padgett, M. J.

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Second-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes,” Phys. Rev. A 56, 4193–4196 (1997).
[Crossref]

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Chapter 5 singular optics: Optical vortices and polarization singularities,” (Elsevier, 2009), pp. 293–363.

Perezhogin, I. A.

I. A. Perezhogin, K. S. Grigoriev, N. N. Potravkin, E. B. Cherepetskaya, and V. A. Makarov, “Transfer efficiency of angular momentum in sum-frequency generation and control of its spin and orbital parts by varying polarization and frequency of fundamental beams,” Laser Phys. Lett. 14, 085401 (2017).
[Crossref]

K. S. Grigoriev, V. A. Makarov, and I. A. Perezhogin, “Polarization singularities in a sum-frequency light beam generated by a bichromatic singular beam in the bulk of an isotropic nonlinear chiral medium,” Phys. Rev. A 92, 023814 (2015).
[Crossref]

Piskarskas, A.

A. Beržanskis, A. Matijošius, A. Piskarskas, V. Smilgevičius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–380 (1998).
[Crossref]

Potravkin, N. N.

I. A. Perezhogin, K. S. Grigoriev, N. N. Potravkin, E. B. Cherepetskaya, and V. A. Makarov, “Transfer efficiency of angular momentum in sum-frequency generation and control of its spin and orbital parts by varying polarization and frequency of fundamental beams,” Laser Phys. Lett. 14, 085401 (2017).
[Crossref]

Rabin, H.

P. Bey, J. Giuliani, and H. Rabin, “Linear and circular polarized laser radiation in optical third harmonic generation,” Phys. Lett. A 26, 128–129 (1968).
[Crossref]

Rojec, B. L.

E. J. Galvez, B. L. Rojec, V. Kumar, and N. K. Viswanathan, “Generation of isolated asymmetric umbilics in light’s polarization,” Phys. Rev. A 89, 031801 (2014).
[Crossref]

Santamato, E.

Smilgevicius, V.

A. Beržanskis, A. Matijošius, A. Piskarskas, V. Smilgevičius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–380 (1998).
[Crossref]

Souza, C. E. R.

W. T. Buono, L. F. C. Moraes, J. A. O. Huguenin, C. E. R. Souza, and A. Z. Khoury, “Arbitrary orbital angular momentum addition in second harmonic generation,” New J. Phys. 16, 093041 (2014).
[Crossref]

Stabinis, A.

A. Beržanskis, A. Matijošius, A. Piskarskas, V. Smilgevičius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–380 (1998).
[Crossref]

Tasgal, R. S.

R. S. Tasgal and Y. B. Band, “Third-harmonic generation in isotropic media by focused pulses,” Phys. Rev. A 70, 053810 (2004).
[Crossref]

Viswanathan, N. K.

E. J. Galvez, B. L. Rojec, V. Kumar, and N. K. Viswanathan, “Generation of isolated asymmetric umbilics in light’s polarization,” Phys. Rev. A 89, 031801 (2014).
[Crossref]

Won, R.

R. Won, “Optical physics: Shaping the topology of light,” Nat. Photon. 8, 8 (2014).
[Crossref]

Appl. Opt. (1)

Laser Phys. Lett. (1)

I. A. Perezhogin, K. S. Grigoriev, N. N. Potravkin, E. B. Cherepetskaya, and V. A. Makarov, “Transfer efficiency of angular momentum in sum-frequency generation and control of its spin and orbital parts by varying polarization and frequency of fundamental beams,” Laser Phys. Lett. 14, 085401 (2017).
[Crossref]

Nat. Photon. (1)

R. Won, “Optical physics: Shaping the topology of light,” Nat. Photon. 8, 8 (2014).
[Crossref]

New J. Phys. (1)

W. T. Buono, L. F. C. Moraes, J. A. O. Huguenin, C. E. R. Souza, and A. Z. Khoury, “Arbitrary orbital angular momentum addition in second harmonic generation,” New J. Phys. 16, 093041 (2014).
[Crossref]

Opt. Commun. (1)

A. Beržanskis, A. Matijošius, A. Piskarskas, V. Smilgevičius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–380 (1998).
[Crossref]

Opt. Express (2)

Phys. Lett. A (1)

P. Bey, J. Giuliani, and H. Rabin, “Linear and circular polarized laser radiation in optical third harmonic generation,” Phys. Lett. A 26, 128–129 (1968).
[Crossref]

Phys. Rev. A (4)

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Second-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes,” Phys. Rev. A 56, 4193–4196 (1997).
[Crossref]

E. J. Galvez, B. L. Rojec, V. Kumar, and N. K. Viswanathan, “Generation of isolated asymmetric umbilics in light’s polarization,” Phys. Rev. A 89, 031801 (2014).
[Crossref]

R. S. Tasgal and Y. B. Band, “Third-harmonic generation in isotropic media by focused pulses,” Phys. Rev. A 70, 053810 (2004).
[Crossref]

K. S. Grigoriev, V. A. Makarov, and I. A. Perezhogin, “Polarization singularities in a sum-frequency light beam generated by a bichromatic singular beam in the bulk of an isotropic nonlinear chiral medium,” Phys. Rev. A 92, 023814 (2015).
[Crossref]

Sci. Adv. (1)

K. E. Ballantine, J. F. Donegan, and P. R. Eastham, “There are many ways to spin a photon: Half-quantization of a total optical angular momentum,” Sci. Adv. 2, e1501748 (2016).
[Crossref] [PubMed]

Other (1)

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Chapter 5 singular optics: Optical vortices and polarization singularities,” (Elsevier, 2009), pp. 293–363.

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Figures (2)

Fig. 1
Fig. 1 Polarization distribution in the waist cross-section (z = 0) of the fundamental beam (a) and the maximum intensity cross-section (z = 1.5zR) of the signal beam (b). C-points with topological charge 1/2 (−1/2) are marked by filled (open) circles. Right-hand polarized ellipses are opened and left-hand ones are filled. The fundamentsl beam parameters are EG/EL = 3, θ = π/4, η = 0, σ = −1, the phase mismatch Δk = 0.
Fig. 2
Fig. 2 Examples of the C-lines in the third-harmonic beam at various signs of the phase mismatch (a) Δk = 5/zR, (b) Δk = −5/zR. C-lines with topological charge 1/2 (−1/2) are marked by filled (open) circles. Two C-lines with positive topological charge are coloured in different tones to be better distinguished from each other. The parameters of the fundamental beam are EG/EL = 3, θ = π/4, η = 0, σ = −1.

Equations (8)

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E ω ( x , y , z ) = 1 β ( z ) ( E G e + E L p ( x + i σ y ) + q ( x i σ y ) 2 w β ( z ) e * ) exp ( x 2 + y 2 w 2 β ( z ) ) ,
P NL ( x , y , z ) = χ ( 3 ) ( E ω E ω ) E ω ,
( z i 2 k 3 ω Δ ) E 3 ω = 2 π i k 3 ω ε 3 ω P NL exp [ i ( 3 k ω k 3 ω ) z ] .
E 3 ω ( x , y , z ) = A { J 1 ( x , y , z ) p ( x + i σ y ) + q ( x i σ y ) 2 w E G e + + [ p q J 2 ( x , y , z ) + ( p ( x + i σ y ) + q ( x i σ y ) 2 w ) 2 J 3 ( x , y , z ) ] E L e * } .
J 1 ( x , y , z ) = z 0 z R ξ exp ( i ν ξ ) β ˜ 2 ( ξ ) B 2 ( ξ , ξ ) exp ( 3 ( x 2 + y 2 ) w 2 B ( ξ , ξ ) ) d ξ ,
J 2 ( x , y , z ) = i z 0 z R ξ ( ξ ξ ) k 3 ω / k ω exp ( i ν ξ ) β ˜ 3 ( ξ ) B 2 ( ξ , ξ ) exp ( 3 ( x 2 + y 2 ) w 2 B ( ξ , ξ ) ) d ξ ,
J 3 ( x , y , z ) = z 0 z R ξ exp ( i ν ξ ) β ˜ 2 ( ξ ) B 3 ( ξ , ξ ) exp ( 3 ( x 2 + y 2 ) w 2 B ( ξ , ξ ) ) d ξ ,
P NL = χ ( 3 ) E G E L β 3 ( z ) { p ( x + i σ y ) + q ( x i σ y ) 2 w β ( z ) E G e + + ( p ( x + i σ y ) + q ( x i σ y ) 2 w β ( z ) ) 2 E L e * } exp ( 3 x 2 + y 2 w 2 β ( z ) ) .

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