Abstract

We investigate nonparaxial spatial solitons in uniaxial media with a cubic nonlinearity, accounting for the fully vectorial electromagnetic perturbation and the longitudinal field component in the self-induced index well. We discuss the effective nonlocality arising in purely Kerr media and soliton self-steering owing to nonlinear changes in walk-off.

© 2011 Optical Society of America

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  1. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 2002).
  2. G. I. Stegeman and M. Segev, Science 286, 1518(1999).
    [CrossRef] [PubMed]
  3. Y. S. Kivshar and G. P. Agrawal, Optical Solitons(Academic, 2003).
  4. J. E. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974).
    [CrossRef]
  5. T. G. Canva, R. A. Fuerst, S. Baboiu, G. I. Stegeman, and G. Assanto, Opt. Lett. 22, 1683 (1997).
    [CrossRef]
  6. B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, Opt. Lett. 32, 154 (2007).
    [CrossRef]
  7. W. Krolikowski, B. Luther-Davies, and C. Denz, IEEE J. Quantum Electron. 39, 3 (2003).
    [CrossRef]
  8. C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. E 72, 066614 (2005).
    [CrossRef]
  9. M. Peccianti and G. Assanto, Phys. Rev. E 65, 035603(2002).
    [CrossRef]
  10. C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 91, 073901 (2003).
    [CrossRef] [PubMed]
  11. D. Suter and T. Blasberg, Phys. Rev. A 48, 4583 (1993).
    [CrossRef] [PubMed]
  12. M. Peccianti, K. Brzadkiewicz, and G. Assanto, Opt. Lett. 27, 1460 (2002).
    [CrossRef]
  13. C. Rothschild, B. Alfassi, O. Cohen, and M. Segev, Nature Phys. 2, 769 (2006).
    [CrossRef]
  14. M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, Nat. Phys. 2, 737 (2006).
    [CrossRef]
  15. M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, Phys. Rev. Lett. 101, 153902 (2008).
    [CrossRef] [PubMed]
  16. A. Barak, O. Pereg, C. Stucchio, A. Soffer, and M. Segev, Phys. Rev. Lett. 100, 153901 (2008).
    [CrossRef] [PubMed]
  17. M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
    [CrossRef]
  18. S. Chi and Q. Guo, Opt. Lett. 20, 1598 (1995).
    [CrossRef] [PubMed]
  19. S. Blair, Chaos 10, 570 (2000).
    [CrossRef]
  20. A. Ciattoni, C. Conti, E. DelRe, P. Di Porto, B. Crosignani, and A. Yariv, Opt. Lett. 27, 734 (2002).
    [CrossRef]
  21. A. W. Snyder and D. J. Mitchell, Science 276, 1538(1997).
    [CrossRef]
  22. D. N. Christodoulides and T. H. Coskun, Opt. Lett. 21, 1220 (1996).
    [CrossRef] [PubMed]
  23. G. Leo, G. Assanto, and W. E. Torruellas, Opt. Lett. 22, 7(1997).
    [CrossRef] [PubMed]
  24. A. Piccardi, A. Alberucci, and G. Assanto, Appl. Phys. Lett. 96, 061105 (2010).
    [CrossRef]
  25. A. E. Kaplan, JETP Lett. 9, 33 (1969).
  26. M. Nazarathy and J. Goodman, J. Opt. Soc. Am. A 3, 523(1986).
    [CrossRef]
  27. W. Krolikowski and O. Bang, Phys. Rev. E 63, 016610(2000).
    [CrossRef]

2010 (1)

A. Piccardi, A. Alberucci, and G. Assanto, Appl. Phys. Lett. 96, 061105 (2010).
[CrossRef]

2008 (2)

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, Phys. Rev. Lett. 101, 153902 (2008).
[CrossRef] [PubMed]

A. Barak, O. Pereg, C. Stucchio, A. Soffer, and M. Segev, Phys. Rev. Lett. 100, 153901 (2008).
[CrossRef] [PubMed]

2007 (1)

2006 (2)

C. Rothschild, B. Alfassi, O. Cohen, and M. Segev, Nature Phys. 2, 769 (2006).
[CrossRef]

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, Nat. Phys. 2, 737 (2006).
[CrossRef]

2005 (1)

C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. E 72, 066614 (2005).
[CrossRef]

2003 (2)

C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef] [PubMed]

W. Krolikowski, B. Luther-Davies, and C. Denz, IEEE J. Quantum Electron. 39, 3 (2003).
[CrossRef]

2002 (3)

2000 (2)

W. Krolikowski and O. Bang, Phys. Rev. E 63, 016610(2000).
[CrossRef]

S. Blair, Chaos 10, 570 (2000).
[CrossRef]

1999 (1)

G. I. Stegeman and M. Segev, Science 286, 1518(1999).
[CrossRef] [PubMed]

1997 (3)

1996 (1)

1995 (1)

1993 (1)

D. Suter and T. Blasberg, Phys. Rev. A 48, 4583 (1993).
[CrossRef] [PubMed]

1986 (1)

1975 (1)

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

1974 (1)

J. E. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974).
[CrossRef]

1969 (1)

A. E. Kaplan, JETP Lett. 9, 33 (1969).

Agrawal, G. P.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons(Academic, 2003).

Alberucci, A.

A. Piccardi, A. Alberucci, and G. Assanto, Appl. Phys. Lett. 96, 061105 (2010).
[CrossRef]

Alfassi, B.

Ashkin, A.

J. E. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974).
[CrossRef]

Assanto, G.

A. Piccardi, A. Alberucci, and G. Assanto, Appl. Phys. Lett. 96, 061105 (2010).
[CrossRef]

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, Phys. Rev. Lett. 101, 153902 (2008).
[CrossRef] [PubMed]

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, Nat. Phys. 2, 737 (2006).
[CrossRef]

C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. E 72, 066614 (2005).
[CrossRef]

C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef] [PubMed]

M. Peccianti, K. Brzadkiewicz, and G. Assanto, Opt. Lett. 27, 1460 (2002).
[CrossRef]

M. Peccianti and G. Assanto, Phys. Rev. E 65, 035603(2002).
[CrossRef]

T. G. Canva, R. A. Fuerst, S. Baboiu, G. I. Stegeman, and G. Assanto, Opt. Lett. 22, 1683 (1997).
[CrossRef]

G. Leo, G. Assanto, and W. E. Torruellas, Opt. Lett. 22, 7(1997).
[CrossRef] [PubMed]

Baboiu, S.

Bang, O.

W. Krolikowski and O. Bang, Phys. Rev. E 63, 016610(2000).
[CrossRef]

Barak, A.

A. Barak, O. Pereg, C. Stucchio, A. Soffer, and M. Segev, Phys. Rev. Lett. 100, 153901 (2008).
[CrossRef] [PubMed]

Bjorkholm, J. E.

J. E. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974).
[CrossRef]

Blair, S.

S. Blair, Chaos 10, 570 (2000).
[CrossRef]

Blasberg, T.

D. Suter and T. Blasberg, Phys. Rev. A 48, 4583 (1993).
[CrossRef] [PubMed]

Brzadkiewicz, K.

Canva, T. G.

Chi, S.

Christodoulides, D. N.

Ciattoni, A.

Cohen, O.

C. Rothschild, B. Alfassi, O. Cohen, and M. Segev, Nature Phys. 2, 769 (2006).
[CrossRef]

Conti, C.

C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. E 72, 066614 (2005).
[CrossRef]

C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef] [PubMed]

A. Ciattoni, C. Conti, E. DelRe, P. Di Porto, B. Crosignani, and A. Yariv, Opt. Lett. 27, 734 (2002).
[CrossRef]

Coskun, T. H.

Crosignani, B.

DelRe, E.

Denz, C.

W. Krolikowski, B. Luther-Davies, and C. Denz, IEEE J. Quantum Electron. 39, 3 (2003).
[CrossRef]

Di Porto, P.

Dyadyusha, A.

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, Phys. Rev. Lett. 101, 153902 (2008).
[CrossRef] [PubMed]

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, Nat. Phys. 2, 737 (2006).
[CrossRef]

Fuerst, R. A.

Goodman, J.

Guo, Q.

Kaczmarek, M.

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, Phys. Rev. Lett. 101, 153902 (2008).
[CrossRef] [PubMed]

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, Nat. Phys. 2, 737 (2006).
[CrossRef]

Kaplan, A. E.

A. E. Kaplan, JETP Lett. 9, 33 (1969).

Kivshar, Y. S.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons(Academic, 2003).

Krolikowski, W.

W. Krolikowski, B. Luther-Davies, and C. Denz, IEEE J. Quantum Electron. 39, 3 (2003).
[CrossRef]

W. Krolikowski and O. Bang, Phys. Rev. E 63, 016610(2000).
[CrossRef]

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Leo, G.

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Luther-Davies, B.

W. Krolikowski, B. Luther-Davies, and C. Denz, IEEE J. Quantum Electron. 39, 3 (2003).
[CrossRef]

Manela, O.

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Mitchell, D. J.

A. W. Snyder and D. J. Mitchell, Science 276, 1538(1997).
[CrossRef]

Nazarathy, M.

Peccianti, M.

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, Phys. Rev. Lett. 101, 153902 (2008).
[CrossRef] [PubMed]

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, Nat. Phys. 2, 737 (2006).
[CrossRef]

C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. E 72, 066614 (2005).
[CrossRef]

C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef] [PubMed]

M. Peccianti and G. Assanto, Phys. Rev. E 65, 035603(2002).
[CrossRef]

M. Peccianti, K. Brzadkiewicz, and G. Assanto, Opt. Lett. 27, 1460 (2002).
[CrossRef]

Pereg, O.

A. Barak, O. Pereg, C. Stucchio, A. Soffer, and M. Segev, Phys. Rev. Lett. 100, 153901 (2008).
[CrossRef] [PubMed]

Piccardi, A.

A. Piccardi, A. Alberucci, and G. Assanto, Appl. Phys. Lett. 96, 061105 (2010).
[CrossRef]

Rothschild, C.

C. Rothschild, B. Alfassi, O. Cohen, and M. Segev, Nature Phys. 2, 769 (2006).
[CrossRef]

Rotschild, C.

Segev, M.

A. Barak, O. Pereg, C. Stucchio, A. Soffer, and M. Segev, Phys. Rev. Lett. 100, 153901 (2008).
[CrossRef] [PubMed]

B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, Opt. Lett. 32, 154 (2007).
[CrossRef]

C. Rothschild, B. Alfassi, O. Cohen, and M. Segev, Nature Phys. 2, 769 (2006).
[CrossRef]

G. I. Stegeman and M. Segev, Science 286, 1518(1999).
[CrossRef] [PubMed]

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 2002).

Snyder, A. W.

A. W. Snyder and D. J. Mitchell, Science 276, 1538(1997).
[CrossRef]

Soffer, A.

A. Barak, O. Pereg, C. Stucchio, A. Soffer, and M. Segev, Phys. Rev. Lett. 100, 153901 (2008).
[CrossRef] [PubMed]

Stegeman, G. I.

Stucchio, C.

A. Barak, O. Pereg, C. Stucchio, A. Soffer, and M. Segev, Phys. Rev. Lett. 100, 153901 (2008).
[CrossRef] [PubMed]

Suter, D.

D. Suter and T. Blasberg, Phys. Rev. A 48, 4583 (1993).
[CrossRef] [PubMed]

Torruellas, W. E.

Yariv, A.

Appl. Phys. Lett. (1)

A. Piccardi, A. Alberucci, and G. Assanto, Appl. Phys. Lett. 96, 061105 (2010).
[CrossRef]

Chaos (1)

S. Blair, Chaos 10, 570 (2000).
[CrossRef]

IEEE J. Quantum Electron. (1)

W. Krolikowski, B. Luther-Davies, and C. Denz, IEEE J. Quantum Electron. 39, 3 (2003).
[CrossRef]

J. Opt. Soc. Am. A (1)

JETP Lett. (1)

A. E. Kaplan, JETP Lett. 9, 33 (1969).

Nat. Phys. (1)

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, Nat. Phys. 2, 737 (2006).
[CrossRef]

Nature Phys. (1)

C. Rothschild, B. Alfassi, O. Cohen, and M. Segev, Nature Phys. 2, 769 (2006).
[CrossRef]

Opt. Lett. (7)

Phys. Rev. A (2)

D. Suter and T. Blasberg, Phys. Rev. A 48, 4583 (1993).
[CrossRef] [PubMed]

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Phys. Rev. E (3)

C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. E 72, 066614 (2005).
[CrossRef]

M. Peccianti and G. Assanto, Phys. Rev. E 65, 035603(2002).
[CrossRef]

W. Krolikowski and O. Bang, Phys. Rev. E 63, 016610(2000).
[CrossRef]

Phys. Rev. Lett. (4)

C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef] [PubMed]

J. E. Bjorkholm and A. Ashkin, Phys. Rev. Lett. 32, 129 (1974).
[CrossRef]

M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, Phys. Rev. Lett. 101, 153902 (2008).
[CrossRef] [PubMed]

A. Barak, O. Pereg, C. Stucchio, A. Soffer, and M. Segev, Phys. Rev. Lett. 100, 153901 (2008).
[CrossRef] [PubMed]

Science (2)

G. I. Stegeman and M. Segev, Science 286, 1518(1999).
[CrossRef] [PubMed]

A. W. Snyder and D. J. Mitchell, Science 276, 1538(1997).
[CrossRef]

Other (2)

Y. S. Kivshar and G. P. Agrawal, Optical Solitons(Academic, 2003).

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 2002).

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Figures (2)

Fig. 1
Fig. 1

First row: (a) | u | 2 and Δ n NL 2 , (b) Re ( e t ) , (c) Im ( e t ) , and (d) Im ( e s ) versus y / λ 0 for w / λ 0 = 0.94 and for n 2 P d β = 1.5 × 10 10 2 (solid black curve and black with triangles) and n 2 P d β = 1 × 10 12 2 (dotted red curve and red with circles). Quantities in (b)–(d) are normalized to the peak of | e t | . Second row: soliton (e) transversality, (f) waist w A , (g) Gaussianity (normalized scalar product of | u | and best-fit Gaussian), and (h) locality (ratio of waist | u | 2 to Δ n NL 2 width) versus excitation n 2 P d β ; blue solid, red with triangles, and green curves with squares correspond to w / λ 0 0 [i.e., G = δ ( y ) ], w / λ 0 = 0.94 , and w / λ 0 = 9.4 , respectively. Here δ = 8 ° , n e ( 0 ) = 1.6 , and β = 2 n e ( 0 ) cos ( δ ) Z 0 1 .

Fig. 2
Fig. 2

(a) Modulus and (b) phase of the soliton profile u versus y / λ 0 . Blue, green, and red curves [from bottom to top in (a) and in the reverse order in (b)] correspond to n 2 P d β = 1 , 3, and 5 × 10 11 2 , respectively. Here δ = 8 ° and n e ( 0 ) = 1.6 .

Equations (10)

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

( z + ϵ y z ϵ z z y ) H x = j ω [ n e 2 ϵ 0 E y ϵ y z ϵ z z P z NL + P y NL ] ,
( z + ϵ y z ϵ z z y ) E y = j ω ϵ z z y 2 H x + j ω μ 0 H x P z NL ϵ z z ,
E z = ϵ y z ϵ z z E y + j ω ϵ z z y H x P z NL ϵ z z ,
z 2 H x 2 z ( ln n e ) [ z H x + j ω cos δ P t NL ] + D y 2 H x + k 0 2 n e 2 H x j ω cos δ [ z P t NL D cos ( δ ) y P z NL ] = 0 ,
E t = z H x j ω ϵ 0 n e 2 cos δ j sin δ ω ϵ z z y H x P t NL ϵ 0 n e 2 cos 2 δ + sin δ ϵ z z P z NL ,
E s = j cos δ ω ϵ z z y H x cos δ ϵ z z P z NL .
k 0 2 ( n NL 2 + 2 n e ( 0 ) n NL ) u + D y 2 u + k 0 2 Δ n e 2 u k 0 2 c ( n e ( 0 ) + n NL ) cos δ p t NL + j k 0 c D y p z NL = 0 ,
S = 1 2 [ s ^ ( n e ( 0 ) + n NL n e 2 cos δ Z 0 | u | 2 + j sin δ ω ϵ z z u * y u + p t NL u * ϵ 0 n e 2 cos δ sin ( δ ) p z NL u * ϵ z z ) + t ^ ( j cos δ ω ϵ z z u * y u cos δ ϵ z z u * p z NL ) ] .
k 0 2 ( n NL 2 + 2 n e ( 0 ) n NL ) u = D y 2 u + k 0 2 Δ n NL 2 u ,
Δ n NL 2 = [ G y y ( y ξ ) | e y | 2 + G z z ( y ξ ) | e z | 2 + 2 G y z ( y ξ ) Re ( e y e z * ) ] d ξ .

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