Abstract

In a way analogous to a light pulse that can be optically delayed via slow light propagation in Kerr-type nonlinear media, we theoretically demonstrate that beam steering and spatial walk-off compensation can be achieved in noncollinear optical parametric amplification. We identify this effect as a result of the quadratic phase shift induced by parametric amplification that leads to the cancellation of the spatial walk-off and collinear propagation of all beams though they have different wavevectors. Experimental evidence is reported of a soliton array steering in a Kerr slab waveguide.

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  1. L. Thévenaz, “Slow and fast light in optical fibers,” Nat. Photon.2, 474–481 (2008).
    [CrossRef]
  2. Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. Boyd, and A. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett.94, 153902 (2005).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  4. J.E. Sharping, Y. Okawachi, and A.L. Gaeta, “Wide bandwidth slow light using a Raman fiber amplifier,” Opt. Express13, 6092–6098 (2005).
    [CrossRef] [PubMed]
  5. D. Dahan and G. Eisenstein, “Tunable all optical delay via slow and fast light propagation in a Raman-assisted fiber optical parametric amplifier: a route to all optical buffering,” Opt. Express13, 6234–6249 (2005).
    [CrossRef] [PubMed]
  6. E. Shumakher, A. Willinger, R. Blit, D. Dahan, and G. Eisenstein, “Large tunable delay with low distortion of 10 Gbit/s data in a slow light system based on narrow band fiber parametric amplification,” Opt. Express14, 8540–8545 (2006).
    [CrossRef] [PubMed]
  7. N. Nasser, G. Fanjoux, E. Lantz, and T. Sylvestre, “Tunable optical delay using parametric amplification in highly birefringent optical fibers,” J. Opt. Soc. Am. B28, 2352–2357 (2011).
    [CrossRef]
  8. M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of anisotropic spatial solitons and modulation instability in liquid crystals,” Nature432, 733–737 (2004).
    [CrossRef] [PubMed]
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    [CrossRef]
  10. L. Torner, W. E. Torruellas, G. I. Stegeman, and C. R. Menyuk, “Beam steering by χ(2) trapping,” Opt. Lett.20, 1952–1954 (1995).
    [CrossRef] [PubMed]
  11. R. L. Carman, R. Y. Chiao, and P. L. Kelley, “Observation of degenerate stimulated four-photon interaction and four-wave parametric amplification,” Phys. Rev. Lett.17, 1281–1283 (1995).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. R. Malendevich, L. Jankovic, G. Stegeman, and J. S. Aitchison, “Spatial modulation instability in a Kerr slab waveguide,” Opt. Lett.26,1879–1881 (2001).
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    [CrossRef]
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    [CrossRef] [PubMed]

2011 (1)

2008 (2)

L. Thévenaz, “Slow and fast light in optical fibers,” Nat. Photon.2, 474–481 (2008).
[CrossRef]

G. Fanjoux, J. Michaud, H. Maillotte, and T. Sylvestre, “Slow-light spatial soliton,” Phys. Rev. Lett.100, 013908 (2008).
[CrossRef] [PubMed]

2007 (1)

2006 (1)

2005 (4)

2004 (1)

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of anisotropic spatial solitons and modulation instability in liquid crystals,” Nature432, 733–737 (2004).
[CrossRef] [PubMed]

2002 (1)

2001 (1)

1996 (1)

W. E. Torruellas, G. Assanto, B. L. Lawrence, R. A. Fuerst, and G. I. Stegeman, “All-optical switching by spatial walk-off compensation and solitary wave locking,” Appl. Phys. Lett.68, 1449–1451 (1996).
[CrossRef]

1995 (2)

L. Torner, W. E. Torruellas, G. I. Stegeman, and C. R. Menyuk, “Beam steering by χ(2) trapping,” Opt. Lett.20, 1952–1954 (1995).
[CrossRef] [PubMed]

R. L. Carman, R. Y. Chiao, and P. L. Kelley, “Observation of degenerate stimulated four-photon interaction and four-wave parametric amplification,” Phys. Rev. Lett.17, 1281–1283 (1995).
[CrossRef]

1993 (2)

A. Penzkofer and H. J. Lehmeier, “Theoretical investigation of noncollinear phase-matched parametric four-photon amplification of ultrashort light pulses in isotropic media,” Opt. Quant. Electron.25, 815–844 (1993).
[CrossRef]

M. Kauranen, A. L. Gaeta, and C. J. McKinstrie, “Transverse instabilities of two intersecting laser beams in a nonlinear Kerr medium,” J. Opt. Soc. Am. B10, 2298–2305 (1993).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, 2001).

Aitchison, J. S.

Assanto, G.

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of anisotropic spatial solitons and modulation instability in liquid crystals,” Nature432, 733–737 (2004).
[CrossRef] [PubMed]

W. E. Torruellas, G. Assanto, B. L. Lawrence, R. A. Fuerst, and G. I. Stegeman, “All-optical switching by spatial walk-off compensation and solitary wave locking,” Appl. Phys. Lett.68, 1449–1451 (1996).
[CrossRef]

Bigelow, M. S.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. Boyd, and A. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett.94, 153902 (2005).
[CrossRef] [PubMed]

Blit, R.

Boyd, R.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. Boyd, and A. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett.94, 153902 (2005).
[CrossRef] [PubMed]

Cambournac, C.

Carman, R. L.

R. L. Carman, R. Y. Chiao, and P. L. Kelley, “Observation of degenerate stimulated four-photon interaction and four-wave parametric amplification,” Phys. Rev. Lett.17, 1281–1283 (1995).
[CrossRef]

Chauvet, M.

Chiao, R. Y.

R. L. Carman, R. Y. Chiao, and P. L. Kelley, “Observation of degenerate stimulated four-photon interaction and four-wave parametric amplification,” Phys. Rev. Lett.17, 1281–1283 (1995).
[CrossRef]

Conti, C.

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of anisotropic spatial solitons and modulation instability in liquid crystals,” Nature432, 733–737 (2004).
[CrossRef] [PubMed]

Dahan, D.

De Luca, A.

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of anisotropic spatial solitons and modulation instability in liquid crystals,” Nature432, 733–737 (2004).
[CrossRef] [PubMed]

Di Trapani, P.

Dubietis, A.

Dudley, J.

Eisenstein, G.

Faccio, D.

Fanjoux, G.

Fuerst, R. A.

W. E. Torruellas, G. Assanto, B. L. Lawrence, R. A. Fuerst, and G. I. Stegeman, “All-optical switching by spatial walk-off compensation and solitary wave locking,” Appl. Phys. Lett.68, 1449–1451 (1996).
[CrossRef]

Gaeta, A.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. Boyd, and A. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett.94, 153902 (2005).
[CrossRef] [PubMed]

Gaeta, A. L.

Gaeta, A.L.

Gauthier, D. J.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. Boyd, and A. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett.94, 153902 (2005).
[CrossRef] [PubMed]

Herráez, M. G.

Jankovic, L.

Kauranen, M.

Kelley, P. L.

R. L. Carman, R. Y. Chiao, and P. L. Kelley, “Observation of degenerate stimulated four-photon interaction and four-wave parametric amplification,” Phys. Rev. Lett.17, 1281–1283 (1995).
[CrossRef]

Lantz, E.

Lawrence, B. L.

W. E. Torruellas, G. Assanto, B. L. Lawrence, R. A. Fuerst, and G. I. Stegeman, “All-optical switching by spatial walk-off compensation and solitary wave locking,” Appl. Phys. Lett.68, 1449–1451 (1996).
[CrossRef]

Lehmeier, H. J.

A. Penzkofer and H. J. Lehmeier, “Theoretical investigation of noncollinear phase-matched parametric four-photon amplification of ultrashort light pulses in isotropic media,” Opt. Quant. Electron.25, 815–844 (1993).
[CrossRef]

Maillotte, H.

Malendevich, R.

McKinstrie, C. J.

Menyuk, C. R.

Michaud, J.

G. Fanjoux, J. Michaud, H. Maillotte, and T. Sylvestre, “Slow-light spatial soliton,” Phys. Rev. Lett.100, 013908 (2008).
[CrossRef] [PubMed]

Nasser, N.

Okawachi, Y.

J.E. Sharping, Y. Okawachi, and A.L. Gaeta, “Wide bandwidth slow light using a Raman fiber amplifier,” Opt. Express13, 6092–6098 (2005).
[CrossRef] [PubMed]

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. Boyd, and A. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett.94, 153902 (2005).
[CrossRef] [PubMed]

Peccianti, M.

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of anisotropic spatial solitons and modulation instability in liquid crystals,” Nature432, 733–737 (2004).
[CrossRef] [PubMed]

Penzkofer, A.

A. Penzkofer and H. J. Lehmeier, “Theoretical investigation of noncollinear phase-matched parametric four-photon amplification of ultrashort light pulses in isotropic media,” Opt. Quant. Electron.25, 815–844 (1993).
[CrossRef]

Piskarskas, A.

Polesana, P.

Schweinsberg, A.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. Boyd, and A. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett.94, 153902 (2005).
[CrossRef] [PubMed]

Sharping, J. E.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. Boyd, and A. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett.94, 153902 (2005).
[CrossRef] [PubMed]

Sharping, J.E.

Shumakher, E.

Song, K. Y.

Stegeman, G.

Stegeman, G. I.

W. E. Torruellas, G. Assanto, B. L. Lawrence, R. A. Fuerst, and G. I. Stegeman, “All-optical switching by spatial walk-off compensation and solitary wave locking,” Appl. Phys. Lett.68, 1449–1451 (1996).
[CrossRef]

L. Torner, W. E. Torruellas, G. I. Stegeman, and C. R. Menyuk, “Beam steering by χ(2) trapping,” Opt. Lett.20, 1952–1954 (1995).
[CrossRef] [PubMed]

Sylvestre, T.

Tamosauskas, G.

Thévenaz, L.

Torner, L.

Torruellas, W. E.

W. E. Torruellas, G. Assanto, B. L. Lawrence, R. A. Fuerst, and G. I. Stegeman, “All-optical switching by spatial walk-off compensation and solitary wave locking,” Appl. Phys. Lett.68, 1449–1451 (1996).
[CrossRef]

L. Torner, W. E. Torruellas, G. I. Stegeman, and C. R. Menyuk, “Beam steering by χ(2) trapping,” Opt. Lett.20, 1952–1954 (1995).
[CrossRef] [PubMed]

Umeton, C.

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of anisotropic spatial solitons and modulation instability in liquid crystals,” Nature432, 733–737 (2004).
[CrossRef] [PubMed]

Valiulis, G.

Valtna, H.

Willinger, A.

Zhu, Z.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. Boyd, and A. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett.94, 153902 (2005).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

W. E. Torruellas, G. Assanto, B. L. Lawrence, R. A. Fuerst, and G. I. Stegeman, “All-optical switching by spatial walk-off compensation and solitary wave locking,” Appl. Phys. Lett.68, 1449–1451 (1996).
[CrossRef]

J. Opt. Soc. Am. B (3)

Nat. Photon. (1)

L. Thévenaz, “Slow and fast light in optical fibers,” Nat. Photon.2, 474–481 (2008).
[CrossRef]

Nature (1)

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of anisotropic spatial solitons and modulation instability in liquid crystals,” Nature432, 733–737 (2004).
[CrossRef] [PubMed]

Opt. Express (5)

Opt. Lett. (2)

Opt. Quant. Electron. (1)

A. Penzkofer and H. J. Lehmeier, “Theoretical investigation of noncollinear phase-matched parametric four-photon amplification of ultrashort light pulses in isotropic media,” Opt. Quant. Electron.25, 815–844 (1993).
[CrossRef]

Phys. Rev. Lett. (3)

G. Fanjoux, J. Michaud, H. Maillotte, and T. Sylvestre, “Slow-light spatial soliton,” Phys. Rev. Lett.100, 013908 (2008).
[CrossRef] [PubMed]

R. L. Carman, R. Y. Chiao, and P. L. Kelley, “Observation of degenerate stimulated four-photon interaction and four-wave parametric amplification,” Phys. Rev. Lett.17, 1281–1283 (1995).
[CrossRef]

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. Boyd, and A. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett.94, 153902 (2005).
[CrossRef] [PubMed]

Other (1)

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, 2001).

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Figures (4)

Fig. 1
Fig. 1

(a) Spatial optical parametric amplification. (b) Scheme of the OPA-induced spatial walk-off compensation between the linear (dashed curves) and the nonlinear (solid curves) regime. (c) Real (black) and imaginary (dashed) parts of the complex parametric gain gc versus the spatial frequency. Red curve shows the induced spatial walk-off. Parameters are the L=7cm, β =1.9×107m−1, γ = 4.10−11 m.W−1, I = 1.8×1012 W.m−2.

Fig. 2
Fig. 2

Numerical simulations: (a) Input (blue) and output (green) parametric gain spectra and theoretical gain (red). (b) Normalized spatial intensity profiles versus propagation distance for signal and idler beams in the linear (pump off) and nonlinear (pump on) regime, respectively. Parameters are the same as in Fig. 1.

Fig. 3
Fig. 3

Spatial walk-off versus the spatial frequency in the linear regime (pump off, dashed blue line) and in the nonlinear regime (pump on, black solid curve for analytical predictions and green curve for numerical results). Propagation length is L = 7 cm and pump intensity is I = 1.8×1012 W.m−2. OPA gain band is also plotted a red dotted curve.

Fig. 4
Fig. 4

Observation of soliton array steering by optical parametric amplification (after Ref. [15]). (a) Numerical simulations: straight propagation of the transverse modulation in the linear regime, I = 1.2×104 W.m−2. (b) Nonlinear regime: soliton array generation and steering by OPA, I = 1.2×1012 W.m−2. (c) Numerical simulations: intensity profiles at the waveguide input (blue) and output (green). (d) Experimental observations. Ω/2π = 8550m−1 and L=7cm.

Equations (5)

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

A z = i 2 β 2 A x 2 + i γ | A | 2 A ,
B s z = i γ I B i * e i κ z ; B i z = i γ I B s * e i κ z ,
g c = g i κ 2 with g 2 = ( γ I ) 2 ( κ 2 ) 2
Δ n g x = c m ( g c ) Ω and X N L = 0 L Δ n g x c d z
Δ n g x = c Ω β and X N L = Ω β L .

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