Abstract

In this Letter we show how encoding techniques for computer-generated holograms may be used to arbitrarily shape a nonlinearly generated spectrum and consequently the temporal shape by modulating the quadratic nonlinear coefficient. We give examples of a modulation pattern and a simple setup that can generate high-order Hermite–Gauss and Airy functions through difference-frequency generation from a transform-limited Gaussian pulse, under practical fabrication considerations.

© 2012 Optical Society of America

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References

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  1. E. Wolf, in Vol. XVI of Progress in Optics (North-Holland, 1978), Chap. 3.
  2. B. R. Brown and A. W. Lohmann, Appl. Opt. 5, 967 (1966).
    [CrossRef]
  3. J. J. Burch, Proc. IEEE 55, 599 (1967).
    [CrossRef]
  4. W. Lee, Appl. Opt. 18, 3661 (1979).
    [CrossRef]
  5. A. Shapira, I. Juwiler, and A. Arie, Opt. Lett. 36, 3015 (2011).
    [CrossRef]
  6. A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, Opt. Lett. 37, 2136 (2012).
    [CrossRef]
  7. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).
  8. G. Imeshev, M. M. Fejer, A. Galvanauskas, and D. Harter, J. Opt. Soc. Am. B 18, 534 (2001).
    [CrossRef]
  9. G. Imeshev, M. A. Arbore, M. M. Fejer, A. Galvanauskas, M. Fermann, and D. Harter, J. Opt. Soc. Am. B 17, 304 (2000).
    [CrossRef]
  10. B. E. A. Saleh and M. C. Teich, in Fundamentals of Photonics (Wiley, 1991), Chap. 3.
  11. I. H. Malitson, J. Opt. Soc. Am. 55, 1205 (1965).
    [CrossRef]
  12. K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, Appl. Phys. Lett. 74, 914 (1999).
    [CrossRef]
  13. A. Zukauskas, G. Strömqvist, V. Pasiskevicius, F. Laurell, M. Fokine, and C. Canalias, Opt. Mater. Express 1, 1319 (2011).
    [CrossRef]
  14. G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
    [CrossRef]
  15. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
    [CrossRef]
  16. T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, Nat. Photonics 3, 395 (2009).
    [CrossRef]
  17. A. Papoulis, in Signal Analysis (McGraw-Hill, 1977), Chap. 8.
  18. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photonics 4, 103 (2010).
    [CrossRef]

2012 (1)

2011 (2)

2010 (1)

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photonics 4, 103 (2010).
[CrossRef]

2009 (1)

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, Nat. Photonics 3, 395 (2009).
[CrossRef]

2007 (2)

G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
[CrossRef]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

2001 (1)

2000 (1)

1999 (1)

K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, Appl. Phys. Lett. 74, 914 (1999).
[CrossRef]

1979 (1)

1967 (1)

J. J. Burch, Proc. IEEE 55, 599 (1967).
[CrossRef]

1966 (1)

1965 (1)

Arbore, M. A.

Arie, A.

A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, Opt. Lett. 37, 2136 (2012).
[CrossRef]

A. Shapira, I. Juwiler, and A. Arie, Opt. Lett. 36, 3015 (2011).
[CrossRef]

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, Nat. Photonics 3, 395 (2009).
[CrossRef]

K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, Appl. Phys. Lett. 74, 914 (1999).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Brown, B. R.

Burch, J. J.

J. J. Burch, Proc. IEEE 55, 599 (1967).
[CrossRef]

Canalias, C.

Chong, A.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photonics 4, 103 (2010).
[CrossRef]

Christodoulides, D. N.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photonics 4, 103 (2010).
[CrossRef]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
[CrossRef]

Dogariu, A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Ellenbogen, T.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, Nat. Photonics 3, 395 (2009).
[CrossRef]

Fejer, M. M.

Fermann, M.

Fokine, M.

Fradkin, K.

K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, Appl. Phys. Lett. 74, 914 (1999).
[CrossRef]

Galvanauskas, A.

Ganany-Padowicz, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, Nat. Photonics 3, 395 (2009).
[CrossRef]

Harter, D.

Imeshev, G.

Juwiler, I.

Laurell, F.

Lee, W.

Lohmann, A. W.

Malitson, I. H.

Papoulis, A.

A. Papoulis, in Signal Analysis (McGraw-Hill, 1977), Chap. 8.

Pasiskevicius, V.

Renninger, W. H.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photonics 4, 103 (2010).
[CrossRef]

Rosenman, G.

K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, Appl. Phys. Lett. 74, 914 (1999).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, in Fundamentals of Photonics (Wiley, 1991), Chap. 3.

Shapira, A.

Shiloh, R.

Siviloglou, G. A.

G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
[CrossRef]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Skliar, A.

K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, Appl. Phys. Lett. 74, 914 (1999).
[CrossRef]

Strömqvist, G.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, in Fundamentals of Photonics (Wiley, 1991), Chap. 3.

Voloch-Bloch, N.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, Nat. Photonics 3, 395 (2009).
[CrossRef]

Wise, F. W.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photonics 4, 103 (2010).
[CrossRef]

Wolf, E.

E. Wolf, in Vol. XVI of Progress in Optics (North-Holland, 1978), Chap. 3.

Zukauskas, A.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, Appl. Phys. Lett. 74, 914 (1999).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (2)

Nat. Photonics (2)

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, Nat. Photonics 3, 395 (2009).
[CrossRef]

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photonics 4, 103 (2010).
[CrossRef]

Opt. Lett. (3)

Opt. Mater. Express (1)

Phys. Rev. Lett. (1)

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Proc. IEEE (1)

J. J. Burch, Proc. IEEE 55, 599 (1967).
[CrossRef]

Other (4)

B. E. A. Saleh and M. C. Teich, in Fundamentals of Photonics (Wiley, 1991), Chap. 3.

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).

A. Papoulis, in Signal Analysis (McGraw-Hill, 1977), Chap. 8.

E. Wolf, in Vol. XVI of Progress in Optics (North-Holland, 1978), Chap. 3.

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Figures (4)

Fig. 1.
Fig. 1.

Generating HG09 spectrum: an input transform-limited 77 fs Gaussian pulse around 810 nm is stretched to 0.5 ns and fed together with a quasi-CW 532 nm pump into an HG09-encoded nonlinear CGH. In-crystal curves: modulated domains (gray), amplitude (blue), and phase (red) of the encoded CGH.

Fig. 2.
Fig. 2.

DFG wave at crystal output shows the reconstructed (a) spectral HG09 measuring 21.2 nm FHWM and (b) its representation in time, measuring 115.6 ps.

Fig. 3.
Fig. 3.

Generating Airy spectrum: a pump and stretched input pulse are fed into a nonlinear crystal, where the encoded nonlinear CGH is a cubically phased Gaussian amplitude (the Fourier transform of is, of course, the finite Airy function). In-crystal curves: modulated domains (gray), amplitude (blue), phase (red) of the encoded CGH.

Fig. 4.
Fig. 4.

DFG pulse at the crystal output. (a) Spectral representation showing an Airy function (red) and the Gaussian approximation of its width [Eq. (8), dashed blue curve]; (b) temporal representation, exhibiting a frequency-to-time mapping.

Equations (9)

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t(x,y)=sign{cos[2πfcx+φ(x,y)]cos[πq(x,y)]},
A3=κd(z)eiΔkzdz,
A3(Δk)=κIFT{D(Δk)}eiΔkzdz,
B<2Δk03.
d(z)=dijsign{cos[Δk0z+φ(z)]cos[πq(z)]},
G(ζ)=H9(ζ)eζ2/2,
u2(ζ(z))=eiπp×ζ3(z/22ln2)eζ2(z)/2,
G(Δk)=e(ΔkΔk02(1+2p)/ω0)2,
B=8(1+2p)×ln2/b,

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