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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 19 — Sep. 17, 2007
  • pp: 11978–11983
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Frequency-selective self-trapping and supercontinuum generation in arrays of coupled nonlinear waveguides

I. Babushkin, A. Husakou, J. Herrmann, and Yuri S. Kivshar  »View Author Affiliations


Optics Express, Vol. 15, Issue 19, pp. 11978-11983 (2007)
http://dx.doi.org/10.1364/OE.15.011978


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Abstract

We study spatiotemporal dynamics of soliton-induced two-octave-broad supercontinuum generated by fs pulses in an array of coupled nonlinear waveguides. We show that after fission of the input pulse into several fundamental solitons, red and blue-shifted nonsolitonic radiation, as well as solitons with lower intensity, spread away in transverse direction, while the most intense spikes self-trap into spatiotemporal discrete solitons.

© 2007 Optical Society of America

Introduction

Self-trapping of quasi-monochromatic beams in an array of coupled nonlinear waveguides and generation of spatial discrete solitons [1

1. D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 13, 794–796 (1988). [CrossRef] [PubMed]

4

4. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003). [CrossRef] [PubMed]

] have been studied extensively in recent years. However, an even more interesting case is the propagation of short pulses in arrays of nonlinear waveguides with anomalous dispersion. The discrete nature of the periodic array was predicted to support stable spatiotemporal solitons, in a sharp contrast to the collapse instability in the corresponding continuum structures [5

5. A. B. Aceves, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Multidimensional solitons in fiber arrays,” Opt. Lett. 19, 329–331 (1994). [CrossRef] [PubMed]

8

8. A. Yulin, D. V. Skryabin, and A. Vladimirov, “Modulational instability of discrete solitons in coupled waveguides with group velocity dispersion,” Opt. Express 14, 12347–12352 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-25-12347. [CrossRef] [PubMed]

]. The frequency-depending spatiotemporal dynamics of polychromatic light was investigated theoretically [9

9. K. Motzek, A. A. Sukhorukov, and Yu. S. Kivshar, “Self-trapping of polychromatic light in nonlinear periodic photonic structures,” Opt. Express 14, 9873–9878 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-21-9873. [CrossRef] [PubMed]

] and also experimentally [10

10. A. A. Sukhorukov, D. N. Neshev, A. Dreischuh, R. Fischer, S. Ha, W. Krolikowski, J. Bolger, A. Mitchell, B. J. Eggleton, and Yu. S. Kivshar, “Polychromatic nonlinear surface modes generated by supercontinuum light,” Opt. Express 14, 11265–11270 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-23-11265. [CrossRef] [PubMed]

], by studying the propagation of coherent supercontinuum light generated externally in a microstructure fiber in an array of LiNbO3 waveguides with a photorefractive nonlinearity.

The discovery of supercontinuum (SC) in photonic crystal fibers with a width of more than one octave generated by fs pulses with nJ energy [11

11. J. K. Ranka, R. S. Windeler, and A. J. Steinz, “Visible continuum generation in air silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. 25, 25–27 (2000). [CrossRef]

] has encouraged large research activities because of its many important applications. The SC generation with an input wavelength selected in the anomalous dispersion range is caused by the fission of the input pulse into several fundamental solitons which emit nonsolitonic radiation in a broad spectral range phase-matched to the corresponding solitons [12

12. A. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901 (2001). [CrossRef] [PubMed]

,13

13. J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett. 88, 173901 (2002). [CrossRef] [PubMed]

]. Besides, depending on the parameters of the system some additional effects, such as Raman soliton frequency shift, four-wave mixing and others can contribute to the resulting shape of supercontinuum (see [14

14. W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. St. J. Russell, F. J. Omenetto, A. Efimov, and A. J. Taylor, “Tanformation and control of ultra-short pulses in dispersion-engineered photonic crystal fibers,” Nature 424, 511–515 (2003). [CrossRef] [PubMed]

17

17. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006). [CrossRef]

] and references therein). Recently, it was shown that soliton-induced SC can also be generated in specifically designed planar rib waveguides [18

18. O. Fedotova, A. Husakou, and J. Herrmann, “Supercontinuum generation in planar rib waveguides enabled by anomalous dispersion,” Opt. Express 14, 1512–1517 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-4-1512. [CrossRef] [PubMed]

].

2. Model and governing equations

We consider an array of planar rib waveguides with the geometry shown in Fig. 1(a) assuming that the waveguides are made of TaFD5 glass placed on SiO2 substrate. Such waveguides exhibit an about 3 times larger nonlinear refractive index than for fused silica. For appropriate parameters the waveguide contribution to dispersion leads to a significant shift of the zero GVD wavelength to shorter wavelengths and anomalous dispersion in the visible, enabling the soliton-induced spectral broadening mechanism [18

18. O. Fedotova, A. Husakou, and J. Herrmann, “Supercontinuum generation in planar rib waveguides enabled by anomalous dispersion,” Opt. Express 14, 1512–1517 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-4-1512. [CrossRef] [PubMed]

].

Fig. 1. (a) Schematic of a waveguide array created by planar rib waveguides. (b) Group-velocity dispersion for an Air-TaFD5-SiO2 rib waveguide (black line) and bulk TaFD5 (red line). Waveguide dimensions are a=4:0 µm, d=1:0 µm, and f=0:5 µm.

We use a generalization of the forward Maxwell equation [12

12. A. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901 (2001). [CrossRef] [PubMed]

] for a waveguide array. This first-order equation is derived from the second-order wave equation neglecting the back-reflected wave but without the slowly-varying-envelope approximation. We suppose that all waveguides operate in the fundamental transverse Ex 00 mode. The complex-valued Fourier transform E j(x;y; z;ω) of the real-valued optical electric field E j(x;y; z; t) in the j-th waveguide can be written as E j(z;ω)F(x;y;ω), where F(x;y;ω) is the transverse fundamental mode profile, z is the coordinate along the waveguides, whereas x and y are the transverse coordinates. Coupling between the waveguides in the array is assumed to be small, and it can be described in the tight-binding approximation [3

3. Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic Press, San Diego, 2003).

]. The evolution of the Fourier transform of the field EjEj(z;ω) along the z coordinate is then governed by the following equation:

iEjz+{β(ω)ων}Ej+κ(Ej1+Ej+1)+μ0α(ω)ω22β(ω)Pnl(j)(z,ω)=0.
(1)

Here β(ω) is the propagation constant of the fundamental mode in the waveguide, κ is the coupling strength between the waveguides, P (j) nl (z;t)=ε 0 χ (3) E 3 j(z;t) is the Kerr nonlinear polarization (without the Raman term for the case of TaFD5 glass), α=∫|F|4 dxdy/∫|F|2 dxdy is the nonlinearity reduction factor, v is the velocity of the moving coordinate frame. The values of β(ω) are calculated using the effective index approach [18

18. O. Fedotova, A. Husakou, and J. Herrmann, “Supercontinuum generation in planar rib waveguides enabled by anomalous dispersion,” Opt. Express 14, 1512–1517 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-4-1512. [CrossRef] [PubMed]

, 19

19. M. J. Adams, An Introduction to Optical Waveguides (John Wiley & Sons, 1981).

]. We notice that Eq. (1) is written directly for the electric field while the generalized nonlinear Schrödinger equation written for the complex amplitude is not numerically faster than Eq. (1) for the case of very broad spectra.

In Fig. 1(b), we show the GVD of a rib waveguide (black curve) with parameters specified in the figure caption, together with the GVD of bulk TaFD5 (red curve). The waveguide contribution to dispersion strongly shifts the zero-GVD wavelength to 1130 nm leading to a wide range of anomalous dispersion. The value κ=6.25×10-5 µm-1 of the linear coupling parameter between the waveguides corresponds to the waveguide separation about 10 µm.

3. Numerical results and discussions

Fig. 2. Evolution of an input 32-fs pulse with peak intensity of 0.5 TW/cm2 and central wavelength λ=1600 nm in an isolated waveguide (a) and the central waveguide of an array (b). Soliton fission appears both in (a) and (b), which leads to formation of three stable solitons in an isolated waveguide (marked by arrows), whereas only one most intense soliton resists diffractive spreading in the array.

Fig. 3. (a,b) Temporal profile and (c) spectrum of pulse in the central [black curve in (a,c)] and neighboring [red curve in (b,c)] waveguide. Parameters of the input pulse are the same as in Fig. 2(b), propagation length is 5 cm. (d) Quadratic mean of the mode diameter W RMS(ω) characterizing the field localization. Nonsolitonic radiation (NSR) and soliton are separated both in (a,b) temporal and (c) spectral domains.

The spectrum of the N=3 soliton is still relatively narrow [Fig. 3(c)]. Now we study the evolution for a longer input pulse (200 fs) with the same intensity corresponding to the soliton number of about N=20, as shown in Fig. 4 by the temporal profile with many soliton spikes and NSR [Fig. 4(b)] and a two-octave spectrum extending from 500 to 3000 nm [Fig. 4(a)]. After 5 cm propagation in the waveguide array, SC is generated in the central waveguide being of the same width and similar shape, as shown in Fig. 4(c) (black curve). A comparison between black and red curves in Fig. 4(c,d) reveals that NSR around 700 nm and 2700 nm is coupled to the neighboring waveguide while the soliton spikes remain self-trapped in the central waveguide, thus demonstrating the frequency-selective self-trapping similar to the short-pulse case. For larger propagation distance [Fig. 4(e,f)], only the strongest spike remains in the central waveguide, and forms a spatiotemporal discrete soliton.

For comparison, we consider also the regime of normal GVD and the input wavelength of 800 nm, illustrated in Fig. 5. For normal GVD solitons can not be generated and, therefore, the pulse does not split but is reshaped towards a top-hat profile [black curve in Fig. 5(b)] and spectral broadening by self-phase modulation yields a much narrower spectrum as can be seen in Fig. 5(a). Therefore, the transverse diffraction occurs equally for all time positions and spectral components, as can be seen in Fig. 5. In this case, for a large enough propagation distance the transverse discrete diffraction destroys the localization of the whole pulse.

Fig. 4. (a,c,e) Spectrum and (b,d,f) temporal shape calculated for a 200 fs input pulse with peak intensity 0.5 TW/cm2 and central wavelength 1600 nm in (a,b) an isolated waveguide and (c-f) waveguide array. Propagation length is (a–d) 5 cm and (e,f) 20 cm, respectively.
Fig. 5. (a) Spectrum and (b) temporal shape calculated for a 200-fs input pulse with peak intensity 0.5 TW/cm2 and central wavelength 800 nm propagating for z=5 cm.

4. Conclusions

References and links

1.

D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 13, 794–796 (1988). [CrossRef] [PubMed]

2.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998). [CrossRef]

3.

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic Press, San Diego, 2003).

4.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003). [CrossRef] [PubMed]

5.

A. B. Aceves, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Multidimensional solitons in fiber arrays,” Opt. Lett. 19, 329–331 (1994). [CrossRef] [PubMed]

6.

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Energy localization in nonlinear fiber arrays: Collapse-effect compresor,” Phys. Rev. Lett. 75, 73–76 (1995). [CrossRef] [PubMed]

7.

D. Cheskis, S. Bar-Ad, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, and D. Ross, “Strong spatiotemporal localization in a silica nonlinear waveguide array,” Phys. Rev. Lett. 91, 223901 (2003). [CrossRef] [PubMed]

8.

A. Yulin, D. V. Skryabin, and A. Vladimirov, “Modulational instability of discrete solitons in coupled waveguides with group velocity dispersion,” Opt. Express 14, 12347–12352 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-25-12347. [CrossRef] [PubMed]

9.

K. Motzek, A. A. Sukhorukov, and Yu. S. Kivshar, “Self-trapping of polychromatic light in nonlinear periodic photonic structures,” Opt. Express 14, 9873–9878 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-21-9873. [CrossRef] [PubMed]

10.

A. A. Sukhorukov, D. N. Neshev, A. Dreischuh, R. Fischer, S. Ha, W. Krolikowski, J. Bolger, A. Mitchell, B. J. Eggleton, and Yu. S. Kivshar, “Polychromatic nonlinear surface modes generated by supercontinuum light,” Opt. Express 14, 11265–11270 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-23-11265. [CrossRef] [PubMed]

11.

J. K. Ranka, R. S. Windeler, and A. J. Steinz, “Visible continuum generation in air silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. 25, 25–27 (2000). [CrossRef]

12.

A. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901 (2001). [CrossRef] [PubMed]

13.

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett. 88, 173901 (2002). [CrossRef] [PubMed]

14.

W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. St. J. Russell, F. J. Omenetto, A. Efimov, and A. J. Taylor, “Tanformation and control of ultra-short pulses in dispersion-engineered photonic crystal fibers,” Nature 424, 511–515 (2003). [CrossRef] [PubMed]

15.

G. Genty, M. Lehtonen, and H. Ludvigsen, “Effect of cross-phase modulation on supercontinuum generated in microstructured fibers with sub-30 fs pulses,” Opt. Express 12, 4614–4624 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-19-4614. [CrossRef] [PubMed]

16.

A. V. Gorbach, D. V. Skryabin, J. M. Stone, and J. C. Knight, “Four-wave mixing of solitons with radiation and quasi-nondispersive wave packets at the short-wavelength edge of a supercontinuum,” Opt. Express 14, 9854–9863 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-21-9854. [CrossRef] [PubMed]

17.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006). [CrossRef]

18.

O. Fedotova, A. Husakou, and J. Herrmann, “Supercontinuum generation in planar rib waveguides enabled by anomalous dispersion,” Opt. Express 14, 1512–1517 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-4-1512. [CrossRef] [PubMed]

19.

M. J. Adams, An Introduction to Optical Waveguides (John Wiley & Sons, 1981).

20.

A. Husakou and J. Herrmann, “Supercontinuum generation in photonic crystal fibers made from highly nonlinear glasses,” Appl. Phys. B 77, 227–234 (2003). [CrossRef]

21.

M. Frosz, P. Falk, and O. Bang, “The role of the second zero-dispersion wavelength in generation of supercontinua and bright-bright soliton-pairs across the zero-dispersion wavelength,” Opt. Express 13, 6181–6192 (2005) http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-16-6181.1. [CrossRef] [PubMed]

OCIS Codes
(060.5530) Fiber optics and optical communications : Pulse propagation and temporal solitons
(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in

ToC Category:
Nonlinear Optics

History
Original Manuscript: June 19, 2007
Revised Manuscript: August 14, 2007
Manuscript Accepted: August 23, 2007
Published: September 5, 2007

Citation
I. Babushkin, A. Husakou, J. Herrmann, and Yuri S. Kivshar, "Frequency-selective self-trapping and supercontinuum generation in arrays of coupled nonlinear waveguides," Opt. Express 15, 11978-11983 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-19-11978


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References

  1. D. N. Christodoulides and R. I. Joseph, "Discrete self-focusing in nonlinear arrays of coupled waveguides," Opt. Lett. 13, 794-796 (1988). [CrossRef] [PubMed]
  2. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998). [CrossRef]
  3. Yu. S. Kivshar, G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic Press, San Diego, 2003).
  4. D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003). [CrossRef] [PubMed]
  5. A. B. Aceves, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, "Multidimensional solitons in fiber arrays," Opt. Lett. 19, 329-331 (1994). [CrossRef] [PubMed]
  6. A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, "Energy localization in nonlinear fiber arrays: Collapse-effect compresor," Phys. Rev. Lett. 75, 73-76 (1995). [CrossRef] [PubMed]
  7. D. Cheskis, S. Bar-Ad, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, and D. Ross, "Strong spatiotemporal localization in a silica nonlinear waveguide array," Phys. Rev. Lett. 91, 223901 (2003). [CrossRef] [PubMed]
  8. A. Yulin, D. V. Skryabin, and A. Vladimirov, "Modulational instability of discrete solitons in coupled waveguides with group velocity dispersion," Opt. Express 14, 12347-12352 (2006), [CrossRef] [PubMed]
  9. K. Motzek, A. A. Sukhorukov, and Yu. S. Kivshar, "Self-trapping of polychromatic light in nonlinear periodic photonic structures," Opt. Express 14, 9873-9878 (2006), [CrossRef] [PubMed]
  10. A. A. Sukhorukov, D. N. Neshev, A. Dreischuh, R. Fischer, S. Ha, W. Krolikowski, J. Bolger, A. Mitchell, B. J. Eggleton, and Yu. S. Kivshar, "Polychromatic nonlinear surface modes generated by supercontinuum light," Opt. Express 14, 11265-11270 (2006). [CrossRef] [PubMed]
  11. J. K. Ranka, R. S. Windeler, and A. J. Steinz, "Visible continuum generation in air silica microstructure optical fibers with anomalous dispersion at 800 nm," Opt. Lett. 25, 25-27 (2000). [CrossRef]
  12. A. Husakou and J. Herrmann, "Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers," Phys. Rev. Lett. 87, 203901 (2001). [CrossRef] [PubMed]
  13. J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, "Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers," Phys. Rev. Lett. 88, 173901 (2002). [CrossRef] [PubMed]
  14. W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. St. J. Russell, F. J. Omenetto, A. Efimov, A. J. Taylor, "Tanformation and control of ultra-short pulses in dispersion-engineered photonic crystal fibers," Nature 424, 511-515 (2003). [CrossRef] [PubMed]
  15. G. Genty, M. Lehtonen, and H. Ludvigsen, "Effect of cross-phase modulation on supercontinuum generated in microstructured fibers with sub-30 fs pulses," Opt. Express 12, 4614-4624 (2004). [CrossRef] [PubMed]
  16. A. V. Gorbach, D. V. Skryabin, J. M. Stone, and J. C. Knight, "Four-wave mixing of solitons with radiation and quasi-nondispersive wave packets at the short-wavelength edge of a supercontinuum," Opt. Express 14, 9854- 9863 (2006). [CrossRef] [PubMed]
  17. J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006). [CrossRef]
  18. O. Fedotova, A. Husakou, and J. Herrmann, "Supercontinuum generation in planar rib waveguides enabled by anomalous dispersion," Opt. Express 14, 1512-1517 (2006). [CrossRef] [PubMed]
  19. M. J. Adams, An Introduction to Optical Waveguides (John Wiley & Sons, 1981).
  20. A. Husakou and J. Herrmann, "Supercontinuum generation in photonic crystal fibers made from highly nonlinear glasses," Appl. Phys. B 77, 227-234 (2003). [CrossRef]
  21. M. Frosz, P. Falk, and O. Bang, "The role of the second zero-dispersion wavelength in generation of supercontinua and bright-bright soliton-pairs across the zero-dispersion wavelength," Opt. Express 13, 6181-6192 (2005). [CrossRef] [PubMed]

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