## Spatiotemporal control of light by Bloch-mode dispersion in multi-core fibers

Optics Express, Vol. 16, Issue 8, pp. 5878-5891 (2008)

http://dx.doi.org/10.1364/OE.16.005878

Acrobat PDF (517 KB)

### Abstract

We study theoretically the dispersion properties of Bloch modes and nonlinearly-induced defect states in two-dimensional waveguide arrays. We define the conditions for achieving anomalous group-velocity dispersion and discuss possibilities for generation of spatiotemporal solitons.

© 2008 Optical Society of America

## 1. Introduction

*positive self-focusing nonlinearity*[2

2. G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: Universality and diversity,” Science **286**, 1518–1523 (1999). [CrossRef] [PubMed]

*spatiotemporal soliton*(STS) or “light bullets” [4–7

4. Y. Silberberg, “Collapse of optical pulses,” Opt. Lett. **15**, 1282–1284 (1990). [CrossRef] [PubMed]

7. B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B: Quantum Semicl. Opt. **7**, R53–R72 (2005). [CrossRef]

*µ*J) ultra-short (femtosecond) pulses. However, even when this necessary condition can be satisfied, it might not be sufficient. In a Kerr-type nonlinear medium, 3D STSs suffer collapse when the input power is above a critical value [4

4. Y. Silberberg, “Collapse of optical pulses,” Opt. Lett. **15**, 1282–1284 (1990). [CrossRef] [PubMed]

8. H. S. Eisenberg, R. Morandotti, Y. Silberberg, S. Bar Ad, D. Ross, and J. S. Aitchison, “Kerr spatiotemporal self-focusing in a planar glass waveguide,” Phys. Rev. Lett. **87**, 043902–4 (2001). [CrossRef] [PubMed]

9. 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**, 223901–4 (2003). [CrossRef] [PubMed]

10. N. I. Nikolov, D. Neshev, O. Bang, and W. Z. Krolikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E **68**, 036614–5 (2003). [CrossRef]

11. X. Liu, L. J. Qian, and F. W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett. **82**, 4631–4634 (1999). [CrossRef]

12. X. Liu, K. Beckwitt, and F. Wise, “Two-dimensional optical spatiotemporal solitons in quadratic media,” Phys. Rev. E **62**, 1328–1340 (2000). [CrossRef]

13. 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]

17. D. Mihalache, D. Mazilu, F. Lederer, and Yu. S. Kivshar, “Stable discrete surface light bullets,” Opt. Express **15**, 589–595 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-2-589. [CrossRef] [PubMed]

## 2. Bandgap structure of infiltrated photonic-crystal fibers

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

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

20. 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]

21. J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. **90**, 023902–4 (2003). [CrossRef] [PubMed]

24. A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, “Discrete propagation and spatial solitons in nematic liquid crystals,” Opt. Lett. **29**, 1530–1532 (2004). [CrossRef] [PubMed]

25. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature **422**, 147–150 (2003). [CrossRef] [PubMed]

26. C. R. Rosberg, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, and Yu. S. Kivshar, “Observation of nonlinear self-trapping in triangular photonic lattices,” Opt. Lett. **32**, 397–399 (2007). [CrossRef] [PubMed]

27. A. Szameit, D. Blömer, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Discrete nonlinear localization in femtosecond laser written waveguides in fused silica,” Opt. Express **13**, 10552–10557 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-26-10552. [CrossRef] [PubMed]

28. F. Couny, F. Benabid, P. J. Roberts, M. T. Burnett, and S. A. Maier, “Identification of Bloch modes in hollow-core photonic crystal fiber cladding,” Opt. Express **15**, 325–338 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-2-325. [CrossRef] [PubMed]

29. C. R. Rosberg, F. H. Bennet, D. N. Neshev, P. D. Rasmussen, O. Bang, W. Krolikowski, A. Bjarklev, and Yu. S. Kivshar, “Tunable diffraction and self-defocusing in liquid-filled photonic crystal fibers,” Opt. Express **15**, 12145–12150 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-19-12145. [CrossRef] [PubMed]

29. C. R. Rosberg, F. H. Bennet, D. N. Neshev, P. D. Rasmussen, O. Bang, W. Krolikowski, A. Bjarklev, and Yu. S. Kivshar, “Tunable diffraction and self-defocusing in liquid-filled photonic crystal fibers,” Opt. Express **15**, 12145–12150 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-19-12145. [CrossRef] [PubMed]

32. F. Luan, A. K. George, T. D. Hedley, G. J. Pearce, D. M. Bird, J. C. Knight, and P. St. Russell, “All-solid photonic bandgap fiber,” Opt. Lett. **29**, 2369–2371 (2004). [CrossRef] [PubMed]

*ε*(

**r**) is the periodic dielectric function, and

*c*is the speed of light in vacuum.

*ω*is the angular frequency, and a harmonic time dependence of the magnetic field has been assumed, i.e.

**H**(

**r**,

*t*)=

**H**

_{ω}(

**r**)exp(-

*iωt*). Since the considered structure is perfectly periodic (no defects), one can use Bloch’s Theorem to calculate the modes

*ε*(

**r**).

*β*is the

*z*-component of the wavevector, while the transverse part of the wavevector is given by

**k**

_{⊥}. The index

*m*in Eq. (2) denotes the eigenvalue number, where the smallest eigenvalue is

*m*=1.

35. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173–190 (2001), http://www.opticsinfobase.org/abstract.cfm?URI=oe-8-3-173. [CrossRef] [PubMed]

*β*=0) there are no bandgaps in the structure, but for out-of-plane propagation bandgaps begin to appear as

*β*increases. In Fig. 1(c) the regions with a photonic bandgap are shown for a fiber with

*d*/Λ=0.50, where

*d*is the hole diameter, and Λ is the pitch (interhole distance). The refractive index of the background material is fixed at 1.44 (corresponding to silica at the wavelength of 1.5

*µ*m), and the refractive index inside the holes is 1.59 corresponding to the refractive index of CS

_{2}at a wavelength of 1.5

*µ*m [36

36. A. Samoc, “Dispersion of refractive properties of solvents: chloroform, toluene, benzene, and carbon disulfide in ultraviolet, visible, and near-infrared,” J. Appl. Phys. **94**, 6167–6174 (2003). [CrossRef]

*λ*/Λ<1.15. The individual waveguides are single-mode as long as the

*V*-parameter [

*V*=

*πd*(

*n*

^{2}

_{co}-

*n*

^{2}

_{cl})

^{1/2}/

*λ*] is less than 2.405. This cutoff is indicated in Fig. 1(c) with a vertical line at

*λ*/Λ=0.44. The dispersion of the lowest lying bands is shown in Fig. 1(d) for a normalized wavelength of

*λ*/Λ=0.5. The fundamental band (

*m*=1) is doubly degenerate, which is analogous to the two polarizations of the fundamental HE

_{11}-mode of an individual high-index rod (step index fiber). The second band is nearly degenerate for long wavelengths, but as the wavelength decreases a polarization splitting appears.

*E*-fields of the modes for

*λ*/Λ=0.5. We see that the top of the first band (Γ-point) has

*LP*

_{01}-like solutions with maximum intensity located inside the high index holes and a flat phase structure. The bottom of the first band (

*K*-point) has a similar intensity distribution as the top of the first band, but the phase varies by ±2

*π*/3 between adjacent holes. The mode at the top of the second band corresponds either to the mode at the

*M*-point for long wavelengths, or to the mode at the

*K*-point for short wavelengths. For the specific structure considered in Fig. 1 this cross-over happens close to

*λ*/Λ≈0.33, as for

*λ*/Λ>0.33 the top of the second band occurs at the

*M*-point, while for

*λ*/Λ<0.33 the top of the second band occurs at the

*K*-point. The modes in the second band have

*LP*

_{11}-like intensity patterns, i.e. they resemble dipoles, with two maxima inside each high-index region. The phases are staggered in the vertical direction, and have

*π*-phase jumps that go through the center of each high-index region. Notice that the fields and phases for the first band are plotted for only the strongest component of the electric field (here the

*x*-component). For the

*M*-point of the second band we have plotted the field and phase along a polarisation direction rotated 60° with respect to the Cartesian coordinate system.

## 3. Group-velocity dispersion of Bloch modes

*K*-point), and at the top of the second band, because they can be easily excited experimentally [26

26. C. R. Rosberg, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, and Yu. S. Kivshar, “Observation of nonlinear self-trapping in triangular photonic lattices,” Opt. Lett. **32**, 397–399 (2007). [CrossRef] [PubMed]

### 3.1. Geometrical dispersion

*ν*=(

_{g}*dβ*/

*dω*)

^{-1}is the group velocity. Here we calculate the group velocity as

*ν*=Re〈

_{g}**E***×

**H**,

**E***×

**H**〉

_{z}/〈

**H**,

**H**〉, which is exact when material dispersion is neglected (the notation 〈·, ·〉 denotes the inner product). The values

*D*<0 correspond to

*normal dispersion*, and

*D*>0 corresponds to

*anomalous dispersion*. In Fig. 2 the normalized dispersion (

*c*Λ

*D*) of the Bloch modes corresponding to the top and bottom of the first band (Γ- and

*K*-point respectively), and the top of the second band are plotted for three different index contrasts (Δ

*n*=10

^{-3}, 5·10

^{-2}, and 10

^{-1}). The index of the silica background is fixed at 1.44, and we consider a structure with

*d*/Λ=0.5. The wavelength at which the fundamental bandgap opens up, depends on the index contrast. For the index contrasts Δ

*n*=10

^{-3}, 5·10

^{-2}, and 10

^{-1}, the fundamental bandgap exists for

*λ*/Λ≤0.11, 0.73, and 0.98, respectively. Additionally, in Fig. 2(a) we show the dispersion of the fundamental HE

_{11}mode of an isolated waveguide in order to compare how the dispersion is significantly changed at long wavelengths in the periodic structure. Note that the dispersion in Fig. 2(a) is plotted as a function of

*λ*/(2

*d*), in order to enable direct comparison with the dispersion for the periodic structure that has

*d*/Λ=0.5. At short wavelengths the dispersion of the HE

_{11}mode of an isolated waveguide, and the modes corresponding to the top and bottom of the first band [Fig. 2(b, c)] are similar. This is also expected since these modes are highly confined inside the high-index inclusions at small wavelengths, and are therefore not affected by the periodic geometry. In Fig. 2(c) the dispersion of the modes lying at the bottom of the first band are shown. We see that for these modes large anomalous dispersion (

*c*Λ

*D*>0) can be achieved for both small and high index contrasts. This anomalous dispersion makes these modes very attractive for temporal localisation of pulses, and we will explore their applicability for spatiotemporal localisation later on in the paper. Note that the dispersion is only shown when a bandgap exists (

*λ*/Λ<0.1) for Δ

*n*=10

^{-3}. In Fig. 2(d) the dispersion of the mode lying at the top of the second band is shown. We see that high anomalous dispersion can be achieved at small wavelengths compared to the pitch (

*λ*/Λ<0.2). The reason why there are discontinuities in the dispersion for Δ

*n*=0.05 and 0.1 at

*λ*/Λ=0.25 and 0.35, respectively is that the mode bounding the top of the second band changes its character at that wavelength due to band crossing. Hence we cannot uniquely define the dispersion of the mode at the top of the second band when the two bands cross.

34. U. Röpke, H. Bartelt, S. Unger, K. Schuster, and J. Kobelke, “Two-dimensional high-precision fiber waveguide arrays for coherent light propagation,” Opt. Express **15**, 6894–6899 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-11-6894. [CrossRef] [PubMed]

_{2}for example, has anomalous dispersion for wavelengths above 1.3

*µ*m, and anomalous chromatic dispersion in an all glass structure can

### 3.2. Chromatic dispersion

_{2}, since this liquid has fast and strong third order nonlinearity. CS

_{2}also has good transmission in the visible and near infrared spectrum [37]. The refractive indices of silica and CS

_{2}are found with the following formulas:

*a*and

_{i}*A*, which are listed in Table 1, are taken from Refs. [38] and [36

_{i}36. A. Samoc, “Dispersion of refractive properties of solvents: chloroform, toluene, benzene, and carbon disulfide in ultraviolet, visible, and near-infrared,” J. Appl. Phys. **94**, 6167–6174 (2003). [CrossRef]

_{2}is

*λ*

_{0}=1.55

*µ*m. We have solved Eq. (1) self consistently by including the wavelength dependent refractive index of the dielectric constant for both silica and CS

_{2}. Here we propose two different designs that have anomalous dispersion for the mode at the top of the second band and the mode at the bottom of the first band. In Fig. 3(a) the chromatic dispersion of the modes at the top of the first (blue line) and second (red line) bands is shown for a design with

*d*/Λ=0.45 and Λ=10

*µ*m corresponding to the parameters of a commercially available LMA-25 fibre (Crystal Fibre A/S, Denmark). The mode at the bottom of the first band [not shown in Fig. 3(a)] experiences almost identical dispersion as the mode at the top of the first band. This is also expected since the pitch is much larger than the wavelength, and therefore the bands are very flat in this region. For the design consisting of a triangular silica matrix infiltrated with CS

_{2}, we have varied the pitch and the relative hole diameter, and found that for the case of Bloch waves at the top of the first band it is not possible to achieve anomalous dispersion in the range of 1.2–1.8

*µ*m. This is a consequence of the strong normal dispersion of the CS

_{2}, which can not be compensated by the weak geometrical anomalous dispersion of the array. For the higher order mode (top of second band) much stronger geometrical anomalous dispersion can be obtained, and anomalous dispersion can be achieved. For the example shown in Fig. 3(a) we find a zero dispersion wavelength (ZDW) of

*λ*

_{ZDW}=1.57

*µ*m, and an anomalous dispersion of 4 ps/(nm·km) at

*λ*=1.6

*µ*m.

*d*/Λ=0.60 and Λ=3.5

*µ*m we can obtain much higher anomalous dispersion of the bottom of the first band, as suggested by the plot of the waveguide dispersion of this mode shown in Fig. 2(c). In Fig. 3(b) the dispersion for a CS

_{2}infiltrated array with

*d*/Λ=0.60 and Λ=3.5

*µ*m is shown. Here

*λ*

_{ZDW}=1.51

*µ*m, and the dispersion at

*λ*=1.60

*µ*m is 20 ps/(nm·km). While in this case anomalous dispersion can be achieved for the bottom of the first band, the dispersion of the top of the first and second bands [not shown in Fig. 3(b)] lies deep into the normal regime. From this investigation of the chromatic dispersion of the Bloch modes, we conclude that for the specific case we are considering, it is necessary to use either the top of the second band, or the bottom of the first band in order to achieve anomalous dispersion.

## 4. Soliton defect modes

*normal*(positive)

*diffraction*correspond to the top of each band, while beams associated with Bloch waves from the bottom of the bands experience

*anomalous*(negative)

*diffraction*. Therefore, pure STS can only exist when the beam is associated with the top of the first or second band for wavelengths

*λ*>1.95

*µ*m and

*λ*>1.57

*µ*m, respectively [see Fig. 3(a)].

_{2}for example, has a strong focusing nonlinearity of the order of

*n*

_{2}≈0.75·10

^{-18}m

^{2}/W in the wavelength range 800–1600nm for 100 fs pulses [39

39. I. P. Nikolakakos, A. Major, J. S. Aitchison, and P. W. E. Smith, “Broadband characterization of the nonlinear optical properties of common reference materials,” IEEE J. Sel. Top. Quantum Electron. **10**, 1164–1170 (2004). [CrossRef]

*n*≈

_{NL}*P*

_{0}

*n*

_{2}/(

*πr*

^{2}), where

*P*

_{0}is the peak power of the laser pulses, and

*r*is the hole radius. If the holes have a radius of 2.25

*µ*m, we can induce a defect of Δ

*n*=10

_{NL}^{-3}with a pulse with a peak power of

*P*

_{0}=21.2 kW, which is typical for a femtosecond fiber laser at a wavelength of 1.5

*µ*m.

*hybrid scheme*for localization. In this scheme two different types of nonlinearity need to be employed for compensation of pulse dispersion and beam diffraction, respectively. A defocusing type of nonlinearity such as slow thermal nonlinearity of the liquid can be employed for spatial localization, while a self-focusing fast electronic response can be responsible for temporal localization. We note that in this case the fast nonlinearity should be weaker than the defocusing, so the total induced defect will always remain negative. A negative defect can be induced by a thermooptic effect, which arises when a small portion of the light is absorbed and heats the liquid. Since CS

_{2}has a relatively large thermooptic coefficient of

*dn*/

*dT*=-7.9·10

^{-4}

*K*

^{-1}[37], index changes of the order of 10

^{-3}could be achieved even at small absorption levels. In this case the negative defect strength will depend not on the peak pulse power, but on the average beam power. Thus there will be two control parameters, which will contribute to the spatiotemporal localization: pulse peak power and average beam power. Further challenges associated with this scheme, including the spatially nonlocal aspect of thermal nonlinearity needs to be investigated.

*ε*(

**r**), which is not periodic. This is done using a full-vectorial commercial finite element mode solver [40

40. COMSOL Multiphysics 3.3 (2007), COMSOL Inc. (http://www.comsol.com/).

*A*=〈

_{eff}**E**,

**E**〉

^{2}/〈|

**E**|

^{2}, |

**E**|

^{2}〉) of the defect modes as a function of normalized wavelength for defects of Δ

*n*=±10

^{-3}. In Fig. 4(a) we show the results for a positive defect, which allows defect modes in the semi-infinite bandgap above the first band (solid line), and in the fundamental bandgap (dashed line). The structure has

*d*/Λ=0.45 such that anomalous dispersion can be achieved for the mode at the top of the second band at 1.6

*µ*m [see Fig. 3(a)]. We see that for an induced index change of only Δ

*n*=10

^{-3}, the defect mode existing in the semi-infinite gap becomes strongly localized on the defect for

*λ*/Λ<0.4, while the defect mode in the fundamental bandgap becomes localized when

*λ*/Λ<0.35.

*d*/Λ=0.45, we see that

*λ*/Λ>0.16 is necessary in order to achieve anomalous dispersion. Therefore such conditions allow for simultaneous spatial localization and anomalous dispersion.

*n*=-10

^{-3}are shown. The structure has

*d*/Λ=0.60, and we see that the defect mode becomes localized when

*λ*/Λ<0.5. Using the result for the structure presented in Fig. 3(b), which also has

*d*/Λ=0.60, we see that

*λ*/Λ>0.43 is necessary in order to achieve anomalous dispersion. Therefore, in this case it is also possible to have an array where spatial localization and anomalous dispersion appear simultaneously.

_{2}infiltrated arrays, that allows for simultaneous spatial localization and anomalous dispersion. In the following section we discuss the criteria that should be satisfied in order to use the anomalous dispersion for temporal localization.

## 5. Spatiotemporal localization

39. I. P. Nikolakakos, A. Major, J. S. Aitchison, and P. W. E. Smith, “Broadband characterization of the nonlinear optical properties of common reference materials,” IEEE J. Sel. Top. Quantum Electron. **10**, 1164–1170 (2004). [CrossRef]

11. X. Liu, L. J. Qian, and F. W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett. **82**, 4631–4634 (1999). [CrossRef]

12. X. Liu, K. Beckwitt, and F. Wise, “Two-dimensional optical spatiotemporal solitons in quadratic media,” Phys. Rev. E **62**, 1328–1340 (2000). [CrossRef]

### 5.1. Coupling length

41. P. G. Kevrekidis, B. A. Malomed, and Y. B. Gaididei, “Solitons in triangular and honeycomb dynamical lattices with the cubic nonlinearity,” Phys. Rev. E **66**, 016609–10 (2002). [CrossRef]

*A*is the amplitude at the lattice point (

_{mn}*m*,

*n*), and

*C*is the coupling constant. Inserting the Bloch wave ansatz

*A*=

_{mn}*A*

_{0}exp(

*i*[

**r**

_{⊥}·

**k**

_{⊥}+

*βz*]) we find the dispersion relation

*k*

_{1},

*k*

_{2}) are the coordinates of the considered

*k*-point in reciprocal space in the basis of the reciprocal lattice vectors. We define the coupling length as the minimum longitudinal distance a mode must propagate before its center moves by one lattice period in the transversal plane. Since the propagation direction of a mode is given by the normal to the diffraction surface defined by

*β*(

*k*,

_{x}*k*), the coupling length is given by

_{y}*L*=2.56/Δ

_{c}*β*, where Δ

*β*is the difference between the smallest and the largest propagation constants at fixed frequency in the first band.

*d*/Λ=0.5 and background index of

*n*=1.44. The coupling length is shown for two different index contrasts, Δ

*n*=0.05 and Δ

*n*=0.15, the latter corresponding to silica and CS

_{2}. We see that the coupling length increases with decreasing wavelength, since the modes become more localized inside the high-index regions at shorter wavelengths, and only weakly overlap with the fields in the neighboring waveguides. The same happens when the index contrast is increased. The coupling length for the mode corresponding to the second band is about one order of magnitude shorter than in the first band. From an experimental point of view it will therefore be advantageous to use the second band, since this will require shorter infiltration lengths.

### 5.2. Dispersion length

*γ*=2

*n*

_{2}

*π*/(

*λ*

*A*), where

_{eff}*A*is the effective area of the given mode. The nonlinear length is then given by

_{eff}*L*=1/(

_{NL}*P*

_{0}

*γ*). This is the propagation length required to achieve a phase shift of 1 in the center of the pulse, when only nonlinear effects are present. The dispersion length is given by

*L*=

_{D}*T*

^{2}

_{0}/|

*β*

_{2}|, where

*β*

_{2}=

*d*

^{2}

*β*/

*dω*

^{2}. This is the length a Gaussian pulse must propagate in order to broaden its temporal width by a factor of √2 in the presence of only linear effects. The soliton number is then defined as

### 5.3. Localization in space and time

_{2}[39

39. I. P. Nikolakakos, A. Major, J. S. Aitchison, and P. W. E. Smith, “Broadband characterization of the nonlinear optical properties of common reference materials,” IEEE J. Sel. Top. Quantum Electron. **10**, 1164–1170 (2004). [CrossRef]

*λ*/Λ~0.15) is that the coupling length would be in the order of meters as shown in Fig. 5, therefore it would be required to infiltrate a very long piece of fiber with CS

_{2}to demonstrate the spatial localization.

_{2}is used to create a negative defect for spatial localization, while the self-focusing Kerr nonlinearity is responsible for temporal pulse confinement. Using the design presented in Fig. 3(b) (

*λ*/Λ~0.4) we find that a pulse with a peak power of only 60Wis sufficient to create a fundamental soliton (

*N*=1), if the temporal width is chosen to be

*T*=100 fs (

_{FWHM}*T*=1.763·

_{FWHM}*T*

_{0}, i.e. we assume a sech-shaped pulse) and the wavelength is 1.6

*µ*m, where the GVD of the defect mode is 40ps/(nm·km). The coupling length for this design would be of the order of one millimeter, and

*L*=

_{D}*L*≈5 cm, hence a 10 cm fiber would be enough to demonstrate both the spatial and temporal localization. The induced index change will then depend on the pulse repetition rate and the light absorption in the liquid, but because of the moderate peak power, the nonlinear index change due to the Kerr effect will be much weaker. The pulse repetition rate and the light absorption can both be controlled independently in experiments. The absorption could for example be adjusted by adding a dye to the liquid before infiltration.

_{NL}## 6. Conclusions

_{2}. It is shown that defect modes with anomalous group velocity dispersion exist in the fundamental bandgap of the structure for both positive and negative induced defects depending on the structure of the array. We have examined the possibilities for realizing light bullets in such structures, and shown that the most experimentally feasible scheme of spatiotemporal localization should involve the fundamental bandgap with a negative induced defect. This can be realized by using a hybrid type of nonlinearity, where a slow defocusing nonlinearity is employed for spatial localization, while fast and weaker positive nonlinearity is responsible for temporal localization.

## Acknowledgments

## References and links

1. | Yu. S. Kivshar and G. P. Agrawal, |

2. | G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: Universality and diversity,” Science |

3. | G. P. Agrawal, |

4. | Y. Silberberg, “Collapse of optical pulses,” Opt. Lett. |

5. | A. B. Blagoeva, S. G. Dinev, A. A. Dreischuh, and A. Naidenov, “Light bullets formation in a bulk media,” IEEE J. Quantum Electron. |

6. | F. Wise and P. Di Trapani, “The hunt for light bullets - spatiotemporal solitons,” Opt. Photon. News |

7. | B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, “Spatiotemporal optical solitons,” J. Opt. B: Quantum Semicl. Opt. |

8. | H. S. Eisenberg, R. Morandotti, Y. Silberberg, S. Bar Ad, D. Ross, and J. S. Aitchison, “Kerr spatiotemporal self-focusing in a planar glass waveguide,” Phys. Rev. Lett. |

9. | 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. |

10. | N. I. Nikolov, D. Neshev, O. Bang, and W. Z. Krolikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E |

11. | X. Liu, L. J. Qian, and F. W. Wise, “Generation of optical spatiotemporal solitons,” Phys. Rev. Lett. |

12. | X. Liu, K. Beckwitt, and F. Wise, “Two-dimensional optical spatiotemporal solitons in quadratic media,” Phys. Rev. E |

13. | A. B. Aceves, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Multidimensional solitons in fiber arrays,” Opt. Lett. |

14. | E. W. Laedke, K. H. Spatschek, and S. K. Turitsyn, “Stability of discrete solitons and quasicollapse to intrinsically localized modes,” Phys. Rev. Lett. |

15. | A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Energy localization in nonlinear fiber arrays - collapse-effect compressor,” Phys. Rev. Lett. |

16. | A. B. Aceves, M. Santagiustina, and C. De Angelis, “Analytical study of nonlinear-optical pulse dynamics in arrays of linearly coupled waveguides,” J. Opt. Soc. Am. B |

17. | D. Mihalache, D. Mazilu, F. Lederer, and Yu. S. Kivshar, “Stable discrete surface light bullets,” Opt. Express |

18. | D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature |

19. | D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled wave-guides,” Opt. Lett. |

20. | H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. |

21. | J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. |

22. | D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings,” Opt. Lett. |

23. | F. Chen, M. Stepić, C. Rüter, D. Runde, D. Kip, V. Shandarov, O. Manela, and M. Segev, “Discrete diffraction and spatial gap solitons in photovoltaic LiNbO3 waveguide arrays,” Opt. Express |

24. | A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, “Discrete propagation and spatial solitons in nematic liquid crystals,” Opt. Lett. |

25. | J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature |

26. | C. R. Rosberg, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, and Yu. S. Kivshar, “Observation of nonlinear self-trapping in triangular photonic lattices,” Opt. Lett. |

27. | A. Szameit, D. Blömer, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Discrete nonlinear localization in femtosecond laser written waveguides in fused silica,” Opt. Express |

28. | F. Couny, F. Benabid, P. J. Roberts, M. T. Burnett, and S. A. Maier, “Identification of Bloch modes in hollow-core photonic crystal fiber cladding,” Opt. Express |

29. | C. R. Rosberg, F. H. Bennet, D. N. Neshev, P. D. Rasmussen, O. Bang, W. Krolikowski, A. Bjarklev, and Yu. S. Kivshar, “Tunable diffraction and self-defocusing in liquid-filled photonic crystal fibers,” Opt. Express |

30. | J. Jasapara, T. H. Her, R. Bise, R. Windeler, and D. J. DiGiovanni, “Group-velocity dispersion measurements in a photonic bandgap fiber,” J. Opt. Soc. Am. B |

31. | P. Mach, M. Dolinski, K. W. Baldwin, J. A. Rogers, C. Kerbage, R. S. Windeler, and B. J. Eggleton, “Tunable microfluidic optical fiber,” Appl. Phys. Lett. |

32. | F. Luan, A. K. George, T. D. Hedley, G. J. Pearce, D. M. Bird, J. C. Knight, and P. St. Russell, “All-solid photonic bandgap fiber,” Opt. Lett. |

33. | T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, “Nonlinearity and disorder in fiber arrays,” Phys. Rev. Lett. |

34. | U. Röpke, H. Bartelt, S. Unger, K. Schuster, and J. Kobelke, “Two-dimensional high-precision fiber waveguide arrays for coherent light propagation,” Opt. Express |

35. | S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

36. | A. Samoc, “Dispersion of refractive properties of solvents: chloroform, toluene, benzene, and carbon disulfide in ultraviolet, visible, and near-infrared,” J. Appl. Phys. |

37. | D. N. Nikogosyan, |

38. | K. Okamoto, |

39. | I. P. Nikolakakos, A. Major, J. S. Aitchison, and P. W. E. Smith, “Broadband characterization of the nonlinear optical properties of common reference materials,” IEEE J. Sel. Top. Quantum Electron. |

40. | COMSOL Multiphysics 3.3 (2007), COMSOL Inc. (http://www.comsol.com/). |

41. | P. G. Kevrekidis, B. A. Malomed, and Y. B. Gaididei, “Solitons in triangular and honeycomb dynamical lattices with the cubic nonlinearity,” Phys. Rev. E |

**OCIS Codes**

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

(230.7370) Optical devices : Waveguides

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: January 30, 2008

Revised Manuscript: April 4, 2008

Manuscript Accepted: April 6, 2008

Published: April 11, 2008

**Citation**

Per D. Rasmussen, Andrey S. Sukhorukov, Dragomir N. Neshev, Wieslaw Krolikowski, Ole Bang, Jesper Lægsgaard, and Yuri S. Kivshar, "Spatiotemporal control of light by Bloch-mode dispersion in multi-core fibers," Opt. Express **16**, 5878-5891 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-8-5878

Sort: Year | Journal | Reset

### References

- Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, 2003).
- G. I. Stegeman and M. Segev, "Optical spatial solitons and their interactions: Universality and diversity," Science 286, 1518-1523 (1999). [CrossRef] [PubMed]
- G. P. Agrawal, Nonlinear Fiber Optics, fourth ed. (Academic Press, New York, 2007).
- Y. Silberberg, "Collapse of optical pulses," Opt. Lett. 15, 1282-1284 (1990). [CrossRef] [PubMed]
- A. B. Blagoeva, S. G. Dinev, A. A. Dreischuh, and A. Naidenov, "Light bullets formation in a bulk media," IEEE J. Quantum Electron. QE-27, 2060 (1991). [CrossRef]
- F. Wise and P. Di Trapani, "The hunt for light bullets - spatiotemporal solitons," Opt. Photon. News 13, 28-32 (2002). [CrossRef]
- B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, "Spatiotemporal optical solitons," J. Opt. B: Quantum Semicl. Opt. 7, R53-R72 (2005). [CrossRef]
- H. S. Eisenberg, R. Morandotti, Y. Silberberg, S. Bar Ad, D. Ross, and J. S. Aitchison, "Kerr spatiotemporal self-focusing in a planar glass waveguide," Phys. Rev. Lett. 87, 043902-4 (2001). [CrossRef] [PubMed]
- 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, 223901-4 (2003). [CrossRef] [PubMed]
- N. I. Nikolov, D. Neshev, O. Bang, and W. Z. Krolikowski, "Quadratic solitons as nonlocal solitons," Phys. Rev. E 68, 036614-5 (2003). [CrossRef]
- X. Liu, L. J. Qian, and F. W. Wise, "Generation of optical spatiotemporal solitons," Phys. Rev. Lett. 82, 4631- 4634 (1999). [CrossRef]
- X. Liu, K. Beckwitt, and F. Wise, "Two-dimensional optical spatiotemporal solitons in quadratic media," Phys. Rev. E 62, 1328-1340 (2000). [CrossRef]
- 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]
- E.W. Laedke, K. H. Spatschek, and S. K. Turitsyn, "Stability of discrete solitons and quasicollapse to intrinsically localized modes," Phys. Rev. Lett. 73, 1055-1059 (1994). [CrossRef] [PubMed]
- A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, "Energy localization in nonlinear fiber arrays - collapse-effect compressor," Phys. Rev. Lett. 75, 73-76 (1995). [CrossRef] [PubMed]
- A. B. Aceves, M. Santagiustina, and C. De Angelis, "Analytical study of nonlinear-optical pulse dynamics in arrays of linearly coupled waveguides," J. Opt. Soc. Am. B 14, 1807-1815 (1997). [CrossRef]
- D. Mihalache, D. Mazilu, F. Lederer, and Yu. S. Kivshar, "Stable discrete surface light bullets," Opt. Express 15, 589-595 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-2-589. [CrossRef] [PubMed]
- D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003). [CrossRef] [PubMed]
- D. N. Christodoulides and R. I. Joseph, "Discrete self-focusing in nonlinear arrays of coupled wave-guides," Opt. Lett. 13, 794-796 (1988). [CrossRef] [PubMed]
- 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]
- J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of discrete solitons in optically induced real time waveguide arrays," Phys. Rev. Lett. 90, 023902-4 (2003). [CrossRef] [PubMed]
- D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, "Spatial solitons in optically induced gratings," Opt. Lett. 28, 710-712 (2003). [CrossRef] [PubMed]
- F. Chen, M. Stepi’c, C. R¨uter, D. Runde, D. Kip, V. Shandarov, O. Manela, and M. Segev, "Discrete diffraction and spatial gap solitons in photovoltaic LiNbO3 waveguide arrays," Opt. Express 13, 4314-4324 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-11-4314. [CrossRef] [PubMed]
- A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, "Discrete propagation and spatial solitons in nematic liquid crystals," Opt. Lett. 29, 1530-1532 (2004). [CrossRef] [PubMed]
- J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature 422, 147-150 (2003). [CrossRef] [PubMed]
- C. R. Rosberg, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, and Yu. S. Kivshar, "Observation of nonlinear self-trapping in triangular photonic lattices," Opt. Lett. 32, 397-399 (2007). [CrossRef] [PubMed]
- A. Szameit, D. Blomer, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, A. Tunnermann, and F. Lederer, "Discrete nonlinear localization in femtosecond laser written waveguides in fused silica," Opt. Express 13, 10552-10557 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-26-10552. [CrossRef] [PubMed]
- F. Couny, F. Benabid, P. J. Roberts, M. T. Burnett, and S. A. Maier, "Identification of Blochmodes in hollow-core photonic crystal fiber cladding," Opt. Express 15, 325-338 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-2-325. [CrossRef] [PubMed]
- C. R. Rosberg, F. H. Bennet, D. N. Neshev, P. D. Rasmussen, O. Bang, W. Krolikowski, A. Bjarklev, and Yu. S. Kivshar, "Tunable diffraction and self-defocusing in liquid-filled photonic crystal fibers," Opt. Express 15, 12145-12150 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-19-12145. [CrossRef] [PubMed]
- J. Jasapara, T. H. Her, R. Bise, R. Windeler, and D. J. DiGiovanni, "Group-velocity dispersion measurements in a photonic bandgap fiber," J. Opt. Soc. Am. B 20, 1611-1615 (2003). [CrossRef]
- P. Mach, M. Dolinski, K. W. Baldwin, J. A. Rogers, C. Kerbage, R. S. Windeler, and B. J. Eggleton, "Tunable microfluidic optical fiber," Appl. Phys. Lett. 80, 4294-4296 (2002). [CrossRef]
- F. Luan, A. K. George, T. D. Hedley, G. J. Pearce, D. M. Bird, J. C. Knight, and P. St. Russell, "All-solid photonic bandgap fiber," Opt. Lett. 29, 2369-2371 (2004). [CrossRef] [PubMed]
- T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tunnermann, and F. Lederer, "Nonlinearity and disorder in fiber arrays," Phys. Rev. Lett. 93, 053901-4 (2004). [CrossRef] [PubMed]
- U. Ropke, H. Bartelt, S. Unger, K. Schuster, and J. Kobelke, "Two-dimensional high-precision fiber waveguide arrays for coherent light propagation," Opt. Express 15, 6894-6899 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-11-6894. [CrossRef] [PubMed]
- S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis," Opt. Express 8, 173-190 (2001), http://www.opticsinfobase.org/abstract.cfm?URI=oe-8-3-173. [CrossRef] [PubMed]
- A. Samoc, "Dispersion of refractive properties of solvents: chloroform, toluene, benzene, and carbon disulfide in ultraviolet, visible, and near-infrared," J. Appl. Phys. 94, 6167-6174 (2003). [CrossRef]
- D. N. Nikogosyan, Properties of Optical and Laser-Related Materials: A Handbook (Wiley, Chichester, UK, 1997).
- K. Okamoto, Fundamentals of optical waveguides (Academic Press, San Diego, 2000).
- I. P. Nikolakakos, A. Major, J. S. Aitchison, and P. W. E. Smith, "Broadband characterization of the nonlinear optical properties of common reference materials," IEEE J. Sel. Top. Quantum Electron. 10, 1164-1170 (2004). [CrossRef]
- COMSOL Multiphysics 3.3 (2007), COMSOL Inc. (http://www.comsol.com/).
- P. G. Kevrekidis, B. A. Malomed, and Y. B. Gaididei, "Solitons in triangular and honeycomb dynamical lattices with the cubic nonlinearity," Phys. Rev. E 66, 016609-10 (2002). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.