Page 1

Observation of two-dimensional defect surface solitons

A. Szameit1, Y. V. Kartashov2, M. Heinrich1, F. Dreisow1, T. Pertsch1, S. Nolte1, A.

Tünnermann1,3, F. Lederer4, V. A. Vysloukh2, and L. Torner2

1Institute of Applied Physics, Friedrich-Schiller-University Jena, Max-Wien-Platz 1,

07743 Jena, Germany

2ICFO-Institut de Ciencies Fotoniques, and Universitat Politecnica de Catalunya,

Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain

3Fraunhofer Institute for Applied Optics and Precision Engineering, Albert-Einstein-

Strasse 7, 07745 Jena, Germany

4Institute for Condensed Matter Theory and Optics, Friedrich-Schiller-University Jena,

Max-Wien-Platz 1, 07743 Jena, Germany

We report on the experimental observation of two-dimensional solitons located in a defect

channel at the surface of a hexagonal waveguide array. The threshold power for the excita-

tion of defect surface solitons existing due to total internal reflection grows with decrease of

the refractive index in negative defects and vanishes for sufficiently strong positive defects.

Negative defects can also support linear surface modes existing due to Bragg-type reflec-

tions.

OCIS codes: 190.0190, 190.6135

Discrete solitons [1,2] may emerge at interfaces of materials yielding different physical

properties. Discrete surface solitons introduced recently for the interface of uniform and pe-

riodic nonlinear media [3,4] are particularly interesting since they combine the characteris-

tics of surface states at uniform interfaces and the unique properties of lattice solitons aris-

ing from the periodicity of underlying medium. Surface solitons may exist not only at focus-

ing but also at defocusing lattice interfaces [5-7]. Very recently the properties of two-

dimensional surface solitons were analyzed theoretically [8-13] and experimentally [14-16]. In

all these settings the discrete surface states were realized at the edge of perfectly periodic

lattices. However, the addition of defects may substantially modify the guiding properties

[17]. Lattices with positive or negative defects allow for linear localized states existing due

1

Page 2

to total internal or Bragg-type reflection [18]. When placed at the edge of a one-dimensional

lattice, the defects strongly alter the properties of surface solitons [19,20]. Nevertheless, in

two-dimensional structures the formation of nonlinear defect surface states and the genera-

tion of thresholdless surface waves were not observed.

In this Letter we show the existence of different types of two-dimensional defect sur-

face solitons, relying on two different guiding mechanisms, and report on their experimental

observation in femtosecond laser written waveguide arrays with hexagonal symmetry. We

show that the threshold power and the excitation dynamics for surface solitons strongly de-

pend on the type and strength of the surface defect.

In the simulations we employ the nonlinear Schrödinger equation for the dimensionless

field amplitude q under the assumption of cw illumination:

22

2

22

1

2

( , ) .

η ζ

pR

q

ξ

qq

iq q

ηζ

⎛

⎜

⎜

⎜

⎝

⎞

⎟

⎟

⎟

⎠

∂

∂

∂

∂

∂

∂

= −+−−

q

aa

d

(1)

Here ξ is the longitudinal and are the transverse coordinates. The parameter stands

for the refractive index modulation depth, while the function describes the refractive

index profile in the array as a superposition of Gaussian functions

modeling the individual waveguides, which are arranged in a hexagonal array with spacing

. The defect waveguide is located in the first row. The refractive index in the defect

waveguide is given by , while in other waveguides it is . For positive defects

one has and for negative ones . We set w

, , and

in accordance with the transverse waveguide dimensions of 4.5

and the spacing of

. We fixed that corresponds to a real modulation depth . In the

simulations we model the array with a finite number of waveguides (according to the ex-

periment) but results hold true for semi-infinite arrays.

, η ζ

p

(

ηζ

w

4

0−

1

( , )

R η ζ

ex

p =

0.45

=

9 μ×

22

p[ ( /

−

)/) ]

ww

ηζ

−

a p

0.9

wζ=

m

s

4

=

2

3

×

∼

s

w

40

d

pp

=

2.8

d

pp

>

d

pp

<

η

m

μ

a p =

The stationary profiles of surface solitons residing in the defect channel are of the form

, where b is the propagation constant. Such solitons exist for b values

above a cutoff and they can be characterized by their total power

.

We found two types of defect surface states: those mediated by total internal reflection and

those emerging from Bragg reflection. The states forming due to total internal reflection

never change their sign between the waveguides [Figs. 1(a)-1(c)]. Their properties are sum-

( , )exp(

η ζ

)

qwib

ξ=

co

b

2dη ζ

∞

−∞

=∫ ∫

Uw

2

Page 3

marized in Figs. 2(a)-2(c). When the dependence for surface solitons is

nonmonotonic [Fig. 2(a), curve 1]. Solitons exist only above the power threshold , which

is also typical for surface solitons in defect-free arrays. The slope of curve tends to in-

finity in the cutoff, where the soliton is most extended. Also, the threshold U

increases

with decreasing [Fig. 2(b)], while the cutoff changes only slightly. In contrast, for

the cutoff changes rapidly [Fig. 2(c)] while the threshold vanishes [Fig. 2(a),

curves 2 and 3], since strong positive surface defects support linear modes [shapes of low-

power solitons close to linear modes are shown in Figs. 1(a) and 1(c)]. Note that for

, when the penetration of the field into the lattice is maximal, increasing or de-

creasing the defect strength results in a dramatic reduction of the light field penetration,

even close to the cutoff. Far from the cutoff the light gradually concentrates in the defect

site [Fig. 1(b)]. Since only surface states on the branches of the U b curve are

linearly stable, there exist narrow instability domains close to the cutoff for

. For the surface solitons become stable in the entire domain of

existence.

d

2.94

p ≤

)

( )

U b

/db

th

U

th

)

m/s

μ

( )

U b

th

U

d

p

/s

d

p

d

2.94

p >

d

2.94

p ≈

d

2.94

p

<<

μ

0

dU

>

(

2.99

m/s

d

2.99

p >

d

( 2.

p <

d

p

Negative defects may support a second type of surface modes mediated by

Bragg-type reflection from the periodic structure. The field in such modes exhibits a compli-

cated staggered structure and changes its sign [Fig. 1(d)]. Nonlinear surface modes sup-

ported by negative defects thus bifurcate from such linear surface modes; hence their power

vanishes in the cutoff [Fig. 2(d)]. Therefore, such states can be dynamically excited with

very small powers. This is in sharp contrast to defect surface solitons mediated by total in-

ternal reflection which feature considerable thresholds at such small values. The degree

of localization of staggered surface modes increases and their narrow domain of existence

expands with decreasing .

8

For the experimental observation of defect surface solitons we used waveguide arrays

in fused silica glass fabricated with a femtosecond direct writing technique [21]. The length

of our samples was 105 mm, while the spacing of the guides was 40 μm. The writing velocity

was 1750

to keep the uniform nonlinear refractive index of the unprocessed material

in the waveguides [22]. The defect sites were written with 1500

and 2000

for a

positive defect and negative defect, respectively. For the excitation of the solitons, we used

a 1 kHz fs laser system (Spectra Tsunami/Spitfire) with a pulse length of about 180 fs at a

central wavelength of 800 nm. To confirm the experimental data by simulations, we solved

m

μ

3

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Eq. (1) with the input conditions

22

d

0

exp[ ()( ) ]

qA

ξ

ηηζζ

==−−−−

d

3)

, where are the

coordinates of the center of the defect surface waveguide. Note that our experiments are

conducted with pulsed pump light, while the numerical simulations assume CW illumina-

tion; thus the comparison serves only to confirm the consistency of observations with soliton

formation around the pulse peak.

dd

,

η ζ

In the array without defect (Fig. 3) the formation of nonlinear surface waves is similar

to that in recent experiments [23]. At low powers one observes strong linear diffraction [Fig.

3(a)]. For intermediate power levels, the light gradually contracts towards the excited sur-

face channel [Fig. 3(b)]. The formation of surface soliton is observed above the threshold

power [Fig. 3(c)]. The picture changes dramatically in the case of a positive surface defect.

Stronger defects result in a gradual suppression of light diffraction even at low powers and,

eventually, in the formation of a linear surface mode. This is accompanied by a strong re-

duction of the power threshold for soliton excitation. Thus, an array with a strong positive

defect supports a well localized linear mode shown in Fig. 4(a). At all power

levels the light remains confined to the surface defect waveguide. An increase of the input

power does not result in considerable changes of the mode profiles [see Figs. 4(b) and 4(c)].

d

( 3.0

p =

We also observed light localization at low powers in the case of a negative surface de-

fect for [Fig. 4(d)], a result that we attribute to the existence of linear defect sur-

face modes caused by Bragg-type reflection from the periodic structure. Notice that in the

absence of defects surface modes, soliton excitation always occurs above a threshold power,

for both total internal or Bragg reflection mechanisms of localization. Increasing the input

power results in a matching of the total refractive index in the defect waveguide and in the

array, which is accompanied by a sudden delocalization at intermediate power levels [Fig.

4(e)]. When the input power is further increased one excites the usual surface soliton exist-

ing due to total internal reflection [Fig. 4(f)]. The threshold power for the excitation of such

solitons is much higher than in the case of defect-free arrays or positive surface defects.

d

2.7

p =

In conclusion, we reported on the existence of two-dimensional defect surface solitons

for both positive and negative defects. The inclusion of positive defects allows for the forma-

tion of surface solitons existing due to total internal reflection at a reduced threshold power.

Negative defects may support surface states mediated by both, total internal reflection and

Bragg-type reflection from the periodic structure.

4

Page 5

References with titles

1.

D. N. Christodoulides and R. I. Joseph, "Discrete self-focusing in nonlinear arrays

of coupled waveguides," Opt. Lett. 13, 794 (1988).

2.

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observa-

tion of two-dimensional discrete solitons in optically induced nonlinear photonic

lattices," Nature 422, 147 (2003).

3.

K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Hache,

"Discrete surface solitons," Opt. Lett. 30, 2466 (2005).

4.

S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Haché, R.

Morandotti, H. Yang, G. Salamo, and M. Sorel, "Observation of discrete surface

solitons," Phys. Rev. Lett. 96, 063901 (2006).

5.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Surface gap solitons," Phys.

Rev. Lett. 96, 073901 (2006).

6.

C. R. Rosberg, D. N. Neshev, W. Krolikowski, A. Mitchell, R. A. Vicencio, M. I.

Molina, and Y. S. Kivshar, "Observation of surface gap solitons in semi-infinite

waveguide arrays," Phys. Rev. Lett. 97, 083901 (2006).

7.

E. Smirnov, M. Stepic, C. E. Rüter, D. Kip, and V. Shandarov, "Observation of

staggered surface solitary waves in one-dimensional waveguide arrays," Opt. Lett.

31, 2338 (2006).

8.

Y. V. Kartashov and L. Torner, "Multipole-mode surface solitons," Opt. Lett. 31,

2172 (2006).

9.

Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, "Surface vortex

solitons," Opt. Express 14, 4049 (2006).

10. K. G. Makris, J. Hudock, D. N. Christodoulides, G. I. Stegeman, O. Manela, and

M. Segev, "Surface lattice solitons," Opt. Lett. 31, 2774 (2006).

11. Y. V. Kartashov, V. A. Vysloukh, D. Mihalache, and L. Torner, "Generation of

surface soliton arrays," Opt. Lett. 31, 2329 (2006).

12. R. A. Vicencio, S. Flach, M. I. Molina, and Y. S. Kivshar, "Discrete surface solitons

in two-dimensional anisotropic photonic lattices," Phys. Lett. A 364, 274 (2007).

5

Page 6

13. H. Susanto, P. G. Kevrekidis, B. A. Malomed, R. Carretero-Gonzalez, and D. J.

Frantzeskakis, "Discrete surface solitons in two dimensions," Phys. Rev. E 75,

056605 (2007).

14. X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G. I.

Stegeman, "Observation of two-dimensional surface solitons," Phys. Rev. Lett. 98,

123903 (2007).

15. A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann,

and L. Torner, "Observation of two-dimensional surface solitons in asymmetric

waveguide arrays," Phys. Rev. Lett. 98, 173903 (2007).

16. A. Szameit, Y. V. Kartashov, F. Dreisow, M. Heinrich, V. Vysloukh, T. Pertsch, S.

Nolte, A. Tünnermann, F. Lederer and L. Torner, "Two-dimensional lattice inter-

face solitons," Opt. Lett. 33, 663 (2008).

17. Z. Chen, "Linear and nonlinear localization in induced waveguide lattices," invited

talk at the 407. Heraeus workshop "Discrete Optics and Beyond" in Bad Honnef

(2008).

18. I. Makasyuk, Z. Chen, J. Yang, "Band-gap guidance in optically induced photonic

lattices with a negative defect," Phys. Rev. Lett. 96, 223903 (2006).

19. M. I. Molina, I. L. Garanovich, A. A. Sukhorukov, and Y. S. Kivshar, "Discrete

surface solitons in semi-infinite binary waveguide arrays," Opt. Lett. 31, 2332

(2006).

20. A. A. Sukhorukov, D. N. Neshev, A. Dreischuh, R. Fischer, S. Ha, W. Krolikowski,

J. Bolger, A. Mitchell, B. J. Eggleton, and Y. S. Kivshar, "Polychromatic nonlinear

surface modes generated by supercontinuum light," Opt. Express 14, 11265 (2006).

21. A. Szameit, F. Dreisow, T. Pertsch, S. Nolte, and A. Tünnermann, "Control of di-

rectional evanescent coupling in fs laser written waveguides," Opt. Express 15,

1579-1587 (2007).

22. D. Blömer, A. Szameit, F. Dreisow, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte,

F. Lederer, and A. Tünnermann, "Nonlinear refractive index of fs-laser-written

waveguides in fused silica," Opt. Express 14, 2151 (2006).

23. A. Szameit, Y. V. Kartashov, V. A. Vysloukh, M. Heinrich, F. Dreisow, T. Pertsch,

S. Nolte, A. Tünnermann, F. Lederer, and L. Torner, "Angular surface solitons in

sectorial hexagonal arrays," Opt. Lett. 33, 1542 (2008).

6

Page 7

References without titles

1.

D. N. Christodoulides and R. I. Joseph, Opt. Lett. 13, 794 (1988).

2.

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422,

147 (2003).

3.

K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Hache,

Opt. Lett. 30, 2466 (2005).

4.

S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Haché, R.

Morandotti, H. Yang, G. Salamo, and M. Sorel, Phys. Rev. Lett. 96, 063901

(2006).

5.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 96, 073901

(2006).

6.

C. R. Rosberg, D. N. Neshev, W. Krolikowski, A. Mitchell, R. A. Vicencio, M. I.

Molina, and Y. S. Kivshar, Phys. Rev. Lett. 97, 083901 (2006).

7.

E. Smirnov, M. Stepic, C. E. Rüter, D. Kip, and V. Shandarov, Opt. Lett. 31, 2338

(2006).

8.

Y. V. Kartashov and L. Torner, Opt. Lett. 31, 2172 (2006).

9.

Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, Opt. Express 14,

4049 (2006).

10. K. G. Makris, J. Hudock, D. N. Christodoulides, G. I. Stegeman, O. Manela, and

M. Segev, Opt. Lett. 31, 2774 (2006).

11. Y. V. Kartashov, V. A. Vysloukh, D. Mihalache, and L. Torner, Opt. Lett. 31,

2329 (2006).

12. R. A. Vicencio, S. Flach, M. I. Molina, and Y. S. Kivshar, Phys. Lett. A 364, 274

(2007).

13. H. Susanto, P. G. Kevrekidis, B. A. Malomed, R. Carretero-Gonzalez, and D. J.

Frantzeskakis, Phys. Rev. E 75, 056605 (2007).

14. X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G. I.

Stegeman, Phys. Rev. Lett. 98, 123903 (2007).

15. A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann,

and L. Torner, Phys. Rev. Lett. 98, 173903 (2007).

7

Page 8

16. A. Szameit, Y. V. Kartashov, F. Dreisow, M. Heinrich, V. Vysloukh, T. Pertsch, S.

Nolte, A. Tünnermann, F. Lederer and L. Torner, Opt. Lett. 33, 663 (2008).

17. Z. Chen, "Linear and nonlinear localization in induced waveguide lattices," invited

talk at the 407. Heraeus workshop "Discrete Optics and Beyond" in Bad Honnef

(2008).

18. I. Makasyuk, Z. Chen, J. Yang, Phys. Rev. Lett. 96, 223903 (2006).

19. M. I. Molina, I. L. Garanovich, A. A. Sukhorukov, and Y. S. Kivshar, Opt. Lett.

31, 2332 (2006).

20. A. A. Sukhorukov, D. N. Neshev, A. Dreischuh, R. Fischer, S. Ha, W. Krolikowski,

J. Bolger, A. Mitchell, B. J. Eggleton, and Y. S. Kivshar, Opt. Express 14, 11265

(2006).

21. A. Szameit, F. Dreisow, T. Pertsch, S. Nolte, and A. Tünnermann, Opt. Express

15, 1579-1587 (2007).

22. D. Blömer, A. Szameit, F. Dreisow, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte,

F. Lederer, and A. Tünnermann, Opt. Express 14, 2151 (2006).

23. A. Szameit, Y. V. Kartashov, V. A. Vysloukh, M. Heinrich, F. Dreisow, T. Pertsch,

S. Nolte, A. Tünnermann, F. Lederer, and L. Torner, Opt. Lett. 33, 1542 (2008).

8

Page 9

Figure captions

Figure 1. Field distributions for defect surface solitons at (a) , ,

(b) , , (c) , , (d) , .

The white dashed lines indicate the interface position.

0.4644

b =

0.274

b =

d

p =

2.96

p =

d

1.7 0.512

b =

d

2.96

p =

0.49

b =

d

3.1

p =

Figure 2. (a) Power versus propagation constant for (curve 1), 2.96 (curve

2), and (curve 3). The circles correspond to solitons in Figs. 1(a) and 1(b).

3

Threshold power (b) and cutoff (c) versus p . Dashed lines indicate the point

. (d) Power versus propagation constant for a surface mode existing

due to Bragg guiding at . The circle corresponds to profile from Fig.

1(d).

d

2.92

p =

d

2.8

d

p =

d

1.7

p =

Figure 3. Comparison of the output intensity distributions for an excitation of a surface

waveguide in array without defect at . Top row - experiment,

bottom row - theory. The input peak power is (a) , (b) 1.3

,

and (c) 3.

.

da

2.8

pp

==

0.18 MWMW

2 MW

Figure 4. Output intensity distributions for an excitation of (a)-(c) positive surface de-

fect waveguide at , or (d)-(f) negative surface defect wave-

guide at , . Input peak power is (a) 0.2

, (b) 1.3

,

(c) 2.7

, (d) , (e) , and (f) .

d

p =

3.03

p =

a

2.8

0.2 MW

a

2.8

p =

1.3 MW

d

2.7

p =

MW

MW

MW

MW

3.2

9

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11

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12

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13