# Observation of discrete quadratic surface solitons.

**ABSTRACT** We report the first observation of discrete quadratic surface solitons in self-focusing and defocusing periodically poled lithium niobate waveguide arrays. By operating on either side of the phase-matching condition and using the cascading nonlinearity, both in-phase and staggered discrete surface solitons were observed. This represents the first experimental demonstration of staggered/gap surface solitons at the interface of a semi-infinite nonlinear lattice. The experimental results were found to be in good agreement with theory.

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**ABSTRACT:**We show that the optical beam evolution equations at the interface between a metal and a biased centrosymmetric photorefractive (CP) crystal can exhibit delocalized surface waves, shock surface waves, and localized surface waves (LSWs). Only LSWs have not oscillating tails in the volume of the CP crystal. For a given value of the propagation constant b , the transverse width and energy of LSWs increases with the order of LSWs. For a given order of LSWs, the energy of LSWs decreases with an increase in |b||b|, the transverse width of LSWs increases with |b||b| when |b||b| is less than the absolute value of a certain threshold bctbct and decreases with an increase in |b||b| when |b|>|bct||b|>|bct|. The stability properties of these LSWs are also discussed in detail.Optics Communications 12/2014; 332:327–331. · 1.54 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We study nonlinear surface modes at the edge of metal–dielectric nanostructured metamaterial with a nonlinear surface layer. We demonstrate that such semi-infinite structures can support transverse electric (TE) polarized surface states with subwavelength localization near the surface, an optical analogue of the Tamm states, even in the cases when the surface modes do not exist in the linear regime. (© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)physica status solidi (RRL) - Rapid Research Letters 01/2012; 6(1):43-45. · 2.39 Impact Factor - S.suntsov, K. G.makris, G. A.siviloglou, R.iwanow, R.schiek, D. N.christodoulides, G. I.stegeman, R.morandotti, H.yang, G.salamo, M.volatier, V.aimez, R.arÈs, M.sorel, Y.min, W.sohler, Xiaoshengwang, Annabezryadina, Zhigangchen[Show abstract] [Hide abstract]

**ABSTRACT:**The recent theoretical predictions and experimental observations of discrete surface solitons propagating along the interface between a one- or two-dimensional continuous medium and a one- or two-dimensional waveguide array are reviewed. These discrete solitons were found in second order (periodically poled lithium niobate) and third order nonlinear media, including AlGaAs, photorefractive media and glass, respectively.Journal of Nonlinear Optical Physics & Materials 01/2012; 16(04). · 0.48 Impact Factor

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Observation of discrete quadratic surface

solitons

Georgios A. Siviloglou, Konstantinos G. Makris, Robert Iwanow, Roland Schiek,

Demetrios N. Christodoulides and George I. Stegeman

College of Optics and Photonics, CREOL & FPCE, University of Central Florida

4000 Central Florida Blvd., Orlando Florida 32816, USA

george@creol.ucf.edu

Yoohong Min and Wolfgang Sohler

University of Paderborn, 33095 Paderborn, Germany

Abstract: We report the first observation of discrete quadratic surface

solitons in self-focusing and defocusing periodically poled lithium niobate

waveguide arrays. By operating on either side of the phase-matching

condition and using the cascading nonlinearity, both in-phase and staggered

discrete surface solitons were observed. This represents the first

experimental demonstration of staggered/gap surface solitons at the

interface of a semi-infinite nonlinear lattice. The experimental results were

found to be in good agreement with theory.

© 2006 Optical Society of America

OCIS codes: (190.2620) Frequency conversion; (190.4350) Nonlinear optics at surfaces;

(190.5530) Pulse propagation and solitons

References and Links

1.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear

waveguide lattices,” Nature 424, 817-823 (2003).

D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,”

Opt. Lett. 13, 794-796 (1988).

T. Peschel, U. Peschel, and F. Lederer, “Discrete bright solitary waves in quadratically nonlinear media,”

Phys. Rev. E 57, 1127-1133 (1998).

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in

photorefractive optically induced photonic lattices,” Phys. Rev. E 66, 046602 (2002).

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).

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, “Observation of

discrete quadratic solitons,” Phys. Rev. Lett. 93, 113902 (2004).

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).

A. Fratalocchi, G. Assanto, K. A. Brzdąkiewicz, and M. A. Karpierz, “Discrete light propagation and self-

trapping in liquid crystals,” Opt. Express. 13, 1808-1815 (2005).

A. A. Maradudin, “Nonlinear surface electromagnetic waves,” in Optical and Acoustic Waves in Solids-

Modern Topics, (World Scientific, Singapore, 1983), Chap. 2.

10. U. Langbein, F. Lederer, and H. E. Ponath, “A new type of nonlinear slab-guided waves,” Opt. Commun.

46, 167-169, (1983).

11. A. D. Boardman, P. Egan, F. Lederer, U. Langbein, and D. Mihalache, “Third-order nonlinear

electromagnetic TE and TM guided waves,” in Nonlinear Surface Electromagnetic Phenomena," edited by

H. E. Ponath, and G. I. Stegeman, (North-Holland, Amsterdam, 1991) p. 73-287.

12. C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations

of nonlinear TE waves guides by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum

Electron. 21, 774-783, (1985).

13. K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Hache, “Discrete surface solitons,”

Opt. Lett. 30, 2466-2468, (2005).

14. S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G.

Salamo, and M. Sorel, “Observation of discrete surface solitons,” Phys. Rev. Lett. 96, 063901 (2006).

15. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface gap solitons,” Phys. Rev. Lett. 96, 073901

(2006).

2.

3.

4.

5.

6.

7.

8.

9.

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16. N. N. Akhmediev, V. I. Korneev, and Y. V. Kuz’menko, Sov. Phys. JETP 61, 62 (1985).

17. R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, “Arrays of weakly

coupled, periodically poled lithium niobate waveguides: beam propagation and discrete spatial quadratic

solitons,” Opto-Electron. Rev. 13, 113-121 (2005).

18. R. Iwanow, G. I. Stegeman, R. Schiek, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, “Highly localized

discrete quadratic solitons,” Opt. Lett. 30, 1033-1035 (2005).

1. Introduction

The interplay between discreteness and nonlinearity has led to a host of new phenomena in

physical sciences. This has been most pronounced in the area of optics where high-quality

discrete structures can be fabricated and the optical power levels required to induce nonlinear

effects can be easily achieved [1]. Discreteness has resulted in the prediction of new classes

of spatial solitons and other phenomena that have no counterparts in continuous systems [2-4].

And indeed, many of these processes have been observed in a variety of Kerr, quadratic,

photorefractive and liquid crystal media [5-8].

Thus far, the arrays used for discrete optics experiments have been fabricated by a variety

of techniques, some of which lend themselves to small and controllable index differences at

the array boundary with continuous media. This feature can now facilitate new experimental

studies in the area of nonlinear surface guided waves which received a great deal of

theoretical attention in the 1980’s and early 1990’s [9-12]. The theoretical feasibility of

guiding waves along an interface between two media, at least one of which exhibits a self-

focusing nonlinearity was discussed extensively. Yet, in spite of these efforts, no successful

experiments have been reported along these lines. Part of the problem was to find media

combinations whose linear index difference was of the order of the maximum index change

allowed by self-focusing nonlinearities, i.e. typically 10-4 and less. For the weakly guiding

arrays currently in use, such small index differences are available at the interface between the

array and the host medium. This can in turn facilitate the observation of interface solitons as

recently suggested by our group [13]. Theory has already shown that such interface guided

waves do exist at the boundary between arrays and continuous media [13], and in fact they

have been observed for the first time in self-focusing Kerr lattices [14].

Discrete quadratic solitons have been previously demonstrated inside arrays governed by

the “cascading” quadratic nonlinearity [6]. One of the unique features of this nonlinearity is

that it can change from effectively self-focusing to defocusing depending on the wavevector

mismatch conditions. Thus both signs of the nonlinearity are accessible in the same sample

just by, for example, changing the temperature. This property has been used to demonstrate

both in-phase and staggered (adjacent fields are π out of phase with each other) spatial

solitons in these arrays [6]. In this paper we show theoretically and experimentally that both

types of quadratic surface discrete solitons exist for both signs of the cascading nonlinearity.

We note that this represents the first observation of gap surface solitons in arrays with

defocusing nonlinearity as earlier predicted [13, 15].

2. Theory

The system shown in Fig. 1 was modeled by employing a coupled mode formulation for

quadratic nonlinear media [3, 6].

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LiNbO3

n=0 n=4

n=3

n=2

n=1

1D Array

Semi-infinite Medium

x

z

y

Harmonic

Index

Fundamental

Fig. 1. (upper) Simple model for the discrete array-continuous medium interface. (lower)

Actual sample structure showing the index distribution, the fundamental fields and the

harmonic fields. The index distribution and fields are overlapped in space but have been

separated for clarity.

In our system, the adjacent waveguides comprising the array are weakly coupled by their

evanescent fields. Given the fact that the second harmonic (SH) TM00-modes are strongly

confined, the coupling process between the SH fields is negligible. Therefore, here we only

consider coupling between the modal fields of the fundamental wave (FW). In physical units,

the pertinent coupled mode equations describing the wave dynamics in a semi-infinite array

are given by the following:

*

0 0

0

1

*

n n

u v

γ

11

2

n

0,0

() 0,1

0,0

n

nn

n

n

u

z

u

z

i cuu v

γ

forn

ic uuforn

v

z

∂

ivu forn

βγ

+−

∂

⎧

⎪⎪ ∂

⎨∂

⎪

⎪ ∂

⎩

∂

++==

+++=≥

−Δ+=≥

(1)

,where un and vn are the FW and SH modal amplitudes in the nth waveguide respectively, c is

the linear coupling constant and γ is the effective quadratic nonlinear coefficient. Furthermore,

Δβ=2β(ω)-β(2ω) is the wavevector mismatch between the FW and SH.

Stationary solutions of the form un=fnexp(icμz) for the FW and vn=snexp(2icμz) for the

SH were numerically determined by applying Newtonian relaxation techniques. Here μ is the

soliton eigenvalue and is related to a nonlinear change in the propagation constant ΔkNL=cμ.

In-phase solitons are possible when 2ΔkNL+Δβ>0, while staggered solitons exist for

2ΔkNL+Δβ<0 [3]. The power versus nonlinear wavevector shift diagrams for both the in-phase

and staggered surface soliton families obtained are shown in Fig. 2 and Fig. 3, respectively,

along with the corresponding typical intensity profiles. Throughout this study we use the

parameters typical of the experiments. More specifically, the coupling length in this array is

taken to be 25 mm and the quadratic nonlinear coefficient is 18 pm/V [6].

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Fig. 2. (a) Surface soliton existence curves for in-phase solitons for 36π (red curve) and -15.5π

(blue curve). (b), (c) Intensity profiles for low, and high powers for FW (blue) and SH (red), in

the case of positive mismatch 36π,respectively, and (d) Intensity profiles for high powers for

FH (blue) and SH (red), in the case of negative mismatch -15.5π. The SH powers of both

solitons are overlapped for large nonlinear wavevector shifts.

Fig. 3. (a) Surface soliton existence curves for staggered solitons for a mismatch of -15.5π (red

curve) and 36π (blue curve). (b), (c) Intensity profiles for low, and high powers for FH (blue)

and SH (red), in the case of negative mismatch -15.5π, respectively, and (d) Intensity profiles

for high powers for FH (blue) and SH (red), in the case of positive mismatch 36π. The SH

powers of both solitons are overlapped for large nonlinear wavevector shifts.

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A number of interesting features are predicted for these quadratic surface solitons.

Different from the infinite arrays case, these surface self-trapped states exist only when their

power exceeds a critical level - a direct consequence of the semi-infinite geometry of the

lattice. This is a feature common to surface solitons at the interface between continuous

media, also found recently for surface solitons propagating due to self-focusing and self-

defocusing nonlinearities in Kerr media [13,15]. As the soliton power increases the fields

become progressively more confined in the n=0 channel. The fraction of power carried by the

SH is decreased as ΔkNL increases.

Furthermore, just as found for discrete solitons in infinite 1D media, the solitons consist of

coupled FW and SH fields. In addition to the expected staggered solutions, in-phase solitons

were also found under negative phase mismatch conditions for 2ΔkNL+Δβ > 0, i.e. with self-

focusing nonlinearities. See the blue curves in Fig. 2(a) for the existence curves and the field

distributions in Fig. 2(d). Note that this family of solitons can only be excited if the SH is

considerably stronger than the FW. Similarly in regions of positive phase-mismatch, both

stable in-phase and staggered (for 2ΔkNL +Δβ < 0, i.e. a self-defocusing nonlinearity for the

blue curves in Fig 3(a) and the fields in [ 3(d)] surface solitons are predicted to exist. This

mirrors the case predicted for infinite quadratically nonlinear 1D arrays [3]. We emphasize

that in all cases the branch associated with the SH wave in the existence curves [see Figs. 2(a)

and 3(a)] does not depend on the value of phase-mismatch Δβ. This can be formally proved

based on the fact that the waveguides are uncoupled for the SH wave.

Finally, we note that stability analysis of Eqs. (1) indicates that the predicted surface

solitons are stable in the regions where the slope of the curve is positive, in accordance with

the Vakhitov-Kolokolov criterion [16].

3. Experiment

The arrays used here consist of channels formed by Ti diffused into the surface of LiNbO3 as

shown in Fig. 1. Phase-matching for second harmonic generation is achieved by periodic

poling of the lithium niobate (PPLN) ferroelectric domains along the propagation direction.

This poling extends beyond the array but in that region the periodicity required for efficient

SHG is different from that required for the channels and the generation of the second

harmonic is very weak and can be neglected. There is no Ti in-diffused outside of the array,

therefore the array boundary corresponds to an interface between the 1D waveguide array and

a semi-infinite half-space.

The samples contained four waveguide arrays each consisting of 101 coupled channel

waveguides with propagation along the X-axis. The spacing between the arrays was

sufficiently large (> 100μm) that the region beyond each array boundary can be considered as

a half-space. Seven cm long waveguides were formed by titanium in-diffusion into the Z-cut

surface. TM00-mode waveguide losses were 0.2dB/cm for the FW at λ ≈1550 nm and

0.4dB/cm for its SH. The center-to-center channel separations was d = 16 μm resulting in a

coupling length of Lc =25 mm for the FW TM00 mode. These were determined from the output

intensity distribution under single waveguide excitation conditions [17,18]. The sample was

periodically poled with a period of 16.75 µm by electric field poling to achieve phase-

matching between the TM00 modes for SH generation at temperatures elevated to the range of

200-250°C. The required wave-vector mismatch was adjusted by varying the sample

temperature T. In our experiments the relation between the phase-mismatch ΔβL and sample

temperature T was measured to be ΔβL=8.1(234-T [0C] ) [17].

A 5-MHz train of bandwidth limited 9-ps-long pulses at a wavelength of 1557 nm was

produced by a modified Pritel fiber laser [17,18]. The pulses were stretched, amplified in a

large area core fiber amplifier, and then recompressed in a bulk grating compressor to give up

to 4 kW of peak power in nearly transform limited pulses 7.5-ps-long. The recompressed

pulses were spatially reshaped into elliptical Gaussian beam with 4.3 μm width (full width at

half maximum, FWHM) and 3.5 μm height which corresponds to the measured intensity

profile of the fundamental mode of a single channel waveguide. In all experiments the n=0

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channel was excited. The sample was heated in an oven both to minimize photorefractive

effects and to adjust the wave-vector mismatch with temperature T. The experimental set-up

used is shown in Fig. 4

Fig.4. Experimental set-up: MO-microscope objective; PPLN-periodically poled lithium

niobate; Pol-Polarizer;λ/2-half-wave plate.

The output of the array was observed with separate cameras for the FW and the SH, and

quantified by measuring temporally averaged output intensities and total powers.

Figure 5 shows the observed FW discrete diffraction pattern obtained at low powers. It is

in good agreement with the theoretical pattern generated from Eqs. (1). of Ref. [13].

Fig. 5. Linear diffraction when only n=0 is excited, experiment (blue) and theory (red).

The evolution of the output intensity distributions versus input peak power of the

fundamental for single channel excitation (n=0) is shown in movies in Fig. 6 (positive phase

mismatch) and Fig. 7 (negative phase mismatch). Increasing the input peak power leads in

both cases to localization into surface solitons, as predicted theoretically. At peak powers of

600W for the focusing case and 500W for the defocusing case the localization is essentially

complete. The observed intensity decay into the array from the boundary out to distances

typical of the low power discrete diffraction pattern is a direct consequence of the pulsed

excitation used which contains a continuum of powers. That is, not the full pulse is trapped as

a surface soliton in the boundary channel and the weaker parts of the pulse appear as part of a

modified linear discrete diffraction pattern. We want to mention here that we controlled the

input powers with a combination of a polarizer and a half-wave plate, and thus the power

scaling is sinusoidal. The weak second harmonic component is localized almost completely in

the n=0 channel, in agreement with theory in Fig. 2(b) and 2(c) and 3(b) and 3(c).

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Fig. 6. The evolution of the output intensity distribution with increasing input peak power of

the fundamental for single channel excitation (n=0) (positive phase mismatch = +36π)

Fig. 7. The evolution of the output intensity distributions versus input peak power of the

fundamental for single channel excitation (n=0) (negative phase mismatch = -15.5π)

An important problem is to verify which field distributions were generated, staggered or

in-phase for each sign of the cascading nonlinearity. Theory has shown the ratio of the FW to

SH powers are very different in the two cases. In order to compare experiment approximately

with theory, the assumed hyperbolic secant temporal profile was decomposed into cw

temporal slices and the pulse response was simulated by adding the slices together. A fourth

order Runge-Kutta method was then used to propagate the fields under the influence of Eq.

(1). Comparing the measured and the calculated ratios of the FW to SH powers at the output,

clearly the observed surface solitons were the staggered ones for negative and the in-phase

ones for positive mismatch since the experimentally measured power ratio FW/SH was much

bigger than unity. It would be necessary to also input the appropriate SH field in order to

excite the other surface solitons.

Using the same numerical approach, the output intensity distributions across the array

were calculated versus input peak power for both positive (+36π) and negative (-15.5π) phase

mismatches for in-phase and staggered solitons respectively. A sampling of these results,

along with the corresponding experimental data is shown in Fig. 8 and Fig. 9:

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Fig. 8

for single channel excitation for two input power levels corresponding to partial collapse into a

surface soliton for FH (first row) and full collapse into a surface soliton for FH (second row)

and SH (last row). Phase mismatch = +36π (self-focusing nonlinearity). The red curves

represent theoretical results (at 435 W and 441 W) and the blue experimental data (at 430 W,

and 600W) FW input powers.

Measured (let-hand-side) and calculated (right-hand-side) output field distributions

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Fig. 9.

for single channel excitation for two input power levels corresponding to partial collapse into a

surface soliton for FH (first row) and full collapse into a surface soliton for FH (second row)

and SH (last row). Phase mismatch = -15.5π (self-defocusing nonlinearity). The red curves

represent theoretical results (at 310 W and 321 W) and the blue experimental data (at 420 W,

and 580W) FW peak input powers.

Measured (let-hand-side) and calculated (right-hand-side) output field distributions

In fact there is good qualitative agreement between experiment and theory considering the

non-ideally hyperbolic secant temporal profile of the input beam and the coupling efficiency

estimated from low power throughput experiments [17,18]. If a coupling efficiency of 50% is

assumed into the input channel, the resulting quantitative agreement is also good

4. Summary

In summary, discrete quadratic solitons guided by the interface between a 1D array and a

semi-infinite medium have been predicted theoretically and observed experimentally. Two of

the four predicted soliton types were generated by exciting the first channel with a beam at the

fundamental frequency. The additional solitons not observed will probably require the

excitation of both fundamental and second harmonic fields. Finally, the results reported here

represent the first observation of staggered discrete surface solitons in any periodic system.

This research was sponsored by NSF.

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