International Journal of Modern Nonlinear Theory and Application, 2012, 1, 130-134
http://dx.doi.org/10.4236/ijmnta.2012.14020 Published Online December 2012 (http://www.SciRP.org/journal/ijmnta)
Modeling of Complex Solitary Waveforms
for Micro-Width Doped ZnO
Rosmin Elsa Mohan1, M. Sivakumar2, K. S. Sreelatha3
1Amrita Vishwa Vidyapeetham, Kollam, India
2Amrita Vishwa Vidyapeetham, Coimbatore, India
3Govt.Polytechnic College, Kottayam, India
Email: firstname.lastname@example.org, email@example.com, firstname.lastname@example.org
Received September 6, 2012; revised November 5, 2012; accepted November 16, 2012
The potential applications of metallic oxides as supporters of nonlinear phenomena are not novel. ZnO shows high
nonlinearity in the range 600 - 1200 nm of the input wavelength . ZnO thus make way to become efficient photolu-
minescent devices. In this paper, the above mentioned property of ZnO is harnessed as the primary material for the fa-
brication of waveguides. Invoking nonlinear phenomena can support intense nonlinear pulses which can be a boost to
the field of communication. The modeling characteristics of undoped and doped ZnO also confirm the propagation of a
solitary pulse . An attempt to generalize the optical pattern of the doped case with varying waveguide widths is car-
ried out in the current investigation. The variations below 6 um are seen to exhibit complex waveforms which resemble
a continuum pulse. The input peak wavelength is kept constant at 600 nm for the modeling.
Keywords: Solitons; Nonlinear Optics; Doped ZnO Waveguides; Continuum
ZnO has recently attracted wide interest for its unique
properties and versatile applications in the fields of pie-
zoelectric devices, light sensors, spintronics [2-10] and
acoustic wave devices [11,12]. Nanostructured electrodes
of ZnO have also been used as solar cells  with its
physical and chemical properties that can be varied by
adequate doping by cationic or anionic substitution. Dop-
ing with B or Mn decreased the resistivity [14,15] or in-
troduced ferromagnetism  respectively. The opto-
electronic properties are generally affected by impurities
and defects. Impurity incorporation thus plays a domi-
nant role in the possible applications of ZnO in the field
The effect of Ag as a Group 1 element is a candidate
acceptor for ZnO. Ag doping could greatly increase the
catalytic doping and photo activity in semiconductors
[17,18]. The silver atoms may be incorporated into the
lattice sites of ZnO only as the substitution of the Zn
atom sites . Thus the doping with Ag requires syste-
matic investigation. The silver ions have novel applica-
tions of shifting the emission spectrum of doped ZnO
beyond the UV-blue region making it a promising can-
didate for communication via the propagation of solitary
The Guiding Phenomena
For any waveguide a refractive index larger than the sur-
roundings is needed. Planar waveguides allow confine-
ment in one direction though diffraction may occur along
the plane of the film. Fiber and channel waveguides al-
low cross-sectional dimensions with the size of confine-
ment to be the order of the wavelength . As a result
higher intensities for a given input power can be supplied
as the effective beam area is minimized in a waveguide.
In effect, the guided wave field is maximum in the region
of high index and decays with distance into the media of
lower index. Nonlinearity can therefore be either in the
core or the surrounding media. However dominance of
the nonlinear phenomena with optimum efficiency is
mostly seen in the core region.
The propagation constants with their corresponding
eigen modes depend on the dimensions of the high-index
(core) region, the geometry of the waveguide structure,
and the refractive index of the wave guiding media. The
modes, TE and TM with E or H in the plane of the sur-
faces, need to be orthogonal to one another and should
occur in two unique polarizations. The two orthogonal
modes dominate though the fields contain contribution
opyright © 2012 SciRes. IJMNTA
R. E. MOHAN ET AL. 131
from all three polarizations. Any degeneracy in the cor-
responding modes can cause birefringence which can be
termed polarization preserving in that specification to
support two orthogonally linearly polarized eigen modes
Wave guiding can also introduce reduction in the spa-
tial degrees of freedom which can in turn limit the propa-
gation wave vectors to two dimensions in planar wave-
guides or to one dimension in fiber and channel guides.
This greatly benefits nonlinear interaction associated
with intensity dependant refractive index. Wave vector
interactions that result can be accomplished by adjusting
the lengths at which these wave vector interactions occur.
It is possible to excite the waveguide modes from the
sides or from the ends of the waveguide. The angle of
incidence is so chosen that only one mode is excited at a
time for plane wave incidence. Optically aligning the
waveguide to the incident beam allows almost all the
guided-wave power to be launched in to lowest-order
mode with appropriate polarization.
phenomena have been demonstrated in
planar waveguides such as second-harmonic generation,
difference frequency generation, optical parametric am-
plification and optical parametric oscillation. SHG has
been widely studied of these, though restricted mainly to
the field of integrated optics. As for the simplest case of
SHG, a single fundamental guided wave is excited at z =
0 propagating to z = L where it leaves the waveguide
along with the second harmonic generated between 0 and
L. The second harmonic power, is given in
terms of the fundamental input power  by
Here is the second harmonic power in
terms of the fundamental input power across
the waveguide length L.
The power scaling of the Second Harmonic generation
is enables characterization and generalization of the
waveguides. Here the waveguide figure of merit is given
n where is the effective index and the
effective waveguide thickness.
Φ is the phase vector, for the simplest SHG condition
in terms of the guided wave vectors, Φ = 0,
where 1 and 2 refer to the funda-
mental and harmonic respectively.
The overlap integral , a concept unique to wave-
guides is given by
where the terms govern the product of the field distribu-
tions across the waveguide; the latter if negative reduces
the value of K in effect.
The overlap integral is usually small for the field dis-
tribution modes. It seems that the existence of modes
with different values of the effective index facilitate phase
matching. In this case, the overlap integral is extremely
reduced or even zero.
In planar waveguides, the angles at which the mode
intersections associated with phase matching occurs must
be very small so as to satisfy the minimum thickness for
the film as required for phase matching. This condition
equally lets the birefringence and material dispersion to
be quite small [20,21].
3. Doped ZnO on Silica Substrates:
Modeling in the Dispersive Regime
The propagation characteristics of the guided wave are
obtained provided the guided-wave field satisfies the pro-
per boundary conditions at the interface of two different
media (i.e., tangential electric and magnetic-field vectors
must be continuous across the boundary) and necessary
radiation conditions.Along a straight line path, every com-
ponent of the electromagnetic wave that propagates may
be represented as
, where z is chosen
as the propagation direction and u, v are orthogonal co-
ordinates in a transverse plane. β is the propagation con-
stant and ω is the frequency of the wave. However, the
fundamental property of a planar waveguide is the relation
between the number and nature of the waveguide modes
propagated and its refractive index .
For nonlinear waveguides, integral representations for
the longitudinal electric and magnetic fields satisfy the
appropriate wave equations and all the necessary bound-
ary conditions. By approximate expansions of these fields
and employing the analytic continuation technique ,
the relevant integral equations may be reduced to linear
algebraic equations which may be solved to obtain the
The field mode distributions of doped ZnO in the 800 -
1200 nm of the input wavelength have shown increased
nonlinear effects. Experimental analysis of Ag doped ZnO
has revealed interesting changes in physical and chemical
properties at the nano scale such as crystallinity, optical
transmittance, absorption and refraction patterns etc. [24,
25]. The doped waveguides can be used for making in-
expensive optical devices. The field modes for a doped
ZnO waveguide structure, n2 = 2.099, show dispersive
behavior in accordance with linear losses and two photon
absorption around 1000 nm. However, we have consi-
dered variations with waveguide width of 0.6 micrometer
and less for an input wavelength of within 600 nm. This
was so chosen so as to minimize the dispersive effects and
Copyright © 2012 SciRes. IJMNTA
R. E. MOHAN ET AL.
oscillatory behavior of solitary pulses beyond these di-
4. Solitary Pulses in Doped ZnO: The Route
The interplay of dispersion and nonlinear self-action in
wave dynamics has been a major area of interest across
many branches of physics since the Fermi-Pasta-Ulam
work. Localized nonlinear waves have been often re-
ferred to as solitary waves, however today the term’ so-
liton’ has been extended over the nonintegrable cases as
well. Optical solitons in fibers have been researched much
over the years as potential information carriers (Molle-
nauer and Gordon 2006; Agrawal in 2007). Octave wide
spectral broadening was later observed in the beginning
of the 21st century which was extensively studied and
came to be known as Supercontinuum.
The first experiments of Supercontinuum inadvertently
marked the presence of solitons in the process. A fiber
with high nonlinearity and the GVD point close to the
pump wavelength has large potential for harnessing. Fi-
bers with silica cores (~1 - 5 µm) of a few microns in
diameter have been studied extensively with a variety of
sources. Investigations with Femtosecond pulses with
wavelength around 800 nm (Ranka et al., 2000) and with
nanosecond microchirp lasers close to 1 µm (Stone and
Knight 2008) gave much promising results. The disper-
sion profile in the latter exhibited intense supercontinua
extending towards the shorter “bluer” wavelengths (Har-
bold et al., 2002; Efimov et al., 2004).
The modeling of ZnO and the doped structure (Ag-ZnO)
confirm the possible passage of a solitary pulse . The
solitons show a self-consistency, characteristic of its self-
guided nature, to be a mode of the linear waveguide it in-
duces [1,2]. The length of the passage may be extended
using a doped form of the initial waveguide. The field
mode distributions are found to vary with the input pow-
er and the wavelength. Solitons if incorporated in wave-
guides can revolutionize optical communication systems
with their ability to carry information over long distances
without a change in shape.
In the current investigation, an index contrast of glass
(1.456) to that of air in addition to the increased nonlin-
earity of the material of the waveguides provides high
modal confinement and significant contribution to dis-
persion. The width of the waveguide structures are varied
within and below the 6 µm scale when a continuum pulse
is seen to propagate (Figures 1(a)-(c)) with variation in
the refractive index governing the waveguide width. The
width variations show a continuum spectra for the silver
doped ZnO structure which confirms the possibility of a
supercontinuum in these structures. The propagating
pulses retain a continuous solitonic path thus enabling
Figure 1. (a) Solitary propagation for w = 1 um at 600 nm
peak input wavelength; (b) for w = 3 um at 600 nm peak
input wavelength; (c) for w = 5 um at 600 nm input wave-
Copyright © 2012 SciRes. IJMNTA
R. E. MOHAN ET AL. 133
Figure 2. Contour plots for the propagating continua pulses
at (a) w = 1 um (b) 3 um (c) 5 um for the input wavelength
of 600 nm.
the basic need of low losses in communication. The re-
fractive index of the doped structure (n2 = 2.099) is par-
ticularly important with relevance to the fact that it pre-
vents the dispersive spreading of the radiation waves
within the short wavelength range of the continuum. The
change in refractive index is seen to exert an inertial
force which further ensures a dispersionless propagation
of the radiation (Figures 2(a)-(c)). Sub-wavelength dia-
meter nanowires have proved to offer effective nonlin-
earities and interesting dispersion profiles . Such
structures enabled ultra-efficient octave spanning for nano-
second and femto second input pulses .
In the Figures 1 and 2 it can seen that an initial pulse,
which has an input wavelength of 600 nm, takes form as
a solitary pulse when it propagates through the waveguide
length. The soliton stability is improved as the width of
wave guide is increased which can be directly seen from
the figures. The amplitude of the pulse can be optimized
using the refractive index variations.
Earlier studies carried out in this regard had confirmed
solitons for an input wavelength between 800 nm and
1000 nm . In the present study we have varied the
waveguide width within a micro range thereby varying
the nature of the continuum passing through the wave-
guide. The doped ZnO waveguide widths below 6 um
were considered in the present investigation. The onset of
a solitary pulse which was confirmed to be stable without
dispersion around 600 nm was exploited. The variation in
the solitary propagation resembles closely to continuum
The author would like to thank University Grants Com-
mission for JRF under the MANF scheme.
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