## Abstract

We present a numerical study of a two dimensional all-optical switching device which consists of two crossed waveguides and a nonlinear photonic band-gap structure in the center. The switching mechanism is based on a dynamic shift of the photonic band edge by means of a strong pump pulse and is modeled on the basis of a two dimensional finite volume time domain method. With our arrangement we find a pronounced optical switching effect in which due to the cross-waveguide geometry the overlay of the probe beam by a pump pulse is significantly reduced.

© Optical Society of America

## 1. Introduction

Due to promising new devices and applications, there has recently been a growing number of theoretical and experimental activities related to photonic crystals. Photonic crystals are periodic arrangements of dielectric scatterers. A substantial property of these structures is the appearance of band-gaps, i.e. a frequency range exists within which wave propagation is exponentially attenuated and waves directed onto the crystal are reflected. For an overview see e.g. [1,2].

In general, the transmission of light through a photonic crystal depends on the
geometry and the index of refraction of the dielectric materials. If one seeks to
dynamically control the optical signal, photonic crystal structures then offer two
possibilities: The first one is to change the geometry of the band-gap structure
e.g. by means of piezo-active materials. A second possibility is the variation of
the index of refraction by means of a nonlinear photonic material and a strong pump
pulse [4,3] (optical switching). Focusing on the latter, we will in the
following analyze the dynamics of the optical switching process in a photonic
band-gap switching device. It consists of a sequence of dielectric layers with
alternating indices of refraction which - via a Kerr nonlinearity -nonlinearly
depend on the light intensity. The transmission of light through the structure is
illustrated in Fig. 1. A probe beam whose frequency is chosen to be outside
the band-gap close to the band edge will then pass the multilayer structure.
However, when we change the dielectric constant
*ϵ*_{h}
, the transmission curve is shifted and the
probe beam no longer passes the structure but instead is nearly completely
reflected.

In our set-up, the change of *ϵ*_{h}
is realized by
means of a second sufficiently strong pump pulse. The principle of this optical
switching has been demonstrated by Scalora and Tran on the basis of a
one-dimensional (1D) model. There, both the probe and the pump pulse are propagating
in the same direction such that the strong pump pulse overlays the probe beam. Here,
we present an alternative set-up with a cross-geometry and take into account the
propagation and dynamic two-dimensional (2D) light field variations. The underlying
configuration is shown in Fig. 2. The photonic switch consists of two crossed
waveguides with a nonlinear photonic band-gap structure in the center. This cross
geometry represents in comparison to the one-dimensional arrangement a more
realistic configuration for an optical switch. In particular, the pump beam
propagates perpendicular to the direction of the probe beam. In our cross-waveguide
optical switch the overlay of the probe beam by the pump beam is significantly
reduced - a fact which may become important in potential technical applications of
all-optical switching devices. Note that the multilayer is structured periodicly
only in the direction of the probe beam and there is no further restriction with
respect to the frequency of the pump pulse.

Presently, most of the numerical methods for the simulation of photonic chrystals are
based either on a treatment of the field equations in the
(*ω*, $\overrightarrow{k}$
) domain or in the
(*ω*,$\overrightarrow{r}$
) domain [2,5]. Another possibility is the use of the slowly varying
envelope approximation [4]. All these methods are based on a fixed frequency. When
nonlinearity and multiple frequencies are incorporated, these methods quickly become
rather complex. To overcome these difficulties, we directly solve
Maxwell’s Equations by numerical integration in the (*t*,
$\overrightarrow{r}$
) domain. The direct integration strategy has
previously been used with nonlinear photonic crystals by Tran in the case of one
spatial dimension and by Reineix in a linear 2D study [6]. To model the photonic switching the method has been
extended to incorporate the 2D nonlinear spatio-temporal variations of the
transmission properties of the photonic chrystal structure. All simulations in this
paper are based on this method. To obtain results in the Fourier domain such as Fig. 1, a Fourier decomposition is performed on the simulated
data in the time domain.

## 2. Basic equations and numerical method

The propagation of electromagnetic waves in periodically structured dielectric media
may be generally described by Maxwell’s Equations. Restricting ourselves
to the TM-polarization^{1}, Maxwell’s equations in the two dimensional (2D)
*x* - *y* plane reduce to^{2}

and the divergence relation

where *c* is the speed of light and *x*,
*y*, *z* label the Cartesian components of the fields.
The interaction between the optical field and the nonlinear dielectric medium is
modeled by a Kerr-type nonlinearity

The spatially varying dielectric constant *ϵ* =
*ϵ*(*x*, *y*), the Kerr
nonlinearity *χ*
_{3} =
*χ*
_{3}(*x*,
*y*), and the nonlinear dielectric function
*ϵ*_{nl}
=
*ϵ*_{nl}
(*x*,
*y*, *E*_{z}
) =
*ϵ*(*x*, *y*)
+ *χ*
_{3}(*x*,
*y*) ∙ ${E}_{z}^{2}$(*x*, *y*, *t*) represent the
nonlinear photonic band-gap structure depicted in Fig. 2. Note that we have assumed possible dispersive effects
to have no critical influence on the switching effects studied here. Further, we
neglect any magnetic response in the material, i.e. *B*
*= H*. Assuming the boundaries of the waveguides to be perfectly
conducting leads to $\overrightarrow{n}$
×
$\overrightarrow{E}$
= 0 and $\overrightarrow{n}$
∙
$\overrightarrow{B}$
= 0, where $\overrightarrow{n}$
is a
vector normal to the boundaries. Then the energy density of the fields reads

Equations (1)–(3) are solved by means of a finite volume solver [7]. The procedure is based on casting the equations in conservative form

which after integration in space and time results in

with an arbitrary domain *G* and $\overrightarrow{V}$
=
1/*c* ∙ (*D*_{z}
,
*B*_{x}
, *B*_{y}
), $\overrightarrow{F}$
_{x} = (-*H*_{y}
,0,-*E*_{z}
) and
$\overrightarrow{F}$
_{y} = (*H*_{x}
, *E*_{z}
,0).

A discretization of the equations is done by decomposing the domain of interest into
a sufficiently large number of subdomains. The field is approximated in any of the
subdomains by means of suitable functions, e.g. polynoms. Using Eq. (7), one can calculate the field values
$\overrightarrow{V}$
_{n+1} at
time *t* =
*t*
_{n+1} =
*t*_{n}
+ Δ*t* from the
existing values $\overrightarrow{V}$
_{n} at time *t* = *t*_{n}
. We note that our
numerical procedure is similar to the frequently used method of time domain finite
differences [8]. It is, however, superior in handling distorted meshs.

## 3. Dynamic optical switching

In the waveguides of the photonic switch depicted in Fig. 2, light can generally propagate in various modes, see
e.g. [9]. In our simulations we take the dominant
*TM*
_{1} modes (or superpositions of these modes) into
account. The explicit functional form of these modes is described in the appendix.
Besides the finite extension in *y*-direction, the multilayer
sequence is periodically structured only along the *x*-direction.
Thus, a photonic band-gap does not exist for all directions of the wave vector
$\overrightarrow{k}$
=
(*k*_{x}
,*k*_{y}
) of the incident
probe beam. A well pronounced band gap is easily obtained, however, by choosing
*k*_{y}
sufficiently small with respect to
*k*_{x}
^{3}.

Before considering the more complicated situation of the dynamic nonlinear switching
process, we first simulate the propagation of the cw probe beam in the waveguides
and through the photonic band-gap structure (without application of the switching
pulse). The frequency of the probe beam is adjusted to *f* =
*f*_{probe}
, marked by the arrow in Fig. 1, and the amplitude *A*_{probe}
is set to 1.0. The resulting energy density distributions are shown in Fig. 3 and Fig. 4 for eh = 2.0 and
*ϵ*_{h}
= 2.1, respectively. As expected, the beam
passes the multilayer structure in the case of
*ϵ*_{h}
= 2.0 and is exponentially attenuated in
the case of *ϵ*_{h}
= 2.1. In addition, the
multilayer structure also serves as an effective waveguide in the center of the
photonic switch where the waveguides are absent: In Fig. 3 and Fig. 4 the portion of the intensity which is being scattered
towards the top and the bottom only amounts to ≈ 0.1%.

While in Fig. 4 we have artificially changed
*ϵ*_{h}
, the change of the dielectric
properties is now provided by a sufficiently strong pump pulse. For specifity, we
assume a Gaussian shaped superposition of the basic *TM*
_{1}
modes (see appendix) resulting in a pulse whose intensity smoothly rises and falls.
The frequency of the pulse is chosen as *f*_{pump}
= 0.483
and its width amounts *σ* = 5.0 ∙
*T*_{pump}
, where *T*_{pump}
=
1/*f*_{pump}
. Its amplitude is set to
*A*_{pump}
= 14.0. In the absence of the probe beam
one obtains the results illustrated as snapshots in Fig. 5 and Fig. 6 which show the spatial distribution of the pump pulse
as it passes the multilayer structure and the distribution of the induced nonlinear
refractive index *ϵ*_{nl}
, respectively. Besides a
small portion which is reflected, the main intensity of the pulse passes the
photonic switch. In Fig. 5 one can clearly see that in the photonic band gap
structure most of its energy is spatially confined to the center, resembling the
initial shape of the incident pulse. As a consequence, the region with a strong
induced change of the (nonlinear) dielectric constant
*ϵ*_{nl}
is thus also located at the center of
the switch. In addition, the distribution of
*ϵ*_{nl}
is wave shaped, according to the wave
length of the pump pulse. In passing we note that this wave length differs between
the waveguide and the area of the photonic band-gap structure. Hence we have a
dynamically induced 2D photonic band-gap structure. Note further that due to the
transverse shape of the pump pulse the number of dielectric layers which will
effectively contribute to a switching process is reduced in comparison to the static
change of the dielectric constant assumed in Fig. 4. In Fig. 5 the loss due to scattering of the pump pulse in the
waveguide amounts to ≈ 0.4% with respect to the incident pump pulse.

In order to study the dynamic optical switching process, we now apply both the cw
probe beam and the pump pulse of Fig.3 and Fig. 5. The dynamics of the resulting optical switching
process is shown in the animation below (Fig. 7). The upper left corner of Fig. 7 shows the spatio-temporal variation of the probe beam.
Note that for an improved visualization of the switching effects a frequency filter
is applied at the frequency of the pump pulse. The effects of the pump pulse are
represented in the upper right corner, which shows the dynamics of the dielectric
constant *ϵ*_{nl}
. The time scale is given in
units of [*λ*
_{0}/*c*]. For optical
wave lengths, this corresponds to 10^{-15} to 10^{-14} seconds.

At time *t* = 0, the intensity of the pump pulse has not yet reached
its maximum, and the change of the *ϵ*_{nl}
is
small. Hence, the probe beam passes the photonic switch. The animation also shows
that the multilayer structure also causes a resonator effect reminiscent of a
distributed feedback arrangement used e.g. in semiconductor lasers. With increasing
time the intensity of the pump pulse rises and
*ϵ*_{nl}
increases. As a consequence, the band
edge is dynamically shifted such that the propagation of the probe beam is
significantly disturbed. At *t* = 10 the maximum of the pump pulse is
reached an the transmitted intensity of the probe beam is continously reduced until
*t* ≈ 20. In the follwing, with the pump pulse having
passed the structure, the probe beam returns to his initial intensity.

## 4. Conclusion

We have presented a numerical simulation of a two dimensional photonic band-gap switch. Our simulations are based on a full vector analysis of Maxwell’s Equations in the time domain using a finite volume time domain integration method which we found to reliably produce results for 2D photonic chrystals. Our nonlinear 2D photonic band-gap switch consists of two crossed waveguides with a nonlinear photonic band-gap structure in the overlap region. With this configuration an effective all optical switching of a cw beam by an optical pulse is demonstrated. Due to the perpendicular directions of propagation of the probe beam and the pump pulse the undesirable overlay of the two signals is strongly reduced in our nonlinear 2D photonic switch.

## Appendix: Beam and pulse shape

Assuming a waveguide extended in x-direction which is bounded by perfectly
conducting walls located at *y* = 0 and *y* =
*a*, the basic guided *TM*
_{1}
modes have the following form:

where *k*_{x}
= ${k}_{x}=\sqrt{\frac{{\omega}^{2}}{{c}^{2}}-{k}_{y}^{2}}$ and *k*_{y}
=
*π*/*a*. The amplitude A and
phase *ϕ*_{x}
may be freely chosen.
Multiplication of these modes with an envelope exp (0.5 ∙
(x* - t -
ϕ)/σ
_{2})
results in a Gaussian shaped superposition. The corresponding modes of the
waveguide in y-direction are obtained by an appropriate
coordinate-rotation.*

*Footnotes*

^{1} | With respect to the z-axis. |

^{2} | In Heaviside-Lorentz units. |

^{3} | In our case we have k_{y}
/k_{x}
≈ 0.1. |

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