Engineered Transitions in Photonic Cavities

doi:10.4236/opj.2012.24030

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Optics and Photonics Journal, 2012, 2, 255-259

http://dx.doi.org/10.4236/opj.2012.24030 Published Online December 2012 (http://www.SciRP.org/journal/opj)

Engineered Transitions in Photonic Cavities

Ali W. Elshaari1, Stefan F. Preble2

1Electrical and Electronic Engineering Department, University of Benghazi, Benghazi, Libya

2Microsystems Engineering Department, Rochester Institute of Technology, Rochester, USA

Email: [email protected]

Received August 25, 2012; revised September 23, 2012; accepted October 15, 2012

ABSTRACT

We demonstrate for the first time, to best of our knowledge, that by engineering the states of a system of cavities it is

possible to control photon transitions using non-adiabatic refractive index tuning. This is used to realize a novel photon

transitions that are independent of the refractive index sign. In particular, we show through coupled mode theory and

FDTD simulations that red shifts are possible in silicon resonators using the free-carrier plasma dispersion refractive

index reduction.

Keywords: Integrated Optics; Resonators; Wavelength Shift

1. Introduction

The engineering and active control of resonant photonic

structures has enabled unique optical functionalities,

from isolators and delay elements [1-3] to 100% efficient

adiabatic wavelength conversion by trapping light while

tuning the state of a resonator [1-7]. However, these

functionalities have been limited by the mechanism used

to realize the refractive index change. This is particularly

the case for adiabatic wavelength conversion on the Sili-

con photonics platform where the free-carrier plasma

dispersion effect (PDE) is used to reduce the refractive

index of a cavity [8-10]. This always results in a wave-

length shift towards the blue, limiting the application of

the effect [7,11]. In contrast, it was shown in [4] that

when a resonator is non-adiabatically perturbed it is pos-

sible to transition photons to other resonant modes—even

towards the red, albeit with a low efficiency. The reason

for this is that the final state of the system couples to a

continuum of output modes—with a dominant excitation

of the adiabatic shift as seen in Figure 1 [7]. Here we

show that by carefully designing the states of a system

of cavities nearly all of the light can be non-adiabati-

cally transitioned to just one state—even towards the red.

This opens the possibility of using the PDE for both blue

and red shifts of light, which will enable rapidly recon-

figurable wavelength converters for use in future on-

chip wavelength-division-multiplexed systems. This is in

contrast to non-linear wavelength converters relying on

Raman scattering (RS) or four wave mixing (FWM)

where high powers are required and wavelength changes

are fixed by the wavelengths used in the system and can

only be reconfigured by slowly tuning an external laser

[12-14].

2. Designing System States

The proposed cavity system is shown in Figure 2. It

consists of one input and output cavity each with large

Free Spectral Range (FSR) and a transition cavity having

closely spaced states (small FSR). By initially aligning

one resonance of the input cavity and one resonance of

the transition cavity it is ensured that the system only has

one allowed state, which we call the input state, as shown

as a double solid line in the figure (the double lines indi-

cate that there is mode splitting). We note that although

there are many internal degrees of freedom for the transi-

tion cavity, they are not allowed because they will all be

off-resonance with respect to the input cavity. This is

ensured by the phase matching and the ortho-normality

of modes. So that only specific transition probabilities

don’t vanish. The probability of a transition between

modes a and b is dictated by the overlap between the

super-modes over the volume of the two rings in both

transverse and longitudinal directions (γ is the coupling

constant, φa,b are the initial and final modes).

volumeaba ab b

 



Furthermore, light cannot couple to the output of the

system because the output cavities resonance is detuned

with respect to the input state (lower energy in Figure 2

[-Eg/2]). As will be shown, this red-shifted output cavity

determines the final state of the system, and consequently

the new wavelength of the light.

To transition the light efficiently to the red-shifted

ouput cavity we induce a non-adiabatic perturbation of t

A. W. ELSHAARI, S. F. PREBLE

256

Figure 1. Light initially excites one state of the resonator (t < t0). When the resonator is switched at a fast rate, multiple

output states are excited (t > t0). Δω is the relative shift of all of the states due to the refractive index change.

Figure 2. Photonic transitions in system consisting of threecoupled-cavities. Initially (t < t0) the system only has one allowed

input state—formed by the alignment of the input and transition cavity resonance (m). The light cannot escape to the output

because the output cavity is in a forbidden state (-Eg/2). By non-adiabatically reducing the refractive index of the transition

cavity (t > t0) the states are shifted by Dw. Now the output cavity is in resonance with the (m–1) mode of the transition cavity.

This is the only allowed state of the system. The remaining transitions lie inside forbidden states in the energy diagram.

the transition resonator through a refractive index re-

ducetion. This will blue-shift all of the states of the tran-

sition resonator, including the initially excited input state.

However, now the output cavity will be on-resonance

with the (m–1) mode of the transition cavity—forming a

newly, and singly, allowed state for the system. Since

this state is at a lower energy than the original input state

the wavelength of the light is efficiently red-shifted.

In order to demonstrate this concept we use time do-

main coupled mode theory to simulate the different mode

dynamics [15-17]. The simulations takes into account

waveguide and carrier losses in the system [9] and has

A. W. ELSHAARI, S. F. PREBLE 257

been successfully applied to analysis of a multitude of

dynamically controlled cavity systems [2,11,17-19]. In

the model we consider realistic parameters—a typical

value of waveguide loss of 3 dB/cm is assumed, and ring

resonators are used as the cavities, with radii of Rinput/output

= 10 μm and Rtransition = 200 μm (FSRs of 8.8 nm and

0.437 nm, respectively). The resonator states are seen in

Figure 3 where we see that the input ring and the transi-

tion ring have the same resonance. The output ring is

placed at the midpoint of the gap between the (m) and

(m–1) mode of the transition resonator. It should be noted

that for clarity the individual ring resonances were cal-

culated separately in order to visualize the position of

different resonances with respect to each other, while in

the actual system they will be coupled—resulting in

mode splitting.

3. Red Transitions with Refractive Index

Reduction

The controlled transition of a pulse of light towards

lower energies (red) is demonstrated in Figure 4 by in-

jecting a carrier density of



4E16 cm%1%nn in

the transition resonator. The carriers are injected in a

time of 100 fs, which is significantly shorter than the

inter-mode coupling time



t1FSR



in order to en-

sure non-adiabatic transitions in the transition ring [4].

We note here that this fast switching speed in not a limi-

tation of the system and can be relaxed by using resona-

tors with a smaller FSR or by using resonant modulation

as discussed below in Section 3. In Figure 4 we see that

most of the light is red shifted to the new (m–1) mode of

the transition cavity and there is negligible excitation of

the other states. An important point to be emphasized

Figure 3. Transmission of different rings in the initial state

of the system. The input resonator and the transition

resonator have the same resonance condition, while the

output resonator is purposely shifted to a lower energy—

one-half FSR away.

Figure 4. States before the switching (blue) and after

switching (red) inside the transition ring. The conversion

efficiency is 96%.

here is the fact that this is a true mode coupling in the

transition ring, not a filtering effect—the spectrum shown

in Figure 4 is measured inside of the transition ring not

at the output port. Lastly, we should note that the posi-

tion of the converted light is not exactly at the half-FSR

point but is slightly blue-shifted by 0.06 nm because of

mode splitting of the coupled transition-output cavity

state. Here we have tuned the resonators coupling con-

stants in order to maximize coupling to only the blue-

shifted split mode. In order to verify the results from

coupled mode theory we numerically simulated the dy-

namic process by solving Maxwell’s equations using the

finite difference time-domain (FDTD) method [20]. The

system has the same configuration as the one described

earlier but with scaling differences in order to speed up

the computation process (Radii of Rtransition = 24 μm and

Rinput/output = 4 μm. These correspond to FSRs of ~4 nm

and ~26 nm, respectively). We see in Figure 5 that with

an index shift of





%9%nn the light is red-shifted

by 2 nm. The double peak in the figure results from the

interference of modes in the ring resonator which was

evident in our previous experimental work [15].

The behavior is qualitatively the same as the results

obtained with coupled mode theory. However, the con-

version efficiency is slightly lower (88%) since the non-

adiabatic transition is not as efficient with the large FSR

used in the FDTD simulation. This is not a fundamental

limitation provided the FSR of the transition ring is small

enough. It all depends on the strength of the wavelength

shift [7]. Provided a large enough refractive index, small

rings with large FSR can be used.

4. Efficiency and State Design

We have shown here that the placement of the cavity

A. W. ELSHAARI, S. F. PREBLE

258

Figure 5. Finite-difference-time-domain verification of red-

shifting results.

states will determine the new modes of the system. In

addition, the maximum conversion efficiency is deter-

mined by the state design. For example, we found that

this is achieved when the output cavities state is a half-

FSR away from the initial transition cavities state (i.e.

halfway between m and m–1). This can be understood

as follows. When this state is closer to the input state

there is unintended adiabatic coupling to original (m)

state. In addition, it would require a larger index change

to achieve red-shifting since the initial (m–1) mode would

be even farther away from the output state. On the other

hand, placing the out- put state closer to the initial (m–1)

state will also reduce the efficiency because the input and

output state are initially further apart, but the refractive

index change would be smaller—resulting in a weaker

non-adiabatic transition. This could be overcome, how-

ever, by using resonant transitions where the resonator is

switched at a rate corresponding to the difference in the

state [2] spacing. We should add that without resonant

transitions it is important to switch the transition cavity

on the order of less than ~30% of 1/FSR (where FSR has

units of frequency) in order to maximize the non-adia-

batic transition process, as simulated in Figure 6.

5. Concluding Remarks

In conclusion, we demonstrated that by engineering the

states of a system of cavities, and using non-adiabatic

transitions, it is possible to obtain wavelength changes

that are independent of the refractive index change sign.

This new phenomena will enable more robust recon-

figurable wavelength conversion systems where refrac-

tive index reductions can be used to both blue and red

shift the frequency of light. In addition, the scheme pro-

posed here for designing the states of a system of cavities

could lead to novel dynamically controlled cavity sys-

Figure 6. The efficiency decreases as switching speed is

slowed due to the enhancement of the adiabatic shift in the

resonator.

tems for optical signal processing.

6. Acknowledgements

We would like to thank Dr. Edwin Hach for helpful dis-

cussions. The authors would also like to thank Dr. Gernot

Pomrenke, of the Air Force Office of Scientific Research

for his support and we thank Joseph Lobozzo II for the

Lobozzo Optics Laboratory.

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