A splitstep timedomain model for the analysis of pulse delay characteristics through ring resonator allpass filters (APF’s) is developed and its accuracy is checked. The dependence of the delay time and pulse distortion on the coupling ratios is investigated using the model. It is observed that the bandwidth of the APF's needs to be wide enough to minimize the pulse distortion.
I. INTRODUCTION
As the demand for highspeed communication both in wired and in wireless systems increases, the optical fiber communication systems have been developed in such a way that the information transmission capacity through an optical fiber exceeds terabits per second. As the data speed increases, the packet switching network is expected to advance from electronic packet switching to alloptical packet switching networks in which electricaltooptical and opticaltoelectrical conversions are not involved. In the alloptical packet switching networks, optical packets carried in a specific wavelength channel can be routed to a destination transparently, thus the network can be expanded flexibly when necessary
[1

2]
. Furthermore, optical packet switching system might be adopted in optical interconnection technology for highspeed signal transmission as the VLSI (Very Large System Integration) circuits become extremely integrated
[3]
.
In the alloptical packet switching or optical interconnection system, optical switches and variable optical delay modules are required. There have been a variety of approaches to realize optical delay modules
[4

16]
. A CROW (Coupled resonator optical waveguide)
[5

8]
has drawn interest for alloptical packet communication or optical signal processing modules because it can reduce the propagation speed variably. The CROW can be implemented in the form of FabryPerot, photonic crystal, or ring resonators, etc. Among them, the CROW based on ring resonators have shown noticeable advances in practical realization in terms of design and fabrication feasibility
[7]
.
Even though several approaches in analytical and numerical analysis of the CROW’s provides lots of information regarding the delay performance of the devices
[17

19]
, the quality of delayed pulse shapes can be directly investigated through a timedomain analysis.
In this paper, an efficient splitstep timedomain approach for the numerical analysis of the ring resonator devices such as CROW devices is presented
[20

21]
. The numerical algorithm is illustrated through the analysis of single, double, triple, and quadruplering resonator allpass filters, which can be extended to general CROWtype devices with arbitrary number of rings. In section II, the splitstep timedomain algorithm is elaborated. In section III, the accuracy of the numerical approach is checked, and the delay characteristics of single, double, triple and quadruplering allpass filter (CROW device) are investigated and some design guidelines are presented. Finally, conclusions are given.
II. FORMULATION OF TIMEDOMAIN MODEL FOR RING RESONATOR ALL PASS FILTER
A doublering resonator allpass filter (APF) shown in
Fig. 1
is considered to explain the timedomain model employed in this paper. For the timedomain analysis of the ring resonator, a splitstep timedomain model (SSTDM) is developed
[20

21]
. In this model, the numerical solution of the timedependent coupledwave equation is found through a splitstep operator approach. The SSTDM consists of the numerical solution of the timedependent coupledwave equations for the forward and reverse waves.
Double ring resonator all pass filter. This structure is considered to illustrate the splitstep timedomain model.
The optical field within waveguides can be written as
where
ω_{0}
is the reference frequency,
β_{0}
is the propagation constant at the reference frequency, and
φ_{i}(x,y)
is the modal function in the bus and ring waveguides.
F_{i}(z,t)
and
R_{i}(z,t)
represent slowly varying amplitudes of the forward and reverse waves propagating along a longitudinal direction in the bus waveguide (
i
=1) and two ring waveguides (
i
=2, 3), respectively. The timedependent coupled wave equations for the simulation of the ring resonator devices can be expressed as follows:
where
c_{g}
is the group velocity, which is assumed to be constant over the frequencies of interest,
G_{i}
and
δ_{i}
are net modal gain/loss and detuning factor in each waveguide,
κ_{ij}
is the coupling coefficient of the directional coupler between waveguide
i
and
j
.
In this paper, passive waveguides are considered and, thus
where
α_{i}
is the waveguide loss.
For the analysis, three sets of timedependent coupledwave equations are used because three waveguides exist in the structure.
Fig. 2
shows the conceptual configuration considered for the simulation. The couplings between waveguides occur at particular sections in three waveguides. The ring (circular) waveguides are assumed to be bus waveguides which require the following periodic boundary conditions :
Conceptual configuration of the spiltstep timedomain model for the double ring APF.
The boundary condition for the bus waveguide is given as
The timedependent coupledwave equations can be numerically solved by dividing the waveguides into a number of small subsections having equal length (Δz) as shown in
Fig. 2.
In the simulation, the calculation in each section (Δz) consists of two operations which are performed for Δt (=Δz/c
_{g}
). First, the forward/reverse waves in three waveguides (a bus and two ring waveguides) are updated to incorporate the attenuation and phase accumulation through a section (Δz) in a time step (Δt). Then, the coupling effects between waveguides are taken into consideration. The splitstep algorithm is computerized through the following procedure:

(1) The waveguides are divided into a number of small subsections having equal length (Δz ).

(2) The photon density in each section of three waveguides is updated using the following timestepping equation:

(3) The coupling effects between waveguides are taken into consideration. For example, the coupling effect between the bus and ring 1 can be incorporated using

Here, the subscripts 1 and 2 represent the field amplitudes in bus and ring 1 waveguides, respectively. In the above equations,zbindicates the position of coupling with ring 1 in the bus waveguide. Similarly,zrlindicates the position of coupling with bus in ring 1. The distributed coupling of the directional coupler is assumed to occur atzbandzrlin a lumped way. The parameterK12is the total coupling ratio through the directional coupler between two waveguides 1 and 2, andTis the total straightthrough ratio given by {}. The coupling between two rings is also incorporated similarly.

(4) The appropriate boundary conditions (4) in the bus and ring waveguides are applied at every time step.
III. NUMERICAL RESULTS AND DISCUSSIONS
The numerical model described in Section II can be checked by comparing the phase and delay responses as a function of wavelength calculated from the numerical model with the analytical and numerical spectral analysis
[17

19]
for a singlering resonator APF. The waveguide is assumed to be lossless throughout the discussions in this section. The perimeter of the ring is set to be 2000
m and the group refractive index of the material 1.5. For this ring, the FSR (Free Spectral Range) is calculated to be 100GHz and the roundthering trip delay along the ring is 10ps. The coupling ratio between the bus and the ring is set to be 0.5 in this case. For the calculation, the continuous wave with varying wavelengths are launched into the singlering APF and the output phase is measured for each wavelength. Then the delay time is calculated using a formula
where Ø is a phase through the APF and
λ
_{1}
is a center wavelength. The results are shown in
Fig. 3.
The calculated FSR is 0.8 nm which is the same as the theoretical one, and the delay time at the resonant wavelength is about 138ps. The phase and delay response calculated from the analytical formula
[17

19]
is also shown in
Fig. 4
which exhibits the same result as shown in
Fig. 3
obtained from the numerical model.
Phase and delay through the singlering APF as a function of wavelength. The results are calculated from the splitstep timedomain numerical model. The busring coupling ratio is 0.5.
Phase and delay through the singlering APF as a function of wavelength. The results are calculated from an analytical formula. The busring coupling ratio is 0.5.
The delayed waveform of the 167pswide (FWHM: Full Width at Half Maximum) input optical pulse through the singlering resonator APF is shown in
Fig. 5.
The input pulse has no chirp. The delay time increases as the coupling ratio decreases. When the coupling ratio is one, the delay time is about 10ps which is the same as the roundthering trip delay along the single ring resonator. When the coupling ratio is 0.5, the delay time is about 110ps, which is smaller than the peak delay time of 138ps shown in
Fig. 3.
The spectral width of the unchirped 167ps (FWHM) optical pulse is about 2.63GHz (0.021 nm)
[22]
. Considering the spectral width of the input optical pulse, the delay time of 110ps seems to be appropriate for the delay time spectrum as shown in
Fig. 3.
When the coupling ratio is 0.4, the delay time is about 145ps while the spurious pulse amplitude is about 17dB of the main pulse. In order to suppress the spurious pulse amplitude below 30dB, the coupling ratio should be larger than 0.5.
Fig. 5
also shows that the spurious pulse amplitude is almost the same as the main pulse when the coupling ratio is 0.2. This can be understood from the phase and delay time calculation as a function of wavelength as shown in
Fig. 6.
The FWHM bandwidth of the delay response is only 0.006 nm (0.75 GHz) which is too narrow for the 167ps pulse to pass through without distortion.
Pulse delay characteristics of a single ring allpass filter for various coupling ratios. The FWHM of the input optical pulse is 167ps (t_{1}=100ps).
Phase and delay through the singlering APF as a function of wavelength. The results are calculated from the splitstep timedomain numerical model. The busring coupling ratio is 0.2.
The delayed waveform for the input optical pulse of 250ps (FWHM) is shown in
Fig. 7.
The delay characteristics are similar to the case of 200ps input pulse, except that the spurious pulse amplitudes are smaller. In
Fig. 8
, the delayed waveform through a double ring resonator APF is shown for various values of the ringring coupling ratios with the busring coupling ratio fixed to be 0.5. When the ringring coupling ratio is 0.1, the maximum delay of 280ps is observed with the spurious pulse amplitude to be 20 dB of the main pulse amplitude. In
Fig. 9
, the delayed waveform through a triple ring resonator APF is shown for various values of the second ringring coupling ratios with the busring coupling ratio and the first ringring coupling ratio fixed to be 0.5 and 0.12, respectively. When the second ringring coupling ratio is 0.08, the maximum delay of 408ps is observed with the spurious pulse amplitude smaller than 20dB of the main pulse amplitude.
Pulse delay characteristics of single ring allpass filter for various coupling ratios. The FWHM of the input optical pulse is 250ps.
Pulse delay characteristics of double ring allpass filter for various coupling ratio pairs (ĸ_{12}, ĸ_{23} ). The FWHM of the input optical pulse is 167ps.
Pulse delay characteristics of the triple ring allpass filter for various coupling ratio combinations (ĸ_{12},ĸ_{23},ĸ_{34} ). The FWHM of the input optical pulse is 167ps.
The delay characteristics for the chirped pulse also need to be investigated. The electric field of linearly chirped optical pulse is expressed as follows
[22]
:
In the above, α is the linewidth enhancement factor and
t
_{1}
is the pulse width. The FWHM pulse width Δt is given by Δt = 1.67
t
_{1 }
[22]
. The pulse delay characteristics for various linewidth enhancement factor values are shown in
Fig. 10
for the case of the coupling ratio being 0.5. As the linewidth enhancement factor increases the pulse shape becomes distorted. When the linewidth enhancement factor is 1, the leading edge portion (zero to 10 % of the pulse peak) is not delayed properly. When the linewidth enhancement factor is 0.5 the pulse is properly delayed except the leading edge corresponding to the time span from zero to 1% of the peak. In order to understand the origin of the pulse distortion, the coupling ratio of 0.6 is also considered and the pulse delay characteristics are shown in
Fig. 11.
In this case, the pulse distortion is relaxed with the sacrifice of the delay time being reduced to 80ps showing that the pulse is properly delayed even when the linewidth enhancement factor is 1. The linewidth enhancement factor is usually smaller than 1 when external modulators are employed for the transmitter
[23]
. The transmitters composed of directmodulated DFBLD’s are not compatible with the ring resonator type delay devices. The delay time as a function of wavelength is calculated for both the coupling ratios of 0.5 and 0.6 and the results are shown in
Fig. 12.
The FWHM bandwidth of delay is about 0.04 nm and 0.06 nm when the coupling ratio is 0.5 and 0.6, respectively. The frequency variation over a time span Δt is given by
Pulse delay characteristics of the singlering allpass filter for various linewidth enhancement factors. The FWHM of the input optical pulse is 167ps. The busring coupling ratio is 0.5.
Pulse delay characteristics of single ring allpass filter for various linewidth enhancement factors. The FWHM of the input optical pulse is 167ps. The busring coupling ratio is 0.6.
Delay as a function of wavelength through the singlering resonator APF. The cases for the busring coupling ratio of 0.5 and 0.6 are compared.
The pulse width corresponding to 1% of the peak is about 500ps for the 167ps (FWHM) optical pulse. For this time span, the frequency variation is 8GHz (0.064 nm) in case of α = 1. This fact illustrates that the bandwidth of the APF with the coupling ratio of 0.5 is not wide enough to delay the leading and trailing edges of the pulse below 1% of the pulse peak. When the coupling ratio is increased from 0.5 to 0.6, the bandwidth is increased from 0.04 nm to 0.06 nm. As a result of the bandwidth increment, the pulse is delayed without severe distortion for the APF with the coupling ratio of 0.6. In both cases, the bandwidth delay product (Δν · Δτ
_{peak}
) is almost the same, which is about 0.72. But, the delay distortion characteristics of the chirped pulse are somewhat different from each other as described above.
An APF composed of four coupled rings are also considered to investigate the effect of the bandwidth on the distortion of the input pulse. The coupling ratios are apodized to increase the bandwidth. The busring coupling ratio is 0.8, the ringring coupling ratios are 0.34, 0.19, and 0.17. The delay characteristics are shown in
Fig. 13.
It is observed that the pulse delay of 175ps is achieved without severe distortion even for α = 2. This result comes from the widened bandwidth as shown in
Fig. 14.
The bandwidth delay product (Δν · Δτ
_{peak}
) is about 3.8 in this case. The bandwidth is about 0.1 nm which is wide enough for the pulse with spectral width of 0.064 nm (8 GHz) to pass through the quadruplering APF.
Pulse delay characteristics of quadruple ring allpass filter for various linewidth enhancement factors. The FWHM of the input optical pulse is 167ps. The busring and ringring coupling ratios are 0.8, 0.34, 0.19, and 0.17.
Delay and phase as a function of wavelength through the quadruplering APF. The results are calculated from the timedomain numerical model. The busring and ringring coupling ratios are 0.8, 0.34, 0.19, and 0.17.
IV. CONCLUSION
A splitstep timedomain modeling approach is developed to analyze the ring resonator delay devices. In this approach, the optical field evolution in terms of phase and loss is incorporated in the small section and then the directional coupling effect is taken into consideration at each time step. For the bus and the rings, appropriate boundary conditions are applied at each time step. The accuracy of the splitstep timedomain analysis is checked by comparing with the spectral analysis using an analytical solution. The timedomain model is applied to analyze the optical pulse transfer characteristics through single, double, and triplering APF’s. In the singlering APF, the pulse delay increases as the coupling ratio decreases. On the other hand, the pulse shape distortion becomes worse as the coupling ratio decreases, which comes from the reduction of bandwidth for lower coupling ratio. As the number of rings are increased with optimum coupling ratios, the delay time scales with the number. For the singlering, doublering, and triplering APF with 100GHz free spectral range, the delay is 140ps, 280ps, and 408ps, respectively. The pulse transfer characteristics through the singlering and apodized quadruplering APF for the chirped pulse are also studied. It is observed that the bandwidth of the APF should be wide enough to minimize the pulse shape distortion. With these examples it is shown that the splitstep timedomain model is useful to efficiently analyze the pulse distortion through the ring resonator delay devices in detail and can be incorporated into an advanced optical communication modeling software in the future.
Acknowledgements
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (NRF2013R1A1A2007276). This work was also partly supported by the Research Grant of Kwangwoon University in 2012.
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