1. REU program sponsored by the
National Science Foundation
Award Number: PHY1156454
Frequencies Can Be Coupled
Oscillators
C. B. Fritz; Dickinson College
R. Roy, T. E. Murphy; University of Maryland
Introduction:
• Optoelectronic feedback loops are the subject of recent research into networks and
synchronization. [1] [2] [3]. Capable of displaying high-dimensional chaos, these nonlinear
oscillators offer rich dynamics for a high-bandwidth frequency range. By analyzing the
nonlinearity, it is hypothesized that the discrete frequencies of this nonlinear oscillator
behave as globally coupled oscillators. This possibility is explored experimentally.
The Experiment:
• In order to test the validity of the theory, an optoelectronic oscillator was limited to three allowable frequencies
using a frequency comb filter:
• The optoelectronic oscillator features a Laser diode fed into a Mach-Zender modulator. The attenuated light signal
is then sent into a photodiode, converting it from optical to electrical. This electrical signal is then filtered and
delayed on a Field Programmable Gate Array (FPGA) . The filtered result is then amplified and fed back into the
Mach-Zender. Figures 2 & 3 show the experimental setup.
The Theory:
• The dynamical behavior of the feedback loop is driven by the nonlinearity of a Mach-Zender modulator:
𝑃𝑃 𝑜𝑜𝑜𝑜𝑜𝑜 = 𝑃𝑃𝑖𝑖 𝑖𝑖 cos 𝑥𝑥 + 𝜑𝜑𝑜𝑜
2
[optical output] = [optical input] * [variable attenuation].
• Using trigonometric identities and a series expansion with 𝜑𝜑 𝑜𝑜 =
𝜋𝜋
4
, the most significant terms become:
𝑃𝑃 𝑜𝑜𝑜𝑜𝑜𝑜 =
𝑃𝑃𝑖𝑖 𝑖𝑖
2
1 − 2𝑥𝑥 +
4
3
𝑥𝑥3 .
• Using a Fourier transform for x:
𝑥𝑥 𝑡𝑡 = ∑𝑗𝑗 𝐴𝐴𝑗𝑗 𝑒𝑒𝑖𝑖𝜔𝜔𝑗𝑗 𝑡𝑡
+ 𝐴𝐴𝑗𝑗
∗
𝑒𝑒𝑖𝑖𝜔𝜔𝑗𝑗
∗
𝑡𝑡
,
𝑃𝑃 𝑜𝑜𝑜𝑜𝑜𝑜 =
𝑃𝑃𝑖𝑖 𝑖𝑖
2
[1 – 2 ∑𝑗𝑗 𝐴𝐴𝑗𝑗 𝑒𝑒𝑖𝑖𝜔𝜔𝑗𝑗 𝑡𝑡
+ 𝐴𝐴𝑗𝑗
∗
𝑒𝑒𝑖𝑖𝜔𝜔𝑗𝑗
∗
𝑡𝑡
+
4
3
∑𝑗𝑗 𝐴𝐴𝑗𝑗 𝑒𝑒𝑖𝑖𝜔𝜔𝑗𝑗 𝑡𝑡
+ 𝐴𝐴𝑗𝑗
∗
𝑒𝑒𝑖𝑖𝜔𝜔𝑗𝑗
∗
𝑡𝑡
3
.
• Each term in the frequency summation of x is coupled to one another by the cubic nonlinearity:
𝑃𝑃 𝑜𝑜𝑜𝑜𝑜𝑜 =
𝑃𝑃𝑖𝑖 𝑖𝑖
2
[
4
3
(𝐴𝐴𝑗𝑗 𝐴𝐴𝑗𝑗+1 𝐴𝐴𝑗𝑗+2)𝑒𝑒𝑖𝑖 𝜔𝜔𝑗𝑗+ 𝜔𝜔𝑗𝑗+1+𝜔𝜔𝑗𝑗+2 𝑡𝑡
+ … ].
• Altering one particular frequency will effect each other frequency, thus the frequencies are globally coupled.
Nonlinearity
Figure 3: A photograph of the physical experiment
setup. There is a polarization controller added to
prevent attenuation of the optical signal due to
polarity differences with the Mach-Zender modulator.
Figure 1: The frequency comb filter allows 12.5, 37.5 and 62.5 MHz frequencies to
pass, while attenuating each other frequency. The red arrows indicate the
coupling effect implied by the system’s nonlinearity.
Figure 2: A block diagram of the experimental setup.
Both the time delay and frequency comb filter were
implemented using an Altera Cyclone III DSP
Development Kit.
The Data:
• Behavior of frequencies qualitatively changes when system feedback is turned on. See figs. 4 &
5.
• Data was collected using an oscilloscope with a 4 GHz sample rate.
• The primary sources of error are quantization error in the filter implementation and
frequency-dependent gain from the amplifier.
• Signal coupling primarily dependent upon time delay , T, and gain G.
Figure 4: A time series showing the magnitude of each frequency allowed by the filter. The laser diode is
turned off, meaning that this is filtered noise from the photodiode; no coupling takes place. The series was
calculated using a sequence of Fast Fourier Transforms (FFT’s) with overlapping windows.
Figure 5: A time series with the laser diode turned on, enabling the frequency coupling. The series
displays more coherence than the uncoupled system. The 37.5 & 62.5 MHz frequencies appear to be
approaching synchrony.
Conclusions & Further Research:
• Frequencies appear to display coupled behavior.
• It is not clear if synchrony will occur or under what conditions.
• Further research questions include:
1. Do similar frequencies tend to have stronger coupling?
2. Will this coupling effect remain or even grow when more frequencies are
allowed?
• The system could potentially resemble a Kuramoto-like scheme of globally coupled oscillators.
Acknowledgements:
• The author acknowledges the wise and insightful guidance of Drs. Roy and Murphy, as well
as the invaluable collaboration with:
1. Joseph Hart
2. Ryan J. Seuss
3. Dr. Aaron Hagerstrom
References:
[1] T. E. Murphy, A. B. Cohen, B. Ravoori, K. R. B. Schmitt, A. V. Setty, F. Sorrentino, C. R. S. Williams, E. Ott
and R. Roy, "Complex dynamics and synchronization of delayed-feedback nonlinear oscillators", Phil. Trans.
R. Soc. A 368(1911) 343-366 (2010)
[2] C. R. S. Williams, T. E. Murphy, R. Roy, F. Sorrentino, T. Dahms and E. Schoell, "Experimental
Observations of Group Synchrony in a System of Chaotic Optoelectronic Oscillators", Phys. Rev. Lett. 110(6)
064104 (2013)
[2] C. R. S. Williams, F. Sorrentino, T. E. Murphy and R. Roy, "Synchronization states and multistability in a
ring of periodic oscillators: Experimentally variable coupling delays", Chaos 23(4) 043117 (2013)