Determination of Phase Noise Spectra in Optoelectronic Microwave Oscillators: A Langevin Approach

Abstract

We introduce a stochastic model for the determination of phase noise in optoelectronic oscillators. After a short overview of the main results for the phase diffusion approach in autonomous oscillators, an extension is proposed for the case of optoelectronic oscillators where the microwave is a limit-cycle originated from a bifurcation induced by nonlinearity and time-delay. This Langevin approach based on stochastic calculus is also successfully confronted with experimental measurements.

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Determination of Phase Noise Spectra in
Optoelectronic Microwave Oscillators:
a Langevin Approach
Y. Kouomou Chembo∗†, K. Volyanskiy‡,
L. Larger§, E. Rubiola¶and P. Colet?
FEMTO-ST Institute
CNRS and Universit´ e de Franche Comt´ e, Besan¸ con, France
May 21, 2008
Abstract
We introduce a stochastic model for the determination of phase noise
in optoelectronic oscillators. After a short overview of the main results
for the phase diffusion approach in autonomous oscillators, an extension
is proposed for the case of optoelectronic oscillators where the microwave
is a limit-cycle originated from a bifurcation induced by nonlinearity and
time-delay. This Langevin approach based on stochastic calculus is also
successfully confronted with experimental measurements.
Keywords:
tor lasers, stochastic analysis.
Optoelectronic oscillators, phase noise, microwaves, semiconduc-
∗FEMTO-ST, Dept. of Optics. Corresponding author, e-mail: yanne.chembo@femto-st.fr
and ckyanne@yahoo.fr.
†Y.K.C. acknowledges a research fellowship from the R´ egion de Franche-Comt´ e in France,
and a financial support from MEC (Spain) and FEDER under projects TEC2006-10009
(PhoDeCC) and FIS2007-60327 (FISICOS).
‡FEMTO-ST, Dept. of Optics, and St. Petersburg State University of Aerospace Instru-
mentation.
§FEMTO-ST, Dept. of Optics.
¶FEMTO-ST, Dept. of Time and Frequency. Home page http://rubiola.org.
?Instituto de F´ ısica Interdisciplinar y Sistemas Complejos IFISC (CSIC-UIB), Palma de
Mallorca, Spain.
1
arXiv:0805.3317v1 [physics.optics] 21 May 2008

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Y. K. Chembo & al.May 21, 2008
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Contents
1Introduction3
2The phase diffusion approach in autonomous oscillators3
3Application of the phase diffusion approach to OEOs: stochastic
delay-differential equations5
4Noise power density spectrum below threshold (γ < 1)8
5 Phase noise spectrum above threshold (γ > 1)10
6Conclusion13
A Determination of the output noise power for γ = 0 14
B Derivation of the stochastic phase equation 15
C An alternative paradigm for phase noise analysis16
References 18

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Y. K. Chembo & al.May 21, 2008
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1Introduction
Optoelectronic oscillators (OEOs) combine a nonlinear modulation of laser light
with optical storage to generate ultra-pure microwaves for lightwave telecom-
munication and radar applications [1, 2]. Their principal specificity is their
extremely low phase noise, which can be as low as −160 dBrad2/Hz at 10 kHz
from a 10 GHz carrier. Despite some interesting preliminary investigations, the
theoretical determination of phase noise in OEOs is still a partially unsolved
problem. The qualitative features of this phase noise spectrum can be recov-
ered using some heuristical guidelines or rough approximations, but however, a
rigorous theoretical background is still lacking.
There are several reasons which can explain that absence of theoretical back-
ground. A first reason is that before refs. [3], there was no time domain model
to describe such systems, so that stochastic analysis could not be used to per-
form the phase noise study. Moreover, unlike most of oscillators, the OEO is a
delay-line oscillator, and very few had been done to study the effect phase noise
on time-delay induced limit-cycles. Finally, the OEO is subjected to multiple
noise sources, which are sometimes non-white, like the flicker (also referred to
as “1/f”) noise which is predominant around the microwave carrier.
The objective of this work is to propose a theoretical study where all these
features are taken into account. The plan of the article is the following. In
Section 2, we present the phase diffusion approach in autonomous systems. It
is a brief review where the fundamental concepts of phase diffusion are recalled,
and where some important earlier contributions are highlighted. Then, we derive
in Section 3 a stochastic delay-differential equation for the phase noise study.
We show that for our purpose, the global interaction of noise with the system can
be decomposed into two contributions, namely an additive and a multiplicative
noise contribution. Section 4 is devoted to the study of the noise spectrum below
threshold. It will appear that the spectrum below threshold will not only be
important to validate the stochastic model, but also that it enables an accurate
calibration of additive noise. In Section 5, we address the problem of phase
noise when there is a microwave output using Fourier analysis, and we show
that it is possible to have an accurate image of the phase noise spectrum in all
frequency ranges. The last section concludes the article.
2 The phase diffusion approach in autonomous
oscillators
2.1Fundamental concepts
For an ideal (noise-free) oscillator, the Fourier spectrum is a collection of Dirac
peaks, standing for the fundamental frequency and its harmonics. The effect of
amplitude white noise is to add a flat background, while the peaks do keep their
zero linewidth; it is the effect of phase noise to widen the linewidth of these
peaks.
Some pioneering papers on the topic of phase noise in autonomous oscillators
using stochastic calculus had been published forty years ago [4]. In particular, it
was demonstrated that a general framework to study the problem of phase noise
in a self-sustained oscillator could be built using some minimalist assumptions.

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Y. K. Chembo & al.May 21, 2008
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Figure 1: Experimental set-up.
The first point is that a strong nonlinearity is an essential necessity in oscillators,
in the sense that nonlinearity can not be regarded as small because it controls
the operating level of the oscillator. The second important point is that the
phase is only neutrally stable, so that quasilinear methods which assume that
fluctuations from some operating point are small (linearization techniques) can
not be applied directly.
The phase is neutrally stable as a consequence of the phase-invariance of
autonomous oscillators. In other words, limit-cycles are stable against amplitude
perturbations, while there is no mechanism able to stabilize the phase to a
given value: hence, phase perturbations are undamped, but they do not diverge
exponentially, though. In a noise free oscillator, the “stroboscopic” state point
on the limit-cycle is immobile, but in the presence of noise, it moves randomly
along the limit-cycle: in other words, the phase of the oscillator undergoes a
diffusion process, in all points similar to a one-dimensional Brownian motion.
In the most simple case, the random fluctuations of the phase are referred to
as a Wiener process, obeying an equation of the kind ˙ ϕ = ξ(t), where ξ is a
Gaussian white noise with autocorrelation ?ξ(t)ξ(t?)? = 2Dδ(t − t?), while D is
a parameter referred to as the diffusion constant. It can be demonstrated that
the phase variance diverges linearly as?ϕ2(t)?= 2Dt, and the single-side band
that D is the unique parameter characterizing all the statistical and spectral
features of phase fluctuations.
phase noise spectrum (in dBc/Hz) explicitly reads L(ω) = 2D/[D2+ ω2], so
2.2The unifying theory of Demir, Mehrota and Roychowd-
hury
On the base of earlier works by Lax [4] and K¨ artner [5], Demir, Mehrota and
Roychowdhury have proposed few years ago a unifying theory of phase noise in
self-sustained oscillators subjected to white noise sources [6]. Their approach,
which had later been extended by Demir to the case colored noise sources [7],
relies on stochastic calculus. The principal point of their contribution was the
introduction of a decomposition of phase and amplitude noise through a pro-
jection onto the periodic time-varying eigenvectors (the so called Floquet eigen-
vectors; also see ref. [8]), and they proved that it provides the correct solution
to the problem.
Demir et al have shown that if the sources of noise are Gaussian and white,

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Y. K. Chembo & al.May 21, 2008
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the phase noise around the fundamental peak (and its harmonics) has a Lorentzian
lineshape, and therefore is fully determined by an “effective” diffusion constant
Deff which is the unique parameter needed for the phase noise determination.
However, if the Demir et al theory has the great and essential advantage of math-
ematical rigorousness, its principal drawback is that exactitude is obtained at
the expense of simplicity: the calculation of Deffis very complex, as it requires
an accurate determination of all the time-varying eigenvectors related to the
autonomous flow. In general this task can only be performed numerically us-
ing quite complicated algorithms, and this lack of flexibility explains why this
method is scarcely used in the phase noise studies available in the literature.
The key challenge for the study of phase noise in OEO would be provide an
accurate description of the phase noise spectrum, while avoiding the determina-
tion of Floquet eigenvectors, which is an extremely complicated task in delayed
system.
3 Application of the phase diffusion approach to
OEOs: stochastic delay-differential equations
The OEO under study is organized in a single-loop architecture as depicted
in Fig. 1. The oscillation loop consists of: (i) A wideband integrated optics
LiNbO3Mach-Zehnder (MZ) modulator, seeded by a continuous-wave semicon-
ductor laser of optical power P; the modulator is characterized by a half-wave
voltage Vπ = 4 V. (ii) A thermalized 4 km fiber performing a time delay of
T = 20 µ on the microwave signal carried by the optical beam; the correspond-
ing free spectral range is ΩT/2π = 1/T = 50 kHz. (iii) A fast photodiode with
a conversion factor S. (iv) A narrow band microwave radio-frequency (RF)
filter, of central frequency F0 = Ω0/2π = 10 GHz, and −3 dB bandwidth of
∆F = ∆Ω/2π = 50 MHz; (v) A microwave amplifier with gain G. (vi) A vari-
able attenuator, in order to scan the gain. (vii) All optical and electrical losses
are gathered in a single attenuation factor κ.
The dynamics of the microwave oscillation can therefore be described in
terms of the dimensionless variable x(t) = πV (t)/2Vπwhose dynamics obeys [3]
?t
where β = πκSGP/2Vπis the normalized loop gain, φ = πVB/2Vπis the Mach-
Zehnder offset phase, while τ = 1/∆Ω and θ = ∆Ω/Ω2
timescale parameters of the bandpass filter. Since we are interested by single-
mode microwave oscillations, the solution of Eq. (1) can be expressed under the
form
x + τdx
dt+1
θ
t0
x(s)ds = β cos2[x(t − T) + φ],
(1)
0are the characteristic
x(t) =1
2A(t)eiΩ0t+1
2A∗(t)e−iΩ0t,
(2)
where A(t) = A(t) exp[iψ(t)] is the slowly varying amplitude of the microwave
x(t). We can significantly simplify the right-hand side term of Eq. (1) because
the cosine of a sinusoidal function of frequency Ω0can be Fourier-expanded in
harmonics of Ω0. In other words, since x(t) is nearly sinusoidal around Ω0, then
the Fourier spectrum of cos2[x(t−T)+φ] will be sharply distributed around the