model the scattering parameters in the chosen frequency range,
and the passivity enforcement phase can become computation-
ally prohibitive. Furthermore, the corresponding ODE in Eq. (4)
will have a large number of equations, and a small time-step (of
the order of femtoseconds) must be adopted to solve it.
In order to tackle these issues, a novel approach based on
baseband equivalent state-space models is proposed in this
contribution.
3. BASEBAND EQUIVALENT STATE-SPACE
MODELS FOR THE TIME-DOMAIN SIMULATION
OF PHOTONIC SYSTEMS
The basic concepts of baseband equivalent signals and systems
are first introduced in Section 3.A, given their importance in
the definition of the proposed modeling approach, which is de-
scribed in Section 3.B.
A. Baseband Equivalent Signals and Systems
The excitation signal of photonic systems is often an amplitude
and/or phase-modulated signal with optical carrier and elec-
tronic modulating signals, which can be written in the follow-
ing form:
utAtcos2πf
c
t ϕt, (5)
where At is the time-varying amplitude or envelope of the
modulated signal, and ϕt is the time-varying phase. In elec-
tronics or radio-frequency (RF) applications, both At and
ϕt relate to electronic signals, such as voltage, current, or
electric field. In photonics, the optical carrier frequency f
c
is much higher than the ban dwidth of At and ϕt, given
that the wavelength of light is much smaller than that of
RF signa ls, so the representation in Eq. (5) is often called a
bandpass signal.
An analytic complex-valued representation of the real-
valued signal in Eq. (5), called the analytic signal, is introduced
here as [24]
u
a
tutjHut Ate
j2πf
c
tϕt
, (6)
where Hut is the Hilbert transform of ut. In the frequency
domain, Eq. (6) becomes
U
a
f 2U f Stepf , (7)
where U
a
f and Uf are the Fourier transform of u
a
t
and ut, respectively, and Stepf is a unit step function de-
fined by
Stepf
8
<
:
1, f>0,
1
2
, f 0,
0, f<0:
(8)
Now, the corresponding baseband equivalent signal of the
bandpass signal is defined as
u
l
tu
a
te
−j2πf
c
Ate
jϕt
, (9)
U
l
f 2U f f
c
Stepf f
c
, (10)
which can be considered as the complex envelope optical signal
representation and is widely used in photonics and optical fiber
communication. The relations between ut, Hut, and
u
l
t in the time and frequency domains are [24]
utRe
u
l
te
j2πf
c
t
, (11)
Hut Im
u
l
te
j2πf
c
t
, (12)
U f
1
2
U
l
−f − f
c
1
2
U
l
f − f
c
, (13)
where the superscrip t * denotes the complex conjugate op-
erator.
In the frequency domain, the concepts of analytic signal and
baseband equivalent signal are intuitive: U f has a symmetric
spectrum with respect to the positive and negative frequencies,
while U
a
f has only a non-zero spectrum in the positive
frequencies around the carrier frequency; shifting the spectrum
of U
a
f in the direction of the negative frequencies of f
c
[or
equivalently in the time domain by multipl ying u
a
t with
e
−j2πf
c
t
] leads to U
l
f . Such relations are illustrated in Fig. 1.
If a system with impulse response ht and frequency re-
sponse H f operates in the bandwidth BW around f
c
, sat-
isfying f
c
≫ BW, then it can be considered as a bandpass
system. Now, the corresponding baseband equivalent system
can be defined by applying the same concepts described for
the baseband signals. Thanks to the relations among bandpass
signals and systems and their baseband equivalents, it can be
proven that the output signal of a bandpass system can be ana-
lytically recovered from the output of the corresponding base-
band system, as illustrated in Fig. 2. A detailed proof is given in
Appendix A.
It is important to remark that performing time-domain sim-
ulations of baseband equivalent systems allows one to efficiently
recover the corresponding bandpass signals, thus avoiding the
Fig. 1. Spectra of bandpass signal U f , analytic signal U
a
f ,
and baseband equivalent signal U
l
f .
Bandpass Input Bandpass System Bandpass Output
Baseband
Equivalent Input
Baseband
Equivalent System
Baseband
Equivalent Output
Fig. 2. Time-domain simulation of the bandpass system and base-
band equivalent system.
562 Vol. 6, No. 6 / June 2018 / Photonics Research
Research Article