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13th International Conference on Microwaves, Radar and Wireless Communications, MIKON’2000 Proceedings,
Vol.2, pp.693-698, Wroclaw, Poland, 22-24 May 2000
693
MICROWAVE HARMONIC GENERATION IN FIBER-OPTICAL LINKS
A.Hilt
ABSTRACT : Optical transmission of microwave (MW) and millimeter-wave (MMW) signals have become an
intensive research area in the last decade. There is a growing interest in optical processing of MW signals [1, 2],
phased array applications [3] and wireless distribution of broadband data in fiber-fed MMW subscriber access
systems [4-9]. This paper extends the existing models of MW/MMW optical links that are based on optical
intensity [10, 11]. The model is suitable for estimating harmonic levels of the MW modulation signal generated
in the optical path. Considering a MW fiber-optic link both the optical transmitter and the receiver are
responsible for harmonic generation. Furthermore, the optical fiber itself inserted between the transmitter and
receiver induces harmonics due to dispersion. Exact modelling of harmonic generation requires a calculation
based on the optical field instead of on a purely intensity basis [12-14].
I. HIGH-SPEED MODULATION OF LIGHT
In interferometric modulators the light of the optical source is splitted into two beams and then interference is
created between these beams (Fig.1). Interferometric optical modulators are usually called Mach-Zehnder
modulators (MZM). A phase modulator is inserted into one branch inducing phase difference between the
beams. If the phase difference is
a rejection of the input optical signal occurs. When the beams interfere
constructively, the output intensity is equal to the input intensity assuming lossless modulator (A=1). If the
power dividers split and recombine the optical power equally, the output intensity is written as :
( )
I t A ItAI t
out in in
( ) cos ( ) cos ( )
= + =
212
2
, where
( )
( ) ( )
mod
tV t
VV V t
V
DC RF
= = +
. (1-2.)
A is the optical loss in the modulator and
(t) is the phase difference between the propagating waves.
Iout(t)
Iin
LiNbO3
Vmod(t)
Iin
Iin/2
Vmod
V
0
Iout(Vmod)
maximum
minimum
quadrature
Figure 1. MZM integrated on LiNbO3
(one arm modulated)
Figure 2. Modulator transmittance as a function
of modulating voltage Vmod
Fig.2 shows the modulator transmittance as a function of the modulation voltage. VDC is the bias voltage of the
modulator. The half-wave voltage V introduces
phase difference between the modulator arms. This voltage is
required to drive the modulator between adjacent minima and maxima. Applying periodic modulation as :
( )
V t V V t
DC RF RFmod( ) cos= +
the intensity becomes
( )
I t It
out in RF
( ) cos cos= + +
21
, (3-4.)
where
= VDC/V and
= VRF/V are the normalized bias and RF signal amplitudes driving the MZM. The
optimal DC bias for linear operation is :
V V iV
DC = +
/2
where i Z. Case of i=0 is called the quadrature.
The output optical intensity can be expressed from Eq.4 by Bessel-function expansion :
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
I t I I J J k t
IJ k t
out in in k
kkRF
in k
kkRF
( ) cos cos
sin cos .
= + + −
− − +
=
+
=
+
+
2 2 2 1 2
22 1 2 1
012
02 1
(5.)
Eq.5 indicates that due to nonlinearity of the modulation function the output intensity contains harmonics of the
modulation frequency
RF in spite of that the modulation voltage is an ideal sinusoid. The double sided intensity
spectrum has been calculated at the MZM output by Fast Fourier Transform (FFT) as a function of the harmonic
: NOKIA Hungary Kft., H-2040 Budaörs, Szabadság út 117./B., Atronyx House, 5th floor., and
BMGE-MHT, Budapest University of Technology and Economics, Dept. of Microwave Telecommunications
H-1111 Budapest, Goldmann György tér 3. V2 épület, Hungary, and
: (+36) 20 936 9486, fax: (+36) 1 463 32 89, e-mail : attila.hilt @ nokia.com
13th International Conference on Microwaves, Radar and Wireless Communications, MIKON’2000 Proceedings,
Vol.2, pp.693-698, Wroclaw, Poland, 22-24 May 2000
694
number k. From the general expression of Eq.5 the special cases are the quadrature (or linear) operation, the
minimum and the maximum transmission modes. If the MZM is biased for linear operation the intensity contains
only odd harmonics and a DC component (Fig.3).
k
-400
-350
-300
-250
-200
-150
-100
-50
0
-15
-10
-5
0
5
10
15
Iopt(kRF) [dB]
k
-15
-10
-5
0
5
10
15
-400
-350
-300
-250
-200
-150
-100
-50
0
Iopt(kRF) [dB]
Figure 3. Calculated double sided optical intensity
spectrum at linear operation (
=0.5 or 1.5) for
=0.25
Figure 4. Calculated double sided optical intensity
spectrum at maximum transmission (=2), for =0.25
In Fig.4 the MZM is biased for maximum transmission. Compared to Fig.3, odd harmonics disappeared and only
even harmonics of the modulation signal are present in the intensity spectrum. The DC term has larger amplitude
but the modulation signal and all its odd harmonics are strongly suppressed. In other words, at this special bias
point the MZM generates second harmonic of the modulation signal (Fig.4). Such special operation modes find
interesting applications in transmission of MMW signals over dispersive fiber as well as in optical generation of
MMW signals. Considering small modulation indices (of VRF(t) << V so
<< 1) and a DC bias for linear
operation with
= 1.5, Eq.4 and Eq.5 simplify to :
( )
I t ItItIm t
out in RF in RF in RF
( ) cos cos sin cos cos= + +
= + +
213
2 2 121
, (6.)
where m =
denotes the optical modulation depth (OMD). Small-signal modulation allows linear
approximation of the sinusoidal modulator transmittance function. Let us suppose now that the MZM is biased at
quadrature for linear operation. Power level of the detected fundamental signal and any odd harmonic can be
calculated as a function of MZM driving voltage :
( ) ( )
P n V RR I RR I A J V
V
DET RF PD out PD in nRF
,= =
50 250 2 2 2 2
8 8
, (7.)
where n=2k+1. R50 stands for the resistive load and n=1 means the detected fundamental signal. Optical
intensities at the modulator input and output are denoted by Iin and Iout, respectively. In Eq.7 a resistive matching
to the 50 load is supposed. Fig.5 and Fig.6 show harmonic levels as a function of MW power and DC bias
driving the modulator, respectively.
PDET [dBm]
MW power driving the MZM [dBm]
-10
0
10
20
30
-120
-100
-80
-60
-40
-20
fundamental
third
harmonic
fifth
harmonic
normalized DC bias
0
1
2
-90
-80
-70
-60
-50
-40
PDET [dBm]
0.5
1.5
DC
fundamental
3rd
harm.
2nd
harm.
4th
harm.
Figure 5. Fundamental and odd harmonics of the
detected optical intensity vs. MW driving power
Iin = 1.2 mW, A = 3dB, V = 5V, RPD = 0.8 A/W
Figure 6. Detected DC and harmonic contents vs.
(calculated with Iopt = 400 W, RPD = 0.356 A/W,
m = 0.586)
13th International Conference on Microwaves, Radar and Wireless Communications, MIKON’2000 Proceedings,
Vol.2, pp.693-698, Wroclaw, Poland, 22-24 May 2000
695
The optical field at the MZM output is :
E t E t t
RF
( ) cos cos cos= +
0 0 2 2
. (8.)
In Eq.8 E0cos
0t is the optical carrier. The optical field expressed from Eq.8 by Bessel-function expansion is :
( )
( ) ( )
E t E J t E J t
E J t E J t
RF
RF RF
( ) cos cos sin cos
cos cos sin cos ...
=
−
+
−
+
+
0 0 0 0 1 0
0 2 0 0 3 0
2 2 2 2
2 2 22 2 3
(9.)
II. QUADRATIC PHOTODETECTION
Usually the photocurrent calculated by Eq.10 and said to be proportional to the modulated optical intensity [15] :
( ) ( )
i t R P t
PD PD opt
=
. (10.)
In Eq.10 the phase information of the optical wave is lost. Since in a coherent model the phase cannot be
neglected, let us consider now the general case when instead of the intensity the optical field is given:
( ) ( ) ( ) ( )
i t R E t R E t E t
PD PD PD
=22 *
. (11.)
< > means time averaging taken over a few optical periods. E(t) represents a real valued function and factor 2 is
chosen for later convenience [16]. Time averaging means the physical fact, that the PD cannot response to rapid
changes at optical frequencies, only the MW/MMW modulation envelope of the optical carrier is detected.
Supposing an incident optical field as a combination of two spectral components having the same polarization :
( )
( ) ( )
( ) ( )
( )
( )
( )
i t R E t E t
RE E E t E t
E E t E E t
R E E E E t
PD PD
PD
PD
+ + + =
=+ + + + +
+ − + − + + + + =
= + + − + −
2
2 2
2 2
2
1 1 1 2 2 2 2
1
22
21
21 1 2
22 2
1 2 1 2 1 2 1 2 1 2 1 2
1
22
21 2 1 2 1 2
cos cos
cos cos
cos cos
cos .
(12.)
The above calculation is referred to as coherent beating of the input optical signals. Terms 2
1, 2
2 and
1+
2
disappeared due to averaging. Remaining terms represent a DC component and a current having a MW
frequency equal to the difference of the input optical frequencies. As seen from Eq.12 coherent beating can be
used to generate MW and MMW signals optically. Let us consider now the optical field present at a MZM
output in case of small OMD. This optical field is approximated now by three spectral lines only. Using the
complex form of Eq.11 we can calculate the photocurrent as :
( )
( )
( ) ( )
( )
( )
( )
( )
i t R E t E t E t
R E e E e E e
E e E e E e
R E E E E
PD PD RF RF RF RF
PD RF j t j t RF j t
RF j t j t RF j t
PD RF RF
RF RF
RF RF
− − + + + + =
= + +
+ +
=
= + +
− − + +
− − − −− + +
2
2 4
0 1 0 0 0 2 2
0
0
0
2 2 02
0 1 00 2
0 1 00 2
cos cos cos
*
cos
−
+ +
+ + +
1 1 2 21 2
2 2 2 2 2
cos cos
RF RF RF
t E t
(13.)
It is seen from Eq.13 that second harmonic of the modulation signal
RF is generated, however small OMD and
ideal photodetector have been supposed. The modulation signal cannot be recovered if the phase difference
2-
1
is equal to (2n+1)
. Generally the optical field E(t) is composed of several spectral lines (see Eq.9). In this case
the exact calculation is difficult, since
RF components arise from the mutual beating of each pair of spectral
lines that are separated by
RF. Similarly, the harmonic n
RF is generated from the beating of any two lines
being separated n times
RF apart. Finally, the photocurrent has a discrete spectrum of :
( ) ( )
( )
i R i k
PD PD RF
k
N
=
−
0
1
, (14.)
where N is the number of the optical field spectral components taken into account. The DC term is given by k=0
and k=N-1 gives the higher order harmonic. Calculating by Eq.13-14 is rather tedious. An easier solution starts
with the optical field E(t) given in time domain and uses the complex form of Eq.11. Then the spectrum of the
photocurrent at the PD output can be simply determined by Fourier transform. For calculation simplicity, this
13th International Conference on Microwaves, Radar and Wireless Communications, MIKON’2000 Proceedings,
Vol.2, pp.693-698, Wroclaw, Poland, 22-24 May 2000
696
method using FFT has been used in our computer simulations :
( ) ( )
)()( *tEtERi PDPD F
. (15.)
III. EFFECT OF CHROMATIC DISPERSION ON MW TRANSMISSION
Considering small modulation index (
<< 1) and optimal modulator bias for linear operation, the optical field at
the MZM output can be approximated by three main spectral components at
0 and
0
RF (Eq.8-9). This field
suffers from dispersion during propagation in a standard singlemode fiber (SMF) exhibiting a chromatic
dispersion factor of D=17ps/km/nm. The optical field at the SMF output is calculated with the fiber transfer
function H(
) approximated by its Taylor series up to the second order :
E E H E e
out in in
j L
( ) ( ) ( ) ( )
' ''
=
− + +
0 0 0
2
2
(16.)
For simplicity we omitted the linear fiber attenuation. In the exponent of Eq.16 the first term results in a phase
delay, meanwhile the second term represents the group delay. These terms are out of interest here. However, the
third term introduces additional phase change due to chromatic dispersion. Inserting the dispersion parameter D
into Eq.16 from
022
'' /= − D c
, supposing an input optical field of Eq.9 and applying Eq.16, the photocurrent
after quadratic photodetection is written as :
( ) ( )
( )
i t R E E E E LD
ct E t
PD PD RF RF RF RF RF RF
= + +
− + −
0
2 2 02 2 2
2 4 42 2cos cos cos
(17.)
where
= 0
'L
is the group delay. As seen in Eq.17 the detected signal is composed of a DC photocurrent, the
fundamental signal delayed by
and its second harmonic. Neglecting DC and harmonic terms, omitting the
delay and inserting ERF =mE0/4 into Eq.17 the photocurrent at the fundamental of the modulation frequency fRF is
:
( )
( )
( )
i t R mE cD L f f t
PD PD RF opt RF
RF
,cos / cos
=
0
22
. (18.)
Based on Eq.18 the electrical power delivered from the matched photodiode to the load is proportional to :
( )
( )
( ) ( )
P f L,D R mE cD L f f cD L f f
RF
dB RF PD RF opt RF opt
[ ] , log cos / logcos /
10 20
0
2222 2
. (19.)
As it is shown in Fig.7-8, the phase difference between the spectral components propagating in the fiber can
result in a complete rejection of the transmitted MW or MMW signals. In Fig.8 results obtained by scalar
measurements on the L=19.2 km long FDDI ring of our University are compared to the theoretical curve [9].
fiber length L [km] resulting in 3 dB C/N penalty
fiber dispersion D [ps/km/nm]
0
4
8
12
16
20
0.1
1
10
100
26 GHz
42 GHz
60 GHz
-50
-40
-30
-20
-10
0
0
5
10
15
20
modulation frequency [GHz]
PRF [dB]
measured
L = 19.2 km
17ps/km nm,
calculated
Figure 7. Maximum SMF length L resulting 3 dB C/N
degradation vs. dispersion parameter D
Figure 8. Measured rejection at fRF =14.2 GHz
for a fiber length of L = 19.2 km.
IV. HARMONICS IN DISPERSIVE TRANSMISSION
In part III. the optical field E(
) present at the SMF input has been approximated by three spectral lines only.
This simplification reduced the calculation difficulties significantly and we were able to derive analytical results.
In the general case however, several optical field spectral lines are present at the fiber input. Detected amplitude
and phase of these optical field spectral components are determined by the LD, by the MZM as well as by
parameters of propagation in the dispersive fiber. Only coherent models can explain properly detected levels of
different harmonics of the modulation signal. Based on the coherent model of the MW optical link we simulated
the effect of chromatic dispersion in the general case of several spectral lines. In this coherent model the
calculation is based on the optical field and not on the optical intensity. Here we present simulation results for
13th International Conference on Microwaves, Radar and Wireless Communications, MIKON’2000 Proceedings,
Vol.2, pp.693-698, Wroclaw, Poland, 22-24 May 2000
697
harmonic generation. Harmonics are generated due to propagation in dispersive fiber. When the MZM is biased
for linear operation, only odd components are present in the optical intensity (Fig.3). In the optical field both
even and odd spectral components are present (Fig.9). When this optical field is launched into a SMF, due to
dispersion even intensity components will appear after propagation. Calculated levels of harmonics are shown in
Fig.10. Since phase of harmonics are rotated faster in the fiber than phase of the fundamental, second harmonic
has two times, third harmonic has three times more rejections between two rejections of the fundamental. We
note that this phenomena cannot be explained by the incoherent model of the MW optical link.
-10
-5
0
5
10
-300
-350
-250
-200
-150
-100
-50
0
|Eopt()| [ dB ]
k
RF
k
0
5
10
15
20
25
30
-200
-150
-100
-50
0
modulation frequency fRF [GHz]
PRF [ dB ]
2nd
3rd
DC
1st
4th
Figure 9. MZM output field at bias for linear
operation, = 0.4, = 0.5.
Figure 10. Detected signals after propagation in
dispersive fiber of L=19.2 km, input field as in Fig.9
On the other hand, if the MZM is biased for minimum transmission (Fig.11), the second harmonic of the
modulation signal is not rejected, even after propagation in a nearly 20 km dispersive fiber (Fig.12). In this case
the phase difference cannot create complete rejection, since the optical carrier is suppressed. The advantage of
the method is that only the subharmonic of the desired MMW signal is desired to drive the optical modulator.
The developed method is rather general, it is suitable for calculating the effect of fiber dispersion simultaneously
with effect of modulator bias in external modulation and chirp of direct modulated laser diodes as well [12-15].
-10
-5
0
5
10
-300
-350
-250
-200
-150
-100
-50
0
|Eopt()| [ dB ]
kRF
k
-80
-60
-40
-20
0
0
5
10
15
20
25
30
PRF [ dB ]
modulation frequency fRF [GHz]
DC
2nd
2nd
4th
Figure 11. MZM output field at bias for
minimum transmission,
= 0.4, = 1
Figure 12. Detected signals after propagation in
dispersive fiber of L = 19.2 km, input field as in Fig.11
V. CONCLUSIONS
Effect of chromatic dispersion on optical transmission of digital baseband signals is well described in the
literature. However, for analogue MW/MMW IM/DD optical links chromatic dispersion has not been fully
analyzed yet. In this paper MW harmonic generation in IM/DD fiber-optical links is discussed. The influence of
chromatic dispersion on the optical transmission of MW/MMW signals in standard singlemode fibers has been
also examined. It was pointed out that standard SMF links operating at
=1550 nm cannot be used for
transmission of MW/MMW signals without encountering the effect of chromatic dispersion. It was shown that
dispersion penalty significantly limits the transmission distance in IM/DD optical links operating above 10 GHz.
A several km long fiber-optical link filters the transmitted MW or MMW signal. As a function of fiber length L
rejections are periodic and these periods are shorter and shorter as the modulation frequency fRF is increased.
13th International Conference on Microwaves, Radar and Wireless Communications, MIKON’2000 Proceedings,
Vol.2, pp.693-698, Wroclaw, Poland, 22-24 May 2000
698
First analytic explanation of the problem was given. Chromatic dispersion results in a difference between phase
states of optical IM field sidebands. These sidebands are beaten coherently on the photodetector. As introduced
in Eq.13 as soon as the phase difference approaches
the modulation signal is lost. Then we presented a general
model to calculate the harmonic levels and the effect of chromatic dispersion numerically. By the presented
coherent model detected harmonics are estimated. To avoid chromatic dispersion problems, one might propose
tailoring the fiber length exactly at the maxima of the penalty function shown in Fig.8. As we demonstrated in
Fig.10 the locations of these maxima do not depend only on the fiber span but also on the IM frequency.
Furthermore, temperature dependence, aging and polarization mode dispersion should be considered too.
ACKNOWLEDGMENTS
This research work was supported by the MOIKIT project of the European Union. The author wishes to thank
the fruitful discussions with Prof.I.Frigyes, Dr.G.Maury, Dr.A.Ho-Quoc, Prof.T.Berceli and T.Marozsák. The
author acknowledges the continuous support of the Hungarian Scientific Research Fund (OTKA No.T017295,
F024113, T030148, T026557).
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