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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

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

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.

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).

REFERENCES

[1] R.Helkey : “Advances in Frequency Conversion Optical Links”, Proc. of the 10th Microwave Colloquium,

MICROCOLL’99, pp.365-369, Budapest, Hungary, March 1999.

[2] A.Hilt : “Transmission et traitement optiques des signaux dans les systèmes de télécommunications

hertziens”, Ph.D. thesis, Institut National Polytechnique de Grenoble, Grenoble, France, May 1999.

[3] I.Frigyes, A.J.Seeds : “Optically Generated True-Time Delay in Phased-Array Antennas”, MTT, Special

Issue on Microwave and Millimeter-wave Photonics, Vol.43, No.9, Part II, pp.2378-2386, September 1995.

[4] T.Berceli, A.Hilt, G.Járó, G.Maury : “Optical Processing and Transmission of Subcarrier Multiplexed

Microwave Signals”, Proc. of the Optical Technologies for Microwave Systems Workshop of the 29th EuMC,

München, Germany, October 1999.

[5] T.Berceli, G.Járó, T.Marozsák, A.Hilt et al.: “Generation of Millimeter Waves for Mobile Radio Systems”,

Proc. of the 10th Microwave Colloquium, MICROCOLL’99, pp.375-378, Budapest, Hungary, March 1999.

[6] A.Hilt, A.Vilcot, T.Berceli, T.Marozsák, B.Cabon : “New Carrier Generation Approach for Fiber-Radio

Systems to Overcome Chromatic Dispersion Problems”, Proc. of the IEEE MTT Symposium, paper TH3C-5,

pp.1525-1528, Baltimore, USA, June 1998.

[7] T.Marozsák, T.Berceli, G.Járó, A.Zólomy, A.Hilt, S.Mihály, E.Udvary, Z.Varga : “A New Optical

Distribution Approach for Millimeter Wave Radio”, Proc. of the IEEE MTT Topical Meeting on Microwave

Photonics, MWP’98, pp.63-66, Princeton, New Jersey, USA, October 1998.

[8] A.Hilt, T.Marozsák, G.Maury, T.Berceli, B.Cabon, A.Vilcot : “Radio-Node Upconversion in Millimeter-

Wave Fiber-Radio Distribution Systems”, Proc. of the International Conference on Microwaves and Radar,

MIKON’98, Vol.1, pp.176-180, Kraków, Poland, May 1998.

[9] A.Hilt, A.Zólomy, T.Berceli, G.Járó, E.Udvary : “Millimeter Wave Synthesizer Locked to an Optically

Transmitted Reference Using Harmonic Mixing”, Technical Digest of the IEEE Topical Meeting on Microwave

Photonics, MWP’97, pp.91-94, Duisburg, Germany, September 1997.

[10] I.Frigyes, I.Habermajer, B.Molnár, A.J.Seeds, F.Som : “Noise and Loss Characteristics of Microwave

Direct Modulated Optical Links”, Proc. of the 27th EuMC, Jerusalem, Israel, 1997.

[11] C.H.Cox III., G.E.Betts, L.M.Johnson : “An Analytic and Experimental Comparison of Direct and External

Modulation in Analog Fiber-Optic Links”, MTT, Vol.38, No.5, pp.501-509, May 1990.

[12] A.Hilt, G.Maury, A.Vilcot, B.Cabon : “Numerical Model of Chromatic Dispersion Effects in Analogue

IM/DD Optical Links”, Proc. of the 2nd International Summer School on Interactions between Microwaves and

Optics, OMW’99, pp.141-142, Autrans, France, July 1999.

[13] A.Hilt, G.Maury, B.Cabon, A.Vilcot, L.Giacotto : “General Approach to Chromatic Dispersion Analysis of

Microwave Optical Link Architectures”, Proc. of the 10th Conference on Microwave Techniques, COMITE’99,

pp.177-180, Pardubice, Czech Republic, October 1999.

[14] G.Maury, A.Hilt, B.Cabon, V.Girod, L.Degoud : “Remote Upconversion in Microwave Fiber-Optic Links

Employing an Unbalanced Mach-Zehnder Interferometer”, Proc. of the SPIE’s 44th Annual Meeting, Terahertz

and Gigahertz Photonics Conference, paper 3795-71, Denver, Colorado, USA, July 1999.

[15] A.Hilt, G.Járó, A.Zólomy et al.: “Microwave Characterization of High-Speed pin Photodiodes”, Proc. of the

9th Conference on Microwave Techniques COMITE’97, pp.21-24, Pardubice, Czech Republic, October 1997.

[16] B.E.A.Saleh, M.C.Teich : “Fundamentals of photonics”, John Wiley and Sons Inc., 1991.