## Abstract

We comprehensively investigate three modulation techniques for the generation of millimeter-wave (mm-wave) using optical frequency quadrupling with a dual–electrode Mach-Zehnder modulator (MZM), i.e. Technique-A, Technique-B and Technique-C. For Technique-A, an RF signal drives the two electrodes of the MZM with maximum transmission bias, and this MZM is used for both the mm-wave generation and signal modulation. Technique-B is the same as Technique-A, but 180° phase shift between the two electrodes is applied. Technique-C is the same as Technique-B, but the MZM is only used for the mm-wave generation without signal modulation. It is found that Technique-B and Technique-C are better for frequency quadrupling than frequency doubling, tripling and sextupling. Both theoretical analysis and simulation show that the generated mm-wave suffers from constructive/destructive interaction due to fiber chromatic dispersion in Technique-A. However, the generated mm-wave is almost robust to fiber chromatic dispersion in Technique-B and Technique- C. It is found that Technique-C is the best in the quality of the generated mm-wave, especially when poor optical filtering is used. In addition, we develop a theory for calculation of Q-factor in an mm-wave over fiber system using the three modulation techniques for mm-wave generation. We consider an RF at 7.5 GHz and obtain an mm-wave at 30 GHz as an example, i.e. a frequency quadrupler. We evaluate the generation and distribution in terms of system Q-factor. The impact of RF modulation index, chromatic dispersion, MZM extinction ratio and optical filtering on Q-factor are investigated.

©2008 Optical Society of America

## 1. Introduction

Radio over fiber (RoF) technique has been considered a cost-effective and reliable solution for the distribution of the future wireless access networks. As a consequence, many research works related to radio over fiber systems have been intensively conducted, such as photonic millimeter-wave (mm-wave) generation and distribution, radio over fiber systems integrated with dense wavelength division multiplexing, and suppression of nonlinear distortion, etc [1-23]. For the future wireless communications, unlicensed mm-wave band has been considered a potential wireless RF carrier band. When RoF technique is used for the distribution of wireless signals, photonic mm-wave signal generation and distribution over fiber link are desired and considered a potential cost-effective solution. To realize the photonic mm-wave generation by frequency up-conversion, many techniques have been reported, such as the frequency up-conversions using four-wave mixing [5] and cross-gain modulation in a semiconductor amplifier [6-8], optical heterodyne detection with optical interleaving [9, 16], frequency doubling using an optical carrier suppression modulation [10-11], and frequency quadrupling and sextupling using optical frequency multiplication (OFM) [24-29].

The principle of OFM is to generate high-order optical harmonics in an electrical to optical (E/O) converter, such as a laser, and an external optical modulator, driven by a low radio frequency (RF) sinusoidal signal. When a sinusoidal RF signal drives an E/O converter, many high-order optical harmonics may be generated due to nonlinear response of the E/O converter. By beating of any two high-order optical harmonics or/and beating of any high-order optical harmonics with the optical carrier in photodetection may generate many mm-wave signals. The OFM technique for a frequency quadrupling and sextupling has been considered a cost-effective solution for photonic generation of mm-waves and delivery of wireless signals to remote antennas [24-30]. This is because a low RF oscillator and a narrow-bandwidth E/O converter are used. The major technical problem in the OFM technique is how to efficiently generate desired optical harmonics and suppress undesired optical harmonics. In other words, high conversion efficiency and less nonlinear distortion impact are desired for the mm-wave generation and distribution.

Generation of mm-waves using the OFM and the application to RoF systems have been already demonstrated experimentally. For example, a unidirectional/bidirectional RoF system, using the sixth-order optical harmonics generated in an optical phase modulator for generating 18 GHz, was demonstrated for transmission over single and multimode fibers for delivery of 16-QAM and 64-QAM [24-27]. Still using the sixth-order optical harmonics, an mm-wave of 40 GHz was generated to deliver 16-QAM and 64-QAM over 25 km single-mode fiber [28].

The OFM technique using an optical phase modulator or a single-electrode Mach-Zehnder modulator combined with an optical Mach-Zehnder interferometer (MZI) as an optical filter was investigated [24-29]. The optical MZI is an optical periodic filter in optical frequency domain. In other words, all optical harmonics that have the same period of optical frequency as the MZI will pass through the MZI. As an example, we consider a 40-GHz mm-wave generation using an RF of 10 GHz that drives an optical modulator. Thus optical harmonics at ±10, ±20, ±30 and ±40 GHz etc may be generated. Thus the generated mm-wave at 40 GHz by photodetection of the above optical harmonics consists of beatings between ± 10 and ∓30 GHz, ±20 and ∓20 GHz, and the optical carrier and ∓40 GHz at least, when an MZI with a period of 10 GHz is used. These beatings may be destructive and constructive, dependent on phase difference of the beatings, and fiber dispersion and fiber length. Therefore when an MZI is used, the generated mm-wave in RF power may fluctuate with fiber length, which is not desired for the distribution of mm-wave signals. In addition, an optical MZI may experience instability due to temperature and thus the mm-wave may be unstable in magnitude since a laser wavelength may not follow the MZI response drift. Therefore, an optical bandpass filter is suggested to be used instead of an optical MZI filter.

To the best of our knowledge, it has not been investigated comprehensively how to effectively generate mm-waves using the frequency quadrupling technique and mm-wave transmission over fiber in performance. In this work, we comprehensively investigate optical modulation techniques for effective mm-wave generation using a dual electrode MZM by theoretical analysis and simulation using VPI-TransmissionMaker. Specifically, we investigate frequency quadrupling using three optical modulation techniques. As an example, we consider 30-GHz mm-wave generation using an RF of 7.5 GHz that drives a dual-electrode MZM and an optical bandpass filter, instead of an optical MZI, namely a frequency quadrupler. The quality of the generated mm-wave for this frequency quadrupler is evaluated in an RoF system considering the impact of modulation index, optical filter bandwidth, and modulation frequency of the RF that drives the MZM as well as MZM extinction ratio. In the end, the limitation imposed by the frequency quadrupling technique combined with fiber dispersion and polarization mode-dispersion (PMD) is discussed.

## 2. Three optical modulation techniques considered

Three optical modulation techniques for mm-wave generation are considered and shown schematically in Fig. 1(a), 1(b), and 1(c), and are named as Technique-A, Technique-B, and Technique-C, respectively. For all the techniques as shown in Fig. 1, a continuous wave (CW) laser is assumed to have a wavelength of *λ*=1553 nm, an output power of *P*=-3 dBm, a linewidth of 10 MHz, and relative intensity noise (RIN) of -150 dB/Hz. A sinusoidal RF signal that drives a dual-electrode MZM, has a frequency *f _{RF}* of 7.5 GHz and voltage of

*V*(

_{RF}*t*)=√2

*V*sin(

_{RF}*ω*+

_{RF}t*ϕ*), where

_{RF}*ω*=2

_{RF}*πf*and

_{RF}*ϕ*- random phase noise of the oscillator. The modulation index for the mm-wave generation is defined by

_{RF}*m*=

_{RF}*V*/

_{RF}*V*, where

_{π}*V*is the

_{π}*π*-phase shift voltage of the MZM. Following the MZM there is an erbium doped fiber amplifier (EDFA), which is used to compensate for all insertion loss and fiber loss, and a noise figure of 5 dB is always assumed. Due to nonlinear transfer function of the MZM, many high-order optical harmonics may be generated if a high modulation index is used. To reduce non-desired optical harmonics, an optical bandpass filter (OBF) is used after the EDFA. The filter has a 5

^{th}order Gaussian transfer function with a bandwidth of 40 GHz and a central wavelength of 1553 nm. It is supposed that the dual-electrode MZM has an extinction ratio of 35 dB,

*V*=5 V and insertion loss of 6 dB for the analysis. The OBF is assumed lossless.

_{π}For Technique-A as shown in Fig. 1(a), the CW light is input into the dual-electrode MZM, biased at *maximum* transmission. An RF sinusoid at 7.5 GHz is mixed with a baseband data signal at 622 Mb/s, and then the output is split into two parts by an electrical splitter and the two outputs directly drives the two electrodes of the MZM. This MZM is used for two functions, i.e. high-order optical harmonic generation and optical signal modulation. Due to nonlinearity of the MZM modulation response, both even- and odd-order optical harmonics are generated as shown in Fig. 1(a) inset (i). Then after the optical filter (bandwidth of 40 GHz), the fourth- and higher- order optical harmonics are removed as shown in Fig. 1 inset (ii), where it is shown that six optical harmonics are left.

Technique-B as shown in Fig. 1(b) is almost identical to Technique-A, but for Technique-B there is 180° phase difference of the applied voltages to the two electrodes of the MZM. The others are the same as in Technique-A. As shown in Fig. 1(b) inset (i), only even-order optical harmonics are generated whereas all odd-order optical harmonics are ideally suppressed. After the optical filter, only two optical harmonics are left as shown in Fig. 1(b) inset (ii).

For Technique-C as shown in Fig. 1(c), the dual-electrode MZM, biased at *maximum* transmission too, is only used for the generation of optical harmonics and not used for optical signal modulation. The optical signal modulation can be obtained by direct modulation of the laser current, or using another external modulator. Note for this technique, that the optical carrier can be suppressed as shown in Fig. 1(c) inset (i) when a modulation index of 76.5% is used. However if the modulation index is not 76.5%, the optical carrier will not be suppressed completely. This is one difference of Technique-C from Technique-B. Note that the dual-electrode MZM is used for both the generation of optical harmonics and data signal modulation in both Technique-A and Technique-B, but only for the generation of optical harmonics in Technique-C.

Distribution of mm-wave signals to remote antennas is realized over a single mode fiber (SMF). It is supposed that the fiber has fiber loss of 0.22 dB/km and chromatic dispersion of 16 *ps*/(*nm*·*km*). After fiber transmission, an optical receiver as shown in Fig. 1(d) is used to evaluate the quality of the generated mm-wave. The optical receiver consists of a photodiode that is characterized by responsivity (ℜ=0.62 A/W at 1553-nm), thermal noise (spectral density of N_{th}=2×10^{-11}A/√Hz) and dark current (I_{d}=2 nA), and an electrical bandpass filter. By photodetection of the optical spectra given in Fig. 1 insets (ii), electrical spectral components at 0, 7.5, 15, 22.5, and 30 GHz etc may be generated. Without loss of generality, we consider, as an example, a generated mm-wave at 30 GHz using the three modulation techniques to investigate the quality of the generated mm-wave at 30 GHz. The electrical bandpass filter has a bandwidth of 1.2 GHz centered at 30 GHz (3^{rd} order Bessel). In practice, this mm-wave is launched into air by using an antenna. For our investigation, the mm-wave signal at 30 GHz is down-converted directly into the baseband signal for the quality evaluation. To obtain the baseband signal from the mm-wave signal at 30 GHz, an electrical mixer driven by a sinusoidal RF at 30 GHz and a low pass band filter (3^{rd} order Bessel) with a bandwidth of 450 MHz are used as shown in Fig. 1(d).

## 3. Theoretical analysis and optimization of RF modulation index

In this section, we present a theoretical analysis of mm-wave generation using the three optical modulation techniques.

For Technique-A, the output electric field after the dual-electrode MZM can be written as

where *γ*=√*ε*-1)/(√*ε*+1), *ε* is the MZM extinction ratio, *d*(*t*)*∈*(0,1) is digital signal, and *J _{n}*(·) is the n

^{th}order Bessel function of first kind.

*P*is the optical power of the laser source, and

*t*is optical insertion loss of the MZM. After optical amplification, optical filtering, and fiber transmission, the incident optical field to the photodiode of the optical receiver is given by

_{ff}where M is the nearest integer to *B _{F}*/

*f*, and 2

_{RF}*B*-3-dB optical bandwidth of the optical filter. After photodetection, the desired spectral component of photocurrent at 4

_{F}*ω*is expressed by

_{RF}$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}+d\left(t\right)\left[{J}_{2}^{2}\left(\pi {m}_{\mathit{RF}}\right){H}_{F}^{2}\left(2{\omega}_{\mathit{RF}}\right)-2{H}_{F}\left({\omega}_{\mathit{RF}}\right){H}_{F}\left(3{\omega}_{\mathit{RF}}\right){J}_{1}\left(\pi {m}_{\mathit{RF}}\right){J}_{3}\left(\pi {m}_{\mathit{RF}}\right)\mathrm{cos}\left(4L{\beta}_{2}{\omega}_{\mathit{RF}}^{2}\right)\right]\mid $$

Equation (2) shows that the optical spectra after the MZM contain even- and odd- order optical harmonics. After the optical filtering with a bandwidth of 40 GHz, the fourth-order optical harmonics and beyond are suppressed. In other words, only the optical carrier and optical harmonics of up to the third order are left as shown in Fig. 1(a) inset (ii). Equation (3) clearly shows that the generated mm-wave signal at 30 GHz mainly consists of two contributions, i.e. a beating between the two optical harmonics at ±2*ω _{RF}*, and the other beating between the optical harmonics at ±3

*ω*and ∓

_{RF}*ω*.

_{RF}For Technique-B, the output optical field after the MZM can be written as

After optical amplification, optical filtering, and fiber transmission, the incident optical field to the photodiode is given by

The desired spectral component of photocurrent at the frequency of 4*ω _{RF}* is expressed by

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\xb7+d\left(t\right)\left[{J}_{2}^{2}\left(\pi {m}_{\mathit{RF}}\right){H}_{F}^{2}\left(2{\omega}_{\mathit{RF}}\right)-\frac{2}{\epsilon}{H}_{F}\left({\omega}_{\mathit{RF}}\right){H}_{F}\left(3{\omega}_{\mathit{RF}}\right){J}_{1}\left(\pi {m}_{\mathit{RF}}\right){J}_{3}\left(\pi {m}_{\mathit{RF}}\right)\mathrm{cos}\left(4L{\beta}_{2}{\omega}_{\mathit{RF}}^{2}\right)\right]\mid $$

Equation (6) clearly shows that the generated mm-wave at 30 GHz mainly consists of the beating of the optical subcarriers at ±2*ω _{RF}* in addition to a small contribution by the beatings between the optical harmonics at ±

*ω*and ∓3

_{RF}*ω*, and between the optical carrier and the optical harmonic at ±4

_{RF}*ω*.

_{RF}For Technique-C, the first MZM is biased at maximum transmission for the mm-wave generation, and the second MZM is biased at quadrature and is driven by the baseband signal, i.e. used for baseband optical modulation. The output optical field after the two MZMs as shown in Fig. 1(c) can be written as

After optical amplification, optical filtering, and fiber transmission, the incident optical field to the photodiode of the optical receiver is given by

Note that for Eq. (8) when *n*=0, i.e. the optical carrier, the term *J*
_{0} (*πm _{RF}*)≈0 is obtained if

*m*=76.5% as shown in Fig. 2(a). This is the reason why the optical carrier in Fig. 1(c) inset (i) is suppressed. The desired spectral component of the photocurrent at the frequency of 4

_{RF}*ω*can be expressed as

_{RF}$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}+2{J}_{0}\left(\pi {m}_{\mathit{RF}}\right){J}_{4}\left(\pi {m}_{\mathit{RF}}\right){H}_{F}\left(0\right){H}_{F}\left(4{\omega}_{\mathit{RF}}\right)\mathrm{cos}\left(8{\beta}_{2}L{\omega}_{\mathit{RF}}^{2}\right)$$

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}-\left(\frac{2}{{\epsilon}_{1}}\right){J}_{1}\left({\pi m}_{\mathit{RF}}\right){J}_{3}\left(\pi {m}_{\mathit{RF}}\right){H}_{F}\left({\omega}_{\mathit{RF}}\right){H}_{F}\left(3{\omega}_{\mathit{RF}}\right)\mathrm{cos}\left(4{\beta}_{2}L{\omega}_{\mathit{RF}}^{2}\right)|$$

Equation (9) shows that the mm-wave at 30 GHz mainly consists of the beating of the two optical harmonics at ∓2*ω _{RF}* in addition to a small contribution by the beatings between the optical harmonics at ±

*ω*and ∓3

_{RF}*ω*, and between the optical carrier and the optical harmonic at ±4

_{RF}*ω*. Note in the above analysis, walk-off time due to fiber chromatic dispersion between any two optical harmonics is ignored.

_{RF}Using Eqs. (4) and (7) we can obtain the average optical power ratio of the optical carrier to the second-order optical harmonics, which are used for the mm-wave generation, at the modulation index of 76.5% (100%) for Technique-B,

and for Technique-C

As is seen that the optical carrier is suppressed by 53.5 (-3.1) dB below the second order harmonics using Technique-C for *m _{RF}*=76.5% (100%), and hence some noise contributions related to the optical carrier are reduced accordingly. In contrast to Technique-C, for Technique-B the optical carrier power is 8.5 (7.7) dB higher than that of the second order harmonics. Higher optical carrier power that beats with other harmonics will introduce data interference to the 30-GHz mm-wave in addition to higher shot noise, signal-amplified spontaneous emission (ASE) beat noise and relative intensity noise (RIN) impact. This is one reason why Technique-C leads to better performance than Technique-B.

Figure 2(a) shows simulated and calculated (using Eqs. (10) and (11)) average optical power ratio of the optical carrier to the second-order optical harmonics versus RF modulation index using Technique-B and Technique-C. It is seen that the simulated and calculated agree well. Maximum optical carrier suppression of ~53 dB below the second-order harmonics is achieved at a modulation index of ~76.5 % using Technique-C.

Conversion efficiency is another important factor for frequency quadruplers. The conversion efficiency can be expressed as
$\eta =\frac{\left(\frac{1}{2}{I}_{4{\omega}_{\mathit{RF}}}^{2}{R}_{L}\right)}{\left(\frac{\frac{1}{2}{m}_{\mathit{RF}}^{2}{V}_{\pi}^{2}}{{R}_{\mathit{in}}}\right)}$
, where *R _{L}* and

*R*are the 50 Ohm load and input MZM impedance, respectively. Assume a 10 dBm maximum optical power incident to a photodetector with responsitivity of 0.62, and Fig. 2(b) shows conversion efficiency with modulation index. The maximum conversion efficiency of ~-30 dB and -45 dB is obtained at an optimum value of 76.5% and 107% for Technique-B and Technique-C, and Technique-A, respectively, as shown in Fig. 2(b). Therefore, it is desirable to use an MZM with low

_{in}*V*and thus a higher modulation index can be easily obtained.

_{π}Using the above theoretical analysis and the parameters given in the above, we calculate Q-factor with the help of Appendix. As an example, we consider a modulation index of 76.5% for the three modulation techniques. Figure 3(a) and 3(b) shows simulated and calculated Q-factor versus fiber length, respectively. The calculated Q-factor almost matches the simulated one, showing that the above theory is accurate. The main difference between the simulated and calculated Q-factors for using Technique-B and Technique-C is that the simulated Q-factor slightly fluctuates and decreases with fiber length. This is because for the calculated Q-factor, the approximated mm-wave current, i.e. only the major beating between the two second-order harmonics, was used and the contributions from the other beatings are small in magnitude and ignored. The other reason is that walk-off between the two second-order optical harmonics induced by fiber chromatic dispersion is not included in the calculation. Moreover Technique-C leads to the best performance in the three modulation techniques. The critical reason for that is the optical carrier using Technique-C is almost suppressed. The higher optical carrier will contribute higher shot noise, RIN impact and signal-ASE beat noise and also beat with the fourth-order harmonic sidebands due to non-perfect optical filtering, and this beating results in interference, which degrades the system performance. It is seen that Technique-A results in a serious fluctuation with fiber length due to chromatic dispersion as shown in Eq. (3). Figure 3 clearly shows that the developed theoretical analysis is accurate for the evaluation of RoF systems in which a frequency quadrupler is used for the mm-wave generation.

## 4. Analysis by simulation and comparison to experiments in generated mm-wave power

In this section, we investigate the impact of modulation index, optical filtering, modulation frequency, and MZM extinction ratio on the generated mm-wave by using simulation. Then we verify the frequency quadrupler using Technique-C by experiments.

We first consider the impact of modulation index for using the three modulation techniques. For Technique-A, Fig. 4(a) shows simulated Q-factor versus fiber length with modulation indexes of *m _{RF}*=25, 50, 76.5 and 100%. It is shown that the Q-factor is poor for the back-to-back system and gradually increases with fiber length of up to ~16.53 km, and then decreases gradually with the increase of fiber length to around 33 km. This is due to the term of cos(4

*β*

_{2}

*Lω*

^{2}

_{RF}) as shown in Eq. (3), representing the impact of chromatic dispersion, which introduces constructive and destructive interactions. At the fiber length of ~16.53 km, we obtain cos(4

*β*

_{2}

*Lω*

^{2}

_{RF})=-1, and thus the mm-wave current at 30 GHz is maximized due to constructive interaction. Similarly at ~33 km, we obtain cos(4

*β*

_{2}

*Lω*

^{2}

_{RF})=1 and thus the mm-wave current at 30 GHz is minimized due to destructive interaction. These constructive and destructive interactions between the two contributions are periodic due to the term of cos(4

*β*

_{2}

*Lω*

^{2}

_{RF}). It is seen that the Q factor is not significantly increased by increasing modulation index. This is because the two contributions with the opposite sign are both increased by increasing modulation index, and thus the mm-wave current does not increase significantly due to the destructive interaction as seen in Eq. (3). A large modulation index may results in a better performance but only at certain fiber lengths where the two contributions interact constructively.

For Technique-B and Technique-C, Fig. 4(b) and 4(c) shows simulated Q-factor versus fiber length, respectively, and clearly indicates that the system performance is greatly improved and does not fluctuate with fiber length compared to Fig. 4(a), and also the Q-factor is enhanced significantly with the increase of modulation index. This is because the spectral component coming from the beating between the first (∓7.5 GHz) and third (∓22.5 GHz) harmonics is strongly reduced (see Eqs. (6) and (9), respectively). Also, the spectral component generated from the beating between the optical carrier and fourth order harmonic (∓30 GHz) can be neglected due to optical filtering. Thus, the mm-wave at 30 GHz is mainly due to the beating of the optical harmonics at ∓2*ω _{RF}* so that constructive and destructive interaction hardly occurs. Thus the generated mm-wave power is almost independent from the fiber dispersion caused fading as indicated by Eqs. (6) and (9). It is seen that with the increase of fiber length, the Q-factor slowly decreases when modulation index of ≥76.5% is used. The reason is the walk-off time due to fiber chromatic dispersion between any two optical harmonics, and the walk-off was ignored in the above analysis and will be discussed later. In addition, Technique-C outperforms Technique-B at all modulation indexes because the optical carrier is highly suppressed in Technique-C than in Technique-B as shown in Fig. 2(a) and for the same reasons stated above in Section 3.

Now we investigate the impact of optical filtering on the generated mm-wave. The modulation index is fixed at 76.5%. Figure 5(a) shows comparison of simulated Q-factor versus fiber length for the three modulation techniques without using optical filtering. It is clearly seen that the Q-factor is very poor over all fiber length for all the three modulation techniques. Because of no optical filtering there are too many beatings between all optical harmonics, and the beatings may be destructive and constructive, e.g. fiber dispersion caused fading (as seen from Fig. 5(c)), to the 30-GHz mm-wave signal for Technique-A. However, Technique-B and Technique-C hardly suffer from dispersion caused fading (as seen from Fig. 5(c)) but mainly from data distortion due to data interference from others components falling within the same mm-wave frequency. The optical filter bandwidth has also an impact as shown in Fig. 5(b), where we compare two optical filters, i.e. one with a bandwidth of 40 GHz and the other with 60 GHz. Compared to the 60-GHz optical filter, it is seen that the 40-GHz optical filter leads to an improvement of the Q-factor for Technique-A and Technique-B, but for Technique-C the improvement is very little. For the 40-GHz optical filtering, the fourth-order optical harmonics and beyond are removed and on the contrary, for the 60 GHz optical filtering the fourth-order optical harmonics are not removed. Thus, for the 60 GHz optical filtering the interaction of the optical carrier and fourth-order optical harmonics will occur, and result in data distortion for Technique-A and Technique-B. However, for Technique-C the interaction is very small because the optical carrier is suppressed as mentioned before (see Fig. 2(a)) if a modulation index of 76.5% is used.

In the above we always consider an RF at 7.5 GHz and mm-wave at 30 GHz, i.e. a frequency quadrupler. Now we investigate the effect of the local oscillator RF on the power of generated mm-wave signal. Figure 6 shows the power variation of the generated mm-wave signal at 30 GHz as a function of fiber length with the local oscillator RF of 5, 7.5, 10 and 15 GHz for the three modulation techniques, corresponding to a frequency sextupler, quadrupler, tripler and doubler, respectively. In Fig. 6, we used a modulation index of 76.5% and an optical filter of 40 GHz bandwidth. For Technique-A, Fig. 6(a) shows that the fluctuation of the mm-wave power with fiber length is reduced with the increase of RF. When RF of 15 GHz is used, i.e. a frequency doubler, the fluctuation is still not negligible. It is seen from Fig. 6(b) and 6(c) for Technique-B and Technique-C that the mm-wave power with fiber length may or may not fluctuate, depending on the RF, and an RF at 7.5 GHz is the best for the generation of 30 GHz mm-wave signal with the highest power level and lowest fluctuation with fiber length. In other words, Technique-B and Technique-C are better for a frequency quadrupling, than a doubling, tripling, and sextupling. This confirms that maximum transmission biased MZM with push-pull drive is efficient for a frequency quadrupler, as reported in [28-29]. The variation of the power level as a function of fiber length follows a trend similar to the Q factor of the system as shown in Fig. 4. Technique-C outperforms Technique-A and Technique-B in terms of produced mm-wave power level, while Technique-A presents the worst performance in terms of transmission stability over fiber length. However, Technique-A outperforms Technique-B and Technique-C for a frequency doubler, such as using *f _{RF}*=15 GHz for 30-GHz generation by the beating of first-order harmonics at ±15 GHz.

In the above analysis, a constant extinction ratio of MZM is used. It is known that optical carrier suppression is impacted by MZM extinction ratio and thus the quality of mm-wave generation may be also affected. To further show the performance of the three modulation techniques, impacted by the extinction ratio of the MZMs, we consider an MZM with an extinction ratio varying from 10 to 35 dB. The RF modulation index and fiber length are fixed at 76.5% and 25 km, respectively. Figure 7 shows the simulated Q-factor as a function of extinction ratio. It is seen from Fig. 7 that the Q-factor, for all three techniques, is almost independent from the extinction ratio if more than 20 dB, which is common in commercially available MZMs. Moreover, Technique-C leads to the best performance for high extinction ratios.

A further comparison of Technique-C and Technique-B is shown in Fig. 8. We consider fiber transmission of 0 and 25 km for comparison of receiver sensitivity. Figure 8 shows simulated receiver sensitivity versus bit error rate (BER). Technique-C leads to an improvement of 7.5 dB in receiver sensitivity at BER=10^{-9} for both back-to-back and 25 km fiber transmission compared to Technique-B. This is mainly due to the higher optical power level of the second-order optical components (±2ω_{RF}) and high suppression of optical carrier using Technique-C. It is seen that fiber dispersion has negligible impact on the generated mm-wave signals for both Technique-B and Technique-C.

The mm-wave generation using Technique-C was investigated experimentally in our previous work [28]. For Technique-C without signal modulation, Fig. 9(a), 9(b) and 9(c) show the measured electrical spectra of the generated mm-wave at 30 GHz after transmission over a 10, 14, and 24 km of single mode fiber, respectively, using a 7.5-GHz local oscillator and a modulation index of 45% with the same parameters used in [28]. In Fig. 10, the measured power level of the generated mm-wave signal at 30 GHz, depicted in Fig. 9(a), 9(b) and 9(c), is plotted with simulated results versus the corresponding fiber length. It can be seen that the power of the generated mm-wave does not suffer from chromatic dispersion but only fiber losses. The simulation shows a good agreement with the measured results. A small discrepancy is due to additional insertion loss in the experiment. This confirms that our analysis is accurate.

## 5. Discussion

In this section, we investigate limitation of the generated mm-wave using the optical frequency quadrupling technique. As an example, we consider Technique-B and Technique-C for the generated 30-GHz mm-wave using an RF at 7.5 GHz, i.e. a frequency quadrupler. Figure 11 shows the simulated Q factor vs. fiber length for Technique-B and Technique-C. The modulation index was set to 76.5%. It is predicted that the Q-factor decreases when fiber transmission distance increases. One reason is due to propagation delay (walk-off) between the two optical harmonics at ±15 GHz. The pulse width of 622 Mb/s signal carried by the optical mm-wave is approximately 1.6 ns. The two optical harmonics with a frequency spacing of Δ*f*=30 GHz will have a walk–off time of Δ*τ*=*DL*Δ*λ*=*DLλ*
^{2}Δ*f*/*c*. This walk-off leads to a pulse width broadening and then results in intersymbol interference and pulse peak power reduction, which degrades the Q-factor and the eye pattern. Comparison of eye pattern at back-to back, 50 km, 100 km and 400 km is shown in Fig. 12. At a fiber length of ~400 km where Δ*τ*=~1.6 ns, i.e. equal to the bit period of the baseband signal, a strong interference is observed in the eye pattern at 400 km as shown in Fig. 12. Therefore the generated mm-wave is limited by fiber chromatic dispersion. The limit of an mm-wave over fiber transmission is that the walk-off must be less than a bit period of the baseband signal transmitted.

In the above we assumed that the impact of fiber PMD is negligible. This assumption is true only when fiber length is short. However, if fiber length is more than 100 km, PMD impact may not be negligible. So the fiber PMD will impose a limit to the generated mm-wave and transmission. The PMD induces polarization state walk-off between the two optical harmonics that are used for the generation of the mm-wave. This walk-off will introduce power fading of the generated mm-wave, and the fading of the mm-wave power is proportional to cos(Δ*τ*·2*π*Δ*f*/2), where Δ*τ*- differential group dispersion (DGD) of the fiber link and Δ*f*=30 GHz. Consider a fiber having a DGD of
$0.1\phantom{\rule{.2em}{0ex}}\frac{\mathit{ps}}{\sqrt{\mathit{km}}}$, and thus polarization walk-off is approximately 3^{0} for 100 km fiber link and 12^{0} for 400 km fiber link. Therefore, fiber PMD also limits the transmission for the generated mm-wave signals.

Another disadvantage of using the optical frequency quadrupling technique is that the generated mm-wave has phase noise that is multiplied by the multiplication factor of the frequency with the phase noise of the RF local oscillator. For example, the generated 30-GHz mm-wave using a frequency quadrupler has phase noise that is four times of the RF local oscillator’s phase noise.

## 6. Conclusion

We have comprehensively investigated three optical modulation techniques for generating mm-wave signal using an MZM to obtain a frequency quadrupler. We first completed theoretical analysis of the mm-wave generation for the three modulation techniques, and the explicit expressions of the mm-wave current and noise are given and can be used for evaluation of RoF system performance such as Q-factor. Then the performance of three modulation techniques using simulation was investigated, considering the impact of modulation index, extinction ratio, optical filtering and modulation frequency as well as the fiber chromatic dispersion. It is found that maximum conversion efficiency is achieved at an RF modulation index of 76.5%, and 107% for Technique-C and Technique-B, and Technique-A, respectively, and the quality of the generated mm-wave is limited by destructive interaction and nonlinear distortion induced interference for Technique-A. Whereas Technique-B and Technique-C suffer from little dispersion caused fading or destructive interaction. By using optical filtering, the quality of generated mm-wave is significantly improved. We have considered an MZM driven by an RF at 7.5 GHz, used for 30-GHz mm-wave generation. For the 30-GHz mm-wave generation, it is found that an optical filter with a bandwidth of ~40 GHz is required to suppress fourth-order optical harmonics and beyond. Moreover Technique-B and Technique-C for mm-wave generation are found better for frequency quadrupling than frequency doubling, tripling and sextupling. Furthermore, the generated mm-wave using those three techniques and transmission over fiber is ultimately limited by chromatic dispersion induced walk-off and PMD fading.

For Technique-A, the mm-wave power was found to fluctuate with fiber length due to constructive and destructive interaction between the two beatings induced by chromatic dispersion, one beating between the two second-order harmonics and the other beating between the third- and first- order harmonics. For Technique-B and Technique-C the generated mm-wave power hardly fluctuates with fiber length, i.e. robust to fiber chromatic dispersion. By comparison to Technique-B, Technique-C results in 7.5 dB improvement of receiver sensitivity. We found that Technique-C is the best in the quality of the generated mm-wave, especially when poor optical filtering is used.

## 7. Appendix

In this appendix we present analytical expressions for calculating Q-factor for the mm-wave signals, which are generated using the three modulation techniques, over fiber transmission.

After transmission over SMF link, the generated photocurrent at the receiver can be expressed as *i*(*t*)=ℜ|*S*(*t*)+*N*(*t*)|^{2}, where *S*(*t*)=√*G*(*h*
_{o}(*t*)⊗*E _{s}*(

*t*)) and

*N*(

*t*)=

*h*(

_{o}*t*)⊗

*E*(

_{ASE}*t*) represent the incident optical fields of the mm-wave signal and noise at the photodetector, respectively.

*E*(

_{s}*t*) represents the optical field of the signal at the input of the optical amplifier,

*h*(

_{o}*τ*)=

*h*(

_{f}*τ*)⊗

*h*(

_{F}*τ*),

*h*(

_{F}*H*(

_{F}*ω*)) and

*h*(

_{f}*H*(

_{f}*ω*)) are the impulse response (transfer function) of the optical filter and linear fiber link, respectively. The symbol ⊗ denotes the convolution.

*G*and

*E*(

_{ASE}*t*) represent the EDFA gain and ASE noise field. The optical transfer function of the optical fiber can be expressed as ${H}_{f}\left(\omega \right)={e}^{-\frac{1}{2}\alpha L\phantom{\rule{.2em}{0ex}}}{e}^{j{\beta}_{1}L\omega}{e}^{j\frac{1}{2}{\beta}_{2}L{\omega}^{2}}$ , where

*L*,

*α*, 1/

*β*

_{1}, and

*β*

_{2}=-

*λ*

^{2}

*D*/2

_{c}*πc*are fiber length, loss, group velocity and chromatic dispersion, respectively,

*D*is the fiber dispersion parameter expressed in unit ps/(km·nm), λ is the wavelength of the light source, and

_{c}*c*is the speed of light in vacuum. The photocurrent may contain many generated spectral components, but only the electrical spectral component at 4

*ω*is considered the radio signal carrier, i.e. a quadrupler. The spectral component at 4

_{RF}*ω*is selected by means of the electrical bandpass filter with an impulse response

_{RF}*h*(

_{e}*τ*) (transfer function

*H*(

_{e}*ω*) centered at 4

*ω*) as shown in Fig. 1(d). The desired spectral component at 4

_{RF}*ω*is down-converted to a baseband signal and filtered by a low passband filter with an impulse response

_{RF}*h*(

_{LP}*t*) (transfer function

*H*(

_{LP}*ω*)). The baseband electrical current

*I*(

_{bb}*t*) contains the distributed signal i.e.

*I*(

_{u}*t*), signal-ASE beating noise, i.e.

*I*

_{S*×N}(

*t*), and ASE-ASE beating noise, i.e.

*I*

_{N×N*}(

*t*) as well as electrical noise

*I*(

_{elec}*t*). Thus, the total current at the output of the optical receiver is given by

where the factor A is a constant and A=0.25 is assumed, *ω _{o}*=2

*πf*

_{o}=4

*ω*is the angular frequency of the local oscillator as shown in Fig. 1(d), and Re(.) and (*) denote the real part and conjugate of complex. The mean and variance of the baseband electrical current can be expressed as

_{RF}where 〈·〉 denotes the statistical mean, *σ*
^{2}
_{X}(*t*) is the variance of a stochastic process *X*(*t*) at an instant t, and *σ*
^{2}
_{elec} is the noise contribution from both optical transmitter and receiver electronic noise, and this is called “back-to-back” system noise (shot noise, thermal noise, and RIN). In the following, we assume that all the filtering does not introduce distortion on the distributed signal. The quantities defined above at an instant t can be expressed as

$$=\frac{1}{2}{A}^{2}{\Re}^{2}G{N}_{o}{B}_{e}{e}^{-2\alpha L}[\left({\mid {T}_{0}\left(d\left(t\right)\right)\mid}^{2}+{\mid {T}_{4}\left(d\left(t\right)\right)\mid}^{2}\right)$$

$$\times {\mid {H}_{F}\left(0\right)\mid}^{2}{\mid {H}_{F}\left(4{\omega}_{\mathit{RF}}\right)\mid}^{2}+{\mid {T}_{0}\left(d\left(t\right)\right)\mid}^{2}{\mid {H}_{F}\left(2{\omega}_{\mathit{RF}}\right)\mid}^{4}$$

$$+({\mid {T}_{1}\left(d\left(t\right)\right)\mid}^{2}+{\mid {T}_{3}\left(d\left(t\right)\right)\mid}^{2}){\mid {H}_{F}\left({\omega}_{\mathit{RF}}\right)\mid}^{2}{\mid {H}_{F}\left(3{\omega}_{\mathit{RF}}\right)\mid}^{2}]\times \frac{\underset{-\infty}{\overset{\infty}{\int}}{\mid {H}_{\mathit{LP}}\left(\omega \right)\mid}^{2}{\mid {H}_{e}\left(\omega +{\omega}_{o}\right)\mid}^{2}d\omega}{\underset{-\infty}{\overset{\infty}{\int}}{\mid {H}_{\mathit{LP}}\left(\omega \right)\mid}^{2}d\omega}$$

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}+{B}_{e}\frac{\underset{-\infty}{\overset{\infty}{\int}}{\mid {H}_{\mathit{LP}}\left(\omega \right)\mid}^{2}{\mid {H}_{e}\left(\omega +{\omega}_{o}\right)\mid}^{2}d\omega}{\underset{-\infty}{\overset{\infty}{\int}}{\mid {H}_{\mathit{LP}}\left(\omega \right)\mid}^{2}d\omega}\times \frac{1}{2}{\mathit{qA}}^{2}\{2\Re \phantom{\rule{.2em}{0ex}}\mathrm{Re}\left[{I}_{4}\left(d\left(t\right)\right)\right]+\Re {N}_{o}{B}_{o}{e}^{-\alpha L}+{I}_{d}\}$$

where
${I}_{4{\omega}_{\mathit{RF}}}$
is the photodetected current component at 4*ω _{RF}*, M is the nearest integer to

*B*/

_{F}*f*, and 2

_{RF}*B*is the 3-dB bandwidth of the optical filter (transfer function

_{F}*H*(

_{F}*ω*)). ${B}_{o}=\frac{1}{2\pi}{\mathrm{\int}}_{-\infty}^{\infty}{\mid {H}_{F}\left(\omega \right)\mid}^{2}d\omega $ and ${B}_{e}=\frac{1}{2\pi}{\int}_{-\infty}^{\infty}{\mid {H}_{\mathit{LP}}\left(\omega \right)\mid}^{2}d\omega $ stand for the equivalent noise bandwidth of the optical filter and low bandpass electrical filter, respectively. q is the electron charge and

*N*/2 represents the single sided ASE noise density for a single polarization expressed by

_{o}*N*/2=

_{o}*Fhc*(

*G*-1)/(2

*λ*), where F is noise figure of the EDFA, and

*h*is the Plank’s constant.

*T*(

_{n}*d*(

*t*)) depends on the modulation techniques and given for Technique-A, Technique-B and Technique-C, respectively as:

and

The Q factor, which is related to bit error rate by $\mathit{BER}=\frac{1}{2}\phantom{\rule{.2em}{0ex}}\mathit{erfc}\left(\frac{Q}{\sqrt{2}}\right)$ , can be written as

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