## Abstract

In this work, we propose and investigate a novel technique for the generation of millimeter-wave (mm-wave), i.e. frequency sixuplexing technique. The proposed technique is comprised of two cascaded Mach-Zehnder modulators (MZMs). The first MZM, biased at maximum transmission, is only used for even-order optical harmonic generation, and then a second MZM, biased at minimum transmission, is used for both optical carrier suppression modulation and data signal modulation. As an example, we consider an RF at 10 GHz, which carries the data signal and drives the MZMs; and an mm-wave signal at 60 GHz, i.e. a frequency sixupler, is obtained. It is found that our proposed sixupler leads to an 8-dB higher RF power at 60 GHz and a 6-dB improvement in receiver sensitivity with comparison to the conventional technique, i.e. optical carrier suppression modulation. The generated mm-wave signal is robust to fiber chromatic dispersion. The proposed technique is verified by experiments.

©2008 Optical Society of America

## 1. Introduction

Due to limited frequency resources and to avoid spectral congestion at low microwave frequencies, millimeter-wave (mm-wave) frequency range of 26~65 GHz has been considered to be used for the future wireless access networks. Photonic mm-wave generation and distribution over single mode fiber (SMF) link has become a potential solution for wireless communications. In other words, radio-over-fiber technique has been considered a cost-effective and reliable solution for the distribution of future wireless access networks. As a consequence, many research works related to radio over fiber systems have been intensively investigated, such as photonic millimeter-wave generation and distribution, radio-over-fiber systems integrated with dense wavelength division multiplexing, optical frequency up-conversion, frequency comb generation and suppression of nonlinear distortion, etc [1–11]. The principle of optical frequency up-conversion is to generate optically high-order optical harmonics using an electrical to optical converter, such as Mach-Zehnder modulators (MZMs) driven by an RF signal. Due to nonlinear response of the electrical to optical converter, many high-order optical harmonics are generated in the optical domain. Beating of two high-order optical harmonics or/and beating of high-order optical harmonics with the optical carrier by photodetection will generate mm-waves [8]. Optical frequency up-conversion technique has potential to provide a cost-effective solution for optically generating mm-waves and delivering wireless signals to remote antennas.

In order to obtain high quality mm-wave generation using frequency up-conversion, it is required that desired optical harmonics are maximized and non-desired optical harmonics are suppressed, so that high up-conversion efficiency is obtained with low nonlinear distortion impact. Recently, a frequency quadrupler for mm-wave signal generation using optical carrier suppression modulation was demonstrated [9]. In this technique two cascaded MZMs were used, and both MZMs are biased at minimum transmission with 90° phase shift between the driving RF signals. Thus, the two second-order optical harmonics are maximized. Therefore, this modulation technique is used better for a frequency quadrupler other than a frequency sixupler.

Basically high-order optical harmonics can be generated using either a single MZM driven by a RF sinusoid, or using more complex techniques in [7, 9], and frequency comb generation technique in [11]. However using the above techniques for a frequency sixupler, not only the frequency up-conversion efficiency is very low but also the mm-wave signal generated suffers from fluctuation in RF power due to fiber chromatic dispersion and nonlinear distortion.

In this work, we propose and investigate a new frequency sixupler for mm-wave generation using two dual-electrode MZMs, one biased at *maximum* transmission and the other biased at *minimum* transmission and performing signal modulation. Two optical sidebands of third-order optical harmonics related to the optical carrier are generated and maximized after the second MZM, and the two sidebands have a frequency spacing of six times of the driving RF to the MZMs.

As an example, we investigate the generation of 60 GHz mm-wave using an RF at 10 GHz, which drives the MZMs. We compare the system performance using this proposed modulation technique with optical carrier suppression modulation technique, referred to the conventional technique in this paper, to generate mm-wave signals. The impact of RF modulation index on generated mm-wave power and systems performance is also investigated and compared. Finally six bias conditions for the two MZMs are compared in generated mm-wave. A conceptual proof is given experimentally.

## 2. Proposed modulation technique for a frequency sixupler

The proposed modulation technique or a frequency sixupler for mm-wave generation is shown schematically in Fig. 1. A continuous wave (CW) laser is assumed to have a wavelength of λ=1552.52 nm, an output optical power of *P*=0 dBm, a linewidth of 10 MHz, and relative intensity noise of -140 dB/Hz. A sinusoidal RF signal has a frequency of *f _{RF}*=10 GHz and voltage of

*V*(

_{RF}*t*)=√2

*V*cos(

_{RF}*ω*+

_{RF}t*ϕ*), where ω

_{RF}*=2π*

_{RF}*f*,

_{RF}*ϕ*- random phase noise of the RF oscillator and √2 is to consider the 3-dB electrical power splitter before the two MZMs. The total RF modulation index for the mm-wave generation is defined by

_{RF}*m*=

_{RF}*V*/

_{RF}*V*, where

_{π}*V*is the modulator’s switching voltage. Two MZMs with a DC extinction ratio of 30 dB and

_{π}*V*=5V are assumed. The insertion loss of each MZM is assumed 6 dB. The CW light is injected to the first MZM, biased at maximum transmission. The dual electrodes are driven by the same RF sinusoid with 180° phase shift. Thus, only even-order optical harmonics are generated whereas all odd-order optical harmonics are suppressed ideally, as seen in Fig. 1 inset (i). Then two coherent optical subcarriers at ±2ω

_{π}_{RF}are modulated in the second MZM by the same RF signal mixed with a baseband data signal at 625 Mb/s. The second intensity MZM modulator is biased at minimum transmission to suppress the optical carrier and even order optical side-bands or subcarriers. As a result, six optical side-bands are generated at ±ω

_{RF}, ±3ω

_{RF}and ±5ω

_{RF}at least as shown in Fig. 1 inset (ii). Two of them (at ±3ω

_{RF}) have a frequency difference that is six times of the frequency of the driving RF signal (ω

_{RF}), i.e. a frequency sixupler. Due to the nonlinear transfer function of the MZM, many high-order optical harmonics can be generated if a high RF modulation index is used, as shown in Fig. 1 inset (ii). Therefore, an optical band-pass filter (OBF) with a bandwidth of 68 GHz (Gaussian of order 3) is used (zero insertion loss assumed). Consequently, the fifth-order optical harmonics and beyond are suppressed, as shown in Fig. 1 inset (iii). The generated mm-wave signal at 60 GHz is due to only the beating of two third-order optical harmonics at ±30 GHz by photodetection. Thus, we can maximize the power level at ±3ω

_{RF}by optimizing the RF modulation index. For example, when

*m*is equal to ~85% (i.e. ~60% for each MZM), the optical spectral components at ±10 GHz, at the output of the second MZM, is further suppressed and the power level of the spectral components at ±30 GHz is maximized. The two third-order optical sidebands situated at ±30 GHz is ~36 dB higher than the two optical sidebands located at ±10 GHz, as shown in Fig. 1 inset (iii).

_{RF}In the conventional technique, the CW light is input to one MZM, biased at minimum transmission and dual-electrodes driven by the same RF signal with 180° of phase shift, i.e. optical carrier suppression modulation [4, 9]. So, only odd-order optical harmonics are generated. Minimum transmission bias (MITB) for the generation of mm-wave at 2ω_{RF} and 6ω_{RF} outperforms the maximum transmission bias (MATB) and quadrature transmission bias (QTB) for any fiber length, as we have reported in [7]. This is the reason why the MZM is biased at minimum transmission.

Then the output signal is amplified by an erbium doped fiber amplifier (EDFA), which is used for compensation for insertion loss of optical modulators and fiber. The EDFA has a noise figure of 5 dB. To remove non-desired optical harmonics and reduce the amplifier noise, an optical filter is used after the EDFA, and the filter has a bandwidth of 68 GHz centered at 1552.52 nm. Finally all the optical harmonics that pass the OBF are transmitted over a single mode fiber, with fiber loss of α=0.23 dB/km, and chromatic dispersion of *D*=16.7 *ps/(nm.km)*. At a base station, all transmitted optical harmonics are detected by a photodiode (with responsivity ℜ=0.7 A/W at 1552.52-nm, spectral density of thermal noise
${N}_{\mathrm{th}}=\frac{2\times {10}^{-11}A}{\sqrt{\mathrm{Hz}}}$
and dark current I_{d}=2 nA). After photodetection, a generated mm-wave signal at 60 GHz is obtained due to only the beating of two third-order optical harmonics at ±30 GHz. To remove other undesired RF carriers or spectral components, an electrical bandpass filter (Bessel of order 3) centered at 60 GHz with a bandwidth of 1.56 GHz is used to recover the mm-wave signal at 60 GHz. For our analysis, the mm-wave signal at 60 GHz is down-converted directly into the baseband signal for radio over fiber system evaluation. To obtain the baseband signal from the mm-wave signal at 60 GHz, an electrical mixer driven by a sinusoidal RF at 60 GHz and a low pass band filter (Bessel of order 3) with a bandwidth of 560 MHz are used, as shown in Fig. 1. All insertion loss of electrical components such as the mixer and filters in Fig. 1 is ignored in analysis.

## 3. Theoretical analysis

The output optical field after the first MZM can be written as

where *ω*
_{0} denotes the optical angular frequency of the light and *P* denotes optical power injected to the first MZM. The output optical field after the second MZM can be written as

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\times \sum _{n=-2}^{2}{j}^{n}{J}_{n}\left(\psi \right){\left(-1\right)}^{n+1}\left({\gamma}_{1}+{\left(-1\right)}^{n}\right){e}^{j\left({\omega}_{0}+n{\omega}_{\mathrm{RF}}\right)t}\sum _{k=-\infty}^{\infty}{j}^{k}{J}_{k}\left(\zeta \right)\left({\gamma}_{2}+{\left(-1\right)}^{k+1}\right){e}^{jk{\omega}_{\mathrm{RF}}t}$$

where *ζ*=*πd*(*t*)*m _{RF}*/√2 and

*ψ*=

*πm*/√2. ${t}_{{\mathrm{ff}}_{1}}$ and ${t}_{{\mathrm{ff}}_{2}}$ are the insertion loss of the two MZMs,

_{RF}*d*(

*t*)∈(0,1) is digital signal, and ${\gamma}_{k}=\frac{\left(\sqrt{{\epsilon}_{k}}-1\right)}{\left(\sqrt{{\epsilon}_{k}}+1\right)}$ ,

*ε*- optical extinction ratio of the two MZMs,

_{k}*k*=1 and 2, i.e. DC extinction ratio (determined by Y-junction splitting ratio). The output optical field after the optical filter of transfer function

*H*(·) can be written

_{F}where c.c. stands for complex conjugate and

The incident optical field to the photodetector is given by

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\times \left\{\begin{array}{c}\left({A}_{1}d\left(t+\frac{1}{6}\Delta \tau \right){e}^{j\left[{\omega}_{\mathrm{RF}}\left(t+{\beta}_{1}L\right)+{\varphi}_{\mathrm{RF}}\right]}+{A}_{1}^{*}d\left(t-\frac{1}{6}\Delta \tau \right){e}^{-j\left[{\omega}_{\mathrm{RF}}\left(t+{\beta}_{1}L\right)+{\varphi}_{\mathrm{RF}}\right]}\right){e}^{j\frac{1}{2}{\beta}_{2}L{\omega}_{\mathrm{RF}}^{2}}\\ \phantom{\rule{.2em}{0ex}}+\left({A}_{3}d\left(t+\frac{1}{2}\Delta \tau \right){e}^{j3\left[{\omega}_{\mathrm{RF}}\left(t+{\beta}_{1}L\right)+{\varphi}_{\mathrm{RF}}\right]}+{A}_{3}^{*}d\left(t-\frac{1}{2}\Delta \tau \right){e}^{-j3\left[{\omega}_{\mathrm{RF}}\left(t+{\beta}_{1}L\right)+{\varphi}_{\mathrm{RF}}\right]}\right){e}^{j\frac{1}{2}{\beta}_{2}L{\left(3{\omega}_{\mathrm{RF}}\right)}^{2}}\end{array}\right\}$$

where 1/*β*
_{1}, and *β*
_{2}=-*λ*
^{2}
*D*/2*πc* are the group velocity and chromatic dispersion of the fiber, respectively. *L* and *G* are fiber length and gain of the optical amplifier, respectively. *α* is the fiber loss. The desired spectral component of photocurrent at the frequency of 6ω_{RF} is expressed by

where Δ*τ*=*β*
_{1}
*L* indicates walk-off due to fiber chromatic dispersion. If ignoring the walk-off impact, the above expression is simplified to

where A_{3} is given by Eq. (3b). Using Eq. (3) the average optical power ratio of third-order to first order optical harmonics is given by

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\times \frac{{\left[\epsilon {J}_{0}\left(\psi \right){J}_{3}\left(\psi \right)+\left(\epsilon -1\right){J}_{1}\left(\psi \right){J}_{2}\left(\psi \right)\right]}^{2}}{{\left[{J}_{2}\left(\psi \right)\left(\left(\epsilon -1\right){J}_{1}\left(\psi \right)-\epsilon {J}_{3}\left(\psi \right)\right)-\epsilon {J}_{0}\left(\psi \right){J}_{1}\left(\psi \right)+\sqrt{\epsilon}{J}_{1}^{2}\left(\psi \right)\right]}^{2}}\approx 35.7\mathrm{dB}$$

for a modulation index of 85%, where *γ*=*γ*
_{1}=*γ*
_{2}=0.939 is assumed. It is seen that the optical power of the third-order optical harmonics is ~36 dB higher than that of the first order optical harmonics. However, the optical power of the third-order optical harmonics is ~5.2 dB lower than that of the first-order optical harmonics for using the conventional technique, the details given in Appendix A. Therefore, the proposed technique will results in a much higher mm-wave power for the same RF oscillator power compared to the conventional technique.

## 4. Results and discussion

Using the parameters given in the above and VPI-TransmissionMaker, we simulate the system as in Fig. 1. Using the proposed modulation technique for mm-wave generation, Fig. 2 shows simulated Q-factor versus fiber length with a modulation index of 70%. Note this modulation index is used for the two MZMs, i.e. total modulation index. For each MZM of the two MZMs, the modulation index is divided by √2. This is because in the conventional technique only one MZM is used, and for fair comparison the total modulation index should be the same so that input RF power driving one MZM for the conventional technique or two MZMs for the proposed technique is the same. For comparison, simulated Q using the conventional modulation technique is also shown in Fig. 2. It is clearly shown that the proposed technique leads to much better performance compared to the conventional technique. This is because the generated mm-wave power is higher and the undesired optical components are more greatly suppressed, which leads to better receiver sensitivity, lower crosstalk and higher Q-factor. Also, the performance is hardly degraded as fiber length increases for the both techniques. This is attributed to the generated mm-wave power that is almost independent of fiber chromatic dispersion as shown by Eq. (5). The simulated Q is slightly degraded with increase of fiber length. The physical reason is due to walk-off. The two optical sidebands at ±3ω_{RF}, both carrying the same signal, have a frequency spacing of Δf=60 GHz, which produces a walk–off time of Δ*τ*=*DL*Δ*λ*=*DLλ*
^{2}Δ*f*/*c*, *c*-light speed. This walk-off will induce pulse width broadening and result in inter-symbol interference (ISI), pulse peak power reduction and thus Q-factor degradation.

Figure 3 shows comparison of simulated Q-factor and mm-wave power vs. modulation index using the proposed technique to the conventional technique, where the signal is transmitted over 25-km of fiber length. In the both techniques, the generated mm-wave signal at 60 GHz is due to only the beating of the two optical subcarriers at ±3ω_{RF}. However, the proposed technique results in higher power level of the generated mm-wave signal than the conventional technique for modulation indexes of up to 86%, as shown in Fig. 3(b). This is the major reason why the proposed technique outperforms the conventional technique for modulation indexes of up to 86% as shown in Fig. 3(a). A Q-factor of 6 can be achieved by using a modulation index of 60% and 74% at least using the proposed and conventional techniques, respectively. This suggests that a smaller RF power is required to achieve the same performance using the proposed technique, i.e. cost-effective. Moreover, it is seen that the generated mm-wave power level using the proposed technique is 8 dB higher than that using the conventional technique (see Fig. 3(b)). This corresponds to 4-dB reduction in optical power. Note the optimum modulation index of 86% for the two MZMs corresponds to modulation index of ~61% for each MZM of the two MZMs.

In theory, electric field amplitude of first order harmonics is given by *E*
_{1}∝*J*
_{2}(*πm*
_{1}) (*J*
_{1}(*πm*
_{2})-*J*
_{3}(*πm*
_{2}))-*J*
_{0}(*πm*
_{1})*J*
_{1}(*πm*
_{2}), and is almost zero for *m*
_{1}=0.6119 and *m*
_{2}=0.6077 of the two MZMs. Thus the beating between first-order and fifth order harmonics is greatly suppressed. On the contrary, the amplitude of third order harmonics is given by *E*
_{3}∝*J*
_{2}(*πm*
_{1})*J*
_{1}(*πm*
_{2})+*J*
_{0}(*πm*
_{1})*J*
_{3}(*πm*
_{2}), which indicates that the two contribution terms always sum up. Thus the optimum modulation index for each MZM is ~61%, and the optimum total modulation index is ~86%.

To further show the advantages of the proposed technique, simulated RF spectra and eye diagrams are shown in Fig. 4(a) and (b) for transmission distances of 0-km (back-to-back), and 50 km, respectively. It is clear from Fig. 4 that the power level of 60 GHz signal was constant at ~-42 dBm for fiber lengths of 0 and 50-km, namely no fading induced by chromatic dispersion. Also, the eye opening before and after transmission is almost identical, implying robustness to fiber dispersion again.

We consider fiber transmission of 0 and 25 km for comparison of receiver sensitivity. Comparison of the proposed technique with the conventional technique is shown in Fig. 5 in simulated receiver sensitivity. The proposed technique leads to an improvement of 6 dB in receiver sensitivity at BER=10^{-9} compared to the conventional technique This is mainly due to the higher optical power level of the third-order optical components (±3ω_{RF}) and suppression of first-order optical components (±ω_{RF}) using the proposed technique. It is seen again that fiber dispersion has negligible impact on the generated mm-wave signals.

To further show the performance of the proposed frequency sixupler, we vary the driving RF to the MZMs from 5 to 15 GHz. Thus the generated mm-wave will be varied from 30 to 90 GHz. The optical filter bandwidth is adjusted accordingly to remove high-order optical harmonics of beyond third order. The RF modulation index is fixed at 70%. Figures 6(a) and (b) shows the simulated Q-factor and the power level of the generated mm-wave signal as a function of the local oscillator RF for fiber lengths of 0 and 25 km using the proposed and conventional techniques. It is seen from Fig. 6(a) and (b) using both techniques that the Q-factor and generated mm-wave power are independent of the local oscillator RF. However, it is shown that the proposed modulation technique leads to higher mm-wave power and thus a much better performance for the generated mm-wave from 30 to 90 GHz, and the performance is hardly degraded by fiber dispersion.

Additionally we show the performance comparison of the proposed modulation technique using MATB/MITB with other bias conditions and RF phase shift between the two MZMs. For the three bias conditions, i.e. QTB, MATB and MITB, we consider six combinations of bias conditions. For each bias combination, we optimize RF modulation index and the RF phase shift or difference between the two MZMs. When the two MZMs are biased at the same conditions, i.e. either MITB/MITB, MATB/MATB or QTB/QTB, the optimum RF phase difference and RF modulation index are found to be (145°, ~85%), (140°, ~106%) and (0°, ~99%), respectively. When the two MZMs are biased at the mixed bias conditions, i.e. either MITB/MATB, QTB/MATB or QTB/MITB, the optimum RF phase difference and RF modulation index are found to be (0°, ~85%), (90°, ~106%) and (90°, ~106%), respectively. Figure 7 shows comparison of simulated Q-factor versus fiber length for the six bias combinations using the optimized RF modulation index and RF phase difference. Note the order of bias conditions does not change the results, for example MITB/MATB is equal to MATB/MITB in results. It is clearly shown from Fig. 7 that our proposed technique using MITB/MATB with the optimized RF phase difference and RF modulation index outperforms all the other bias conditions for mm-wave generation with a sixupler. Since a smaller modulation index suggests lower RF input power to the MZMs, the bias condition of MITB/MATB provides a more efficient up-conversion than the other bias conditions.

Why does the MATB/MITB lead to the best performance? This is because the MATB is the best for even order harmonics generation and MITB is the best for odd-order harmonics generation. Thus the highest second-order harmonics are generated over all other even-order harmonics from the first MZM with MATB. The two highest second-order harmonics are injected into the second MZM with MITB, so that the highest first-order harmonics related to the above two highest second-order harmonics are generated. Thus these two highest first-order harmonics are the third-order harmonics relative to the optical carrier. Consequently the highest mm-wave power is generated and also other non-desired harmonics are suppressed. This is the reason why the proposed technique leads to better up-conversion efficiency and less nonlinear distortion impact.

In order to verify the proposed technique, an experimental proof was conducted and an experimental setup is shown Fig. 1. A continuous wave light source at 1552.52 nm with a 0 dBm of output optical power is modulated by an MZMs driven by a 5 GHz RF signal. The switching voltage for the first and second MZM is 5 V and 3.26 V, respectively. The RF input power applied to each electrode of the two MZMs is 12 dBm (1.26 V), which gives an RF modulation index of 25.2% and 38.6% for the first and second MZMs, respectively. The first and second MZMs have insertion loss of 6 dB and 7 dB, respectively. Insertion loss of the first MZM is compensated for by an optical amplifier. In the conventional technique, we used the second MZM of 3.26 V switching voltage and kept the same input optical power (0 dBm) and RF driven voltage (1.26 V). The modulation index is much smaller than that in the above simulation due to limitation of our instruments. Figures 8(a) and (b) shows the measured optical spectra after the two cascaded MZMs for the proposed technique and one MZM for the conventional technique. The experimental results show that the power difference between the first-order and third-order optical harmonics is 17 dB and 23 dB for the proposed and conventional techniques, respectively. It is clearly seen that the proposed technique leads to a 6-dB improvement in the power level compared to the conventional technique. This is mainly attributed to the increase in optical power of the third-order optical sidebands. Also, compared to the conventional technique, the proposed technique induces a further suppression of 8 and 3 dB in the optical carrier and first-order optical sidebands, respectively.

Using the same parameters as the experiments, Figures 9(a) and (b) shows simulated optical spectra for the proposed and conventional techniques, respectively. It is seen that the power level difference between the third-order and first-order optical harmonics is 16.5 dB and 23 dB for the proposed and conventional techniques, respectively, in a good agreement with the experiments. The discrepancy of 0.5 dB between the measurement and simulation for using the proposed technique is mainly due to the fact that the simulated MZMs were assumed to have an identical DC extinction ratio of 30 dB. However the two MZMs used in the experiments have different extinction ratios.

## 5. Conclusion

In this work, we have proposed and investigated a novel optical modulation technique for a frequency sixupler. This frequency sixupler, using two cascaded MZMs driven by a low frequency oscillator, can be used for generating mm-wave signals. Using this technique, desired third-order optical harmonics are maximized and non-desired optical harmonics are greatly suppressed, so that high frequency up-conversion efficiency is obtained with less nonlinear distortion impact. It is found that the bias condition of MATB/MITB for the two MZMs is the best, leading to the best up-conversion efficiency and less nonlinear distortion impact. With this best bias condition, the optimum modulation index is found ~86% in total and ~61% for each of the two MZMs. In addition, the best RF phase difference between the two MZMs is found to be zero.

As an example, we analyze 60-GHz generation using a 10-GHz RF. It has been shown that the proposed technique leads to much higher mm-wave power and thus better performance for using RF modulation indexes of 60–86%. By comparison to the conventional technique, the proposed technique results in an 8-dB increase of 60-GHz power and 6-dB improvement of receiver sensitivity. The concept proof of the proposed technique was verified by experiments.

## Appendix A

In this appendix we give the theoretical analysis of optical mm-wave generation using optical carrier suppression modulation, i.e. the conventional technique. The output optical field after the MZM can be written as follows,

The incident optical field to the photodetector is given by

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\times \left\{\begin{array}{c}{J}_{1}\left(\pi {m}_{\mathrm{RF}}\right)\left(\begin{array}{c}d\left(t+\frac{1}{6}\Delta \tau \right){H}_{F}\left({\omega}_{\mathrm{RF}}\right){e}^{j\left[{\omega}_{\mathrm{RF}}\left(t+{\beta}_{1}L\right)+{\varphi}_{\mathrm{RF}}\right]}\\ \phantom{\rule{.2em}{0ex}}+d\left(t-\frac{1}{6}\Delta \tau \right){H}_{F}^{*}\left({\omega}_{\mathrm{RF}}\right){e}^{-j\left[{\omega}_{\mathrm{RF}}\left(t+{\beta}_{1}L\right)+{\varphi}_{\mathrm{RF}}\right]}\end{array}\right){e}^{j\frac{1}{2}{{\beta}_{2}L\left({\omega}_{\mathrm{RF}}\right)}^{2}}\\ -{J}_{3}\left(\pi {m}_{\mathrm{RF}}\right)\left(\begin{array}{c}d\left(t+\frac{1}{2}\Delta \tau \right){H}_{F}\left(3{\omega}_{\mathrm{RF}}\right){e}^{j3\left[{\omega}_{\mathrm{RF}}\left(t+{\beta}_{1}L\right)+{\varphi}_{\mathrm{RF}}\right]}\\ \phantom{\rule{.2em}{0ex}}+d\left(t-\frac{1}{2}\Delta \tau \right){H}_{F}^{*}\left(3{\omega}_{\mathrm{RF}}\right){e}^{-j3\left[{\omega}_{\mathrm{RF}}\left(t+{\beta}_{1}L\right)+{\varphi}_{\mathrm{RF}}\right]}\end{array}\right){e}^{j\frac{1}{2}{\beta}_{2}L{\left(3{\omega}_{\mathrm{RF}}\right)}^{2}}\end{array}\right\}$$

Equation (8) clearly shows that the mm-wave current at 60 GHz generated by photodetection mainly consists of the beating of the optical subcarriers at ±3ω_{RF}, and the desired spectral component of photocurrent at the frequency of 6ω_{RF} can be expressed by

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\times {\mid {H}_{F}\left(3{\omega}_{\mathrm{RF}}\right)\mid}^{2}\mathrm{cos}\left[6{\omega}_{\mathrm{RF}}\left(t+\Delta \tau \right)\right]$$

Using Eq. (8) an average optical power ratio of the third-order optical harmonics to the first-order optical harmonics at an RF modulation index of 85% is given by

This indicates that optical power of third-order optical harmonics is ~5.2 dB lower than that of first-order optical harmonics.

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