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Millimeter-wave wireless communications networks can take advantage of frequency multiplication schemes that generate high-frequency signals from lower-frequency sources. In one case, frequency multiplication for radio-over-fiber links can be achieved by means of Mach- Zehnder-modulator (MZM) based schemes. With three MZMs biased at their minimum transmission points, and a tunable optical delay line placed between two of the modulators and adjusted for the appropriate group delay, the output of the third MZM will produce a large number of multiplied-frequency signals. By beating these frequencies together, it is possible to obtain output frequencies that are multiples of the input frequency. By using a microwave bandpass filter, a desired output signal can be extracted from the total output spectrum and used as a practical signal source for high-capacity point-topoint microwave communications. These millimeter-wave links, being increasingly implemented at bands such as 71 to 76 GHz and 81 to 86 GHz, provide the bandwidth needed in support of transferring high-speed data and high-definition video over short-haul distances, such as remote broadcast installations.

This novel frequency quadrupling technique¹ for radio-over-fiber links achieves the desired frequency multiplication by properly adjusting the time delay between the first two MZMs. Frequency upconversion is realized by feeding the output of the second MZM to a third MZM to which an intermediate-frequency (IF) signal is applied. The output of the third MZM is a frequency-upconverted millimeter-wave signal. The figure shows a simplified schematic diagram of the proposed scheme for frequency quadrupling and frequency upconversion. Following the procedure described in ref. 1, and assuming that the first modulator (MZM1) and the second modulator (MZM2) are biased at their minimum transmission points, that is, the bias voltage applied to MZM1 and MZM2, V_b, is equal to their half-wave voltage, V_π , the output of the second modulator, MZM2, can be expressed as shown in Eq. 1 (in the box below), where

E₀ = the intensity of the electrical field and
Ω₀ = the optical angular frequency of the incident light wave from the laser diode, φ(t) = mcos(Ω_LOt),
m = pV_s/V_p = the modulation index,
Ω_LO = 2pfLO = the frequency of the modulating signal,
Vs = the signal voltage,
t = the group delay introduced by the optical delay line between modulation MZM1 and modulator MZM2, and
φ_cc = p + [φ(t – t) + φ(t)]/2 = the

residual phase of the optical carrier. Under small signal conditions, with m << 1, and using the approximation sinθ ≈θ, Eq. 1 reduces to Eq. 2: (in the box above),

where
φ_RF = Ω_LOt = the phase shift introduced by the group delay of the tunable optical delay line.

If the group delay of the tunable optical delay line is properly selected so that

φ_RF = Ω_LOt = 2lp + p/2,
where
l = an integer value,
then Eq. 2 reduces to Eq. 3¹.

Inspection of Eq. 3 clearly shows that the spectrum at the output of modulator MZM2 comprises two components with frequencies equal to Ω₀ ± 2Ω_LO. By beating together these two signal components via a photodetector, the resulting electrical signal would have a frequency equal to 4Ω_LO. Thus, a frequency that is four times the input microwave frequency can be obtained. Moreover, it appears that the optical carrier and the firstorder sidebands are totally suppressed. This conclusion is, however, based on the assumption that the signal is sufficiently small, with m << 1.

By virtue of its derivation, Eq. 3 cannot be used to describe the performance of the MZM frequency-multiplication scheme under large-signal conditions. The major intention of this report is, therefore, to extend the analysis presented in ref. 1 to investigate the large-signal performance of the MZM frequency-multiplication scheme, and to explore the feasibility of obtaining frequency multiplication with factors other than four.

LARGE-SIGNAL ANALYSIS
In general, Eq. 1 can be rewritten as Eq. 4. Applying the trigonometric identity to Eq. 4 yields Eq. 5, and Eq. 4 can further be reduced to Eq. 6. By selecting Ω0t = 2lp + p/2, Eq. 6 can be reduced to Eq. 7.

Inspection of Eq. 7 shows that the spectrum at the output of modulator MZM2 comprises a huge number of frequency components for different combinations of n and k. The table summarizes some of the output frequency components with their amplitudes. Referring to the table shows that at the output of modulator MZM2, the amplitude of the components with frequencies equal to Ω_O ± 2Ω_LO can be expressed as Eq. 8.

In fact, these two frequency components are the same as those considered in ref. 1 where the amplitude reduces to 0.5(m/2)² when m << 1. Beating together these two frequencies by means of a photodetector would generate a frequency that is four times the frequency of the input microwave signal.

A study of the table shows that at the output of MZM2, the amplitude of the components with frequencies Ω_O ± 4Ω_LO can be expressed as Eq. 9.

Similarly, at the output of modulator MZM2, the amplitude of the components of frequencies Ω_O ± 6Ω_LO can be expressed as Eq. 10.

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