By using several Mach-Zehnder modulators and an optical delay line, it is possible to provide the frequency multiplication needed for creating millimeter-wave frequencies.

*See sidebar:* What Is Radio Over Fiber?

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 technique^{1} 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

E_{0} = the intensity of the electrical field and

Ω_{0} = the optical angular frequency of the incident light wave from the laser diode, φ(t) = mcos(Ω_{LO}t),

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 + /2 = the

residual phase of the optical carrier. Under small signal conditions, with m Eq. 1 reduces to Eq. 2: (in the box above),

where

φ_{RF} = Ω_{LO}t = 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_{LO}t = 2lp + p/2,

where

l = an integer value,

then Eq. 2 reduces to Eq. 3^{1}.

Inspection of Eq. 3 clearly shows that the spectrum at the output of modulator MZM2 comprises two components with frequencies equal to Ω_{0} 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

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

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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Beating the output components of modulator MZM2 with frequencies equal to ω_{O} 6ω_{LO}, and the amplitudes give by Eq. 10, by means of a photodetector, would generate a frequency that is 12 times the input microwave frequency.

It is worth mentioning here that a frequency that is 12 times the input microwave frequency can be produced by beating together other frequencies at the photodetector. For example, beating together frequencies equal to ω_{O} 4ω_{LO} and frequencies equal to ω_{O} 8ω_{LO} would produce frequencies that are also 12 times the input microwave frequency. However, because of the relative magnitudes of the ordinary Bessel functions, the contribution of beating frequencies equal to ω_{O} 6ω_{LO} will be dominant. Furthermore, beating together MZM2 output components with frequencies ω_{O} 2ω_{LO} and ω_{O} 4ω_{LO}, and amplitudes given by Eqs. 8 and 9, respectively, in a photodetector, would generate frequencies that are two and six times, respectively, the frequency of the input microwave frequency.

Beating together other frequency components would also produce frequencies that are two and four times the frequency of the input microwave signal. However, because of the relative magnitudes of the ordinary Bessel functions, the contribution of beating together frequencies ω_{O} 2ω_{LO} and ω_{O} 4ω_{LO} will be dominant. These three cases reveal that for large amplitudes of the input microwave signal, the output of a photodetector will comprise a large number of multiples of the input microwave frequency. With a microwave bandpass filter, it is possible to select a signal frequency of interest from this large number.

Although the design of a microwave bandpass filter with prespecified parameters may not be an easy task, optimization techniques for such filters are available in the literature; see, for example, refs. 2 and 3 and the references cited therein. Thus, the scheme shown in the figure can provide not only an output with frequency equal to four times the input microwave frequency as concluded in ref. 1, but also several outputs with different multiplication factors subject to the condition that the driving microwave signal must be relatively large in amplitude.

In conclusion, the MZM-based frequency multiplication scheme proposed in ref. 1 for quadrupling the frequency of a relatively low-level microwave signal can also be used for multiplying microwave signals over a wide range of multiplication factors. By driving an MZM with a relatively large-amplitude microwave signal, rather than the low-level signal cited in ref. 1, multiplication factors other than four can be easily obtained. This extension of the proposed MZM frequency multiplication technique has enormous potential for wireless communication networks operating at millimeter-wave frequencies.

REFERENCES

1. H. Chi and J. Yao, "Frequency quadrupling and upconversion in a radio over fiber link," *Journal of Lightwave Technology*, Vol. 26, 2008, pp. 2706-2711.

2. D. Budimir, "Optimized E-plane bandpass filters with improved stop band performance,"* IEEE Transactions on Microwave Theory and Techniques*, Vol. MTT-45, 1997, pp. 212-220.

3. D. Budimir and G. Goussetis, "Design of asymmetrical RF and microwave bandpass filters by computer optimization," *IEEE Transactions on Microwave Theory and Techniques*, Vol. 51, 2003, pp. 1174-1178.