## Abstract

In this paper the effect of mismatch and loss of the input lossy impedance matching network on the noise figure (NF) of microwave photonic links (MWPLs) operating under large radio frequency (RF) signal modulation is theoretically investigated. An intensity-modulation with direct-detection (IMDD) MWPL in which the external modulator is a Mach-Zehnder modulator (MZM) is studied here. The nonlinear input-output relationship of the MZM under large RF signal modulation can lead this link to operate in the nonlinear large-signal regime. The main goal of this paper is to investigate and find the input impedance mismatch conditions for minimizing large-signal NF of MWPLs. To the best of our knowledge, this is the first study on this subject. It is found that large-signal NF of an IMDD MWPL depends on the input impedance mismatch factor and input applied RF power. It is shown that for $\text{M}\simeq \text{0}\text{.39}$ the NF can be minimized that is approximately $0.95dB$ lower compared to the NF at the perfect match ($\text{M=1}$). Regarding a generic optoelectronic oscillator (OEO) containing a MWPL at saturation, the value of this analytical approach is to study the effect of such large-signal NF on the OEO phase noise performance.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Microwave photonics is an interdisciplinary field that has been grown rapidly in recent years. Indeed, this area of research emerged from long experience and in-depth knowledge of traditional disciplines like optics, photonics, microwave engineering, electronics, and so forth. A combination of various theories and techniques common in these disciplines, are used to design a typical microwave photonic system. The superiority of microwave photonic systems over their traditional counterparts has been corroborated by their ability to make it attainable to have sophisticated functionalities such as ultra-low phase noise microwave oscillation, microwave photonic signal processing, frequency comb generation, radio-over fiber, and biomedical applications [1–8].

Microwave photonic link, abbreviated to MWPL, plays a pivotal role in the vast majority of microwave photonic systems, in particular optoelectronic oscillators, abbreviated to OEOs [4–7]. Generally speaking, a MWPL is comprised of a modulation part, a detection part, and an optical medium that transmits the modulated light to a detector. Such link can be categorized according to the type of modulation and detection used. The topic of interest in this paper is an IMDD MWPL which is shown in Fig. 1. The MZM modulates the intensity of the laser light according to the electrical (or RF) signal applied to its electrodes. This light traveling across an optical medium, such as an optical fiber, an optical resonator, a combination of both, etc., reaches the photodetector (PD). Then, the PD recovers the electrical signal from the incident light.

Input impedance conditions for minimizing the small-signal NF of an IMDD MWPL has already been studied in [9]. Besides, large-signal NF of an IMDD MWPL has already been studied in [10] with no discussion on input impedance conditions. In this paper input impedance mismatch effects on the large-signal NF of MWPLs and its application to OEOs are theoretically investigated. To the best of our knowledge, this is the first study on this subject.

## 2. MWPL operating under large RF signal modulation

The main purpose in this section is to investigate MWPL operating under large RF signal conditions. In this section, without loss of generality, we assume that the nonlinearity is just due to the MZM and all other components are operating in their linear regime. If the electrical signal applied to the MZM consists of a bias voltage and a time-varying voltage, that is,

then, the RF signal received by the PD can be expressed as [11]In fact, Eq. (2) obviously shows a nonlinear input-output relationship governing the MWPL which signal plus noise in Eq. (3) are applied to its input. Such a MWPL has already been investigated theoretically through a calculation of the power spectral density (PSD) of the link output by taking the Fourier transform of the autocorrelation function of its output. It has been shown that the signal power gain can be different from the noise power gain due to this nonlinearity [10]. The signal power gain can be written as follows if the input of the MWPL is complex conjugate impedance matched [10],

where ${P}_{in}={V}_{m}^{2}/2{R}_{L}$ is RF input power applied to the input of the MWPL and ${h}_{01}={V}_{ph}{J}_{1}\left(\pi {V}_{m}/{V}_{\pi}\right)\mathrm{sin}\left(\pi {V}_{B}/{V}_{\pi}\right){e}^{-{\sigma}^{2}{\pi}^{2}/2{V}_{\pi}^{2}}$ in which ${\sigma}^{2}$ is the variance of the input voltage noise. The method introduced in [10] to obtain the noise power gain yieldsUsing Eqs. (4) and (5), these two power gains are shown in Fig. 2 versus RF input power. To plot this figure, ${T}_{FF}=0.5$, ${P}_{I}=400\text{\hspace{0.17em}}\text{\hspace{0.17em}}mW$, and ${V}_{\pi}=3.1\text{\hspace{0.17em}}\text{\hspace{0.17em}}V$ are considered to be typical values for a $LiNb{O}_{3}$ MZM. Moreover, in this numerical simulation we assume that ${R}_{L}=50\text{\hspace{0.17em}}\text{\hspace{0.17em}}\Omega $, ${V}_{ph}=3.2\text{\hspace{0.17em}}\text{\hspace{0.17em}}V$. This figure is an overwhelming evidence to substantiate the fact that the signal power gain diverges from the noise power gain in large-signal nonlinear regime.

#### 2.1 NF definition

According to [12–14], the noise factor ($F$) is defined as the ratio of the input signal-to-noise (SNR) to the output SNR when the input noise is the thermal noise generated at the standard temperature ($290\text{\hspace{0.17em}}\text{\hspace{0.17em}}\xb0K$). NF (in dB) is defined as $NF=10\mathrm{log}\left(F\right)$, that is,

## 3. Mismatch effect on the large-signal NF of a MWPL with input lossy matching network

Equivalent circuit model for an IMDD MWPL with input lossy impedance matching network is shown in Fig. 3. According to the equivalent circuit model shown in this figure, ${Z}_{M}={R}_{M}+j{X}_{M}$ and ${Z}_{D}={R}_{D}+j{X}_{D}$ are impedances representing MZM and the PD respectively. Moreover, in this circuit model, a voltage-controlled current source is used to describe the photocurrent generated because of the modulated light incident on the PD. The dotted boxes are used to distinguish the input and the output impedance matching networks. In addition, ${R}_{1}$ and ${R}_{2}$ are responsible for ohmic losses in the input impedance matching circuit.

Three dominant noise sources in this IMDD MWPL are shot noise, relative intensity noise (RIN), and thermal noise. In this circuit model, ${i}_{shot}$ and ${i}_{rin}$represent photodetector shot noise and laser RIN respectively. Furthermore, ${v}_{th,m}$ and ${v}_{th,d}$ are thermal noises due to ohmic losses existing in MZM and PD respectively.

The impedance matching networks act like bridges to couple the electrical signal into/out of the MWPL. The main goal of this section is to find out the effect of inserting mismatch and losses into the input impedance matching network on the NF of this MWPL.

According to the circuit model in Fig. 3, when there is a mismatch at the input of the link, in order to obtain the total power gain, it is necessary to multiply Eq. (4) and (5) by a mismatch factor, that is [15],

Moreover, the fact that the matching circuit is lossy should be considered, that is,

Consequently, using Eqs. (4) and (5) and Eqs. (7) and (8), the total signal and noise power gains, considering impedance mismatch and loss of matching networks, are as follows,

Hence considering this fact that ${G}_{c}$ is not equal to ${G}_{n}$ and using Fig. 3, ${N}_{add}$ can be written as follows,

Finally, substituting Eqs. (9)-(11) in Eq. (6) yields

Using Eq. (12), NF versus input impedance mismatch factor (M) for two values of RF input powers (${P}_{in}$) are calculated and shown in Fig. 4. In addition, NF versus RF input power for several values of input impedance mismatch factor are calculated and shown in Fig. 5.

It is clear from Fig. 4 that NF under large-signal condition (${P}_{in}=10\text{\hspace{0.17em}}dBm$) is larger than that in small-signal regime (${P}_{in}=-10\text{\hspace{0.17em}}dBm$). Besides, it is clear from Fig. 5 that NF increases by increasing RF input power. In addition, it is clear from Fig. 4 that NF is a function of impedance mismatch factor and can be minimized by appropriate selection of the mismatch factor.

According to Fig. 2, ${G}_{nm}$ is less than ${G}_{cm}$ in nonlinear regime. Thus, the terms first through fourth of Eq. (12) reduced, compared to the linear regime. Nonetheless, since ${G}_{cm}$ is decreasing going toward nonlinear regime, the last term of Eq. (12) increases and this leads to a higher NF under nonlinear conditions which is evident in Fig. 4 and Fig. 5. To plot this figures, we assume that ${g}_{m}=0.7dB$, ${Z}_{link}=10.3-j19.3\Omega $ and${G}_{n}=24.5dB$.

Based on the fact that in this paper the matching circuits are passive, M cannot become greater than 1. A snap judgment is that the NF increases under mismatch condition ($M<1$) due to the rise in value of the second, fourth and fifth terms of Eq. (12). However, a further investigation reveals that, for certain values of M, ${R}_{M}^{2}/{\left|{R}_{M}+{Z}_{M}{}^{\prime}\right|}^{2}$ becomes less than 1 and this can lead to a lower NF compared to the NF under perfect match condition. Figure 4 shows that for $\text{M}\simeq \text{0}\text{.39}$ the NF can be minimized that is approximately $0.95dB$ lower compared to the NF at the perfect match ($\text{M=1}$).

Figure 2 depicts that under large-signal condition, not only the signal power gain is not equal to noise power gain, but also these two power gains depend on the input power. Hence, the level of NF reduction for different values of M is dependent on the input power. Figure 4 and Fig. 5 obviously demonstrate this fact. The impedance associated with the optimum input impedance mismatch factor can be realized by inserting a matching circuit, as shown in Fig. 6, at the input of the MWPL. Using Smith char and Fig. 6 or various other topologies, the desired mismatch factor can be realized [16].

## 4. Application to OEOs

OEOs are amongst state-of-the-art technologies to generate ultra-low phase noise microwave oscillation [4–7]. A typical schematic of an OEO is shown in Fig. 7. According to Fig. 7, the electrical signal at the output of the PD is filtered, amplified, and then, fed back to the input of the MZM in order to close the loop. This process happens repeatedly until achieving the sustained oscillation. It is worth mentioning the fact that a phase adjustment is needed to fulfill the Barkhausen's phase criterion.

The MWPL encircled by dashed lines in Fig. 7, is an essential part of a typical OEO. The dominant amplitude-limiting mechanism controlling the amplitude of oscillation is the nonlinearity inherent in MZM transfer function. This fact should be taken into account while calculating the gain and the NF of a MWPL existing inside the OEO. Another important issue is the impedance matching networks used to couple this MWPL to the electrical part of the OEO's loop. The primary objective is to investigate the effects of mismatch and lossy elements existing in the input matching network of this MWPL on its NF and ultimately on the phase noise of the OEO containing this link.

According to Leeson's model for the phase noise of a feedback oscillator [17], the phase noise of an OEO considering the NF of the MWPL existing in its loop is as follows [10],

It is clear from Eq. (13) that NF of the MWPL can alter phase noise of an OEO specially can degrade its far-from-carrier phase noise (or noise floor). Since the MWPL operates in nonlinear large-signal regime in OEO, so using Fig. 5 and choosing the appropriate input impedance mismatch factor $\text{M}\simeq \text{0}\text{.39}$ for minimizing $\text{NF}\simeq \text{22}\text{\hspace{0.17em}}\text{dB}$ of MWPL can reduce the noise floor of OEO about 1 dB. Although, this phase noise improvement is not so much for this example but it is shown that appropriate input impedance mismatch factor can minimize NF of the MWPL results in phase noise improvement of an OEO.

It is worth mentioning here that, not only the NF is affected by the input impedance mismatch, but also the gain, bandwidth, power efficiency and input reflection of the MWPLs are closely related to the input impedance mismatch. But this paper focuses on the NF improvement. In future studies, we aim to investigate other effects of the mismatch factor on the phase noise of an OEO containing such MWPL like the one shown in Fig. 7, however, there are several issues regarding this topic.

The first one is that while the input and the output impedances of the electrical part in the OEO's loop are at perfect match, this MWPL can easily be connected to the electrical part of the OEO's loop with no concern for the reflections due to coupling the signals from the electrical part to the MWPL, and vice versa. Nonetheless, the losses existing inside the output matching network lead to a mismatch. Thus, there will be reflected signals needed to be considered to reach a sustained oscillation.

The second issue is that a lossy matching network can modify the total quality factor of this MWPL which should be considered in calculating the phase noise.

The third topic of debate is the phase shifts that the signal experiences because of these matching networks. These shifts in phase should be considered while determining the oscillation frequency and also the phase noise.

While minimizing the NF of a MWPL could be useful to lower the phase noise of an OEO containing such links but according to the above-mentioned issues, the effects of impedance matching on the phase noise of an OEO need further investigations.

## 5. Conclusion

In this paper the effect of the impedance mismatch of the input lossy matching network on the NF of a MWPL operating under large RF signal modulation was investigated. It has been shown that large-signal NF of a MWPL mainly depends on the input impedance mismatch factor and input applied RF power. So for a given input RF power, the NF of a MWPL can be minimized by choosing an appropriate mismatch factor. The main conclusion was that the NF of a MWPL operating under large-signal conditions can potentially be reduced for certain values of the mismatch factor. The numerical results showed that for $M\approx 0.39$ at ${P}_{in}=10dBm$, the NF is approximately $0.95dB$ lower compared to the NF at the perfect match ($M=1$). Moreover, the NF under large-signal conditions varies according to the input RF power. Hence, both the mismatch factor and the input RF power affect the NF of a MWPL. Besides, its application to OEOs was investigated. This study can also be important in other future applications of MWPLs operating under large-signal modulation.

## References

**1. **J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics **1**(6), 319–330 (2007). [CrossRef]

**2. **S. Iezekiel, *Microwave Photonics: Devices and Applications* (Wiley, 2009).

**3. **J. Yao, “Microwave photonics,” J. Lightwave Technol. **27**(3), 314–335 (2009). [CrossRef]

**4. **X. S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” J. Opt. Soc. Am. **13**(8), 1725–1735 (1996). [CrossRef]

**5. **S. E. Hosseini, A. Banai, and F. X. Kartner, “Tunable low-jitter low-drift spurious-free transposed-frequency optoelectronic oscillator,” IEEE Trans. Microw. Theory Tech. **65**(7), 2625–2635 (2017). [CrossRef]

**6. **S. E. Hosseini, A. Banai, and F. X. Kartner, “Low-drift optoelectronic oscillator based on a phase modulator in a sagnac loop,” IEEE Trans. Microw. Theory Tech. **65**(7), 2617–2624 (2017). [CrossRef]

**7. **K. Saleh, R. Henriet, S. Diallo, G. Lin, R. Martinenghi, I. V. Balakireva, P. Salzenstein, A. Coillet, and Y. K. Chembo, “Phase noise performance comparison between optoelectronic oscillators based on optical delay lines and whispering gallery mode resonators,” Opt. Express **22**(26), 32158–32173 (2014). [CrossRef] [PubMed]

**8. **S. E. Hosseini, S. Shojaeddin, and H. Abiri, “Theoretical investigation of an ultra-low phase noise microwave oscillator based on an IF crystal resonator-amplifier and a microwave photonic frequency transposer,” J. Opt. Soc. Am. B **35**(6), 1422–1432 (2018). [CrossRef]

**9. **E. I. Ackerman, C. Cox, G. Betts, H. Roussell, F. O’Donnell, and K. Ray, “Input impedance conditions for minimizing the noise figure of an analog optical link,” IEEE Trans. Microw. Theory Tech. **46**(12), 2025–2031 (1998). [CrossRef]

**10. **S. E. Hosseini and A. Banai, “Noise figure of microwave photonic links operating under large-signal modulation and its application to optoelectronic oscillators,” Appl. Opt. **53**(28), 6414–6421 (2014). [CrossRef] [PubMed]

**11. **C. H. Cox, *Analog Optical Links: Theory and Practice* (Cambridge University Press, 2006).

**12. **H. T. Friis, “Noise figures of radio receivers,” Proc. IRE, 419–422 (1944).

**13. **“IRE standards on methods of measuring noise in linear two ports,” Proc. IRE48(1), 60–68 (1959).

**14. **“IEEE Standard 100” The Authoritative Dictionary of IEEE Standards Terms 7th ed. (2000).

**15. **E. Robert Collin, *Foundations for Microwave Engineering* (John Wiley & Sons, 2007).

**16. **D. M. Pozar, *Microwave Engineering* (Wiley, 1997).

**17. **E. Rubiola, *Phase Noise and Frequency Stability in Oscillators,* 1 ed. (Cambridge University Press, 2009).