Effect of input impedance mismatch on the large-signal noise figure of microwave photonic links

Mandana Jahanbozorgi; S. Esmail Hosseini

doi:10.1364/OSAC.1.000564

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

Fig. 1 General schematic of externally intensity-modulated with direct-detection microwave photonic link, PD: Photodetector., CW: Continuous wave, MZM: Mach–Zehnder modulator, BPF: Bandpass filter.

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. 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,

v_{i n} (t) = V_{B} + x (t)

then, the RF signal received by the PD can be expressed as [11]

y (t) = V_{p h} [1 + \cos (\frac{π V_{B}}{V_{π}} + \frac{π x (t)}{V_{π}})] = g (x)

where

V_{π}

is the MZM half-wave voltage,

V_{B}

is its bias voltage and

V_{p h} = r_{d} R_{L} T_{F F} P_{I} / 2

, where

T_{F F}

is the insertion loss of the MZM,

P_{I}

is its input optical power,

r_{d}

is the responsivity of the photodetector and

R_{L}

is its load impedance. Suppose that the MWPL RF input

x (t)

consists of a sinusoidal signal plus zero-mean, stationary white Gaussian noise as

x (t) = V_{m} \cos (ω_{R F} t) + n (t)

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],

G_{c m} (P_{i n}) = \frac{2 h_{01}^{2} / R_{L}}{P_{i n}}

where

P_{i n} = V_{m}^{2} / 2 R_{L}

is RF input power applied to the input of the MWPL and

h_{01} = V_{p h} J_{1} (π V_{m} / V_{π}) \sin (π V_{B} / V_{π}) e^{- σ^{2} π^{2} / 2 V_{π}^{2}}

in which

σ^{2}

is the variance of the input voltage noise. The method introduced in [10] to obtain the noise power gain yields

G_{n m} (P_{i n}) = \frac{\int_{B} S_{y, N} (f) d f}{σ^{2} / R_{L}}

where

S_{y, N}

describes PSD of the output noise, which depends on the interaction of the input signal, and noise in large-signal regime, a dependence that is not readily apparent under small-signal conditions [10].

Using Eqs. (4) and (5), these two power gains are shown in Fig. 2

Fig. 2 Signal power gain ( $G_{c m}$ ) and noise power gain ( $G_{n m}$ ) versus input RF power.

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versus RF input power. To plot this figure,

T_{F F} = 0.5

,

P_{I} = 400 m W

, and

V_{π} = 3.1 V

are considered to be typical values for a

L i N b O_{3}

MZM. Moreover, in this numerical simulation we assume that

R_{L} = 50 Ω

,

V_{p h} = 3.2 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 ( $290 ° K$ ). NF (in dB) is defined as $N F = 10 \log (F)$ , that is,

N F = 10 \log [{{(\frac{S}{N})}_{i n} / {(\frac{S}{N})}_{o u t} |}_{T_{0} = 290 ° K}] = 10 \log [\frac{S_{i n}}{S_{o u t}} \frac{N_{o u t}}{k T_{0}}] = 10 \log [\frac{1}{G_{c m}} \frac{G_{n m} k T_{0} + N_{a d d}}{k T_{0}}]

where

S_{i n}

and

S_{o u t}

are input and output signal power respectively,

k = 1.38 \times 10^{- 23} J / ° K

is the Boltzmann's constant, and

N_{a d d}

is the output noise power density added by the link. Based on the divergence between the noise power gain and the signal power gain, which is clear from Fig. 2, the NF of this link under large-signal condition depends on the input power and also is different from the case where these two power gains considered the same.

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

Fig. 3 Equivalent circuit model for an IMDD MWPL with input lossy matching network [‎9].

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. 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, $i_{s h o t}$ and $i_{r i n}$ represent photodetector shot noise and laser RIN respectively. Furthermore, $v_{t h, m}$ and $v_{t h, d}$ 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],

M = 4 R_{l i n k} R_{s} / {| Z_{l i n k} + Z_{s} |}^{2}

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

g_{m} = (R_{l i n k} - R_{1}) / (R_{l i n k} + R_{1})

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,

G_{c} (P_{i n}) = M g_{m} G_{c m} (P_{i n})

G_{n} = M g_{m} G_{n m}

Hence considering this fact that $G_{c}$ is not equal to $G_{n}$ and using Fig. 3, $N_{a d d}$ can be written as follows,

\begin{array}{l} N_{a d d} = k T G_{n} \frac{4}{g_{m} M} \frac{R_{M}^{2}}{{| R_{M} + Z_{M}^{'} |}^{2}} + k T G_{n} \frac{1 - g_{m}}{1 + g_{m}} \frac{R_{l i n k}}{R_{S}} \\ + k T G_{n} \frac{1 - g_{m}}{1 + g_{m}} \frac{1}{M g_{m}} \frac{{[(1 + g_{m}) R_{S} + (1 - g_{m}) R_{l i n k}]}^{2} + {(1 + g_{m})}^{2} X_{l i n k}^{2}}{{(R_{l i n k} + R_{S})}^{2} + X_{l i n k}^{2}} + k T + (i_{r i n}^{2} + i_{s n}^{2}) \frac{{| Z_{D} |}^{2}}{4 R_{D}} \end{array}

where the first term describes the MZM thermal noise. Moreover, the second and the third terms describe the thermal noise due to

R_{1}

and

R_{2}

, respectively. The forth term is the photodetector thermal noise and the last term show the laser RIN and the photodiode shot noise.

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

\begin{array}{l} N F = 10 \log [\frac{G_{n m}}{G_{c m}} + \frac{G_{n m}}{G_{c m}} \frac{4}{g_{m} M} \frac{R_{M}^{2}}{{| R_{M} + Z_{M}^{'} |}^{2}} + \frac{G_{n m}}{G_{c m}} \frac{1 - g_{m}}{1 + g_{m}} \frac{R_{l i n k}}{R_{S}} \\ + \frac{G_{n m}}{G_{c m}} \frac{1 - g_{m}}{1 + g_{m}} \frac{1}{M g_{m}} \frac{{[(1 + g_{m}) R_{S} + (1 - g_{m}) R_{l i n k}]}^{2}}{{(R_{l i n k} + R_{S})}^{2} + X_{l i n k}^{2}} \\ + \frac{G_{n m}}{G_{c m}} \frac{1 - g_{m}}{1 + g_{m}} \frac{1}{M g_{m}} \frac{{(1 + g_{m})}^{2} X_{l i n k}^{2}}{{(R_{l i n k} + R_{S})}^{2} + X_{l i n k}^{2}} + \frac{(i_{r i n}^{2} + i_{s n}^{2}) \frac{{| Z_{D} |}^{2}}{4 R_{D}} + k T}{M g_{m} G_{c m} k T}] \end{array}

Using Eq. (12), NF versus input impedance mismatch factor (M) for two values of RF input powers ( $P_{i n}$ ) are calculated and shown in Fig. 4

Fig. 4 Noise figure versus impedance mismatch factor (M) for two values of RF input power ( $P_{i n}$ ).

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. In addition, NF versus RF input power for several values of input impedance mismatch factor are calculated and shown in Fig. 5

Fig. 5 Noise figure versus RF input power ( $P_{i n}$ ) for several values of impedance mismatch factor (M).

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.

It is clear from Fig. 4 that NF under large-signal condition ( $P_{i n} = 10 d B m$ ) is larger than that in small-signal regime ( $P_{i n} = - 10 d B m$ ). 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, $G_{n m}$ is less than $G_{c m}$ in nonlinear regime. Thus, the terms first through fourth of Eq. (12) reduced, compared to the linear regime. Nonetheless, since $G_{c m}$ 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 $g_{m} = 0.7 d B$ , $Z_{l i n k} = 10.3 - j 19.3 Ω$ and $G_{n} = 24.5 d B$ .

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, $R_{M}^{2} / {| R_{M} + Z_{M}^{'} |}^{2}$ 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 $M ≃ 0 .39$ the NF can be minimized that is approximately $0.95 d B$ lower compared to the NF at the perfect match ( $M=1$ ).

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

Fig. 6 Input impedance matching circuit for realizing appropriate input impedance mismatch factor.

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, 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

Fig. 7 General schematic of an OEO, CW: continuous wave, MZM: Mach Zehnder modulator, PD: photodetector, BPF: bandpass filter, PS: phase shifter, Amp: amplifier . In this figure $τ_{i n}$ and $τ_{e}$ determine the internal losses and the coupling losses respectively. The optical medium can be a long optical fiber or a whispering gallery mode (WGM) resonator.

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. 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],

L (f) = (1 + {(\frac{f_{L}}{f})}^{2}) (\frac{F_{a m p} k_{B} T_{0} + G F_{M W P L} k_{B} T_{0}}{P_{R F}} + \frac{κ}{f})

in which

f

is the offset frequency from the frequency of oscillation and

P_{R F}

is the average power of the output microwave signal generated by this OEO, and

κ

is the flicker noise parameter of the amplifier. Moreover,

F_{a m p}

and

F_{M W P L}

are the noise factor of the amplifier and the MWPL, respectively. In addition,

f_{L} = f_{R F} / 2 Q_{e q}

is the Leeson's frequency where

f_{R F}

is the oscillation frequency and

Q_{e q}

is the equivalent quality factor.

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 $M ≃ 0 .39$ for minimizing $NF ≃ 22 dB$ 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 $P_{i n} = 10 d B m$ , the NF is approximately $0.95 d B$ 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.

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Effect of input impedance mismatch on the large-signal noise figure of microwave photonic links

Abstract

1. Introduction

2. MWPL operating under large RF signal modulation

2.1 NF definition

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

4. Application to OEOs

5. Conclusion

References

Cited By

Figures (7)

Equations (13)

OSA Continuum