# Analysis of Trade-offs in RF Photonic Links based on Multi-Bias Tuning of
Silicon Photonic Ring-Assisted Mach Zehnder Modulators

Md Jubayer Shawon, and Vishal Saxena The authors are with the Department of
Electrical and Computer Engineering, University of Delaware, Newark, DE, 19716
USA e-mail<EMAIL_ADDRESS>
###### Abstract

Recent progress in silicon-based photonic integrated circuits (PICs) have
opened new avenues for analog circuit designers to explore hybrid integration
of photonics with CMOS ICs. Traditionally, optoelectronic systems are designed
using discrete optics and electronics. Silicon photonic (SiP) platforms
provide the opportunity to realize these systems in a compact chip-scale form
factor and alleviate long-standing challenges in optoelectronics. In this
work, we analyze multi-bias tuning in Ring-Assisted Mach Zehnder Modulator
(RAMZM) and resulting trade-offs in analog RF photonic links realized using
RAMZMs. Multi-bias tuning in the rings and the Mach-Zehnder arms allow
informed trade-offs between link noise figure and linearity. We derive
performance metrics including gain, noise figure, and linearity metrics
associated with tuning of multiple bias settings in RAMZM based links and
present resulting design optimization. Compared to MZM, an improvement of 18
dB/Hz${}^{\frac{2}{3}}$ in SFDR is noted when RAMZM is linearized. We also
propose a biasing scheme for RAMZM that provides 6x improvement in slope
efficiency, or equivalently, 15.56dB in power Gain over MZMs (single drive)
while still providing similar SFDR performance ($\sim$ 109
dB/Hz${}^{\frac{2}{3}}$) as MZMs. Moreover, a method to improve gain in
photodiode saturation limited links is presented and studied.

###### Index Terms:

Analog optical link, Silicon Photonics, Photonic Integrated Circuit (PIC),
Ring-Assisted Mach Zehnder Modulator (RAMZM), RF Photonics, RF-to-optical
modulator.

##  I Introduction

The emergence of silicon-based photonic integrated circuits (PICs) and the
ability to fabricate them on foundry multi-project wafer (MPW) platforms is
now allowing analog circuit designers to integrate entire optoelectronic
system in the same chip-scale package. Radio-frequency (RF) photonics is an
established field where RF signals are processed and then transmitted over
optical fibers using discrete lasers, modulators (photo)detectors and
associated electronics. In addition to improved size, weight, power, and cost
(SWaPc) metrics, large-scale PICs can be designed to alleviate longstanding
limitations of the discrete optoelectronic systems.

Figure 1: An analog optical link using RAMZI modulator. Here, $P_{I}$ and
$P_{M,o}$ are the input and output optical power at the modulator, and
$P_{o,d}$ is the optical input power to the detector. $P_{s,a}$ and $P_{L}$
are the available input power and output powers of the link.

As shown in Fig. 1, an analog RF photonic optical link employs either a direct
modulated laser (DML) or a Mach-Zehnder Modulator (MZM) which modulates a
continuous-wave (CW) laser source (typically at the 1550nm telecommunication
wavelength) using an RF signal. The RF signal in the optical domain undergoes
optical signal processing (e.g. filtering, phased-array processing, etc.) and
transmitted over optical fiber(s). The optical-domain RF signal is finally
converted back into RF domain using a (photo)detector, which is optionally
followed by a low-noise transimpedance amplifier (LNTIA). In Fig. 1, $P_{s,a}$
is the input RF source power, $P_{M,o}$ is the optical power at the modulator
output, $P_{o,d}$ is the optical power at the detector input, $P_{L}$ is the
RF power at the load. Since the detected RF power depends upon the square of
optical power, the optical gain (loss) is squared in the link gain ($G$),
determined as

$G=G_{M}\cdot G_{ch}^{2}\cdot G_{det}$ (1)

where $G_{M}\triangleq\frac{P_{M,o}^{2}}{P_{s,a}}$,
$G_{ch}\triangleq\frac{P_{o,d}}{P_{M,o}}$,
$G_{det}\triangleq\frac{P_{L}}{P_{o,d}^{2}}$ are the individual available
power gains (or losses) for the modulator, optical channel, and the detector
respectively.

In prior art, the whole link is modeled end-to-end and the overall small-
signal gain, noise figure (NF), and distortion metrics are determined [1, 2].
Though MZM-based links have been here for the past several decades, their
analog performance is limited by their inherent raised-cosine optical non-
linearity.

Recently, ring-assisted Mach-Zehnder modulator (RAMZM) has been investigated
with significantly improved linearity performance[3, 4]. However, the
performance of RAMZM in an analog photonic link has not been studied in
detail. The authors provided a preliminary analysis in [5] and this article
expands on this analysis while incorporating novel design trade-offs based on
the tuning of multiple optical biasing in the RAMZM.

In this article, we present analysis for analog RF photonic links using RAMZM
for high linearity. We derive the small-signal gain, noise figure, and
linearity metrics associated with the RAMZM based link and present design
trade-offs. The rest of this article is organized as follows: Section II
describes the RAMZM and derives small-signal gain expression. In section III,
the noise figure of the entire optical link is derived and simulated. Section
IV summarizes link distortion analysis followed by conclusion.

##  II Ring-Assisted Mach-Zehnder Modulator

###  II-A MZM and Microring Modulators

A linear RF-to-optical modulation is required before optical-domain signal
processing is performed. Discrete lithium niobate (LiNbO3) MZMs have been
widely used in analog photonic links and are limited to $<95$ dB/Hz2/3 spur-
free dynamic range (SFDR) at 1GHz. Linearization of MZMs has been a long-
studied topic in RF photonics [1, 6]. Active electronic linearization using a
CMOS pre-distortion circuit has been investigated using third or fifth-order
active predistortion [7, 8]. The active linearization techniques exhibit lower
bandwidths ($<1$GHz), and incur noise penalty and supply voltage limitation. A
third-order intermodulation (IM3) cancellation technique was recently
demonstrated with $<120$ dB/Hz2/3 SFDR at 9GHz [9]. However, this technique is
narrowband around the 9GHz center frequency.

Silicon photonics based analog MZMs have been designed where DC Kerr non-
linearity was used to compensate for the pn-junction non-linearity to achieve
a broadband SFDR linearity of $\sim 106$ dB/Hz2/3 over a 13GHz bandwidth [10,
11]. However, such analog modulators require a long traveling-wave electrode
design ($<2mm$) which incurs higher noise figure due to electrode losses [12].

Microring modulators (MRM) allow modulation with a very small form-factor
($\sim 5\mu m$ radius) and lumped capacitive interface, but typically suffer
both even as well as odd-order distortion due to the Lorentzian optical
transmission of the microring. Consequently, SiP MRMs exhibit an SFDR of $63$
dB/Hz2/3 with single-ended drive [13]. Linearization of MRMs has been
attempted but the resulting performance is limited below $98$ dB/Hz2/3 up to
13 GHz [14].

Figure 2: Conceptual schematic of a silicon photonic Ring Assisted Mach
Zehnder Modulator (RAMZM), interfaced with an RF source and a CW laser source.

###  II-B RAMZM Modulators

In the last decade, ring-assisted linearized MZ (RAMZI) modulators have been
explored where a differential configuration, similar to MZM, is employed but
the two (upper and lower) phase shifters are replaced by ring modulators as
shown in Fig. 2. By exploiting the Lorentzian transmission characteristics,
upper and lower rings can be optically biased such that their voltage-to-phase
responses are expansive. This expansive response can be enhanced by a suitable
value of coupling coefficient ($\kappa$) and then used to compensate for the
compressive phase-to-intensity non-linearity of a Mach-Zehnder interferometer
[15, 4, 16, 3]. The MZ push-pull configuration suppresses and even-order non-
linearity.

Fig. 2 depicts the schematic of a SiP RAMZM and its interface with the RF
input source using a differential balun. Fig. 3 shows the chip micrograph of
an RAMZM fabricated by the authors using IMEC’s ISIPP50G SiP process [17]. As
shown in Fig. 2, the output of the CW laser is equally split into two arms of
the MZM. RF signals are applied to the ring modulators in a differential push-
pull configuration around a DC bias ($V_{M}$). These differential RF signals
are translated into different phase changes in each arms of the MZI. When the
two optical waves combine at the output of the MZM, optical modulation is
achieved via constructive or destructive interference of light [18].

Figure 3: Chip micrograph of a RAMZM fabricated in IMEC’s ISIPP50G SiP
process.

###  II-C RAMZM Transmission Characteristics

We now derive the transmission characteristics of the RAMZM. The optical wave
electric field at the output of a RAMZM can be expressed as [19]

$\displaystyle E_{out}$
$\displaystyle=\frac{E_{in}}{2}\Big{(}\left|a_{r1}(\theta)\right|e^{-j.(knL_{1}+\phi_{bias}+\angle
a_{r1}(\theta))}+\cdots$ (2)
$\displaystyle+\left|a_{r2}(\theta)\right|e^{-j.(knL_{2}+\angle
a_{r2}(\theta))}\Big{)}$

$\displaystyle a_{ri}(\theta)=\frac{\tau-\alpha e^{-j\theta}}{1-\tau\alpha
e^{-j\theta}}$ (3)

where the variables are defined below. Here, index term $i=1$ refers to upper
while $i=2$ refers to the lower arm of RAMZM.
$E_{in}$: input electric field
$L_{i}$: length of the RAMZM arms
$\left|a_{ri}(\theta)\right|$: microring magnitude response
$\angle a_{ri}(\theta)$: microring phase response
$\phi_{bias}$: phase difference between the upper and lower arms introduced by
thermal phase shifter
$\tau=\sqrt{1-\kappa^{2}}$: transmission coefficient of the microring couplers
$\alpha$: microring loss factor
$\theta=\theta(V)$: microring roundtrip phase shift, which depends on the
applied voltage in the phase-modulators.

If we neglect losses in the short MZI arms, the transmission function,
$T(\theta)=\frac{I_{out}}{I_{in}}=\Big{|}\frac{E_{out}}{E_{in}}\Big{|}^{2}$,
can be written as

$\displaystyle T(\theta)$
$\displaystyle=\frac{1}{4}\left|e^{-j.(knL_{1}+\phi_{bias}+\angle
a_{r1}(\theta))}+e^{-j.(knL_{2}+\angle a_{r2}(\theta))}\right|^{2}$ (4)

When the optical path of the RAMZM arms have equal length ($L_{2}=L_{1}$), the
transfer function can be simplified as

$\displaystyle T(\theta)$
$\displaystyle=\frac{1}{2}\left[1+\cos\Big{(}\phi_{bias}+\angle
a_{r1}(\theta)-\angle a_{r2}(\theta)\Big{)}\right]$ (5)

$\displaystyle\phi_{i}=\angle
a_{ri}(\theta)=\tan^{-1}\left(\frac{\alpha(1-\tau^{2})\sin\theta}{\tau(1+\alpha^{2})-\alpha(1+\tau^{2})\cos\theta}\right)$
(6)

As mentioned earlier, $\theta$ is the round-trip phase delay in the
microrings. By applying differential signal
($\theta=\theta_{DC}\pm\theta_{mod}$) in the microrings, one can achieve
electro-optic modulation. Here, $\theta_{DC}$ is the microring optical DC bias
point, whereas $\theta_{mod}$ is the modulating RF input signal. Now, if we
assume low-loss rings, i.e. $\alpha\approx 1$, and apply differential
modulating signal, trigonometric manipulation of $T(\theta)$ and expanding its
series around $\theta_{mod}$ gives us Eq. 12 to Eq. 17 [1, 20].

$P_{M,o}(\theta_{mod})=P_{I}T(\theta)=\frac{P_{I}\gamma_{0}}{2}-\gamma_{1}P_{I}\theta_{mod}-\gamma_{2}P_{I}\theta_{mod}^{2}+\gamma_{3}P_{I}\theta_{mod}^{3}+\gamma_{4}P_{I}\theta_{mod}^{4}+\mathcal{O}(\theta_{mod}^{5})$
(12)

$\displaystyle\gamma_{0}$ $\displaystyle=1+\cos(\phi_{bias})$ (13)
$\displaystyle\gamma_{1}$
$\displaystyle=\frac{(\tau^{2}-1)\cdot\sin(\phi_{bias})}{\tau^{2}-2\tau\cos(\theta_{DC})+1}$
(14) $\displaystyle\gamma_{2}$
$\displaystyle=\frac{(\tau^{2}-1)^{2}\cdot\cos(\phi_{bias})}{(\tau^{2}-2\tau\cos(\theta_{DC})+1)^{2}}$
(15) $\displaystyle\gamma_{3}$
$\displaystyle=\frac{(\tau^{2}-1)(2\cos(\theta_{DC})^{2}\tau^{2}+\cos(\theta_{DC})\tau^{3}+2\tau^{4}+\cos(\theta_{DC})\tau-8\tau^{2}+2)\cdot\sin(\phi_{bias})}{3(\tau^{2}-2\tau\cos(\theta_{DC})+1)^{3}}$
(16) $\displaystyle\gamma_{4}$
$\displaystyle=\frac{(\tau^{2}-1)^{2}(4\cos(\theta_{DC})^{2}\tau^{2}+2\cos(\theta_{DC})\tau^{3}+\tau^{4}+2\cos(\theta_{DC})\tau-10\tau^{2}+1)\cdot\cos(\phi_{bias})}{(\tau^{2}-2\tau\cos(\theta_{DC})+1)^{4}}$
(17)

In these equations, $P_{I}$ is the modulator optical input power. Upon
observing Eq. 12, we can easily deduce that even-order distortions can
entirely be eliminated simply by setting $\phi_{bias}=\frac{\pi}{2}$. This is
achieved by tuning thermal heaters on the RAMZM arms [18]. On the other hand,
the third-order nonlinearity can be cancelled by setting the rings at the
anti-resonance point (i.e. optical bias $\theta_{DC}=\pi$) [4]. This is
achieved by tuning the thermal phase shifters of the microrings while the
driver voltage is set to a common-mode voltage, $V_{M}$ DC. The differential
modulating signals, $v_{m}$, are applied around this common-mode voltage, so
that the individual ring drive voltages are $V_{M}\pm v_{m}$. Therefore, the
round-trip phase shift at the upper and lower ring will be
$\pi+\theta_{mod}(v_{m})$ and $\pi-\theta_{mod}(v_{m})$, respectively.

Figure 4: Spectral response of RAMZM (fabricated in IMEC’s ISIPP50G SiP
process) when arm bias ($\phi_{bias}$) is tuned by applying 0-14V DC on a
$n$-doped resistive heater placed around the arm.

###  II-D RAMZM Optical Biasing

As mentioned before, biasing schemes are implemented on a SiP RAMZM by means
of thermal phase shifters. The SiP RAMZM presented in this work utilizes
$n$-doped resistive heaters [21] placed around the arms and rings to tune
$\phi_{bias}$ (arm bias) and $\theta_{DC}$ (ring bias), respectively. The
change in spectral response of the RAMZM with various arm bias voltages (0-14V
DC) is shown in Fig. 4. Here, a laser source was coupled into the on-chip
RAMZM by means of a fiber array and grating coupler [18]. The RAMZM output was
taken off the chip via another grating coupler and fiber array. The fiber
array was terminated at a butt-coupled photodetector.

Similarly, by applying 0-7V DC on the ring heaters, the resonances of the
lower (Fig. 5) and upper (Fig. 6) ring were tuned. Here, it can be seen that
as the tuning voltage is increased, multiple resonance peaks start appearing
simply due to the resonance mismatch between the upper and lower rings. To
linearize RAMZM, both microrings need to have their resonances matched. This
can easily be achieved by ‘detuning’ one or both of the rings [22]. It is
important to note that the spectral response shown in Fig. 4 \- Fig. 6 has
been deembedded and normalized for the optical response of grating couplers.

Figure 5: Spectral response of RAMZM (fabricated in IMEC’s ISIPP50G SiP
process) when lower microring bias ($\theta_{DC}$) is tuned by applying 0-7V
on a $n$-doped resistive heater placed around the ring. Figure 6: Spectral
response of RAMZM (fabricated in IMEC’s ISIPP50G SiP process) when upper
microring bias ($\theta_{DC}$) is tuned by applying 0-7V on a $n$-doped
resistive heater placed around the ring.

The aforementioned bias conditions give us the Taylor series coefficients
$\gamma_{1}=\frac{1-\tau}{1+\tau}$ and
$\gamma_{3}=\frac{2\tau^{3}-7\tau^{2}+7\tau-2}{3(1+\tau)(\tau^{2}+2\tau+1)}$.
Interestingly, the third-order nonlinearity can be set to zero by enforcing
$\gamma_{3}=0$, which is achieved by setting $\tau=\frac{1}{2}$. This ensures
that third-order distortion is suppressed, leaving the modulator with fifth or
higher odd-order distortions, which are very miniscule in practical
application scenarios. The optical power transfer function of a linearized
$(\\{\phi_{bias},\theta_{DC},\tau\\}=\\{\pi/2,\pi,0.5\\})$ RAMZM along with
the transfer functions of MZMs (both single and push-pull drive [23]) are
plotted in Fig. 7. As can be seen, the RAMZM is significantly linear compared
to the MZMs when biased at this regime.

Figure 7: RAMZM, MZM (single drive) and MZM (push-pull drive) Optical Transfer
Function as a function of RF voltage induced relative phase shift,
$\theta_{mod}$. Here, the RAMZM is biased at
$\\{\phi_{bias},\theta_{DC},\tau\\}=\\{\frac{\pi}{2},\pi,\frac{1}{2}\\}$ and
both MZMs are quadrature biased.

On chip, modulation signal is applied on the high speed p-n phase shifters
placed around the microring waveguide as shown in Fig. 2. The input RF signal
modulates the carriers present within the microring waveguide, thus modulating
the phase of light. This phenomena is known as plasma dispersion effect [18].
When the antiresonance point ($\theta_{DC}=\pi$) is determined at a particular
wavelength (1555.55nm in our case), the input signal will modulate the output
optical power of RAMZM (as shown in the inset of Fig. 8). In this work, the
p-n phase shifters on microrings are biased at 1.125V and differential
modulating signal was applied on the rings. The corresponding detected
photodiode power exhibits linear response.

Figure 8: RAMZM spectral response when both upper and lower ring p-n phase
shifters are biased at $1.125V$ and applying modulation signal,
$\theta_{mod}(v_{m})$, on both microring p-n phase shifters in a differential
manner. The inset plot on the right shows the modulation of photodiode output
power based on $\theta_{mod}(v_{m})$ at 1555.55nm. The left inset presents
ideal RAMZM transfer function as a function of $\theta_{mod}(v_{m})$.

###  II-E RAMZM Gain and Slope Efficiency

Now, assuming that a plasma dispersion effect based (depletion or forward-bias
mode) phase modulator with linear response is realized (combined effect of
plasma dispersion and DC Kerr effect [11]), we express the phase modulation as
$\theta_{mod}=\frac{\pi v_{m}}{V_{\pi}}$. Here, $V_{\pi}$ is defined as the
voltage required to achieve $\pi$ phase shift in the given length of the phase
modulator. Now, neglecting higher-order modulation terms, we obtain the RAMZM
small signal output optical power as [1]-

$P_{M,o}=\frac{\pi\gamma_{1}P_{I}v_{m}}{V_{\pi}L}$ (13)

Here, the term ‘L’ is used to include the effects of optical loss. If lossy
impedance matching [1] at the input side (as shown in Fig. 11) is implemented,
the differential voltage across the modulator is related to the available
power by

$v_{m}=\frac{v_{s}}{2}=\sqrt{4P_{s,a}R_{s}}$ (14)

Consequently, the small-signal available power gain is given by [1] -

$\displaystyle G_{M}$
$\displaystyle=\frac{P_{M,o}^{2}}{P_{s,a}}=\frac{s_{rmz}^{2}}{R_{s}}=\Big{[}\frac{\pi\gamma_{1}P_{I}R_{s}}{V_{\pi}L}\Big{]}^{2}\frac{1}{R_{s}}$
(15)

where
$s_{rmz}\triangleq\frac{P_{M,o}}{v_{m}}=\frac{\pi\gamma_{1}P_{I}R_{s}}{V_{\pi}L}=\frac{\pi
P_{I}R_{s}}{3V_{\pi}L}$ (when $\tau=\frac{1}{2},\gamma_{1}=\frac{1}{3}$) is
the RAMZM slope efficiency. The effective RAMZM gain can be increased by
increasing the input optical power $P_{I}$ (from a CW laser), reducing the
insertion loss $L$, or decreasing modulator’s $V_{\pi}$. To put that in
perspective, MZM operating in singe and push-pull drive has the slope
efficiencies of $s_{mz}=\frac{\pi P_{I}R_{s}}{2V_{\pi}L}$ and
$s_{mz}=\frac{\pi P_{I}R_{s}}{V_{\pi}L}$, respectively.

###  II-F Lumped Capacitive Drive

Since, in most of the cases, RAMZM is treated as a lumped capacitive element
from the electrical point of view, the voltage across the capacitance when
driven using an external transmission line (of characteristic impedance
$Z_{0}$) can be expressed as

$v_{m}=v_{+}+v_{-}=(1+|\Gamma_{m}|)v_{s}$ (16)

where

$\Gamma_{m}=\frac{Z_{m}-Z_{0}}{Z_{m}+Z_{0}}=\frac{r_{m}-Z_{0}+\frac{1}{j\omega
C_{m}}}{r_{m}+Z_{0}+\frac{1}{j\omega C_{m}}}$ (17)

is the frequency-dependent reflection coefficient. At low frequencies,
$|\Gamma_{m}|\approx 1$ and thus $v_{m}=2v_{+}=2(v_{s}/2)=v_{s}$. On the
contrary, for a transmission-line drive MZM with impedance matching, we have
$v_{m}=v_{s}/2$ [1]. Therefore, for a lumped capacitive drive the small-signal
power gain is

$\displaystyle G_{M}$
$\displaystyle=\Big{[}\frac{2\pi\gamma_{1}P_{I}R_{s}}{V_{\pi}L}\Big{]}^{2}\frac{1}{R_{s}}$
(18)

where $s_{rmz}\triangleq\frac{P_{M,o}}{v_{m}}=\frac{2\pi
P_{I}R_{s}}{3V_{\pi}L}$.

Although lumped capacitive treatment of RAMZM yields better slope efficiency
(2$\times$), we will use $G_{M}$ and $s_{rmz}$ derived in the previous
subsection throughout the rest of this work for a fair comparison between
RAMZM and MZM.

###  II-G Improving RAMZM Gain

It’s evident from the previous analysis and Fig. 7 that the linear response of
RAMZM comes at the cost of the small-signal power gain. While RAMZM provides
excellent distortion cancellation upto 5th order, its slope efficiency is only
67% and 33% of that of an MZM driven in single and push-pull manner,
respectively. To improve the slope efficiency of the RAMZM, the derivative of
the transfer function with respect to $\theta$ can be evaluated to find the
bias point where slope efficiency is the maximum for a given $\tau$. Here, we
propose a biasing scheme where $\theta_{DC}=0$, instead of $\pi$ while keeping
$\phi_{bias}=\frac{\pi}{2}$. The transfer function presented in Eq. 4 can be
simplified and expanded as

$P_{M,o}(\theta_{mod})=P_{I}T(\theta)=\frac{P_{I}}{2}-\gamma_{1}P_{I}\theta_{mod}+\gamma_{3}P_{I}\theta_{mod}^{3}+\mathcal{O}(\theta_{mod}^{5})$
(19)

Here, $\gamma_{1}=\frac{1+\tau}{1-\tau}$ and
$\gamma_{3}=\frac{2\tau^{3}+7\tau^{2}+7\tau+2}{3(\tau-1)(\tau^{2}-2\tau+1)}$.
To optimize the gain and noise performance of the modulator, $\gamma_{1}$
needs to be maximized (i.e. $\tau\to 1$) whereas $\gamma_{3}$ is to be
minimized ($\tau\to 0$). This contradicting criteria for selecting $\tau$
presents a design trade-off between linearity and gain. By choosing
$\tau=\frac{1}{2}$ (i.e $\gamma_{1}=3$), the small signal power gain can be
calculated as

$\displaystyle G_{M}$ $\displaystyle=\Big{[}\frac{3\pi
P_{I}R_{s}}{V_{\pi}L}\Big{]}^{2}\frac{1}{R_{s}}$ (20)

where $s_{rmz}\triangleq\frac{3\pi P_{I}R_{s}}{V_{\pi}L}$. The optical power
transfer function of this RAMZM
$(\\{\phi_{bias},\theta_{DC},\tau\\}=\\{\pi/2,0,0.5\\})$ along with the
transfer functions of MZMs (both single and push-pull drive) are plotted in
Fig. 9. As can be seen, the RAMZM has significantly higher slope efficiency
compared to the MZMs. Here, slope efficiency is 3$\times$ (9.54 dB in power
gain) better than that of MZIs driven in push-pull manner and 6$\times$ (15.56
dB in power Gain) better than that of MZMs operated in single drive [1, 20].
This means, for the same link gain, this biasing scheme will allow upto
6$\times$ reduction in phase shifter length compared to MZMs, enabling lumped
drive, smaller form factor, smaller energy footprint and excellent noise
performance (discussed in Section III). However, this significantly higher
gain and noise performance arrives at the cost of linearity. The linearity
penalty for this biasing scheme will be discussed later in Section IV of this
article.

Figure 9: RAMZM, MZM (single drive) and MZM (push-pull drive) Optical Transfer
Function as a function of RF voltage induced relative phase shift,
$\theta_{mod}$. Here, the RAMZM is biased at
$\\{\phi_{bias},\theta_{DC},\tau\\}=\\{\frac{\pi}{2},0,\frac{1}{2}\\}$ and
both MZMs are quadrature biased.

##  III RAMZM Link Noise Analysis

An analog optical link using RAMZM is shown in Fig. 10. Here, $P_{I}$ and
$P_{M,o}$ are the optical power at the modulator input and output
respectively. $P_{s,a}$ is the input RF power, $P_{o,d}$ and $P_{L}$ are the
optical and then detected RF power at the receiver.

Figure 10: An analog optical link using RAMZI modulator (RAMZM).

Next, the RAMZM based analog optical link gain & noise contribution will be
derived and then utilized to determine the link noise figure. Fig. 11 shows
the small-signal noise model for the analog optical link seen in Fig. 10. The
link parameters used for the subsequent numerical analysis are provided in
Table I. These link parameters are on par with the Silicon Photonic components
that are available from foundry PDKs [24, 12].

TABLE I: Analog Optical Link Parameters Parameter | Value | Unit
---|---|---
Laser input power ($P_{I}$) | 13 | dBm
Laser Relative intensity noise (RIN) | -145 | dB/Hz
Modulator $V_{\pi}$ | 5 | V
Modulator Insertion loss (IL) | 10 | dB
Source Impedance ($R_{s}$) | 50 | Ohm
Detector responsivity ($r_{d}$) | 1.1 | A/W
Load Impedance ($R_{L}$) | 50 | Ohm
BW | 1 | Hz

###  III-A RAMZM Noise Modeling

As seen in Fig. 11, the two rings in the modulator present differential lumped
capacitive load, $C_{M}$, with series electrode resistance, $r_{M}$. Thermal
noise is contributed by the input source resistance ($R_{s}$) with noise power
$kT\Delta f$. The series electrode resistances, $r_{M}$, also contributes
thermal noise. The resulting mean square noise voltage across each $C_{M}$ is
simply $\frac{kT}{C_{M}}$. The net input-referred mean square noise voltage is

$\overline{v_{n,in}^{2}}=\overline{v_{nM,p}^{2}}+\overline{v_{nM,m}^{2}}=\frac{kT}{C_{M}}+\frac{kT}{C_{M}}=\frac{2kT}{C_{M}}$
(21)

The corresponding noise power at the RAMZM output would be
$s_{rmz}^{2}\frac{2kT}{R_{s}C_{d}}$. This can be interpreted as if net thermal
noise is equal to the input noise power $kT\Delta f$, with the corresponding
bandwidth $\Delta f=\frac{1}{2\pi(R_{s}+2r_{M})\cdot
C_{M}/2}=\frac{1}{\pi(R_{s}+2r_{M})\cdot C_{M}}$.

Figure 11: Small-signal noise model for the analog optical link employing the
RAMZM as seen in Fig. 10.

###  III-B Optical Channel

Once the RF signal is converted into the optical domain, it can be processed
by optical signal processing circuits, such SiP filters [25] and phased-arrays
[26], and then transported over long distances on a single-mode optical fiber.
Here, we can safely assume that the optical device/channel is passive and
noiseless. However, on-chip or in-package optical gain using a semiconductor
optical amplifier (SOA) or erbium-doped fiber amplifier (EDFA) is also
possible with the associated noise figure [27]. The optical channel power gain
is given as $G_{ch}\triangleq\frac{P_{o,d}}{P_{M,o}}=\frac{1}{L_{ch}}$, where
$L_{ch}$ is the optical power loss.

###  III-C Photodetector

Photo-detection in silicon-based PICs is realized using on-chip Germanium
detectors [28]. The detector is either interfaced with a passive load $R_{L}$,
or a transimpedance amplifier (TIA). In this work, only passive load is
considered. The detected power in the load is
$P_{L}=i_{d}^{2}R_{L}=(r_{d}\cdot P_{o,d})^{2}R_{L}$ and thus detector gain is
[1]

$G_{det}\triangleq\frac{P_{L}}{p_{o,d}^{2}}=\frac{(r_{d}P_{o,d})^{2}R_{L}}{P_{o,d}^{2}}=r_{d}^{2}R_{L}$
(22)

where $r_{d}$ is the detector responsivity.

###  III-D RAMZM Link Gain

In the previous section, we developed the gain of RAMZM. To be able to find
the gain of the entire link, the product of modulator and detector gain must
be evaluated. For simplicity, it is assumed that there is no optical loss/gain
element ($\frac{p_{o,d}^{2}}{P_{M,o}^{2}}=1$) between modulator & detector and
$R_{s}=R_{L}$. Therefore, the entire link gain can be expressed as-

$G=\frac{P_{M,o}^{2}}{P_{s,a}}\cdot\frac{p_{o,d}^{2}}{P_{M,o}^{2}}\cdot\frac{P_{L}}{p_{o,d}^{2}}=\Big{[}\frac{\pi\gamma_{1}P_{I}R_{s}r_{d}}{V_{\pi}L}\Big{]}^{2}$
(23)

Fig. 12 shows the contour plot for the RAMZM link gain, $G$, for various bias
conditions $(\theta_{DC},\tau)$. As described in the previous section, the
link gain is the highest when the thermal phase shifters on the RAMZM arm are
tuned in such way that $\theta_{DC}=0$.

Figure 12: Contour plot of RAMZM-based analog optical link Gain at different
bias conditions $(\theta_{DC},\tau)$ while keeping
$\phi_{bias}=\frac{\pi}{2}$. The link parameters employed are listed in Table
I.

###  III-E RAMZM Link Noise Figure and Ring Biasing

The noise figure (NF) for the entire link is given by [1]

$NF=10\cdot log_{10}\Big{(}\frac{N_{out}}{GN_{i}}\Big{)}$ (24)

where $N_{out}$ is the total noise power at the link output, $N_{i}$ is the
input noise, and $G$ is the link gain. The input noise power is due to the
source resistance and given by $N_{i}=kT\Delta f$ [1]. By observing Fig. 11,
the noise components of the link can be summarized as [1]

  1. 1.

Laser relative intensity noise (RIN), $\overline{i_{rin}^{2}}\Delta f$

  2. 2.

Detector shot noise, $\overline{i_{sn}^{2}}\Delta f$

  3. 3.

Thermal noise from the load matched resistance of the detector
($R_{L}=R_{pd}$), $kT\Delta f$

RIN of the laser source is expressed in dB/Hz and the resulting RIN noise
current spectral density (CSD) is given by
$\overline{i_{RIN}^{2}}=\frac{\overline{I_{D}}^{2}}{2}10^{\frac{RIN}{10}}$
where $\overline{I_{D}}=r_{d}P_{av}=\frac{r_{d}P_{I}}{2L}[1+cos(\phi_{bias})]$
is average detector current [1]. It is important to note that
$P_{av}=\frac{P_{I}}{2L}[1+cos(\phi_{bias})]$ comes from Eq. 12, weighted by
optical loss L. The detector shot noise current spectral density is described
as $\overline{i_{sn}^{2}}=2q\overline{I_{D}}$. Thus, RIN and shot noise CSD
depend upon $\overline{I_{D}^{2}}$ and $\overline{I_{D}}$ respectively.

When lossy impedance matching is implemented at the input and output side, the
total noise for the link is derived as

$N_{out}=\frac{1}{4}(\overline{i_{rin}^{2}}+\overline{i_{sn}^{2}})\Delta
fR_{L}+(1+G)kT\Delta f$ (25)

Consequently, the NF of the link is expressed as

$NF=10\cdot
log_{10}\Big{[}1+\frac{R_{L}}{4kTG}\Big{(}\frac{\overline{I_{D}}^{2}}{2}10^{\frac{RIN}{10}}+2q\overline{I_{D}}\Big{)}+\frac{1}{G}\Big{]}$
(26)

The corresponding plot of NF for the different bias conditions
$(\theta_{DC},\tau)$ is illustrated in Fig. 13. Here, it is evident that for
$\theta_{DC}=0$, NF is maximized due to the large link gain.

Figure 13: Contour plot of RAMZM-based analog optical link NF at different
bias conditions $(\theta_{DC},\tau)$ while keeping
$\phi_{bias}=\frac{\pi}{2}$. The link parameters employed are the same as in
Table I.

On the other hand, if a passive transformer with turn ratio, $N_{D}$, is used
to match diode to the load, $R_{L}$, link NF in Eq. 24 is derived as-

$NF=10\cdot
log_{10}\Big{[}1+\frac{N_{D}^{2}R_{s}}{kTG}\Big{(}\frac{\overline{I_{D}}^{2}}{2}10^{\frac{RIN}{10}}+2q\overline{I_{D}}\Big{)}+\frac{1}{G}\Big{]}$
(27)

Fig. 14 plots Eq. 27 with respect to the small-signal power gain, $G$. If the
detector noise terms are ignored, then for large optical attenuation (i.e.
$G\ll 0~{}dB$), the load thermal noise dominates the detector noise terms and
Eq. 27 reduces to the passive attenuation limit
$NF=log_{10}\Big{(}1+\frac{1}{G}\Big{)}$. For $G\gg 0~{}dB$, the NF
asymptotically approaches the lossless match limit of 0 dB. One can observe
that the link NF is rather high due to the detector noise, as
$\overline{I_{D}}$ depends upon $\frac{P_{I}}{2L}[1+cos(\phi_{bias})]$. This
can be improved by biasing the arms of the modulator at different angle,
$\phi_{bias}>\frac{\pi}{2}$ [1, 20], or by filtering out the optical carrier
[29, 20] using an on-chip SiP filter [30].

Figure 14: NF versus gain for a RAMZM-based analog optical link. Here, $P_{I}$
and $L_{ch}$ are varied to sweep the gain. Also, $P_{I}<30dBm$ while other
link parameters are from Table I.

As discussed before, the slope efficiency of RAMZM, when perfect third-order
nonlinearity cancellation is achieved, is $\frac{1}{3}$ (or 9.54 dB lower
power gain) compared to the MZMs (push-pull drive). This lower gain also
result in higher NF in RAMZM, approximately by the same proportion. On the
other hand, the gain enhanced RAMZM will have 9.54 dB improvements in both
power gain and NF.

###  III-F Null Biasing of RAMZM

As discussed earlier, RIN and shot noise CSD depend upon
$\overline{I_{D}^{2}}$ and $\overline{I_{D}}$ respectively. This means,
reducing $\overline{I_{D}}$ improves noise performance of the link, especially
when the link is RIN limited ($\overline{I_{D}^{2}}$ dependence). Since
$\overline{I_{D}}=\frac{r_{d}P_{I}}{2L}[1+cos(\phi_{bias})]$, by changing
$\phi_{bias}$ (null or low biasing [31, 32]) we can reduce the overall noise
as shown in Fig. 15.

Figure 15: RAMZM Noise PSD vs arm bias angle ($\phi_{bias}$) for different
input optical power, $P_{I}$. Other link parameters employed are from Table I.

Although the link gain reduces along with noise in null biasing technique
(also known as low-biasing in the context of MZMs), the overall advantage in
NF can be achieved, especially when a high laser power ($P_{I}$) is used as
illustrated in Fig. 16.

Figure 16: RAMZM NF vs arm bias angle ($\phi_{bias}$) for different input
optical power. The link parameters for these plots are from Table I.

On the other hand, although mode evolution based photodiodes (PDs) provide
higher saturation power over butt-coupled PDs [28], the use of high optical
input power in integrated photonics is still limited due to poor saturation
power performances of the PDs. In such cases, null biasing can be implemented
in the RAMZM based analog optical link. When the link gain is limited by PD
saturation, by employing null biasing one can increase the link gain without
increasing PD current just by increasing the laser power (as shown in Fig.
17).

Figure 17: Contour plot of Link Gain for various arm bias ($\phi_{bias}$) and
input optical power. The link parameters for the analysis are from Table I.

In Fig. 17, a constant photocurrent of 15.5mA [28] is plotted over the input
laser power and RAMZM arm bias versus Gain. It can be observed that by
increasing laser power from 25 to 30dBm, it’s possible to achieve a Link Gain
advantage of $>$6dB when RAMZM arm bias is adjusted without having to increase
the PD current.

##  IV RAMZM Link Distortion Analysis

Fig. 10 had two active components that contribute distortion in the RF
photonic link, namely the modulator and the detector. Unlike RAMZM, detector
distortion does not arise from the optical to photo-current transfer
characteristics, but from the higher-order effects [1]. These experimentally
characterized effects depend upon the reverse bias voltage, $V_{b,pd}$, and
$\overline{I_{D}}$, and can be minimized by using a sufficiently large
$V_{b,pd}$ and avoiding detector saturation (by restricting $\overline{I_{D}}$
using null biasing technique) [1]. Since the distortion in an analog photonic
link is dominated by the modulator nonlinearity, this study of link distortion
analysis only focuses on RAMZM. Also, since harmonic distortions only appear
in very broadband links, here we limit our discussion only to intermodulation
distortions.

###  IV-A Second and Third-order Intercept Point

From Eq. 12, the optical power of the third-order intermodulation distortion
for a two-tone input, $\theta_{mod}=\frac{\pi
V_{m}}{V_{\pi}}\cdot[sin(\omega_{1}t)+sin(\omega_{2}t)]$, can be written as
$P_{optical-im3}=\frac{P_{I}}{L}\cdot\frac{3\gamma_{3}}{4}\cdot(\frac{\pi
V_{m}}{V_{\pi}})^{3}cos(2\omega_{1}\pm\omega_{2})$ at
$2\omega_{1}\pm\omega_{2}$ [1]. On the other hand, its fundamental counterpart
can be expressed as $P_{optical-
fund}=\frac{P_{I}}{L}\cdot\gamma_{1}\cdot(\frac{\pi
V_{m}}{V_{\pi}})cos(\omega_{1})$ at $\omega_{1}$. Since a lossy impedance
matching is considered at the photodiode output, the output electrical powers
are expressed as

$P_{fund}=\big{(}\frac{1}{2}r_{d}P_{optical-fund}\big{)}^{2}R_{L}$ (28)

$P_{im3}=\big{(}\frac{1}{2}r_{d}P_{optical-im3}\big{)}^{2}R_{L}$ (29)

Using $V_{m}=\sqrt{2P_{in}R_{s}}$ in Eq. 28 and 29, we can express all
electrical output powers in terms of input electrical power, $P_{in}$ as-

$P_{fund}=\Big{(}\frac{\pi^{2}r_{d}^{2}P_{I}^{2}R_{s}R_{L}\gamma_{1}^{2}}{4V_{\pi}^{2}L^{2}}\Big{)}\cdot
P_{in}$ (30)

$P_{im3}=\Big{(}\frac{9\pi^{6}r_{d}^{2}P_{I}^{2}R_{s}^{3}R_{L}\gamma_{3}^{2}}{16V_{\pi}^{6}L^{2}}\Big{)}\cdot
P_{in}^{3}$ (31)

Now, the $P_{in}$ for which $P_{fund}=P_{im3}$ is called the third order
intercept point (IIP3) [33]. Therefore, equating Eqs. 30 and 31 and solving
for $P_{in}$ results in

$IIP_{3}=\frac{2V_{\pi}^{2}\big{[}\tau^{2}-2\cos(\theta_{DC})\tau+1\big{]}^{2}}{\pi^{2}R_{s}\big{[}2\cos(\theta_{DC})^{2}\tau^{2}+(\tau^{3}+\tau)\cos(\theta_{DC})+2\tau^{4}-8\tau^{2}+2\big{]}}$
(32)

Similarly, it can be shown that

$IIP_{2}=\frac{V_{\pi}^{2}\big{[}\tau^{2}-2\cos(\theta_{DC})\tau+1\big{]}^{2}}{2\pi^{2}R_{s}(\tau^{2}-1)^{2}}\cdot\tan(\phi_{bias})^{2}$
(33)

From these expressions, it can be noted that both IIP2 and IIP3 are strong
functions of $V_{\pi}$. Also, the IIP3 of an RAMZM is independent of the arm
bias ($\phi_{bias}$), but strongly dependent upon ring bias ($\theta_{DC}$)
and the coupling ratio. On the other hand, unlike IIP3, IIP2 strongly depends
upon arm biasing.

To show the dynamic characteristics, relative power of the fundamental, IM2
and IM3 are plotted against the modulation angle of the RAMZM in Fig. 18.
Here, 1st, 2nd and 3rd derivatives of the RAMZM transfer function are
calculated to find the relative power of the fundamental, IM2 and IM3
distortions considering
$\\{\phi_{bias},\theta_{DC},\tau\\}=\\{\frac{\pi}{2},\pi,\frac{1}{2}\\}$.
Here, we see that for small modulation stimulus, the IM2 and IM3 are virtually
non-existent.

Figure 18: RAMZM Fundamental, second and third-order distortion components vs
ring modulation angle, $\theta_{mod}$. Here, the RAMZM is biased at
$\\{\phi_{bias},\theta_{DC},\tau\\}=\\{\frac{\pi}{2},\pi,\frac{1}{2}\\}$

To provide a comprehensive view of the link gain and total noise, SFDR is
evaluated and plotted in Fig. 19) using the method described in [33] and link
parameters listed in Table I. Here, we see that the highest SFDR of $\sim 128$
dB/Hz2/3 is obtained when the rings are biased in anti-resonance and
$\tau=\frac{1}{2}$. As ring bias are tuned away from anti-resonance, a narrow
band for coupling ratio $\tau$ exists for highest SFDR. The generated
countours in Figs. 12, 13 and 19 can be used together to make design trade-
offs between the link NF and distortion.

Figure 19: Contour plot of RAMZM-based analog optical link SFDR in
$dB/Hz^{\frac{2}{3}}$ at different bias conditions, keeping
$\phi_{bias}=\frac{\pi}{2}$. The link parameters employed are listed in Table
I.

###  IV-B Linearity of RAMZM and MZM

As discussed before, MZM suffers from higher nonlinearity due to its
sinusoidal transfer function. To quantify the linearity of both MZM and
linearized RAMZM, $SFDR_{3}$ is calculated in Fig. 20. Here, RAMZM and MZM
have SFDR values of 128.16 and 109.94 $dB/Hz^{\frac{2}{3}}$ at 1GHz,
respectively. This is 18dB improvement over regular MZMs, which makes RAMZM
very attractive for high performance RF photonic applications.

Figure 20: SFDR performance of RAMZM and MZM in a Photonic Link. Here, RAMZM
and MZM bias conditions are
$\\{\phi_{bias},\theta_{DC},\tau\\}=\\{\frac{\pi}{2},\pi,\frac{1}{2}\\}$ and
$\\{\phi_{bias}\\}=\\{\frac{\pi}{2}\\}$, respectively. Also, the link
parameters are from Table I.

###  IV-C Linearity of Gain Enhanced (GE) RAMZM

In section II, we proposed a new biasing scheme where slope efficiency of
RAMZM is upto 6$\times$ higher than that of MZMs. However, as previously
discussed, this gain enhancement comes at the expense of linearity since
$\theta_{DC}$ has to be shifted from $\pi$ towards 0. In Fig. 21, we show that
the SFDR of GE RAMZM is about 109.57 $dB/Hz^{\frac{2}{3}}$ at 1GHz. It is
interesting to note that the GE RAMZM provides upto 6$\times$ improvement in
slope efficiency over MZM while still providing the same SFDR performance as
an MZM.

Figure 21: Comparison of SFDR performance of RAMZMs in a Photonic Link with
different bias conditions. Here, Linearized and Gain Enhanced (GE) RAMZM bias
conditions are
$\\{\phi_{bias},\theta_{DC},\tau\\}=\\{\frac{\pi}{2},\pi,\frac{1}{2}\\}$ and
$\\{\frac{\pi}{2},0,\frac{1}{2}\\}$, respectively. Also, the link parameters
are from Table I.

##  V Conclusion

In this work, we presented analysis for silicon photonic RAMZM-based analog
optical link that provides significant improvement in linearity over the MZM-
based links. It is shown that the linearized RAMZM-based link provides
18dB/H$z^{\frac{2}{3}}$ higher SFDR than its MZM counterpart. Using analytical
expressions, we have also shown how link and RAMZM parameters affect the link
Gain, Noise, Linearity performances. The associated link design trade-offs are
also explored. Using an on-chip RAMZM, fabricated in IMEC’s ISIPP50G Silicon
Photonic process, the multi-biasing process is described and explained. We
have also proposed a novel biasing scheme which improves link gain and NF by
upto 15.5dB (compared to MZM-based links). This will enable the use of smaller
depletion mode/forward biased phase-shifters (lumped capacitive element) in
Silicon Photonic technology. This will significantly reduce the complexity of
modulator driver design since transmission line drive will not be required.
Moreover, null biasing technique is explored for photodiode saturation limited
RAMZM-based optical link. This study will provide the designers with in-depth
understanding into the design tradeoffs and nuances of high performance
Silicon Photonic modulator based link design.

##  VI Acknowledgment

The authors gratefully acknowledge the generous funding support from the Air
Force Office of Sponsored Research (AFOSR) YIP Award FA9550- 17-1-0076 and the
National Science Foundation (NSF) CAREER Award EECS-2014109.

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