PHYSICAL REVIEW APPLIED 15, 034056 (2021) Microwave Frequency Demodulation Using two Coupled Optical Resonators with Modulated Refractive Index Adam Mock * School of Engineering and Technology, Central Michigan University, Mount Pleasant, Michigan 48859, USA (Received 16 October 2020; revised 1 February 2021; accepted 10 February 2021; published 18 March 2021) Traditional electronic frequency demodulation of a microwave frequency voltage is challenging because it requires complicated phase-locked loops, narrowband filters with fixed passbands, or large footprint local oscillators and mixers. Herein, a different frequency demodulation concept is proposed based on refractive index modulation of two coupled microcavities excited by an optical wave. A frequency- modulated microwave frequency voltage is applied to two photonic crystal microcavities in a spatially odd configuration. The spatially odd perturbation causes coupling between the even and odd supermodes of the coupled-cavity system. It is shown theoretically and verified by finite-difference time-domain sim- ulations how careful choice of the modulation amplitude and frequency can switch the optical output from on to off. As the modulating frequency is detuned from its off value, the optical output switches from off to on. Ultimately, the optical output amplitude is proportional to the frequency deviation of the applied voltage making this device a frequency-modulated-voltage to amplitude-modulated-optical- wave converter. The optical output can be immediately detected and converted to a voltage that would result in a frequency-demodulated voltage signal. Or the optical output can be fed into a larger radio- over-fiber optical network. In this case the device presents a compact, low power, and tunable route for multiplexing frequency-modulated voltages with amplitude-modulated optical communication systems. The resulting system requires modest modulation amplitudes and operates at frequencies relevant for modern communication systems. The cavity designs have realistic quality factors that are well within the range of experimental implementation. The role of modulation sidebands in reducing switching contrast is explored, and two methods for mitigating these effects are demonstrated. DOI: 10.1103/PhysRevApplied.15.034056 I. INTRODUCTION The manipulation of eigenmodes in coupled optical res- onators is a well-studied component of photonic device design [1–3]. Spatiotemporal modulation of the cavity material offers an additional mechanism by which photonic signal processing capabilities can be implemented [4–6]. For example, spatiotemporal modulation can be used to break time-reversal symmetry resulting in optical isolation [7–9], optical circulation [10], and optical orbital angu- lar momentum generation [11]. On the other hand, the interplay of microwave frequency electronic signals with visible and near-infrared photonic signals underpins many of the capabilities of modern microwave photonic devices [12,13]. This work combines these two perspectives to * [email protected]Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Fur- ther distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. propose a compact microwave-frequency-controlled pho- tonic switch based on time modulation of two coupled pho- tonic resonators. The overall concept is shown in Fig. 1. A continuous wave (cw) optical carrier is coupled into one cavity resonator of a two-coupled-cavity resonator system. The refractive indices of the two cavities are modulated at a microwave frequency and 180 ◦ out of phase. The ampli- tude of the resultant output optical wave is then modulated in response to the frequency of the modulating voltage. Control of an optical output intensity by the frequency of an applied voltage is unique to this device topology and is not an effect that can be readily achieved using existing devices such as a standard Mach-Zehnder intensity modu- lator. Two tandem phase modulators could potentially be used to achieve the same effect though at the expense of larger foot print and higher electrical drive power [14,15]. Frequency modulation (FM) is a classic modulation format recognized for its improved noise immunity over that of amplitude modulation (AM) that comes at the expense of larger bandwidth usage and more complicated modulation and demodulation hardware [16]. However, because FM signals are transmitted at a constant maxi- mized power level, need for power regularization circuitry 2331-7019/21/15(3)/034056(15) 034056-1 Published by the American Physical Society
Transcript
PHYSICAL REVIEW APPLIED 15, 034056 (2021)
Microwave Frequency Demodulation Using two Coupled Optical Resonators withModulated Refractive Index
Adam Mock *
School of Engineering and Technology, Central Michigan University, Mount Pleasant, Michigan 48859, USA
(Received 16 October 2020; revised 1 February 2021; accepted 10 February 2021; published 18 March 2021)
Traditional electronic frequency demodulation of a microwave frequency voltage is challenging becauseit requires complicated phase-locked loops, narrowband filters with fixed passbands, or large footprintlocal oscillators and mixers. Herein, a different frequency demodulation concept is proposed based onrefractive index modulation of two coupled microcavities excited by an optical wave. A frequency-modulated microwave frequency voltage is applied to two photonic crystal microcavities in a spatiallyodd configuration. The spatially odd perturbation causes coupling between the even and odd supermodesof the coupled-cavity system. It is shown theoretically and verified by finite-difference time-domain sim-ulations how careful choice of the modulation amplitude and frequency can switch the optical outputfrom on to off. As the modulating frequency is detuned from its off value, the optical output switchesfrom off to on. Ultimately, the optical output amplitude is proportional to the frequency deviation ofthe applied voltage making this device a frequency-modulated-voltage to amplitude-modulated-optical-wave converter. The optical output can be immediately detected and converted to a voltage that wouldresult in a frequency-demodulated voltage signal. Or the optical output can be fed into a larger radio-over-fiber optical network. In this case the device presents a compact, low power, and tunable route formultiplexing frequency-modulated voltages with amplitude-modulated optical communication systems.The resulting system requires modest modulation amplitudes and operates at frequencies relevant formodern communication systems. The cavity designs have realistic quality factors that are well within therange of experimental implementation. The role of modulation sidebands in reducing switching contrastis explored, and two methods for mitigating these effects are demonstrated.
DOI: 10.1103/PhysRevApplied.15.034056
I. INTRODUCTION
The manipulation of eigenmodes in coupled optical res-onators is a well-studied component of photonic devicedesign [1–3]. Spatiotemporal modulation of the cavitymaterial offers an additional mechanism by which photonicsignal processing capabilities can be implemented [4–6].For example, spatiotemporal modulation can be used tobreak time-reversal symmetry resulting in optical isolation[7–9], optical circulation [10], and optical orbital angu-lar momentum generation [11]. On the other hand, theinterplay of microwave frequency electronic signals withvisible and near-infrared photonic signals underpins manyof the capabilities of modern microwave photonic devices[12,13]. This work combines these two perspectives to
Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license. Fur-ther distribution of this work must maintain attribution to theauthor(s) and the published article’s title, journal citation, andDOI.
propose a compact microwave-frequency-controlled pho-tonic switch based on time modulation of two coupled pho-tonic resonators. The overall concept is shown in Fig. 1.A continuous wave (cw) optical carrier is coupled into onecavity resonator of a two-coupled-cavity resonator system.The refractive indices of the two cavities are modulated ata microwave frequency and 180◦ out of phase. The ampli-tude of the resultant output optical wave is then modulatedin response to the frequency of the modulating voltage.Control of an optical output intensity by the frequency ofan applied voltage is unique to this device topology andis not an effect that can be readily achieved using existingdevices such as a standard Mach-Zehnder intensity modu-lator. Two tandem phase modulators could potentially beused to achieve the same effect though at the expense oflarger foot print and higher electrical drive power [14,15].
Frequency modulation (FM) is a classic modulationformat recognized for its improved noise immunity overthat of amplitude modulation (AM) that comes at theexpense of larger bandwidth usage and more complicatedmodulation and demodulation hardware [16]. However,because FM signals are transmitted at a constant maxi-mized power level, need for power regularization circuitry
2331-7019/21/15(3)/034056(15) 034056-1 Published by the American Physical Society
FIG. 1. Depiction of the frequency demodulation concept. An optical carrier is inputted from the left. Two coupled optical resonatorsare modulated by a frequency-modulated microwave (MW) frequency voltage VMW(t). The frequency modulation shown is binaryFSK. The output optical signal is an amplitude-modulated version of the FSK electrical signal. This optical output can be combinedwith a larger optical communication system or it can be directly fed to a photodiode for electrical demodulation of the FM signal.
is reduced. Digital signaling is implemented in FM sys-tems by assigning a binary zero to an assigned frequencyf0 = fc − fd/2 (“space” frequency) and a binary one to adifferent assigned frequency f1 = fc + fd/2 (“mark” fre-quency), where fc is the carrier frequency and fd is thedifference frequency. The minimum difference frequencyto maintain orthogonal signal waveforms is fd = 1/(2T),where T is the signal period, and most traditional demodu-lation approaches require fd = 1/T. Overall, such a schemeis referred to as binary frequency shift keying (FSK) andcan be extended to M -ary modulation formats as well. Itsconstant power transmission and enhanced immunity tonoise make FSK popular in radio frequency identification(RFID) technologies [17], biomedical implant communi-cation [18], Bluetooth [19], and a variety of home wirelesstechnologies [20]. By the same token, FSK is attractive invisible light digital communication systems to avoid flickerwhen the carrier wave doubles as a room lighting source[21,22]. FSK is also prevalent in radar systems in whichranging and doppler shifts are more accurately measuredthan in pulsed time-of-flight implementations [23].
The microwave-frequency-controlled switch depicted inFig. 1 amplitude modulates a cw optical carrier wave inresponse to an applied FM voltage. If the message signalembedded in the FM voltage is a generic analog signal thenthe output optical wave is amplitude modulated with thesame analog message. If the FM voltage is a FSK signalthen the digital information is imparted on the output opti-cal wave in the form of amplitude shift keying (ASK) oron-off keying (OOK). If a photodiode is used immediatelyto convert the output optical wave into an electrical signal,the proposed switch is a frequency demodulator, producinga demodulated electrical message signal from an electri-cal frequency-modulated input signal. As discussed later inthis work, this electrical-optical-electrical FM demodula-tor approach is dynamically tunable, so that a single devicecan be used for different frequency ranges and frequency
deviations which makes it advantageous over traditionalapproaches of FSK demodulation that use narrowband fil-ters or local oscillators with fixed center frequencies. Inthe case of general FM demodulation it avoids the needfor a phase-locked loop (PLL) and works for virtuallyany microwave frequency. Primary power requirements ofthe proposed electrical-optical-electrical demodulator areto drive the chip-scale laser source that is similar to that ofan electronic PLL or local oscillator.
If the output optical wave is not immediately detectedand converted to a voltage, it can be multiplexed witha larger radio-over-fiber optical communication network.In this implementation, the proposed device functional-ity is attractive as few if any existing compact devicesperform the specific task of directly amplitude modulat-ing an optical wave in response to the frequency of anapplied voltage. Use of radio over fiber is ubiquitous inwireless communication [24], and microwave and mil-limeter wave wireless frequencies are a significant andgrowing component in 5G communication systems [25].Using conventional demodulation approaches, a FM orFSK electronic signal would need to be converted to anamplitude-modulated electronic signal before it can ampli-tude modulate an optical carrier to join the larger opticalcommunication network. The proposed device obviates theneed for FM to AM conversion in the electrical domain,thereby improving the power efficiency and SNR of theprocess.
In the following, a coupled mode theory (CMT) modelis used to develop the device concept, and results fromfull-wave finite-difference time-domain (FDTD) electro-dynamic simulations confirm device properties derivedvia CMT. It is shown how the device may be imple-mented in a one-dimensional (1D) photonic crystal (PC)platform. The 1D PC consists of circular air holes etchedinto a semiconductor ridge waveguide of width less than400 nm. The device length is 12.5 μm. Such a small
device footprint is advantageous for dense integration andfor enhanced light-matter interaction. A refractive indexchange of �n = 1.48 × 10−3 with a modulation frequencyfm = 23.8 GHz are shown numerically to produce an on-offratio of 26.4 dB. These results show that this device designis consistent with well-established experimental capabili-ties [26–32]. Furthermore, it is shown that there are a widerange of modulation amplitudes and frequencies that maybe used to switch the optical output on and off in a sin-gle device design. The effect of modulation sidebands isaddressed, and effective techniques for their suppressionare introduced.
II. COUPLED MODE THEORY ANALYSIS ANDSCATTERING MATRIX
A. Modes of the two-coupled-cavity system
Consider the two-cavity system depicted in Fig. 2(a). Toobtain design constraints on the proposed system, coupledmode theory analysis is used to find the scattering spectraand their dependence on device parameters [10,33–35]. Itis assumed that the fields of the coupled two-cavity sys-tem may be constructed from the fields of the isolatedresonators according to [36]
|F(t)〉 = a1(t) |F1〉 + a2(t) |F2〉 , (1)
where |F1〉 and |F2〉 represent the spatial distributions ofthe vector fields in resonators 1 and 2, respectively, inthe absence of coupling; and a1(t) and a2(t) denote thetime-dependent expansion coefficients. Inserting Eq. (1)into Maxwell’s equations, and viewing resonator 1 (2) as asmall perturbation to resonator 2 (1) results in
da1(t)dt
= (−iω0 − γ0)a1 − iκa2, (2a)
da2(t)dt
= (−iω0 − γ0)a2 − iκa1. (2b)
A detailed derivation of Eqs. (2) is provided in Ref. [37].Letting ak(t) = ak e−iωt for k = 1, 2 results in
−iωa1 = (−iω0 − γ0)a1 − iκa2, (3a)
−iωa2 = (−iω0 − γ0)a2 − iκa1. (3b)
These equations can be written as a matrix
−iω[
a1a2
]=
[−iω0 − γ0 −iκ−iκ −iω0 − γ0
] [a1a2
], (4)
from which the eigenfrequencies of the two-resonatorsystem are found to be ω± = ω0 ± κ − iγ0. This showsthat the nominal resonance frequency ω0 is split to valuesslightly above and below this value with a similar effecton the loss rate. The corresponding eigenvectors are givenby a+ = [1, 1]T/
√2 and a− = [1, −1]T/
√2, and the full
eigenstates are given by
|F±(t)〉 = 1√2
[|F1〉 ± |F2q〉] e−iω±t. (5)
These eigenstates can be viewed as even (+) and odd (−)supermodes, as depicted in Fig. 2(b).
Figure 2(c) depicts the 1D PC cavity platform in whichthe two-coupled-cavity system is implemented. The peri-odic array of air holes creates a photonic bandgap whosecenter frequency is tuned with hole size and waveguidewidth. A resonant cavity is formed by removing holesfrom the array. In this case, three holes are removed toform a cavity with a high quality factor and small footprint[38–41]. A second cavity is coupled to the first one asdepicted in the figure. The gray regions denote semicon-ductor material with refractive index n = 3.1 consistentwith electrorefractive materials suitable for microwavephotonics such as silicon and indium phosphide [26,27].The hole radii are r = 0.3w, where w is the width of thewaveguide, and the hole spacing is equal to the waveguide
ω0 ω0κ
21
x
x
(a)
(b)
ω0 ω0κ
21
x
y
(c)
(d)
w
+max
–max
0
Λ
|F+
|F–
FIG. 2. (a) Two optical resonators defined by a spatially varying electric permittivity ε(r) whose proximity results in couplingdenoted by κ . The resonators are assumed identical with a nominal resonance frequency ω0. (b) Schematic depiction of the even |F+〉and odd |F−〉 supermodes along the x direction. (c) One-dimensional photonic crystal geometry with two defect cavities. (d) Spatialdistribution of the magnetic field Hz(x, y) for the even and odd supermodes of the system schematically depicted in (b) calculatedusing the 2D FDTD method.
034056-3
ADAM MOCK PHYS. REV. APPLIED 15, 034056 (2021)
width � = w. A value of w = 0.372 μm tunes the deviceto operate at λ0 = 1.55 μm. The coupling rate κ may betuned by the number of air holes between the two cavi-ties; for example, κ increases as the cavities are broughtcloser together. Overall, these geometrical specificationsare inline with well-established fabrication capabilities[28,42–44].
It is worth pointing out that the demodulator conceptcan be implemented in any geometry with waveguides andtwo coupled cavities having nondegenerate resonances.This includes defect waveguides and cavities in a two-dimensional PC lattice [10,45]. However, the nanoscaleridge waveguide with a localized 1D PC shown in Fig. 3(b)is easier to fabricate, easier to design, and will likely havelower surface scattering than a full two-dimensional PClattice while offering the same functionality. Microring res-onators are another popular and well-understood cavitysystem with potential to implement the proposed demod-ulation concept; however, due to the degenerate (or nearlydegenerate) counterpropagating whispering gallery modesin a single resonator, a separate analysis would need to beundertaken to determine the modulation format to obtainsimilar functionality.
Figure 2(d) shows the spatial distribution of the mag-netic field Hz(x, y) for the even and odd supermodes of thesystem depicted in Fig. 2(c) calculated using the 2D FDTDmethod. The figure shows the field in the x-y plane, wherethe [Ex(x, y), Ey(x, y), Hz(x, y)] polarization is used. Theeven and odd parity along the x direction is apparent alongwith the mode confinements along x and y.
B. Coupling to the two-coupled-cavity system withinput and output waveguides
Light energy will be inserted into the two-resonator sys-tem via an input waveguide, and the output energy will be
collected via an output waveguide as shown in Fig. 3(a).To account for these energy transfers, Eqs. (3) are modifiedaccording to
−iωa1(ω) = [−iω0 − γ0 − γc1]a1(ω)
− iκa2(ω) + d1si1(ω), (6a)
−iωa2(ω) = [−iω0 − γ0 − γc2]a2(ω)
− iκa1(ω) + d2si2(ω), (6b)
where dl is the coupling rate from waveguide l and sil is theinput amplitude in waveguide l [1,10,33–35]. In this work,waveguides 1 and 2 are the input and output waveguides,respectively, and are symmetrically coupled, so d1 = d2 ≡d. The waveguide-loaded cavities have a decay rate aug-mented by γc1 = γc2 ≡ γc. The coupling rate and decayrate are related via 2γc = d2. In the waveguide-coupled-cavity system, the frequency ω is the excitation frequency,and the amplitudes al(ω) are continuous functions of ω.To quantify the energy output, the mode amplitudes arerelated to the output waves via srl(ω) = −sil(ω) + dal(ω)
for l = 1, 2. Figure 3(b) depicts waveguides 1 and 2 imple-mented in the 1D PC ridge waveguide configuration.Similar to the way κ is controlled by the number of airholes between the cavities, the waveguide coupling rateγc is controlled by the number of holes between the uni-form ridge waveguide and the defect. Figure 3(c) showsthe field distribution when the system is excited at ω±.The relation to the even and odd supermodes is visible,but the amount of energy in the two cavities is no longerequal, which is consistent with theory predictions whenthe energy is incident from only one side. Figure 3(d)shows the output scattering spectra for waveguides 1 and2 assuming input in waveguide 1 for typical values ofω0, κ , d, and γ0. The scattering parameters are defined
ω0 ω0κ
21(a) (d)d d1 2 1.0
0.8
0.6
0.4
0.2
0.0
Sca
tterin
g co
effic
ient
192.1 192.9 193.7 194.5 195.3Frequency (THz)
|S11|2 CMT
|S11|2 FDTD
|S21|2 CMT
|S21|2 FDTD
ω0 ω0
κd d21 21(b)
si1
sr1 sr2
21 +max
–max
0
(c)
ω– ω+
ω
ω
si1
sr1 sr2ω0
x
y
_
+
FIG. 3. (a) Diagram showing resonators 1 and 2 coupled to waveguides 1 and 2, respectively. Here si1 and sr1 denote the incident andreflected wave amplitudes in waveguide 1, and sr2 denotes the reflected wave amplitude in waveguide 2. The system is excited fromwaveguide 1, which makes waveguide 1 the input waveguide and waveguide 2 the output waveguide. (b) Two photonic crystal defectcavities coupled to waveguides. (c) Spatial distribution of the magnetic field Hz(x, y) for excitation at ω− and ω+ calculated using thetwo-dimensional finite-difference time-domain method. (d) Scattering spectra for the two-cavity system coupled to waveguides. Solidlines are obtained from CMT analysis. Circles are obtained from 2D FDTD simulation. Cavity parameters are ω0 = 2π(193.5 THz) =2π(c0/1.55 μm), γ0 = 1.26 × 1011 s−1, γc = 1.34 × 1012 s−1, and κ = (2.70 + i0.0114) × 1012 s−1.
FIG. 4. (a) Depiction of the two resonator system shown in Fig. 3(b) but with metal electrodes shown for modulating the refractiveindex in the cavity regions. (b) A plot of Eq. (15) for parameters γ = 1.461 × 1012 s−1 and κ = 2.70 × 1012 s−1. The minimumδω along with ωm,min consistent with Eq. (16) is indicated. (c) Scattering spectra for the modulated two-cavity system coupled towaveguides when the modulation is turned on showing the reduction of |S21(ω0)|2. Solid lines are obtained from CMT analysis. Circlesare obtained from 2D FDTD simulation. The same parameters as Fig. 3(d) are used in the CMT fit. The modulation parameters are�n = 0.028 55 and ωm = 2π(683 GHz). (d) Spatial distribution of the magnetic field Hz(x, y) for excitation at ω0 when the modulationis on (top) and off (bottom) calculated using 2D FDTD.
as S11(ω) = sr1(ω)/si1(ω) and S21(ω) = sr2(ω)/si1(ω).The split resonance shape corresponding to the frequenciesω± is apparent.
C. Time-modulated two-coupled-cavity system
Now it is imagined that metallic electrical contacts areconnected to the resonators by which a time-dependentvoltage V(r, t) = Vm(r) cos(ωmt) is applied to the mate-rial as depicted in Fig. 4(a). In this expression, ωm = 2π fmis the modulation frequency and Vm(r) is the spatiallydependent amplitude related to the geometry of the con-tact electrodes. In this way the refractive index changesin proportion to the voltage via the electrorefractive effectaccording to n(r, t) = n0(r) + �n(r, t), where n0(r) is thestatic two-resonator geometry, and it is assumed that�n(r, t) depends linearly on the applied voltage V(r, t)where values up to |�n| ∼ 10−2 are achievable in siliconand indium phosphide [26,27]. The time-dependent changein refractive index produces a corresponding change inthe resonance frequency of the isolated resonator ω(t) =ω0 + δω cos(ωmt), and the change in resonance frequencyis related to the change in refractive index by δω =(�n/n0)ω0, where �n and n0 are effective indices anddepend on the resonance mode of interest.
If one were to modulate both cavities in phase, theresulting effect would be an overall phase modulation ofthe system producing signal components at ω + sωm forinteger s. This could be useful if the objective is to phasemodulate a carrier wave inputted to the system. However,the demodulation device concept proposed here relies onmodulating the cavities with a 180◦ phase shift. This isa spatially odd time-dependent perturbation that causescoupling between the even and odd supermodes of the two-cavity system. Adding waveguide coupling and the twophase-shifted time-dependent frequency terms to Eqs. (2)results in
da1(t)dt
= [−iω0 − iδω cos(ωmt) − γ ]a1(t)
− iκa2(t) + dsi1(t), (7a)
da2(t)dt
= [−iω0 + iδω cos(ωmt) − γ ]a2(t)
− iκa1(t) + dsi2(t), (7b)
where γ ≡ γ0 + γc. Equations (7) can be written in matrixform according to
Obtaining relations for ωm and δω is facilitated by trans-forming to the supermode basis. To that end, Eq. (8) ismultiplied from the left by U, where U is constructed from
the eigenvectors of the two-resonator system
U = 1√2
[1 11 −1
](9)
034056-5
ADAM MOCK PHYS. REV. APPLIED 15, 034056 (2021)
and UU† = 1. The result is
ddt
[a+(t)a−(t)
]
=[−i(ω0 + κ) − γ −iδω cos(ωmt)
−iδω cos(ωmt) −i(ω0 − κ) − γ
] [a+(t)a−(t)
]
+ d√2
[si1(t) + si2(t)si1(t) − si2(t)
], (10)
where [a+(t)a−(t)
]= U
[a1(t)a2(t)
], (11)
and the coupling between the even and odd supermodesdue to the spatially odd time-dependent perturbation isclear from its location in the off-diagonal positions of thematrix.
The optical excitation is incident in waveguide 1 withsi2 = 0 and time dependence given by si1(t) = si1(ω) e−iωt.The presence of the δω cos(ωmt) modulation terms willintroduce harmonic frequency components spaced by ωm,so solutions of the form
a±(t) =+∞∑
s=−∞as,±(ω) e−i(ω+sωm)t (12)
are assumed. The resulting equations for the s componentsafter matching frequency terms are
− [i(ω + sωm − ω0 − κ) − γ ]as,+(ω)
+ iδω
2[as+1,−(ω) + as−1,−(ω)]
= d√2
si1(ω)δs0, (13a)
− [i(ω + sωm − ω0 + κ) − γ ]as,−(ω)
+ iδω
2[as+1,+(ω) + as−1,+(ω)]
= d√2
si1(ω)δs0, (13b)
where δs0 is the Kronecker delta. These equations are aninfinite set of coupled linear equations for the as,± compo-nents. However, in order to obtain usable relations for ωmand δω, the Fourier series [Eq. (12)] is truncated to keeponly the fundamental and first harmonics s = −1, 0, 1.This truncation is an approximation that improves as ωmincreases or δω decreases relative to κ and γ [37]. Solvingfor a0,+ and a0,− at ω = ω0 yields
a0,+(ω0) = d√2
ω2m − κ2 + γ 2 + i2κγ
γ [ω2m + κ2 + γ 2 + (δω)2/2] + iκ[ω2
m − κ2 − γ 2 − (δω)2/2], (14a)
a0,−(ω0) = d√2
ω2m − κ2 + γ 2 − i2κγ
γ [ω2m + κ2 + γ 2 + (δω)2/2] − iκ[ω2
m − κ2 − γ 2 − (δω)2/2]. (14b)
The proposed device concept hinges on extinguish-ing the optical output by choosing specific values ofωm and δω. To that end, values of ωm and δω aresought that make S21(ω0) = sr2(ω0)/si1(ω0) = 0. Since
sr2(ω0) = da2(ω0) = (d/√
2)[a0,+(ω0) − a0,−(ω0)], mak-ing a0,+(ω0) = a0,−(ω0) will cause S21(ω0) = 0. Aftersome manipulation of Eqs. (14) and neglecting the imagi-nary part of κ , choosing
δω =[
2κ2(ω2
m − κ2 − γ 2) − ω2m(ω2
m − κ2 + γ 2) − γ 2(ω2m + κ2 + γ 2)
κ2 + γ 2 − ω2m
]1/2
, (15)
along with ω2m > κ2 + γ 2, will force S21(ω0) = 0.
Figure 4(b) shows the dependence of δω on ωm forcavity parameters γ = 1.461 × 1011 s−1 and κ = 2.70 ×1012 s−1. The minimum value of ωm is 2π(489 GHz). Theminimum value of δω and the value of ωm,min for which itoccurs can be found by differentiating Eq. (15) with respectto ωm and setting the result equal to zero. The result is
ωm,min = [κ2 + γ 2 + 2γ√
κ2 + γ 2]1/2. (16)
In the present case, the minimum modulation amplitudeof δω/ω0 = 6 × 10−3 occurs at ωm,min = 2π(683 GHz), asshown in Fig. 4(b). Figure 4(c) shows the scattering spec-tra obtained from CMT where the reduction in |S21(ω0)|2due to the applied modulation is apparent. Figure 4(c)also shows the scattering spectra obtained from 2D FDTDnumerical simulation. For the simulation, a �n value of0.028 55 is used to produce the desired δω. The good qual-itative agreement between CMT and FDTD validates thedesign constraints obtained from CMT. When the proposedmodulation scheme is applied in the FDTD simulation,|S21(ω0)|2 is reduced to 8 × 10−3 at ω0 = 2π(193.5 THz).When the modulation is turned off, |S21(ω0)|2 = 0.582, asdepicted in Fig. 3(d), so applying the modulation swingsthe output by 18.6 dB. Figure 4(d) depicts the spatialdistribution of the magnetic field Hz(x, y) for excitationat ω0 when the modulation is on and off calculated via2D FDTD. It is apparent that the energy in the outputwaveguide is minimized when the modulation is appliedin accordance with Eqs. (15) and (16). Additionally, theenergy in cavity 2 is also minimized, consistent with theCMT predictions described above.
In a very general sense, actuating the output powerin this way bears some similarity to electromagneticallyinduced transparency in the sense that it uses two simulta-neous excitations at two frequencies [in this case, one opti-cal (“probe”) and one microwave (“dressing”)] to induceinterference between eigenstates of the system [46]. How-ever, in this device, output extinction, rather than trans-parency, is the objective. More specifically, according toEq. (11), the condition a0,+(ω0) = a0,−(ω0) that leads to|S21(ω0)|2 = 0 also leads to a0,2(ω0) = 0. So the modula-tion scheme couples the eigenstates a0,+(ω0) and a0,−(ω0)
in such a manner as to cancel the energy in resonator 2. Butthe energy in resonator 2 only comes from coupling to res-onator 1, so the condition ω2
m > κ2 + γ 2 makes intuitivesense if interpreted to mean that the engineered energyexchange rate associated with frequency ωm must be largerthan (or dominate over) the “natural” energy exchangerates characterized by κ and γ .
Further insight is obtained by inspecting the spectra inthe “on” [Fig. 3(d)] and “off” [Fig. 4(c)] states. As men-tioned previously, without index modulation, a+(ω) anda−(ω) are resonant at ω0 + κ and ω0 − κ , respectively.When the modulation is applied, a0,+(ω) and a0,−(ω)
retain their resonances at frequencies ω0 + κ and ω0 − κ ,
but they also acquire resonance features shifted by sωm.Figure 5 shows the scattering spectra calculated via CMTusing the same cavity parameters as used in Fig. 4(c). Themodulation amplitude is also the same as that in Fig. 4(c),and the modulation frequency ωm0 corresponds to thatused in Fig. 4(c). Figures 5(a)–5(e) depict the scatteringspectra when the modulation frequency takes on valuesfrom 5 down to 1.5 times ωm0 while keeping δω con-stant. When the modulation frequency is set to ωm = 5ωm0,as in Fig. 5(a), the spectra in the vicinity of ω+ and ω−bear resemblance to the spectra when the modulation is off[Fig. 4(c)]. Additionally, there are shallower replicas of thedouble peak resonance features shifted by ±5ωm0 [thereare small peaks in |S21|2 at these frequencies, but they aredifficult to see in Fig. 5(a)]. As the modulation frequencyis decreased, the resonance features in the shifted repli-cas become deeper and move closer to ω+ and ω−. At thesame time, the transmission peaks at ω+ and ω− decreasewhile the transmission features in the shifted replicasgrow. Figures 5(a)–5(f) show that the output extinctionresults from interference between the fundamental reso-nances at ω+ and ω− and their first harmonics shifted by±ωm. The proper choice of ωm (i.e., ωm = ωm0) producesspectral overlap of these multiple resonances. Specifically,Figs. 5(e) and 5(f) show that the transmission extinctionat ω0 results from destructive interference between thetransmission peak at ω+ and the shifted replica featureat ω− + ωm0 and the destructive interference between thetransmission peak at ω− and the shifted replica featureat ω+ − ωm0. Basically, this is a graphical depiction ofwhat Eq. (13) already says: a0,+(ω) is coupled to the oddsupermode through its first-order harmonics a±1,−(ω), anda0,−(ω) is coupled to the even supermode through its first-order harmonics a±1,+(ω). The spectra shown in Fig. 5(a),in which ωm is significantly detuned from ωm0, is reminis-cent of a basic phase modulation effect. As the resonantfeatures in the first-order harmonics overlap those of thefundamental, destructive interference occurs, resulting inextinction.
Figure 5(f) corresponds to the “off” state (ωm = ωm0).The residual peaks that appear in this transmission spec-trum come from the uncanceled shifted replicas at ω+ +ωm0 and ω− − ωm0. Figures 5(g) and 5(h) show how thespectra evolve as ωm is reduced below ωm0. Ultimately,the various cancelations between the eigenstates and theirharmonics are no longer balanced, and a peak at ω0reemerges.
III. OPTIMIZING THE CAVITY DESIGN
Now that the basic device functionality has been illus-trated, it is shown how the device can be optimized in the1D PC cavity platform to meet desired spectral shapes orto use desired modulation parameters. First, increasing theoptical output amplitude in the “on” state is investigated.
FIG. 5. Scattering spectra calculated using CMT for the same cavity and parameters as Fig. 4. The modulation amplitude is the sameas that used in Fig. 4, specifically, �n = 0.028 55. The modulation frequency for producing the “off” state is ωm0 = 2π(683 GHz).Panels (a)–(e) show spectra when the modulation frequency is set larger than ωm0, with values of (a) 5ωm0, (b) 4ωm0, (c) 3ωm0,(d) 2ωm0, and (e) 1.5ωm0. (f) Scattering spectra for the “off” state when ωm = ωm0, which is the same as Fig. 4(c). (g) Scattering spectrawhen ωm = 0.5ωm0. (h) Scattering spectra when ωm = 0.0.
When the modulation is off,
|S21(ω0)| = 2γcκ
γ 2 + κ2 , (17)
where the imaginary part of κ is ignored. This func-tion is maximized when κ = γ . Figure 6(a) shows thescattering spectra for different values of κ , where othercavity parameters are the same as those used in Fig. 3(d).Figure 6(b) confirms the optimal value of κ = γ that maxi-mizes |S21(ω0)|2, which in this case reaches 0.84. As notedin Fig. 3(b), the number of air holes between the cav-ities determines κ that quantifies the spectral separationbetween the resonance peaks in the transmission spectrum.Decreasing κ by increasing the number of holes betweenthe cavities draws the two resonant peaks closer together.This increases the throughput energy at ω0 when the mod-ulation is off by making the valley between the two peaksshallower. Decreasing κ beyond its optimal value causesdestructive interference between the two resonance peaksthat lowers the throughput at ω0.
From Eq. (17) one sees that increasing γc also increasesthe throughput when the voltage modulation is off.Figure 6(c) shows the effect on the scattering spectra fordifferent values of γc, where other cavity parameters arethe same as those in Fig. 3(d). The throughput amplitudeincreases as expected as does the spectral width of the
scattering spectra. As noted in Fig. 3(b), d (and thereforeγc) can be increased by decreasing the number of air holesbetween the cavities and the input-output waveguides.
In the plots shown in Fig. 6, γ0 is kept constant atγ0 = 1.26 × 1011 s−1, so these results should be consid-ered with respect to this value of γ0. Lossless materialsare used in the simulations, so γ0 is determined only byradiative loss and is related to the cavity quality (Q) factorvia γ0 = ω0/(2Q). The value γ0 = 1.26 × 1011 s−1 corre-sponds to a Q factor of 4840. With this value of γ0 and withγc = 1.34 × 1011 and κ = (1.45 + i0.0613) × 1011, mod-ulation parameters fm = 510 GHz and δω/ω0 = 0.0528 areneeded to switch the optical output off. If γ0 were reduced(Q increased) by modifying the cavity design, then γcand κ could be scaled accordingly using the design con-cepts discussed above. The scattering spectra would havethe same shape but with reduced spectral widths. Becausethe minimum values of ωm and δω are tied to κ andγ , the required modulation parameters also scale with γ0.Figures 7(a) and 7(b) show the scattering spectra when γ0,γc, and κ are all reduced by a factor of 10. This corre-sponds to a higher passive Q factor of Q = 48 400. Thespectral shapes are similar to those shown in previous fig-ures but narrower by a factor of 10. And the modulationparameters used to switch off |S21(ω0)|2 are a factor of10 smaller, fm = 51 GHz and δω/ω0 = 5.28 × 10−3. Of
FIG. 6. (a) Scattering spectra for the two-coupled-cavity system shown in Fig. 3(b) with no modulation applied for different valuesof κ . Resonator parameters used in the plot are ω0 = 2π(193.5 THz) = 2π(c0/1.55 μm), γ0 = 1.26 × 1011 s−1, γc = 1.34 × 1012 s−1.The three κ values are κ = (9.17 + i0.0385) × 1011 s−1, κ = (1.45 + i0.006 13) × 1012 s−1, and κ = (2.27 + i0.009 55) × 1012 s−1.(b) Plots of |S11(ω0)|2 and |S21(ω0)|2 as a function of κ for the same resonator parameters as in (a). The real part of κ is shown on thehorizontal axis, but the imaginary part is swept simultaneously while maintaining Im[κ] = 0.004 20 Re[κ]. (c) Scattering spectra forthe two-coupled-cavity system shown in Fig. 3(b) with no modulation applied for different values of γc. Resonator parameters used inthe plot are ω0 = 2π(193.5 THz) = 2π(c0/1.55 μm), γ0 = 2.00 × 1010 s−1, κ = (1.45 + i0.006 13) × 1011 s−1. The three γc valuesare γc = 6.72 × 1011 s−1, γc = 1.34 × 1012 s−1, and γc = 3.35 × 1012 s−1.
particular interest is the lower modulation frequency thatis closer to values relevant to communication systems.Ultimately, the modulation frequency can be tuned to anyvalue, assuming that the requisite Q factor can be obtainedin a given material platform and cavity design. Figure 7(c)displays the required modulation frequency and modula-tion amplitude as a function of the Q factor. For operationat the communication frequency fm = 40 GHz, a Q fac-tor of 6.2 × 104 or greater and a modulation amplitude ofδω/ω0 = 4.14 × 10−4 (corresponding to �n ≈ 1.3 × 103)or greater is required. Passive Q factors of this magni-tude are routinely demonstrated using modern fabricationcapabilities [28]. The requisite �n value is achievable indoped silicon [26] and InP [27].
Though they are not included in the simulations, mate-rial losses can be included in the model in a straightforwardmanner. As mentioned previously, γ0 is the rate of loss dueto all mechanisms other than waveguide coupling; theseinclude radiative losses as well as linear material absorp-tion and scattering. For silicon, reported material lossesare in the range of 0.1–1 dB/cm [26,29,31], which woulddecrease the throughput by 10%–50%, but this reduc-tion could be mitigated by increasing γc and increasingthe drive voltage and frequency concomitantly. InP haslarger waveguide losses in the range of 2–3 dB/cm, plac-ing it at a disadvantage to silicon in terms of loss [30];however, its amenability to on-chip light sources could,nevertheless, render it useful from a system-wide
(a) 1.0
0.8
0.6
0.4
0.2
0.0
Sca
tterin
g co
effic
ient
Frequency (THz)193.4 193.5 193.6 193.7
(c)(b)
193.8
1.0
0.8
0.6
0.4
0.2
0.0
Sca
tterin
g co
effic
ient
Frequency (THz)193.4 193.5 193.6 193.7 193.8
1013
1012
1011
1010
10–3
10–2
10–5
10–4
103 104 105 106
|S11|2 CMT
|S21|2 CMT f m
(Hz) δω
/ω0
Intrinsic quality factor
FIG. 7. (a) Scattering spectra for the two-coupled-cavity system with resonance frequency ω0 = 2π(193.5 THz) = 2π(c0/1.55 μm)
and loss parameters γ0 = 1.26 × 1010 s−1, γc = 1.34 × 1011 s−1, and κ = (1.45 + i0.0613) × 1011 s−1 that are a factor of 10 lowerthan those used in Fig. 6(a). Here κ is chosen to maximize |S21(ω0)| according to Eq. (17) and Fig. 6(b). (b) Scattering spectra whenthe modulation is applied. Modulation parameters are fm = 51 GHz and δω/ω0 = 5.28 × 10−3. (c) Dependence of fm and δω/ω0 onthe intrinsic quality (Q) factor of the cavity via Eqs. (15) and (16). The quality factor Q is related to γ0 via γ0 = ω0/(2Q). The ratiosγc = 10.6γ0 and Re[κ] = γ are maintained in constructing the plot. Green (purple) line refers to the left (right) axis.
034056-9
ADAM MOCK PHYS. REV. APPLIED 15, 034056 (2021)
perspective. Recent advances in the microfabrication oflow loss lithium niobate waveguides and resonators makesit a prospective material. Lithium niobate is attractive dueto its low absorption and large electro-optic effect [47,48],though tight control of photonic crystal hole size andlocation could still be challenging to fabricate. The pro-posed demodulator device could also be implemented inheterogeneous material systems, such as graphene [49],ferroelectric lead zirconate titanate [50], or lithium nio-bate films [51] on low-loss silicon nitride, which may offeradditional design flexibility but are not explicitly designedfor here.
In addition to its ability to tune κ and γc in a straightfor-ward manner, the 1D PC cavity design offers an ability totune the Q factor where values of 1.47 × 105 have beenexperimentally measured [28]. The design is based onreducing radiation loss from the cavity by tapering the sizeof the holes nearest the cavity. The tapered hole size allowsthe localized field to extend further along the x direc-tion, thereby making its k-space distribution more local-ized. This reduces k-space overlap with the radiation conedefined by total internal reflection at the air-semiconductorinterfaces that make up the sides of the ridge waveguide.Ultimately, the extended field along x results in reducedradiation loss along y. Using this design methodology,the Q factor of the cavity shown in Fig. 8(a) is increasedto Q = 1.69 × 105 by tapering the size of three holes oneither side of both cavities. The hole radii are tapered from
the nominal size of r = 0.3w = 111.6 nm to 89.3, 55.8,39.1 nm. This corresponds to radii scaled by 0.8, 0.5, and0.35. The cavity width is reduced to 3.5w = 1.30 μm in theoptimized version from 4w = 1.49 μm in the cavity shownin Fig. 3(b) to counteract the red shift in resonant frequencycaused by the increased effective index introduced by thehole taper. A nominal cavity size of 4w corresponds toremoving three air holes from a 1D PC. The number ofholes between the two cavities is increased to 14 to obtainthe optimal value for κ .
Figure 8(b) shows the scattering spectra when the mod-ulation is off. At ω0, the power throughput is |S21(ω0)|2 =0.875. Figure 8(c) shows the scattering spectra whenthe modulation is on with �n = 1.48 × 10−3 and ωm =2π(23.8 GHz). The throughput power is reduced to|S21(ω0)|2 < 2 × 10−3, which represents a 26.4 dB swingfrom on to off. With the increased Q factor of the pas-sive cavity compared to that of the nominal cavity designshown in Fig. 4(a), the magnitude of the modulationparameters required for output extinction is decreased bya similar factor. Figure 8(d) displays the relation betweenωm and δω determined by Eq. (15). The modulation ampli-tude is well within realistic parameter ranges for InP andSi [26,27]. The modulation frequency range is consistentwith that of practical communication systems operatingat 40 GHz. Therefore, the device functionality displayedin Fig. 8 is both experimentally feasible and tuned forpractical applications.
1.0
0.8
0.6
0.4
0.2
0.0
Sca
tterin
g co
effic
ient
193.496 193.545 194.593 193.641Frequency (THz)
1.0
0.8
0.6
0.4
0.2
0.0
Sca
tterin
g co
effic
ient
193.496 193.545 194.593 193.641Frequency (THz)
|S11|2 CMT
|S11|2 FDTD
|S21|2 CMT
|S21|2 FDTD
(d)
0.0
0.2
0.4
0.6
0.8
15 20 25 30 35
fm (GHz)
103
δω/ω
0
1.0
40 45
(c)(b)
(a)
~ V(t)
–1
ω0 + δω(t) ω0 – δω(t)1 2si1
sr1 sr 2
ωm,min
Tapered hole size
x
y
FIG. 8. (a) Cavity design reducing γ0 by tapering the photonic crystal air hole size on either side of the defect cavities. The threeholes adjacent to the cavities are reduced by factors of 0.8, 0.5, and 0.35 compared to the nominal air hole size. The tapered regionon the right-hand side of cavity 2 is denoted by a dashed box. Resonator parameters are ω0 = 2π(193.5 THz) = 2π(c0/1.55 μm),γ0 = 3.59 × 109 s−1, γc = 6.13 × 1011 s−1, and κ = (7.35 + i0.0165) × 1010 s−1. This value of γ0 corresponds to a passive Q factorof 169 000. (b) Scattering spectra for the optimized two-cavity system when the modulation is turned off. (c) Scattering spectra for theoptimized two-cavity system when the modulation is turned on, showing the reduction of |S21(ω0)|2. The modulation parameters are�n = 0.001 48 and ωm = 2π(23.8 GHz). In both (b) and (c) solid lines are obtained from CMT analysis. Circles are obtained from 2DFDTD simulation. (d) A plot of Eq. (15) for parameters γ = γ0 + γc and κ = 7.35 × 1010 s−1. The minimum δω along with ωm,minconsistent with Eq. (16) is indicated.
Figure 9 shows the dependence of the output scatteringamplitude on fm = ωm/(2π) for different values of δω. Theblue curve corresponds to the lowest δω as determined byEqs. (15) and (16). As ωm is increased above the valuegiven by Eq. (16) while keeping δω unchanged, the out-put increases from near zero to a value near its value whenno modulation is applied at all (approximately 0.875). Thehorizontal dashed line corresponds to half of the maximumvalue (approximately 0.875/2 = 0.4375), which definesthe threshold frequency value for binary symbol determi-nation denoted by fm0. For modulation frequencies abovefm0, the output is high, corresponding to a binary one. Like-wise, for modulation frequencies below fm0, the output islow, corresponding to a binary zero.
Figure 9 displays another advantageous feature of thisdevice concept, which is its tunability. From the previ-ous analysis, it was found that ωm can be freely chosenso long as ω2
m > κ2 + γ 2, and δω is then set accordingto Eq. (15). If the modulation amplitude is increased to alarger value, the threshold frequency fm0 increases as well.The red and green curves in Fig. 9 show the dependence of|S21(ω0)|2 on fm = ωm/(2π) for two higher values of δω.In all three cases the same static two-cavity system is used.
20 40 60 80 100fm (GHz)
0410210
0.8
0.6
0.4
0.2
0.0
|S21
(ω0)|
2
1.0
fm < fm0Binary 0
fm > fm0Binary 1
f m0 =
33
GH
z
f m0 =
50
GH
z
f m0 =
73
GH
z
Increasing δω
FIG. 9. The dependence of |S21(ω0)|2 on fm = ωm/(2π) forthree values of δω. The blue curve corresponds to δω/ω0 =1.196 × 10−4; the red curve corresponds to δω/ω0 = 1.825 ×10−4; the green curve corresponds to δω/ω0 = 2.658 × 10−4.Cavity parameters are ω0 = 2π(193.5 THz) = 2π(c0/1.55 μm),γ0 = 1.80 × 109 s−1, γc = 3.07 × 1011 s−1, and κ = 3.68 ×1010 s−1. Horizontal dashed line denotes the half-max at 0.4375.Vertical dashed lines denote the symbol detection frequencythresholds for binary FSK demodulation corresponding to thedifferent values of δω.
This shows how a given fabricated two-cavity system canbe used for binary FSK demodulation with symbol detec-tion frequency thresholds in the range 33–73 GHz (andhigher).
Returning to the application introduced in Fig. 1 wherethe device functionality is envisioned to be used to converta FSK electronic signal into an OOK optical signal that canthen be converted to a voltage for direct FSK demodulationor be fed into a radio-over-fiber optical communicationsystem, Fig. 10 displays the time-domain optical outputsignal |Hz(t)|2 from the demodulator shown in Fig. 8when the frequency of the modulating voltage is switchedbetween fm = 60.0 GHz and fm = 23.8 GHz. As noted pre-viously, the frequency fm = 23.8 GHz drives the output tozero as prescribed by Eq. (16) for the cavity design shownin Fig. 8. This produces the “off” state. Detuning the modu-lation frequency to fm = 60.0 GHz allows the input opticalsignal to couple to the output waveguide, resulting in the“on” state. Figure 10 illustrates how the FSK signal is con-verted to an optical ASK output by causing the opticaloutput to switch between on and off states.
In addition to the basic functionality of the proposeddemodulator, Fig. 10 illustrates other important temporalfeatures of the device. Figures 10(b) and 10(c) show theresponse times of the optical output switch as the modu-lating voltage frequency is changed. Figure 10(b) showsthat to switch the optical output from high to low takes16.8 ps. The photon lifetime of the system due to radiationand waveguide coupling is τp = 1/γ = 15.4 ps. That thefall time is approximately equal to τp is reasonable sincethe energy will presumably radiate out of the cavity sys-tem via external radiation and coupling to both the inputand output waveguides when the output is switched offusing the microwave voltage. Figure 10(c) shows that toswitch the optical output from low to high takes 51.5 ps,which is 3.34τp . It is hypothesized that the system has alonger rise time because energy must first build up in cav-ity 2 before it can couple to the output waveguide. Energyin cavity 2 couples from cavity 1 at a rate defined by κ witha coupling time of 1/κ = 13.6 ps. Furthermore, energy isentering the system only from waveguide 1, whereas τpconsiders coupling to both waveguides. With these con-siderations, an approximate theoretical prediction of therise time would be 1/(γ0 + γc/2) + 1/κ = 42.8 ps, a valueapproaching the observed time of 51.5 ps. Ultimately, thelonger response time limits the maximum bit rate for thedevice, which is 1/51.5 ps = 19.4 Gbps. As mentionedin the Introduction, the minimum bit period typicallyneeded in traditional electronic demodulation schemes isT = 1/fd, where fd is the difference frequency between thefrequencies used for zero and one. In the present analysis,this comes to T = 27.6 ps. Therefore, a rise time of 51.5 psreduces the maximum bit rate achievable in the presentdevice by 1.87 compared to traditional FSK demodula-tion. Nevertheless, in situations in which at least two full
034056-11
ADAM MOCK PHYS. REV. APPLIED 15, 034056 (2021)
0.0 0.1 0.2 0.3 0.4 0.5 0.6Time (ns)
0.2
0.4
0.6
0.8
0.00.155 0.165
Time (ns)0.175 0.32
Time (ns)0.380.350.29
Vmax
0.0
–Vmax
|Hz(t
)|2V M
W(t)
(c)(b)(a)
16.8 ps 51.5 ps
FIG. 10. Top: microwave frequency modulating voltage as a function of time. The frequency switches between fm = 60.0 GHz andfm = 23.8 GHz. Bottom: optical output of the demodulator versus time as the applied microwave signal frequency is changed. Herefm = 23.8 GHz corresponds to the off state defined by Eqs. (15) and (16), so the output is low during these frequency intervals. Whenfm = 60.0 GHz, the output is high since Eqs. (15) and (16) are not satisfied by this frequency. (a) Depiction of two on-off cycles of thedemodulator. (b) Enlarged view of the first transition from on to off, showing the fall time as 16.8 ps. (c) Enlarged view of a transitionfrom off to on, showing the rise time as 51.5 ps.
periods of each microwave waveform for zero and one areused for signaling, the observed rise time would not posea limitation since two periods of a 23.8 GHz sinusoid is84.0 ps, which is longer than 51.5 ps.
V. REDUCING SIDEBAND ENERGY
Another feature displayed in Fig. 10 is the presence ofripples both in the “on” and “off” states. Especially inthe off state, the ripples can reach to 20% of the on stateamplitude, which significantly reduces the amplitude con-trast between on and off. These ripples are due to energyin the modulation sidebands at frequencies ω0 + sωm (fornonzero integer s). Even if the throughput amplitude isdriven to zero at ω0, as shown spectrally in Fig. 8(c), therewill be energy modulated into sideband components thatwill ultimately be coupled to the output waveguide. Sincethis energy is detuned from ω0, one way to remove it isvia passive filtering. To this end, Fig. 11 displays how apassive filter with a passband similar to that of the demod-ulator cavities may be attached to waveguide 2 to removethe ripples. The filter itself has the same geometry as theisolated cavities and is placed a few wavelengths beyondthe right cladding of cavity 2, as shown in the inset ofFig. 11. The plot in Fig. 11 shows the time domain out-put after passage through the passive filter. The ripples inthe off state are significantly reduced, which shows theeffectiveness of this strategy to reduce sideband energy.
The on state pulses are reduced in amplitude and lesssharp than those shown in Fig. 10, but a more sophisti-cated filter design could alleviate those issues. Specifically,
~ V(t)–1 21
si1
sr1 sr2 Sideband passive filter
3
sr3Modulated cavities
2.0
1.5
1.0
0.5
0.0
2.5
0.1 0.2 0.3 0.4 0.5 0.6Time (ns)
Voltage signal set to off state|Hz(t
)|2
FIG. 11. Optical output of the demodulator shown in Fig. 8after passing through a passive filter to remove sideband energy.The modulation amplitude is �n = 1.48 × 10−3. From 0 to0.155 ns, the modulation frequency is ωm = 2π(60.0 GHz),which produces the on state. From 0.155 to 0.465 ns, the modu-lation frequency is set to ωm = 2π(23.8 GHz), which producesthe off state. At 0.465 ns, the modulation frequency is set back toωm = 2π(60.0 GHz) to produce the on state. The inset shows thegeometry modeled using FDTD. The filter has the same cavitydesign as that shown in Fig. 8 except it is a single cavity withfive cladding holes beyond the taper region.
the filter can be optimized to produce a sharper roll offbetween ω0 and ω0 ± ωm to suppress the sidebands morecompletely [2]. A ring resonator filter could also be attrac-tive for isolating the signal at ω0 while also reducing backreflection not just at ω0 but at ω0 ± ωm and higher-orderharmonics as well. Nevertheless, as a proof of principle,the time trace shown in Fig. 11 shows the efficacy of asimple passive filter to increase the contrast between theon and off states.
Looking again at Fig. 9, one notes that, as δω/ω0 isincreased, the output port can be switched off for a range ofmodulation frequencies below that described by Eq. (16).That is, if a (ωm, δω) combination is chosen according toEqs. (15) and (16), then δω can be increased while keep-ing ωm the same, and the output will remain turned off. Forone, this means that the demodulator can be used with awide range of parameter values. Additionally, Fig. 12(a)shows the output spectrum |S21(ω)|2 as the modulatingvoltage amplitude is increased to 1.75 and 2.5 times thatdescribed by Eq. (15) while keeping the modulation fre-quency the same at fm = 23.8 GHz. Though the outputdoes not go exactly to zero using these other modula-tion amplitudes, it remains low nevertheless. Interestingly,it is observed that the range of optical frequencies that
are switched off increases as the modulation amplitudeincreases. Figure 12(d) shows that the CMT predictionsare in qualitative agreement with the FDTD modeling evenas the modulation amplitude is increased. This suggestsa route for dynamic tuning the overall bandwidth of thedemodulator.
Additional inspection of Fig. 12(a) reveals that in addi-tion to a widening of the off state bandwidth as δω/ω0 isincreased, the height of the spectral peaks on either wide ofω0 are reduced as well. This offers a potential route towardreducing the ripples caused by sideband energy that werediscussed in relation to Fig. 10. Returning to Eqs. (13), itis noted that the sideband energy amplitudes a±1,±(ω) arerelated to the fundamental energy amplitudes a0,±(ω) via
a±1,+(ω) = iδω
2a0,−(ω) + a±2,−(ω)
i(ω ± ωm − ω0 − κ) − γ
≈ iδω
2a0,−(ω)
i(ω ± ωm − ω0 − κ) − γ, (18a)
a±1,−(ω) = iδω
2a0,+(ω) + a±2,+(ω)
i(ω ± ωm − ω0 + κ) − γ
≈ iδω
2a0,+(ω)
i(ω ± ωm − ω0 + κ) − γ, (18b)
0.16
193.50 193.55 193.60 193.65
0.12
0.08
0.04
0.00 193.50 193.55 193.60 193.65
0.07
0.05
0.03
0.01
193.50 193.55 193.60 193.65
0.07
0.05
0.03
0.01
Sca
tterin
g co
effic
ient
Frequency (THz)
δω/ω0 = 2.4×10–4
δω/ω0 = 4.2×10–4
δω/ω0 = 6.0×10–4
Sca
tterin
g co
effic
ient
Frequency (THz) Frequency (THz)
193.50 193.55 193.60 193.65Frequency (THz)
0.04
0.03
0.02
0.01
|S21|2 CMT
|S21|2 FDTD
0.0 0.1 0.2 0.3 0.4 0.5 0.6Time (ns)
0.1
0.2
0.3
0.4
0.0
|Hz(t
)|2
(c)(b)(a)
(e)(d)
0.0 0.1 0.2 0.3 0.4 0.5 0.6Time (ns)
0.1
0.2
0.3
0.4
0.0
|Hz(t
)|2
(f)
0.5
Sca
tterin
g co
effic
ient
0.00
FIG. 12. (a)–(c) Output scattering spectra |S21(ω)|2 for different voltage modulation amplitudes while keeping the modulationfrequency at fm = 23.8 GHz. Here δω/ω0 = 2.4 × 10−4 corresponds to the modulation configuration shown in Fig. 8; δω/ω0 =4.2 × 10−4 and δω/ω0 = 6.0 × 10−4 correspond to 1.75 and 2.5 times larger amplitudes, respectively. The required index change forthe three cases is �n = 1.48 × 10−3, 2.59 × 10−3, and 3.70 × 10−3, respectively. (a) Depiction of the output spectra at the fundamen-tal frequency. (b) Depiction of the output spectra in the s = −1 harmonic. (c) Depiction of the output spectra in the s = +1 harmonic.(d) Comparison between CMT and FDTD simulations with δω/ω0 = 6.0 × 10−4. (e) Same as Fig. 10(a) except that δω/ω0 =6.0 × 10−4, and the “on” frequency is increased to fm = 120 GHz. (f) Same as Fig. 10(a) except that δω/ω0 = 8.4 × 10−4 (3.5 timesthat in Fig. 8), and the “on” frequency is increased to fm = 180 GHz.
034056-13
ADAM MOCK PHYS. REV. APPLIED 15, 034056 (2021)
where the approximation neglects harmonics of secondorder and higher. The amplitude of the output waveresulting from modulation into the first-order harmonicsis given by sr2,±1(ω) = (d/
√2)[a±1,+(ω0) − a±1,−(ω0)],
and the corresponding scattering coefficients are definedas S21,±1(ω) = sr2,±1(ω)/si1(ω). This analysis shows thedirect proportionality between the wave energy at the fun-damental frequency and the wave energy in the first-orderharmonics. So if the amplitude of the side peaks is reducedin the fundamental harmonic as shown in Fig. 12(a), thenone should expect the amplitudes in the first harmonics todecrease as well. Figures 12(b) and 12(c) show the scatter-ing spectra for the s = −1 and s = +1 harmonics, respec-tively, and the decrease in amplitude as δω/ω0 increasesis exhibited. Figures 12(e) and 12(f) show |Hz(t)|2 forδω/ω0 = 6.0 × 10−4 and δω/ω0 = 8.4 × 10−4, respec-tively. In Fig. 12(e), the ratio of ripple height in the offstate to the on state amplitude is 15%, which is improvedfrom the nominal ratio of 20% for the signal depicted inFig. 10. In Fig. 12(f), the ratio is further decreased to 7%.Though, in both of these cases, a larger detuning frequencyis required to reach the on state. Ultimately, this analysisshows how the microwave voltage can be used to dynam-ically control or suppress the sideband energy in a singlefabricated device.
VI. CONCLUSION
This work presents a frequency demodulator conceptbased on refractive index modulation of two coupledmicrocavities excited by an optical carrier wave. Anapplied microwave frequency voltage changes the refrac-tive indices of the microcavities via the electorefrac-tive effect. The modulating voltage is phase shifted 180◦between the two cavities that induces coupling between theeven and odd supermodes. Through CMT and FDTD, itis shown that certain combinations of modulation ampli-tude and frequency cause the optical output to go tozero at the resonant frequency of the cavities. Requiredmodulation amplitudes and frequencies are tunable andfall within experimentally realizable ranges. As the mod-ulation frequency is detuned from the value prescribedto turn the optical output off, the output turns on andapproaches 100% power transmission for large detuning.This configuration results in amplitude modulation of theoptical signal in response to the frequency deviation ofan applied microwave voltage. The device functionality isdemonstrated through simulation of a high Q factor 1DPC cavity with small footprint. The device can be usedfor direct frequency demodulation or for multiplexing aFSK microwave voltage into a larger amplitude-modulatedradio-over-fiber optical system. For this last applicationparticularly, this demodulator has advantages over existingdevices in terms of power, size, and dynamic tunability.
[1] B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine,Microring resonator channel dropping filters, J. LightwaveTechnol. 15, 998 (1997).
[2] C. K. Madsen and J. H. Zhao, Optical Filter Designand Analysis: A Signal Processing Approach (Wiley,New York, 1999).
[3] H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides,and M. Khajavikhan, Parity-time-symmetric microringlasers, Science 346, 975 (2014).
[4] P. Dong, S. F. Preble, J. T. Robinson, S. Manipatruni, andM. Lipson, Inducing Photonic Transitions between DiscreteModes in a Silicon Optical Microcavity, Phys. Rev. Lett.100, 033904 (2008).
[5] H. Lira, Z. Yu, S. Fan, and M. Lipson, Electrically DrivenNonreciprocity Induced by Interband Photonic Transitionon a Silicon Chip, Phys. Rev. Lett. 109, 033901 (2012).
[6] D. L. Sounas and A. Alù, Non-reciprocal photonics basedon time modulation, Nat. Photonics 11, 774 (2017).
[7] Z. Yu and S. Fan, Complete optical isolation created byindirect photonic transitions, Nat. Photonics 3, 91 (2009).
[8] D. L. Sounas and A. Alù, Angular-momentum-biasednanorings to realize magnetic-free integrated optical isola-tion, ACS Photonics 1, 198 (2014).
[9] A. Zarif, K. Mehrany, M. Memarian, and H. Heydarian,Optical isolation enabled by two time-modulated point per-turbations in a ring resonator, Opt. Express 28, 16805(2020).
[10] A. Mock, D. Sounas, and A. Alù, Magnet-free circulatorbased on spatiotemporal modulation of photonic crystaldefect cavities, ACS Photonics 6, 2056 (2019).
[11] A. Mock, D. L. Sounas, and A. Alù, Tunable Orbital Angu-lar Momentum Radiation from Angular Momentum-BiasedMicrocavities, Phys. Rev. Lett. 121, 103901 (2018).
[12] J. Capmany and D. Novak, Microwave photonics combinestwo worlds, Nat. Photonics 1, 319 (2007).
[13] D. Marpaung, J. Yao, and J. Capmany, Integratedmicrowave photonics, Nat. Photonics 13, 80 (2019).
[14] C. R. Doerr, N. Dupris, and L. Zhang, Optical isolatorusing two tandem phase modulators, Opt. Lett. 36, 4293(2011).
[15] Q. Lin, J. Wang, and S. Fan, Compact dynamic opticalisolator based on tandem phase modulators, Opt. Lett. 44,2240 (2019).
[16] J. G. Proakis and M. Salehi, Fundamentals of Communica-tion Systems (Pearson, New Jersey, 2014), 2nd ed.
[17] M. M. Islam, J. Song, K. Rasilainen, and V. Viikari,Optimization of RFID sensor with frequency modulation,IEEE Sens. J. 16, 5993 (2016).
[18] M. Ghovanloo and K. Najafi, A wideband frequency-shiftkeying wireless link for inductively powered biomedicalimplants, IEEE Trans. Circuits Syst. 51, 2374 (2004).
[19] E. Main and D. Coffing, An FSK demodulator for bluetoothapplications having no external components, IEEE Trans.Circuits Syst. II: Analog and Digit. Signal Process. 49, 373(2002).
[20] C. Gomez and J. Paradells, Wireless home automation net-works: A survey of architectures and technologies, IEEECommun. Mag. 48, 92 (2010).
with frequency-shift-keying for visible light communica-tion, Opt. Express 26, 12530 (2018).
[22] A. W. Azim, A. Rullier, Y. L. Guennec, L. Ros, andG. Maury, Energy efficient M-ary frequency-shift keying-based modulation techniques for visible light commu-nication, IEEE Trans. Cogn. Commun. Netw. 5, 1244(2019).
[23] M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, USA, 2002), 3rd ed.
[24] R. M. Borges, T. R. R. Marins, M. S. B. Cunha, H. R. D.Filgueiras, I. F. da Costa, R. N. da Silva, D. H. Spadoti,L. L. Mendes, and J. Arismar Cerqueira Sodré, Integra-tion of a GFDM-based 5G transceiver in a GPON usingradio over fiber technology, J. Lightwave Technol. 36, 4468(2018).
[25] Y. Niu, Y. Li, D. Jin, L. Su, and A. V. Vasilakos, A sur-vey of millimeter wave communications (mmWave) for5G: Opportunities and challenges, Wireless Netw. 21, 2657(2015).
[26] R. Soref and B. R. Bennett, Electrooptical effects in silicon,IEEE J. Quantum Electron. QE-23, 123 (1987).
[27] B. R. Bennet, R. A. Soref, and J. A. Del Alamo,Carrier-induced change in refractive index of InP. GaAs,and InGaAsP, IEEE J. Quantum Electron. 26, 113(1990).
[28] A. R. M. Zain, N. P. Johnson, M. Sorel, and R. D. L.Rue, Ultra high quality factor one dimensional photoniccrystal/photonic wire micro-cavities in silicon-on-insulator(SOI), Opt. Express 16, 12084 (2008).
[29] Y. Zhang, S. Yang, A. E.-J. Lim, G.-Q. Lo, C. Galland,T. Baehr-Jones, and M. Hochberg, A compact and low lossY-junction for submicron silicon waveguide, Opt. Express21, 1310 (2013).
[30] E. Bitincka, G. Gilardi, and M. K. Smi, On-wafer opti-cal loss measurements using ring resonators with inte-grated sources and detectors, IEEE Photonics J. 6, 6601212(2010).
[31] M. A. Tran, D. Huang, T. Komljenovic, J. Peters, A.Malik, and J. E. Bowers, Ultra-low-loss silicon waveguidesfor heterogeneously integrated silicon/III-V photonics,Appl. Sci. 8, 1139 (2018).
[32] M. Burla, C. Hoessbacher, W. Heni, C. Haffner, Y. Fedo-ryshyn, D. Werner, T. Watanabe, H. Massler, D. L. Elder,L. R. Dalton, and J. Leuthold, 500 GHz plasmonic Mach-Zehnder modulator enabling sub-THz microwave photon-ics, APL Photonics 4, 056106 (2019).
[33] W. Suh, Z. Wang, and S. Fan, Temporal coupled-mode the-ory and the presence of non-orthogonal modes in losslessmultimode cavities, IEEE J. Quantum Electron. 40, 1511(2004).
[34] N. A. Estep, D. L. Sounas, J. Soric, and A. Alù, Magnetic-free non-reciprocity and isolation based on parametricallymodulated coupled-resonator loops, Nat. Phys. 10, 923(2014).
[35] N. A. Estep, D. L. Sounas, and A. Alù, Magnetlessmicrowave circulators based on spatiotemporally modu-lated rings of coupled resonators, IEEE Trans. Microw.Theory Tech. 64, 502 (2016).
[36] B. Vial and Y. Hao, A coupling model for quasi-normalmodes of photonic resonators, J. Opt. 18, 115004 (2016).
[37] See Supplemental Material http://link.aps.org/supplemental/10.1103/PhysRevApplied.15.034056, which includesRef. [52], for additional details on derivation of Eq. (2) anda discussion of the error in the truncation of Eq. (12).
[38] T. Yang, A. Mock, J. D. O’Brien, S. Lipson, and D. G.Deppe, Lasing characteristics of InAs quantum dot micro-cavity lasers as a function of temperature and wavelength,Opt. Express 15, 7281 (2007).
[39] M. Nomura, N. Kumagai, S. Iwamoto, Y. Ota, andY. Arakawa, Photonic crystal nanocavity laser with singlequantum dot gain, Opt. Express 17, 15975 (2009).
[40] M. Galli, D. Gerace, K. Welna, T. F. Krauss, L. O’Faolain,G. Guizzetti, and L. C. Andreani, Low-power continuous-wave generation of visible harmonics in silicon photoniccrystal nanocavities, Opt. Express 18, 26613 (2010).
[41] S. Noda, Photonic crystal lasers–ultimate nanolasers andbroad-area coherent lasers, J. Opt. Soc. Am. B 27, B1(2010).
[42] P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan,and M. Loncar, Coupled photonic crystal nanobeam cavi-ties, Appl. Phys. Lett. 95, 031102 (2009).
[43] B. C. Richards, J. Hendrickson, J. D. Olitzky, R. Gibson,M. Gehl, K. Kieu, U. K. Khankhoje, A. Homyk, A. Scherer,J.-Y. Kim, Y.-H. Lee, G. Khitrova, and H. M. Gibbs, Char-acterization of 1D photonic crystal nanobeam cavities usingcurved microfiber, Opt. Express 18, 20558 (2010).
[44] Y. Chen, W. S. Fegadolli, W. M. Jones, A. Scherer, andM. Li, Ultrasensitive gas-phase chemical sensing basedon functionalized photonic crystal nanobeam cavities,ACS Nano 8, 522 (2014).
[45] L. Lu, A. Mock, and J. O’Brien, Efficient couplingbetween a photonic crystal nanocavity and a waveguidewith directional end-facet emission, J. Opt. 14, 055502(2012).
[46] M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Elec-tromagnetically induced transparency: Optics in coherentmedia, Rev. Mod. Phys. 77, 633 (2005).
[47] R. Wu, J. Zhang, N. Yao, W. Fang, L. Qiao, Z. Chai,J. Lin, and Y. Cheng, Lithium niobate micro-disk resonatorsof quality factors above 107, Opt. Lett. 43, 4116 (2018).
[48] M. Zhang, C. Wang, R. Cheng, A. Shams-Ansari, andM. Loncar, Monolithic ultra-high-Q lithium niobate micror-ing resonator, Optica 4, 1536 (2018).
[49] C. T. Phare, Y.-H. D. Lee, J. Cardenas, and M. Lipson,Graphene electro-optic modulator with 30 GHz bandwidth,Nat. Photonics 9, 511 (2015).
[50] K. Alexander, J. P. George, J. Verbist, K. Neyts, B. Kuyken,D. V. Thourhout, and J. Beeckman, Nanophotonic Pockelsmodulators on a silicon nitride platform, Nat. Commun. 9,3444 (2018).
[51] S. Jin, L. Xu, H. Zhang, and Y. Li, LiNbO3 thin-film modu-lators using silicon nitride surface ridge waveguides, IEEEPhotonics Technol. Lett. 28, 736 (2016).
[52] J. N. Winn, S. Fan, J. D. Joannopoulos, and E. P. Ippen,Interband transitions in photonic crystals, Phys. Rev. B 59,1551 (1999).
