Microwave Frequency Demodulation Using Two Coupled Optical Resonators with Modulated Refractive Index

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 simulations 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 propose a compact microwave-frequency-controlled photonic switch based on time modulation of two coupled pho- The manipulation of eigenmodes in coupled optical res- tonic resonators. The overall concept is shown in Fig. 1. onators is a well-studied component of photonic device A continuous wave (cw) optical carrier is coupled into one design [1–3]. Spatiotemporal modulation of the cavity cavity resonator of a two-coupled-cavity resonator system. material offers an additional mechanism by which photonic The refractive indices of the two cavities are modulated at signal processing capabilities can be implemented [4–6]. a microwave frequency and 180◦ out of phase. The ampli- For example, spatiotemporal modulation can be used to tude of the resultant output optical wave is then modulated break time-reversal symmetry resulting in optical isolation in response to the frequency of the modulating voltage. [7–9], optical circulation [10], and optical orbital angu- Control of an optical output intensity by the frequency of lar momentum generation [11]. On the other hand, the an applied voltage is unique to this device topology and interplay of microwave frequency electronic signals with is not an effect that can be readily achieved using existing visible and near-infrared photonic signals underpins many devices such as a standard Mach-Zehnder intensity modu- of the capabilities of modern microwave photonic devices lator. Two tandem phase modulators could potentially be [12,13]. This work combines these two perspectives to 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 *[email protected] format recognized for its improved noise immunity over that of amplitude modulation (AM) that comes at the Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Fur- expense of larger bandwidth usage and more complicated ther distribution of this work must maintain attribution to the modulation and demodulation hardware [16]. However, author(s) and the published article’s title, journal citation, and because FM signals are transmitted at a constant maxi- DOI. mized power level, need for power regularization circuitry

2331-7019/21/15(3)/034056(15) 034056-1 Published by the American Physical Society ADAM MOCK PHYS. REV. APPLIED 15, 034056 (2021)

V (t) MW 101100 FM or FSK voltage f ≈ 40 GHz t

V (t) MW

–1

I (t) opt 101100 Optical carrier f = 194 THz ω ω λ = 1.55 μm 0 0 0 t

FIG. 1. Depiction of the frequency demodulation concept. An optical carrier is inputted from the left. Two coupled optical resonators are modulated by a frequency-modulated microwave (MW) frequency voltage VMW(t). The frequency modulation shown is binary FSK. The output optical signal is an amplitude-modulated version of the FSK electrical signal. This optical output can be combined with 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- deviations which makes it advantageous over traditional tems by assigning a binary zero to an assigned frequency approaches of FSK demodulation that use narrowband fil- f0 = fc − fd/2 (“space” frequency) and a binary one to a ters or local oscillators with fixed center frequencies. In different assigned frequency f1 = fc + fd/2 (“mark” fre- the case of general FM demodulation it avoids the need quency), where fc is the carrier frequency and fd is the for a phase-locked loop (PLL) and works for virtually difference frequency. The minimum difference frequency any microwave frequency. Primary power requirements of to maintain orthogonal signal waveforms is fd = 1/(2T), the proposed electrical-optical-electrical demodulator are where T is the signal period, and most traditional demodu- to drive the chip-scale laser source that is similar to that of lation approaches require fd = 1/T. Overall, such a scheme an electronic PLL or local oscillator. is referred to as binary frequency shift keying (FSK) and If the output optical wave is not immediately detected can be extended to M-ary modulation formats as well. Its and converted to a voltage, it can be multiplexed with constant power transmission and enhanced immunity to a larger radio-over-fiber optical communication network. noise make FSK popular in radio frequency identification In this implementation, the proposed device functional- (RFID) technologies [17], biomedical implant communi- ity is attractive as few if any existing compact devices cation [18], Bluetooth [19], and a variety of home wireless perform the specific task of directly amplitude modulat- technologies [20]. By the same token, FSK is attractive in ing an optical wave in response to the frequency of an visible light digital communication systems to avoid flicker applied voltage. Use of radio over fiber is ubiquitous in when the carrier wave doubles as a room lighting source wireless communication [24], and microwave and mil- [21,22]. FSK is also prevalent in radar systems in which limeter wave wireless frequencies are a significant and ranging and doppler shifts are more accurately measured growing component in 5G communication systems [25]. than in pulsed time-of-flight implementations [23]. Using conventional demodulation approaches, a FM or The microwave-frequency-controlled switch depicted in FSK electronic signal would need to be converted to an Fig. 1 amplitude modulates a cw optical carrier wave in amplitude-modulated electronic signal before it can ampli- response to an applied FM voltage. If the message signal tude modulate an optical carrier to join the larger optical embedded in the FM voltage is a generic analog signal then communication network. The proposed device obviates the the output optical wave is amplitude modulated with the need for FM to AM conversion in the electrical domain, same analog message. If the FM voltage is a FSK signal thereby improving the power efficiency and SNR of the then the digital information is imparted on the output opti- process. cal wave in the form of amplitude shift keying (ASK) or In the following, a coupled mode theory (CMT) model on-off keying (OOK). If a photodiode is used immediately is used to develop the device concept, and results from to convert the output optical wave into an electrical signal, full-wave finite-difference time-domain (FDTD) electro- the proposed switch is a frequency demodulator, producing dynamic simulations confirm device properties derived a demodulated electrical message signal from an electri- via CMT. It is shown how the device may be imple- cal frequency-modulated input signal. As discussed later in mented in a one-dimensional (1D) photonic crystal (PC) this work, this electrical-optical-electrical FM demodula- platform. The 1D PC consists of circular air holes etched tor approach is dynamically tunable, so that a single device into a semiconductor ridge waveguide of width less than can be used for different frequency ranges and frequency 400 nm. The device length is 12.5 μm. Such a small

034056-2 MICROWAVE FREQUENCY DEMODULATION. . . PHYS. REV. APPLIED 15, 034056 (2021) device footprint is advantageous for dense integration and A detailed derivation of Eqs. (2) is provided in Ref. [37]. −iωt for enhanced light-matter interaction. A refractive index Letting ak(t) = ak e for k = 1, 2 results in change of n = 1.48 × 10−3 with a modulation frequency − ω = (− ω − γ ) − κ fm = 23.8 GHz are shown numerically to produce an on-off i a1 i 0 0 a1 i a2,(3a) ratio of 26.4 dB. These results show that this device design −iωa2 = (−iω0 − γ0)a2 − iκa1. (3b) is consistent with well-established experimental capabilities [26–32]. Furthermore, it is shown that there are a wide These equations can be written as a matrix range of modulation amplitudes and frequencies that may be used to switch the optical output on and off in a sin- a −iω − γ −iκ a −iω 1 = 0 0 1 ,(4) gle device design. The effect of modulation sidebands is a2 −iκ −iω0 − γ0 a2 addressed, and effective techniques for their suppression are introduced. from which the eigenfrequencies of the two-resonator system are found to be ω± = ω0 ± κ − iγ0. This shows II. COUPLED MODE THEORY ANALYSIS AND that the nominal resonance frequency ω0 is split to values SCATTERING MATRIX slightly above and below this value with a similar effect on the loss rate. The√ corresponding eigenvectors√ are given A. Modes of the two-coupled-cavity system T T by a+ = [1, 1] / 2anda− = [1, −1] / 2, and the full Consider the two-cavity system depicted in Fig. 2(a).To eigenstates are given by obtain design constraints on the proposed system, coupled mode theory analysis is used to find the scattering spectra 1 −iω±t |F±(t)=√ [|F1±|F2q] e .(5) and their dependence on device parameters [10,33–35]. It 2 is assumed that the fields of the coupled two-cavity system may be constructed from the fields of the isolated These eigenstates can be viewed as even (+) and odd (−) resonators according to [36] supermodes, as depicted in Fig. 2(b). Figure 2(c) depicts the 1D PC cavity platform in which | ( )= ( ) | + ( ) | F t a1 t F1 a2 t F2 ,(1)the two-coupled-cavity system is implemented. The peri- odic array of air holes creates a photonic bandgap whose | | where F1 and F2 represent the spatial distributions of center frequency is tuned with hole size and waveguide the vector fields in resonators 1 and 2, respectively, in width. A resonant cavity is formed by removing holes ( ) ( ) the absence of coupling; and a1 t and a2 t denote the from the array. In this case, three holes are removed to time-dependent expansion coefficients. Inserting Eq. (1) form a cavity with a high quality factor and small footprint into Maxwell’s equations, and viewing resonator 1 (2) as a [38–41]. A second cavity is coupled to the first one as small perturbation to resonator 2 (1) results in depicted in the figure. The gray regions denote semiconductor material with refractive index n = 3.1 consistent da1(t) = (−iω0 − γ0)a1 − iκa2,(2a)with electrorefractive materials suitable for microwave dt photonics such as silicon and indium phosphide [26,27]. da (t) = 2 = (−iω − γ )a − iκa . (2b) The hole radii are r 0.3w, where w is the width of the dt 0 0 2 1 waveguide, and the hole spacing is equal to the waveguide

(a) (c) 1 2 ω κ ω Λ 0 0 w κ ω ω 0 0 y (b) (d) 1 2 |F +max + x 0 |F – x –max x

FIG. 2. (a) Two optical resonators deﬁned by a spatially varying electric permittivity ε(r) whose proximity results in coupling denoted 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) Spatial distribution of the magnetic ﬁeld Hz(x, y) for the even and odd supermodes of the system schematically depicted in (b) calculated using 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 device collected via an output waveguide as shown in Fig. 3(a). to operate at λ0 = 1.55 μm. The coupling rate κ may be To account for these energy transfers, Eqs. (3) are modified tuned by the number of air holes between the two cavi- according to ties; for example, κ increases as the cavities are brought closer together. Overall, these geometrical specifications −iωa1(ω) = [−iω0 − γ0 − γc1]a1(ω) are inline with well-established fabrication capabilities − κ (ω) + (ω) [28,42–44]. i a2 d1si1 ,(6a) It is worth pointing out that the demodulator concept −iωa2(ω) = [−iω0 − γ0 − γc2]a2(ω) can be implemented in any geometry with waveguides and − iκa (ω) + d s (ω), (6b) two coupled cavities having nondegenerate resonances. 1 2 i2 This includes defect waveguides and cavities in a two- dimensional PC lattice [10,45]. However, the nanoscale where dl is the coupling rate from waveguide l and sil is the ridge waveguide with a localized 1D PC shown in Fig. 3(b) input amplitude in waveguide l [1,10,33–35]. In this work, is easier to fabricate, easier to design, and will likely have waveguides 1 and 2 are the input and output waveguides, = ≡ lower surface scattering than a full two-dimensional PC respectively, and are symmetrically coupled, so d1 d2 lattice while offering the same functionality. Microring res- d. The waveguide-loaded cavities have a decay rate aug- γ = γ ≡ γ onators are another popular and well-understood cavity mented by c1 c2 c. The coupling rate and decay γ = 2 system with potential to implement the proposed demod- rate are related via 2 c d . In the waveguide-coupled- ω ulation concept; however, due to the degenerate (or nearly cavity system, the frequency is the excitation frequency, (ω) ω degenerate) counterpropagating whispering gallery modes and the amplitudes al are continuous functions of . in a single resonator, a separate analysis would need to be To quantify the energy output, the mode amplitudes are (ω) =− (ω) + (ω) undertaken to determine the modulation format to obtain related to the output waves via srl sil dal = similar functionality. for l 1, 2. Figure 3(b) depicts waveguides 1 and 2 imple- Figure 2(d) shows the spatial distribution of the mag- mented in the 1D PC ridge waveguide configuration. Similar to the way κ is controlled by the number of air netic field Hz(x, y) for the even and odd supermodes of the system depicted in Fig. 2(c) calculated using the 2D FDTD holes between the cavities, the waveguide coupling rate γ method. The figure shows the field in the x-y plane, where c is controlled by the number of holes between the uni- the [E (x, y), E (x, y), H (x, y)] polarization is used. The form ridge waveguide and the defect. Figure 3(c) shows x y z ω even and odd parity along the x direction is apparent along the field distribution when the system is excited at ±. with the mode confinements along x and y. The relation to the even and odd supermodes is visible, but the amount of energy in the two cavities is no longer equal, which is consistent with theory predictions when B. Coupling to the two-coupled-cavity system with the energy is incident from only one side. Figure 3(d) input and output waveguides shows the output scattering spectra for waveguides 1 and Light energy will be inserted into the two-resonator sys- 2 assuming input in waveguide 1 for typical values of tem via an input waveguide, and the output energy will be ω0, κ, d,andγ0. The scattering parameters are defined

(a) 1 2 s (d) 1 i1 d κ d 2 1.0 ω ω ω ω – + 0 0 s s 0.8 (b) r1 r2 s |S |2 CMT i1 ddκ 0.6 11 1 122 2 ω ω |S | FDTD 0 0 ω 11 0 |S |2 CMT s s 0.4 21 r1 r2 |S |2 FDTD (c) 21 +max 1 2 ω_ 0.2 Scattering coefficient

y ω 0 0.0 + 192.1 192.9 193.7 194.5 195.3

–max Frequency (THz) x

FIG. 3. (a) Diagram showing resonators 1 and 2 coupled to waveguides 1 and 2, respectively. Here si1 and sr1 denote the incident and reflected wave amplitudes in waveguide 1, and sr2 denotes the reflected wave amplitude in waveguide 2. The system is excited from waveguide 1, which makes waveguide 1 the input waveguide and waveguide 2 the output waveguide. (b) Two photonic crystal defect cavities coupled to waveguides. (c) Spatial distribution of the magnetic field Hz(x, y) for excitation at ω− and ω+ calculated using the two-dimensional finite-difference time-domain method. (d) Scattering spectra for the two-cavity system coupled to waveguides. Solid lines are obtained from CMT analysis. Circles are obtained from 2D FDTD simulation. Cavity parameters are ω0 = 2π(193.5 THz) = 11 −1 12 −1 12 −1 2π(c0/1.55 μm), γ0 = 1.26 × 10 s , γc = 1.34 × 10 s ,andκ = (2.70 + i0.0114) × 10 s .

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V(t) (a) s ~ (b) 2.0 (c) 1.0 |S |2 CMT i1 11 1 –1 2 S 2 | 11| FDTD ω + δω(t) ω – δω(t) 2 0 0 |S | CMT 0.8 21 s s 1.5 |S |2 FDTD

r r 0 21 (d) 1 2 1 2 0.6 Modulation off 1.0 ω ω δω/ω m 0 +max ,min Input 2 0.4

10 0.5 0.2 ω Modulation on 0 0 ω Scattering coefficient Input 0 y 0.0 0.0 –max 0.6 0.8 1.0 1.2 1.4 192.1 192.9 193.7 194.5 195.3 f Frequency (THz) x m (THz)

FIG. 4. (a) Depiction of the two resonator system shown in Fig. 3(b) but with metal electrodes shown for modulating the refractive index 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 minimum δω along with ωm,min consistent with Eq. (16) is indicated. (c) Scattering spectra for the modulated two-cavity system coupled to 2 waveguides when the modulation is turned on showing the reduction of |S21(ω0)| . Solid lines are obtained from CMT analysis. Circles are 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 modulation is on (top) and off (bottom) calculated using 2D FDTD.

(ω) = (ω)/ (ω) (ω) = (ω)/ (ω) as S11 sr1 si1 and S21 sr2 si1 . If one were to modulate both cavities in phase, the The split resonance shape corresponding to the frequencies resulting effect would be an overall phase modulation of ω± is apparent. the system producing signal components at ω + sωm for integer s. This could be useful if the objective is to phase C. Time-modulated two-coupled-cavity system modulate a carrier wave inputted to the system. However, Now it is imagined that metallic electrical contacts are the demodulation device concept proposed here relies on ◦ connected to the resonators by which a time-dependent modulating the cavities with a 180 phase shift. This is voltage V(r, t) = Vm(r) cos(ωmt) is applied to the mate- a spatially odd time-dependent perturbation that causes rial as depicted in Fig. 4(a). In this expression, ωm = 2πfm coupling between the even and odd supermodes of the two- is the modulation frequency and Vm(r) is the spatially cavity system. Adding waveguide coupling and the two dependent amplitude related to the geometry of the con- phase-shifted time-dependent frequency terms to Eqs. (2) tact electrodes. In this way the refractive index changes results in in proportion to the voltage via the electrorefractive effect ( ) ( ) = ( ) + ( ) ( ) da1 t according to n r, t n0 r n r, t , where n0 r is the = [−iω0 − iδω cos(ωmt) − γ ]a1(t) static two-resonator geometry, and it is assumed that dt n(r, t) depends linearly on the applied voltage V(r, t) − iκa2(t) + dsi1(t),(7a) where values up to |n|∼10−2 are achievable in silicon da2(t) and indium phosphide [26,27]. The time-dependent change = [−iω0 + iδω cos(ωmt) − γ ]a2(t) dt in refractive index produces a corresponding change in − κ ( ) + ( ) the resonance frequency of the isolated resonator ω(t) = i a1 t dsi2 t , (7b) ω + δω cos(ω t), and the change in resonance frequency 0 m γ ≡ γ + γ is related to the change in refractive index by δω = where 0 c. Equations (7) can be written in matrix form according to (n/n0)ω0, where n and n0 are effective indices and depend on the resonance mode of interest.

d a (t) −iω − iδω cos(ω t) − γ −iκ a (t) s (t) 1 = 0 m 1 + d i1 .(8) dt a2(t) −iκ −iω0 + iδω cos(ωmt) − γ a2(t) si2(t)

Obtaining relations for ωm and δω is facilitated by trans- the eigenvectors of the two-resonator system forming to the supermode basis. To that end, Eq. (8) is multiplied from the left by U, where U is constructed from = √1 11 U − (9) 2 1 1

034056-5 ADAM MOCK PHYS. REV. APPLIED 15, 034056 (2021) and UU† = 1. The result is are assumed. The resulting equations for the s components after matching frequency terms are d a+(t) ( ) dt a− t − (ω + ω − ω − κ) − γ (ω) [i s m 0 ]as,+ −i(ω0 + κ) − γ −iδω cos(ωmt) a+(t) δω = + (ω) + (ω) −iδω cos(ωmt) −i(ω0 − κ) − γ a−(t) i [as+1,− as−1,− ] 2 d s (t) + s (t) + √ i1 i2 = √d (ω)δ ( ) − ( ) , (10) si1 s0, (13a) 2 si1 t si2 t 2 where − [i(ω + sωm − ω0 + κ) − γ ]as,−(ω) δω a+(t) a (t) + (ω) + (ω) = U 1 , (11) i [as+1,+ as−1,+ ] a−(t) a2(t) 2 = √d (ω)δ and the coupling between the even and odd supermodes si1 s0, (13b) 2 due to the spatially odd time-dependent perturbation is clear from its location in the off-diagonal positions of the where δs0 is the Kronecker delta. These equations are an matrix. infinite set of coupled linear equations for the as,± compo- The optical excitation is incident in waveguide 1 with nents. However, in order to obtain usable relations for ω = ( ) = (ω) −iωt m si2 0 and time dependence given by si1 t si1 e . and δω, the Fourier series [Eq. (12)] is truncated to keep δω (ω ) The presence of the cos mt modulation terms will only the fundamental and first harmonics s =−1, 0, 1. introduce harmonic frequency components spaced by ω , m This truncation is an approximation that improves as ωm so solutions of the form increases or δω decreases relative to κ and γ [37]. Solving +∞ for a0,+ and a0,− at ω = ω0 yields −i(ω+sωm)t a±(t) = as,±(ω) e (12) s=−∞

ω2 − κ2 + γ 2 + κγ d m i2 a +(ω ) = √ , (14a) 0, 0 γ ω2 + κ2 + γ 2 + (δω)2/ + κ ω2 − κ2 − γ 2 − (δω)2/ 2 [ m 2] i [ m 2]

ω2 − κ2 + γ 2 − κγ d m i2 a −(ω ) = √ . (14b) 0, 0 γ ω2 + κ2 + γ 2 + (δω)2/ − κ ω2 − κ2 − γ 2 − (δω)2/ 2 [ m 2] i [ m 2]

√ The proposed device concept hinges on extinguish- sr2(ω0) = da2(ω0) = (d/ 2)[a0,+(ω0) − a0,−(ω0)], making the optical output by choosing speciﬁc values of ing a +(ω ) = a −(ω ) will cause S (ω ) = 0. After ω δω ω δω 0, 0 0, 0 21 0 m and . To that end, values of m and are some manipulation of Eqs. (14) and neglecting the imagi- (ω ) = (ω )/ (ω ) = sought that make S21 0 sr2 0 si1 0 0. Since nary part of κ, choosing

κ2(ω2 − κ2 − γ 2) − ω2 (ω2 − κ2 + γ 2) − γ 2(ω2 + κ2 + γ 2) 1/2 δω = 2 m m m m , (15) κ2 + γ 2 − ω2 m

ω2 >κ2 + γ 2 (ω ) = along with m , will force S21 0 0.

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Figure 4(b) shows the dependence of δω on ωm for but they also acquire resonance features shifted by sωm. cavity parameters γ = 1.461 × 1011 s−1 and κ = 2.70 × Figure 5 shows the scattering spectra calculated via CMT 12 −1 10 s . The minimum value of ωm is 2π(489 GHz).The using the same cavity parameters as used in Fig. 4(c).The minimum value of δω and the value of ωm,min for which it modulation amplitude is also the same as that in Fig. 4(c), occurs can be found by differentiating Eq. (15) with respect and the modulation frequency ωm0 corresponds to that to ωm and setting the result equal to zero. The result is used in Fig. 4(c). Figures 5(a)–5(e) depict the scattering spectra when the modulation frequency takes on values 2 2 2 2 1/2 ω δω ωm,min = [κ + γ + 2γ κ + γ ] . (16) from 5 down to 1.5 times m0 while keeping constant. When the modulation frequency is set to ωm = 5ωm0, In the present case, the minimum modulation amplitude as in Fig. 5(a), the spectra in the vicinity of ω+ and ω− −3 of δω/ω0 = 6 × 10 occurs at ωm,min = 2π(683 GHz),as bear resemblance to the spectra when the modulation is off shown in Fig. 4(b). Figure 4(c) shows the scattering spec- [Fig. 4(c)]. Additionally, there are shallower replicas of the 2 tra obtained from CMT where the reduction in |S21(ω0)| double peak resonance features shifted by ±5ωm0 [there 2 due to the applied modulation is apparent. Figure 4(c) are small peaks in |S21| at these frequencies, but they are also shows the scattering spectra obtained from 2D FDTD difficult to see in Fig. 5(a)]. As the modulation frequency numerical simulation. For the simulation, a n value of is decreased, the resonance features in the shifted repli- 0.028 55 is used to produce the desired δω. The good qual- cas become deeper and move closer to ω+ and ω−. At the itative agreement between CMT and FDTD validates the same time, the transmission peaks at ω+ and ω− decrease design constraints obtained from CMT. When the proposed while the transmission features in the shifted replicas modulation scheme is applied in the FDTD simulation, grow. Figures 5(a)–5(f) show that the output extinction 2 −3 |S21(ω0)| is reduced to 8 × 10 at ω0 = 2π(193.5 THz). results from interference between the fundamental reso- 2 When the modulation is turned off, |S21(ω0)| = 0.582, as nances at ω+ and ω− and their first harmonics shifted by depicted in Fig. 3(d), so applying the modulation swings ±ωm. The proper choice of ωm (i.e., ωm = ωm0) produces the output by 18.6 dB. Figure 4(d) depicts the spatial spectral overlap of these multiple resonances. Specifically, distribution of the magnetic field Hz(x, y) for excitation Figs. 5(e) and 5(f) show that the transmission extinction at ω0 when the modulation is on and off calculated via at ω0 results from destructive interference between the 2D FDTD. It is apparent that the energy in the output transmission peak at ω+ and the shifted replica feature waveguide is minimized when the modulation is applied at ω− + ωm0 and the destructive interference between the in accordance with Eqs. (15) and (16). Additionally, the transmission peak at ω− and the shifted replica feature energy in cavity 2 is also minimized, consistent with the at ω+ − ωm0. Basically, this is a graphical depiction of CMT predictions described above. what Eq. (13) already says: a0,+(ω) is coupled to the odd In a very general sense, actuating the output power supermode through its first-order harmonics a±1,−(ω),and in this way bears some similarity to electromagnetically a0,−(ω) is coupled to the even supermode through its first- induced transparency in the sense that it uses two simulta- order harmonics a±1,+(ω). The spectra shown in Fig. 5(a), neous excitations at two frequencies [in this case, one opti- in which ωm is significantly detuned from ωm0, is reminis- cal (“probe”) and one microwave (“dressing”)] to induce cent of a basic phase modulation effect. As the resonant interference between eigenstates of the system [46]. How- features in the first-order harmonics overlap those of the ever, in this device, output extinction, rather than trans- fundamental, destructive interference occurs, resulting in parency, is the objective. More specifically, according to extinction. Eq. (11), the condition a0,+(ω0) = a0,−(ω0) that leads to Figure 5(f) corresponds to the “off” state (ωm = ωm0). 2 |S21(ω0)| = 0 also leads to a0,2(ω0) = 0. So the modula- The residual peaks that appear in this transmission spec- tion scheme couples the eigenstates a0,+(ω0) and a0,−(ω0) trum come from the uncanceled shifted replicas at ω+ + in such a manner as to cancel the energy in resonator 2. But ωm0 and ω− − ωm0. Figures 5(g) and 5(h) show how the the energy in resonator 2 only comes from coupling to res- spectra evolve as ωm is reduced below ωm0. Ultimately, ω2 >κ2 + γ 2 onator 1, so the condition m makes intuitive the various cancelations between the eigenstates and their sense if interpreted to mean that the engineered energy harmonics are no longer balanced, and a peak at ω0 exchange rate associated with frequency ωm must be larger reemerges. than (or dominate over) the “natural” energy exchange rates characterized by κ and γ . III. OPTIMIZING THE CAVITY DESIGN Further insight is obtained by inspecting the spectra in the “on” [Fig. 3(d)] and “off” [Fig. 4(c)] states. As men- Now that the basic device functionality has been illus- tioned previously, without index modulation, a+(ω) and trated, it is shown how the device can be optimized in the a−(ω) are resonant at ω0 + κ and ω0 − κ, respectively. 1D PC cavity platform to meet desired spectral shapes or When the modulation is applied, a0,+(ω) and a0,−(ω) to use desired modulation parameters. First, increasing the retain their resonances at frequencies ω0 + κ and ω0 − κ, optical output amplitude in the “on” state is investigated.

034056-7 ADAM MOCK PHYS. REV. APPLIED 15, 034056 (2021)

(a) (b) (c) (d) ω – 5ω ω – 5ω ω ω ω + 5ω ω +5ω ω – 4ω ω – 4ω ω ω ω + 4ω ω + 4ω ω – 3ω ω – 3ω ω ω ω + 3ω ω + 3ω ω ω ω ω ω ω – m0 + m0 – + – m0 + m0 – m0 + m0 – + – m0 + m0 – m0 + m0 – + – m0 + m0 – 2 m0 + 2 m0 1.0 1.0 1.0 1.0 + – + –

0.8 0.8 0.8 0.8 ω – 2ω ω ω – m0 ++ 2 m0 0.6 0.6 0.6 0.6

0.4 0.4 0.4 0.4

0.2 0.2 0.2 0.2

0.0 0.0 0.0 0.0 188 190 192 194 196 198 188 190 192 194 196 198 188 190 192 194 196 198 188 190 192 194 196 198 Frequency (THz) Frequency (THz) Frequency (THz) Frequency (THz) (e) (f) (g) (h) ω ω ω ω ω ω ω ω ω ω ω ω – 1.5 ω ω + 1.5 ω – ω ω + ω – 0.5 – + + 0.5 ω ω 1.0 + m0 – + – m0 1.0 + m0 – + – m0 1.0 + m0 – m0 1.0 – +

0.8 0.8 0.8 0.8 ω ω ω ω ω ω ω ω ω ω ω ω –– m0 ++ m0 – 0.5 + 0.5 –– 1.5 m0 ++ 1.5 m0 – m0 + m0 0.6 0.6 0.6 0.6

0.4 0.4 0.4 0.4

0.2 0.2 0.2 0.2

0.0 0.0 0.0 0.0 188 190 192 194 196 198 188 190 192 194 196 198 188 190 192 194 196 198 188 190 192 194 196 198 Frequency (THz) Frequency (THz) Frequency (THz) Frequency (THz) S 2 | 11| S 2 | 21|

FIG. 5. Scattering spectra calculated using CMT for the same cavity and parameters as Fig. 4. The modulation amplitude is the same as 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 spectra when ωm = 0.5ωm0. (h) Scattering spectra when ωm = 0.0.

When the modulation is off, scattering spectra. As noted in Fig. 3(b), d (and therefore γ ) can be increased by decreasing the number of air holes 2γ κ c |S (ω )|= c , (17) between the cavities and the input-output waveguides. 21 0 γ 2 + κ2 In the plots shown in Fig. 6, γ0 is kept constant at γ = × 11 −1 where the imaginary part of κ is ignored. This func- 0 1.26 10 s , so these results should be consid- γ tion is maximized when κ = γ . Figure 6(a) shows the ered with respect to this value of 0. Lossless materials γ scattering spectra for different values of κ, where other are used in the simulations, so 0 is determined only by cavity parameters are the same as those used in Fig. 3(d). radiative loss and is related to the cavity quality (Q) factor γ = ω /( ) γ = × 11 −1 Figure 6(b) confirms the optimal value of κ = γ that maxi- via 0 0 2Q . The value 0 1.26 10 s corre- 2 γ mizes |S21(ω0)| , which in this case reaches 0.84. As noted sponds to a Q factor of 4840. With this value of 0 and with 11 11 in Fig. 3(b), the number of air holes between the cav- γc = 1.34 × 10 and κ = (1.45 + i0.0613) × 10 , mod- ities determines κ that quantifies the spectral separation ulation parameters fm = 510 GHz and δω/ω0 = 0.0528 are between the resonance peaks in the transmission spectrum. needed to switch the optical output off. If γ0 were reduced Decreasing κ by increasing the number of holes between (Q increased) by modifying the cavity design, then γc the cavities draws the two resonant peaks closer together. and κ could be scaled accordingly using the design con- This increases the throughput energy at ω0 when the mod- cepts discussed above. The scattering spectra would have ulation is off by making the valley between the two peaks the same shape but with reduced spectral widths. Because shallower. Decreasing κ beyond its optimal value causes the minimum values of ωm and δω are tied to κ and destructive interference between the two resonance peaks γ , the required modulation parameters also scale with γ0. that lowers the throughput at ω0. Figures 7(a) and 7(b) show the scattering spectra when γ0, From Eq. (17) one sees that increasing γc also increases γc,andκ are all reduced by a factor of 10. This corre- the throughput when the voltage modulation is off. sponds to a higher passive Q factor of Q = 48 400. The Figure 6(c) shows the effect on the scattering spectra for spectral shapes are similar to those shown in previous fig- different values of γc, where other cavity parameters are ures but narrower by a factor of 10. And the modulation 2 the same as those in Fig. 3(d). The throughput amplitude parameters used to switch off |S21(ω0)| are a factor of −3 increases as expected as does the spectral width of the 10 smaller, fm = 51 GHz and δω/ω0 = 5.28 × 10 .Of

034056-8 MICROWAVE FREQUENCY DEMODULATION. . . PHYS. REV. APPLIED 15, 034056 (2021)

(a) (b) (c) 1.0 0 1.0 S 2 1.0 ω | 11| CMT |S |2 CMT 0.8 0.8 21 0.8 Incr. κ Incr. γ 0.6 0.6 0.6 c

0.4 0.4 0.4 Incr. γc 0.2 Incr. κ 0.2 0.2 Scattering coefficient Scattering coefficient

0.0 at Scattering coefficient 0.0 0.0 192 193 194 195 0 12345×1012 192 193 194 195 Frequency (THz) κ (s –1) Frequency (THz)

FIG. 6. (a) Scattering spectra for the two-coupled-cavity system shown in Fig. 3(b) with no modulation applied for diﬀerent values 11 −1 12 −1 of κ. Resonator parameters used in the plot are ω0 = 2π(193.5 THz) = 2π(c0/1.55 μm), γ0 = 1.26 × 10 s , γc = 1.34 × 10 s . The three κ values are κ = (9.17 + i0.0385) × 1011 s−1, κ = (1.45 + i0.006 13) × 1012 s−1,andκ = (2.27 + i0.009 55) × 1012 s−1. 2 2 (b) Plots of |S11(ω0)| and |S21(ω0)| as a function of κ for the same resonator parameters as in (a). The real part of κ is shown on the horizontal axis, but the imaginary part is swept simultaneously while maintaining Im[κ] = 0.004 20 Re[κ]. (c) Scattering spectra for the two-coupled-cavity system shown in Fig. 3(b) with no modulation applied for diﬀerent values of γc. Resonator parameters used in 10 −1 11 −1 the plot are ω0 = 2π(193.5 THz) = 2π(c0/1.55 μm), γ0 = 2.00 × 10 s , κ = (1.45 + i0.006 13) × 10 s .Thethreeγc values 11 −1 12 −1 12 −1 are γc = 6.72 × 10 s , γc = 1.34 × 10 s ,andγc = 3.35 × 10 s . particular interest is the lower modulation frequency that Though they are not included in the simulations, mate- is closer to values relevant to communication systems. rial losses can be included in the model in a straightforward Ultimately, the modulation frequency can be tuned to any manner. As mentioned previously, γ0 is the rate of loss due value, assuming that the requisite Q factor can be obtained to all mechanisms other than waveguide coupling; these in a given material platform and cavity design. Figure 7(c) include radiative losses as well as linear material absorp- displays the required modulation frequency and modulation and scattering. For silicon, reported material losses tion amplitude as a function of the Q factor. For operation are in the range of 0.1–1 dB/cm [26,29,31], which would at the communication frequency fm = 40 GHz, a Q fac- decrease the throughput by 10%–50%, but this reduc- 4 tor of 6.2 × 10 or greater and a modulation amplitude of tion could be mitigated by increasing γc and increasing −4 3 δω/ω0 = 4.14 × 10 (corresponding to n ≈ 1.3 × 10 ) the drive voltage and frequency concomitantly. InP has or greater is required. Passive Q factors of this magni- larger waveguide losses in the range of 2–3 dB/cm, plac- tude are routinely demonstrated using modern fabrication ing it at a disadvantage to silicon in terms of loss [30]; capabilities [28]. The requisite n value is achievable in however, its amenability to on-chip light sources could, doped silicon [26]andInP[27]. nevertheless, render it useful from a system-wide

(a) (b) (c) 1.0 1.0 1013 –2 0.8 0.8 10

10 δω/ω 0.6 –3 S 2 0.6 10 | 11| CMT

2 (Hz) |S | CMT 11 m

21 0 0.4 0.4 f 10 10–4

0.2 0.2 1010 Scattering coefficient Scattering coefficient 10–5 0.0 0.0 193.4 193.5 193.6 193.7 193.8 193.4 193.5 193.6 193.7 193.8 103 104 105 106 Frequency (THz) Frequency (THz) 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) 10 −1 11 −1 11 −1 and loss parameters γ0 = 1.26 × 10 s , γc = 1.34 × 10 s ,andκ = (1.45 + i0.0613) × 10 s that are a factor of 10 lower than those used in Fig. 6(a).Hereκ is chosen to maximize |S21(ω0)| according to Eq. (17) and Fig. 6(b). (b) Scattering spectra when −3 the modulation is applied. Modulation parameters are fm = 51 GHz and δω/ω0 = 5.28 × 10 . (c) Dependence of fm and δω/ω0 on the 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 ratios γc = 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 of the nominal size of r = 0.3w = 111.6 nm to 89.3, 55.8, low loss lithium niobate waveguides and resonators makes 39.1 nm. This corresponds to radii scaled by 0.8, 0.5, and it a prospective material. Lithium niobate is attractive due 0.35. The cavity width is reduced to 3.5w = 1.30 μm in the to its low absorption and large electro-optic effect [47,48], optimized version from 4w = 1.49 μm in the cavity shown though tight control of photonic crystal hole size and in Fig. 3(b) to counteract the red shift in resonant frequency location could still be challenging to fabricate. The pro- caused by the increased effective index introduced by the posed demodulator device could also be implemented in hole taper. A nominal cavity size of 4w corresponds to heterogeneous material systems, such as graphene [49], removing three air holes from a 1D PC. The number of ferroelectric lead zirconate titanate [50], or lithium nio- holes between the two cavities is increased to 14 to obtain bate films [51] on low-loss silicon nitride, which may offer the optimal value for κ. additional design flexibility but are not explicitly designed Figure 8(b) shows the scattering spectra when the mod- 2 for here. ulation is off. At ω0, the power throughput is |S21(ω0)| = In addition to its ability to tune κ and γc in a straightfor- 0.875. Figure 8(c) shows the scattering spectra when −3 ward manner, the 1D PC cavity design offers an ability to the modulation is on with n = 1.48 × 10 and ωm = tune the Q factor where values of 1.47 × 105 have been 2π(23.8 GHz). The throughput power is reduced to 2 −3 experimentally measured [28]. The design is based on |S21(ω0)| < 2 × 10 , which represents a 26.4 dB swing reducing radiation loss from the cavity by tapering the size from on to off. With the increased Q factor of the pas- of the holes nearest the cavity. The tapered hole size allows sive cavity compared to that of the nominal cavity design the localized field to extend further along the x direc- shown in Fig. 4(a), the magnitude of the modulation tion, thereby making its k-space distribution more local- parameters required for output extinction is decreased by ized. This reduces k-space overlap with the radiation cone a similar factor. Figure 8(d) displays the relation between defined by total internal reflection at the air-semiconductor ωm and δω determined by Eq. (15). The modulation ampli- interfaces that make up the sides of the ridge waveguide. tude is well within realistic parameter ranges for InP and Ultimately, the extended field along x results in reduced Si [26,27]. The modulation frequency range is consistent radiation loss along y. Using this design methodology, with that of practical communication systems operating the Q factor of the cavity shown in Fig. 8(a) is increased at 40 GHz. Therefore, the device functionality displayed to Q = 1.69 × 105 by tapering the size of three holes on in Fig. 8 is both experimentally feasible and tuned for either side of both cavities. The hole radii are tapered from practical applications.

y V(t) ~ Tapered hole size s –1 (a) 1 i1 x 2 ω δω t ω δω t 0 + ( ) 0 – ( ) s s r1 r 2 (b) (c) (d) 1.0 1.0 1.0

0.8 0.8 0.8 0 S 2 | 11| CMT 0.6 S 2 0.6 0.6 | 11| FDTD 2

S δω/ω | 21| CMT

0.4 S 2 0.4 3 0.4 ω | 21| FDTD m,min 0.2 0.2 10 0.2 Scattering coefficient Scattering coefficient 0.0 0.0 0.0 193.496 193.545 194.593 193.641 193.496 193.545 194.593 193.641 15 20 25 30 35 40 45 Frequency (THz) Frequency (THz) f m (GHz)

FIG. 8. (a) Cavity design reducing γ0 by tapering the photonic crystal air hole size on either side of the defect cavities. The three holes 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 region on 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), 9 −1 11 −1 10 −1 γ0 = 3.59 × 10 s , γc = 6.13 × 10 s ,andκ = (7.35 + i0.0165) × 10 s . This value of γ0 corresponds to a passive Q factor of 169 000. (b) Scattering spectra for the optimized two-cavity system when the modulation is turned oﬀ. (c) Scattering spectra for the 2 optimized two-cavity system when the modulation is turned on, showing the reduction of |S21(ω0)| . 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 2D 10 −1 FDTD simulation. (d) A plot of Eq. (15) for parameters γ = γ0 + γc and κ = 7.35 × 10 s . The minimum δω along with ωm,min consistent with Eq. (16) is indicated.

034056-10 MICROWAVE FREQUENCY DEMODULATION. . . PHYS. REV. APPLIED 15, 034056 (2021)

IV. FREQUENCY CONTROLLED SWITCHING This shows how a given fabricated two-cavity system can be used for binary FSK demodulation with symbol detec- Figure 9 shows the dependence of the output scattering tion frequency thresholds in the range 33–73 GHz (and amplitude on f = ω /(2π)for different values of δω.The m m higher). blue curve corresponds to the lowest δω as determined by Returning to the application introduced in Fig. 1 where Eqs. (15) and (16).Asω is increased above the value m the device functionality is envisioned to be used to convert given by Eq. (16) while keeping δω unchanged, the out- a FSK electronic signal into an OOK optical signal that can put increases from near zero to a value near its value when then be converted to a voltage for direct FSK demodulation no modulation is applied at all (approximately 0.875). The or be fed into a radio-over-fiber optical communication horizontal dashed line corresponds to half of the maximum system, Fig. 10 displays the time-domain optical output value (approximately 0.875/2 = 0.4375), which defines signal |H (t)|2 from the demodulator shown in Fig. 8 the threshold frequency value for binary symbol determi- z when the frequency of the modulating voltage is switched nation denoted by f . For modulation frequencies above m0 between f = 60.0 GHz and f = 23.8 GHz. As noted pre- f , the output is high, corresponding to a binary one. Like- m m m0 viously, the frequency f = 23.8 GHz drives the output to wise, for modulation frequencies below f , the output is m m0 zero as prescribed by Eq. (16) for the cavity design shown low, corresponding to a binary zero. in Fig. 8. This produces the “off” state. Detuning the modu- Figure 9 displays another advantageous feature of this lation frequency to f = 60.0 GHz allows the input optical device concept, which is its tunability. From the previ- m signal to couple to the output waveguide, resulting in the ous analysis, it was found that ω can be freely chosen m “on” state. Figure 10 illustrates how the FSK signal is con- so long as ω2 >κ2 + γ 2,andδω is then set according m verted to an optical ASK output by causing the optical to Eq. (15). If the modulation amplitude is increased to a output to switch between on and off states. larger value, the threshold frequency f increases as well. m0 In addition to the basic functionality of the proposed The red and green curves in Fig. 9 show the dependence of demodulator, Fig. 10 illustrates other important temporal |S (ω )|2 on f = ω /(2π) for two higher values of δω. 21 0 m m features of the device. Figures 10(b) and 10(c) show the In all three cases the same static two-cavity system is used. response times of the optical output switch as the modulating voltage frequency is changed. Figure 10(b) shows that to switch the optical output from high to low takes 16.8 ps. The photon lifetime of the system due to radiation f f f f and waveguide coupling is τp = 1/γ = 15.4 ps. That the m < m > 0 m m0 fall time is approximately equal to τ is reasonable since Binary 0 Binary 1 p = 50 GHz = 33 GHz = 73 GHz 0 0 0 the energy will presumably radiate out of the cavity sys- m m m f f 1.0 f tem via external radiation and coupling to both the input and output waveguides when the output is switched off 0.8 using the microwave voltage. Figure 10(c) shows that to switch the optical output from low to high takes 51.5 ps, 2 τ

)| which is 3.34 p . It is hypothesized that the system has a 0 0.6 Increasing δω

ω longer rise time because energy must ﬁrst build up in cav- (

21 ity 2 before it can couple to the output waveguide. Energy

S 0.4 | in cavity 2 couples from cavity 1 at a rate defined by κ with a coupling time of 1/κ = 13.6 ps. Furthermore, energy is 0.2 entering the system only from waveguide 1, whereas τp considers coupling to both waveguides. With these con- 0.0 siderations, an approximate theoretical prediction of the 0 20 40 60 80 100 120 140 rise time would be 1/(γ + γ /2) + 1/κ = 42.8 ps, a value f (GHz) 0 c m approaching the observed time of 51.5 ps. Ultimately, the longer response time limits the maximum bit rate for the | (ω )|2 = ω /( π) FIG. 9. The dependence of S21 0 on fm m 2 for device, which is 1/51.5 ps = 19.4 Gbps. As mentioned δω δω/ω = three values of . The blue curve corresponds to 0 in the Introduction, the minimum bit period typically −4 1.196 × 10 ; the red curve corresponds to δω/ω0 = 1.825 × −4 −4 needed in traditional electronic demodulation schemes is 10 ; the green curve corresponds to δω/ω0 = 2.658 × 10 . T = 1/fd, where fd is the difference frequency between the Cavity parameters are ω0 = 2π(193.5 THz) = 2π(c0/1.55 μm), γ = × 9 −1 γ = × 11 −1 κ = × frequencies used for zero and one. In the present analysis, 0 1.80 10 s , c 3.07 10 s ,and 3.68 = 1010 s−1. Horizontal dashed line denotes the half-max at 0.4375. this comes to T 27.6 ps. Therefore, a rise time of 51.5 ps Vertical dashed lines denote the symbol detection frequency reduces the maximum bit rate achievable in the present thresholds for binary FSK demodulation corresponding to the device by 1.87 compared to traditional FSK demodula- different values of δω. tion. Nevertheless, in situations in which at least two full

034056-11 ADAM MOCK PHYS. REV. APPLIED 15, 034056 (2021)

V (a) max (b) (c) ) t ( 0.0 MW V

V – max 0.8 16.8 ps 51.5 ps

0.6 2 )| t ( z 0.4 H |

0.2

0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.155 0.165 0.175 0.29 0.32 0.35 0.38 Time (ns) Time (ns) Time (ns)

FIG. 10. Top: microwave frequency modulating voltage as a function of time. The frequency switches between fm = 60.0 GHz and fm = 23.8 GHz. Bottom: optical output of the demodulator versus time as the applied microwave signal frequency is changed. Here fm = 23.8 GHz corresponds to the off state defined by Eqs. (15) and (16), so the output is low during these frequency intervals. When fm = 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 the demodulator. (b) Enlarged view of the first transition from on to off, showing the fall time as 16.8 ps. (c) Enlarged view of a transition from off to on, showing the rise time as 51.5 ps. periods of each microwave waveform for zero and one are The on state pulses are reduced in amplitude and less used for signaling, the observed rise time would not pose sharp than those shown in Fig. 10, but a more sophisti- a limitation since two periods of a 23.8 GHz sinusoid is cated filter design could alleviate those issues. Specifically, 84.0 ps, which is longer than 51.5 ps.

V t s ~ ( ) i V. REDUCING SIDEBAND ENERGY 1 1 –1 2 3 s s s 2.5 r r2 r Another feature displayed in Fig. 10 is the presence of 1 Sideband passive filter 3 Modulated cavities ripples both in the “on” and “off” states. Especially in 2.0 the off state, the ripples can reach to 20% of the on state 2 )| t ( amplitude, which significantly reduces the amplitude con- z 1.5 H trast between on and off. These ripples are due to energy | Voltage signal set to off state 1.0 in the modulation sidebands at frequencies ω0 + sωm (for nonzero integer s). Even if the throughput amplitude is 0.5 driven to zero at ω0, as shown spectrally in Fig. 8(c), there 0.0 will be energy modulated into sideband components that 0.1 0.2 0.3 0.4 0.5 0.6 will ultimately be coupled to the output waveguide. Since Time (ns) this energy is detuned from ω0, one way to remove it is via passive filtering. To this end, Fig. 11 displays how a FIG. 11. Optical output of the demodulator shown in Fig. 8 passive filter with a passband similar to that of the demod- after passing through a passive filter to remove sideband energy. = × −3 ulator cavities may be attached to waveguide 2 to remove The modulation amplitude is n 1.48 10 .From0to 0.155 ns, the modulation frequency is ω = 2π(60.0 GHz), the ripples. The filter itself has the same geometry as the m which produces the on state. From 0.155 to 0.465 ns, the modu- isolated cavities and is placed a few wavelengths beyond lation frequency is set to ωm = 2π(23.8 GHz), which produces the right cladding of cavity 2, as shown in the inset of the off state. At 0.465 ns, the modulation frequency is set back to Fig. 11. The plot in Fig. 11 shows the time domain out- ωm = 2π(60.0 GHz) to produce the on state. The inset shows the put after passage through the passive filter. The ripples in geometry modeled using FDTD. The filter has the same cavity the off state are significantly reduced, which shows the design as that shown in Fig. 8 except it is a single cavity with effectiveness of this strategy to reduce sideband energy. five cladding holes beyond the taper region.

034056-12 MICROWAVE FREQUENCY DEMODULATION. . . PHYS. REV. APPLIED 15, 034056 (2021) the filter can be optimized to produce a sharper roll off are switched off increases as the modulation amplitude between ω0 and ω0 ± ωm to suppress the sidebands more increases. Figure 12(d) shows that the CMT predictions completely [2]. A ring resonator filter could also be attrac- are in qualitative agreement with the FDTD modeling even tive for isolating the signal at ω0 while also reducing back as the modulation amplitude is increased. This suggests reflection not just at ω0 but at ω0 ± ωm and higher-order a route for dynamic tuning the overall bandwidth of the harmonics as well. Nevertheless, as a proof of principle, demodulator. the time trace shown in Fig. 11 shows the efficacy of a Additional inspection of Fig. 12(a) reveals that in addi- simple passive filter to increase the contrast between the tion to a widening of the off state bandwidth as δω/ω0 is on and off states. increased, the height of the spectral peaks on either wide of Looking again at Fig. 9, one notes that, as δω/ω0 is ω0 are reduced as well. This offers a potential route toward increased, the output port can be switched off for a range of reducing the ripples caused by sideband energy that were modulation frequencies below that described by Eq. (16). discussed in relation to Fig. 10. Returning to Eqs. (13),it That is, if a (ωm, δω) combination is chosen according to is noted that the sideband energy amplitudes a±1,±(ω) are Eqs. (15) and (16), then δω can be increased while keep- related to the fundamental energy amplitudes a0,±(ω) via ω ing m the same, and the output will remain turned off. For δω (ω) + (ω) one, this means that the demodulator can be used with a (ω) = a0,− a±2,− a±1,+ i (ω ± ω − ω − κ) − γ wide range of parameter values. Additionally, Fig. 12(a) 2 i m 0 shows the output spectrum |S (ω)|2 as the modulating δω (ω) 21 ≈ a0,− voltage amplitude is increased to 1.75 and 2.5 times that i , (18a) 2 i(ω ± ωm − ω0 − κ) − γ described by Eq. (15) while keeping the modulation fre- δω (ω) + (ω) quency the same at f = 23.8 GHz. Though the output (ω) = a0,+ a±2,+ m a±1,− i (ω ± ω − ω + κ) − γ does not go exactly to zero using these other modula- 2 i m 0 tion amplitudes, it remains low nevertheless. Interestingly, δω +(ω) ≈ a0, it is observed that the range of optical frequencies that i , (18b) 2 i(ω ± ωm − ω0 + κ) − γ

(a) (b) (c) 0.16 0.07 0.07

δω ω –4 0.12 0.05 / 0 = 2.4×100.05 δω ω –4 / 0 = 4.2×10 0.08 δω/ω = 6.0×10–4 0.03 0 0.03 0.04 0.01 0.01 Scattering coefficient 0.00 Scattering coefficient 193.50 193.55 193.60 193.65 193.50 193.55 193.60 193.65 193.50 193.55 193.60 193.65 Frequency (THz) Frequency (THz) Frequency (THz) (d) (e) (f) S 2 0.04 | 21| CMT 0.4 S 2 0.5 | 21| FDTD 0.03 0.3 0.4 2 2 )| )| t t ( (

z z 0.3 0.02 0.2 H H | | 0.2 0.01 0.1 0.1

Scattering coefficient 0.00 0.0 0.0 193.50 193.55 193.60 193.65 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Frequency (THz) Time (ns) Time (ns)

2 FIG. 12. (a)–(c) Output scattering spectra |S21(ω)| for diﬀerent voltage modulation amplitudes while keeping the modulation −4 frequency at fm = 23.8 GHz. Here δω/ω0 = 2.4 × 10 corresponds to the modulation conﬁguration shown in Fig. 8; δω/ω0 = −4 −4 4.2 × 10 and δω/ω0 = 6.0 × 10 correspond to 1.75 and 2.5 times larger amplitudes, respectively. The required index change for the 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 fundamental frequency. (b) Depiction of the output spectra in the s =−1 harmonic. (c) Depiction of the output spectra in the s =+1 harmonic. −4 (d) Comparison between CMT and FDTD simulations with δω/ω0 = 6.0 × 10 . (e) Same as Fig. 10(a) except that δω/ω0 = −4 −4 6.0 × 10 , and the “on” frequency is increased to fm = 120 GHz. (f) Same as Fig. 10(a) except that δω/ω0 = 8.4 × 10 (3.5 times that in Fig. 8), and the “on” frequency is increased to fm = 180 GHz.

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034056-14 MICROWAVE FREQUENCY DEMODULATION. . . PHYS. REV. APPLIED 15, 034056 (2021)

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