Optical Properties of Direct and Indirect Coupling of ... - OSA Publishing

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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 32, NO. 21, NOVEMBER 1, 2014

Optical Properties of Direct and Indirect Coupling of Cascaded Cavities in Resonator-Waveguide Systems Jianhong Zhou, Wenbo Han, Ying Meng, Hongfei Song, Da Mu, and Wounjhang Park

Abstract—By using the coupled-mode theory-based transfer matrix, we analyze the optical properties of direct and indirect coupling of cascaded cavities in the resonator-waveguide system. The dispersion relation of the system, bandgap opening mechanisms, and anticrossing behavior are investigated theoretically and numerically. We also derive the transmission coefficients of the direct and indirect coupling system with multiple cavities and demonstrate the interaction between the modes, which leads to Fano resonance with an asymmetric transmission line shape. All the results obtained by the transfer matrix method are confirmed by the finite-difference time-domain method in photonic crystals. Index Terms—Optical coupling, optical waveguides, photonic integrated circuits.

I. INTRODUCTION HE OPTICAL properties of resonator-waveguide coupling systems have been intensively researched in recent years due to their fundamental importance in design of optical devices in photonic integrated circuits, such as channel drop filters [1]–[3], power splitters [4], [5] and intersections [6], [7]. The directly coupled system, called coupled-resonator optical waveguide (CROW) where the nearest resonators are coupled with each other via an evanescent optical field was proposed and analyzed by Yariv group [8]. And the indirectly coupled system, or indirect CROW, where the multiple resonators are coupled together by the propagating modes of a waveguide have also been studied in detail for the mode coupling mechanisms and the transmission properties [9]. However, resonator-waveguide systems with both direct and indirect couplings, or direct and indirect CROW, are seldom studied yet. In this work, we present a comprehensive analysis of the modes in direct and indirect coupling resonator-waveguide systems using the transfer matrix method based on the temporal coupled-mode theory. Using this novel method, the dispersion

T

Manuscript received February 26, 2014; revised July 26, 2014; accepted August 6, 2014. Date of publication August 14, 2014; date of current version September 17, 2014. This work was supported by the National Natural Science Foundation of China under Grant 11104020 & 11474041, and Science and Technology Innovation Funds of the Changchun University of Science and Technology (No. XJJLG-2011-07). J. Zhou, W. Han, Y. Meng, H. Song, and D. Mu are with the School of Photoelectric Engineering, Changchun University of Science and Technology, Changchun 130022, China (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]). W. Park is with the Department of Electrical, Computer and Energy Engineering, University of Colorado, Boulder, CO 80309 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JLT.2014.2346789

Fig. 1. Illustration of the direct and indirect coupled waveguide-resonator system.

relation of the system is obtained, and two kinds of bandgap opening mechanisms and anti-crossing behavior are investigated theoretically. Finally, we investigate the optical transmission through a waveguide directly and indirectly coupled with multiple cavities. Our analysis shows that the interaction between the modes through direct and indirect coupling leads to mode frequency splitting and Fano resonance. All the characteristics of the coupling systems are also confirmed numerically by the finite-difference time-domain method (FDTD) in photonic crystals. II. “TRANSFER MATRIX” ANALYSIS In order to analyze the problem of direct and indirect coupling of cascaded cavities in resonator-waveguide system, temporal coupled-mode theory [10] based transfer matrix method is used to get the scalability of the system. Fig. 1 shows the periodic structure of the coupling system, which consists of a cascade of cavities side-coupled to the waveguide, where both the cavity and waveguide are monomodal. The normalized amplitude of the ith cavity, the amplitudes of the incoming and outgoing waves into the cavity are denoted by ai , S+ij and S−ij (j = 1, 2), respectively. The optical mode properties of such a system can be calculated by the coupled-mode theory in time domain [10]. When the electromagnetic wave at a frequency ω in the form of ej ω t is incident upon the system, the time evolution of the amplitudes of the cavities in steady state can be described as 2 dai = jω0 − ai − jμai+1 − jμai−1 dt τ 2 (S+i1 + S+i2 ), i = · · · , −1, 1, 2, · · · (1) +j τ 2 S−i1 = S+i2 − j ai , (2) τ 2 S−i2 = S+i1 − j ai , (3) τ

0733-8724 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information.

ZHOU et al.: OPTICAL PROPERTIES OF DIRECT AND INDIRECT COUPLING OF CASCADED CAVITIES IN RESONATOR-WAVEGUIDE SYSTEMS

where ω0 is the center frequency of the cavity resonance and μ is the mutual coupling coefficient which describes a direct evanescent tunneling process between the cavities, 2/τ is the decay rate of the cavity mode amplitude ai into the waveguide. Due to the indirect coupling via waveguide between cavities nearby, the incoming waves to the cavities should satisfy the relationships S+i1 = S−(i+1)2 exp(−jβR)

(5)

where β is the wave propagation constant in the waveguide and R is the distance from ith cavity to (i + 1)th cavity. So Eq. (1)–Eq. (3) can be rewritten in transfer matrix form as φi = M φi−1

(6)

where the transfer matrix M and normalized mode amplitudes φi , respectively, are ⎤ ⎡ ω−ω ej β R 2 e−j β R 2 −1 ⎢− μ μ τ μ τ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 1 0 0 0 ⎥ ⎢ ⎥ ⎢ (7) M =⎢ ⎥ 2 j β R ⎥ ⎢ −j 0 e 0 ⎥ ⎢ ⎥ ⎢ τ ⎦ ⎣ 2 0 0 e0j β R j τ and

M = φi = ai+1

ai

S+i1

S−i1

(8)

In the method, the characteristics of the direct and indirect coupling in the resonator-waveguide system are described by the transfer matrix M . And the input and output modes are described by the normalized amplitudes of the resonant and propagation modes φi , which are supported by unit cavity segments. The input and output properties are connected by transfer matrix to determine the properties of the system. III. DISPERSION RELATIONS Since the cavity segments of the periodic structure are coupled through the evanescent Bloch waves and the waveguide modes satisfy the Bloch theorem [11], the complex amplitude of the ith cavity φi can be expressed as φi = Ae−j (q iR −ω q t)

(9)

where A and ωq are the amplitude and the resonant frequency of the eigenmodes, respectively. q is the “lattice wave” vector (Block wave vector) and qiR is the resonant phase of ith cavity segment. Substituting Eq. (9) into Eq. (6) and solving the corresponding secular equation, we obtain the dispersion relation for the direct and indirect CROW modes to be

sin(βR) =0 μτ

cos(qR) = cos(βR) +

2 sin(βR) τ ωq − ω0

(10)

where cos θ = (ω0 − ωq ) / 2μ. From Eq. (10), the dispersion relation of the system depends on many parameters, such as

(11)

which was discussed in Ref. [9]. For 2 / τ = 0, which means that there is no coupling between cavities and waveguide, i.e., the optical behaviors of the mutual coupled cavities and the waveguide mode oscillate independently, we can taken the optical system as a CROW and a normal waveguide, respectively. From Eq. (10), the dispersion relation can be written as ωq − ω0 (12) cos(qR) = − cos(θ) = 2μ cos(qR) = cos(βR).

(13)

Eq. (12) is the dispersion relation of CROW [8] and Eq. (13) is the dispersion relation of a normal waveguide with a continuous mode. For βR ≈ nπ, the optical oscillations in the coupled cavities and the waveguide are in phase or out of phase. From Eq. (10), the dispersion relation can be obtained ⎧ ωq − ω0 ⎪ ⎨ cos(qR) = − cos θ = , (βR ≈ 2nπ) 2μ ω − ω0 ⎪ ⎩ cos(qR) = cos θ = − q , (βR ≈ (2n + 1)π). 2μ (14) In general, from Eq. (10), the dispersion relation can be further simplified as cos(qR) =

cos θ + cos(βR) 2 1 4 sin(βR) ± . [cos θ − cos(βR)]2 + 2 μτ

(15)

From Eq. (15), we find that the direct and indirect coupling waveguide-resonator systems open two types of bandgaps where no propagating modes exist. The condition for opening the first type of badgaps is [cos θ − cos(βR)]2 +

4 sin(βR) < 0. μτ

(16)

This type of bandgaps opens in the vicinity of resonant frequency of the cavity ω0 and arises from the mixing between the cavity modes and the waveguide modes, where the phase shift βR should be (2n − 1)π < βR < 2nπ. The condition for opening the second type of bandgaps is |cos(qR)| > 1.

cos2 (qR) − [cos θ + cos(βR)] cos(qR) + cos θ cos(βR) −

coupling coefficient μ, the decay rate 2 / τ and the phase shift βR incurred as the waveguide mode travels the distance of unit segment R in the waveguide. For μ = 0, which means that the direct coupling between cavities can be ignored and the system can be taken as indirect CROW [9], we obtain the dispersion relation

(4)

S+(i+1)2 = S−i1 exp(−jβR)

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(17)

This type of bandgaps opens at the edges of the Brillouin zone and is the result of periodicity as commonly observed in photonic crystals. In order to examine the physics of the dispersion relation described by Eq. (15), the dependences of the relation on βR

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Fig. 3. The unit supercell of the direct and indirect waveguide in photonic crystal where radius of the point defect is rx . The supercell is consisting of five periods of the crystal (R = 5a).

Fig. 2. The dispersion relations of the direct and indirect coupling systems calculated from Eq. (15) where we choose parameters υ g = 0.3c (c is the light velocity in vacuum), 2 / τ = 0.02 Ω, μ = 5 / τ and R/υ g = 5/0.3 Ω −1 . We plot the frequencies in arbitrary unit of Ω. (a) β0 R = 2nπ − 0.4π, (b) β 0 R = 2nπ, (c) β0 R = 2nπ + 0.4π, where n is a integer. The solid lines are plot by (+) in Eq. (15) and dash lines are by (−). The dot lines are dispersion relation of unperturbed waveguide mode plotted by Eq. (18).

are plot in Fig. 2, where we assume a linear dispersion relation for the unperturbed waveguide mode [9] β = β0 +

Δω υg

(18)

where υg is the group velocity of the unperturbed waveguide mode and Δω = ω − ω0 . A typical value of υg is υg = 0.3c (c is the light velocity in vacuum) for many photonic crystal waveguide [9]. From Fig. 2, some results can be obtained: (1) near the resonant frequency (Δω = 0), anti-crossing occurs [11]–[13]; (2) the first kind of bandgap is opened near the resonant frequency (Δω = 0), while the second kind of bandgap is opened at the edge of the Brillouin zone which is far from the resonant frequency as shown in Fig. 2 (a); (3) only the second kind of bandgap is opened at the edges of the Brillouin zone and the first kind of bandgap is closed due to the frequency overlap of the (+) and (−) states in Eq. (15) since the waveguide modes extend to frequency region of anti-crossing, as shown in Fig. 2(c); (4) from Fig. 2(b), the bandgap just above resonant frequency of the cavity is cancelled by the (−)states in Eq. (15), while the bandgap just below the resonant frequency, which is the first type of bandgap, exists; (5) when the frequency is far form the resonant frequency of the cavity, the direct and indirect CROW modes approach the folded states of the unperturbed waveguide modes. In order to validate the results obtained from the theoretical analysis, we compare the theoretical analysis to FDTD simulations in photonic crystals. The crystal is made up of a square

array of dielectric rods in air with lattice constant a, which possesses a bandgap for transverse magnetic (electric field parallel to the rods) modes [14]. The rods have radius of 0.20a and a dielectric constant of 11.56. The waveguide is formed by removing a row of rods in the crystal. Cavities located 3a away from the center of the waveguide are created by reducing the radii of rods to rx , which can be used to change the cavity resonant frequencies by adjusting the radii of the point defects. Fig. 3 shows the unit supercell of the direct and indirect CROW in photonic crystal. In order to obtain the dispersion relation of the system as shown in Fig. 3, 2-D FDTD with perfectly matched layer boundaries imposed on the top and bottom of the supercell is carried out [15]–[17]. The dispersion relations of the system are shown in Fig. 4 for different resonant frequencies of the cavities. From Fig. 4(a) and (c), we find that near the resonant frequency of the cavity, the two bands that are expected to intersect are instead coupled to one another and repel, and the two band modes are mutually mixed and anti-crossing occurs. For Fig. 4(b) where the phase-matching condition β0 R = 2nπ is satisfied, the mode splits into three bands at the edge of the Brillouin zone q = 0. The split of the upper and lower states which come from the folded waveguide mode, is due to the periodicity of the structure and a bandgap opens, while the center state is due to oscillation of the cascaded cavities as shown in Eq. (14) except that anticrossing mixes the mode behavior, which may lead the mode coupling between the cavity mode and the waveguide mode. Comparing Fig. 4 with Fig. 2, an excellent agreement has been found with the theoretical analysis. In order to get more insight on the mixing due to the anticrossing behavior, we plot the electric field distributions for the direct and indirect CROW as shown in Fig. 5. For the dispersion relation as shown in Fig. 4(c), the states of the modes exchange their field distribution patterns due to anti-crossing: the fields in Fig. 5(g) and (j) are strongly localized in the cavities due to the mode frequencies near the cavity resonant frequency, and compared to Fig. 5(g) and (j), Fig. 5(h) and (i) have additional pair of nodes in the vertical direction. For the dispersion relation as shown in Fig. 4(b), the mixing is more complicated. For Fig. 5(d) and (f), the PC waveguide modes are dominant and for Fig. 5(e) is the cavity mode. For


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Fig. 6. Illustration of the direct and indirect coupling of n cascaded cavities in waveguide-resonantor system. The two side cavities are the auxiliary cavities with a n + 1 = 0 and a 0 = 0.

IV. OPTICAL TRANSMISSION THROUGH A DIRECT AND INDIRECT WAVEGUIDE WITH MULTIPLE CAVITIES Fig. 6 shows the structure of the coupling system with finite cavities, which consists of a cascade of n cavities side-coupled to a waveguide. The relationships between the inputs and the outputs of the system can be described as φn = M n φ0

(19)

where φ0 and φn are the start and termination conditions described as φ0 = [ S−12 exp(−jβR) φn

= [ S+n 1 exp(jβR)

S+12 exp(jβR) 0 S−n 1

an

0 ].

a1 ]

(20) (21)

In the start and termination conditions, the amplitudes of the two auxiliary cavities an +1 = 0 and a0 = 0 are used to make Eq. (19) solvable. Eq. (19) can be written as Fig. 4. The dispersion relations of the direct and indirect waveguide at different resonant frequencies. (a) rx = 0.10a, (b) rx = 0.0795a,(c) rx = 0.05a, which corresponds to β0 R < 2nπ, β0 R = 2nπ and β0 R > 2nπ, respectively. The inserts are the same plots except for the frequency range to exhibit further details of the dispersion line shapes near the resonant frequency of the cavity.

Fig. 5 Electric field distributions for the direct and indirect CROW, with (a–j) corresponding to the labeled points of Fig. 4.

Fig. 5(a) and (c), the mixing of the waveguide mode and the cavity mode are very strong. It is interesting that, for Fig. 5(b), just the waveguide mode exists and no cavity mode is excited even it is at the resonant frequency. This is because at the edge of the Brillouin zone q = 0, the waveguide mode is a standing wave with odd symmetry with respect to the vertical mirror plane [6], [7].

φn = Dφ0

(22)

D = Mn.

(23)

where

Eq. (23) can be solved by using the Hamilton–Cayley theorem [18]. As an example, we consider the case in the frequency vicinity satisfied the phase matching condition β(ω)R ≈ 2mπ (m is an integer). When the electromagnetic wave is incident upon the system from the left port, i.e., S+12 = 0, as shown in Fig. 6, the transmission of the system can be expressed as 2 2 S−12 2 D44 1 = = Tn = S+n 1 D11 D44 − D41 D14 1 + jσn (ω) (24) where 1 σn (ω) = 2τ μ sin2 θ2 ⎞ ⎛ n+1 2 nθ sin θ sin ⎟ ⎜ sin nθ 2 2 + − n⎟ ·⎜ ⎠ (25) ⎝2 n+1 sin θ θ sin θ cos 2 cos θ = (ω0 − ω) / 2μ.

(26)

Eq. (26) describes the dispersion relation of CROW [8]. Examining Eq. (24), we note that the cascade of n cavities in the directly and indirectly coupling system can be treated as a single composite cavity with the detuning function σn (ω)[16], [17], as described as Eq. (25), side-coupled to the waveguide. When

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Fig. 8. Structure of direct and indirect coupling system in photonic crystals. The crystal consists of a square lattice of rods with a dielectric constant of 11.56 and a radius of 0.2a in air (where a is the lattice constant). Waveguide is formed by removing rows or columns of rods in the crystal and cavities are created by reducing the radii of rods. The cascaded cavities, created by reducing the radii of rods to 0.0795a are placed at the interval of 5a.

Fig. 7. Theoretical transmission spectra of the optical systems as shown in Fig. 6 with different number of cavities. The solid lines are the spectra calculated from Eq. (12) with (a) n = 2, (b) n = 3, (c) n = 4 and (d) n = 10. The dashed lines represent the transmission spectra with one cavity. We plot the frequencies in arbitrary unit of Ω. The resonant frequencies of the cavities are ω 0 = 0.5 Ω. The decay rate of the cavity mode amplitude a i into the waveguide is 2 / τ = 0.01 Ω. The mutual coupling coefficient between the cavities is μ = 0.02 Ω.

σn (ω) → ∞, i.e., cos[(n + 1)θ/2] = 0, zero transmission of the system can be achieved at frequencies 2m + 1 π , (27) ω = ω0 − 2μ cos n+1 where m is an integer and n > 2m. From Eq. (27), we notice that the resonance dips are located between the frequency range [ω0 − 2μ, ω0 + 2μ], this is because the cascaded cavities forms CROW which opens the photonic bandgap between the range where the resonant states can be excited [8]. In order to analyze the transmission characteristics of the structure with different number of cavities, in Fig. 7 we plot the theoretical spectra calculated from Eq. (24). As a comparison, we also show in Fig. 7 as dashed lines the transmission spectra for the same structure with only one cavity. Examining Fig. 7(a)– (c), we note that with the successive increase of the number of cavities n, the number of the transmission dips equals n/2 (the nearest integer less than or equal to n/2) which is determined by the condition n > 2m in Eq. (27). From Fig. 7(a), the composite cavity consisting of two mutually coupled cavities shifts the resonant frequency and broadens the line profile. While for three- and four-cavity systems, there exists a transmission peak, which is referred to as the Fano resonance [13], as shown in Fig. 7(b) and (c). The reason for the phenomena is that under the phase-matching condition, only the even modes can be excited. For a system with cavity number less than two, only one even mode can excited, while for a system with cavity number more than three, more than one even modes (= (n + 1)/2) can be exited and the interaction between these modes lead to Fano resonance, where the lowest order mode forms the spectrum line shape background and the higher order modes form the transmission peaks in the transmission spectrum. From Eq. (24), with further increasing the number of cavities, the frequency position of the left (right) dip approaches to ω0 −

2μ (ω0 + 2μ), which means that the Fano resonance occurs in the CROW bandgap range, as shown in Fig. 7(d). The frequency positions of the transmission dips and the intervals between them can be determined by Eq. (27). We also note that the spectra consist of Fano resonance features superimposed upon a Lorentzian line shape background which is not defined by the single cavity resonance, where the background line profile is broadened. The direct coupling of the cavities forms different modes, which comes from the modulation by the Block wave and each mode are side-coupled into the waveguide. The broadening of the background profiles comes from the lowest mode (first-order mode) of the coupling system [9], [19]. The interaction between the higher order modes and the broadened lowest mode leads to resonance peaks known as the Fano resonance. This can also be seen from Eq. (24) and (25), where the resonant frequency of the composite cavity splits due to the discontinuity of the detuning function at the frequencies described by Eq. (27). It is interesting that except n = 1 and n = 2, in all other cases, the system exhibits an asymmetric line shape and the widths of the resonant dips decrease with the increase of the frequency because the resonant frequencies of the modes are all different and higher order modes with higher resonant frequencies have smaller decay rates. To validate the results of the transfer matrix method mentioned above, we compare the theoretical analysis to FDTD simulations in photonic crystals. Fig. 8 shows the designed direct and indirect coupling resonator-waveguide systems in a photonic crystal. In order to meet the phase-matching condition β(ω)R ≈ 2mπ, distance between cavities is set to R = 5a and the radii of rods are set to 0.0795a with 3a away from the center of the waveguide. The normalized transmission coefficients are obtained as shown as solid lines in Fig. 9 with different number of cavities. In comparison, we also plot in Fig. 9 (dashed lines) the transmission spectra for the same structure with only one cavity. The mutual coupling coefficients can be determined to be μ = 2.05 × 10−4 (2πc/a) from the transmission dips between one cavity and two-cavity systems in Fig. 9(a) or the two transmission dips in the three-cavity system in Fig. 4(b) by using Eq. (27). Comparing Fig. 9(a)–(c), we note that the spectra are asymmetric with respect to the center frequency ω0 = 0.34576(2πc/a) and the width of transmission line profiles become broader with the increase of the cavity number. Especially, in Fig. 9(b), the frequency difference between


Fig. 9. Solid lines are the transmission spectra of the structure (as shown in Fig. 8) with different number of cavities: (a) n = 2, (b) n = 3 and (c) n = 4. The dashed line is the transmission spectra for the same structure with only one cavity.

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and (e), respectively. The cascaded cavities can be regarded as a segment of CROW, and the modes supported by the segments are modulated by the Block wave and form different order of modes. The first-order modes, as shown in Fig. 10(b) and (d) have large decay rates and form broad resonant dips which form a Lorentzian line shape background in the transmission spectra, as shown in Fig. 9(b) and (c), respectively, while the higher order modes have small decay rates [as shown in Fig. 10(c) and (e)] and interact with first-order modes. The first-order modes can be treated as the bright modes and the higher order modes as dark modes. The interaction between the bright modes and dark modes in the waveguide leads to Fano resonance [13], [15]. As a result, the resonant frequency of the composite cavity splits and forms asymmetry peaks on the background line shape, which referred as the Fano resonance. Since the frequencies of these modes are different, that is why the line shapes of the composite cavities are always asymmetry. V. CONCLUSION In conclusion, we have investigated an optical system with direct and indirect coupling between cavities and waveguide by using transfer matrix based on temporal coupled-mode theory. The dispersion relation, bandgap opening mechanisms, anti-crossing behavior and Fano resonance are theoretically analyzed and numerically demonstrated by the FDTD and excellent agreements have been found with the present theoretical analysis. REFERENCES

Fig. 10. Steady-state electric field distributions in the resonatorwaveguide structures. (a) n = 2, ω = 0.34555(2πc / a) (b) n = 3, ω = 0.34547(2πc / a) and (c) n = 3, ω = 0.34602(2πc / a); (d) n = 4, ω = 0.345406(2πc / a); (e) n = 4, ω = 0.345854(2πc / a).

the peak and the right dip, which can be used to switch the system from complete reflection to complete transmission, is 2.25 × 10−5 (2πc/a). The frequency shift is as small as 0.11μ, which can rapidly reduce the optical power to switch the system. All the transmission properties simulated by the FDTD completely agree with the theoretical analysis. In Fig. 9(a), only one resonant dip is shown in the spectrum of two cavity system. In general, in two-coupled-cavity system, the resonance mode splits into one odd mode and one even mode, so there are two resonant dips in the spectrum [20]. However, in our system, we just consider the case in which the phase-matching condition β(ω)d ≈ 2mπ is satisfied. Under this condition, only the even mode is excited and the odd mode is cancelled, as shown in Fig. 10(a). This feature is also conserved for all other cases as shown in Fig. 10 where just the even modes are excited. In Fig. 10, the steady state electric field distributions of the three and four cavity systems are shown where the lowest modes of the composite cavity are shown in Fig. 10(b) and (d) and the higher modes are shown in Fig. 10(c)

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Authors’ biographies not available at the time of publication.