QCSE Tuned Embedded Ring Modulator

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 32, NO. 1, JANUARY 1, 2014 107

QCSE Tuned Embedded Ring ModulatorViswas Sadasivan and Utpal Das, Senior Member, IEEE

Abstract—A micro embedded ring resonator modulator usingquantum confined stark effect (QCSE) has been modeled and stud-ied in this paper. Semi analytical design methods for various com-ponents like bend, coupler, and the whole embedded ring have beenused here. Time domain and frequency domain analysis methodshave been developed and studied. Using these methods a 50 Gbps,11 dB extinction ratio, 5 μm outer radius embedded ring, QCSEtuned optical modulator design has been proposed.

Index Terms—Embedded ring resonator, InGaAsP/InP, microring resonator, modulator, quantum confined stark effect (QCSE),state equation, taper coupler, transient analysis.

I. INTRODUCTION

IN recent years, ring resonators have arouse significant in-terest because of their small size [1], [2], cavity power en-

hancement [3], [4] and narrow line widths [5]. These qualitiesmake micro rings suitable for various high speed switching ap-plications in communication [6] and signal processing [3], [7].

Speed of switching resonant structures is often limited byswitching mechanisms [6], [8] and cavity lifetimes [9]. Vari-ous fast tuning methods in single ring [10], [8] and tuning ofring resonators in general were theoretically and experimen-tally studied in [9], [11], [12]. The switching speeds obtainableby single rings were proposed to be surpassed by using “Elec-tromagnetically induced transparency” (EIT) like spectrum ofembedded ring resonators [13], but they remained speed limitedby free carrier switching mechanisms.

InGaAsP/InP hetrostructures are widely used in optical com-munication band [14]. The availability of strong quantumconfined stark effect (QCSE) and diverse high speed photoniccomponents like sources and detectors in this material systemmakes InGaAsP/InP excellent material for making optoelec-tronic modulators. The opportunities offered by QCSE tuning inembedded ring resonators has not yet been studied and reported.

Block reducibility of problems is a desirable quality in pho-tonic device simulation. Full volume simulation methods likeFDTD [15], FDM [16], [17] etc. provide accurate field patternsof diverse geometries. But, due to their high computational de-mands and lack of block reducibility, building a complex systemmodel with them is difficult. A fully analytical method for mod-eling devices also has limited use, owing to their poor accuracyand lack of generality. In case of embedded ring resonators,

Manuscript received January 5, 2013; revised August 17, 2013 and October8, 2013; accepted October 30, 2013. Date of publication November 5, 2013;date of current version December 2, 2013.

The authors are with the Indian Institute of Technology Kanpur, UP 208016,India (e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JLT.2013.2289324

Fig. 1. Embedded Ring Resonator.

the mid course solution of semi analytical method has not beenreported before.

Here, we have presented a semi-analytical modeling methodfor embedded ring resonators tuned using QCSE, extendableto general coupled ring structures. The modeling problem isdivided into two sections, theoretical formulation in Section IIand its computational adaptation in Section III. Each of thesesections is subdivided into building blocks of the resonator. InSection IV, the results are given and a particular design for ahigh speed modulator is proposed and analyzed.

II. THEORETICAL MODEL

The embedded ring modulator consists of bent, straight opti-cal waveguides and four waveguide couplers as shown in Fig. 1.The theoretical models used for analyzing each of them, for inte-grating these blocks to form an embedded ring and for calculat-ing the system’s wavelength, transient and modulator responsesare given in this section. For simplicity, polarization depen-dency and reflections are ignored during waveguide and couplerdesign. Nearly TE like propagation was assumed although thesame procedures can be extended to TM like modes. At theend of this section, the models used for bulk semiconductor andQCSE are discussed.

A. Straight and Bent Waveguides

The straight and bend multilayer rib or ridge waveguide hasbeen solved using transfer matrix method (TMM) followed byeffective index approximation [18]. The bent rib waveguideproblem is reduced into horizontal and vertical bent slab prob-lems [19], [20]. The vertical bent slab waveguide’s TE field canbe expressed by scalar Bessel equation of the form

∂2Ez

∂r2 +1r

∂Ez

∂r+ [k2n2

i −γ2R2

r2 ]Ez = 0 (1)

Here Ez is the transverse electric field, k is the free spacepropagation constant, γ = β − jα is the complex propagationconstant which also account for the losses, ni are the refractiveindices of regions as defined in bend of Fig. 2. R is the outerradius of the bend and r, θ are defined in Fig. 2.

0733-8724 © 2013 IEEE

108 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 32, NO. 1, JANUARY 1, 2014

Fig. 2. The bent coupler structure. r, θ define the polar position vector.

The field solution for (1) is given by [21], [22] as

Ez = ei(ωt−ν θ)

×

⎧⎨

⎩

M11Jv [k1x] + M12Yv [k1x]; for R − a < x < R

M2H(2)ν [k2x] for x > R

M3Jv [k3x] for x < R − a

(2)

Hθ = − j

ωμ

∂Ez

∂x, Hr=

ν

rμωEz (3)

Here, ν = γr , ki = nik and Mi(j ) are the coefficients.These Bessel (Yν or Jν ) or Henkel (Hν ) functions are com-plex order functions. On matching fields at the boundaries, amatrix equation can be obtained using (2), (3) as⎡

⎢⎢⎢⎣

Jν (Rk1 ) Yν (Rk1 ) −H( 2 )ν (Rk3 ) 0

Jν [(R −a)k1 ] Yν [(R −a)k 1 ] 0 −Jν [(R −a)k5 ]

k1 J ′ν (Rk1 ) k1 Yν (Rk1 ) −k3 H ′ν ( 2 ) (Rk3 ) 0

k1 J ′ν [(R −a)k1 ] k1 Y ′

ν [(R −a)k 1 ] 0 −k5 J ′ν [(R −a)k5 ]

⎤

⎥⎥⎥⎦

×

⎡

⎢⎢⎢⎣

M 1 1

M 1 2

M 2

M 3

⎤

⎥⎥⎥⎦

=

⎡

⎢⎢⎢⎣

0

0

0

0

⎤

⎥⎥⎥⎦

(4)

B. Coupler

The coupler was modeled by vector coupled mode theorybased on [17], [22], [23] to maintain generality. As the prop-agation direction for the bend is along θ in polar coordinatesand along y for straight waveguide in rectangular coordinatesshown in Fig. 2, a set of unit vector relations as given below arerequired for interchanging between them,

r = (xx + yy)/ |r| , θ = (−yx + xy)/ |r| (5)

A mode in pth waveguide whose refractive index profile isnp(x, y, z) (or np ) has electric and magnetic fields Ep, Hp

respectively. The total fields E, H in the coupling region maybe expressed as a linear combination of individual mode fields,

E = A(z)EI + B(θ)EII (6)

H = A(z)HI + B(θ)HII (7)

Here a straight waveguide is indexed as I and a bent wave-guide as II. A(z), B(θ) are the field envelopes. Using (6), (7) inMaxwell’s equations for the coupled system, one gets,

∇× [A(z)EI + B(θ)EII ] = −jωμ0 [A(z)HI + B(θ)HII ]

(8)

∇× [A(z)HI + B(θ)HII ]

= jωε0n(x, y, z)2 [A(z)EI + B(θ)EII ] (9)

Here n(x,y,z) (or n) represents the total waveguide refractiveindex profile. It can be shown that

∇×A

{EI

HI

}

=A

{−jωμ0HI

jωε0n21EI

}

+ y∂A

∂y×

{EI

HI

}

(10)

∇×B

{EII

HII

}

=B

{−jωμ0HII

jωε0n22EII

}

+θ

|r|∂B

∂θ×

{EII

HII

}

(11)

using these in (8) and (9), and on simplification, the mixedcoordinate coupled mode equations for bent to straight couplercan be obtained as,

∂A

∂y+

1|r|

∂B

∂θCI ,I I + j A XI + j B KI,I I = 0 (12)

and,

1|r|

∂B

∂θ+

∂A

∂yCII ,I + j B XI I + j AKII ,I = 0 (13)

where,

Cpq =1

Dp

∫ ∞

−∞

∫ ∞

−∞np .[E

∗p×Hq−Eq×H

∗p]drdz (14.a)

Xp =1

Dpωε0

∫ ∞

−∞

∫ ∞

−∞(n2 − n2

p) E∗p.Ep drdz (14.b)

Kpq =1

Dpωε0

∫ ∞

−∞

∫ ∞

−∞(n2 − n2

q ) E∗p.Eq drdz (14.c)

Dp =∫ ∞

−∞

∫ ∞

−∞np .[E

∗p × Hp + Ep × H

∗p]drdz (15)

p, q ={

I, straight waveguideII, bent waveguide

, np ={

y, p = I

θ, p = II

Throughout the calculation, the phase of the normal modewas retained, so that the envelopes A(z), B(θ) were not affectedby phase arising from length of coupling region. Hence, thisinherently is a point coupler model which can be placed at thesmallest separation point of the coupler and not a distributedmodel which needs phase correction.

C. Embedded Ring - Steady State Model

Steady state and transient analysis of ring resonator systemsrequire the impulse response of the system to be computed.Approximate, direct analytical [24] and coupled resonator [5]models are available. Here in this study, such complex exactanalytical solution of equations is avoided and a direct matrixinversion method with fewer approximations is developed.

SADASIVAN AND DAS: QCSE TUNED EMBEDDED RING MODULATOR 109

Fig. 3. Embedded ring variables for (a) static and (b) dynamic models.

Fig. 4. Conversion of bidirectional power flow model to unidirectional one.

The field envelopes at couplers of embedded ring can bewritten in matrix form, by inspection of Fig. 3(a) as

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

b1

b2

b3

b4

b5

b6

b7

b8

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

︸︷︷︸B

=

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

τ11 κ11 0 0 0 0 0 0

κ21 τ21 0 0 0 0 0 0

0 0 τ13 κ13 0 0 0 0

0 0 κ23 τ23 0 0 0 0

0 0 0 0 τ12 κ12 0 0

0 0 0 0 κ22 τ22 0 0

0 0 0 0 0 0 τ14 κ14

0 0 0 0 0 0 κ24 τ24

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

︸︷︷︸M

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

a1

a2

a3

a4

a5

a6

a7

a8

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

︸︷︷︸A

(16)

Here [τ1i κ1i

κ2i τ2i]is the coupling matrix of ith coupler (29).

Using the field envelope relation in each branch as

a2 = b7e−jϕ1 , a3 = b2e

−jϕ1 , a4 = b8e−jϕ2

a6 = b3e−jϕ1 , a7 = b6e

−jϕ1 , a8 = b4e−jϕ2 (17)

where, the field in ith ring’s branch of length Li and propagationconstant (βi-j αi), acquires a phase given by,

ϕi = (βi − jαi)Li (18)

From (16) one gets

A = M−1B (19)

and then using (17) in (19) one gets,⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

a1

000a5

000

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

︸︷︷︸A ′

=M−1B−

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 0 0 0 00 0 0 0 0 0 e−jϕ 1 00 e−jϕ 1 0 0 0 0 0 00 0 0 0 0 0 0 e−jϕ 2

0 0 0 0 0 0 0 00 0 e−jϕ 1 0 0 0 0 00 0 0 0 0 e−jϕ 1 0 00 0 0 e−jϕ 2 0 0 0 0

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

︸︷︷︸P

B

(20)

on multiplying with (M−1 − P )−1 on both sides, one gets,

B = (M−1 − P )−1A′ (21)

This expression when evaluated gives steady state field en-velopes at each branch; the knowledge of which can be veryuseful in understanding features as EIT like response [13]. Thebus input and output field envelopes are defined as,

a1 = aie−jϕb , at = b2e

−jϕb

ad = b5e−jϕb , a5 = aport3e

−jϕb (22)

Also, aport3 will be zero when input is given only from port 1.Using these, the transfer functions Hdrop,ER (f) and

Hthrough,ER (f) at drop and through port respectively, can beobtained as:

Hdrop,ER(f) =ad(f)ai(f)

and Hthrough,ER(f) =at(f)ai(f)

(23)

The method discussed here for embedded ring, can be ex-tended to more general coupled rings. Structures like one shownin Fig. 4 which has both forward and backward propagation inthe same branch, can be split into separate forward and back-ward sections as shown in the figure and solved using the abovemethod.

D. Embedded Ring - Pulse Input and Modulator Model

The three physical mechanisms restricting the maximum bitrate of a filter or modulator employing ring resonators are (a)long cavity lifetime, leading to tailing of pulses, (b) long cavitypath length, leading to ringing and (c) the band limited nature ofthe filter leading to ripples in pulses. A direct multiplication ofthe Fourier transform of the input signal with the ring resonatortransfer function followed by inverse transform models all theselimitations in pulse response. Thus,

ad(t) = F−1( F(s(t)) Hdrop(f) )

at(t) = F−1( F(s(t)) Hthrough(f) ) (24)

Here F is the Fourier operator and s(t) the input signal.In order to model the modulation process in ring resonators,

where system property itself changes with time, leaving residualfield in the cavity, a time domain analysis using a differential


equation [5] or a complex analytical model [12] was used inprevious works. In [5], the cavity lifetime and resonance fre-quencies are explicitly specified. Also, it uses the energy of thecavity as a uniform quantity and hence features like EIT andunequal cavity power distribution caused by phase gets ignored.A more complete, yet simple model was achieved by writingstate equation for the ring and iteratively calculating the timeevolution as discussed in Section III.

E. Layer Structure and Quantum Confined Stark Effect

The multilayer III-V semiconductor epitaxial structure designinvolves optimization of the optical confinement, losses, num-ber of modes and vertical core size. It has been designed as aPIN-hetrostructure with embedded quantum wells (QW) in thewaveguide core. For getting QCSE [25], a reverse bias needsto be applied across the QW. The I-region width posts a trade-off of electric field and the QW-to-light interaction resultingin good tuning, against loss. Losses in heavily doped claddingand contact metal may decrease the filter roll-off rates. Addi-tional I-InGaAsP wave guiding layers and then InP claddinglayers with 2 × 1016 cm−1 doping were chosen to improve lightcoupling without significantly increasing the losses. For deeplyetched structures, contact pads may have to be made away fromthe device, hence a thick N+ bottom contact layer on a semiinsulating (SI) substrate was chosen. The parallel plate capaci-tance is calculated using I-region thickness as the gap and P+

contact area. To minimize capacitance, the P+ contact line andpad should be on additional dielectric layer except in the devicecontact region.

The QW with an added electric field term has been mod-eled using effective mass approximation of Schrodinger waveequation. For obtaining significant refractive index change, thedevice is being operated near the band edge where role of ex-citons is very important. Hence a separate excitonic calculationbased on [26] is used. In the bulk semiconductor layers, compo-sition variation has been used for band gap engineering so thatthe waveguide remain transparent for optical propagation. Theenergy gap of InGaAsP has been calculated using [27], with lat-tice matching condition to InP. The relation for refractive indexof undoped InGaAsP was obtained from [28]. With these theabsorption coefficient of the material is also obtained [29].

III. COMPUTATIONAL SOLUTION

The computational model is an extension of the theoreticalframe work described in Section II. Discretization of equationsfor computational purpose and conditions of the model if any,are mentioned component wise, in this section.

A. Straight and Bent Waveguides

Straight waveguides are computationally solved using TMMand effective index method. Available complex order Besselfunctions [30] have been used to solve the bend problem. For(4) to have a nontrivial solution, the determinant of the 4 ×4 matrix will have to be equal to zero. This condition whenenforced forms a complex valued characteristic equation in α

and β. The α and β eigen values have been then obtained bymeshing and locating the intersection of real and imaginaryparts of characteristic equation close to zero. The coefficientsMi , obtained by placing these α and β in (4) have been thenused in (2) to calculate the fields.

B. Coupler

The coupled mode equations given in (12)–(15) were discre-tised along the radius and along the coupling region to solvethem. Here a simple finite difference discretisation has beenused in y and θ. Replacing ∂y by Δy and ∂θ by Δθ, (12) and(13) may be written as

An+1 − An

Δθ

−1|r| cosθn+1/2

+Bn+1 − Bn

|r|ΔθCI ,I I

+ jAn+1XI + jBn+1KI,I I = 0 (25)

Bn+1 − Bn

|r|Δθ+

An+1 − An

Δθ

−1|r| cosθn+1/2

CII ,I

+ jBn+1XI I + jAn+1KII ,I = 0 (26)

Δθ=θn+1 − θn , θn+1/2 = (θn+1 + θn )/2, Δθ ∼= Δy/ |r|

(25), (26) written in matrix form after simplification becomes[ −sec θn+1/2 + jΔθ |r| XI CI ,I I + jΔθ |r| KI,I I

−CII ,I sec θn+1/2 + jΔθ |r|KII ,I 1 + jΔθ |r|XI I

]

︸︷︷︸D1 , n

×[

An+1

Bn+1

]

=

[ −sec θn+1/2 CI ,I I

−CII ,I sec θn+1/2 1

]

︸︷︷︸D2 , n

[An

Bn

]

(27)

or,[

An+1

Bn+1

]

= D−11,nD2,n

︸︷︷︸Dn

[An

Bn

]

(28)

Cascading this recursive equation, for N steps the first andlast field envelopes can be written as

[AN

BN

]

=[

τ1 κ1

κ2 τ2

] [A1

B1

]

,

[τ1 κ1

κ2 τ2

]

= DN −1DN −2 . . .D2D1 (29)

The fields have been numerically integrated along the radialdirection at each θ step to find the coefficients Kpq ,Xp and Cqp .Steps in θ alone indicates unchanging radial field-mesh, but achanging field-mesh in the straight waveguide is used, avoid-ing recalculation of the bent fields and it provided significantcomputational advantage.

C. Embedded Ring - Steady State

The embedded ring transfer function has been obtained bynumerically evaluating (21) for each wavelength. This sameequation can be used with single ring dual bus resonator when


Fig. 5. Quantum well (a) absorption and (b) dispersion for different fields.

coupling coefficients of couplers 3, 4 in Fig. 3(a) are zero andcorresponding transmissions set to unity.

D. Embedded Ring - Pulse Input and Modulator

For finding step response, a pulse sequence of arbitrary shapecan be given as baseband input. Then, Fourier domain calcu-lation using FFT in (24) was used to obtain the pulse shape atoutput ports of resonator.

For time domain analysis of the modulator-a time varyingsystem, from Fig. 3(b), the amplitude state equation connect-ing nth and (n+1)th time step can be written as below anditeratively solved to get the field evolution.⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

a1 ,2

a1 ,5

a1 ,4

a1 ,7

a1 ,6

a1 ,9

a1 ,8

a1 ,10

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

n +1

=

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

τ11 κ11 0 0 0 0 0 0

κ21 τ21 0 0 0 0 0 0

0 0 τ12 κ12 0 0 0 0

0 0 κ22 τ22 0 0 0 0

0 0 0 0 τ13 κ13 0 0

0 0 0 0 κ23 τ23 0 0

0 0 0 0 0 0 τ14 κ14

0 0 0 0 0 0 κ24 τ24

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

aN 1 ,1

aN 8 ,1

aN 3 ,3

aN 6 ,6

aN 5 ,5

aN 1 0 ,10

aN 7 ,7

aN 9 ,9

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

n

(30)

where

ah+1,m |n+1 = ah,m |n e−j φ mN m −1 , h = 1, 2, 3 ... , (Nm − 1)

(31)

a1,1 |n + 1

= ai |n , a1,3 |n+1 = 0,

at |n + 1= aN2 ,2 |n , ad |n+1 = aN4 ,4 |n (32)

If one uses uniform space steps, there will be discretisationerror at the couplers since signals are looped over different pathlengths. Here to avoid this difficulty without having to use veryfine step, each (mth) arm is divided into integer number of steps(Nm ), thus making every coupler a node. Step length of shortestarm is chosen such that it is much smaller than slowly varying

Fig. 6. (a) Dispersion of straight (450 nm wide) and bent (5 μm radius, 700 nmwide) ridge waveguides for different electric fields. (b) Calculated complexpropagation constants for different bent waveguide radii and widths. 1554 nmwavelength and 1.535 refractive index of polymer insulation are used.

Fig. 7. Variation of power coupling and transmission coefficients against(a) wavelength for R = 100 μm (800 nm bend, 680 nm straight waveguidewidths and g = 500 nm) and R = 5 μm (700 nm bend and 450 nm straightwaveguide widths and g = 100 nm). (b) waveguide separation at 1554 nmwavelength.

signal period and at least two nodes are there per arm. Other Nm

are chosen such that their steps are same or shorter in length.

E. Layer Structure and Quantum Confined Stark Effect

Discretisation of Schrodinger equation and its solving hasbeen performed using three point FDM method [31], [32]. Theexciton energy has been calculated by minimizing the excitonenergy variational parameter. Wave functions which grow con-tinuously or tunnel through without getting confined in the wellwere discarded to prevent such wave functions from interferingwith the band bending model solution.

The refractive indices of the layers at each wavelength ofinterest were obtained by numerical integration of absorptioncoefficients as per Kramers-Kronig relations. The values of ef-fective masses were obtained by interpolation from the binary


Fig. 8. Simulated coupler cross sectional fields for g = 100 nm, 700 nmwide 2.5 μm radius bend coupled to (a) (above) 400 nm straight waveguide(power coupling coefficient, κ2 = 8.6% at 1554 nm), (below) 300 nm straightwaveguide (κ2 = 15% at 1554 nm). (b) Proposed taper coupler structure witha taper (κ2 = 12% at 1554 nm for W1 = 400 nm and W2 =300nm).

composition values [33] and constants in absorption coefficientcalculation, were obtained from [29].

IV. RESULTS AND DISCUSSION

The dispersion of bulk InGaAsP for various compositions anddoping values were studied. Rapid variation of refractive indexwas observed around 1017 cm−3 of doping and hence dopinglevels between 8 × 1016 cm−3and 1018 cm−3 were avoided.A P+ -InGaAs (50 nm) contact layer, P+ -InP (200 nm),P -InP (300 nm), I-InGaAsP (200 nm), I-multi QW (135 nm),I-InGaAsP (150 nm), N -InP (200 nm), N+ -InP (2 μm) andSI-InP (substrate) structure is chosen. The small size of ringresonator makes their electrical bandwidth much larger com-pared to their QCSE based Mach–Zehnder counterparts. Herethe RC time constant for a 5 um radius embedded ring has beencalculated as 0.4 ps indicating 400 GHz electrical bandwidth.

The QW model is favorably compared with experimentallyreported values [34]. For communication band operation, theQW absorption was designed to peak at 1337 nm. QW withIn0.70Ga0.30As0.64P0.36 (15 nm) well and InP (15 nm) barrierwas used. With I-InGaAsP (1.05 eV band gap) wave guidinglayer on either side, the total I region width is 485 nm. Effectiveindex tuning of 1.7 × 10−3 is calculated for 63 kV·cm−1 bias.The quantum well absorption and dispersion plots are shown inFig. 5. Internal 0 V field of 21 kV/cm was assumed.

Validity of the bend waveguide model was verified using [22].Also, the calculated neff matched well with measured valuefor a similar geometry in [10]. For very large index contrastwaveguides with large radii, the accuracy of this method fallsbecause of loss of field continuity calculated from the complexorder Bessel functions [35]. Since ring with large radius doesnot require large index contrast, and short radius waveguidehas sufficient radiation loss even with large index contrast, thislimitation could be circumvented.

For 100 μm radius 800 nm wide bends, ridge etch depth of830 nm and in short radius structures >860 nm (typically 1.5–2 μm) has been calculated to obtain radiation losses <1 dBcm−1 ,ignoring surface scattering. For deeply etched structures, surfacescattering losses dominate, and need special techniques for mod-eling [36]. Such scattering calculation is beyond the scope ofthis study. Therefore, additional losses of 2 dBcm−1for 100 μm

Fig. 9. Embedded ring resonator wavelength response and tuning (a) for103 μm outer and ∼101 μm inner ring radii, and two 35 μm straight waveg-uides in outer ring. Pointed arrows mark the outer ring resonance and diamondmarkers indicate inner ring resonance. (b) for 5 μm outer, 4.45 μm inner ringradii and 3.383 μm straight waveguide segments in outer ring.

radius and 20 dBcm−1 for 5 μm radius bend [10], [37] were as-sumed. This loss is in addition to the 15 dB/cm waveguide losscalculated in quantum well and doped cladding layers for shortradius structures. The calculated dispersion of bent and straightwaveguides is shown in Fig. 6 for various waveguide widths andbend radii. Straight waveguides are kept narrow for better fieldprofile match with the bend and control over the mode count.

The accuracy of coupling coefficient calculation has been ver-ified using measured results [37]. The calculated power couplingcoefficient variation with wavelength and waveguide separation(g) is shown in Fig. 7. It can be seen that the power couplingshowed strong sensitivity to waveguide separation. For given awaveguide separation, the coupling coefficients are larger fornarrower straight waveguides; because, they have effective fieldpeak closer to the bend as shown in Fig. 8(a); and hence alarger overlap integral. A coupler with narrow waveguide widththroughout will have very large scattering loss. Hence, a ta-per like in Fig. 8(c) was introduced in the straight waveguide,increasing the coupling coefficient by ∼40%. Phase matchingrequirement in this non-racetrack coupler is less stringent, sincethe coupling region is short.

The transfer functions of an ∼100 μm radius embedded ringis shown in Fig. 9(a). For large g the resonance is very narrowbut for smaller g, the resonances became broader and the passband amplitudes increases. Non-uniform energy distribution isobserved in the rings at resonance, similar to earlier reports[25], with inner ring resonating only when inner and outer ringresonances overlap. Rapid changes in the spectrum caused byopening of alternate light path with different phase, throughthe inner ring, were observed. The FSR and 3 dB-line widthsobtained are tabulated in Table I.

For short radii rings, bringing both the ring resonances to-gether is a difficult task because of their large free spectralranges (FSR) and sensitivity to symmetry. Hence the ring


TABLE IBANDWIDTH, FSR, AND FREQUENCY DOMAIN SPEED ESTIMATE

Fig. 10. A 5 μm radius embedded ring’s. (a) Drop port rectangular pulse inputresponse, (b) through port modulation response and signal Q(dB) at 60, 50, and40 Gbps (at 1526.90, 1526.96, 1527.02 nm center wavelengths).

dimensions are to be precisely made for a particular operationwavelength. This difficulty can be further overcome by employ-ing an initial temperature tuning or initial bias of the device tillit is brought to intended resonance wavelength.

Operation of the modulator: In Fig. 9(b), the ∼5 μm radiusembedded ring shows sharp roll off between the outer and innerring resonances. With OFF state bias, the operating wavelengthlied within the stop band and input CW optical power was cut-off from the through port. With ON state bias applied, the passband shifted to input CW wavelength allowing power to reachthe through port. Fast transitions between OFF and ON statestook place because of the steep spectral roll-off and fast QCSEtuning, allowing high data rate modulation. The exact operationpoint and electric field values depended on the detuning betweenrings and hence the ripple in the spectrum. In this study, 0 V wasused for OFF state and 2 V for ON state, applied simultaneouslyon both the rings.

The optical pulse responses of embedded ring filters in thepass band were calculated using frequency domain method andare shown in Fig. 10(a). The bit rate obtained from pass bandpulse response is a good estimate for the modulator limit. But,it did not account for residual cavity power in the modulatorand hence the time domain method was used. The modulatorresponse with electrical input obtained using time domain modelis given in Fig. 10(b) and the best for BER<10−9 was 50 Gbpswith 11 dB extinction ratio. The validity of frequency and timedomain methods has been confirmed by their agreement in thepulsed optical signal responses.

V. CONCLUSION

A new theoretical model and a semi-analytical computationalmodel for designing embedded ring resonators or general cou-pled rings in steady state, pulsed and modulator modes are devel-oped. A novel method for increasing bent to straight waveguidecoupling is proposed. Using embedded ring resonance, taper

coupler and QCSE, a 50 Gbps modulator with an extinctionratio of 11 dB is proposed.

ACKNOWLEDGMENT

Discussions with Dr. R. K. Sonkar, Dr. T. Bhowmick, S. Das,Raghunandan M. R., Prof. R. Vijaya, S. Dhongdi, V.P. Singh,R. Ranjan, R. K. Chaudhary and others have greatly helped inthis study.

REFERENCES

[1] M. K. Chin, C. Youtsey, W. Zhao, T. Pierson, Z. Ren, S. L. Wu et al.,“GaAs microcavity channel-dropping filter based on a race-rrack res-onator,” IEEE Photon. Technol. Lett., vol. 11, no. 12, pp. 1620–1622, Dec1999.

[2] Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale siliconelectro-optic modulator,” Nature, vol. 435, pp. 325–327, 2005.

[3] V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P. T. Ho,“Optical signal processing using nonlinear semiconductor microring res-onators,” IEEE J. Sel. Topics Quantum Electron., vol. 8, no. 3, pp. 6524–6540, May/Jun. 2002.

[4] M. F. Morichetti and M. Martinelli, “Four-wave mixing and wavelengthconversion in coupled-resonator optical waveguides,” J. Opt. Soc. Amer.B, vol. 25, no. 12, pp. C87–C97, Dec. 2008.

[5] B. E. Little, “Microring resonator channel dropping filters,” J. Lightw.Tech., vol. 15, no. 6, pp. 998–1005, 1997.

[6] M. S. Nawrocka, T. Liu, X. Wang, and R. R. Panepucci, “Tunable siliconmicroring resonator with wide free spectral range,” Appl. Phys. Lett.,vol. 89, no. 071110, pp. 1–3, 2006.

[7] L. Zhang, R. Ji, Y. Tian, L. Yang, P. Zhou, Y. Lu et al., “Simultaneousimplementation of XOR and XNOR operations using a directed logiccircuit based on two microring resonators,” Opt. Exp., vol. 19, no. 7,pp. 6524–6540, Mar. 2011.

[8] Q. Xu, S. Manipatruni, B. Schmidt, J. Shakya, and M. Lipson, “12.5 Gbit/scarrier-injection-based silicon micro-ring silicon modulators,” Opt. Exp.,vol. 15, no. 2, pp. 430–436, Jan. 2007.

[9] I.-L. Gheorma and R. M. Osgood, Jr., “Fundamental limitations of opticalresonator based high-speed EO modulators,” IEEE Photon. Technol. Lett.,vol. 14, no. 6, pp. 795–797, Jun. 2002.

[10] R. Grover, T. A. Ibrahim, S. Kanakaraju, L. Lucas, L. C. Calhoun, andP.-T. Ho, “A tunable GaInAsP–InP optical microring notch filter,” IEEEPhoton. Technol. Lett., vol. 16, no. 2, pp. 467–469, Feb. 2004.

[11] Yariv, “Critical coupling and its control in optical waveguide-ring res-onator systems,” IEEE Photon. Technol. Lett., vol. 14, no. 4, pp. 483–485,Apr. 2002.

[12] W. D. Sacher and J. K. S. Poon, “Dynamics of microring resonator mod-ulators,” Opt. Exp., vol. 16, no. 20, pp. 15741–15753, 2008.

[13] L. Zhang, M. Song, T. Wu, L. Zou, R. G. Beausoleil, and A. E. Willner,“Embedded ring resonators for microphotonic applications,” Opt. Lett.,vol. 33, no. 17, pp. 1978–1980, Sep. 2008.

[14] P. Bhattacharya, Semiconductor Optoelectronic Devices, 2nd ed. NewDelhi, India: PHI, 2009.

[15] Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2 nd ed. Norwood, MA, USA: ArtechHouse, 2000.

[16] K. Okamoto, Fundamentals of Optical Waveguides, 2 nd ed. New York,NY, USA: Academic, 2006.

[17] P. Lusse, P. Stuwe, J. Schule, and H.-G. Unger, “Analysis of vectorialmode fields in optical waveguides by a new finite difference method,” J.Lightw. Technol., vol. 12, no. 3, pp. 487–494, Mar 1994.

[18] H. Kogelnik, Theory of dielectric waveguides”, Guided-Wave Optoelec-tronics. Berlin, Germany: Springer-Verlag, 1988, pp. 7–88.

[19] E. Marcatili, “Bends in optical dielectric guides,” Bell Syst. Tech. J.,vol. 48, no. 7, pp. 2103–2132, 1969.

[20] K. R. Hiremath, “Analytic approach to dielectric optical bent slab waveg-uides,” J. Opt. Soc. Amer. A, vol. 26, pp. 2321–2326, 2004.

[21] D. Marcuse, “Bending losses of the asymmetric slab waveguide,” BellSyst. Tech. J., vol. 50, no. 8, pp. 2103–2132, 1971.

[22] K. Hiremath, “Modeling of 2D cylindrical integrated optical microres-onators,” M.Sc. Thesis, Univ. of Twente, Enschede, The Netherlands,2003.

[23] A. Arbabi, Y. M. Kang, and L. L. Goddard, “Cylindrical coordinates cou-pled mode theory,” IEEE J. Quantum Electron., vol. 46, no. 12, pp. 1769–1774, Dec. 2010.


[24] K. Oda, N. Takato, and H. Toba, “A wide-FSR waveguide double-ringresonator for optical FDM transmission systems,” J. Lightw. Technol.,vol. 9, no. 6, pp. 728–736, Jun. 1991.

[25] A. J Moseley, D. J. Robbins, A. C. Marshall, M. Q. Kearley, andJ. I. Davies, “Quantum confined Stark effect in InGaAs/InP single quan-tum wells investigated by photocurrent spectroscopy,” Semicond. Sci.Technol., vol. 4, no. 184, 1989.

[26] T. Bhowmick and U. Das, “Integrated MQW intermixed InGaAsP/InPwaveguide photodiodes,” Opt. Quant. Electron., vol. 42, pp. 109–120,2010.

[27] R. E. Nahory, M. A. Pollack, W. D. Johnston, and R. L. Barns,“Band gap versus composition and demonstration of Vegard’s law forIn1−xGax Asy P1−y lattice matched to InP,” Appl. Phys. Lett., vol. 33,no. 659, pp. 659–661, 1978.

[28] Broberg and S. Lindgren, “Refractive-index of In1-xGaxAsyP1-y layersand InP in the transparent wavelength region,” J. Appl. Phys., vol. 55,no. 9, pp. 3376–3381, May 1984.

[29] B. R. Bennett, R. A. Soref, and J. A. Del Alamo, “Carrier-induced changein refractive index of InP, GaAs and InGaAsP,” IEEE J. Quantum Elec-tron., vol. 26, no. 1, pp. 113–122, Jan. 1990.

[30] M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions(Applied Mathematics Series 55),” National Bureau of Standards, Wash-ington, DC, USA, 1964.

[31] V. Barve and U. Das, “Design of a grating-assisted lateral directional cou-pler by impurity-induced quantum-well intermixing of InGaAs/GaAs,” J.Lightw. Tech., vol. 25, no. 9, pp. 2448–2455, 2007.

[32] R. K. Sonkar and U. Das, “Quantum well intermixed waveguide grating,”Opt. Quantum Electron., vol. 42, no. 9–10, pp. 631–643, Sep. 2011.

[33] S. Adachi, Properties of Group-IV, III-V and II-VI Semiconductors (Seriesin materials for Electronics and Optoelectronic Applications). Hoboken,NJ, USA: Wiley, 2005.

[34] I. Bar Joseph, C. Klingshirn, D. A. B. Miller, D. S. Chemla, U. Koren,and B. I. Miller, “Quantum-confined Stark effect in InGaAs/InP quantumwells grown by organometallic vapor phase epitaxy,” Appl. Phy. Lett.,vol. 50, no. 15, pp. 1010–1012, 1987.

[35] M. Hammer, K. R. Hiremath, and R. Stoffer, “Analytical approaches to thedescription of optical microresonator devices,” in Proc. MicroresonatorsBuilding Blocks VLSI Photon., Erice, Italy, 18–25, 2003, pp. 48–71.

[36] C. G. Poulton, C. Koos, M. Fujii, A. Pfrang, T. Schimmel, J. Leuthold,and W. Freude, “Radiation modes and roughness loss in high index-contrast waveguides,” IEEE J. Sel. Topics Quantum Electron., vol. 12,no. 6, pp. 1306–1321, Nov./Dec. 2006.

[37] D. Rafizadeh, J. P. Zhang, R. C. Tiberio, and S. T. Ho, “Propagation lossmeasurements in semiconductor microcavity ring and disk resonators,” J.Lightw. Tech., vol. 16, no. 7, pp. 1308–1314, Jul. 1998.

Viswas Sadasivan received the B.Tech. degree in electronics and communica-tion engineering from Rajiv Gandhi Institute of Technology, Kottayam, India in2006 and M.E. degree in communication engineering from the Birla Institute ofTechnology and Science Pilani, Pilani, India in 2009. He is currently workingtoward the Ph.D degree in electrical engineeringfrom the Indian Institute ofTechnology Kanpur, Kanpur, India.

His current research interests include theoretical and experimental study ofphotonic integrated circuits on III–V semiconductors.

Utpal Das (M’93–SM’97) received the B.Tech. degree in radio physics andelectronics from the University of Calcutta, Kolkata, India, in 1979, the M.S.degree from Oregon State University, Corvallis, OR, USA, in 1983, and thePh.D. degree from the University of Michigan, Ann Arbor, MI, USA, in 1987.

In 1988, he was with the University of Florida, Gainesville, as an AssistantProfessor. In 1994, he joined the Indian Institute of Technology, Kanpur, India,where he is a Professor with the Department of Electrical Engineering.

His current research interests include areas of nanostructured semiconductoroptoelectronic devices and optoelectronic integration.

Documents

QCSE Tuned Embedded Ring Modulator