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CHAPTER7Modulators

by Wei Shi, Lukas Chrostowski, Tom Baehr-Jones,Ran Ding, et al.

This chapter describes how optical modulation is achievedusing carrier injection in a PN junction, followed by the modeland design of a ring modulator.

7.1 PN Junction Phase Shifter

7.1.1 Silicon, carrier density dependance

In 1987, Soref and Bennett predicted the change in silicon’srefraction index due to carriers [1]. This is termed the plasmadispersion effect, and is used for silicon modulators where the

concentration of carriers is varied either by injecting or remov-ing carriers from the device. The commonly used (e.g., [2])phenomenological expressions are as follows:

The change in index of refraction is phenomenologically de-scribed by

∆n (at 1550nm) = −8.8× 10−22∆N − 8.5× 10−18∆P 0.8∆n (at 1310nm) = −6.2× 10−22∆N − 6× 10−18∆P 0.8

(7.1)

The change in absorption is described by

∆α (at 1550nm) = 8.5× 10−18∆N + 6× 10−18∆P [cm−1]∆α (at 1310nm) = 6× 10−18∆N + 4× 10−18∆P [cm−1]

(7.2)104 Silicon Photonics Design, c©2012 Lukas Chrostowski, Draft: February 9, 2013

7.1. PN Junction Phase Shifter

where ∆N,∆P are the carrier densities of electrons and holes[cm−3].

It should be noted that holes have a smaller absorption ascompared with electrons, whereas they have a larger indexshift. Thus, holes are most effective for providing an indexshift with minimal absorption hence modulators typically useholes for phase-shifted designs (e.g., Mach-Zehnder modula-tor).

1017

1018

1019

1020

10−4

10−3

10−2

Free electrons

Free holes

Carrier Density [cm−3]

−∆n

∆N, 1550 nm∆N, 1310 nm∆P, 1550 nm∆P, 1310 nm

Figure 7.1: Change in index of refraction versus carrier density,Eq. 7.3.

Equations 7.1–7.2 have been updated in 2011 using morerecent experimental data as follows [3]:

∆n (at 1550nm) = −5.4× 10−22∆N1.011 − 1.53× 10−18∆P 0.838∆n (at 1310nm) = −2.98× 10−22∆N1.016 − 1.25× 10−18∆P 0.835

(7.3)

1017

1018

1019

1020

100

101

102

103

Carrier Density [cm−3]

∆α

[cm

−1 ]

∆N, 1550 nm∆P, 1550 nm∆N, 1310 nm∆P, 1310 nm

Figure 7.2: Change in absorption versus carrier density, Eq. 7.4.

The change in absorption is described by

∆α (1550nm) = 8.88× 10−21∆N1.167 + 5.84× 10−20∆P 1.109∆α (1310nm) = 3.48× 10−22∆N1.229 + 1.02× 10−19∆P 1.089

(7.4)

The wavelength dependance can be introduced by consider-ing the theoretical free-carrier (Drude) model, in which boththe index and absorption vary as λ2. Taking into accountthe wavelength dependance, Eqs. 7.1-7.2 are extended withwavelength-dependant fitting parameters:

∆n(λ) = −3.64× 10−10λ2∆N − 3.51× 10−6λ2∆P 0.8∆α(λ) = 3.52× 10−6λ2∆N + 2.4× 10−6λ2∆P [cm−1]

(7.5)

where λ is the wavelength [m]. The wavelength dependanceis plotted in Figure 7.3.

Silicon Photonics Design, c©2012 Lukas Chrostowski, Draft: February 9, 2013 105

7. Modulators

1.2 1.4 1.6 1.8 2

10−4

10−3

10−2

∆N=1e+17

∆N=1e+18

∆N=1e+19

∆P=1e+17

∆P=1e+18

∆P=1e+19

Wavelength [µm]

−∆n

(a) Absorption

1.2 1.4 1.6 1.8 2

100

101

102

∆N=1e+17

∆N=1e+18

∆N=1e+19

∆P=1e+17

∆P=1e+18

∆P=1e+19

Wavelength [µm]

∆α

[cm

−1 ]

(b) Index Change

Figure 7.3: a) Change in index of refraction versus wavelength. b)Change in absorption versus wavelength. Eq. 7.5 with data points(X) taken from Eqs. 7.1-7.2. Update data from Ref. [3] is included(circles, triangles).

7.1.2 PN Junction Carrier Distribution

The impurity and carrier distributions in a carrier-depletionphase modulator are illustrated in Fig. 7.4. We have the fol-lowing assumptions or approximations for the p-n junction:

• The diffused p-n junction is approximated by a stepor abrupt junction where the impurity profile changesabruptly across the mask-defined doping boundary;

• The width of the p-n junction is much shorter than thediffusion length, therefore, a linear distribution in minor-ity carrier densities are assumed between the depletionregion and the heavily doped region.

The width of the depletion region, Wd, is determined by theimpurity densities (NA and ND), as well as the applied voltage

0x

p++ p n++n

xn++xp++

p, n

Wd

pn0np0

xnxp

NDNA

0

w

xoffset

NA++ NA ND++ND

dn++dp++

x0

p

n

(a)

(b)

(c)

Figure 7.4: p-n junction in a rib waveguide: (a) cross-sectional viewof the impurity distribution assumed in the abrupt junction model;(b) cross-sectional view of the carrier distribution; (c) 1D free-carrierdistributions.

106 Silicon Photonics Design, c©2012 Lukas Chrostowski, Draft: February 9, 2013


(V), and is given by

Wd =

√2�0�s(NA +ND)(Vbi − V )

qNAND(7.6)

where �s is the relative permittivity and Vbi is the built-in ordiffusion potential of the junction given by

Vbi =kBT

qlnNANDn2i

(7.7)

The boundaries of the depletion region are given by

xp = xoffset −Wd

1 +NA/ND(7.8a)

xn = xoffset +Wd

1 +ND/NA(7.8b)

The carrier densities, ∆N = n(x, V ), and ∆P = p(x, V ),are given in Eqs. 7.9-7.10, where np0 and pn0 are given by

np0 =n2iNA

(7.11a)

pn0 =n2iND

(7.11b)

Using the above equations, the carrier distributions in thewaveguide can be solved using the Matlab code 7.1.

% by Wei Shi, UBC, Nov. 2012%function [n, p, x, xn, xp, Rj, Cj]=pn_depletion(wg_width,

pn_offset, ds_npp, ds_ppp, T, V, pts)% usage e.g., [n, p, x, xn, xp, Rj, Cj]=pn_depletion(500e-9,

50e-9, 1e-6, 1e-6, 25, -1, 100)%% N_D, N_A: doping densities

% V: applied voltage; positive for forward bias; negative forreverse bias

% ds_npp: distance of the n++ boundary to the pn junction centre% ds_ppp: distance of the p++ boundary to the pn junction centre% Rj: junction resistance in ohms% Cj: junction capacitance in F/m

epsilon0=8.854187817620e-12; epsilon_s=11.68;q=1.60217646e-19; kB= 1.3806503e-23;% Boltzmann constantT=T+273.15; VT=kB*T/q;

%material constantsNA=5e17*1e6;% cm^-3*1e6ND=3e17*1e6;ni=1e10*1e6;Rs_rib_n=2.5e3;Rs_rib_p=4.0e3;Rs_slab_n=0.6e4;Rs_slab_p=1e4;

% waveguide heighth_rib=220e-9; h_slab=90e-9;

Vbi=VT*log(NA*ND/ni^2); % built-in or diffusion potentialWd=sqrt(2*epsilon0*epsilon_s*(NA+ND) / (q*NA*ND) *(Vbi-V)); %

depletion widthxp=-Wd/(1+NA/ND)+pn_offset;xn=Wd/(1+ND/NA)+pn_offset;

del_x=wg_width/(pts-1);x_ppp=-ds_ppp+pn_offset; x_npp=ds_npp+pn_offset;x_NA=x_ppp:del_x:xp-del_x;x_dep=xp:del_x:xn;% for the depletion regionx_ND=xn+del_x:del_x:x_npp;x=[x_NA, x_dep, x_ND];

n0_NA=ni^2/NA; p0_ND=ni^2/ND;

% Long-base assumption% Lp=sqrt(Dp*tau_p);% Ln=sqrt(Dn*tau_n);% del_n_NA=n0_NA*(exp(q*V/(kB*T))-1)* exp(-abs(x_NA-xp)/Ln); %

minority electron density in p(NA) region% del_p_ND=p0_ND*(exp(q*V/(kB*T))-1)* exp(-abs(x_ND-xn)/Lp); %

minority hole density in n(ND) region


7. Modulators

n(x, V ) =

np0[1 + (1− xp−xxp−xp++ )e

( qVkBT

−1)] for xp++ < x < xp0 for xp < x < xn

ND for xn++ > x > xn

(7.9)

p(x, V ) =

NA for xp++ < x < xp

0 for xp < x < xn

pn0[1 + (1− x−xnxp++−xp )e( qVkBT

−1)] for xn++ > x > xn

(7.10)

% Short-base assumptiondel_n_NA=n0_NA*(exp(q*V/(kB*T))-1)*

(1-abs((x_NA-xp)/(xp-x_ppp))); % minority electron densityin p(NA) region

del_p_ND=p0_ND*(exp(q*V/(kB*T))-1)*(1-abs((x_ND-xn)/(x_npp-xn))); % minority hole density inn(ND) region

n_NA=n0_NA+del_n_NA; p_ND=p0_ND+del_p_ND;p_dep=zeros(1, length(x_dep)); n_dep=zeros(1, length(x_dep));p_NA=ones(1, length(x_NA))*NA; % majority holes in p(NA) regionn_ND=ones(1, length(x_ND))*ND; % majority electrons in n(ND)

region

n=[n_NA, n_dep, n_ND]; p=[p_NA, p_dep, p_ND];

Rj=(wg_width/2-xn)* Rs_rib_n+(wg_width/2+xp)*Rs_rib_p+(-wg_width/2-x_ppp)* Rs_slab_p+(x_npp-wg_width/2)*Rs_slab_n;

Cj=sqrt(q*epsilon0*epsilon_s/2/ (1/ND+1/NA)/(Vbi-V))*h_rib;

Listing 7.1: PN junction depletion, Matlab model, http://siepic.ubc.ca/book/ch actives/pn depletion.m

7.1.3 Optical Phase Response

The changes in the silicon refractive index, nco, and opticalloss, at λ = 1.55µm, due to the free carriers are given byEquations 7.1-7.2.

Then the effective index, neff , and the optical loss due tothe free-carrier absorption, αpn, as functions of applied voltageare given by

neff (V ) = neff,i +

∫E∗(x) · 4n(x, V ) E(x) dx∫

E∗(x) · E(x) dx ·dneffdnco

αpn(V ) =

∫E∗(x) · 4α(x, V ) E(x) dx∫

E∗(x) · E(x) dx(7.12)

where neff,i is the effective index of the waveguide without anydoping and dneff/dnco (change of mode effective index versuschange in the waveguide core effective index) is typically veryclose to 1 (including silicon strip and rib waveguides). E(x) isthe 1D field profile found using the effective index method, seeMatlab function 4.13. Then the voltage-dependent changes ineffective index and phase are given by


http://siepic.ubc.ca/book/ch_actives/pn_depletion.mhttp://siepic.ubc.ca/book/ch_actives/pn_depletion.m


0 2 4 6 8 100

0.5

1

1.5

2

2.5

3x 10

−4

∆ n

eff

Voltage (V)0 2 4 6 8 10

5

6

7

8

9

10

11

αp

n (d

B/c

m)

Figure 7.5: Changes in effective index and free-carrier-caused opticalloss as functions of applied voltage (reverse biased).

4neff (V ) = neff (V )− neff (0)

4φ(V ) [π · cm−1] = 0.024 neff (V )λ

(7.13)

Using the above equations (Matlab code 7.2), we calculatethe changes in effective index and optical loss due to the free-carrier absorption for a design with the following waveguideparameters: rib width w = 500 nm, rib thickness, t = 220 nm,and slab thickness, tslab = 90 nm. Holes have stronger effecton the effective index than electrons do, as revealed by Eq. 7.1,therefore, an offset of the junction to the waveguide centre canbe used to optimize the modulation efficiency. A 50 nm dopingoffset is used in the calculation.

0 2 4 6 8 100

0.5

1

1.5

2

2.5

3

3.5

4

Voltage (V)

∆ φ

(π

/cm

)

[1.6V π]

Figure 7.6: Phase change as a function of applied voltage (reversebiased).

% Function to find waveguide field profile, PN junction indexperturbabtion due to carriers, and calculate fieldperturbation to deterimine modified waveguide properties

% example [del_neff alpha Rj Cj]=neff_V(1.55e-6, 220e-9,500e-9, 90e-9, 3.8, 1.44, 1.44, 50e-9, 1e-6, 1e-6, 25, 1,500, 3)

function [neff alpha Rj Cj]=neff_V(lambda, t, w, t_slab,n_core, n_clad, n_oxide, pn_offset, ds_n_plus, ds_p_plus,T, V, pts, M)

[xwg, TM_E_TEwg, neff0]=wg_TElike_1Dprofile_neff(lambda, t, w,t_slab, n_core, n_clad, n_oxide, pts, M);

Ewg=TM_E_TEwg(:,1)’;[n, p, xdoping, xn, xp, Rj, Cj]=pn_depletion(w, pn_offset,

ds_n_plus, ds_p_plus, T, V, pts);

pts_x=length(xwg); dxwg=zeros(1, pts_x);dxwg(1)=xwg(2)-xwg(1); dxwg(pts_x)=xwg(pts_x)-xwg(pts_x-1);for i=2:pts_x-1

dxwg(i)=xwg(i+1)/2-xwg(i-1)/2;


7. Modulators

end

n_wg=interp1(xdoping, n, xwg); p_wg=interp1(xdoping, p, xwg);

del_ne=-8.8e-22*sum(conj(Ewg).*n_wg*1e-6.*Ewg.*dxwg)/sum(conj(Ewg).*Ewg.*dxwg);

del_nh=-8.5e-18*sum(conj(Ewg).*(p_wg*1e-6).^0.8.*Ewg.*dxwg)/sum(conj(Ewg).*Ewg.*dxwg);

del_neff=del_ne+del_nh; neff=neff0+del_neff;

del_alpha_e=8.5e-18*sum(conj(Ewg).*n_wg*1e-6.*Ewg.*dxwg)/sum(conj(Ewg).*Ewg.*dxwg);

del_alpha_h=6e-18*sum(conj(Ewg).*p_wg*1e-6.*Ewg.*dxwg)/sum(conj(Ewg).*Ewg.*dxwg);

alpha=del_alpha_e+del_alpha_h;

Listing 7.2: PN junction effective index and optical loss, Matlabmodel, http://siepic.ubc.ca/book/ch actives/neff V.m

The calculated results are shown in Fig. 7.5 (Matlab code7.3). The effective index reduces, while the optical loss re-duces, as the voltage increases, because the carriers are re-moved from the waveguide by the applied voltage. The changein the phase is plotted in Fig. 7.6. A voltage of 1.6 V is neededto make a phase shift of π for a 1-cm-long waveguide, indicat-ing a Vπ · L product of 1.6 V·cm.% example:% [neff alpha delta_neff delta_phi fc]=neff_V_plot(1.5e-6,

220e-9, 500e-9, 90e-9, 3.47, 1.44, 1.44, 50e-9, 10e-6,10e-6, 25, V, 500, 3);

function [neff alpha delta_neff delta_phi fc] =neff_V_plot(lambda, t, w, t_slab, n_core, n_clad, n_oxide,pn_offset, ds_n_plus, ds_p_plus, T, V, pts, M);

neff=zeros(1, length(V)); alpha=zeros(1, length(V));Rj=zeros(1, length(V)); Cj=zeros(1, length(V));for i=1:length(V);

[neff(i) alpha(i) Rj(i) Cj(i)]=neff_V(lambda, t, w, t_slab,n_core, n_clad, n_oxide, pn_offset, ds_n_plus,ds_p_plus, T, V(i), pts, M);

end[neff_v0 alpha_v0]=neff_V(lambda, t, w, t_slab, n_core, n_clad,

n_oxide, pn_offset, ds_n_plus, ds_p_plus, T, 0, pts, M);delta_neff=neff-neff_v0;alpha_dB=-10*log10(exp(-alpha));

figure; plot(-V, delta_neff)figure; plot(-V, alpha_dB);

% Phase shift per cmdelta_phi=2*pi/lambda*delta_neff*1e-2/pi;% per cmfigure; plot(-V, delta_phi, ’linewidth’, 2);

% Cut-off frequencyfc=1./(2*pi*Rj.*Cj)*1e-9;% in GHzfigure; plot(-V, fc, ’linewidth’, 2);

Listing 7.3: PN junction depletion, Matlab model, http://siepic.ubc.ca/book/ch actives/neff V plot.m

7.1.4 Small-Signal Response

The resistance and capacitance of the p-n junction are givenby

Rj [Ω ·m] = (w

2+ xp)Rsrp + (

w

2− xn)Rsrn

− (w2

+ xp++)Rssp + (xn++ −w

2)Rssn

Cj [F/m] = trib

√q�0�s

2(1/ND + 1/NA)(Vbi − V )

(7.14)

where Rsrn, Rsrp, Rssn, and Rssp are the sheet resistances ofthe n-doped rib, p-doped rib, n-doped slab, and p-doped slab,respectively.


http://siepic.ubc.ca/book/ch_actives/neff_V.mhttp://siepic.ubc.ca/book/ch_actives/neff_V_plot.mhttp://siepic.ubc.ca/book/ch_actives/neff_V_plot.m

7.2. Ring resonators

Then, the RC-constant determined cutoff (3 dB) frequencycan be found by

fc =1

2πRjCj(7.15)

Using the parameters given above, fc is calculated to be 35GHz and 51 GHz at 0 and 1 V, respectively. As shown inFig. 7.7, fc increases as the applied DC voltage increases dueto the simultaneously reduced Rj and Cj, as a result of theexpanded depletion region. We can see that the frequency re-sponse of the p-n junction can easily go beyond tens of GHz,therefore, the intrinsic RC of the junction is typically not alimiting factor of silicon optical modulators. The RC becomesa limitation in cases where a long junction is used (hence alarge capacitance) together with a large source impedance,e.g., 50 Ω. This is the case for a basic Mach-Zehnder modu-lator or phase shifter; to avoid this RC limit, structures suchas travelling wave electrodes need to be employed.

7.2 Ring resonators

For a description of ring resonators, please see the review pa-per by W. Bogaerts at Ghent University [4]. The reader is alsoreferred to numerous papers on ring modulators, e.g., [5, 6].

A high-Q optical resonator has strong wavelength selectiv-ity, performing as a narrow-bandwidth filter. The resonantwavelength is determined by the round-trip phase of the res-onator. Therefore, when operating at a wavelength close tothe renounce of a optical resonator, the transmission wouldbe very sensitive to the phase change of the cavity. Basedon this effect, one can obtain a very efficient modulator by,e.g., integrating a p-n junction into the resonator cavity and

0 2 4 6 8 1040

60

80

100

120

140

160

Voltage (V)

f c (G

Hz)

Figure 7.7: Cutoff frequency as a function of applied voltage (reversebiased)

modulating the phase though the plasma effect as describedin the previous section.

Two coupled-resonator configurations, i.e., all-pass andadd-drop filters, can be used to obtain a microring modula-tor, as shown in Fig. 7.9. The microring cavity generally hasa racetrack shape, consisting of two 180o circular waveguidesand two straight waveguides (for directional couplers), with aroundtrip length given by

Lrt = 2πr + 2Lc (7.16)

where r and Lc are the bend radius and the coupler length,respectively. When Lc = 0, the microring is point coupledand the cavity becomes a complete circle.


7. Modulators

Figure 7.8: Microscope image of a ring resonator modulator [6]. Three electrical pads are shown on the left, for Ground-Signal-Ground (GSG);the ring is adjacent; it is optically characterized using two grating couplers on the right side. The couplers are separated by 0.5 mm from themicrowave probes for ease of testing. Additional pairs of grating couplers are included in the layout as characterization test structures.

Because the micoring modulator can only operate withina narrow spectral window around its resonant wavelength,wavelength stabilization is usually needed for practical appli-cations. For example, as shown in Fig. 7.9(a), a quarter of theoptical cavity is integrated with a resistor heater for thermaltuning and wavelength stability. As a result of compromise,the modulation efficiency becomes lower as compared to thefully modulated cavity.

7.2.1 Transfer Function

The through-port response of an all-pass microring resonator,shown in Fig. 7.9(a), is given by

Ethru =−√A+ te−iφrt

−√At∗ + e−iφrt

(7.17)

where t is the straight-through coupling coefficient of the opti-cal field; φrt and A are the round-trip optical phase and powerattenuation, respectively, and are given by

φrt = βLrt (7.18)

andA = e−αLrt (7.19)



SiEPIC Workshop – Active Silicon PhotonicsSi-EPIC CREATE

Two devices available in the PDK

1

A B

In Thru

In Thru

Drop

• Ring or disk with varied R, W, G, offset, coupler length, doping distance• A: Add-drop; 100% doping profile for the highest efficiency• B: Add-pass; thermal tuning; tunable duty cycles for heater and modulator• Wavelength: 1310 nm (McGill) 1490 nm, 1550 nm

(a) all-pass (integrated with a heater for wavelength tuning)


Two devices available in the PDK

1

A B

In Thru

In Thru

Drop

• Ring or disk with varied R, W, G, offset, coupler length, doping distance• A: Add-drop; 100% doping profile for the highest efficiency• B: Add-pass; thermal tuning; tunable duty cycles for heater and modulator• Wavelength: 1310 nm (McGill) 1490 nm, 1550 nm

(b) add-drop (fully modulated)

Figure 7.9: Mask layouts of microring modulators.

The add-drop filter, shown in Fig. 7.9(b), has two outputs,i.e., the through-port signal, Ethru, and the drop-port signal,Edrop, which are given by

EthruEin

=t1 − t∗2

√Aeiφrt

1−√At∗1t

∗2eiφrt

(7.20)

EdropEin

=−κ∗1κ∗2

√Aeiφrt/2

1−√At∗1t

∗2eiφrt

(7.21)

where t1, κ1, t2, and κ2 are the straight-though and cross-overcoupling coefficients of the input and drop couplers, respec-tively, which are typically identical for a symmetric design,i.e., t1 = t2 = t and κ1 = κ2 = κ. Assuming lossless couplers(i.e., the optical losses of the couplers are incorporated intothe round trip loss of the entire optical cavity), t and κ havethe relationship

|κ|2 + |t|2 = 1 (7.22)The transfer function of the microring resonator is imple-

mented using Matlab code 7.4.

% RingMod.m: Ring modulator 1D model% Usage, e.g.,% [Ethru Edrop Qi Qc Rj Cj]=RingMod(1.55e-6, ’all-pass’,

10e-6, 0, 2*pi*10e-6, 220e-9, 500e-9, 90e-9, 3.47, 1.44,1.44, 50e-9, 1e-6, 1e-6, 25, -1, 500);

%% Wei Shi, UBC, 2012, [email protected]

function [Ethru Edrop Qi Qc Rj Cj]=RingMod(lambda, Filter_type,r, Lc, L_pn, w, pn_offset, ds_n_plus, ds_p_plus, T, V);

%% type: "all-pass" or "add-drop"% r: radius% Lc: coupler length% Lh: heater length%


7. Modulators

% neff_pn, alpha_pn: effective index and free-carrierobsorption of the phase modulator

% Rj, Cj: junction resistance and capacitance of the phasemodulator

% parameterst=220e-9; t_slab=90e-9; n_core=3.47; n_clad=1.44; n_oxide=1.44;

pts=200;

[neff_pn alpha_pn Rj Cj]=neff_V(lambda, t, w, t_slab, n_core,n_clad, n_oxide, pn_offset, ds_n_plus, ds_p_plus, pts, T,V);

% undoped waveguide mode and effective index[xwg0 TM_E_TEwg0 neff0]=wg_TElike_1Dprofile_neff(lambda, t, w,

t_slab, n_core, n_clad, n_oxide, pts, 2);neff_exc=neff0;del_lambda=0.1e-9;[xwg1 TM_E_TEwg1

neff0_1]=wg_TElike_1Dprofile_neff(lambda+del_lambda, t, w,t_slab, n_core, n_clad, n_oxide, pts, 2);

ng=neff0-(neff0_1-neff0)/del_lambda*lambda;

alpha_wg_dB=5; % optical loss of intrinsic optical waveguide,in dB/cm

alpha_wg=-log(10^(-alpha_wg_dB/10));% converted to /cmalpha_pn=alpha_wg+alpha_pn;alpha_exc=alpha_wg; % optical loss of the ring cavity excluding

the phase modulator

L_rt=Lc*2+2*pi*r;L_exc=L_rt-L_pn;phi_pn=(2*pi/lambda)*neff_pn*L_pn;phi_exc=(2*pi/lambda)*neff_exc*L_exc;phi_rt=phi_pn+phi_exc;

A_pn=exp(-alpha_pn*100*L_pn); % attunation due to pn juncitonA_exc=exp(-alpha_exc*100*L_exc); % attunation over L_excA=A_pn*A_exc; % round-trip optical power attenuation

alpha_av=-log(A)/L_rt;% average loss of the cavityQi=2*pi*ng/lambda/alpha_av;

%coupling coefficientsk=0.2; t=sqrt(1-k^2);

if (Filter_type==’all-pass’)Ethru=(-sqrt(A)+t*exp(-1i*phi_rt)) /

(-sqrt(A)*conj(t)+exp(-1i*phi_rt));Edrop=0;Qc=-(pi*L_rt*ng)/(lambda*log(abs(t)));elseif (Filter_type==’add-drop’)Ethru=(t-conj(t)*sqrt(A)*exp(1i*phi_rt)) /

(1-sqrt(A)*conj(t)^2*exp(1i*phi_rt));Edrop=-conj(k)*k*sqrt(A)*exp(1i*phi_rt) /

(1-sqrt(A)*conj(t)^2*exp(1i*phi_rt));Qc=-(pi*L_rt*ng)/(lambda*log(abs(t)))/2;else

error(1, ’The’’Filter_type’’ has to be ’’all-pass’’ or’’add-drop’’.\n’);

end

Listing 7.4: Ring modulator, Matlab model, http://siepic.ubc.ca/book/ch actives/RingMod.m

% wg_TElike_1Dprofile.m - Effective Index Method - 1D modeprofile

% Lukas Chrostowski, 2012% modified by Wei Shi, 2012

% usage, e.g.:% [xwg, TM_E_TEwg]=wg_TElike_1Dprofile (1.55e-6, 0.22e-6,

0.5e-6, 90e-9,3.47, 1, 1.44, 100, 2);% figure; plot(xwg, TM_E_TEwg(:,1))

function [xwg, TM_E_TEwg,neff_TEwg_1st]=wg_TElike_1Dprofile_neff (lambda, t, w,t_slab, n_core, n_clad, n_oxide, pts, M)

Fontsize=15;

% TE (TM) modes of slab waveguide (core and slab portions):[nTE,nTM]=wg_1D_analytic2 (lambda, t, n_oxide, n_core, n_clad);if t_slab>0

[nTE_slab,nTM_slab]=wg_1D_analytic2 (lambda, t_slab,n_oxide, n_core, n_clad);

elsenTE_slab=n_clad; nTM_slab=n_clad;

end[xslab, TE_Eslab, TE_Hslab, TM_Eslab, TM_Hslab]=

wg_1D_mode_profile (lambda, t, n_oxide, n_core, n_clad,


http://siepic.ubc.ca/book/ch_actives/RingMod.mhttp://siepic.ubc.ca/book/ch_actives/RingMod.m


pts, M);

% TE-like modes of the etched waveguide (for fundamental slabmode):

[nTE,nTM]=wg_1D_analytic2 (lambda, w, nTE_slab(1), nTE(1),nTE_slab(1));

neff_TEwg_1st=nTM(1);[xwg, TE_E_TEwg, TE_H_TEwg, TM_E_TEwg, TM_H_TEwg]=

wg_1D_mode_profile (lambda, w, nTE_slab(1), nTE(1),nTE_slab(1), pts, M);

Listing 7.5: Waveguide mode profile and effective indexcalculation, Matlab model, http://siepic.ubc.ca/book/ch actives/wg TElike 1Dprofile neff.m

7.2.2 Ring resonator experimental results

In this section, passive optical characterization results for afabricated ring modulator are presented. The ring modulatoris shown in Fig. 7.8. The ring has a radius of 15 µm, uses arib waveguide with 500 nm width and 90 nm slab, is a double-bus design with straight bus waveguides (a single bus deviceis shown in Fig. 7.8), and has a small racetrack section (0.1µm) for the directional couplers. The optical spectrum ofthe through port is plotted in Fig. 7.10a, measured using apair of fibre grating couplers [7]. The resonant wavelengthsare identified using a peak finding algorithm [8]. The free-spectral range (FSR, ∆λ), defined as the wavelength differencebetween resonances, is plotted in Fig. 7.10b, and is in generalwavelength dependant. Using the FSR, and given the round-trip length of the resonator, L, the group index, ng, of thewaveguide can be found:

ng =λ2

L∆λ(7.23)

The results are plotted and compared with the waveguidemodel in Fig. 4.17b.

7.2.3 Ring Tuneability

Here we discuss the microring modulator with a reverse-biasedp-n junction. Incorporating the p-n junction model (Matlabcodes 7.2 and 7.1) with the microring resonator transfer func-tion (Equations 7.21-7.20), we can simulate its spectrum as afunction of applied voltage (Matlab codes 7.6 and 7.7). Thisexample microring modulator is implemented based on thestructure shown in Fig. 7.11.

% Wei Shi UBC, 2012% [email protected]

function [Ethru Edrop Qi Qc Rj Cj]=RingMod_spectrum(lambda,Filter_type, r, Lc, L_pn, w, pn_offset, ds_n_plus,ds_p_plus, T, V);

%Ethru=zeros(1, length(lambda));Edrop=zeros(1, length(lambda));%for i=1:length(lambda)

[Ethru(i) Edrop(i) Qi(i) Qc(i) Rj Cj]=RingMod(lambda(i),Filter_type, r, Lc, L_pn, w, pn_offset, ds_n_plus,ds_p_plus, T, V);

end

Listing 7.6: Ring modulator spectrum, Matlab model, http://siepic.ubc.ca/book/ch actives/RingMod spectrum.m

function RingMod_spectrum_plot();lambda=1e-9*[1530:0.1:1560];Filter_type=’add-drop’;r=10e-6; Lc=0; L_pn=2*pi*r; w=500e-9; % WG parameterspn_offset=0; ds_n_plus=1e-6; ds_p_plus=1e-6; % pn-junction

designT=25; V0=0; % temperature and voltage


http://siepic.ubc.ca/book/ch_actives/wg_TElike_1Dprofile_neff.mhttp://siepic.ubc.ca/book/ch_actives/wg_TElike_1Dprofile_neff.mhttp://siepic.ubc.ca/book/ch_actives/RingMod_spectrum.mhttp://siepic.ubc.ca/book/ch_actives/RingMod_spectrum.m

7. Modulators

1.51 1.52 1.53 1.54 1.55 1.56 1.57−40

−35

−30

−25

−20

−15

−10

Wavelength [µm]

Tra

nsm

issi

on (

dB)

(a) Through-port spectrum of ring resonator

1.51 1.52 1.53 1.54 1.55 1.56

3.1

3.15

3.2

3.25

3.3

3.35

Wavelength [µm]

FS

R [n

m]

(b) Free spectral range (FSR) of ring resonator

Figure 7.10: Experimental spectra and free spectral range (FSR) of aring resonator modulator [OpSIS-IME] [6].

[Ethru0 Edrop0 Qi0 Qc0 Rj0 Cj0] = RingMod_spectrum (lambda,Filter_type, r, Lc, L_pn, w, pn_offset, ds_n_plus,ds_p_plus, T, V0);

% zoom at one peak wavelengthlambda_zoom=1e-9*[1540.7:0.0025:1541];V=-4:1:0;lenV = length(V); lenLZ = length(lambda_zoom);Ethru=zeros(lenV, lenLZ); Edrop=zeros(lenV, lenLZ);A=zeros(lenV, lenLZ);Qi=zeros(lenV, lenLZ); Qc=zeros(lenV, lenLZ);Cj=zeros(lenV,1); Rj=zeros(lenV,1);for i=1:lenV[Ethru(i,:) Edrop(i,:) Qi(i,:) Qc(i,:) Rj(i,:)

Cj(i,:)]=RingMod_spectrum(lambda_zoom, Filter_type, r,Lc, L_pn, w, pn_offset, ds_n_plus, ds_p_plus, T, V(i));

end

Qt=1./(1./Qi+1./Qc); % total Qtp=Qt./(3e8/1541e-9*2*pi); % photon lifetimetp_av=sum(tp, 2)/ (length(lambda_zoom)); % average tp over the

spectrumfcq=1./(2*pi*tp_av); fcj=1./(2*pi*Rj.*Cj);fc=1./(1./fcq+1./fcj);

figure; plot(lambda*1e9, [10*log10(abs(Ethru0).^2);10*log10(abs(Edrop0).^2)], ’linewidth’, 2);

figure; plot(lambda_zoom*1e9, 10*log10(abs(Ethru).^2),’linewidth’, 2);

figure; plot(lambda_zoom*1e9, 10*log10(abs(Edrop).^2),’linewidth’, 2);

figure; plot(-V, [fcq fcj fc]*1e-9, ’linewidth’, 2);xlabel(’Voltage (V)’); ylabel(’Cutoff frequency (GHz)’);

Listing 7.7: Ring modulator spectrum plot, Matlab, http://siepic.ubc.ca/book/ch actives/RingMod spectrum plot.m

We consider a fully modulated (i.e., no thermal tuning isused), add-drop microring modulator using point couplers(i.e., Lc = 0) with r = 10 µm and default values for other


http://siepic.ubc.ca/book/ch_actives/RingMod_spectrum_plot.mhttp://siepic.ubc.ca/book/ch_actives/RingMod_spectrum_plot.m



Matlab solver

6

function pn_depletion

1D electrical solver for the reverse-biased p-n junction

function RingMod_spectrum_plot

Voltage scanSmall-signal response bandwidth

function RingMod_spectrum

Optical spectrum (transfer functions)

function RingMod

Design parameters as the inputsTransmission @ single wavelength, single voltage

function neff_V

Mode overlapCarrier-depletion phase modulator

function wg_TElike_1Dprofile_nefffunction wg_1D_mode_profilefunction wg_1D_analytic2

!D optical solver for the TE-like mode

Figure 7.11: Structure diagram of a 1D analytic model for microringmodulators.

parameters given in Matlab code 7.7. The calculated through-port and drop-port responses are shown in Fig. 7.12. We cansee that the through-port power transmission is more sensitiveto the change of the roundtrip phase and, therefore, should beused as the modulator output. As shown in Fig. 7.12(a), thecentral wavelength shifts by 0.016 nm/V. Due to the highquality factor (about 10,000), this relatively small spectralshift results in a considerable change in the power transmis-sion. For example, for the 3-dB insertion loss wavelength atzero bias (∼1540.9 nm), when the applied voltage (reverse bi-ased) changes from 0 to 4V, the transmission drops by about8 dB. In order to improve the modulation efficiency, we can

increase the quality factor by, e.g, reducing the coupling (i.e.,reducing κ) to make the transmission notch narrower. How-ever, a higher Q means a longer photon lifetime, which willlimit the frequency response of the modulator, as we will seein the next.

7.2.4 Small-Signal Modulation Response

The cutoff frequency (3 dB), fc, of the small-signal response ofa microring modulator is determined by both the RC constantof the reverse-biased p-n junction and the photon lifetime, τp,of the optical cavity, i.e.,

1

fc=

1

fτp+

1

fRC(7.24)

The τp determined cutoff frequency is given by

fτp =1

2πτp(7.25)

where τp is related to the total quality factor, Qt, of the opticalcavity and is given by

τp =Qtωo

(7.26)

where ωo is the optical frequency and Qt is determined byboth the coupling and propagation losses, i.e.,

1

Qt=

1

Qc+

1

Qi(7.27)

where the intrinsic quality factor is given by

Qi =2πngλα

(7.28)


7. Modulators


Through-port response

9

1540.7 1540.75 1540.8 1540.85 1540.9 1540.95 1541−16

−14

−12

−10

−8

−6

−4

−2

0

λ (nm)

Tra

nsm

issio

n (

dB

)

4V

3V

2V

1V

0V

• Vpp=2V• 5 dB ER, 4dB IL

• Vpp=4V• 8 dB ER, 3dB IL

• Efficiency as size

0.016 nm/V

(a) Through-port

1540.7 1540.75 1540.8 1540.85 1540.9 1540.95 1541−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

λ (nm)

Tran

smis

sion

(dB)

4V3V2V1V0V

(b) Drop-port

Figure 7.12: Through-port (a) and drop-port (b) spectra with variedapplied voltage (reverse biased).

For the all-pass filter, the coupling-determined quality factor,Qc, is given by

Qc = −πLrtngλloge|t|

(7.29)

If the add-drop configuration is used, the coupling-determinedquality factor should be decided by 2 since two couplers areused in this case.

For the same design as described for the DC performance,we can predict the cutoff frequency of the microring modulatorusing above equations. The calculated results are shown inFig. 7.13. In this case, the RC constant of the p-n junctionhas a cutoff frequency (i.e., fRC) of over 40 GHz, while the τpdetermined cutoff frequency (i.e., fτp) is about 20 GHz. As asa result, the total cutoff frequency, fc, is 15 GHz at the biasof 1V, mainly limited by the photon lifetime.

References

[1] R. Soref and B. Bennett. “Electrooptical effects insilicon”. Quantum Electronics, IEEE Journal of 23.1(1987), pp. 123–129 (cit. on p. 104).

[2] GT Reed et al. “Silicon optical modulators”. Naturephotonics 4.8 (2010), pp. 518–526 (cit. on p. 104).

[3] M. Nedeljkovic, R. Soref, and G.Z. Mashanovich. “Free-Carrier Electrorefraction and Electroabsorption Modu-lation Predictions for Silicon Over the 1-14 micron In-frared Wavelength Range”. Photonics Journal, IEEE



0 0.5 1 1.5 2 2.5 3 3.5 410

20

30

40

50

60

70

80

90

100

Voltage (V)

Cu

toff

fre

qu

en

cy (

GH

z)

τp determined

p−n junction determinedfc

Figure 7.13: Structure diagram of the 1D model of a microring mod-ulator.

3.6 (2011), pp. 1171 –1180. issn: 1943-0655. doi:10.1109/JPHOT.2011.2171930 (cit. on pp. 105, 106).

[4] W. Bogaerts et al. “Silicon microring resonators”. Laser& Photonics Reviews (2012) (cit. on p. 111).

[5] Xi Xiao et al. “25 Gbit/s silicon microring modu-lator based on misalignment-tolerant interleaved PNjunctions”. Opt. Express 20.3 (2012), pp. 2507–2515. doi: 10 . 1364 / OE . 20 . 002507. url: http :/ / www . opticsexpress . org / abstract . cfm ? URI = oe - 20 -3-2507 (cit. on p. 111).

[6] Tom Baehr-Jones et al. “A 25 Gb/s Silicon PhotonicsPlatform”. arXiv:1203.0767v1 (2012) (cit. on pp. 111,112, 116).

[7] A. Mekis et al. “A Grating-Coupler-Enabled CMOSPhotonics Platform”. Selected Topics in Quantum Elec-tronics, IEEE Journal of 17.3 (2011), pp. 597 –608.issn: 1077-260X. doi: 10 . 1109 / JSTQE . 2010 . 2086049(cit. on p. 115).

[8] url: http://terpconnect.umd.edu/∼toh/spectrum/PeakFindingandMeasurement.htm (cit. on p. 115).

http://dx.doi.org/10.1109/JPHOT.2011.2171930http://dx.doi.org/10.1364/OE.20.002507http://www.opticsexpress.org/abstract.cfm?URI=oe-20-3-2507http://www.opticsexpress.org/abstract.cfm?URI=oe-20-3-2507http://www.opticsexpress.org/abstract.cfm?URI=oe-20-3-2507http://dx.doi.org/10.1109/JSTQE.2010.2086049http://terpconnect.umd.edu/~toh/spectrum/PeakFindingandMeasurement.htmhttp://terpconnect.umd.edu/~toh/spectrum/PeakFindingandMeasurement.htm

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