of W . The slab waveguide is oriented vertically instead of horizontally as in Fig. 4(b). To account for this change

in orientation, the eigenvalue equation for the TM should be used with nTE eff,I and nTE

eff,II for deriving the overall

effective index and the field distributions of the TE mode (Eq. 4). Similarly, to obtain the effective index of the

TM mode, nTM eff,I and nTM

eff,II should be used in Eq. 3.

Fig. 5(a) and (b) illustrate the temperature dependence of the effective index of a Si channel waveguide as

a function of the waveguide width W , assuming the waveguide height is fixed at H = 220 nm. Similar to the

observation in Fig. 3, dneff

dT varies with the waveguide width due to the change in the confinement of the optical

mode inside of the core region. In the extreme case of zero waveguide width (W =0),theeffectiveTOcoefficients

of both the fundamental TE and TM modes approach to the material TO coefficient of the cladding. As the

waveguide width increases, the effective TO coefficients of the TE and TM modes reach that of the 220-nm thick

Si slab modes. Fig. 5 shows that the athermal condition (i.e., dneff

dT = 0) can be achieved by simply choosing an

appropriate waveguide width.

4. DESIGN CRITERIA FOR ATHERMAL PERFORMANCE

This section investigates the athermal conditions of a channel waveguide. In the simplest geometry of a channel

waveguide (ncore) surrounded by a single cladding material (ncl), the waveguide effective index is related to the

confinement factor: neff =Γ ncore + dncore

dT (T − T0) +(1− Γ) ncl + dncl

dT (T − T0) . By setting dneff

dT =0,the

athermal condition for a channel waveguide is then:

dT (ncore − ncl)+ dncl

dT +

Γ+ dΓ

(T − T0)

dT

dncore

dT

dncl

− =0, (8)

dT

whereΓdefinesthepercentageoftheopticalmodethatresidesinthewaveguidecoreregion.Sincethetemper-

ature dependence of the confinement factor ( dΓ

dT ) is usually on the order of 10−4 ,andtheTOcoefficientsofthe

dT ) are on the order of 10−5 to 10 −4 , the left side of Eq. 8 can be simplified to just

the first two terms. In other words, the cladding TO coefficient dncl

dT must be negative for a waveguide core with

a positive dncore

dT . It is evident that an athermal operation can be realized by choosing specific material systems

such that the TO coefficient of the cladding fully compensates the temperature dependence of the waveguide

confinement factor. However, this condition does not apply to the basic channel structure as shown in Fig. 4 due

to the geometrical asymmetry. It may be impossible to derive an equivalent simple expression as in Eq. 8. Here

we derive the athermal condition numerically.

The waveguide geometry is designed such that only the fundamental TE mode propagates in the waveguide.

The material TO coefficient of the top cladding is chosen to be opposite than that of the waveguide core.

This design ensures that the negative thermal effects experienced by the optical mode in the cladding material

effectively compensates the positive thermal effects on the mode from the waveguide core. The confinement

factor of the optical mode is then varied by tuning the index contrast between the waveguide core/cladding

materials.

We plot in Fig. 6 the athermal conditions for various material systems. The confinement factor is a function

of the index and TO coefficient of the core/cladding, the waveguide geometry, as well as temperature. In Fig. 6,

the confinement factor is simulated at the room temperature, which implies that the Γ’s on the design curves

only depend on the intrinsic index ratio ncore/ncl and the waveguide geometry. The indices and TO coefficients

of the required cladding materials shown in Fig. 6 are within the range of the expected material properties in

the commercially available polymers. Our simulations conclude that the polymer cladding approach is feasible

for practical athermal silicon waveguides.

Empirical equations have been derived for the general athermal condition of an SOI channel waveguide:

ncl

dT

dT Γ2 + BΓ 2 + C dncore

dncore

Γ+DΓ+E

dT dT

+ F, (9)

where A, B, C, D, E, F are numerical coefficients: A = −4.854065 × 10 1 ; B = −2.169000 × 10 −5 ; C =5.206713 ×

10 1 ; D =2.326577 × 10 −5 ; E = −1.589253 × 10 1 ; F =4.575161 × 10 −6 .

Proc. of SPIE Vol. 6897 68970S-6

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