The Q factors of waveguide-coupled micro-rings can be classified as two components:

*Q*_{intrinsic} and

*Q*_{couple}, where the former is the Q of an isolated resonator and the latter takes into account of the coupling loss to the bus waveguide.

*Q*_{intrinsic} is primarily limited by the optical losses in an isolated resonator, and can be attributed to these loss mechanisms: radiation loss, surface scattering loss, material absorption loss. Therefore the micro-resonator’s overall Q factor can be expressed as:

where

*Q*_{rad},

*Q*_{scatt}, and

*Q*_{abs} are radiation loss-related Q, surface scattering loss-related Q, and absorption loss-related Q, respectively. The total loss-related Q and coupling related Q can be extracted from the fitted transmission spectrum [

23], the radiation related Q can be obtained from the simulation, the absorption loss-related Q can be measured from a thermal bi-stability effect (will be described in more detail below), and finally the scattering related Q can be extracted from

Eq. (1).

and show the transmission spectrum of a polymer micro-ring imprinted using the silicon mold without and with resist reflow process, respectively. shows the total Q of around 1 × 10

^{4}, and shows the total Q of around 3 × 10

^{4}. The amplitude attenuation factor can be obtained from the micro-ring transmission equation, which can be expressed as:

where

*τ* is the field coupling coefficient between the input and output port,

*a* is the amplitude attenuation factor, and

*ϕ* is the round trip phase. The amplitude attenuation is due to the various optical losses in the micro-ring, and therefore can be used to obtain the

*Q*_{intrinsic}. The amplitude attenuation factor

*a* is related to the intrinsic Q by the following equation [

24]:

where R is the radius of the micro-ring, n

_{eff} is the effective refractive index of the resonance mode and λ

_{0} is the resonance wavelength. By fitting with

Eq. (2), we can get the field coupling coefficient between the input and output port

*τ* = 0.990 and the amplitude attenuation factor

*a* = 0.925 for the micro-ring device fabricated using the mold without the resist reflow process. The calculated intrinsic Q according to

Eq. (3) is around 1.2 × 10

^{4}, corresponding to a propagation loss of 22.7 dB/cm. For the device created from the mold with the resist reflow process, we find that the field coupling coefficient between the input and output port

*τ* = 0.996 and the amplitude attenuation factor

*a* = 0.975. The calculated intrinsic Q is around 3.5 × 10

^{4}, corresponding to the propagation loss of 7.7 dB/cm. Therefore the resist reflow process can reduce the optical propagation loss by a factor of 3. shows the transmission spectrum of the waveguide coupled micro-ring devices fabricated from the mold made with the process including the resist reflow and thermal oxidation steps. The total Q is fitted to be around 1.1 × 10

^{5}, and the field coupling coefficient between the input and output port is found to be

*τ* = 0.99946 while the amplitude attenuation factor is

*a* = 0.994. The intrinsic Q is calculated to be~1.5 × 10

^{5}, which represents propagation loss of 1.8 dB/cm. The thermal oxidation step further helps with reducing surface roughness leading to a total reduction of propagation loss by more than one order of magnitude.

COMSOL multi-physics software was used to simulate the radiation loss of the polymer micro-ring devices [

25]. By taking the advantage of the axial-symmetry of the resonance modes in the micro-ring, the 3-dimensional eigenvalue problem can be transformed to an equivalent 2-dimensional problem. The exact dimensions of the micro-rings which are used in the simulation were obtained from scanning electron microscope (SEM) while the refractive index of polymer was found using spectroscopic ellipsometry. By solving the eigenvalue problem in COMSOL, we obtained the eigen frequency for the transverse electrical mode’s in the micro-ring with real part equal to 1.930625 × 10

^{14} and imagery part equal to 1.00754 × 10

^{8} Hz. Therefore the radiation loss-related Q = Re(ω)/2Im(ω) [

26] is 9.6 × 10

^{5}.

To determine the absorption loss-related Q factor in the polymer micro-ring, we adopted a method based on the thermal-instability phenomenon of polymer resonators. We measured the transmission spectrum for different input powers, and the results are shown in
. When increasing the input power, we found that the transmission spectra exhibit two notable changes: (1) the resonance peak shifts to a shorter wavelength, which is mainly due to the negative value of the opto-thermal coefficient of the polymer, and (2) distortion of resonance line shape, which is due to the material absorption induced thermal bi-stability effect [

27]. The absorption loss-related Q can be extracted from the linear relation between the internal cavity energy and absorbed power by following the method described in Reference

28. Assuming steady-state condition, the absorbed power in the cavity can be expressed P

_{abs} = ΔT/R

_{th}, where ΔT is the temperature change in the cavity, and R

_{th} is the effective thermal resistance, which includes the thermal resistance of the heat sink from the polymer cavity to ambient and thermal resistance of cavity itself. This was modeled using COMSOL multi-physics and found to be 9.5 × 10

^{3} W/K. The absorbed power then can be expressed as:

where

*n* is the refractive index of the polymer,

*λ* is the resonance wavelength,

*dn*/

*dT* is the thermo-optical coefficient,

*α* is the thermal expansion coefficient and Δ

*λ* is the resonance wavelength shift.
shows the power dependence of the absorption effect from of the polymer micro-ring. The intra-cavity energy (U

_{c}) is calculated according to the Reference

28. The linear absorption related coefficient γ

_{lin} = P

_{abs}/U

_{c} was extracted and found to be 5.6 × 10

^{9} Hz, which can be used to calculate the absorption loss-related Q = ω/γ

_{lin} = 2.1 × 10

^{5} [

28]. This corresponds to a material absorption loss of 1.3dB/cm for the polystyrene used in our devices, which is consistent with the published data [

29].

Finally, the surface scattering loss-related Q can be calculated from the

Eq. (1) and is ~1.2 × 10

^{6}, and the surface scattering loss can be shown as low as 0.23 dB/cm. From this analysis, we know that the dominating loss in polymer micro-rings at 1.55 μm wavelength range is the material absorption loss (~1.3dB/cm), which is attributed to the carbon-hydrogen bonds harmonic absorption in the near-IR range. Such absorption loss can be minimized by replacing H with F atoms. Fortunately, in the visible wavelength range, polymer materials can have an absorption loss as low as 0.004dB/cm [

30]. Therefore, we believe that our polymer micro-ring’s total Q can be greatly increased by moving the working wavelength from the NIR to the visible range.