1. Introduction

With the growth of optical communication networks to the homes and businesses, polymer optical waveguides have been receiving increasing attention due to their low cost, ease of processing, simplicity of integration onto desired surfaces, and energy-saving property (e.g., large thermo-optic coefficient) [1,2]. For applications in optical communications, it is necessary that the performance of the optical devices be insensitive to the polarization state of light [3,4]. However, stress-induced effect remains a key issue for the design of such polarization-insensitive devices [5–10].

Stresses in polymer waveguide are generated mainly during the fabrication process due to polymerization shrinkage and thermal-expansion mismatch in the constituent materials. The generated stresses, when anisotropic, cause anisotropic change in the refractive index of waveguide materials (i.e., material birefringence), which then makes the device polarization-dependent. Such stress-induced effect is usually not considered in the design of waveguide devices, which is mainly due to the difficulties in measuring the stresses in tiny optical waveguides. Uncertainties in the accurate prediction of the stresses involving complex behavior of polymer material may be another important reason why the stress-induced effect is not incorporated into the design. As a result, the performance of the fabricated device often differs from that of theoretical designs [7–10]. In such cases, researchers usually follow the trial and error process involving successive design, fabrication, and characterization until the desired performance is achieved. This process is however time-consuming, costly, and cannot provide sufficient information or insight on the practical phenomena for further improvements. In considering these aspects, the design of new waveguide devices need to be considered from an integrated perspective, taking into account the process-induced stresses.

Recently, a number of works demonstrated the potential of designing various silicon (Si) based waveguide devices considering the stress-induced effect [10–15]. Most of these works use either analytical methods [11,12] or simplified finite element (FE) models in numerical approaches [10,13–15] for the estimation of process-induced thermal stresses. Although this approach might be reasonable for inorganic material-based waveguides in which only two different materials are involved, its application can give erroneous results for multilayer polymeric devices with diverse materials, and thus can seriously mislead the optical designs [16]. The few researchers, who used the simplified model to investigate the polymeric waveguides [17] or photonic packaging consisting of polymers [18], have only discussed their results qualitatively. An approach for the accurate analysis of thermal stresses in polymeric waveguides and thus the stress-induced characteristics of polymer rib waveguides has been reported in [16]. The design results of polymer waveguide devices taking the stress-effects into account have been briefly presented in [19]. However, it is necessary to present the details of such a design approach for its generalized applications.

This work describes the details of a systematic approach and some of its important aspects for the accurate design of polarization independent polymer waveguide devices considering stress-induced effects. The potential and validity of this design approach are demonstrated through the accurate design of Bragg gratings on a benzocyclobutene (BCB) strip channel waveguide. This work also presents an extensive investigation of stress-induced birefringence characteristics of various strip waveguide structures. The waveguide stresses are found first through the finite-element (FE) solution of the thermo-mechanical problem. Our analysis is based on the more accurate model in the FE analysis which considers the entire history of the waveguide fabrication process. The details of the implementation of the process modeling and a general discussion of the features included in our model are presented in Section 2.

Once the stress distributions are determined, combining them with stress-optic coefficients, the perturbed material indices are derived. Then, the stress-induced characteristics of material birefringence are investigated thoroughly for different geometries of strip waveguides. Finally, the stress-induced effect is incorporated in the modal analysis using the wave matching method (WMM) [20] and the analysis results are compared with our published experimental data [9]. A good agreement is achieved between the design and experimental results, which implies the potential of our approach for the efficient design of polymeric optical waveguide devices. The design results and associated analysis are presented in Section 3.

2. Methodology

2.1 Finite element analysis of thermal stress

Figure 1 shows the schematic of a Bragg grating on strip waveguide (width, w; and height, h). As the material chosen in this work, the waveguide is comprised of benzocyclobutene (BCB) for the core layer and epoxy (OPTOCAST 3507) for the lower cladding layer.

 

Fig. 1 Schematic of a Bragg grating on strip channel waveguide. The strip width and height are denoted by w and h, respectively, and $\Lambda$ is the pitch of the grating.

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To predict the residual stresses in the waveguide, important thermomechanical steps of the fabrication process should be strictly considered and analyzed. In the fabrication process [9], epoxy was first spin-coated on a silicon wafer, cured under UV light at room temperature (24 °C) and post-cured at 120 °C. The BCB was next spin coated on the epoxy to a thickness of 2.9 µm and cured at 250 °C. The BCB film was then patterned into rectangular core by photolithography and reactive-ion etching. Finally, a Bragg gratings with pitch ($\Lambda$) of 0.515 µm and corrugation depth of 200 nm was formed on the surface of BCB core via laser ablation using excimer laser (248 nm).

In this sequential fabrication process, each layer of polymer material is deposited at different temperatures and finally cooled down to room temperature. Therefore, stress can be easily induced by curing shrinkage and thermal expansion mismatch. On the other hand, stress relaxation occurs due to viscoelastic nature of polymer. The stress can also be relaxed due to the etching process. Empirical formulas or simple numerical models are not suitable to capture such stress generations over the described process of polymer waveguide fabrication and in consequence, may mislead the optical design. To accurately predict the stresses, we employed our developed FE model [16] in MSC Marc software. Comparing to the conventional models, this model is more realistic for the thermo-mechanical analysis of polymer optical waveguides over the entire sequential fabrication process.

In the simulation, the geometry of the final structure is modeled at the very beginning and only the substrate is considered to be stress free at room temperature. The BCB layer is deactivated (i.e., exhibits no stiffness) till the end of curing of the epoxy. Thereafter, it is activated and this addition of material elements is done in a “strain-free” manner. Since the BCB is liquid in the uncured condition, it is activated with very low elastic modulus (multiplied by 10⁻⁶). Its stiffness and thus the stress begin to develop from the point of 60% of the total curing at temperature 250 °C [21]. The stress induced by the curing shrinkage is considered in our model by using the coupled cure-thermal-mechanical approach. In the modeling, the curing process (i.e., cure rate, degree of cure) is controlled following the typical curing process of epoxy and BCB [22,23]. The curing shrinkage is considered to be 3% and 5% for epoxy and BCB, respectively. To give the pattern of strip channel waveguide, etching process is simulated by deactivating the selected elements which demonstrate zero stresses and strains on the post elements. The pattern of grating is ignored for the stress analysis due to its insignificant influence on stress.

The stress relaxation due to the viscoelastic nature of polymer is an important factor in determining the accuracy of thermal stress calculations. The relaxation process is considered in the FE analysis using the creep model$\dot{\epsilon }=C{\sigma }^n{t}^m$ where, $\dot{\epsilon }$ is creep strain rate, $\sigma$ is stress, $t$ is time, $C$ and $n$ are material constants and $m$ is time constant, for 10000 seconds. The material creep parameters ($C$ = 2 × 10⁻²⁵, $n$ = 2.42 and $m$ = 1) have been fitted by optimization comparing the simulated and the measured stress in BCB thin films. For the stress calculation, all the waveguides are allowed to relax by this model before patterning the channel. A summary of the properties of fully cured materials used in the simulation are shown in Table 1 [24–27]. The other parameters (e. g., time, temperature) considered in the FE analysis are the same as those described in the fabrication process. However, the model is generic in nature and can be equally employed in the modeling of typical polymer waveguide fabrication.

Table 1. Material properties used in the FE analysis

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The strip waveguide analyzed in this paper is of 2.9 μm in height and has lower cladding of thickness 10 μm on Si substrate (500 μm thick). For the stress calculation, the waveguide geometry having dimension of 5 mm × 512.9 μm × 5 mm is used in the FE model which consists of over 8000 hex elements and 10000 nodes with required thermal and mechanical boundary condition (BC). The temperature that is considered for thermal BC is stated in the description of the fabrication process earlier in this section. For the mechanical BCs, the structure is constrained at the center to avoid unwanted rigid motion, and all elements were free to move in any direction responding to thermal expansion and induced stress.

2.2 Stress-optic theory and waveguide design

In the elastic range of the material, the principal refractive indices are related to the principle stresses by the Neumann-Maxwell equations [28]:

To analyze the influence of induced stress on the design of Bragg grating, the perturbed material indices are incorporated in the modal analysis to calculate the wavelength difference of Bragg grating. The Bragg wavelength (${\lambda }_B$), a key parameter of Bragg grating devices, can be determined by the phase matching condition as ${\lambda }_B=2n_{eff}\Lambda$, where $n_{eff}$ is the effective index of the guided mode, and $\Lambda$ is the pitch of the grating. Then, the wavelength difference between the two orthogonally polarized modes, horizontally polarized mode (quasi-TE) and vertically polarized mode (quasi-TM), can be obtained as $\Delta {\lambda }_B$ = ${\lambda }_B(TE)-{\lambda }_B(TM)$. The WMM method is used in this case for the solution of effective indices (i.e., propagation constants) of strip waveguide.

3. Results and discussions

To facilitate the design of the optical devices considering stress-effects, the stress and its induced characteristics in the optical waveguides must be well understood. This section firstly explains the stress distributions and their induced characteristics in strip waveguides. Then, the design results of the Bragg grating on such strip waveguides are demonstrated extensively.

3.1 Stress distributions

From the simulation model, the principal stresses (${\sigma }_x$, ${\sigma }_y$ and ${\sigma }_z$) are noted at room temperature for two process steps: before etching and after etching. Before etching, the stresses in the planar BCB film (core layer) are obtained as ${\sigma }_{x,film}$ = ${\sigma }_{z,film}$ = 44.1 MPa and ${\sigma }_{y,film}$ = 0.657 MPa for the in-plane and out-of-plane directions, respectively. Such in-plane stress often causes out-of-plane deformation i.e., warpage in the structure. Thus, the film stress can be verified easily in this stage by measuring the warpage of the structure. The stress is related to the warpage by the Stoney’s formula [29] as,

For the measurement purpose, few samples of different sizes are prepared with a BCB film of about 6 μm thick on Si substrate following the same procedure of waveguide fabrication, and also simulated for stress estimation. The warpage is measured by the stylus profiler Ambios XP-2. Figure 2 shows the measured and simulated results of warpage for a sample of 3.5 × 3.5 cm² in size. The results are taken along the diagonal line starting from the center of the sample, shown in the figure as an inset. The corresponding measured stress (in-plane) is 45 ± 3 MPa, which is in good agreement with the simulated result (44.1 MPa). To understand the influence of time-dependent behavior of polymer on this stress, the sample was kept at room temperature for 14 days, and the stress was then measured in the same way. However, no significant change was found in the stress value due to the time factor. This confirms the stability of design results concerning the time dependent properties of polymer. Since the structure is planar before etching, the stress is uniform all over the BCB layer.

 

Fig. 2 Measured and simulated warpage of BCB thin film sample of size 3.5 × 3.5 cm².

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After etching to the strip channel, the film stress is changed significantly. However, it is very difficult, sometimes impractical, to measure that stress in the tiny channel waveguides. Figure 3 shows the FE analysis results of typical stress distributions in the strip channel consisting of width $w$ = 3 µm and height $h$ = 2.9 µm. Contrary to the case of the planar (slab) waveguide, the stress distributions in the patterned structure (channel waveguide) are highly non-uniform. The stresses in the lower cladding (epoxy) regions near the strip are also changed significantly. Figure 3(a) reveals that the in-plane stress (${\sigma }_x$) in the strip channel is greatly relieved due to etching. The attained maximum in-plane stress is about 50% of the film stress, but only in an extremely narrow area adjacent to the lower cladding or substrate. This is due to the existence of the vertical sidewall of the strip; on the wall surface, the traction-free boundary condition has to be satisfied [29]. Apart from the in-plane stress (${\sigma }_x$), in Fig. 3(b), a significant amount of out-of-plane stress (${\sigma }_y$) also exists in the channel. Note that the calculated value of ${\sigma }_y$ is very small for the planar film (before etching). However, after etching, the amount of ${\sigma }_y$ in the channel is comparable to the in-plane stress. Herein, positive stress is tensile, whereas negative stress is compressive. All these stress components are invariant along the propagation direction of light (i.e, z axis). Since the stress-optic coefficient of epoxy is zero, no stress is noted for the epoxy layer.

 

Fig. 3 Contour plot of stress distribution in the strip channel of width, w = 3 µm and height, h = 2.9 µm (a) in-plane stress, ${\sigma }_x$ and (b) out-of-plane stress, ${\sigma }_y$.

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3.2 Stress-induced characteristics

Once the stress distributions are obtained, the perturbed material indices are calculated using Eq. (1). Thus, the stress-induced anisotropy i.e., birefringence ($\Delta n$) is also determined for the waveguide material as$\Delta n=n_x-n_y=(C_1-C_2)({\sigma }_x-{\sigma }_y)$. Before etching, the birefringence (n_x - n_y) obtained for the BCB film was 0.003 (n_x = 1.5565, n_y = 1.5535), which is uniform all over the BCB layer. In this case, it was a slab waveguide and the material index of BCB can be measured easily by prism coupler. The measured birefringence was 0.0034 (n_x = 1.5565, n_y = 1.5531) for BCB at 1550 nm wavelength [9], which is comparable to the calculated result of 0.003. Such agreement primarily confirms the validity of our approach in the application of optical waveguide design.

Figure 4 illustrates the stress-induced variation of material in-plane index (n_x) along the height of BCB strip waveguide (i.e., after etching). It is calculated along the height (y direction) of the core considering the stresses at the center of width (x = 0), shown on the figure as an inset. The corresponding birefringence retained in the waveguide is shown in Fig. 5. It is observed that significant amount of birefringence remains in the structure. The birefringence is higher near the lower cladding or substrate, with its maximum value of about 50% of the birefringence (0.003) remained in the film before etching.

 

Fig. 4 Material in-plane index (n_x) as a function of the position (y) along the height of a strip channel (h = 2.9 µm) for different widths (w = 2.25 µm, 3.0 µm and 4.5 µm).

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Fig. 5 Variation of material birefringence (n_x - n_y) along the height of a strip channel (h = 2.9 µm) for different widths (w) of 2.25 µm, 3.0 µm and 4.5 µm.

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Figure 5 shows that the birefringence decreases exponentially to the zero value as it approaches the top of the channel. As explained earlier, the material birefringence in the planar waveguide is induced mainly by the in-plane stress. In the etching process, however, the in-plane stress relaxes remarkably, while more of the out-of-plane stress (comparable to in-plane stress) is induced in the strip region. This is the reason why the material birefringence decreases sharply as it approaches the top of the channel. As the results presented for different width in Fig. 5, the birefringence at a particular height slightly increases with the increase in strip width.

Figure 6 shows the variation of material birefringence along the width (x) at different heights (y) of 0.5, 1.25, and 2.0 µm of a strip waveguide with w = 3 µm and h = 2.9 µm. The birefringence does not vary significantly along the width of the strip channel. It is mentioned that apart from the quantitative values which are derived for the BCB waveguide, the stress-birefringence characteristics would be applicable for other polymers as well.

 

Fig. 6 Material birefringence along the width at three different heights of 0.5 µm, 1.25 µm, and 2.0 µm of a strip waveguide with dimension of w = 3 µm and h = 2.9 µm.

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3.3 Design of Bragg waveguide grating

As obtained, stress-induced material birefringence (anisotropy) is not uniform in the channel waveguide structure. Birefringence along the height of the channel varies continuously from its maximum value to zero; whereas it remains nearly unchanged along the width. To incorporate such non-uniform material index in the optical design, a multilayer core with different refractive indices, as shown in Fig. 7, is considered in WMM for the solution of effective index (n_eff). The refractive index, as well as material birefringence, is considered constant and uniform in each layer. The total number of layers ($N$) and their thickness ($t$) are chosen according to the variation of material birefringence in the waveguide, and optimized to produce the invariant results. For example, smaller thicknesses are chosen near the lower cladding of a strip waveguide since the change of birefringence is steeper in this region. Then, the Bragg wavelength is calculated as λ_B = 2 n_eff Λ, as described in Section 2.

 

Fig. 7 Multiple-layered approximation of a strip waveguide.

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Figure 8 shows a comparison of the simulated and the experimental wavelength differences, $\Delta {\lambda }_B$ = ${\lambda }_B(TE)-{\lambda }_B(TM)$, as a functions of core width $w$ at wavelength of 1550 nm. As demonstrated in [9], the refractive indices of lower cladding epoxy for TE and TM polarizations are measured as 1.5108 and 1.5103, respectively, and an index-matching liquid with index of 1.511 is considered for upper cladding. The liquid has no stress-effects on the waveguide. The dashed line in Fig. 8 is obtained considering isotropic (stress free) index of 1.5565 for BCB, as like the traditional work. This result reflects only the influence of geometric birefringence, the modal birefringence caused by the waveguide geometry. The solid line plots$\Delta {\lambda }_B$, considering the previously estimated stress-induced material birefringence (Fig. 5). The design results with and without considering stress clearly show different values of $w$ that give a polarization-independent wavelength ($\Delta {\lambda }_B$ = 0).

 

Fig. 8 Bragg wavelength difference ($\Delta {\lambda }_B$) between TE and TM polarizations as a function of core width ($w$). The core height ($h$) is 2.9 µm.

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The dots in Fig. 8 show the experimental results, reported in [9], measured in two different ways. It has been explained that while Δλ_B can be measured directly from the transmission spectra of the grating, it can also be obtained from the modal birefringence (B) measured by the polarimetric method as Δλ_B = BΛ. The simulated Δλ_B considering stress-effects (solid line) agrees very well with the experimental results (dots). In contrast, the simulated results following the traditional approach (dashed line) are largely deviated from the experimental results. Here, the scattering in the experimental results is mostly due to the practical aspects of the fabrication process, and lies on average within the range ± 0.1 nm in reference to the design result (solid line). On the other hand, the difference between the results, with considering stress (solid line) and without considering stress (dashed line), estimates the stress induced effects. This difference is found to be about 0.25 nm for various widths which is higher than the fabrication induced effects. Thus, the stress-induced birefringence plays a critical role in the design of waveguide devices. The agreement between the design and the experimental results confirms the feasibility of our approach in the accurate design of polarization-insensitive waveguide devices. This approach is generic in nature and would enable the optimization of any integrated-optical design from the standpoint of material systems, waveguide geometry, and process parameters.

4. Conclusions

A methodology for the efficient design of polarization insensitive polymer optical waveguide devices is demonstrated considering stress-induced effects. The feasibility of this method is verified by the accurate design of Bragg gratings in a BCB strip channel waveguide. In the used material systems, the stress-induced waveguide birefringence mainly results from the anisotropy in the core material because the cladding material has relatively negligible stress-optic effect. The stress-induced characteristics of the strip waveguide are presented in detail. Very good agreement is achieved between the simulation results considering stress-effects and the experimental results, whereas the stress free assumption completely fails. This implies that the proposed approach provides a more accurate way for the design and analysis of any complex polymer waveguide device. Being generic in nature, this approach also enables optimized waveguide design from the standpoint of material systems, waveguide geometry, and process parameters.