1. Introduction

Stress, which is unavoidably created in the fabrication process, has long been a crucial issue for the yield improvement of many optical waveguide devices [1–6]. The stress is generated in the waveguide materials primarily because of the thermal expansion mismatch among the constituent materials. In polymer based optical waveguides, the stress can also be generated due to the polymerization shrinkage even at room temperature when cured under UV light [7]. Due to the photo-elastic effect, this stress may result in anisotropic changes in the refractive index of the material, referred to as material birefringence, causing the waveguide to become polarization dependent. The problem becomes more complicated in case of polymer based optical waveguide devices, since the thermal expansion coefficients of the polymer and the inorganic substrate usually differ greatly, and hence a significant amount of stress can be generated in the polymer films [8]. To tackle this problem efficiently, one of the effective ways is to include the stress-induced effects at the early stages of the design works [9–11]. Therefore, an accurate estimation of stress and knowledge on its influence on the optical properties of the waveguides are important. In recent years, polymer materials have received increasing popularity for optical waveguide devices because of their ease of integration onto the required surfaces, simplicity of processing, low cost, as well as energy-saving characteristic (e.g., large thermo-optic coefficient) [8,12].

Significant works have been devoted to investigate the stress-induced (i.e., photo-elastic) material birefringence in optical waveguide devices [3–6,13,14]. Most of these works used either analytical [3,4] or numerical methods [6,13,14] for the stress analysis, since the measurement of stress is difficult in tiny waveguides or inside the components. Available analytical methods are simple but insufficient to capture the complex behavior of polymer materials, and thus to estimate the stress accurately in polymer waveguides [14]. Numerical analysis based on finite element (FE) method is, therefore, often employed for the investigation of stress-birefringence in optical waveguides. Recently, we have proposed a more realistic FE model to accurately estimate the stress in polymer waveguides [10,14]. Thus, the birefringence has been demonstrated for the benzocyclobutene (BCB) strip [10] and rib waveguides [14] by using this model with good accuracy. An approach for the more accurate design of polarization insensitive polymer waveguide devices has also been demonstrated based on the stress analysis [10]. However, such accurate prediction of stress requires extensive material properties (e.g., elasto-plastic and viscoelastic behavior of polymers, chemical kinetics for cure reaction, etc) which are usually unavailable in literature for many optical polymers, and also difficult to characterize in thin films.

This work presents a generalized characteristic of photo-elastic birefringence in polymer strip waveguide through the detailed numerical investigation of thermal stress in various waveguide structures using the realistic FE model. As mentioned, the birefringence characteristics of BCB strip waveguide have been presented previously. In this work, such characteristics are further investigated for a wide range of material properties, such as elastic modulus, viscoelastic stress relaxation and coefficient of thermal expansion (CTE), covering very low (~6 MPa) to high stress (~56 MPa) polymers. The influences of the lower cladding material on the stress which is built up in the core layer are discussed in detail. The obtained characteristics are then generalized and expressed by an empirical relation. The developed relation can be employed to specify the stress-birefringence in an air-clad strip waveguide of various polymers avoiding the need of extensive numerical analysis. Since the strip waveguide has been widely used in fabricating various optical devices [9,15], the presented results will be useful for the treatment of stress-induced problems in many of those optical devices.

2. Stress analysis and photo-elastic theory

2.1 Stress build-up process

A strip waveguide’s cross section is illustrated in Fig. 1. During the fabrication stage of such waveguide, the polymer layers are spin coated on silicon (Si) wafer, and treated at elevated temperature (120–300 °C) one after another. In this process, when the structure is cooled down to room temperature, the mismatch of CTE between the involving materials induces the stress in the polymer films. At the same time, a significant amount of stress is relaxed also due to the viscoelastic nature of polymers. The stress is further relaxed in the core layer when it is patterned to form tiny strip channel. However, the total amount of stress remaining in the film depends on the material properties (elastic modulus, relaxation modulus, CTE, etc.) and the temperature difference between the deposition to room temperature [16].

2.2 Finite element analysis of stress

To precisely estimate the stress, the authors’ FE model [10] developed in MSC Marc software is utilized. This model is more realistic than the conventional models for analyzing the thermo-mechanical characteristics of polymer optical waveguides in different steps of the fabrication process. We modeled the geometry of the structure at the very beginning of our simulation where the substrate was assumed to be stress free at room temperature. Until the completion of the curing of lower cladding layer, the core layer was deactivated exhibiting no stiffness. Subsequently, the core layer was activated in a “strain-free” fashion. We simulated the etching step through elimination of the selected material elements which signify zero stress and strain on the post elements.

For the demonstration, the deposition temperature is considered here as 250 °C. The Thermo-rheologically-simple viscoelastic model is employed to simulate the stress relaxation behavior of polymers. The master curves of various stress relaxation modulus as used with different polymers are shown in Fig. 2 at a reference temperature of 30 °C. The time-temperature shift in the viscoelastic model is considered according to Williams-Landel-Ferry with coefficients of 10 and 67 at the reference temperature. The detail of such viscoelastic modeling has been described in [16,17]. The effect of the temperature dependence of the material properties on the stress development is mostly accounted by such a model. A wide range of material properties as used in the FE stress analysis are listed in Table 1. In Table 1, the properties of material R2 are corresponding to the epoxy Vilax 81 [16]. The properties of other materials are considered arbitrarily to produce different values of thermal stress in its film. The properties include a range of low (2.1 GPa) to high (3.8 GPa) values of elastic modulus under the materials R1 and R5, respectively. The elastic modulus of materials R2, R3 and R4 are very close (~2.9 GPa), however, their relaxation modulus are different (RM2 > RM3 > RM4). The polymer (R2) with relaxation behavior RM2 is highly viscous than others, and therefore, relaxes more stress. On the other hand, the material E1 and E2 are fully elastic (i.e., no viscoelastic relaxation) with modulus of 3.8 and 2.9 GPa, respectively.

Table 1. Material characteristics employed in the stress analysis

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Table 1 also lists the simulated film stress corresponding to each of these materials when deposited on the silicon wafer. Here, the standard isotropic condition is assumed with a constant Poisson’s ratio of 0.34 for all polymers and 0.27 for Si. The obtained stress lies in a wide range of very low to high values (6.2 – 55.7 MPa) as in optical polymers. Thus, the considered properties will cover various polymers (very low stress epoxy [15] to medium stress SU-8, PMMA [18] to high stress benzocyclobutene [10], polyimide [19] etc.) used for the fabrication of optical waveguide devices.

The strip waveguide studied in this work is of various dimensions including heights of 2 – 4 µm and width of 2.25 – 4.5 µm, and has a lower cladding thickness of 10 µm on a 500 µm thick Si substrate. The waveguide dimension along the x- and z-directions is considered as 5 mm × 5 mm for the stress simulation. The FE model of the waveguide comprises more than 8000 hex elements as well as 10000 nodes involving necessary thermal plus mechanical boundary conditions [10]. The element sizes are refined (smaller) at the corners and interfaces where the stress is highly non-uniform.

2.3. Photo-elastic theory

Considering the material’s elastic range, the principal refractive indices are associated with the principal stress through the Neumann-Maxwell equations [20]:

 

Fig. 1 Cross-section of a strip channel waveguide. The width and height of the strip are indicated by w and h. The aspect ratio R is defined as w/h.

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3. Results and discussions

Based on our simulation, the principal stress (, , and ), and thus the photo-elastic birefringence are considered at room temperature for the two fabrication stages: before and after etching. These are described in the following sections followed by their generalized representations and validations.

3.1 Photo-elastic birefringence: before etching

In the before etching stage, the structure is like a planar (slab) waveguide in which the stresses are uniform over the entire polymer layers. Fig. 3 illustrates the typical stress distribution in a planar film of height, h = 3.0 μm and consisting of the materials in Case 5 (Table 2).The value of in-plane stress (${\sigma }_{x,film}$) is found as 18.5 MPa in the core layer against the corresponding out-of-plane stress (${\sigma }_{y,film}$) of 0.71 MPa. Table 2 lists the in-plane film stress (${\sigma }_{x,film}$ = ${\sigma }_{z,film}$ = ${\sigma }_{film}$) in the core layer for several combinations (Case 1 - 11) of core and lower cladding materials. As presented in Table 2, for example for Case 1, the material R1-E1 represents a waveguide consisting of material R1 (Table 1) as the core and E1 as the lower cladding layer over Si substrate. Similarly, the materials considered for different cases in Table 2 correspond to the properties given in Table 1. Here, the various combinations (Case 1 - 11) in Table 2 are chosen to present the stress analysis for the conditions of very similar core and cladding materials to large mismatch in their properties. This produces very low to high value of stresses in the polymer film and helps to investigate the effects of various material properties on the photo-elastic birefringence, as presented in the following parts.

 

Fig. 2 Master curves of various stress relaxation modulus as used for different polymers.

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Fig. 3 Typical distributions of a) in-plane ${\sigma }_x$, and b) out-of-plane ${\sigma }_y$ stress in a planar film (i.e., before etching) consisting film thickness of 3.0 µm.

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Table 2. Various material combinations considered for the waveguide, and the corresponding stress and birefringence retained in the planar film (before etching)

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As observed in Case 9 (E1-R3), the polymer E1 having higher value of elastic modulus (3.8 GPa) and no relaxation produces the highest value of film stress (50.9 MPa). On the other hand, the lowest value of film stress (6.3 MPa) is observed for Case 1 (R1-E1), where the core material R1 has low elastic modulus (2.1 GPa) and high viscoelastic relaxation property (RM1). However, the lower cladding material also has significant influence on this stress generation. By comparing Case 6 (E1-R1) with Case 9 (E1-R3), where the core material E1 is common, the low stress cladding material (R1) induces lower film stress (37.1 MPa) than that (50.9 MPa) with high stress clad (R3). Similar results are observed by comparing Case 4 (R4-R2), Case 5 (R4-R3), and Case 8 (R4-Si), where the film of same material R4 induces different stresses (16.8 MPa, 18.5 MPa, and 24.3 MPa, respectively) over the lower clad materials of R2, R3 and Si, respectively. This indicates that the lower clad with low elastic modulus and/or high relaxation helps reduce the film (in-plane) stress in the core layer. The stress in the out-of-plane direction (${\sigma }_{y,film}$) is not shown here since it is insignificant (two order less) compared to the corresponding in-plane stress.

After obtaining the stress distributions, the photo-elastic birefringence ($\Delta n$) is calculated for the waveguide material by using Eq. (2). For the demonstration, the values of C₁ and C₂ in Eq. (2) are considered as 99 × 10⁻¹² Pa⁻¹ and 31 × 10⁻¹² Pa⁻¹, respectively, corresponding to the BCB thin film at an optical wavelength of 1550 nm [21]. Before etching, the birefringence values calculated in the core film ($\Delta n_{film}$) are listed also in Table 2 besides the film stress of different cases. These values lie between the range 0.00043 – 0.00346 for the analyzed material combinations (Case 1 - 11). As observed in Table 2, the higher values of film stress produced the higher values of birefringence. Similar to the film stress, however, the birefringence is also uniform over the film layer. For any practical polymer waveguide, the value of $\Delta n_{film}$ can be obtained by measuring the refractive indices for the two orthogonal polarizations (TE and TM) using the simple prism coupler technique. The birefringence in the lower cladding layer is not analyzed here since its effect on the waveguide performance is less significant (one order less) than the effect of birefringence in the core layer [14].

3.2 Photo-elastic birefringence: after etching

After etching the slab structure to form the strip channel, the stress was relaxed, and turned into non-uniform. Figures 4(a)-(c) display the typical contours of the principal stress (${\sigma }_x$, ${\sigma }_y$, and ${\sigma }_z$) in the strip channel and around the lower cladding near the strip. The results are obtained for the waveguide consisting of the materials in Case 5 (Table 2), and the dimensions of width, w = 3.0 μm, and height, h = 3.0 μm. The stress values are normalized by the corresponding in-plane stress retained in the film (${\sigma }_{film}$) before etching, as given in Table 2. Such normalization will help us understand the general feature of the birefringence characteristics. Unlike the planar (slab) waveguide, highly non-uniform stress distributions can be observed in the channel waveguide.

 

Fig. 4 Typical distributions of a) in-plane ${\sigma }_x$, b) out-of-plane ${\sigma }_y$, and c) z-component ${\sigma }_z$of stress in the strip waveguide consisting width w = 3.0 µm and height h = 3.0 µm.

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According to Fig. 4(a), the in-plane stress (${\sigma }_x$) in the strip channel is significantly relieved. The rest of the maximum in-plane stress is around 50% of the film stress, however, only in an exceptionally narrow region nearby the lower cladding or the substrate. The values of ${\sigma }_x$ are about 2–15% of the ${\sigma }_{film}$ within the most parts of the strip area. This is because of the absence of constraining force in the free standing part of strip structure compare to the bottom [22]. As can be seen in Fig. 4(b), a great portion of out-of-plane stress (${\sigma }_y$) also exists in the channel which is now in the same order of the in-plane stress. Note that the value of ${\sigma }_y$ was very small (two order less than${\sigma }_x$) for the planar film (before etching). We define positive stress as tensile, and negative stress as compressive. As shown in Fig. 4(c), the extent of stress relaxation is not very large along the direction of light propagation. The values of ${\sigma }_z$ are less than 80% of the ${\sigma }_{film}$ within the most parts of the strip area, but remain higher than 60% in overall. All these stress components are invariant along the propagation direction (z-axis) of light.

Figure 5 illustrates the photo-elastic birefringence ($\Delta n$) as a function of the position (y) alongside the height of the strip channel for various waveguide materials. These are obtained for a waveguide consisting of width, w = 3 µm, and height, h = 3 µm. Due to the effect of stress relaxation in the etching, the birefringence exponentially reduced to the zero value as the position y reaches the top of the channel. Given the stress-optic constants, the channel birefringence (Fig. 5) is usually higher for the materials with higher values of film birefringence or film stress (Table 2). However, while the film birefringence for Case 9 (E1-R3) is higher (0.00346) than that (0.00276) for Case 10 (R5-Si), the channel birefringence for Case 9 is lower than that for Case 10. This is due to: the cladding with lower elastic modulus supports more stress relaxation in the core over the etching process. Since the modulus of R3 (2.98 GPa) is much lower than that of Si (130 GPa), the core material E1 over R3 (Case 9) relaxes more stress than the core R5 over Si (Case 10). Ultimately, the materials in Case 9 produce the lower strip birefringence. Similar results are observed for Case 6, 7 and 8, where the channel birefringence values (Fig. 5) are in the reverse order compared to their order in the film birefringence (0.0025, 0.0021and 0.0016, respectively).

 

Fig. 5 Calculated photo-elastic birefringence ($\Delta n$) over the height of a strip channel for various combination of materials with dimensions of width w = 3.0 µm, and height h = 3.0 µm.

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The birefringence experiences only little variations along the strip channel width. In particular, at the center area of the core where the light is mostly confined, the variation along the width becomes negligible [10]. However, to consider the influence of this little variation, the birefringence presented here is taken from the position of little aside (w/4) the center position, representing its average value along the width.

To examine the influence of strip geometries on the evolution of birefringence, numerical analysis is performed for a broad range of strip dimensions (2.25– 4.5 µm in width and 2.0–4.0 µm in height). Figure 6 shows the birefringence ($\Delta n$) characteristics for various dimensions of waveguides corresponding to the materials given by Case 11 (E2-R3) in Table 2. For a given width, the birefringence retaining in the strip waveguide is constant, irrespective of the channel height. This is appeared in Fig. 6(a) with the essentially identical lines for width w = 3 µm, and heights h = 2, 3 and 4 µm. The significant birefringence remains only up to the height of about 2 µm regardless of the dimensions of the strip. Since the stress is an issue particularly for the single mode design, the dimensions analyzed here are around the typical dimensions of the single mode waveguide. However, Fig. 6(b) shows that the birefringence at a given height marginally rises with the upturn in the width of the strip. This is expected, since the free standing boundary moves apart with the increasing width and, therefore, the constraining force in the free standing part increases (i.e., stress relaxation in the core decreases) producing more stress and more birefringence.

 

Fig. 6 Variation of birefringence ($\Delta n$) over the height of a strip channel (a) for different heights (h = 2 – 4 µm), width w = 3.0 µm and (b) for different widths (w = 2.25 – 4.5 µm), and height h = 3 µm.

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3.3 Generalized birefringence

Based on the aforementioned observations, a generalized characteristic of the material birefringence can be obtained by plotting the normal birefringence against the position (y/h) normalized by the height of the respective waveguide. Similar to the waveguide stress (Fig. 4), birefringence is also normalized by the value that retained in the planar film, $\Delta n_{film}$ (i.e., before etching). Figure 7 shows the re-illustrated curves of Fig. 5, in which the x-axis is replaced by the normal position (normalized by the maximum height of the corresponding waveguide) instead of the actual position along the height. It is found that the maximum birefringence adjacent to the polymer lower cladding is around 50% stayed in the film prior to etching, whereas it is about 70% (Case 8 and Case 10) for the polymer film over inorganic (Si) substrate. However, a number of curves are produced in Fig. 7 corresponding to various combinations of waveguide materials, as listed in Table 2, with different elastic modulus, CTE and stress relaxation. Figure 7(a) shows the normalized birefringence for several combinations of materials to investigate the effects of elastic modulus of the core and cladding materials. On the other hand, Fig. 7(b) plots the results for some other combinations of materials to observe the effects of CTE and stress relaxation.

 

Fig. 7 Normalized photo-elastic birefringence in the strip waveguide for a wide range of material properties. a) Effects of elastic modulus of core and cladding materials, and b) effects of CTE and stress relaxation on the birefringence characteristics.

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3.3.1 Influence of elastic modulus

As observed in Fig. 7(a), the normalized birefringence curve is unique for Case 2 (R3-R3) and Case 7 (R5-R5), although their film stress (Table 2) and the channel birefringence (Fig. 5) are significantly different. Here, the elastic modulus of the core/clad materials for the two cases are different (2.9 and 3.8 GPa, respectively), while, the ratio of the elastic modulus of core and cladding layer (E_n = E_core / E_clad) for both case is unity. Similarly, the normalized birefringence is almost same for the other two cases, Case 3 (R4-R1) and Case 9 (E1-R3), consisting materials of closer modulus ratio (E_n = 1.38 and 1.28, respectively). However, by observing Case 2 (E_n = 1), Case 9 (E_n = 1.28) and Case 6 (E_n = 1.81), the normalized birefringence becomes lower (shift downward) for the increasing modulus ratio. Here, the ‘increasing modulus ratio’ means that the elastic modulus of the cladding material is relatively decreasing. Such low elastic bottom clad allows more stress to be relaxed in the constrained region of the core through the local strain within itself. This produces less birefringence in this case compared to the waveguide with high modulus clad, and ultimately makes the normalized birefringence downward. In contrast, the normalized birefringence curve moves upward for smaller modulus ratio, i.e., for higher elastic modulus of clad, as found in Case 1 (E_n = 0.75).

The polymer over the inorganic lower cladding, for example Si, produces the upper limit of the normalized birefringence, whereas, the lower limit is observed in our analyses for the lower clad with very low modulus (2.1 GPa) corresponding to material R1. Since the elastic modulus of polymer is very small than that of Si (E_polymer << E_Si), the difference of E_n corresponding to different polymers over Si is not significant to vary their normalized birefringence. Therefore, the normalized birefringence of polymer core over Si is unique as observed for Case 8 (R4-Si) and Case 10 (R5-Si) in Fig. 7(a). Thus, the normalized birefringence curves in Fig. 7(a) become different depending on the degree of mismatch (E_n) of the elastic modulus of the core and the lower cladding layer, irrespective of their absolute values (E_core and E_clad).

3.3.2 Influence of stress relaxation and CTE

In Fig. 7(b), considering Case 5 (R4-R3) and Case 11 (E2-R3), the material R4 in Case 5 has some finite stress relaxation than the fully elastic material E2 in Case 11, whereas their elastic moduli are same (2.9 GPa). However, due to the common modulus ratio (E_n = 0.97), their (Case 5 and 11) normalized birefringence curve is very similar (almost overlaps). It means that the viscoelastic relaxation of core material has negligible effect on the normalized birefringence characteristics. Similarly, very close results for Case 4 (R4-R2) and Case 5 (R4-R3), where the two lower clad materials (R2 and R3) have different stress relaxation, indicate that the normalized birefringence is less influenced by the stress relaxation of clad also.

Furthermore, Fig. 7(b) compares the normalized birefringence for two different values of CTE (57 and 40 ppm, respectively) considered for the core layer in Case 5. Very close curves for these two different conditions again implies the negligible influence of CTE on the normalized birefringence. This is due to the fact that, for a given elastic modulus, the viscoelastic relaxation or CTE defines the stress development before etching only, and these have no influence on the stress relaxation over the etching process. Thus, the channel birefringence depends linearly on the film birefringence producing same normalized result. Due to the same reason, the glass transition temperature (T_g), the deposition temperature and the cooling rate have no effects on the normalized birefringence. This is in contrast to the influence of elastic modulus, which defines the stress relaxation over the etching process in addition with the development of film stress, as explained by Fig. 7(a).

3.3.3 Influence of photo-elastic coefficients

As shown in Eq. (1) and (2), the photo-elastic coefficients C1 and C2 define the amount of birefringence as the product with a given value of stress. These coefficients are a function of the wavelength of light (i.e., dispersive) also. However, they have no contribution in the stress development and relaxation. Therefore, the influence of those constants or the wavelength dependence of birefringence is cancelled out, when the strip (after etching) birefringence is normalized by the film (before etching) birefringence.

3.3.4 Influence of waveguide dimensions

To generalize the geometric influence, Fig. 8 re-illustrates the result of Fig. 6 through normalizing the values as in Fig. 7. As can be observed, the normalized channel birefringence for a given aspect ratio (R = w/h) lies on the same line irrespective of the waveguide dimensions (w and h) and the amount of film birefringence ($\Delta n_{film}$). Thus, the photo-elastic birefringence remaining in a strip waveguide is now simply related to such birefringence in the planar film ($\Delta n_{film}$) and the aspect ratio R of the strip channel.

 

Fig. 8 Normalized photo-elastic birefringence for various aspect ratio (R = 0.75 to 1.5) of strip waveguides. The birefringence is normalized by the value of that remains in the film before etching, and the position is normalized by the height of the respective strip channel.

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In summary, the generalized birefringence characteristics can be described as:

Using the curve fitting technique, the obtained generalized characteristics can be expressed by the following empirical relation:

Thus, by knowing the stress-induced birefringence before etching ($\Delta n_{film}$), and the elastic modulus of the waveguide materials, the birefringence remaining in a polymer strip waveguide can be easily determined using Eqs. (3) and (4). As previously mentioned, the film birefringence can be measured with good accuracy using the prism coupler. The elastic modulus of the bulk polymers can also be characterized easily, for example, by using the nanoindentation technique. For the polymers, such values are sometimes available in the literature, and will be accepted in Eq. (4). In case the moduli are not available, it is shown below that the developed relation can still produce the channel birefringence with good accuracy for most of the material combinations. To ignore the elastic modulus, consider E_n = 1, and for the inorganic lower cladding material, the value of $k_0$ = 1.

Figure 9 compares the material birefringence in strip channel obtained through the FE method and the developed empirical relation in Eq. (3). Figure 9(a) shows the formulated birefringence with considering the elastic modulus in Eq. (3), whereas Fig. 9(b) shows the similar result but without considering the modulus (E_n = 1). By knowing the elastic modulus, the formulated results agree very well with the FE method results for all material combinations, as in Fig. 9(a). The results even satisfy for most of the material combinations when the elastic modulus is unknown. A little difference is found only for the lower clad materials with very low value of elastic modulus producing E_n ~1.8, as observed for Case 6 (E1-R1) in Fig. 9(b). Since the normalized birefringence is independent of or less influenced by the CTE, viscoelastic relaxation and stress-optic constants, as previously explained, the developed formula can produce the respective channel birefringence accurately for the polymers with different values of them. Thus the empirical relation in Eq. (3) is applicable to any isotropic polymers producing very low (Case 1 in Table 2) to high value (Case 9 in Table 2) of film stress/birefringence.

 

Fig. 9 Material birefringence (Δn) along the strip height obtained through the FE method (line with markers) and the empirical relation (smoothed line) in Eqs. (3) and (4) for various combinations of polymers. The empirical relation is deployed for two conditions: a) with considering the elastic modulus and b) without considering the elastic modulus.

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The formulated birefringence for three different waveguides (w = 2.25 µm, h = 3.0 µm; w = 4.5 µm, h = 4.5 µm and w = 3.0 µm, h = 2.0 µm) having different aspect ratio (R) of 0.75, 1.0 and 1.5, respectively, are shown in Fig. 10.To further show the generalized application of the developed formula, a different material combination of R4-R3 (Case 5 in Table 2) is considered in Fig. 10 than the material E2-R3 (Case 11) considered in Figs. 5 and 7. It shows that the results for various dimensions of waveguide agree very well with those obtained using the FE method. Thus, the obtained birefringence characteristics, which are caused by the photo-elastic effect in the strip channel, will remain valid for a broad range of strip dimensions, in particular, for the single mode design with any isotropic polymers.

 

Fig. 10 Material birefringence (Δn) along the strip height obtained through the FE method (line with markers) and the empirical relation (smoothed line) in Eqs. (3) and (4) for various dimensions of waveguide and corresponding to the materials R4-R3 (Case 5 in Table 2).

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In case of the waveguides with upper cladding layer, the remaining stress and thus the birefringence in the core layer will again be altered due to the stress exerted by the upper cladding layer [11]. For a given upper clad stress, therefore, the ultimate birefringence will depend on the birefringence remaining in the core before deposition of the upper clad. Thus, the results of this work will form the foundation for further investigation of channel birefringence consisting of upper clad layer. However, considering the limited scope of the paper, this work presents the birefringence characteristics for air clad strip waveguide only.

3.4 Validation

To justify the validity of the formulation in estimating the photo-elastic birefringence and thus to use it in the design of polarization insensitive devices, the material birefringence ($\Delta n$) is predicted for benzocyclobutene (BCB) strip waveguide. For the BCB waveguide of height 2.5 µm over the epoxy clad, the film birefringence ($\Delta n_{film}=n_{TE}-n_{TM}$) has been reported as 0.0036 [15]. This value of $\Delta n_{film}$ has been obtained by measuring the refractive indices for the two orthogonal polarizations (1.5755 for TE and 1.5719 for TM) using the prism coupler technique at 1550 nm wavelength. Corresponding to this measured $\Delta n_{film}$, the estimated channel birefringence by using Eq. (3) is shown in Fig. 11(a) for various widths of strip channel. Nonetheless, the measurement of stress-birefringence in a tiny channel of optical waveguides is very challenging, and sometimes impractical. Therefore, the estimated photo-elastic birefringence is considered in the modal analysis ($n_{eff}$) of the respective waveguide following the similar approach as described in [10], and then the results are compared with the previously published, as in [15], experimental results.

 

Fig. 11 a) Estimated photo-elastic birefringence (Δn) using the developed empirical relation for various widths of BCB strip channel, and b) comparison of modal birefringence simulated with (smoothed line) and without (dashed line) considering the photo-elastic birefringence of BCB and the previously published experimental results (points) for various dimensions of strip waveguide. The material birefringence is derived for the film birefringence (Δn_film) of 0.0036.

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Figure 11(b) plots the simulated (line) and the experimental (points) modal birefringence ($\Delta n_{eff}=n_{eff}^{TE}-n_{eff}^{TM}$) as a function of core width (w). The lower cladding index is considered as 1.5238 for TE and 1.5220 for TM, as reported in [15] for epoxy. As observed, the simulated results (smoothed line) considering the estimated photo-elastic birefringence agree very well with the experimental results, whereas the simulation results without considering the stress-induced effects (dashed line) are completely different. Such agreement implies that the photo-elastic birefringence for the BCB waveguide is well predicted. However, the conformity between the results for BCB waveguides in Fig. 11, together with the comprehensive analysis for various material combinations, indicates that the developed formula will be able to predict the photo-elastic birefringence in many other polymeric optical strip waveguides.

Polymer materials may have other factors (e.g., spinning, moisture absorption etc) causing optical anisotropy in the film. But most of them are usually not significant for reliable optical polymers and therefore not considered in our investigation. Here, only the photo-elastic effect is considered in the analyses. The birefringence characteristics will also be applicable to a material with negative birefringence (i.e., negative photo-elastic constant), like PMMA [23]. In such a case, the birefringence values will be multiplied by a negative sign only.

4. Conclusions

This work presents a comprehensive numerical exploration of material birefringence induced by thermal stress in a polymeric optical strip waveguide. The stress in the waveguide material is determined first through the finite element analysis of the thermo-mechanical problem for various waveguide materials and dimensions. By merging the stress distributions with the photo-elastic constants, the material birefringence is obtained in two process steps: before etching (planar film) and after etching (strip channel). Unlike the planar film, the birefringence is found to be non-uniform in the strip channel. The maximum birefringence in the core layer is experienced adjacent to the lower cladding decreasing to zero towards the top of the channel. The birefringence slightly enhances with the rise in the width of the strip. However, upon normalization by the film birefringence, the obtained channel birefringence is unique for a given ratio of elastic modulus of core and cladding layer. This normalized birefringence is less influenced by the viscoelastic relaxation, CTE and stress-optic coefficients of the constituent materials. Considering the geometry, the normalized birefringence is also common for a given aspect ration of the waveguide. Such stress-induced characteristics are presented and discussed by a generalized empirical relation, allowing for their applicability to other polymer materials. Using this empirical relation, the photo-elastic birefringence of a polymer strip waveguide can now be easily obtained, which can then be included in the optical design of waveguide devices for the better management of stress-induced problems.