Hollow Bragg waveguides fabricated by controlled buckling of Si/SiO<sub>2</sub> multilayers

doi:10.1364/OE.18.024917

Hollow Bragg waveguides fabricated by controlled buckling of Si/SiO₂ multilayers

E. Epp, N. Ponnampalam, W. Newman, B. Drobot, J. N. McMullin, A. F. Meldrum, and R. G. DeCorby »View Author Affiliations

Optics Express, Vol. 18, Issue 24, pp. 24917-24925 (2010)
http://dx.doi.org/10.1364/OE.18.024917

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– Abstract
1. Introduction
2. Design and fabrication of silicon-based hollow waveguides
3. Optical characterization
4. Discussion and conclusions
– References and links

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Abstract

We describe integrated air-core waveguides with Bragg reflector claddings, fabricated by controlled delamination and buckling of sputtered Si/SiO₂ multilayers. Thin film deposition parameters were tailored to produce a desired amount of compressive stress, and a patterned, embedded fluorocarbon layer was used to define regions of reduced adhesion. Self-assembled air channels formed either spontaneously or upon heating-induced decomposition of the patterned film. Preliminary optical experiments confirmed that light is confined to the air channels by a photonic band-gap guidance mechanism, with loss ~5 dB/cm in the 1550 nm wavelength region. The waveguides employ standard silicon processes and have potential applications in MEMS and lab-on-chip systems.

1. Introduction

Integrated air-core waveguides are attracting interest for lab-on-chip [1

H. Schmidt, J. P. Dongliang Yin, Barber, and A. R. Hawkins, “Hollow-core waveguides and 2-D waveguide arrays for integrated optics of gases and liquids,” IEEE J. Sel. Top. Quantum Electron. 11(2), 519–527 (2005). [CrossRef]

] and optical interconnect [2

Y. Zhou, V. Karagodsky, B. Pesala, F. G. Sedgwick, and C. J. Chang-Hasnain, “A novel ultra-low loss hollow-core waveguide using subwavelength high-contrast gratings,” Opt. Express 17(3), 1508–1517 (2009). [CrossRef] [PubMed]

T. C. Shen, Y. S. Kim, J. A. Bur, and S.-Y. Lin, “Optical characterization of bending efficiency in on-chip hollow-core Bragg waveguides at λ = 1.5 μm,” J. Lightwave Technol. 28(11), 1714–1719 (2010). [CrossRef]

] applications. Amongst other attributes, they have potential for widely tunable operation using MEMS-based actuation techniques [4

M. Kumar, T. Sakaguchi, and F. Koyama, “Wide tunability and ultralarge birefringence with 3D hollow waveguide Bragg reflector,” Opt. Lett. 34(8), 1252–1254 (2009). [CrossRef] [PubMed]

E. Epp, N. Ponnampalam, J. N. McMullin, and R. G. Decorby, “Thermal tuning of hollow waveguides fabricated by controlled thin-film buckling,” Opt. Express 17(20), 17369–17375 (2009). [CrossRef] [PubMed]

]. Strategies for the fabrication of hollow waveguides include sacrificial etching [1

], wafer bonding [6

S.-S. Lo, M.-S. Wang, and C.-C. Chen, “Semiconductor hollow optical waveguides formed by omni-directional reflectors,” Opt. Express 12(26), 6589–6593 (2004). [CrossRef] [PubMed]

], and chemical vapor deposition in a pre-defined trench [3

]. We recently reported an alternative approach [7

R. G. DeCorby, N. Ponnampalam, H. T. Nguyen, M. M. Pai, and T. J. Clement, “Guided self-assembly of integrated hollow Bragg waveguides,” Opt. Express 15(7), 3902–3915 (2007). [CrossRef] [PubMed]

N. Ponnampalam and R. G. Decorby, “Self-assembled hollow waveguides with hybrid metal-dielectric Bragg claddings,” Opt. Express 15(20), 12595–12604 (2007). [CrossRef] [PubMed]

] based on the controlled formation of straight-sided delamination buckles [9

M.-W. Moon, K.-R. Lee, K. H. Oh, and J. W. Hutchinson, “Buckle delamination on patterned substrates,” Acta Mater. 52(10), 3151–3159 (2004). [CrossRef]

] within a multilayer thin film stack. Those waveguides were fabricated using a combination of a chalcogenide glass and a commercial polymer. Here, we describe work on an analogous process, but using silicon-based materials and MEMS-compatible fabrication steps. Silicon-based processing should expand the scope for practical application of these self-assembled waveguides.

2. Design and fabrication of silicon-based hollow waveguides

In principle, the buckling self-assembly technique [7

N. Ponnampalam and R. G. Decorby, “Self-assembled hollow waveguides with hybrid metal-dielectric Bragg claddings,” Opt. Express 15(20), 12595–12604 (2007). [CrossRef] [PubMed]

] can be adapted to any Bragg reflector material system given: (i) means to control the compressive stress of the layers, and (ii) means to create patterned regions of reduced adhesion at a desired interface. If the refractive index contrast between the materials is sufficient to produce an omnidirectional band gap, enhanced functionality such as low-loss bends [3

] and wavelength-dependent, out-of-plane coupling from tapers [10

N. Ponnampalam and R. G. DeCorby, “Out-of-plane coupling at mode cutoff in tapered hollow waveguides with omnidirectional reflector claddings,” Opt. Express 16(5), 2894–2908 (2008). [CrossRef] [PubMed]

,11

R. G. DeCorby, N. Ponnampalam, E. Epp, T. Allen, and J. N. McMullin, “Chip-scale spectrometry based on tapered hollow Bragg waveguides,” Opt. Express 17(19), 16632–16645 (2009). [CrossRef] [PubMed]

] is possible.

2.1. Development of compressively stressed a-Si/SiO₂ Bragg mirrors

The index contrast between amorphous silicon (a-Si) and SiO₂ enables a large omnidirectional band gap [12

B. Gallas, S. Fisson, E. Charron, A. Brunet-Bruneau, G. Vuye, and J. Rivory, “Making an omnidirectional reflector,” Appl. Opt. 40(28), 5056–5063 (2001). [CrossRef]

], which has motivated their use for hollow waveguides operating in the 1550 nm wavelength region [3

S.-S. Lo, M.-S. Wang, and C.-C. Chen, “Semiconductor hollow optical waveguides formed by omni-directional reflectors,” Opt. Express 12(26), 6589–6593 (2004). [CrossRef] [PubMed]

]. The optical losses in a-Si are reasonably low for wavelengths above 800 nm (κ < 0.1, where κ is the extinction coefficient), and for hydrogenated a-Si (a-Si:H) they can be several orders of magnitude lower [13

H. Piller, “Silicon (amorphous) (a-Si),” in Handbook of Optical Constants of Solids , E.D. Palik, ed. (Academic Press, San Diego, 1998).

]. Moreover, a tradeoff between omnidirectional bandwidth and transparency is possible by deposition of SiO_x/SiO₂ multilayers with varying oxygen content, x [14

Q. Song, F. Huang, M. Li, B. Xie, H. Wang, Y. Jiang, and Y. Song, “Graded refractive-index SiO_x infrared filters prepared by reactive magnetron sputtering,” J. Vac. Sci. Technol. A 26(2), 265–269 (2008). [CrossRef]

]. In the literature, high quality a-Si/SiO₂ Bragg reflectors have been reported using e-beam evaporation [12

B. Gallas, S. Fisson, E. Charron, A. Brunet-Bruneau, G. Vuye, and J. Rivory, “Making an omnidirectional reflector,” Appl. Opt. 40(28), 5056–5063 (2001). [CrossRef]

] or sputtering [15

H.-Y. Lee, H. Makino, T. Yao, and A. Tanaka, “Si-based omnidirectional reflector and transmission filter optimized at a wavelength of 1.55 μm,” Appl. Phys. Lett. 81(24), 4502–4504 (2002). [CrossRef]

] of Si and SiO₂ targets, and reactive sputtering of Si targets [14

]. For the present work, the a-Si:H/SiO₂ Bragg reflectors reported by Yoda et al. [16

H. Yoda, K. Shiraishi, Y. Hiratani, and O. Hanaizumi, “a-Si:H/SiO₂ multilayer films fabricated by radio-frequency magnetron sputtering for optical filters,” Appl. Opt. 43(17), 3548–3554 (2004). [CrossRef] [PubMed]

] had particularly desirable properties including low loss (κ < 10⁻⁴), high index contrast, and high compressive stress for both the a-Si:H and SiO₂ layers. Those multilayers were deposited by reactive RF magnetron sputtering in the presence of O₂ and H₂. We employed a similar sputtering system.

It is well known that dense sputtered films tend to exhibit compressive stress, and that the stress can be varied within some range by varying the deposition parameters [17

H. Windischmann, “Intrinsic stress in sputter-deposited thin films,” Crit. Rev. Solid State Mater. Sci. 17(6), 547–596 (1992). [CrossRef]

]. Here, Bragg reflectors were deposited in a 3-source magnetron sputtering system, using silicon targets (n-type). Both a-Si and SiO₂ layers were deposited at a pressure of 3.5 mTorr, using a pulsed source (square wave at 150 kHz). For the a-Si layers, substrate bias was 535 V, source power was 200 W, and a Ti target (biased at 50 V and coupled to a 50 W DC source) was used as a getter to reduce oxygen contamination of the growing films. For SiO₂ deposition, substrate bias was 370 V and source power was again 200 W. Furthermore, the SiO₂ layers were reactively sputtered, with argon and oxygen flow rates of 50 and 2.5 sccm, respectively. The deposition rates for a-Si and SiO₂ were ~10 nm/min and ~17 nm/min, respectively. Bragg reflectors were deposited on cleaned Si substrates held at 150 °C, by sequential deposition of SiO₂ and Si layers and without breaking vacuum. The Bragg reflectors exhibit good optical properties and moderately high compressive stress, as shown in Fig. 1 .

Fig. 1 Results for a 4-period Bragg reflector are shown. For the theoretical curves, layer thicknesses of 102 nm and 260 nm were used for a-Si and SiO₂, respectively. (a) Reflectance at near normal incidence as measured with a spectrophotometer (blue symbols) and as modeled (green line). To illustrate the predicted omnidirectional band for TE polarized light, the modeled reflectance for TE polarized light at 88 degrees incidence angle is also shown (red dotted line). (b) Reflectance at 20 degrees incidence for TM polarization, as measured by a VASE instrument (blue symbols) and as modeled (green line). The discontinuity and discrepancy in the experimental data above ~1400 nm is due to optical loss in the VASE instrument. Modeled reflectance for TM polarized light at 88 degrees incidence angle is also shown (red dotted line). (c) Optical constants from Ref [13

H. Piller, “Silicon (amorphous) (a-Si),” in Handbook of Optical Constants of Solids , E.D. Palik, ed. (Academic Press, San Diego, 1998).

]. for a-Si, used in the modeling. Also shown is the index extracted from fitting to experimental data. (d) Net stress of the multilayer measured at a series of increasing and then decreasing temperatures, after several days of storage in air. The stress measured soon after deposition was −235 MPa.

Reflectance was measured using both a spectrophotometer and a variable angle spectroscopic ellipsometry (VASE) instrument; typical results are shown in Figs. 1(a) and 1(b). In addition to 4-period Bragg multilayers, single-layer SiO₂ and a-Si films were studied. SiO₂-like films with typical index ~1.47 in the near infrared were verified. For the a-Si films, optical constants were extracted from fitting experimental data. Both the refractive index [see Fig. 1(c)] and the extinction coefficient were in good agreement with the data for evaporated or sputtered a-Si layers reported in ref [13

H. Piller, “Silicon (amorphous) (a-Si),” in Handbook of Optical Constants of Solids , E.D. Palik, ed. (Academic Press, San Diego, 1998).

]. The theoretical curves shown in Figs. 1(a) and 1(b) were obtained using a standard transfer matrix model and the optical constants from Ref [13

H. Piller, “Silicon (amorphous) (a-Si),” in Handbook of Optical Constants of Solids , E.D. Palik, ed. (Academic Press, San Diego, 1998).

]. The fit to experimental data is good, although the width of the experimental stop band is slightly less than predicted. This is likely due to a lower than expected index contrast (by ~0.1) in the multilayers, consistent with the lower index of the a-Si films grown here (n ~3.41 at 1550 nm, versus n ~3.5 reported in Ref [13

H. Piller, “Silicon (amorphous) (a-Si),” in Handbook of Optical Constants of Solids , E.D. Palik, ed. (Academic Press, San Diego, 1998).

].). Inter-diffusion between Si and SiO₂ layers might also play a role in reducing the index contrast [12

B. Gallas, S. Fisson, E. Charron, A. Brunet-Bruneau, G. Vuye, and J. Rivory, “Making an omnidirectional reflector,” Appl. Opt. 40(28), 5056–5063 (2001). [CrossRef]

]. A more detailed study of the a-Si films (including the effects of hydrogenation) is ongoing.

Stress in both single films and multilayers was measured using a Flexus 2320 system, which has the capability of substrate heating. For multilayers, the effective-medium stress (~235 MPa compressive immediately after deposition) was well corroborated by measurements on single a-Si and SiO₂ layers, which revealed compressive stress ~420 MPa and ~160 MPa, respectively (note that the mirror is ~29% a-Si and ~71% SiO₂). Figure 1(d) shows the stress variation versus temperature for a multilayer heated at a rate ~15 °C/min up to 300 °C, and then allowed to cool back to room temperature at a rate ~5 °C/min. The stress was fairly constant for moderate temperature increases, consistent with the similar thermal expansion coefficients expected for the a-Si and SiO₂ layers and the Si substrate. However, heating above ~200 °C resulted in partial relaxation of the compressive stress. This implies that the morphology (peak height, etc.) of delamination buckles might depend to some extent on the temperature at which they form.

2.2. Development of patterned layer with low adhesion and decomposability

The buckling self-assembly process relies on a patterned low-adhesion layer (LAL) to define the regions of delamination. The LAL should satisfy some practical requirements, as follows: First, if the LAL is to remain on the inner surface of the hollow waveguide (i.e. as a surface layer on the upper or lower cladding mirror), it should be thin, have low roughness, and be optically transparent in the wavelength range of interest. Second, the LAL must be amenable to photolithographic patterning, either by etching or liftoff. Third, the patterned LAL must survive deposition of the upper Bragg mirror, subject to the deposition temperature that optimizes the stress, mechanical properties, and optical properties of the mirror.

Control over the adhesion energy of Si-based films is a topic of great importance to the microelectronics and MEMs communities [18

B. Bhushan, “Adhesion and stiction: mechanisms, measurement techniques, and methods for reduction,” J. Vac. Sci. Technol. B 21(6), 2262–2296 (2003). [CrossRef]

]. For MEMs devices in particular, low adhesion surfaces have been sought in the interest of reducing the friction and stiction of moving parts [19

Y. X. Zhuang and A. Menon, “On the stiction of MEMS materials,” Tribol. Lett. 19(2), 111–117 (2005). [CrossRef]

]. Various methods for reducing the adhesion of Si and SiO₂ surfaces have been developed, mostly by addition of an ultrathin film or by chemical surface modification. Popular coatings include hard nitride- or carbon-based films (TiN, DLC, TiC, SiC, etc.), fluorocarbon films (such as PTFE-like films), and organic self-assembled monolayers (SAMs) [18

B. Bhushan, “Adhesion and stiction: mechanisms, measurement techniques, and methods for reduction,” J. Vac. Sci. Technol. B 21(6), 2262–2296 (2003). [CrossRef]

,19

Y. X. Zhuang and A. Menon, “On the stiction of MEMS materials,” Tribol. Lett. 19(2), 111–117 (2005). [CrossRef]

]. Of particular interest here, vapor-phase deposited fluorocarbon films have been widely studied [20

H. V. Jansen, J. G. E. Gardeniers, J. Elders, H. A. C. Tilmans, and M. Elwenspoek, “Applications of fluorocarbon polymers in micromechanics and micromachining,” Sens. Actuators A Phys. 41(1-3), 136–140 (1994). [CrossRef]

,21

B. K. Smith, J. J. Sniegowski, G. LaVigne, and C. Brown, “Thin Teflon-like films for eliminating adhesion in released polysilicon microstructures,” Sens. Actuators A Phys. 70(1-2), 159–163 (1998). [CrossRef]

] and shown to have excellent uniformity, durability, and biocompatibility. Furthermore, C-F based films tend to have good transparency in the near infrared.

Here, we employed thin fluorocarbon layers deposited using the passivation process in an inductively coupled plasma reactive ion etch (ICP-RIE) system (Surface Technologies Systems) [22

A. A. Ayon, D.-Z. Chen, R. Khanna, R. Braff, H. H. Sawin, and M. A. Schmidt, “A novel integrated MEMS process using fluorocarbon films deposited with a deep reactive ion etching (DRIE) tool,” Mater. Res. Soc. Symp. Proc. 605, 141–147 (2000). [CrossRef]

]. Under suitable deposition conditions, these films exhibit low surface energy (water contact angle > 110 ° [23

J. Han, J. Yeom, G. Mensing, D. Joe, R. I. Masel, and M. A. Shannon, “Surface energy approach and AFM verification of the (CF)_n treated surface effect and its correlation with adhesion reduction in microvalves,” J. Micromech. Microeng. 19(8), 085017 (2009). [CrossRef]

]), low coefficient of friction, low pinhole density [22

], and good thermal stability [24

Y. X. Zhuang and A. Menon, “Wettability and themal stability of fluorocarbon films deposited by deep reactive ion etching,” J. Vac. Sci. Technol. A 23(3), 434–439 (2005). [CrossRef]

]. Films were deposited at a base pressure of 1 mTorr, RF power of 300 W, and using C₄F₈ as a source gas with flow rate of 60 sccm. The hydrophobic nature of the films was confirmed by observation of water contact angle. Films with thicknesses ~10 to 40 nm showed little variation in hydrophobic properties or amenability to patterning by liftoff.

Figure 2(a) shows typical results for a liftoff patterning process, which was tested on polished silicon wafers and on sputtered or e-beam evaporated a-Si layers. In all cases, the fluorocarbon layers showed excellent adhesion to the underlying base layer and no visible damage from the photoresist removal process. Thermal stability of the patterned fluorocarbon layer was tested by heating samples on a hot plate in air. For the results shown in Fig. 2(b), samples were heated to a series of increasing temperatures. For each end-point temperature, the total heating time (including ramp up and ramp down to room temperature) was ~1 hour. Remaining film thickness was measured at room temperature after each step, using a contact profilometer (Alphastep). Consistent with the detailed study reported in Ref [24

Y. X. Zhuang and A. Menon, “Wettability and themal stability of fluorocarbon films deposited by deep reactive ion etching,” J. Vac. Sci. Technol. A 23(3), 434–439 (2005). [CrossRef]

], rapid decomposition of the layers was observed for temperatures exceeding ~300 °C. It is important to note that these experiments were conducted for uncovered fluorocarbon films; in the case of an embedded fluorocarbon strip discussed below, the decomposition behavior is expected to depend on the permeability of the covering layers and other factors [25

P. J. Joseph, H. A. Kelleher, S. A. B. Allen, and P. A. Kohl, “Improved fabrication of micro air-channels by incorporation of a structural barrier,” J. Micromech. Microeng. 15(1), 35–42 (2005). [CrossRef]

Fig. 2 (a) A microscope image of a patterned fluorocarbon layer (~15 nm thick) is shown. Five strips, each 80 μm wide, are faintly visible. Alignment mark features (crosses and squares) are also visible, near the top and bottom of the image. (b) Remaining thickness of fluorocarbon layers with two different starting thicknesses is plotted versus annealing temperature (see main text for details).

2.3. Buckling self-assembly process

In our process, controlled thin film buckling is exploited for the self-assembly of a three dimensional air core waveguide using otherwise planar (two-dimensional) processing steps. The process starts with a piranha cleaned silicon wafer, and follows the sequence of steps shown in Fig. 3 .

Fig. 3 Schematic showing the process steps used to fabricate hollow waveguides: (a) a 4-period Bragg reflector was deposited, (b) a fluorocarbon LAL layer was deposited and patterned by liftoff, (c) a second 4-period Bragg mirror (with net compressive stress) was deposited, and (d) the sample was heated to promote loss of adhesion in the regions of the LAL.

First, a four-period Bragg reflector was deposited as the bottom cladding of the hollow waveguides. After photoresist patterning, a fluorocarbon LAL (typically 10 to 30 nm thick) was deposited as described in Section 2.2. The LAL was patterned by liftoff, thereby defining regions for subsequent delamination of the upper Bragg mirror. Next, the same sputtering system was used to deposit another four period Bragg reflector, which eventually acts as the upper cladding of the hollow waveguides.

Over large areas (2 cm x 1 cm) of patterned LAL, some spontaneous buckling of the upper mirror was observed immediately upon removal from the sputtering chamber (or within a few days). To induce buckling over the small LAL features, wafers were heated on a hotplate in an air or nitrogen atmosphere to ~200 °C. Heating in this range does not significantly affect the compressive stress of the films, as discussed in Section 2.1. However, we speculate that the heating results in a partial decomposition of the fluorocarbon layer [24

Y. X. Zhuang and A. Menon, “Wettability and themal stability of fluorocarbon films deposited by deep reactive ion etching,” J. Vac. Sci. Technol. A 23(3), 434–439 (2005). [CrossRef]

], further reducing the adhesion between the upper and lower mirror. It is also possible that the out-gassing of volatile C-F compounds exerts force on the upper mirror, providing additional driving energy for buckle formation. The precise mechanisms involved are the subject of ongoing study; in any case, a high yield of straight-sided delamination buckles was realized. Microscope images of typical waveguides with 80 μm base width are shown in Figs. 4(a) and 4(b). An SEM image of the cleaved facet of a guide with 40 μm base width is shown in Fig. 4(c). The peak height of the delamination buckles (and thus the air cores) is dependent on the buckle base width [7

N. Ponnampalam and R. G. Decorby, “Self-assembled hollow waveguides with hybrid metal-dielectric Bragg claddings,” Opt. Express 15(20), 12595–12604 (2007). [CrossRef] [PubMed]

] and in the present case varied between ~1 and ~4 μm. Yield varied between samples, but was as high as ~80% in some cases. Small defects were observed in most guides, as highlighted in Fig. 4(a), possibly associated with points where the Euler buckle joined after nucleating and propagating from opposite sides of the defect.

Fig. 4 Images of buckled waveguides are shown. (a) Microscope image of an array of five waveguides with 80 μm base width. The red circle indicates a typical defect in one of the guides. (b) Higher magnification image of two of the guides from part a. (c) SEM image of the cleaved facet of a waveguide with 40 μm base width.

The process might be viewed as a hybrid between two previously reported approaches for fabricating enclosed microchannels: the compressive stress of the upper mirror drives the formation of Euler buckles [7

], and the fluorocarbon acts in part as a sacrificial decomposition layer [25

]. However, the fluorocarbon layer is extremely thin in the present case, and we postulate that the morphology of the air-channel is determined primarily by the stress-driven buckling of the upper mirror. To assess this, we used elastic buckling theory [8

N. Ponnampalam and R. G. Decorby, “Self-assembled hollow waveguides with hybrid metal-dielectric Bragg claddings,” Opt. Express 15(20), 12595–12604 (2007). [CrossRef] [PubMed]

] and approximated the upper mirror as an effective-medium equivalent layer. For a-Si thin films, the Young’s modulus has been reported to be ~100 GPa [26

M. M. de Lima, R. G. Lacerda, J. Vilcarromero, and F. C. Marques, “Coefficient of thermal expansion and elastic modulus of thin films,” J. Appl. Phys. 86(9), 4936–4942 (1999). [CrossRef]

]. SiO₂ films have Young’s modulus in the range ~40 to 70 GPa [27

W. D. Nix, “Mechanical properties of thin films,” Metall. Mater. Trans., A Phys. Metall. Mater. Sci. 20(11), 2217–2245 (1989). [CrossRef]

], with lower values typical for films grown at low temperature (as is the case here). Moreover, the modulus of a thin film is known to depend greatly on deposition parameters (including growth temperature), and can be lower than the corresponding bulk modulus by as much as 80% [26

M. M. de Lima, R. G. Lacerda, J. Vilcarromero, and F. C. Marques, “Coefficient of thermal expansion and elastic modulus of thin films,” J. Appl. Phys. 86(9), 4936–4942 (1999). [CrossRef]

]. Using 70 GPa for SiO₂ results in an effective medium modulus Y ~80 GPa (as mentioned, the mirror is ~29% a-Si and ~71% SiO₂), which can be considered an upper bound for the multilayers described.

Figure 5(a) shows an optical profilometer scan on a typical buckled waveguide. Line scans along the axis of the buckles produced estimates for RMS roughness as low as ~0.5 nm. Figure 5(b) shows the peak buckle height versus base width as predicted by the elastic buckling theory using Y = 80 GPa or Y = 40 GPa, and an effective medium compressive stress of 235 MPa (see Sec. 2.1). Also shown are average measured buckle heights, revealing good agreement with the theoretical predictions obtained using the lower effective modulus. The experimental average was based on ~10 waveguides of each width on a particular sample, and the variation in peak height was ~0.1 μm in each case. Also consistent with the predictions of the elastic buckling theory, 10 and 20 μm wide LAL patterns on the same wafer did not buckle. Independent verification of the thin film elastic properties by other measurements would be desirable, but is left for future work.

Fig. 5 (a) Topographical scan is shown for a waveguide with 80 μm base width and ~3.6 μm peak height, as obtained using an optical profilometer. (b) Predicted buckle height according to elastic theory given net compressive stress of 235 MPa and effective medium Young’s modulus of Y = 40 GPa (solid line) or Y = 80 GPa (dashed line). The symbols show the mean values measured experimentally.

3. Optical characterization

Samples were cleaved to facilitate light guiding experiments, and light from either a tunable laser (Santec) or a supercontinuum source (Koheras SuperK Red) was coupled via fiber-based polarization control optics and a tapered fiber (Oz Optics) with a nominal focal spot size of 6.5 μm. An objective lens was used to collect light from the output facet, for delivery to a photodetector, an optical spectrum analyzer (Anritsu), or an infrared camera. In some cases, the infrared camera was mounted on a microscope in order to capture images of light scattered from the top surface of the waveguides. Further details on the experimental setup can be found in Refs [7

N. Ponnampalam and R. G. Decorby, “Self-assembled hollow waveguides with hybrid metal-dielectric Bragg claddings,” Opt. Express 15(20), 12595–12604 (2007). [CrossRef] [PubMed]

,10

,11

]. Waveguides with both 60 and 80 μm base width were studied, but the results presented below pertain to 60 μm guides with ~2.5 μm peak core height. The small peak core height for the 40 μm guides implies propagation near cutoff [10

], and in that case very high propagation loss was verified experimentally.

Owing to the low reflectance of the cladding mirrors for TM polarized light at high incidence angles (see Fig. 1), all of the waveguides exhibit high loss for TM polarized light. On the other hand, due to their low height-to-width aspect ratio, the waveguides support multiple TE (in-plane) polarized modes [7

,10

]. Figure 6(a) shows the four lowest-order TE modes at a wavelength of 1560 nm, selectively excited by adjusting the position of the tapered fiber relative to the input facet of a waveguide.

Fig. 6 TE light guiding results are shown for a waveguide with 60 μm base width and peak height ~2.5 μm. (a) Waveguide end facet images captured by an infrared camera via a 60x objective lens, for varying input coupling conditions of a 1560 nm laser source. (b) Scattered light image captured by an infrared camera for a waveguide ~6 mm in length, with supercontinuum light coupled at left. The bright spot at right is the output facet. (c) Relative power captured by the infrared camera versus distance along the waveguide axis, for an input wavelength of 1560 nm. The peaks near 3.4 mm and the bright scattering point in the image of part (b) correspond to the same location on the waveguide. The red line is a linear fit to the data.

Figure 6(b) shows a top view of a waveguide, ~0.6 cm in length, excited by the supercontinuum source. The input and output facets are visible as the bright scattering points at the left and right, respectively. Similar images captured using the tunable laser source produced only faintly-visible light streaks, apparently due to low radiation and scattering of light from the top surface of the waveguides. Nevertheless, images were sufficiently bright to enable loss estimation from the decay of the scattered light streak. Figure 6(c) shows the result of one such measurement, for a waveguide with base width of 60 μm and peak height ~2.5 μm, and with the input coupling conditions adjusted for preferential excitation of the fundamental TE mode. The estimated loss of 5.1 dB/cm is slightly higher than the loss reported for a-Si/SiO₂-based hollow waveguides in Ref [6

S.-S. Lo, M.-S. Wang, and C.-C. Chen, “Semiconductor hollow optical waveguides formed by omni-directional reflectors,” Opt. Express 12(26), 6589–6593 (2004). [CrossRef] [PubMed]

], although those devices employed 6-period cladding mirrors. Overall insertion loss (determined by first measuring the throughput of the experimental system without the sample in place) was as low as ~10 to 12 dB, with a significant portion attributable to input coupling loss between the tapered fiber and the hollow waveguide.

A key signature of a photonic band bap guidance mechanism is the wavelength dependence of the optical transmission. To predict the low-loss guidance band of the waveguides, we used a transfer-matrix-based slab waveguide model described in detail elsewhere [10

,11

,28

S. Dasgupta, A. Ghatak, and B. P. Pal, “Analysis of Bragg reflection waveguides with finite cladding: an accurate matrix method formulation,” Opt. Commun. 279(1), 83–88 (2007). [CrossRef]

]. Propagation loss versus wavelength was calculated for the fundamental TE mode of a symmetric, air-core slab waveguide, clad by mirrors with the parameters specified in Fig. 1. The slab model is reasonably valid for the buckled waveguides, due to their small height-to-width aspect ratio and tapered lateral profile [7

]. Moreover, the low-loss modes all exhibit a single lobe in the vertical direction [see Fig. 6(a)] and are well-approximated by the fundamental mode from the slab model.

In the model, the a-Si layers were assigned the refractive index data shown in Fig. 1(c), and were treated either as lossless or as having the extinction coefficient data taken from Ref [13

H. Piller, “Silicon (amorphous) (a-Si),” in Handbook of Optical Constants of Solids , E.D. Palik, ed. (Academic Press, San Diego, 1998).

]. and shown in Fig. 1(c). The results of the simulations are shown in Fig. 7(a) . For lossless layers, a transmission band in the 1000-1600 nm range with loss ~1 dB/cm is predicted. With the silicon loss included, minimum propagation loss ~7 dB/cm is predicted and the transmission band is narrowed. Given the experimental loss estimate from Fig. 6(c), it is possible that the extinction coefficient of our a-Si layers is somewhat lower than that reported in Ref [13

H. Piller, “Silicon (amorphous) (a-Si),” in Handbook of Optical Constants of Solids , E.D. Palik, ed. (Academic Press, San Diego, 1998).

]. This might indicate the presence of oxygen in the layers, consistent with their lower refractive index discussed in Section 2. As mentioned, a more detailed study of the optical constants of these films is the subject of ongoing work.

Fig. 7 (a) Predicted propagation loss for the fundamental TE mode of a slab Bragg waveguide with 4-period mirrors and air-core height 2.5 μm, assuming lossless Si layers (green dotted curve) or with Si loss included (blue solid curve). (b) Experimentally measured transmission spectrum for two waveguide samples, each with peak core height ~2.5 μm. The apparent transmission below ~900 nm is an artifact arising from aliasing in the diffraction-grating based OSA.

Experimentally, the wavelength-dependent transmission was measured using the supercontinuum source and the OSA. Figure 7(b) shows raw transmission spectra for two waveguides with peak core height ~2.5 μm, but from two different samples with slightly different mirror parameters. A broad transmission band between ~1100 nm ~1700 nm was observed, in reasonable agreement with the theoretical prediction. As discussed elsewhere [8

N. Ponnampalam and R. G. Decorby, “Self-assembled hollow waveguides with hybrid metal-dielectric Bragg claddings,” Opt. Express 15(20), 12595–12604 (2007). [CrossRef] [PubMed]

], waveguide measurements with a supercontinuum source are complicated by spectral variation in coupling efficiency, especially if the waveguide supports multiple low-loss modes. Most of the random variations and dips in the transmission band can be attributed to these factors. However, each waveguide tested exhibited a deep and relatively sharp dip in transmission near ~1350 nm. The exact spectral location of this dip varied between samples (with slightly different cladding mirrors) and between waveguides (with different air-core size) on a given sample. This implies that the dip is due to a geometrically-dependent resonant effect, as opposed to a material absorption resonance. We postulate that it arises from the interaction of in-plane polarized light with the ‘sidewalls’ of the upper, buckled cladding mirror. As shown in Fig. 1(b), the mirrors exhibit a notch in reflectance at a similar wavelength, for TM-polarized light at high incidence angles. This effect has been discussed previously [7

], and can be mitigated by tailoring the thickness of the first a-Si layer in the upper mirror [1

M. Kumar, T. Sakaguchi, and F. Koyama, “Wide tunability and ultralarge birefringence with 3D hollow waveguide Bragg reflector,” Opt. Lett. 34(8), 1252–1254 (2009). [CrossRef] [PubMed]

] or by metal termination of the upper mirror [8

N. Ponnampalam and R. G. Decorby, “Self-assembled hollow waveguides with hybrid metal-dielectric Bragg claddings,” Opt. Express 15(20), 12595–12604 (2007). [CrossRef] [PubMed]

]. We hope to explore such improvements in future work.

Finally, it is worth mentioning that polarization control of the input light had little effect on the shape of the transmission spectrum. This is due to the extremely high polarization dependent loss of the waveguides. As mentioned above, and predicted by slab waveguide simulations for TM-polarized light (not shown), the TM-polarized modes exhibit high loss. Thus, only TE-polarized modes carry significant power at the output facet. Adjusting the input polarization simply causes a variation in the fraction of power launched into the TE-polarized modes, so that only the relative power (not the shape) of the transmission spectrum is affected. To support this conclusion, we used polarization optics to verify that only TE-polarized light is present at the output facet.

4. Discussion and conclusions

A MEMS-compatible process for fabricating silicon-based, air-core waveguides was described. To date, the yield of the process (in terms of percentage of LAL strip areas that buckle without defects) has typically been in the 50 to 80% range. We believe that the main factor limiting the yield is the uniformity of the fluorocarbon LAL. Efforts to improve this uniformity, by varying the deposition and annealing conditions, are under study. Nevertheless, arrays of low-defect waveguides with length ~1 cm have already been realized. The waveguides exhibited loss as low as ~5 dB/cm (in the 1550 nm wavelength region), partly attributable to absorption by a-Si layers in the cladding mirrors. The loss is comparable to that of other air-core integrated waveguides reported in the literature, and might be reduced further by the use of hydrogenated a-Si.

We believe the waveguides described are a promising alternative to conventional air-core waveguides fabricated by wafer bonding or sacrificial etching techniques. The self-assembled nature of the waveguides results in smooth sidewall interfaces, which is beneficial for guiding light and might also benefit the efficient flow of fluids through the air channel. Moreover, complex geometries including bends and tapers [7

] can be fabricated in parallel. Because of these attributes, and given their silicon-based processing, the waveguides might find application in optofluidic and lab-on-chip analysis systems [1

Acknowledgements

The work was supported by the Natural Sciences and Engineering Research Council of Canada and by TRLabs. Devices were fabricated at the Nanofab of the University of Alberta.

References and links

1.	H. Schmidt, J. P. Dongliang Yin, Barber, and A. R. Hawkins, “Hollow-core waveguides and 2-D waveguide arrays for integrated optics of gases and liquids,” IEEE J. Sel. Top. Quantum Electron. 11(2), 519–527 (2005). [CrossRef]
2.	Y. Zhou, V. Karagodsky, B. Pesala, F. G. Sedgwick, and C. J. Chang-Hasnain, “A novel ultra-low loss hollow-core waveguide using subwavelength high-contrast gratings,” Opt. Express 17(3), 1508–1517 (2009). [CrossRef] [PubMed]
3.	T. C. Shen, Y. S. Kim, J. A. Bur, and S.-Y. Lin, “Optical characterization of bending efficiency in on-chip hollow-core Bragg waveguides at λ = 1.5 μm,” J. Lightwave Technol. 28(11), 1714–1719 (2010). [CrossRef]
4.	M. Kumar, T. Sakaguchi, and F. Koyama, “Wide tunability and ultralarge birefringence with 3D hollow waveguide Bragg reflector,” Opt. Lett. 34(8), 1252–1254 (2009). [CrossRef] [PubMed]
5.	E. Epp, N. Ponnampalam, J. N. McMullin, and R. G. Decorby, “Thermal tuning of hollow waveguides fabricated by controlled thin-film buckling,” Opt. Express 17(20), 17369–17375 (2009). [CrossRef] [PubMed]
6.	S.-S. Lo, M.-S. Wang, and C.-C. Chen, “Semiconductor hollow optical waveguides formed by omni-directional reflectors,” Opt. Express 12(26), 6589–6593 (2004). [CrossRef] [PubMed]
7.	R. G. DeCorby, N. Ponnampalam, H. T. Nguyen, M. M. Pai, and T. J. Clement, “Guided self-assembly of integrated hollow Bragg waveguides,” Opt. Express 15(7), 3902–3915 (2007). [CrossRef] [PubMed]
8.	N. Ponnampalam and R. G. Decorby, “Self-assembled hollow waveguides with hybrid metal-dielectric Bragg claddings,” Opt. Express 15(20), 12595–12604 (2007). [CrossRef] [PubMed]
9.	M.-W. Moon, K.-R. Lee, K. H. Oh, and J. W. Hutchinson, “Buckle delamination on patterned substrates,” Acta Mater. 52(10), 3151–3159 (2004). [CrossRef]
10.	N. Ponnampalam and R. G. DeCorby, “Out-of-plane coupling at mode cutoff in tapered hollow waveguides with omnidirectional reflector claddings,” Opt. Express 16(5), 2894–2908 (2008). [CrossRef] [PubMed]
11.	R. G. DeCorby, N. Ponnampalam, E. Epp, T. Allen, and J. N. McMullin, “Chip-scale spectrometry based on tapered hollow Bragg waveguides,” Opt. Express 17(19), 16632–16645 (2009). [CrossRef] [PubMed]
12.	B. Gallas, S. Fisson, E. Charron, A. Brunet-Bruneau, G. Vuye, and J. Rivory, “Making an omnidirectional reflector,” Appl. Opt. 40(28), 5056–5063 (2001). [CrossRef]
13.	H. Piller, “Silicon (amorphous) (a-Si),” in Handbook of Optical Constants of Solids , E.D. Palik, ed. (Academic Press, San Diego, 1998).
14.	Q. Song, F. Huang, M. Li, B. Xie, H. Wang, Y. Jiang, and Y. Song, “Graded refractive-index SiO_x infrared filters prepared by reactive magnetron sputtering,” J. Vac. Sci. Technol. A 26(2), 265–269 (2008). [CrossRef]
15.	H.-Y. Lee, H. Makino, T. Yao, and A. Tanaka, “Si-based omnidirectional reflector and transmission filter optimized at a wavelength of 1.55 μm,” Appl. Phys. Lett. 81(24), 4502–4504 (2002). [CrossRef]
16.	H. Yoda, K. Shiraishi, Y. Hiratani, and O. Hanaizumi, “a-Si:H/SiO₂ multilayer films fabricated by radio-frequency magnetron sputtering for optical filters,” Appl. Opt. 43(17), 3548–3554 (2004). [CrossRef] [PubMed]
17.	H. Windischmann, “Intrinsic stress in sputter-deposited thin films,” Crit. Rev. Solid State Mater. Sci. 17(6), 547–596 (1992). [CrossRef]
18.	B. Bhushan, “Adhesion and stiction: mechanisms, measurement techniques, and methods for reduction,” J. Vac. Sci. Technol. B 21(6), 2262–2296 (2003). [CrossRef]
19.	Y. X. Zhuang and A. Menon, “On the stiction of MEMS materials,” Tribol. Lett. 19(2), 111–117 (2005). [CrossRef]
20.	H. V. Jansen, J. G. E. Gardeniers, J. Elders, H. A. C. Tilmans, and M. Elwenspoek, “Applications of fluorocarbon polymers in micromechanics and micromachining,” Sens. Actuators A Phys. 41(1-3), 136–140 (1994). [CrossRef]
21.	B. K. Smith, J. J. Sniegowski, G. LaVigne, and C. Brown, “Thin Teflon-like films for eliminating adhesion in released polysilicon microstructures,” Sens. Actuators A Phys. 70(1-2), 159–163 (1998). [CrossRef]
22.	A. A. Ayon, D.-Z. Chen, R. Khanna, R. Braff, H. H. Sawin, and M. A. Schmidt, “A novel integrated MEMS process using fluorocarbon films deposited with a deep reactive ion etching (DRIE) tool,” Mater. Res. Soc. Symp. Proc. 605, 141–147 (2000). [CrossRef]
23.	J. Han, J. Yeom, G. Mensing, D. Joe, R. I. Masel, and M. A. Shannon, “Surface energy approach and AFM verification of the (CF)_n treated surface effect and its correlation with adhesion reduction in microvalves,” J. Micromech. Microeng. 19(8), 085017 (2009). [CrossRef]
24.	Y. X. Zhuang and A. Menon, “Wettability and themal stability of fluorocarbon films deposited by deep reactive ion etching,” J. Vac. Sci. Technol. A 23(3), 434–439 (2005). [CrossRef]
25.	P. J. Joseph, H. A. Kelleher, S. A. B. Allen, and P. A. Kohl, “Improved fabrication of micro air-channels by incorporation of a structural barrier,” J. Micromech. Microeng. 15(1), 35–42 (2005). [CrossRef]
26.	M. M. de Lima, R. G. Lacerda, J. Vilcarromero, and F. C. Marques, “Coefficient of thermal expansion and elastic modulus of thin films,” J. Appl. Phys. 86(9), 4936–4942 (1999). [CrossRef]
27.	W. D. Nix, “Mechanical properties of thin films,” Metall. Mater. Trans., A Phys. Metall. Mater. Sci. 20(11), 2217–2245 (1989). [CrossRef]
28.	S. Dasgupta, A. Ghatak, and B. P. Pal, “Analysis of Bragg reflection waveguides with finite cladding: an accurate matrix method formulation,” Opt. Commun. 279(1), 83–88 (2007). [CrossRef]

OCIS Codes
(130.3120) Integrated optics : Integrated optics devices
(130.5296) Integrated optics : Photonic crystal waveguides

ToC Category:
Integrated Optics

History
Original Manuscript: August 5, 2010
Revised Manuscript: October 29, 2010
Manuscript Accepted: October 31, 2010
Published: November 15, 2010

Citation
E. Epp, N. Ponnampalam, W. Newman, B. Drobot, J. N. McMullin, A. F. Meldrum, and R. G. DeCorby, "Hollow Bragg waveguides fabricated by controlled buckling of Si/SiO₂ multilayers," Opt. Express 18, 24917-24925 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-24-24917

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References

H. Schmidt, J. P. Dongliang Yin, Barber, and A. R. Hawkins, “Hollow-core waveguides and 2-D waveguide arrays for integrated optics of gases and liquids,” IEEE J. Sel. Top. Quantum Electron. 11(2), 519–527 (2005). [CrossRef]
Y. Zhou, V. Karagodsky, B. Pesala, F. G. Sedgwick, and C. J. Chang-Hasnain, “A novel ultra-low loss hollow-core waveguide using subwavelength high-contrast gratings,” Opt. Express 17(3), 1508–1517 (2009). [CrossRef] [PubMed]
T. C. Shen, Y. S. Kim, J. A. Bur, and S.-Y. Lin, “Optical characterization of bending efficiency in on-chip hollow-core Bragg waveguides at λ = 1.5 μm,” J. Lightwave Technol. 28(11), 1714–1719 (2010). [CrossRef]
M. Kumar, T. Sakaguchi, and F. Koyama, “Wide tunability and ultralarge birefringence with 3D hollow waveguide Bragg reflector,” Opt. Lett. 34(8), 1252–1254 (2009). [CrossRef] [PubMed]
E. Epp, N. Ponnampalam, J. N. McMullin, and R. G. Decorby, “Thermal tuning of hollow waveguides fabricated by controlled thin-film buckling,” Opt. Express 17(20), 17369–17375 (2009). [CrossRef] [PubMed]
S.-S. Lo, M.-S. Wang, and C.-C. Chen, “Semiconductor hollow optical waveguides formed by omni-directional reflectors,” Opt. Express 12(26), 6589–6593 (2004). [CrossRef] [PubMed]
R. G. DeCorby, N. Ponnampalam, H. T. Nguyen, M. M. Pai, and T. J. Clement, “Guided self-assembly of integrated hollow Bragg waveguides,” Opt. Express 15(7), 3902–3915 (2007). [CrossRef] [PubMed]
N. Ponnampalam and R. G. Decorby, “Self-assembled hollow waveguides with hybrid metal-dielectric Bragg claddings,” Opt. Express 15(20), 12595–12604 (2007). [CrossRef] [PubMed]
M.-W. Moon, K.-R. Lee, K. H. Oh, and J. W. Hutchinson, “Buckle delamination on patterned substrates,” Acta Mater. 52(10), 3151–3159 (2004). [CrossRef]
N. Ponnampalam and R. G. DeCorby, “Out-of-plane coupling at mode cutoff in tapered hollow waveguides with omnidirectional reflector claddings,” Opt. Express 16(5), 2894–2908 (2008). [CrossRef] [PubMed]
R. G. DeCorby, N. Ponnampalam, E. Epp, T. Allen, and J. N. McMullin, “Chip-scale spectrometry based on tapered hollow Bragg waveguides,” Opt. Express 17(19), 16632–16645 (2009). [CrossRef] [PubMed]
B. Gallas, S. Fisson, E. Charron, A. Brunet-Bruneau, G. Vuye, and J. Rivory, “Making an omnidirectional reflector,” Appl. Opt. 40(28), 5056–5063 (2001). [CrossRef]
H. Piller, “Silicon (amorphous) (a-Si),” in Handbook of Optical Constants of Solids, E.D. Palik, ed. (Academic Press, San Diego, 1998).
Q. Song, F. Huang, M. Li, B. Xie, H. Wang, Y. Jiang, and Y. Song, “Graded refractive-index SiOx infrared filters prepared by reactive magnetron sputtering,” J. Vac. Sci. Technol. A 26(2), 265–269 (2008). [CrossRef]
H.-Y. Lee, H. Makino, T. Yao, and A. Tanaka, “Si-based omnidirectional reflector and transmission filter optimized at a wavelength of 1.55 μm,” Appl. Phys. Lett. 81(24), 4502–4504 (2002). [CrossRef]
H. Yoda, K. Shiraishi, Y. Hiratani, and O. Hanaizumi, “a-Si:H/SiO2 multilayer films fabricated by radio-frequency magnetron sputtering for optical filters,” Appl. Opt. 43(17), 3548–3554 (2004). [CrossRef] [PubMed]
H. Windischmann, “Intrinsic stress in sputter-deposited thin films,” Crit. Rev. Solid State Mater. Sci. 17(6), 547–596 (1992). [CrossRef]
B. Bhushan, “Adhesion and stiction: mechanisms, measurement techniques, and methods for reduction,” J. Vac. Sci. Technol. B 21(6), 2262–2296 (2003). [CrossRef]
Y. X. Zhuang and A. Menon, “On the stiction of MEMS materials,” Tribol. Lett. 19(2), 111–117 (2005). [CrossRef]
H. V. Jansen, J. G. E. Gardeniers, J. Elders, H. A. C. Tilmans, and M. Elwenspoek, “Applications of fluorocarbon polymers in micromechanics and micromachining,” Sens. Actuators A Phys. 41(1-3), 136–140 (1994). [CrossRef]
B. K. Smith, J. J. Sniegowski, G. LaVigne, and C. Brown, “Thin Teflon-like films for eliminating adhesion in released polysilicon microstructures,” Sens. Actuators A Phys. 70(1-2), 159–163 (1998). [CrossRef]
A. A. Ayon, D.-Z. Chen, R. Khanna, R. Braff, H. H. Sawin, and M. A. Schmidt, “A novel integrated MEMS process using fluorocarbon films deposited with a deep reactive ion etching (DRIE) tool,” Mater. Res. Soc. Symp. Proc. 605, 141–147 (2000). [CrossRef]
J. Han, J. Yeom, G. Mensing, D. Joe, R. I. Masel, and M. A. Shannon, “Surface energy approach and AFM verification of the (CF)n treated surface effect and its correlation with adhesion reduction in microvalves,” J. Micromech. Microeng. 19(8), 085017 (2009). [CrossRef]
Y. X. Zhuang and A. Menon, “Wettability and themal stability of fluorocarbon films deposited by deep reactive ion etching,” J. Vac. Sci. Technol. A 23(3), 434–439 (2005). [CrossRef]
P. J. Joseph, H. A. Kelleher, S. A. B. Allen, and P. A. Kohl, “Improved fabrication of micro air-channels by incorporation of a structural barrier,” J. Micromech. Microeng. 15(1), 35–42 (2005). [CrossRef]
M. M. de Lima, R. G. Lacerda, J. Vilcarromero, and F. C. Marques, “Coefficient of thermal expansion and elastic modulus of thin films,” J. Appl. Phys. 86(9), 4936–4942 (1999). [CrossRef]
W. D. Nix, “Mechanical properties of thin films,” Metall. Mater. Trans., A Phys. Metall. Mater. Sci. 20(11), 2217–2245 (1989). [CrossRef]
S. Dasgupta, A. Ghatak, and B. P. Pal, “Analysis of Bragg reflection waveguides with finite cladding: an accurate matrix method formulation,” Opt. Commun. 279(1), 83–88 (2007). [CrossRef]

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