Abstract

We describe the thermal tuning of air-core Bragg waveguides, fabricated by controlled formation of delamination buckles within a multilayer stack of chalcogenide glass and polymer. The upper cladding mirror is a flexible membrane comprising high thermal expansion materials, enabling large tuning of the air-core dimensions for small changes in temperature. Measurements on the temperature dependence of feature heights showed good agreement with theoretical predictions. We applied this mechanism to the thermal tuning of modal cutoff conditions in waveguides with a tapered core profile. Due to the omnidirectional nature of the cladding mirrors, these tapers can be viewed as waveguide-coupled, tunable Fabry-Perot filters.

©2009 Optical Society of America

1. Introduction

Air-core integrated waveguides have been studied as optical interconnects and for applications in sensing [13]. They also provide the unique possibility of giant device tuning through micro-mechanical actuation of one of the cladding mirrors [4]. In principle, actuation schemes can be adapted from the extensive literature on tuning of Fabry-Perot filters (with two Bragg mirrors separated by an air cavity) using micro-electro-mechanical systems (MEMS) [511]. However, previous experimental work on the tuning of hollow waveguides has mostly relied on external piezoelectric actuators for alignment of separately fabricated upper and lower cladding mirrors [4,12]. Furthermore, the work has mainly employed slab waveguides while channel waveguides are preferable or even necessary for many applications [13]. Thus, a desirable goal is the monolithic fabrication of air-core channel waveguides, combined with an integrated actuation mechanism (electrostatic, electro-thermal, etc.).

Air-core filters and waveguides have traditionally been fabricated by surface or bulk micromachining processes, such as sacrificial etching or wafer bonding. We recently described [14] an alternative method based on controlled buckling delamination within a multilayer stack. This method produces self-assembled air cavities and channels sandwiched by Bragg reflectors, where the upper cladding mirror is a flexible membrane. Of related interest, tethered [5,6], clamped [7], and buckled [8] membranes have been widely studied in the context of micro-electro-optical-mechanical systems (MOEMS). Here, we demonstrate tuning of the buckle waveguides through thermal deflection of the upper mirror. This deflection modifies the dimensions of the hollow core, thereby modifying the loss, dispersion and other properties of the guide. As a specific example, we describe thermal tuning of the modal cutoff conditions in a tapered waveguide [15]. These tapered guides are essentially side-coupled Fabry-Perot cavities [16], and in the present context can be viewed as a new geometry for the implementation of tunable optical filters.

2. Thermo-mechanical analysis

The self-assembly fabrication process used to produce the air-core structures is discussed in detail elsewhere [14]. The cladding mirrors are gold-terminated Bragg reflectors, in turn comprising low index polyamide-imide (PAI) layers and high index Ge33As12Se55 (IG2 chalcogenide glass) layers. The upper and lower mirrors possess an overlapping band of omnidirectional reflection in the 1450-1650 nm range [15]. The buckled regions are defined by creating areas of low adhesion, through patterning of an embedded Ag layer and use of the photodoping phenomenon [14]. Buckle formation is driven by a controlled amount of net compressive stress in the upper cladding mirror, such that straight-sided (Euler) delamination buckles are formed within a desired range of feature sizes.

Within the limits of elastic deformation, the peak height of a straight-sided delamination buckle (the Euler buckle) is given by [17]:

δ=h43(σ0σC1)=h43((bbmin)21)2hb3bmin,
where h and σ0 are the thickness and intrinsic stress, respectively, of the buckled film or multilayer, b is the half-width of the buckle (see Fig. 1 ), bmin is the minimum half-width for the onset of buckling (given h and σ0), and σC is the critical compressive stress for the onset of buckling (given h and b):

 

Fig. 1 Shown is a schematic illustration of the change in height of a buckled hollow waveguide (end facet view) driven by an increase in temperature. A positive temperature change increases the compressive stress in the upper mirror, since it comprises materials with higher thermal expansion coefficient than the underlying silicon substrate. The added compressive stress results in a slight change in shape and increased peak height for the buckle.

Download Full Size | PPT Slide | PDF

σC=π212E(1ν2)h2b2.

Here, E and ν are the Young’s modulus and Poisson’s ratio of the film (or effective medium values for a buckled multilayer). Note that the approximation in Eq. (1) is restricted to the range b>>bmin.

Consider a pre-existing delamination buckle subject to a change in temperature. Due to thermal expansion mismatch between the film and substrate, the change in temperature modifies the biaxial compressive stress of the buckled feature according to [18]:

Δσ=E(1ν)ΔαΔT,
where Δα is the difference in coefficient of thermal expansion (CTE) between the film and substrate and ΔT is the change in temperature. Furthermore, the change in stress modifies the peak buckle height. We can approximate this from Eq. (1) as follows:
(δ+Δδ)2=4h23[(σ0+Δσ)σC1],
which reduces to:

Δδ2h23δΔσσC.

Combining Eq. (2), Eq. (3), and Eq. (5), an expression for the rate of change of the height with temperature is obtained:

ΔδΔT=8π2(1ν2)(1ν)b2δΔα.

Note that since δ is approximately proportional to b (provided b>>bmin, see Eq. (1)), it follows that the change in buckle height with temperature is approximately proportional to the starting buckle height.

To analyze buckling of a thin-film stack, an effective medium approach can be used [17]. For the present case, the buckled upper mirror comprises ~61% PAI polymer, ~37% IG2 glass, and ~2% Au. Using the known mechanical properties of these materials [19], the effective medium parameters are as follows: Young’s modulus E~11 GPa, Poisson’s coefficient ν~0.3, and CTE mismatch with silicon substrate Δα~20x10−6 K−1. Given these values, the predictions of Eq. (6) are in good agreement with experimental results as shown below.

3. Experimental results and analysis

In initial experiments, samples were placed on a thermo-electric cooler and an optical profilometer (Zygo) was used to measure the height change. Figure 2(a) shows a typical result for a straight-sided buckle waveguide with base width 2b ~60 μm and peak height δ ~3.4 μm. Using these dimensions and the effective medium parameters cited above, Eq. (6) predicts (Δδ/ΔT) ~5.6 nm/K. As shown in Fig. 2(b), this estimate is in good agreement with the experimental data.

 

Fig. 2 (a) Optical profilometer scans for a buckled waveguide with 60 μm base width, at a series of fixed temperatures. The inset plot shows the top of the curves in greater detail. (b) Plot of the measured change in peak height versus change in temperature. The straight line is a linear fit to the data.

Download Full Size | PPT Slide | PDF

Such changes in core height can produce significant changes in the properties of a leaky guided mode. To illustrate, we describe the tuning of the mode cutoff position in waveguides with a tapered core profile. In these tapered guides [15], out-of-plane coupling at mode cutoff arises due to the omnidirectional nature of the cladding mirrors. As shown schematically in Fig. 3(a) , these guides can thus be viewed as side-coupled Fabry-Perot cavities [16]. From a slab waveguide model δm ~δm-1 + λ0/2, where δm is the cutoff height for mode order m = 0,1,2…, and λ0 is the free-space wavelength. For an increase in temperature, the increase in core height causes the cutoff point for each mode to shift towards the smaller end of the taper. For each vertical mode order, a family of lateral modes exist [15]. However, preferential excitation of the fundamental lateral modes is possible by careful alignment of the input fiber.

 

Fig. 3 (a) Cross-sectional schematic of a tapered air-core waveguide with omnidirectional claddings, near a mode cutoff point. The black arrows depict the ray-optics model of vertical radiation at cutoff, and the red curve represents the vertical field profile (m = 1 case shown) at the cutoff point. An increase in core height due to increase in temperature causes a positional shift of the cutoff point. (b) Images captured by an infrared camera via a microscope, showing the shift in the m = 1 to 5 cutoff positions with temperature, for a wavelength of 1550 nm. (c) The experimental shift in out-coupling position plotted versus change in temperature, for mode orders 1 to 4. The straight lines are linear fits to the data.

Download Full Size | PPT Slide | PDF

The microscope images in Fig. 3(b) show the out-coupling positions of 5 vertical mode orders at a wavelength of 1550 nm, for a waveguide with width (2b) linearly tapered from 80 to 10 μm over a distance of 5 mm (i.e. Δb/Δz is ~7 μm/mm). The arrows indicate the final out-coupling point of the fundamental lateral mode for each vertical mode order. Secondary spots to the left of these points are mainly due to standing-wave effects and radiation of higher-order lateral modes [15], both of which are sensitive to input coupling conditions and vary slightly as the temperature is changed. In the analysis below, we focus only on the position of the final cutoff points for each vertical mode order. For a purely elastic deformation and for b>>bmin, Eq. (1) predicts that the peak buckle height will also exhibit a linear profile. This prediction is supported by the approximately equal spacing of the cutoff positions for the m = 2 to 5 modes, since the difference in core height between adjacent orders at cutoff is ~λ0/2. However, within our process the buckle formation is influenced by both elastic and plastic deformation [15]. The larger spacing between the m = 2 and m = 1 spots indicates a lower slope at the small end of the taper. Also note that, due to the particular mirror design used [15], the fundamental (m = 0) vertical mode order does not exhibit cutoff.

Using a slab model for the waveguides, the shift in out-coupling point with temperature can be estimated as follows:

ΔzΔTΔzΔδΔδΔT=1STΔδΔT,
where ST is the taper slope. From profilometer measurements and the spacing of cutoff points, we estimated ST~1.1-1.3 μm/mm for the main portion of the taper (i.e. for the m = 2 to m = 5 mode orders, as discussed above) and ST~0.8-9 μm/mm near the narrow end of the taper (i.e. for the m = 1 mode order). Assuming a fixed slope, and since Δδ is approximately proportional to δ as discussed in Section 2, Eq. (6) and Eq. (7) predict that the shift in out-coupling position scales with the mode order. This is corroborated by the data plotted in Fig. 3(c). For m = 4 (3), δ~2.7 (2.0) μm, 2b~48 (32) μm and ST~1.2 (1.3) μm/mm were estimated, so that Eqs. (6) and (7) produce Δz/ΔT ~3.8 (2.1) μm/K, in good agreement with experimental observations. However, the theory was found to slightly overestimate the positional shift for the lower mode orders, possibly due to the neglect of plastic deformation in the derivation of Eq. (6).

These tapers could be used as a new type of tunable Fabry-Perot filter, by extracting the vertically radiated light from a fixed position and varying the temperature. The wavelength shift can be approximated as follows:

Δλ0ΔTΔλ0ΔzΔzΔT=1DTΔzΔT,
where DT = Δz/Δλ0 is the spatial dispersion provided by the taper. Neglecting the wavelength-dependent penetration depth of the cladding mirrors, DT~(m + 1)/(2ST). Since both DT and Δz/ΔT scale approximately with mode order (m + 1), the wavelength shift per unit temperature change is expected to be approximately the same for all vertical mode orders. The upper limit on tuning range is set either by the free spectral range (FSR) between mode orders or by the omnidirectional bandwidth of the cladding mirrors. For the low mode orders employed here, FSR~λm/(m + 2), where λm is the resonant wavelength. Thus, both the FSR and omnidirectional cladding bandwidth are on the order of several hundred nm. Practical limitations on tuning range include the maximum temperature variation that can be induced and the thermal stability range of the cladding materials, which is >300 °C for the present case [19].

Spectral line-width is another important parameter for an optical filter. As discussed in detail elsewhere [20], the line-width for the cutoff-based out-coupling mechanism depends on several factors, including the spatial dispersion of the taper, the presence of multiple lateral modes within each vertical mode family, and modal back-reflection and standing-wave formation near the cutoff point. However, it is possible to minimize the multimode and standing wave effects by use of appropriate input and output coupling optics [20]. In that case, line-width can be estimated using a vertical Fabry-Perot model to describe the out-coupling mechanism:

ΔλFWHMzPDT+λ0(m+1)π(R1R)   ,
where zP is the length of taper from which light is collected and R is the normal-incidence reflectance of the cladding mirrors (assumed equal).

To test these predictions, light from a tunable laser (Santec model TSL-320) was input to a tapered hollow waveguide via a polarization controller and a tapered (lensed) optical fiber (Oz Optics, nominal spot size ~5 μm), enabling preferential coupling of the fundamental lateral modes. The results below are for TE polarization, because the TM modes exhibit somewhat higher loss [14]. However, it is important to note that the cutoff-based coupling mechanism is nearly independent of polarization since TE and TM modes become degenerate at cutoff [15,20]. This implies the potential for a polarization-independent tunable filter, provided the polarization-dependent loss (PDL) of the waveguides is minimized.

The out-coupled light was collected by a cleaved single-mode fiber (Corning SMF-28, NA~0.13) placed in close proximity to the tapered waveguide surface, and delivered to a calibrated photodetector. The pick-up fiber was kept at a fixed position, and wavelength scans were obtained for a series of fixed temperatures. This collection setup implies that zP~9 μm, i.e. the effective collection length is approximately the core diameter of the pickup fiber. As explained in detail elsewhere [20], the use of a low NA pickup fiber reduces the impact of satellite peaks on the short wavelength side of the main spectral feature. Figure 4(a) shows typical results for the m = 2 mode, with the output spectrum exhibiting a temperature-dependent pass-band. The inset shows that the out-of-band extinction exceeds 10 dB, limited on the short wavelength side by secondary peaks arising from standing waves formed in advance of the mode cutoff point [20]. The measured FWHM line-width is ~4 nm, although it varied between ~3 and ~5 nm with temperature. This variation might be partly due to slight variations in the local taper slope with temperature, given that the slope relates directly to the dispersion [20]. Instabilities in our experimental setup also contribute, based on observed line-width variations of ~0.5 nm for measurements repeated at fixed temperature.

 

Fig. 4 (a) Spectral scans obtained at a fixed out-coupling point corresponding to the m = 2 mode, for a series of fixed temperatures. The inset shows the 16.5 °C data on a logarithmic scale. (b) Plot of peak out-coupling wavelength versus temperature, revealing a wavelength shift of ~0.45 nm/°C.

Download Full Size | PPT Slide | PDF

From measurements with a tunable laser [15], we estimated (1/DT)~450 nm/mm for the m = 2 mode (at room temperature) near 1550 nm. Thus, with zP~9 μm and R~0.997 [15], Eq. (9) predicts ΔλFWHM~4.5 nm, in reasonable agreement with the results of Fig. 4(a). Using the same value for taper dispersion and the experimentally determined out-coupling shift for the m = 2 mode (Δz/ΔT~1 μm/K, see Fig. 3(c)), Eq. (8) predicts Δλ0/ΔT~0.45 nm/K, in good agreement with the results of Fig. 4(b).

The demonstrated tuning range (~10 nm) was limited by the experimental setup, which employed an external thermo-electric stage for temperature variation. Integration of thermal actuators is expected to enable a much wider tuning range. Bragg mirrors of the present type have been shown [19] to survive temperature excursions between −196 °C and + 300 °C. However, the buckle guides exhibit damage (wrinkling, etc.) for temperatures above ~150 °C [14]. Interesting comparisons can be made with tunable filters based on hollow waveguides with a Bragg reflector defined on one of the cladding mirrors [1113]. A tuning range of ~150 nm was recently reported [13], but requiring a core thickness variation (controlled by external piezoelectric actuators) in the 5 to 10 μm range. A much smaller tuning range (2.6 nm) was reported for a similar device driven by an integrated thermal actuator [11]. Since MEMS actuators typically have restricted range of motion, a reasonable figure of merit for tunable hollow waveguide filters is the wavelength shift per unit change in core height (Δλ0/Δδ). Using a hard mirror approximation for the Fabry-Perot, Δλ0/Δδ = 2/(m + 1), or 667 nm/μm for the m = 2 case. From the results above, the actual value is somewhat lower (~400 nm/μm), but still more than an order of magnitude higher than for the device reported in reference [13] (~30 nm/μm).

Finally, we note that the measured insertion loss was approximately 30 dB, and is mainly caused by the use of a fairly opaque (~50 nm thick) outer Au layer on the cladding mirrors [14]. As described in detail previously [15], the light radiated from the taper is essentially an asymmetric diverging beam. The insertion loss for the data in Fig. 4(a) varied slightly with temperature (by ~1 dB), probably due to experimental instabilities that affected the coupling efficiency to the pickup fiber. With the use of all-dielectric mirrors, optimized taper geometry, and improved coupling optics, it should be possible to reduce this loss by more than 2 orders of magnitude [20].

4. Summary and conclusions

We described the thermal tuning of hollow waveguides fabricated by controlled buckling delamination within a multilayer stack. The tuning is made possible by the thermal expansion mismatch between the upper buckled mirror and the underlying substrate. This mechanism was used to demonstrate a tunable filter based on a tapered hollow waveguide with omnidirectional claddings. In these filters, the input is waveguide-coupled and the output is collected from the top of the waveguide by surface-normal coupling optics. Furthermore, the input and output are interchangeable. Many integrated optical systems could benefit from this unique geometry. Furthermore, the mechanism was shown to be highly sensitive in terms of wavelength shift per unit change in core height, promising compatibility with integrated (electrostatic or electrothermal) actuation schemes. We hope to explore this possibility in future work.

Acknowledgements

The work was supported by the Natural Sciences and Engineering Research Council of Canada, the Microsystems Technology Research Initiative (MSTRI), and by TRLabs. Devices were fabricated at the Nanofab of the University of Alberta.

References and Links

1. E. J. Lunt, P. Measor, B. S. Phillips, S. Kühn, H. Schmidt, and A. R. Hawkins, “Improving solid to hollow core transmission for integrated ARROW waveguides,” Opt. Express 16(25), 20981–20986 (2008). [CrossRef]   [PubMed]  

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. H.-K. Chiu, F.-L. Hsiao, C.-H. Chan, and C.-C. Chen, “Compact and low-loss bent hollow waveguides with distributed Bragg reflector,” Opt. Express 16(19), 15069–15073 (2008). [CrossRef]   [PubMed]  

4. F. Koyama, T. Miura, and Y. Sakurai, “Tunable hollow waveguides and their applications for photonic integrated circuits,” Electronics and Communications in Japan 29(Part 2), 9–19 (2006).

5. A. T. T. D. Tran, Y. H. Lo, Z. H. Zhu, D. Haronian, and E. Mozdy, “Surface micromachined Fabry-Perot tunable filter,” IEEE Photon. Technol. Lett. 8(3), 393–395 (1996). [CrossRef]  

6. P. Tayebati, P. Wang, M. Azimi, L. Maflah, and D. Vakhshoori, “Microelectromechanical tunable filter with stable half symmetric cavity,” Electron. Lett. 34(20), 1967–1968 (1998). [CrossRef]  

7. Y. Yi, P. Bermel, K. Wada, X. Duan, J. D. Joannopoulos, and L. C. Kimerling, “Tunable multichannel optical filter based on silicon photonic band gap materials actuation,” Appl. Phys. Lett. 81(22), 4112–4114 (2002). [CrossRef]  

8. H. Halbritter, M. Aziz, F. Riemenschneider, and P. Meissner, “Electrothermally tunable two-chip optical filter with very low-cost and simple concept,” Electron. Lett. 38(20), 1201–1202 (2002). [CrossRef]  

9. S. Irmer, J. Daleiden, V. Rangelov, C. Prott, F. Romer, M. Strassner, A. Tarraf, and H. Hillmer, “Ultralow biased widely continuously tunable Fabry-Perot filter,” IEEE Photon. Technol. Lett. 15(3), 434–436 (2003). [CrossRef]  

10. C. F. R. Mateus, M. C. Y. Huang, and C. J. Chang-Hasnain, “Micromechanical tunable optical filters: general design rules for wavelengths from near-IR up to 10 μm,” Sensors and Actuators A 119(1), 57–62 (2005). [CrossRef]  

11. Y. Sakurai, H. Yamakawa, Y. Yokota, A. Matsutani, T. Sakaguchi, and F. Koyama, “Hollow Waveguide Distributed Bragg Reflector for Widely Tunable Optical Devices,” in Optical Fiber Communication Conference, Technical Digest (CD) (Optical Society of America, 2006), paper OTuM4.

12. Y. Sakurai and F. Koyama, “Tunable hollow waveguide distributed Bragg reflectors with variable air core,” Opt. Express 12(13), 2851–2856 (2004). [CrossRef]   [PubMed]  

13. 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]  

14. 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]  

15. 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]  

16. M. L. Povinelli, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, “Slow-light enhancement of radiation pressure in an omnidirectional-reflector waveguide,” Appl. Phys. Lett. 85(9), 1466–1468 (2004). [CrossRef]  

17. 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]  

18. S. R. Choi, J. W. Hutchinson, and A. G. Evans, “Delamination of multilayer thermal barrier coatings,” Mech. Mater. 31(7), 431–447 (1999). [CrossRef]  

19. R. G. DeCorby, N. Ponnampalam, H. T. Nguyen, and T. J. Clement, “Robust and flexible free-standing all-dielectric omnidirectional reflectors,” Adv. Mater. 19(2), 193–196 (2007). [CrossRef]  

20. R.G. DeCorby, N. Ponnampalam, E. Epp, T. Allen, J.N. McMullin, “Chip-scale spectrometry based on tapered hollow Bragg waveguides,” submitted for publication.

References

  • View by:
  • |
  • |
  • |

  1. E. J. Lunt, P. Measor, B. S. Phillips, S. Kühn, H. Schmidt, and A. R. Hawkins, “Improving solid to hollow core transmission for integrated ARROW waveguides,” Opt. Express 16(25), 20981–20986 (2008).
    [Crossref] [PubMed]
  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. H.-K. Chiu, F.-L. Hsiao, C.-H. Chan, and C.-C. Chen, “Compact and low-loss bent hollow waveguides with distributed Bragg reflector,” Opt. Express 16(19), 15069–15073 (2008).
    [Crossref] [PubMed]
  4. F. Koyama, T. Miura, and Y. Sakurai, “Tunable hollow waveguides and their applications for photonic integrated circuits,” Electronics and Communications in Japan 29(Part 2), 9–19 (2006).
  5. A. T. T. D. Tran, Y. H. Lo, Z. H. Zhu, D. Haronian, and E. Mozdy, “Surface micromachined Fabry-Perot tunable filter,” IEEE Photon. Technol. Lett. 8(3), 393–395 (1996).
    [Crossref]
  6. P. Tayebati, P. Wang, M. Azimi, L. Maflah, and D. Vakhshoori, “Microelectromechanical tunable filter with stable half symmetric cavity,” Electron. Lett. 34(20), 1967–1968 (1998).
    [Crossref]
  7. Y. Yi, P. Bermel, K. Wada, X. Duan, J. D. Joannopoulos, and L. C. Kimerling, “Tunable multichannel optical filter based on silicon photonic band gap materials actuation,” Appl. Phys. Lett. 81(22), 4112–4114 (2002).
    [Crossref]
  8. H. Halbritter, M. Aziz, F. Riemenschneider, and P. Meissner, “Electrothermally tunable two-chip optical filter with very low-cost and simple concept,” Electron. Lett. 38(20), 1201–1202 (2002).
    [Crossref]
  9. S. Irmer, J. Daleiden, V. Rangelov, C. Prott, F. Romer, M. Strassner, A. Tarraf, and H. Hillmer, “Ultralow biased widely continuously tunable Fabry-Perot filter,” IEEE Photon. Technol. Lett. 15(3), 434–436 (2003).
    [Crossref]
  10. C. F. R. Mateus, M. C. Y. Huang, and C. J. Chang-Hasnain, “Micromechanical tunable optical filters: general design rules for wavelengths from near-IR up to 10 μm,” Sensors and Actuators A 119(1), 57–62 (2005).
    [Crossref]
  11. Y. Sakurai, H. Yamakawa, Y. Yokota, A. Matsutani, T. Sakaguchi, and F. Koyama, “Hollow Waveguide Distributed Bragg Reflector for Widely Tunable Optical Devices,” in Optical Fiber Communication Conference, Technical Digest (CD) (Optical Society of America, 2006), paper OTuM4.
  12. Y. Sakurai and F. Koyama, “Tunable hollow waveguide distributed Bragg reflectors with variable air core,” Opt. Express 12(13), 2851–2856 (2004).
    [Crossref] [PubMed]
  13. 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]
  14. 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]
  15. 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]
  16. M. L. Povinelli, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, “Slow-light enhancement of radiation pressure in an omnidirectional-reflector waveguide,” Appl. Phys. Lett. 85(9), 1466–1468 (2004).
    [Crossref]
  17. 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]
  18. S. R. Choi, J. W. Hutchinson, and A. G. Evans, “Delamination of multilayer thermal barrier coatings,” Mech. Mater. 31(7), 431–447 (1999).
    [Crossref]
  19. R. G. DeCorby, N. Ponnampalam, H. T. Nguyen, and T. J. Clement, “Robust and flexible free-standing all-dielectric omnidirectional reflectors,” Adv. Mater. 19(2), 193–196 (2007).
    [Crossref]
  20. R.G. DeCorby, N. Ponnampalam, E. Epp, T. Allen, J.N. McMullin, “Chip-scale spectrometry based on tapered hollow Bragg waveguides,” submitted for publication.

2009 (2)

2008 (3)

2007 (2)

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]

R. G. DeCorby, N. Ponnampalam, H. T. Nguyen, and T. J. Clement, “Robust and flexible free-standing all-dielectric omnidirectional reflectors,” Adv. Mater. 19(2), 193–196 (2007).
[Crossref]

2006 (1)

F. Koyama, T. Miura, and Y. Sakurai, “Tunable hollow waveguides and their applications for photonic integrated circuits,” Electronics and Communications in Japan 29(Part 2), 9–19 (2006).

2005 (1)

C. F. R. Mateus, M. C. Y. Huang, and C. J. Chang-Hasnain, “Micromechanical tunable optical filters: general design rules for wavelengths from near-IR up to 10 μm,” Sensors and Actuators A 119(1), 57–62 (2005).
[Crossref]

2004 (3)

Y. Sakurai and F. Koyama, “Tunable hollow waveguide distributed Bragg reflectors with variable air core,” Opt. Express 12(13), 2851–2856 (2004).
[Crossref] [PubMed]

M. L. Povinelli, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, “Slow-light enhancement of radiation pressure in an omnidirectional-reflector waveguide,” Appl. Phys. Lett. 85(9), 1466–1468 (2004).
[Crossref]

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]

2003 (1)

S. Irmer, J. Daleiden, V. Rangelov, C. Prott, F. Romer, M. Strassner, A. Tarraf, and H. Hillmer, “Ultralow biased widely continuously tunable Fabry-Perot filter,” IEEE Photon. Technol. Lett. 15(3), 434–436 (2003).
[Crossref]

2002 (2)

Y. Yi, P. Bermel, K. Wada, X. Duan, J. D. Joannopoulos, and L. C. Kimerling, “Tunable multichannel optical filter based on silicon photonic band gap materials actuation,” Appl. Phys. Lett. 81(22), 4112–4114 (2002).
[Crossref]

H. Halbritter, M. Aziz, F. Riemenschneider, and P. Meissner, “Electrothermally tunable two-chip optical filter with very low-cost and simple concept,” Electron. Lett. 38(20), 1201–1202 (2002).
[Crossref]

1999 (1)

S. R. Choi, J. W. Hutchinson, and A. G. Evans, “Delamination of multilayer thermal barrier coatings,” Mech. Mater. 31(7), 431–447 (1999).
[Crossref]

1998 (1)

P. Tayebati, P. Wang, M. Azimi, L. Maflah, and D. Vakhshoori, “Microelectromechanical tunable filter with stable half symmetric cavity,” Electron. Lett. 34(20), 1967–1968 (1998).
[Crossref]

1996 (1)

A. T. T. D. Tran, Y. H. Lo, Z. H. Zhu, D. Haronian, and E. Mozdy, “Surface micromachined Fabry-Perot tunable filter,” IEEE Photon. Technol. Lett. 8(3), 393–395 (1996).
[Crossref]

Azimi, M.

P. Tayebati, P. Wang, M. Azimi, L. Maflah, and D. Vakhshoori, “Microelectromechanical tunable filter with stable half symmetric cavity,” Electron. Lett. 34(20), 1967–1968 (1998).
[Crossref]

Aziz, M.

H. Halbritter, M. Aziz, F. Riemenschneider, and P. Meissner, “Electrothermally tunable two-chip optical filter with very low-cost and simple concept,” Electron. Lett. 38(20), 1201–1202 (2002).
[Crossref]

Bermel, P.

Y. Yi, P. Bermel, K. Wada, X. Duan, J. D. Joannopoulos, and L. C. Kimerling, “Tunable multichannel optical filter based on silicon photonic band gap materials actuation,” Appl. Phys. Lett. 81(22), 4112–4114 (2002).
[Crossref]

Chan, C.-H.

Chang-Hasnain, C. J.

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]

C. F. R. Mateus, M. C. Y. Huang, and C. J. Chang-Hasnain, “Micromechanical tunable optical filters: general design rules for wavelengths from near-IR up to 10 μm,” Sensors and Actuators A 119(1), 57–62 (2005).
[Crossref]

Chen, C.-C.

Chiu, H.-K.

Choi, S. R.

S. R. Choi, J. W. Hutchinson, and A. G. Evans, “Delamination of multilayer thermal barrier coatings,” Mech. Mater. 31(7), 431–447 (1999).
[Crossref]

Clement, T. J.

R. G. DeCorby, N. Ponnampalam, H. T. Nguyen, and T. J. Clement, “Robust and flexible free-standing all-dielectric omnidirectional reflectors,” Adv. Mater. 19(2), 193–196 (2007).
[Crossref]

Daleiden, J.

S. Irmer, J. Daleiden, V. Rangelov, C. Prott, F. Romer, M. Strassner, A. Tarraf, and H. Hillmer, “Ultralow biased widely continuously tunable Fabry-Perot filter,” IEEE Photon. Technol. Lett. 15(3), 434–436 (2003).
[Crossref]

DeCorby, R. G.

Duan, X.

Y. Yi, P. Bermel, K. Wada, X. Duan, J. D. Joannopoulos, and L. C. Kimerling, “Tunable multichannel optical filter based on silicon photonic band gap materials actuation,” Appl. Phys. Lett. 81(22), 4112–4114 (2002).
[Crossref]

Evans, A. G.

S. R. Choi, J. W. Hutchinson, and A. G. Evans, “Delamination of multilayer thermal barrier coatings,” Mech. Mater. 31(7), 431–447 (1999).
[Crossref]

Halbritter, H.

H. Halbritter, M. Aziz, F. Riemenschneider, and P. Meissner, “Electrothermally tunable two-chip optical filter with very low-cost and simple concept,” Electron. Lett. 38(20), 1201–1202 (2002).
[Crossref]

Haronian, D.

A. T. T. D. Tran, Y. H. Lo, Z. H. Zhu, D. Haronian, and E. Mozdy, “Surface micromachined Fabry-Perot tunable filter,” IEEE Photon. Technol. Lett. 8(3), 393–395 (1996).
[Crossref]

Hawkins, A. R.

Hillmer, H.

S. Irmer, J. Daleiden, V. Rangelov, C. Prott, F. Romer, M. Strassner, A. Tarraf, and H. Hillmer, “Ultralow biased widely continuously tunable Fabry-Perot filter,” IEEE Photon. Technol. Lett. 15(3), 434–436 (2003).
[Crossref]

Hsiao, F.-L.

Huang, M. C. Y.

C. F. R. Mateus, M. C. Y. Huang, and C. J. Chang-Hasnain, “Micromechanical tunable optical filters: general design rules for wavelengths from near-IR up to 10 μm,” Sensors and Actuators A 119(1), 57–62 (2005).
[Crossref]

Hutchinson, J. W.

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]

S. R. Choi, J. W. Hutchinson, and A. G. Evans, “Delamination of multilayer thermal barrier coatings,” Mech. Mater. 31(7), 431–447 (1999).
[Crossref]

Ibanescu, M.

M. L. Povinelli, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, “Slow-light enhancement of radiation pressure in an omnidirectional-reflector waveguide,” Appl. Phys. Lett. 85(9), 1466–1468 (2004).
[Crossref]

Irmer, S.

S. Irmer, J. Daleiden, V. Rangelov, C. Prott, F. Romer, M. Strassner, A. Tarraf, and H. Hillmer, “Ultralow biased widely continuously tunable Fabry-Perot filter,” IEEE Photon. Technol. Lett. 15(3), 434–436 (2003).
[Crossref]

Joannopoulos, J. D.

M. L. Povinelli, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, “Slow-light enhancement of radiation pressure in an omnidirectional-reflector waveguide,” Appl. Phys. Lett. 85(9), 1466–1468 (2004).
[Crossref]

Y. Yi, P. Bermel, K. Wada, X. Duan, J. D. Joannopoulos, and L. C. Kimerling, “Tunable multichannel optical filter based on silicon photonic band gap materials actuation,” Appl. Phys. Lett. 81(22), 4112–4114 (2002).
[Crossref]

Johnson, S. G.

M. L. Povinelli, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, “Slow-light enhancement of radiation pressure in an omnidirectional-reflector waveguide,” Appl. Phys. Lett. 85(9), 1466–1468 (2004).
[Crossref]

Karagodsky, V.

Kimerling, L. C.

Y. Yi, P. Bermel, K. Wada, X. Duan, J. D. Joannopoulos, and L. C. Kimerling, “Tunable multichannel optical filter based on silicon photonic band gap materials actuation,” Appl. Phys. Lett. 81(22), 4112–4114 (2002).
[Crossref]

Koyama, F.

Kühn, S.

Kumar, M.

Lee, K.-R.

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]

Lo, Y. H.

A. T. T. D. Tran, Y. H. Lo, Z. H. Zhu, D. Haronian, and E. Mozdy, “Surface micromachined Fabry-Perot tunable filter,” IEEE Photon. Technol. Lett. 8(3), 393–395 (1996).
[Crossref]

Lunt, E. J.

Maflah, L.

P. Tayebati, P. Wang, M. Azimi, L. Maflah, and D. Vakhshoori, “Microelectromechanical tunable filter with stable half symmetric cavity,” Electron. Lett. 34(20), 1967–1968 (1998).
[Crossref]

Mateus, C. F. R.

C. F. R. Mateus, M. C. Y. Huang, and C. J. Chang-Hasnain, “Micromechanical tunable optical filters: general design rules for wavelengths from near-IR up to 10 μm,” Sensors and Actuators A 119(1), 57–62 (2005).
[Crossref]

Measor, P.

Meissner, P.

H. Halbritter, M. Aziz, F. Riemenschneider, and P. Meissner, “Electrothermally tunable two-chip optical filter with very low-cost and simple concept,” Electron. Lett. 38(20), 1201–1202 (2002).
[Crossref]

Miura, T.

F. Koyama, T. Miura, and Y. Sakurai, “Tunable hollow waveguides and their applications for photonic integrated circuits,” Electronics and Communications in Japan 29(Part 2), 9–19 (2006).

Moon, M.-W.

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]

Mozdy, E.

A. T. T. D. Tran, Y. H. Lo, Z. H. Zhu, D. Haronian, and E. Mozdy, “Surface micromachined Fabry-Perot tunable filter,” IEEE Photon. Technol. Lett. 8(3), 393–395 (1996).
[Crossref]

Nguyen, H. T.

R. G. DeCorby, N. Ponnampalam, H. T. Nguyen, and T. J. Clement, “Robust and flexible free-standing all-dielectric omnidirectional reflectors,” Adv. Mater. 19(2), 193–196 (2007).
[Crossref]

Oh, K. H.

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]

Pesala, B.

Phillips, B. S.

Ponnampalam, N.

Povinelli, M. L.

M. L. Povinelli, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, “Slow-light enhancement of radiation pressure in an omnidirectional-reflector waveguide,” Appl. Phys. Lett. 85(9), 1466–1468 (2004).
[Crossref]

Prott, C.

S. Irmer, J. Daleiden, V. Rangelov, C. Prott, F. Romer, M. Strassner, A. Tarraf, and H. Hillmer, “Ultralow biased widely continuously tunable Fabry-Perot filter,” IEEE Photon. Technol. Lett. 15(3), 434–436 (2003).
[Crossref]

Rangelov, V.

S. Irmer, J. Daleiden, V. Rangelov, C. Prott, F. Romer, M. Strassner, A. Tarraf, and H. Hillmer, “Ultralow biased widely continuously tunable Fabry-Perot filter,” IEEE Photon. Technol. Lett. 15(3), 434–436 (2003).
[Crossref]

Riemenschneider, F.

H. Halbritter, M. Aziz, F. Riemenschneider, and P. Meissner, “Electrothermally tunable two-chip optical filter with very low-cost and simple concept,” Electron. Lett. 38(20), 1201–1202 (2002).
[Crossref]

Romer, F.

S. Irmer, J. Daleiden, V. Rangelov, C. Prott, F. Romer, M. Strassner, A. Tarraf, and H. Hillmer, “Ultralow biased widely continuously tunable Fabry-Perot filter,” IEEE Photon. Technol. Lett. 15(3), 434–436 (2003).
[Crossref]

Sakaguchi, T.

Sakurai, Y.

F. Koyama, T. Miura, and Y. Sakurai, “Tunable hollow waveguides and their applications for photonic integrated circuits,” Electronics and Communications in Japan 29(Part 2), 9–19 (2006).

Y. Sakurai and F. Koyama, “Tunable hollow waveguide distributed Bragg reflectors with variable air core,” Opt. Express 12(13), 2851–2856 (2004).
[Crossref] [PubMed]

Schmidt, H.

Sedgwick, F. G.

Strassner, M.

S. Irmer, J. Daleiden, V. Rangelov, C. Prott, F. Romer, M. Strassner, A. Tarraf, and H. Hillmer, “Ultralow biased widely continuously tunable Fabry-Perot filter,” IEEE Photon. Technol. Lett. 15(3), 434–436 (2003).
[Crossref]

Tarraf, A.

S. Irmer, J. Daleiden, V. Rangelov, C. Prott, F. Romer, M. Strassner, A. Tarraf, and H. Hillmer, “Ultralow biased widely continuously tunable Fabry-Perot filter,” IEEE Photon. Technol. Lett. 15(3), 434–436 (2003).
[Crossref]

Tayebati, P.

P. Tayebati, P. Wang, M. Azimi, L. Maflah, and D. Vakhshoori, “Microelectromechanical tunable filter with stable half symmetric cavity,” Electron. Lett. 34(20), 1967–1968 (1998).
[Crossref]

Tran, A. T. T. D.

A. T. T. D. Tran, Y. H. Lo, Z. H. Zhu, D. Haronian, and E. Mozdy, “Surface micromachined Fabry-Perot tunable filter,” IEEE Photon. Technol. Lett. 8(3), 393–395 (1996).
[Crossref]

Vakhshoori, D.

P. Tayebati, P. Wang, M. Azimi, L. Maflah, and D. Vakhshoori, “Microelectromechanical tunable filter with stable half symmetric cavity,” Electron. Lett. 34(20), 1967–1968 (1998).
[Crossref]

Wada, K.

Y. Yi, P. Bermel, K. Wada, X. Duan, J. D. Joannopoulos, and L. C. Kimerling, “Tunable multichannel optical filter based on silicon photonic band gap materials actuation,” Appl. Phys. Lett. 81(22), 4112–4114 (2002).
[Crossref]

Wang, P.

P. Tayebati, P. Wang, M. Azimi, L. Maflah, and D. Vakhshoori, “Microelectromechanical tunable filter with stable half symmetric cavity,” Electron. Lett. 34(20), 1967–1968 (1998).
[Crossref]

Yi, Y.

Y. Yi, P. Bermel, K. Wada, X. Duan, J. D. Joannopoulos, and L. C. Kimerling, “Tunable multichannel optical filter based on silicon photonic band gap materials actuation,” Appl. Phys. Lett. 81(22), 4112–4114 (2002).
[Crossref]

Zhou, Y.

Zhu, Z. H.

A. T. T. D. Tran, Y. H. Lo, Z. H. Zhu, D. Haronian, and E. Mozdy, “Surface micromachined Fabry-Perot tunable filter,” IEEE Photon. Technol. Lett. 8(3), 393–395 (1996).
[Crossref]

Acta Mater. (1)

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]

Adv. Mater. (1)

R. G. DeCorby, N. Ponnampalam, H. T. Nguyen, and T. J. Clement, “Robust and flexible free-standing all-dielectric omnidirectional reflectors,” Adv. Mater. 19(2), 193–196 (2007).
[Crossref]

Appl. Phys. Lett. (2)

M. L. Povinelli, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, “Slow-light enhancement of radiation pressure in an omnidirectional-reflector waveguide,” Appl. Phys. Lett. 85(9), 1466–1468 (2004).
[Crossref]

Y. Yi, P. Bermel, K. Wada, X. Duan, J. D. Joannopoulos, and L. C. Kimerling, “Tunable multichannel optical filter based on silicon photonic band gap materials actuation,” Appl. Phys. Lett. 81(22), 4112–4114 (2002).
[Crossref]

Electron. Lett. (2)

H. Halbritter, M. Aziz, F. Riemenschneider, and P. Meissner, “Electrothermally tunable two-chip optical filter with very low-cost and simple concept,” Electron. Lett. 38(20), 1201–1202 (2002).
[Crossref]

P. Tayebati, P. Wang, M. Azimi, L. Maflah, and D. Vakhshoori, “Microelectromechanical tunable filter with stable half symmetric cavity,” Electron. Lett. 34(20), 1967–1968 (1998).
[Crossref]

Electronics and Communications in Japan (1)

F. Koyama, T. Miura, and Y. Sakurai, “Tunable hollow waveguides and their applications for photonic integrated circuits,” Electronics and Communications in Japan 29(Part 2), 9–19 (2006).

IEEE Photon. Technol. Lett. (2)

A. T. T. D. Tran, Y. H. Lo, Z. H. Zhu, D. Haronian, and E. Mozdy, “Surface micromachined Fabry-Perot tunable filter,” IEEE Photon. Technol. Lett. 8(3), 393–395 (1996).
[Crossref]

S. Irmer, J. Daleiden, V. Rangelov, C. Prott, F. Romer, M. Strassner, A. Tarraf, and H. Hillmer, “Ultralow biased widely continuously tunable Fabry-Perot filter,” IEEE Photon. Technol. Lett. 15(3), 434–436 (2003).
[Crossref]

Mech. Mater. (1)

S. R. Choi, J. W. Hutchinson, and A. G. Evans, “Delamination of multilayer thermal barrier coatings,” Mech. Mater. 31(7), 431–447 (1999).
[Crossref]

Opt. Express (6)

Opt. Lett. (1)

Sensors and Actuators A (1)

C. F. R. Mateus, M. C. Y. Huang, and C. J. Chang-Hasnain, “Micromechanical tunable optical filters: general design rules for wavelengths from near-IR up to 10 μm,” Sensors and Actuators A 119(1), 57–62 (2005).
[Crossref]

Other (2)

Y. Sakurai, H. Yamakawa, Y. Yokota, A. Matsutani, T. Sakaguchi, and F. Koyama, “Hollow Waveguide Distributed Bragg Reflector for Widely Tunable Optical Devices,” in Optical Fiber Communication Conference, Technical Digest (CD) (Optical Society of America, 2006), paper OTuM4.

R.G. DeCorby, N. Ponnampalam, E. Epp, T. Allen, J.N. McMullin, “Chip-scale spectrometry based on tapered hollow Bragg waveguides,” submitted for publication.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1 Shown is a schematic illustration of the change in height of a buckled hollow waveguide (end facet view) driven by an increase in temperature. A positive temperature change increases the compressive stress in the upper mirror, since it comprises materials with higher thermal expansion coefficient than the underlying silicon substrate. The added compressive stress results in a slight change in shape and increased peak height for the buckle.
Fig. 2
Fig. 2 (a) Optical profilometer scans for a buckled waveguide with 60 μm base width, at a series of fixed temperatures. The inset plot shows the top of the curves in greater detail. (b) Plot of the measured change in peak height versus change in temperature. The straight line is a linear fit to the data.
Fig. 3
Fig. 3 (a) Cross-sectional schematic of a tapered air-core waveguide with omnidirectional claddings, near a mode cutoff point. The black arrows depict the ray-optics model of vertical radiation at cutoff, and the red curve represents the vertical field profile (m = 1 case shown) at the cutoff point. An increase in core height due to increase in temperature causes a positional shift of the cutoff point. (b) Images captured by an infrared camera via a microscope, showing the shift in the m = 1 to 5 cutoff positions with temperature, for a wavelength of 1550 nm. (c) The experimental shift in out-coupling position plotted versus change in temperature, for mode orders 1 to 4. The straight lines are linear fits to the data.
Fig. 4
Fig. 4 (a) Spectral scans obtained at a fixed out-coupling point corresponding to the m = 2 mode, for a series of fixed temperatures. The inset shows the 16.5 °C data on a logarithmic scale. (b) Plot of peak out-coupling wavelength versus temperature, revealing a wavelength shift of ~0.45 nm/°C.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

δ=h43(σ0σC1)=h43((bbmin)21)2hb3bmin,
σC=π212E(1ν2)h2b2.
Δσ=E(1ν)ΔαΔT,
(δ+Δδ)2=4h23[(σ0+Δσ)σC1],
Δδ2h23δΔσσC.
ΔδΔT=8π2(1ν2)(1ν)b2δΔα.
ΔzΔTΔzΔδΔδΔT=1STΔδΔT,
Δλ0ΔTΔλ0ΔzΔzΔT=1DTΔzΔT,
ΔλFWHMzPDT+λ0(m+1)π(R1R)   ,

Metrics