## Bragg waveguides with low-index liquid cores |

Optics Express, Vol. 20, Issue 1, pp. 48-62 (2012)

http://dx.doi.org/10.1364/OE.20.000048

Acrobat PDF (1359 KB)

### Abstract

The spectral properties of light confined to low-index media by binary layered structures is discussed. A novel phase-based model with a simple analytical form is derived for the approximation of the center of arbitrary bandgaps of binary layered structures operating at arbitrary effective indices. An analytical approximation to the sensitivity of the bandgap center to changes in the core refractive index is thus derived. Experimentally, significant shifting of the fundamental bandgap of a hollow-core Bragg fiber with a large cladding layer refractive index contrast is demonstrated by filling the core with liquids of various refractive indices. Confirmation of these results against theory is shown, including the new analytical model, highlighting the importance of considering material dispersion. The work demonstrates the broad and sensitive tunability of Bragg structures and includes discussions on refractive index sensing.

© 2011 OSA

## 1. Introduction

1. P. Yeh and A. Yariv, “Bragg reflection waveguides,” Opt. Commun. **19**, 427–430 (1976). [CrossRef]

3. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. **68**, 1196–1201 (1978). [CrossRef]

4. H. Schmidt and A. Hawkins, “Optofluidic waveguides: I. Concepts and implementations,” Microfluid. Nanofluid. **4**, 3–16 (2008). [CrossRef] [PubMed]

6. D. Yin, H. Schmidt, J. P. Barber, E. J. Lunt, and A. R. Hawkins, “Optical characterization of arch-shaped ARROW waveguides with liquid cores,” Opt. Express **13**, 10564–10570 (2005). [CrossRef] [PubMed]

7. M. Skorobogatiy, “Microstructured and Photonic Bandgap Fibers for Applications in the Resonant Bio- and Chemical Sensors,” J. Sensors **2009**, 1–20 (2009). [CrossRef]

8. S. Campopiano, R. Bernini, L. Zeni, and P. M. Sarro, “Microfluidic sensor based on integrated optical hollow waveguides,” Opt. Lett. **29**, 1894–1896 (2004). [CrossRef] [PubMed]

9. P. Measor, S. Kühn, E. J. Lunt, B. S. Phillips, A. R. Hawkins, and H. Schmidt, “Multi-mode mitigation in an optofluidic chip for particle manipulation and sensing,” Opt. Express **17**, 24342–24348 (2009). [CrossRef]

10. B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO_{2} laser transmission,” Nature **420**, 650–653 (2002). [CrossRef] [PubMed]

11. K. Kuriki, O. Shapira, S. Hart, G. Benoit, Y. Kuriki, J. Viens, M. Bayindir, J. Joannopoulos, and Y. Fink, “Hollow multilayer photonic bandgap fibers for NIR applications,” Opt. Express **12**, 1510–1517 (2004). [CrossRef] [PubMed]

12. H. T. Bookey, S. Dasgupta, N. Bezawada, B. P. Pal, A. Sysoliatin, J. E. McCarthy, M. Salganskii, V. Khopin, and A. K. Kar, “Experimental demonstration of spectral broadening in an all-silica Bragg fiber,” Opt. Express **17**, 17130–17135 (2009). [CrossRef] [PubMed]

13. O. Shapira, K. Kuriki, N. D. Orf, A. F. Abouraddy, G. Benoit, J. F. Viens, A. Rodriguez, M. Ibanescu, J. D. Joannopoulos, Y. Fink, and M. M. Brewster, “Surface-emitting fiber lasers,” Opt. Express **14**, 3929–3935 (2006). [CrossRef] [PubMed]

14. J. Scheuer and X. Sun, “Radial Bragg resonators,” in *Photonic Microresonator Research and Applications*, I. Chremmos, O. Schwelb, and N. Uzunoglu, eds. (Springer Series in Optical Sciences, 2010), Chap. 15. [CrossRef]

15. D. Zhou and L. Mawst, “High-power single-mode antiresonant reflecting optical waveguide-type vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. **38**, 1599–1606 (2002). [CrossRef]

7. M. Skorobogatiy, “Microstructured and Photonic Bandgap Fibers for Applications in the Resonant Bio- and Chemical Sensors,” J. Sensors **2009**, 1–20 (2009). [CrossRef]

4. H. Schmidt and A. Hawkins, “Optofluidic waveguides: I. Concepts and implementations,” Microfluid. Nanofluid. **4**, 3–16 (2008). [CrossRef] [PubMed]

6. D. Yin, H. Schmidt, J. P. Barber, E. J. Lunt, and A. R. Hawkins, “Optical characterization of arch-shaped ARROW waveguides with liquid cores,” Opt. Express **13**, 10564–10570 (2005). [CrossRef] [PubMed]

9. P. Measor, S. Kühn, E. J. Lunt, B. S. Phillips, A. R. Hawkins, and H. Schmidt, “Multi-mode mitigation in an optofluidic chip for particle manipulation and sensing,” Opt. Express **17**, 24342–24348 (2009). [CrossRef]

7. M. Skorobogatiy, “Microstructured and Photonic Bandgap Fibers for Applications in the Resonant Bio- and Chemical Sensors,” J. Sensors **2009**, 1–20 (2009). [CrossRef]

8. S. Campopiano, R. Bernini, L. Zeni, and P. M. Sarro, “Microfluidic sensor based on integrated optical hollow waveguides,” Opt. Lett. **29**, 1894–1896 (2004). [CrossRef] [PubMed]

16. R. Bernini, S. Campopiano, and L. Zeni, “Design and analysis of an integrated antiresonant reflecting optical waveguide refractive-index sensor,” Appl. Opt. **41**, 70–73 (2002). [CrossRef] [PubMed]

4. H. Schmidt and A. Hawkins, “Optofluidic waveguides: I. Concepts and implementations,” Microfluid. Nanofluid. **4**, 3–16 (2008). [CrossRef] [PubMed]

5. A. Hawkins and H. Schmidt, “Optofluidic waveguides: II. Fabrication and structures,” Microfluid. Nanofluid. **4**, 17–32 (2008). [CrossRef]

8. S. Campopiano, R. Bernini, L. Zeni, and P. M. Sarro, “Microfluidic sensor based on integrated optical hollow waveguides,” Opt. Lett. **29**, 1894–1896 (2004). [CrossRef] [PubMed]

17. G. Testa, Y. Huang, P. M. Sarro, L. Zeni, and R. Bernini, “High-visibility optofluidic Mach-Zehnder interferometer,” Opt. Lett. **35**, 1584–1586 (2010). [CrossRef] [PubMed]

*both*layer types, as described by Ref. [18

18. K. J. Rowland, S. Afshar V., and T. M. Monro, “Bandgaps and antiresonances in integrated-ARROWs and Bragg fibers; a simple model,” Opt. Express **16**, 17935–17951 (2008). [CrossRef] [PubMed]

18. K. J. Rowland, S. Afshar V., and T. M. Monro, “Bandgaps and antiresonances in integrated-ARROWs and Bragg fibers; a simple model,” Opt. Express **16**, 17935–17951 (2008). [CrossRef] [PubMed]

21. F. Benabid, P. J. Roberts, F. Couny, and P. S. Light, “Light and gas confinement in hollow-core photonic crystal fibre based photonic microcells,” J. Eur. Opt. Soc. **4**, 1–9 (2009). [CrossRef]

18. K. J. Rowland, S. Afshar V., and T. M. Monro, “Bandgaps and antiresonances in integrated-ARROWs and Bragg fibers; a simple model,” Opt. Express **16**, 17935–17951 (2008). [CrossRef] [PubMed]

22. J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Light-wave Technol. **11**, 416–423 (1993). [CrossRef]

*e.g.*, Refs. [4

**4**, 3–16 (2008). [CrossRef] [PubMed]

5. A. Hawkins and H. Schmidt, “Optofluidic waveguides: II. Fabrication and structures,” Microfluid. Nanofluid. **4**, 17–32 (2008). [CrossRef]

9. P. Measor, S. Kühn, E. J. Lunt, B. S. Phillips, A. R. Hawkins, and H. Schmidt, “Multi-mode mitigation in an optofluidic chip for particle manipulation and sensing,” Opt. Express **17**, 24342–24348 (2009). [CrossRef]

17. G. Testa, Y. Huang, P. M. Sarro, L. Zeni, and R. Bernini, “High-visibility optofluidic Mach-Zehnder interferometer,” Opt. Lett. **35**, 1584–1586 (2010). [CrossRef] [PubMed]

23. S. Kühn, P. Measor, E. J. Lunt, A. R. Hawkins, and H. Schmidt, “Particle manipulation with integrated optofluidic traps,” Digest of the IEEE/LEOS Summer Topical Meetings , pp. 187–188 (2008). [CrossRef]

**29**, 1894–1896 (2004). [CrossRef] [PubMed]

16. R. Bernini, S. Campopiano, and L. Zeni, “Design and analysis of an integrated antiresonant reflecting optical waveguide refractive-index sensor,” Appl. Opt. **41**, 70–73 (2002). [CrossRef] [PubMed]

**16**, 17935–17951 (2008). [CrossRef] [PubMed]

*ñ*) and apply it to the case of variable core indices in multilayer waveguides. We derive a simple expression for the approximation of the center of arbitrary bandgaps, and their sensitivities, over arbitrary

*ñ*; the expression is simple in that it can be evaluated directly with input of only the layer refractive indices and thicknesses and the orders of the desired cladding bandgaps/resonances. These approximate analytical expressions are then confirmed against a full Bloch bandgap analysis. The new theory can be applied to any system with oblique incidence upon a binary structure with arbitrary layer indices from an arbitrary refractive index or incidence angle.

## 2. Experiment – liquid filled Bragg fiber

10. B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO_{2} laser transmission,” Nature **420**, 650–653 (2002). [CrossRef] [PubMed]

11. K. Kuriki, O. Shapira, S. Hart, G. Benoit, Y. Kuriki, J. Viens, M. Bayindir, J. Joannopoulos, and Y. Fink, “Hollow multilayer photonic bandgap fibers for NIR applications,” Opt. Express **12**, 1510–1517 (2004). [CrossRef] [PubMed]

*λ*≈ 700 nm. A 15 cm length of this fiber was used in the following experiments. The hollow core had diameter

*t*

_{core}≈ 330

*μ*m and was surrounded by a periodic cladding of concentric rings with 9 pairs of layers consisting of Arsenic Trisulphide (As

_{2}S

_{3}) chalcogenide glass and Poly-ether Imide (PEI) polymer of thicknesses

*t*

_{1}≈ 76 nm and

*t*

_{0}≈ 124 nm, respectively. The cladding was terminated by a thick protective jacket of PEI producing a total outer diameter (OD) of 585

*μ*m. The first and final As

_{2}S

_{3}layers of the cladding were half-thickness so as to minimize guided surface states (which, through coupling, can introduce loss features in the core transmission spectrum).

_{2}S

_{3}and PEI have non-negligible material dispersion. Figure 2 shows the refractive indices of the two materials over the wavelength range of interest. The curves are fits to experimental datapoints (measured via an ellipsometric technique [31

31. MIT Photonics Bandgap Fibers and Devices Group material database, http://mit-pbg.mit.edu/Pages/DataBase.html.

^{th}-order Gaussian series (

*a*and

_{n}*b*) to fit the data to within a 95% confidence interval. The results are two continuous interpolation functions

_{n}*n*

_{As2S3}(

*λ*) and

*n*

_{PEI}(

*λ*). This fit was used instead of a conventional Sellmeier fit (a series of inverse powers of wavelength) for reasons of convenience (a fitting routine was readily available). It is assumed that the refractive indices of the layers do not change from these distributions during the fiber drawing process – a reasonable approximation for these fibers [11

11. K. Kuriki, O. Shapira, S. Hart, G. Benoit, Y. Kuriki, J. Viens, M. Bayindir, J. Joannopoulos, and Y. Fink, “Hollow multilayer photonic bandgap fibers for NIR applications,” Opt. Express **12**, 1510–1517 (2004). [CrossRef] [PubMed]

^{™}Compact). The liquids used to fill the fiber were ‘Immersion Liquids’ from Cargille

^{™}with refractive indices 1.4019, 1.4620 and 1.5780 (all standardized at a wavelength of

*λ*= 589.3 nm at a temperature of 25°

*C*). According to the product data, the chromatic dispersion of the liquids was negligible compared to the dispersive properties of the fiber materials, modes and bandgap edges and so was not considered here. All liquids used were relatively transparent over the entire visible range so that, compared to the waveguide losses, the liquid material losses were negligible over the considered spectrum. The light transmitted through the fiber was subsequently coupled into a spectrum analyser (Fig. 3) with spectral resolution

*δλ*= 0.05 nm. Each trace of the output spectrum was point-averaged over 200 samples to increase the signal to noise ratio.

*n*

_{core}≈ 1, and when filled with each liquid,

*n*

_{core}≈ 1.4018,1.4720,1.5780. Each spectrum is normalized to its own maximum value,

*i.e.*, no relative loss information is contained in this representation. There is a clear trend followed by the set of peaks: as the core index increases, the transmission wavelengths monotonically decrease, shifting across almost the entire visible spectrum; from right to left in Fig. 4, the peaks have maxima at wavelengths of

*λ*

_{peak}≈ 700 nm,555 nm,533 nm and 500 nm, respectively. The trend is almost linear (Fig. 4,

*right*) due to the layers’ material dispersion having the effect of ‘straightening out’ the bandgap edges (discussed later, § 3.1). Also, the peak width appears to initially decrease and then increase again: from right to left, the peaks have full widths at half-maximum of 30 nm, 21 nm, 33 nm and 40 nm, respectively. This behavior coincides with what is qualitatively expected of the fundamental TM bandgap and the associated Brewster condition, discussed below. These results are summarized in Table 1.

*n*

_{core}, and thus the sensitivity of a sensor based on this mechanism, say, is not linear due to the dispersive properties of the band edges. Here, however, as explained later in § 3.1, the dispersion of the layer materials themselves has the effect of ‘straightening out’ the band edges; this explains why the shifting of the peaks follows an approximately linear trend, as shown in Fig. 4 (

*right*). Given this, the average shifting of the transmission peaks with core index here corresponds to a linear sensitivity of

*∂λ*

_{peak}/

*∂n*

_{core}≈ 330 nm/RIU. This sensitivity value is comparable with the I-ARROW refractive index sensor architecture of Ref. [8

**29**, 1894–1896 (2004). [CrossRef] [PubMed]

*e.g.*, Refs. [7

**2009**, 1–20 (2009). [CrossRef]

27. H. Qu and M. Skorobogatiy, “Liquid-core low-refractive-index-contrast Bragg fiber sensor,” Appl. Phys. Lett. **98**, 201114 (2011). [CrossRef]

27. H. Qu and M. Skorobogatiy, “Liquid-core low-refractive-index-contrast Bragg fiber sensor,” Appl. Phys. Lett. **98**, 201114 (2011). [CrossRef]

*n*

_{core}≈ 1.4018 to 1.5780 (Δ

*n*

_{core}≈ 0.176) vs.

*n*

_{core}= 1.33 to 1.38 (Δ

*n*

_{core}≈ 0.05); about a 3.5 times larger dynamic range. This increase in dynamic range is aided by the omnidirectional nature of the bandgap used here: the gaps are open for all effective indices (Fig. 5, § 3.1), save for the TM gap closure at the Brewster condition (§ 3.1), allowing the core index to continuously deviate far below

*n*

_{0}without forcing all guided light out of the bandgap due to complete gap closure [18

**16**, 17935–17951 (2008). [CrossRef] [PubMed]

## 3. Background theory and comparison with experiment

*θ*in Fig. 1. For a given plane wave wavevector with amplitude

*k*=

_{i}*n*(Fig. 1), where

_{i}k*k*= 2

*π/λ*(

*λ*is the free-space wavelength) and

*n*is the dielectric refractive index the wave is supported in (

_{i}*n*

_{core}≤

*n*

_{0}<

*n*

_{1}, Fig. 1), both

*n*and

_{i}*θ*affect the amplitudes of the wavevector components normal and parallel to the layers:

*k*

_{i⊥}=

*n*cos

_{i}k*θ*and

*k*

_{i}_{||}=

*n*sin

_{i}k*θ*, respectively. The effective refractive index of the wave is defined as

*ñ*=

*k*

_{i}_{||}/

*k*. Note that

*k*

_{i}_{||}is often represented in the literature as

*β*and called the

*propagation constant*(having the same value in all regions).

### 3.1. Bloch wave bandgaps in 1D photonic crystals

3. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. **68**, 1196–1201 (1978). [CrossRef]

30. W. J. Hsueh, S. J. Wun, and T. H. Yu, “Characterization of omnidirectional bandgaps in multiple frequency ranges of one-dimensional photonic crystals,” J. Opt. Soc. Am. B **27**, 1092–1098 (2010). [CrossRef]

*ϕ*=

_{i}*k*

_{⊥}

*t*is the phase accumulated by the plane wave component transverse to the stack within the

_{i}*i*

^{th}layer type. The transverse electric (TE) and transverse magnetic (TM) polarization forms of Λ are Λ

_{TE}= [

*k*

_{1⊥}/

*k*

_{0⊥}+

*k*

_{0⊥}/

*k*

_{1⊥}]/2 and

*ζ*values in the range

*ζ*= [−1,1] (

*e.g.*, Refs. [3

3. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. **68**, 1196–1201 (1978). [CrossRef]

30. W. J. Hsueh, S. J. Wun, and T. H. Yu, “Characterization of omnidirectional bandgaps in multiple frequency ranges of one-dimensional photonic crystals,” J. Opt. Soc. Am. B **27**, 1092–1098 (2010). [CrossRef]

*ζ*| ≤ 1 and bandgaps for |

*ζ*| > 1.

*n*=

_{i}*n*(

_{i}*λ*= 700 nm) – Fig. 5,

*top*] and of layer material dispersion [

*n*=

_{i}*n*(

_{i}*λ*) – Fig. 5,

*bottom*]. These Bloch spectra are based on the layer parameters (

*t*and

_{i}*n*; see § 2) of the fiber used in the experiment above. Figure 5 also shows plots of the refractive indices of the core liquids, low-index layer (the high-index value is beyond the axis limits) and the effective index of the Brewster condition:

_{i}**16**, 17935–17951 (2008). [CrossRef] [PubMed]

*n*

_{0}and Brewster – dashed lines) are curved in the material dispersion case due to the index behaviour shown in Fig. 2, but the core indices (solid lines) are straight in both cases due to negligible material dispersion of the liquids used. In both cases, the Brewster line coincides with the TM gap closure, as expected [3

**68**, 1196–1201 (1978). [CrossRef]

**16**, 17935–17951 (2008). [CrossRef] [PubMed]

*ñ*; this is demonstrated in Fig. 4 (

*right*). Note that the TM bandgaps appear to predominantly determine the transmission spectra peak positions, as may be expected due to the random polarization of the input light and the large number of modes supported for the length of fiber used (it was not within the effective few- or single-moded regime which is dominated by the low loss TE

_{01}mode [29

29. K. J. Rowland, S. Afshar V., and T. M. Monro, “Novel low-loss bandgaps in all-silica Bragg fibers,” J. Light-wave Technol. **26**, 43–51 (2008). [CrossRef]

**16**, 17935–17951 (2008). [CrossRef] [PubMed]

*ñ*is defined here to be the mean of wavenumber (

*k*= 2

*π/λ*) values of the band edges (|

*ζ*| = 1) either side of a bandgap at a given value of

*ñ*: where

*k*at which a given

*ñ*line intersects the bandgap edges. Note that

*λ*values aren’t averaged since the gaps (or more precisely, the resonances of a given layer [18

**16**, 17935–17951 (2008). [CrossRef] [PubMed]

*k*(and frequency

*ω*, etc.) not

*λ*. The Bloch gap central wavelength

*ñ*

_{B}as it must, aiding the increase in dynamic range as discussed above.

### 3.2. Layer resonances and the SPARROW model

*k*

_{mi}and

*k*

_{mj}for {

*i, j*} = {1,0}) at a given

*ñ*, instead of two band edges (

*k*

_{mi,j}(

*ñ*) are the resonance curves [given by Eq. (3)] of the high- (order

*m*

_{1}) and/or low-index (order

*m*

_{0}) layers forming adjacent sides of the bounding region of interest;

*i*and

*j*take values 1 or 0 depending upon which curves form the bounding region at the given

*ñ*[18

**16**, 17935–17951 (2008). [CrossRef] [PubMed]

*k*′

_{c}in Ref. [18

**16**, 17935–17951 (2008). [CrossRef] [PubMed]

*k*

_{m1}(

*ñ*) and

*k*

_{m0}(

*ñ*) (at an intersection point) over a range of

*ñ*.

## 4. New theory and comparison with experiment

*ñ*≲

*n*

_{0}is derived in a simple analytical form.

*ñ*and is here applied to the case of multilayer waveguides with variable core refractive indices. The theory produces a simple analytical approximate expression for the calculation of the gap center – simple in that the expression requires only input of the layer refractive indices and thicknesses and the order of the layer bandgaps/resonances of interest. From this, an analytical expression for the sensitivity to changes in core index is derived. Both the bandgap center and sensitivity expressions are shown to be a good approximation when compared to the calculated Bloch gap center, even for non-negligible material dispersion. The above experimental results are used as a specific example for validation, where it is also shown how material dispersion of the layers must be considered in order for this model (as for the bandgap spectra above) to agree with the observed filled Bragg fiber transmission spectra.

*e.g.*,

*P*of the SPARROW model [18

**16**, 17935–17951 (2008). [CrossRef] [PubMed]

*arbitrary*single-pass phase accumulation within each layer, not just the cases of resonance (integer multiples of

*π*) or antiresonance (half-integer multiples of

*π*). By then enforcing the restriction that the

*combined*phase accumulation of both layer types is a constant integer multiple of

*π*(not

*ϕ*individually),

_{i}*ϕ*

_{1}+

*ϕ*

_{0}=

*π*,2

*π*,... = const.,

*P*will sweep out a parametric curve within the bound region and hence within the associated bandgap (since, as discussed, each bound region houses a bandgap). Thus,

*P*defines a ‘central curve’ through a given bandgap.

*P*is expressed as: where

*ρ*=

_{i}*m*(

_{i}_{μ}/t_{i}*i*= 1,0) and

*η*=

*ρ*

_{1}/

*ρ*

_{0}. The parametrized phase orders are defined here as

*m*

_{1}

*=*

_{μ}*m*

_{1}–

*μ*and

*m*

_{0}

*=*

_{μ}*m*

_{0}+

*μ*with

*m*∈

_{i}^{+}and

*μ*∈

^{+}, representing (via

*μ*) arbitrary phase accumulation within the high- and low-index layers, respectively. The addition of the accumulated phases in each layer thus produces a constant value

*ϕ*

_{1}+

*ϕ*

_{0}= (

*m*

_{1}

*+*

_{μ}*m*

_{0}

*)*

_{μ}*π*= (

*m*

_{1}+

*m*

_{0})

*π*=

*π,*2

*π*,... = const. ∀ (

*k, ñ*) on

*P*, as asserted above. Note that this expression is simple in that it returns the

_{μ}*k*along

*P*at an effective index

_{μ}*ñ*given only the values of the layer properties

*t*and

_{i}*n*, the order of the bandgap of interest 〈

_{i}*m*

_{1},

*m*

_{0}〉 (cf. Ref. [18

**16**, 17935–17951 (2008). [CrossRef] [PubMed]

*μ*which effectively determines the position along the

*P*curve. In practice, this expression can be easily evaluated over a range of

_{μ}*μ*to define a curve through a given gap over (

*λ,ñ*) as shown in Fig. 5.

*P*is identical to

_{μ}*P*defined in Ref. [18

**16**, 17935–17951 (2008). [CrossRef] [PubMed]

*m*are generalized to be continuous (

_{iμ}*μ*∈

^{+}) instead of discrete integers or half integers (

*μ*= 0 or ½), representing arbitrary phase accumulation within each layer instead of just resonance and antiresonance, respectively. The integer terms

*m*∈

_{i}^{+}correspond, via

*P*, to the local bound region of order 〈

_{μ}*m*

_{1},

*m*

_{0}〉 (a nomenclature suggested previously [18

**16**, 17935–17951 (2008). [CrossRef] [PubMed]

*m*

_{1},

*m*

_{0}〉,

*P*traces out a curve within the region for

_{μ}*μ*= 0 → 1, starting at the maximal bounding point

*P*

_{0}[

*P*of order (

_{μ}*m*

_{1}

*, m*

_{0})], passing through the central point

*P*

_{½}[

*P*of order (

_{μ}*m*

_{1}– ½,

*m*

_{0}+ ½)] and finishing at the minimal bounding point

*P*

_{1}[

*P*of order (

_{μ}*m*

_{1}

*–*1

*,m*

_{0}+ 1)].

*P*for varying

_{μ}*μ*passes through the closure points (

*P*

_{0}and

*P*

_{1}) and the approximate central point (

*P*

_{½}) of an arbitrary bandgap (when these points exist in the domain 0 ≤

*ñ*≤

*n*

_{0}for a given bound region 〈

*m*

_{1},

*m*

_{0}〉).

*P*thus provides an approximation to the central

_{μ}*λ*or

*k*of a given bandgap for arbitrary

*ñ*within the gap. Like the SPARROW model it is inherited from, this generalized central curve definition holds for any alteration in

*ñ*(or

*k*), such as when the core size or shape is altered, higher-order modes are considered or when, as for the case here, the core refractive index of a Bragg waveguide is changed directly.

*P*provides an agreement to better than 0.5% for the TM bandgap and 4% for the TE bandgap central wavelengths for all core refractive indices considered. The agreement is shown in Fig. 5 (

_{μ}*top*) where the

*P*curves appear to lie on top of the calculated Bloch bandgap center curves (

_{μ}*bottom*) produces slightly larger wavelength differences of 0.8% and 5% for the TM and TE gaps, respectively. In both cases, the TE gap center begins to deviate for higher

*ñ*due to the fact that

*P*appears to intercept the TM gap closure point (

_{μ}*P*

_{B}, due to the Brewster effect [3

**68**, 1196–1201 (1978). [CrossRef]

**16**, 17935–17951 (2008). [CrossRef] [PubMed]

*P*approximation to the gap center breaks down as

_{μ}*ñ*→

*n*

_{0}for gaps terminating on the low-index line

*ñ*=

*n*

_{0}, where

*e.g.*, Ref. [27

27. H. Qu and M. Skorobogatiy, “Liquid-core low-refractive-index-contrast Bragg fiber sensor,” Appl. Phys. Lett. **98**, 201114 (2011). [CrossRef]

*P*and the Bloch bandgap center,

_{μ}*P*also shows a reasonable agreement with the experimentally measured transmission peak positions, compared directly in Fig. 4.

_{μ}### 4.1. Sensitivity to refractive index

*λ*to changes in

*ñ*along the

*P*curve is now derived, extending the theory developed above. For large core waveguides, where

_{μ}*ñ*≈

*n*

_{core}, this is thus a measure of the sensitivity of the transmission peak central wavelength to changes in the core index. As for the definition of

*P*, this treatment is applicable to arbitrary bandgaps of arbitrary binary stacks (waveguide or otherwise). Using this general expression, an example is given based on the fundamental bandgap of the Bragg fiber cladding structure considered above.

_{μ}*ñ*versus the wavenumber

_{μ}*k*(hence frequency or wavelength):

_{μ}*∂ñ*. The inverse of this,

_{μ}/∂k_{μ}*∂k*, is thus a measure of the sensitivity of a bandgap center to changes in the effective refractive index of the guided light. Note that this closed form derivation assumes the refractive indices are locally flat over the spectrum,

_{μ}/∂ñ_{μ}*i.e.*, at a wavelength

*λ*′, the index takes value

*n*(

_{i}*λ*) =

*n*(

_{i}*λ*′) according to it’s material dispersion (

*e.g.*, Fig. 2) but one assumes

*∂n*= 0. Later, this approximation is compared to the full numerical derivative which inherently includes the material dispersion derivative (

_{i}/∂λ*∂n*≠ 0). The approximate closed form of the sensitivity is now derived.

_{i}/∂λ*λ*= 2

_{μ}*π/k*, implying

_{μ}*λ*component of the

_{μ}*P*point for changes in

_{μ}*ñ*[as does Eq. (10) for the sensitivity of

_{μ}*k*]. It thus also provides an approximation to the sensitivity of an arbitrary Bloch bandgap center to changes in

_{μ}*ñ*. In the large-core regime here where

*ñ*≈

*n*

_{core}, this is thus an approximation to the sensitivity of the Bloch bandgap center (and hence waveguide transmission peak) to changes in the core refractive index. This expression for the sensitivity can be evaluated for any given point (any value of

*μ*) on the central curve

*P*= (

_{μ}*k*) and obviates the need for numerical derivation of

_{μ}, ñ_{μ}*P*or

_{μ}*k, ñ*) (keeping in mind the approximation of locally flat material dispersion:

*∂n*= 0).

_{i}/∂λ*n*= const., Eq. (11) provides an excellent approximation to the Bloch bandgap sensitivity

_{i}*top*) demonstrates this for the Bragg fiber considered above. The Bloch gap sensitivity is calculated numerically, directly from

*P*and

_{μ}*n*=

_{i}*n*(

_{i}*λ*) are permitted but

*∂n*= 0 is enforced] it can be used as a reasonable analytic approximation to the Bloch bandgap center sensitivity when material dispersion cannot be neglected, demonstrated in Fig. 6 (

_{i}/∂λ*bottom*). In this case,

*k*and

_{μ}*ñ*are calculated from Eq. (5) via root-finding and used to evaluate Eq. (11) directly,

_{μ}*i.e.*,

*P*is evaluated with material dispersion, but the evaluation of the derivative assumes the dispersion of the layer indices is locally flat (

_{μ}*∂n*= 0). The agreement deviates significantly toward shorter wavelengths as may be expected – this is the region where the material indices fluctuate most (cf. Fig. 2) – but is still within 50% of the TM Bloch gap sensitivity for

_{i}/∂λ*λ*≈ 570 nm, and much better for longer

*λ*.

*∂n*≠ 0) results in more complex expressions. One straightforward alternative is to use numerical root-finding of the

_{i}/∂λ*P*coordinates (as above) but then also numerically calculate the local slope, rather than approximating it analytically as per Eq. (11); this inherently includes the spectral derivatives of the layer materials but requires a significant number of calculated points, and hence iterations, for sufficient precision. The evaluation of the Bloch gap sensitivity in Fig. 6 was also calculated numerically in this fashion and hence also inherently includes the full material dispersion. Figure 6 (

_{μ}*bottom*) shows the comparison between these numerical derivatives of

*P*and the Bloch bandgap central wavelength when the layers’ material dispersion is considered. As expected from their good agreement in absolute value as per Fig. 5, their sensitivities also agree well: below 1% for

_{μ}*λ*≳ 575 nm and better than 9.5% over all wavelengths considered for the TM gap centre.

*P*(both including material dispersion) agree well with experiment. From four data points, the experimental results implied an approximately linear sensitivity of 330 nm/RIU, § 2. As Fig. 6 shows, from a continuous range of points over the wavelengths of interest (

_{μ}*λ*=700 nm–500 nm), the numerical calculations predict sensitivities of

*∂λ*/

_{μ}*∂ñ*≈ 200–422 nm/RIU. The analytical approximation of the latter [Eq. (11)] agrees well with these values for longer wavelengths but deviates, increasing to ≈719 nm/RIU at

*λ*=500 nm, due to the appreciable material dispersion at shorter

*λ*(Fig. 2). In regimes of non-negligible dispersion, Eq. (11) is thus useful as a rapid design tool, with the full numerical values required for more precise calculations.

### 4.2. Sensitivity trends

*P*[Eq. (5)] and hence

_{μ}*∂λ*[Eq. (11)] allows some fundamental physical observations to be made with respect to the sensitivity. The case considered here has layers of a high refractive index contrast. The

_{μ}/∂ñ*∂λ*/

_{μ}*∂ñ*implies that layers with a lower refractive index contrast would produce a more sensitive response to

_{μ}*ñ*(core index here). Also, the

*ñ*variations closer to the low layer index (

*ñ*=

*n*

_{0}) will induce more sensitive spectral shifts. One can see from Fig. 5, for example, that this is the case since the gaps generally flatten out as

*ñ*→

*n*

_{0}over (

*λ,ñ*), and is demonstrated explicitly by the sensitivity curves of Fig. 6.

**98**, 201114 (2011). [CrossRef]

*arbitrary*refractive index, not just to the regime of low index contrast, say. It is thus useful as an analysis and design tool for many platforms and devices of interest today with arbitrary layer indices (of high or low contrast) and arbitrary core (

*n*

_{core}) or effective mode indices (

*ñ*) – up to the aforementioned

*ñ*→

*n*

_{0}approximation limit (§ 4) – such as most modern hollow Bragg fibers and I-ARROWs that can be filled with liquids.

## 5. Discussion and conclusion

*∂λ*

_{peak}/

*∂n*

_{core}≈ 330 nm/RIU, which is comparable with the results of a similar I-ARROW based architecture (which relies on detection of a transmission minimum, not maximum as used here) [8

**29**, 1894–1896 (2004). [CrossRef] [PubMed]

**98**, 201114 (2011). [CrossRef]

*P*[Eq. (5)]: a generalized, parametric, version of the intersection point

_{μ}*P*of the SPARROW model [18

**16**, 17935–17951 (2008). [CrossRef] [PubMed]

*n*and

_{i}*t*and the bandgap/resonance order 〈

_{i}*m*

_{1},

*m*

_{0}〉 of interest.

*P*was shown to be a close approximation to the central frequency of the considered bandgap spectra. For the Bragg fiber cladding considered here,

_{μ}*P*approximated the TM Bloch bandgap central wavelength to better than 0.8% for all cases considered. For large core waveguides (where

_{μ}*ñ*≈

*n*

_{core}), this analytic expression can be used to analyse and design binary layer waveguides with low-index cores for arbitrary layer parameters, core indices, and bandgaps/resonances.

*P*allowed an analytic expression for its derivative to be found (

_{μ}*∂λ*), thus describing the sensitivity of the approximate bandgap central frequency to changes in the effective index

_{μ}/∂ñ_{μ}*ñ*(by altering the core index, say). Good agreement between the analytic

*P*sensitivity and the numerically calculated Bloch bandgap center sensitivity was shown (Fig. 6). The sensitivity expressions also agreed well with the measured sensitivity of the filled Bragg fiber considered (≈ 330 nm/RIU), which included the effects of nontrivial layer material dispersion while maintaining analyticity (by assuming the layer indices, while variable with

_{μ}*λ*, are everywhere locally flat). The expression was used to show how the sensitivity could be enhanced by using low refractive index contrasts between cladding layers and/or between the core and cladding layers or, alternatively, how the sensitivity could be reduced by using high contrasts to produce devices with invariant spectral properties under fluctuating sample indices.

13. O. Shapira, K. Kuriki, N. D. Orf, A. F. Abouraddy, G. Benoit, J. F. Viens, A. Rodriguez, M. Ibanescu, J. D. Joannopoulos, Y. Fink, and M. M. Brewster, “Surface-emitting fiber lasers,” Opt. Express **14**, 3929–3935 (2006). [CrossRef] [PubMed]

15. D. Zhou and L. Mawst, “High-power single-mode antiresonant reflecting optical waveguide-type vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. **38**, 1599–1606 (2002). [CrossRef]

## References and links

1. | P. Yeh and A. Yariv, “Bragg reflection waveguides,” Opt. Commun. |

2. | P. Yeh, A. Yariv, and C. S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. |

3. | P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. |

4. | H. Schmidt and A. Hawkins, “Optofluidic waveguides: I. Concepts and implementations,” Microfluid. Nanofluid. |

5. | A. Hawkins and H. Schmidt, “Optofluidic waveguides: II. Fabrication and structures,” Microfluid. Nanofluid. |

6. | D. Yin, H. Schmidt, J. P. Barber, E. J. Lunt, and A. R. Hawkins, “Optical characterization of arch-shaped ARROW waveguides with liquid cores,” Opt. Express |

7. | M. Skorobogatiy, “Microstructured and Photonic Bandgap Fibers for Applications in the Resonant Bio- and Chemical Sensors,” J. Sensors |

8. | S. Campopiano, R. Bernini, L. Zeni, and P. M. Sarro, “Microfluidic sensor based on integrated optical hollow waveguides,” Opt. Lett. |

9. | P. Measor, S. Kühn, E. J. Lunt, B. S. Phillips, A. R. Hawkins, and H. Schmidt, “Multi-mode mitigation in an optofluidic chip for particle manipulation and sensing,” Opt. Express |

10. | B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO |

11. | K. Kuriki, O. Shapira, S. Hart, G. Benoit, Y. Kuriki, J. Viens, M. Bayindir, J. Joannopoulos, and Y. Fink, “Hollow multilayer photonic bandgap fibers for NIR applications,” Opt. Express |

12. | H. T. Bookey, S. Dasgupta, N. Bezawada, B. P. Pal, A. Sysoliatin, J. E. McCarthy, M. Salganskii, V. Khopin, and A. K. Kar, “Experimental demonstration of spectral broadening in an all-silica Bragg fiber,” Opt. Express |

13. | O. Shapira, K. Kuriki, N. D. Orf, A. F. Abouraddy, G. Benoit, J. F. Viens, A. Rodriguez, M. Ibanescu, J. D. Joannopoulos, Y. Fink, and M. M. Brewster, “Surface-emitting fiber lasers,” Opt. Express |

14. | J. Scheuer and X. Sun, “Radial Bragg resonators,” in |

15. | D. Zhou and L. Mawst, “High-power single-mode antiresonant reflecting optical waveguide-type vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. |

16. | R. Bernini, S. Campopiano, and L. Zeni, “Design and analysis of an integrated antiresonant reflecting optical waveguide refractive-index sensor,” Appl. Opt. |

17. | G. Testa, Y. Huang, P. M. Sarro, L. Zeni, and R. Bernini, “High-visibility optofluidic Mach-Zehnder interferometer,” Opt. Lett. |

18. | K. J. Rowland, S. Afshar V., and T. M. Monro, “Bandgaps and antiresonances in integrated-ARROWs and Bragg fibers; a simple model,” Opt. Express |

19. | M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO |

20. | N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. |

21. | F. Benabid, P. J. Roberts, F. Couny, and P. S. Light, “Light and gas confinement in hollow-core photonic crystal fibre based photonic microcells,” J. Eur. Opt. Soc. |

22. | J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Light-wave Technol. |

23. | S. Kühn, P. Measor, E. J. Lunt, A. R. Hawkins, and H. Schmidt, “Particle manipulation with integrated optofluidic traps,” Digest of the IEEE/LEOS Summer Topical Meetings , pp. 187–188 (2008). [CrossRef] |

24. | J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, |

25. | P. Yeh, |

26. | K. J. Rowland, S. Afshar V., A. Stolyarov, Y. Fink, and T. M. Monro, “Spectral properties of liquid-core Bragg fibers”, Conference on Lasers and Electro-Optics (CLEO), Baltimore, Maryland, US, June 2–4 2009. |

27. | H. Qu and M. Skorobogatiy, “Liquid-core low-refractive-index-contrast Bragg fiber sensor,” Appl. Phys. Lett. |

28. | D. Yin, H. Schmidt, J. Barber, and A. Hawkins, “Integrated ARROW waveguides with hollow cores,” Opt. Express |

29. | K. J. Rowland, S. Afshar V., and T. M. Monro, “Novel low-loss bandgaps in all-silica Bragg fibers,” J. Light-wave Technol. |

30. | W. J. Hsueh, S. J. Wun, and T. H. Yu, “Characterization of omnidirectional bandgaps in multiple frequency ranges of one-dimensional photonic crystals,” J. Opt. Soc. Am. B |

31. | MIT Photonics Bandgap Fibers and Devices Group material database, http://mit-pbg.mit.edu/Pages/DataBase.html. |

**OCIS Codes**

(060.2280) Fiber optics and optical communications : Fiber design and fabrication

(060.2400) Fiber optics and optical communications : Fiber properties

(230.1480) Optical devices : Bragg reflectors

(230.7370) Optical devices : Waveguides

(310.4165) Thin films : Multilayer design

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: October 14, 2011

Revised Manuscript: November 25, 2011

Manuscript Accepted: November 28, 2011

Published: December 19, 2011

**Citation**

Kristopher J. Rowland, Shahraam Afshar, Alexander Stolyarov, Yoel Fink, and Tanya M. Monro, "Bragg waveguides with low-index liquid cores," Opt. Express **20**, 48-62 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-1-48

Sort: Year | Journal | Reset

### References

- P. Yeh and A. Yariv, “Bragg reflection waveguides,” Opt. Commun.19, 427–430 (1976). [CrossRef]
- P. Yeh, A. Yariv, and C. S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am.67, 423–437 (1977). [CrossRef]
- P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am.68, 1196–1201 (1978). [CrossRef]
- H. Schmidt and A. Hawkins, “Optofluidic waveguides: I. Concepts and implementations,” Microfluid. Nanofluid.4, 3–16 (2008). [CrossRef] [PubMed]
- A. Hawkins and H. Schmidt, “Optofluidic waveguides: II. Fabrication and structures,” Microfluid. Nanofluid.4, 17–32 (2008). [CrossRef]
- D. Yin, H. Schmidt, J. P. Barber, E. J. Lunt, and A. R. Hawkins, “Optical characterization of arch-shaped ARROW waveguides with liquid cores,” Opt. Express13, 10564–10570 (2005). [CrossRef] [PubMed]
- M. Skorobogatiy, “Microstructured and Photonic Bandgap Fibers for Applications in the Resonant Bio- and Chemical Sensors,” J. Sensors2009, 1–20 (2009). [CrossRef]
- S. Campopiano, R. Bernini, L. Zeni, and P. M. Sarro, “Microfluidic sensor based on integrated optical hollow waveguides,” Opt. Lett.29, 1894–1896 (2004). [CrossRef] [PubMed]
- P. Measor, S. Kühn, E. J. Lunt, B. S. Phillips, A. R. Hawkins, and H. Schmidt, “Multi-mode mitigation in an optofluidic chip for particle manipulation and sensing,” Opt. Express17, 24342–24348 (2009). [CrossRef]
- B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature420, 650–653 (2002). [CrossRef] [PubMed]
- K. Kuriki, O. Shapira, S. Hart, G. Benoit, Y. Kuriki, J. Viens, M. Bayindir, J. Joannopoulos, and Y. Fink, “Hollow multilayer photonic bandgap fibers for NIR applications,” Opt. Express12, 1510–1517 (2004). [CrossRef] [PubMed]
- H. T. Bookey, S. Dasgupta, N. Bezawada, B. P. Pal, A. Sysoliatin, J. E. McCarthy, M. Salganskii, V. Khopin, and A. K. Kar, “Experimental demonstration of spectral broadening in an all-silica Bragg fiber,” Opt. Express17, 17130–17135 (2009). [CrossRef] [PubMed]
- O. Shapira, K. Kuriki, N. D. Orf, A. F. Abouraddy, G. Benoit, J. F. Viens, A. Rodriguez, M. Ibanescu, J. D. Joannopoulos, Y. Fink, and M. M. Brewster, “Surface-emitting fiber lasers,” Opt. Express14, 3929–3935 (2006). [CrossRef] [PubMed]
- J. Scheuer and X. Sun, “Radial Bragg resonators,” in Photonic Microresonator Research and Applications, I. Chremmos, O. Schwelb, and N. Uzunoglu, eds. (Springer Series in Optical Sciences, 2010), Chap. 15. [CrossRef]
- D. Zhou and L. Mawst, “High-power single-mode antiresonant reflecting optical waveguide-type vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron.38, 1599–1606 (2002). [CrossRef]
- R. Bernini, S. Campopiano, and L. Zeni, “Design and analysis of an integrated antiresonant reflecting optical waveguide refractive-index sensor,” Appl. Opt.41, 70–73 (2002). [CrossRef] [PubMed]
- G. Testa, Y. Huang, P. M. Sarro, L. Zeni, and R. Bernini, “High-visibility optofluidic Mach-Zehnder interferometer,” Opt. Lett.35, 1584–1586 (2010). [CrossRef] [PubMed]
- K. J. Rowland, S. Afshar, and T. M. Monro, “Bandgaps and antiresonances in integrated-ARROWs and Bragg fibers; a simple model,” Opt. Express16, 17935–17951 (2008). [CrossRef] [PubMed]
- M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett.49, 13–15 (1986). [CrossRef]
- N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett.27, 1592–1594 (2002). [CrossRef]
- F. Benabid, P. J. Roberts, F. Couny, and P. S. Light, “Light and gas confinement in hollow-core photonic crystal fibre based photonic microcells,” J. Eur. Opt. Soc.4, 1–9 (2009). [CrossRef]
- J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Light-wave Technol.11, 416–423 (1993). [CrossRef]
- S. Kühn, P. Measor, E. J. Lunt, A. R. Hawkins, and H. Schmidt, “Particle manipulation with integrated optofluidic traps,” Digest of the IEEE/LEOS Summer Topical Meetings, pp. 187–188 (2008). [CrossRef]
- J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).
- P. Yeh, Optical Waves in Layered Media (John Wiley & Sons Inc., 2005).
- K. J. Rowland, S. Afshar, A. Stolyarov, Y. Fink, and T. M. Monro, “Spectral properties of liquid-core Bragg fibers”, Conference on Lasers and Electro-Optics (CLEO), Baltimore, Maryland, US, June 2–4 2009.
- H. Qu and M. Skorobogatiy, “Liquid-core low-refractive-index-contrast Bragg fiber sensor,” Appl. Phys. Lett.98, 201114 (2011). [CrossRef]
- D. Yin, H. Schmidt, J. Barber, and A. Hawkins, “Integrated ARROW waveguides with hollow cores,” Opt. Express12, 2710–2715 (2004). [CrossRef] [PubMed]
- K. J. Rowland, S. Afshar, and T. M. Monro, “Novel low-loss bandgaps in all-silica Bragg fibers,” J. Light-wave Technol.26, 43–51 (2008). [CrossRef]
- W. J. Hsueh, S. J. Wun, and T. H. Yu, “Characterization of omnidirectional bandgaps in multiple frequency ranges of one-dimensional photonic crystals,” J. Opt. Soc. Am. B27, 1092–1098 (2010). [CrossRef]
- MIT Photonics Bandgap Fibers and Devices Group material database, http://mit-pbg.mit.edu/Pages/DataBase.html .

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