## Bandgaps and antiresonances in integrated-ARROWs and Bragg fibers; a simple model

Optics Express, Vol. 16, Issue 22, pp. 17935-17951 (2008)

http://dx.doi.org/10.1364/OE.16.017935

Acrobat PDF (2189 KB)

### Abstract

We consider the spectral properties of dielectric waveguides with low refractive index cores and binary layered claddings, such as Bragg fibers and integrated-ARROWs. We show that the full, nontrivial, 2-D spectrum of Bloch bands (hence bandgaps) of such claddings correspond, in structure and topology, to the dispersion properties of both constituent layer types; quantitatively demonstrating an intimate relationship between the bandgap and antiresonance guidance mechanisms. The dispersion functions of these layers, and the interactions thereof, thus form what we coin the Stratified Planar Anti-Resonant Reflecting OpticalWaveguide (SPARROW) model, capable of quantitative, analytic, descriptions of many nontrivial bandgap and antiresonance properties. The SPARROW model is useful for the spectral analysis and design of Bragg fibers and integrated-ARROWs with cores of arbitrary refractive index (equal to or less than the lowest cladding index). Both waveguide types are of interest for sensing and microfluidic applications due to their natural ability to guide light within low-index cores, permitting low-loss guidance within a large range of gases and liquids. A liquid-core Bragg fiber is discussed as an example, demonstrating the applicability of the SPARROW model to realistic and important waveguide designs.

© 2008 Optical Society of America

## 1. Introduction

1. P. Yeh, A. Yariv , and C. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. **67**, 423–438 (1977). [CrossRef]

8. T. Katagiri, Y. Matsuura, and M. Miyagi, “All-solid single-mode bragg fibers for compact fiber devices,” J. Lightwave Technol. **24**, 4314–4318 (2006). [CrossRef]

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

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

1. P. Yeh, A. Yariv , and C. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. **67**, 423–438 (1977). [CrossRef]

14. Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science **282**, 1679–1682 (1998). [CrossRef] [PubMed]

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

15. M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO_{2}-Si multilayer structures,” Appl. Phys. Lett. **49**, 13–15 (1986). [CrossRef]

10. D. Yin, J.P. Barber, A. R. Hawkins, and H. Schmidt, “Waveguide loss optimization in hollow-core ARROW waveguides,” Opt. Express **13**, 9331–9336 (2005). [CrossRef] [PubMed]

12. F. Poli, M. Foroni, A. Cucinotta, and S. Selleri, “Spectral behavior of integrated antiresonant reflecting hollow-core waveguides,” J. Lightwave Technol. **25**, 2599–2604 (2007). [CrossRef]

15. M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO_{2}-Si multilayer structures,” Appl. Phys. Lett. **49**, 13–15 (1986). [CrossRef]

*equal*to the lowest of the cladding indices [15

15. M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO_{2}-Si multilayer structures,” Appl. Phys. Lett. **49**, 13–15 (1986). [CrossRef]

21. N. M. Litchinitser, S. C. Dunn, P. E. Steinvurzel, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. M. de Sterke, “Application of an ARROW model for designing tunable photonic devices,” Opt. Express **12**, 1540–1550 (2004). [CrossRef] [PubMed]

*level-core*waveguides. Indeed, this line of thought saw the ARROWmodel successfully applied to (non-layered cladding) photonic crystal fibers (PCFs) [20

20. N. M. Litchinitser, S. C. Dunn, B. Usner, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. M. de Sterke, “Resonances in microstructured optical waveguides,” Opt. Express **11**, 1243–1251 (2003). [CrossRef] [PubMed]

21. N. M. Litchinitser, S. C. Dunn, P. E. Steinvurzel, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. M. de Sterke, “Application of an ARROW model for designing tunable photonic devices,” Opt. Express **12**, 1540–1550 (2004). [CrossRef] [PubMed]

19. 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]

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

*any value up to*that of the lowest cladding index; we will call these

*depressed-core*waveguides.

*All*cladding layer resonances in depressed-core waveguides influence the spectral behaviour of the core modes [9

**11**, 416–423 (1993). [CrossRef]

10. D. Yin, J.P. Barber, A. R. Hawkins, and H. Schmidt, “Waveguide loss optimization in hollow-core ARROW waveguides,” Opt. Express **13**, 9331–9336 (2005). [CrossRef] [PubMed]

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

_{2}-Si multilayer structures,” Appl. Phys. Lett. **49**, 13–15 (1986). [CrossRef]

18. A. K. Abeeluck, N. M. Litchinitser, C. Headley, and B. J. Eggleton, “Analysis of spectral characteristics of photonic bandgap waveguides,” Opt. Express **10**, 1320–1333 (2002). [PubMed]

21. N. M. Litchinitser, S. C. Dunn, P. E. Steinvurzel, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. M. de Sterke, “Application of an ARROW model for designing tunable photonic devices,” Opt. Express **12**, 1540–1550 (2004). [CrossRef] [PubMed]

*Stratified Planar*Anti-Resonant Reflecting OpticalWaveguide (SPARROW) model to distinguish it from both the original Archambault-ARROW model itself [9

**11**, 416–423 (1993). [CrossRef]

_{2}-Si multilayer structures,” Appl. Phys. Lett. **49**, 13–15 (1986). [CrossRef]

19. 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]

**12**, 1540–1550 (2004). [CrossRef] [PubMed]

**11**, 416–423 (1993). [CrossRef]

10. D. Yin, J.P. Barber, A. R. Hawkins, and H. Schmidt, “Waveguide loss optimization in hollow-core ARROW waveguides,” Opt. Express **13**, 9331–9336 (2005). [CrossRef] [PubMed]

22. D. Yin, H. Schmidt, J. P. Barber, and A. R. Hawkins, “Integrated ARROW waveguides with hollow cores,” Opt. Express **12**, 2710–2715 (2004). [CrossRef] [PubMed]

3. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. Engeness, M. Soljačić, S. Jacobs, J. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express **9**, 748–779 (2001). [CrossRef] [PubMed]

4. 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,” Nature **420**, 650–653 (2002). [CrossRef] [PubMed]

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

5. G. Vienne, Y. Xu, C. Jakobsen, H. Deyerl, J. Jensen, T. Sørensen, T. Hansen, Y. Huang, M. Terrel, R. Lee, N. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and Amnon Yariv, “Ultra-large bandwidth hollow-core guiding in all-silica Bragg fibers with nano-supports,” Opt. Express **12**, 3500–3508 (2004). [CrossRef] [PubMed]

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

**11**, 416–423 (1993). [CrossRef]

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

*both*the high- and low-index layer resonances must be taken into account in order to describe the chosen region of the 2-D bandgap spectrum. This follows directly from the definition of the

*general antiresonance point*(Section 4.2) which explicitly accommodates this behaviour. As mentioned, further analysis of the curve interactions (Section 4) leads to expressions describing nontrivial properties of the associated bandgaps, such as: the positions of all bandgap closure points and, from them, a consistent nomenclature for arbitrary bandgap spectra (Section 4.1); the

*central gap point*(Section 4.4) - a special case of the general antiresonance point; and a quantitative measure of the topology of arbitrary bandgap spectra - the number of gaps within a specific domain (Section 4.5). Concluding remarks are given in Section 5.

## 2. Background theory

### 2.1. Bragg stack Bloch analysis

1. P. Yeh, A. Yariv , and C. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. **67**, 423–438 (1977). [CrossRef]

3. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. Engeness, M. Soljačić, S. Jacobs, J. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express **9**, 748–779 (2001). [CrossRef] [PubMed]

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

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

*n*

_{1}and

*n*

_{0}and core refractive index

*n*

_{core}, such that

*n*

_{1}>

*n*

_{0}≥

*n*

_{core}≥1 (a ‘depressed-core’ waveguide as defined above), with layer thicknesses

*t*

_{1}and

*t*

_{0}and core thickness

*t*

_{core}, Fig. 1.

**67**, 423–438 (1977). [CrossRef]

*A*

_{TE,TM}(

*k*,

*ñ*) [1

**67**, 423–438 (1977). [CrossRef]

*n*and

_{i}*t*) and wave parameters (

_{i}*k*and

*ñ*). Only solutions satisfying the condition [1

**67**, 423–438 (1977). [CrossRef]

*x*-direction in Fig. 1). The evaluation of Inequality 1 for a particular layer configuration is straightforward since

*k*and

*ñ*are the only free parameters. The regions in the 2-dimensional space (

*k*,

*ñ*) satisfying Ineq. (1) are allowed bands (waves allowed to propagate in the cladding) for either TE or TM waves (in Fig. 2, black represents where the TE and TM bands overlap, so for

*ñ*<

*n*

_{0}all TE bands are black since the TM bands are larger, thus both black and grey). For

*ñ*>

*n*

_{0}the TE and TM bands diverge [1

**67**, 423–438 (1977). [CrossRef]

*n*

_{core}≤

*n*

_{0}) must have both

*k*and

*ñ*within a bandgap. All figures within plot over (Λ/

*λ*,

*ñ*)=(

*Λk*/2

*π*,

*ñ*) so that both axes are unitless, naturally representing the scale-invariance of the system.

*θ*=

*π*/2) produces

*β*=

*k*⇒

_{i}*ñ*=

*n*, referred to as the ‘

_{i}*n*-light-line’. For waveguides of sufficiently large core radius, core-bound modes of low order exhibit dispersion curves

_{i}*ñ*(

*k*) close to those of plane waves of glancing incidence, hence

*ñ*(

*k*)≈

*n*. Core modes of the waveguides considered here will thus typically only exist where the

_{i}*n*

_{core}-light line intercepts the relevant bandgap; namely, TE modes within TE gaps and TM and HE modes within TM gaps (since HE modes contain components of both TE and TM rays [27]).

30. A. Argyros, “Guided modes and loss in Bragg fibres,” Opt. Express **10**, 1411–1417 (2002). [PubMed]

*ñ*[30

30. A. Argyros, “Guided modes and loss in Bragg fibres,” Opt. Express **10**, 1411–1417 (2002). [PubMed]

*ñ*<

*n*

_{0}, the TM gaps always lie within the TE gaps [23

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

*n*

_{B}in the case of Fig. 2 is coincidental (discussed further in Sections 3 and 4.1).

4. 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,” Nature **420**, 650–653 (2002). [CrossRef] [PubMed]

*n*

_{0}-light-line, as recently discussed in [23

**26**, 43–51 (2008). [CrossRef]

*ñ*=0, as is the case for conventional Bragg fibers [3

3. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. Engeness, M. Soljačić, S. Jacobs, J. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express **9**, 748–779 (2001). [CrossRef] [PubMed]

4. 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,” Nature **420**, 650–653 (2002). [CrossRef] [PubMed]

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

**26**, 43–51 (2008). [CrossRef]

*ñ*=

*n*

_{0}, as is the case for the more recently discussed types of single-material Bragg fibers [5

5. G. Vienne, Y. Xu, C. Jakobsen, H. Deyerl, J. Jensen, T. Sørensen, T. Hansen, Y. Huang, M. Terrel, R. Lee, N. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and Amnon Yariv, “Ultra-large bandwidth hollow-core guiding in all-silica Bragg fibers with nano-supports,” Opt. Express **12**, 3500–3508 (2004). [CrossRef] [PubMed]

**26**, 43–51 (2008). [CrossRef]

7. T. Katagiri, Y. Matsuura, and M. Miyagi, “Photonic bandgap fiber with a silica core and multilayer dielectric cladding,” Opt. Lett. **29**, 557–559 (2004). [CrossRef] [PubMed]

8. T. Katagiri, Y. Matsuura, and M. Miyagi, “All-solid single-mode bragg fibers for compact fiber devices,” J. Lightwave Technol. **24**, 4314–4318 (2006). [CrossRef]

**13**, 9331–9336 (2005). [CrossRef] [PubMed]

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

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

22. D. Yin, H. Schmidt, J. P. Barber, and A. R. Hawkins, “Integrated ARROW waveguides with hollow cores,” Opt. Express **12**, 2710–2715 (2004). [CrossRef] [PubMed]

**13**, 9331–9336 (2005). [CrossRef] [PubMed]

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

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

22. D. Yin, H. Schmidt, J. P. Barber, and A. R. Hawkins, “Integrated ARROW waveguides with hollow cores,” Opt. Express **12**, 2710–2715 (2004). [CrossRef] [PubMed]

*ñ*≈

*n*

_{core}[27] and will move to lower

*ñ*as the core size decreases (e.g. [23

**26**, 43–51 (2008). [CrossRef]

*ñ*and hence intercept different bandgap regions (important for exploiting modal-loss discrimination [3

**9**, 748–779 (2001). [CrossRef] [PubMed]

**26**, 43–51 (2008). [CrossRef]

29. I. M. Bassett and A. Argyros, “Elimination of polarization degeneracy in round waveguides,” Opt. Express **10**, 1342–1346 (2002). [PubMed]

30. A. Argyros, “Guided modes and loss in Bragg fibres,” Opt. Express **10**, 1411–1417 (2002). [PubMed]

*n*

_{core}or

*t*

_{core}, the confined modes’

*ñ*will be able to intercept this richly structured bandgap region, making analyses based on the SPARROW model increasingly important. An explicit example of a liquid-core Bragg fiber is presented in Section 3.1.

### 2.2. ARROW models

_{2}-Si multilayer structures,” Appl. Phys. Lett. **49**, 13–15 (1986). [CrossRef]

**11**, 416–423 (1993). [CrossRef]

*n*

_{core}≤min{

*n*}. By considering the transverse phase accumulated by a propagating ray (Fig. 1) per round trip (

_{i}*including*that from traversing the core, via the V-parameter) and equating it to 2

*π*(resonance), the Archambault-ARROW model determines the wavelengths at which the

*i*

^{th}cladding layer will be resonant with the guided light [9

**11**, 416–423 (1993). [CrossRef]

**12**, 2710–2715 (2004). [CrossRef] [PubMed]

*individual*cladding layers are found between the cladding layer resonance points by considering half-integer layer mode orders:

*m*→

_{i}*m*+½[9

_{i}**11**, 416–423 (1993). [CrossRef]

**13**, 9331–9336 (2005). [CrossRef] [PubMed]

**13**, 10564–10570 (2005). [CrossRef] [PubMed]

**12**, 2710–2715 (2004). [CrossRef] [PubMed]

*m*=1 (e.g. [22

_{i}**12**, 2710–2715 (2004). [CrossRef] [PubMed]

*k*of interest). As will be shown in Section 4.2, a general analysis requires the resonances of all layer types to be considered together, leading to our definition of the

*general antiresonance point*

*k*′

*which is more suited for general analyses of arbitrary cladding configurations and strictly necessary for arbitrary*

_{c}*ñ*and

*m*.

_{i}*U*

_{∞}→

*π*/2 (planar core with

*p*=0) in Eq. (2), the Duguay-ARROW resonance condition [15

_{2}-Si multilayer structures,” Appl. Phys. Lett. **49**, 13–15 (1986). [CrossRef]

*i*label redundant). This substitution is equivalent to assuming the core-bound rays only make glancing incidence with the cladding layer [15

_{2}-Si multilayer structures,” Appl. Phys. Lett. **49**, 13–15 (1986). [CrossRef]

*n*

_{core}=

*n*

_{0}<

*n*

_{1}) ARROWs (e.g. [15

_{2}-Si multilayer structures,” Appl. Phys. Lett. **49**, 13–15 (1986). [CrossRef]

17. T. Baba and Y. Kokubun, “Dispersion and radiations loss characteristics of antiresonant reflecting optical waveguides - numerical results and analytical expressions,” J. Quantum Electron. **28**, 1689–1700 (1992). [CrossRef]

19. 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]

**12**, 1540–1550 (2004). [CrossRef] [PubMed]

18. A. K. Abeeluck, N. M. Litchinitser, C. Headley, and B. J. Eggleton, “Analysis of spectral characteristics of photonic bandgap waveguides,” Opt. Express **10**, 1320–1333 (2002). [PubMed]

**27**, 1592–1594 (2002). [CrossRef]

20. N. M. Litchinitser, S. C. Dunn, B. Usner, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. M. de Sterke, “Resonances in microstructured optical waveguides,” Opt. Express **11**, 1243–1251 (2003). [CrossRef] [PubMed]

**12**, 1540–1550 (2004). [CrossRef] [PubMed]

**27**, 1592–1594 (2002). [CrossRef]

*m*→

*m*+½ is required for the rod-cladding case, to accommodate for the modal cut-off frequencies of cylinders instead of layers [20

20. N. M. Litchinitser, S. C. Dunn, B. Usner, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. M. de Sterke, “Resonances in microstructured optical waveguides,” Opt. Express **11**, 1243–1251 (2003). [CrossRef] [PubMed]

**12**, 1540–1550 (2004). [CrossRef] [PubMed]

*n*

_{core}=

*n*

_{0}) waveguides is that the resonance effects of the cladding on the core-guided modes is dominated by the high-index inclusions, independent of the low-index region between them [19

**27**, 1592–1594 (2002). [CrossRef]

*ñ*=

*n*

_{core}=

*n*

_{0}in the SPARROW model.

18. A. K. Abeeluck, N. M. Litchinitser, C. Headley, and B. J. Eggleton, “Analysis of spectral characteristics of photonic bandgap waveguides,” Opt. Express **10**, 1320–1333 (2002). [PubMed]

**27**, 1592–1594 (2002). [CrossRef]

*n*

_{0}>

*n*

_{core}=1) Bragg fibers, where Bloch-wave analysis (as in Section 2.1) is more common [1

**67**, 423–438 (1977). [CrossRef]

**9**, 748–779 (2001). [CrossRef] [PubMed]

**12**, 1510–1517 (2004). [CrossRef] [PubMed]

**26**, 43–51 (2008). [CrossRef]

**11**, 416–423 (1993). [CrossRef]

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

*ñ*. This then permits a bandgap-style analysis of the layer resonances (Section 3) ultimately showing a close relationship between the bandgap and antiresonance guidance mechanisms and, more importantly, also allowing the derivation of some very simple expressions for nontrivial and useful properties of arbitrary bandgap spectra (Section 4).

### 2.3. Dispersion analysis of slab waveguides below the light-line

*n*

_{a}and

*n*

_{b}, respectively. There are thus two types of slab waveguide to consider: one with a high slab index

*n*

_{a}=

*n*

_{1}with a lower cladding index

*n*

_{b}=

*n*

_{0}(

*n*

_{a}>

*n*

_{b}), and one with a low slab index

*n*

_{a}=

*n*

_{0}with a high cladding index

*n*

_{b}=

*n*

_{1}(

*n*

_{a}<

*n*

_{b});

*n*

_{1}>

*n*

_{0}as before (Fig. 1). For this analysis, the same wave-vector decomposition as in Fig. 1 is used, with the wavevector

*k*=

_{i}*k*

_{a}incident at angle

*θ*to the normal within the dielectric of refractive index

_{a}*n*

_{a}[the Bragg stack schematic in Fig. 2, though, is replaced by a homogeneous region with the equivalent vector diagram for a transmitted ray (

*a*→

*b*)].

*ñ*

_{a}

*below*the

*n*

_{0}-light-line (

*ñ*

_{a}<

*n*

_{0}), all slab modes are inherently leaky [27]. This is because for the former (high-index) slab

*ñ*

_{a}=

*β*

_{1}/

*k*=

*n*

_{1}sin

*θ*

_{a}<

*n*

_{0}, so

*θ*

_{a}<

*θ*

_{c}=sin

^{-1}(

*n*

_{b}/

*n*

_{a}) (the critical angle) and no total internal reflection occurs, thus relying on regular (lossy) reflection. In other words, the modes must exist below the bound mode cut-off [27]. The latter (low-index) slab relies on inherently lossy Fresnel reflection for all

*θ*

_{a}since

*n*

_{a}<

*n*

_{b}. The loss characteristics are considered here only insofar as they aid the phase analysis via the Fresnel reflection coefficients [27, 28]:

*A*

_{TE,TM}and

*A*′

_{TE,TM}respectively, by

*A*

_{TE,TM}=Γ

_{TE,TM}

*A*′

_{TE,TM}.

*ñ*

_{a}<

*n*

_{0}guidance, Snell’s law implies the high-index (

*n*>

_{a}*n*) slab must have

_{b}*θ*

_{a}<

*θ*. Since cos(

_{c}*θ*)=

_{i}*k*/

_{ix}*k*and 0≤

_{i}*θ*<

_{a}*θ*

_{c}≤

*π*/2 (ℤ0≤

*θ*

_{b}≤

*π*/2) then cos(

*θ*

_{a})>cos(

*θ*

_{b}) and thus

*k*

_{ax}>

*k*

_{bx}. Equation (4) then implies Γ

_{TE,TM}∈ℝ

^{+}so that sign(

*A*

*)=sign(*

_{TE,TM}*A*′

*) and no phase shift occurs upon reflection. However, for the low-index (*

_{TE,TM}*n*

_{a}<

*n*

_{b}) slab,

*θ*>

_{a}*θ*with no critical angle threshold, so similar reasoning implies

_{b}*k*

_{ax}<

*k*

_{bx}. Thus Γ

_{TE,TM}∈ℝ

^{-}, meaning sign(

*A*

*)=-sign(*

_{TE,TM}*A*′

*) such that a*

_{TE,TM}*π*phase shift occurs upon reflection at the interface. Note that the TE and TM modes are thus degenerate for

*ñ*<

*n*

_{0}due to their equivalence in this phase analysis (Γ

_{TE,TM}produce the same reflective phase conditions for both

*n*>

_{a}*n*and

_{b}*n*<

_{a}*n*for

_{b}*ñ*<

_{a}*n*

_{0}guidance).

*π*. For both slabs, the transverse phase accumulated by traversing the slab region once is

*k*

_{ax}

*t*

_{a}. The forms of the low- and high-index slabs’ phase relations thus differ only in their reflection terms. Equating the cumulative phase shifts to

*m*2

*π*(

*m*∈ℤ

^{+}), a dispersion relation for each waveguide is derived [27, 32

32. P. K. Tien, “Light waves in thin films and integrated optics,” Appl. Opt. **10**, 2395–2413 (1971). [CrossRef] [PubMed]

*m*=0 is obviously not allowed for the high-index slab, implying that the

*m*=0 bound mode has no leaky counterpart [27]. By rearranging the phase relations and setting a→1 and b→0 for the high-index (

*n*

_{a}=

*n*

_{1}>

*n*

_{b}=

*n*

_{0}) slab and a→0 and b→1 for the low-index (

*n*

_{a}=

*n*

_{0}<

*n*

_{b}=

*n*

_{1}) slab, we find a unified dispersion relation:

*m*

_{1}=

*m*and

*m*

_{0}=

*m*+1. Groups of dispersion curves for a range of mode orders are plotted in Fig. 2 (right) and subsequently in Figs. 3, 4 and 5.

*m*

_{1}=0 refers to the

*ñ*-axis (

*k*=0) and

*m*

_{0}=0 to the

*n*

_{0}-light-line (

*ñ*=

*n*

_{0}). It is easily shown that, while not representative of physical modes, these definitions still satisfy Eqs. 5 and 6. Hereon the ‘SPARROW curves’ will refer to both the physical slab dispersion curves (

*m*

_{1,0}∈ℕN)

*and*these

*m*

_{1,0}=0 lines, unless otherwise specified. The lower limit line

*ñ*=0 is also important but its inclusion in this set is not required, as will soon be evident.

32. P. K. Tien, “Light waves in thin films and integrated optics,” Appl. Opt. **10**, 2395–2413 (1971). [CrossRef] [PubMed]

*n*, implying that, below the

_{i}*n*

_{0}-light-line, the dispersion properties of each slab depend only on the slab refractive index, not that of the medium surrounding it.

## 3. Formal SPARROWmodel definition

**11**, 416–423 (1993). [CrossRef]

*λ*,

*ñ*) plot (e.g. Figs. 2,3,4 and 5); and the freedom to consider core modes of arbitrary

*ñ*. The latter point requires that, for the SPARROWmodel to accurately predict core-mode spectral features, sensible values for a core mode’s

*ñ*must be provided independently. For large-core waveguides, this is trivial, since most low-order modes will lie very close to the

*n*

_{core}-light-line (demonstrated in Section 3.1). However, it has also been shown that, at least for air-core Bragg fibers, it is possible to infer the (real part of the) core modes’

*ñ*solely from the core geometry [23

**26**, 43–51 (2008). [CrossRef]

**10**, 1411–1417 (2002). [PubMed]

*λ*) waveguides.

*ñ*or

*k*dimension, meaning that the discrete bandgaps within the stack bandgap spectrum are each

*enclosed by a subset of the SPARROW curves*. We thus define the concept of a

*bounding region*for any particular bandgap: the (

*k*,

*ñ*) region enclosed by the curves surrounding a particular bandgap. This behaviour implies the nontrivial discrete bandgap spectrum of the stratified cladding is replicated in position and topology by the equivalent SPARROW curves (including the physical limits

*k*=0,

*ñ*=0 and

*ñ*=

*n*

_{0}, as discussed). Consequences and applications of this are discussed in Section 4.

### 3.1. Confirmation via FEM analysis

*and*low-index resonances are evident. We model a realistically achievable configuration: a fabricable (as demonstrated by Temelkuran et al. [4

**420**, 650–653 (2002). [CrossRef] [PubMed]

**420**, 650–653 (2002). [CrossRef] [PubMed]

*As*

_{2}

*Se*

_{3}chalcogenide glass (

*n*≈2.8) and the polymer polyethersulphone (PES,

*n*≈1.55) with thicknesses of 270

*nm*and 900

*nm*respectively.

*n*

_{core}=1.45 is chosen here simply because it produces modes revealing interesting structure in the bandgap spectrum. We choose a smaller core diameter (20µ

*m*) than the cited fabricated fiber (≈700µ

*m*), and fewer rings (4 pairs of layers instead of 9), due to numerical restrictions of the method employed (the relevant discretisations are stored in finite computer memory), but note that the bandgap behaviour would be very similar between the two structures regardless (like the 700µ

*m*core, a 20µ

*m*core also produces modes close to the

*n*

_{core}-light-line, Fig. 4).

*ñ*, and hence for the confinement loss CL=20log

_{10}(

*e*)

*k*Im{

*ñ*} [34

34. V. Finazzi, T. M. Monro, and D. J. Richardson, “Small-core silica holey fibers: nonlinearity and confinement loss trade-offs,” J. Opt. Soc. Am. B **20**, 1427–1436 (2003). [CrossRef]

_{01}mode exists only within the cladding bandgaps, with

*ñ*having no solutions within the allowed bands, as expected. Note that the core-mode dispersion curve lies relatively close to the

*n*

_{core}-light-line for this

*t*

_{core}.

*n*

_{core}) via Eq. (7)) are approached. This example clearly demonstrates how the SPARROW model is required to describe the resonant properties of both high- and low-index inclusions for arbitrary

*ñ*. Note how the minimum CL for each gap falls close to mid-way (in frequency) between the adjacent cladding resonance points (this important point is discussed further in Section 4.2). Of course, it is well-known that the absolute CL depends on the number of cladding layers; more layers produce a lower loss. Incidentally, the CL spectra of the next two higher-loss modes, TM

_{01}and HE

_{11}, have values above the domain presented in Fig. 4 (bottom), unsurprisingly (due to the narrow TM bandgap via the Brewster effect, Fig. 2) commensurate with the effectively-single-mode behaviour of Bragg fibers [3

**9**, 748–779 (2001). [CrossRef] [PubMed]

**10**, 1411–1417 (2002). [PubMed]

## 4. Further analysis of the SPARROWmodel

### 4.1. Curve intersections and gap nomenclature

*i*={1,0}) can be found by equating either

*ñ*or

*k*. The intersection point of arbitrary

*k*) (high-index) and

*k*) (low-index) curves is found to be:

*bounding points*.

*m*

_{1},

*m*

_{0}〉 gap that close up within the 0<

*ñ*<

*n*

_{0}region will do so at the intersection points

*P*(

*m*

_{1},

*m*

_{0}) and

*P*(

*m*

_{1}-1,

*m*

_{0}+1), should these points exist within the domain for the given gap. If these points don’t exist, the gap must then be terminated by the

*ñ*=0 or

*n*

_{0}lines, leaving it open (seen explicitly in Figs. 3 and 5). Incidentally, we believe that this gap closure behaviour helps explain the mode suppression phenomena observed in [35

35. J. Li and K. S. Chiang, “Disappearance of modes in planar Bragg waveguides,” Opt. Lett. **32**, 2369–2371 (2007). [CrossRef] [PubMed]

### 4.2. The general antiresonance point

*k*′

*, at arbitrary*

_{c}*ñ*=

*ñ*′, as the arithmetic mean of the bounding curve values

*k*value):

*p*and

*q*refer to the adjacent bounding curve types in the

*k*-dimension (i.e.

*p*,

*q*∈{1,0}) and

*m*and

_{q}*m*to their order. Note that the curves adjacent (in the

_{p}*k*-dimension) to

*k*′

*will change as*

_{c}*ñ*′ varies within the bounding region. For example,

*ñ*′ in the middle section of the 〈

*m*

_{1},

*m*

_{0}〉 bounding region (i.e. between

*P*(

*m*

_{1}-1,

*m*

_{0}) and

*P*(

*m*

_{1},

*m*

_{0}+1) in the

*ñ*-dimension) implies

*k*′

*is between curves with either*

_{c}*m*=

_{p}*m*

_{1}and

*m*=

_{q}*m*

_{1}-1 or

*m*=

_{p}*m*

_{0}and

*m*=

_{q}*m*

_{0}+1 (depending on the type of gap), due to the monotonicity of the curves. For

*ñ*′ in the top section of the bounding region (above the middle section but below

*P*(

*m*

_{1},

*m*

_{0})),

*k*′

*will be in between curves with*

_{c}*m*=

_{p}*m*

_{1}and

*m*=

_{q}*m*

_{0}, whereas for

*ñ*′ near the bottom of the gap (below the middle section but above

*k*′

*is between the*

_{c}*m*=

_{p}*m*

_{1}-1 and

*m*=

_{q}*m*

_{0}+1 curves.

*k*′

*naturally predicts the approximate position of lowest CL for a core mode of the fiber discussed in Section 3.1. The reason CL reaches a minimum near*

_{c}*k*′

*is that, at that point, the guided wave is maximally antiresonant with the pair of slabs producing the bounding dispersion curves (*

_{c}*m*and

_{p}*m*); as one or the other bounding curve is approached, the wave becomes more resonant with the associated slab, thus allowing greater coupling from the core to the cladding. This is why the resonances of all layer types must be considered

_{q}*together*, rather than separately, as discussed earlier, and is inherent in the definition of

*k*′

*. The general antiresonance point is thus a powerful tool for waveguide design and analysis, allowing immediate determination of the approximate point of minimum CL at arbitrary*

_{c}*ñ*within an arbitrary bandgap.

*k*′

*with respect to all waveguide parameters (*

_{c}*∂k*′

*/*

_{c}*∂n*,

_{i}*∂k*′

*/*

_{c}*∂t*, etc.) can be easily derived, but are omitted here for brevity. These derivatives are ideal for the direct calculation of fabrication tolerances in waveguide design, or sensitivities to core materials for sensing, for example.

_{i}### 4.3. Special cases

*ñ*=

*n*

_{0}and

*ñ*=0. It is easily shown that the former actually reduces to the large-core limit of the Duguay-ARROW model, since by setting

*i*=1 and

*ñ*=

*n*

_{0}resonances are shown in Fig. 5 (magenta circles). Thus, the SPARROW model also explains why the Duguay-ARROW model in the large-core regime (Litchinitser et al. [20

**11**, 1243–1251 (2003). [CrossRef] [PubMed]

*ñ*=

*n*

_{0}and hence never intercept it; their resonant features can never appear on the

*n*

_{0}-light-line (the region of applicability of Eq. (3)).

*ñ*=0 (the zero-line), is derived in much the same way. However, in this case, both high- and low-index curves intercept the zero-line. Setting

*ñ*=0 we derive:

_{i}*i*∈{1,0}.

*ñ*=0 corresponds to rays normally incident to the layered cladding, similar to high-order modes within hollow-core solid-cladding Bragg fibers [30

**10**, 1411–1417 (2002). [PubMed]

*ñ*≤1.

### 4.4. The central gap point

*i*={1,0} (when expressed in terms of

*k*and

*ñ*). An example is shown in Fig. 5 (green dotted curves). Thus, we define the

*central gap point*as the intersection point of the half-order curves within a particular gap. In the nomenclature defined in Section 4.1, the curves producing the central gap point for the 〈

*m*

_{1},

*m*

_{0}〉 gap are the

*P*is entirely commensurate with the general antiresonance point (

_{c}*k*′

*) of Eq. (9) such that:*

_{c}*ñ*always lies between the middle two intersection points

_{c}*k*′

*is always bound on either side by either the*

_{c}*m*=

_{p}*m*

_{1}and

*m*=

_{q}*m*

_{1}-1 curves or the

*m*=

_{p}*m*

_{0}and

*m*=

_{q}*m*

_{0}+1 curves (as discussed in Section 4.2). An example based on the fiber cladding examined in [5

5. G. Vienne, Y. Xu, C. Jakobsen, H. Deyerl, J. Jensen, T. Sørensen, T. Hansen, Y. Huang, M. Terrel, R. Lee, N. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and Amnon Yariv, “Ultra-large bandwidth hollow-core guiding in all-silica Bragg fibers with nano-supports,” Opt. Express **12**, 3500–3508 (2004). [CrossRef] [PubMed]

**26**, 43–51 (2008). [CrossRef]

*ñ*=

*n*

_{0}and/or

*ñ*=0 lines are open (above and below, respectively), this definition of the center point is insufficient when

*P*reaches these lines, or where

_{c}*P*simply doesn’t exist (as is the case for all 〈

_{c}*m*

_{1},0〉 gaps). In these cases, we use the general antiresonance point (Eq. (9)) to define

*P*=(

_{c}*k*′

*(*

_{c}*ñ*′),

*ñ*′) where

*ñ*′=

*n*

_{0}or 0 as appropriate. Thanks to Eq. (12), this gap center formalism is thus consistent for all bandgaps.

*P*will determine the approximate point of lowest modal CL for a given bandgap in not just the

_{c}*k*-dimension (via the relation to

*k*′

*from Eq. (12)) but also the*

_{c}*ñ*-dimension. This is to be expected from the work of [30

**10**, 1411–1417 (2002). [PubMed]

*P*dictates the point at which light is maximally antiresonant with both cladding layer types, it is reasonable to expect that the CL is thus minimum near this point. While omitted here for brevity, a quantitative verification of this would be straight-forward, requiring the CL spectra of the modes of interest to be calculated for a range of

_{c}*ñ*. This could be achieved by iterating the spectral analysis (via a FEM as in Section 3.1, for example) over a range of

*n*

_{core}to generate modes within the entire domain of the bandgap of interest (i.e. to calculate CL in both the

*k*- and

*ñ*-dimensions).

*P*with respect to all cladding parameters (

_{c}*∂k*/

_{c}*∂n*,

_{i}*∂k*/

_{c}*∂t*,

_{i}*∂ñ*/

_{c}*∂n*and

_{i}*∂ñ*/

_{c}*∂t*) can also be easily derived, and would also be ideal for similar applications.

_{i}### 4.5. Bandgap topology and the bounding region

*n*

_{m1}(

*k*), which have orders

*m*

_{1}-1 (left curve in the

*k*-dimension) and

*m*

_{1}(right curve). We define

*ñ*=

*n*

_{0}and

*ñ*=0 lines. We focus on the TE bandgaps here, since the TM gaps have the same topology save for the simple gap closure induced by the Brewster effect (Section 2.1, Figs. 2 and 3).

*n*

_{m0}(

*k*)] with the rightmost bounding curve [

*n*

_{m1}(

*k*)], excluding the

*n*

_{0}-light-line (

*m*

_{0}=0). To show this analytically, we must enforce upon the intersection point expression (Eq. (8)) the physical condition:

*P*(

*m*

_{1},

*m*

_{0})∈ℝ

^{2}. By enforcing

*k*∈ℝ, the square-root requires (

*m*

_{1}/

*t*

_{1})

^{2}-(

*m*

_{0}/

*t*

_{0})

^{2}>0⇒

*η*>1 (where

*η*is defined with Eq. (8)). This can be used to find an upper limit

*m*

_{0}<

*m*

_{1}(

*t*

_{0}/

*t*

_{1}), but a more strict limit is found by enforcing the second physical condition

*ñ*∈ℝ: since

*η*>1, 1-

*η*

^{2}<1 so that the numerator (within the square-root) must also be negative:

*n*

^{2}

_{1}-

*n*

^{2}

_{0}

*η*

^{2}<1⇒

*n*

_{1}/

*n*

_{0}<

*η*. This last inequality gives the most strict range physically imposed on

*m*

_{0}, namely:

*m*

_{0}-curve within

*m*

_{0}-curves within

*ñ*={0,

*n*

_{0}} lines). The number of TE bandgaps within

*m*

^{max}

_{0}+1; the ‘+1’ accounting for the ever present 〈

*m*

_{1},0⎚ gaps, bound above by the

*n*

_{0}-light-line, whose maximal bounding point

*m*

^{max}

_{0}by definition, as discussed.

*on*the zero-line (

*ñ*=0) such that the point exists but the associated gap does not (the bounding region becomes singular). In this case, the number of gaps within

*k*) (including the

*ñ*=0 bounding point). Quantitatively, the condition for this behaviour can be deduced from the SPARROW model’s

*ñ*=0 special case (Eq. (10)) by setting

*equality*(i.e. the floor function of Eq. (14) becomes redundant).

*m*

_{1}(required to define

*ñ*=

*n*

_{B}(Section 2.1). The only exception is when

*n*

_{B}coincides with a TE gap closure point,

*P*at

_{c}*ñ*=0 case above) and the number of gaps within

## 5. Concluding remarks

*below*the low-cladding-index light-line, implying that Bragg fibers and integrated-ARROWs guide by fundamentally the same principles. This was done by developing an antiresonance model, the SPARROW model, which describes the resonances of an arbitrary waveguide’s cladding layers, independent of the core properties. The SPARROW model is a generalisation of the Archambault-ARROW model [9

**11**, 416–423 (1993). [CrossRef]

*m*

_{1},

*m*

_{0}〉); the approximate position of lowest core-mode confinement loss of any gap via the general antiresonance point (

*k*′

*); the precise closure points of a given gap (*

_{c}*ñ*- and

*k*-dimensions via the central gap point (

*P*); and the number of bandgaps within a specific domain (e.g.

_{c}**4**, 3–16 (2008). [CrossRef] [PubMed]

## Acknowledgments

## References and links

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

2. | Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” J. Lightwave Technol. |

3. | S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. Engeness, M. Soljačić, S. Jacobs, J. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express |

4. | 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,” Nature |

5. | G. Vienne, Y. Xu, C. Jakobsen, H. Deyerl, J. Jensen, T. Sørensen, T. Hansen, Y. Huang, M. Terrel, R. Lee, N. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and Amnon Yariv, “Ultra-large bandwidth hollow-core guiding in all-silica Bragg fibers with nano-supports,” Opt. Express |

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

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8. | T. Katagiri, Y. Matsuura, and M. Miyagi, “All-solid single-mode bragg fibers for compact fiber devices,” J. Lightwave Technol. |

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

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17. | T. Baba and Y. Kokubun, “Dispersion and radiations loss characteristics of antiresonant reflecting optical waveguides - numerical results and analytical expressions,” J. Quantum Electron. |

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34. | V. Finazzi, T. M. Monro, and D. J. Richardson, “Small-core silica holey fibers: nonlinearity and confinement loss trade-offs,” J. Opt. Soc. Am. B |

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**OCIS Codes**

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

(130.0130) Integrated optics : Integrated optics

(230.4170) Optical devices : Multilayers

(230.7370) Optical devices : Waveguides

(060.4005) Fiber optics and optical communications : Microstructured fibers

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: August 29, 2008

Revised Manuscript: October 15, 2008

Manuscript Accepted: October 19, 2008

Published: October 21, 2008

**Virtual Issues**

Vol. 3, Iss. 12 *Virtual Journal for Biomedical Optics*

**Citation**

Kristopher J. Rowland, Shahraam Afshar V., and Tanya M. Monro, "Bandgaps and antiresonances in
integrated-ARROWs and Bragg fibers;
a simple model," Opt. Express **16**, 17935-17951 (2008)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-16-22-17935

Sort: Year | Journal | Reset

### References

- P. Yeh, A. Yariv and C. Hong, "Electromagnetic propagation in periodic stratified media. I. General theory," J. Opt. Soc. Am. 67, 423-438 (1977). [CrossRef]
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