# Associate Professor Tim Langtry

**Associate of the Faculty,**School of Mathematical and Physical Sciences

BAppSc (NSWIT), BA (Hons) (UNSW), MAppSc (NSWIT), PhD (UNSW)

Member, Australian Mathematical Society

**Phone**

+61 2 9514 2258

**ORCID**

### Research Interests

My main research interests are in numerical analysis and computational mathematics. Particular areas of interest include quasi-Monte Carlo methods for numerical multiple integration (eg rank 1 lattice rules) and, more recently, computational photonics.

My work in photonics has focussed on computational modelling of various aspects of photonic crystals - composite materials which exhibit a periodic variation in their refractive index, leading to novel and interesting optical properties. An example of some results of this work may be seen in the paper Effects of disorder in two-dimensional photonic crystal waveguides.

**Can supervise:**Yes

## Chapters

Botten, L.C., McPhedran, R.C., de Sterke, C.M., Nicorovici, N.A., Asatryan, A.A., Smith, G.H., Langtry, T., White, T., Fussell, D.P. & Kuhlmey, B. 2006, 'From Multipole Methods to Photonic Crystal Device modeling' in Yasumoto, K. (ed),

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*Electromagnetic Theory and Applications for Photonic Crystals*, Taylor & Francis, Florida, USA, pp. 47-122.View/Download from: UTS OPUS

NA

Botten, L.C., McPhedran, R.C., Nicorovici, N.A., Asatryan, A.A., de Sterke, C.M., Robinson, P.A., Busch, K., Smith, G. & Langtry, T. 2003, 'Rayleigh multipole methods for photonic crystal calculations' in Priou, A. & Itoh, T. (eds),

*Electromagnetic applications of photonic band gap materials and structures*, Massachusetts Institute of Technology, USA, pp. 21-60. Langtry, T. 1998, 'A generalisation of ratios of Fibonacci numbers with application to numerical quadrature' in Bergum, G.E., Philippou, A.N. & Horadam, A.F. (eds),

*Applications of Fibonacci Nymbers: Volume 7*, Kluwer Academic Publishers, Dordrecht, pp. 239-253. Langtry, T. 1997, 'Bounds on the figure of merit of some lattice rules' in Niederreiter, H., Hellekalek, P., Larcher, G. & Zinterhof, P. (eds),

*Monte Carlo and Quasi-Monte Carlo Methods 1996*, Springer Verlag, New York, pp. 308-320. Langtry, T. 1993, 'Applications of Fibonacci Numbers: Volume 5' in Bergum, G.E., Philippou, A.N. & Horadam, A.F. (eds),

*Applications of Fibonacci Numbers: Volume 5*, Kluwer Academic Publishers, Dordrecht, pp. 331-343.## Conferences

Zinder, Y., Nicorovici, N.A. & Langtry, T. 2011, 'Mathematica based platform for self-paced learning',

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*Proceedings of the 10th International Conference on Technology in Mathematics Teaching (ICTMT10)*, ICTMT10, University of Portsmouth, Portsmouth, United Kingdom, pp. 203-208.View/Download from: UTS OPUS

One of the major challenges in teaching applied mathematics is the large amount of calculation involved in many practical mathematical methods. On one hand, mastering these methods requires students to gain experience in performing all steps of a calculation. This experience is crucial for gaining an understanding of the methods, their capabilities and limitations, and cannot be replaced by black box type commercial software which simply displays the results of calculations and gives no insight into the nature of the operations performed. On the other hand, the calculations involved in many modern mathematical methods are tedious and time consuming with fatal results caused by even minor computational mistakes. The advent of computer algebra systems such as Mathematica opened new horizons in teaching mathematics. The ability to perform symbolic calculations in combination with powerful graphics and programming capabilities makes it possible to develop software that provides students with the opportunity for step by step exploration of mathematical procedures. We present the results of an ongoing research project aimed at the development of a Mathematica based platform which allows academics to develop software for self-paced learning.

Zinder, Y., Nicorovici, N.A. & Langtry, T. 2011, 'Mathematica based platform for self-paced learning',

*Proceedings of EDULEARN 10*, EDULEARN, International Association of Technology, Education and Development, Spain, pp. 323-330. One of the major challenges in teaching applied mathematics is the large amount of calculation involved in many practical mathematical methods. On one hand, mastering these methods requires students to gain experience in performing all steps of a calculation. This experience is crucial for gaining an understanding of the methods, their capabilities and limitations, and cannot be replaced by black box type commercial software which simply displays the results of calculations and gives no insight into the nature of the operations performed. On the other hand, the calculations involved in many modern mathematical methods are tedious and time consuming with fatal results caused by even minor computational mistakes. The advent of computer algebra systems such as Mathematica opened new horizons in teaching mathematics. The ability to perform symbolic calculations in combination with powerful graphics and programming capabilities makes it possible to develop software that provides students with the opportunity for step by step exploration of mathematical procedures. We present the results of an ongoing research project aimed at the development of a Mathematica based platform which allows academics to develop software for self-paced learning.

Maher, A., Lake, M., Langtry, T. & Hill, C. 2007, 'A proteomics laboratory system and digital mentor',

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*APAC07*, Conference and Exhibition on Advanced Computing, Grid Application and eResearch, Australian Partnership for Advanced Computing, Perth, Australia, pp. 1-19.View/Download from: UTS OPUS

The typical small biological laboratory is cmposed of electronic scientific equipment and wet laboratory equipment where staff record reesults into paper workbooks. There is oftern a lack of expert staff to guide new users, and hard won experience is locked away with individual staff and their workbooks. The Proteomics Laboratory System and Digital Mentor aims to: facilitate the capture, analysis and sharing of results; facilitate the creation and documentation of standardised work flows and experimental procedures; and mentore inexperienced users in best practice when engaged in particular procedures in the laboratory. In this paper we discuss the design of the system and report on the initial production release. This is nased on a relational database and web-server framework. A novel aspect of the system is the exploitation of custom extensions to a wiki-style interface that add functionality for non-expert end-users.

Asatryan, A.A., Botten, L.C., McPhedran, R.C., de Sterke, C.M., Langtry, T. & Nicorovici, N.A. 2004, 'Conductance of photons and Anderson localization of light',

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*Conference on Lasers and Electro-Optics/International Quantum Electronics Conference and Photonic Applications Systems Technologies*, Conference on Lasers and Electro-Optics/International Quantum Electronics Conference and Photonic Applications Systems Technologies, Optical Society of America, San Francisco, USA, pp. 860-861.View/Download from: UTS OPUS

Conductance properties of photons in disordered two-dimensional photonic crystals is calculated using exact multipole expansions technique. The Landauers two-terminal formula is used to calculate average of the conductance, its variance and the probability density distribution.

Langtry, T., Botten, L.C., Asatryan, A.A., Byrne, M.A. & Bourgeois, A. 2003, 'Localisation and disorder in the design of 2D photonic crystal devices',

*ANZIAM Journal - 11th Biennial Computational Techniques and Applications Conference: CTAC-2003*, Computational Techniques and Applications Conference, Cambridge University Press, Sydney, Australia, pp. 744-758. Photonic crystals are meta-materials that can inhibit the propagation of light in all directions for specific wavelength ranges. Material or structural defects can be introduced into the crystal to cause localised modes, providing the ability to mould the flow of light on the wavelength scale and allowing the development of miniaturised, integrated photonic devices. For this reason, photonic crystals will likely be key building blocks for future micro-optical and communication technology. In this paper, we examine the Bloch mode modelling of 2D photonic crystal structures with application to the analysis of photonic crystal waveguides and their susceptibility to disorder, which provides a framework for studying fabrication tolerances in realistic devices.

Botten, L.C., Asatryan, A.A., White, T.P., McPhedran, R.C., De Sterke, C.M. & Langtry, T.N. 2004, 'Modelling of extended photonic crystal devices using scattering matrix techniques',

*PIERS 2004 - Progress in Electromagnetics Research Symposium, Extended Papers Proceedings*, pp. 13-16. A rigorous semi-analytic approach to the modelling of coupling, guiding and propagation in complex microstructures embedded in photonic crystals is presented. The method, based on Bloch modes and generalized Fresnel coefficients, is outlined and a variety of applications of the design tool are presented.

Asatryan, A.A., Botten, L.C., McPhedran, R.C., De Sterke, C.M., Langtry, T. & Nicorovici, N.A. 2004, 'Conductance of photons and anderson localization of light',

*OSA Trends in Optics and Photonics Series*, pp. 989-990. The conductance properties of photons in disordered two-dimensional photonic crystals is calculated using exact multipole expansions technique. The Landauer's two-terminal formula is used to calculate the average of the conductance, its variance and the probability density distribution. © 2003 Optical Society of America.

Asatryan, A.A., Botten, L.C., White, T.P., de Sterke, C.M., McPhedran, R.C. & Langtry, T. 2003, 'Modelling of complex waveguides structures embedded in photonic crystals',

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*COIN/ACOFT 2003. Proceedings on the Optical Internet and Optical Fibre Technology*, Conference on the Optical Internet/Australian Conference on Optical Fibre Technology 2003, COIN/ACOFT 2003, Melbourne, Australia, pp. 145-148.View/Download from: UTS OPUS

de Sterke, C.M., Botten, L.C., White, T.P., McPhedran, R.C., Asatryan, A.A. & Langtry, T. 2003, 'Photonic bandgap effects as a basis for novel compact devices',

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*COIN/ACOFT 2003. Conference Proceedings on the Optical Internet and Optical Fibre Technology*, Conference on the Optical Internet / Australian Conference on Optical Fibre Technology 2003, COIN/ACOFT 2003, Melbourne, Australia, pp. 133-136.View/Download from: UTS OPUS

Langtry, T., Botten, L.C., Asatryan, A.A., Byrne, M.A., Bourgeois, A. & McPhedran, R.C. 2003, 'Computational modelling of photonic crystals',

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*Proceedings of the APAC Conference and Exhibition on Advanced Computing, Grid Application and eResearch*, Conference and Exhibition on Advanced Computing, Grid Application and eResearch, Australian Partnership for Advanced Computing (APAC, Gold Coast, Australia, pp. 1-19.View/Download from: UTS OPUS

Botten, L.C., Asatryan, A.A., Langtry, T.N., White, T.P., De Sterke, C.M. & McPhedran, R.C. 2003, 'An analytic treatment of propagation in straight and bent photonic crystal waveguides',

*OSA Trends in Optics and Photonics Series*, pp. 1021-1022. A semi-analytic model is developed for coupling and guiding in nano-structured waveguides embedded in two-dimensional photonic crystals. The method, based on Bloch modes and generalized Fresnel coefficients, is applied to waveguide junctions and bends. © 2003 Optical Society of America.

White, T.P., Botten, L.C., De Sterke, C.M., McPhedran, R.C., Asatryan, A.A. & Langtry, T.N. 2003, 'Ultra-compact devices in photonic crystals: Optical behaviour and semi-analytic models',

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*LEOS Summer Topical Meeting*, pp. 17-18.View/Download from: Publisher's site

© 2003 IEEE. An efficient and powerful method recently developed for calculating the transmission properties of photonic crystal waveguides to design an ultra-compact optical device that combines mode coupling in parallel waveguides and Fabry-Perot effects to reduce the coupling length and create a sharp spectral response. The general approach for joining multiple PC waveguide structures can be applied to study many different optical devices and coupling problems.

Botten, L.C., Asatryan, A.A., Langtry, T., de Sterke, C.M. & McPhedran, R.C. 2002, 'Propagation of Photonic Crystal Waveguides',

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*Proceedings of the Australian Institute of Physics 15th Biennial Congress 2002.*, Congress of the Australian Institute of Physics, Australian Institute of Physics, Sydney, Australia, pp. 289-291.View/Download from: UTS OPUS

Langtry, T., Botten, L.C., de Sterke, C.M., Asatryan, A.A. & McPhedran, R.C. 2002, 'Effects of Disorder in Photonic Crystal Waveguides',

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*Proceedings of the Australian Institute of Physics 15th Biennial Congress 2002.*, Congress of the Australian Institute of Physics, Australian Institute of Physics, Sydney, Australia, pp. 310-312.View/Download from: UTS OPUS

Asatryan, A.A., McPhedran, R.C., De Sterke, C.M., Langtry, T.N. & Botten, L.C. 2002, 'Analysis of waveguides in finite photonic crystals',

*Conference on Quantum Electronics and Laser Science (QELS) - Technical Digest Series*, p. 124. The design of a high-quality waveguide in a photonic crystal is outlined. This design requires careful considerations of the end effects, as these may lead to significant Fabry-Perot like interference. The problem can be alleviated by introducing tapers at the ends of the guide.

Langtry, T.N., Botten, L.C., De Sterke, C.M., Asatryan, A.A. & McPhedran, R.C. 2002, 'Disorder in two-dimensional photonic crystal waveguides',

*Conference on Quantum Electronics and Laser Science (QELS) - Technical Digest Series*, p. 125. The effect of disorder in the photonic crystal on the quality of the waveguide was studied. For this purpose, a two-dimensional photonic crystal and a disordered structure were considered. Evidence was obtained suggesting a similar degree of sensitivity to perturbations of refractive index, but a greater tolerance to perturbations of cylinder position.

Asatryan, A.A., McPhedran, R.C., Martijn de Sterke, C., Langtry, T.N. & Botten, L.C. 2002, 'Analysis of waveguides in finite photonic crystals',

*Pacific Rim Conference on Lasers and Electro-Optics, CLEO - Technical Digest*, pp. 308-309. Waveguides in finite photonic crystals (PC) were analyzed. The transmission through a PC of length 11d, with a line defect of width h=d, driven by a line antenna situated on the waveguide's central axis, at a distance d from the structure was shown. The density plot of the internal field intensity in the channel centre versus k for a guide of length 20d was presented.

Langtry, T.N., Botten, L.C., Martijn de Sterke, C., Asatryan, A.A. & McPhedran, R.C. 2002, 'Disorder in two-dimensional photonic crystal waveguides',

*Pacific Rim Conference on Lasers and Electro-Optics, CLEO - Technical Digest*, pp. 309-310. Effect of disorder in two-dimensional photonic crystal waveguides was studied. The waveguide was excited by a light source that was parallel to the cylinders and was located close to an entrance of the waveguide. A disordered structure was achieved by randomizing cylinder positions, refractive indices and radii. Results indicated that disorder in the structure decreased the field intensity by an order of magnitude at far end of the guide.

Langtry, T., Botten, L.C., Asatryan, A.A. & McPhedran, R.C. 2001, 'Local Density of States Calculations for Photonic Crystals',

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*Proceedings of HPC Asia 2001*, International Conference and Exhibition on High Performance Computing in the Asia-Pacific Region, Queensland Parallel Supercomputing Facilty and Australian Partnership for Advasnced Computing, Gold Coast, pp. 0-0.View/Download from: UTS OPUS

Langtry, T. 1998, 'A discrepancy-based analysis of figures of merit for lattice rules',

*Monte Carlo and Quasi-Monte Carlo Methods 1998*, Springer-Verlag, Claremont (California), pp. 296-310. Groen, L., Coupland, M., Memar, J. & Langtry, T.N. 2014, 'The past, present and future student of Mathematics – mastery learning to address the assumed mathematics knowledge gap, encourage learning and reflection, and future-proof academic performance', The Australia New Zealand Mathematics Convention, Melbourne, Australia.

First year students and academic staff in Science, Technology, Engineering and Mathematics (STEM) disciplines currently face many challenges. Failure rates at UTS are high in first year undergraduate Mathematics subjects for STEM programs. These high failure rates are particularly pronounced in students who studied General Mathematics this includes the subject Foundation Mathematics, a subject designed to address any gap in assumed knowledge and skills of first year students. Attrition is also a concern, with around 10% attrition after one semester and an additional 15% after two semesters. UTS is not alone in facing these challenges – under-preparedness for tertiary mathematics is a problem world-wide. When this problem first came to light more than a decade ago, UTS introduced the Readiness Survey (diagnostic test) to assess the extent to which the 'Assumed Knowledge could indeed be assumed. This assessment of assumed knowledge and the associated pre-teaching could be effective but as the failure rates demonstrate, success has been mixed. A meeting of first year Mathematics academics in 2013 decided to trial a different and historically successful approach – Mastery Learning. Mastery Learning endorses the belief that aptitude relates to the amount of time it takes someone to learn, rather than necessarily capability to master content. The research literature indicates positive effects of Mastery Learning on students, especially in achievement, attitudes towards learning and retention of content. This paper describes the learning design and positives and negatives of implementing Mastery Learning in first year Mathematics subjects.

## Journal articles

Groen, L., Coupland, M., Langtry, T., Memar, J., Moore, B.J. & Stanley, J. 2015, 'The Mathematics Problem and Mastery Learning for First-Year, Undergraduate STEM Students',

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*International Journal of Learning, Teaching and Educational Research*, vol. 11, no. 1, pp. 141-160.View/Download from: UTS OPUS

In the 2014 academic year Mastery Learning was implemented in four first-year mathematics subjects in an effort to address a lack of preparedness and poor outcomes of increasing numbers of undergraduate students in science, engineering and mathematics programs. This followed partial success in the use of diagnostic testing and pre-teaching, active learning, and a greater emphasis on problem solving in context - under-prepared students were still more likely to fail the pre-teaching subject and to struggle with subsequent mathematics subjects. Also, failure rates overall were higher than benchmarks required. This paper describes the learning design used, and the outcomes achieved, with implementing Mastery Learning – the positive: improved academic success, time management, and attitudes towards learning and Mathematics, an increased sense of independence, confidence and retention of content, and reduced stress and anxiety; and the negative: students having a sense of being taught how to pass a test rather than having a deeper understanding of the content. It will be seen that this negative is a consequence of a small but important difference in implementation.

Groen, L., Coupland, M., Memar, J. & Langtry, T. 2015, 'Mastery Learning to Address the Assumed Mathematics Knowledge Gap, Encourage Learning and Reflection, and Future-proof Academic Performance',

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*International Journal of Innovation in Science and Mathematics Education*, vol. 23, no. 2, pp. 65-79.View/Download from: UTS OPUS

UTS Science, Engineering and Mathematics students who have studied General Mathematics at high school are far more likely to fail their first undergraduate mathematics subject compared to their counterparts who meet the non-compulsory 'Assumed Knowledge of 2 unit Mathematics. This problem has been growing in recent years as an increasing number of students seek to improve their tertiary entrance score by taking the no-calculus General Mathematics at the Higher School Certificate. This problem is not unique to the University of Technology, Sydney - mathematical under-preparedness is a problem world-wide, with a decade, or more, long history. For some years, UTS has used diagnostic testing and pre-teaching to assist under-prepared students. Unfortunately, students who studied General Mathematics are also more likely to fail the pre-teaching subject. This suggested something more was required. Mastery Learning was chosen as a potential solution. Results to date have been promising with improvements in academic success for under-prepared students. Students have also reported increased satisfaction, confidence and retention of content. However, some students felt all Mastery Learning taught them was how to pass the Mastery Tests. Differences in student experience appear to be due to differences in how Mastery Learning was implemented.

Langtry, T.N., Giokaris, P., Milthorpe, B.K. & Lord, M.S. 2012, 'Parameter estimation for a model of fibronectin adsorption onto hydroxylapatite, oxidised polystyrene and nanostructured silica',

*ANZIAM Journal*, vol. 54, no. SUPPL. Fibronectin is a protein present in blood and the extracellular matrix which has important roles in cell adhesion and migration, wound healing and blood clotting. Three models of fibronectin adsorption, on various substrates of interest to biochemists, are compared. The first model (of Langmuir) is expressed explicitly as a time dependent function for the mass of protein adsorbed. The second model is a modification of the scaled particle theory of Reiss et al. [J. Chem. Phys., 31:369,380, 1959] and takes into account the probability of a molecule finding a sufficiently large vacant area on the adsorbing substrate surface. The third model extends the second model to the case in which molecules may expand the radius of their contact with the substrate upon adsorption. We used datasets obtained from experiments to compare the models. The Langmuir model is straightforward to fit to a dataset. The remaining models are fitted using a steepest descent method to minimise least squares error. We describe initial estimates for parameters for this procedure and compare the quality of fit of the models. © Austral. Mathematical Soc. 2013.

Langtry, T.N., Chew, K.L., Zinder, Y., Yu, Q. & Li, L. 2012, 'Estimation of biochemical parameters from leaf photosynthesis',

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*ANZIAM Journal*, vol. 53, no. EMAC2011, pp. C218-C235.View/Download from: UTS OPUS

Langtry, T., Giokaris, P., Milthorpe, B. & Lord, M.E. 2012, 'Parameter estimation for a model of fibronectin adsorption onto hydroxylapatite, oxidised polystyrene and nanostructured silica',

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*ANZIAM Journal : Electronic Supplement*, vol. 54, pp. c429-c445.View/Download from: UTS OPUS or Publisher's site

Fibronectin is a protein present in blood and the extracellular matrix which has important roles in cell adhesion and migration, wound healing and blood clotting. Three models of fibronectin adsorption, on various substrates of interest to biochemists, are compared. The first model (of Langmuir) is expressed explicitly as a time dependent function for the mass of protein adsorbed. The second model is a modification of the scaled particle theory of Reiss et al. [J. Chem. Phys., 31:369--380, 1959] and takes into account the probability of a molecule finding a sufficiently large vacant area on the adsorbing substrate surface. The third model extends the second model to the case in which molecules may expand the radius of their contact with the substrate upon adsorption. We used datasets obtained from experiments to compare the models. The Langmuir model is straightforward to fit to a dataset. The remaining models are fitted using a steepest descent method to minimise least squares error. We describe initial estimates for parameters for this procedure and compare the quality of fit of the models

Langtry, T.N., Chew, K.L., Zinder, Y., Yu, Q. & Li, L. 2012, 'Estimation of biochemical parameters from leaf photosynthesis',

*ANZIAM Journal*, vol. 53, no. EMAC2011, pp. C218-C235. Jay, K., Chaumet, P.C., Langtry, T. & Rahmani, A. 2010, 'Optical binding of electrically small magnetodielectric particles',

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*Journal of Nanophotonics*, vol. 4, pp. 1-9.View/Download from: UTS OPUS or Publisher's site

An ensemble of spherical particles with arbitrary dielectric permittivity and magnetic penneability was considered in the dipole approximation. Each particle was described by complex electric and magnetic polarizabilities. A computational approach based on the coupled dipole method, also called the discrete dipole approximation, was used to derive the optical force experienced by each particle due to an incident electromagnetiG..Ji.eld and the fields scattered by all other particles. This approach is general and can handle material dispersion and losses. In order to illustrate this approach, we studied the case of two spherical particles separated by a distance d, and illuminated by an incident plane wave whose wave vector is normal to the axis of the particles. We computed the optical force experienced by each particle in the direction of the beam (radiation pressure), and perpendicular to the beam (optical binding) for particles with positive and negative refractive indices. We also considered the effect of material losses.

Jay, K., Chaumet, P.C., Langtry, T.N. & Rahmani, A. 2010, 'Optical binding of electrically small magnetodielectric particles',

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*JOURNAL OF NANOPHOTONICS*, vol. 4.View/Download from: Publisher's site

Asatryan, A.A., Botten, L.C., Byrne, M.A., Langtry, T.N., Nicorovici, N.A., McPhedran, R.C., de Sterke, C.M. & Robinson, P.A. 2005, 'Conductance of photons in disordered photonic crystals',

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*PHYSICAL REVIEW E*, vol. 71, no. 3.View/Download from: Publisher's site

Botten, L.C., White, T.P., de Sterke, C.M., McPhedran, R.C., Asatryan, A.A. & Langtry, T. 2004, 'Photonic crystal devices modelled as grating stacks: matrix generalisations of thin film optics',

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*Optics Express*, vol. 12, no. 8, pp. 1592-1604.View/Download from: UTS OPUS or Publisher's site

A rigorous semi-analytic approach to the modelling of coupling guiding and propagation in complex microstructures embedded in two-dimensional photonic crystals is presented. The method, which is based on Bloch mode expansions an generalised Fresnel coefficients is shown to be able to treat photonic crystal devices in ways which are analogous to those used in thin film optics with uniform media. Asymptotic methods are developed and exemplified through the study of a serpentine waveguide, a potential slow wave device.

White, T.P., Botten, L.C., de Sterke, C.M., McPhedran, R.C., Asatryan, A.A. & Langtry, T. 2004, 'Bloch mode scattering matrix methods for modelling extended photonic crystal structures. II: Applications',

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*Physical Review E*, vol. 70, no. 5, pp. 1-10.View/Download from: UTS OPUS or Publisher's site

The Bloch mode scattering matrix method is applied to several photonic crystal waveguide structures and devices, including waveguide dislocations, a Fabry-Perot resonator, a folded directional coupler, and a Y-junction design. The method is an efficient tool for calcuating the properties of extended photonic crystal (PC) devices, in particular whenthe device consists of a small number of distict photonic crystal structures, or for long propagation lengths through uniform PC waveguides. The physical insight provided by the method i sused to derive simple, semianalytic models that allow fast and efficient calculations of complex photonic crystal structures. We discuss the situations in which such simplifications can be made and provide examples.

Langtry, T., Botten, L.C., Asatryan, A.A., Byrne, M.A. & Bourgeois, A. 2004, 'Localisation and disorder in the design of 2D photonic crystal devices',

View/Download from: UTS OPUS

*ANZIAM Journal*, vol. 45, pp. 744-758.View/Download from: UTS OPUS

Photonic crystals are meta-materials that can inhibit the propagation of light in all directions for specific wavelength ranges. Material or structural defects can be introduced into the crystal to cause localised modes, providing the ability to mould the flow of light on the wavelength scale and allowing the development of miniaturised, integrated photonic devices. For this reason, photonic crystals will likely be key building blocks for future micro-optical and communication technology. In this paper, we examine the Bloch mode modelling of 2D photonic crystal structures with application to the analysis of photonic crystal waveguides and their susceptibility to disorder, which provides a framework for studying fabrication tolerances in realistic devices.

Botten, L.C., White, T.P., Asatryan, A.A., Langtry, T., de Sterke, C.M. & McPhedran, R.C. 2004, 'Bloch Mode Scattering Matrix Methods For Modeling Extended Photonic Crystal Structures. I. Theory',

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*Physical Review E*, vol. 70, no. 5, pp. 1-13.View/Download from: UTS OPUS

We present a rigorous Bloch mode scattering matrix method for modeling two-dimensional photonic crystal structures and discuss the formal properties of the formulation. Reciprocity and energy conservation considerations lead to modal orthogonality relati

White, T.P., Botten, L.C., de Sterke, C.M., McPhedran, R.C., Asatryan, A.A. & Langtry, T.N. 2004, 'Bloch mode scattering matrix methods for modeling extended photonic crystal structures. II. Applications.',

*Physical review. E, Statistical, nonlinear, and soft matter physics*, vol. 70, no. 5 Pt 2, p. 056607. The Bloch mode scattering matrix method is applied to several photonic crystal waveguide structures and devices, including waveguide dislocations, a Fabry-Pérot resonator, a folded directional coupler, and a Y-junction design. The method is an efficient tool for calculating the properties of extended photonic crystal (PC) devices, in particular when the device consists of a small number of distinct photonic crystal structures, or for long propagation lengths through uniform PC waveguides. The physical insight provided by the method is used to derive simple, semianalytic models that allow fast and efficient calculations of complex photonic crystal structures. We discuss the situations in which such simplifications can be made and provide examples.

Botten, L.C., White, T.P., Asatryan, A.A., Langtry, T.N., de Sterke, C.M. & McPhedran, R.C. 2004, 'Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory.',

*Physical review. E, Statistical, nonlinear, and soft matter physics*, vol. 70, no. 5 Pt 2, p. 056606. We present a rigorous Bloch mode scattering matrix method for modeling two-dimensional photonic crystal structures and discuss the formal properties of the formulation. Reciprocity and energy conservation considerations lead to modal orthogonality relations and normalization, both of which are required for mode calculations in inhomogeneous media. Relations are derived for studying the propagation of Bloch modes through photonic crystal structures, and for the reflection and transmission of these modes at interfaces with other photonic crystal structures.

Asatryan, A.A., Robinson, P.A., McPhedran, R.C., Botten, L.C., de Sterke, C.M., Langtry, T. & Nicorovici, N.A. 2003, 'Diffusion and anomalous diffusion of light in two-dimensional photonic crystals',

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*Physical Review E*, vol. 67, no. 3, pp. 1-8.View/Download from: UTS OPUS

Langtry, T., Coupland, M.P. & Moore, B.J. 2003, 'Mathematica in context',

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*International Journal of Mathematical Education in Science & Technology*, vol. 34, no. 5, pp. 699-718.View/Download from: UTS OPUS or Publisher's site

Langtry, T., Botten, L.C., Asatryan, A.A. & McPhedran, R.C. 2003, 'Monte Carlo modelling of imperfections in two-dimensional photonic crystals',

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*Mathematics And Computers In Simulation*, vol. 62, no. 3-6, pp. 385-393.View/Download from: UTS OPUS or Publisher's site

In this paper we describe a Monte Carlo simulation of imperfections in photonic crystals, a new class of materials with optical properties that offer promise in a range of potential applications in the areas of information and communications technology. We describe the relevant physical and structural properties of these materials and outline the derivation of a theoretical model. We then present a Monte Carlo investigation of the tolerance of these materials to fabrication defects.

Botten, L.C., Asatryan, A.A., Langtry, T., White, T.P., de Sterke, C.M. & McPhedran, R.C. 2003, 'Semianalytic treatment for propagation in finite photonic crystal waveguides',

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*Optics Letters*, vol. 28, no. 10, pp. 854-856.View/Download from: UTS OPUS or Publisher's site

e present a semianalytic theory for the properties of two-dimensional photonic crystal waveguides of finite length. For single-mode guides, the transmission spectrum and field intensity can be accurately described by a simple two-parameter model. Analogies are drawn with FabryPerot interferometers, and generalized Fresnel coefficients for the interfaces are calculated.

Langtry, T.N., Asatryan, A.A., Botten, L.C., de Sterke, C.M., McPhedran, R.C. & Robinson, P.A. 2003, 'Effects of disorder in two-dimensional photonic crystal waveguides',

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*PHYSICAL REVIEW E*, vol. 68, no. 2.View/Download from: Publisher's site

Asatryan, A.A., Robinson, P.A., McPhedran, R.C., Botten, L.C., Martijn de Sterke, C., Langtry, T.L. & Nicorovici, N.A. 2003, 'Diffusion and anomalous diffusion of light in two-dimensional photonic crystals',

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*, vol. 67, no. 3 2. The transport velocity v E /c introduced by Brillouin was calculated from first principles for a random medium with an intermediate filling fraction f=0.5. It was found that the transport velocity of light v E in ordered finite-size photonic crystals can be substantially lower than the free space value for a gap wavelength. The effects of the disorder on the transport velocity were also considered and it was shown that v E /c can be more than 15 times less than in free space, which was in agreement with the experimental results reported.

Langtry, T.N., Asatryan, A.A., Botten, L.C., De Sterke, C.M., McPhedran, R.C. & Robinson, P.A. 2003, 'Effects of disorder in two-dimensional photonic crystal waveguides',

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*, vol. 68, no. 2 2. The effects of disorder on the waveguiding properties of two-dimensional photonic crystals were studied. The guide was excited by a line source parallel to the cylinders, located close to the entry of the guide. The resulting field intensity and local density of states (LDOS) were calculated using a highly accurate multipole method. Quantitative results characterizing the effects of fabrication defects for a TM-polarized field were obtained by Monte Carlo simulation.

Thornton, B.S., Nguyen, H.T., Hung, A., Hirst, C., Thornton-Benko, E. & Langtry, T. 2001, 'Breast Screening Outcomes: Communications Problems, Chaos Relationship and Control Theory',

*Canadian Applied Mathematics Quarterly*, vol. 9, no. 4, pp. 377-401. Langtry, T. 2001, 'A construction of higher-rank lattice rules',

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*Mathematics and Computers in Simulation*, vol. 55, no. 1-3, pp. 103-111.View/Download from: UTS OPUS or Publisher's site

Lattice rules are quasi-Monte Carlo methods for numerical multiple integration that are based on the selection of an s-dimensional integration lattice. The abscissa set is the intersection of the integration lattice with the unit hypercube. It is well-known that the abscissa set of a lattice rule can be generated by a number of fixed rational vectors. In general, different sets of generators produce different integration lattices and rules, and a given rule has many different generator sets. The rank of the rule is the minimum number of generators required to span the abscissa set. A lattice rule is usually specified by a generator set, and the quality of the rule varies with the choice of generator set. This paper describes a new method for the construction of generator sets for higher-rank rules that is based on techniques arising from the theory of simultaneous Diophantine approximation. The method extends techniques currently applied in the rank 1 case.

Langtry, T. 1999, 'Lattice rules of minimal and maximal rank with good figures of merit',

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*Journal of Computational and Applied Mathematics*, vol. 112, no. 1-2, pp. 147-164.View/Download from: UTS OPUS or Publisher's site

For periodic integrands with unit period in each variable, certain error bounds for lattice rules are conveniently characterised by the figure of merit rho, which was originally introduced in the context of number theoretic rules. The problem of finding good rules of order N (that is, having N distinct nodes) then becomes that of finding rules with large values of rho. This paper presents efficient search methods for the discovery of rank 1 rules, and of maximal rank rules of high order, which possess good figures of merit.

Langtry, T. 1996, 'An application of Diophantine approximation to the construction of rank-1 lattice quadrature rules',

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*Mathematics Of Computation*, vol. 65, no. 216, pp. 1635-1662.View/Download from: UTS OPUS or Publisher's site

Lattice quadrature rules were introduced by Frolov (1977), Sloan (1985) and Sloan and Kachoyan (1987). They are quasi-Monte Carlo rules for the approximation of integrals over the unit cube in R(s) and are generalizations of 'number-theoretic' rules introduced by Korobov (1959) and Hlawka (1962)-themselves generalizations, in a sense, of rectangle rules for approximating one-dimensional integrals, and trapezoidal rules for periodic integrands. Error bounds for rank-1 rules are known for a variety of classes of integrands. For periodic integrands with unit period in each variable, these bounds are conveniently characterized by the figure of merit rho, which was originally introduced in the context of number-theoretic rules. The problem of finding good rules of order N (that is, having N nodes) then becomes that of finding rules with large values of rho. This paper presents a new approach, based on the theory of simultaneous Diophantine approximation, which uses a generalized continued fraction algorithm to construct rank-1 rules of high order.

Langtry, T. 1995, 'The determination of canonical forms for lattice quadrature rules',

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*Journal of Computational and Applied Mathematics*, vol. 59, no. 2, pp. 129-143.View/Download from: Publisher's site

Lattice rules are equal weight numerical quadrature rules for the integration of periodic functions over the s-dimensional unit hypercube U-s = [0, 1)(s). For a given lattice rule, say Q(L), a set of points L (the integration lattice), regularly spaced in all of R(s), is generated by a finite number of rational vectors. The abscissa set for Q(L) is then P(Q(L))=L boolean AND U-s. It is known that P(Q(L)) is a finite Abelian group under addition module the integer lattice Z(s), and that QL(f) may be written in the form of a nonrepetitive multiple sum, known as a canonical form, in which + denotes addition module Z(s). In this form, z(i) is an element of Z(S), m is called the rank and n(1), n(2),..., n(m) are called the invariants of Q(L), and n(i+1)\n(i) for i = 1,2,..., m-1 . The rank and invariants are uniquely determined for a given lattice rule. In this paper we provide a construction of a canonical form for a lattice rule Q(L), given a generator set for the lattice L. We then show how the rank and invariants of QL may be determined directly from the generators of the dual lattice L perpendicular to.

Thornton, B.S. & Langtry, T. 1988, 'Prescheduling graphic displays for optimal cancer therapies to reveal possible tumour regression or stabilisation',

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*Journal of Medical Systems*, vol. 12, no. 1, pp. 31-41.View/Download from: UTS OPUS

The paper describes an adaptive control approach to the problem of the treatment of solid tumors. The evolution with time t of the state of a tumor is modelled by a two-compartment system, governed by two differential equations forming an autonomous system under therapy control u, dy1/dt = f1(y1,y2;u) dy2/dt = f2(y1,y2;u), where y1 and y2 are the number of proliferating and nonproliferating cells, respectively. The output is analyzed in the phase plane y1y2. The control problem is that of restricting the tumor state to a predetermined region of the plane by selecting a suitable change in therapy control u, e.g., modality and dosage, when the state solution intersects the boundary of this region and the ratio y1/y2 of proliferating to nonproliferating cells is displayed together with an elapsed time scale. Then, consequent selection of a suitable therapeutic sequence may be assisted by the use of a data base as part of an expert system. The process is repeated at each intersection of the prescribed boundary. Such sequences may lead to stabilization of the system through the appearance on a computer display screen of a stable equilibrium point or a limit cycle.