Bend performance-enhanced photonic crystal fibers with anisotropic numerical aperture

Benjamin G. Ward

doi:10.1364/OE.16.008532

Bend performance-enhanced photonic crystal fibers with anisotropic numerical aperture

Benjamin G. Ward

Optics Express
Vol. 16,
Issue 12,
pp. 8532-8548
(2008)
•https://doi.org/10.1364/OE.16.008532

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Benjamin G. Ward, "Bend performance-enhanced photonic crystal fibers with anisotropic numerical aperture," Opt. Express 16, 8532-8548 (2008)

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Related Topics
Table of Contents Category
- Photonic Crystal Fibers
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Fiber design and fabrication (060.2280)
- Fiber optics (060.2310)
- Fiber optics amplifiers and oscillators (060.2320)
- Photonic crystal fibers (060.5295)
- Lasers, fiber (140.3510)
- Waveguides (230.7370)

About this Article
History
- Original Manuscript: April 10, 2008
- Revised Manuscript: May 21, 2008
- Manuscript Accepted: May 23, 2008
- Published: May 27, 2008

Accessible

Open Access

Abstract

Numerical calculations predict that particular birefringent photonic crystal fiber designs exhibit slightly better performance in a coiled configuration than non-birefringent step-index fiber designs with respect to higher order mode suppression for the realization of large mode area effectively single transverse mode fibers. The passive losses of the fundamental and first few higher order modes of a birefringent photonic crystal fiber design with a 41µm diameter core incorporating stress applying parts (SAP) were calculated using an integrated electromechanical finite element method. Minimum higher order mode losses of up to 5.5 dB/m were predicted for fundamentalmode losses of only 0.0014 dB/m. The bend performance of this PCF design was predicted to be relatively insensitive to manufacturing tolerances with respect to air hole size and device assembly tolerances with respect to coiling diameter based on the calculated dependence of the mode losses on these parameters. The positions and refractive index of the SAP render the numerical aperture of the core anisotropic allowing further tailoring of the bend performance by adjusting the angle between the coiling plane of the fiber and the orientation of the SAP within the cladding. Fundamental and higher-order mode losses are calculated for step-index fiber (SIF) designs with a 40µm diameter core for comparison. The step-index fiber designs were predicted to exhibit slightly inferior bend loss mode discrimination and higher sensitivity to packaging configuration compared to the photonic crystal fiber designs presented.

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Corrections

Benjamin Ward, "Bend performance-enhanced photonic crystal fibers with anisotropic numerical aperture: errata," Opt. Express 20, 7966-7972 (2012)
http://proxy.osapublishing.org/oe/abstract.cfm?uri=oe-20-7-7966

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References

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[Crossref]
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J. Limpert, O. Schmidt, J. Rothhardt, F. Rser, T. Schreiber, A. Tnnermann, S. Ermeneux, P. Yvernault, and F. Salin, “Extended single-mode photonic crystal fiber lasers,” Opt. Express 14, 2715–2720 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-7-2715.
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R. T. Schermer and J. H. Cole, “Improved Bend Loss Formula Verified for Optical Fiber by Simulation and Experiment,” IEEE J. Quantum Electron. 43, 899–909 (2007).
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Robert D. Cook, Finite Element Modeling for Stress Analysis (Wiley, New York, 1995).
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[Crossref]
Kazuaki Sakoda, Optical Properties of Photonic Crystals (Springer, New York, 2001).
T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B 19, 2322–2330 (2002).
[Crossref]
B. T. Kuhlmey, University of Sydney Australian Research Counsel Centre of Excellence for Ultrahigh-bandwidth Devices for Optical Systems Microstructured Optical Fibre Utilties Software 2004.
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[Crossref]
J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, pp. 185–200, (1994).
[Crossref]
T. Schreiber, F. Rser, O. Schmidt, J. Limpert, R. Iliew, F. Lederer, A. Petersson, C. Jacobsen, K. Hansen, J. Broeng, and A. Tnnermann, “Stress-induced single-polarization single-transverse mode photonic crystal fiber with low nonlinearity,” Opt. Express 13, 7621–7630 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-19-7621.
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K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on finite element scheme: Application to photonic crystal fibers,” IEEE J. Quantum Electron. 38, 927–933 (2002).
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J. Olszewski, M. Szpulak, and W. Urbanczyk, “Effect of coupling between fundamental and cladding modes on bending losses in photonic crystal fibers,” Opt. Express 13, 6015–6022 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-16-6015.
[Crossref] [PubMed]
Jeremy Kepner, “Parallel Programming with MatlabMPI,” in Proceedings of the High Performance Embedded Computing (HPEC 2001) workshop,(MIT Lincoln Laboratory, Lexington, MA, 2001).
Satish Balay, Kris Buschelman, Victor Eijkhout, William D. Gropp, Dinesh Kaushik, Matthew G. Knepley, Lois Curfman McInnes, Barry F. Smith, and Hong Zhang, PETSc Users Manual ANL-95/11 - Revision 2.1.5, Argonne National Laboratory, 2004.
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[Crossref]

2008 (2)

T. -w Wu, L. Dong, and H. Winful, “Bend performance of leakage channel fibers,” Opt. Express 16, 4278–4285 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-6-4278.
[Crossref] [PubMed]

B. G. Ward, “Finite Element Analysis of Photonic Crystal Rods with Anisotropic Inhomogeneous Refractive Index Tensor,” IEEE J. Quantum Electron. 44, 150–156 (2008).
[Crossref]

2007 (1)

R. T. Schermer and J. H. Cole, “Improved Bend Loss Formula Verified for Optical Fiber by Simulation and Experiment,” IEEE J. Quantum Electron. 43, 899–909 (2007).
[Crossref]

2002 (2)

K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on finite element scheme: Application to photonic crystal fibers,” IEEE J. Quantum Electron. 38, 927–933 (2002).
[Crossref]

T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B 19, 2322–2330 (2002).
[Crossref]

2001 (1)

S. Hfer, A. Liem, J. Limpert, H. Zellmer, A. Tnnermann, S. Unger, S. Jetschke, H. -R. Mller, and I. Freitag, “Single-frequency master-oscillator fiber power amplifier system emitting 20 Wof power,” Opt. Lett. 26, 1326–1328 (2001), http://www.opticsinfobase.org/abstract.cfm?URI=ol-26-17-1326.
[Crossref]

2000 (1)

J. P. Koplow, D. A. V. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. 25, 442–444 (2000), http://www.opticsinfobase.org/abstract.cfm?URI=ol-25-7-442.
[Crossref]

1999 (1)

I. Zawischa, K. Plamann, C. Fallnich, H. Welling, H. Zellmer, and A. Tnnermann, “All-solid-state neodymiumbased single-frequency master-oscillator fiber power-amplifier system emitting 5.5 W of radiation at 1064 nm,” Opt. Lett. 24, 469–471 (1999), http://www.opticsinfobase.org/abstract.cfm?URI=ol-24-7-469.
[Crossref]

1998 (1)

F. L. Teixeira and W. C. Chew, “General Closed-Form PML Constitutive Tensors to Match Arbitrary Bianisotropic and Dispersive Linear Media,” IEEE Microwave Guid. Wave Lett. 8, 223–225 (1998).
[Crossref]

1994 (1)

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, pp. 185–200, (1994).
[Crossref]

Alegria, C.

Alvarez-Chavez, J. A.

Balay, Satish

Satish Balay, Kris Buschelman, Victor Eijkhout, William D. Gropp, Dinesh Kaushik, Matthew G. Knepley, Lois Curfman McInnes, Barry F. Smith, and Hong Zhang, PETSc Users Manual ANL-95/11 - Revision 2.1.5, Argonne National Laboratory, 2004.

Berenger, J.-P.

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, pp. 185–200, (1994).
[Crossref]

Botten, L. C.

Broeng, J.

Buschelman, Kris

Chew, W. C.

Chryssou, C. E.

Codemard, C. A.

Cole, J. H.

R. T. Schermer and J. H. Cole, “Improved Bend Loss Formula Verified for Optical Fiber by Simulation and Experiment,” IEEE J. Quantum Electron. 43, 899–909 (2007).
[Crossref]

Cook, Robert D.

Robert D. Cook, Finite Element Modeling for Stress Analysis (Wiley, New York, 1995).

de Sterke, C. M.

Dong, L.

Dupriez, P.

Eijkhout, Victor

Ermeneux, S.

Fallnich, C.

Fini, J.

Frede, M.

Freitag, I.

Goldberg, L.

Gropp, William D.

Hansen, K.

Hernandez, Vicente

Vicente Hernandez, Jose E. Roman, and Vicente Vidal, “SLEPc: A Scalable and Flexible Toolkit for the Solution of Eigenvalue Problems,” ACM Transactions on Mathematical Software, 31, 351–362 (2005).
[Crossref]

Hfer, S.

Hickey, L. M. B.

Hildebrandt, M.

Horley, R.

Iliew, R.

Jacobsen, C.

Jeong, Y.

Jetschke, S.

Kaushik, Dinesh

Kepner, Jeremy

Jeremy Kepner, “Parallel Programming with MatlabMPI,” in Proceedings of the High Performance Embedded Computing (HPEC 2001) workshop,(MIT Lincoln Laboratory, Lexington, MA, 2001).

Kliner, D. A. V.

Knepley, Matthew G.

Koplow, J. P.

Koshiba, M.

Kracht, D.

Kuhlmey, B. T.

B. T. Kuhlmey, University of Sydney Australian Research Counsel Centre of Excellence for Ultrahigh-bandwidth Devices for Optical Systems Microstructured Optical Fibre Utilties Software 2004.

Kwee, P.

Lederer, F.

Liem, A.

Limpert, J.

Maystre, D.

McInnes, Lois Curfman

McLaughlin, J. M.

McPhedran, R. C.

Mller, H. -R.

Nilsson, J.

Olszewski, J.

Payne, D. N.

Peng, X.

Petersson, A.

Plamann, K.

Renversez, G.

Roman, Jose E.

Rothhardt, J.

Rser, F.

Sahu, J. K.

Saitoh, K.

Sakoda, Kazuaki

Kazuaki Sakoda, Optical Properties of Photonic Crystals (Springer, New York, 2001).

Salin, F.

Schermer, R. T.

R. T. Schermer and J. H. Cole, “Improved Bend Loss Formula Verified for Optical Fiber by Simulation and Experiment,” IEEE J. Quantum Electron. 43, 899–909 (2007).
[Crossref]

Schmidt, O.

Schreiber, T.

Smith, Barry F.

Soh, D. B. S.

Szpulak, M.

Teixeira, F. L.

Tnnermann, A.

Turner, P. W.

Unger, S.

Urbanczyk, W.

Vidal, Vicente

Wanzcyk, L.

Ward, B. G.

B. G. Ward, “Finite Element Analysis of Photonic Crystal Rods with Anisotropic Inhomogeneous Refractive Index Tensor,” IEEE J. Quantum Electron. 44, 150–156 (2008).
[Crossref]

Welling, H.

White, T. P.

Willke, B.

Winful, H.

Wong, W. S.

Wu, T. -w

Yvernault, P.

Zawischa, I.

Zellmer, H.

Zhang, Hong

IEEE J. Quantum Electron. (3)

R. T. Schermer and J. H. Cole, “Improved Bend Loss Formula Verified for Optical Fiber by Simulation and Experiment,” IEEE J. Quantum Electron. 43, 899–909 (2007).
[Crossref]

B. G. Ward, “Finite Element Analysis of Photonic Crystal Rods with Anisotropic Inhomogeneous Refractive Index Tensor,” IEEE J. Quantum Electron. 44, 150–156 (2008).
[Crossref]

IEEE Microwave Guid. Wave Lett. (1)

J. Comput. Phys. (1)

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, pp. 185–200, (1994).
[Crossref]

J. Opt. Soc. Am. B (1)

Opt. Express (6)

Opt. Lett. (6)

Other (6)

B. T. Kuhlmey, University of Sydney Australian Research Counsel Centre of Excellence for Ultrahigh-bandwidth Devices for Optical Systems Microstructured Optical Fibre Utilties Software 2004.

Kazuaki Sakoda, Optical Properties of Photonic Crystals (Springer, New York, 2001).

Robert D. Cook, Finite Element Modeling for Stress Analysis (Wiley, New York, 1995).

Jeremy Kepner, “Parallel Programming with MatlabMPI,” in Proceedings of the High Performance Embedded Computing (HPEC 2001) workshop,(MIT Lincoln Laboratory, Lexington, MA, 2001).

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Fig. 1. Photonic crystal fiber structure. Th coloring was created by rendering the real part of the diagonal refractive index tensor components in the finite element model as a red green blue (RGB) triplet.

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Fig. 2. Meshing scheme (a) capillary cell (b) solid cell (c) node locations on element boundary.

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Fig. 3. Plots of the stress tensor components within the fiber cross-section for a coiling diameter of 16.8 cm showing bend-induced symmetry breaking. The scale on the right applies only to σ_xx , σ_yy , and σ_zz .

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Fig. 4. Calculated propagation losses as a function of air hole diameter to lattice pitch ratio(d/Λ)for the first four modes of the PCF with a coil diameter of 47.6 cm and coiling in the x direction. The lines serve only as a guide to the eye for all mode loss plots presented in this work.

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Fig. 5. Calculated modal intensity distributions for the four lowest-order fiber modes for coiling in the x direction with a diameter of 47.6 cm and a hole size d/Λ=0.12.

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Fig. 6. Calculated propagation losses as a function of air hole diameter to lattice pitch ratio (d/Λ)for the first four modes of the PCF with a coil diameter of 47.6 cm and coiling in the y direction.

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Fig. 7. Calculated modal intensity distributions for the four lowest-order fiber modes for coiling in the y direction with a diameter of 95.2 cm and a hole size d/Λ=0.12.

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Fig. 8. Calculated propagation losses as a function of coiling plane angle relative to the 00-plane of the SAP for the first three modes of the PCF with a coil diameter of 47.6 cm and d/Λ=0.12.

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Fig. 9. Calculated propagation losses as a function of coiling diameter for the first four modes of the PCF with d/Λ=0.12 and coiling in the x direction. The modal intensity plot insets are logarithmic in scale and show an example of how the SAP influence the guided modes depending on the coiling diameter.

Download Full Size | PPT Slide | PDF

Fig. 10. Calculated propagation losses as a function of coiling diameter for the first four modes of the PCF with d/Λ=0.12 and coiling in the y direction. The modal intensity plot insets are logarithmic in scale and show an example of how the guided modes change with coiling diameter.

Download Full Size | PPT Slide | PDF

Fig. 11. Calculated propagation losses as a function of coiling radius for the first three modes of a step-index fiber with 40µm core diameter and a numerical aperture of 0.06.

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Fig. 12. Bend-distorted mode field intensity plots of the first three modes of a step-index fiber with 40µm core diameter and a numerical aperture of 0.06 coiled to a diameter of 6 cm. The circle on each plot indicates the extent of the core.

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Fig. 13. Calculated propagation losses as a function of coiling diameter for the first three modes of a step-index fiber with 40µm core diameter and a numerical aperture of 0.035. The losses were averaged over the two polarization states of each mode.

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Tables (1)

Table 1. A comparison of LP₀₁ and LP₁₁ modal losses for a PCF with d/Λ=0.12 and 41µm core diameter and a SIF with a numerical aperture of 0.035 and a core diameter of 40µm. The 48 cm coil is in the x direction for the PCF. The optimal coiling diameter for the PCF is 95 cm in the y direction. The optimal coiling diameter for the SIF is 56 cm. All losses are in units of dB/m.

View Table

Equations (20)

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

U = \int_{Ω} [\frac{1}{2} ε (r) \cdot C (r) \cdot ε (r) - R_{th} (r) \cdot x (r)] d^{2} r

ε_{z} = \frac{x}{r_{x}} - \frac{y}{r_{y}}

U = \int_{Ω} [\frac{1}{2} ε_{t} (r) \cdot C_{t} (r) \cdot ε_{t} (r) + \sum_{i = x, y} ε_{z} C_{zi} ε_{i} - R_{th} (r) \cdot x (r)] d^{2} r

U = \int_{Ω} [\frac{1}{2} ε_{t} (r) \cdot C_{t} (r) \cdot ε_{t} (r) - R_{t} (r) \cdot x (r)] d^{2} r

R_{t} (r) = ε_{z} C_{zt} \nabla_{t} - R_{th} (r)

K_{t} \cdot x_{t} = R_{t}

S_{i} = \int_{Ω_{i}} [(\vec{\nabla} \times {\vec{H}}^{*}) \cdot {\overset{\leftrightarrow}{\tilde{ε}}}_{i}^{- 1} \cdot (\vec{\nabla} \times \vec{H}) - ω^{2} {\vec{H}}^{*} \cdot {\overset{\leftrightarrow}{μ}}_{i} \cdot \vec{H}] d^{2} \vec{r}

{\overset{\leftrightarrow}{\tilde{ε}}}_{i} = T_{i}^{- 1} \cdot {\overset{\leftrightarrow}{ε}}_{i} \cdot T_{i} (1 + 2 \frac{{\vec{r}}_{i} \cdot \vec{R}}{R^{2}})

ε_{i, 11} = ε_{0} (n_{0}^{2} - 2 (C_{1} σ_{1} + C_{2} σ_{2} + C_{2} σ_{3}))

ε_{i, 22} = ε_{0} (n_{0}^{2} - 2 (C_{2} σ_{1} + C_{1} σ_{2} + C_{2} σ_{3}))

ε_{i, 33} = ε_{0} (n_{0}^{2} - 2 (C_{2} σ_{1} + C_{2} σ_{2} + C_{1} σ_{3}))

ε_{i, jk} = 0, j \neq k

{\overset{\leftrightarrow}{\tilde{ε}}}_{i} = S_{i} \cdot T_{i}^{- 1} \cdot {\overset{\leftrightarrow}{ε}}_{i} \cdot T_{i} \cdot S_{i} (1 + 2 \frac{{\vec{r}}_{i} \cdot \vec{R}}{R^{2}})

{\overset{\leftrightarrow}{\tilde{μ}}}_{i} = S_{i} \cdot {\overset{\leftrightarrow}{μ}}_{i} \cdot S_{i}

S_{i} = \sqrt{s_{ix} s_{iy} s_{iz}} (\begin{matrix} s_{ix}^{- 1} & 0 & 0 \\ 0 & s_{iy}^{- 1} & 0 \\ 0 & 0 & s_{iz}^{- 1} \end{matrix}) .

S_{i} = U_{i}^{- 1} \cdot S_{ix} \cdot U_{i}

{\overset{\leftrightarrow}{\tilde{ε}}}_{i} = U_{i}^{- 1} \cdot S_{i} \cdot U_{i} . T^{- 1} \cdot {\overset{\leftrightarrow}{ε}}_{i} \cdot T \cdot U_{i}^{- 1} \cdot S_{i} \cdot U_{i} (1 + 2 \frac{{\overset{\leftrightarrow}{r}}_{i} \cdot \overset{\leftrightarrow}{R}}{R^{2}})

{\overset{\leftrightarrow}{\tilde{μ}}}_{i} = U_{i}^{- 1} \cdot S_{i} \cdot U_{i} \cdot {\overset{\leftrightarrow}{μ}}_{i} \cdot U_{i}^{- 1} \cdot S_{i} \cdot U_{i}

[K - β^{2} M] \overset{\leftrightarrow}{ϕ} = 0

L (\frac{dB}{m}) = 8.686 \times Im [β]

	48cm coil LP₀₁ loss	48cm coil LP₁₁ loss	optimal coil LP₀₁ loss	optimal coil LP₁₁ loss
PCF	0.0016	1.1	0.0014	5.5
SIF	0.14	6.8	0.0058	1.7

Abstract

Corrections

References

2008 (2)

2007 (1)

2006 (2)

2005 (5)

2003 (1)

2002 (2)

2001 (1)

2000 (1)

1999 (1)

1998 (1)

1994 (1)

Alegria, C.

Alvarez-Chavez, J. A.

Balay, Satish

Berenger, J.-P.

Botten, L. C.

Broeng, J.

Buschelman, Kris

Chew, W. C.

Chryssou, C. E.

Codemard, C. A.

Cole, J. H.

Cook, Robert D.

de Sterke, C. M.

Dong, L.

Dupriez, P.

Eijkhout, Victor

Ermeneux, S.

Fallnich, C.

Fini, J.

Frede, M.

Freitag, I.

Goldberg, L.

Gropp, William D.

Hansen, K.

Hernandez, Vicente

Hfer, S.

Hickey, L. M. B.

Hildebrandt, M.

Horley, R.

Iliew, R.

Jacobsen, C.

Jeong, Y.

Jetschke, S.

Kaushik, Dinesh

Kepner, Jeremy

Kliner, D. A. V.

Knepley, Matthew G.

Koplow, J. P.

Koshiba, M.

Kracht, D.

Kuhlmey, B. T.

Kwee, P.

Lederer, F.

Liem, A.

Limpert, J.

Maystre, D.

McInnes, Lois Curfman

McLaughlin, J. M.

McPhedran, R. C.

Mller, H. -R.

Nilsson, J.

Olszewski, J.

Payne, D. N.

Peng, X.

Petersson, A.

Plamann, K.

Renversez, G.

Roman, Jose E.

Rothhardt, J.

Rser, F.

Sahu, J. K.

Saitoh, K.

Sakoda, Kazuaki

Salin, F.

Schermer, R. T.

Schmidt, O.