1. www.cudos.org.au Modes in Microstructured Optical Fibres Martijn de Sterke, Ross McPhedran, Peter Robinson, CUDOS and School of Physics, University of Sydney, Australia Boris Kuhlmey, Gilles Renversez, Daniel Maystre Institut Fresnel, Université Aix Marseille III, France

2. www.cudos.org.au Outline Microstructured optical fibres (MOFs) Modal cut-off in MOFS―what is issue? Analysis MOF modes―Bloch transform Modal cut-off of MOF modes –Second mode –Fundamental mode Conclusion

3. www.cudos.org.au MOFs: Holes Silica matrix Core: - air hole - silica cladding, n J Conventional fibres: core n C >n J Total internal reflection MOFs and conventional fibres

4. www.cudos.org.au Cladding, n J Conventional fibres: Core, n C >n J Total reflection MOFs:  d Holes Silica matrix Core: - air hole - silica MOFs and conventional fibres

5. www.cudos.org.au Key MOF properties “Endlessly single-modedness” (Birks et al, Opt. Lett. 22, 961 (1997)) Unique dispersion

6. www.cudos.org.au MOFs and structural losses Finite number of rings  always losses

7. www.cudos.org.au Dilemma of Modes in MOFs (1) Conventional fibre: number of modes is number of bound modes (without loss) In a MOF, all modes have loss Want: way to select small set of preferred MOF modes, to get a mode number

8. www.cudos.org.au Dilemma of Modes in MOFs (2) The answer lies in the difference between bound modes and extended modes Few bound modes: sensitive to core details, loss decreases exponentially with fibre size Many extended modes: insensitive to core details, loss decreases algebraically with fibre size

9. www.cudos.org.au Properties of Modes in MOFs Mode properties have been studied using the vector multipole method This enables calculation of confinement loss accurately, down to very small levels The form of modal fields is also calculated, and symmetry/degeneracy properties can be incorporated into the method JOSA B 19, 2322 & 2331 (2002)

10. www.cudos.org.au Bloch Transform Bloch transform enables post processing of each mode to clarify structure better Combine quantities B n (describe field amplitude at each cylinder centred at c l ) Define: If fields at all holes are in phase: peaks at k=0

11. www.cudos.org.au Bloch Transform: properties Peaks at Bloch vectors associated with mode Periodic in k-space (if holes on lattice) Knowledge in first Brillouin zone suffices Other properties as for Fourier transform –Heisenberg-like relation – Parseval-like relation

12. www.cudos.org.au Bloch Transform: benefit Understand and recognize modes x y |Sz||Sz| kxkx kyky Bloch Transform Max Min Real space Reciprocal space

13. www.cudos.org.au Extended modes : dependence on cladding shape |S z | (real space) Bloch Transform (reciprocal lattice)

14. www.cudos.org.au Centred CoreDisplaced CoreNo Core Extended modes: weak dependence on core

15. www.cudos.org.au Defect modes |Sz||Sz|Bloch Transform

16. www.cudos.org.au Defect Modes: weak dependence on cladding shape

17. www.cudos.org.au Centred CoreDisplaced CoreNo Core Defect Modes: strong dependence on defect ? Very strong losses!

18. www.cudos.org.au Cutoff of second mode: from multimode to single mode In modal cutoff studies, operate at λ=1.55  m; follow modal changes as rescale period  and hole diameter d, keeping ratio constant.

19. www.cudos.org.au Cutoff of second mode: localisation transition Mode size Loss 1 10 -4 10 -8 10 -12 Loss  d/  =0.55,  m

20. www.cudos.org.au 8 10 rings 8 The transition sharpens Mode size Loss 1 10 -4 10 -8 10 -12 Loss  d/  =0.55,  m 4

21. www.cudos.org.au Zero-width transition for infinite number of rings Number of rings Transition Width (on period)

22. www.cudos.org.au Without the cut-off moving Cut-off wavlength (on period) Number of rings

23. www.cudos.org.au Phase diagram of second mode  multimode monomode “endlessly monomode” d 

24. www.cudos.org.au Cutoff of second mode: experimental verification From J. R. Folkenberg et al., Opt. Lett. 28, 1882 (2003).

25. www.cudos.org.au Fundamental mode transition? Conventional fibres: no cut-off W Fibres : cut-off possible, cut-off wavelength proportional to jacket size MOF’s ?

26. www.cudos.org.au Hint of fundamental mode cut-off d/  =0.3,  m Loss

27. www.cudos.org.au 33 4 3 4 5 3 4 56 3 4 8 56 Transition sharpens d/  =0.3,  m Loss

28. www.cudos.org.au But keeps non-zero width Number of rings Transition width (on period)

29. www.cudos.org.au      Transition of finite width: transition region d/  =0.3,  m Loss  Q Confined Extended Cut-off Transition

30. www.cudos.org.au Phase diagram and operating regimes Homogenisation

31. www.cudos.org.au Simple interpretation of second mode cut-off From Mortensen et al., Opt. Lett. 28, 1879 (2003)

32. www.cudos.org.au Conclusions Both fundamental and second MOF modes exhibit transitions from extended to localized behaviour, but the way this happens differs Number of MOF modes may be regarded as number of localized modes MOF modes behave substantially differently than in conventional fibres only where they change from extended to localized