The design and modeling of microstructured optical fiber

1. The design and modeling ofmicrostructured optical fiber Steven G. Johnson MIT / Harvard University

2. Outline • What are these fibers (and why should I care)? • The guiding mechanisms: index-guiding and band gaps • Finding the guided modes • Small corrections (with big impacts)

3. Outline • What are these fibers (and why should I care)? • The guiding mechanisms: index-guiding and band gaps • Finding the guided modes • Small corrections (with big impacts)

4. losses ~ 0.2 dB/km at l=1.55µm (amplifiers every 50–100km) more complex profiles to tune dispersion “high” index doped-silica core n ~ 1.46 “LP01” confined mode field diameter ~ 8µm protective polymer sheath but this is ~ as good as it gets… Optical Fibers Today(not to scale) silica cladding n ~ 1.45 [ R. Ramaswami & K. N. Sivarajan, Optical Networks: A Practical Perspective ]

5. Loss: amplifiers every 50–100km …limited by Rayleigh scattering (molecular entropy) …cannot use “exotic” wavelengths like 10.6µm Nonlinearities: after ~100km, cause dispersion, crosstalk, power limits (limited by mode area ~ single-mode, bending loss) also cannot be made (very) large for compact nonlinear devices The Glass Ceiling:Limits of Silica Radical modifications to dispersion, polarization effects? …tunability is limited by low index contrast High Bit-Rates Compact Devices Long Distances Dense Wavelength Multiplexing (DWDM)

6. Bragg fiber 1000x better loss/nonlinear limits [ Yeh et al., 1978 ] (from density) 1d crystal + omnidirectional = OmniGuides 2d crystal PCF (You can also put stuff in here …) [ Knight et al., 1998 ] Breaking the Glass Ceiling:Hollow-core Bandgap Fibers Photonic Crystal

7. Breaking the Glass Ceiling:Hollow-core Bandgap Fibers Bragg fiber [ Yeh et al., 1978 ] [ figs courtesy Y. Fink et al., MIT ] + omnidirectional = OmniGuides white/grey = chalco/polymer silica [ R. F. Cregan et al., Science285, 1537 (1999) ] 5µm PCF [ Knight et al., 1998 ]

8. Breaking the Glass Ceiling:Hollow-core Bandgap Fibers Guiding @ 10.6µm (high-power CO2 lasers) loss < 1 dB/m (material loss ~ 104 dB/m) [ figs courtesy Y. Fink et al., MIT ] white/grey = chalco/polymer [ Temelkuran et al., Nature420, 650 (2002) ] silica Guiding @ 1.55µm loss ~ 13dB/km [ R. F. Cregan et al., Science285, 1537 (1999) ] 5µm [ Smith, et al., Nature424, 657 (2003) ] OFC 2004: 1.7dB/km BlazePhotonics

9. Breaking the Glass Ceiling II:Solid-core Holey Fibers solid core holey cladding forms effective low-index material Can have much higher contrast than doped silica… strong confinement = enhanced nonlinearities, birefringence, … [ J. C. Knight et al., Opt. Lett.21, 1547 (1996) ]

10. Breaking the Glass Ceiling II:Solid-core Holey Fibers nonlinear fibers endlessly single-mode [ Wadsworth et al., JOSA B19, 2148 (2002) ] [ T. A. Birks et al., Opt. Lett.22, 961 (1997) ] polarization -maintaining low-contrast linear fiber (large area) [ K. Suzuki, Opt. Express9, 676 (2001) ] [ J. C. Knight et al., Elec. Lett.34, 1347 (1998) ]

11. Outline • What are these fibers (and why should I care)? • The guiding mechanisms: index-guiding and band gaps • Finding the guided modes • Small corrections (with big impacts)

12. Universal Truths: Conservation Laws an arbitrary-shaped fiber (1) Linear, time-invariant system: (nonlinearities are small correction) z frequency w is conserved cladding (2) z-invariant system: (bends etc. are small correction) wavenumber b is conserved core electric (E) and magnetic (H) fields can be chosen: E(x,y) ei(bz– wt), H(x,y) ei(bz– wt)

13. Sequence of Computation Plot all solutions of infinite cladding as w vs. b 1 w “light cone” b empty spaces (gaps): guiding possibilities Core introduces new states in empty spaces — plot w(b) dispersion relation 2 Compute other stuff… 3

14. Conventional Fiber: Uniform Cladding uniform cladding, index n b kt (transverse wavevector) w light cone light line: w = cb / n b

15. Conventional Fiber: Uniform Cladding uniform cladding, index n b w light cone higher-order core with higher index n’ pulls down index-guided mode(s) fundamental w = cb / n' b

16. PCF: Periodic Cladding periodic cladding e(x,y) Bloch’s Theorem for periodic systems: fields can be written: b a E(x,y) ei(bz+ktxt – wt), H(x,y) ei(bz+ktxt – wt) transverse (xy) Bloch wavevector kt periodic functions on primitive cell primitive cell where: satisfies eigenproblem (Hermitian if lossless) constraint:

17. w b PCF: Cladding Eigensolution Finite celldiscrete eigenvalues wn Want to solve for wn(kt, b), & plot vs. b for “all” n,kt constraint: where: H(x,y) ei(bz+ktxt – wt) Limit range ofkt: irreducibleBrillouin zone 1 Limit degrees of freedom: expand H in finitebasis 2 Efficiently solve eigenproblem: iterative methods 3

18. ky kx PCF: Cladding Eigensolution Limit range ofkt: irreducible Brillouin zone 1 —Bloch’s theorem: solutions are periodic in kt K M first Brillouin zone = minimum |kt| “primitive cell” G irreducible Brillouin zone: reduced by symmetry Limit degrees of freedom: expand H in finite basis 2 Efficiently solve eigenproblem: iterative methods 3

19. PCF: Cladding Eigensolution Limit range ofkt: irreducible Brillouin zone 1 Limit degrees of freedom: expand H in finite basis 2 — must satisfy constraint: Planewave (FFT) basis Finite-element basis constraint, boundary conditions: Nédélec elements [ Nédélec, Numerische Math. 35, 315 (1980) ] constraint: nonuniform mesh, more arbitrary boundaries, complex code & mesh, O(N) uniform “grid,” periodic boundaries, simple code, O(N log N) [ figure: Peyrilloux et al., J. Lightwave Tech. 21, 536 (2003) ] Efficiently solve eigenproblem: iterative methods 3

20. PCF: Cladding Eigensolution Limit range ofkt: irreducible Brillouin zone 1 Limit degrees of freedom: expand H in finite basis (N) 2 solve: finite matrix problem: Efficiently solve eigenproblem: iterative methods 3

21. PCF: Cladding Eigensolution Limit range ofkt: irreducible Brillouin zone 1 Limit degrees of freedom: expand H in finite basis 2 Efficiently solve eigenproblem: iterative methods 3 Slow way: compute A & B, ask LAPACK for eigenvalues — requires O(N2) storage, O(N3) time Faster way: — start with initial guess eigenvector h0 — iteratively improve — O(Np) storage, ~O(Np2) time for p eigenvectors (psmallest eigenvalues)

22. PCF: Cladding Eigensolution Limit range ofkt: irreducible Brillouin zone 1 Limit degrees of freedom: expand H in finite basis 2 Efficiently solve eigenproblem: iterative methods 3 Many iterative methods: — Arnoldi, Lanczos, Davidson, Jacobi-Davidson, …, Rayleigh-quotient minimization

23. PCF: Cladding Eigensolution Limit range ofkt: irreducible Brillouin zone 1 Limit degrees of freedom: expand H in finite basis 2 Efficiently solve eigenproblem: iterative methods 3 Many iterative methods: — Arnoldi, Lanczos, Davidson, Jacobi-Davidson, …, Rayleigh-quotient minimization for Hermitian matrices, smallest eigenvalue w0minimizes: minimize by conjugate-gradient, (or multigrid, etc.) “variational theorem”

24. dimensionless units: Maxwell’s equations are scale-invariant n=1.46 2r PCF: Holey Silica Cladding a r = 0.1a light cone w (2πc/a) w = bc b (2π/a)

25. n=1.46 2r PCF: Holey Silica Cladding a r = 0.17717a light cone w (2πc/a) w = bc b (2π/a)

26. n=1.46 2r PCF: Holey Silica Cladding a r = 0.22973a light cone w (2πc/a) w = bc b (2π/a)

27. n=1.46 2r PCF: Holey Silica Cladding a r = 0.30912a light cone w (2πc/a) w = bc b (2π/a)

28. n=1.46 2r PCF: Holey Silica Cladding a r = 0.34197a light cone w (2πc/a) w = bc b (2π/a)

29. n=1.46 2r PCF: Holey Silica Cladding a r = 0.37193a light cone w (2πc/a) w = bc b (2π/a)

30. n=1.46 2r PCF: Holey Silica Cladding a r = 0.4a light cone w (2πc/a) w = bc b (2π/a)

31. n=1.46 2r PCF: Holey Silica Cladding a r = 0.42557a light cone w (2πc/a) w = bc b (2π/a)

32. gap-guided modes go here index-guided modes go here n=1.46 2r PCF: Holey Silica Cladding a r = 0.45a light cone w (2πc/a) w = bc b (2π/a)

33. above air line: guiding in air core is possible below air line:surface states of air core n=1.46 2r PCF: Holey Silica Cladding a r = 0.45a light cone w (2πc/a) air light linew = bc b (2π/a)

34. Bragg Fiber Cladding Bragg fiber gaps (1d eigenproblem) at large radius, becomes ~ planar w nhi = 4.6 nlo = 1.6 b radial kr (Bloch wavevector) 0 by conservation of angular momentum kf wavenumber b b = 0: normal incidence

35. Omnidirectional Cladding Bragg fiber gaps (1d eigenproblem) w omnidirectional (planar) reflection e.g. light from fluorescent sources is trapped for nhi / nlo big enough and nlo > 1 [ J. N. Winn et al, Opt. Lett.23, 1573 (1998) ] wavenumber b b b = 0: normal incidence

36. Outline • What are these fibers (and why should I care)? • The guiding mechanisms: index-guiding and band gaps • Finding the guided modes • Small corrections (with big impacts)

37. Sequence of Computation Plot all solutions of infinite cladding as w vs. b 1 w “light cone” b empty spaces (gaps): guiding possibilities Core introduces new states in empty spaces — plot w(b) dispersion relation 2 Compute other stuff… 3

38. New considerations: Boundary conditions 1 Leakage (finite-size) radiation loss 2 Interior eigenvalues 3 Computing Guided (Core) Modes Same differential equation as before, …except no kt constraint: — can solve the same way where: magnetic field =H(x,y) ei(bz– wt)

39. Computing Guided (Core) Modes Boundary conditions 1 computational cell Only care about guided modes: — exponentially decaying outside core Effect of boundary cond. decays exponentially — mostly, boundaries are irrelevant! periodic (planewave), conducting, absorbing all okay Leakage (finite-size) radiation loss 2 Interior eigenvalues 3

40. Guided Mode in a Solid Core small computation: only lowest-w band! (~ one minute, planewave) holey PCF light cone flux density 1.46 – bc/w = 1.46 – neff fundamental mode (two polarizations) n=1.46 2r endlessly single mode: Dneff decreases with l a r = 0.3a l /a

41. group velocity = power / (energy density) (a.k.a. Hellman-Feynman theorem, a.k.a. first-order perturbation theory, a.k.a. “k-dot-p” theory) Fixed-frequency Modes? Here, we are computing w(b'), but we often want b(w') — l is specified No problem! Just find root of w(b') – w', using Newton’s method: (Factor of 3–4 in time.)

42. …except when we want (small) finite-size losses… Computing Guided (Core) Modes Boundary conditions 1 computational cell Only care about guided modes: — exponentially decaying outside core Effect of boundary cond. decays exponentially — mostly, boundaries are irrelevant! periodic (planewave), conducting, absorbing all okay Leakage (finite-size) radiation loss 2 Interior eigenvalues 3

43. Or imaginary-distance BPM: [ Saitoh, IEEE J. Quantum Elec. 38, 927 (2002) ] in imaginary z, largest b (fundamental) mode grows exponentially Computing Guided (Core) Modes Boundary conditions 1 Leakage (finite-size) radiation loss 2 Use PML absorbing boundary layer perfectly matched layer [ Berenger, J. Comp. Phys.114, 185 (1994) ] …with iterative method that works for non-Hermitian (dissipative) systems: Jacobi-Davidson, … Interior eigenvalues 3

44. Computing Guided (Core) Modes Boundary conditions 1 n=1.45 Leakage (finite-size) radiation loss 2 d imaginary-distance BPM L [ Saitoh, IEEE J. Quantum Elec. 38, 927 (2002) ] 2 rings 3 rings Interior eigenvalues 3

45. [ J. Broeng et al., Opt. Lett.25, 96 (2000) ] 2.4 Gap-guided modes lie above continuum (~ N states for N-hole cell) fundamental & 2nd order guided modes 2.0 air light line w (2πc/a) 1.6 fundamental air-guided mode …but most methods compute smallest w (or largest b) 1.2 bulk crystal continuum 0.8 1.11 1.27 1.43 1.59 1.75 1.91 2.07 2.23 2.39 b (2π/a) Computing Guided (Core) Modes Boundary conditions 1 Leakage (finite-size) radiation loss 2 Interior eigenvalues 3

46. ii Use interior eigensolver method— …closest eigenvalues to w0 (mid-gap) Jacobi-Davidson, Arnoldi with shift-and-invert, smallest eigenvalues of (A–w02)2 … convergence often slower Other methods: FDTD, etc… iii Computing Guided (Core) Modes Boundary conditions 1 Leakage (finite-size) radiation loss 2 Interior (of the spectrum) eigenvalues 3 Compute N lowest states first: deflation (orthogonalize to get higher states) i [ see previous slide ] Gap-guided modes lie above continuum (~ N states for N-hole cell) …but most methods compute smallest w (or largest b)

47. Interior Eigenvalues by FDTD finite-difference time-domain Simulate Maxwell’s equations on a discrete grid, + PML boundaries + eibzz-dependence • Excite with broad-spectrum dipole ( ) source Dw Response is many sharp peaks, one peak per mode signal processing complexwn [ Mandelshtam, J. Chem. Phys.107, 6756 (1997) ] decay rate in time gives loss: Im[b] = – Im[w] / dw/db

48. Interior Eigenvalues by FDTD finite-difference time-domain Simulate Maxwell’s equations on a discrete grid, + PML boundaries + eibzz-dependence • Excite with broad-spectrum dipole ( ) source Dw Response is many sharp peaks, one peak per mode narrow-spectrum source mode field profile

49. An Easier Problem:Bragg-fiber Modes In each concentric region, solutions are Bessel functions: c Jm (kr) + d Ym(kr) eimf “angular momentum” At circular interfaces match boundary conditions with 4  4 transfer matrix …search for complexb that satisfies: finite at r=0, outgoing at r= [ Johnson, Opt. Express9, 748 (2001) ]

50. metal waveguide modes frequency w 1970’s microwave tubes @ Bell Labs wavenumber b modes are directly analogous to those in hollow metal waveguide Hollow Metal Waveguides, Reborn OmniGuide fiber modes wavenumber b