Binary radially-chirped Bragg and graded-index Fresnel fibre lenses for singlemode fibre to photonic crystal coupling Michael C. Parker and Makiko Hisatomi
Fujitsu Laboratories of Europe, Columba House, Adastral Park, Ipswich, IP5 3RE, U.K. Tel: +44(0)1473 614143, Fax:+44(0)1206 762916, email:
[email protected]
Stuart D. Walker
University of Essex, Department of Electronic Systems Engineering, Wivenhoe Park, Colchester, CO4 3SQ, U.K.
Abstract: Binary radially-chirped Bragg and graded-index Fresnel fibre lens couplers are compared. With 41µm focal length, and 2.3 average refractive index, a radially-chirped Bragg fibre lens reduces spot-size from 4.5µm to 600nm incurring 0.4dB loss. © 2004 Optical Society of America
OCIS codes: (060.0060) Fiber Optics and optical communications; (060.2280) Fiber design and fabrication; (060.2310) Fiber optics; (060.2400) Fiber properties
1. Introduction Coupling of light between photonic crystals (PhCs) with high refractive index (RI) of approximately n » 3.5 , and low RI ( n » 1.5 ) single mode fibre (SMF) is problematic due to the large index mismatch, causing strong Fresnel reflections at the interface and highly mismatched spatial modes, i.e. SMF has a spot-size radius of about 4.5mm and PhCs have typical waveguide dimensions of the order of 0.5mm´0.5mm. Various techniques have been suggested to overcome this problem, e.g. a butt-jointable PhC waveguide [1], grating coupler structures [2], tapered couplers [3], and spot-size conversion [4]. Here we discuss a novel binary radially-chirped Bragg fibre (RCBF) solution, and compare it with an equivalent graded-index (GRIN) RCBF design. Recently, photonic crystal fibres (PCF’s), such as Bragg [5,6] and other microstructured fibres (MSF’s) [7] have been the focus of increasing scientific and technological interest, with Fresnel fibre also recently suggested as a possible waveguide [8,9]. In this paper, we describe an innovative radially-chirped PCF, which offers unique optical guiding based on refractive index gradients, rather than the conventional total internal reflection of SMF, or photonic bandgap confinement effects. The fibre lens’s geometry can be regarded as the longitudinal embodiment of a Fresnel zone plate. The binary embodiment of our design is a generalisation of the well-known Bragg fibre geometry, which consists of concentric rings of different refractive indices, where the incremental radius of each ring is equivalent to a constant l/4 path length change, such that light at the Bragg wavelength is reflected back into the core by the photonic bandgap effect. However, unlike conventional Bragg fibre, our binary RCBF guides light by periodically refocussing it within the fibre structure, analogous to a conventional continuous GRIN parabolic RI profile fibre. Since a quarter-period of the RCBF is analogous to a focal length, our structure therefore provides an elegant means for transferring light between SMF and PhC.
SMF n=
2.35+ 2.25 = 2.3 2
9 mm
0.5 mm f
n=1.5
RCBF
PhC
n=3.5
Figure 1: Schematic of radially-chirped Bragg fibre (RCBF) lens acting to match both spatialmodes and impedances (refractive indices) between SMF and PhC.
Figure 1 shows a schematic of a quarter-length of a RCBF lens interposed between a length of SMF and a PhC waveguide. The RCBF lens causes spatial mode conversion, between a spot-size of radius 4.5mm, and a spot-size radius close to 250nm, for efficient spatial modal overlap. In addition, it acts as an impedance match to minimise Fresnel reflections at the SMF/RCBF and RCBF/PhC interfaces.
2. Graded-Index Radially-Chirped Bragg Fibre Lens Figures 2(a) and 2(b) show a schematic diagram of the proposed GRIN RCBF lens and its RI profile. The GRIN RCBF lens guides light in a similar way to conventional GRIN fibre, but employs RI discontinuities at the Fresnel zone boundaries in order to keep the overall refractive index contrast (RIC) low. Propagation of light in the GRIN RCBF is analysed using Fermat’s principle of least time, with equation (1) describing the trajectory of a light ray in a medium with RI distribution n( x) as a function of space x [10], where t is the (scalar) trajectory of the ray and z is the longitudinal co-ordinate: dæ dx ö d 2r 1 dn (1) (2) = ç n( x) ÷ = Ñn( x) dt è dt ø dz 2 n(r ) dr We approximate equation (1) using the paraxial approximation, assuming only radial (r) variation of the RI, to yield equation (2). A Gaussian GRIN RI profile leads to a completely aberration-free sinusoidal trajectory [11]. However, to make it more practical to fabricate, we zone the Gaussian RI profile with M=9 zones and RIC=0.1, where the radius of the mth zone is given by mr1 , with the central zone radius r1=1.67mm. The trajectories associated with the GRIN RCBF are shown in Fig.2(c) for n1=2.35 and n2=2.25. Such refractive indices intermediate to the SMF and PhC indices can be achieved with heavy metal oxide or chalcogenide glasses. The on-axis trajectory for standard SMF with its smaller numerical aperture (NA) is shown as a bold-dotted line, the quarter-period focal length is 8.9mm and the spot size radius is converted from 4.5mm down to 315nm, thus indicating good capability for spatial mode conversion between SMF and PhC dimensions. f =8.88ì = 8.9 microns m
rin
r
f
(a)
(a)
z
2 0 -2 -4 -6
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n1=2.35 n2=2.25
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5
n1=2.35 n2=2.25
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Radial coordinate (microns)
Radial coordinate (microns)
6
3 2 1
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woo=315nm 2w =0.63ì m
-3 -4
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cladding
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2 0 -2 -4 -6
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wo=600nm
n1=2.35 n2=2.25
4
n2
n1
Refractive index
Electric Field Amplitude [a.u.]
rm
Radial coordinate (microns)
f = 41 microns
(f)
Figure 2: (a) Schematic diagram of graded-index RCBF, (b) zoned GRIN refractive index profile n(r), (c) Sinusoidal trajectories of light, featuring periodic focussing, with 8.9mm quarter-period, and spot-size radius conversion from 4.5mm down to 315nm, (d) Cross-section diagram of binary RCBF, (e) zoned binary refractive index profile n(r), (f) Modal evolution along length of binary RCBF featuring periodic focussing and refocussing, with 41mm quarter-period, and spot size radius conversion from 4.5mm down to 600nm.
3. Binary Radially-Chirped Bragg Fibre Lens Figure 2(d) shows a cross-section of the binary RCBF lens, where again the radius of the mth zone is given by mr1 , with the central zone radius r1=1.67mm, M=9 zones. The two refractive indices are n1=2.35, and n2=2.25, and the binary RCBF radial index profile is shown in fig. 2(e). Solving Maxwell’s equations, we have performed an eigenmode analysis of the binary RCBF structure to simulate the electric field evolution along a length of the waveguide. Our design highlights the fact that a strictly radially-periodic dielectric structure (e.g. as for conventional Bragg fibre) is not necessary for optical confinement and light waveguiding. The device structure facilitates modal coupling, such that a tightly-confined spatial mode expands to a weakly-confined mode, which then reverts to the original tight structure on a periodic basis, as shown in fig. 2(f). With the horizontal lines indicating the zone boundaries, fig. 2(f) shows evolution of a 600nm radius Gaussian input mode (l=1550nm) into a 4.5mm width mode after a longitudinal distance of 41mm, and then back again. Qualitatively similar behaviour of the light is evident on comparison of figures 2(c) and 2(f), with both figures featuring periodic modal field ‘breathing’. However, in the binary RCBF case, the quarter-period of the oscillation is 41mm, compared with only 8.9mm for the GRIN RCBF case. In addition, due in part to its longer focal length, the binary RCBF can only focus light down to a 600nm radius spot size, compared with 315nm for the GRIN case. The much longer quarterperiod of the binary case, as compared with the graded-index case can be attributed to the coarse approximation that the binary RI profile (fig. 2e) is to the ideal zoned GRIN RI profile (fig. 2b). Since an index gradient is the primary basis for light focussing, the binary radially-chirped case is only able to approximately perform this at the zone boundaries, where there is an index discontinuity. However, the index gradient needs to be carefully controlled to avoid profile dispersion, whereas the binary RCBF offers more straightforward manufacturability than the zoned, GRIN case. Both RCBF lens designs offer an average refractive index of 2.3, that is close to the optimum impedance-matching condition nSMFnPhC = 2.29 for minimal Fresnel reflections between the SMF, RCBF lens, and PhC, leading to a reflection loss at each of the SMF/RCBF and RCBF/PhC interfaces of 0.2dB, given by: (3) 0.2dB = 10log10 éë1 - {(nRCBF - nSMF ) /(nRCBF + nSMF )}2 ùû = 10log10 éë1- {(nPhC - nRCBF ) /(nPhC + nRCBF )}2 ùû , with an overall insertion loss of 0.4dB. 4. Conclusions We have analysed light propagation in both graded-index and binary radially-chirped Bragg fibre lenses, suitable for coupling light between SMF and PhC devices. The use of radially-chirped Bragg fibre offers an alternative light waveguiding technique as compared with total-internal reflection, or photonic bandgap confinement. We have found that both devices allow periodic spatial-mode conversion between at least 4.5mm and 600nm dimensions, with 0.4dB insertion loss. However, the binary case (which offers greater ease of fabrication) requires a quarter-period length of 41mm, whereas the GRIN case is almost five times shorter at 8.9mm, and offers tighter spot-size focussing. Further work is concerned with the optimisation of the binary geometry, to achieve a shorter quarter-period length and a smaller focussed spot-size radius. 5. References [1] T. Saito et al, “Fibre Butt-jointable Waveguide and Wavelength Filter Consisting of Photonic Crystals”, in ECOC 2002, pp. 4.4.2. (2002) [2] M.E. Potter and R.W. Ziolkowski, “Two compact structures for perpendicular coupling of optical signals between dielectric and photonic crystal waveguides”, Optics Express, 10, No.15, 691-698 (2002) [3] A. Mekis and J.D. Joannopoulas, “Tapered couplers for efficient interfacing between dielectric and photonic crystal waveguides”, J.Lightwave Tech, 19, No.6, 861-865 (2001) [4] T. Shoji et al, “Spot-size converter for low-loss coupling between 0.3-mm-square Si wire waveguides and single-mode fibers”, in IEEE LEOS Annual Meeting, pp.289-290, 2002 [5] P. Yeh, A. Yariv and E. Marom, “Theory of Bragg fibre”, J. Opt.Soc.Am., 68, No.9, 1196-1201 (1978). [6] M. Ibanescu et al., “Analysis of mode structure in hollow dielectric waveguide fibers”, Phys.Rev. E, 67(4), article No. 046608 (2003). [7] J.C. Knight et al., “All-silica single mode fibre with photonic crystal cladding”, Opt.Lett. 21, 1547-1549 (1996). [8] J. Canning, “Diffraction-free mode generation and propagation in optical waveguides”, Opt Comms, 207, 35-39 (2002). [9] M. Hisatomi, M.C. Parker and S.D. Walker, “Zoned microstructure fibre for low dispersion waveguiding and coupling to photonic crystals”, submitted to Optics Letters [10] M. Born, E. Wolf, Principles of Optics, (Pergamon Press, Oxford, 6th ed., 1984), Chap. 3. [11] D.W. Berreman, “Growth of Oscillations of a ray about the irregularly wavy axis of a lens light guide”, BSTJ, 44, 2117-2132 (1965).
