1054 OPTICS LETTERS / Vol. 29, No. 10 / May 15, 2004

Zoned microstructure fiber for low-dispersion waveguidingand coupling to photonic crystals

Makiko Hisatomi and Michael C. Parker

Fujitsu Laboratories of Europe, Columba House, Adastral Park, Ipswich IP5 3RE, UK

Stuart D. Walker

Department of Electronics Systems Engineering, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, UK

Received October 31, 2003

We describe a zoned microstructure fiber that exhibits low dispersion and virtually zero spherical aberrationbecause of its optimized piecewise Gaussian index profile. We present results of a nine-zone design that hasan average refractive index of 2.3, a refractive-index contrast of 0.1, a first zone radius of 1.67 mm, and amaximum core radius of 5 mm. It has an in-fiber focal length of 8.88 mm and can focus light to a spot sizeof radius 315 nm, facilitating efficient coupling between single-mode fiber and photonic crystals. © 2004Optical Society of America

OCIS codes: 060.0060, 060.2280, 060.2310, 060.2400.

Recently, microstructured fibers (MSFs), such asBragg fibers1,2 and photonic crystal fibers (PCFs),3

have been the focus of increasing scientific and techno-logical interest, with a parabolic-phase Fresnel fiberalso recently suggested as a possible waveguide.4,5 Inthis Letter we present a variational calculus analysisof light-ray propagation in a zoned MSF that has anoptimized piecewise Gaussian graded-index (GRIN)refractive-index (RI) profile that exhibits less aber-ration and dispersion than a parabolic Fresnel fiber.As well as offering novel waveguiding capability(compared with photonic bandgap light conf inementand total internal ref lection), our structure offers ameans for the efficient coupling of light between asingle-mode fiber (SMF) with a core diameter of 9 mmand photonic crystals (PhCs), whose typical wave-guiding dimensions are 0.3 mm 3 0.3 mm. Figure 1shows a schematic of our proposed Gaussian-zoned(GZ) MSF waveguide and its RI profile. Our GZ MSFguides light similarly to a GRIN medium but employsRI discontinuities at the zone boundaries to keep theRI contrast (RIC) low. Fabrication of our design canbe achieved by coaxial assembly of concentric hollowpreforms, each corresponding to a zone, followed bydrawing of the composite preform to the appropriatedimensions, e.g., as described in Ref. 6. The Gauss-ian index gradient in each hollow preform can beachieved by use of appropriate diffusion techniques,as used in the fabrication of conventional GRIN fiber.

Propagation of light in the GZ MSF is analyzed byuse of Fermat’s principle of least time. This approachnot only has the advantage of reduced computationaleffort compared with beam propagation modal analysisbut also yields a deeper insight into the waveguidingproperties and aberration (dispersive) characteristicsof our device. Because diffraction is the spatial analogof dispersion in time, we treat aberration as the spa-tial equivalent of waveguide dispersion. The equationthat describes the trajectory of a light ray in a medium

0146-9592/04/101054-03$15.00/0

with a spatially varying RI is7

ddt

∑n�x�

dxdt

∏� =n�x� , (1)

where n�x� is the RI distribution as a function of spacex and t is the (scalar) trajectory of the ray. Usingthe paraxial approximation and assuming only radialvariation in RI permits Eq. (1) to be simplif ied to asecond-order differential equation as a function of lon-gitudinal coordinate z and radial coordinate r:

d2rdz2

�1

n�r�dndr

. (2)

Substitution into Eq. (2) of an unzoned Gaussian GRINprofile n�r� � n1 exp�2k1

2r2�2� leads to an exact equa-tion of simple-harmonic motion8:

d2rdz2

� 2k12r . (3)

Fig. 1. (a) Schematic of GZ MSF waveguide, (b) piecewiseGZ RI prof ile n�r�.

© 2004 Optical Society of America

May 15, 2004 / Vol. 29, No. 10 / OPTICS LETTERS 1055

Equation (3)describes a completely aberration-free, i.e.,dispersion-free, sinusoidal trajectory, r � rin cos k1z 1�rin0�k1�sin k1z, where rin is the input radius of thetrajectory and rin0 is the initial slope of the trajectory(a prime indicates differentiation with respect toz). The parameter for the Gaussian RI profile, k1,therefore also determines the quarter-period focallength f of the GRIN lens, such that f � p�2k1.Figure 2 shows the trajectories of light in such aGaussian RI profile, with an index n1 � 2.35 atthe waveguide center; the maximum core radius isa � 5 mm, and k1 � 0.197 mm21 such that the RIis n2 � 1.45 at the maximum core radius. Thisgives a calculated focal length of f � 7.99 mm, ingood agreement with the simulation results. Fig-ure 2(b) is a zoomed-in diagram of the on-axis focalpoint that shows virtually no aberration, as wouldbe expected. We estimate the spot-size diameter oflight focused from the SMF to be 2w0 � 0.57 mmby considering the range of angles input from aSMF [i.e., corresponding to its numerical aperture(NA)]. An example on-axis trajectory for standardSMF with its NA (smaller than the NA for the MSF)is shown as bold dashed curves in Fig. 2(a). How-ever, if the Gaussian GRIN lens were used to couplelight between the SMF and the PhC, the lens’s largeRIC would cause high ref lections at the interfaces andreduce coupling eff iciency.

To reduce ref lections at the PhC, MSF, and SMFinterfaces, the RIC must be reduced, as one canachieve by adopting a zoning strategy. A commonapproximation to a Gaussian RI profile is a parabola,n�r� � n1�1 2 k1

2r2�2�, and this can be straightfor-wardly zoned such that the radius of the mth zoneis given by rm �

pmr1. However, Fig. 3 shows the

higher aberration (compared with that in Fig. 2) thatis due to such a Fresnel f iber, with parameters N � 9zones, maximum radius a � 5 mm, n1 � 2.35 at thewaveguide center, and n2 � 2.25 at the zone edge. Thelarge variation in focal lengths, approximately 90 nmfor the on-axis focal point as in Fig. 3(b), is apparent,with the spot size estimated to be 2w0 � 0.64 mm.However, Eq. (3) indicates that a Gaussian RI profileexhibits the lowest aberration possible. Hence wehave adopted the ideal Gaussian profile but introducedzoning to keep the RIC low. We designed the GZ RIprofile to have N � 9 zones and a RIC of 0.1, similarto the Fresnel f iber above. The equation for the RIvariation in the mth zone of a piecewise GZ MSF isgiven by

nm�r� � n1 exp�2km2r2�2� 1 Dnm , (4)

where Dnm is related to the index discontinuity at thezone boundary, which is optimized along with Gauss-ian parameter km and radius rm of the mth zone suchthat substitution of Eq. (4) into Eq. (2) yields

d2rdz2

�2km

2r1 1 �Dnm�n1�exp�km

2r2�2�� 2k̃1

2r . (5)

The values for the parameters rm, km, and Dnm areoptimized such that in each zone their combination

in Eq. (5) is nearly equal to a constant, k̃1, equal toGaussian parameter k1 for the center zone. Thismeans that light trajectories in all zones closely obeythe same equation of simple harmonic motion. Thisensures that all the ray trajectories (independentlyof slope angle) in each zone describe a sinusoid ofthe same spatial frequency k̃1, remain in phase, andhave a common focus, thus minimizing aberrationand dispersion. The values for optimized parametersare listed in Table 1, where to further minimizeaberration we optimized k̃1 for the midpoint of eachzone. Figure 4 shows the ray trajectories for ournovel GZ MSF design for n1 � 2.35 and n2 � 2.25. At1550 nm such RIs can be achieved by use of heavymetal oxide or chalcogenide glasses.9 The on-axisfocal length now varies by less than 10 nm, indicatingvery low aberration and dispersion. In this GZ case,k1 � 0.177 mm21, which gives an estimated focallength of f � 8.88 mm, which matches the simulationresults well. Figure 4(b) also now indicates extremelylow spherical aberration of the GZ MSF lens. Thespot-size diameter for our GZ MSF is estimatedto be 2w0 � 0.63 mm. Naturally, deviations fromthe ideal RI profile will cause an overall degrada-tion in performance, leading to profile dispersion.10

However, assuming that the imperfections are ran-dom (i.e., not systematic), the profile dispersionwill make itself evident by a larger spot size andgreater spherical aberration. The average GZ MSFrefractive index of 2.3 is close to the theoretical RIimpedance-matching condition �nSMFnPhC�1�2 � 2.29for minimum Fresnel ref lection among SMF(nSMF � 1.5), GZ MSF, and PhC (nPhC � 3.5),leading to a reduced ref lection loss of 0.2 dB �10 log10�1 2 ��nGZ MSF 2 nPhC���nGZ MSF 1 nPhC��2� �10 log10�1 2 ��nSMF 2 nGZ MSF ���nSMF 1 nGZ MSF ��2� at

Fig. 2. (a) Trajectories for Gaussian GRIN fiber,(b) zoomed-in on-axis diagram.

Fig. 3. (a) Trajectories for parabolic (Fresnel) zoned f iber,(b) zoomed-in on-axis diagram.

1056 OPTICS LETTERS / Vol. 29, No. 10 / May 15, 2004

Table 1. Optimized Values for rm, km, and Dnm

Zone Number �m�

Optimized Parameter 1 2 3 4 5 6 7 8 9

rm �mm� 1.67 2.36 2.89 3.33 3.73 4.08 4.41 4.72 5.00km �mm21� 0.177 0.181 0.186 0.191 0.198 0.206 0.216 0.231 0.264

Dnm 0 0.105 0.215 0.332 0.459 0.598 0.757 0.952 1.268k̃1 �mm21� 0.177 0.177 0.177 0.177 0.177 0.177 0.177 0.177 0.177

Fig. 4. (a) Trajectories for GZ MSF, (b) zoomed-in on-axisdiagram.

each of the two interfaces compared with the loss forunzoned fiber. The use of a piecewise Gaussian RIprofile rather than a piecewise parabolic (Fresnel) RItherefore produces much reduced aberration and lowerdispersion.

In conclusion, we have presented a novel Gaussianzoned microstructure fiber waveguide design witha discontinuous piecewise Gaussian refractive-indexprofile based on a carefully implemented optimizationstrategy. Overall, our design offers both reducedwaveguide dispersion and enhanced light-couplingefficiency between single-mode fiber and photoniccrystals.

M. Hisatomi’s e-mail address is m.hisatomi@f le.fujitsu.com.

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