Coupling losses between standard single-mode fibers and rectangular waveguides for integrated optics
Lars O. Lierstuen and Aasmund Sv. Sudbø

The butt-coupling loss between different tapered rectangular waveguides and a standard single-mode optical fiber has been calculated. Losses as low as 0.16 dB can be reached for waveguides with a refractive-index contrast in the range of 0.5% to 1.96%. The fabrication tolerances are such that practical devices with coupling losses below 0.25 dB are feasible. Key words: Coupling loss, optical waveguides, integrated optics.

1. Introduction

In optical systems that contain integrated optical components, mode-field mismatch between the fundamental mode of the waveguides of an integrated optical circuit and the optical fiber that is connected to the circuit may cause unacceptable power losses. This problem becomes more important as the refractive-index contrast of the integrated optical waveguides, D, which is given by the core refractive index nco and the cladding refractive index ncl,

D5

nco2 2 ncl2 2nco
2

<

nco 2 ncl nco

,

112

is increased. To achieve a good match to optical fibers, silica waveguides are usually designed to have D in the range 0.3–0.7%. Coupling loss as low as 0.05 dB has been reported for such waveguides.1 Increasing D, however, may be preferable because it makes a reduction of the size 1and hence the price2 of an optical component possible.2 The size reduction is mainly due to shorter coupling lengths in directional couplers and reduced radius of curvature in S bends. Systematic calculation of the coupling loss between waveguides and optical fibers has to our knowledge

L. O. Lierstuen is with the Center for Technology at Kjeller, P.O. Box 70, N-2007 Kjeller, Norway. A. Sv. Sudbø is with Norwegian Telecom Research, P.O. Box 83, N-2007 Kjeller, Norway. Received 30 August 1993; revised manuscript received 6 July 1994. 0003-6935@95@061024-05$06.00@0. r 1995 Optical Society of America. 1024 APPLIED OPTICS @ Vol. 34, No. 6 @ 20 February 1995

only been done for coupling between multimode silica waveguides and multimode fiber,3 using ray optics analysis, between strip-loaded waveguides and optical fibers,4 using Fourier analysis, and between rib waveguides and optical fiber,5 using three different methods for numerical analysis. The second and third analyses dealt with semiconductor waveguides. Similar calculations have also been performed in Ref. 6 for coupling between a diode laser and a rectangular waveguide, using Fourier analysis, but are not brought to an optimum design. High-D waveguides with square core cross sections have smaller mode fields than standard single-mode optical fibers, and this results in high coupling loss between such waveguides and a fiber. Some kind of mode-field converter must consequently be incorporated in the design of a low-loss optical component to use these waveguides. Several techniques for making mode-field converters have been proposed. These include thermal diffusion of dopants,7 modified flame hydrolysis deposition,8 and dual tapered waveguides.9 A mode-field converter can also be realized by the use of a wedge structure6 by the addition of two extra processing steps, one lithographic and one etching, to the waveguide fabrication. Figure 1 shows this type of taper schematically. A sufficiently long and smooth taper will have negligible loss, so that the coupling loss between a high-D waveguide and a single-mode fiber through a taper will be dominated by the mode-field mismatch at the taper–fiber interface. Hence the losses of the waveguide taper itself are not addressed in this paper. The purpose of this paper is to show how coupling loss between a tapered waveguide and an optical fiber

Fig. 1. Wedge taper to improve the coupling between square waveguides and optical fibers.

is affected by the waveguide cross section at the waveguide–fiber interface and by the waveguide D. The optimum cross sections are given for different D, and the fabrication tolerances are estimated.
2. Waveguide and Fiber Model

The fiber model is designed to represent a typical telecom fiber: a matched cladding structure that is single mode with cutoff wavelength lc 5 1300 nm and a core radius a 5 5 µm. The cladding refractive index is chosen to be ncl 5 1.46. Using V 5 12pncoa@lc2 Œ2D 5 2.405 1Ref. 102 as the upper limit for singlemode operation, we calculate the core refractive index of the fiber model to be nco 5 1.4634. To design a numerical technique that is capable of treating arbitrarily shaped waveguides, we have approximated the circular fiber core by a total of 78 rectangles, with equal refractive index but different size, per quadrant. The division of the fiber core is shown in Fig. 2. A cutoff wavelength lc 5 1300 nm has been verified numerically for the fiber model. The effective indices neff 5 b@k0 of the fundamental mode at 1300 and 1550 nm are found to be 1.46181041 and 1.46143824, respectively. b is the angular repetency of the mode, k0 5 2p@l is the free-space angular repetency, and l is the free-space wavelength. The corresponding numbers for a circular fiber with equal refractive indices and core area, calculated from

the analytically derived eigenvalue equation,10 are 1.46181086 and 1.46143860. Our integrated optical waveguide model is designed to represent the silica-on-silicon waveguides. Even though the cores of such waveguides are separated from the silicon substrate and the surrounding air by thick low-index layers, a slight asymmetry is present. This asymmetry gives rise to a birefringence that may be compensated for by a slight flattening of the waveguide cross section. Because this birefringence is a function of the thickness of the buffer layer that separates the waveguide from the substrate, and the thickness of the cladding layer that separates the waveguide from the air, we have chosen to neglect this effect and have designed waveguides with square cross sections. Nine different waveguides are designed to have lc 5 1300 nm and cross sections from 3 µm 3 3 µm to 12 µm 3 12 µm. From these specifications, and ncl 5 1.46, the core refractive indices nco are calculated. The waveguide parameters are summarized in Table 1. The method of film-mode matching11 1previously developed into the transverse resonance method122 is used in the scalar approximation for the calculation of neff and the mode fields of the waveguide structures and the overlap integral between the fundamental mode of the waveguide and the fiber. The scalar approximation fails for large refractive-index contrasts, but the error introduced in the overlap integrals by the scalar approximation in our calculations can safely be assumed to be less than the index contrast D. With reference to Fig. 2, the mode-matching method consists, briefly, of treating each stack of rectangles with equal width in the model as a planar film waveguide and computing a number of film modes within each stack. The fiber mode field is expanded in these film modes. Our fiber model consists of 13 stacks per quadrant, each with 13 rectangles 1a rectangular waveguide can of course be modeled by many fewer rectangles2. For the purpose of calculating overlap integrals between well-bound modes, 30 film modes are sufficient for accurate results. Calculations that involved modes near cutoff were performed with 100 film modes.

Table 1.

Parameters for Square Waveguides Discussed in the Texta

Dimensions 1µm 3 µm2 333 434 535 636 737 838 939 10 3 10 12 3 12
Fig. 2. Subdivision of the fiber core for the numerical calculations. The dimensions are in micrometers. aRefractive nco 1.4895 1.4765 1.4706 1.4674 1.4654 1.4641 1.4633 1.4626 1.4618

D 1%2 1.96 1.11 0.72 0.50 0.37 0.28 0.23 0.18 0.12

neff at 1550 nm 1.4723013167 1.4668172816 1.4643803758 1.4630657180 1.4622232990 1.4616760357 1.4613680652 1.4610544281 1.4607279214

index of the cladding is ncl 5 1.46. 1025

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To reduce processing time, only one quadrant of the waveguide structure is used in calculating neff and overlap integrals. The four walls that govern the boundary conditions constitute a 65 µm 3 67.5 µm rectangle whose lower left corner is the waveguide center. The distance from the waveguide center to the right and upper walls is sufficient for accurate calculations of neff relatively near nclk0, i.e., near cutoff. This has been verified by investigation of plots of the field distribution of modes near cutoff. The number of film modes and the distance to the walls both affect the precision of the calculations, and there is a tradeoff between simulation time and precision. Figure 3 shows how the number of film modes and the distance to the walls affects the coupling loss at the optimum coupling between the D 5 1.96% waveguide and the fiber. As can be seen from Fig. 3, the improvement in accuracy that is gained when the number of film modes is increased or the boundary rectangle is expanded is negligible. Reducing the size of the boundary rectangle yields faster convergence of the coupling loss as a function of the number of film modes. In the case of Fig. 3 the walls of the smaller rectangle are located too close to the waveguides, and this yields an erroneous result for the coupling loss. The reflectance of a plane wave that is perpendicularly incident upon an interface between media with refractive indices n1 5 1.4634 and n2 5 1.4895 is 241 dB. Reflection losses for two butt-coupled waveguides with core refractive indices n1 and n2 should be less than 241 dB and much smaller than the losses induced by mode-field mismatch in our calculations. Reflection loss is consequently neglected in our calculations. Low coupling loss between a rectangular waveguide and an optical fiber can only be achieved when the

mode field of the waveguide closely matches the mode field of the fiber. Additional effects that increase the coupling loss, such as angular misalignment and core-axis offset, may in this case be treated by the use of the theory of fiber-to-fiber splicing,13 and hence are not addressed in this paper.
3. Search for Optimum Coupling

Minimum coupling losses are to be found for each D by variation of the dimension of the waveguide cross section at the waveguide–fiber interface. All optimizations are carried out for a wavelength l 5 1550 nm. Three combinations of heights 1h2 and widths 1w2 of the rectangular waveguide will provide the local minima for the coupling loss. These are 1a2 w@h 5 1, 1b2 w@h . 1 and w comparable with the fiber core diameter, 1c2 w@h , 1 and h comparable with the fiber core diameter. The combinations w@h . 1 and w@h , 1 are equivalent, because of the rotation symmetry of the fiber mode-field distribution. w@h , 1 is impractical to fabricate for the optimum w@h ratios and consequently is not considered further in this paper. A waveguide core with a square cross section also yields an optimum coupling for the fundamental mode when the core cross section is comparable in size with the fiber core and when the core cross section is much smaller than the fiber core. For high-D waveguides the first structure is highly multimoded, and the latter is difficult to fabricate because of its small dimensions. These two structures are, consequently, not studied further in this paper. We search for optimum w@h combinations, where w@h . 1 and w is comparable with the fiber core diameter. Such a waveguide has a mode-field that is

Fig. 3. Coupling loss between the optimum-sized 1D 5 1.96%2 waveguide and an optical fiber as a function of the number of film modes used in the calculation. Solid curve, boundary rectangle of 67.5 µm 3 65 µm; dashed curve, boundary rectangle of 12.5 µm 3 10 µm; dashed–dotted curve, boundary rectangle of 127.5 µm 3 125 µm.
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Fig. 4. Coupling loss between fiber and square waveguides with a cutoff wavelength of 1300 nm as a function of refractive-index contrast D 3see Eq. 1124 and at a wavelength of 1550 nm.

Fig. 5. Coupling loss between fiber and rectangular waveguides, as a function of the width of the core of the guide, for four core heights. For each core height, the refractive-index contrast D 3see Eq. 1124 is chosen to yield a cutoff wavelength of 1300 nm for a square waveguide with that height.

Fig. 6. Sensitivity of coupling loss to deviations from optimum waveguide dimensions. The crosshair centers are at the points of minimum coupling loss, and the extent of the crosshair indicates a loss within 0.1 dB of the minimum for the 1550-nm wavelength.

confined to the waveguide core in the horizontal direction and loosely bound vertically.
4. Results

First the coupling losses between the square waveguides and the fiber are calculated. The results are given in Fig. 4 as a function of the refractive-index contrast D of the waveguides. Optimization of the coupling efficiency between the waveguide and the fiber is then carried out for the four waveguides with the highest D. To improve the coupling efficiency, the waveguides are tapered in one dimension; i.e., the waveguide width is increased. This is a convenient way of tapering a waveguide, because it can be realized without adding new processing steps to the waveguide fabrication. The calculations are limited to waveguides with height-to-width ratios less than one, except for the 6 µm 3 6 µm waveguide, which varies in the range from 6 µm 3 1 µm to 6 µm 3 18 µm, to show the existence of two points with minimum coupling loss. The results of these calculations are shown in Fig. 5.

As a final step toward a high coupling efficiency, we have found the optimum cross section for twodimensional tapers; i.e., the waveguide is permitted to change both the height and the width toward the fiber–waveguide interface, as illustrated in Fig. 1. The optimum dimensions are given in Table 2, together with the associated coupling losses at 1300 and 1550 nm. Note that the losses are considerably higher at 1300 nm 1the fiber cutoff wavelength2 than at the optimum 1550-nm wavelength. The sensitivity of the coupling loss to a change in waveguide dimensions from the optimum is important if one is to realize such a taper. This sensitivity is illustrated in Fig. 6. The optimum w@h combination lies at the crossing of the vertical and horizontal lines for each D. Varying waveguide width and height in the range indicated in Fig. 6 by the horizontal and vertical lines, respectively, does not increase the coupling loss by more than 0.1 dB. Deviation in width in the range from 158 to 228% or height in the range from 118 to 214% still maintains a coupling loss of less than 0.26 dB for D 5 1.96%.
5. Conclusion

Table 2. Dimensions, Cutoff Wavelengths, and Coupling Losses to Optical Fibers for Rectangular Waveguides Optimized for Minimum Coupling Loss at the 1550-nm Wavelengths by Varying the Width AwB and Height AhB

D 1%2 1.96 1.11 0.72 0.50

Optimum h3w 1µm 3 µm2 0.43 3 7.70 0.79 3 7.90 1.30 3 7.85 2.05 3 7.59

Coupling Loss 1dB2 Cutoff Wavelength 1nm2 1150 1148 1134 1109 11300 nm2 0.40 0.36 0.32 0.28 11550 nm2 0.16 0.15 0.14 0.13

We have calculated the coupling loss between tapered waveguides for integrated optics and a standard single-mode optical fiber for various cross sections of a rectangular waveguide at the waveguide–fiber interface. A one-dimensional taper yields only a little improvement of the coupling loss but can be added to the design of low-D waveguides because it does not affect the number of processing steps. The only way to make a low-loss connection between a high-D waveguide and a standard optical fiber is to insert a two-dimensional taper at the waveguide end. This can be done quite simply with a wedge taper.9 The cross-sectional area of the waveguide at the fiber interface is not critical.
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Our minimum coupling loss calculated for D 5 1.96% is higher than the 0.1 dB measured in Ref. 6 for D 5 2%, with a taper made by outdiffusion of dopants. This method of tapering the waveguide has the advantage of smoothly converting the rectangular core cross section to a circular cross section with larger area, thus allowing one to realize a very good match to the mode field of the fiber. The wedge taper, on the other hand, provides both flexibility and reproducibility, because it can be fabricated with the standard processes for integrated optics, namely, photolithography and dry etching, and the high-temperature processing associated with diffusion is avoided.
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5. M. J. Robertson, S. Ritchie, and P. Dayan, ‘‘Semiconductor waveguides: analysis of coupling between rib waveguides and optical fibres,’’ in Integrated Optical Circuit Engineering II, S. Sriram, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 578, 184–191 119852. 6. Y. Shani, C. H. Henry, R. C. Kistler, K. J. Orlowsky, and D. A. Ackerman, ‘‘Efficient coupling of a semiconductor laser to an optical fiber by means of a tapered waveguide on silicon,’’ Appl. Phys. Lett. 55, 2389–2391 119892. 7. M. Yanagisawa, Y. Yamada, and M. Kobayashi, ‘‘Low-loss and large-tolerance fiber coupling of high-D silica waveguides by local mode-field conversion,’’ IEEE Photon. Technol. Lett. 4, 433–435 119832. 8. H. Yanagawa, T. Shimizu, and I. Ohyama, ‘‘Index-anddimensional taper and its application to photonic devices,’’ J. Lightwave Technol. 10, 587–592 119922. 9. N. Yamaguchi and Y. Kokubun, ‘‘Spot size convertor by overlapping of two tapered waveguides,’’ Electron. Lett. 25, 128–130 119892. 10. A. W. Snyder and J. D. Love, Optical Waveguide Theory 1Chapman and Hall, London, 19832, Chap. 14, pp. 301–335. 11. A. S. Sudbø, ‘‘Film mode matching: a versatile method for mode field calculations in dielectric waveguides,’’ Pure Appl. Opt. 2, 211–233 119932. 12. R. Sorrentino, ‘‘Transverse resonance technique,’’ in Numerical Techniques for Microwave and Millimeter Wave Passive Structures, T. Itoh, ed. 1Wiley, New York, 19882, Chap. 11, 637–693. 13. M. Artiglia, G. Coppa, P. Di Vita, and M. Potenza, ‘‘Theory of propagation in optical fibers,’’ in Fiber Optic Communications Handbook, 2nd ed., F. Tosco, ed. 1TAB Professional and Reference Books, Blue Ridge Summit, Pa., 19902, pp. 215–218.

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