The authors would like to acknowledge L. F. Mollenauer and A. M. DelGaudio for guidance with the relative index measurements, G. D. Boyd, R. E. Nahory and W. D. Johnston for helpful suggestions, and J. A. Rentschler for technical assistance. for the material closest in composition to InP is omitted, since the very small fringe amplitude could not be read with sufficient accuracy. The implied bandgap values, the lattice mismatch (Aa/a), and a summary of the Nahory and Pollack predictions are also listed. The reasonable agreement con- Table 1 PARAMETERS FOR G a x I n i _ x A s , P , _ y TEST FILMS y X K (Ref. 1) Aa/a n(k = kg) (Ref. 3) n(/ = kg) (eqn. 5) k{k=Xg) (pm)"1 /o 0-415 0-411 0-282 0125 1-000 0-884 0-614 0-276 dnidk{k = kg) 0076 0003 0008 0006 1-67 1 56 1-34 109 3-58 + 002 3-56 + 001 3-50 + 003 — 3-56 3-55 3-52 3-46 vinced us that it was valid to calculate deff from the predicted n(x, kg) of Nahory and Pollack. Fig. 1 plots the resulting index data as a function of wavelength using the calculated thicknesses for both transmission and reflection. The index dispersion dn/dk at k = kg is estimated from the data and listed in Table 1. Also shown for comparison are theoretical dispersion curves of Utaka et al.5 based on the theory of Afromowitz.6 The agreement is fair over most of the wavelength range, but the singularity at the bandgap (k = kg) in the theory renders it of little use for predicting dispersion in this important region. The simpler singleoscillator theory of Wemple and DiDomenico 7 does not have this problem, and it is probably of greater utility for k ~ kg. The larger y value films all showed a maximum (or at least a shoulder) just below the absorption edge wavelength. Previous measurements on the binaries 8 have given similar features, and we believe they are real in our cases. In summary, we have used interference fringes from thin films of GalnAsP to measure its index as a function of composition and wavelength. The measurements are in reasonable agreement with theory for wavelengths larger than the absorption edge, but at and below the band edge the simple theories do not predict the measured slopes and other details. Illumination levels were low, so that band-filling effects should not be present. Future experiments under heavy visible pumping illumination may produce slight shifts in the index level, particularly near the bandgap. The index dispersion (dn/dk) may change more significantly in this case, however. 0-2 0-6 0-7 10 007 007 007 007 P. CHANDRA L. A. COLDREN K. E. STREGE 19th November 1980 Bell Laboratories Holmdel, NJ 07733, USA References 1 NAHORY, R. E., POLLACK, M. A., JOHNSTON, W. D., JUN., a n d BARNES, R. L.: 'Bandgap versus composition and demonstration of Vegard's law for I n ^ j G a ^ A S j , ? , . , , lattice-matched to InP', Appl. Phys. Lett., 1978, 33, pp. 659-661 2 STREGE, K. E., JOHNSTON, w. D., and BALLMAN, A. A.: "Vapor phase epitaxial growth of material for low threshold I n ^ ^ G a ^ A S y P , ^ diode lasers at 1-52 ^m wavelength', submitted to J. Elec. Math. 3 NAHORY, R. E., and POLLACK, M. A.: 'Threshold dependence on active-layer thickness in InGaAsP/InP d.h. lasers', Electron. Lett., 1978, 14, pp. 727-729 4 MOLLENAUER, L. F., and OLSON, D. H.: 'Simple optical absorption spectrometer suitable for measurements at low temperature', Rev. Sci. lnstrum., 1975, 46, pp. 677-679 5 6 7 UTAKA, K., SUEMATSU, Y., KOBAYASHI, K., a n d KAWANISHI, H.: 'GalnAsP/InP integrated twin-guide lasers with first-order distributed Bragg reflectors at 1-3 fim wavelength', Japan. J. Appl. Phys., 1980, 19, pp. L137-L140 AFROMOWITZ, M. A.: 'Refractive index of Ga 1 _ x Al x As > , Solid-State Commun., 1974, 15, pp. 59-63 WEMPLE, s. H., and DIDOMENICO, M., JUN.: 'Behavior of the electronic dielectric constant in covalent and ionic materials', Phys. Rev. B, 1971, 3, pp. 1338-1351 8 WILLIARDSON, R. K., and BEER, A. c. (EDS.): 'Semiconductors and semimetals; vol. 3' (Academic Press, New York, 1967), pp. 138, 530-532 37 Y = 0-884 0013-5194/811010006-02$!. 50/0 Y=10 36 Y=0-6U 3-5 - Y=0 276 EXTINCTION CROSS-SECTION MEASUREMENTS ON SINGLE FALLING WATER DROPS AT 100 GHz 3-4 - Y = 00 (InP) Indexing terms: propagation 33 Microwaves, Radiowave An open resonator with 100 GHz source is used to determine the extinction cross-section of water drops falling freely through the atmosphere. Results show a very close agreement with the theoretical values computed using classical Mie scattering theory and values of complex refractive index from an empirical model. 32 3-1 05 Attenuation, 0-9 1-7 1-3 wavelength A, pm 21 2-5 1809/11 of GaxIn1_xAsfPl-y for several Fig. 1 Refractive-index/wavelength values of y Dashed curve, theory. 5 - 6 Solid data points for reflection (connected by curve); open data points for transmission; triangular points maxima; circular points minima. Data point height represents uncertainty ELECTRONICS LETTERS 8th January 1981 Vol.17 In an earlier letter 1 we showed that an open resonator can be used to measure the extinction cross-section of scattering objects without the need for physically translating them axially by x/4 to eliminate the effect of the standing wave between the mirrors. The alternative is to make the resonator sufficiently long to enable two adjacent fundamental resonances to be displayed within the allowable frequency scan of the oscillator. No. 1 Our open resonator uses two spherical mirrors, inner surface coated and with a radius of curvature of 2-44 m, and having 4 mm diameter coupling holes. A mirror separation of 2-87 m allowed the two resonances to be displayed in the 100 MHz frequency sweep of the impatt oscillator, and gave conditions well removed from degeneracy. The first higher-order mode, 5-9 MHz displaced from the fundamental, was allowed to exist to provide calibration of the resonance curve. The diameter of the mirrors was 0-23 m. Results were given, in our earlier letter, of extinction cross-sections for a range of sizes of stationary conducting and paraffin wax spheres from 1-6 to 8 mm diameter at 100 GHz (normalised radii Ina/X from approximately 2 to 8), showing a close correlation with both Mie theory values and other experimental values taken at lower frequencies. The work has now been extended to include measurements on water drops falling freely through the atmosphere. An optical laser with beam splitters and photocells with fine apertures allowed the impatt to be swept when a water drop was central in the open-resonator beam, the water drop vertical diameter to be measured, and the falling velocity to be determined. A number of nozzles of different diameters enabled the drop size to be changed. The mean diameter was also measured by weighing a known number of drops falling through the resonator. The falling velocity was limited to about 2 ms" 1 , corresponding to little deformation of the drop from spherical. A computer program was written based on Mie theory for spherical drops to give the predicted extinction cross-section, using the drop size and calculated values of the complex refractive index based on an empirical model by Ray.2 The theoretical variation with normalised drop size is shown in Fig. 1, based on a complex refractive index of m = 3-282 — jl-864. Also shown are experimental values of extinction cross-section for falling water drops with diameters from 2-8 to 5 8 mm. canting angle, can then be determined. Initial tests show that such measurements can be made with the resonator system. The support of the Science Research Council for this work is acknowledged. D. J. HARRIS 13th November 1980 T. M. TEO Department of Physics, Electronics and Electrical Engineering University of Wales, Institute of Science and Technology Cardiff, CF1 3NU, Wales References 1 HARRIS, D. J., and TEO, T. M.: 'Measurement of extinction crosssection of single particles at 100 GHz', Electron. Lett., 1980,16, pp. 500-501 2 RAY, P. s.: 'Broadband complex refractive indices of ice and water', Appl. Opt., 1972, 11, pp. 1836-1844 0013-5194/81/010007-02$!.50/0 ASYMPTOTIC EIGENVALUES OF VECTOR WAVE EQUATION FOR GUIDED MODES IN GRADED-INDEX FIBRE Indexing terms: Optical fibres, Wave propagation Asymptotic eigenvalues of the vector wave equation—a system of coupled second-order differential equations—are calculated by the perturbation method in which we use the exact solutions of the scalar wave equation as the unperturbed term. We examine a graded-index fibre with a polynomial profile core and obtain the propagation constants of the vector modes in analytic form. The results coincide with those found by a sophisticated method. *: 30 b Z 25 The propagation characteristics of multimode graded-index fibres can be well analysed by solving the scalar wave equation. 1 Scalar wave analysis does not give accurate results for lower-order modes propagation characteristics in such guides as single and quasi-single mode fibres. So we need to solve the eigenvalue problem for the vector wave equation. The purpose of this letter is to present an analytical expression for the propagation constants of vector modes on weakly guiding fibres. We consider a radially inhomogeneous fibre in which the refractive index is described by 20 Z 1-5 •o 10 n(r) = n,[l - h(r) '2 10 20 30 40 50 normalised radius 2 IT a/A 60 Fig. 1 Comparison of experimental and theoretical results Mie theory (m = 3-282 - jl-864) * Average experimental values I Range of experimental values Measurements were all made with vertical polarisation for both excitation and detection. The correlation between theory and experiment is good, and confirms that Mie theory using values of complex refractive index from the Ray expressions can be used with confidence to predict the total loss of radiation by an individual water drop at these short wavelengths. The axial symmetry of the resonator system allows the plane of polarisation of the input wave to be rotated, so that measurements can be made for both vertically and horizontally polarised waves. By rotating a short section of dominant mode guide between the output aperture and the detector, it is possible to differentiate between vertically and horizontally polarised components of the fields in the resonator. Thus crosspolar signals which arise from a scattering object can be measured. The extent of depolarisation by nonsymmetrical scattering objects, e.g. an oblate spheroid with 8 where «i is the refractive index at the centre of the core and h(r) is a monotonically increasing function of r satisfying h(0) = 0. The guided modes propagate along the waveguide axis z according to exp {j(a>t + nd - Pz)}. If n =£ 0, we have a system of coupled wave equations, that is, a vector wave equation such as 0>i + (l/r)Oi + {k2(b - h)- {n - l ) 2 / / - 2 ) ^ + (/i +fi +h)®i/2 + (/i ~fi +/3) ( I ) 2/2 = 0 I1) 2 2 Q>"2 + (l/r)(D'2 + {k\b - fi) - (n + l) /r )0) 2 + (/i - / 2 - / 3 ) O 2 / 2 + (/, + / 2 - / 3 ) O ! / 2 = 0 with j? = /c(l - b) 1/2 , /c = konu k0 = 2n/X, where 0>! and 0>2 are related to the transverse magnetic fields Hr and Hg in the following manner: * i = 0 i + <p H9 =j(l - h) exp {J((ot + n9- Hr = <f>2 exp + nO - Pz)} ELECTRONICS LETTERS 8th January 1981 Vol.17 No. 1

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