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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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