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ADACHI, s., et al.: 'A new gate structure vertical GaAs F E T , IEEE
Electron Device Lett., 1985, EDL-6, p. 264
RATHMAN, D. D. : The microwave silicon permeable base transistor'.
IEDM tech. dig., 1982, p. 650
VESCAN, L., et al.: 'Sharp doping profiles in silicon grown by
advanced LPVPE'. To be presented at int. symp. on trends and
new appl. in thin films, Strasbourg, France, March 1987
When the two input signals are different in phase or of
unequal amplitude, the output power oscillates between both
guides. The condition to be used for a splitter also provides
the simplest boundary condition for evaluation of the device:
£i(0) = 1 and E2(0) = 0. Then the power varies as
= (jJ sin
Indexing terms: Optical fibres, Fibre optics, Single-mode fibre,
Branching devices, Splitters, Optical couplers, Optical multi/
The fused-head-end technique, used successfully until now for
producing passive fibre components in multimode fibre, can
also be applied to produce single-mode fibre couplers and
splitters. The device is based on evanescent mode coupling
and has no pigtail problem. The first experimental results are
presented, together with the theoretical interpretation. Some
expected system applications and further improvements are
Introduction: In multinoded short-haul fibre-optic networks
passive fibre components (PFCs) play a dominant role. A
technique has been described, called fused-head-end (FHE), in
which a variety of 1-to-N splitters/couplers, directional couplers and star couplers can be made in multimode fibre.1
Detailed investigations were reported in subsequent papers. In
this letter it is shown that the FHE technique can also be
applied to produce a similar device in single-mode fibre. The
theory presented illustrates the agreement of the first experimental results with the model, based on evanescent mode
Technology: The techniques used for single- and multimode
fibre are essentially identical.1 A number of bare fibres are
etched over a length of about lcm to a diameter slightly
larger than the core diameter 2a = 9 fim, so that a cladding
thickness b of about 2 fim remains. A conically etched section,
also about lcm long, forms the transition to the unetched
fibre. Two such fibres are introduced into a silica capillary of
inner diameter such that the unetched fibres fit closely. The
bottom end is closed with a flame and the top end connected
to a small container where the two fibre tails can be accommodated. This is evacuated and the capillary appendix is
introduced into a small furnace. Owing to the vacuum force
the weak capillary contracts to form a fused end with the two
cores at a distance d, slightly smaller than twice the remaining
cladding thickness b, about 3-4 fim. The result is a device with
one polished end and with two fibre pigtails at the other end.
Theory: The coupling of two modes in two fibres through the
overlapping evanescent fields has been described before.2"4
For two dominant modes in two different fibres the equations
relating the twofieldsEt and E2 read
where /?j and /?2 are the propagation constants and Kx and
K2 are the two complex coupling factors. It can be shown
from the conservation of energy that Kl = K%, so that
KtK2 = KXKX = K2. The solutions have a common factor
P [—J(Pi + /?2)z/2]- F° r both fields the remaining dependence on z yields a solution periodic in /?oz, where £x(0) and
E2(0) are the two complex field strengths at z = 0 and /?0
combines the two characteristic physical parameters as
Vol. 23
No. 9
In the experiments described here two identical fibres were
used; hence, the depth of modulation is K2/fil = 1. The
advantage gained by using two different fibres is discussed
Experiment: For all experiments single-mode fibres were used
with step-index profile, 2a = 9-9-5 fim and An = 3 x 10~3;
hence V = 2 4 at Ac = 1 -28 fim. Using the two fibres as output
ports, a single fibre of the same type excited with white light
can easily be butt-coupled to the polished end. The result is a
three-port device, as shown in the inset of Fig. 1.
-0 2
11 A ,pm
Fig. 1 Output of the two branches of an FHE evanescent mode-coupled
device as a function of wavelength X with single branch excitation
Configuration is shown in inset
The parameter K = fi0 can be determined by measuring the
output as a function of f30 z, and hence for various lengths z at
fixed fi0. In principle, the same measurement is done on the
assembled three-port device by scanning /?oz at fixed z,
exploiting the dependence of K on the wavelength X. Because
the spot radius of the dominant mode increases with X, the
coupling K increases with X, more steeply with smaller distance d. The full relation K{X) is known analytically for stepindex fibres3 and numerically for graded-index fibres.4
The single fibre is excited using a white light source, and the
output power of the two fibres is scanned with a monochromator, referred to the single-fibre input power and plotted. A
number of FHEs were made, with remaining cladding thickness b ranging from about 6 fim to zero and hence with d from
about 5 fim to zero. Quite evidently, the A-range over which
the power output of one branch moves from maximum to
zero decreases with decreasing b. Fig. 1 shows a typical plot of
the output of the two ports for what in an FHE turned out to
be the best condition: thefibredistance equal to about 2 fim.
Comparison to the theory: On the basis of the dependence of
the power on X in the region of the minimum of port 2, the
dependence of K on X can be determined. Making the approximation K = Ko + K'X + K"X2, the shape of the minimum at
1-378fim yields K'L = 4-6pm~l and K"L = 0; hence K' =
5-3 x 10~4/mi~2 with L = 8-6mm. The wavelengths at the
two extrema and the two 50/50 positions confirm this value,
K! = 5-7 x 10~4/im~2, over the range 1 < X < 1-6/mi.
Experimentally, the value of Ko has to be determined from
the L-dependence by cutting the device. With a reduced length
of L = 7-3mm, the same device yields K' = 5-5 x 10~4/im~2.
Apparently Ko = 0 is within the precision of this experiment.
Applications: The three-port device described can fulfil most
functions commonly required in single-mode fibre-optic net449
works of any nature. Because no losses are involved in eqn. 1,
the device is essentially lossless. Some loss occurs due to
manufacturing imperfections: the maxima do not reach unity.
For X below Ac some deficiency appears due to competition of
the next higher-order mode.
With the single-fibre input it is a demultiplexer for the
wavelengths 108 and 1-38 /un. With the single-fibre output it
is a multiplexer when power at 1 08 jrni is fed into the coupled
port and power at 1-38 fim is fed into the through port. It is
therefore also a wavelength-selective lossless splitter/coupler
required for two-wavelength full-duplex transmission over a
single fibre. Power at 1-38 ftm flows over the through ports
and power at l-08jim over the coupled ports, in both directions. For the wavelengths of 1-21 fim and 1-52/im the device
is a 50/50 splitter/coupler, to be used, for instance, in an interferometric sensor arrangement. At a 20 nm wavelength difference it is a 40/60 splitter/coupler. When two identical FHEs
are joined the device is a directional coupler. This four-port
application demands much stronger control of the manufacturing reproducibility than the three-port applications mentioned.
The laser linewidth is then given by the Schawlow-Townes
(ST) formula1 provided the g-factor of the cold cavity is large,
or else provided the laser medium is spatially homogeneous.
Otherwise, the ST result must be multiplied by the K-factor
introduced by Petermann.2 A simple expression for K, applicable to any electromagnetic cavity, has been reported in References 3-5.
Usually, lasers operate in the saturated regime, and the
laser linewidth is only half that predicted by the ST formula
because amplitude fluctuations are suppressed. On the other
hand, the linewidth is increased by a factor approximately
equal to 1 + a2, where a denotes the ratio of changes in the
real to imaginary parts of the active medium refractive index
caused by carrier fluctuations, as Henry has shownfirst.6(The
so-called 'adiabatic' approximation is made throughout this
letter. Effects related to relaxation oscillations are not considered.)
Arnaud,7 however, has shown that the ST result should be
multiplied by the more accurate factor
Discussion: In systems the applications mentioned above
demand that the critical points are located at the desired
wavelengths of 1-3 and 1-5/mi. Tailoring the curves of Fig. 1
to this effect requires that the values of d and L which yield
the correct values of Ko L and K'L are computed, and that the
etching and fusing processes are controlled carefully enough
to achieve these values. For the 50/50 splitter/coupler, at the
point where the two curves cross, the splitting ratio is most
A-dependent. When two different fibres are used so that the
depth of modulation K2/0o = 0-5, and hence fit - p2 = 2K,
then a 50/50 splitter is made at the point of zero slope. With
Px — f}2 = 27i tsnJX, it turns out that this effect can be reached
when Ane ~ 10~4, i.e. when the two fibres are only slightly
different in a or NA.
where aA is the ratio of the real to imaginary parts of the laser
cavity complex resonant frequency caused by a carrier density
change. In some circumstances, this factor differs significantly
from Henry's a-factor. Furuya8 introduced earlier a factor ac,
defined as the ratio of changes of the real and imaginary parts
of the propagation constant at a fixed real frequency.
However, one can show (using the fact that the propagation
constant is a regular function of frequency) that <xA coincides
with <xe only when the wave gain is independent of frequency.
This is not the case in general. Furthermore, the ae-factor
makes sense when the laser cavity incorporates a uniform
waveguide section, but not for an arbitrary cavity.
The purpose of this letter is to show that our result in eqn. 1
is in fact identical to a result given in 1971 by Thaler et al.9 in
connection with impatt oscillators. The agreement would not
be obtained if aA in eqn. 1 were replaced by a or ae, or if the
K-factor were omitted.
The derivation in Reference 9 is exceedingly simple. The
active medium is represented by an admittance — Y0(n), where
n is the carrier density (or any other relevant parameter, such
as field strength or temperature), in parallel with a linear
admittance Y(f), where / denotes the (real) optical frequency.
The frequency dependence of Yo is neglected here for simplicity, without much loss of generality.
If there were no noise source the circuit equation
Acknowledgment: The efforts of A. P. Severijns and C. M. van
Bommel in making the device are gratefully acknowledged.
23rd February 1987
Philips Research Laboratories
PO Box 80.000
5600 JA Eindhoven, The Netherlands
SEVERIN, p. j . , SEVERIJNS, A. p., and VAN BOMMEL, c. M. : 'Passive fibre
components for multimode fibre networks', J. Lightwave TechnoL,
1986, LT-4, pp. 490-496
2 HAUS, H. A.: 'Waves and fields in optoelectronics' (Prentice Hall,
New Jersey, 1984)
SNYDER, A. w., and LOVE, j . D.: 'Optical waveguide theory'
TEWARI, R., and THYAGARAJAN, K. : 'Analysis of tunable single-mode
(Chapman & Hall, New York, 1983)
fiber directional couplers using simple and accurate relations', J.
Lightwave Technol., 1986, LT-4, pp. 386-390
Indexing terms: Optics, Semiconductor lasers
The classical Schawlow-Townes formula for the linewidth of
a laser should be multiplied by a factor K' = (1 + aA)K/2. In
this formula, <xA is the ratio of changes in the real and imaginary parts of the complex resonant frequency resulting
from a change in carrier density, and K is Petermann's
factor. It is shown here that this factor is identical to a result
reported in 1971 by other authors in connection with impatt
K' = (1 + a2)K/2
ro("o) " Y(f0) = 0
would hold, where n0 and/ 0 are constant carrier density and
resonating frequency, respectively. In the following, we set
*o = Go + iB0 and Y = G + iB. The fluctuation-dissipation
(or optical Nyquist) theorem, however, tells us that a white
Gaussian current source is associated with any active conductance (complete population inversion and zero temperature
are assumed, for simplicity). For narrowband operation about
the frequency / 0 , this noise current can be written as
c(t) + is{i), where c and s are white Gaussian uncorrelated
processes whose spectral densities are equal to 4/i/0G0.
Because of this current source, the circuit equation becomes
where the voltage V across the circuit has been replaced on
the right hand side of eqn. 2b by its RMS value Vo. In other
words, as in Reference 7, we postulate that the voltage fluctuations are suppressed by saturation.
To first order, eqns. 2 give, separating real and imaginary
GOn 5n(t) - Gf 5f(t) = c{t)/V0
BOn Sn(t) - B, Sf(t) = s(t)/V0
When a laser operates in the unsaturated regime the populations of the upper and lower levels are independent of time.
where the subscripts n,/denote differentiations with respect to
these variables, the resulting quantities being evaluated at n0,
f0. Also, bn and Sf refer to small variations of n and/
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