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 (3) SINGLE-MODE-FIBRE PASSIVE COMPONENTS MADE IN THE FUSED-HEAD-END TECHNIQUE = (jJ sin Indexing terms: Optical fibres, Fibre optics, Single-mode fibre, Branching devices, Splitters, Optical couplers, Optical multi/ demultiplexers 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 indicated. 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 coupling. 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 (1) dE2 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 ex 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 (2) ELECTRONIC LETTERS 23rd April 1987 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 later. 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. 10 08 0-6 -04 -0 2 09 11 A ,pm 1-3 15 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. P. J. SEVERIN 23rd February 1987 Philips Research Laboratories PO Box 80.000 5600 JA Eindhoven, The Netherlands References 1 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) 3 SNYDER, A. w., and LOVE, j . D.: 'Optical waveguide theory' 4 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 ROLE OF PETERMANN'S AC-FACTOR IN SEMICONDUCTOR LASER OSCILLATORS—A FURTHER NOTE 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 oscillators. K' = (1 + a2)K/2 (1) (2a) 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 (2b) 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 parts, GOn 5n(t) - Gf 5f(t) = c{t)/V0 (3a) BOn Sn(t) - B, Sf(t) = s(t)/V0 (36) 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/ 450 ELECTRONIC LETTERS 23rd April 1987 Vol.23 No. 9

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