Modelling EM transient propagation over irregular dispersive boundary
small grazing angles 1, all incident monochromatic plane waves with
ELECTRONICS LETTERS 4th July2002
Fig. 2 Initial ulse shape and receivedpulse waveformsfor and o = 1O-'S/m Receiver range and elevation: x = 1 !an, z - h = 20 m
0=
S/m
The condition providing perfect transparency of the artificial upper boundary z= b is derived using the general approach developed in [7] (see also [SI):
Both nonlocal BCs ( 6 ) and (8) are easily discretised (see [9]) yielding, together with FD equation (4), a three-diagonal matrix equation to find
Vol. 38 No. 14
69 1
~
~
l
the only unknown solution vector at each step of (x,s) marching. Here we present an example of ultra-wideband pulse propagation over an irregular terrain with realistic soil parameters E = 10, D = IO-*lop3 S/m. The initial waveform is a damped sinusoid f ( c t ) = sin(act) exp(-bct) with b = a and the spatial pulse length A - x / a = 30 m. Fig. 1 shows a series of snapshots giving a general view of the pulse evolution. Using the transparency BC (8) we avoid spurious reflection from the upper boundary The effect of bottom dispersion manifests itself in some delay of the reflected pulse. The comparison of the received waveforms for two different values of D (Fig. 2) shows a significant dependence of the signal tail on the ground conductivity, which can be used for ecological monitoring (water pollution, earthquakes, etc.). Acknowledgments: This work has been supported by the German Research Society (DFG grant La 484/21-1) and the Russian Foundation for Basic Research (RFFI grant 01-02-04003).
0 IEE 2002 Electronics Letters Online No: 20020426 DOZ: IO. 1049/el:20020426
I O January 2002
A.V. Popov and VV. Kopeikin (Institute of Terrestrial Magnetism, Ionosphere and Radiowave Propagation, IZMIRAN, Troitsk, Moscow region, 142190 Russia) Ning Yan Zhu and EM. Landstorfer (Znstitut fuer Hochfrequenztechnik, Universitaet Stuttgart Pfaffenwaldring 47, 0-70550 Stuttgart, Germany) E-mail:
[email protected]
laser. While several methods have been developed to combine laser diodes [ l , 21 or solid-state lasers [3, 41, only few optical waveguide devices have been the subject of experimental work. Spatial and spectral beam combining of two fibre lasers was only obtained in the past with the use of a specific half singlemode coupler [5]. In this Letter, we report efficient coherent combining of two fibre lasers by means of an interferometric arrangement which includes only standard components (see Fig. 1). The two arms of the active interferometer are two independent fibre amplifiers connected to a 2 x 2 coupler which performs a field coherent addition. This coherent combining occurs by using one of the coupler input ports to close the laser cavity. In the case where the two arms of the interferometer are balanced in length, an active stabilisation will be required to preserve the inphase coupling on the mirror port and hence laser oscillation. To avoid this expensive and inconvenient solution, an imbalanced interferometer is preferred. The reflectivity transfer function in intensity of such an interferometer is dwxibed by R(w) = sin2(o.2A1/c 40) where AI denotes the difference in effective length between the two arms and $0 is a constant phase difference. Assuming an equal gain on each arm of the interferometer, the net gain profile provided by the setup is modulated by R(w) with a 3 dB modulation depth. It is always possible to get several peaks in the gain bandwidth of the amplifier and even in the reflectivity bandwidth of the end mirrors (supposed to be almost similar) of each branch by adjustment of AI. Therefore, the laser spontaneously oscillates on the gain peaks which correspond to coherent combining. Changes in the interferometer phase-shift $, due to the lack of active control arc compensated by an automatic adjustment of the oscillating frequency. In the following, we describe the coherent combining of two fibre lasers in a Michelson-type configuration, then in a Mach-Zehnder-type configuration. SO mW SM output
+
at 1550 nm
References
7
TAPPERT, ED.: ‘The parabolic approximation method’ in KELLER, J.B. and PAPADAKIS, J.S. (Eds.): ‘Wave propagation and underwater acoustics’
(Springer, New York, 1977), (Lect. Notes Phys., Vol. 70) ‘Parabolic equation method for electromagnetic wave propagation’, IEE Electromagn. Wave Ser, 2000, 45 FOCK, VA.: ‘Electromagnetic diffraction and propagation problems’ (Pergamon Press, Oxford, UK, 1965) CLAERBOUT, J.F.: ‘Fundamentals of geophysical data processing’ (McGraw-Hill, New York, 1976) MURPHY, J.E.: ‘Finite-difference treatment of a time-domain parabolic equation: theory’, 1 Acoust. Soc. Am., 1985, 77, (5), pp. 1958-1960 COLLINS, M.D.: ‘The time-domain solution of the wide-angle parabolic equation including the effect of sediment dispersion’, 1 Acoust. SOC.Am., 1988, 84, (6), pp. 2114-2125 LEVY, M.F.:
POPOV, A.V:
‘Accurate modeling of transparent boundaries in quasi-
optics’, Radio Sei., 1996, 31, (7), pp. 1781-1790 COURANT,R., and HILBERT, D.: ‘Methoden der mathematischen physik, Vol. 2’ (Springer-Verlag,Berlin, 1937) BASKAKOY VA., and POPOY A.V: ‘Implementation of transparent boundaries for numerical solution of the Schroedinger equation’, Wave Motion, 1991, 14, (l), pp. 123-128
Power scaling of fibre lasers with all-fibre interferometric cavity D. Sabourdy, Y Kermene, A. Desfarges-Berthelemot, L. Lefort, A. Barthklemy, C. Mahodaux and D. Pureur Two interferometric laser cavities are proposed for efficient combining of two fibre lasers. A 3 dB gain in power by coherent coupling of two identical erbium-doped fibre lasers is demonstrated.Scaling in power is envisaged through cascading of a coherent combination of elementary lasers. Introduction: All-fibre lasers have many advantages such as high conversion efficiency, compactness, and reliability. Scaling in output power usually comes from the use of high-power pump laser diodes associated with double clad doped fibres. An alternative approach is offered by coherent combining of several lasers which permits one to extend the use of an available technology and of an already optimised
692
41.5 mW c _
angle
/3c
\
/
-
EDF
WDM
CFBG
EDF
WDM
CFBG
coupler
cleave
Fig. 1 Experimental setup of Michelson jibre laser LD1, LD2: 980 nm pump laser diodes; EDF: erbium-doped fibre; CFBG: chirped
fibre Bragg grating at 1550 nm; WDM: wavelength division multiplexer
Michelson-type conjguration: In the past, this resonator architecture has been investigated for producing a single-frequency output from gaseous lasers [6]. We have adapted the principle of this configuration to the all-fibre lasers. Our experimental device, shown in Fig. I , consists of two independent erbium-doped fibre amplifiers (EDFAs) which are core pumped by two pigtailed laser diodes emitting at 980nm with a power up to 1I5 mW. The erbium-doped fibres had equivalent lengths of -17 m, corresponding to almost complete pump absorption. The two arms of the active interferometer were ended by two identical chirped fibre Bragg gratings (CFBG) with high reflectivity (198%) at 1550 and 2 nm bandwidth. These hvo erbium-doped fibres were spliced to a 50/f;0 coupler. The output coupler of the Michelson laser had a 4% reflectivity and was obtained by cleaving one fibre of the coupler input ports. It could have been replaced by an additional FBG with an optimised reflectivity to provide a fibre output. The second input port of the coupler was angle cleaved to avoid any reflection. We also inserted a 1 m-length singlemode fibre (SMF) in one arm to prevent the interferometric instabilities. We have measured the conversion efficiency of the Michelson fibre laser (MFL). We have also compared the performances obtained with the MFL with those obtained with an individual fibre laser (IFL) based on the components forming ore branch of the MFL. Fig. 2 shows that the MFL threshold of 20 mW was twice the IFL one; this is because in each arm of the MFL the population inversion occurs at the same pumping level as in the IFL. The MFL and the IFL had almost the same slope efficiency of 44%. This means that the combining efficiency was maximum, which was confirmed by the very low power leakage measured on the angle cleave output (
