Strong coupling in a 2x2 directional coupler leads to cutoff of the antisymmetric supermode. This phenomenon produces unique dichroic splitting properties which have been observed experimentally and can be exploited in twin-core fiber sensors. A variational calculation of the supermode dispersion relations shows that the antisymmetric supermode for cores with V parameter unity cutoff when the core separation is four times the core diameter.
Keywords: Twin-cored fiber, directional coupler, antisymmetric mode
Coupled mode theory and supermode theory are sometimes presented as alternative ways of describing the behavior of optical fiber and integrated directional couplers. On the contrary, a correct development of coupled mode theory requires symmetric and antisymmetric combinations of single core modes to be introduced from the start unless asymmetric structures or very weak coupling regimes only are considered. A 2x2 directional coupler with cross section as shown in Figure 1 can be analyzed using coupled mode theory for the modes of the separated cores but on account of the symmetry of the cores this is a form of degenerate perturbation theory1. It is well known that degenerate perturbation theory is necessarily divergent at higher order unless the matrix of the coupling is diagonalised in the space of the degenerate unperturbed functions (modes). The linear combinations of modal fields which diagonalise the coupling matrix are the symmetric and antisymmetric supermodes referred to in the literature2. While the fundamental symmetric supermode of a 2x2 coupler is necessarily guided the corresponding anti-symmetric supermode may be cutoff with dramatic effect on the dichroic splitting properties of the device.
Calculation of Cutoff Condition
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With r the variational parameter. The variation solution in the weak guiding approximation was performed with analytic evaluation of the infinite integrals and a Monte Carlo evaluation of the integral of ∆ (n2) = nco2 - ncl2 over the cores. In the approximation given, the result of bringing the cores close together can be expressed in reduced form by plotting dispersion curves of the reduced propagation constant, U =a √nco2ω2/c2 - β2, against the reduced frequency (fiber parameter), V = a ω/c √∆(n2) , for different values of the reduced core spacing D = d/a. These curves are shown in Figure 2. The use of the variational method permits solution of the guiding problem for arbitrary core refractive index profiles. To check the numerical results comparison can be made with the analytic results of symmetric-antisymmetric mode splitting in step index and Gaussian index cores given by Meltz3. The antisymmetric mode cutoff referred to in reference 3 requires evaluation of direct, as well as mixing terms of the mode coupling matrix and is provide here for the first time. A further check on the accuracy of our numerical solution is the dispersion curves for symmetric and antisymmetric modes for large Vwhich converge on the single core HEll dispersion curve data for which have been tabulated4
Twin-cored fibers were prepared by the rod and tube method from laboratory Pyrex cladding (ncl = 1.474) and Schott ZKN7 zinc crown optical glass core (nco = 1.506). The core diameters were such as to satisfy the single mode condition for each core considered separately, that is to say that the fiber parameter, V, was less than 2.405. A phase contrast image of twin-cored fiber is shown in Figure 3. The cores in the first specimens were sufficiently well separated for their fundamental modes to be phase mismatched so that significant coupling between cores was avoided (Figure 4 a). In these specimens the expected sine-squared-theta fringes were observed crossing the far field mode spot (Figure 4 b) when the cores of the fiber were equally excited with a HeNe laser.
Some fibers were then made with somewhat smaller core separations. In these fibers the antisymmetric supermode was cutoff but, in addition, the variations of fiber diameter in the hand drawn fibers caused the symmetric and antisymmetric modes to be coupled. These fibers would not guide red HeNe laser light at all. It can be inferred that the symmetric-antisymmetric mode coupling length was not too different from the antisymmetric mode radiation length at this wavelength for it can be shown by coupled mode theory that the attenuation of a guided mode caused by coupling to a radiative mode is greatest in these circumstances5. On the other hand, if the radiation length of the radiative mode is much shorter than the coupling length the radiative mode amplitude never becomes large enough to attenuate the guided mode appreciably. These fibers guide the green and blue fluorescence, albeit imperfectly coupled from the laser, with high efficiency, allowing the spectrum to be recorded at the output end of the fiber.
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The next step was the introduction of a directional coupling biconical taper in a fiber with well separated cores, the waist diameter in the taper being chosen to ensure cutoff of the antisymmetric mode. Since the tapering process leaves the ratio of core diameter to separation unchanged, the cutoff of the antisymmetric mode is brought about by reduction of the V-parameter for each core. The biconical taper provides 3dB splitting between the two cores of the symmetric part of the input excitation. In order to verify the expected behavior it was necessary to couple light into only one core so that the observation of equal core excitation at the output would confirm the 3dB splitting. Launching into one or other core at the input end of the fiber was achieved by scanning the fiber across a line focus produced by placing a cylindrical lens of focal length 6cm between the laser and the 40x input objective. The near field was imaged by a CCTV camera and the 50-50 splitting could be verified by displaying the appropriate line scan from the raster on an oscilloscope (see Figure 6).
The guided power from the fiber showed the expected double maximum as the fiber was transversely scanned through the line focus of the laser beam. The transfer characteristics of the antisymmetric mode shedding coupler are quite different from those of normal 2x2 directional couplers. The latter are optical 4-ports with polarization dependent insertion loss between any pair of ports which for monochromatic excitation can be characterized by an 8x8 complex unitary matrix (or a 4x4 matrix if reflections are neglected). This is effectively an extended Jones matrix of the type that is used in beam optics. The corresponding transfer matrix for the couplers described here is intrinsically nonunitary.
The experimental results reported in sections 3 and 4 provide firm evidence for the feasibility of fabricating in-fiber splitting (and combining) components for single fiber interferometers6. The inclusion of reference and measured paths within a single fiber offers the prospect of fiber interferometry with much reduced sensitivity to temperature induced noise and drift. Ultimately it may even be possible to incorporate fluorescent source and detector sections within a single fiber and so achieve truly measurement systems within a self-contained fiber structure.
- D.Marcuse, Bell SystTech.J., 50, 1791-1816, (1971).
- A.Yariv, "Optical Electronics", Holt-Saunders, New York, (1985), p442.
- G.Meltz, J.R.Dunphy, W.W.Morey and E.snitzer, Appl.Optics, 22, 464-477, (1983).
- Snyder and Love Optical Fibre Waveguides
- R.J.Potton, A.D.Boardman, G.S.Cooper and P.Papadopoulos, Proc.Soc.Photo., 630, 245-248, (1986).