Fiber Bragg grating as an optical filter tuned by a magnetic field

We describe a novel tunable optical filter for use in optical-frequency-domain multiplexed communication systems. The shift in the Bragg condition of a fiber Bragg grating as a result of magnetically induced circular birefringence is calculated with coupled-mode theory on the basis of circular states of polarization, and the values obtained for silica and terbium-doped optical fibers are compared.  1997 Optical Society of America

Fiber Bragg gratings (FBG's) are currently used in sensing (strain and temperature, 1 magnetic field 2 ) and optical communication systems.In-fiber Bragggrating sensors have attracted considerable interest because of their intrinsic nature and wavelengthencoded operation.In this Letter we demonstrate the ability to implement tunable optical filters that use such devices.These new tunable optical filters are based on the Bragg-condition shift of a fiber-optic Bragg grating without linear birefringence 3 perturbed by variable magnetic fields.
An electromagnetic f ield in anisotropic single-mode fibers (perturbed systems) can be represented as a linear combination of round-core f iber (unperturbedsystem) fields 4 by E ͓A͑z͒e 1 1 B͑z͒e 2 ͔exp͑ jvt͒, H ͓A͑z͒h 1 1 B͑z͒h 2 ͔exp͑ j vt͒ . ( Here ͕e 1 , h 1 ͖ and ͕e 2 , h 2 ͖ denote two eigenfunctions in the unperturbed fiber and satisfy an orthonormalization relationship to each other.The coefficients A͑z͒ and B͑z͒ are the field amplitudes, which depend on the fiber-axis coordinate z. The dielectric-constant tensor De represents a deviation from the unperturbed system.We derived the fields in the perturbed system by substituting Eqs.(1) into Maxwell's equations and the material equations and regarding De as a perturbation.
The behavior of A͑z͒ and B͑z͒ is represented by the simple coupled-mode equations with where the asterisks indicate complex conjugation, b 0 is the propagation constant in the unperturbed sys-tem, e 0 is the electric permittivity in vacuum, v is the angular frequency, and S is the cross section of the fiber.Coupled-mode theory 5 is a powerful tool for studying the change in light propagation caused by perturbations of geometry of material properties, and it is most suitable for describing the superposition of different perturbations in single-mode fibers, given by their respective dielectric tensors, which will change the polarization properties of the fiber.For the correct application of coupled-mode theory, a natural frame has to be used to represent the basic modes of the nonperturbed fiber and the dielectric tensor of the perturbation.Tang showed 6 that there is a natural frame in a nonperturbed (isotropic) waveguide that follows an arbitrarily curved line in space, for which the solutions of the wave equation become orthogonal.
When a magnetic f ield H z in the direction of the light propagation, z, is applied to an optical fiber, the Faraday effect induces a circular-birefringence perturbation given by the dielectric tensor with De F 2jn 0 VH z ͞k, where V is the Verdet constant and k v͞c.The typical values for the Verdet constant in single-mode f ibers can be approximately expressed as V 0.14 3 10 232 c 2 ͞l 2 rad͑͞Gauss m͒.
The basis of the fundamental modes e 1 and e 2 is usually chosen to describe orthogonal linear polarization; this choice is arbitrarily adapted to the anisotropy involved.Since the Faraday effect induces circular birefringence, and since the eigenmodes of the rotation effects are circular, it is most convenient to express E on the basis of orthogonal left-and right-circular polarizations.The coupled-mode approach now assumes that the electric f ield E in the perturbed medium is given by a linear superposition of the nonperturbed modes: Then, assuming single-mode fibers and two orthogonal circular polarizations, R and L, we can write the coupled equations for the amplitudes C 1 and C 2 of two such modes as where k ij ͑i, j 1, 2͒ are the coupling coefficients.From Eqs. ( 3) and ( 4), on a circular basis, one finds the coupling coefficients for the Faraday effects for light propagating in opposite directions, 1z and 2z: where H z is the magnetic-field component in the direction of the z axis.For both directions of the propagation the two circular polarizations are not coupled, but they will propagate with different speeds.
The refractive indices for the R mode n 1 and the L mode n 2 , for small perturbation, are given by 7 where g is the magnetogyration coefficient of the medium.
The specific rotatory power in a medium with cular birefringence (the rotation per unit length of the linearly polarized electric field that propagates in the medium with optical activity) is given by and, as the specific rotation for the Faraday effect is often written as r VH z , the circular birefringence induced by the Faraday effect becomes then When a magnetic f ield is applied to a FBG (a periodically layered medium induced in the core of the singlemode optical fiber), the circular birefringences n 1 1f , n 2 1f and n 1 2f , n 2 2f are induced in layers 1 and 2, respectively, of that FBG (Fig. 1).
Bragg ref lection is a typical example of contradirectional coupling. 7Therefore we can use the coupledmode theory to treat the optical properties of a shallow Bragg ref lector perturbed by a magnetic field.The normal modes in the FBG with H z are plane waves exp͑2jk ?r͒ with circular states of polarization and wave numbers given by where n1 and n2 are the geometric average refractive indices of the medium with the Faraday effect.
Since both perturbed and nonperturbed dielectric constants are scalar, mode coupling between R and L waves does not exist.Consequently, only coupling between waves of the same polarization state can take place (Fig. 2).This is possible only in the contradirectional coupling, because the phase-matching condition for codirectional coupling can never be satisf ied; Db b 1 2 b 2 2 m͑2p͞L͒ 2m͑2p͞L͒ 0 for f inite L. Maximum ref lectivity occurs at the Bragg wavelength, when Db 0. b 1 and b 2 are the propagation constants of the circular-polarization modes.Thus, in the presence of magnetic fields applied to the FBG, the index is changed for the two circular polarizations, and consequently two Bragg conditions 8 are observed (Fig. 3): Based on this effect, it is possible to implement a new tunable optical filter.The tuning of the new tunable optical filter that is proposed is based on the change in index induced by a magnetic f ield applied to a FBG.The range for tuning, when L is given, is then dependent on parameters V and H z .Thus, it is possible for one to select the central wavelength of the proposed f ilter by modifying these parameters: In silica f ibers with l 1.3 mm, the Verdet constant is V ഠ 8 3 10 25 rad͑͞G m͒, so a FBG ͑L ഠ 1 mm͒     needs high magnetic fields, H z ഠ 750 G, for a Braggcondition shift of 10 MHz ͑ϳ 0.00005 nm͒, and its dependence on H z is represented for different pitches in Fig. 4. One solution is to use a terbium-doped optical fiber with V ഠ 2.5 3 10 24 rad͑͞G m͒; in this case, for the same FBG and Bragg-condition shift, H z ഠ 240 G, at the same wavelength (Fig. 5).Another way of creating a larger Bragg-condition shift is to use a multilayer Bragg grating, whose pitches can be larger than FBG pitches, that is composed of paramagnetic or ferromagnetic optical materials with a larger Verdet constant than silica fibers.
The spectral response of this device is shown in Fig. 6, in which the left-polarization -circularpolarization converter is an optical circuit that transforms the incident linearly polarized light into the right and left circularly polarized modes that interact with the FBG exposed to the magnetic field.
In conclusion, a new tunable optical f ilter based on Faraday effect on Bragg gratings, useful for wavelength fine tuning in optical circuits, for sensors, and for optical communication systems has been studied.Our results have shown that its implementation in FBG necessitates high magnetic fields.However, by use of optical materials with a high Verdet constant, its further development is possible.

Fig. 1 .
Fig. 1.Circular birefringence induced by the Faraday effect on each FBG layer.SOP, state of polarization.

Fig. 2 .
Fig. 2. Powers of the incident and ref lected circularly polarized modes in a periodic layered medium when Db 0.

Fig. 3 .
Fig. 3. Wavelength shift induced by the Faraday effect in the FBG.

Fig. 4 .
Fig. 4. Bragg condition shift versus magnetic f ield in a standard optical fiber.
This work was sponsored by the Spanish Ministry of Education and Science through Comisión Interministerial de Ciencia y Tecnología project TIC95-0631-C04-01.