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A full vector complex coupled mode theory (CMT) for the analysis of tilted fiber gratings is presented. With the combination of the perfectly matched layer (PML) and the perfectly reflecting boundary (PRB), the continuous radiation modes are well represented by a set of discrete complex modes. Simulation of coupling to radiation modes is greatly simplified and may be treated in the same fashion as guided modes. Numerical results of the tilted fiber Bragg gratings (TFBGs) with outer-cladding index equal, lower and higher than that of the inner-cladding indicate that the complex coupled mode approach is highly effective in the simulation of couplings to cladding and radiation modes in tilted fiber gratings. The reflective TFBGs are investigated by the proposed approach in detail.
©2010 Optical Society of America
1. Introduction
Recently, tilted fiber Bragg gratings (TFBGs) have attracted more and more attention due to its stronger and more efficient cladding/radiation mode couplings and polarization dependency compared with the traditional non-tilted fiber gratings i.e. fiber Bragg gratings (FBGs) and long period fiber gratings (LPFGs). Numerous applications based on TFBGs have been reported, for example, by utilizing its strong cladding/radiation mode couplings, temperature-insensitive displacement/strain/vibration sensors, refractometer [1–8], in-fiber spectrometer [9], erbium-doped fiber amplifier gain flattening [10], in-fiber polarimeter (by utilizing its polarization dependency) [11–13], twist sensor [14], polarization-dependent loss equalizer [15,16], etc.
The basic principle of fiber gratings is to facilitate efficient mode- and wavelength-selective couplings among the core modes, the guided cladding modes, and the unguided radiation modes. In non-tilted fiber gratings, only the couplings between the vector fiber modes with identical azimuthal order μ are allowed (usually μ = 1), and the non-tilted fiber gratings are polarization independent if they are written in circular fibers. By contrast, for the tilted fiber gratings, the coupling between the vector fiber modes with dissimilar azimuthal order μ = 0,1,2,⋯ are allowed, and the coupling is polarization dependent even if the gratings are written in circular fibers. The enhanced cladding/radiation mode resonances and the polarization dependenc lead to new potential applications.
Similar to the non-tilted fiber gratings, the characteristics of mode couplings depend critically on the refractive indices of the inner and outer cladding as well as their difference with that of the fiber core. Assume that the gratings are written in a fiber with core index n co, inner cladding index n cl, and outer cladding index n s as shown in Fig. 1. When n s < n cl, both guided cladding and unguided radiation modes are supported, hence the couplings from the core mode to both guided and radiative cladding modes exist. When n s ≥ n cl, no guided cladding modes are supported, the core mode may couple only to the radiation modes which will give rise to the leakage of power from the fiber. In the special case of infinite cladding when n s = n cl, there is no reflection at the inner-outer cladding interface so that the transmission spectrum is smooth over the entire transmission band. When n s > n cl, the partially reflected radiated fields from the inner-outer cladding interface will induce a Fabry-Perot like interference and form guided-mode-like leaky modes with complex propagation constants. Consequently, resonance peaks will appear in the radiation mode coupling spectra. With the increase of index difference between the inner and outer cladding, these resonance peaks will be more pronounced, the behavior of radiation mode resonances will be more like the guided cladding modes.
There are several approaches for the analysis of TFBGs, i.e. the coupled mode theory (CMT) [17–22], the free space model based on antenna theory method [23], the near-field model [24,25] and the volume current method (VCM) [26–28]. The CMT approach is an intuitive and effective method for the analysis of mode couplings in fiber gratings and has been applied to investigate the mode interations in TFBGs such as the Bragg scattering, the radiation mode couplings with scalar LP or vector HE/EH -like radiation modes and the coupling strength for guided cladding mode resonances [17–20]. The free space model has been utilized to investigate the properties of the wavelength spectra of sidetap filters [23,29]. In the free space model, a tilted grating is considered to be equivalent to an infinite set of gratings that are perpendicular to the fiber axis and concentric with the fiber core, and the filter properties are calculated by antenna theory in free space. In the VCM and the near field model, the scattered field calculated by considering the index variation inside a waveguide as an induced effective current source. The VCM has been verified by CMT with HE/EH like radiation modes [20]. The free space model, the near-field model and the VCM are better suited for the analysis of non-paraxial scatterings (i.e. θ ~ 45°) compared with CMT. However in all these approaches the waveguide boundaries are neglected, they are not appropriate for the analysis of shallow tilt-angle gratings (i.e. θ ~ 0° or θ ~ 90°), in which the waveguide has a dominant influence, and the leaky modes give rise to a peak close to the Bragg wavelength in the loss spectrum [20,23,29].
All the investigations about the radiation mode couplings in the approaches discussed above are based on the assumption that the TFBGs are written in weakly guiding fibers with infinite homogeneous cladding (i.e. n cl = n s and r cl → ∞), which is only an approximation to practical situations. A comprehensive model which can handle both the couplings to guided cladding and unguided radiation modes for the cases of the outer-cladding index n s smaller, equal or higher the inner-cladding index n cl is still absent. The difficulties for such a rigorous model are due to several factors. Firstly, the index difference between inner- and outer-cladding (i.e. n cl and n s) may be large, weakly guiding approximation (LP modes) may lose its accuracy. Hence the full vector fiber modes should be considered. The couplings between the vector fiber modes with dissimilar azimuthal order μ = 0,1,2,⋯ are allowed. The resonances between the core mode and the vector cladding modes with even and odd azimuthal order are very close in TFBGs. Hence, it requires a highly accurate mode solver to be capable of calculating a large number of modes very efficiently, and one single resonance in TFBGs usually corresponds to many vector cladding modes, thus the simple two-mode analytical formulation is unapplicable for TFBGs [35]. Secondly, coupling to the radiation modes of continuous spectrum must be accounted for. Theoretically, the complete radiation modes of continuous spectrum can be used to derived the coupled mode equations, which can be solved with different approximations [17,20–22,30,31]. However, the expressions for radiation modes are extremely complicated, especially for optical fiber with inhomogeneous cladding (i.e. n cl ≠ n s case). To solve the coupled mode equations, one may have to resort to successive numerical integration which is computationally intensive. Discrete leaky modes [32] and quasi-modes [33] can be used to approximate a sub-set of radiation modes. However, their application is limited by the availability of analytical solutions [35,36]. The application of leak/quasi modes in the analysis of tilted fiber gratings do not appear to be trivial nor intuitive.
Recently, an improved full vector complex mode solver for circular optical waveguides with arbitrary radial index profile was developed and reported [34]. It was demonstrated that the improved mode solver is very powerful in solving real and complex modes in circular fibers with high degree of accuracy and efficiency. Further, a complex coupled mode theory (CMT) for optical waveguides based on a waveguide model enclosed by perfectly matched layer (PML) and perfectly reflecting boundary (PRB) is proposed [35] In the complex coupled mode theory (CMT), the continuous radiation modes are discretized into a set of discrete complex quasi-leaky modes so that they can be treated in a similar fashion as the guided modes. It was demonstrated in the case of non-tilted Bragg gratings in slab waveguides that the complex CMT is highly accurate and efficient in predicting the transmission spectra for couplings to both guided and non-guided cladding modes.
In this paper, we extend the work in [35] to tilted fiber gratings. The fiber gratings are enclosed by perfectly matched layer (PML) and perfectly reflecting boundary (PRB). With the help of PML and PRB the radiation fields can be represented by a set of well-behaved discrete complex modes so that the simulation of coupling to the radiation modes is greatly simplified and may be treated in the same fashion as guided modes. A comprehensive approach for tilted fiber gratings with refractive index of the outer cladding lower, equal and higher than that of the inner cladding is proposed. The structure under investigation and the definition of parameters are given in Fig. 1.
The remainder part of this paper is organized as follows. Section 2 describes mode coupling mechanism in TFBGs which includes the phase matching conditions and the classification of TFBGs. The limitation of the proposed method is also discussed. The formulation of CMT for tilted fiber gratings and the solving technique of coupled mode equations are given in section 3. The simulation results of reflective TFBGs with different outer-cladding indices are shown in section 4. The conclusions are given in section 5.
2. Mode couplings in tilted fiber gratings
Fig. 1. Diagram of a tilted fiber grating written in a three layer optical fiber and the definitions for the analysis.
The index modulation pattern of tilted fiber gratings is slanted with respect to the fiber axis as shown in Fig. 1. Due to the tilt angle of the gratings, the polarization effect arises. Hence two specific polarization states should be considered i.e. p/s-polarized modes or p/s-modes. The oscillation of electric field of p-modes is in parallel with x -axis in the x - z plane, and that of s-modes is perpendicular to the x - z plane. It will be shown later that the coupling coefficients between p- and s-modes are null so that we may treat them independently.
The the strongest interaction occurs when the wave vectors satisfies the phase matching condition
which is indicated by Fig. 2. Usually the gratings are written in the core of weakly guiding fiber (n co ≈ n cl). We can derive the relation between the phase matching wavelength and the grating period from Fig. 2 as
where n eff ≈ n co ≈ n cl. The TFBGs can be classified as reflective or transmissive type for tilt angle θ < 45° and θ > 45°, respectively. For the reflective/transmissive TFBGs, the contra/co-propagating couplings occur. Under the special case of θ ~ 45°, the optical power will couple to the radiative field whose propagating direction is almost perpendicular to the fiber axis at which the contra- and co-directional couplings will merge. The mode couplings for θ < 45°, θ ~ 45°, θ > 45° are illustrated in Fig. 3 intuitively, the circles represent the excited core mode at the input i.e. Re(β 01) ; the rectangles indicate the range which core mode will couple to. For the θ ~ 45° case, both the forward and backward modes should be included as shown in Fig. 3(b) and the coupled mode equations become a difficult boundary value problem to solve. We can drop the forward/backward modes for reflective/transmissive TFBGs, the θ = 45° can be considered as either reflective or transmissive type [22]. Such approximation will induce an underestimation of transmissivity in the shorter/longer wavelength range for reflective/transmissive TFBGs. Another issues related to the simulation for the θ ~ 45° case is due to the fact that evanescent radiation modes with Re(β) ~ 0 as shown in Fig. 3(b). As the mode fields are fast oscillating along the radial direction, fine mesh is required to solve for the modes for numerical solutions based on radial discretization, which means that the computational burden will increase.
Fig. 3. Mode couplings in TFBGs with different tilted angles. (a) Reflective TFBGs. (c) Transmissive TFBGs.
3 Coupled mode formulation
3.1 General coupled mode equations for tilted fiber gratings
The gratings under investigation are written in circular fibers. Due to the circular symmetry, the azimuthal dependence of the mode fields are known to be sinusoidal functions. In tilted fiber gratings, the source polarization effect arises due to the grating tilt. The full vector modes of circular fibers can be classified into p-modes and s-modes which correspond to x-and y- polarized linear polarized modes under weakly guiding conditions, respectively. For the m -th p-mode the mode fields can be written as
where μ = 0,1,2,⋯ is the azimuthal order. Let cos → sin and sin → -cos in Eq. (3), the expressions for s-modes can be obtained. A single superscript “p” is used to identify the p/s-modes, “+” denotes the p-modes; “-” denotes the s-modes. The transverse fields in the tilted fiber gratings can be expanded as in terms of the power normalized fields of the modes as
Insert Eq. (4) into the Maxwell's equations and utilize the orthogonality of the modes (i.e. ⟨e ptm, h qtn⟩ = 0 when m ≠ n or p ≠ q, ⟨e,h⟩ = 1/2∫∫(e × h⟩ · ž dS), we can derive the amplitude equation for apm(z) as
The coupling coefficients are given by
where n̄ (r) is the refractive index distribution of the fiber; P(r) is the r dependence of the index modulation (e.g. P(r) = 1 inside the perturbed area, P(r) = 0 outside the perturbed area); Δn(z’) is the index variation in the fiber core which is given by Eq. (7). The only assumption in Eq. (6) is that the variation in the refractive index is small compared with the index of the ideal fiber (i.e. Δn ≪ n̄), which is valid for fiber gratings. The index variation in the fiber core is given by [17]
where x = rcos(ϕ); and σ(z) = σ̄(zcosθ), χ(z) = χ̄(zcosθ), ϕ(z) = ϕ̄(zcosϕ) describe the slow varying DC perturbation in the background refractive index of the grating, grating amplitude and chirp, respectively. Since the length of the grating is much larger than the diameter of the fiber core, it is permissible in the slowly varying functions in Eq. (7) to put z’ = zcos(θ) + xsin(θ) ≈ zcos(θ).
By insertion of Eq. (3) into Eq. (6), the following relations can be proved rigorously
which means the two sets of modes (i.e. p-mode and s-mode) cannot couple to each other in tilted fiber gratings. The p- and s-mode solutions are independent and can be considered separately. Thus, the superscripts and “Σp=±” in Eqs. (4)–(6) can be dropped. For the simplicity of discussion, we assume that the azimuthal order of m -th mode is μ and that of n -th mode is v. The coupling coefficients for the p-modes are derived by insertion of Eq. (3) into Eq. (6) as
where
The functions of C(r) and S(r) can be calculated numerically by patterson quadrature. The relations in Eq. (11) is derived from the fact that C = (-1)μ+v C * and S = (-1)μ+v S *, where the superscript “*” denotes the complex conjugate. For the s-modes, the coupling coefficients are obtained by making swap C ↔ S in the coupling coefficients of the p-modes.
3.2 Reduced coupled mode equations for tilted fiber gratings
Assume the first mode is excited by external source. Introduce new variables u(z), un(z), v(z), vn(z)
where n ≠ 1 in Eq. (15) and β̄n = Re(βn). Actually, the variable substitution takes two steps indicated by “†” and “‡” in Eq. (14) and Eq. (15), respectively. The “†” step extracts the fast oscillating terms; the “†” step simplifies the coupled mode equations. Insert Eq. (14) and Eq. (15) into Eq. (5) and only keep the phase matched terms, the amplitude equation for u(z), un(z), v(z), vn(z) are derived as
where
The propagation constant for the phase matched mode is estimated by Δ± n ~ 0 and all the modes in the field expansion are searched around β̄n.
From the phase matching conditions Δ± n ~ 0, the couplings in reflective and transmissive TFBGs are {u} ↔ {v, vn} and {u} ↔ {un}, respectively. For reflective TFBGs un ≡ 0, the remaining equations for u(z), v(z), vn(z) can be solved by Runge-Kutta integration in the interval [-L/2,+L/2] with initial condition u(L/2) = 1,v(L/2) = vn(L/2) = 0 ; for transmissive TFBGs v ≡ 0 , vn ≡ 0, the remaining equations for u(z) and un(z) can be solved in the interval [-L/2,+L/2] with initial condition u(-L/2) = 1, un(-L/2) = 0, where L is the grating length, N is the number of modes used in the field expansion.
In order to increase the efficiency of the algorithm, the syntonical approximation is utilized to deal with the fast oscillating terms, which is implemented by evaluating the exponential function as
where ζ is a positive number. Equation (21) means if the length of the integral path L is larger than ζ times of the period of exp(jϕz), the corresponding term is ignored as fast oscillating term (the amplitudes are slow varying over L distance). Usually ζ = 100 is accurate enough for the calculation. After u(z), v(z), un(z), vn(z), (n = 2,3,⋯,N) are solved, Eq. (14) and Eq. (15) are utilized to solve for an(z), bn(z), (n = 1,2,⋯,N).
The reflection and transmission coefficient of the TFBGs are calculated as
The grating loss is calculated by L = 1 - R - T. Note that the power orthogonality is not applicable for complex modes, Eq. (22) is accurate as long as the complex modes are almost power orthogonal as shown in [35]. Such criteria is usually valid for the modeling of a really system. If the complex modes are very lossy, the mode amplitude usually decay to zero at the input/output end of the gratings (i.e. an(+L/2) = 0 or bn(-L/2) = 0 in the numerator), thus the “orthogonality” issue has no influence of the accuracy of Eq. (22); if the mode loss is small, the power conservation is quasi-satisfied, thus “the quasi-power orthogonal” criteria comes into existence automatically. For the really applications, the gratings are usually written in single mode fibers, and we are only interested in the fundamental mode, other modes are usually filtered by introducing a “C” bend i.e. to make sure an(+L/2) = 0 and bn(-L/2) = 0 . Thus Eq. (22) is widely appliable in the modelling of fiber grating applications.
4. Numerical results and discussions
In this section, the transmission spectra of reflective TFBGs are calculated by the complex CMT. As for the transmissive TFBGs, the solving technique is identical, which is indicated by the analysis given in section 2 and the symmetry of the formulation derived in section 3. For the sake of conciseness, the transmissive TFBGs is not considered here. The sketch of the fiber grating under investigation and the definition of the parameters are given Fig. 1. The TFBGs are written in a single mode fiber with r co = 4.1μm , r cl = 62.5μm , r s = 75μm, n cl = 1.466, Δ = (n 2 co - n 2 cl) / (2n 2 co) = 0.36%. The mesh size is Δr = 0.06μm . The TFBGs has a Gaussian modulation with full width half maximum (FWHM) equals to 5mm, and 100% modulation is assumed i.e. max(σ) = 2max(χ). The grating period is fixed as Λ = 0.5278μm. The parameters of the PML are chosen as d PML = 10μm, R = 10-12, M = 2 [34].
4.1 Bragg scattering
The Bragg scattering is due to the coupling from the core mode to the same mode with reverse propagating direction. The mode fields of the core modes have negligible overlap with the inner-outer cladding interface. The influence of the index difference between inner- and outer-cladding can be ignored. the Fig. 4(a) shows a calculation of peak reflectivity versus tilt angle θ, for TFBGs with index modulation max(ρ) equals to 2.0 × 10-3 , 1.0 × 10-3, 0.5 × 10-3. Nulls and peaks can be observed in curve of reflectivity versus the tilted angle. In Fig. 4(b), the reflection spectra for tiltled angle θ belongs to the first to the fourth peaks in Fig. 4(a). The results in Fig. 4 are for s-modes, the curves for p-modes are indistinguishable. The results in Fig. 4 fits well with those in the literatures [17,21]. Here we only present the numerical results for verification of our model, as for the more comprehensive analysis we refer to [17,21].
Fig. 4. (a) The calculated maximum Bragg reflectivity versus tilt angle for TFBGs with different values of index modulation. (b) The calculated reflection spectra for TFBGs with different tilt angle..
4.2 Cladding and radiation mode couplings
The cladding and radiation modes are sensitive to the inner-outer cladding index difference, hence the characteristics of the transmission spectra are highly depends on that index difference. In this section the transmission spectra of TFBGs with different outer-cladding index, i.e. (a) equal (n s = n cl), (b) smaller (n s = 1.0 < n cl), (c) higher (n s = n cl + 0.1) than the inner-cladding index are calculated. The index modulation strength is max (ρ) = 1.0 × 10-3, other parameters are identical with the TFBGs investigated above. For the case (a) n s = n cl, the modes with azimuthal 0 ≤ μ ≤ 11 for θ ≤ 10 , 0 ≤ μ ≤ 19 for 10 < θ ≤ 30 are included in the field expansion and for each azimuthal order, ten “phase matched” modes (with |Δ± n| ~ 0) are chosen. For the case (b) n s = 1.0 < n cl and (c) n s = n cl + 0.1, the modes with azimuthal 0 ≤ μ ≤ 15 for θ ≤ 10, 0 ≤ μ ≤ 19 for 10 < θ ≤ 30 are included in the field expansion and for each azimuthal order, five “phase matched” modes are chosen. Further addition of modes in the field expansion does not affect the results which indicates the convergence of the algorithm.
Figure 5 gives the transmission spectrum evolution, Fig. 6 gives the minimum transmission and the corresponding peak wavelength for the case (a). Due to the truncation of the number of modes in the field expansion, there are still fringes in the calculated transmission spectra [35,36], which can be smoothed by moving average algorithm. Figure 7–8 and Fig. 9–10 give the results for the case (b) and (c), respectively. In Fig. 7 and the leftside of Fig. 9, the s-mode is assumed, the p-mode solutions take the similar shape with s-mode under identical .θ.. For the case (b), the transmission spectra for 10° < θ < 22° take similar shape with θ ≡ 10°. For the case (c), the transmission spectra for 16° < θ ≤ 30° take similar shape with θ = 16°. The minimum transmissivity and corresponding peak wavelength of the spectra (indicated by circles in Fig. 7 and Fig. 9) for the not given cases are indicated by Fig. 8 and Fig. 10.
For the case (a) n s = n cl and (c) n s = n cl + 0.1, no guided cladding modes are supported. Except for the Bragg scattering, the transmission loss is due to the couplings to the radiation modes. For the case (a), smooth transmission spectra can be observed. The case (c) has a similar characteristics with case (a), but with resonance peaks appear in the transmission spectra. These peaks derive from the couplings to the leaky modes which are formed by the interference of the partially reflected radiated-field off the inner-outer cladding interface. For the case (b) n s = 1.0 < n cl, both guided cladding modes and unguided radiation modes are supported. The problem becomes more complicated. When 0° ≤ θ < 22° (27° ≤ θ ≤ 30°), the transmission losses are mainly due to the couplings to the cladding (radiation) modes. When 22° ≤ θ < 27°, both the couplings to the cladding and radiation modes are pronounced, the divergence between the couplings to the cladding and radiation modes will induce an abrupt change in the transmissivity around θ = 22° as shown in Fig. 8.
The results show that for all the three cases, the envelops of the transmission spectra share the same evolution characteristics with the increase of tilted angle θ, except for the abrupt change in the transmissivity around θ = 22° (see Fig. 8) in case (b). The phase matched wavelength is well predicted by Eq. (2) as shown in Fig. 6, Fig. 8 and Fig. 10. The polarization dependency increases with the increase of tilted angle and will reach its maximum at θ = 45° due to the Brewster's angle [21,22,26]. The strongest resonance peak occurs for the shallow tilted-angle gratings with θ ~ 3° for all the cases. Such resonance peaks derive from the couplings to a set of cladding/radiation modes with Re(β) ~ Re(β01). Because we cannot find a real single mode corresponds to these peaks, usually these peaks are considered to be induced by couplings to the “Ghost mode” which is actually a set of modes with similar Re(β). The “Ghost mode” resonance always appear for the shallow tilted-angle gratings, the core mode will couple to a region of the mode spectrum where the mode density is higher and the resonances of different modes overlap with each other, thereby induce the strongest resonance. The mode density is defined as the number of modes per unit change in Re(β). The light is likely distributed in the high index region, so the mode density in the mode spectrum is higher when Re(β) ~ k 0 n co than Re(β) ~ 0. This can be confirmed by the sharpness of the envelop of the transmission spectra for case (a), (b) and (c) or the separation of the transmissive peaks for case (b) and (c).
Fig. 9. Transmission spectrum evolution of TFBG with the increase of tilt angle. (n cl < n s = n cl + 0.1)
5. Conclusions
A full vector complex coupled mode theory (CMT) for tilted fiber gratings is developed and presented. The radiation modes are expaned by complex modes solved from an improved full-vector waveguided mode solver with perfectly matched layer (PML) and the perfectly reflecting boundary (PRB). With the help of PML and PRB, an ideal waveguide model is introduced for which the physical domain of the waveguide is not affected by the reflection from the edge of the computation window and yet the model fields are well defined in a finite and closed simulation environment. Thus, the continuous radiation modes can be well represented by a set of discrete complex modes so that simulation of coupling to radiation modes is greatly simplified and may be treated in the same fashion as guided modes.
The coupled mode formulations based on vector fiber modes for tilted fiber gratings is derived and solved numerically. In particular, the cases when the outer cladding index is lower, equal and higher than the inner cladding index are examinzed. By using the complex CMT, all the cases can be treated readily in the same fashion. We believe that the complex CMT extended the scope of the conventional guided mode based CMT so that the treatment of continuous radiation modes in typical optical waveguides is much more easier than the conventional CMT which involves the whole set of continuous radiation mode spectrum.
Acknowledgements
This research is sponsered by the National Natural Science Foundation of China under Grant No.60607001. Y.-C. Lu would like to acknowledge the support from the National High Technology Research and Development Program (“863” Program) of China (under Grant No. 2008AA01Z215) and the National Natural Science Foundation of China (under Grant No. 60837002 and 60707007).
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