1. Introduction

Coupled-mode theory (CMT) is a general theory to describe the couplings among the waves or vibrations. Over the past decades, it has been successfully applied to the analysis of interactions between electromagnetic waves or oscillations in the microwaves and optics. In the coupled-mode analysis, by expanding the unknown fields in non-ideal transmission or oscillation system in terms of the known solutions of a reference system, the governing Maxwell equations are transferred into a set of coupled differential equations, i.e., the coupled-mode equations (CMEs) in space or time. These equations intuitively describe how the amplitudes of the individual modes interact and vary with the spatial or temporal variable of z or t in the system. In principle, CMEs are equivalent to the Maxwell’s equations as long as the complete set of modes in the properly selected reference system is used. In practice, however, many problems are well treated with remarkable accuracy by considering only finite or even two guided modes close to phase matching conditions. The concision in mathematical form and the intuition in physical concept thus become the true spirit of the coupled-mode analysis in real-life applications [1].

To effectively apply the coupled-mode theory, a key step is to properly choose an expansion basis which consists of a set of accurately solved and properly selected modes in a suitable reference system. Traditionally, in most coupling problems describing transmission process in open waveguides in optics, the modes taking dominant effect are guided modes which can be dealt with within the conventional coupled mode formalism. In this respect, the radiation modes are normally neglected or treated with simplifying approximations. This has been a major limitation for the application of coupled mode theory, especially for open waveguides in which the effects of radiation modes are significant. As modes of continuum, the radiation modes are difficult to be solved or even normalized, especially for waveguides with complicated structures. An alternative treatment is to approximate radiation modes by discrete leaky modes [2]. However, the leaky modes are not proper solutions of the modal equations. They are difficult to be solved and used in the mode expansion. To circumvent this problem, an equivalent closed waveguide model has been proposed and demonstrated in which the open waveguide is enclosed with a properly set perfectly matched layer (PML) [3,4] terminated by a perfect electric conductor (PEC). With this model, the radiation fields are represented by a set of discrete, orthogonal, and normalizable quasi leaky modes in the solution system of complex modes. The idea to expand the radiation field in terms of quasi leaky modes was successfully utilized and intensively studied in the mode matching method (MMM) [5–10]. By introducing the expansion of the equivalent leaky modes into coupled-mode analysis, all guided and non-guided modes can be considered in a unified frame, and the interactions between the guided and radiation modes can be treated in a similar way to those between the guided modes, free from considering the cumbersome radiation modes. This is not only a unified and simplified treatment in the coupled-mode analysis, but also a novel way to extend the scope of the applications for this powerful method.

Though the idea was demonstrated in [11] for slab waveguides and in [12–14] for fibers, the general principle of how to effectively choose equivalent leaky modes to approximate radiation modes is not quite clear. Besides, the applicability and accuracy of the reduced coupled-mode model is rather ambiguous [13]. The objectives of this paper are to reveal the principle of quasi leaky mode approximations and the implementation on the unified coupled-mode analysis for more general applications, including how to set PML parameters to develop an equivalent closed waveguide model, how to set up an equivalent expansion basis to well approximate radiation modes, how to understand the mode coupling and power exchanging between complex modes and when to use the reduced coupled-mode model suggested in [11,13].

For the purpose, we give a detailed analysis on the characteristics of complex modes in the closed reference waveguide with different refractive index of outer-cladding. Then according to the different characteristics of quasi leaky modes in complex modes, we explicitly classify them into two kinds, which will facilitate a proper choice for an equivalent expansion basis in different cases. Further, since the focus of this work is to investigate and analyze the interactions of the guided core and the non-guided radiation modes by using equivalent complex modes, the roles of complex modes, especially the contributions of the modal losses and complex coupling coefficients in the couplings are examined to gain a clear understanding.

Fiber grating is taken here as example. Traditionally, it has been a classical example that is solved successfully by using coupled-mode analysis [15,16] when only guided core and cladding modes take effect. Recently, couplings to radiation modes in fiber grating have become a subject of more intensive research [17–22], and an approach which is well known and easy to be implemented is preferred. On the other hand, the theoretical and experimental results reported in literature will serve as useful benchmarks for our present trials [15,17,19,20]. Our present study on the couplings to radiation modes in fiber gratings will be a stepping stone to successfully apply the unified coupled-mode analysis to more applications. In this work, we will focus on the couplings to radiation modes in two cases where the refractive index of the surround is higher than or equal to that of the cladding. When the refractive index of the surround is lower than that of the cladding, the concern on couplings between guided modes in LPGs (long-period gratings) or FBGs (fiber Bragg gratings) has been well addressed in previous work [6,16]. It is demonstrated in recent research, couplings to radiation modes in the intermediate-period chiral fiber gratings lead to interesting spectral characteristics and applications [21], to which the principle and implementation to be presented here is also applicable.

The remaining part of the paper is organized as follows. In Section 2, the theoretical model for fiber gratings is described and the equivalent closed reference waveguide model is introduced for the unified coupled-modes analysis. In Section 3, the characteristics of the complex modes in the closed reference waveguides with an outer-cladding of different refractive indices are expounded, and the equivalence relationship between the quasi leaky modes and radiation modes in the original open waveguides are discussed. Based on an expansion of equivalent complex modes, the unified coupled-mode equations for fiber gratings are derived in Section 4. Characteristics of the complex coupling coefficients are also studied. In Section 5, before doing simulations on the transmission spectra of fiber gratings with different surrounds, mode coupling and power exchanging between two complex modes are discussed. Conclusions are drawn in the final section.

2. Theoretical model of fiber grating

Consider a fiber grating with a length of L as shown in Fig. 1(a) . It is written in the core of a conventional single-mode fiber and surrounded by material with a refractive index of n_oc. Here the surround is seen as an outer-cladding in the following discussions. The refractive indices and radii of the core and inner-cladding are n_co and n_ic, r_co and r_ic, respectively. The periodically modulated refractive index in the core along the longitudinal direction of the grating is described as usual,

 

Fig. 1 Theoretical model of fiber grating and its reference waveguide models. (a) sketch of fiber grating; (b) index profile of open reference waveguide; (c) index profile of equivalent closed reference waveguide.

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Similar to the conventional coupled-mode analysis, we consider the couplings of modes in the unperturbed waveguide due to the periodically perturbed refractive index in the grating. Instead of an open reference waveguide used in conventional coupled-mode analysis as shown in Fig. 1(b), here we introduce an equivalent closed reference waveguide model for the unified coupled-mode analysis, as shown in Fig. 1(c), which is a closed waveguide model obtained by enclosing the open waveguide with a PML terminated by PEC [3–14,23]. As proved in previous studies, if properly choosing the parameters of the PML such as the starting position denoted by the outer radius of the outer-cladding or the internal radius of the PML, namely, r_oc, the thickness d_PML and the reflection coefficient R_PML, the PML cannot only perfectly match with the outer-cladding, but also rapidly attenuate the out-going field in the PML to effectively eliminate the additional field reflected by PEC. This model is thus at most time equivalent to the original open waveguide. The principle of how to set PML parameters to obtain an equivalent waveguide model will be discussed in the following sections. Meanwhile, the modes in the equivalent waveguide model will be examined before being selected as the expansion basis in the unified coupled-mode analysis.

3. Characteristics of complex modes in equivalent reference waveguide model

Consider the waveguide as shown in Fig. 1(c) with structure parameters as follows, the core radius r_co = 2.5μm, the inner-cladding radius r_ic = 62.5μm, the core refractive index n_co = 1.458 and the inner-cladding refractive index n_ic = 1.45, which are chosen to be same as those in [5] for comparison and all the following simulations will be based on. The refractive index of the outer-cladding n_oc will be chosen to be higher than or equal to n_ic for different cases. The main model parameters of PML are respectively r_oc = 80.0μm, d_PML = 10.0μm, and R_PML = 10⁻¹², which will be used in the following simulations if no special declarations. The working wavelength is set to be 1.55μm.

For the use in coupled-mode analysis, the propagation constants and field patterns of complex modes are solved by an improved full-vector complex mode solver [23] which is currently the most efficient and accurate mode solver for circular waveguides to the best of our knowledge. We apply the PML following what suggested in [4] to ‘ensure the analytic continuation of the frequency-domain Maxwell’s equations to complex space’, which is actually an important basis for extending coupled-mode theory to complex space. As suggested in previous references [5,6], the solved complex modes can be classified into guided, quasi leaky and PML modes by compared with the modes in the original open waveguides. The fields of the PML modes are mainly distributed in PML, and the overlap integral with the field of the core mode is almost zero. Since the PML modes have practically no contributions to the couplings between the guided and radiation modes we are focusing on, we pay little attention on them. Two cases of different n_oc will be successively discussed as follows.

Case 1 n_oc is higher than n_ic

Mode Spectrum of the waveguide with n_oc = 1.46 are shown in Fig. 2 , where the horizontal and vertical abscissas represent the real and imaginary parts of the effective refractive indices, respectively. There are several obviously divided branches which can be well defined by simultaneously looking into their field patterns.

 

Fig. 2 Mode spectrum of the closed waveguide with an outer-cladding of n_oc = 1.46.

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In Fig. 2, the guided fundamental HE₁₁ mode is denoted by a diamond. Its amplitude and phase are shown in Fig. 3(a) . As seen from Fig. 3(a), its amplitude is the same as that in the open waveguide only in the core and inner-cladding regions. In the outer-cladding and PML region, the magnitude of the guided mode is negligible, and the phase increases rapidly.

 

Fig. 3 Amplitudes and phases of the radial components of the electric fields Er, for the complex modes with an outer-cladding of n_oc = 1.46. (a) the guided core mode HE₁₁; (b) the inner-cladding leaky mode HE₁₆; (c) the lowest order outer-cladding leaky mode; (d) a higher order outer-cladding leaky mode.

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The PML modes are denoted by stars in Fig. 2, which are easily identified as their positions in the mode spectrum are highly sensitive to the change of the PML parameters and their fields are predominately localized in PML region [5,6].

Further, there are two branches of quasi-leaky modes as indicated by circles and triangles in Fig. 2, which we call inner-cladding and outer-cladding leaky modes, respectively. If modifying PML parameters to strengthen the absorbing effect of PML, the inner-cladding leaky modes in the left branch rapidly approach to stable positions. For example, we have successively reduced the PML reflection coefficients from R_PML = 1 to R_PML = 10⁻³⁰, and intercepted a part of mode spectrum interested in the following simulations to clearly show the convergent process of effective refractive indices with the variation of R_PML, as illustrated in Fig. 4 . It is observed that the results become convergent when R_PML = 10⁻¹². In addition, we also plotted field pattern of a typical inner-cladding leaky mode, namely, HE₁₆ as shown in Fig. 3(b). Due to partial reflection at the inner- and outer-cladding interface, the leaky modes are guided-mode-like in the inner-cladding region with sufficient thickness as compared to the wavelength. Meanwhile, the exponentially growing field in the outer-cladding is effectively attenuated by PML, and smoothly transit to vanish before reaching PEC.

 

Fig. 4 Variations of the effective refractive indices of the inner-cladding leaky modes in the closed waveguide with an outer-cladding of n_oc = 1.46 with the change of the absorption coefficient of PML, R_PML.

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Different from the inner-cladding leaky modes represented by the left branch, the behaviors of the outer-cladding leaky modes indicated by the right branch in Fig. 2 are much more sensitive to the PML parameters and hence difficult to be convergent. This is not difficult to understand as in the case of open waveguide, there is no mechanism for field confinement in the outer-cladding and a true leaky mode will exhibit exponential growth in the outer-cladding. In the closed waveguide model, however, the perfectly reflecting boundary condition at the edge of the computation domain causes 100% artificial reflection. In order to re-create a pure outer-cladding leaky mode as described above, the PML has to attenuate 100% the reflected field, which is indeed difficult if not impossible. Figure 3(c) shows the field patterns of the lowest outer-cladding leaky mode solved with PML reflection coefficient R_PML = 10⁻¹² and R_PML = 10⁻⁶⁰. It is noted that, despite the fact that the PML reflection is extremely small, the residual reflection effect from the PEC still plays some role in confining the field in the outer-cladding region, leading to a quasi-standing wave pattern, which is especially obvious for a higher order outer-cladding leaky mode, as illustrated in Fig. 3(d).

Though sensitive to the PML parameters, the outer-cladding leaky modes may not be seen as PML modes. When enlarging r_oc, more and more outer-cladding leaky modes emerge in the right branch. Compared with the guide-mode-like inner-cladding leaky modes, outer-cladding leaky modes are more radiation-mode-like. Meanwhile, they move left and upside, even to the left of the left branch if further enlarging r_oc, but distinguished from inner-cladding ones, as shown in Fig. 5 . Actually, in the case of n_oc<n_ic, the leaky modes can also be classified as the inner- and outer-cladding modes in a similar way, though the distributions of the two kinds of leaky modes are different from those in the present case of n_oc>n_ic. In both cases, the difference between the inner- and outer-cladding modes decreases with the decrease of the thickness of the inner-cladding or the index difference between the inner- and outer-cladding until degenerates to the next case of n_oc = n_ic.

 

Fig. 5 Variations of effective refractive indices of inner- and outer-cladding leaky modes in the closed waveguide with an outer-cladding of n_oc = 1.46 with the change of the starting position of PML, r_oc.

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Case 2 n_oc is equal to n_ic

Figure 6 gives the mode spectrum of the waveguide with an infinite cladding. There are no special points for the core and PML modes. As for the leaky modes, with only a single cladding and without the partial reflection supported by the inner- and outer-cladding interface, they are all outer-cladding ones in one branch. Similar to those in Case 1, the solutions of the effective refractive indices and field patterns are difficult to approach convergence when modifying PML parameters. If the PML is set near the fiber core, for example, r_oc = 80.0μm, there are only a few leaky modes denoted by stars in the mode spectrum, as shown in Fig. 7 which is also intercepted from the mode spectra. With the location of PML moving to be far away from the core, much more leaky modes appear and tend to be a continuum; meanwhile, they move left and upside to approach to the radiation modes in open waveguide, as shown by circles in Fig. 7, where r_oc = 500.0μm. Intuitively, the latter will be a better approximation to the radiation modes in the original waveguide with infinite cladding, which will be validated in the following simulations for the transmission spectra of fiber gratings with index-matched outer-claddings.

 

Fig. 6 Mode spectrum of the closed waveguide with an outer-cladding of n_oc = 1.45.

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Fig. 7 Variations of effective refractive indices of the closed waveguide with an outer-cladding of n_oc = 1.45 with the change of the starting position of PML r_oc.

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Besides propagation constants and field patterns, the orthogonal relationship of the complex modes in the equivalent waveguide model is very important in the following derivation for coupled-mode equations. A conjugate-formed orthogonal relationship is usually used in the conventional coupled-mode analysis as,

4. Unified coupled-mode equations for fiber gratings

Based on mode characteristics well studied above, the unified coupled-mode equations or the complex coupled-mode equations referred to in [11] is derived with an expansion of the complex modes in the equivalent waveguide as follows [11],

Here the expansion basis includes N guided modes and M leaky modes. a_n and b_n are respectively the expansion coefficients or the complex amplitudes of the fields of the forward and backward propagating modes. ${\beta }_m$ is the propagation constant of the m-th complex mode. The coupling coefficients between co-propagating modes and contra-propagating modes κ_mn and χ_mn are respectively expressed as follows,

Further, we separate the slowly varying envelopes and the fast oscillating carriers of the mode amplitudes,

Inserting Eq. (6) into Eq. (4), we can have,

For fiber gratings with periodic index perturbations along the longitudinal directions described as Eq. (1), we can further derive the coupling coefficients by substituting Eq. (1) into Eq. (5) as

For a conventional single-mode-fiber-based grating, the fundamental guided mode or core mode is labeled as 1, the guided cladding modes and the leaky modes are labeled accordingly as 2, 3,… N + M. The integral term of the longitudinal fields in Eq. (8) is ignored since it is much smaller than that of the transverse fields. The self couplings of leaky modes and mutual couplings between leaky modes are also ignored since they are too weak in the fiber core [16]. Then the coupling coefficients in Eq. (8) can be simplified as follows,

For LPGs, the coupled-mode equations describing co-propagating interactions are further expressed as follows,

For FBGs, the coupled-mode equations describing counter-propagating interactions are further expressed as follows,

It is clear that the differences of the unified coupled-mode equations from the conventional ones are that the coupling modes, the propagation constants, and the coupling coefficients are all complex. The equivalence of the complex modes has been discussed in the previous section. For complex propagation constants, note that the real and the imaginary parts of the complex propagation constant respectively represent the modal phase and loss. Hence the phase matching condition is satisfied when the real part of the phase detuning factor δ vanishes, that is,

The difference of the definition of the phase detuning factor in Eq. (14) from that in Eq. (11) or Eq. (13) results from the self coupling of the core mode. c usually equals 2, except for the coupling to the counter-propagating core mode, where c = 1 [16]. Normally in practice, the reference waveguide are chosen so as to the self coupling coefficient for the guided modes $K_{11}$ will be zero [25].

Then we are interested in the characteristics of the complex coupling coefficients. Since the couplings can only occur between modes with the same angular order for non-tilted gratings to be discussed here, the coupling coefficients between the core mode and the lower 120th radial order inner-cladding leaky modes with the angular order of m = 1 are examined. Figure 8 gives a comparison of the coupling strengths between odd leaky modes of HE_1n and even leaky modes of EH_1n in the case of a higher-index outer-cladding. As seen from the figure, when the mode order is lower, the coupling strength of the core mode to even leaky modes is very weak compared with that to odd leaky modes of the same order. However, it increases with the increase of the radial order, and becomes comparable with that to the same order odd one when the radial order is larger than 40 in this case. The variation trends are common for waveguides with different structure and PML model parameters, except that the variation curves change with the PML model parameters in the case of infinite cladding. Meanwhile, the variation trends shown in Fig. 8 are similar to those observed in the couplings between the core and guided cladding modes in [16], which will be helpful for choosing effective leaky modes involved in couplings in the following simulations.

 

Fig. 8 Comparison of coupling strengths between the core mode and leaky modes with angular order m = 1 at λ = 1.55μm (n_oc = 1.48).

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5. Numerical results and discussions for fiber gratings

The transmission spectra of fiber gratings with different surrounds are studied in this section. We focus on LPGs with the same waveguide structure as that described in Section 3 and grating parameters of Λ = 312.0μm, Δn = 2.4 × 10⁻⁴, and L = 2.5cm, which are also chosen to be same as those in [5] for comparison.

As the first step to apply the unified coupled-mode analysis, we investigate the mode coupling and power exchanging between the guided and leaky modes, which is treated similarly to those between two guided modes yet exhibit different behaviors. Figure 9 show the powers carried by the guided core mode and a leaky mode. As seen from Fig. 9(a), at the resonant wavelength of λ = 1.23μm determined by Eq. (14), the powers of the two modes periodically oscillate with decreasing envelops due to the leakage of the leaky mode. The period of the power exchanging is inversely proportional to the real part of the coupling coefficient in this case of phase matching. When λ gradually increases from 1.23μm, as shown in Fig. 9(b)–9(d), with the increase of the phase detuning factor δ, the power percentage of the core mode taking part in the power exchanging decreases, meanwhile, the period of the power exchanging decreases. The envelops of the periodically oscillating powers of the two modes both decrease at first, and then gradually change to monotonic attenuations, though the power of the leaky mode is not comparable with that of the core mode. From the power evolutions in Fig. 9, the roles of complex propagation constants and complex coupling coefficients on mode coupling and power exchanging can be intuitively understood. The differences from those of non-complex ones can be attributed to the leakage, which can be further understood by analogy to a simple harmonic vibration with damping. For the applications in fiber gratings, the roles will result in a larger power loss of the core mode at a certain coupling wavelength and length. The rules of power exchanging in the coupling between complex modes can be used in the design and optimization of devices, for example, properly setting parameters such as grating length to realize required performances such as coupling ratio or properly setting leakage loss to effectively shorten the coupling length. Generally, the roles of complex modes in the mode couplings have relation to multiple factors, which will not be further discussed in this paper due to the limited space.

 

Fig. 9 Power exchanging between the core mode and a leaky mode at different wavelength (n_oc = 1.75).

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Based on the above discussions, we can now simulate the couplings of the core mode to radiation modes in LPG by using the unified coupled-mode analysis.

Case 1 n_oc is higher than n_ic

The simulated transmission spectra are shown in Fig. 10(a) –10(c). As seen from the figures, there are seven obvious resonant dips in each spectrum, corresponding to the couplings of the core mode to seven lowest order odd inner-cladding leaky modes. The resonant loss of the guided core mode coupling to a certain leaky mode increases with the increase of the surrounding index. This is because with the increase of the index difference between the cladding and the surround, the leaky modes will be better confined, and their couplings with the core mode become stronger consequently. The trend agrees very well with those obtained experimentally [17] and theoretically by using conventional coupled-mode analysis with a direct integral of the radiation modes [19]. Meanwhile, in Fig. 10(b), we show the comparison of the present simulation results denoted by solid line with those obtained by complex mode-matching method (CMMM) [5] and denoted by the scattered dots. They exactly agree with each other, which indicates the present algorithm is correctly implemented.

 

Fig. 10 Simulated transmission spectra of LPGs with different surrounding indices.

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In our simulations, the couplings between the core mode and the lower order even modes are neglected since the coupling strengths are very weak as shown in the last section. Generally, at a certain wavelength, the coupling of the core mode to only one odd inner-cladding leaky mode which is most close to phase matching is dominant, especially for distinctly separate resonances. Thus a reduced coupled-mode model is suggested in [11] and [13]. To evaluate the accuracy of the approximate model, the convergence of the simulation results is examined by including more than one leaky mode in the mode expansion. From each side of this dominant leaky mode in the mode spectrum, one or two leaky modes that nearly satisfy phase matching conditions are selected for mode expansion. Since more leaky modes need to be used in the mode expansion if the index difference between the inner- and out-cladding tends to zero, the case of n_oc = 1.46 is taken as example from the three cases discussed successively in Fig. 10(a)–10(c), as shown in Fig. 10(a). As seen from the figure, convergent results are obtained with only three leaky modes used in the mode expansion, and the results obtained by the reduced two-mode model can be taken as a good approximation. Therefore, it is obvious that in this case of n_oc>n_ic, the present coupled-mode analysis is greatly simplified compared with the conventional expansion in terms of the guided core mode and radiation modes.

Case 2 n_oc is equal to n_ic

In the case of an index-matched surround, as known from experiments, the transmission spectrum is broad and flat [17,18]. It indicates more leaky modes are required to take into account. If choosing a few leaky modes nearly satisfying phase matching condition from those denoted by stars in Fig. 7, there will be fluctuant resonances in the spectrum, as shown by the dotted line in Fig. 10(d). At certain wavelength range, we choose 11 outer-cladding leaky modes from those obtained by enlarging the radius of the outer-cladding to at most 500.0μm, then convergent transmission spectrum is obtained, as shown by the solid line in Fig. 10(d), which agree very well with the results obtained in [19]. Obviously, the reduced two-mode coupling model will not be applicable.

It needs to point out, when enlarging r_oc, the radial order from which the coupling strength to the odd and even one becomes comparable also greatly increases, thus the even modes can still be neglected to simplify the simulations.

In summary, in the case of n_oc>n_ic, the radiation modes in couplings are well approximated by at most 3 inner-cladding leaky modes. It is thus suggested that a moderate PML with which inner-cladding leaky modes can be convergent is enough for mode solving, such as R = 10⁻¹² as seen in Fig. 4, while an even stronger PML will induce much more useless outer-cladding leaky modes and PML modes, and then unnecessary complication in the whole analysis. In the case of n_oc = n_ic, the convergent spectra owe to the contribution of about 11 radiation-mode-like and nearly phase-matching outer-cladding modes obtained by using an enlarged r_oc instead of a stronger PML with a less R_PML. It is thus evident the establishing of equivalence relations between guided-mode-like inner- or radiation-mode-like outer-cladding leaky modes and radiation modes for different cases is also helpful for suitably setting PML parameters of the equivalent waveguide model.

To further validate the equivalence principle of quasi leaky mode approximations and the implementation on unified coupled mode analysis presented above, we also take a brief look at the simulated transmission spectra of FBGs with Λ = 533.66nm, Δn = 9.0 × 10⁻⁴, and L = 5.0mm. The Bragg resonant wavelength between two contra-propagating core modes is designed at λ_B = 1.55μm.

Figure 11(a) is for the case that the refractive index of the surround is a little lower than that of the cladding. The couplings of the core mode with the contra-propagating core mode, guided cladding modes are also simulated here to validate the unified coupled-mode analysis. In the wavelength range of 1.5435μm<λ<1.5490μm, though there are 13 odd order cladding modes respectively satisfying the phase matching conditions with the core mode, the resonant dips corresponding to the couplings with the lower order cladding modes are submerged in the side band of the Bragg resonance due to the limited coupling strengths. In the range of λ<1.5435μm, the couplings to inner-cladding leaky modes result in fringes in the transmission spectrum. These fringes are shallower than the resonant dips caused by the couplings to the cladding modes. Figure 11(b) corresponds to the case of FBG with an index-matched surround. Similar to the treatment for LPG in the same case, the smooth transmission spectrum is obtained by considering more outer-cladding leaky modes which are solved with an enlarged r_oc and distributed more concentrated. In this case, when r_oc is increased to 200.0μm, well convergent results are obtained with 11 outer-cladding leaky modes taken into mode expansion, as seen in Fig. 11(b). Figure 11(c) and 11(d) show the cases that the refractive indices of the surrounds are higher than those of the claddings. Seen from the transmission spectra, the dips correspond to the couplings between the guided core mode and the inner-cladding leaky modes. The variation trend of a certain dip with the change of n_oc is similar to that in LPGs of the same case.

 

Fig. 11 Simulated transmission spectra of FBGs with different surrounding indices.

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The simulation results for FBGs also agree very well with both the experimental [15,16] and theoretical results obtained by using coupled-mode analysis with the expansion of the guided core mode and the radiation modes achieved by direct integral [20]. Compared with the treatments in LPG, 2 more modes are usually taken into mode expansion when the space between resonances is narrower. Meanwhile, at the short wavelength range, couplings with the even complex modes are not negligible, since the order of the leaky modes dominant in couplings is very high and the coupling strength to the even modes is comparable with that to the odd modes.

The applicability and accuracy of the reduced coupled-mode model are also examined. Figure 11(b) and 11(c) give comparisons of the simulation results obtained with different number of complex modes included in the mode expansion in the case of n_oc = n_ic and n_oc>n_ic, respectively. The reduced two-mode model may be an acceptable approximation in the case of n_oc>n_ic, however, in the case of n_oc = n_ic, at least 9 leaky modes need to be included to approximate radiation modes.

6. Conclusion

The principle of quasi leaky mode approximations is clearly identified and successfully implemented by applying a unified coupled-mode analysis to analyze LPGs and FBGs with different surrounding indices. With the present approach, traditional problems of couplings between guided modes are analyzed without additional difficulties; while with the new insight, the new problems of couplings to radiation modes are conveniently treated in the unified fashion. The principle and the implementation are quite flexible and well adapted to general applications. The novel findings pave the way for effectively applying the unified coupled-mode analysis to the couplings with radiation modes which are difficult to be solved in complicated structures. Also, the new results reported here shed more light on the salient features of the unified coupled mode analysis as a powerful method to general optical waveguide problems, especially when the couplings to radiation modes are significant.