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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 20 — Sep. 27, 2010
  • pp: 20595–20609
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New insight into quasi leaky mode approximations for unified coupled-mode analysis

Li Yang, Lin-Lin Xue, Yu-Chun Lu, and Wei-Ping Huang  »View Author Affiliations


Optics Express, Vol. 18, Issue 20, pp. 20595-20609 (2010)
http://dx.doi.org/10.1364/OE.18.020595


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Abstract

To facilitate the analysis of radiation mode couplings, quasi leaky mode approximations were utilized in coupled-mode analysis. The key to effectively and accurately apply this approach is how to well approximate radiation modes by the quasi leaky modes in an equivalent closed waveguide model. In this paper, the principle, applicability and accuracy of the approximations are demonstrated, and the detailed implementation is also suggested by applying a unified coupled-mode analysis to fiber gratings. First of all, based on a thorough study on the characteristics of the complex modes, for the first time, quasi leaky modes are classified into guided-mode-like inner-cladding and radiation-mode-like outer-cladding leaky modes so as to explicitly establish equivalence relationships between the discrete leaky modes and the continuous radiation modes. With this new insight, the whole analysis process especially for some of the practically tricky issues such as the criteria for developing the proper equivalent waveguide model and the subsequent mode expansion basis are better understood and easier to be dealt with for different problems where radiation modes come into play. Moreover, as essential preconditions to extend the conventional coupled mode analysis to the present unified one, the couplings between the guided core mode and a leaky mode are studied in a systematic and consistent manner. An intuitive and then a deep understanding on the roles of complex modes on mode coupling and power exchanging are thus gained for further simulations. Lastly, the transmission spectra of fiber gratings with different surrounding indices are simulated. The simulated results agree well with those obtained theoretically and experimentally in the literatures, which strongly validate the principle of quasi leaky mode approximations and its implementation on the unified coupled-mode analysis expounded in this paper.

© 2010 OSA

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

1. J. R. Qian, “Coupled-mode theory,” Textbook for Graduation in University of Science and Technology of China.

].

Though the idea was demonstrated in [11

11. W. P. Huang and J. W. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express 17(21), 19134–19152 (2009). [CrossRef]

] for slab waveguides and in [12

12. Y. C. Lu, W. P. Huang, and S. S. Jian, “Full vector complex coupled mode theory for tilted fiber gratings,” Opt. Express 18(2), 713–726 (2010). [CrossRef] [PubMed]

14

14. L. Yang and L. L. Xue, “Simplified treatment for radiation mode in coupled-mode analysis,” ICMMT 2010, Chengdu, May, 2010.

] 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

13. N. Song, J. W. Mu, and W. P. Huang, “Application of the complex coupled-mode theory to optical fiber grating structures,” J. Lightwave Technol. 28(5), 761–767 (2010). [CrossRef]

]. 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

11. W. P. Huang and J. W. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express 17(21), 19134–19152 (2009). [CrossRef]

,13

13. N. Song, J. W. Mu, and W. P. Huang, “Application of the complex coupled-mode theory to optical fiber grating structures,” J. Lightwave Technol. 28(5), 761–767 (2010). [CrossRef]

].

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)
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.
. It is written in the core of a conventional single-mode fiber and surrounded by material with a refractive index of noc. 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 nco and nic, rco and ric, respectively. The periodically modulated refractive index in the core along the longitudinal direction of the grating is described as usual,
n(z)=nco+Δn[1+νcos(2πΛz)],
(1)
where Λ is the grating period, Δn is the index change spatially averaged over a period, and ν is the fringe visibility of the index change.

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 rco = 2.5μm, the inner-cladding radius ric = 62.5μm, the core refractive index nco = 1.458 and the inner-cladding refractive index nic = 1.45, which are chosen to be same as those in [5

5. Y. C. Lu, L. Yang, W. P. Huang, and S. S. Jian, “Unified approach for coupling to cladding and radiation modes in fiber Bragg and long-period gratings,” J. Lightwave Technol. 27(11), 1461–1468 (2009). [CrossRef]

] for comparison and all the following simulations will be based on. The refractive index of the outer-cladding noc will be chosen to be higher than or equal to nic for different cases. The main model parameters of PML are respectively roc = 80.0μm, dPML = 10.0μm, and RPML = 10−12, which will be used in the following simulations if no special declarations. The working wavelength is set to be 1.55μm.

Case 1 noc is higher than nic

Mode Spectrum of the waveguide with noc = 1.46 are shown in Fig. 2
Fig. 2 Mode spectrum of the closed waveguide with an outer-cladding of noc = 1.46.
, 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.

In Fig. 2, the guided fundamental HE11 mode is denoted by a diamond. Its amplitude and phase are shown in Fig. 3(a)
Fig. 3 Amplitudes and phases of the radial components of the electric fields Er, for the complex modes with an outer-cladding of noc = 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.
. 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.

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

5. Y. C. Lu, L. Yang, W. P. Huang, and S. S. Jian, “Unified approach for coupling to cladding and radiation modes in fiber Bragg and long-period gratings,” J. Lightwave Technol. 27(11), 1461–1468 (2009). [CrossRef]

,6

6. H. Derudder, F. Olyslager, D. De Zutter, and S. Van den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers,” IEEE Trans. Antenn. Propag. 49(2), 185–195 (2001). [CrossRef]

].

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 RPML = 1 to RPML = 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 RPML, as illustrated in Fig. 4
Fig. 4 Variations of the effective refractive indices of the inner-cladding leaky modes in the closed waveguide with an outer-cladding of noc = 1.46 with the change of the absorption coefficient of PML, RPML.
. It is observed that the results become convergent when RPML = 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.

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 RPML = 10−12 and RPML = 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).

Case 2 noc is equal to nic

Figure 6
Fig. 6 Mode spectrum of the closed waveguide with an outer-cladding of noc = 1.45.
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, roc = 80.0μm, there are only a few leaky modes denoted by stars in the mode spectrum, as shown in Fig. 7
Fig. 7 Variations of effective refractive indices of the closed waveguide with an outer-cladding of noc = 1.45 with the change of the starting position of PML roc.
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 roc = 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.

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,
12(etm×htn*)z^dA=δmn,
(2)
which is derived from power conservation law. Though it is approximately valid between the core mode and the high order complex modes, it is not valid between the high order leaky modes, especially when they are with sufficiently large leakage losses [11

11. W. P. Huang and J. W. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express 17(21), 19134–19152 (2009). [CrossRef]

,14

14. L. Yang and L. L. Xue, “Simplified treatment for radiation mode in coupled-mode analysis,” ICMMT 2010, Chengdu, May, 2010.

]. For the derivation of unified coupled-mode equations based on the expansion of complex modes, we use a non-conjugate-formed orthogonal relationship derived from the reciprocal principle which is satisfied in general media characterized by symmetrical tensor permeability and permittivity [24

24. R. E. Collin, Field Theory of Guided Waves, (A John Wiley & Sons. Inc., Publication, IEEE Press 1990), Chapter 1.

],
12(etm×htn)z^dA=0         mn,
(3a)
while for m = n, we defined the integral coefficient for the m-th mode as
12(etm×htm)z^dA=Nm,
(3b)
which will be used in the unified coupled-mode equations.

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

11. W. P. Huang and J. W. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express 17(21), 19134–19152 (2009). [CrossRef]

] is derived with an expansion of the complex modes in the equivalent waveguide as follows [11

11. W. P. Huang and J. W. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express 17(21), 19134–19152 (2009). [CrossRef]

],

damdz+jβmam=jn=1N+Mκmnanjn=1N+Mχmnbn,dbmdzjβmbm=+jn=1N+Mχmnan+jn=1N+Mκmnbn.
(4)

Here the expansion basis includes N guided modes and M leaky modes. an and bn are respectively the expansion coefficients or the complex amplitudes of the fields of the forward and backward propagating modes. βm is the propagation constant of the m-th complex mode. The coupling coefficients between co-propagating modes and contra-propagating modes κmn and χmn are respectively expressed as follows,
κmn=ωε04Nm(n2n02)(etmetnn02n2ezmezn)ds,χmn=ωε04Nm(n2n02)(etmetn+n02n2ezmezn)ds,
(5)
where e tn and h tn, e zn and h zn are the transverse and longitudinal mode functions of complex modes in the reference waveguide, respectively. n0 and n are respectively index profiles of the reference waveguide and the structure under consideration which is fiber grating in the following discussions. Nm is defined in Eq. (3b).

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

an=Anexp(jβnz),bn=Bnexp(jβnz).
(6)

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

dAmdz=jn=1N+MκmnAnexp[j(βnβm)z]jn=1N+Mχmnexp[+j(βn+βm)z],dBmdz=+jn=1N+MκmnBnexp[+j(βnβm)z]+jn=1N+Mχmnexp[j(βn+βm)z].
(7)

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

κmn=ωε0ncoΔn2Nm[1+νcos(2πΛz)]core(etmetnn02n2ezmezn)ds,χmn=ωε0ncoΔn2Nm[1+νcos(2πΛz)]core(etmetn+n02n2ezmezn)ds.
(8)

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

16. T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14(8), 1760–1773 (1997) Errata: T. Erdogan, J. Opt. Soc. Am. A, 17(11), 2113–2113, (2000). Errata: Yinquan Yuan, J. Opt. Soc. Am. A, 26(10), 2199–2201, (2009). [CrossRef]

]. Then the coupling coefficients in Eq. (8) can be simplified as follows,

κmn=χmn=ωε0ncoΔn2Nmcoreetmetnds[1+νcos(2πΛz)]              =Kmn[1+νcos(2πΛz)].
(9)

For LPGs, the coupled-mode equations describing co-propagating interactions are further expressed as follows,
dA1dz=jK11A1jn=2N+Mν2K1nAnexp(j2Δβnz),dAndz=jν2Kn1A1exp(j2Δβnz),n=2,3,N+M,
(10)
where Δβn is the phase detuning factor and defined as

Δβn=12(β1βn2πΛ).
(11)

For FBGs, the coupled-mode equations describing counter-propagating interactions are further expressed as follows,
dA1dz=jK11A1jn=1N+Mν2K1nBnexp(j2Δβnz),dB1dz=jK11B1+jν2K11A1exp(j2Δβ1z),dBndz=jν2Kn1A1exp(j2Δβnz),n=2,3,N+M,
(12)
where phase detuning factor Δβn is defined as

Δβn=12(β1+βn2πΛ).
(13)

Re{δ}=Re{Δβn+K11c}=0.
(14)

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

16. T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14(8), 1760–1773 (1997) Errata: T. Erdogan, J. Opt. Soc. Am. A, 17(11), 2113–2113, (2000). Errata: Yinquan Yuan, J. Opt. Soc. Am. A, 26(10), 2199–2201, (2009). [CrossRef]

]. Normally in practice, the reference waveguide are chosen so as to the self coupling coefficient for the guided modes K11 will be zero [25

25. W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. 13(4), 134–141 (1977). [CrossRef]

].

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

5. Y. C. Lu, L. Yang, W. P. Huang, and S. S. Jian, “Unified approach for coupling to cladding and radiation modes in fiber Bragg and long-period gratings,” J. Lightwave Technol. 27(11), 1461–1468 (2009). [CrossRef]

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

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 noc is higher than nic

The simulated transmission spectra are shown in Fig. 10(a)
Fig. 10 Simulated transmission spectra of LPGs with different surrounding indices.
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

17. H. Patrick, A. Kersey, and F. Bucholtz, “Analysis of the response of long period fiber gratings to external index of refraction,” J. Lightwave Technol. 16(9), 1606–1612 (1998). [CrossRef]

] and theoretically by using conventional coupled-mode analysis with a direct integral of the radiation modes [19

19. Y. Koyamada, “Numerical analysis of core-mode to radiation-mode coupling in long-period fiber gratings,” IEEE Photon. Technol. Lett. 13(4), 308–310 (2001). [CrossRef]

]. 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

5. Y. C. Lu, L. Yang, W. P. Huang, and S. S. Jian, “Unified approach for coupling to cladding and radiation modes in fiber Bragg and long-period gratings,” J. Lightwave Technol. 27(11), 1461–1468 (2009). [CrossRef]

] and denoted by the scattered dots. They exactly agree with each other, which indicates the present algorithm is correctly implemented.

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

11. W. P. Huang and J. W. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express 17(21), 19134–19152 (2009). [CrossRef]

] and [13

13. N. Song, J. W. Mu, and W. P. Huang, “Application of the complex coupled-mode theory to optical fiber grating structures,” J. Lightwave Technol. 28(5), 761–767 (2010). [CrossRef]

]. 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 noc = 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 noc>nic, 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 noc is equal to nic

In the case of an index-matched surround, as known from experiments, the transmission spectrum is broad and flat [17

17. H. Patrick, A. Kersey, and F. Bucholtz, “Analysis of the response of long period fiber gratings to external index of refraction,” J. Lightwave Technol. 16(9), 1606–1612 (1998). [CrossRef]

,18

18. D. B. Stegall and T. Erdogan, “Leaky cladding mode propagation in long-period fiber grating devices,” IEEE Photon. Technol. Lett. 11(3), 343–345 (1999). [CrossRef]

]. 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

19. Y. Koyamada, “Numerical analysis of core-mode to radiation-mode coupling in long-period fiber gratings,” IEEE Photon. Technol. Lett. 13(4), 308–310 (2001). [CrossRef]

]. Obviously, the reduced two-mode coupling model will not be applicable.

It needs to point out, when enlarging roc, 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 noc>nic, 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 noc = nic, 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 roc instead of a stronger PML with a less RPML. 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 λB = 1.55μm.

Figure 11(a)
Fig. 11 Simulated transmission spectra of FBGs with different surrounding indices.
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 roc and distributed more concentrated. In this case, when roc 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 noc is similar to that in LPGs of the same case.

The simulation results for FBGs also agree very well with both the experimental [15

15. T. Erdogan, “Fiber gratings spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997). [CrossRef]

,16

16. T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14(8), 1760–1773 (1997) Errata: T. Erdogan, J. Opt. Soc. Am. A, 17(11), 2113–2113, (2000). Errata: Yinquan Yuan, J. Opt. Soc. Am. A, 26(10), 2199–2201, (2009). [CrossRef]

] 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

20. Y. Koyamada, “Analysis of core-mode to radiation-mode coupling in fiber Bragg gratings with finite cladding radius,” J. Lightwave Technol. 18(9), 1220–1225 (2000). [CrossRef]

]. 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 noc = nic and noc>nic, respectively. The reduced two-mode model may be an acceptable approximation in the case of noc>nic, however, in the case of noc = nic, 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.

Acknowledgments

Li Yang acknowledges the support from the National Natural Science Foundation of China (NNSFC) (under Grant No. 60807023) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. The authors acknowledge Prof. Jing-Ren Qian, Dr. Cheng-Lin Xu, and Ms. Jing Li for all the valuable discussions, and acknowledge the editor and reviewers for all the comments and suggestions helpful for improving our manuscript.

References and links

1.

J. R. Qian, “Coupled-mode theory,” Textbook for Graduation in University of Science and Technology of China.

2.

S. L. Lee, Y. Chung, L. A. Coldren, and N. Dagli, “On leaky mode approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31(10), 1790–1802 (1995). [CrossRef]

3.

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994). [CrossRef]

4.

F. Teixeira and W. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microw. Guid. Wave Lett. 7(11), 371–373 (1997). [CrossRef]

5.

Y. C. Lu, L. Yang, W. P. Huang, and S. S. Jian, “Unified approach for coupling to cladding and radiation modes in fiber Bragg and long-period gratings,” J. Lightwave Technol. 27(11), 1461–1468 (2009). [CrossRef]

6.

H. Derudder, F. Olyslager, D. De Zutter, and S. Van den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers,” IEEE Trans. Antenn. Propag. 49(2), 185–195 (2001). [CrossRef]

7.

H. Rogier and D. De Zutter, “Berenger and leaky modes in optical fibers terminated with a perfectly matched layer,” J. Lightwave Technol. 20(7), 1141–1148 (2002). [CrossRef]

8.

K. Jiang and W. P. Huang, “Finite-differene-based mode-matching method for 3-D waveguide structures under semivectorial approximation,” J. Lightwave Technol. 23(12), 4239–4248 (2005). [CrossRef]

9.

J. Mu and W. P. Huang, “Simulation of three-dimensional waveguide discontinuities by a full-vector mode-matching method based on finite-difference schemes,” Opt. Express 16(22), 18152–18163 (2008). [CrossRef] [PubMed]

10.

D. Vande Ginste, H. Rogier, and D. De Zutter, “Efficient computation of TM- and TE-polarized leaky modes in multilayered circular waveguides,” J. Lightwave Technol. 28(11), 1661–1669 (2010). [CrossRef]

11.

W. P. Huang and J. W. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express 17(21), 19134–19152 (2009). [CrossRef]

12.

Y. C. Lu, W. P. Huang, and S. S. Jian, “Full vector complex coupled mode theory for tilted fiber gratings,” Opt. Express 18(2), 713–726 (2010). [CrossRef] [PubMed]

13.

N. Song, J. W. Mu, and W. P. Huang, “Application of the complex coupled-mode theory to optical fiber grating structures,” J. Lightwave Technol. 28(5), 761–767 (2010). [CrossRef]

14.

L. Yang and L. L. Xue, “Simplified treatment for radiation mode in coupled-mode analysis,” ICMMT 2010, Chengdu, May, 2010.

15.

T. Erdogan, “Fiber gratings spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997). [CrossRef]

16.

T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14(8), 1760–1773 (1997) Errata: T. Erdogan, J. Opt. Soc. Am. A, 17(11), 2113–2113, (2000). Errata: Yinquan Yuan, J. Opt. Soc. Am. A, 26(10), 2199–2201, (2009). [CrossRef]

17.

H. Patrick, A. Kersey, and F. Bucholtz, “Analysis of the response of long period fiber gratings to external index of refraction,” J. Lightwave Technol. 16(9), 1606–1612 (1998). [CrossRef]

18.

D. B. Stegall and T. Erdogan, “Leaky cladding mode propagation in long-period fiber grating devices,” IEEE Photon. Technol. Lett. 11(3), 343–345 (1999). [CrossRef]

19.

Y. Koyamada, “Numerical analysis of core-mode to radiation-mode coupling in long-period fiber gratings,” IEEE Photon. Technol. Lett. 13(4), 308–310 (2001). [CrossRef]

20.

Y. Koyamada, “Analysis of core-mode to radiation-mode coupling in fiber Bragg gratings with finite cladding radius,” J. Lightwave Technol. 18(9), 1220–1225 (2000). [CrossRef]

21.

V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, “Chiral fiber gratings,” Science 305(5680), 74–75 (2004). [CrossRef] [PubMed]

22.

Y. Y. Shevchenko and J. Albert, “Plasmon resonances in gold-coated tilted fiber Bragg gratings,” Opt. Lett. 32(3), 211–213 (2007). [CrossRef] [PubMed]

23.

Y. C. Lu, L. Yang, W. P. Huang, and S. S. Jian, “Improved full-vector finite-difference complex mode solver for optical waveguides of circular symmetry,” J. Lightwave Technol. 26(13), 1868–1876 (2008). [CrossRef]

24.

R. E. Collin, Field Theory of Guided Waves, (A John Wiley & Sons. Inc., Publication, IEEE Press 1990), Chapter 1.

25.

W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. 13(4), 134–141 (1977). [CrossRef]

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(060.2310) Fiber optics and optical communications : Fiber optics
(230.7370) Optical devices : Waveguides
(350.2770) Other areas of optics : Gratings
(350.5610) Other areas of optics : Radiation

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: July 22, 2010
Revised Manuscript: September 5, 2010
Manuscript Accepted: September 7, 2010
Published: September 14, 2010

Citation
Li Yang, Lin-Lin Xue, Yu-Chun Lu, and Wei-Ping Huang, "New insight into quasi leaky mode approximations for unified coupled-mode analysis," Opt. Express 18, 20595-20609 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-20-20595


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References

  1. J. R. Qian, “Coupled-mode theory,” Textbook for Graduation in University of Science and Technology of China.
  2. S. L. Lee, Y. Chung, L. A. Coldren, and N. Dagli, “On Leaky Mode Approximations for modal expansion in multilayer open waveguides,” IEEE J. Quantum Electron. 31(10), 1790–1802 (1995). [CrossRef]
  3. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994). [CrossRef]
  4. F. Teixeira and W. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microw. Guid. Wave Lett. 7(11), 371–373 (1997). [CrossRef]
  5. Y. C. Lu, L. Yang, W. P. Huang, and S. S. Jian, “Unified approach for coupling to cladding and radiation modes in fiber Bragg and long-period gratings,” J. Lightwave Technol. 27(11), 1461–1468 (2009). [CrossRef]
  6. H. Derudder, F. Olyslager, D. De Zutter, and S. Van den Berghe, “Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers,” IEEE Trans. Antenn. Propag. 49(2), 185–195 (2001). [CrossRef]
  7. H. Rogier and D. De Zutter, “Berenger and leaky modes in optical fibers terminated with a perfectly matched layer,” J. Lightwave Technol. 20(7), 1141–1148 (2002). [CrossRef]
  8. K. Jiang and W. P. Huang, “Finite-differene-based mode-matching method for 3-D waveguide structures under semivectorial approximation,” J. Lightwave Technol. 23(12), 4239–4248 (2005). [CrossRef]
  9. J. Mu and W. P. Huang, “Simulation of three-dimensional waveguide discontinuities by a full-vector mode-matching method based on finite-difference schemes,” Opt. Express 16(22), 18152–18163 (2008). [CrossRef] [PubMed]
  10. D. Vande Ginste, H. Rogier, and D. De Zutter, “Efficient computation of TM- and TE-polarized leaky modes in multilayered circular waveguides,” J. Lightwave Technol. 28(11), 1661–1669 (2010). [CrossRef]
  11. W. P. Huang and J. W. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express 17(21), 19134–19152 (2009). [CrossRef]
  12. Y. C. Lu, W. P. Huang, and S. S. Jian, “Full vector complex coupled mode theory for tilted fiber gratings,” Opt. Express 18(2), 713–726 (2010). [CrossRef] [PubMed]
  13. N. Song, J. W. Mu, and W. P. Huang, “Application of the Complex Coupled-Mode Theory to Optical Fiber Grating Structures,” J. Lightwave Technol. 28(5), 761–767 (2010). [CrossRef]
  14. L. Yang and L. L. Xue, “Simplified Treatment for Radiation Mode in Coupled-mode Analysis”, ICMMT 2010, Chengdu, May, 2010.
  15. T. Erdogan, “Fiber gratings spectra,” J. Lightwave Technol. 15(8), 1277–1294 (1997). [CrossRef]
  16. T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14(8), 1760–1773 (1997) Errata: T. Erdogan, J. Opt. Soc. Am. A, 17(11), 2113–2113, (2000). Errata: Yinquan Yuan, J. Opt. Soc. Am. A, 26(10), 2199–2201, (2009). [CrossRef]
  17. H. Patrick, A. Kersey, and F. Bucholtz, “Analysis of the response of long period fiber gratings to external index of refraction,” J. Lightwave Technol. 16(9), 1606–1612 (1998). [CrossRef]
  18. D. B. Stegall and T. Erdogan, “Leaky cladding mode propagation in long-period fiber grating devices,” IEEE Photon. Technol. Lett. 11(3), 343–345 (1999). [CrossRef]
  19. Y. Koyamada, “Numerical analysis of core-mode to radiation-mode coupling in long-period fiber gratings,” IEEE Photon. Technol. Lett. 13(4), 308–310 (2001). [CrossRef]
  20. Y. Koyamada, “Analysis of core-mode to radiation-mode coupling in fiber Bragg gratings with finite cladding radius,” J. Lightwave Technol. 18(9), 1220–1225 (2000). [CrossRef]
  21. V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, “Chiral fiber gratings,” Science 305(5680), 74–75 (2004). [CrossRef] [PubMed]
  22. Y. Y. Shevchenko and J. Albert, “Plasmon resonances in gold-coated tilted fiber Bragg gratings,” Opt. Lett. 32(3), 211–213 (2007). [CrossRef] [PubMed]
  23. Y. C. Lu, L. Yang, W. P. Huang, and S. S. Jian, “Improved full-vector finite-difference complex mode solver for optical waveguides of circular symmetry,” J. Lightwave Technol. 26(13), 1868–1876 (2008). [CrossRef]
  24. R. E. Collin, Field Theory of Guided Waves, (A John Wiley & Sons. Inc., Publication, IEEE Press 1990), Chapter 1.
  25. W. Streifer, D. R. Scifres, and R. D. Burnham, “Coupled wave analysis of DFB and DBR lasers,” IEEE J. Quantum Electron. 13(4), 134–141 (1977). [CrossRef]

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