## Full vector complex coupled mode theory for tilted fiber gratings

Optics Express, Vol. 18, Issue 2, pp. 713-725 (2010)

http://dx.doi.org/10.1364/OE.18.000713

Acrobat PDF (1335 KB)

### Abstract

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

1. G. Laffont and P. Ferdinand, “Tilted short-period fibre-bragg-grating-induced coupling to cladding modes for accurate refractometry,” Meas. Sci. Technol. **12**(7), 765–770 (2001). [CrossRef]

9. K. Feder, P. Westbrook, J. Ging, P. Reyes, and G. Carver, “In-fiber spectrometer using tilted fiber gratings,” IEEE Photon. Technol. Lett. **15**(7), 933–935 (2003). [CrossRef]

10. R. Kashyap, R. Wyatt, and R. Campbell, “Wideband gain flattened erbium fibre amplifier using a photosensitive fibre blazed grating,” Electron. Lett. **29**(2), 154–156 (1993). [CrossRef]

11. K. Zhou, G. Simpson, X. Chen, L. Zhang, and I. Bennion, “High extinction ratio in-fiber polarizers based on 45° tilted fiber Bragg gratings,” Opt. Lett. **30**(11), 1285–1287 (2005). [CrossRef] [PubMed]

14. X. Chen, K. Zhou, L. Zhang, and I. Bennion, “In-fiber twist sensor based on a fiber bragg grating with 81° tilted structure,” IEEE Photon. Technol. Lett. **18**(24), 2596–2598 (2006). [CrossRef]

16. P. Ivanoff, D. C. Reyes, and P. S. Westbrook, “Tunable pdl of twisted-tilted fiber gratings,” IEEE Photon. Technol. Lett. **15**(6), 828–830 (2003). [CrossRef]

*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.

*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.

17. T. Erdogan and J. E. Sipe, “Tilted fiber phase gratings,” J. Opt. Soc. Am. A **13**(2), 296–313 (1996). [CrossRef]

23. M. Holmes, R. Kashyap, and R. Wyatt, “Physical properties of optical fiber sidetap grating filters: free-space model,” IEEE J. Sel. Top. Quantum Electron. **5**(5), 1353–1365 (1999). [CrossRef]

24. C. Juregui and J. M. Lpez-Higuera, “Near-field theoretical model of radiation from a uniform-tilted fiber-bragg grating,” Microw. Opt. Technol. Lett. **37**(2), 124–127 (2003). [CrossRef]

25. C. Juregui, A. Cobo, and J. M. Lpez-Higuera, “3d near-field model for uniform slanted fiber gratings,” Microw. Opt. Technol. Lett. **38**(5), 428–432 (2003). [CrossRef]

26. Y. Li, M. Froggatt, and T. Erdogan, “Volume current method for analysis of tilted fiber gratings,” J. Lightwave Technol. **19**(10), 1580–1591 (2001). [CrossRef]

17. T. Erdogan and J. E. Sipe, “Tilted fiber phase gratings,” J. Opt. Soc. Am. A **13**(2), 296–313 (1996). [CrossRef]

23. M. Holmes, R. Kashyap, and R. Wyatt, “Physical properties of optical fiber sidetap grating filters: free-space model,” IEEE J. Sel. Top. Quantum Electron. **5**(5), 1353–1365 (1999). [CrossRef]

20. Y. Li and T. G. Brown, “Radiation modes and tilted fiber gratings,” J. Opt. Soc. Am. B **23**(8), 1544–1555 (2006). [CrossRef]

*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

20. Y. Li and T. G. Brown, “Radiation modes and tilted fiber gratings,” J. Opt. Soc. Am. B **23**(8), 1544–1555 (2006). [CrossRef]

23. M. Holmes, R. Kashyap, and R. Wyatt, “Physical properties of optical fiber sidetap grating filters: free-space model,” IEEE J. Sel. Top. Quantum Electron. **5**(5), 1353–1365 (1999). [CrossRef]

*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

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

17. T. Erdogan and J. E. Sipe, “Tilted fiber phase gratings,” J. Opt. Soc. Am. A **13**(2), 296–313 (1996). [CrossRef]

20. Y. Li and T. G. Brown, “Radiation modes and tilted fiber gratings,” J. Opt. Soc. Am. B **23**(8), 1544–1555 (2006). [CrossRef]

30. 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]

31. 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]

*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

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

33. Y. Jeong, B. Lee, J. Nilsson, and D. Richardson, “A quasi-mode interpretation of radiation modes in long-period fiber gratings,” IEEE J. Quantum Electron. **39**(9), 1135–1142 (2003). [CrossRef]

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

36. 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]

34. 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]

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

**17**(21), 19134–19152 (2009). [CrossRef]

## 2. Mode couplings in tilted fiber gratings

*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.

## 3 Coupled mode formulation

### 3.1 General coupled mode equations for tilted fiber gratings

*m*-th p-mode the mode fields can be written as

*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

**e**

*,*

^{p}_{tm}**h**

*⟩ = 0 when*

^{q}_{tn}*m*≠

*n*or

*p*≠

*q*, ⟨

**e,h**⟩ = 1/2∫∫(

**e**×

**h**⟩ · ž

*dS*), we can derive the amplitude equation for

*a*(

^{p}_{m}*z*) as

*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); Δ

*n*(

*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. Δ

*n*≪

*n̄*), which is valid for fiber gratings. The index variation in the fiber core is given by [17

**13**(2), 296–313 (1996). [CrossRef]

*x*=

*r*cos(ϕ); and

*σ*(

*z*) =

*σ̄*(

*z*cos

*θ*), χ(

*z*) =

*χ̄*(

*z*cos

*θ*),

*ϕ*(

*z*) =

*ϕ̄*(

*z*cos

*ϕ*) 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*’ =

*z*cos(

*θ*) +

*x*sin(

*θ*) ≈

*z*cos(

*θ*).

_{p=±}” 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

*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

*u*(

*z*),

*u*(

_{n}*z*),

*v*(

*z*),

*v*(

_{n}*z*)

*n*≠ 1 in Eq. (15) and

*β̄*= Re(

_{n}*β*). 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

_{n}*u*(

*z*),

*u*(

_{n}*z*),

*v*(

*z*),

*v*(

_{n}*z*) are derived as

^{±}

_{n}~ 0 and all the modes in the field expansion are searched around

*β̄*.

_{n}^{±}

_{n}~ 0, the couplings in reflective and transmissive TFBGs are {

*u*} ↔ {

*v, v*} and {

_{n}*u*} ↔ {

*u*}, respectively. For reflective TFBGs

_{n}*u*≡ 0, the remaining equations for

_{n}*u*(

*z*),

*v*(

*z*),

*v*(

_{n}*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) =

*v*(

_{n}*L*/2) = 0 ; for transmissive TFBGs

*v*≡ 0 ,

*v*≡ 0, the remaining equations for

_{n}*u*(

*z*) and

*u*(

_{n}*z*) can be solved in the interval [-

*L*/2,+

*L*/2] with initial condition

*u*(-

*L*/2) = 1,

*u*(-

_{n}*L*/2) = 0, where

*L*is the grating length,

*N*is the number of modes used in the field expansion.

*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*),

*u*(

_{n}*z*),

*v*(

_{n}*z*), (

*n*= 2,3,⋯,

*N*) are solved, Eq. (14) and Eq. (15) are utilized to solve for

*a*(

_{n}*z*),

*b*(

_{n}*z*), (

*n*= 1,2,⋯,

*N*).

*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

**17**(21), 19134–19152 (2009). [CrossRef]

*a*(+

_{n}*L*/2) = 0 or

*b*(-

_{n}*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

*a*(+

_{n}*L*/2) = 0 and

*b*(-

_{n}*L*/2) = 0 . Thus Eq. (22) is widely appliable in the modelling of fiber grating applications.

## 4. Numerical results and discussions

*r*

^{co}= 4.1μm ,

*r*

_{cl}= 62.5

*μ*m ,

*r*

_{s}= 75

*μ*m,

*n*

_{cl}= 1.466, Δ = (

*n*

^{2}

_{co}-

*n*

^{2}

_{cl}) / (2

*n*

^{2}

_{co}) = 0.36%. The mesh size is Δ

*r*= 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

34. 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]

### 4.1 Bragg scattering

*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

**13**(2), 296–313 (1996). [CrossRef]

21. O. Xu, S. Lu, Y. Liu, B. Li, X. Dong, L. Pei, and S. Jian, “Analysis of spectral characteristics for reflective tilted fiber gratings of uniform periods,” Opt. Commun. **281**(15–16), 3990–3995 (2008). [CrossRef]

**13**(2), 296–313 (1996). [CrossRef]

21. O. Xu, S. Lu, Y. Liu, B. Li, X. Dong, L. Pei, and S. Jian, “Analysis of spectral characteristics for reflective tilted fiber gratings of uniform periods,” Opt. Commun. **281**(15–16), 3990–3995 (2008). [CrossRef]

### 4.2 Cladding and radiation mode couplings

*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.

**17**(21), 19134–19152 (2009). [CrossRef]

36. 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]

*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.

*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.

## 5. Conclusions

## Acknowledgements

## References and links

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2. | X. Chen, K. Zhou, L. Zhang, and I. Bennion, “Optical chemsensor based on etched tilted bragg grating structures in multimode fiber,” IEEE Photon. Technol. Lett. |

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9. | K. Feder, P. Westbrook, J. Ging, P. Reyes, and G. Carver, “In-fiber spectrometer using tilted fiber gratings,” IEEE Photon. Technol. Lett. |

10. | R. Kashyap, R. Wyatt, and R. Campbell, “Wideband gain flattened erbium fibre amplifier using a photosensitive fibre blazed grating,” Electron. Lett. |

11. | K. Zhou, G. Simpson, X. Chen, L. Zhang, and I. Bennion, “High extinction ratio in-fiber polarizers based on 45° tilted fiber Bragg gratings,” Opt. Lett. |

12. | T. Erdogan, T. A. Strasser, and P. S. Westbrook, “In-line polarimeter using blazed fiber gratings,” IEEE Photon. Technol. Lett. |

13. | J. Peupelmann, E. Krause, A. Bandemer, and C. Schaffer, “Fibre-polarimeter based on grating taps,” Electron. Lett. |

14. | X. Chen, K. Zhou, L. Zhang, and I. Bennion, “In-fiber twist sensor based on a fiber bragg grating with 81° tilted structure,” IEEE Photon. Technol. Lett. |

15. | S. Mihailov, R. Walker, T. Stocki, and D. Johnson, “Fabrication of a tilted fiber-grating polarization-dependent loss equalizer,” in Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides. Optical Society of America, 2001, p. BFD2. |

16. | P. Ivanoff, D. C. Reyes, and P. S. Westbrook, “Tunable pdl of twisted-tilted fiber gratings,” IEEE Photon. Technol. Lett. |

17. | T. Erdogan and J. E. Sipe, “Tilted fiber phase gratings,” J. Opt. Soc. Am. A |

18. | L. Dong, B. Ortega, and L. Reekie, “Coupling characteristics of cladding modes in tilted optical fiber bragg gratings,” Appl. Opt. |

19. | K. S. Lee and T. Erdogan, “Fiber mode coupling in transmissive and reflective tilted fiber gratings,” Appl. Opt. |

20. | Y. Li and T. G. Brown, “Radiation modes and tilted fiber gratings,” J. Opt. Soc. Am. B |

21. | O. Xu, S. Lu, Y. Liu, B. Li, X. Dong, L. Pei, and S. Jian, “Analysis of spectral characteristics for reflective tilted fiber gratings of uniform periods,” Opt. Commun. |

22. | S. Lu, O. Xu, S. Feng, and S. Jian, “Analysis of radiation-mode coupling in reflective and transmissive tilted fiber bragg gratings,” J. Opt. Soc. Am. A |

23. | M. Holmes, R. Kashyap, and R. Wyatt, “Physical properties of optical fiber sidetap grating filters: free-space model,” IEEE J. Sel. Top. Quantum Electron. |

24. | C. Juregui and J. M. Lpez-Higuera, “Near-field theoretical model of radiation from a uniform-tilted fiber-bragg grating,” Microw. Opt. Technol. Lett. |

25. | C. Juregui, A. Cobo, and J. M. Lpez-Higuera, “3d near-field model for uniform slanted fiber gratings,” Microw. Opt. Technol. Lett. |

26. | Y. Li, M. Froggatt, and T. Erdogan, “Volume current method for analysis of tilted fiber gratings,” J. Lightwave Technol. |

27. | Y. Li, S. Wielandy, G. E. Carver, P. I. Reyes, and P. S. Westbrook, “Scattering from nonuniform tilted fiber gratings,” Opt. Lett. |

28. | R. B. Walker, S. J. Mihailov, P. Lu, and D. Grobnic, “Shaping the radiation field of tilted fiber Bragg gratings,” J. Opt. Soc. Am. B |

29. | M. Holmes, R. Kashyap, R. Wyatt, and R. Smith, “Ultra narrow-band optical fibre sidetap filters,” in Proc. ECOC’98, vol. 1, Sep 1998, 137–138. |

30. | Y. Koyamada, “Analysis of core-mode to radiation-mode coupling in fiber Bragg gratings with finite cladding radius,” J. Lightwave Technol. |

31. | Y. Koyamada, “Numerical analysis of core-mode to radiation-mode coupling in long-period fiber gratings,” IEEE Photon. Technol. Lett. |

32. | D. Stegall and T. Erdogan, “Leaky cladding mode propagation in long-period fiber grating devices,” IEEE Photon. Technol. Lett. |

33. | Y. Jeong, B. Lee, J. Nilsson, and D. Richardson, “A quasi-mode interpretation of radiation modes in long-period fiber gratings,” IEEE J. Quantum Electron. |

34. | 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. |

35. | W.-P. Huang and J. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express |

36. | 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. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(060.0060) Fiber optics and optical communications : Fiber optics and optical communications

(060.2310) Fiber optics and optical communications : Fiber optics

(230.7370) Optical devices : Waveguides

(350.5610) Other areas of optics : Radiation

(060.3735) Fiber optics and optical communications : Fiber Bragg gratings

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: October 26, 2009

Revised Manuscript: December 9, 2009

Manuscript Accepted: December 9, 2009

Published: January 5, 2010

**Citation**

Yu-Chun Lu, Wei-Ping Huang, and Shui-Sheng Jian, "Full vector complex coupled mode theory for tilted fiber gratings," Opt. Express **18**, 713-725 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-2-713

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### References

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