## Pseudo-spectral analysis of radially-diagonalized Maxwell’s equations in cylindrical co-ordinates

Optics Express, Vol. 11, Issue 23, pp. 3048-3062 (2003)

http://dx.doi.org/10.1364/OE.11.003048

Acrobat PDF (493 KB)

### Abstract

We present a robust and accuracy enhanced method for analyzing the propagation behavior of EM waves in *z*-periodic structures in (*r,ϕ,z*)-cylindrical co-ordinates. A cylindrical disk, characterized by the radius a and the periodicity length *L*_{z}
, defines the fundamental cell in our problem. The permittivity of the dielectric inside this cell is characterized by an arbitrary, single-valued function *ε*(*r,ϕ,z*) of all three spatial co-ordinates. We consider both open and closed boundary problems. Irrespective of the type of the boundary conditions on the surface *r*=*a*, our method requires the discretization of the fields in the interior of the disk only. Inside the disk volume, we expand the fields in terms of planewaves on discrete cylindrical surfaces *r*_{i}
=*i*Δ, with Δ being the discretization step length. The fields on the nested surfaces *r*_{i}
=*i*Δ in the interior of the simulation domain are interrelated by the application of a simple, yet, powerful finite difference scheme. In free space outside the disk, the fields are expanded in terms of the closed-form eigensolutions of the Maxwell’s equations in cylindrical co-ordinates. In order to uniquely determine the involved unknown coefficients, the solutions in the interior- and exterior domains are matched on the disk’s bounding surface. Our formulation utilizes a radially-diagonalized form of Maxwell’s equations, and merely involves four (out of the six) field components. It is demonstrated that our formulation is perfectly suited, but by no means limited, to cylindrical symmetric problems.

© 2003 Optical Society of America

## 1. Introduction

2. J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. **21**, 1547–1549 (1996). [CrossRef] [PubMed]

4. C.T Chan, Q. L. Yu, and K. M. Ho, “Order-N spectral method for electromagnetic waves,” Phys. Rev. B **51**, 16635–16642 (1995). [CrossRef]

5. Z.-Y. Li and L.-L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E **67**, 046607 (2003). [CrossRef]

6. Elson J. Merle and P. Tran, “Dispersion in photonic media and diffraction from gratings: a different modal expansion for the R-matrix propagation technique,” J. Opt. Soc. Am. A **12**, 1765–1771 (1995). [CrossRef]

7. L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A , **10**, 2581–2591 (1993). [CrossRef]

8. S. G. Johnson and J.D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173–190 (2001) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173. [CrossRef] [PubMed]

9. P. Bienstman, *Rigorous and Efficient Modelling of Wavelength Scale Photonic Components*, doctoral disseration 2001, Ghent University, Belgium. Available online at http://photonics.intec.rug.ac.be/download/phd 104.pdf

10. Z.-Y. Li and K.-M. Ho, “Analytic modal solution to light propagation through layer-by-layer metallic photonic crystals,” Phys. Rev. B. **67**, 165104 (2003). [CrossRef]

*r*=

*a*and the axis

*r*=0. Instead, we include the field components on nested cylindrical surfaces

*r*

_{i}=

*i*Δ, with 0<

*r*

_{i}<

*a*, as explicit unknowns. In this manner we obtain sparse matrices, which can be solved, according to our extensive numerical experiments, with greater robustness and accuracy.

## 2. Outline of the method

*z*-periodic geometry can be handled straightforwardly. Needless to say that for cylindrical symmetric problems, and in particular for

*z*-independent structures, the number of basis functions required reduces substantially. For problems which are homogeneous in the

*z*-direction, only one harmonic function suffices to describe the propagating mode. For cylindrical symmetric problems, the eigenmodes are Bessel functions and the radial dependence can be described in terms of two appropriately chosen planewaves.

- Realistic boundary conditions can be implemented easily and efficiently.
- Iterative solvers are utilized which require modest computational resources.
- The wavenumber and the frequency are both input parameters allowing targeted and efficient computations.
- The method is applicable to eigenmode- as well as field excitation problems.
- In case of open boundary problems, the availability of the fields in closed-form in free space allows us to easily express the fields anywhere in space once the solution in the interior domain is known.

## 3. Diagonalization of the Maxwell’s equations

- Only the “transversal” field components, i.e., those involved in the interface conditions on cylindrical surfaces (
*r*=constant) enter our calculations. - The operator
*ℒ*involves spatial derivatives*∂/∂ϕ*and*∂/∂z*, and material parameters which are defined on*r*=constant surfaces. - The derivative of the transversal fields with respect to
*r*appears at the RHS only. - In view of (1) we realize that once we have determined the transversal field distribution on a certain cylindrical surface (
*r*=constant), we progressively can compute the transversal fields on any other neighboring cylindrical surface. - A further property of
*ℒ*is that repeated application of*ℒ*to (1) results in higher-order*r*-derivatives of the transversal fields. This can be shown fairly easily: Assume (2) is given:

*ℒ*=

*ℒ*(

*∂/∂ϕ,∂/∂z*). (For notational simplicity material parameters are not shown explicitly in (2).) Applying

*ℒ*to both sides of (2), writing

*ℒ*

^{(2)}for

*ℒℒ*at the LHS, and interchanging the order of

*ℒ*and

*∂/∂r*at the RHS, we obtain:

*∂Ψ⃗*/

*∂r*for

*ℒΨ⃗*at the RHS of (3) results in

*n*:

*r*constant (

*r*=

*r*

_{0}), we can formally build the derivatives of

*Ψ⃗*with respect to

*r*to any order desired. Thereby, we merely need to know the function

*Ψ⃗*(

*r*

_{0},

*ϕ,z*).

*Ψ⃗*(

*r*

_{0}+

*h,ϕ,z*) at

*r*=

*r*

_{0}we can write

*jωt*)):

**u**

_{r},

*r*

**u**

_{ϕ}, and

**u**

_{z}, respectively, denoting the unit vectors in the

*r*-,

*ϕ*-, and

*z*-directions. It is advantageous to use the variable substitution defined in (11) for all

*ϕ*-directional field components. This substitution not only simplifies the manipulations, but also enhances the accuracy of the computations.

**E**↔

**H**, and

*ε*↔ -

*µ*. Therefore, it is sufficient to perform our manipulations on (8) only; results associated with (9) can be obtained by the aforementioned replacements.

*r*, a moment’s reflection on Eqs. (12) reveals the following facts: (i) The first equation in (12) does not involve any

*r*-derivatives. Therefore, it will not appear in our diagonalized form explicitly. (ii) The second and third equations, respectively, involve the

*r*-derivative of

*ẽ*

_{ϕ}and

*e*

_{z}. This fact implies that these latter field components are the “essential” field components in our equations, and, therefore,

*e*

_{r}has to be eliminated. (iii) Due to the symmetry of our problem we conclude that our diagonalized form will ultimately also involve the variables

*h̃*

_{ϕ}and

*h*

_{z}, and that the component

*h*

_{r}has to be eliminated. The first equation in (12) serves to expressing the undesirable field component

*h*

_{r}in terms of the “essential” field components

*e*

_{z}and

*ẽ*

_{ϕ}. Likewise using the second curl equation

*e*

_{r}can be expressed in terms of the “essential” field components

*h*

_{z}and

*h̃*

_{ϕ}. More explicitly, we obtain:

_{ij}and

_{ij}in (16) are summarized below:

*ẽ*

_{ϕ},

*e*

_{z},

*h̃*

_{ϕ}, and

*h*

_{z}, have been determined, the remaining “normal” field components

*e*

_{r}and

*h*

_{r}can be solved from

*r*-derivatives. (In (25) the superscript

^{T}denotes transposition.)

## 4. Field expansions

*z,ϕ*)-cylindrical surfaces we expand the fields in terms of planewaves with general (

*r*-dependent) expansion coefficients

*f*

_{m,n}(

*r*)

*k*

_{m}=2

*πm/L*

_{z}+

*K*

_{z}with

*K*

_{z}denoting the Bloch phasing factor in the

*z*-direction. Details concerning the discretization in

*r*-variable will be explained below.

## 5. Discretization in the radial direction

*r*-derivative of the

*e*-field is given by the

*h*-field components alone, and vice versa. This property suggests the discretization of the electric- and magnetic fields on interlaced cylindrical surfaces. Stated more precisely, we discretize

*h*

_{z}and

*h*

_{ϕ}on surfaces

*r*

_{i}=

*i*Δ, and discretize

*e*

_{z}and

*e*

_{ϕ}on surfaces

*r*

_{i+1/2}=(

*i*+1/2)Δ, where Δ stands for the discretization step length in the radial direction. In the following we will refer to a specific

*r*-position by providing an appropriate multiplier of Δ.

**h**

_{z},

**h̃**

_{ϕ},

**e**

_{z}, and

**ẽ**

_{ϕ}comprise expansion coefficients of dimension

*M*×

*N*of the corresponding fields.

*r*-derivative of the Fourier coefficients on a given cylindrical surface. In this manner we can formulate the

*r*-dependence of the Fourier coefficients in terms of finite differences, exactly the same way as we would operate with real domain fields.

## 6. Boundary conditions

*r*=0), and two conditions on outer bounding surface (

*r*=

*a*).

### 6.1. Axis of the cylinder: r=0

*r*=

*i*Δ, including

*r*=0. Based on the variable substitution in (11) we require that

*r*=0 axis, is slightly more complicated. Naively, it would be tempting to set

**0**; however, the transversal electric fields in the discrete system are defined on the layers

*r*=(

*i*+1/2)Δ, rather than on the center axis of the simulation disk. However, setting

*a priori*unknown coefficients. Utilizing the orthogonality of the harmonic basis functions we obtain an adequate number of equations for the determination of the unknowns.

*l*:

*r*=Δ/2]. On this path, the involved scalar product becomes

*z*-axis. The integrand becomes

*h*

_{z}to be constant over the surface: 0<

*r*<Δ/2, 0<

*ϕ*<2

*π*,

*z*=

*z*

_{0}. Upon this assumption, the integral at the RHS assumes the form:

*δ*[

*n*] is the Kronecker delta symbol.

*ϕ*-dependent harmonics can be used to create a sufficient number of equations. We multiply both sides of (34) by exp(-

*jn̂ϕ*) for all

*N*harmonics in the basis and integrate over the periodicity interval with the length 2

*π*. These steps result in the following set of equations for various

*n̂*

*jk*

_{m̂}

*z*) for

*M*distinct planewaves, and, consecutively, integrating over the periodicity interval with the length

*L*

_{z}. This procedure results in the desired set of equations:

### 6.2. Free space boundary condition

*r*=

*a*we utilize the free space eigenvectors of the diagonalized form given in (16).

15. S. Liu, L. W. Li, M. S. Leong, and T. S. Yeo, “Theory of Gyroelectric Waveguides,” PIER , **29**, 231–259 (2000). [CrossRef]

*H*

_{n}(·) denotes Hankel function of order

*n*and the first kind, and

*ω*

^{2}

*εµ*-

*ℜ*(

*λ*

_{m})=0 and ℑ(

*λ*

_{m})>0.

*r*>

*a*) can be expanded in terms of the eigenvectors in (38) and (39):

*r,ϕ,z*) comprises the transversal field components in real space, with

*a*

_{m,n}and

*b*

_{m,n}being expansion coefficients. Note that the basis functions are harmonic in the variables

*ϕ*and

*z*, and thus orthogonal in (

*ϕ,z*)-domain. Utilizing this property, we can establish relationships between

*a*

_{m,n}and

*b*

_{m,n}and the expansion coefficients of the fields in the interior domain. There is, however, an important detail which we should be aware of while implementing these ideas: Inside the simulation disk, the magnetic- and electric fields are defined on interlaced rather than immediate neighboring surfaces.

*r*by the distance Δ/2 in the radial direction. Note that in our formulation this can be accomplished fairly easily, since the functional form of the eigensolutions in the

*r*-direction is known.

### 6.3. Closed boundary

## 7. Constructing the system matrix

*z*-corrugated fiber, it is instructive to collect all the occurring expansion coefficients into one vector, say,

**f**. Next we set up the system matrix

**M**, by evaluating (27) and (28) at individual (discrete) cylindrical surfaces in the simulation domain, and matching the solutions. We obtain the following homogeneous system

*ω*and

*K*

_{z}have been written explicitly in order to emphasize their role as input parameters.

**M**is very well structured. This fact is instrumental in numerical calculations: The equations resulting from (27) and (28) create two unity side diagonals, having opposite signs, and being offset from the main diagonal by ±2

*M*×

*N*elements, with

*M*and

*N*denoting the number of planewaves in the

*z*- and

*ϕ*-directions, respectively. Between these two diagonals the matrix operators

**A**

_{m,n}and

**B**

_{m,n}generate three block diagonals of the size

*M*×

*N*. The equations originating from free space eigenvectors can be written in a form which preserves this structure. Unfortunately, however, the equations resulting from the boundary conditions in the proximity of the center axis, do not have this desirable structure. Nonetheless, the matrix elements arising from these equations are sparse, and structured systematically in a specific way.

## 8. Solving the system of equations

### 8.1. Excitation problems

**f̂**numerically. While, real space derivatives can be effectively evaluated in Fourier domain, multiplications by functions, e.g.

*ε*can be preferably carried out in real domain. Therefore, we will Fourier and inverse Fourier transform the coefficient vector, in order to move back and forth between the two spaces whenever necessary. Direct discretization would create

*MN*by

*MN*complex-valued matrices and multiplications involving them would require

*O*((

*MN*)

^{3}) operations. In the proposed way, the matrices involved are all diagonal, and the resulting dominating factor

*O*(

*MN*ln(

*MN*)) stems from the Fast Fourier Transform. Proceeding along these lines, we can construct the matrix product piece by piece. The actual matrix

**M**is never constructed explicitly.

16. R. W. Freund, “A Transpose-Free Quasi-Minimal Residual Algorithm for Non-Hermitian Linear Systems,” SIAM Journal of Scientific Computing , **14**, 470–482 (1993). [CrossRef]

**M̂**, which corresponds to a simplified problem defined by

*∂*ε ^
/∂ϕ=

*∂*ε ^
/∂z≡0. On each cylindrical surface, the value of

*can be obtained by averaging the*ε ^

*ε*of the original problem over the corresponding cylindrical surface. The resulting simplified problem, leads to a sparse and well-structured matrix

**M̂**, the direct factorization of which is computationally affordable.

**M**, and a routine which computes

**M̂**

^{-1}

**g**for a given vector

**g**are available, the preconditioned system can be solved as described in [17

17. N. Nachtigal, R.W. Freund, and J. C. Reeb, “QMRPACK user’s guide,” ORNL Technical Report ORNL/TM-12807, August 1994, also available online at http://www.cs.utk.edu/~santa/homepage/12807.ps.gz

*r*>

*a*.

### 8.2. Eigenproblems

*ω,K*

_{z}), which make the system matrix

**M**singular. There are a variety of methods in our disposal for determining the singular points of a matrix. The common feature of most of these “traditional” methods is that they rely on the direct factorization of the system matrix. Here, we propose an alternative way, which is based on the field solution to a randomly, yet, suitably chosen excitation. To illuminate the details consider a (

*ω,K*

_{z})-point for which the matrix

**M**in (42) approaches a singularity. (For the sake of simplicity we limit ourselves to non-degenerate cases.) Then, one of the eigenvalues of

**M**approaches zero. Assume that

**y**is the corresponding eigenvector. Under these conditions,

**f**, the solution of (42) approaches

**y**, unless the algebraic scalar product

**y**

^{T}·

*ρ*is vanishingly small. At the same time, the

**L**

_{2}norm ∥

**f**∥

_{2}approaches infinity. This theorem can be easily verified if both the matrix product

**Mf**and the excitation

*ρ*are thought to be expanded in terms of the eigenvectors of

**M**, as is done in [18].

*ω*for a given

*K*

_{z}(or vice versa), solve fields in response to a random excitation, and use the solution norm as a measure for the singularity.

## 9. Numerical results

### 9.1. Bragg fiber

*z*-periodic Bragg fiber with the following properties: one unit cell is constructed by stacking two cylindrical elements on top of each other. In addition, a sector of 90 degrees has been removed from both elements as sketched in Fig. 1. The refractive indices of the elements are

*n*

_{1}=1.5 and

*n*

_{2}=2. The radius of the fiber is

*r*=0.5, the height of both pieces constituting the unit cell is

*d*=0.5 and the height of the unit cell is

*L*

_{z}=1. Outside the fiber is free space.

8. S. G. Johnson and J.D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173–190 (2001) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173. [CrossRef] [PubMed]

### 9.2. Bragg fiber with cylindrical symmetry

*ε*

_{r1}=2.25 and

*ε*

_{r2}=4. The radius of the fiber is

*r*=0.5 and the heights of both disks are

*h*

_{1}=

*h*

_{2}=0.5. The medium outside the fiber is free space. Due to the cylindrical symmetry of the problem, we only use 4 planewaves in the

*ϕ*-direction. This allows us to expand the modes in terms of the Bessel functions up to the first order. We solve one mode only with

*ω*=0.4×2

*πc*/

*L*

_{z}and compare the results with those obtained from the eigenmode expansion technique [9

9. P. Bienstman, *Rigorous and Efficient Modelling of Wavelength Scale Photonic Components*, doctoral disseration 2001, Ghent University, Belgium. Available online at http://photonics.intec.rug.ac.be/download/phd 104.pdf

*k*

_{z}=0.4673591×2

*π*/

*L*

_{z}, while for the eigenmode expansion method being

*k*

_{z}=0.4673595×2

*π*/

*L*

_{z}, with the relative difference compared with the latter being 8.068×10

^{-7}. It should be pointed out that due to the intrinsic differences between the two methods, the discretization sizes are not directly comparable.

### 9.3. Fields in a fiber with air holes

*r*

_{f}=1, the assumed relative dielectric constant is

*ε*

_{r}=4, and outside the fiber is free space. The centers of the air holes are located on a circle with

*r*

_{c}=0.5

*L*(for some length scale

*L*), and the radii of the air holes are

*r*

_{a}=0.1

*L*. There is no variation in the

*z*-direction. We compute the lowest-order eigenmode for the frequency

*ω*=0.3×2

*πc/L*and find its propagation constant along the

*z*-axis to be

*k*

_{z}=0.33035×2

*π/L*. The eigenmode is found by the alternative method outlined above: solve the field pattern associated with a random excitation and seek the maximum of the solution norm. In this way the eigenmode field pattern is obtained immediately by inverse Fourier transform of the resulting set of expansion coefficients. Since the simulation is two dimensional we have used a single planewave in the

*z*-direction only. In the

*r*-direction, we discretized the fields on 64 concentric cylindrical surfaces, while in the

*ϕ*-direction we have used 64 planewaves. The resulting

*z*-directional fields are shown in Fig. 4.

## 10. Conclusion

## Acknowledgments

## References and links

1. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, “Photonic Crystals: Molding the Flow of Light,” Princeton, September 1995. |

2. | J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. |

3. | A. Cucinotta, A. Selleri, L. Vincetti, and M. Zoboli, “Holey Fiber Analysis Through the Finite-Element Method,” IEEE Photon. Technol. Lett. |

4. | C.T Chan, Q. L. Yu, and K. M. Ho, “Order-N spectral method for electromagnetic waves,” Phys. Rev. B |

5. | Z.-Y. Li and L.-L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E |

6. | Elson J. Merle and P. Tran, “Dispersion in photonic media and diffraction from gratings: a different modal expansion for the R-matrix propagation technique,” J. Opt. Soc. Am. A |

7. | L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A , |

8. | S. G. Johnson and J.D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

9. | P. Bienstman, |

10. | Z.-Y. Li and K.-M. Ho, “Analytic modal solution to light propagation through layer-by-layer metallic photonic crystals,” Phys. Rev. B. |

11. | K. Varis and A. R. Baghai-Wadji, “Hybrid Planewave/Finite Difference Transfer Method for Solving Photonic Crystals in Finite Thickness Slabs,” EDMO, November 15–16, 2001, Vienna, Austria, pp. 161–166. |

12. | K. Varis and A.R. Baghai-Wadhi, “Z-diagonalized Planewave/FD Approach for Analyzing TE Polarized Waves in 2D Photonic Crystals,” ACES, March 24–28, 2003, Monterey, CA, USA. |

13. | K. Varis and A. R. Baghai-Wadji, “A Novel 2D Pseudo-Spectral Approach of Photonic Crystal Slabs,” ACES Special Issue, (submitted). |

14. | K. Varis and A.R. Baghai-Wadji, “A Novel 3D Pseudo-spectral Analysis of Photonic Crystal Slabs,” ACES Special Issue, (submitted). |

15. | S. Liu, L. W. Li, M. S. Leong, and T. S. Yeo, “Theory of Gyroelectric Waveguides,” PIER , |

16. | R. W. Freund, “A Transpose-Free Quasi-Minimal Residual Algorithm for Non-Hermitian Linear Systems,” SIAM Journal of Scientific Computing , |

17. | N. Nachtigal, R.W. Freund, and J. C. Reeb, “QMRPACK user’s guide,” ORNL Technical Report ORNL/TM-12807, August 1994, also available online at http://www.cs.utk.edu/~santa/homepage/12807.ps.gz |

18. | W. H. Press, S. A. Teukolsky, W. T. Wetterling, and B. P. Flannery, |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(000.4430) General : Numerical approximation and analysis

**ToC Category:**

Research Papers

**History**

Original Manuscript: September 4, 2003

Revised Manuscript: October 27, 2003

Published: November 17, 2003

**Citation**

Karri Varis and A. Baghai-Wadji, "Pseudo-spectral Analysis of Radially-Diagonalized Maxwell's Equations in Cylindrical Co-ordinates," Opt. Express **11**, 3048-3062 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-23-3048

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### References

- Joannopoulos J. D., Meade R. D., Winn J. N., �??�??Photonic Crystals: Molding the Flow of Light,�??�?? Princeton, September 1995
- Knight J. C., Birks T. A., Russell P. St. J., Atkin D. M., �??�??All-silica single-mode optical fiber with photonic crystal cladding,�??�?? Opt. Lett. 21, 1547-1549 (1996). [CrossRef] [PubMed]
- Cucinotta A., Selleri A., Vincetti L., Zoboli M., �??�??Holey Fiber Analysis Through the Finite-Element Method,�??�?? IEEE Photon. Technol. Lett. 3, 147-149 (1991).
- Chan, C.T, Yu Q. L., Ho K. M., �??�??Order-N spectral method for electromagnetic waves,�??�?? Phys. Rev. B 51, 16635- 16642 (1995). [CrossRef]
- Li Z.-Y., Lin L.-L., �??�??Photonic band structures solved by a plane-wave-based transfer-matrix method,�??�?? Phys. Rev. E 67, 046607 (2003). [CrossRef]
- Merle Elson J., Tran P., �??�??Dispersion in photonic media and diffraction from gratings: a different modal expansion for the R-matrix propagation technique, �??�?? J. Opt. Soc. Am. A 12, 1765-1771 (1995). [CrossRef]
- Li L., �??�??Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,�??�?? J. Opt. Soc. Am. A, 10, 2581-2591 (1993). [CrossRef]
- Johnson S. G., Joannopoulos J.D., �??�??Block-iterative frequency-domain methods for Maxwell�??s equations in a planewave basis,�??�?? Opt. Express 8, 173-190 (2001)<a href=" http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173</a> [CrossRef] [PubMed]
- </a>Bienstman P., Rigorous and Efficient Modelling of Wavelength Scale Photonic Components, doctoral dissertation 2001, Ghent University, Belgium. Available online at <a href="http://photonics.intec.rug.ac.be/download/phd 104.pdf">http://photonics.intec.rug.ac.be/download/phd 104.pdf</a>
- Li Z.-Y., Ho K.-M., �??�??Analytic modal solution to light propagation through layer-by-layer metallic photonic crystals,�??�?? Phys. Rev. B. 67, 165104 (2003). [CrossRef]
- Varis K., Baghai-Wadji A. R., �??�??Hybrid Planewave/Finite Difference Transfer Method for Solving Photonic Crystals in Finite Thickness Slabs,�??�?? EDMO, November 15-16, 2001, Vienna, Austria, pp. 161-166
- Varis K., Baghai-Wadhi A.R., �??�??Z-diagonalized Planewave/FD Approach for Analyzing TE Polarized Waves in 2D Photonic Crystals,�??�?? ACES, March 24-28, 2003, Monterey, CA, USA
- Varis K., Baghai-Wadji A. R., �??�??A Novel 2D Pseudo-Spectral Approach of Photonic Crystal Slabs,�??�?? ACES Special Issue, (submitted).
- Varis K., Baghai-Wadji A.R., �??�??A Novel 3D Pseudo-spectral Analysis of Photonic Crystal Slabs,�??�?? ACES Special Issue, (submitted).
- Liu S., Li L. W., Leong M. S., Yeo T. S., �??�??Theory of Gyroelectric Waveguides,�??�?? PIER, 29, 231-259 (2000). [CrossRef]
- Freund R. W., �??�??A Transpose-Free Quasi-Minimal Residual Algorithm for Non-Hermitian Linear Systems,�??�?? SIAM Journal of Scientific Computing, 14, 470-482 (1993). [CrossRef]
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- Press W. H., Teukolsky S. A., Wetterling W. T., Flannery B. P., Numerical recipes in C: the Art of Scientific Computing, Cambridge University Press, Cambridge, pp. 493-495

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