Abstract

Computing the eigenmodes of a photonic crystal (PhC) slab waveguide is computationally expensive, since it leads to eigenvalue problems in three-dimensional domains that are large compared with the wavelength. In this paper, a procedure is developed to reduce the eigenvalue problem for PhC slab waveguides to a nonlinear problem defined on a small surface in the waveguide core. The reduction process is efficiently performed based on the so-called Dirichlet-to-Neumann maps of the unit cells. The nonlinear eigenvalue problem can be efficiently solved by standard root-finding methods, such as the secant method.

© 2011 Optical Society of America

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  1. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University Press, 2008).
  2. T. Baba, N. Fukaya, and Y. Yonekura, “Observation of light propagation in photonic crystal optical waveguides with bends,” Electron. Lett. 35, 654–655 (1999).
    [CrossRef]
  3. S. Y. Lin, E. Chow, S. G. Johnson, and J. D. Joannopoulos, “Demonstration of highly efficient waveguiding in a photonic crystal slab at the 1.5 μm wavelength,” Opt. Lett. 25, 1297–1299 (2000).
    [CrossRef]
  4. M. Loncar, D. Nedeljkovic, T. Doll, J. Vuckovic, A. Scherer, and T. P. Pearsall, “Waveguiding in planar photonic crystals,” Appl. Phys. Lett. 77, 1937–1939 (2000).
    [CrossRef]
  5. A. Chutinan and S. Noda, “Waveguides and waveguide bends in two-dimensional photonic crystal slabs,” Phys. Rev. B 62, 4488–4492 (2000).
    [CrossRef]
  6. S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
    [CrossRef]
  7. G. R. Hadley, “Out-of-plane losses of line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 14, 642–644 (2002).
    [CrossRef]
  8. P. Lalanne, “Electromagnetic analysis of photonic crystal waveguides operating above the light core,” IEEE J. Quantum Electron. 38, 800–804 (2002).
    [CrossRef]
  9. C. Sauvan, P. Lalanne, J. Rodier, J. Hugonin, and A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 15, 1243–1245 (2003).
    [CrossRef]
  10. Y. Huang, Y. Y. Lu, and S. Li, “Analyzing photonic crystal waveguides by Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. B 24, 2860–2867 (2007).
    [CrossRef]
  11. S. Li and Y. Y. Lu, “Efficient method for computing leaky modes in two-dimensional photonic crystal waveguides,” J. Lightwave Technol. 28, 978–983 (2010).
    [CrossRef]
  12. Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps,” J. Lightwave Technol. 24, 3448–3453 (2006).
    [CrossRef]
  13. L. Yuan and Y. Y. Lu, “Dirichlet-to-Neumann map method for analyzing hole arrays in a slab,” J. Opt. Soc. Am. B 27, 2568–2579 (2010).
    [CrossRef]
  14. J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  15. W. C. Chew and W. H. Weedon, “A 3-D perfectly matched medium from modified Maxwell’s equations with stretching coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
    [CrossRef]
  16. Y. Y. Lu, “Minimizing the discrete reflectivity of perfectly matched layers,” IEEE Photon. Technol. Lett. 18, 487–489(2006).
    [CrossRef]

2010 (2)

2007 (1)

2006 (2)

Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps,” J. Lightwave Technol. 24, 3448–3453 (2006).
[CrossRef]

Y. Y. Lu, “Minimizing the discrete reflectivity of perfectly matched layers,” IEEE Photon. Technol. Lett. 18, 487–489(2006).
[CrossRef]

2003 (1)

C. Sauvan, P. Lalanne, J. Rodier, J. Hugonin, and A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 15, 1243–1245 (2003).
[CrossRef]

2002 (2)

G. R. Hadley, “Out-of-plane losses of line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 14, 642–644 (2002).
[CrossRef]

P. Lalanne, “Electromagnetic analysis of photonic crystal waveguides operating above the light core,” IEEE J. Quantum Electron. 38, 800–804 (2002).
[CrossRef]

2000 (4)

M. Loncar, D. Nedeljkovic, T. Doll, J. Vuckovic, A. Scherer, and T. P. Pearsall, “Waveguiding in planar photonic crystals,” Appl. Phys. Lett. 77, 1937–1939 (2000).
[CrossRef]

A. Chutinan and S. Noda, “Waveguides and waveguide bends in two-dimensional photonic crystal slabs,” Phys. Rev. B 62, 4488–4492 (2000).
[CrossRef]

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

S. Y. Lin, E. Chow, S. G. Johnson, and J. D. Joannopoulos, “Demonstration of highly efficient waveguiding in a photonic crystal slab at the 1.5 μm wavelength,” Opt. Lett. 25, 1297–1299 (2000).
[CrossRef]

1999 (1)

T. Baba, N. Fukaya, and Y. Yonekura, “Observation of light propagation in photonic crystal optical waveguides with bends,” Electron. Lett. 35, 654–655 (1999).
[CrossRef]

1994 (2)

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

W. C. Chew and W. H. Weedon, “A 3-D perfectly matched medium from modified Maxwell’s equations with stretching coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Baba, T.

T. Baba, N. Fukaya, and Y. Yonekura, “Observation of light propagation in photonic crystal optical waveguides with bends,” Electron. Lett. 35, 654–655 (1999).
[CrossRef]

Berenger, J.

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

Chew, W. C.

W. C. Chew and W. H. Weedon, “A 3-D perfectly matched medium from modified Maxwell’s equations with stretching coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Chow, E.

Chutinan, A.

A. Chutinan and S. Noda, “Waveguides and waveguide bends in two-dimensional photonic crystal slabs,” Phys. Rev. B 62, 4488–4492 (2000).
[CrossRef]

Doll, T.

M. Loncar, D. Nedeljkovic, T. Doll, J. Vuckovic, A. Scherer, and T. P. Pearsall, “Waveguiding in planar photonic crystals,” Appl. Phys. Lett. 77, 1937–1939 (2000).
[CrossRef]

Fan, S.

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

Fukaya, N.

T. Baba, N. Fukaya, and Y. Yonekura, “Observation of light propagation in photonic crystal optical waveguides with bends,” Electron. Lett. 35, 654–655 (1999).
[CrossRef]

Hadley, G. R.

G. R. Hadley, “Out-of-plane losses of line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 14, 642–644 (2002).
[CrossRef]

Huang, Y.

Hugonin, J.

C. Sauvan, P. Lalanne, J. Rodier, J. Hugonin, and A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 15, 1243–1245 (2003).
[CrossRef]

Joannopoulos, J. D.

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

S. Y. Lin, E. Chow, S. G. Johnson, and J. D. Joannopoulos, “Demonstration of highly efficient waveguiding in a photonic crystal slab at the 1.5 μm wavelength,” Opt. Lett. 25, 1297–1299 (2000).
[CrossRef]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University Press, 2008).

Johnson, S. G.

S. Y. Lin, E. Chow, S. G. Johnson, and J. D. Joannopoulos, “Demonstration of highly efficient waveguiding in a photonic crystal slab at the 1.5 μm wavelength,” Opt. Lett. 25, 1297–1299 (2000).
[CrossRef]

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University Press, 2008).

Lalanne, P.

C. Sauvan, P. Lalanne, J. Rodier, J. Hugonin, and A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 15, 1243–1245 (2003).
[CrossRef]

P. Lalanne, “Electromagnetic analysis of photonic crystal waveguides operating above the light core,” IEEE J. Quantum Electron. 38, 800–804 (2002).
[CrossRef]

Li, S.

Lin, S. Y.

Loncar, M.

M. Loncar, D. Nedeljkovic, T. Doll, J. Vuckovic, A. Scherer, and T. P. Pearsall, “Waveguiding in planar photonic crystals,” Appl. Phys. Lett. 77, 1937–1939 (2000).
[CrossRef]

Lu, Y. Y.

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University Press, 2008).

Nedeljkovic, D.

M. Loncar, D. Nedeljkovic, T. Doll, J. Vuckovic, A. Scherer, and T. P. Pearsall, “Waveguiding in planar photonic crystals,” Appl. Phys. Lett. 77, 1937–1939 (2000).
[CrossRef]

Noda, S.

A. Chutinan and S. Noda, “Waveguides and waveguide bends in two-dimensional photonic crystal slabs,” Phys. Rev. B 62, 4488–4492 (2000).
[CrossRef]

Pearsall, T. P.

M. Loncar, D. Nedeljkovic, T. Doll, J. Vuckovic, A. Scherer, and T. P. Pearsall, “Waveguiding in planar photonic crystals,” Appl. Phys. Lett. 77, 1937–1939 (2000).
[CrossRef]

Rodier, J.

C. Sauvan, P. Lalanne, J. Rodier, J. Hugonin, and A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 15, 1243–1245 (2003).
[CrossRef]

Sauvan, C.

C. Sauvan, P. Lalanne, J. Rodier, J. Hugonin, and A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 15, 1243–1245 (2003).
[CrossRef]

Scherer, A.

M. Loncar, D. Nedeljkovic, T. Doll, J. Vuckovic, A. Scherer, and T. P. Pearsall, “Waveguiding in planar photonic crystals,” Appl. Phys. Lett. 77, 1937–1939 (2000).
[CrossRef]

Talneau, A.

C. Sauvan, P. Lalanne, J. Rodier, J. Hugonin, and A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 15, 1243–1245 (2003).
[CrossRef]

Villeneuve, P. R.

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

Vuckovic, J.

M. Loncar, D. Nedeljkovic, T. Doll, J. Vuckovic, A. Scherer, and T. P. Pearsall, “Waveguiding in planar photonic crystals,” Appl. Phys. Lett. 77, 1937–1939 (2000).
[CrossRef]

Weedon, W. H.

W. C. Chew and W. H. Weedon, “A 3-D perfectly matched medium from modified Maxwell’s equations with stretching coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University Press, 2008).

Yonekura, Y.

T. Baba, N. Fukaya, and Y. Yonekura, “Observation of light propagation in photonic crystal optical waveguides with bends,” Electron. Lett. 35, 654–655 (1999).
[CrossRef]

Yuan, L.

Appl. Phys. Lett. (1)

M. Loncar, D. Nedeljkovic, T. Doll, J. Vuckovic, A. Scherer, and T. P. Pearsall, “Waveguiding in planar photonic crystals,” Appl. Phys. Lett. 77, 1937–1939 (2000).
[CrossRef]

Electron. Lett. (1)

T. Baba, N. Fukaya, and Y. Yonekura, “Observation of light propagation in photonic crystal optical waveguides with bends,” Electron. Lett. 35, 654–655 (1999).
[CrossRef]

IEEE J. Quantum Electron. (1)

P. Lalanne, “Electromagnetic analysis of photonic crystal waveguides operating above the light core,” IEEE J. Quantum Electron. 38, 800–804 (2002).
[CrossRef]

IEEE Photon. Technol. Lett. (3)

C. Sauvan, P. Lalanne, J. Rodier, J. Hugonin, and A. Talneau, “Accurate modeling of line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 15, 1243–1245 (2003).
[CrossRef]

Y. Y. Lu, “Minimizing the discrete reflectivity of perfectly matched layers,” IEEE Photon. Technol. Lett. 18, 487–489(2006).
[CrossRef]

G. R. Hadley, “Out-of-plane losses of line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 14, 642–644 (2002).
[CrossRef]

J. Comput. Phys. (1)

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

J. Lightwave Technol. (2)

J. Opt. Soc. Am. B (2)

Microw. Opt. Technol. Lett. (1)

W. C. Chew and W. H. Weedon, “A 3-D perfectly matched medium from modified Maxwell’s equations with stretching coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. B (2)

A. Chutinan and S. Noda, “Waveguides and waveguide bends in two-dimensional photonic crystal slabs,” Phys. Rev. B 62, 4488–4492 (2000).
[CrossRef]

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

Other (1)

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University Press, 2008).

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Figures (3)

Fig. 1
Fig. 1

PhC slab waveguide formed by filling one row of air holes in a PhC slab: (a) 3D view; (b) top view.

Fig. 2
Fig. 2

Top view of the truncated domain S m ( m = 4 ) for one period of a PhC slab waveguide. Also shown are the unit cells Ω j , the lateral surfaces Γ j and the special surface Γ.

Fig. 3
Fig. 3

Dispersion relations of the guided modes of a PhC slab waveguide calculated by solving the nonlinear Eq. (11) for (top) given frequencies or (bottom) given Bloch wavenumbers .

Tables (2)

Tables Icon

Table 1 Numerical Convergence of Vertical Slab Modes and a PhC Slab Waveguide Mode with Respect to the Number of Points for Discretizing the Vertical Direction

Tables Icon

Table 2 Numerical Convergence of a PhC Slab Waveguide Mode with Respect to the Number of Sampling Points on Each Edge of the Hexagon Cross Section of Regular Unit Cells

Equations (25)

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× E = i k 0 μ H , × H = i k 0 ε E ,
[ E H ] = [ Φ Ψ ] e i β x ,
[ E ( x + L , y , z ) H ( x + L , y , z ) ] = ρ [ E ( x , y , z ) H ( x , y , z ) ] ,
S = { ( x , y , z ) :     0 < x < L , < y < , < z < } .
S [ w + | x = 0 w | x = L ] = [ S 11 S 12 S 21 S 22 ] [ w + | x = 0 w | x = L ] = [ w | x = 0 w + | x = L ] .
[ S 11 I S 21 0 ] [ w + | x = 0 w | x = 0 ] = ρ [ 0 S 12 I S 22 ] [ w + | x = 0 w | x = 0 ] .
G [ w | x = 0 w | x = L ] = [ G 11 G 12 G 21 G 22 ] [ w | x = 0 w | x = L ] = [ x w | x = 0 x w | x = L ] .
w | x = L = ρ w | x = 0 , x w | x = L = ρ x w | x = 0 .
[ G 11 I G 21 0 ] [ w | x = 0 x w | x = 0 ] = ρ [ 0 G 12 I G 22 ] [ w | x = 0 x w | x = 0 ] .
B ( ω , β ) w | Γ = 0 ,
λ 1 ( B ) = 0 ,
Λ j w = ν w on Ω j ,
M j [ w j w j 1 ] = [ M 11 ( j ) M 12 ( j ) M 21 ( j ) M 22 ( j ) ] [ w j w j 1 ] = [ ν w j ν w j 1 ]
Q j w j = ν w j
Q j = M 11 ( j ) + M 12 ( j ) ( Q j 1 M 22 ( j ) ) 1 M 21 ( j ) , j = 2 , 3 , , m .
Λ m + 1 [ w m w m + 1 w | x = L w | x = 0 ] = [ Λ 11 ( m + 1 ) Λ 12 ( m + 1 ) Λ 13 ( m + 1 ) Λ 14 ( m + 1 ) Λ 21 ( m + 1 ) Λ 22 ( m + 1 ) Λ 23 ( m + 1 ) Λ 24 ( m + 1 ) Λ 31 ( m + 1 ) Λ 32 ( m + 1 ) Λ 33 ( m + 1 ) Λ 34 ( m + 1 ) Λ 41 ( m + 1 ) Λ 42 ( m + 1 ) Λ 43 ( m + 1 ) Λ 44 ( m + 1 ) ] [ w m w m + 1 w | x = L w | x = 0 ] = [ ν w m ν w m + 1 x w | x = L x w | x = 0 ] ,
U [ w | x = L w | x = 0 ] = [ U 11 U 12 U 21 U 22 ] [ w | x = L w | x = 0 ] = [ x w | x = L x w | x = 0 ] ,
U = [ Λ 31 ( m + 1 ) Λ 32 ( m + 1 ) Λ 41 ( m + 1 ) Λ 42 ( m + 1 ) ] N + [ Λ 33 ( m + 1 ) Λ 34 ( m + 1 ) Λ 43 ( m + 1 ) Λ 44 ( m + 1 ) ] ,
N = [ Λ 11 ( m + 1 ) Q m Λ 12 ( m + 1 ) Λ 21 ( m + 1 ) Λ 22 ( m + 1 ) Q m + 1 ] 1 [ Λ 13 ( m + 1 ) Λ 14 ( m + 1 ) Λ 23 ( m + 1 ) Λ 24 ( m + 1 ) ] .
B = U 12 ρ U 22 + ρ U 11 ρ 2 U 21 .
z ^ = 0 z s ( τ ) d τ , s ( z ) = { 1 + σ 0 [ ( c 0 z ) / ( c 0 d 0 ) ] q , d 0 z < c 0 , 1 + σ 1 [ ( z c 1 ) / ( d 1 c 1 ) ] q , c 1 < z d 1 , 1 , otherwise ,
μ s d d z ( 1 μ s d ϕ ( 1 ) d z ) + k 0 2 ε μ ϕ ( 1 ) = [ η ( 1 ) ] 2 ϕ ( 1 ) , d 0 < z < d 1 ,
ϕ ( 1 ) = 0 , z = d 0 and d 1 ,
ε s d d z ( 1 ε s d ϕ ( 2 ) d z ) + k 0 2 ε μ ϕ ( 2 ) = [ η ( 2 ) ] 2 ϕ ( 2 ) , d 0 < z < d 1 ,
d ϕ ( 2 ) d z = 0 , z = d 0 and d 1 .

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