Abstract

An approximate analytical approach for calculating the dispersion relations of two-dimensional photonic crystals, which was earlier developed for E-polarization, is extended for H-polarization (which is usually problematic for analytical treatment, because of field discontinuities). Useful analytical formulas, for calculating the dispersion relations and the magnetic fields, are developed. We show that the presented approach and the derived expressions provide a good approximation for a wide range of photonic crystal parameters. The results are also compared with accurate numerical calculations to check the validity of the approximations. This approach provides not only a fast way for photonic crystal calculations, but it also can be useful for the investigation of various physical effects as well as for the design and analysis of new photonic crystal devices.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).
  2. S. A. Schelkunoff, Electromagnetic Waves (D. Van Nostrand C., 1948).
  3. K. Sakoda, Optical Properties of Photonic Crystals2nd ed. (Springer-Verlag, 2005).
  4. S. Kawakami, “Analytically solvable model of photonic crystal structures and novel phenomena,” J. Lightwave Technol. 20, 1644-1650 (2002).
    [CrossRef]
  5. I. Ponomarev, “Separation of variables in the computation of spectra in 2-D photonic crystals,” SIAM J. Appl. Math. 61, 1202-1218 (2000).
    [CrossRef]
  6. K. Samokhvalova, C. Chen, and B.-L. Qian, “Analytical and numerical calculations of the dispersion characteristics of two-dimensional dielectric photonic band gap structures,” J. Appl. Phys. 99, 63104 (2006).
    [CrossRef]
  7. W. Axmann, P. Kuchment, and L. Kunyansky, “Asymptotic methods for thin high-contrast two-dimensional PBG materials,” J. Lightwave Technol. 17, 1996-2007 (1999).
    [CrossRef]
  8. T. J. Shepherd, P. J. Roberts, and R. Loudon, “Soluble two-dimensional photonic crystal model,” Phys. Rev. E 55, 6024-6038 (1997).
    [CrossRef]
  9. A. Figotin and P. Kuchment, “Band gap structure of spectra of periodic dielectric and acoustic media. II. Two-dimensional photonic crystals,” SIAM J. Appl. Math. 56, 1561-1620 (1996).
    [CrossRef]
  10. I. Nusinsky and A. A. Hardy, “Approximate analysis of two-dimensional photonic crystals with rectangular geometry. I. E-polarization,” J. Opt. Soc. Am. B 25, 1135-1143 (2008).
    [CrossRef]
  11. A. H. Nayfeh, Introduction to Perturbation Techniques (Wiley, 1993).
  12. W. Magnus and S. Winkler, Hill's Equation (Wiley, 1966).
  13. I. Nusinsky and A. A. Hardy, “Band gap-analysis of one-dimensional photonic crystals and conditions for gap closing,” Phys. Rev. B 73, 125104 (2006).
    [CrossRef]
  14. D. J. Griffiths, Introduction to Quantum Mechanics (Prentice-Hall, 1995).
  15. C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics Vol. 1 (Wiley, 1977), p. 406-411.

2008 (1)

2006 (2)

K. Samokhvalova, C. Chen, and B.-L. Qian, “Analytical and numerical calculations of the dispersion characteristics of two-dimensional dielectric photonic band gap structures,” J. Appl. Phys. 99, 63104 (2006).
[CrossRef]

I. Nusinsky and A. A. Hardy, “Band gap-analysis of one-dimensional photonic crystals and conditions for gap closing,” Phys. Rev. B 73, 125104 (2006).
[CrossRef]

2002 (1)

2000 (1)

I. Ponomarev, “Separation of variables in the computation of spectra in 2-D photonic crystals,” SIAM J. Appl. Math. 61, 1202-1218 (2000).
[CrossRef]

1999 (1)

1997 (1)

T. J. Shepherd, P. J. Roberts, and R. Loudon, “Soluble two-dimensional photonic crystal model,” Phys. Rev. E 55, 6024-6038 (1997).
[CrossRef]

1996 (1)

A. Figotin and P. Kuchment, “Band gap structure of spectra of periodic dielectric and acoustic media. II. Two-dimensional photonic crystals,” SIAM J. Appl. Math. 56, 1561-1620 (1996).
[CrossRef]

Axmann, W.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

Chen, C.

K. Samokhvalova, C. Chen, and B.-L. Qian, “Analytical and numerical calculations of the dispersion characteristics of two-dimensional dielectric photonic band gap structures,” J. Appl. Phys. 99, 63104 (2006).
[CrossRef]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics Vol. 1 (Wiley, 1977), p. 406-411.

Diu, B.

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics Vol. 1 (Wiley, 1977), p. 406-411.

Figotin, A.

A. Figotin and P. Kuchment, “Band gap structure of spectra of periodic dielectric and acoustic media. II. Two-dimensional photonic crystals,” SIAM J. Appl. Math. 56, 1561-1620 (1996).
[CrossRef]

Griffiths, D. J.

D. J. Griffiths, Introduction to Quantum Mechanics (Prentice-Hall, 1995).

Hardy, A. A.

I. Nusinsky and A. A. Hardy, “Approximate analysis of two-dimensional photonic crystals with rectangular geometry. I. E-polarization,” J. Opt. Soc. Am. B 25, 1135-1143 (2008).
[CrossRef]

I. Nusinsky and A. A. Hardy, “Band gap-analysis of one-dimensional photonic crystals and conditions for gap closing,” Phys. Rev. B 73, 125104 (2006).
[CrossRef]

Kawakami, S.

Kuchment, P.

W. Axmann, P. Kuchment, and L. Kunyansky, “Asymptotic methods for thin high-contrast two-dimensional PBG materials,” J. Lightwave Technol. 17, 1996-2007 (1999).
[CrossRef]

A. Figotin and P. Kuchment, “Band gap structure of spectra of periodic dielectric and acoustic media. II. Two-dimensional photonic crystals,” SIAM J. Appl. Math. 56, 1561-1620 (1996).
[CrossRef]

Kunyansky, L.

Laloe, F.

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics Vol. 1 (Wiley, 1977), p. 406-411.

Loudon, R.

T. J. Shepherd, P. J. Roberts, and R. Loudon, “Soluble two-dimensional photonic crystal model,” Phys. Rev. E 55, 6024-6038 (1997).
[CrossRef]

Magnus, W.

W. Magnus and S. Winkler, Hill's Equation (Wiley, 1966).

Nayfeh, A. H.

A. H. Nayfeh, Introduction to Perturbation Techniques (Wiley, 1993).

Nusinsky, I.

I. Nusinsky and A. A. Hardy, “Approximate analysis of two-dimensional photonic crystals with rectangular geometry. I. E-polarization,” J. Opt. Soc. Am. B 25, 1135-1143 (2008).
[CrossRef]

I. Nusinsky and A. A. Hardy, “Band gap-analysis of one-dimensional photonic crystals and conditions for gap closing,” Phys. Rev. B 73, 125104 (2006).
[CrossRef]

Ponomarev, I.

I. Ponomarev, “Separation of variables in the computation of spectra in 2-D photonic crystals,” SIAM J. Appl. Math. 61, 1202-1218 (2000).
[CrossRef]

Qian, B.-L.

K. Samokhvalova, C. Chen, and B.-L. Qian, “Analytical and numerical calculations of the dispersion characteristics of two-dimensional dielectric photonic band gap structures,” J. Appl. Phys. 99, 63104 (2006).
[CrossRef]

Roberts, P. J.

T. J. Shepherd, P. J. Roberts, and R. Loudon, “Soluble two-dimensional photonic crystal model,” Phys. Rev. E 55, 6024-6038 (1997).
[CrossRef]

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals2nd ed. (Springer-Verlag, 2005).

Samokhvalova, K.

K. Samokhvalova, C. Chen, and B.-L. Qian, “Analytical and numerical calculations of the dispersion characteristics of two-dimensional dielectric photonic band gap structures,” J. Appl. Phys. 99, 63104 (2006).
[CrossRef]

Schelkunoff, S. A.

S. A. Schelkunoff, Electromagnetic Waves (D. Van Nostrand C., 1948).

Shepherd, T. J.

T. J. Shepherd, P. J. Roberts, and R. Loudon, “Soluble two-dimensional photonic crystal model,” Phys. Rev. E 55, 6024-6038 (1997).
[CrossRef]

Winkler, S.

W. Magnus and S. Winkler, Hill's Equation (Wiley, 1966).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

J. Appl. Phys. (1)

K. Samokhvalova, C. Chen, and B.-L. Qian, “Analytical and numerical calculations of the dispersion characteristics of two-dimensional dielectric photonic band gap structures,” J. Appl. Phys. 99, 63104 (2006).
[CrossRef]

J. Lightwave Technol. (2)

J. Opt. Soc. Am. B (1)

Phys. Rev. B (1)

I. Nusinsky and A. A. Hardy, “Band gap-analysis of one-dimensional photonic crystals and conditions for gap closing,” Phys. Rev. B 73, 125104 (2006).
[CrossRef]

Phys. Rev. E (1)

T. J. Shepherd, P. J. Roberts, and R. Loudon, “Soluble two-dimensional photonic crystal model,” Phys. Rev. E 55, 6024-6038 (1997).
[CrossRef]

SIAM J. Appl. Math. (2)

A. Figotin and P. Kuchment, “Band gap structure of spectra of periodic dielectric and acoustic media. II. Two-dimensional photonic crystals,” SIAM J. Appl. Math. 56, 1561-1620 (1996).
[CrossRef]

I. Ponomarev, “Separation of variables in the computation of spectra in 2-D photonic crystals,” SIAM J. Appl. Math. 61, 1202-1218 (2000).
[CrossRef]

Other (7)

A. H. Nayfeh, Introduction to Perturbation Techniques (Wiley, 1993).

W. Magnus and S. Winkler, Hill's Equation (Wiley, 1966).

D. J. Griffiths, Introduction to Quantum Mechanics (Prentice-Hall, 1995).

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics Vol. 1 (Wiley, 1977), p. 406-411.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

S. A. Schelkunoff, Electromagnetic Waves (D. Van Nostrand C., 1948).

K. Sakoda, Optical Properties of Photonic Crystals2nd ed. (Springer-Verlag, 2005).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Distribution of the dielectric function ϵ ( x , y ) of 2D rectangular photonic crystal (top view).

Fig. 2
Fig. 2

Distribution of the dielectric function in the elementary cell of 2D rectangular photonic crystals. (a) The main problem, ϵ ( x , y ) . (b) The modified function ϵ ̃ ( x , y ) , (c) ϵ ͌ ( x , y ) .

Fig. 3
Fig. 3

Distribution of 1D functions in the elementary cell (a) ϵ ̃ x ( x ) and ϵ ̃ y ( y ) . (b) ϵ ͌ x ( x ) and ϵ ͌ y ( y ) .

Fig. 4
Fig. 4

Band structure of photonic crystal with following parameters: ϵ 1 = 1 , and (a) ϵ 2 = n 2 2 = 2.1 2 = 4.41 , a x = a y = 0.85 L x = 0.85 L y , (b) ϵ 2 = 3.4 2 = 11.56 , a x = a y = 0.8 L x = 0.8 L y . The solid curves present numerical solution, calculated by plane wave expansion method. Dashed curves are for zero-order approximation. The filled and empty circles are for first- and second-order approximation, respectively.

Fig. 5
Fig. 5

Band structure of photonic crystal with following parameters: ϵ 2 = 1 , and (a) ϵ 1 = 4.41 , a x = a y = 0.85 L x = 0.85 L y . (b) ϵ 1 = 3.4 2 = 11.56 , a x = a y = 0.8 L x = 0.8 L y . The notations are same as in Fig. 4.

Equations (51)

Equations on this page are rendered with MathJax. Learn more.

x 1 ϵ ( x , y ) H z ( x , y ) x + y 1 ϵ ( x , y ) H z ( x , y ) y = ω 2 c 2 H z ( x , y ) ,
x 1 ϵ ͌ ( x , y ) H ̃ z ( x , y ) x + y 1 ϵ ͌ ( x , y ) H ̃ z ( x , y ) y = ω ̃ 2 c 2 ϵ ̃ ( x , y ) ϵ ͌ ( x , y ) H ̃ z ( x , y ) .
ϵ ̃ ( x , y ) = ϵ ̃ x ( x ) + ϵ ̃ y ( y ) ,
ϵ ͌ ( x , y ) = ϵ ͌ x ( x ) ϵ ͌ y ( y ) .
e = ( ϵ 1 + ϵ 2 ) b x b y + ϵ 2 ( a x b y + a y b x ) b x L y + b y L x .
v = exp ( ( a x b x ) ln e ̃ + b x ln ( ϵ 1 2 e ̃ ϵ 1 ) L x ) ,
u = exp ( ( a y b y ) ln e ̃ + b y ln ( ϵ 1 2 e ̃ ϵ 1 ) L y ) .
cos ( K x B L x ) = cos ( k 1 x a x ) cos ( k 2 x b x ) 1 2 ( k 1 x γ x k 2 x + γ x k 2 x k 1 x ) sin ( k 1 x a x ) sin ( k 2 x b x ) ,
cos ( K y B L y ) = cos ( k 1 y a y ) cos ( k 2 y b y ) 1 2 ( k 1 y γ y k 2 y + γ y k 2 y k 1 y ) sin ( k 1 y a y ) sin ( k 2 y b y ) ,
k 1 x 2 = ω ̃ 2 c 2 ( 0.5 ϵ 1 β 2 ) ,
k 2 x 2 = ω ̃ 2 c 2 ( e 0.5 ϵ 1 β 2 ) ,
k 1 y 2 = ω ̃ 2 c 2 ( 0.5 ϵ 1 + β 2 ) ,
k 2 y 2 = ω ̃ 2 c 2 ( e 0.5 ϵ 1 + β 2 ) ,
ω n ( K x B , K y B ) = ω ̃ n ( K x B , K y B ) + ω n ( 1 ) ( K x B , K y B ) + ω n ( 2 ) ( K x B , K y B ) ,
H z n ( x , y ) = H ̃ z n ( x , y ) + H z n ( 1 ) ( x , y ) + H z n ( 2 ) ( x , y ) ,
ω n ( 1 ) ( K x B , K y B ) = ω ̃ n ( K x B , K y B ) 0 L x 0 L y Δ ϵ ( x , y ) ϵ ͌ ( x , y ) H ̃ z n 2 d x d y + c 2 ω ̃ n ( K x B , K y B ) 0 L x 0 L y H ̃ z n * D H ̃ z n 2 0 L x 0 L y ϵ ̃ ( x , y ) ϵ ͌ ( x , y ) H ̃ z n 2 d x d y ,
D = [ x ( ln ϵ ͌ ϵ ) 1 ϵ ͌ x + y ( ln ϵ ͌ ϵ ) 1 ϵ ͌ y ] .
0 L x 0 L y H ̃ z n * D H ̃ z n = 0 L y ( H ̃ z n * 1 ϵ ͌ H ̃ z n x x = a x + H ̃ z n * 1 ϵ ͌ H ̃ z n x x = L x ) d y + 0 L x ( H ̃ z n * 1 ϵ ͌ H ̃ z n y y = a y + H ̃ z n * 1 ϵ ͌ H ̃ z n y y = L y ) d x .
ω n ( 1 ) = c 2 2 ω ̃ n W n n + ω ̃ n 2 c 2 V n n N n n ,
V n n = ( ϵ 2 e ) ( f a b n n u + f b a n n v ) + ( ϵ 1 + ϵ 2 2 e ) ϵ 1 u v f b b n n ,
W n n = h y n n ( 1 u g a x n n ln u ϵ 2 + ϵ 1 u v g b x n n ln u ϵ 1 ) + h x n n ( 1 v g a y n n ln v ϵ 2 + ϵ 1 u v g b y n n ln v ϵ 1 ) ,
N n n = f a a n n + e ( f a b n n u + f b a n n v ) + ( 2 e ϵ 1 ) ϵ 1 u v f b b n n ,
T = ( ω ̃ m 1 0 0 0 0 0 0 ω ̃ m M ) c 2 2 ( ω ̃ m 1 N m 1 m 1 0 0 ω ̃ m M N m M m M ) 1 ( P m 1 m 1 P m 1 m M P m M m 1 P m M m M ) ,
P i j = ω ̃ i 2 c 2 V i j + W i j ,
ω ̃ ± + ω ± ( 1 ) = ω ̃ n + ω ̃ m 2 c 2 2 ( P n n ω ̃ n N n n + P m m ω ̃ m N m m ) ± ( ( ω ̃ n ω ̃ m + c 2 P n n 2 ω ̃ n N n n c 2 P m m 2 ω ̃ m N m m ) 2 + 4 P m n P n m ω ̃ n ω ̃ m N n n N m m ) .
ω ̃ ± + ω ± ( 1 ) = ω ̃ n c 2 2 ω ̃ n ( P n n N n n + P m m N m m ± ( P n n N n n P m m N m m ) 2 + 4 P m n P n m N n n N m m ) .
ω n ( 2 ) ( K x B , K y B ) = ω n ( 1 ) 0 L x 0 L y Δ ϵ ( x , y ) ϵ ͌ ( x , y ) H ̃ z n 2 d x d y + ω ̃ n 2 0 L x 0 L y H ̃ z n * ( Δ ϵ ( x , y ) ϵ ͌ ( x , y ) + c 2 ω ̃ n 2 D ) H z n ( 1 ) d x d y 0 L x 0 L y ϵ ̃ ( x , y ) ϵ ͌ ( x , y ) H ̃ z n 2 d x d y ( ω n ( 1 ) ) 2 2 ω ̃ n ,
H z n ( 1 ) ( x , y ) = m n C m n H ̃ z m ( x , y ) ,
H z n ( 1 ) ( x , y ) = m n c 2 H ̃ z m ( x , y ) ω ̃ m 2 ω ̃ n 2 P m n N m m .
ω n ( 2 ) = ω n ( 1 ) V n n N n n c 4 2 ω ̃ n m n ( ω ̃ n 2 c 2 V n m + W n m ) ( ω ̃ n 2 c 2 V m n + W m n ) ( ω ̃ m 2 ω ̃ n 2 ) N n n N m m ( ω n ( 1 ) ) 2 2 ω ̃ n ,
H z ( x , y ) = [ X 1 ( x ) + A x X 2 ( x ) ] [ Y 1 ( y ) + A y Y 2 ( y ) ] .
X 1 ( x ) = { cos ( k 1 x x ) 0 x a x cos ( k 1 x a x ) cos ( k 2 x ( x a x ) ) k 1 x k 2 x γ x sin ( k 1 x a x ) sin ( k 2 x ( x a y ) ) a x x L x , }
Y 1 ( y ) = { cos ( k 1 y y ) 0 y a y cos ( k 1 y a y ) cos ( k 2 y ( y a y ) ) k 1 y k 2 y γ y sin ( k 1 y a y ) sin ( k 2 y ( x a y ) ) a y y L y , }
X 2 ( x ) = { sin ( k 1 x x ) k 1 x 0 x a x sin ( k 1 x a x ) k 1 x cos ( k 2 x ( x a x ) ) + cos ( k 1 x a x ) k 2 x γ x sin ( k 2 x ( x a x ) ) a x x L x , }
Y 2 ( y ) = { sin ( k 1 y y ) k 1 y 0 y a y sin ( k 1 y a y ) k 1 y cos ( k 2 y ( y a y ) ) + cos ( k 1 y a y ) k 2 y γ y sin ( k 2 y ( y a y ) ) a y y L y , }
A x = X 1 ( L x ) p x X 2 ( L x ) ,
A y = Y 1 ( L y ) p y Y 2 ( L y ) ,
p x = exp ( i K x B L x ) ,
p y = exp ( i K y B L y ) ,
f a a m n = 0 a x 0 a y H ̃ z m * H ̃ z n d x d y = g a x m n g a y m n ,
f b a m n = a x L x 0 a y H ̃ z m * H ̃ z n d x d y = g b x m n g a y m n ,
f a b m n = 0 a x a y L y H ̃ z m * H ̃ z n d x d y = g a x m n g b y m n ,
f b b m n = a x L x a y L y H ̃ z m * H ̃ z n d x d y = g b x m n g b y m n ,
g a i m n = a i 2 ( 1 + A ¯ i m A i n k 1 i m k 1 i n ) sinc ( a i ( k 1 i m k 1 i n ) ) + a i 2 ( 1 A ¯ i m A i n k 1 i m k 1 i n ) sinc ( a i ( k 1 i m + k 1 i n ) ) + ( A i n k 1 i n A ¯ i m k 1 i m ) cos ( a i ( k 1 i m k 1 i n ) ) 1 2 ( k 1 i m k 1 i n ) ( A i n k 1 i n + A ¯ i m k 1 i m ) cos ( a i ( k 1 i m + k 1 i n ) ) 1 2 ( k 1 i m + k 1 i n ) , i = x , y ,
g b i m n = b i 2 ( B ¯ i m B i n + C ¯ i m C i n k 2 i m k 2 i n ) sinc ( b i ( k 2 i m k 2 i n ) ) + b i 2 ( B ¯ i m B i n C ¯ i m C i n k 2 i m k 2 i n ) sinc ( b i ( k 2 i m + k 2 i n ) ) + ( B ¯ i m C i n k 2 i n C ¯ i m B i n k 2 i m ) cos ( b i ( k 2 i m k 2 i n ) ) 1 2 ( k 2 i m k 2 i n ) ( B ¯ i m C i n k 2 i n + C ¯ i m B i n k 2 i m ) cos ( b i ( k 2 i m + k 2 i n ) ) 1 2 ( k 2 i m + k 2 i n ) ,
h i m n = B ¯ i m C i n ( B ¯ i m cos ( k 2 i m b i ) + C ¯ i m sin ( k 2 i m b i ) k 2 i m ) ( k 2 i n B i n sin ( k 2 i n b i ) + C i n cos ( k 2 i n b i ) ) ,
B i n = cos ( k 1 i n a i ) + A i n sin ( k 1 i n a i ) k 1 i n , i = x , y ,
C i n = k 1 i n sin ( k 1 i n a i ) + A i n cos ( k 1 i n a i ) γ i , i = x , y ,
V m n = ( ϵ 2 e ) ( f a b m n u + f b a m n v ) + ( ϵ 1 + ϵ 2 2 e ) ϵ 1 u v f b b m n ,
W m n = h y m n ( 1 u g a x m n ln u ϵ 2 + ϵ 1 u v g b x m n ln u ϵ 1 ) + h x m n ( 1 v g a y m n ln v ϵ 2 + ϵ 1 u v g b y m n ln v ϵ 1 ) ,
N m n = f a a m n + e ( f a b m n u + f b a m n v ) + ( 2 e ϵ 1 ) ϵ 1 u v f b b m n .

Metrics